An extended large sieve for Maaß cusp forms

Abstract

For a certain big family of Maaß cusp forms, which in a way extends beyond the Hecke congruence subgroup, we establish a large sieve inequality. The set of functions under consideration is constructed by summing specific families of Maaß cusp forms for the Hecke congruence subgroup of odd prime level N with respect to Dirichlet characters of the modulus of the level. The result hinges on a suitable version of the Bruggeman- Kuznetsov formula, upon which we build our argument, proving in a first step an asymptotic formula for a weighted L^2 sum, featuring the Fourier coefficients of the functions from the family we consider. The inequality we finally conclude in the main theorem, describes an upper bound for this weighted L^2 sum, that in general does not meet the expectation from theoretical considerations of the large sieve.