  We rewrite the worldsheet torus partition function of the Thermal AdS CFT by
isolating the boundary parameters. Using this, we show that the spectrum of the
Euclidean BTZ black hole and Lorentzian AdS3 can be extracted -- the latter as
a zero temperature limit. A similar procedure recovers the Lorentzian BTZ
spectrum proposed in an earlier work. We then use our expression to construct a
boundary modular invariant expression as a Poincar\'e series.
