On multiplicativity of Fourier coefficients at cusps other than infinity

Details On multiplicativity of Fourier coefficients at cusps other than infinity On multiplicativity of Fourier coefficients at cusps other than infinity

2011-02-11

arxiv.org/abs/1102.2295v1

Mathematics

Number Theory

Scientific paper

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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