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We revisit the modular flavor symmetry from a more general perspective. Thescalar modular forms of principal congruence subgroups are extended to thevector-valued modular forms, then we have more possible finite modular groupsincluding $\Gamma_N$ and $\Gamma'_N$ as the flavor symmetry. The theory ofvector-valued modular forms provide a method of differential equation toconstruct the modular multiplets, and it also reveals the simple structure ofthe modular invariant mass models. We review the theory of vector-valuedmodular forms and give general results for the lower dimensional vector-valuedmodular forms. The general finite modular groups are listed up to order 72. Weapply the formalism to construct two new lepton mass models based on the finitemodular groups $A_4\times Z_2$ and $GL(2,3)$.

Modular flavor symmetry and vector-valued modular forms