Computing actions on cusp forms

For positive integers $k$ and $N$, we describe how to compute the natural action of $S{L}_2(\mathbb{Z})$ on the space of cusp forms $S_k(\mathrm{\Gamma }(N))$, where a cusp form is given by sufficiently many terms of its $q$-expansion. This will reduce to computing the action of the Atkin–Lehner operator on $S_k(\mathrm{\Gamma })$ for a congruence subgroup ${\mathrm{\Gamma }}_1(N)\subseteq \mathrm{\Gamma }\subseteq {\mathrm{\Gamma }}_0(N)$. Our motivating application of such fundamental computations is to compute explicit models of some modular curves $X_G$.