  Turbulence, magnetic fields and radiation feedback are key components that
shape the formation of stars, especially in the metal-free environments at high
redshifts where Population III stars form. Yet no 3D numerical simulations
exist that simultaneously take all of these into account. We present the first
suite of radiation-magnetohydrodynamics (RMHD) simulations of Population III
star formation using the adaptive mesh refinement (AMR) code FLASH. We include
both turbulent magnetic fields and ionizing radiation feedback coupled to
primordial chemistry, and resolve the collapse of primordial clouds down to few
au. We find that dynamically strong magnetic fields significantly slow down
accretion onto protostars, while ionizing feedback is largely unable to
regulate gas accretion because the partially ionized \ion{H}{ii} region gets
trapped near the star due to insufficient radiative outputs from the star. The
maximum stellar mass in the HD and RHD simulations that only yield one star
exceeds $100\,\rm{M_{\odot}}$ within the first $5000\,\rm{yr}$. However, in the
corresponding MHD and RMHD runs, the maximum mass of Population III star is
only $60\,\rm{M_{\odot}}$. In other realizations where we observe widespread
fragmentation leading to the formation of Population III star clusters, the
maximum stellar mass is further reduced by a factor of few due to
fragmentation-induced starvation. We thus conclude that magnetic fields are
more important than ionizing feedback in regulating the mass of the star, at
least during the earliest stages of Population III star formation, in typical
dark matter minihaloes at $z \approx 30$.
