 Solution

Applying the transform to the coset state gives amplitudefor $|c⟩$. The geometric sum vanishes unless ${\omega }^{3c}=1$, which means $c\equiv 0(\mathrm{mod}4)$. For $c=0,4,8$ its magnitude is $4/(2\sqrt{12})=1/\sqrt{3}$. Thus the quantum Fourier transform of a periodic coset state gives