  A classical result from analytic number theory by Rademacher gives an exact
formula for the Fourier coefficients of modular forms of non-positive weight.
We apply similar techniques to study the spectrum of two-dimensional unitary
conformal field theories, with no extended chiral algebra and $c>1$. By
exploiting the full modular constraints of the partition function we propose an
expression for the spectral density in terms of the light spectrum of the
theory. The expression is given in terms of a Rademacher expansion, which
converges for spin $j \neq 0$. For a finite number of light operators the
expression agrees with a variant of the Poincare construction developed by
Maloney, Witten and Keller. With this framework we study the presence of
negative density of states in the partition function dual to pure gravity, and
propose a scenario to cure this negativity.
