  The frequency spectra of the gravito-electromagnetic perturbations of the
Kerr-Newman (KN) black hole with the slowest decay rate have been computed
recently. It has been found that KN has two families $-$ the photon sphere and
the near-horizon families $-$ of quasinormal modes (QNMs), which display the
interesting phenomenon of eigenvalue repulsion. The perturbation equations, in
spite of being a coupled system of two PDEs, are amenable to an analytic
solution using the method of separation of variables in a near-horizon
expansion around the extremal KN black hole. This leads to an analytical
formula for the QNM frequencies that provides an excellent approximation to the
numerical data near-extremality. In the present manuscript we provide an
extended study of these properties that were not detailed in the original
studies. This includes: 1) a full derivation of a gauge invariant system of two
coupled PDEs that describes the perturbation equations \cite{Dias:2015wqa}, 2)
a derivation of the eikonal frequency approximation
\cite{Zimmerman:2015trm,Dias:2021yju} and its comparison with the numerical QNM
data, 3) a derivation of the near-horizon frequency approximation
\cite{Dias:2021yju} and its comparison with the numerical QNMs, and 4) more
details on the phenomenon of eigenvalue repulsion (also known as \emph{level
repulsion}, \emph{avoided crossing} or \emph{Wigner-Teller effect}) and a first
principles understanding of it that was missing in the previous studies.
Moreover, we provide the frequency spectra of other KN QNM families of interest
to demonstrate that they are more damped than the ones we discuss in full
detail.
