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We propose a minimal extension of the Standard Model by incorporating sterileneutrinos and a QCD axion to account for the mass and mixing hierarchies ofquarks and leptons and to solve the strong CP problem, and by introducing$G_{\rm SM}\times \Gamma_N\times U(1)_X$ symmetry. We demonstrate that theK{\"a}hler transformation corrects the weight of modular forms in thesuperpotential and show that the model is consistent with the modular and$U(1)_X$ anomaly-free conditions. This enables a simple construction of amodular-independent superpotential for scalar potential. Using minimalsupermultiplets, we demonstrate a level 3 modular form-induced superpotential.Sterile neutrinos explain small active neutrino masses via the seesaw mechanismand provide a well-motivated $U(1)_X$ breaking scale, whereas gauge singletscalar fields play crucial roles in generating the QCD axion, heavy neutrinomass, and fermion mass hierarchy. The model predicts a range for the $U(1)_X$breaking scale from $10^{13}$ GeV to $10^{15}$ GeV for $1\,\mbox{TeV}<m_{3/2}<10^6\,\mbox{TeV}$. In the supersymmetric limit, all Yukawa coefficientsin the superpotential are given by complex numbers with an absolute value ofunity, implying a democratic distribution. Performing numerical analysis, westudy how model parameters are constrained by current experimental results. Inparticular, the model predicts that the value of the quark Dirac CP phase fallsbetween $38^\circ$ to $87^\circ$, which is consistent with experimental data,and the favored value of the neutrino Dirac CP phase is around $250^\circ$.Furthermore, the model can be tested by ongoing and future experiments on axionsearches, neutrino oscillations, and $0\nu\beta\beta$-decay.

Simple Modular invariant model for Quark, Lepton, and flavored-QCD axion