  Modular invariance strongly constrains the spectrum of states of two
dimensional conformal field theories. By summing over the images of the modular
group, we construct candidate CFT partition functions that are modular
invariant and have positive spectrum. This allows us to efficiently extract the
constraints on the CFT spectrum imposed by modular invariance, giving
information on the spectrum that goes beyond the Cardy growth of the asymptotic
density of states. Some of the candidate modular invariant partition functions
we construct have gaps of size (c-1)/12, proving that gaps of this size and
smaller are consistent with modular invariance. We also revisit the partition
function of pure Einstein gravity in AdS3 obtained by summing over geometries,
which has a spectrum with two unphysical features: it is continuous, and the
density of states is not positive definite. We show that both of these can be
resolved by adding corrections to the spectrum which are subleading in the
semi-classical (large central charge) limit.
