Seventh power moments of Kloosterman sums
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Evaluations of the n-th power moments S n of Kloosterman sums are known only for n ⩽ 6. We present here substantial evidence for an evaluation of S 7 in terms of Hecke eigenvalues for a weight 3 newform on ΓO(525) with quartic nebentypus of conductor 105. We also prove some congruences modulo 3, 5 and 7 for the closely related quantity T 7, where T n is a sum of traces of n-th symmetric powers of the Kloosterman sheaf.
KeywordsDirichlet Character Symmetric Power Quadratic Character Power Moment Congruence Modulo
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