Higher Symmetric Powers over Arithmetic Progressions and Applications

We investigate higher moments of symmetric power Fourier coefficients of holomorphic cusp forms in arithmetic progressions. Using explicit L-function decompositions together with subconvexity and hybrid subconvexity bounds, we obtain asymptotic formulas for the third and fourth symmetric power moments. As an application, we derive corresponding asymptotic formulas for weighted shifted convolution sums of higher symmetric power coefficients with respect to a kernel function.