LEVEL 593
GRITSENKO LIFTS
Number of wt 2 Gritsenko lifts: 19
WEIGHT 4 CUSP FORMS
The weight 4 space of cusp forms (and the plus and minus parts) have dimensions:
dim S4(K(593)) = 674
dim S4(K(593))+ = 518
dim S4(K(593))- = 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 19 wt 2 Gritsenko lifts.
See the list of wt 2 theta blocks below.
We compute the products of these wt 2 Gritsenko lifts, and these give us
190 wt 4 plus forms.
-
We compute the Hecke operators T2, T3, T4, T5 on the above products
of wt 2 Gritsenko lifts.
This gives us an additional 760 wt 4 plus forms.
We computed the initial 1692 Fourier coefficients
of these forms (which is up through determinant 751/4).
(This of course required much longer expansions of the Fourier series
before computing any Hecke operators - actually 220506 Fourier coefficients were
computed.)
and the rank of these 950 truncated series was computed
mod 19 to be 518.
So these forms span at least
518 linearly independent
plus forms.
Hence dim S4(K(593))+ ≥ 518.
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 528 combinations of P,Q∈A4.
We truncated the Fourier series to the first
512 coefficients
(which corresponded to going up to determinant 886/4),
and the rank of these truncated series was computed
mod 19 to be 156,
and this proved
dim S4(K(593))- ≥ 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(593)) = 674.
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(593))+,
we could only compute that
dim H(2)''+ ≤ 11
which did not imply anything about the nonexistence of nonlifts.
Instead, we computed H(2)'+ which had dimension at most
132 and we used the
"g1, g2" method by taking
g1 = G12
g2 = G15
and showed that there were no nonzero solutions to
h2*g1=h1*g2
for h1,h2 ∈ H(2)'+.
This proves that there are no plus wt 2 nonlifts.
Using the spanning set above for S4(K(593))-,
we could only compute that
dim H(2)- ≤ 26
which did not imply anything about the nonexistence of nonlifts.
We used the
"g1, g2" method by taking
g1 = G12
g2 = G15
and showed that there were no nonzero solutions to
h2*g1=h1*g2
for h1,h2 ∈ H(2)-.
This proves that there are no minus wt 2 nonlifts.
Dimensions of subspaces of S4(K(593))+
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(593))+
| dimension |
W | 190 |
Above along with T2(W) | 380 |
Above along with T3(W) | 513 |
Above along with T4(W) | 517 |
Above along with T5(W) | 518 |
Weight 2 Theta Blocks
G1 = Grit(THBK2(3,4,6,7,7,9,11,13,16,20))
G2 = Grit(THBK2(3,4,6,6,7,10,11,13,17,19))
G3 = Grit(THBK2(3,4,5,6,7,11,11,14,17,18))
G4 = Grit(THBK2(3,4,4,7,7,11,11,15,16,18))
G5 = Grit(THBK2(3,4,4,6,7,9,11,13,17,20))
G6 = Grit(THBK2(3,3,5,6,8,9,11,12,16,21))
G7 = Grit(THBK2(3,3,4,7,8,11,12,15,15,18))
G8 = Grit(THBK2(3,3,4,6,9,9,12,12,15,21))
G9 = Grit(THBK2(3,3,4,6,7,10,11,14,17,19))
G10 = Grit(THBK2(3,3,3,6,6,9,11,14,17,20))
G11 = Grit(THBK2(2,5,5,6,7,11,12,13,17,18))
G12 = Grit(THBK2(2,4,6,7,9,9,10,13,17,19))
G13 = Grit(THBK2(2,4,6,6,8,10,11,14,17,18))
G14 = Grit(THBK2(2,4,5,6,7,9,11,13,18,19))
G15 = Grit(THBK2(2,4,4,6,7,10,11,14,18,18))
G16 = Grit(THBK2(2,3,7,7,9,10,10,12,17,19))
G17 = Grit(THBK2(2,3,5,7,8,11,13,14,15,18))
G18 = Grit(THBK2(2,3,5,7,8,9,11,12,17,20))
G19 = Grit(THBK2(2,3,5,6,8,9,11,14,17,19))
The set A4 of
4x4 matrices used in theta tracing.
Here |A4|=32.
{{2,0,-1,-1},{0,2,0,-1},{-1,0,2,0},{-1,-1,0,100}}
{{2,0,-1,-1},{0,2,0,-1},{-1,0,4,2},{-1,-1,2,44}}
{{2,0,-1,-1},{0,2,1,0},{-1,1,4,0},{-1,0,0,50}}
{{2,0,-1,-1},{0,2,0,-1},{-1,0,6,0},{-1,-1,0,28}}
{{2,0,-1,-1},{0,2,1,0},{-1,1,8,2},{-1,0,2,22}}
{{2,0,-1,-1},{0,2,1,0},{-1,1,12,0},{-1,0,0,14}}
{{2,1,-1,-1},{1,2,0,-1},{-1,0,4,0},{-1,-1,0,60}}
{{2,1,-1,-1},{1,2,0,-1},{-1,0,8,2},{-1,-1,2,28}}
{{2,0,-1,-1},{0,4,0,-1},{-1,0,4,0},{-1,-1,0,22}}
{{2,0,-1,-1},{0,4,1,0},{-1,1,4,2},{-1,0,2,24}}
{{2,0,-1,-1},{0,4,2,-1},{-1,2,4,2},{-1,-1,2,32}}
{{2,0,-1,0},{0,4,2,-1},{-1,2,8,1},{0,-1,1,12}}
{{2,1,-1,-1},{1,4,0,-2},{-1,0,4,1},{-1,-2,1,26}}
{{2,1,0,-1},{1,4,1,-2},{0,1,6,0},{-1,-2,0,16}}
{{2,1,0,-1},{1,4,1,0},{0,1,6,2},{-1,0,2,16}}
{{2,1,-1,-1},{1,4,0,-1},{-1,0,8,0},{-1,-1,0,12}}
{{2,1,-1,-1},{1,4,0,1},{-1,0,8,4},{-1,1,4,14}}
{{2,1,-1,-1},{1,4,1,-2},{-1,1,8,4},{-1,-2,4,16}}
{{2,1,-1,-1},{1,4,1,1},{-1,1,8,3},{-1,1,3,14}}
{{2,1,0,-1},{1,4,0,2},{0,0,8,3},{-1,2,3,14}}
{{2,1,0,-1},{1,4,1,0},{0,1,8,2},{-1,0,2,12}}
{{2,0,-1,-1},{0,6,1,-2},{-1,1,8,1},{-1,-2,1,8}}
{{2,0,-1,0},{0,6,1,1},{-1,1,8,3},{0,1,3,8}}
{{2,1,-1,0},{1,6,0,-1},{-1,0,8,2},{0,-1,2,8}}
{{2,0,-1,-1},{0,8,2,3},{-1,2,8,4},{-1,3,4,8}}
{{4,0,-2,-1},{0,4,1,-1},{-2,1,4,0},{-1,-1,0,14}}
{{4,0,-1,-2},{0,4,1,-1},{-1,1,4,1},{-2,-1,1,12}}
{{4,0,-1,-2},{0,4,1,-1},{-1,1,6,0},{-2,-1,0,8}}
{{4,1,-1,-2},{1,4,0,-1},{-1,0,6,0},{-2,-1,0,8}}
{{4,1,-1,-1},{1,4,1,-1},{-1,1,6,1},{-1,-1,1,8}}
{{4,1,-2,-2},{1,4,0,-2},{-2,0,8,3},{-2,-2,3,8}}
{{4,0,-1,1},{0,6,1,2},{-1,1,6,2},{1,2,2,6}}