On Signs of eigenvalues of Modular forms satisfying Ramanujan Conjecture

Let $F \in S_{k_1}(\Gamma^{(2)}(N_1))$ and $G \in S_{k_2}(\Gamma^{(2)}(N_2))$ be two Siegel cusp forms over the congruence subgroups $\Gamma^{(2)}(N_1)$ and $\Gamma^{(2)}(N_2)$ 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, $\{ p : \lambda_F(p) \lambda_G(p) < 0 \}.$