  We generalize the modular invariance approach to include the half-integral
weight modular forms. Accordingly the modular group should be extended to its
metaplectic covering group for consistency. We introduce the well-defined
half-integral weight modular forms for congruence subgroup $\Gamma(4N)$ and
show that they can be decomposed into the irreducible multiplets of finite
metaplectic group $\widetilde{\Gamma}_{4N}$. We construct concrete expressions
of the half-integral/integral modular forms for $\Gamma(4)$ up to weight 6 and
arrange them into the irreducible representations of $\widetilde{\Gamma}_4$. We
present three typical models with $\widetilde{\Gamma}_4$ modular symmetry for
neutrino masses and mixing, and the phenomenological predictions of each model
are analyzed numerically.
