Multipliers of a Family of Almost Modular Functions

Abstract

The goal of this paper is to complete an investigation begun by Cohn and Knopp in their 1994 paper, “Application of Dedekind eta-multipliers to modular equations.” The paper concerned Λ k (z), a family of modular forms on Γ0(N) (N a positive integer) with possibly non-trivial multiplier systems. Cohn and Knopp defined new functions Ψ k (z) and a new group containing Γ0(N) and proved that for all S in the larger group and for all k, Λ k (Sz) = M k(S)Ψ k (z), where M k(S)24 = 1. This yielded interesting invariance properties of Λ k , dependent on the values of M k(S). Fixing a constant integer e, independent of k, Cohn and Knopp proved that for all k and all S in the larger group, M k(S) e = (±1) e . They determined the sign of M k(S) e in many, but not all, cases. In this paper, we give a complete determination of the values of M k(S) e in the remaining cases.