Quasinormal modes of Type I perturbations of dilaton-Euler-Heisenberg black holes

We study the type I quasinormal modes (QNMs) of magnetically charged dilaton–Euler–Heisenberg (dEH) black holes, where axial gravitational and polar electromagnetic perturbations are coupled. Direct integration and Chebyshev pseudospectral methods give consistent QNM frequencies for the fundamental branches continued from Schwarzschild at $\mathrm{\ell }=1,2,3$. Varying $\alpha$ at fixed $\zeta =\alpha -\beta$ changes the spectrum on the same background. The damping response depends on charge and multipole and can have opposite signs for the two coupled branches. For $\zeta <0$, dipole damping decreases with charge toward horizon merger. For $\zeta >0$, the Type I charge range extends beyond the complementary polarization’s local angular-gradient bound. All tracked modes remain damped in the nonextremal ranges, supporting the stability of these branches.