= Solution
Let $\Gamma$ be a <congruence subgroup>. A <modular form> of integral weight $k$ and level $\Gamma$ is a holomorphic function $f:\mathfrak h\to\mathbb C$ such that
$$
f(\gamma\tau)=(c\tau+d)^kf(\tau)
$$
for every $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma$, and such that $f$ is holomorphic at every cusp. In terms of the <slash operator for modular forms>, the first condition is $f|_k\gamma=f$; the second says that $f|_k\sigma$ has a <Fourier expansion of a modular form> with no negative powers in the local parameter whenever $\sigma\in SL_2(\mathbb Z)$ sends infinity to a cusp.
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