  The existence of a primordial magnetic field (PMF) would affect both the
temperature and polarization anisotropies of the cosmic microwave background
(CMB). It also provides a plausible explanation for the possible disparity
between observations and theoretical fits to the CMB power spectrum. Here we
report on calculations of not only the numerical CMB power spectrum from the
PMF, but also the correlations between the CMB power spectrum from the PMF and
the primary curvature perturbations. We then deduce a precise estimate of the
PMF effect on all modes of perturbations. We find that the PMF affects not only
the CMB TT and TE modes on small angular scales, but also on large angular
scales. The introduction of a PMF leads to a better fit to the CMB power
spectrum for the higher multipoles, and the fit at lowest multipoles can be
used to constrain the correlation of the PMF with the density fluctuations for
large negative values of the spectral index. Our prediction for the BB mode for
a PMF average field strength $|B_\lambda| =4.0$ nG is consistent with the upper
limit on the BB mode deduced from the latest CMB observations. We find that the
BB mode is dominated by the vector mode of the PMF for higher multipoles. We
also show that by fitting the complete power spectrum one can break the
degeneracy between the PMF amplitude and its power spectral index.
