  We explore the large spin spectrum in two-dimensional conformal field
theories with a finite twist gap, using the modular bootstrap in the lightcone
limit. By recursively solving the modular crossing equations associated to
different $PSL(2,\mathbb{Z})$ elements, we identify the universal contribution
to the density of large spin states from the vacuum in the dual channel. Our
result takes the form of a sum over $PSL(2,\mathbb{Z})$ elements, whose leading
term generalizes the usual Cardy formula to a wider regime. Rather curiously,
the contribution to the density of states from the vacuum becomes negative in a
specific limit, which can be canceled by that from a non-vacuum Virasoro
primary whose twist is no bigger than $c-1\over16$. This suggests a new upper
bound of $c-1\over 16$ on the twist gap in any $c>1$ compact, unitary conformal
field theory with a vacuum, which would in particular imply that pure AdS$_3$
gravity does not exist. We confirm this negative density of states in the pure
gravity partition function by Maloney, Witten, and Keller. We generalize our
discussion to theories with $\mathcal{N}=(1,1)$ supersymmetry, and find similar
results.
