Cusp form

Remarks

For a congruence subgroup $\Gamma$, a modular form $f$ for $\Gamma$ of weight $k$ is a cusp form if $(c\tau + d)^{-k} f(\gamma \tau) \to 0$ as $\operatorname{Im} \tau \to \infty$, for every $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\Z)$. Example: $\Delta = (E_4^3 - E_6^2) / 1728$ is a cusp form of weight $12$; the constant terms of $E_4^3$ and $E_6^2$ are both $1$, so they cancel. Non-example: the Eisenstein series $E_4 = 1 + 240 q + \cdots$ is a modular form but not a cusp form, since its constant term is $1$. For $\mathrm{SL}_2(\Z)$, $S_k = 0$ for $k < 12$ and $S_{12} = \C \Delta$.