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We study the modular symmetry of zero-modes on $T_1^2 \times T_2^2$ andorbifold compactifications with magnetic fluxes, $M_1,M_2$, where modulusparameters are identified. This identification breaks the modular symmetry of$T^2_1 \times T^2_2$, $SL(2,\mathbb{Z})_1 \times SL(2,\mathbb{Z})_2$ to$SL(2,\mathbb{Z})\equiv\Gamma$. Each of the wavefunctions on $T^2_1 \timesT^2_2$ and orbifolds behaves as the modular forms of weight 1 for the principalcongruence subgroup $\Gamma$($N$), $N$ being 2 times the least common multipleof $M_1$ and $M_2$. Then, zero-modes transform each other under the modularsymmetry as multiplets of double covering groups of $\Gamma_N$ such as thedouble cover of $S_4$.

Modular symmetry by orbifolding magnetized $T^2\times T^2$: realization of double cover of $\Gamma_N$