  We study criteria for freeness and for the existence of a vanishing line for
modules over certain Hopf subalgebras of the motivic Steenrod algebra over
$\mathrm{Spec}(\mathbb{C})$ at the prime 2. These turn out to be determined by
the vanishing of certain Margolis homology groups in the quotient Hopf algebra
$\mathcal{A}/\tau$.
