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

2018/01/31 by Gun, Sanoli, Kumar, Balesh, Paul, Biplab · 1 citation
#11F11 #11F30 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1801.10590

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 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(pⁿ)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.

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