Title: On M-functions associated with modular forms,
Abstract: Let $f$ be a primitive cusp form of weight $k$ and level $N,$ let $\chi$ be a
Dirichlet character of conductor coprime with $N,$ and let
$\mathfrak{L}(f\otimes \chi, s)$ denote either $\log L(f\otimes \chi, s)$ or
$(L'/L)(f\otimes \chi, s).$ In this article we study the distribution of the
values of $\mathfrak{L}$ when either $\chi$ or $f$ vary. First, for a
quasi-character $\psi\colon \mathbb{C} \to \mathbb{C}^\times$ we find the limit
for the average $\mathrm{Avg}\_\chi \psi(L(f\otimes\chi, s)),$ when $f$ is
fixed and $\chi$ varies through the set of characters with prime conductor that
tends to infinity. Second, we prove an equidistribution result for the values
of $\mathfrak{L}(f\otimes \chi,s)$ by establishing analytic properties of the
above limit function. Third, we study the limit of the harmonic average
$\mathrm{Avg}^h\_f \psi(L(f, s)),$ when $f$ runs through the set of primitive
cusp forms of given weight $k$ and level $N\to \infty.$ Most of the results are
obtained conditionally on the Generalized Riemann Hypothesis for
$L(f\otimes\chi, s).$