A larger GL2 large sieve in the level aspect

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.