Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 3 / a / 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 thatfor every $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma$, 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 $\sigma \in S{L}_2(\mathbb{Z})$ sends infinity to a cusp.