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