The dispersion relations of parallel propagating modes (Langmuir mode,right and left handed circular polarized waves) in the weak magnetic field limit ｜ω-k· v ｜ (>>) Ω are considered for ultra-relativistic arbitrary degenerate electron plasma.The results are presented in terms of moments of Fermi-Dirac distribution.The increase in the electron equilibrium number density from negative large (weakly degenerate) to positive large (highly degenerate) values of μ/Te is observed (where μ is the electron chemical potential and Te is the electron thermal energy).As a result,shifting of the cutoff points in all the real dispersion branches towards the higher values and increasing in the band gap between unmagnetized longitudinal and transverse modes in k-space are examined.Also,the suppression of the weak magnetic field effects in weakly magnetized right handed and left handed circular polarized waves and a decrease in the longitudinal and transverse screening effects are observed in the graphical pattes due to an increase in the equilibrium number density.