Holomorphic modular form

Let $\Gamma$ be a finite-index subgroup of $\operatorname{SL}_2(\mathbb Z)$ and $k\ge0$ an integer. A holomorphic modular form of weight $k$ for $\Gamma$, with trivial character, is a holomorphic function $f$ on $\mathcal H=\{z\in\mathbb C:\operatorname{Im}z>0\}$ such that

and $f$ is holomorphic at every cusp. Explicitly, for every $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\operatorname{SL}_2(\mathbb Z)$,

has a convergent expansion $\sum_{n\ge0}a_n e^{2\pi inz/h}$ near infinity, for some positive integer period $h$. This is the cusp condition at $\gamma\infty$.