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May 28 at 16:14 comment added Danny Ruberman As Oscar says, the $\hat{A}$-genus isn't defined in odd dimensions. But you can find the Pontrjagin classes by the same method as for projective spaces. see Ewing, John; Moolgavkar, Suresh; Smith, Larry; Stong, R. E. Stable parallelizability of lens spaces. J. Pure Appl. Algebra10 (1977/78), no.2, 177–191. At that point, you can work out the A-polynomial, which is perhaps what you mean.
May 28 at 6:59 comment added Oscar Randal-Williams Isn't the A-hat genus zero for manifolds whose dimension is not divisible by 4, more or less by definition?
May 27 at 23:20 history asked Nicolas Boerger CC BY-SA 4.0