  The crossing equations of a conformal field theory can be systematically
truncated to a finite, closed system of polynomial equations. In certain cases,
solutions of the truncated equations place strict bounds on the space of all
unitary CFTs. We describe the conditions under which this holds, and use the
results to develop a fast algorithm for modular bootstrap in 2d CFT. We then
apply it to compute spectral gaps to very high precision, find scaling
dimensions for over a thousand operators, and extend the numerical bootstrap to
the regime of large central charge, relevant to holography. This leads to new
bounds on the spectrum of black holes in three-dimensional gravity. We provide
numerical evidence that the asymptotic bound on the spectral gap from spinless
modular bootstrap, at large central charge $c$, is $\Delta_1 \lesssim c/9.1$.
