We prove, in respect of an arbitrary Hecke congruence subgroup Γ = Γ 0 (q 0 ) ≤ SL(2, Z[i]), some new upper bounds (or 'spectral large sieve inequalities') 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 = ∞). For this reason, our proof is reliant upon an extension, to arbitrary cusps, of the spectral-Kloosterman sum formula for Γ\SL(2, C) obtained by Hristina Lokvenec-Guleska in her doctoral thesis (generalising the sum formulae of Roelof Bruggeman and Yoichi Motohashi for P SL(2, Z[i])\P SL(2, C) in several respects, though not as regards the choice of cusps). A proof of the required extension of the sum formula is given in an appendix.