  A high-order convergent and robust numerical solver is constructed and used
to find complex eigenwavenumbers and electromagnetic eigenfields of dielectric
objects with axial symmetry. The solver is based on Fourier--Nystr\"om
discretization of combined integral equations for the transmission problem and
can be applied to demanding resonance problems at microwave, terahertz, and
optical wavelengths. High achievable accuracy, even at very high wavenumbers,
makes the solver ideal for benchmarking and for assessing the performance of
general purpose commercial software.
