Ramanujan - type results for Siegel cusp forms of degree 2

A result of Chai–Faltings on Satake parameters of Siegel cusp forms together with the classification of unitary, unramified, irreducible, admissible representations of GSp4 over a p-adic field, imply that the local components of the automorphic representation of GSp4 attached to a cuspidal Siegel eigenform of degree 2 must lie in certain families. Applications include estimates on Hecke eigenvalues, an improved domain of convergence of the standard L-function, and a new characterization of the Maaß space.