If $Q:\mathbb{Z}^{2k}\to \mathbb{Z}$ is any positive definite integer -valued quadratic form in $2k$ variables, then it is well known, that the $\textbf{theta series}$ $\theta_Q(z):=\sum_{m\in\mathbb{Z}^{2k}}q^{Q(m)}\ (q=e^{2\pi i z})$ is a modular form of weight $k$ on the congruence group $\Gamma_0(N)$ ( for some integer $N$ ) with some character $\chi$ $\bmod$ $N$, i.e. $\theta_Q \left(\frac{az+b}{cz+d}\right)=\chi(d)(cz+d)^k \theta_Q(z)$ for all $\big(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\big)\in \Gamma_0(N)$. Let us denote the space of modular forms of weight $k$ on $\Gamma_0(N)$ with character $\chi$ by M$_k(\Gamma_0(N),\chi)$. Let S$_k(\Gamma_0(N),\chi)$ be the subspace of cusp forms and $\mathcal{E}_k(\Gamma_0(N),\chi)$ be the eisenstein subspace. We know $\text{M}_k(\Gamma_0(N),\chi)=\mathcal{E}_k(\Gamma_0(N),\chi)\oplus\text{S}_k(\Gamma_0(N),\chi)$ and this decomposition is orthogonal under the Petersson inner product. My Question is: Is there any explicit formula for the eisenstein part of the the theta function $\theta_Q$? Perhaps a simple form, when $\chi$ is the trivial character?!

  • $\begingroup$ For the trivial character, the Eisenstein series are $E_{N,\alpha}(z) = \sum_{\gamma \in \Gamma_0(N) / \langle T \rangle} \frac{d}{dz} (\gamma \alpha z)^k$ where $T z = z+1$ and $\alpha \in SL_2(\mathbb{Z})$ represents a cusp $\endgroup$ – reuns Dec 25 '17 at 2:18

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