  In the first paper we proved that on the cyclic groups of odd order d, there
exist non zero functions whose convolution square f*f(2t) is proportional to
their square f(t)^2 when the proportionality constant is an odd algebraic
integer of norm d whose both real and imaginary part are square roots of
integers. We show here that the function f can be chosen to be equal to the
conjugate of its Fourier transform.
