  We present a new texture of neutrino mass matrix having three complex
relations among its elements and study in detail the phenomenological
implications. A characteristic feature of the resulting neutrino mass matrix is
that the atmospheric neutrino mixing angle is predicted to lie in a very narrow
region near $45^{\circ}$. We illustrate how such a form of the neutrino mass
matrix can be realized using the non-Abelian flavor symmetry $A_4$ in the
framework of type-I+II seesaw mechanism.
