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 k 1, k 2≥ 2 and levels N 1, N 2≥ 1 respectively. Also let a f (n) and a g (n) be the n-th Fourier-coefficients of f and g respectively. In this article, we investigate the first sign change of the sequence {a f (p α) a g (p α)} p α∈ N, α≤ 2, where p is a prime number. We further study the non-vanishing of the sequence {a f (n) a g (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 {a f (p n) a g (p m)} 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.