The first multimegawatt (4 MW, /spl eta/=8%) harmonic (/spl omega/=s/spl Omega//sub c/, s=2,3) relativistic gyrotron traveling-wave tube (gyro-twt) amplifier experiment has been designed, built, and tested. Results from this experimental setup, including the first ever reported third-harmonic gyro-twt results, are presented. Operation frequency is 17.1 GHz. Detailed phase measurements are also presented. The electron beam source is SNOMAD-II, a solid-state nonlinear magnetic accelerator driver with nominal parameters of 400 kV and 350 A. The flat-top pulsewidth is 30 ns. The electron beam is focused using a Pierce geometry and then imparted with transverse momentum using a bifilar helical wiggler magnet. The imparted beam pitch is a /spl alpha//spl equiv//spl beta//sub /spl perp////spl beta//sub /spl par///spl ap/1. Experimental operation involving both a second-harmonic interaction with the TE/sub 21/ mode and a third-harmonic interaction with the TE/sub 31/ mode, both at 17 GHz, has been characterized. The third-harmonic interaction resulted in 4-MW output power and 50-dB single-pass gain, with an efficiency of up to /spl sim/8% (for 115-A beam current). The best measured phase stability of the TE/sub 31/ amplified pulse was /spl plusmn/10/spl deg/ over a 9-ns period. The phase stability was limited because the maximum RF power was attained when operating far from wiggler resonance. The second harmonic, TE/sub 21/ had a peak amplified power of 2 MW corresponding to 40 dB single-pass gain and 4% efficiency. The second-harmonic interaction showed stronger superradiant emission than the third-harmonic interaction. Characterizations of the second- and third-harmonic gyro-twt experiments presented here include measurement of far-field radiation patterns, gain and phase versus interaction length, phase stability, and output power versus input power.