Nebentypus character ($f{|}_k\gamma =\psi (d)f,\gamma \in {\Gamma }_0(N)$)

A Dirichlet character $\psi$ modulo $N$ specifying the displayed transformation law is the nebentypus of a modular form. The action of $-I$ forces $\psi (-1)=(-1{)}^k$ for a nonzero form. It describes a character eigenspace of the diamond action rather than an additional analytic condition at modular cusps.