LEVEL 277
|
PROOF OF EXISTENCE OF NONLIFT
Define L, L' ∈ S2(K(277))+
and Q,Q' ∈ S4(K(277))+ as follows.
(See bottom of page for definitions of the Theta Blocks Gritsenko lifts
Gi,
etc.)
- L = − G1 − 3G2 + 2G3 − G4 + G6 + 2G7 + G8 − G9
- Q = − 14 G12 + 15 G1 G10 + 20 G1 G2 − 30 G10 G2 + 6 G22 + 45 G1 G3 − 30 G10 G3 − 28 G2 G3 − 34 G32 − 2 G1 G4 + 23 G2 G4 + 8 G3 G4 − 3 G42 − 10 G1 G5 + 15 G2 G5 + 20 G3 G5 + 5 G4 G5 − 13 G1 G6 + 7 G2 G6 + 22 G3 G6 + 6 G4 G6 − 5 G5 G6 − 3 G62 + 39 G1 G7 − 30 G10 G7 − 31 G2 G7 − 56 G3 G7 + 17 G4 G7 + 20 G5 G7 + 13 G6 G7 − 22 G72 − 7 G1 G8 − 20 G2 G8 + 10 G3 G8 − 3 G4 G8 − 5 G5 G8 + 3 G6 G8 + G7 G8 + 6 G82 − 16 G1 G9 + 15 G10 G9 + 11 G2 G9 + 21 G3 G9 − 5 G4 G9 − 10 G5 G9 − 10 G6 G9 + 15 G7 G9 − 4 G8 G9 − 2 G92
- L' = 15G1 + G10 + 6G2 − 5G3 + 5G4 − 6G5 − 5G6 − 13G7 + 7G8 − 8G9
- Q' = − 196 G12 + 211 G1 G10 + 17 G1 G2 − 148 G10 G2 + 8 G22 + 396 G1 G3 − 163 G10 G3 − 156 G2 G3 − 404 G32 − 34 G1 G4 − 3 G10 G4 + 154 G2 G4 + 37 G3 G4 − 24 G42 − 151 G1 G5 + 5 G10 G5 − 2 G2 G5 + 333 G3 G5 + 73 G4 G5 − 55 G52 + 49 G1 G6 + 63 G10 G6 − 4 G2 G6 + 113 G3 G6 − 12 G4 G6 − 73 G5 G6 − 24 G62 + 235 G1 G7 − 94 G10 G7 − 19 G2 G7 − 245 G3 G7 + 160 G4 G7 + 89 G5 G7 − 10 G6 G7 − 54 G72 − 71 G1 G8 + 9 G10 G8 − 13 G2 G8 + 83 G3 G8 + 123 G4 G8 − 109 G5 G8 − 138 G6 G8 − 148 G7 G8 − 63 G82 + 26 G1 G9 − 62 G10 G9 + 145 G2 G9 − 58 G3 G9 + 17 G4 G9 + 52 G5 G9 + 58 G6 G9 − 18 G7 G9 − 5 G8 G9 + 12 G92
If we can prove that the weight 8 plus form
F = Q2 + L Q L' + L2Q'
is identically zero,
then by Theorem (see paper), it would follow that
the form
f = Q/L
would be a holomorphic cusp form.
And because we know that there is at most one nonlift
(see here),
then it would follow that
dim S2(K(277))=
11
and we can compute the action of the Hecke operators
to see that actually this f is an eigenform.
Its Fourier coefficients can be found
here.
Theorem: The above weight 8 cusp form F is zero.
Proof: By the discussion below on Weight 8 cusp forms,
it follows that we only need to check that its first
3990 Fourier coefficients are zero.
We have checked that the first 5000
coefficients are zero. End of proof.
INTEGRALITY
Theorem: The above eigenform f is integral.
Proof: Because f=Q/L with both Q and L
and L can be checked to have content 1 by looking at its Fourier coefficients..
CONGRUENCES
Assuming that the nonlift f exists,
then its first Fourier-Jacobi coefficient is φ where
Grit(φ) = − G1 − 2G2 − 2G3 + 3G4 − 5G5 − 3G6 + 4G7 + 6G8 + 2G9
Assuming that the nonlift exists and is integral,
then by considering the maximal minors of the matrix of
Fourier coefficients of f and the wt 2 Gritsenko lifts
given by the listed theta blocks,
we find that the GCD of the maximal minors must be a factor of
15,
which proves that any nontrivial congruence relation involving
f and the wt 2 Gritsenko lifts must be modulo a factor of
15.
After solving for all possible congruences
modulo 15,
we find that the only possible congruence relation is
f ≡ Grit(φ) mod 15
Continuing to assume that the nonlift exists and is integral,
we can prove that
f ≡ Grit(φ) mod 15
because
f − Grit(φ)
= (15( G12 − G1 G10 − G1 G2 + 2 G10 G2 − 3 G1 G3 + 2 G10 G3 + 2 G2 G3 + 2 G32 − 2 G2 G4 + G1 G5 − 2 G3 G5 + G1 G6 − 2 G3 G6 − 3 G1 G7 + 2 G10 G7 + G2 G7 + 4 G3 G7 − G4 G7 − 2 G5 G7 − G6 G7 + 2 G72 + G7 G8 + G1 G9 − G10 G9 − G2 G9 − G3 G9 + G5 G9 + G6 G9 − G7 G9))/( G1 + 3 G2 − 2 G3 + G4 − G6 − 2 G7 − G8 + G9)
and note that this is a multiple of 15
because the content of the denominator is 1,
and because the numerator is obviously a multiple of 15.
WEIGHT 8 CUSP FORMS
In an attempt to find manageable set of determining coefficients
for the weight 8 space of cusp forms, we will attempt to find a spanning set for it.
The weight 8 space of cusp forms has dimension
dim S8(K(277)) = 2529
We attempt to find cusp forms in the plus and minus parts separately
and hope that the dimensions add up to the above 2529.
We use the following wt 8 plus forms in an attempt to fill out the wt 8 plus space of cusp forms.
-
We make the 56 wt 4 Gritsenko lifts.
We compute the products of these wt 4 Gritsenko lifts, and these give us
1596 wt 8 plus forms.
-
We compute the Hecke operators T2, T3 on the above products
of wt 4 Gritsenko lifts.
This gives us an additional 3192 wt 8 plus forms.
-
We compute the products of all wt 4 plus cusp forms with one theta trace to get
161 cusp forms.
For each such product h, we compute μ(h)+h and
we get
161 plus cusp forms.
(See "proof of at most one lift in level 277"
for how the wt 4 plus space was spanned.)
[This category of wt 8 forms yielded 57
more forms beyond what was made from the previous above categories.]
We computed the initial 6998
(or fewer, in the case of multiplying a theta trace with
a wt 4 form)
Fourier coefficients
of these forms (which is up through determinant 1939/4).
(This of course required much longer expansions of the Fourier series
before computing any Hecke operators.)
and the rank of these 4949 truncated series was computed
mod 19 to be 1817.
So these forms span at least
1817 linearly independent
plus forms.
Hence dim S8(K(277)) ≥ 1817.
For the wt 8 minus space, we use the following forms:
-
We take the 17 wt 4 minus forms
(
We computed Tr(ϑPϑQ)
for all 21 combinations of P,Q∈A4.)
and multiply them by the 161 wt 4 plus forms.
It turns out these gave us at least
595
dimensions of wt 8 minus forms.
-
We apply the Hecke operator T2 to the above wt 8 minus forms,
and obtained at least
57
more dimensions of wt 8 minus forms.
and this proved
dim S8(K(277))- ≥ 712.
Because the plus and minus spanned dimensions
add up to the dimension of the whole space,
then we have succeeded in finding a spanning set of the weight 8 space.
And in fact
dim S8(K(277))+ = 1817
dim S8(K(277))- = 712
An investigation of the coefficients of the spanning set reveals that
a determining set of coefficients may be taken to be the first
3990 Fourier coefficients.
Dimensions of subspaces of S8(K(277))+
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 4 Gritsenko lifts.
Subspace of S8(K(277))+
| dimension |
W | 1495 |
Above along with T2(W) | 1756 |
Above along with T3(W) | 1760 |
Weight 2 Theta Blocks
(Number of wt 2 Gritsenko lifts: 10)
G1 = Grit(THBK2(2,4,4,4,5,6,8,9,10,14))
G2 = Grit(THBK2(2,3,4,5,5,7,7,9,10,14))
G3 = Grit(THBK2(2,3,4,4,5,7,8,9,11,13))
G4 = Grit(THBK2(2,3,3,5,6,6,8,9,11,13))
G5 = Grit(THBK2(2,3,3,5,5,8,8,8,11,13))
G6 = Grit(THBK2(2,3,3,5,5,7,8,10,10,13))
G7 = Grit(THBK2(2,3,3,4,5,6,7,9,10,15))
G8 = Grit(THBK2(2,2,4,5,6,7,7,9,11,13))
G9 = Grit(THBK2(2,2,4,4,6,7,8,10,11,12))
G10 = Grit(THBK2(2,2,3,5,6,7,9,9,11,12))
Weight 4 Theta Blocks
(Number of wt 4 Gritsenko lifts: 56)
C1 = Grit(THBK4(1,1,1,1,1,1,8,22))
C2 = Grit(THBK4(1,1,1,1,1,2,4,23))
C3 = Grit(THBK4(1,1,1,1,1,2,16,17))
C4 = Grit(THBK4(1,1,1,1,1,4,7,22))
C5 = Grit(THBK4(1,1,1,1,1,7,10,20))
C6 = Grit(THBK4(1,1,1,1,1,8,14,17))
C7 = Grit(THBK4(1,1,1,1,1,9,12,18))
C8 = Grit(THBK4(1,1,1,1,2,4,13,19))
C9 = Grit(THBK4(1,1,1,1,2,5,11,20))
C10 = Grit(THBK4(1,1,1,1,2,11,13,16))
C11 = Grit(THBK4(1,1,1,1,3,6,8,21))
C12 = Grit(THBK4(1,1,1,1,4,5,5,22))
C13 = Grit(THBK4(1,1,1,1,4,7,14,17))
C14 = Grit(THBK4(1,1,1,1,4,13,13,14))
C15 = Grit(THBK4(1,1,1,1,8,11,13,14))
C16 = Grit(THBK4(1,1,1,2,3,4,9,21))
C17 = Grit(THBK4(1,1,1,2,4,11,11,17))
C18 = Grit(THBK4(1,1,1,2,7,7,7,20))
C19 = Grit(THBK4(1,1,1,2,7,11,11,16))
C20 = Grit(THBK4(1,1,1,3,5,6,15,16))
C21 = Grit(THBK4(1,1,1,3,9,11,12,14))
C22 = Grit(THBK4(1,1,1,6,9,11,12,13))
C23 = Grit(THBK4(1,1,2,2,4,4,16,16))
C24 = Grit(THBK4(1,1,2,2,4,8,8,20))
C25 = Grit(THBK4(1,1,2,3,3,3,11,20))
C26 = Grit(THBK4(1,1,2,4,4,4,4,22))
C27 = Grit(THBK4(1,1,2,5,5,7,7,20))
C28 = Grit(THBK4(1,1,2,8,11,11,11,11))
C29 = Grit(THBK4(1,1,2,9,9,11,11,12))
C30 = Grit(THBK4(1,1,4,5,5,6,15,15))
C31 = Grit(THBK4(1,1,4,7,7,10,13,13))
C32 = Grit(THBK4(1,1,5,8,10,11,11,11))
C33 = Grit(THBK4(1,2,2,2,2,2,2,23))
C34 = Grit(THBK4(1,2,2,2,2,2,7,22))
C35 = Grit(THBK4(1,2,2,2,2,5,16,16))
C36 = Grit(THBK4(1,2,2,2,3,8,12,18))
C37 = Grit(THBK4(1,2,2,2,6,9,10,18))
C38 = Grit(THBK4(1,2,2,2,7,10,14,14))
C39 = Grit(THBK4(1,2,4,5,6,6,6,20))
C40 = Grit(THBK4(1,2,7,10,10,10,10,10))
C41 = Grit(THBK4(1,3,3,3,4,10,11,17))
C42 = Grit(THBK4(1,5,8,8,10,10,10,10))
C43 = Grit(THBK4(1,6,6,6,6,6,7,18))
C44 = Grit(THBK4(1,8,8,8,8,8,8,13))
C45 = Grit(THBK4(2,2,2,2,2,5,5,22))
C46 = Grit(THBK4(2,2,2,2,2,13,13,14))
C47 = Grit(THBK4(2,2,4,4,4,4,11,19))
C48 = Grit(THBK4(2,3,3,3,3,3,8,21))
C49 = Grit(THBK4(2,3,3,3,4,13,13,13))
C50 = Grit(THBK4(2,3,4,5,5,5,15,15))
C51 = Grit(THBK4(2,7,7,7,7,7,7,16))
C52 = Grit(THBK4(2,8,9,9,9,9,9,9))
C53 = Grit(THBK4(3,3,3,3,3,3,4,22))
C54 = Grit(THBK4(5,5,6,6,6,6,6,18))
C55 = Grit(THBK4(6,6,6,6,6,6,13,13))
C56 = Grit(THBK4(6,7,8,9,9,9,9,9))
The set A4 of
4x4 matrices used in theta tracing.
Here |A4|=6.
{{16,5,5,-5},{5,16,7,1},{5,7,18,3},{-5,1,3,26}}
{{14,1,-7,-6},{1,20,3,-6},{-7,3,20,5},{-6,-6,5,22}}
{{12,3,1,-5},{3,14,6,-2},{1,6,20,1},{-5,-2,1,30}}
{{14,4,1,4},{4,14,6,5},{1,6,18,2},{4,5,2,30}}
{{14,1,-5,3},{1,14,0,-1},{-5,0,16,6},{3,-1,6,32}}
{{16,5,5,-3},{5,16,7,-8},{5,7,18,0},{-3,-8,0,28}}