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The even weight modular forms of level $N$ can be arranged into the commonirreducible representations of the inhomogeneous finite modular group$\Gamma_N$ and the homogeneous finite modular group $\Gamma'_N$ which is thedouble covering of $\Gamma_N$, and the odd weight modular forms of level $N$transform in the new representations of $\Gamma'_N$. We find that the abovestructure of modular forms can naturally generate texture zeros of the fermionmass matrices if we properly assign the representations and weights of thematter fields under the modular group. We perform a comprehensive analysis forthe $\Gamma'_3\cong T'$ modular symmetry. The three generations of left-handedquarks are assumed to transform as a doublet and a singlet of $T'$, we find sixpossible texture zeros structures of quark mass matrix up to row and columnpermutations. We present five benchmark quark models which can produce verygood fit to the experimental data. These quark models are further extended toinclude lepton sector, the resulting models can give a unified description ofboth quark and lepton masses and flavor mixing simultaneously although theycontain less number of free parameters than the observables.

Modular symmetry origin of texture zeros and quark lepton unification