Cusp of a modular group

A cusp of a finite-index subgroup $\Gamma \le S{L}_2(\mathbb{Z})$ is an orbit in $\Gamma \backslash {\mathbb{P}}^1(\mathbb{Q})$. If $\sigma \in S{L}_2(\mathbb{Z})$ sends infinity to a cusp, behavior there is studied through $f{|}_k\sigma$.

The width of a cusp represented by $\sigma \in S{L}_2(\mathbb{Z})$ is the least positive $h$ such that $\sigma T^h{\sigma }^{-1}$ lies in the subgroup, up to its center. Its local parameter is $q_h=e^{2\pi i\tau /h}$.

A modular function is holomorphic at a cusp when its expansion in the corresponding local parameter has no negative powers. It vanishes at the cusp when that expansion has zero constant term.