Average behavior of Fourier coefficients of cusp forms over sum of two squares

Let λf(n) denote the nth normalized Fourier coefficient of the classical holomorphic cusp form of even integral weight k≥2 for the full modular group SL(2,Z). In this paper, we investigate the average behavior of the power sum∑a2+b2≤xλf(a2+b2)ℓ for x≥1, 2≤ℓ≤8 and a,b,ℓ∈Z.