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In this paper, we prove some divisibility results for the Fouriercoefficients of reduced modular forms of sign vectors. More precisely, wegeneralize a divisibility result of Siegel on constant terms when the weight isnon-positive, which is related to the weight of Borcherds lifts when the weightis zero. By considering Hecke operators for the spaces of weakly holomorphicmodular forms with sign vectors, and obtain divisibility results in an"orthogonal" direction on reduced modular forms.

Divisibility properties for weakly holomorphic modular forms with sign vectors