  The redshifted 21 cm signal from neutral hydrogen is a direct probe of the
physics of the early universe and has been an important science driver of many
present and upcoming radio interferometers. In this study we use a single night
of observations with the New Extension in Nan\c{c}ay Upgrading LOFAR (NenuFAR)
to place upper limits on the 21 cm power spectrum from cosmic dawn at a
redshift of $z$ = 20.3. NenuFAR is a new low-frequency radio interferometer,
operating in the 10-85 MHz frequency range, currently under construction at the
Nan\c{c}ay Radio Observatory in France. It is a phased array instrument with a
very dense uv coverage at short baselines, making it one of the most sensitive
instruments for 21 cm cosmology analyses at these frequencies. Our analysis
adopts the foreground subtraction approach, in which sky sources are modeled
and subtracted through calibration and residual foregrounds are subsequently
removed using Gaussian process regression. The final power spectra are
constructed from the gridded residual data cubes in the uv plane. Signal
injection tests are performed at each step of the analysis pipeline, the
relevant pipeline settings are optimized to ensure minimal signal loss, and any
signal suppression is accounted for through a bias correction on our final
upper limits. We obtain a best 2$\sigma$ upper limit of $2.4\times 10^7$
$\text{mK}^{2}$ at $z$ = 20.3 and $k$ = 0.041 $h\,\text{cMpc}^{-1}$. We see a
strong excess power in the data, making our upper limits two orders of
magnitude higher than the thermal noise limit. We investigate the origin and
nature of this excess power and discuss further improvements to the analysis
pipeline that can potentially mitigate it and consequently allow us to reach
thermal noise sensitivity when multiple nights of observations are processed in
the future.
