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A spectral transmission-line method (TLM) is developed for computing dispersion diagrams and eigenmodes of photonic-bandgap structures. By terminating the unit cell with periodic boundary conditions and allowing complex-valued voltages on the mesh, the system's modes may be found as solutions to an eigenvalue equation describing the system at steady state. A combination of sparse matrix techniques that exploit the spectral properties of the TLM scattering matrices enable efficient calculations despite the dimensionality of the eigenvalue equation. In contrast to conventional applications of the TLM, this formulation does not require selection of arbitrary mesh points to compute band diagrams and produces eigenmodes directly from band structure calculations without requiring a time history of the entire mesh and its Fourier transform, or additional frequency-domain computation.