On multiplicativity of Fourier coefficients at cusps other than infinity

TL;DR

This paper investigates the conditions under which Fourier coefficients of Maass-Hecke newforms at non-infinity cusps are multiplicative, establishing necessary divisibility criteria related to the level and matrix entries.

Contribution

It proves that Fourier coefficients are not multiplicative unless the level divides 576 times the product of specific matrix entries, refining previous results.

Findings

Fourier coefficients are multiplicative only if N divides 576cd.

If Hecke eigenvalues vanish less than half the time, the divisibility condition reduces to N dividing 48cd.

Established precise divisibility conditions for multiplicativity at non-infinity cusps.

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.