  We study $q$-series-valued invariants of 3-manifolds that depend on the
choice of a root system $G$. This is a natural generalization of the earlier
works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu
[arXiv:1904.06057] where they focused on $G={\rm SU}(2)$ case. Although a full
mathematical definition for these ''invariants'' is lacking yet, we define
$\hat{Z}^G$ for negative definite plumbed 3-manifolds and $F_K^G$ for torus
knot complements. As in the $G={\rm SU}(2)$ case by Gukov and Manolescu, there
is a surgery formula relating $F_K^G$ to $\hat{Z}^G$ of a Dehn surgery on the
knot $K$. Furthermore, specializing to symmetric representations, $F_K^G$
satisfies a recurrence relation given by the quantum $A$-polynomial for
symmetric representations, which hints that there might be HOMFLY-PT analogues
of these 3-manifold invariants.
