I would like to know if there's a $q$-expansion principle for $\Gamma(N)$.
Namely, let $f$ be a weight $k$ weakly holomorphic modular form for $\Gamma(N)$ whose $q$-expansion at infinity has integral coefficients. Is it true that for every $M\in\Gamma_0(N)$ (or perhaps $\Gamma_1(N)$), the $q$-expansion of $f|_k M$ also has integral coefficients?