On higher moments of Dirichlet coefficients attached to symmetric square L -functions over certain sparse sequence

Let 2≤ j≤ 8 be any fixed positive integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ =SL(2,ℤ) . Denote by λ _sym^2f(n) the n th normalized coefficient of the Dirichlet expansion of the symmetric square L -function L(sym^2f,s) attached to f . In this paper, we are interested in the average behaviour of the following summatory function ∑ _[ a^2 + b^2 + c^2 + d^2≤ x; (a,b,c,d)∈ℤ^4 ]λ _sym^2f^j(a^2+b^2+c^2+d^2) for x≥ x_0 (sufficiently large), which improves and generalizes the recent works of Sharma and Sankaranarayanan (Res Number Theory 8:19, 2022, Rend Circ Mat Palermo II Ser 72:1399–1416, 2023).