A simple and beautiful idea of Poincaré on Poincaré series in automorphic functions can be applied to an arbitrary ring $R$ acted by a group $G$. When $G$ is finite, the key is to look at the 0-dimensional Tate cohomology of $(G, R)$ twisted by the 1-cohomology class of the group of units of $R$. As a simplest case, we examine when $R$ is the ring of integers of a quadratic field.
Takashi Ono. "A note on Poincaré sums for finite groups." Proc. Japan Acad. Ser. A Math. Sci. 79 (4) 95 - 97, April 2003. https://doi.org/10.3792/pjaa.79.95