  We consider unitary, modular invariant, two-dimensional CFTs which are
invariant under the parity transformation $P$. Combining $P$ with modular
inversion $S$ leads to a continuous family of fixed points of the $SP$
transformation. A particular subset of this locus of fixed points exists along
the line of positive left- and right-moving temperatures satisfying $\beta_L
\beta_R = 4\pi^2$. We use this fixed locus to prove a conjecture of Hartman,
Keller, and Stoica that the free energy of a large-$c$ CFT$_2$ with a suitably
sparse low-lying spectrum matches that of AdS$_3$ gravity at all temperatures
and all angular potentials. We also use the fixed locus to generalize the
modular bootstrap equations, obtaining novel constraints on the operator
spectrum and providing a new proof of the statement that the twist gap is
smaller than $(c-1)/12$ when $c>1$. At large $c$ we show that the operator
dimension of the first excited primary lies in a region in the
$(h,\overline{h})$-plane that is significantly smaller than
$h+\overline{h}<c/6$. Our results for the free energy and constraints on the
operator spectrum extend to theories without parity symmetry through the
construction of an auxiliary parity-invariant partition function.
