LEVEL 227
GRITSENKO LIFTS
Number of wt 2 Gritsenko lifts: 5
WEIGHT 4 CUSP FORMS
The weight 4 space of cusp forms (and the plus and minus parts) have dimensions:
dim S4(K(227)) = 109
dim S4(K(227))+ = 91
dim S4(K(227))- = 18
Theta tracing was used to span the whole space of cusp forms, both plus and minus.
The quadratic forms used in Theta Tracing are listed at
the end of this web page.
We computed Tr(ϑPϑQ)
for all 231 combinations of P,Q∈A4.
We truncated the Fourier series to the first
246 coefficients
(which corresponded to going up to determinant 211/4)
and this proved
dim S4(K(227))+ ≥ 91
and
dim S4(K(227))- ≥ 18
which sufficed to prove the wt 4 dimensions as claimed above
because we know a priori that
dim S4(K(227)) = 109.
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(227))+,
we could only compute that
dim H(2)''+ ≤ 1
which did not imply anything about the nonexistence of nonlifts.
Instead, we computed H(2)'+ which had dimension at most
3 and we used the
"g1, g2" method by taking
g1 = G3
g2 = G4
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 a spanning set for S4(K(227))- computed above,
we calculated that
dim H(2)- = 0.
Then dim H(2)- < (number of wt 2 Gritsenko lifts
= 5)
implies there are no
wt 2 minus form,
and in particular no wt 2 minus nonlift.
Weight 2 Theta Blocks
G1 = Grit(THBK2(2,3,4,4,4,6,7,8,10,12))
G2 = Grit(THBK2(2,3,3,4,5,6,7,9,9,12))
G3 = Grit(THBK2(2,2,3,5,5,7,7,8,9,12))
G4 = Grit(THBK2(1,3,4,4,5,7,7,8,9,12))
G5 = Grit(THBK2(1,3,4,4,5,6,7,9,10,11))
The set A4 of
4x4 matrices used in theta tracing.
Here |A4|=21.
{{8,1,-1,-3},{1,20,6,-9},{-1,6,22,2},{-3,-9,2,22}}
{{8,1,-2,-1},{1,10,0,-5},{-2,0,12,3},{-1,-5,3,60}}
{{12,1,3,-5},{1,14,6,0},{3,6,14,3},{-5,0,3,32}}
{{12,1,-1,-4},{1,14,7,-2},{-1,7,20,9},{-4,-2,9,26}}
{{10,4,3,2},{4,12,1,-1},{3,1,14,5},{2,-1,5,40}}
{{8,3,3,0},{3,10,2,-2},{3,2,14,3},{0,-2,3,58}}
{{6,1,-1,-3},{1,12,5,-3},{-1,5,28,9},{-3,-3,9,34}}
{{6,1,-2,-1},{1,8,0,-4},{-2,0,20,5},{-1,-4,5,60}}
{{10,4,-1,-4},{4,12,5,-5},{-1,5,16,3},{-4,-5,3,42}}
{{6,1,-1,2},{1,8,1,-3},{-1,1,10,4},{2,-3,4,118}}
{{6,1,-3,-1},{1,12,4,-2},{-3,4,16,3},{-1,-2,3,58}}
{{6,2,-1,-2},{2,8,3,3},{-1,3,12,2},{-2,3,2,116}}
{{4,0,-1,0},{0,10,2,5},{-1,2,12,1},{0,5,1,116}}
{{8,1,-4,-2},{1,8,2,1},{-4,2,10,5},{-2,1,5,116}}
{{4,1,-1,-1},{1,6,0,-3},{-1,0,20,10},{-1,-3,10,120}}
{{16,0,-5,-2},{0,16,2,-5},{-5,2,16,0},{-2,-5,0,16}}
{{14,0,-4,-3},{0,14,3,-4},{-4,3,18,0},{-3,-4,0,18}}
{{14,0,-5,0},{0,14,0,-5},{-5,0,18,0},{0,-5,0,18}}
{{12,0,-3,-2},{0,12,2,-3},{-3,2,20,0},{-2,-3,0,20}}
{{12,0,-3,-2},{0,12,2,3},{-3,2,20,1},{-2,3,1,20}}
{{16,0,-6,-5},{0,16,5,-6},{-6,5,18,0},{-5,-6,0,18}}