  We show that the action of conformal vector fields on functions on the sphere
determines the spectrum of the Laplacian (or the conformal Laplacian), without
further input of information. The spectra of intertwining operators (both
differential and non-local) with principal part a power of the Laplacian
follows as a corollary. An application of the method is the sharp form of
Gross' entropy inequality on the sphere. The same method gives the spectrum of
the Dirac operator on the sphere, as well as of a continuous family of nonlocal
intertwinors, and an infinite family of odd-order differential intertwinors.
