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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?

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The general formula can be found in Ouyang - Geometric Invariants For Seifert Fibred 3-Manifolds.

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 or Szczepanski.

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