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We present 41 bursts from the first repeating fast radio burst discovered(FRB 121102). A deep search has allowed us to probe unprecedentedly low burstenergies during two consecutive observations (separated by one day) using theArecibo telescope at 1.4 GHz. The bursts are generally detected in less than athird of the 580-MHz observing bandwidth, demonstrating that narrow-band FRBsignals may be more common than previously thought. We show that the bursts arelikely faint versions of previously reported multi-component bursts. There is astriking lack of bursts detected below 1.35 GHz and simultaneous VLAobservations at 3 GHz did not detect any of the 41 bursts, but did detect onethat was not seen with Arecibo, suggesting preferred radio emission frequenciesthat vary with epoch. A power law approximation of the cumulative distributionof burst energies yields an index $-1.8\pm0.3$ that is much steeper than thepreviously reported value of $\sim-0.7$. The discrepancy may be evidence for amore complex energy distribution. We place constraints on the possibility thatthe associated persistent radio source is generated by the emission of manyfaint bursts ($\sim700$ ms$^{-1}$). We do not see a connection between burstfluence and wait time. The distribution of wait times follows a log-normaldistribution centered around $\sim200$ s; however, some bursts have wait timesbelow 1 s and as short as 26 ms, which is consistent with previous reports of abimodal distribution. We caution against exclusively integrating over the fullobserving band during FRB searches, because this can lower signal-to-noise.

A sample of low energy bursts from FRB 121102