In this paper we study the orthogonality of Fourier coefficients of holomorphic cusp forms in the sense of large sieve inequality. We investigate the family of GL 2 cusp forms modular with respect to the congruence subgroups Γ1(q), with additional averaging over the levels q ∼ Q. We obtain the orthogonality in the range N ≪ Q 2−δ for any δ > 0, where N is the length of linear forms in the large sieve.

Classical Kloosterman sums have a prominent role in the study of automorphic forms on GL${}_{2}$ and further they have numerous applications in analytic number theory. In recent years, various problems in analytic theory of automorphic forms on GL${}_{3}$ have been considered, in which analogous GL${}_{3}$-Kloosterman sums (related to the corresponding Bruhat decomposition) appear. In this note we investigate the first four power-moments of the Kloosterman sums associated with the group SL${}_{3}\left(\mathbb{Z}\right)$. We give formulas for the...

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