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We report upper-limits on the Epoch of Reionization (EoR) 21 cm powerspectrum at redshifts 7.9 and 10.4 with 18 nights of data ($\sim36$ hours ofintegration) from Phase I of the Hydrogen Epoch of Reionization Array (HERA).The Phase I data show evidence for systematics that can be largely suppressedwith systematic models down to a dynamic range of $\sim10^9$ with respect tothe peak foreground power. This yields a 95% confidence upper limit on the 21cm 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 themost sensitive to-date by over an order of magnitude. While we find evidencefor residual systematics at low line-of-sight Fourier $k_\parallel$ modes, athigh $k_\parallel$ modes we find our data to be largely consistent with thermalnoise, an indicator that the system could benefit from deeper integrations. Theobserved systematics could be due to radio frequency interference, cablesub-reflections, or residual instrumental cross-coupling, and warrant furtherstudy. This analysis emphasizes algorithms that have minimal inherent signalloss, although we do perform a careful accounting in a companion paper of thesmall forms of loss or bias associated with the pipeline. Overall, theseresults are a promising first step in the development of a tuned,instrument-specific analysis pipeline for HERA, particularly as Phase IIconstruction is completed en route to reaching the full sensitivity of theexperiment.

First Results from HERA Phase I: Upper Limits on the Epoch of Reionization 21 cm Power Spectrum