  Applying the $A_4$ symmetry in the scenario of unimodular second scheme of
trimaximal $TM_2$ mixing matrix, where the charged lepton mass matrix is
diagonal and the nature of neutrinos are Majorana, we investigate and analyze
feasible two zeros neutrino mass matrices. Among the seven possible two-zero
textures with $A_4$ symmetry, we have found that only two textures, namely the
texture with $(e, e)$ and $(e,\mu)$ vanishing element of mass matrix and its
permutation, are consistent with the experimental data in the non-perturbation
method. We also obtain new significant relations between phases of our model,
namely $\rho+\sigma=\phi\pm\pi$ and $\sin^2{\theta_{13}}=\frac{2}{3}R_\nu$
where $R_\nu=\frac{\delta m^2}{\Delta m^2}$. Subsequently, by admitting the
experimental ranges of $R_\nu$, we retrieve the allowed range of the unknown
phase $\phi$. Such a procedure assist us to determine the ranges of all the
neutrino observable parameters, the masses of neutrinos, the CP-violating
phases and $J$ parameter as well as to predict the normal hierarchy for the
neutrino mass. Finally, we show that our predictions with respect to our
herewith reported specific textures are consistent with the corresponding data
reported from neutrino oscillation, cosmic microwave background and
neutrinoless double beta decay experiments.
