In this talk, we present our recent work on the periods for cusp forms on . Let be an irreducible globally generic cuspidal automorphic representation of with trivial central character. We proved the algebraicity of the critical values of certain automorphic -functions for in terms of the Whittaker periods associated to . By comparing with the results of Januszewski and Morimoto, when is stable, we obtain an automorphic analogue of Yoshida's period relation for the conjectural motivic periods attached to . As an application of the period relation, we proved the algebraicity of the the symmetric sixth power -functions of elliptic modular forms in terms of certain Petersson norms of holomorphic cusp forms.