  General Relativity predicts the existence of quadratic quasi-normal modes at
second order in perturbation theory. Building on our recent work, we compute
the amplitudes and polarizations of these modes for non-rotating black holes,
showing that they are completely determined by the amplitudes and polarizations
of linear modes. We obtain the ratio of quadratic to linear amplitudes, which
still depends on the initial conditions of the merger through the polarization
of linear modes. However, we demonstrate that this dependence is captured by
four fundamental numbers, independent of initial conditions, representing four
different combinations of linear modes parities. Additionally, we prove two
selection rules regarding the vanishing of classes of quadratic modes. Our
results are available online as a package which provides the ratio of
amplitudes across a broad spectrum of angular momenta.
