A Is the overlap of coherent states circular symmetric?

[Danny Boy]

What I am interested in doing, is considering the angular momentum eigenstate for a spin $1$ system: $|J=1, M=1\rangle = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, forming the coherent state $|CS \rangle = \begin{bmatrix} 0.5 \\ -\frac{i}{\sqrt{2}} \\ -0.5 \end{bmatrix}$ by taking the rotation $\text{exp}(-\frac{i \pi}{2}\hat{J}_x)\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0.5 \\ -\frac{i}{\sqrt{2}} \\ -0.5 \end{bmatrix}$.

I then want to work out the modulus squared of the overlap of the coherent state with a rotation of the coherent state about the $x$-axis and the $z$-axis for various values of $\theta$ and $\phi$: $$|\langle CS| \text{exp}(-i \phi \hat{J}_z) \text{exp}(-i \theta \hat{J}_x)|CS\rangle|^2.$$ The result I get is http://farm5.staticflickr.com/4211/34692246053_e0fd2d7cd8_b.jpg. Is this the type of plot you would have predicted? I would have expected something more circular symmetric?
Thanks for any assistance.

 

[thephystudent]

Looks quite circular symmetric to me, given that the origin is not in the middle of the figure.