We present a set of diagonal matrices which index enough Fourier coefficients for a complete characterization of all Siegel cusp forms of degree 2, weight k, level N and character chi, where k is an even integer >= 4, N is an odd, squarefree positive integer, and chi has conductor equal to N. As an application, we show that the Koecher-Maass series of any F is an element of S-k(2) twisted by the set of Maass waveforms whose eigenvalues are in the continuum spectrum of the hyperbolic Laplacian determines F. We also generalize a result due to Skogman about the non-vanishing of all theta components of a Jacobi cusp form of even weight and prime index, which may have some independent interest. (C) 2019 Elsevier Inc. All rights reserved.