  We construct Hecke operators acting on Maass waveforms of integer non-zero
weight and transforming according to a non-trivial multiplier system on the
modular group. Using these Hecke operators we obtain multiplicativity relations
for the Fourier coefficients of such Maass waveforms. These relations
generalize the usual weight zero relations. We also obtain an unexpected
relation between coefficients with positive and negative indices with a
constant of proportionality involving the Laplace eigenvalue. Numerical
examples of multiplicativity relations are given at the end of the paper.
