A modular $A_4$ symmetric model of dark matter and neutrino

We propose a model based on modular $A_4$ symmetry containing a dark matter candidate, realizing radiatively induced neutrino mass at one-loop level. One finds that stability of dark matter candidate can be assured by nonzero value of modular weight and heavy neutral fermion mass hierarchies, which include dark matter under the $A_4$ triplet, are uniquely determined; $M_X\ll M_2<M_3$. Therefore we clearly identify single dark matter field which can discriminate from the other models with $A_4$ modular symmetry. Then we discuss several phenomenological aspects and show predictions on the lepton sector.Especially, we find $0.56\lesssim {\sin }^2{\theta }_{23}\lesssim 0.624$, which could have an advantage of this narrow region and be also discriminated from the other modular $A_4$ models.