What are some specific values of Eisenstein Series? 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.
 A: *

*$E_4(i)$ 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 $\Bbb{Q}[i,E_4(i)]$)
