= Cusp holomorphy under rational slash operators
If $f$ is a <modular form on a finite-index subgroup> and $\gamma$ is rational with positive <determinant>, then $f|_k\gamma$ is bounded near infinity. Choose $\sigma\in SL_2(\mathbb Z)$ with $\sigma\infty=\gamma\infty$. The <matrix> $\sigma^{-1}\gamma$ is upper triangular and sends $z$ to $Az+B$ with $A>0$. Thus the existing cusp expansion of $f|_k\sigma$ stays bounded under this substitution. <Rational conjugation of finite-index modular subgroups> gives a positive period for the translate, so boundedness gives a <removable singularity> in its cusp parameter. Apply this to $\gamma\rho$ for every $\rho\in SL_2(\mathbb Z)$ to obtain holomorphy at every cusp of the new subgroup. Vanishing at the cusps is preserved too.
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