  Results concerning the existence and spectral stability and instability of
multiple periodic wave solutions for the nonlinear Schr\"odinger system with
\textit{dnoidal} and \textit{cnoidal} profile will be determined in this
manuscript. The spectral analysis for the corresponding linearized operator is
established by using the comparison theorem and tools of Floquet theory. The
main results are determined by applying the spectral stability theory in
\cite{KapitulaKevrekidisSandstedeI} and \cite{KapitulaKevrekidisSandstedeII}
via Krein signature.
