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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**History:**(expand/hide all)

- 2019-04-29 09:06:01 by David Farmer (Reviewed)
- 2019-04-28 21:44:10 by David Farmer (Reviewed)
- 2019-04-28 20:37:25 by David Farmer
- 2019-04-28 18:01:26 by David Farmer
- 2019-04-28 18:00:05 by David Farmer
- 2019-04-28 17:57:33 by David Farmer
- 2019-04-28 17:43:02 by David Farmer
- 2019-04-28 17:41:22 by David Farmer
- 2018-12-08 09:38:02 by Andrew Sutherland

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