CONJECTURE: Let A= (c0,c1,..,cn) be a circulant matrix, i.e if (c0,c1,..,cn) is the first column of A then the i-th column of A is obtained by applying the permutation (1,2,..,n)^{n-1}. Assume  A in GL_n(Z), i.e A with integer entries and determinant=+-1 and moreover c0+c1+..+cn=+-1.
Then there exists one j such that cj=+-1 and ci=0 for all i diffrent from j.

IS IT TRUE?

What if we add the assumption that n=p a prime?
Thanks for any idea!
Fabienne