# Modular form

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A modular form of weight $k$ for $\mathrm{SL}(2,\mathbb{Z})$ is a function $f \colon \mathbb{H} \rightarrow \mathbb{C}$, where $\mathbb{H}$ is a the upper half-plane that satisfies three conditions:
1. $f$ is holomorphic on $\mathbb{H}$,
2. for any $z \in \mathbb{H}$ and $\begin{bmatrix} a&b \\ c&d \end{bmatrix} \in \mathrm{SL}(2,\mathbb{Z})$, $$f \left( \dfrac{az+b}{cz+d} \right) = (cz+d)^k f(z),$$ and
3. $f$ is holomorphic at the cusp.