We constrain the properties of the progenitor system of the highly reddened Type Ia supernova (SN) 2014J in Messier 82 (M82; d ~ 3.5 Mpc). We determine the SN location using Keck-II K-band adaptive optics images, and we find no evidence for flux from a progenitor system in pre-explosion near-ultraviolet through near-infrared Hubble Space Telescope (HST) images. Our upper limits exclude systems having a bright red giant companion, including symbiotic novae with luminosities comparable to that of RS Ophiuchi. While the flux constraints are also inconsistent with predictions for comparatively cool He-donor systems (T < ~35,000 K), we cannot preclude a system similar to V445 Puppis. The progenitor constraints are robust across a wide range of R_V and A_V values, but significantly greater values than those inferred from the SN light curve and spectrum would yield proportionally brighter luminosity limits. The comparatively faint flux expected from a binary progenitor system consisting of white dwarf stars would not have been detected in the pre-explosion HST imaging. Infrared HST exposures yield more stringent constraints on the luminosities of very cool (T < 3000 K) companion stars than was possible in the case of SN Ia 2011fe.
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