  Any unitary compact two-dimensional CFT with $c>1$ and no symmetries beyond
Virasoro has a parametrically large density of primary states at large spin for
$\bar{h}>\bar{h}_\text{extr}\sim \frac{c-1}{24}$, of a universal form
determined by modular invariance. By including the contribution of light
primary operators and multi-twist composites constructed from them in the
modular bootstrap, we find that $\bar{h}_\text{extr}$ receives corrections in a
large spin expansion, which we compute at finite $c$. The analysis uses a
formulation of the modular S-transform as a Fourier transform acting on the
density of primary states. For theories with gravitational duals,
$\bar{h}_\text{extr}$ is interpreted as the extremality bound of rotating BTZ
black holes, receiving quantum corrections which we compute at one loop by
prohibiting naked singularities in the quantum-corrected geometry. This gravity
result is reproduced by modular bootstrap in a semiclassical $c\to\infty$
limit.
