Can we construct a free action of the cyclic group $ \Bbb Z/3\Bbb Z$ on $ S^n \times \cdots \times S^n$ ($ n$ is odd) without multiplying any coordinate by $ e^{2\pi i/3} $ ? In other words, do I Need to multiply at least one coordinate by a 3rd root of unity to get a free action of $ \Bbb Z/3\Bbb Z$ ? I have tried to construct such examples but could not find any.

Thank you so much in advance.