Funct. Approx. Comment. Math. 48 (2), 213-376, (June 2013) DOI: 10.7169/facm/2013.48.2.4
KEYWORDS: Spectral theory, large sieve, mean value, Hecke congruence group, Gaussian number field, Gaussian integers, sum formula, automorphic form, cusp form, non-holomorphic modular form, Fourier coefficient, Kloosterman sum, inverse Bessel transform, eigenvalue conjecture, grössencharakter, Hecke character, 11F30, 11F37, 11F70, 11F72, 11L05, 11L07, 11M41, 11N13, 11N35, 11R42, 11R44, 22E30, 33C10, 44A15

We prove new bounds for weighted mean values of sums involving Fourier coefficients of cusp forms that are automorphic with respect to a~Hecke congruence subgroup $\Gamma\leq SL(2,{\mathbb Z}[i])$, and correspond to exceptional eigenvalues of the Laplace operator on the space $L^2(\Gamma\backslash SL(2,{\mathbb C})/SU(2))$. These results are, for certain applications, an effective substitute for the generalised Selberg eigenvalue conjecture. We give a~proof of one such application, which is an upper bound for a~sum of generalised Kloosterman sums (of significance in the study of certain mean values of Hecke zeta-functions with groessencharakters). Our proofs make extensive use of Lokvenec-Guleska's generalisation of the Bruggeman-Motohashi summation formulae for $PSL(2,{\mathbb Z}[i])\backslash PSL(2,{\mathbb C})$. We also employ a~bound of Kim and Shahidi for the first eigenvalues of the relevant Laplace operators, and an `unweighted' spectral large sieve inequality (our proof of which is to appear separately).