  We report upper-limits on the Epoch of Reionization (EoR) 21 cm power
spectrum at redshifts 7.9 and 10.4 with 18 nights of data ($\sim36$ hours of
integration) from Phase I of the Hydrogen Epoch of Reionization Array (HERA).
The Phase I data show evidence for systematics that can be largely suppressed
with systematic models down to a dynamic range of $\sim10^9$ with respect to
the peak foreground power. This yields a 95% confidence upper limit on the 21
cm power spectrum of $\Delta^2_{21} \le (30.76)^2\ {\rm mK}^2$ at $k=0.192\ h\
{\rm Mpc}^{-1}$ at $z=7.9$, and also $\Delta^2_{21} \le (95.74)^2\ {\rm mK}^2$
at $k=0.256\ h\ {\rm Mpc}^{-1}$ at $z=10.4$. At $z=7.9$, these limits are the
most sensitive to-date by over an order of magnitude. While we find evidence
for residual systematics at low line-of-sight Fourier $k_\parallel$ modes, at
high $k_\parallel$ modes we find our data to be largely consistent with thermal
noise, an indicator that the system could benefit from deeper integrations. The
observed systematics could be due to radio frequency interference, cable
sub-reflections, or residual instrumental cross-coupling, and warrant further
study. This analysis emphasizes algorithms that have minimal inherent signal
loss, although we do perform a careful accounting in a companion paper of the
small forms of loss or bias associated with the pipeline. Overall, these
results are a promising first step in the development of a tuned,
instrument-specific analysis pipeline for HERA, particularly as Phase II
construction is completed en route to reaching the full sensitivity of the
experiment.
