# Properties

 Label 684.3.g.a Level $684$ Weight $3$ Character orbit 684.g Analytic conductor $18.638$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 684.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.6376500822$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( -2 - 2 \beta_{1} ) q^{4} + ( 2 - 2 \beta_{2} ) q^{5} + ( -\beta_{1} - \beta_{3} ) q^{7} -8 q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( -2 - 2 \beta_{1} ) q^{4} + ( 2 - 2 \beta_{2} ) q^{5} + ( -\beta_{1} - \beta_{3} ) q^{7} -8 q^{8} + ( 2 - 4 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{10} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{11} + ( -12 + 3 \beta_{2} ) q^{13} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{14} + ( -8 + 8 \beta_{1} ) q^{16} + ( 4 - \beta_{2} ) q^{17} + ( \beta_{1} + 2 \beta_{3} ) q^{19} + ( -4 - 8 \beta_{1} + 4 \beta_{2} - 12 \beta_{3} ) q^{20} + ( -16 - 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{22} + ( -7 \beta_{1} - 15 \beta_{3} ) q^{23} + ( 35 - 4 \beta_{2} ) q^{25} + ( -12 + 15 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} ) q^{26} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{28} + ( 12 + 5 \beta_{2} ) q^{29} -16 \beta_{3} q^{31} + ( 16 + 16 \beta_{1} ) q^{32} + ( 4 - 5 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{34} + ( -12 \beta_{1} - 4 \beta_{3} ) q^{35} + ( -2 - 8 \beta_{2} ) q^{37} + ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{38} + ( -16 + 16 \beta_{2} ) q^{40} + ( 26 + 10 \beta_{2} ) q^{41} + ( 4 \beta_{1} - 16 \beta_{3} ) q^{43} + ( -32 + 12 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{44} + ( -6 - 7 \beta_{1} + 15 \beta_{2} - 15 \beta_{3} ) q^{46} + ( -28 \beta_{1} - 8 \beta_{3} ) q^{47} + ( 43 + \beta_{2} ) q^{49} + ( 35 - 39 \beta_{1} - 4 \beta_{2} - 12 \beta_{3} ) q^{50} + ( 24 + 30 \beta_{1} - 6 \beta_{2} + 18 \beta_{3} ) q^{52} + ( 4 - 7 \beta_{2} ) q^{53} + ( -40 \beta_{1} - 32 \beta_{3} ) q^{55} + ( 8 \beta_{1} + 8 \beta_{3} ) q^{56} + ( 12 - 7 \beta_{1} + 5 \beta_{2} + 15 \beta_{3} ) q^{58} + ( 39 \beta_{1} + 7 \beta_{3} ) q^{59} + ( -30 - 4 \beta_{2} ) q^{61} + ( 16 + 16 \beta_{2} - 16 \beta_{3} ) q^{62} + 64 q^{64} + ( -108 + 24 \beta_{2} ) q^{65} + ( -7 \beta_{1} + 33 \beta_{3} ) q^{67} + ( -8 - 10 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{68} + ( -32 - 12 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{70} + ( -2 \beta_{1} + 22 \beta_{3} ) q^{71} + ( -88 + 7 \beta_{2} ) q^{73} + ( -2 - 6 \beta_{1} - 8 \beta_{2} - 24 \beta_{3} ) q^{74} + ( 2 - 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{76} + ( -20 + 6 \beta_{2} ) q^{77} + ( 62 \beta_{1} - 2 \beta_{3} ) q^{79} + ( -16 + 32 \beta_{1} + 16 \beta_{2} + 48 \beta_{3} ) q^{80} + ( 26 - 16 \beta_{1} + 10 \beta_{2} + 30 \beta_{3} ) q^{82} + ( 26 \beta_{1} - 34 \beta_{3} ) q^{83} + ( 36 - 8 \beta_{2} ) q^{85} + ( 28 + 4 \beta_{1} + 16 \beta_{2} - 16 \beta_{3} ) q^{86} + ( 48 \beta_{1} + 16 \beta_{3} ) q^{88} + ( 2 - 14 \beta_{2} ) q^{89} + ( 27 \beta_{1} + 15 \beta_{3} ) q^{91} + ( -12 + 14 \beta_{1} + 30 \beta_{2} + 30 \beta_{3} ) q^{92} + ( -76 - 28 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{94} + ( 20 \beta_{1} + 2 \beta_{3} ) q^{95} + ( 2 - 28 \beta_{2} ) q^{97} + ( 43 - 42 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} - 8 q^{4} + 4 q^{5} - 32 q^{8} + O(q^{10})$$ $$4 q + 4 q^{2} - 8 q^{4} + 4 q^{5} - 32 q^{8} + 4 q^{10} - 42 q^{13} - 6 q^{14} - 32 q^{16} + 14 q^{17} - 8 q^{20} - 60 q^{22} + 132 q^{25} - 42 q^{26} - 12 q^{28} + 58 q^{29} + 64 q^{32} + 14 q^{34} - 24 q^{37} - 32 q^{40} + 124 q^{41} - 120 q^{44} + 6 q^{46} + 174 q^{49} + 132 q^{50} + 84 q^{52} + 2 q^{53} + 58 q^{58} - 128 q^{61} + 96 q^{62} + 256 q^{64} - 384 q^{65} - 28 q^{68} - 120 q^{70} - 338 q^{73} - 24 q^{74} - 68 q^{77} - 32 q^{80} + 124 q^{82} + 128 q^{85} + 144 q^{86} - 20 q^{89} + 12 q^{92} - 288 q^{94} - 48 q^{97} + 174 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 15$$$$)/10$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 9 \nu + 5$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{3} - 2 \nu^{2} + 2 \nu + 25$$$$)/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 5 \beta_{1} + 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{3} - 2 \beta_{1} + 7$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$343$$ $$533$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
343.1
 2.13746 + 0.656712i −1.63746 − 1.52274i 2.13746 − 0.656712i −1.63746 + 1.52274i
1.00000 1.73205i 0 −2.00000 3.46410i −6.54983 0 1.31342i −8.00000 0 −6.54983 + 11.3446i
343.2 1.00000 1.73205i 0 −2.00000 3.46410i 8.54983 0 3.04547i −8.00000 0 8.54983 14.8087i
343.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −6.54983 0 1.31342i −8.00000 0 −6.54983 11.3446i
343.4 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 8.54983 0 3.04547i −8.00000 0 8.54983 + 14.8087i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.g.a 4
3.b odd 2 1 76.3.b.a 4
4.b odd 2 1 inner 684.3.g.a 4
12.b even 2 1 76.3.b.a 4
24.f even 2 1 1216.3.d.a 4
24.h odd 2 1 1216.3.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.b.a 4 3.b odd 2 1
76.3.b.a 4 12.b even 2 1
684.3.g.a 4 1.a even 1 1 trivial
684.3.g.a 4 4.b odd 2 1 inner
1216.3.d.a 4 24.f even 2 1
1216.3.d.a 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 2 T_{5} - 56$$ acting on $$S_{3}^{\mathrm{new}}(684, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( -56 - 2 T + T^{2} )^{2}$$
$7$ $$16 + 11 T^{2} + T^{4}$$
$11$ $$3136 + 188 T^{2} + T^{4}$$
$13$ $$( -18 + 21 T + T^{2} )^{2}$$
$17$ $$( -2 - 7 T + T^{2} )^{2}$$
$19$ $$( 19 + T^{2} )^{2}$$
$23$ $$1140624 + 2139 T^{2} + T^{4}$$
$29$ $$( -146 - 29 T + T^{2} )^{2}$$
$31$ $$1048576 + 2816 T^{2} + T^{4}$$
$37$ $$( -876 + 12 T + T^{2} )^{2}$$
$41$ $$( -464 - 62 T + T^{2} )^{2}$$
$43$ $$614656 + 3296 T^{2} + T^{4}$$
$47$ $$2027776 + 4064 T^{2} + T^{4}$$
$53$ $$( -698 - T + T^{2} )^{2}$$
$59$ $$12588304 + 8027 T^{2} + T^{4}$$
$61$ $$( 796 + 64 T + T^{2} )^{2}$$
$67$ $$12362256 + 13659 T^{2} + T^{4}$$
$71$ $$3211264 + 5612 T^{2} + T^{4}$$
$73$ $$( 6442 + 169 T + T^{2} )^{2}$$
$79$ $$141324544 + 23852 T^{2} + T^{4}$$
$83$ $$3136 + 22076 T^{2} + T^{4}$$
$89$ $$( -2768 + 10 T + T^{2} )^{2}$$
$97$ $$( -11028 + 24 T + T^{2} )^{2}$$
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