LEVEL 251
GRITSENKO LIFTS
Number of wt 2 Gritsenko lifts: 4
WEIGHT 4 CUSP FORMS
The weight 4 space of cusp forms (and the plus and minus parts) have dimensions:
dim S4(K(251)) = 131
dim S4(K(251))+ = 113
dim S4(K(251))- = 18
We use the following wt 4 plus forms in an attempt to fill out the wt 4 plus space of cusp forms.
-
We make the 4 wt 2 Gritsenko lifts.
We compute the products of these wt 2 Gritsenko lifts, and these give us
10 wt 4 plus forms.
-
We compute the Hecke operators T2, T3, T4, T5, T6, T7, T8, T9, T10, T11, T12 on the above products
of wt 2 Gritsenko lifts.
This gives us an additional 110 wt 4 plus forms.
We computed the initial 330 Fourier coefficients
of these forms (which is up through determinant 236/4).
(This of course required much longer expansions of the Fourier series
before computing any Hecke operators - actually 540461 Fourier coefficients were
computed.)
and the rank of these 120 truncated series was computed
mod 19 to be 113.
So these forms span at least
113 linearly independent
plus forms.
Hence dim S4(K(251))+ ≥ 113.
Theta tracing was used only to span the space of minus cusp forms.
The quadratic forms used in Theta Tracing are listed at
the end of this web page.
We computed Tr(ϑPϑQ)
for all 78 combinations of P,Q∈A4.
We truncated the Fourier series to the first
85 coefficients
(which corresponded to going up to determinant 111/4),
and the rank of these truncated series was computed
mod 19 to be 18,
and this proved
dim S4(K(251))- ≥ 18.
The two inequalities on the plus and minus subspaces
suffice to prove the wt 4 dimensions as claimed
because we know a priori that
dim S4(K(251)) = 131.
PROVING UPPER BOUND ON NUMBER OF WEIGHT 2 NONLIFTS
We claim:
At most number of wt 2 plus nonlifts: 0
At most number of wt 2 minus nonlifts: 0
Using the spanning set above for S4(K(251))+,
we compute that
dim H(2)''+ = 0
which implies that there are no plus wt 2 nonlifts.
Using a spanning set for S4(K(251))- computed above,
we calculated that
dim H(2)- = 0.
Then dim H(2)- < (number of wt 2 Gritsenko lifts
= 4)
implies there are no
wt 2 minus form,
and in particular no wt 2 minus nonlift.
Dimensions of subspaces of S4(K(251))+
as we Hecke smear.
Just for curiosity, here is a table that shows how the dimensions of
the subspaces progress as we increase the number of Hecke smears.
Denote W=span of the products of wt 2 Gritsenko lifts.
| Subspace of S4(K(251))+
| dimension |
|---|
| W | 10 |
| Above along with T2(W) | 20 |
| Above along with T3(W) | 30 |
| Above along with T4(W) | 40 |
| Above along with T5(W) | 50 |
| Above along with T6(W) | 60 |
| Above along with T7(W) | 70 |
| Above along with T8(W) | 80 |
| Above along with T9(W) | 90 |
| Above along with T10(W) | 100 |
| Above along with T11(W) | 110 |
| Above along with T12(W) | 113 |
The set A4 of
4x4 matrices used in theta tracing.
Here |A4|=12.
{{10,1,0,3},{1,12,4,-3},{0,4,14,1},{3,-3,1,44}}
{{10,1,-3,-4},{1,12,2,-2},{-3,2,14,3},{-4,-2,3,44}}
{{10,1,-5,1},{1,12,1,-2},{-5,1,28,14},{1,-2,14,30}}
{{12,5,2,-2},{5,12,4,3},{2,4,14,3},{-2,3,3,44}}
{{8,1,-2,-3},{1,10,5,-4},{-2,5,16,2},{-3,-4,2,66}}
{{8,2,-3,-2},{2,12,4,-1},{-3,4,14,6},{-2,-1,6,66}}
{{8,1,-4,-2},{1,8,0,0},{-4,0,10,5},{-2,0,5,128}}
{{4,1,0,-2},{1,6,1,0},{0,1,22,11},{-2,0,11,132}}
{{6,3,-1,-2},{3,8,2,-3},{-1,2,14,6},{-2,-3,6,130}}
{{4,1,-2,-1},{1,8,2,1},{-2,2,18,9},{-1,1,9,130}}
{{16,0,-2,-1},{0,16,1,-2},{-2,1,16,0},{-1,-2,0,16}}
{{14,0,-1,0},{0,14,0,-1},{-1,0,18,0},{0,-1,0,18}}