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(\underline 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(\underline x)\} $ and obtain the analytic density and the natural density of the above sets. These results generalize previous ones.