  We investigate the effects of the violation of the Ginsparg-Wilson (GW)
relation in the M\"obius domain-wall fermion formulation on the lattice with
finite fifth dimension. Using a decomposion in terms of the eigenmodes of its
four-dimensional effective Dirac operator, we isolate the GW-violating terms
for various physical quantities including the residual mass and the meson
susceptibilities relevant for the effective restoration of the axial U(1)
symmetry at finite temperature. Numerical result shows that the GW-violating
effect is more significant, or even overwhelming, for the quantities that are
dominated by the low-lying eigenmodes.
