Cuspidal modular form (reviewed)

Let $k$ be a positive integer and let $\Gamma$ be a finite index subgroup of the modular group $\SL(2,\Z)$.

A cusp form of weight $k$ on $\Gamma$ is a modular form $f\in M_k(\Gamma)$ that vanishes at all cusps of $\Gamma$. In particular, the constant term in the Fourier expansion of $f$ about any cusp is zero.

The cusp forms in $M_k(\Gamma)$ form a subspace $S_k(\Gamma)$. For each Dirichlet character $\chi$ of modulus $N$ the cusp forms in $M_k(N,\chi)$ form a subspace $S_k(N,\chi)$; these are the cusp forms of weight $k$, level $N$, and character $\chi$.