LEVEL 563
GRITSENKO LIFTS
Number of wt 2 Gritsenko lifts: 15
WEIGHT 4 CUSP FORMS
The weight 4 space of cusp forms (and the plus and minus parts) have dimensions:
dim S4(K(563)) = 599
dim S4(K(563))+ = 443
dim S4(K(563))- = 156
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 15 wt 2 Gritsenko lifts.
We compute the products of these wt 2 Gritsenko lifts, and these give us
120 wt 4 plus forms.
-
We compute the Hecke operators T2, T3, T4 on the above products
of wt 2 Gritsenko lifts.
This gives us an additional 360 wt 4 plus forms.
We computed the initial 1502 Fourier coefficients
of these forms (which is up through determinant 691/4).
(This of course required much longer expansions of the Fourier series
before computing any Hecke operators - actually 98887 Fourier coefficients were
computed.)
and the rank of these 480 truncated series was computed
mod 19 to be 443.
So these forms span at least
443 linearly independent
plus forms.
Hence dim S4(K(563))+ ≥ 443.
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 210 combinations of P,Q∈A4.
We truncated the Fourier series to the first
820 coefficients
(which corresponded to going up to determinant 464/4),
and the rank of these truncated series was computed
mod 19 to be 156,
and this proved
dim S4(K(563))- ≥ 156.
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(563)) = 599.
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(563))+,
we compute that
dim H(2)''+ = 0
which implies that there are no plus wt 2 nonlifts.
Using a spanning set for S4(K(563))- computed above,
we calculated that
dim H(2)- ≤ 14.
Then dim H(2)- < (number of wt 2 Gritsenko lifts
= 15)
implies there are no
wt 2 minus form,
and in particular no wt 2 minus nonlift.
Dimensions of subspaces of S4(K(563))+
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(563))+
| dimension |
W | 120 |
Above along with T2(W) | 240 |
Above along with T3(W) | 360 |
Above along with T4(W) | 443 |
The set A4 of
4x4 matrices used in theta tracing.
Here |A4|=20.
{{18,0,-1,-4},{0,20,1,-7},{-1,1,22,3},{-4,-7,3,44}}
{{20,1,-5,-1},{1,22,2,-1},{-5,2,22,4},{-1,-1,4,36}}
{{18,3,7,8},{3,20,6,4},{7,6,36,15},{8,4,15,36}}
{{20,2,-1,-5},{2,24,6,7},{-1,6,30,14},{-5,7,14,32}}
{{16,1,0,3},{1,28,8,6},{0,8,30,13},{3,6,13,32}}
{{20,3,-6,-9},{3,20,0,-7},{-6,0,22,1},{-9,-7,1,46}}
{{26,3,-11,-13},{3,28,5,-2},{-11,5,28,14},{-13,-2,14,30}}
{{18,0,-1,-4},{0,20,9,-3},{-1,9,26,2},{-4,-3,2,42}}
{{18,7,8,3},{7,24,11,-1},{8,11,30,2},{3,-1,2,36}}
{{22,9,8,7},{9,24,11,10},{8,11,26,4},{7,10,4,40}}
{{20,6,7,1},{6,22,3,2},{7,3,22,6},{1,2,6,42}}
{{14,2,-3,0},{2,18,1,-1},{-3,1,28,9},{0,-1,9,50}}
{{24,11,9,-7},{11,26,4,-8},{9,4,28,3},{-7,-8,3,30}}
{{18,2,1,-3},{2,24,5,-10},{1,5,30,11},{-3,-10,11,36}}
{{14,3,-1,1},{3,20,7,7},{-1,7,36,17},{1,7,17,44}}
{{14,3,3,-5},{3,20,4,-2},{3,4,22,5},{-5,-2,5,60}}
{{14,0,-3,-7},{0,20,5,-2},{-3,5,22,1},{-7,-2,1,60}}
{{16,1,-2,-2},{1,24,11,-8},{-2,11,26,1},{-2,-8,1,44}}
{{18,1,-7,-4},{1,22,5,-3},{-7,5,24,8},{-4,-3,8,44}}
{{14,3,1,0},{3,16,3,5},{1,3,32,14},{0,5,14,54}}