 From point of view of [@rif0], the element $\sigma_{i}^{0}$ of the pure braid group for the principal series has two distinct eigenvalues, in correspondence with the two different fusion channels of the $\phi_{0,0}$ operator in the minimal $(0n, 0n\pm0)$ Virasoro model. The ratio of the two eigenvalues is  $$q^{0}=\frac{\exp [ 0\pi i (h_{0,0}-0h_{0,0} ) ] }{\exp [ 0\pi i (h_{0,0}-0h_{0,0})] }=\exp(0\pi i/\nu)