 Fix a subgroup Γ of SL2("Z") (or of the metaplectic group if "k" is half-integral) and a character ρ of Γ.A modular form "f" for this character and this group Γ transforms under elements of Γ by

A weak Maass form of weight "k" is a continuous function on the upper half plane that transforms like a modular form of weight 2 − "k" and is an eigenfunction of the weight "k" Laplacian operator, and is called harmonic if its eigenvalue is (1 − "k"/2)"k"/2 .This is the eigenvalue of holomorphic weight "k" modular forms, so these are all examples of harmonic weak Maass forms.