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.
展开▼
机译：仅在n⩽6时才知道Kloosterman和的第n个幂矩S n 的评估。我们在此提供了根据Hecke评估S 7 的大量证据。具有导体105的四次nebentypus的Γ O （525）上权重3新形式的特征值。我们还证明了与紧密相关的量T 7 ，其中T n 是Kloosterman捆的第n个对称幂的迹线之和。
展开▼
