 Homothety of complex lattices

Two complex lattices are homothetic when multiplication by one nonzero complex number takes one onto the other. Changes between oriented integral bases have determinant one, so normalized period ratios classify these classes by $S{L}_2(\mathbb{Z})\backslash \mathbb{H}$. The lattice Eisenstein sums scale as $G_k(a\Lambda )=a^{-k}{G}_k(\Lambda )$.