A proof of Hecke’s formula for binary quadratic forms

Abstract

In Mathematische Werke, Hecke defines the operator [Formula: see text] and describes their utility in conjunction with theta series of quadratic forms. In particular, he shows that the image of theta series associated to classes of binary quadratic forms in CL[Formula: see text] is again a theta series associated to a collection of forms in CL[Formula: see text]. We state and prove an explicit formula for the action of [Formula: see text] on a binary quadratic form of negative discriminant.