Hecke eigenforms of level one among Siegel cusp forms of degree two are determined by the second Hecke eigenvalue under assumption and can be distinguished by L-functions.
Siegel modular forms and representations
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On distinguishing Siegel cusp forms of degree two
Hecke eigenforms of level one among Siegel cusp forms of degree two are determined by the second Hecke eigenvalue under assumption and can be distinguished by L-functions.