A note on average behaviour of the Fourier coefficients of jth symmetric power L-function over certain sparse sequence of positive integers

Let j >= 2 be a given integer. Let H-k* be the set of all normalized primitive holomorphic cusp forms of even integral weight k >= 2 for the full modulo group SL(2, Z). For f is an element of H-k*, denote by lambda(symj f)(n) the nth normalized Fourier coefficient of jth symmetric power L-function (L(s, sym(j) f)) attached to f. We are interested in the average behaviour of the sum & sum;(n=a12+a22+a32+a42+a52+a62 <= x(a1,a2,a3,a4,a5,a6)is an element of Z6) lambda(2)(symj f)(n), where x is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).