Waves in a magnetized electron–positron plasma, supporting a large-amplitude electric field E0 of superluminal phase speed, are considered. The case of perturbations with the same phase speed as the large-amplitude wave can be treated exactly, and we restrict our attention to this case, obtaining analytical results. The exact analytical results provide insight into the effect of the magnetic field and the large-amplitude wave on the harmonic structure of the perturbations. Three solutions are found for waves polarized perpendicular to E0. Waves are amplitude-modulated for weak magnetic fields (relative to the strength of the large-amplitude wave) and frequency-modulated for strong magnetic fields. This suggests that frequency modulation may be relevant to pulsars.