  We study the statistical properties (such as shape and spin) of high-z halos
likely hosting the first (PopIII) stars with cosmological simulations including
detailed gas physics. In the redshift range considered ($11 < z < 16$) the
average sphericity is $<s> = 0.3 \pm 0.1$, and for more than 90% of halos the
triaxiality parameter is $T \lesssim 0.4$, showing a clear preference for
oblateness over prolateness. Larger halos in the simulation tend to be both
more spherical and prolate: we find $s \propto M_h^{\alpha_s}$ and $T \propto
M_h^{\alpha_T}$, with $\alpha_s \approx 0.128$ and $\alpha_T= 0.276$ at z = 11.
The spin distributions of dark matter and gas are considerably different at
$z=16$, with the baryons rotating slower than the dark matter. At lower
redshift, instead, the spin distributions of dark matter and gas track each
other almost perfectly, as a consequence of a longer time interval available
for momentum redistribution between the two components. The spin of both the
gas and dark matter follows a lognormal distribution, with a mean value at z=16
of $<\lambda> =0.0184$, virtually independent of halo mass. This is in good
agreement with previous studies. Using the results of two feedback models (MT1
and MT2) by McKee & Tan (2008) and mapping our halo spin distribution into a
PopIII IMF, we find that at high-$z$ the IMF closely tracks the spin lognormal
distribution. Depending on the feedback model, though, the distribution can be
centered at $\approx 65 M_\odot$ (MT1) or $\approx 140 M_\odot$ (MT2). At later
times, model MT1 evolves into a bimodal distribution with a second prominent
peak located at $35-40 M_\odot$ as a result of the non-linear relation between
rotation and halo mass. We conclude that the dark matter halo properties might
be a key factor shaping the IMF of the first stars.
