We study the properties of the magnetic field {vec}(B) of a dynamo by expanding the field in a special set of base functions, namely the eigenfunctions of its dynamo equation [{vec}(B)={SIGMA}sigma_csigma^(t){vec}(b)sigma^({vec}(r))]. We prove that the time evolution of the mean and the frequency spectrum of csigma^(t) are completely determined by the eigenvalue lambdasigma_ of the dynamo equation. The spectrum is a Lorentzian with central frequency omegasigma_=Imlambdasigma_ and width gammasigma_=-Relambdasigma_. This property had been conjectured by Hoyng (1988), and is now shown to hold for arbitrary dynamos. The proof is relatively straightforward, and we point out the error in the analysis of Hoyng (1988). Finally, we illustrate how this property sheds light on the physical meaning of the dynamo equation, and we compare the predicted spectra qualitatively with recent studies of the global properties of the solar magnetic field.