On multiplicativity of Fourier coefficients at cusps other than infinity

This paper treats the problem of determining conditions for the Fourier coefficients of a Maass–Hecke newform at cusps other than infinity to be multiplicative. To be precise, the Fourier coefficients are defined using a choice of matrix in \(\mathit{SL}(2, \mathbb{Z})\) which maps infinity to the cusp in question. Let c and d be the entries in the bottom row of this matrix, and let N be the minimal level. In earlier work with Dorian Goldfeld and Min Lee, we proved that the coefficients will be multiplicative whenever N divides 2cd. This paper proves that they will not be multiplicative unless N divides 576cd. Further, if one assumes that the Hecke eigenvalue vanishes less than half the time, then this number drops to 4cd, and a precise condition governing the case where N divides 4cd and not 2cd is obtained.