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Efficiently generated spaces of classical Siegel modular forms and the Boecherer conjecture

We state and verify up to weight 172 a conjecture on the existence of a certain generating set for spaces of classical Siegel modular forms. This conjecture is particularly useful for calculations involving Fourier expansions. Using this generating set we verify the Boecherer conjec- ture for non-rational eigenforms. As one further application we verify another conjectures for weights up to 150 and investigate an analogue of the Victor-Miller basis. Additionally, we describe some arithmetic properties of the basis we found.