How can I calculate the Eisenstein series of a complex lattice?

Question

Suppose I have a lattice $\Lambda = \mathbb{Z}+\frac{3}{2}i\mathbb{Z}$. How would I go about calculating $G_{2n}(\Lambda)$ for a given $n \in \mathbb{N}$?

Answer 1

Up to the $\zeta(4),\zeta(6)$ constants this is $E_4,E_6$, some modular forms with rational coefficients.

$E_6(i)=0$ and $E_4(i)$ is given in term of the beta function $\int_0^\infty \frac{dx}{\sqrt{x^3+x}}$, then $E_{2k}(z)\in \Bbb{Z}[E_4(z),E_6(z),\Delta(z)]$ so $E_{2k}(i\frac32)\in \Bbb{Z}[\frac16,E_4(i\frac32),E_6(i\frac32)]$

and $E_4(i\frac32),E_6(i\frac32)$ are algebraic over $\Bbb{Z}[E_6(i)]$, this is because the coefficients of $$\prod_{\gamma \in \Gamma_0(6)\backslash SL_2(\Bbb{Z})} (X-E_6(\frac32 \gamma(z)))$$ are modular forms $\in M_{(d-n)6}(SL_2(\Bbb{Z}))$ with rational coefficients, so they are in $\Bbb{Q}[E_4(z),E_6(z)]$.