The 21 cm transition of neutral hydrogen is opening an observational window into the Cosmic Dawn of the universe-the epoch of first star formation. We use 28 hr of data from the Owens Valley Radio Observatory Long Wavelength Array to place upper limits on the spatial power spectrum of 21 cm emission at z 18.4 ( ), and within the absorption feature reported by the EDGES experiment. In the process we demonstrate the first application of the double Karhunen-Loève transform for foreground filtering, and diagnose the systematic errors that are currently limiting the measurement. We also provide an updated model for the angular power spectrum of low-frequency foreground emission measured from the northern hemisphere, which can be used to refine sensitivity forecasts for next-generation experiments.