  We study flavor phenomenologies in a basis of a double covering of modular
$A_4$ group with a hidden $SU(2)$ symmetry, in which we work on regions at
nearby two fixed points and three special points. These special points of
$SL(2,\mathbb{Z})$ moduli space are statistically favored in flux
compactifications of Type IIB string theory. Neutrino masses are approximately
obtained by using the seesaw mechanism due to our two additional symmetries. We
perform chi square numerical analysis for each of the fixed/special points in
the normal and inverted hierarchies and demonstrate predictions for each case.
Then, we briefly discuss the possible signature of hidden gauge bosons at
collider experiments from the hidden $SU(2)$ symmetry.
