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Starspot evolution, Differential Rotation and Correlation between Chromospheric and Photospheric Activities on Kepler-411

We present an analysis of the starspot evolution, the surface differential rotation (SDR), the correlation between chromospheric activity indicators and the spatial connection between chromospheric and photospheric activities on the active star Kepler-411, using time series photometry over 4 years from Kepler, and spectroscopic data from Keck I 10-m telescope and Lijiang 2.4-m telescope. We constructed the light curve by re-performing photometry and reduction from the Target Pixel Files and Cotrending Basis Vectors with a manually redefined aperture using the software PyKE3. An efficient program, GEMC_LCM, was developed to apply a two-spots model to chosen light curve segments with three spot groups at fixed latitudes (30, 45), (30, 60) and (45, 60). We found a periodic variation of the starspots at period of about 660 days which independs on spot latitudes, and estimated the lower limit of SDR as $P_{\text{eq}} = 9.7810(0.0169)$ days and $α= 0.1016(0.0023)$. Simultaneously, the relative variations of chromospheric activity indicators were derived by subtracting the overall mean spectrum from individual spectrum. It is found that Ca II H and K emissions are strongly correlated with each other, and there also exists a correlation between H$α$ and Ca II H & K emissions, with large dispersion, in accordance with previous results. Furthermore we find the correlation between Ca II H and K emissions is different in 2011 and 2012. The chromospheric emission variation shows a highly spatial anti-correlation with the light curve, suggesting a spatial connection between the chromospheric active region and spot region.