\section{SYMMETRY REDUCES THE ORDER OF ANY SECOND ORDER ODE}

Lie showed how the invariance of a second-order ordinary differential
equation (ODE) under a point symmetry leads constructively to a first-order
ODE plus a quadrature.  If the symmetry were a variational symmetry of the
Action Principle, Noether's well-known theorem would lead to a conservation
law \cite{Olver2,Boyce,JordonSmith}.

We consider the \emph{scaling symmetry} $\xi \rightarrow A\xi,~\theta_n
(\xi) \rightarrow \theta_{nA} (\xi)$ of the second-order ODE
\emph{Lane-Emden equation} (Section III) which describes the hydrostatic
equilibrium of a gaseous sphere or star in local thermodynamic equilibrium
(Section II). Because this is not a variational symmetry, but only a
symmetry of the Lane-Emden equation, scaling symmetry leads only to a
\emph{non-conservation law } \cite{BludKenI,BludKenII}.  This is a
first-order differential equation for scale invariants $u_n,~v_n$, which
can be solved for $v_n (u_n)$ for given boundary conditions (Section III).
From these first integrals, quadrature finally leads to solutions of the
original second-order equation (Section IV).

Sections II and III will
consider only \emph{regular solutions} which have density finite at the
origin and apply only to complete polytropes.
Section IV will generalize to \emph{irregular solutions} which have densities
infinite at the origin (F-solutions) or vanish away from the origin
(M-solutions) and can apply only to stellar envelopes.

\section{HYDROSTATIC EQUILIBRIUM OF SELF-GRAVITATING SPHERES}

An adiabatic sphere in hydrostatic equilibrium obeys the equations of
equilibrium between gravitational and internal (pressure)forces and of mass
continuity \be -d P/\rho dr= G m/r^2 , \quad d m/d r=4\pi r^2 \rho \quad,
\label{eq:masspressbalance} \ee where the local pressure, mass density, and
included mass $P(r),~\rho(r),~m(r)$ depend on radius $r$.  In terms of the
the gravitational potential $V(r)=\int_\infty^r Gm/r^2 dr$ and thermostatic
potential (specific enthalpy, ejection energy) $H(r)=\int_{0}^{P(r)}
dP/\rho $, (\ref{eq:masspressbalance}) and its integrated form \be -d H/d
r=d V/d r ,\quad V(r)+H(r)=-\frac{GM}{R}\quad, \ee

\begin{figure}[t]
\includegraphics[scale=0.65]{LEFunctionsOrder}
\caption{Emden Functions (Regular Solutions of the Lane-Emden Equation) of
order n=1,2,3,4,5}
\end{figure}

\begin{figure}[t]
\includegraphics[scale=0.65]{MassIncluded}
\caption{'Mass' included inside radius $\xi$ for complete polytropes of
order n=1,2,3,4,5 terminating at $\xi_{1n}=3.141,4.353,
6.897,14.97,\infty$.}
\end{figure}

expresses the conservation of the specific energy as sum of gravitational
and internal energies, in a star of mass $M$ and radius $R$. The two
first-order equations (\ref{eq:masspressbalance}) can be combined into the
second-order Poisson's Law \be \frac{1}{r^2}\frac{d}{dr} \Big( r^2\frac{d
H}{dr} \Big) + 4 \pi G \rho(H)=0 \quad , \label{eq:secondorder} \ee
in terms of the enthalpy $H(r)$.

The two equations (\ref{eq:masspressbalance}) can always be written
\be
d\log{u}/d\log{r}=3-u(r)-n(r) v(r)\quad ,\quad
 d\log{w}/d\log{r}=u-1+v(r)-d\log{[1+n(r)]}/d\log{r}\quad, \label{eq:equil}
\ee
in terms of the logarithmic derivatives
\be
u(r):= d\log{m}/d\log{r} \\
w(r):= n(r) v(r)= -d\log{\rho}/d\log{r} \\
v(r):= -d\log{(P/\rho)}/d\log{r}
\ee
and an index $n(r)$ \be n(r):=d\log{\rho}/d \log{(P/\rho)} \quad,\quad
1+1/n(r):=d\log{P} /d\log{\rho}\quad ,\ee which depends on the local adiabatic
equation of state $P=P(\rho)$.

For solutions regular at the origin, spherical symmetry requires that $dP/dr=0$ and that $\rho(r),~P(r),~H(r)$ be even functions of $r$. Mass
continuity requires, to order $r^2$, \be \rho(r)\approx\rho_c (1-Ar^2),
~~m(r)\approx\frac{4 \pi r^3}{3}\cdot(1-\frac{3}{5}A r^2)\approx\frac{4 \pi
r^3}{3} \cdot \rho_c^{2/5} \rho(r) ^{3/5} \quad . \label{eq:origin} \ee
Thus, near the origin, the average mass density inside radius
$\bar{\rho}(r):=\frac{m(r)}{4\pi r^3/3}\approx\rho_c^{2/5} \rho(r) ^{3/5}$
and $u(0)=3, v(0)=0$. At the stellar surface, $u(R)=0, v(R)=\infty$.
Here
$\bar{\rho}(r):=\frac{m(r)}{4\pi r^3/3}$ is the average mass density inside radius, so that
$u(r)=3\rho/\bar{\rho}$ decreases from $u(0)=3$ at the origin to $u(R)=\infty$ at the stellar surface. The ratio
$v(r):=\frac{3}{2}(-Gm/r)/(\text{P} / \rho )=\frac{3}{2}
(\text{'gravitational energy'})/(\text{'internal energy of ideal gas'})$ increases from $v(0)=0$ at the origin to $v(R)=\infty$ at the stellar surface.

\section{SCALE TRANSFORMATIONS ON                                                                                SOLUTIONS OF THE LANE-EMDEN EQUATION}

The symmetry we consider is scaling symmetry, the most general
simplification that one can make for any dynamical system.  If a scale
transformation $r\rightarrow Ar$ transforms $m,~\rho, ~P$ multiplicatively,
the logarithmic derivatives $u,~v$ will be scale invariant.  The
structural equations (5) will then be autonomous, if and only if
$n=constant$, so that, $P(r)=K\rho(r)^{1+\frac{1}{n}}$, with the constant
$K$ determined by the constant specific entropy.  When this is so, scaling
symmetry leads to the two coupled autonomous equations for the scale
invariants \be du_n/d\log{r}=u_n (3-u_n-n v_n)\quad ,\quad
dv_n/d\log{r}=v_n(u_n-1+v_n)\quad, \label{eq:chareqns12} \ee
implying the first-order equation
\be
dv_n/du_n=v_n(u_n-1+v_n)/u_n (3-u_n-n v_n)\quad. \label{eq:chareqns13} \ee
This reduced equation for $v_n(u_n)$ incorporates all the consequences of scaling
symmetry, from which all the properties of polytropes follow.

In this and the next section, we consider only \emph{regular solutions}
(Emden or E-solutions) of the Lane-Emden equation, which have finite
density at the origin, and are applicable only to complete polytropes.
At the origin, the initial conditions on regular solutions of \ref{eq:chareqns13} are
$u_n (0)=3, v_n (0)=0$. (We defer to Section IV the irregular solutions $v_n (u_n)$, where the density at the origin is not finite.)

Figure 3 shows how, moving radially outwards, $u_n(r)=3
\rho(r)/\bar{\rho(r)}$ decreases from $3$ at the origin to $0$ at the outer
boundary $R$ and $v_n(r)$ increases from $0$ to $\infty$. The combinations $\omega_n(r):=(u v_n^n)^{1/(n-1)}\equiv
-\xi^{1+\tilde{\omega}_n} \theta_n '$ approach the finite values
$_0\omega_n (R)$ and characterize each n-polytrope.

\begin{figure}[t]
\includegraphics[scale=0.60]{ScaleTransformThetaLogLogfig3}
\caption{Emden function $\theta_3(\xi,A)$ for three scales A=4 (blue),
1 (green), .25 (red).  On a log-log plot, rescaling simply transforms the
Lane-Emden functions along lines of constant
$v_3:=-d\log{(P/\rho)}/d\log{r}$. The three dashed lines connect homologous
points for the three illustrative values $v_3=0.081, 0.25, 2.3$.}
\end{figure}

In terms of the dimensional central density, pressure $\rho_c,~P_c=K \rho_c ^{1+1/n}
$ and constant $\alpha ^2:=((n+1)/{4\pi G})K\rho_c ^{1/n-1},
~H_c:=(n+1) (P/\rho)_{c} \equiv (n+1) K \rho^{1/n}_c $, the dimensional
radius, enthalpy, mass density and included mass are \be
r=\alpha \xi,\quad H=H_c\theta_n, \quad \rho=\rho_c \theta_n ^n,\quad
m(r)=(4 \pi \rho_c \alpha ^3 )(-\xi^2 \theta_n ') \quad ,\ee where prime
designates the derivative $':=d/d \xi$. The scale invariants are
\be u_n:=-\xi {\theta_n}^n/\theta_n' , \qquad v_n := -\xi\theta_n '/\theta_n  \qquad \omega_n:=(uv^n)^{1/(n-1)}=-\xi^{\frac{n=1}{n-1}}\quad,\ee
where $\tilde{\omega_n}:=2/(n-1)$.

In these dimensionless units,                                ,
Poisson's Law (\ref{eq:secondorder}) becomes the {\emph {Lane-Emden
equation}} \cite{Chandra} \be \frac{d}{d\xi}\Bigl( \xi^2
\frac{d\theta_n}{d\xi}\Bigr) + \xi^2\theta_n ^n = 0
\label{eq:lane-emden-1}, \ee whose regular solutions
have
\be \theta_n=const, \qquad \theta_n ' (0) =0,\qquad u(0)=3,\qquad v_n(0) =0 \quad , \ee
The \emph{normalized regular solutions} with $\theta_n=1$
define the {\em Emden Functions} for the n-polytrope radial evolution
(Figure 1). Their dimensionless normalized 'mass' $-\theta^2 \theta_n '
(\xi) $ is shown in Figure 2. The first order equations (\ref{eq:chareqns12}) have analytic solutions
listed in Table 1 for $n=0, 1, 5$, but must be integrated numerically for any
other polytropic index $n$.
\begin{figure}[t]
\includegraphics[scale=0.60]{MassIncludedLogLogfig4}
\caption{'Mass' included inside radius $\xi$ for n=3 polytrope. On a
log-log plot, rescaling simply transforms the 'mass' along lines of
constant $v_3:=-d\log{(P/\rho)}/d\log{r}$. The dashed lines connect
homologous points for the three illustrative values $v_3=0.081, 0.25, 2.3$.}
\end{figure}

The Lane-Emden equation (\ref{eq:lane-emden-1}) is invariant under the
scale transformation $\xi\rightarrow A\xi,~t\rightarrow t+\log{A},
~\theta_n(\xi)\rightarrow A^{\tilde{\omega}_n} \theta_n(A~\xi):=\theta_{n
A} (\xi) $. Besides the Emden Function $\theta_n(\xi)$, normalized so that
$\theta_{n1} (\xi)\equiv\theta_n (\xi)$, the Lane-Emden equation has
rescaled regular solutions $\theta_{nA} (\xi)$, whose value at the origin
is $A^{\tilde{\omega_n}}$. Figures 3 and 4 show log-log plots of the
rescaled n=3 function $\theta_{3A}(\xi)$ and rescaled 'mass' $-\theta'^2
\vartheta_\xi \theta_{3A} (\xi)$ for three different rescalings A=4 (blue), 1
(green), 0.25 (red). On a log-log plot, rescaling appears as a translation
along the dashed lines of constant $v_n$ and $u_n$, which connect homologous
points. All the familiar properties of polytropes \cite{Chandra} follow from this scaling
symmetry incorporated in the reduced equation for $v_n(u_n)$.

\begin{table}[h]
\caption {\textbf{Scaling Invariants and normalized Emden Functions for n=0, 1, 5}}
\begin{tabular}{|l||l||l|l||l|l||l|r|}
\hline\hline
$n$                        &$v_n(u_n)$                        &$u_n(\xi)$                   &$v_n(\xi)$          &$\theta_n(\xi)$      &$-\xi^2 \theta_n'(\xi)=\xi^{\frac{(n+1)}{(n-1)}}\omega_n$   &$\xi_{1n}$                &$ _0 \omega_n$  \\

\hline\hline
0                          &$u=3$                             &3                            &$2\xi^2/(6-\xi^2)$  &$1-\xi^2/6$          &$\xi^3/3$                                              &2.45                      &0.333          \\
1                          &parametric $u_n(\xi),v_n(\xi)$    &$\xi^2/(1-\xi\cot{\xi})$     &$1-\xi\cot{\xi}$    &$\sin{\xi}/\xi$      &$\sin{\xi}-\xi\cos{\xi}$
&3.14                      &$\cdots$       \\
5                          &$1-u_5/3$                         &$3/(1+\xi^2 /3)$               &$\xi^2 /(3+\xi^2)$&$(1+\xi^2 /3)^{-1/2}$&$\xi^3/3(1+\xi^2 /3)^{3/2}$
&$\infty$                  &0              \\
\hline\hline
\end{tabular}
\end{table}

\begin{figure}[t]
\includegraphics[scale=0.60]{Montage_uv_omega_499fig5}
\caption{The Emden scale invariants $u_n(r)=d\log{m}/d\log{r}, v_n(r)=-d\log{P}/d\log{r}$ and
$\omega_n (r)=(uv^n)^{1/(n-1)}$. While $v_n(r)$ diverges at the stellar radius $R$, $\omega_n$ (in green) approaches the finite values $_0 \omega_2=10.49,  ~_0 \omega_3=2.02,
~_0 \omega_4=0.73,
~_0 \omega_{4.99} \approx 0$ characteristic of each n-polytrope.
The curves in the $n=4.99$ figure are excellent approximations to the $n=5$ functions on the bottom line of Table I for which $R=\infty$.}
\end{figure}

\section{PHASE DIAGRAM FOR ALL FIRST INTEGRALS}

Equation (\ref{eq:chareqns13}) and the integral curves $v_n (u_n)$ and $u_n (v_n)$ are scale invariant functions of the scale invariant (r/R) plotted in
Figure 5 for n=2,~3,~4,~4.99.
All the dependence on scale $R$ is contained in (\ref{eq:chareqns12})which reads \be \log{(r/R)}=\int^\infty_0
\frac{dv_n} {v_n(u_n(v_n)-1+v_n)}=\int^0_3 \frac{du_n} {u_n(3-u_n-n v_n
(u_n))} \quad \ee
for the \emph{regular solutions} obeying $u_n(0)=3, v_n(0)=0$ at the
origin.

The \emph{irregular solutions} of (\ref{eq:chareqns13}) apply only to incomplete
polytropes, the scale-invariant envelopes of stars.  As shown in Figure 6, these
are \emph{F-solutions} (green curves, $u_n (0)>3$) which have infinite density at the
origin and \emph{M-solutions }(blue curves, $v_n (0)$) which have density vanishing
before the origin ($v(0)>0$). These irregular solutions would obtain by
integrating in from the boundary values $_F \omega_n > _0 \omega_n$ and  $_0 \omega_n> _M
\omega_n $. Their separatrices (red curves) are the regular (Emden) solutions, which have finite density
at the origin, $u_n(0)=3,~v_n(0)=0$ and boundary values $_0 \omega_n $.

\section{CONCLUDING SUMMARY}

A symmetry of the equations of
motion generally implies, not a conservation law, but a still
useful reduction of order of the equations of motion to a
first-order equation between invariants, which can then be integrated by
quadrature.

For the Lane-Emden equation, the reduced equation is the first-order
equation (\ref{eq:chareqns13}), whose first integrals (Figure 6) encapsulate
all the properties of index-n polytropes.  From this reduced equation
(Figure 5), we obtained the regular (Emden) solutions (Figures 1, 2) and
simply demonstrated their scale dependence on log-log plots (Figures 3, 4).

\begin{figure}[t]
\includegraphics[scale=0.60]{Montage_Phase5fig6}
\caption{Critical Points and Phase Diagrams for $v_n(u_n)$ for n=2, 3, 4, 5.  The green curves
(F-solutions) have infinite density at the stellar center.  The blue curves (M-solutions) have
vanishing density away from the origin.  Their separatrix, the red curve (Emden solutions), has finite
density at the origin. }
\end{figure}


\section{Introduction}
Recently a single right-handed neutrino scenario  (but not  with a single $\eta$) was proposed by D. Hehn and A. Ibarra group \cite{Hehn:2012kz} and discussed promising dark matter (DM) candidates in the framework of radiative seesaw model \cite{Ma:2006km}. One of the promising achievement of a single right-handed neutrino is to introduce gauged $B-3L_i$ proposed by E. Ma \cite{Ma:1997nq, Ma:1998dp, Ma:1998dr}.

Combining appropriately the radiative seesaw model and gauged $B-3L_i$ model, in our paper,  we try to analyze two DM candidates ; that is, fermionic one (right-handed neutrino) or bosonic one (neutral inert Higgs).
The neutrino sector can be mainly led in the five-dimensional operator without any right-handed neutrinos. One also finds that these DMs can be independently analyzed from the form of the lepton mass matrix because DMs do not affect to the matrix, which is unlike the typical radiative seesaw model.

This paper is organized as follows. In Section 2, we set up our model and explain the lepton sector and Higgs sector.
In Section 3, we discuss the DM properties in the fermionic case and the bosonic case.
In Section 4, we  conclude the paper.

\section{The Model}

\begin{table}[thbp]
\centering {\fontsize{10}{12}
\begin{tabular}{||c|c|c|c|c|c|c|c||c|c|c||}
\hline\hline ~~Particle~~ & ~~$L_e$~~ & ~~$L_\mu$ ~~ & ~~ $L_\tau $~~ & ~~$e^c$~~ & ~~$\mu^c$ ~~ & ~~ $\tau^c$~~ & ~~ $N^c$~~
 & ~~$\Phi~~ $& ~~$\eta~~ $ & $\chi $\\\hline
$SU(2)_L$&$\bm{2}$ & $\bm{2}$ & $\bm{2}$ & $\bm{1}$ & $\bm{1}$ &
$\bm{1}$ & $\bm{1}$ & $\bm{2}$ & $\bm{2}$ & $\bm{1}$\\\hline
$Y_{B-3L_e}$ & $3$ & $0$ & $0$ & $-3$ & $0$ & $0$ & $3$ & $0$ & $0$ & $-3$\\\hline
$\mathbb{Z}_2$ & $+$  & $+$& $+$ & $+$ & $+$ & $+$ & $-$ & $+$ & $-$ & $+$\\
\hline
\end{tabular}%
} \caption{The field content and the charges. Here we take $i=e$ of  $B-3L_i$ as an example. } \label{tab:b-3l}
\end{table}
The seesaw mechanism can be realized by introducing the $B-3L_i$ symmetry which is
spontaneously broken at TeV scale.
The field content of our model is shown in Table \ref{tab:b-3l} and the Lagrangian in the lepton sector is
\begin{eqnarray}
{\cal L}&=&
y_e L_e\Phi e^c + y_\mu L_\mu\Phi \mu^c  + y_\tau L_\tau\Phi \tau^c  + y'_{\mu\tau} L_\mu\Phi\tau^c  + y_{\mu\tau} L_\tau\Phi \mu^c \\
&+& y'_{e\mu} L_e\Phi \mu^c\chi + y'_{e\tau} L_e\Phi \tau^c\chi+ y_{e\mu} L_\mu\Phi e^c\chi^\dag + y_{e\tau} L_\tau\Phi e^c\chi^\dag\\
&+&
y_\nu  L_e\eta N^c + \frac{y_N}{\Lambda}\chi^2 N^cN^c  \\
&+&
{\lambda^\nu_1 \over \Lambda} (L_\mu\phi)(L_\tau\phi) + {\lambda^\nu_2 \over \Lambda} (L_\mu\phi)^2 + {\lambda^\nu_3 \over \Lambda} (L_\tau\phi)^2
\\
&+&
 {\lambda^\nu_4 \over \Lambda^2} (L_e\phi)(L_\mu\phi)\chi +
{\lambda^\nu_5 \over \Lambda^2} (L_\tau\phi)(L_e\phi)\chi
\\
&+&{\lambda^\nu_6 \over \Lambda^3} (L_e\phi)^2\chi^2 +
 \mathrm{h.c.},
\label{eq:lagrangian}
\end{eqnarray}
where $\Lambda$ is a cut-off scale to be ${\cal O}(10^{7\sim8})$ GeV to maintain $\lambda^\nu_{i}\le{\cal O}(1)$. As a result
we obtain the right-handed mass range as follows; ${\cal O}$(1) GeV $\le m_{N^C}\le$ ${\cal O}$(10) GeV, where $m_{N^C}=y_Nv'^2/\Lambda$ and $v'$ is set to be $12$ TeV to satisfy the LEP bound \cite{carena}.

After the symmetry breaking, that is $\chi=(\chi^0+v')/\sqrt2$,
$\Phi=(\phi^+,\phi)^T$  with $\phi=(\phi^0+v)/\sqrt2$ and $v=246\
{\rm GeV}$, the neutrino sector in the flavor basis can be written as
\begin{equation}
{\cal M}_\ell\simeq\left(
\begin{array}{ccc}
y_ev & \frac{y'_{e\mu}}{\Lambda}vv' & \frac{y'_{e\tau}}{\Lambda}vv' \\
\frac{y_{e\mu}}{\Lambda}vv' & y_{\mu}v & y'_{\mu\tau}v\\
\frac{y_{e\tau}}{\Lambda}vv' & y_{\mu\tau}v & y_{\tau}v
\end{array}
\right),\quad
{\cal M}_\nu \simeq
\left(
\begin{array}{ccc}
\delta m_{\nu}+\frac{\lambda^\nu_6}{\Lambda^3}v^2v'^2 &\frac{\lambda^\nu_4}{\Lambda^2}v^2v' & \frac{\lambda^\nu_5}{\Lambda^2}v^2v'\\
\frac{\lambda^\nu_4}{\Lambda^2}v^2v' & \frac{\lambda^\nu_2}{\Lambda}v^2 &  \frac{\lambda^\nu_1}{\Lambda}v^2\\
\frac{\lambda^\nu_5}{\Lambda^2}v^2v' &   \frac{\lambda^\nu_1}{\Lambda}v^2 &  \frac{\lambda^\nu_3}{\Lambda}v^2
\end{array}
\right),\label{chgd-mass}
\end{equation}
where  $\delta m_{\nu}$ is obtained through the radiative seesaw mechanism in Ref. \cite{Ma:2006km} \footnote{Here we take $i=e$ of  $B-3L_i$ as an example, however the other choices are possible because they do not change the result of the lepton mass and mixing by introducing an appropriate rotating matrix. Only the difference is the final state of the DM annihilation.}.
Here notice that $\eta\equiv(\eta^+,(\eta_R+i\eta_I)/\sqrt2)^T$ with ($\eta^0=(\eta_R+i\eta_I)/\sqrt2)$) does not
have any vacuum expectation values (VEV). Apparently these mass matrices can successfully lead the observed neutrino mass and mixing \cite{t2k, daya, reno}, and our DM candidate $N^c$ or $\eta^0$ is independent of the form of the these matrices. Hence we are apart from the discussion of lepton sector. Notice furthermore that there is no constraint of $N^c$, that is DM, from LFV violation, since the first charged lepton flavor eigenstate $L_e$ is almost  the same as the  mass eigenstate at the leading order as can be seen in Eq. (\ref{chgd-mass}).

\subsection{Higgs Sector}
The Higgs potential of this model is given by \footnote{In details, see Ref. \cite{Okada:2012np}. }
\begin{eqnarray}
V \!\!\!&=&\!\!\!
 m_1^{2} \Phi^\dagger \Phi + m_2^{2} \eta^\dagger \eta  + m_3^{2} \chi^\dagger \chi +
\lambda_1 (\Phi^\dagger \Phi)^{2} + \lambda_2
(\eta^\dagger \eta)^{2} + \lambda_3 (\Phi^\dagger \Phi)(\eta^\dagger \eta)
+ \lambda_4 (\Phi^\dagger \eta)(\eta^\dagger \Phi)
\nonumber \\ &&\!\!\!
+
\lambda_5 [(\Phi^\dagger \eta)^{2} + \mathrm{h.c.}]+
\lambda_6 (\chi^\dagger \chi)^{2} + \lambda_7  (\chi^\dagger \chi)
(\Phi^\dagger \Phi)  + \lambda_8  (\chi^\dagger \chi) (\eta^\dagger \eta),
\end{eqnarray}
where $\lambda_5$ has been chosen to be real without any loss of
generality.
Inserting the tadpole conditions; $m^2_1=-\lambda_1v^2-\lambda_7v'^2/2$
and $m^2_3=-\lambda_6v'^2 - \lambda_7v^2/2$,
the resulting mass matrice are given by
\begin{eqnarray}
m^{2} (\phi^{0},\chi^0) = \left(%
\begin{array}{cc}
  2\lambda_1v^2 & \lambda_7vv' \\
  \lambda_7vv' & 2\lambda_6v'^2 \\
\end{array}%
\right) &=& \left(\begin{array}{cc} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{array}\right)
\left(\begin{array}{cc} m^2_{h} & 0 \\ 0 & m^2_{H}  \end{array}\right)
\left(\begin{array}{cc} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{array}\right), \nonumber\\\\
m^{2} (\eta^{\pm}) &=& m_2^{2} + \frac12 \lambda_3 v^{2} + \frac12 \lambda_8 v'^{2}, \\
m^2_R\equiv m^{2} ( Re \eta^{0}) &=& m_2^{2} + \frac12 \lambda_8 v'^{2}
 + \frac12 (\lambda_3 + \lambda_4 + 2\lambda_5) v^{2}, \\
m^2_I\equiv m^{2} ( Im \eta^{0}) &=& m_2^{2} + \frac12 \lambda_8 v'^{2}
 + \frac12 (\lambda_3 + \lambda_4 - 2\lambda_5) v^{2},
\end{eqnarray}
where $h$ implies standard model (SM)-like Higgs and $H$ is an additional Higgs mass
eigenstate. Here we identify $m_R$ as DM mass ($m_{\rm DM}$).
The masses of $\phi^0$ and $\chi^0$ are rewritten in terms of the mass eigenstates of $h$ and $H$
as
\begin{eqnarray}
\phi^0 &=& h\cos\alpha + H\sin\alpha, \nonumber\\
\chi^0 &=&- h\sin\alpha + H\cos\alpha.
\label{eq:mass_weak}
\end{eqnarray}

\section{Dark Matter}
In our model, there are two DM candidates; that is, the right-handed neutrino $N^C$ or the neutral inert Higgs boson $\eta_R/\eta_I$. In bosonic case, however, we assume $\eta_R$ is DM. The mass $m_N^c$ is given by $y_Nv'^2/\Lambda$, as can be seen in the previous section, and the mass range should be ${\cal O}$(1) GeV $\le m_{\chi}\le$ ${\cal O}$(10) GeV in order to maintain $\lambda^\nu_{i}\le{\cal O}(1)$.
Hereafter, we symbolize DM as $\chi$  and strict ourselves the DM mass region of $1-10$ GeV for both cases.
Below we analyze each of DM candidate.

\subsection{Fermionic Dark Matter}
In case of fermionic DM($N^C\equiv \chi$), the effective Lagrangian is given by
 \begin{eqnarray}
{\cal L}\simeq
y_\nu(\ell_e\eta^+-\nu_e \eta^o) \chi + \frac{m_\chi}{\sqrt2v'} \chi\chi (-h\sin\alpha+H\cos\alpha)
+ \frac{y_f}{\sqrt2}(h\cos\alpha+H\sin\alpha)f\bar f,
 \end{eqnarray}
where the first term plays an  role in explaining the observed relic density reported by WMAP \cite{wmap}.
The other terms are relevant to direct detection experiments \footnote{Notice that  our $\chi$ has a coupling of $\chi-\chi-Z'$, where $Z'$ is an extra neutral gauge boson. However the process vanishes in the spin independent scattering since $\chi$ is the Majorana type particle. }.

The annihilation cross section to explain WMAP is obtained through the $t$ and $u$ channels with the final states of an $e^--e^+$ pair and a $\nu_e-\bar\nu_e$ pair as follows \cite{d6-rad}:
\begin{equation}
\sigma v_{ eff}\simeq \frac{|y_\nu|^2}{24\pi}\frac{m^2_\chi(m^4_\chi+m^4_\eta)}{(m^2_\chi+m^2_\eta)^4}v^2_{ eff},
\label{f-sigma}
\end{equation}
where $v_{eff}$ is the relative velocity and we assume that these masses of the final states are taken to be zero.

In the direct detection, the spin independent elastic cross section $\sigma_{SI}$ with nucleon $N$ is given by
\begin{equation}
\sigma_{SI}\simeq \frac{\mu^2_{\rm DM}}{\pi}\left(\frac{1}{m^2_h}-\frac{1}{m^2_H}\right)^2
\left(\frac{m_\chi m_N\sin\alpha\cos\alpha}{vv'}\sum_q f^p_q \right)^2,
\end{equation}
where $\mu_{\rm DM}=\left(m_\chi^{-1}+m_N^{-1}\right)^{-1}$ is the
DM-nucleon reduced mass.
The parameters $f_q^p$ are determined by the pion-nucleon sigma term as follows:
\begin{eqnarray}
&&f_u^p=0.023,\quad
f_d^p=0.032,\quad
f_s^p=0.020,
\end{eqnarray}
for the light quarks and $f_Q^p=2/27\left(1-\sum_{q\leq 3}f_q^p\right)$ for
the heavy quarks $Q(=c,b,t)$ \cite{pdg} where $q\leq3$ implies the summation of the light
quarks.
As a result, we obtain the elastic scattering cross section as
 \begin{equation}
1.0\times10^{-52}({\rm cm^2})\lesssim\sigma_{SI}\lesssim 1.0\times10^{-50}({\rm cm^2}).
\end{equation}

In the view of indirect detections, we should consider a constraint of no excess of the antiproton flux, which is proportional to $\sigma v_{eff.}/m^2_\chi$, reported by PAMELA \cite{Adriani:2010rc}, since our $m_\chi$ is tiny and couples to quarks.
Moreover if the $\sigma v_{eff.}$ has $s$-wave, the amount of the cross section is retained to the current universe with no suppression of the relative velocity \cite{Okada:2012cc}. As a result antiproton might be overproduced, depending on the profile \cite{Evoli:2011id}.
In other words, our DM is completely safe, because our $\sigma v_{eff.}$ does not have $s$-wave in Eq.(\ref{f-sigma}).

The LHC experiment recently reported an invisible decay of the SM-like Higgs that the branching ratio ${\rm Br}_{inv}$ is excluded to the region $0.4\le{\rm Br}_{inv}$ \cite{Giardino:2012dp}, when $m_\chi<m_h/2$.
 Our invisible decay width is given by
\begin{eqnarray}
\Gamma(h\rightarrow2\mathrm{\chi})
\simeq
\frac{m_h}{2\pi}\left(\frac{ m_\chi\sin\alpha}{v'}\right)^2 \sqrt{\frac14-\left(\frac{m_\chi}{m_h}\right)^2}
\left[\frac12-\left(\frac{m_\chi}{m_h}\right)^2 \right]
,
\end{eqnarray}
where we fix  the mass of SM-like Higgs to be $125~\mathrm{GeV}$ \cite{Higgsd1,Higgsd2}\footnote{See the Ref. \cite{Dittmaier:2012vm} for the total branching ratio of SM Higgs.}.
As a result, we find our ${\rm Br}_{inv}$ is at most less than $10^{-3}$, which is safe for the LHC experiment.

As a conclusion, we obtain the elastic scattering cross section as
\begin{equation}
1.0\times10^{-52}({\rm cm^2})\lesssim\sigma_{SI}\lesssim 1.0\times10^{-50}({\rm cm^2})\
{\rm with}\ 1\ {\rm GeV}\lesssim {m_\chi}\lesssim 10\ {\rm GeV},
\end{equation}
where we take the limit of the maximal mixing of $\sin\alpha$ for simplicity. Notice here that the above result is obtained by the direct detection , since the annihilation cross section to explain WMAP has two independent parameters $y_\nu$ and $m_\eta$  and the invisible decay constraint of LHC satisfies our model.
It implies that future experiments could be tested.

\subsection{Bosonic Dark Matter}
In case of bosonic DM($\eta_R\equiv \chi$),
the mass $m_\chi$ should be always less than the right-handed mass $m_{N^c}(\lesssim{\cal O}(10)\ {\rm GeV})$ in order to forbid the too fast decay through term $y_\nu L_e\eta N^c$, and
there are two annihilation modes;
$t$ and $u$ channel of $2N^c\rightarrow \eta^{0(\pm)}\rightarrow 2\nu(\ell\bar\ell)$
and $s$-channel of  $2\eta\rightarrow h/H\rightarrow f\bar f$. However since the $t$ and $u$ channel does not have $s$ wave contribution, the dominant cross section is given only through the $s$ channel as follows \cite{Kajiyama:2012xg}:
\begin{eqnarray}
\sigma v_{eff.}
&\simeq&
\sum_{f}\frac{c_fy^2_f}{2\pi}
\left|\frac{\lambda_h\cos\alpha}{4m_{\chi}^2-m_{h}^2+im_{h}\Gamma_h}
+\frac{\lambda_H\sin\alpha}{4m_{\chi}^2-m_H^2+im_{H}\Gamma_H}\right|^2,
\label{b-sigma}\\
2\lambda_h&\equiv& (\lambda_3+2\lambda_4+2\lambda_5)v \cos\alpha-\lambda_8v'\sin\alpha,\nonumber\\
2\lambda_H&\equiv& (\lambda_3+2\lambda_4+2\lambda_5)v \sin\alpha+\lambda_8v'\cos\alpha,\nonumber
\end{eqnarray}
where $m_{DM}=m_R$ is DM mass,
$y_f$ is Yukawa coupling of SM matter particle,  $\Gamma_h=4.1\times 10^{-3}$ GeV \cite{higgsdecay}, $\Gamma_H=\sum_{f}\frac{c_fy_{f}^2\cos^2\alpha}{16\pi}m_H\left(1-\frac{4m_{f}^2}{m_H^2}\right)^{3/2}$, and
the color factor $c_f$ is $3$ for quarks and $1$ for leptons.

Since our DM is less than $m_h/2$, the invisible decay constraint should be considered.
Our invisible decay width is give by
$\Gamma(h\rightarrow 2\chi)\simeq\frac{\lambda_h^2}{16\pi m_h}\sqrt{1-4m_{\chi}^2/m_h^2}$, which is the same result as in Ref. \cite{Kajiyama:2012xg}.

In the direct detection, the spin independent elastic cross section $\sigma_{SI}$ with nucleon $N$ is given by
\begin{equation}
\sigma_{SI}=\frac{\mu_{\mathrm{DM}}^2}{\pi}\frac{m_N^2}{m_{\chi}^2v^2}
\left(\frac{\lambda_h\cos\alpha}{m_h^2}+\frac{\lambda_H\sin\alpha}{m_H^2}\right)^2
\left(\sum_qf_q^p\right)^2,
\end{equation}
where $\mu_{\rm DM}$ and $f_q^p$ has been defined in the fermionic part.

Considering all these constraints, we find the elastic scattering cross section\footnote{See the Fig.3 in ref. \cite{Kajiyama:2012xg}} as
\begin{equation}
5.0\times10^{-41}({\rm cm^2})\lesssim\sigma_{SI}\lesssim 5.0\times10^{-40}({\rm cm^2})\
{\rm with}\ 1\ {\rm GeV}\lesssim {m_\chi}\lesssim 10\ {\rm GeV}.
\end{equation}
The range of the
cross section is consistent with  CRESSTII~\cite{cresst}, CoGeNT~\cite{cogent}, and DAMA~\cite{dama}.

\section{Conclusions}
We studied the Dark matter features of right-handed fermion $N^c$ and neutral inert boson $\eta_R$ in
a local $B-3L_i$ partially radiative model. Due to the perturbative theory of the Higgs quartic coupling, both of DMs masses should be within ${\cal O}$(10) GeV. In case of the fermionic DM, we find that the maximum scattering cross section is found to be
$
1.0\times10^{-52}({\rm cm^2})\lesssim\sigma_{SI}\lesssim 1.0\times10^{-50}({\rm cm^2})\
{\rm with}\ 1\ {\rm GeV}\lesssim {m_\chi}\lesssim 10\ {\rm GeV},
$
It implies that future experiments could be tested.
On the other hand, in case of the bosonic DM,  the elastic scattering cross section is found to be
$
5.0\times10^{-41}({\rm cm^2})\lesssim\sigma_{SI}\lesssim 5.0\times10^{-40}({\rm cm^2})\
{\rm with}\ 1\ {\rm GeV}\lesssim {m_\chi}\lesssim 10\ {\rm GeV}.
$
It tells us that the range of the
cross section is consistent with  CRESSTII~\cite{cresst}, CoGeNT~\cite{cogent}, and DAMA~\cite{dama}.

\section*{Acknowledgments}
Author thanks to Dr Yuji Kajiyama and Takashi Toma for fruitful discussion.


\section{Introduction}
Motivated by Ap\'{e}ry's proof of irrationality of $\zeta(3)$, Don Zagier studies in~\cite{Zag} the question of finding those triples of rational numbers $(A,B,\lambda)$ for which the sequence obtained by the recursive formula
\[
(n+1)^2 u_{n+1} \- (An^2 + A n + \lambda)u_n \+ B n^2 u_{n-1} \= 0
\]
starting with $u_0=1$ has all integer terms, i.e. $u_n \in \mathbb Z$. The generating function $\phi_0(t)=1+u_1 t + u_2 t^2 + \dots$ is the normalized  analytic at $t=0$ solution of the differential equation ${\mathcal L} \phi_0 \= 0$ with
\begin{equation}\label{D2zag}
{\mathcal L}(t) \= D^2 \- t \, (A D^2 + A D + \lambda) \+ B \, t^2 \, (D+1)^2   \,,
\end{equation}
where we use the notation $D \= t \frac {d}{dt}$ throughout the paper. We will refer later on to~\eqref{D2zag} as the Beukers-Zagier differential operator since it appeared in the works of these two authors. A table of respective  triples  $(A,B,\lambda)$   is obtained in~\cite{Zag} by searching in a large range of values. It appears that all degenerate  cases in the table, i.e. those
with either $A^2 = 4 B$ or $ B = 0$
come as members of infinite families of triples $(A,B,\lambda)$ with $\phi_0(t) \in \mathbb Z[[t]]$.
On the contrary, imposing the assumption
\begin{equation}\label{nondeg}
A^2 \ne  4 B, \;  B \ne 0
\end{equation}
one arrives at 14 ``sporadic'' cases with no obvious pattern. Remarkably, in all those sporadic cases the corresponding differential equation can be parametrized by modular forms. Namely, one can find a modular function $t(\tau)$ that vanishes at $\infty$ and a modular form $f(\tau)$ of weight~1  such that $\phi_0(t(\tau))=f(\tau)$ for all $\tau$ in the upper half-plane with large enough imaginary part. These cases are listed in the table below.

\begin{center}
\begin{tabular}{ccc|cc|cccccc}
&&&&&&&&&&\\
$A$&$B$& $\lambda$&$t(\tau)$ & $f(\tau)$ &$u_0$& $u_1$&$u_2$& $u_3$&$u_4$ & $u_5$\\
\hline
&&&&&&&&&&\\
7&-8&2&  ${\bf \frac{1^36^9}{2^33^9}}$&${\bf \frac{2^13^6}{1^26^3}}$& 1 &2 &10 &56&346&2252\\
&&&&&&&&&&\\
9&27&3&  ${\bf \frac{1^34^318^9}{2^99^336^3}}$&${\bf \frac{2^93^112^1}{1^34^36^3}}$& 1 &3 &9 &21&9&-297\\
&&&&&&&&&&\\
10&9&3&  ${\bf \frac{1^46^8}{2^83^4}}$&${\bf \frac{2^63^1}{1^36^2}}$& 1 &3 & 15&93&639&4653\\
&&&&&&&&&&\\
11&-1&3& $q \prod\limits_{n=1}^{\infty}(1-q^n)^{5 (\frac{n}{5})}$ &$\Bigl(\frac1{t(\tau)}\cdot {\bf \frac{5^5}{1^1}}\Bigr)^{1/2}$&1  &3 &19 &147&1251&11253\\
&&&&&&&&&&\\
12&32&4&  ${\bf \frac{1^44^28^4}{2^{10}}}$&${\bf \frac{2^{10}}{1^44^4}}$& 1 &4 &20 &112&676&4304\\
&&&&&&&&&&\\
17&72&6&  ${\bf \frac{1^53^14^56^212^1}{2^{14}}}$&${\bf \frac{2^{15}3^212^2}{1^64^66^5}}$& 1 &6 &42 &312&2394&18756\\
&&&&&&&&&&\\
0&-16&0&  ${\bf \frac{2^48^8}{4^{12}}}$&${\bf \frac{4^{10}}{2^48^4}}$&  1&0 &4 &0&36&0\\
\end{tabular}
\end{center}

\noindent
The products in the forth and fifth columns stand for eta-products, e.g. ${\bf \frac{1^36^9}{2^33^9}} = \frac{\eta(\tau)^3\eta(6\tau)^9}{\eta(2\tau)^3\eta(3\tau)^9}$. We use this notation throughout the paper. For each of the first 6 rows one should also consider $(\tilde{A},\tilde{B},\tilde{\lambda})=(-A,B,-\lambda)$ with the corresponding $\tilde{u}_n=(-1)^n u_n$, $\tilde{t}(\tau)=-t(\tau)$, $\tilde{f}(\tau)=f(\tau)$. For the last row there is also a triple $(0,16,0)$ leading to $\tilde{u}_n=(-1)^{n/2} u_n$. In total, the table gives us  14 triples $(A,B,\lambda)$ satisfying~\eqref{nondeg} with $\phi_0(t)\in\mathbb Z[[t]]$. Zagier conjectures that there are no more such cases, or if there are, they all will have modular parametrization.

We observe in this paper that the differential operator~\eqref{D2zag} satisfying the assumption~\eqref{nondeg} is a specific form of the so called determinantal differential operator of order~2. The necessary definitions and properties will be recalled in Section~\ref{sec:DN}. We deal with determinantal differential operators of order~2 in Sections~\ref{sec:D2} and~\ref{sec:D2modular} and recover Zagier's list above in a new context. Then we proceed  to obtain an analog of this list for determinantal differential operators of order~3, namely, the complete list of D3's that satisfy a multiplicativity property that we discuss later.

\section{Determinantal differential equations}\label{sec:DN}

Determinantal differential equations of order $N$ were defined in~\cite{GS}.
A DN equation is obtained from  an $(N+1)\times(N+1)$ matrix $A=(a_{ij})_{i,j=0}^N$
that satisfies
\begin{equation}\label{asA}\begin{aligned}
&a_{ij}\=0\,, \quad i-j>1 \\
&a_{ij}\=1\,, \quad i-j=1 \\
&a_{ij}\=a_{N-j,N-i}\,, \quad i-j<1 \\
\end{aligned}\end{equation}
The respective differential operator is then defined as
\[
{\mathcal L}_{A,\infty}(z) \= \det\,_{\mathrm{right}} \,\Bigl( \delta_{ij} z \frac{d}{dz} - a_{ij} \bigl(\frac{d}{dz}\bigr)^{j-i+1} \Bigr) \, \bigl(\frac{d}{dz}\bigr)^{-1}
\]
where $\delta_{ij}$ is the Kronecker symbol and $\det_{\mathrm{right}}$ refers to the way of expanding the determinant of a matrix with non-commuting entries with respect to the rightmost column.

The matrix $A$ can be reconstructed from the coefficients of the differential operator (\cite{GS}, Corollary 3.3). Assume in addition that all eigenvalues of $A$ are distinct. Then obviously $A$ is diagonalizable. In fact, for a matrix satisfying~\eqref{asA} the two conditions are equivalent: $A$ is diagonalizable if and only if all eigenvalues of $A$ are distinct. It follows immediately if one observes that $A$ cannot have an eigenvector whose  last component iz zero.
According to Corollary 6.4 in~\cite{GS} the singularities of the differential operator ${\mathcal L}_{A,\infty}(z)$ are regular singular points located at $\infty$ and the eigenvalues $\lambda_0,\dots,\lambda_N$ of $A$. Moreover, the differential equation has maximal unipotent monodromy at $z=\infty$ and the valuation of its analytic solution at $z=\infty$ is equal to $1$. This motivates the following notation.

\begin{definition} A differential operator of order $N$ is of type $DN_{\infty,1}$ if it equals ${\mathcal L}_{A,\infty}(z)$ for some matrix $A$ satisfying~\eqref{asA}.\end{definition}

\bigskip

We denote the characteristic polynomial of $A$ by $F(z)=\det(z-A)$
throughout the paper.  It will be convenient to also use the variable $t = \frac1z$. Namely, consider the operator
\[
{\mathcal L}_{A,0}(t) \= (-1)^N \, {\mathcal L}_{A,\infty}\Bigl( \frac1t \Bigr) t\,.
\]
The respective differential equation has maximal unipotent monodromy at $t=0$ and the valuation of its analytic solution at this point equals $0$.

\begin{definition} A differential operator of order $N$ is of type $DN_{0,0}$ if it equals ${\mathcal L}_{A,0}(t)$ for some matrix $A$ satisfying~\eqref{asA}.
\end{definition}

\bigskip

By DN we mean either $DN_{0,0}$ or $DN_{\infty,1}$ the case being clear from the context. Observe that the following operations  with the defining  matrices
\[\begin{aligned}
& A \mapsto A' = A + \varepsilon \\
& A \mapsto A'' = \Bigl( \lambda^{j-i+1} a_{ij} \Bigr)
\end{aligned}\]
lead to the substitutions in the differential equations
\begin{equation}\label{shiftDN}
{\mathcal L}_{A',\infty}(z) \= {\mathcal L}_{A,\infty}(z - \varepsilon)\,, \qquad {\mathcal L}_{A',0}(t) \= {\mathcal L}_{A,0}\Bigl( \frac{t}{1-\varepsilon t}\Bigr) \, (1-\varepsilon t)
\end{equation}
and
\begin{equation}\label{scaleDN}
{\mathcal L}_{A'',\infty}(z) \= \lambda \, {\mathcal L}_{A,\infty}\Bigl(\frac{z}{\lambda}\Bigr)\,,\qquad {\mathcal L}_{A'',0}(t) \= {\mathcal L}_{A,0}( \lambda t)
\end{equation}
respectively.

\section{The Beukers-Zagier equation as a D2 equation}\label{sec:D2}

Let us consider D2 equations in detail. According to our definitions one has
\[\begin{aligned}
&{\mathcal L}_{A,\infty}(z) \= \det\,_{right} \, \begin{pmatrix}
(z-a_{00})\frac{d}{dz} & -a_{01}\bigl(\frac{d}{dz}\bigr)^2 & -a_{02}\bigl(\frac{d}{dz}\bigr)^3 \\
-1 & (z-a_{11})\frac{d}{dz} & -a_{01}\bigl(\frac{d}{dz}\bigr)^2 \\
0 & -1 & (z-a_{00})\frac{d}{dz}\\
\end{pmatrix} \; \bigl(\frac{d}{dz}\bigr)^{-1}\\
&\= -a_{02}\bigl(\frac{d}{dz}\bigr)^2 -a_{01}\bigl(\frac{d}{dz}\bigr)^2 (z-a_{00}) + (z-a_{00})\frac{d}{dz}\Bigl( (z-a_{11})\frac{d}{dz}(z-a_{00}) -a_{01}\frac{d}{dz}\Bigr)\\
&\= F(z) \, \bigl(\frac{d}{dz}\bigr)^2 \+ F'(z) \, \frac{d}{dz} \+ (z-a_{00})
\end{aligned}\]
where
\[\begin{aligned}
F(z) &\= \det\bigl(z-A\bigr) \= z^3 \+ \alpha_2 z^2 \+ \alpha_1 z \+ \alpha_0 \\
&\alpha_2 \= -a_{11}-2a_{00}\\
&\alpha_1 \= 2 a_{00} a_{11} + a_{00}^2 - 2 a_{01}\\
&\alpha_0 \= 2 a_{00} a_{01} - a_{00}^2 a_{11} - a_{02}\\
\end{aligned}\]
A $D2_{\infty,1}$ differential operator is then any operator of the form
\[
F(z) \, \bigl(\frac{d}{dz}\bigr)^2 \+ F'(z) \, \frac{d}{dz} \+ (z- \beta)
\]
with a cubic monic polynomial with distinct roots $F(z) \= z^3 \+ \alpha_2 z^2 \+ \alpha_1 z \+ \alpha_0$. One can recover the matrix $A$ from $\alpha_i$ and $\beta$ via
\[\begin{aligned}
&a_{00} \= \beta \\
&a_{11} \=  - 2 \beta - \alpha_2 \\
&a_{01} \= -\frac32 \beta^2  - \beta \alpha_2 -\frac12 \alpha_1 \\
&a_{02} \= -\beta^3-\beta^2\alpha_2 -\beta a_1 -\alpha_0
\end{aligned}\]

\bigskip

The generic equation of type $D2_{0,0}$ is then
\[\begin{aligned}
&F\Bigl(\frac1t\Bigr) \Bigl(-t^2 \frac{d}{dt}\Bigr)^2 t \+ F'\Bigl(\frac1t\Bigr) \Bigl(-t^2 \frac{d}{dt}\Bigr)t \+ \Bigl(\frac1t - \beta\Bigr) t \\
& \= t^5 F\Bigl(\frac1t\Bigr) \Bigl(\frac{d}{dt}\Bigr)^2 \+ \Bigl(4 t^4 F\Bigl(\frac1t\Bigr) - t^3 F'\Bigl(\frac1t\Bigr) \Bigr) \frac{d}{dt}
+  2 t^3 F\Bigl(\frac1t\Bigr) - t^2 F'\Bigl(\frac1t\Bigr) + 1 - \beta t \\
& \= t G(t) \Bigl(\frac{d}{dt}\Bigr)^2 + t G'(t) \frac{d}{dt} + t H(t) \\
\end{aligned}\]
with
\[\begin{aligned}
& G(t) \= t + \alpha_2 t^2 + \alpha_1 t^3 + \alpha_0 t^4 \\
& H(t) \= -\beta + \alpha_1 t + 2 \alpha_0 t^2
\end{aligned}\]
With the notation $D = t \frac{d}{dt}$ we can further rewrite it as
\begin{equation}\label{D2}\begin{aligned}
(1 + \alpha_2 t + \alpha_1 t^2 + \alpha_0 t^3) (D^2-D) &\+ (1 + 2\alpha_2 t + 3 \alpha_1 t^2 + 4 \alpha_0 t^3)D \\
& -\beta t + \alpha_1 t^2 + 2 \alpha_0 t^3 \\
\= D^2 \+ t (\alpha_2 D^2 + \alpha_2 D - \beta) &\+ \alpha_1 t^2 (D+1)^2 + \alpha_0 t^3 (D+1)(D+2)
\end{aligned}\end{equation}

Notice that putting $\alpha_0=0$ we obtain precisely operator~\eqref{D2zag} with $A=-\alpha_2$, $B=\alpha_1$ and $\lambda=\beta$.

\section{Modular  equations D2}\label{sec:D2modular}

Recall that a D2 differential equation depends on 4 parameters $(\alpha_2,\alpha_1,\alpha_0,\beta)$. It determines a local system of rank~2 over the base
\[
\mathbb{P}^1(\mathbb C) \setminus \{ \infty,\, \text{the roots of } z^3+\alpha_2 z^2 +\alpha_1 z+\alpha_0 \}\,.
\]
Consider also the basis in the space of solutions of D2 near $t=0$ which consists on normalized analytic and logarithmic solutions:
\[\begin{aligned}
\phi_0(t) &\= 1 + \beta t + \Bigl(-\frac12\alpha_2 \beta + \frac14 \beta^2 - \frac14\alpha_1\Bigr)t^2 + \dots \\
\phi_1(t) &\= \log t \, \phi_0(t) \+ \Bigl(-\alpha_2 - 2\beta\Bigr)t + \Bigl(\frac12\alpha_2^2 + \frac12\alpha_2\beta - \frac34\beta^2 - \frac14\alpha_1\Bigr) t^2 + \dots
\end{aligned}\]

\begin{definition}  We say that an equation D2 with parameters
$(\alpha_2,\alpha_1,\alpha_0, \beta) \in \mathbb Q^4$ is modular if the analytic continuation of
\[
\tau \= \frac1{2 \pi i} \frac{\phi_1(t)}{\phi_0(t)}
\]
gives uniformization of the base by the upper halfplane with the group of deck transformations being a congruence subgroup of $\rm{SL}(2,\mathbb Z)$ and the function $\tau \mapsto \phi_0(t(\tau))$ is a modular form of weight~1.
\end{definition}

In this case  $t(\tau)$ is a modular function whose $q$-expansion can be written explicitly. Indeed, inverting the series
\[
q = \exp\Bigl( \frac{\phi_1(t)}{\phi_0(t)}\Bigr) \= t \+ (- \alpha_2 - 2 \beta)t^2 + (\alpha_2^2 + \frac72 \alpha_2 \beta + \frac{13}4 \beta^2 - \frac14 \alpha_1)t^3 + \dots
\]
one gets
\[
t \=  q \+ (\alpha_2 + 2 \beta)q^2 + (\alpha_2^2 + \frac92 \alpha_2 \beta + \frac{19}4 \beta^2 + \frac14 \alpha_1)q^3 + \dots
\]
Further, substituting this expansion into $\phi_0(t)$ one obtains
\[
f \= \phi_0(t(\tau)) \= 1 \+  \beta q \+ (\frac12 \alpha_2\beta + \frac94 \beta^2 - \frac14 \alpha_1) q^2 \+ \dots
\]
This must be a modular form of weight~1.

\bigskip

Put $Q=q^{\frac12}$ and consider the series
\begin{equation}\label{cndef}
t^{\frac12}  \sqrt{1+\alpha_2 t +\alpha_1 t^2+\alpha_0 t^3} \, \phi_0(t)^2 \= \sum_{n=1}^{\infty} c_n Q^n
\end{equation}
whose coefficients $c_n=c_n(\vec{\alpha},\beta)$ can be determined explicitly as follows. One writes
\[
Q \= \exp\Bigl( \frac12 \frac{\phi_1(t)}{\phi_0(t)}\Bigr) \= t^{\frac12} \Bigl(1 + (-\frac12 \alpha_2 - \beta)t + (\frac38 \alpha_2^2 + \frac54 \alpha_2 \beta + \frac98 \beta^2 - \frac18 \alpha_1)t^2 + \dots \Bigr)
\]
and inverts this series in order to get
\[
t^{\frac12} \= Q \Bigl(1 + (\frac12 \alpha_2 + \beta)Q^2 + (\frac38 \alpha_2^2 + \frac74 \alpha_2 \beta + \frac{15}8 \beta^2 + \frac18 \alpha_1)Q^4 + \dots \Bigr)
\]
which can be substituted into the left-hand side of~\eqref{cndef}. We have
\begin{equation}\label{cnformulas}\begin{aligned}
& c_1 \= 1 \\
& c_2 \= 0 \\
& c_3 \= \alpha_2 + 3 \beta \\
& c_4 \= 0 \\
& c_5 \= \alpha_2^2 + \frac{25}{4} \alpha_2 \beta + \frac{75}{8} \beta^2 + \frac18 \alpha_1 \\
& \dots
\end{aligned}\end{equation}
It is not hard to see that all even coefficients in fact vanish.

\begin{theorem} \label{LfunD2}
Assume one is given a nondegenerate modular D2  with parameters $\alpha_2,\alpha_1,\alpha_0,\beta \in \mathbb Q$ . Consider the  weight~2 modular form  $\sum_{n=1}^{\infty} c_n q^n$ with $c_n$  determined from the expansion~\eqref{cndef}. If in addition it is a newform then
\[
L(s) \= \sum_{n=1}^{\infty} \frac{c_n}{n^s}
\]
is the L-function of the elliptic curve
\begin{equation}\label{ellcurve}
y^2 = z^3 +\alpha_2 z^2 +\alpha_1 z+\alpha_0 \,.
\end{equation}
\end{theorem}
By being a newform we
mean that the modular form belongs to the subspace on
 newforms of certain level.
 We do not require it to be a Hecke eigenform a priori; rather,   the Hecke--eigen
property is a consequence of the theorem. In particular, we have the following

\begin{corollary}
If $\sum_{n=1}^{\infty} c_n q^n$ is a newform, then its coefficients~\eqref{cnformulas} are multiplicative, i.e.
\begin{equation}\label{multrel}
c_{mn}(\vec{\alpha},\beta) \= c_{m}(\vec{\alpha},\beta) \, \cdot \, c_{n}(\vec{\alpha},\beta)
\end{equation}
as soon as $m$ and $n$ are coprime.
\end{corollary}

We will solve equations~\eqref{multrel} with respect to the parameters $\alpha_2,\alpha_1,\alpha_0,\beta$ later in this section. It appears that modulo a certain transformation which preserves both the sequence $\{ c_n; n \ge 1\}$ and the $L$-function of~\eqref{ellcurve} there are  finitely many cases.

\bigskip

The proof of Theorem~\ref{LfunD2} will rely on the following result.

\bigskip

{\bf Theorem} (Atkin $\&$ Swinnerton-Dyer congruences, Theorem~4 in~\cite{ASD})
\emph{
Let $p \ne 2,3$, and let $y^2=z^3+Bz+C$ be an elliptic curve over  $\mathbb{Z}_p$ with good reduction. Choose a local parameter at $0$ so that  $z = \xi^{-2} + \sum_{n=-1}^{\infty} d_n \xi^n$ and $y=\xi^{-3}+\dots$ are the respective expansions , and write
\[
- \frac12 \frac{dz}{y} \= \Bigl(\sum_{n=1}^{\infty} c_n \xi^n \Bigr) \frac{d\xi}{\xi}\,.
\]
If $B,C,d_n,c_n$ are $p$-adic integers, then
\begin{equation}\label{ASDcong}
c_{np} \- a_p \, c_{n} \+ p \, c_{\frac{n}p} \; \equiv\;  0 \mod p^{{\rm ord}_p(n)+1}
\end{equation}
where
\[
a_p \= - \sum_{m=0}^{p-1} \Bigl( \frac{m^3+Bm+C}{p}\Bigr)\,.
\]
}

Notice that this theorem can be applied to an elliptic curve defined over $\mathbb Q$ with  good reduction at $p$  as soon as the coefficients $B$, $C$, $d_n$ and $c_n$ do not contain $p$ in their denominators. Moreover, $a_p = p+1-\#E({\mathbb F}_p)$ is then the $p$-th coefficient of the $L$-function of this elliptic curve.

\begin{proof} [Proof of Theorem~\ref{LfunD2}.] Let $a_n$, $n \ge 1$ be the  coefficients of the $L$-function $L(s)=\sum_n \frac{a_n}{n^s}$ of the elliptic curve~\eqref{ellcurve}. One can check that
\[
\phi_1'(t) \phi_0(t) - \phi_1(t) \phi_0'(t) \= \frac1{t(1+\alpha_2 t+\alpha_1 t^2 + a_0 t^3)}\=  t^{-4} F\Bigl(\frac1t\Bigr)^{-1}\,,
\]
hence
\[\begin{aligned}
& \sum_{n=1}^{\infty} c_n(\vec{\alpha},\beta) Q^n \; \frac{dQ}{Q} \= t^2 F\Bigl(\frac1t\Bigr)^{\frac12} \phi_0(t)^2 \; d \Bigl( \frac12 \frac{\phi_1(t)}{\phi_0(t)}\Bigr) \\
&\= \frac12 t^2 F\Bigl(\frac1t\Bigr)^{\frac12} \Bigl(\phi_1'(t) \phi_0(t) - \phi_1(t) \phi_0'(t)\Bigr) \, dt \= \frac12 t^{-2} F\Bigl(\frac1t\Bigr)^{-\frac12} dt \= -\frac12 \frac{dz}{y}
\end{aligned}\]
where we substitute $z = 1/t$, $y^2=F(z)$. This is a holomorphic differential on the curve~\eqref{ellcurve}, and since $Q \sim t^{\frac12}$ for small $t$ we conclude that $Q$ is a local parameter on the curve near the origin. Moreover, $z \sim Q^{-2}$ and $y \sim Q^{-3}$ and therefore the theorem of Atkin and Swinnerton-Dyer would be applicable for every prime $p$ not dividing the conductor of the curve as soon as all $c_n$ and $d_n$ defined from the expansion $z = \frac1{t} = Q^{-2} + \sum_{n=-1}^{\infty} d_n Q^n$ do not contain $p$ in their denominators. First we show that this is indeed the case for all but finitely many primes $p$ using the assumption of modularity.

We have $Q(t(\tau)) = q^{\frac12}$. Looking at~\eqref{cndef} we see that $\sum_{n=1}^{\infty} c_n q^n$ is the $q$-expansion of the modular form $\Bigl[t^{\frac12}  \sqrt{1+\alpha_2 t +\alpha_1 t^2+\alpha_0 t^3} f^2 \Bigr] (2 \tau)$ of weight~2. It follows that possibly after  multiplication by an integer all $c_n$ become integers simultaneously. The same holds for $d_n$ since $z(\tau)$ is a modular function. Therefore for all but finitely many prime numbers $p$ we have congruences~\eqref{ASDcong}. Another consequence of modularity of $\sum_{n=1}^{\infty} c_n q^n$ is that
\begin{equation}\label{cnass}
c_n = o(n)\,, \qquad n \to \infty\,.
\end{equation}
Our next step is to show that~\eqref{ASDcong} together with~\eqref{cnass} imply that $c_n=a_n$ for all $n$ not divisible by a finite set of primes. Since $a_n = o(n^{\frac12 + \varepsilon})$ for any $\varepsilon > 0$ and $c_n=o(n)$ there is a number $N$ such that
\[
\Big|\frac{c_n}{n} \Big| < \frac12\,, \quad \Big|\frac{a_n}{n} \Big| < \frac12
\]
for all $n > N$. Obviously we can assume that~\eqref{ASDcong} is true for all $p>N$ increasing $N$ if necessary. From~\eqref{ASDcong} with $n=1$ we get
\[
c_p \equiv a_p \mod p\,.
\]
Since for all $p>N$ also $|c_p|, |a_p| < \frac{p}2$ we conclude that $c_p=a_p$. Suppose $p>N$ and we have proved that $c_{p^m}=a_{p^m}$ for all $m\le M$. Since
\[\begin{aligned}
& a_{p^{M+1}} \- a_p a_{p^M} + p a_{p^{M-1}} \=0\\
& c_{p^{M+1}} \- a_p c_{p^M} + p c_{p^{M-1}} \equiv 0 \mod p^{M+1}\\
\end{aligned}\]
we conclude that
\[
c_{p^{M+1}} \equiv a_{p^{M+1}} \mod p^{M+1}\,.
\]
Therefore $c_{p^{M+1}}=a_{p^{M+1}}$ because $p^{M+1}>N$ again. It follows now by induction that $c_{p^m}=a_{p^m}$ for all $m \ge 0$ and $p>N$. Our next step is to show that $c_n=a_n$ for all $n$ not divisible by finite number of primes $p \le N$. Let $n$ be such a number and suppose that for every proper divisor $n'|n$ we have already proved that $a_{n'}=c_{n'}$. By~\eqref{ASDcong} with a prime divisor $p|n$ and  $n'=\frac{n}{p}$ instead of $n$ we have
\[
c_n \- a_p c_{n'} + p c_{\frac{n'}{p}} \= c_n \- a_n \equiv 0 \mod p^m
\]
where $m={\rm ord}_p(n)$. Since this is true for every prime dividing $n$ we conclude that $c_n \equiv a_n \mod n$, and therefore $c_n=a_n$ again by our estimate.

Consider both newforms $\sum_n c_n q^n$ and $\sum_n a_n q^n$ on the intersection of the corresponding congruence subgroups which is again a congruence subgroup. Since $c_n=a_n$ for all $n$ not divisible by primes form a certain finite set, it follows that both forms have the same eigenvalues for infinitely many Hecke operators. Therefore by multiplicity one theorem these forms are just equal, and our theorem is proved.
\end{proof}

Now we can  substitute the polynomials~\eqref{cnformulas} into the multiplicativity relations~\eqref{multrel} and solve the resulting equations. We do not expect finitely many solutions because the shifts~\eqref{shiftDN} preserve modularity. Under the shift $A \mapsto A'=A+\varepsilon$ the parameters become
\[
(\alpha_2',\alpha_1',\alpha_0',\beta') \= (\alpha_2-3 \varepsilon, 3\varepsilon^2 - 2\alpha_2\varepsilon + \alpha_1,-\varepsilon^3 + \alpha_2\varepsilon^2 - \alpha_1\varepsilon + \alpha_0,\beta+\varepsilon) \,.
\]
Zagier's choice $\alpha_0=0$ is not natural from this point of view because one can make $\alpha_0=0$ only if $F(z)$ has a rational root. It is more natural to choose the equation with $\beta=0$ as a unique representative of the orbit of the shifts.

Solving the first few relations with Gr\"{o}bner bases ( we used computer algebra system~\cite{MAGMA})
\[\begin{aligned}
& c_6\=c_2 \cdot c_3\,,\quad c_{10}\=c_2 \cdot c_5\,,\quad c_{12}\=c_4 \cdot c_3\,,\quad c_{14}\=c_2 \cdot c_7\,,\\
& c_{15}\=c_3 \cdot c_5\,,\quad c_{18}\=c_2 \cdot c_9\,,\quad c_{21}\=c_3 \cdot c_7\,,\quad c_{22}\=c_2 \cdot c_{11}\\
\end{aligned}\]
we obtain 8 points $(\alpha_2,\alpha_1,\alpha_0)$ plus two one-parametric families $(0,0,\alpha_0)$ and $(0,\alpha_1,0)$. In order to show that there are actually finitely many cases in these families we used more relations by considering about 200 further coefficients. The results are given in the table below.

\bigskip

\begin{center}
\begin{tabular}{ccc|l|l}
$\alpha_2$&$\alpha_1$& $\alpha_0$ & $F(z)$ & $(A,B,\lambda)$\\
\hline
1&0&0&$z^2(z+1)$&$(-1,0,0),(2,1,1)$\\
-1&0&0&$z^2(z-1)$&$(1,0,0),(-2,1,-1)$\\
-4&-80&-192&$(z-12)(z+4)^2$&$(-32,256,-12),(16,0,4)$\\
4&-80&192&$(z+12)(z-4)^2$&$(32,256,12),(-16,0,-4)$\\
\hline
-2&-40&-75&$(z+3)(z^2-5z-25)$&$(11,-1,3)$\\
2&-40&75&$(z-3)(z^2+5z-25)$&$(-11,-1,-3)$\\
-1&-24&-36&$(z-6)(z+2)(z+3)$&$(-17,72,-6),(7,-8,2),(10,9,3)$\\
1&-24&36&$(z+6)(z-2)(z-3)$&$(17,72,6),(-7,-8,-2),(-10,9,-3)$\\
0& $16 \zeta_4$ &0&$z(z^2 - 16 \zeta_4)$&$(0,\pm16,0),(12,32,4),(-12,32,-4)$\\
0&0& $27 \zeta_6$ &$z^3-27 \zeta_6$&$(9,27,3),(-9,27,-3)$\\
\end{tabular}
\end{center}
\bigskip

\noindent
The first three columns contain all solutions of a few first multiplicativity equations.
The fourth column shows the roots of the respective polynomial $F(z)=z^3+\alpha_2 z^2 + \alpha_1 z+\alpha_0$. Polynomials in the first 4 rows appear to have multiple roots meaning that the respective differential operator is degenerate. In the last column we shift the differential operator by various roots of $F(z)$ in order to obtain operators with $\alpha_0=0$, the respective values of the parameters $(A,B,\lambda)=(-\alpha_2,\alpha_1,\beta)$ being listed. The last 6 rows give us D2 equations, and shifting by various rational roots we obtain precisely Zagier's table. One can easily check at this point that in each case the statement of Theorem~\ref{LfunD2} holds, and therefore the respective $(\alpha_2,\alpha_1,\alpha_0)$ indeed solve all multiplicativity equations. The triples corresponding to the degenerate differential equations from the first four rows can be found in~\cite{Zag} as $\#1,\#3, \#19$ and $\#11$. On the other hand, the degenerate triples $\# 14, \#20$ and $\#25$ are also modular but do not appear on our list.

\section{Differential equations of type D3}\label{sec:D3}

Our goal in this section is to write the generic form of a D3 equation by making exactly the same steps as in Section~\ref{sec:D2} but now with $N=3$. We get

\[\begin{aligned}
&{\mathcal L}_{A,\infty} \= \det\,_\mathrm{right} \, \begin{pmatrix}
(z-a_{00})\frac{d}{dz} & -a_{01}\bigl(\frac{d}{dz}\bigr)^2 & -a_{02}\bigl(\frac{d}{dz}\bigr)^3 & -a_{03}\bigl(\frac{d}{dz}\bigr)^4 \\
-1 & (z-a_{11})\frac{d}{dz} & -a_{12}\bigl(\frac{d}{dz}\bigr)^2 & -a_{02}\bigl(\frac{d}{dz}\bigr)^3 \\
0 & -1 & (z-a_{11})\frac{d}{dz} & -a_{01}\bigl(\frac{d}{dz}\bigr)^2\\
0 & 0 & -1 & (z-a_{00})\frac{d}{dz}\\
\end{pmatrix} \; \bigl(\frac{d}{dz}\bigr)^{-1}\\
&\= F(z) \bigl(\frac{d}{dz}\bigr)^3 \+ \frac32 F'(z) \bigl(\frac{d}{dz}\bigr)^2 \+ \Bigl(\frac12 F''(z) + G(z) \Bigr) \frac{d}{dz} + \frac12 G'(z)
\end{aligned}\]
with
\[\begin{aligned}
F(z) &\= \det\bigl(z-A\bigr) \= z^4 \+ \alpha_3 z^3 \+ \alpha_2 z^2 \+ \alpha_1 z \+ \alpha_0\\
&\alpha_3 \= -2 a_{11} - 2 a_{00}\\
&\alpha_2 \= 4 a_{00} a_{11} + a_{00}^2 + a_{11}^2 - 2 a_{01} - a_{12}\\
&\alpha_1 \= -2 a_{02} -2a_{11}a_{00}^2 + 2 a_{00}(a_{01} -a_{11}^2 + a_{12}) + 2 a_{11}a_{01}\\
&\alpha_0 \= -a_{03} + 2 a_{00}a_{02} + (a_{11}^2 - a_{12})a_{00}^2 - 2 a_{11}a_{01}a_{00} + a_{01}^2\\
G(z) &\= z^2 \+ \beta_1 z + \beta_0\\
&\beta_1 \= -2 a_{00} \\
&\beta_0 \= 2 a_{00} a_{11} - a_{11}^2 - 2 a_{01} + a_{12}\\
\end{aligned}\]

Recall that  a $D3_{\infty,1}$ differential equation is called non--degenerate whenever the roots of $F(z)$ are distinct. Notice that our order $3$  differential operator is the symmetric square of the order $2$ operator
\[
F(z) \bigl(\frac{d}{dz}\bigr)^2 \+ \frac12 F'(z) \frac{d}{dz} \+ \frac14 G(z)\,.
\]

\bigskip

 We also compute the generic $D3_{0,0}$. We have
\[\begin{aligned}
&-F\Bigl(\frac1t\Bigr) \Bigl(-t^2 \frac{d}{dt}\Bigr)^3 t \- \frac32 F'\Bigl(\frac1t\Bigr) \Bigl(-t^2 \frac{d}{dt}\Bigr)^2 t \-
\Bigl( \frac12 F''\Bigl(\frac1t\Bigr) + G\Bigl(\frac1t\Bigr)\Bigr) \Bigl(-t^2 \frac{d}{dt}\Bigr)t \- \frac12 G'\Bigl(\frac1t\Bigr) t\\
& \= t \Bigl[ H(t) \bigl(\frac{d}{dt}\bigr)^3 \+ \frac32 H'(t) \bigl(\frac{d}{dt}\bigr)^2 \+ \Bigl(\frac12 H''(t) + U(t) \Bigr) \frac{d}{dt} + \frac12 U'(t) \Bigr]
\end{aligned}\]
with
\[\begin{aligned}
&H(t) \= t^6 F \Bigl(\frac1t\Bigr) \= t^2 + \alpha_3 t^3 + \alpha_2 t^4 + \alpha_1 t^5 + \alpha_0 t^6 \\
&U(t) \= t^2 G \Bigl(\frac1t\Bigr) \+ 3 t^4 F \Bigl(\frac1t\Bigr) \- t^3 F' \Bigl(\frac1t\Bigr)  \= 3 \alpha_0 t^4 + 2 \alpha_1 t^3 + (\alpha_2 + \beta_0) t^2 + \beta_1 t
\end{aligned}\]

Finally, this differential operator can be written as
\begin{equation}\label{D3}\begin{aligned}
D^3 &\+ t\bigl(D + \frac12\bigr) \bigl(\alpha_3 (D^2 + D) \+ \beta_1\bigr) \\
&\+ t^2 (D+1) \bigl( \alpha_2 (D+1)^2 \+ \beta_0  \bigr) \\
&\+ \alpha_1 t^3 (D+2)\bigl(D+\frac32\bigr)(D+1)\\
&\+ a_0 t^4 (D+3)(D+2)(D+1)
\end{aligned}\end{equation}

\section{ Nondegenerate modular equations D3}

In this section we will prove the analog of Theorem~\ref{LfunD2} for D3 equations. In order to state  it we first associate to such an equation an appropriate elliptic curve. Recall that $F(z)=\det\bigl(z-A\bigr)$ has distinct roots, so the discriminant of $F$ is nonzero. For a $D3$ equation we have $F(z) \= z^4 \+ \alpha_3 z^3 \+ \alpha_2 z^2 \+ \alpha_1 z \+ \alpha_0$. Consider the curve
\begin{equation}\label{C}
w^2 \= z^4 \+ \alpha_3 z^3 \+ \alpha_2 z^2 \+ \alpha_1 z \+ \alpha_0.
\end{equation}
 Put it into the Weierstrass form.

\begin{definition}\label{spec_ell} The spectral elliptic curve of a D3 equation is
\begin{equation}\label{spec_ell_D3}\begin{aligned}
y^2 \= x^3 \+ \Big( \alpha_1\alpha_3 - & \frac13 \alpha_2^2 - 4 \alpha_0 \Bigr) x \\
& \+ \Bigl( \alpha_0\alpha_3^2-\frac13\alpha_1\alpha_2\alpha_3+\frac2{27}\alpha_2^3-\frac83\alpha_0\alpha_2+\alpha_1^2 \Bigr) \,.
\end{aligned}\end{equation}
\end{definition}

\noindent
It is indeed an elliptic curve because the discriminant of the cubic polynomial in the right-hand side is equal to the discriminant of $F(z)$ (as a function of $\alpha_i$), and therefore the right-hand side has 3 distinct roots.

\begin{lemma}\label{curves_lemma} Curves~\eqref{C} and~\eqref{spec_ell_D3} are birational over the splitting field of the polynomial $F(z)$. Moreover, the holomorphic differential $\frac{dx}{y}$ on the spectral elliptic curve transforms into $-\frac{dz}{w}$ on~\eqref{C} under this birational equivalence.
\end{lemma}
\begin{proof}
Let $F(z_0)=0$. Then
\[
w^2 \= (z-z_0)^4 \+ \tilde{\alpha}_3 (z-z_0)^3 \+ \tilde{\alpha}_2 (z-z_0)^2 \+ \tilde{\alpha}_1 (z-z_0)
\]
with $\tilde{\alpha}_i=\frac1{i!}F^{(i)}(z_0)$, and
\[
\Bigl(\frac{\tilde{\alpha}_1 \, w}{(z-z_0)^2}\Bigr)^2 \= \tilde{\alpha}_1^2 \+ \tilde{\alpha}_1\tilde{\alpha}_3 \Bigl(\frac{\tilde{\alpha}_1 }{z-z_0}\Bigr) \+ \tilde{\alpha}_2 \Bigl(\frac{\tilde{\alpha}_1 }{z-z_0}\Bigr)^2 \+ \Bigl(\frac{\tilde{\alpha}_1 }{z-z_0}\Bigr)^3 \,.
\]
Hence the variables
\begin{equation}\label{xyzw}
x \= \frac{\tilde{\alpha}_1 }{z-z_0}+\frac{\tilde{\alpha}_2}{3}\,,\quad y \= \frac{\tilde{\alpha}_1 \, w}{(z-z_0)^2}
\end{equation}
satisfy $y^2=x^3+Bx+C$ with
\[\begin{aligned}
B &\= \tilde{\alpha}_1\tilde{\alpha}_3 - \frac{\tilde{\alpha}_2^2}{3} \= \alpha_1\alpha_3 -  \frac13 \alpha_2^2 - 4 \alpha_0\,,\\
C &\= \tilde{\alpha}_1^2 + \frac2{27}\tilde{\alpha}_2^3 - \frac13  \tilde{\alpha}_1\tilde{\alpha}_2\tilde{\alpha}_3 \= \alpha_0\alpha_3^2-\frac13\alpha_1\alpha_2\alpha_3+\frac2{27}\alpha_2^3-\frac83\alpha_0\alpha_2+\alpha_1^2\,.
\end{aligned}\]
The equality of the differentials follows immediately.
\end{proof}

\bigskip

Consider the normalized analytic and logarithmic solutions of D3 near $t=0$:
\begin{equation}\label{frbas}\begin{aligned}
\phi_0(t)  &\= 1  \- \frac12 \beta_1 t \+ \Bigl(\frac3{16}\alpha_3\beta_1+\frac3{32}\beta_1^2-\frac18\alpha_2 -\frac18\beta_0\Bigr) t^2 + \dots\\
\phi_1(t)  &\= \log t \, \phi_0(t) + \psi(t) \= \log t \, \phi_0(t) \+ \Bigl(-\frac12\alpha_3+\frac12 \beta_1\Bigr)t \+ \dots\\\
\end{aligned}\end{equation}

Again, the 6 parameters naturally split into two groups. Parameters $\alpha_3, \alpha_2,\alpha_1,\alpha_0$ determine the base
\[
\mathbb{P}^1(\mathbb C) \setminus \{ \infty,\, \text{the roots of } z^4+\alpha_3 z^3+\alpha_2 z^2 +\alpha_1 z+\alpha_0 \}
\]
and also the spectral elliptic curve~\eqref{spec_ell_D3}. Of course the solutions~\eqref{frbas} and the respective local system of rank~3 depend also on the remaining parameters $\beta_1$ and $\beta_0$.

\begin{definition} We say that an equation D3 with  $\alpha_3,\alpha_2,\alpha_1,\alpha_0, \beta_1, \beta_0 \in \mathbb{Q}$  is modular if the analytic continuation of
\begin{equation}\label{tau_map}
\tau \= \frac1{2 \pi i} \frac{\phi_1(t)}{\phi_0(t)}
\end{equation}
gives uniformization of the base by the upper halfplane with the group of deck transformations being a congruence subgroup of $\rm{SL}(2,\mathbb Z)$ and the function $\tau \mapsto \phi_0(t(\tau))$ is a modular form of weight~2.
\end{definition}

Consider the power series
\[
q(t) \= \exp \Bigl(\frac{\phi_1(t)}{\phi_0(t)} \Bigr) \= t \exp\Bigl(\frac{\psi(t)}{\phi_0(t)} \Bigr) \= t + \Bigl(-\frac12\alpha_3 + \frac12 \beta_1\Bigr)t^2  + \dots
\]
We can invert this expansion in order to write $t$ as a power series in $q$, and for accessory values of parameters this must be then the $q$-expansion of a modular function. Analogously, $\phi_0(t)$ written as a power series in $q$ will be the $q$-expansion of a modular form of weight~2 since the local system is of rank~3. We will be specifically interested in the coefficients of the modular form
\begin{equation}\label{fgl}
t \, \phi_0(t) \= \sum_{n=1}^{\infty} c_n q^n
\end{equation}
which can be computed explicitly as polynomials in the initial parameters
\begin{equation}\label{cnformulasD3}\begin{aligned}
& c_1 \= 1 \\
& c_2 \= \frac12 \alpha_3 - \beta_1 \\
& c_3 \= \frac3{16}\alpha_3^2 - \frac{27}{32}\alpha_3 \beta_1 + \frac{57}{64}\beta_1^2 + \frac1{16}\alpha_2 - \frac3{16}\beta_0 \\
& \dots
\end{aligned}\end{equation}

\bigskip

\begin{theorem}\label{Lfun} Assume we are given a non—degenerate modular D3. If the modular form~\eqref{fgl} of weight~2  is  a newform then
\[
\sum_{n=1}^{\infty} \frac{c_n}{n^s}
\]
is the L-function of the spectral elliptic curve~\eqref{spec_ell_D3}.
\end{theorem}

\bigskip

\begin{proof}
We will use the same method as for D2. Let $a_n, n \ge 1$ be the coefficients of the $L$-function $L(s)=\sum_n a_n \, n^{-s}$  of the spectral elliptic curve~\eqref{spec_ell_D3}.  One can check that
\begin{equation}\label{holdif}\begin{aligned}
t \, \phi_0(t) \, \frac{d q}{q} &\= t \, \phi_0(t) \, d \log q \= t \frac{\phi_1'(t) \phi_0(t) - \phi_1(t) \phi_0'(t)}{\phi_0(t)} dt \\
&\= F\Bigl(\frac1t\Bigr)^{-\frac12} t^{-2} \, dt \= -\frac{dz}{w}
\end{aligned}\end{equation}
where we substitute $z = 1/t$, $w^2=F(z)$.
Now using the birational transformation~\eqref{xyzw} we see that $t$ (hence also $q$) is a local parameter on the spectral elliptic curve near the point $P=\Bigl(\frac13 \tilde{\alpha}_2,\tilde{\alpha}_1\Bigr)$ where $\tilde{\alpha}_i=\frac1{i!}F^{(i)}(z_0)$ and $z_0$ is a chosen root of $F(z)=0$ as in the proof of Lemma~\ref{curves_lemma}. Since our differential equation is modular, composition of modular uniformization with the birational transform gives a map from a modular curve to the spectral elliptic curve.
The preimages of both points $ \pm P=\Bigl(\frac13 \tilde{\alpha}_2,\pm \tilde{\alpha}_1\Bigr)$ are cusps because $t$ is zero there and this is a cuspidal value as we know.
Therefore by the Manin-Drinfeld theorem their difference $P-(-P)=2P$ and hence also $P$ is a point of finite order.
Let us write the spectral curve~\eqref{spec_ell_D3} as $y^2 = x^3 + B x + C$. The differential $- \frac{dz}{w}$ in~\eqref{holdif} transforms to $\frac{dx}{y}$ according to Lemma~\ref{curves_lemma}.
One can find an isogenous curve $\tilde y^2 = \tilde x^3 + B \tilde x + C$ where $P$ is mapped to the origin and the differential is mapped to $-\frac12\frac{d\tilde x}{\tilde y}$. Then $q$ is a local parameter on the latter curve near the origin, and the $L$-function of this curve is again $L(s)=\sum_n a_n \, n^{-s}$ since isogenous curves have equal $L$-functions.
One can check that since the expansion $-\frac12\frac{d\tilde x}{\tilde y} \= \Bigl( \sum_{n=1}^{\infty} c_n q^n \Bigr) \frac{dq}{q}$ starts with $c_1=1$ then $\tilde x \sim q^{-2}$. All the coefficients $c_n$ in \eqref{fgl} and $d_n$ in the expansion $\tilde x = q^{-2} + \sum_{n=-1}^{\infty} d_n q^n$ do not contain $p$ in denominators for all but finitely many primes $p$ because these are $q$-expansions of a modular form of weight~2 and modular function respectively. The rest of the proof goes exactly like in Theorem~\ref{LfunD2}. Namely, one has Atkin and Swinnerton--Dyer congruences for all but finitely many primes and together with $c_n = o(n)$, which is another consequence of modularity, this implies that $a_n=c_n$ for all $n$ non divisible by a finite set of primes.
The normalized newforms $\sum_n a_n q^n$ and $\sum_n c_n$ are then in one Hecke-eigenspace and therefore are equal by multiplicity one theorem for the space of newforms.
\end{proof}

By this theorem, if $\sum_{n=1}^{\infty} c_n q^n$ is a newform then the coefficients $c_n$ are the coefficients of the $L$-function of an elliptic curve, and we have the following consequence.

\begin{corollary}
If $\sum_{n=1}^{\infty} c_n q^n$ is a newform, its coefficients~\eqref{cnformulasD3} are multiplicative, i.e.
\begin{equation}\label{multrel1}
c_{mn}(\vec{\alpha},\beta) \= c_{m}(\vec{\alpha},\beta) \, \cdot \, c_{n}(\vec{\alpha},\beta)
\end{equation}
as soon as $m$ and $n$ are coprime.
\end{corollary}

Let us solve equations~\eqref{multrel1} for the parameters $(\vec{\alpha},\beta)$.

\begin{lemma}\label{shift_lemma_D3} A D3 equation has the following properties with respect to the shift~\eqref{shiftDN}:
\begin{itemize}
\item[(i)] the parameters change according to the rule
\[\begin{aligned}
(&\alpha_3',\alpha_2',\alpha_1',\alpha_0')\\
&\=(\alpha_3 - 4\varepsilon,\alpha_2 -3\varepsilon\alpha_3 + 6\varepsilon^2,\alpha_1 - 2 \varepsilon\alpha_2 + 3 \varepsilon^2\alpha_3 - 4\varepsilon^3,\alpha_0-\varepsilon\alpha_1 +\varepsilon^2 \alpha_2 -\varepsilon^3\alpha_3 +\varepsilon^4) \\
(&\beta_1',\beta_0') \= (\beta_1-2\varepsilon,\beta_0-\beta_1 \varepsilon+\varepsilon^2)
\end{aligned}\]
\item[(ii)] the coefficients of the spectral elliptic curve~\eqref{spec_ell_D3} do not change
\item[(iii)] all $c_n$ in~\eqref{cnformulasD3} do not change
\item[(iv)] modular differential equations transform into modular ones (assuming $\varepsilon \in \mathbb Q$)
\end{itemize}
\end{lemma}

\begin{proof}
Indeed, since for $A'=A+\varepsilon$ one has  ${\mathcal L}_{A',\infty}(z) \= {\mathcal L}_{A,\infty}(z-\varepsilon)$ the formulas in~(i) follow from
\[\begin{aligned}
z^4 + \alpha_3' z^3 & + \alpha_2' z^2 + \alpha_1' z + \alpha_0' \\
& \= (z-\varepsilon)^4 + \alpha_3 (z-\varepsilon)^3 + \alpha_2 (z-\varepsilon)^2 \+ \alpha_1 (z-\varepsilon) + \alpha_0 ,\\
z^2 + \beta_1' z & + \beta_0' \= (z-\varepsilon)^2 + \beta_1(z-\varepsilon) + \beta_0 \,.
\end{aligned}\]
Part (ii) follows by a tedious computation. Next, from ${\mathcal L}_{A',0}(t) \= {\mathcal L}_{A,0}\Bigl( \frac{t}{1-\varepsilon t}\Bigr) \, (1-\varepsilon t)$ we conclude that Frobenius bases must simply transform as $\phi_i'(t) \= (1-\varepsilon t)\phi_i\Bigl( \frac{t}{1-\varepsilon t}\Bigr)$ for $i=0,1,2$. Therefore the uniformization maps~\eqref{tau_map} differ by the transformation $t \mapsto \frac{t}{1-\varepsilon t}$ of the base, from where~(iii) and~(iv) follow immediately.
\end{proof}

It follows from Lemma~\ref{shift_lemma_D3} that it suffices to solve equations~\eqref{multrel1} for a single representative of every orbit of shifts.
There are two natural choices, $\alpha_3=0$ and $\beta_1=0$. We will use the latter one. It appears that the system of equations~\eqref{multrel1} has finitely many solutions with $\beta_1=0$. We will give the complete list later, but first   we list the rational solutions that correspond to nondegenerate D3, i.e. such that the roots of $F(z)$ are all distinct. There are exactly 18 of them. We list the $\alpha_i$'s In the table below.  These determine the spectral elliptic curve~\eqref{spec_ell_D3} which we denote by $\mathcal{E}$. Then we give its $j$-invariant, its level $N$ and the newform $g(\tau)$ of level $N$
whose Mellin transform is the $L$-function of $\mathcal{E}$.
We give the value of $\beta_0$ in the last column.

\begin{center}
\begin{tabular}{cccc|c|c|c|c}
$\alpha_3$&$\alpha_2$&$\alpha_1$& $\alpha_0$ &$j(\mathcal{E})$ &$N(\mathcal{E})$&  $g_{\mathcal{E}}(\tau) $ & $\beta_0$\\
\hline
&&&&&&\\
0&-44&0&-16&$-\frac{20720464}{15625}$& 20& ${\bf 2^2 \cdot 10^2}$&-4  \\
0&44&0&-16&& 80 &${\bf 4^6 \cdot 20^6 / 2^2 \cdot 8^2 \cdot 10^2 \cdot 40^2}$ &4 \\
0&-28&0&-128&$\frac{207646}{6561}$& 24 & ${\bf 2\cdot 4\cdot 6\cdot 12}$&-4 \\
0&28&0&-128&& 48 & ${\bf 4^4 \cdot 12^4 / 2\cdot6\cdot8\cdot24}$&4 \\
0&-40&0&144&$\frac{35152}{9}$& 24 &${\bf 2\cdot 4\cdot 6\cdot 12}$&-8  \\
0&40&0&144& & 48 & ${\bf 4^4 \cdot 12^4 / 2\cdot6\cdot8\cdot24}$&8\\
0&-68&0&1152& $\frac{3065617154}{9}$ & 48 & ${\bf 4^4 \cdot 12^4 / 2\cdot6\cdot8\cdot24}$&-28 \\
0&68&0&1152& &24 &${\bf 2\cdot 4\cdot 6\cdot 12}$&28  \\
0&-48&0&512& 287496& 32 & ${\bf 4^2\cdot 8^2}$& -16 \\
0&48&0&512& &32 & ${\bf 4^2\cdot 8^2}$&16 \\
0&0&0&-256& 1728 & 32 & ${\bf 4^2\cdot 8^2}$&0  \\
0&0&0&256& & 64 & ${\bf 8^8 / 4^2 \cdot 16^2}$&0  \\
0&-36&0&432& 54000&144 & ${\bf 12^{12} / 6^4\cdot 24^4}$&-12  \\
0&36&0&432& & 36 & ${\bf 6^4}$&12  \\
-4&-88&-300&-304 & $-\frac{122023936}{161051}$&11&${\bf 1^2 \cdot 11^2 }$&-8\\
0&0&-108&0 &0&27&${\bf 3^2 \cdot 9^2}$&0\\
-2&-43&-156&-216&$\frac{4733169839}{3515625}$& 15& $ {\bf1 \cdot 3 \cdot 5 \cdot 15}$&-5\\
&&&&&&\\
-2&-59&-136&-80&$\frac{4956477625}{941192}$ &14&${\bf 1 \cdot 2 \cdot 7 \cdot 14}$ &-5 \\
\end{tabular}
\end{center}

In the proof of the Theorem~\ref{Lfun} we have constructed the point $P=\Bigl(\frac13 \tilde{\alpha}_2,\tilde{\alpha}_1\Bigr)$ of finite order on the spectral elliptic curve. The order of $P$ is 4 in the first 14 cases and for the last 4 rows it is 5,3,8 and 3 respectively.

\section{All solutions of the multiplicativity equations for D3}

The goal of this section is to list all the solutions to the multiplicativity equations. We have obtained them via Gr\"{o}bner bases with the aid of the computer algebra system \cite{MAGMA}, doing computations over several finite fields and lifting solutions afterwards. Apart from the non--degenerate cases which we listed in the previous section, there are solutions defined over number fields and also solutions with degenerate $F(z)$. In order to list them in an efficient way we consider the twists $A \mapsto A'' = \Bigl( \lambda^{j-i+1} a_{ij} \Bigr)$ which lead to the simple variable change in the differential equation ${\mathcal L}_{A'',0}(t) \= {\mathcal L}_{A,0}( \lambda t)$.

\begin{lemma}\label{twist_lemma_D3}
A D3 equation has the following properties with respect to the twist ${\mathcal L}_{A'',0}(t) \= {\mathcal L}_{A,0}( \lambda t)$:
\begin{itemize}
\item[(i)] the parameters transform according to the rule
\[\begin{aligned}
\vec{\alpha}'' \= (\lambda \alpha_3, \lambda^2 \alpha_2, \lambda^3 \alpha_1, \lambda^4 \alpha_0)\,, \quad \vec{\beta}'' \= (\lambda \beta_1, \lambda^2 \beta_0)
\end{aligned}\]
\item[(ii)] the spectral curve transforms via
\[
y^2 \= x^3 \+ \lambda^4 B x \+ \lambda^6 C
\]
\item[(iii)] the coefficients $c_n(\vec{\alpha}, \vec{\beta})$ transform via $c_n'' \= \lambda^{n-1} c_n$
\item[(iv)] for the function $\tau(t) \= \frac1{2 \pi i} \frac{\phi_1(t)}{\phi_0(t)}$, one has
\[
\tau'' \= \tau \- \frac{\log \lambda}{2 \pi i}\,,
\]
\end{itemize}
\end{lemma}

The proof is straightforward.
The way the solutions transform under the twists  described in ~(iii) and~(iv) of Lemma~\ref{twist_lemma_D3}   shows that only finitely many twists are possible for a given D3  that preserve the multiplicativity property, and the only twists possible
are those by roots of unity.
We give the complete list of solutions to~\eqref{multrel1} below. We list only one representative in every family of twists  and give all possible twists in the right-most column. We start with nondegenerate cases.  These have already been given  in the previous section up to the twists by roots of unity.

\begin{center}
\begin{tabular}{cccc|c|c|c|c|c}
$\alpha_3$&$\alpha_2$&$\alpha_1$& $\alpha_0$ &$j(\mathcal{E})$ &$N(\mathcal{E})$&  $g_{\mathcal{E}}(\tau) $ & $\beta_0$& twists \\
\hline
&&&&&&&\\
0&-44&0&-16&$-\frac{20720464}{15625}$& 20& ${\bf 2^2 \cdot 10^2}$&-4 & $\lambda^2 = \pm1$ \\
0&-28&0&-128&$\frac{207646}{6561}$& 24 & ${\bf 2\cdot 4\cdot 6\cdot 12}$&-4& $\lambda^2 = \pm1$ \\
0&-40&0&144&$\frac{35152}{9}$& 24 &${\bf 2\cdot 4\cdot 6\cdot 12}$&-8 & $\lambda^2 = \pm1$ \\
0&68&0&1152& $\frac{3065617154}{9}$ &24 &${\bf 2\cdot 4\cdot 6\cdot 12}$&28 & $\lambda^2 = \pm1$\\
0&48&0&512& 287496 &32 & ${\bf 4^2\cdot 8^2}$&16 & $\lambda^{16} = 1$ \\
0&0&0&-256& 1728 & 32 & ${\bf 4^2\cdot 8^2}$&0 & $\lambda^{16} = 1$\\
0&36&0&432& 54000& 36 & ${\bf 6^4}$&12 & $\lambda^{36} = 1$\\
-4&-88&-300&-304 & $-\frac{122023936}{161051}$&11&${\bf 1^2 \cdot 11^2 }$&-8& $\lambda = \pm1$ \\
0&0&-108&0 &0&27&${\bf 3^2 \cdot 9^2}$&0&$\lambda^{18}=1$\\
-2&-43&-156&-216&$\frac{4733169839}{3515625}$& 15& $ {\bf1 \cdot 3 \cdot 5 \cdot 15}$&-5& $\lambda = \pm1$\\
-2&-59&-136&-80&$\frac{4956477625}{941192}$ &14&${\bf 1 \cdot 2 \cdot 7 \cdot 14}$ &-5& $\lambda = \pm1$\\
\end{tabular}
\end{center}

In addition, there is one more nondegenerate solution over $\mathbb Q(\sqrt{5})$

\begin{center}
\begin{tabular}{cccc|c|c|c}
$\alpha_3$&$\alpha_2$&$\alpha_1$& $\alpha_0$ &$j(\mathcal{E})$ & $\beta_0$& twists\\
\hline
&&&&&&\\
0&$22-30\sqrt5$&0&$1000-440\sqrt5$ & $\notin \mathbb Q$ &  $18 - 10\sqrt{5}$& $\lambda^2 = \pm 1$ \\
\end{tabular}
\end{center}

\noindent
which appears to give the same modular form $t(\tau) \phi_0(\tau) = {\bf 2^2 \cdot 10^2}$ as in the first row of the above table.

\bigskip

The solutions of~\eqref{multrel1} with degenerate polynomial $F(z) \= z^4+\alpha_3 z^3+\alpha_2 z^2 + \alpha_1 z + \alpha_0$ are given in the table below. Remarkably, some of them are still modular.

\begin{center}
\begin{tabular}{ccccc|c|c}
$\alpha_3$&$\alpha_2$&$\alpha_1$& $\alpha_0$ & $\beta_0$ &  $t(\tau) \phi_0(\tau) $ &  twists\\
\hline
&&&&&&\\
0&0&0&0&0 & $q$ & --- \\
0&4&0&0&-4 & $q/(1-q^2)$& $\lambda^2 = \pm 1$\\
0&-8&0&16&-8&$q/(1-q^2)$ & $\lambda^2 = \pm 1$ \\
0&128&0&4096&64& ${\bf 2^4\cdot 8^4 / 4^4 }$ & $\lambda^2 = \pm 1$ \\
0&64&0&0&0& ${\bf 4^8 / 2^4}$ &$\lambda^2 = \pm 1$\\
4&0&0&0&0&$q/(1-q)^2$& $\lambda = \pm 1$\\
2&1&0&0&-1&$q/(1-q)$ & $\lambda = \pm1$\\
-2&-3&0&0&3&$q/(1+q+q^2)$& $\lambda = \pm1$ \\
-6&-135&-540&-648&-9&${\bf 1^3 \cdot 9^3/ 3^2}$&$\lambda = \pm1$\\
4&-80&192&0&-16&${\bf 2^4\cdot 6^4 /1^2 \cdot 3^2}$&$\lambda = \pm1$\\
2&9&-216&432&-9&${\bf 3^3\cdot 6^3 /1 \cdot 2}$&$\lambda = \pm1$\\
2&-55&-100&1000&-25&${\bf 5^5 /1}$ & $\lambda = \pm 1$\\
8&-176&768&-1024&-16 & ${\bf 2^{10}\cdot 8^4 /1^4 \cdot 4^6}$ & $\lambda = \pm 1$\\
\end{tabular}
\end{center}

\bigskip

\section{From D2's to D3's}

Let $\phi_0(t)=1+u_1 t + u_2 t^2 + \dots$ be the solution of the differential equation~\eqref{D2} with $\alpha_0=0$. As we already mentioned, this is exactly the class of D2's considered by Zagier in~\cite{Zag} where his parameters $A,B,\lambda$ are our $-\alpha_2,\alpha_1,\beta$ correspondingly. One then has
\[
(n+1)^2 u_{n+1} + (\alpha_2 n^2+\alpha_2 n-\beta)u_n + \alpha_1n^2u_{n-1}=0\,,
\]
and we observe that the sequence $u_n'=\binom{2n}{n}u_n$ satisfies
\[
(n+1)^3 u_{n+1} + 2 (2n+1)(\alpha_2 n^2+\alpha_2 n-\beta)u_n + 4\alpha_1(2n+1)(2n-1)nu_{n-1}=0\,.
\]
In other words, $\phi_0'(t)=\sum_{n=0}^{\infty}\binom{2n}{n}u_nt^n$ is a solution of
\[\begin{aligned}
D^3 &\+ t\bigl(D + \frac12\bigr) \bigl(4 \alpha_2 (D^2 + D) \- 4\beta\bigr) \\
&\+ t^2 (D+1) \bigl( 16\alpha_1 (D+1)^2 \- 4\alpha_1  \bigr) \\
\end{aligned}\]
which turns out to be a D3 equation  with parameters
\[
(\alpha'_3,\alpha'_2,\alpha'_1,\alpha'_0)\=(4 \alpha_2,16\alpha_1,0,0)\,, \quad(\beta_1',\beta_0')=(-4\beta,-4\alpha_1)\,.
\]
This equation is degenerate as its symbol has double roots. Passing  to  $\phi_0''(t)=\sum_{n=0}^{\infty}\binom{2n}{n}u_nt^{2n}$, which is a solution of
\[
D^3 \+ t^2 (D+1) \bigl( 4\alpha_2 (D+1)^2 - 16 \beta - 4 \alpha_2  \bigr) \+ 16 \alpha_1 t^4 (D+3)(D+2)(D+1) \,,
\]
we thus come to a D3 equation whose parameters are
\[
(\alpha''_3,\alpha''_2,\alpha''_1,\alpha''_0)\=(0,4\alpha_2,0,16 a_1)\,, \quad(\beta_1'',\beta_0'')=(0,- 16 \beta - 4 \alpha_2 )\,.
\]
This D3 is nondegenerate if the initial D2 was nondegenerate. Indeed, the symbol $F''(z)=z^4+4\alpha_2 z^2+16\alpha_1$ has 4 distinct roots if and only if $\alpha_1 \ne 0$ and $\alpha_2^2\ne 4\alpha_1$, which are exactly the conditions for the roots of the symbol $F(z)=z^3+\alpha_2z^2+\alpha_1z$ of the initial D2 to be all distinct. Therefore we have a map from nondegenerate D2's with $\alpha_0=0$ to nondegenerate D3's with $\alpha_3=\alpha_1=\beta_1=0$ given by
\begin{equation}\label{D2toD3}
\begin{aligned} \vec{\alpha} &= (\alpha_2,\alpha_1,0) \\
\vec{\beta} &= (\beta) \end{aligned}
\quad \mapsto \quad
\begin{aligned} \vec{\alpha} &= (0,4\alpha_2,0,16 \alpha_1) \\
\vec{\beta} &= (0,- 16 \beta - 4 \alpha_2 ) \end{aligned} \\
\end{equation}
This map is obviously bijective. The analytic solution at $t=0$ transforms as
\[
\sum_{n=0}^{\infty}u_n\,t^{n}
\quad \mapsto \quad
\sum_{n=0}^{\infty}\binom{2n}{n} \, u_n \, t^{2n}\,.
\]
We find that under the map~\eqref{D2toD3} Zagier's triples all go into D3's from our list of accessory equations and, moreover, they exhaust all D3's in our list with $\alpha_3=\alpha_1=0$. There 14 cases on both lists, below we show half of them as we already did in Introduction. Namely, for every D2 listed in the table with the parameters $(\alpha_2,\alpha_1,\alpha_0)$ there is also $(-\alpha_2,\alpha_1,-\alpha_0)$. We denote the spectral elliptic curves of the D2 and D3 differential equations by $\mathcal{E}_2$ and $\mathcal{E}_3$ respectively.

\begin{center}
\begin{tabular}{ccccc|ccccc}
&&$D2$&&&&&$D3$&&\\
&&&&&&&&&\\
$\alpha_2$&$\alpha_1$& $\beta$&$j(\mathcal{E}_2)$ & $N(\mathcal{E}_2)$ & $\alpha_2$&$\alpha_0$& $\beta_0$&$j(\mathcal{E}_3)$ & $N(\mathcal{E}_3)$\\
\hline
&&&&&&&&&\\
-7&-8&2&  $\frac{1556068}{81}$&$24$&  -28& -128& -4&$\frac{207646}{6561}$&$24$\\
&&&&&&&&&\\
-9&27&3&  $0$&$144$&  -36& 432& -12&$54000$&$144$\\
&&&&&&&&&\\
-10&9&3&  $\frac{1556068}{81}$&$24$&  -40& 144& -8&$\frac{35152}{9}$&$24$\\
&&&&&&&&&\\
-11&-1&3& $\frac{488095744}{125}$ &$20$&  -44& -16& -4&$-\frac{20720464}{15625}$&$20$\\
&&&&&&&&&\\
-12&32&4&  $1728$&$32$&  -48& 512& -16&$287496$&$32$\\
&&&&&&&&&\\
-17&72&6&  $\frac{1556068}{81}$&$48$&  -68& 1152& -28&$\frac{3065617154}{9}$&$48$\\
&&&&&&&&&\\
0&-16&0&  $1728$&$32$&  0& -256& 0&$1728$&$32$\\
\end{tabular}
\end{center}

\bigskip
\noindent
The spectral elliptic curves appear to be non--isomorphic in general but their levels coincide.
In fact, the respective spectral curves are isogenous, and therefore their $L$-functions are equal. Indeed, the spectral curve of D2
\[
\mathcal{E}_2 \,: \quad y^2 \= z^3 + \alpha_2 z^2 + \alpha_1 z
\]
is isogenous to the elliptic curve
\[
\mathcal{E}_2' \,: \quad y^2 \= z^3 - 2 \alpha_2 z^2 + (\alpha_2^2-4 \alpha_1)z\,,
\]
the isogeny of degree 2  given by
\[\begin{aligned}
 \mathcal{E}_2 &\rightarrow \mathcal{E}_2' \\
 (z,y) &\mapsto \Bigl( \frac{y^2}{z^2}, \frac{y(\alpha_1 - z^2)}{z^2} \Bigr)
\end{aligned}\]
(see Example 4.5 in \cite{Silverman}). This latter curve is in turn isomorphic to the spectral elliptic curve of the respective D3
\[
\mathcal{E}_3 \,: \quad y^2 \= z^3 + ( -\frac{16}3 \alpha_2^2 - 64 \alpha_1) z + (\frac{128}{27}\alpha_2^3 - \frac{512}3\alpha_1\alpha_2)
\]
with the map
\[\begin{aligned}
 \mathcal{E}_2' &\rightarrow \mathcal{E}_3 \\
 (z,y) &\mapsto \Bigl(\frac14 z+\frac23\alpha_2, \frac18 y \Bigr)\,.
\end{aligned}\]