We present NuSTAR observations of the powerful radio galaxy Cygnus A, focusing on the central absorbed active galactic nucleus (AGN). Cygnus A is embedded in a cool-core galaxy cluster, and hence we also examine archival XMM-Newton data to facilitate the decomposition of the spectrum into the AGN and intracluster medium (ICM) components. NuSTAR gives a source-dominated spectrum of the AGN out to >70keV. In gross terms, the NuSTAR spectrum of the AGN has the form of a power law (Gamma~1.6-1.7) absorbed by a neutral column density of N_H~1.6x10^23 cm^-2. However, we also detect curvature in the hard (>10keV) spectrum resulting from reflection by Compton-thick matter out of our line-of-sight to the X-ray source. Compton reflection, possibly from the outer accretion disk or obscuring torus, is required even permitting a high-energy cutoff in the continuum source; the limit on the cutoff energy is E_cut>111keV (90% confidence). Interestingly, the absorbed power-law plus reflection model leaves residuals suggesting the absorption/emission from a fast (15,000-26,000km/s), high column-density (N_W>3x10^23 cm^-2), highly ionized (xi~2,500 erg cm/s) wind. A second, even faster ionized wind component is also suggested by these data. We show that the ionized wind likely carries a significant mass and momentum flux, and may carry sufficient kinetic energy to exercise feedback on the host galaxy. If confirmed, the simultaneous presence of a strong wind and powerful jets in Cygnus A demonstrates that feedback from radio-jets and sub-relativistic winds are not mutually exclusive phases of AGN activity but can occur simultaneously.
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