Is the eta invariant of spherical space form $\eta(S^3/\Gamma)$ always nonegative?
Can we calculate it with the information of $\Gamma\in SO(4)$ explicitly? 


In fact, i need a reference for the calculation of eta invariant. Can some one give me some advice or download the following paper for me? Thank you!


(1) Hitchin, N. J.(4-CAMB)
Einstein metrics and the eta-invariant. (Italian summary) 
Boll. Un. Mat. Ital. B (7) 11 (1997), no. 2, suppl., 95–105. 


(2) G.W. Gibbons, C.N. Pope
Index theorem boundary terms for gravitational instantons
Nuclear Physics B Volume 157, Issue 3, 1 October 1979, Pages 377–386.


Email: xuyiyan@math.pku.edu.cn



Eta Invariant of Spherical Space Form 
This is not a full answer, but for the lens spaces (i.e., $\Gamma$ cyclic), the eta invariant is computed in the [second paper](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2075856) of Atiyah, Patodi and Singer. In case you're outside the paywall, you can try Proposition 5.2 in [this eprint](http://arxiv.org/abs/math/0602231).


**Edit**


There are formula for the general case in [this Inventiones paper](http://www.springerlink.com/content/k41u1r623p776852/) of Peter Gilkey's. 


