We report the first detections of the repeating fast radio burst source FRB 121102 above 5.2 GHz. Observations were performed using the 4$-$8 GHz receiver of the Robert C. Byrd Green Bank Telescope with the Breakthrough Listen digital backend. We present the spectral, temporal and polarization properties of 21 bursts detected within the first 60 minutes of a total 6-hour observations. These observations comprise the highest burst density yet reported in the literature, with 18 bursts being detected in the first 30 minutes. A few bursts clearly show temporal sub-structures with distinct spectral properties. These sub-structures superimpose to provide enhanced peak signal-to-noise ratio at higher trial dispersion measures. Broad features occur in $sim 1$ GHz wide subbands that typically differ in peak frequency between bursts within the band. Finer-scale structures ($sim 10-50$ MHz) within these bursts are consistent with that expected from Galactic diffractive interstellar scintillation. The bursts exhibit nearly 100% linear polarization, and a large average rotation measure of 9.359$pm$0.012 $times$ 10$^{rm 4}$ rad m$^{rm -2}$ (in the observers frame). No circular polarization was found for any burst. We measure an approximately constant polarization position angle in the 13 brightest bursts. The peak flux densities of the reported bursts have average values (0.2$pm$0.1 Jy), similar to those seen at lower frequencies ($<3$ GHz), while the average burst widths (0.64$pm$0.46 ms) are relatively narrower.
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