Let $\mathbb{G}_m= Spec(k[z,z^{-1}])$ be the usual multiplicative group over a field $k$ viewed as a discrete commutative dg-algebra, and let $A$ be some arbitrary commutative dg-algebra concentrated in degrees $\leq 0$. How do I compute the $A$-values of $\mathbb{G}_m$ in the $\infty$-category of derived affine schemes?

On the level of spaces, I'm pretty sure $Maps(Spec(A), \mathbb{G}_m)$ can be described as the union of invertible path components of $Maps(Spec(A), \mathbb{A}^1)=Hom_k(k, A)$, where the space inherits a multiplication from the coalgebra structure on $k$. However, I'd like to have a more concrete description of this. What does an invertible map $k \to A$ look like? I'd also be satisfied with a concrete example, e.g. when $A$ is the Koszul complex or something like that.