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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?!

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  • $\begingroup$ Some danger to say that Eisenstein series are "orthogonal" to cuspforms, since Eisenstein series are not in $L^2$... $\endgroup$ – paul garrett Apr 14 '14 at 18:14
  • $\begingroup$ You're right, Eisenstein series are not in $L^2$, but this fact does not make any difficulties. The Petersson product $\langle f,\ g\rangle$ is well defined if $f$ or $g$ is a cusp form. $\endgroup$ – Abdullah.Y Apr 14 '14 at 19:06
  • $\begingroup$ It may create difficulties if one wants to "project", e.g., there is no "projector" to the space of Eisenstein series of the form $f\to \sum_i \langle f,E_i\rangle\cdot E_i$. $\endgroup$ – paul garrett Apr 14 '14 at 23:37
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I believe the formula you're looking for is a formula of Siegel that expresses the $n$th Fourier coefficient of an Eisenstein series as a product of local densities. There are lots of papers giving formulas for these local densities in terms of the arithmetic of $Q$ in the literature. My two favorite sources are

$\bullet$ ''Local densities and explicit bounds for representability by a quadratic form'' by Jonathan Hanke, Duke Math. J. (2004), no. 2, 351-388, and

$\bullet$ ''An explicit formula for local densities of quadratic forms,'' by Tonghai Yang, J. Number Theory 72 (1998), 309-356.

The formulas you get simplify quite a bit if $\chi = 1$, and even more if the integer $n$ you want to represent is coprime to $N$.

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    $\begingroup$ Thanks Mr. Rouse. That's exactly what I'm looking for. It's something funny that you of all people have answered me, couse currently I read your paper "Explicit bounds for sums of squares" and try to understand how you get the term $\rho_{2k}(n)=\frac{2k(-1)^{k/2+1}}{(2^k-1)B_k}\big(\sigma_{k-1}(n)+(-1+(-1)^{k/2+1})\sigma_{k-1}(n/2)+(-1)^{k/2}2^k\sigma_{k-1}(n/4)\big)$. $\endgroup$ – Abdullah.Y Apr 14 '14 at 19:49
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    $\begingroup$ That is a bit surprising, although less surprising than you might guess. When $N = 4$, $\chi = 1$, and the weight $k$ is even (and greater than $2$) things are simpler - $E_{k}(z)$, $E_{k}(2z)$ and $E_{k}(4z)$ span the $3$-dimensional space of Eisenstein series. Then it's a matter of matching up the value of the theta series at the three cusps of $\Gamma_{0}(4)$ with the values of $E_{k}(z)$, $E_{k}(2z)$ and $E_{k}(4z)$. $\endgroup$ – Jeremy Rouse Apr 14 '14 at 20:05
  • $\begingroup$ Oh crap, that's a very natural response-how stupid of me. If I specify a basis for $\mathcal{E}_k(\Gamma_0(N),\ \chi)$(generalized Eisenstein series) then it works same. $\endgroup$ – Abdullah.Y Apr 14 '14 at 20:27
  • $\begingroup$ But how do you evaluate the value of the theta series at the three cusps? $\endgroup$ – Abdullah.Y Apr 16 '14 at 18:38
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    $\begingroup$ There are many ways - one is to express the theta series in terms of the Dedekind eta function and use that representation to compute its values at cusps. $\endgroup$ – Jeremy Rouse Apr 16 '14 at 19:02
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The general statement is the Siegel-Weil theorem ... specialized to the case of real-anisotropic orthogonal group in an even number of variables paired against $SL_2$ or $Sp_n$.

The Siegel-Weil formula directly expresses certain Eisenstein series as linear combinations of holomorphic theta series, and the manifest fact that Eisenstein series are finite linear combinations of those formed from local data that factors over primes gives the factorization into local data.

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