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We discuss the modular $A_4$ invariant model of leptons combining with thegeneralized CP symmetry. In our model, both CP and modular symmetries arebroken spontaneously by the vacuum expectation value of the modulus $\tau$. Thesource of the CP violation is a non-trivial value of ${\rm Re} [\tau]$ whileother parameters of the model are real. The allowed region of $\tau$ is in verynarrow one close to the fixed point $\tau=i$ for both normal hierarchy (NH) andinverted ones (IH) of neutrino masses. The CP violating Dirac phase$\delta_{CP}$ is predicted clearly in $[98^\circ,\,110^\circ]$ and$[250^\circ,\,262^\circ]$ for NH at $3\,\sigma$ confidence level. On the otherhand, $\delta_{CP}$ is in $[95^\circ,100^\circ]$ and $[260^\circ,\,265^\circ]$for IH at $5\,\sigma$ confidence level. The predicted $\sum m_i$ is in$[82,\,102]$ meV for NH and $\sum m_i=[134,\,180]$ meV for IH. The effectivemass $\langle m_{ee}\rangle$ for the $0\nu\beta\beta$ decay is predicted in$[12.5,\,20.5]$ meV and $[54,\,67]$ meV for NH and IH, respectively.

Spontaneous CP violation by modulus $\tau$ in $A_4$ model of lepton flavors