The average behaviour of Hecke eigenvalues over certain sparse sequence of positive integers

Let j >= 2 be a given integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Gamma = SL(2, Z). Denote by lambda(symjf)(n) the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function L(sym(j)f, s) attached to f. In this paper, we are interested in the average behaviour of the following sum Sigma(a2+b2+c2+d2 <= x(a,b,c,d)is an element of Z4) lambda(2)(symjf)(a(2) + b(2) + c(2) + d(2)), where x is sufficiently large, which improves and generalizes the recent work of Sharma and Sankaranarayanan [52]. By analogy, we also consider the analogous results for higher moments of normalized Fourier coefficients and the second moment of normalized coefficients of two symmetric power L-functions attached to two distinct cusp forms of the same sequence.