The Hurwitz integers are $$ \mathcal H= \{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}. $$ I want to know if there is a formula, for $m\in\mathbb Z$, for the number of elements $\alpha\in\mathcal H$ such that $|\alpha|^2=m$.

This is equivalent to known the number of vectors $v$ in the lattice $F_4$ such that $\|v\|^2=m$.

I think this formula already exist but I can find it. An appropiate reference would be appreciated. Thanks.-.