The first simultaneous sign change and non-vanishing of Hecke eigenvalues of newforms

Abstract

Let f and g be two distinct newforms which are normalized Hecke eigenforms of weights k1,k2≥2 and levels N1,N2≥1 respectively. Also let af(n) and ag(n) be the n-th Fourier-coefficients of f and g respectively. In this article, we investigate the first sign change of the sequence {af(pα)ag(pα)}pα∈N,α≤2, where p is a prime number. We further study the non-vanishing of the sequence {af(n)ag(n)}n∈N and derive bounds for first non-vanishing term in this sequence. We also show, using ideas of Kowalski–Robert–Wu and Murty–Murty, that there exists a set of primes S of natural density one such that for any prime p∈S, the sequence {af(pn)ag(pm)}n,m∈N has no zero elements. This improves a recent work of Kumari and Ram Murty. Finally, using B-free numbers, we investigate simultaneous non-vanishing of coefficients of m-th symmetric power L-functions of non-CM forms in short intervals.