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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?

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  • $\begingroup$ You might have problems at primes dividing N but the usual proof goes through otherwise, right? Do you care about primes dividing N? And what does weakly holomorphic mean? You're allowing poles? Where? $\endgroup$
    – znt
    Jun 7, 2016 at 17:49
  • $\begingroup$ Actually I think $\Gamma_0(N)$ preserves the component of $X(N)$ containing the cusp infinity even at primes dividing $N$ (maybe this is Katz-Mazur 13.10.3(3)) so maybe you're OK even at primes dividing $N$. You can't beef it up to all of $SL(2,Z)$ though because then the components get moved around and you can have forms on the integral model which have poles along a component in char p not containing infinity but in the SL(2,Z) orbit of infinity. $\endgroup$
    – znt
    Jun 7, 2016 at 17:59

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