  In this paper, we investigate the double covering of modular $\Gamma^{}_5
\simeq A^{}_5$ group and derive all the modular forms of weight one for the
first time. The modular forms of higher weights are also explicitly given by
decomposing the direct products of weight-one forms. For the double covering
group $\Gamma^\prime_5 \simeq A^\prime_5$, there exist two inequivalent
two-dimensional irreducible representations, into which we can assign two
right-handed neutrino singlets in the minimal seesaw model. Two concrete models
with such a salient feature have been constructed to successfully explain
lepton mass spectra and flavor mixing pattern. The allowed parameter space for
these two minimal scenarios has been numerically explored, and analytically
studied with some reasonable assumptions.
