Even if one knows everything for a conformal field theory on the infinite plane, new data are required to place the same theory in finite geometries. This is a physically relevant question for finite-temperature/finite-size critical systems. I will show in this talk how to apply an 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. The main result is 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. I will argue that this pattern unveils the large-N vacuum structure of the corresponding theories at zero temperature.