  Using modular bootstrap we show the lightest primary fields of a unitary
compact two dimensional conformal field theory(with $c, \bar{c}>1$) has a
conformal weight $h_1\le \frac{c}{12}+\mathcal{O}(1)$.This implies that the
upper bound on the dimension of the lightest primary fields depends on their
spin. In particular if the set of lightest primary fields includes extremal or
near extremal states whose spin to dimension ratio $\frac{j}{\Delta}\approx 1$,
the corresponding dimension is $\Delta\le \frac{c}{12}+\mathcal{O}(1)$. From
AdS/CFT correspondence, we obtain an upper bound on the spectrum of black hole
in three dimensional gravity. Our results show that if the first primary fields
have large spin, the corresponding three dimensional gravity has extremal or
near extremal BTZ black hole.
