Protoplanetary disks form through angular momentum conservation in collapsing dense cores. In this work, we perform the first simulations with a maximal resolution down to the astronomical unit (au) of protoplanetary disk formation, through the collapse of 1000 M_⊙ clumps, treating self-consistently both non-ideal magnetohydrodynamics with ambipolar diffusion as well as radiative transfer in the flux-limited diffusion approximation including stellar feedback. Using the adaptive mesh-refinement code RAMSES, we investigate the influence of the magnetic field on the disks properties with three models. We show that, without magnetic fields, a population dominated by large disks is formed that is not consistent with Class 0 disk properties as estimated from observations. The inclusion of magnetic field leads, through magnetic braking, to a very different evolution. When it is included, small <50 au disks represent about half the population. In addition, about 70% of the stars have no disk in this case, which suggests that our resolution is still insufficient to preserve the smaller disks. With ambipolar diffusion, the proportion of small disks is also prominent and we report a flat mass distribution around 0.01-0.1M_⊙ and a typical disk-to-star mass ratios of ~10^-2-10^-1. This work shows that the magnetic field and its evolution plays a prominent role in setting the initial properties of disk populations.