Population III (pop III) stars were born in halos characterised by a pristine gas composition. In such a halo, once the gas density reaches n$_{\mathrm{H}} \sim$ 1 cm$^{-3}$, molecular cooling leads to the collapse of the gas and the birth of pop III stars. Halo properties, such as the chemical abundances, mass, and angular momentum can affect the collapse of the gas, thereby leading to the pop III initial mass function (IMF) of star formation. We want to study the properties of primordial halos and how halos that host early star formation differ from other types of halos. The aim of this study is to obtain a representative population of halos at a given redshift hosting a cold and massive gas cloud that enables the birth of the first stars. We investigated the growth of primordial halos in a $\Lambda$CDM Universe in a large cosmological simulation. We used the hydrodynamic code RAMSES and the chemical solver KROME to study halo formation with non-equilibrium thermochemistry. We then identified structures in the dark and baryonic matter fields, thereby linking the presence or absence of dense gas clouds to the mass and the physical properties of the hosting halos. In our simulations, the mass threshold for a halo for hosting a cold dense gas cloud is $\simeq 7 \times 10^5 M_{\odot}$ and the threshold in the H$_2$ mass fraction is found to be $\simeq 2 \times 10^{-4}$. This is in agreement with previous works. We find that the halo history and accretion rate play a minor role. Here, we present halos with higher HD abundances, which are shown to be colder, as the temperature in the range between $10^2 - 10^4 \, \mathrm{cm^{-3}}$ depends on the HD abundance to a large extent. The higher fraction of HD is linked to the higher spin parameter that is seen for the dense gas.