  For positive integers $k$ and $N$, we describe how to compute the natural
action of $SL_2(\mathbb{Z})$ on the space of cusp forms $S_k(\Gamma(N))$, 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 $S_k(\Gamma)$
for a congruence subgroup $\Gamma_1(N)\subseteq \Gamma \subseteq \Gamma_0(N)$.
Our motivating application of such fundamental computations is to compute
explicit models of some modular curves $X_G$.
