Dimension bound for modular forms on a finite-index subgroup ($\dim M_k(\Gamma )\le \lfloor k[S{L}_2(\mathbb{Z}):\Gamma ]/12\rfloor +1$)

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 $k[S{L}_2(\mathbb{Z}):\Gamma ]/12$. 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.