A double-modulation effect detected in a double-mode high-amplitude $\delta$ Scuti star: KIC 10284901

In this paper, we present an analysis of the pulsating behavior of $Kepler$ target KIC 10284901. The Fourier transform of the high-precision light curve reveals seven independent frequencies for its light variations. Among them, the first two frequencies are main pulsation modes: F0 = 18.994054(1) $\rm{day^{-1}}$ and F1 = 24.335804(4) $\rm{day^{-1}}$, the ratio of F0/F1 = 0.7805 classifies this star as a double-mode high-amplitude $\delta$ Scuti (HADS) star; another two frequencies, $f_{m1}$ = 0.4407 day$^{-1}$ and $f_{m2}$ = 0.8125 day$^{-1}$, are detected directly, and the modulations of $f_{m1}$ and $f_{m2}$ to F0 and F1 modes (seen as quintuplet structures centered on these two modes in the frequency spectrum) are also found. This is the first detection of a double-modulation effect in the HADS stars. The features of the frequency pattern and the ratio ($f_{m1}$/$f_{m2}$ $\approx$ 1:2), as well as the cyclic variation of amplitude of the two dominant pulsation modes, which seem to be similar to that in Blazhko RR Lyrae stars, indicate this modulation might be related to the Blazhko effect. A preliminary analysis suggests that KIC 10284901 is in the bottom of the HADS instability strip and situated in the main sequence.