Density results on the coefficients of the triple product $L$-functions

Let $H_k^{\ast }$ denote the set of normalized primitive holomorphic Hecke cusp forms of even integral weight $k$ for the full modular group. Denote by ${\lambda }_{f\times f\times f}(n)$ the $n$th coefficient of the triple product $L$-function $L(f\times f\times f,s)$ attached to $f\in H_k^{\ast }$. Suppose $Q(\underset{\_}{x})$ is a primitive integral positive-definite binary quadratic form of fixed discriminant $D<0$ with the class number $h(D)=1$. In this paper, we study the distribution of ${\lambda }_{f\times f\times f}(n)$ on the set of all primes and its subset $\{p:p=Q(\underset{\_}{x})\}$ and obtain the analytic density and the natural density of the above sets. These results generalize previous ones.