  We study the large $ N$ sign oscillation of the twisted indices for 3D
theories of class $\mathcal{R}$ obtained from M5-branes wrapped on a hyperbolic
3-manifold. Holographically, the oscillatory behavior can be understood from
the imaginary part of on-shell actions for the two Euclidean supergravity
solutions, Bolt$_{\pm}$ with $p=0$, which are Wick rotation of magnetically
charged AdS$_4$ black holes. The two solutions have the same imaginary part
with opposite sign. The imaginary part comes from the $F\wedge F$-term in the
supergravity and the coefficient is proportional to the Chern-Simons invariant
of 3-manifold. Combining the holographic computation with 3D-3D relation for
twisted indices, we propose a non-trivial mathematical conjecture regarding the
phase factor of a twisted Reidemeister-Ray-Singer torsion on hyperbolic
3-manifold.
