Lattice Eisenstein sum ($G_k(\Lambda )={\sum }_{\omega \in \Lambda \setminus \{0\}}{\omega }^{-k}$)

For a period lattice $\Lambda \subset \mathbb{C}$ and an integer $k>2$, this sum converges absolutely. Pairing $\omega$ and $-\omega$ makes it zero when $k$ is odd. For even weights it supplies the Laurent coefficients of the Weierstrass elliptic function. Scaling the period lattice gives $G_k(a\Lambda )=a^{-k}{G}_k(\Lambda )$.

Equal sums give equal Klein j-invariants, hence the complex lattices are homothetic. If both sums are nonzero, the scaling factor obeys $a^4=a^6=1$ and is $\pm 1$. If only $G_6$ vanishes, the lattice is homothetic to the square lattice and the possible fourth roots of unity already preserve it. If only $G_4$ vanishes, it is homothetic to the hexagonal lattice and the possible sixth roots of unity preserve it. The two sums cannot vanish together because the elliptic discriminant is nonzero.