Coupled by Design: Computing Kerr-Newman Quasinormal Modes with a Hybrid SpectralPINN Solver

We extend our \texttt{SpectralPINN} solver to the computation of Kerr-Newman quasinormal modes by applying it to solve the system of two coupled master PDEs -- advancing from the single, separable equation of the uncharged Kerr limit to a genuinely coupled two-field problem. The coupling between the gravitational and electromagnetic fields gives rise to two families of solutions: the photon-sphere, connecting with Kerr; and near-horizon, disconnected from the Kerr limit, each with two branches of solutions depending on the leading field: the gravitational- and vector-led. Benchmarking against publicly available datasets shows relative frequency errors of $\sim 10^{-4}$ worst case and $\sim10^{-7}$ for most cases. The computed public dataset spans five photon-sphere modes, both gravitational- and vector-led, up to $\ell=4$, as well as two fundamental gravitational-led near-horizon modes. The vector-led photon-sphere branch is computed and systematically characterized for the first time. We apply the dataset and observe the onset of the eigenvalue repulsion reported by Dias \textit{et al.}, to exclude an inter-polarization repulsion within the resolved domain, and to forecast Einstein Telescope constraints on the black-hole charge-to-mass ratio.