On multiplicativity of Fourier coefficients at cusps other than infinity

2011/02/11 by Hundley, Joseph
#11F30 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1102.2295

Abstract

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 SL(2, 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 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 48cd.

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