  Modular flavor symmetry can be used to explain the quark and lepton flavor
structures. The SUSY partners of quarks and leptons, which share the same
superpotential with the quarks and leptons, will also be constrained by the
modular flavor structure and show a different flavor(mixing) pattern at the GUT
scale. So, in realistic modular flavor models with SUSY completion, constraints
from the collider and DM constraints can also be used to constrain the possible
values of the modulus parameter. In the first part of this work, we discuss the
possibility that the $S_3$ modular symmetry can be preserved by the fixed
points of $T^2/Z_N$ orbifold, especially from $T^2/Z_2$. To illustrate the
additional constraints from collider etc on modular flavor symmetry models, we
take the simplest UV SUSY-completion $S_3$ modular invariance SU(5) GUT model
as an example with generalized gravity mediation SUSY breaking mechanism. We
find that such constraints can indeed be useful to rule out a large portion of
the modulus parameters. Our numerical results show that the UV-completed model
can account for both the SM (plus neutrino) flavor structure and the collider,
DM constraints. Such discussions can also be applied straightforwardly to other
modular flavor symmetry models, such as $A_4$ or $S_4$ models.
