On signs of Fourier coefficients of cusp forms

Abstract

We consider two problems concerning signs of Fourier coefficients of classical modular forms, or equivalently Hecke eigenvalues: first, we give an upper bound for the size of the first sign-change of Hecke eigenvalues in terms of conductor and weight; second, we investigate to what extent the signs of Fourier coefficients determine an unique modular form. In both cases we improve recent results of Kowalski, Lau, Soundararajan and Wu. A part of the paper is also devoted to generalized rearrangement inequalities which are utilized in an alternative treatment of the second question.

A variant of multiplicity one theorems for half-integral weight modular forms

“…For primitive forms f of integral weight, the extent to which the signs of Hecke eigenvalues at primes p determine f uniquely has been first studied by Kowalski et al. [KLSW10] (and also by Matomäki [Mat12], who refined some of their results). A natural question to ask if similar results continue to hold in the case of half-integral weight modular forms?…”

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:…”

Additive bases with coefficients of newforms

“…Let 1 ≤ s ≤ 2k be such that q k+t = p s . It holds that q t , q k+t−1 ≤ p s−1 , and from (6), (8) and the Ramanujan-Petersson-Deligne estimate |a(p s−1 )| ≤ 2p…”