On modular signs

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.Comment: 23 pages, 1 figure; new version with new coauthor and strong improvements of two of the two main results

On signs of Fourier coefficients of cusp forms

“…We will look for bounds for the size of the first sign change as well as investigate to what extent the signs of the Fourier coefficients determine an unique modular form. In both cases we improve recent results of Kowalski, Lau, Soundararajan and Wu [14]. More background and motivation for the study can be found from the introduction of their paper.…”

“…Most recently Kowalski, Lau, Soundararajan and Wu [14] have shown that n f Q 9/20 . In the paper, they point out that 9/20 should not be the limit of their method; here we will push the method further to get the significantly better exponent 3/8.…”

“…The same strategy has been used in the previous works on the subject. Of particular importance is that the authors of [14] showed that the Hecke relation (1·1) allows one to replace λ f (n) by a step function when considering the lower bound.…”

Comparing Hecke eigenvalues of newforms

“…The extent to which the signs of λ f (p) at primes p determine f uniquely has been first studied by Kowalski, Lau, Soundararajan and Wu [3] (and also by Matomäki [4], who refined some of their results). We shall concern ourselves with the following related question:…”

Bagchi’s Theorem for Families of Automorphic Forms

“…This is a well-known consequence of the Petersson formula, see e.g. [10,Prop. 8], [12,Appendix] or [3, Prop.…”