On the image of the Saito-Kurokawa lifting over a totally real number field and the Maass relation

We give a formula for the Fourier coefficients of the image of the Saito-Kurokawa lifting over a totally real number field $K$ , and we prove the image of the lifting satisfies a generalization of the Maass relation. We also give an explicit form of a Siegel series of degree 2 for any finite extension of ${\mathbb {Q}}_p$ .