Estimate for Higher Moments of Cusp Form Coefficients Over Sum of Two Squares Coefficients Over Sum of Two Squares

Let f and g be two distinct primitive holomorphic cusp forms of even integral weights k1 and k2 for the full modular group Γ = SL(2, Z), respectively. Denote by λf (n) and λg(n) the nth normalized Fourier coefficients of f and g, respectively. In this paper, we consider the summatory function X n=a2+b2≤x λf (n)iλg(n)j , for x ≥ 2, where a, b ∈ Z and i, j ≥ 1 are positive integers.