Spectral large sieve inequalities for Hecke congruence subgroups of SL(2,Z[i])

We prove, in respect of an arbitrary Hecke congruence subgroup Γ=Γ0(q0)⩽SL(2,Z[i]), some new upper bounds for sums involving Fourier coefficients of Γ-automorphic cusp forms on SL(2,C). The Fourier coefficients in question may arise from the Fourier expansion at any given cusp c of Γ (our results are not limited to the case c=∞). Our proof utilises an extension, to arbitrary cusps, of a spectral-Kloosterman summation formula for Γ\SL(2,C) that was obtained by Lokvenec-Guleska (in her doctoral thesis).