We show the index 0 Fredholm property of the spectral family of the azimuthal
harmonics of the charged Klein-Gordon operator on general cosmological slowly
accelerating and rotating charged black hole type spacetimes, including the De
Sitter-Kerr-Newman family, using a parametrix construction for abstract totally
characteristic operators. We then present a numerical scheme to compute the
meromorphic poles of the inverse of the spectral family and provide an explicit
estimate of the numerical error.