Computing Classical Modular Forms for Arbitrary Congruence Subgroups

In this paper, we prove the existence of an efficient algorithm for the computation of $q$-expansions of modular forms of weight $k$ and level $\mathrm{\Gamma }$, where $\mathrm{\Gamma }\subseteq S{L}_2(\mathbb{Z})$ is an arbitrary congruence subgroup. We also discuss some practical aspects and provide the necessary theoretical background.