  The response of black holes to small perturbations is known to be partially
described by a superposition of quasinormal modes. Despite their importance to
enable strong-field tests of gravity, little to nothing is known about what
overtones and quasinormal-mode amplitudes are like for black holes in
extensions to general relativity. We take a first step in this direction and
study what is arguably the simplest model that allows first-principle
calculations to be made: a nonrotating black hole in an effective-field-theory
extension of general relativity with cubic-in-curvature terms. Using a
phase-amplitude scheme that uses analytical continuation and the Pr\"ufer
transformation, we compute, for the first time, the quasinormal overtone
frequencies (in this theory) and quasinormal-mode excitation factors (in any
theory beyond general relativity). We find that the overtone quasinormal
frequencies and their excitation factors are more sensitive than the
fundamental mode to the lengthscale $l$ introduced by the higher-derivative
terms in the effective field theory. We argue that a description of all
overtones cannot be made within the regime of validity of the effective field
theory, and we conjecture that this is a general feature of any extension to
general relativity that introduces a new lengthscale. We also find that a
parametrization of the modifications to the general-relativistic quasinormal
frequencies in terms of the ratio between $l$ and the black hole's mass is
somewhat inadequate, and we propose a better alternative. As an application, we
perform a preliminary study of the implications of the breakdown, in the
effective field theory, of the equivalence between the quasinormal mode spectra
associated to metric perturbations of polar and axial parity of the
Schwarzschild black hole in general relativity. We also present a simple
justification for the loss of isospectrality.
