= Dimension bound for modular forms on a finite-index subgroup
{title2=$\dim M_k(\Gamma)\leq\lfloor k[SL_2(\mathbb Z):\Gamma]/12\rfloor+1$}
For nonnegative integer $k$ and a finite-index subgroup $\Gamma$, a nonzero <modular form> has order at infinity, measured in a genuine periodic cusp parameter, at most $k[SL_2(\mathbb Z):\Gamma]/12$. The product of its slash translates over the left cosets is a nonzero level-one form; the translates along the infinity-cusp orbit contribute its full local order to this product. The <valence formula for the modular group> gives the bound. Initial coefficients through that bound therefore give an injective linear map.
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