  We calculate the contributions of a general non-vacuum conformal family to
R\'enyi entropy in two-dimensional conformal field theory (CFT). The primary
operator of the conformal family can be either non-chiral or chiral, and we
denote its scaling dimension by $\Delta$. For the case of two short intervals
on complex plane, we expand the R\'enyi mutual information by the cross ratio
$x$ to order $x^{2\Delta+2}$. For the case of one interval on torus with the
temperature being low, we expand the R\'enyi entropy by $q=\exp(-2\pi\beta/L)$,
with $\beta$ being the inverse temperature and $L$ being the spatial period, to
order $q^{\Delta+2}$. To make the result meaningful, we require that the
scaling dimension $\Delta$ cannot be too small. For two intervals on complex
plane we need $\Delta>1$, and for one interval on torus we need $\Delta>2$. We
work in small Newton constant limit in gravity side and so large central charge
limit in CFT side, and find matches of gravity and CFT results.
