Twisted Maass-Koecher series and spinor zeta functions

It is shown that a Siegel-Hecke eigenform of integral weight k and genus 2 is uniquely determined by its Fourier coefficients indexed by nT where T runs over all half-integral positive definite primitive matrices of size 2 and n over all squarefree positive integers, The proof uses analytic arguments involving Koecher-MaaB series and spinor zeta functions.