Let $\Gamma$ be a finite subgroup of SU(2) and consider the quotient of $S^3$ by $\Gamma$ via its left action. Pick a simply connected compact Lie group $G$ and take a flat connection on this quotient. Or equivalently, choose a homomorphism $\Gamma \to G$. The Chern-Simons action functional should assign a value in $\mathbb{R}/\mathbb{Z}$. How do I compute it?

The paper http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002334585 by Kronheimer and Nakajima has the formula for the eta invariant which is closely related, but I think it gives the CS functional only as a value in $\mathbb{R}/(\mathbb{Z}/h^\vee(G))$.