  The current study aimed to investigate the special case of two zeros in a
Majorana neutrino mass matrix based on $A_4$ symmetry, where charged lepton
mass matrix is diagonal. The texture $M_\nu^{S_7}$ with $(\mu, \mu)$ and
$(\tau,\tau)$ vanishing element of mass matrix has magic and $\mu-\tau$
symmetry, with a tribimaximal form of the mixing matrix which leads to
$\theta_{13}=0$ that it is not consistent with experimental data and does not
seem to be allowed. Since $\theta_{13}$ is small compared to other neutrino
mixing angles, we show that $\theta_{13}$, and $\delta$ could be obtained by
using a complex symmetrical perturbation in the mass basis and also could be
shown that $\delta m^2\equiv~m_2^2-m^2_1\neq0$ affecting the atmospheric mixing
angle.
  We find that for the complex perturbation mass matrix, only the results of
the case I, $\Delta<0$ and $\textit{Re}(\alpha)<0$, are consistent with
experimental data. Furthermore, the allowed range of our parameter space and
complex element of perturbation are found which led to finding the deviation of
$\theta_{23}$ from $45^\circ$ where this deviation is in line with the
experimental data which indicate the accuracy of our model and its results. Our
prediction is inverted mass ordering in the Case I. The results of the case II,
$\Delta>0$ and $\textit{Re}(\alpha)>0$, are ruled out.
