We have discovered a new, rare triple-mode RR Lyr star, EPIC 201585823, in the Kepler K2 mission Campaign 1 data. This star pulsates primarily in the fundamental and first-overtone radial modes, and, in addition, a third nonradial mode. The ratio of the period of the nonradial mode to that of the first-overtone radial mode, 0.616285, is remarkably similar to that seen in 11 other triple-mode RR Lyr stars, and in 260 RRc stars observed in the Galactic Bulge. This systematic character promises new constraints on RR Lyr star models. We detected subharmonics of the nonradial mode frequency, which are a signature of period doubling of this oscillation; we note that this phenomenon is ubiquitous in RRc and RRd stars observed from space, and from ground with sufficient precision. The nonradial mode and subharmonic frequencies are not constant in frequency or in amplitude. The amplitude spectrum of EPIC 201585823 is dominated by many combination frequencies among the three interacting pulsation mode frequencies. Inspection of the phase relationships of the combination frequencies in a phasor plot explains the `upward shape of the light curve. We also found that raw data with custom masks encompassing all pixels with significant signal for the star, but without correction for pointing changes, is best for frequency analysis of this star, and, by implication, other RR Lyr stars observed by the K2 mission. We compare several pipeline reductions of the K2 mission data for this star.
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