Multiplicative relations for Fourier coefficients of degree 2 Siegel eigenforms

We prove multiplicative relations between certain Fourier coefficients of degree 2 Siegel eigenforms. These relations are analogous to those for elliptic eigenforms. We also provide two sets of formulas for the eigenvalues of degree 2 Siegel eigenforms. The first evaluates the eigenvalues in terms of the form’s Fourier coefficients, in the case $a(I)\mathrm{\ne }0$. The second expresses the eigenvalues of index $p$ and $p^2$, for $p$ prime, solely in terms of $p$ and $k$, the weight of the form, in the case $a(0)\mathrm{\ne }0$. From this latter case, we give simple expressions for the eigenvalues associated to degree 2 Siegel Eisenstein series.