What are some specific values of Eisenstein Series?

Question

Just as the person who made this question, I am looking for specific values of Eisenstein series on certain values. Specifically, it would be nice to have the values of $E_k(\tau)$ for $k = 2,4$ and $\tau = i,2i$ or even generally $ni$ if possible! The references that I have found refer to only $E_{4k}$ and Hurwitz numbers, both of which don't really help for my goals.

I'm looking for these values because I need to determine how to express certain weight 2 and 4 modular forms, $q_2 \in \mathcal{M}_2(\Gamma_0(2))$ and $q_4 \in \mathcal{M}_4(\Gamma_0(2))$, on $\Gamma_0(2)$ in terms of Eisenstein bases or, even better, q-expansions. My group and I suspect that $q_2 = \lambda G_{2,2}$ for some lambda, but we're having trouble figuring out the specifics. Knowing certain values, or even general relations would be integral for our project going forward.

All these Eisenstein series can be evaluated in terms of singular moduli and elliptic integrals corresponding to these singular moduli. More generally they can be evaluated for $\tau=i\sqrt{n} $ where $n$ is a positive rational number in terms of values of Gamma function at rational points and certain algebraic numbers. — Paramanand Singh, Apr 05 '21 at 13:30
@ParamanandSingh That sounds perfect for our goals, do you happen to have a source that describes the methods for how to perform these evaluations? — Garrett Credi, Apr 05 '21 at 15:30
You will need to study theta functions, elliptic integrals and related topics to see the connection. I have described the theory in some of my blog posts. Will post some relevant links after some time. — Paramanand Singh, Apr 05 '21 at 16:11
You may have a look at this post and the next one to get the connection between Eisenstein series and elliptic integrals. — Paramanand Singh, Apr 06 '21 at 01:48
Also see this answer about Chowla Selberg formula which relates the values of eta function to Gamma function. — Paramanand Singh, Apr 06 '21 at 01:51

reuns · Answer 1 · 2021-03-30T01:36:47.063

$E_4(i)$ is found by looking at $\int_0^1 dz=\int_0^1\frac{d\wp_i(z)}{\wp_i'(z)}=\int_C \frac{dx}{\sqrt{4x^3-g_2(i)x}}$ on the complex torus $\Bbb{C/(Z+iZ)} \cong E:y^2=4x^3-g_2(i)x$, it will reduce to $\int_0^1\frac{dx}{\sqrt{x^3-x}}$ having a closed in term of $\Gamma(1/4)$ from the beta function $B(1/4,1/2)$.
$E_4(ni)$ is algebraic over $\Bbb{Q}[E_4(i),E_6(i)]=\Bbb{Q}[E_4(i)]$, this is because $E_4(nz)$ is a level $n$ modular form so the coefficients of $\prod_{ad=n,b\bmod d} (T-d^{-2}E_4(n\frac{az+b}{d}))$ are $SL_2(\Bbb{Z})$ modular forms with rational coefficients, they are in $\Bbb{Q}[E_4(z),E_6(z)]$. The coefficients of the polynomial $\prod_{ad=n,b\bmod d} (T-d^{-2}E_4(n\frac{ai+b}{d}))\in \Bbb{Q}[E_4(i)][T]$ are found from the first few Fourier coefficients.
$E_2(i)$ is found from $z^{-2}G_2(-1/z)= G_2(z) - \frac{2\pi i}{z}$
$E_2(i)-n E_2(ni)$ is algebraic over $\Bbb{Q}[E_4(i)]$ because $E_2(z)-n E_2(nz)$ is a level $n$ modular form and the same reason as above.

The polynomials involved in the algebraic parts are solvable: this is due to that $i$ is a CM point, the roots of the polynomial $\prod_{ad=n,b\bmod d} (T-d^{-2}E_4(n\frac{ai+b}{d}))$ are grouped by ideal classes of $\Bbb{Z}[ni]$ giving an abelian Galois group (over $\Bbb{Q}[i,E_4(i)]$)

This is perfect, thank you! Do you know of any sources that cover these facts in detail? — Garrett Credi, Mar 30 '21 at 17:55
Also, how could I go about figuring the expressions for $E_4(ni)$ and $E_2(i) - nE_2(ni)$ over $E_4(i)$? — Garrett Credi, Mar 30 '21 at 22:54
I second the question about sources that cover these facts in detail. — nilo de roock, May 23 '21 at 12:41

What are some specific values of Eisenstein Series?

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