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Gyrotrons are typically driven by electron beams produced by magnetron-type thermionic electron guns operating in the regime of temperature limited emission. Very often, the current density in such annular electron beams is azimuthally nonuniform. To describe the effect of this nonuniformity on gyrotron operation, the code MAGY [M. Botton et al., IEEE Trans. Plasma Sci. 26, 882 (1998)], which is widely used for modeling of slow and fast microwave sources, was properly modified. The results of numerical simulations demonstrate the effect of azimuthal inhomogeneity of the emission on the excitation of low- and high-frequency satellites of the operating mode and on the efficiency degradation. The calculations are done for parameters typical for megawatt-class, long-pulse, millimeter-wave gyrotrons, which are currently under development for electron cyclotron plasma heating and current drive experiments in controlled fusion reactors.