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We study quark and lepton mass matrices with the common modulus $\tau$ in the$A_4$ modular symmetry. The viable quark mass matrices are composed of modularforms of weights $2$, $4$ and $6$. It is remarked that the modulus $\tau$ isclose to $i$, which is a fixed point in the fundamental region of SL$(2,Z)$,and the CP symmetry is not violated. Indeed, the observed CP violation isreproduced at $\tau$ which is deviated a little bit from $\tau=i$. The chargedlepton mass matrix is also given by using modular forms of weights $2$, $4$ and$6$, where five cases have been examined. The neutrino mass matrix is generatedin terms of the modular forms of weight $4$ through the Weinberg operator.Lepton mass matrices are also consistent with the observed mixing angles at$\tau$ close to $i$ for NH of neutrino masses. Allowed regions of $\tau$ ofquarks and leptons overlap each other for all cases of the charged lepton massmatrix. However, the sum of neutrino masses is crucial to test the common$\tau$ for quarks and leptons. The minimal sum of neutrino masses $\sum m_i$ is$140$meV at the common $\tau$.  The inverted hierarchy of neutrino masses is unfavorable in our framework. Itis emphasized that our result suggests the residual symmetry$\mathbb{Z}_2^{S}=\{ I, S \}$ in the quark and lepton mass matrices.

Quark and lepton flavors with common modulus $\tau$ in $A_4$ modular symmetry