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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 (see Theorem 3.1 (2)–(4) and §5). Numerical examples of multiplicativity relations are given at the end of the paper. [ABSTRACT FROM PUBLISHER]