We calculate the contributions of a general non-vacuum conformal family to Rényi entropy in twodimensional 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 △.For the case of two short intervals on a complex plane,we expand the Rényi mutual information by the cross ratio x to order x2△+2.For the case of one interval on a torus with low temperature,we expand the Rényi entropy by q =exp(-2πβ/L),with β being the inverse temperature and L being the spatial period,to order q△+2.To make the result meaningful,we require that the scaling dimension △ cannot be too small.For two intervals on a complex plane we need △ ＞ 1,and for one interval on a torus we need △ ＞ 2.We work in the small Newton constant limit on the gravity side and so a large central charge limit on the CFT side,and find matches of gravity and CFT results.