I was wondering how much is known about the integral motivic cohomology groups of $\mathrm{Spec}\, k$, $H^{n,p}_{\mathrm{mot}}(\mathrm{Spec}\, k,\mathbb{Z})$. One knows that $H^{n,n}_{\mathrm{mot}}(\mathrm{Spec}\, k,\mathbb{Z})\simeq K^M_n(k)$, the Milnor $K$-theory of the field $k$. Also, because of the isomorphism to higher Chow groups $H^{n,p}_{\mathrm{mot}}(X,\mathbb{Z})\simeq CH^p(X,2p-n)$ when $k$ is perfect, we get $H^{2p,p}_{\mathrm{mot}}(\mathrm{Spec}\, k,\mathbb{Z})\simeq CH^p(\mathrm{Spec}\, k)=0$ if $p\neq 0$ and $\mathbb{Z}$ otherwise. This also implies $H^{n,p}_{\mathrm{mot}}(\mathrm{Spec}\, k,\mathbb{Z})$ vanishes for $n>p$.

For which $(n,p)$ is $H^{n,p}_{\mathrm{mot}}(\mathrm{Spec}\, k,\mathbb{Z})$ known to vanish? In other words, is there any $n\neq p$ for which the group is non-trivial?