  In this study, we explore the application of the $\Gamma(3)$ modular group,
which is isomorphic to the $A_{4}$ symmetric group, in developing a model for
neutrino mass. We construct a left-right asymmetric model using modular
symmetry, which offers the advantage of eliminating the need for additional
particles known as "flavons." The Yukawa coupling term is expressed in terms of
modular forms denoted as $Y_{1}$, $Y_{2}$, and $Y_{3}$. To effectively
implement the extended see-saw process in this model, we introduce one fermion
singlet for each generation. We compute the effective mass and the associated
half-life of $0\nu\beta\beta$ by accounting for both standard and non-standard
contributions. Additionally, our study investigates the non-unitary effects and
CP-violation arising from non-unitarity in this context. The model predicts
values for the sum of neutrino masses and neutrino oscillation parameters are
constraints with experiments. Furthermore, it yields satisfactory results in
calculating the effective mass and half-life of $0\nu\beta\beta$ decay. These
findings underscore the viability and utility of employing modular symmetry in
neutrino mass model construction.
