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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 atnearby two fixed points and three special points. These special points of$SL(2,\mathbb{Z})$ moduli space are statistically favored in fluxcompactifications of Type IIB string theory. Neutrino masses are approximatelyobtained by using the seesaw mechanism due to our two additional symmetries. Weperform chi square numerical analysis for each of the fixed/special points inthe normal and inverted hierarchies and demonstrate predictions for each case.Then, we briefly discuss the possible signature of hidden gauge bosons atcollider experiments from the hidden $SU(2)$ symmetry.

Novel moduli space in modular flavor models : a case study for modular $T'$ seesaw model with hidden $SU(2)$ gauge symmetry