On multiplicativity of Fourier coefficients at cusps other than infinity
Number Theory
2011-02-14 v1
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.
Cite
@article{arxiv.1102.2295,
title = {On multiplicativity of Fourier coefficients at cusps other than infinity},
author = {Joseph Hundley},
journal= {arXiv preprint arXiv:1102.2295},
year = {2011}
}