Zee model in a non-holomorphic modular $A_4$ symmetry

We study a Zee model in a non-holomorphic modular $A_4$ flavor symmetry in which we find good predictions in both the cases of normal and inverted hierarchy. Parameter reduction on neutrino sector occurs due to large mass hierarchies between charged-leptons mass eigenvalues and new singly-charged bosons in addition to this flavor symmetry. As a result, we have two complex free parameters including modulus $\tau$. We show several predictions in terms of verifiable observables such as Dirac CP, Majorana phases, sum of the neutrino masses, and the effective mass for neutrino double beta decay in addition to demonstrating allowed regions for our input parameters.