LEVEL 53

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GRITSENKO LIFTS

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Number of wt 2 Gritsenko lifts: 1
Number of wt 4 Gritsenko lifts: 9

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WEIGHT 4 CUSP FORMS

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The weight 4 space of cusp forms (and the plus and minus parts) have dimensions:
     dim S⁴(K(53)) = 10
     dim S⁴(K(53))⁺ = 10
     dim S⁴(K(53))^- = 0
Here, there is one wt 2 Gritsenko lift and 9 wt 4 Gritsenkow lifts. Because the square of the wt 2 Gritsenko lift is necessarily linearly independent of the wt 4 Gritsenko lifts, then the square of the wt 2 Gritsenko lift along with the wt 4 Gritsenko lifts must span the the 10-dimesnional wt 4 cusp forms, implying that all the wt 4 cusp forms are plus forms.

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PROVING UPPER BOUND ON NUMBER OF WEIGHT 2 NONLIFTS

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We claim:
     At most number of wt 2 plus nonlifts: 0
     At most number of wt 2 minus nonlifts: 0
We use the theorem that if there is only one wt 2 Gritsenko lift and if dim S⁴(K(53)) equals 1 plus the number of wt 4 Gritsenko lifts, then we have that
     dim H(2)''⁺ = 0
which implies that there are no plus wt 2 nonlifts.
Because S⁴(K(53))^-=0 and there is at least one wt 2 Gritsenko lift, then there cannot be any wt 2 minus form, and in particular no wt 2 minus nonlift.