# S-transformation of generalized Eisenstein series

I'm currently using generalized Eisenstein series to construct weight 2 modular forms under $\Gamma_1(N)$. They are defined as

$E_2^{\psi,\phi}(\tau) = \delta(\psi) L(-1,\phi) + 2\sum_{n=1}^{\infty} \sigma^{\psi,\phi}(n) q^n$, $q=e^{2\pi i \tau}$, where $\psi, \phi$ are Dirichlet characters (The Definition is taken from Diamond & Shurman).

Maybe I'm just not seeing it due to lack of experience (I'm a physicist) but now I would like to calculate the S-transformation ($\tau\mapsto{-1\over\tau}$) of these functions. Can somebody point me in the right direction?

Thanks!

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Personally, I think it is difficult to understand how the Eisenstein series transforms using the $q$-expansion. If I were you (in the notation of D-S, Section 4.6), I'd be stuck with noting that $E_2^{\psi,\phi}$ is a multiple of $G_2^{\psi,\phi}$, which is a weighted sum of $G_2^{\bar v}$, which is the sum of two functions, $f^{\bar v}_2(\tau)$ and $G_2(\tau)/N$. Maybe someone else knows a smarter way. –  B R Dec 10 '11 at 20:45
@phoboid: Perhaps the $\sigma^{\psi,\phi}$ notation is not universal... can it be explained briefly? This might enable further answers. –  paul garrett Dec 10 '11 at 23:09
@paul garrett: Sorry for being unclear, the $\sigma$'s are defined as $\sigma^{\psi,\phi}(n) = \sum_{m\mid n, m>0} \psi(n/m) \phi(m) m$, so it is basically a sum-of-divisors function weighted by the Dirichlet characters. –  phoboid Dec 11 '11 at 8:29

## 1 Answer

As BR notes, the $q$-expansion presentation of Eisenstein series is unhelpful for determining what happens under $z\rightarrow -1/z$. (The weight-two aspect creates some additional complications, but these are not the crucial ones here. Let's ignore convergence issues, at least for a while.)

Another way to describe level-$N$ Eisenstein series is as $$E_{c_o,d_o,N}(z) = \sum_{c=c_o (N),\;d=d_o (N)} {1\over (cz+d)^{2k}}$$ where $2k>2$ for convergence. It's immediate that $z\rightarrow -1/z$ interchanges $c_o$ and $d_o$, so the transformation of such Eisenstein series is clear.

For weights $>2$, the Fourier coefficients are readily computed, as in many standard sources (probably Diamond-Shurman, too). Various devices (often called "Hecke summation") extend this to the weight-two case.

Suitable linear combinations of these, summed over $c_o,d_o$ mod $N$, give the type desired. But there are complications depending on whether the characters are primitive or not mod $N$, probably a necessity of inverting matrices of Dirichlet $L$-function values, things like that. Do-able, but detail-y.

(That book-keeping is very nicely packaged by rewriting everything as modular forms on adele groups. This may be too pricey for your purposes.)

D. Bump's book on automorphic forms also does some such computations.

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Thanks a lot for your answer, I think this definitely points me in the right direction! –  phoboid Dec 12 '11 at 7:26