I was wondering how to compute the eta invariant $\eta(T^3)$ of a flat torus $T^3$, with respect to the signature operator.


In general, how can we compute the $\eta(T^3/\Gamma)$ of a finite quotient of a flat torus?



How to compute the eta invariant of torus
The general formula can be found in [Ouyang - Geometric Invariants For Seifert Fibred 3-Manifolds](https://www.ams.org/journals/tran/1994-346-02/S0002-9947-1994-1257644-9/).


In particular, for $\Gamma \cong 1, \mathbb{Z}\_2, \mathbb{Z}\_3, \mathbb{Z}\_4, \mathbb{Z}\_6, \mathbb{Z}\_2^2$, we have $\eta(T^3/\Gamma) = 0, 0, \frac{-2}{3}, -1, \frac{-4}{3}, 0$, respectively. See also [Long and Reid](https://projecteuclid.org/euclid.gt/1513883282) or [Szczepanski](https://link.springer.com/article/10.1007/s10455-011-9274-0).


