A formalism is developed for determining the possibility of operation of a gyrotron at both the fundamental, w = wc in a TEmpq mode and at the second harmonic, w' = 2wc, in a TEm'p'q' mode. This formalism is then applied to two physical problems. First, the conditions are derived for conversion of a gyrotron from operation at w = wc to operation at w' = 2wc. Such a conversion is shown to be most practical when the mode at w = wc is a higher order mode (m or p large) and/or the beam position is at a higher radial maximum. Several specific examples are considered. Second, the conditions are derived for oscillation of a parasitic mode at w = wc in a gyrotron designed for operation at w' = 2wc. Again, this problem is more severe for higher order modes and beams at larder radii. For a whispering gallery mode TEm'11 oscillating at w' = 2wc, if m' is odd, there is a resonant para-sitic whispering gallery mode TEmll at w = wc, with m = 0.5 (m' - 1). This would be expected to spoil the 2wc operation of all TEmy 11 modes, with m' odd.