Higher symmetric power L-functions and their Fourier coefficients

Let Hₖ be the set of all normalized primitive holomorphic cusp forms of even integral weight k≥ 2 for the full modular group SL(2, Z), and let j≥ 3 be any fixed integer. For f∈ Hₖ, we write λ_symʲ f(n) for the nᵗh normalized Fourier coefficient of L(s,symʲ f). In this article, we establish an asymptotic formula for the sum equation ∑_n=a₁²+a₂²+…+a₆²≤ x\\ (a₁,a₂,…, a₆)∈ Z⁶ λ_symʲ f²(n), equation with an improved error term.