Let $DM^{eff}_{gm}(k)$ be the triangulated category of effective geometric motives over some field $k$ (in the sense of Voevodsky), and let $DM^{eff}_{gm}(k)(1)$ be its full triangulated subcategory consisting of those effective motives Tate-twisted once.

Consider $C:=DM^{eff}_{gm}(k)/DM^{eff}_{gm}(k)(1)$.

Is there an explicit description of this category? What I actually want to understand is $K_0(\cal{C})$. More precisely, I have a nonzero object in $C$, and I want to show that its class in $K_0(C)$ is nonzero by computing its image under some function $K_0(C)\rightarrow A$ into some abelian group $A$...probably coming from some realization functor.