gr.group theory - Elements living in the conjugacy class and in the centralizer of an m-cycle in Am

Thank you, Douglas.

With the notation giving above and that giving in the paper of Marek Szyjewski (that you refered me), the following statements are equivalent:
1) x^j in C,
2) sgn(lambda_ j )=1,
3) J(j,m)=1, J the Jacobi symbol.

1) <=> 2) is easy (I have chequed).
2) <=> 3) is Theor. 1 of the paper of Marek Szyjewski. This is an unplubished article yet. I had no time to chequed all of it; I have only chequed Case 1, but I guess that Case 2 and 3 are correct.(?)

I am interested in the case m=3 p, with p>3 prime. I need to prove that there exist j, with j mod 3 =2, such that x^j in C.
This amounts to prove that there exists j, 0< j < m, such that:
-) ( j,m)=1,
-) j mod 3 =2 (i.e. J( j,3)= -1),
-) J( j,p)= -1,
because J( j,m)=J( j,3) J( j,p).

Do you have any clue for that?

Thank you in advance.