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