Spaces of modular forms (reviewed)

Suppose $N$ is a positive integer and let $\Gamma$ be either $\Gamma_0(N)$ or $\Gamma_1(N)$.

If $k$ is a positive integer then the modular forms of weight $k$ for $\Gamma$ forms a complex vector space $M_k(\Gamma)$ of finite dimension.

Modular forms that vanish at the cusps of $\Gamma$ are called cusp forms. The subspace of cusp forms in $M_k(\Gamma)$ is denoted $S_k(\Gamma)$. The Eisenstein series in $M_k(\Gamma)$ form a subspace $E_k(\Gamma)$ orthogonal to $S_k(\Gamma)$ and we have \[ M_k(\Gamma) = E_k(\Gamma)\oplus S_k(\Gamma) . \] Those subspaces are orthogonal with respect to the Petersson inner product

Both $E_k(\Gamma)$ and $S_k(\Gamma)$ are direct sums of subspaces of old forms and new forms.

The space of cusp forms of weight $k$ for $\Gamma_0(N)$ with character $\chi$ is denoted either by $S_k(\Gamma_0(N), \chi)$ or $S_k(N,\chi)$, and we have the decomposition \[ S_k(\Gamma_1(N)) = \bigoplus_{\chi \mod N} S_k(N, \chi) . \] There is a similar decomposition for newforms.