On modular signs

TL;DR

This paper investigates the signs of Hecke eigenvalues of modular forms, exploring their uniqueness in determining forms and analyzing the first sign-change, with improved estimates on its size.

Contribution

It provides new results on how signs of eigenvalues can determine modular forms and offers significantly improved bounds on the first sign-change of these eigenvalues.

Findings

Signs of eigenvalues can sometimes uniquely determine modular forms.

New statistical and individual results on eigenvalue signs.

Significantly improved bounds on the first sign-change size.

Abstract

We consider some questions related to the signs of Hecke eigenvalues or Fourier coefficients of classical modular forms. One problem is to determine to what extent those signs, for suitable sets of primes, determine uniquely the modular form, and we give both individual and statistical results. The second problem, which has been considered by a number of authors, is to determine the size, in terms of the conductor and weight, of the first sign-change of Hecke eigenvalues. Here we improve significantly the recent estimate of Iwaniec, Kohnen and Sengupta.