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 ₒₘ₌⋑ 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 ₍=₀䃑ℂ+₀䃒ℂ++₀䃖ℂ ₗ\\ (₀䃑, ₀䃒, , ₀䃖) ₙ䃆 ₒₘ₌ʲ f² (n), equation with an improved error term.