On Signs of eigenvalues of Modular forms satisfying Ramanujan Conjecture

Let $F\in S_{k_1}({\mathrm{\Gamma }}^{(2)}(N_1))$ and $G\in S_{k_2}({\mathrm{\Gamma }}^{(2)}(N_2))$ be two Siegel cusp forms over the congruence subgroups ${\mathrm{\Gamma }}^{(2)}(N_1)$ and ${\mathrm{\Gamma }}^{(2)}(N_2)$ respectively. Assume that they are Hecke eigenforms in different eigenspaces and satisfy the Generalized Ramanujan Conjecture. Let ${\lambda }_F(p)$ denote the eigenvalue of $F$ with respect to the Hecke operator $T(p)$. In this article, we compute a lower bound for the density of the set of primes, $\{p:{\lambda }_F(p){\lambda }_G(p)<0\}\mathrm{.}$