The free metaplectic transform (FMT), an n-dimensional nonseparable extension of linear canonical transform (LCT), has applications in harmonic analysis, quantum mechanics, and filter design. Despite its importance, its spectrum related theory remains largely unexplored. In this paper, we establish two important spectrum related theorems for the FMT. First we prove Riemann–Lebesgue lemma and other results that form the foundation for real Paley–Wiener and Boas theorems. These results provide new spectrum related characterizations in the FMT domain and advance the theoretical understanding of the transform.