Representation growth of Fuchsian groups and modular forms

Let $\mathrm{\Gamma }$ be a cocompact, oriented Fuchsian group which is not on an explicit finite list of possible exceptions and $q$ a sufficiently large prime power not divisible by the order of any non-trivial torsion element of $\mathrm{\Gamma }$. Then $\mathrm{\mid }\mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{\Gamma },{\mathrm{G}\mathrm{L}}_n(q))\mathrm{\mid }\sim c_{q,n}{q}^{(1-\chi (\mathrm{\Gamma }))n^2}$, where $c_{q,n}$ is periodic in $n$. As a function of $q$, $c_{q,n}$ can be expressed as a Puiseux series in $1\mathrm{/}q$ whose coefficients are periodic in $n$ and $q$. Moreover, this series is essentially the $q$-expansion of a meromorphic modular form of half-integral weight.