Hecke eigenvalues and relations for degree 2 Siegel Eisenstein series

We evaluate the action of Hecke operators on Siegel Eisenstein series of degree 2, square-free level N and arbitrary character χ, without using knowledge of their Fourier coefficients. From this we construct a basis of simultaneous eigenforms for the full Hecke algebra, and we compute their eigenvalues. As well, we obtain Hecke relations among the Eisenstein series. Using these Hecke relations, we discuss how to generate the Fourier series of Eisenstein series in a basis from the Fourier series of one basis element.