  In a class of neutrino mass models with modular flavor symmetries, it has
been observed that CP symmetry is preserved at the fixed point (or stabilizer)
of the modulus parameter $\tau = {\rm i}$, whereas significant CP violation
emerges within the neighbourhood of this stabilizer. In this paper, we first
construct a viable model with the modular $A^\prime_5$ symmetry, and explore
the phenomenological implications for lepton masses and flavor mixing. Then, we
introduce explicit perturbations to the stabilizer at $\tau = {\rm i}$, and
present both numerical and analytical results to understand why a small
deviation from the stabilizer leads to large CP violation. As low-energy
observables are very sensitive to the perturbations to model parameters, we
further demonstrate that the renormalization-group running effects play an
important role in confronting theoretical predictions at the high-energy scale
with experimental measurements at the low-energy scale.
