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

Let $f$ and $g$ be two distinct newforms which are normalized Hecke eigenforms of weights $k_1, k_2 \ge 2$ and levels $N_1, N_2 \ge 1$ 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 $\{a_f(p^{\alpha})a_g(p^{\alpha}) \}_{p^{\alpha} \in \N, \alpha \le 2}$, where $p$ is a prime number. We further study the non-vanishing of the sequence $\{a_f(n)a_g(n) \}_{n \in \N}$ 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 $p \in S$, the sequence $\{a_f(p^n)a_g(p^m) \}_{n,m \in \N}$ 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.