  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 $\alpha = 0.1016(0.0023)$. 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$\alpha$ 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.
