We explore a new parametrization scheme, known as exponential parametrization, for the Majorana neutrino mass matrix. In this framework, the elements of the mass matrix are represented in terms of their absolute values and arguments. We propose a neutrino mass matrix texture that highlights four correlations among its elements under this parametrization scheme. For normal ordering, the mixing scheme derived from the proposed texture aligns with experimental observations, whereas the texture is ruled out for inverted ordering. Additionally, we derive the proposed texture from the seesaw mechanism, associated with the \(SU(2)_L \times U(1)_Y \times A_4 \times Z_{10}\) group.