  The modular bootstrap program for 2d CFTs could be seen as a systematic
exploration of the physical consequences of consistency conditions at the
elliptic points and at the cusp of their toruspartition function. The study at
$\tau=i$, the elliptic point stabilized by the modular inversion $S$, was
initiated by Hellerman, who found a general upper bound for the most relevant
scaling dimension $\Delta$. Likewise, analyticity at $\tau=i\infty$, the cusp
stabilized by the modular translation $T$, yields an upper bound on the twist
gap. Here we study consistency conditions at $\tau=\exp[2i\pi/3]$, the elliptic
point stabilized by $S T$. We find a much stronger upper bound in the large-c
limit, namely $\Delta<\frac{c-1}{12}+0.092$, which is very close to the minimal
mass threshold of the BTZ black holes in the gravity dual of $AdS_3/CFT_2$
correspondence.
