Cohomology of toric blowup

Let $ n\geq2$ . Let $ G$ be a linear automorphisms group of prime order on $ \mathbb{C}^n$ . We assume that $ 0$ is the unique fixed point of $ G$ . I consider the quotient $ \mathbb{C}^n/G$ . It is a toric variety, so I can consider the toric blowup: $ \widetilde{\mathbb{C}^{n}/G}$ . I am interesting in the integral cohomology (singular cohomology) of $ \widetilde{\mathbb{C}^{n}/G}$ . Especially, I would like to prove that $ H^{2k}(\widetilde{\mathbb{C}^{n}/G},\mathbb{Z})$ is torsion free for $ k\leq n-1$ . Do someone know an efficient method to deal with this kind of problems? Has this cohomology been studied somewhere?

I managed to prove that $ H^2(\widetilde{\mathbb{C}^{n}/G},\mathbb{Z})$ and $ H^{2n-2}(\widetilde{\mathbb{C}^{n}/G},\mathbb{Z})$ are torsion free using Danilov combinatorial result. However for the other cohomology groups, it becomes too technical to be done by hand.