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.

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