  Experimental data have provided stringent constraints on neutrino mixing
parameters. In the standard parameterization the mixing angle $\theta_{23}$ is
close to $\pi/4$. There are also evidences show that the CP violating phase is
close to $-\pi/2$. We study neutrino mass matrix reconstructed using this
information and find several interesting properties. We show that a theoretical
model based on the $A_4$ symmetry naturally predicts $\delta = -\pi/2$ and
$\theta_{23} = \pi/4$ when the Yukawa couplings and scalar vacuum expectation
values are real reaching a $\mu-\tau$ exchange and CP conjugate symmetry limit.
In this case CP violation solely comes from the complex group theoretical
Clebsh-Gordan coefficients. The model also predicts $|V_{e2}|=1/\sqrt{3}$
consistent with data. With complex Yukawa couplings the values for $\delta$ and
$\theta_{23}$ can be significantly deviate away from the symmetry values
$-\pi/2$ and $\pi/4$, respectively. But $|V_{e2}|= 1/\sqrt{3}$ is not altered.
This matrix is an excellent lowest order approximation for theoretical model
buildings of neutrino mass matrix.
