The discovery that many classical novae produce detectable GeV $gamma$-ray emission has raised the question of the role of shocks in nova eruptions. Here we use radio observations of nova V809 Cep (Nova Cep 2013) with the Jansky Very Large Array to show that it produced non-thermal emission indicative of particle acceleration in strong shocks for more than a month starting about six weeks into the eruption, quasi-simultaneous with the production of dust. Broadly speaking, the radio emission at late times -- more than a six months or so into the eruption -- is consistent with thermal emission from $10^{-4} M_odot$ of freely expanding, $10^4$~K ejecta. At 4.6 and 7.4 GHz, however, the radio light-curves display an initial early-time peak 76 days after the discovery of the eruption in the optical ($t_0$). The brightness temperature at 4.6 GHz on day 76 was greater than $10^5 K$, an order of magnitude above what is expected for thermal emission. We argue that the brightness temperature is the result of synchrotron emission due to internal shocks within the ejecta. The evolution of the radio spectrum was consistent with synchrotron emission that peaked at high frequencies before low frequencies, suggesting that the synchrotron from the shock was initially subject to free-free absorption by optically thick ionized material in front of the shock. Dust formation began around day 37, and we suggest that internal shocks in the ejecta were established prior to dust formation and caused the nucleation of dust.
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