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Let be a given integer. Let be the set of all normalized primitive holomorphic cusp forms of even integral weight for the full modulo group . For , denote by the th normalized Fourier coefficient of th symmetric power -function () attached to . We are interested in the average behaviour of the sum
where is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).
Let be a nonnormal cubic extension which is given by an irreducible polynomial . Denote by the Dedekind zeta-function of the field and the number of integral ideals in with norm . In this note, by the higher integral mean values and subconvexity bound of automorphic -functions, the second and third moment of is considered, i.e.,
where , are polynomials of degree 1, 4, respectively, is an arbitrarily small number.
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