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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.

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• Review status: reviewed
• Last edited by David Farmer on 2019-04-29 09:06:01
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