  We apply the OPE inversion formula to thermal two-point functions of bosonic
and fermionic CFTs in general odd dimensions. This allows us to analyze in
detail the operator spectrum of these theories. We find that nontrivial thermal
CFTs arise when the thermal mass satisfies an algebraic transcendental equation
that ensures the absence of an infinite set of operators from the spectrum. The
solutions of these gap equations for general odd dimensions are in general
complex numbers and follow a particular pattern. We argue that this pattern
unveils the large-$N$ vacuum structure of the corresponding theories at zero
temperature.
