A fully relativistic treatment of Bernstein waves in an electron-positron pair plasma has remained too formidable a task, owing to the very complex nature of the problem. In this paper, we perform contour integration of the dielectric-response function and numerically compute the dispersion curves for a uniform magnetized relativistic electron-positron pair plasma. The behavior of the dispersion solution for several cases with different plasma temperatures is highlighted. In particular, we find two wave modes that exist only for large wavelengths and frequencies similar to the cyclotron frequency in a moderately relativistic pair plasma. The results presented here have important implications for the study of those objects where a hot magnetized electron-positron plasma plays a fundamental role in generating the observed radiation.