LEVEL 353

EXISTENCE OF NONLIFT

Define L, L' ∈ S²(K(353))⁺ and Q,Q' ∈ S⁴(K(353))⁺ as follows. (See bottom of page for definitions of the Theta Blocks Gritsenko lifts G_i, etc.)
L = − G₁₁ + G₃

Q = − G₁₀ G₁₁ − 2 G₁₁ G₂ + G₁₀ G₃ + 4 G₁₁ G₃ + 2 G₂ G₃ − 4 G₃² − 5 G₁₁ G₄ + 5 G₃ G₄ + G₁₁ G₅ − G₃ G₅ + 6 G₁₁ G₆ − 6 G₃ G₆ − 11 G₁ G₇ + 11 G₁₀ G₇ − 9 G₁₁ G₇ + 9 G₃ G₇ + 11 G₄ G₇ − 11 G₆ G₇ + 11 G₇² + 11 G₁ G₈ − 11 G₁₀ G₈ + 9 G₁₁ G₈ − 9 G₃ G₈ − 11 G₄ G₈ + 11 G₆ G₈ − 22 G₇ G₈ + 11 G₈² − 2 G₁₁ G₉ + 2 G₃ G₉

L' = 9G₁₀ − 3G₁₁ − 4G₂ + G₄ + G₆ + 7G₉

Q' = 22 G₁² − 165 G₁ G₁₀ + 133 G₁₀² + 385 G₁ G₁₁ − 646 G₁₀ G₁₁ + 407 G₁₁² + 224 G₁₀ G₂ − 214 G₁₁ G₂ + 125 G₂² − 286 G₁ G₃ + 330 G₁₀ G₃ − 562 G₁₁ G₃ + 121 G₂ G₃ + 127 G₃² − 165 G₁ G₄ + 230 G₁₀ G₄ − 491 G₁₁ G₄ + 119 G₂ G₄ + 330 G₃ G₄ + 113 G₄² − 220 G₁ G₅ + 231 G₁₀ G₅ − 25 G₁₁ G₅ − 22 G₂ G₅ + 113 G₃ G₅ + 110 G₄ G₅ − 100 G₅² + 121 G₁ G₆ − 177 G₁₀ G₆ + 103 G₁₁ G₆ − 123 G₂ G₆ − 165 G₃ G₆ − 60 G₄ G₆ + 110 G₅ G₆ − 30 G₆² − 572 G₁ G₇ + 594 G₁₀ G₇ − 479 G₁₁ G₇ + 99 G₂ G₇ + 457 G₃ G₇ + 330 G₄ G₇ − 103 G₅ G₇ − 66 G₆ G₇ + 205 G₇² + 473 G₁ G₈ − 495 G₁₀ G₈ + 644 G₁₁ G₈ − 220 G₂ G₈ − 501 G₃ G₈ − 352 G₄ G₈ − 18 G₅ G₈ + 66 G₆ G₈ − 432 G₇ G₈ + 227 G₈² + 242 G₁ G₉ − 214 G₁₀ G₉ + 213 G₁₁ G₉ − 377 G₂ G₉ − 198 G₃ G₉ + 132 G₅ G₉ + 64 G₆ G₉ − 242 G₇ G₉ + 185 G₈ G₉ + 224 G₉² − 955 C₁ − 1419 C₁₀ − 1133 C₁₁ − 178 C₁₂ + 862 C₁₃ − 1375 C₁₄ + 242 C₁₅ − 1012 C₁₆ − 42 C₁₇ − 185 C₁₈ + 983 C₁₉ + 213 C₂ + 79 C₂₀ + 442 C₂₁ + 57 C₂₂ − 471 C₂₃ − 770 C₂₄ + 528 C₂₅ + 321 C₂₇ + 35 C₂₈ + 299 C₂₉ − 1197 C₃ + 35 C₃₀ + 143 C₃₁ − 442 C₃₂ − 143 C₃₃ + 1104 C₃₅ + 143 C₃₆ + 143 C₃₇ − 385 C₃₈ + 719 C₄₀ + 684 C₄₁ − 143 C₄₂ + 143 C₄₄ − 684 C₄₆ + 728 C₅ + 420 C₅₀ + 143 C₅₂ + 385 C₅₃ + 385 C₅₆ + 799 C₆ + 528 C₆₀ + 143 C₇ + 1012 C₈ + 420 C₉ + 264 D₁ D₂ + 528 D₂ D₃ − 143 D₂ D₄ − 178 D₃ D₄ + 264 D₁ D₅ + 528 D₂ D₅ + 528 D₃ D₅ − 143 D₄ D₅ − 242 D₂ D₆ − 242 D₃ D₆ − 242 D₅ D₆ − 143 D₁ D₇ + 35 D₃ D₇ − 35 D₄ D₇ − 143 D₁ D₈ − 264 D₃ D₈ + 143 D₄ D₈ − 143 D₇ D₈

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(353))= 12
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.

Conjecture: The above weight 8 cusp form F is zero.
Evidence: We have checked that the first 29782 coefficients are zero. By the discussion below on Weight 8 cusp forms, it is very likely that this is way more than sufficiently many vanishing Fourier coefficients to show that F=0.

Theorem: If the first 29782 Fourier coefficients determine a weight 8 cusp form, then the above weight 8 cusp form F is zero.

INTEGRALITY

Theorem: If the above f is a holomorphic cusp form, then it is integral.
Proof: This follows because f=Q/L where both Q and L are integral and because 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) = G₁₀ + 2G₂ − 4G₃ − 6G₄ − G₅ + 5G₆ − 2G₇ + 2G₈ + 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 11, which proves that any nontrivial congruence relation involving f and the wt 2 Gritsenko lifts must be modulo a factor of 11.
After solving for all possible congruences modulo 11, we find that the only possible congruence relation is
     f ≡ Grit(φ) mod 11

Continuing to assume that the nonlift exists and is integral, we can prove that
     f ≡ Grit(φ) mod 11
because
     f − Grit(φ) = (11( G₁₁ G₄ − G₃ G₄ − G₁₁ G₆ + G₃ G₆ + G₁ G₇ − G₁₀ G₇ + G₁₁ G₇ − G₃ G₇ − G₄ G₇ + G₆ G₇ − G₇² − G₁ G₈ + G₁₀ G₈ − G₁₁ G₈ + G₃ G₈ + G₄ G₈ − G₆ G₈ + 2 G₇ G₈ − G₈²))/( G₁₁ − G₃)
and note that this is a multiple of 11 because the content of the denominator is 1, and because the numerator is obviously a multiple of 11.

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(353)) = 4032
We attempt to find cusp forms in the plus and minus parts separately and hope that the dimensions add up to the above 4032.

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 69 wt 4 Gritsenko lifts. We compute the products of these wt 4 Gritsenko lifts, and these give us 2415 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 4830 wt 8 plus forms.

We computed the initial 10357 (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 2500/4). (This of course required much longer expansions of the Fourier series before computing any Hecke operators.)
and the rank of these 7245 truncated series was computed mod 541 to be 2500.
So these forms span at least 2500 linearly independent plus forms. Hence dim S⁸(K(353)) ≥ 2500.

There has not yet been an attempt to span S⁸(K(353))^-.

Because the plus and minus spanned dimensions do not add up the the dimension of the whole space, we cannot yet deduce anything about a set of determining Fourier coefficients.

Dimensions of subspaces of S⁸(K(353))⁺ 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(353))⁺	dimension
W	2210
Above along with T₂(W)	2599
Above along with T₃(W)	2602

Weight 2 Theta Blocks (Number of wt 2 Gritsenko lifts: 11)

     G₁ = Grit(THBK₂(3,4,4,4,6,7,8,10,12,16))
     G₂ = Grit(THBK₂(3,3,4,4,5,7,7,10,12,17))
     G₃ = Grit(THBK₂(2,3,5,5,6,7,9,10,11,16))
     G₄ = Grit(THBK₂(2,3,5,5,6,7,8,10,13,15))
     G₅ = Grit(THBK₂(2,3,5,5,5,7,10,10,12,15))
     G₆ = Grit(THBK₂(2,3,4,6,6,7,9,9,13,15))
     G₇ = Grit(THBK₂(2,3,4,6,6,7,8,10,14,14))
     G₈ = Grit(THBK₂(2,3,4,5,6,7,9,11,13,14))
     G₉ = Grit(THBK₂(2,3,4,5,5,7,8,9,12,17))
     G₁₀ = Grit(THBK₂(2,3,4,4,6,8,10,11,12,14))
     G₁₁ = Grit(THBK₂(2,3,3,5,6,9,9,11,12,14))

Weight 4 Theta Blocks (Number of wt 4 Gritsenko lifts: 69)

     C₁ = Grit(THBK₄(1,1,1,1,1,2,11,24))
     C₂ = Grit(THBK₄(1,1,1,1,1,2,16,21))
     C₃ = Grit(THBK₄(1,1,1,1,1,3,4,26))
     C₄ = Grit(THBK₄(1,1,1,1,1,4,18,19))
     C₅ = Grit(THBK₄(1,1,1,1,1,5,10,24))
     C₆ = Grit(THBK₄(1,1,1,1,1,8,14,21))
     C₇ = Grit(THBK₄(1,1,1,1,1,11,16,18))
     C₈ = Grit(THBK₄(1,1,1,1,1,12,14,19))
     C₉ = Grit(THBK₄(1,1,1,1,2,3,8,25))
     C₁₀ = Grit(THBK₄(1,1,1,1,2,5,12,23))
     C₁₁ = Grit(THBK₄(1,1,1,1,2,9,16,19))
     C₁₂ = Grit(THBK₄(1,1,1,1,3,6,9,24))
     C₁₃ = Grit(THBK₄(1,1,1,1,3,8,10,23))
     C₁₄ = Grit(THBK₄(1,1,1,1,3,12,15,18))
     C₁₅ = Grit(THBK₄(1,1,1,1,4,5,6,25))
     C₁₆ = Grit(THBK₄(1,1,1,1,4,6,17,19))
     C₁₇ = Grit(THBK₄(1,1,1,1,5,14,15,16))
     C₁₈ = Grit(THBK₄(1,1,1,1,9,13,14,16))
     C₁₉ = Grit(THBK₄(1,1,1,2,3,11,13,20))
     C₂₀ = Grit(THBK₄(1,1,1,2,5,7,7,24))
     C₂₁ = Grit(THBK₄(1,1,1,3,3,3,10,24))
     C₂₂ = Grit(THBK₄(1,1,1,3,4,5,13,22))
     C₂₃ = Grit(THBK₄(1,1,1,3,10,12,15,15))
     C₂₄ = Grit(THBK₄(1,1,1,4,5,7,17,18))
     C₂₅ = Grit(THBK₄(1,1,1,4,11,11,11,18))
     C₂₆ = Grit(THBK₄(1,1,1,5,5,5,12,22))
     C₂₇ = Grit(THBK₄(1,1,1,5,6,7,8,23))
     C₂₈ = Grit(THBK₄(1,1,1,5,12,13,13,14))
     C₂₉ = Grit(THBK₄(1,1,1,10,11,12,13,13))
     C₃₀ = Grit(THBK₄(1,1,2,2,2,6,16,20))
     C₃₁ = Grit(THBK₄(1,1,2,2,2,8,12,22))
     C₃₂ = Grit(THBK₄(1,1,2,3,3,3,12,23))
     C₃₃ = Grit(THBK₄(1,1,2,3,3,4,15,21))
     C₃₄ = Grit(THBK₄(1,1,2,3,4,5,5,25))
     C₃₅ = Grit(THBK₄(1,1,2,3,7,11,11,20))
     C₃₆ = Grit(THBK₄(1,1,3,3,3,9,14,20))
     C₃₇ = Grit(THBK₄(1,1,3,7,7,7,8,22))
     C₃₈ = Grit(THBK₄(1,1,4,5,5,5,17,18))
     C₃₉ = Grit(THBK₄(1,1,4,7,7,7,10,21))
     C₄₀ = Grit(THBK₄(1,1,4,9,10,13,13,13))
     C₄₁ = Grit(THBK₄(1,1,7,10,11,11,12,13))
     C₄₂ = Grit(THBK₄(1,2,2,2,2,2,3,26))
     C₄₃ = Grit(THBK₄(1,2,2,2,2,2,18,19))
     C₄₄ = Grit(THBK₄(1,2,2,2,2,3,14,22))
     C₄₅ = Grit(THBK₄(1,2,2,2,2,6,13,22))
     C₄₆ = Grit(THBK₄(1,2,2,2,2,7,8,24))
     C₄₇ = Grit(THBK₄(1,2,2,2,3,10,10,22))
     C₄₈ = Grit(THBK₄(1,2,2,4,4,11,12,20))
     C₄₉ = Grit(THBK₄(1,2,2,8,8,8,8,21))
     C₅₀ = Grit(THBK₄(1,2,3,4,13,13,13,13))
     C₅₁ = Grit(THBK₄(1,2,5,5,5,5,5,24))
     C₅₂ = Grit(THBK₄(1,2,7,7,7,7,8,21))
     C₅₃ = Grit(THBK₄(1,3,8,10,10,12,12,12))
     C₅₄ = Grit(THBK₄(1,4,4,4,4,4,15,20))
     C₅₅ = Grit(THBK₄(1,4,5,5,5,6,17,17))
     C₅₆ = Grit(THBK₄(1,6,8,11,11,11,11,11))
     C₅₇ = Grit(THBK₄(2,2,2,2,2,5,6,25))
     C₅₈ = Grit(THBK₄(2,2,2,2,2,6,11,23))
     C₅₉ = Grit(THBK₄(2,2,2,3,8,8,14,19))
     C₆₀ = Grit(THBK₄(2,2,3,3,4,4,18,18))
     C₆₁ = Grit(THBK₄(2,2,3,10,10,10,10,17))
     C₆₂ = Grit(THBK₄(2,4,7,7,7,7,7,21))
     C₆₃ = Grit(THBK₄(2,9,10,10,10,10,10,11))
     C₆₄ = Grit(THBK₄(4,4,4,4,4,4,13,21))
     C₆₅ = Grit(THBK₄(4,4,4,4,4,5,5,24))
     C₆₆ = Grit(THBK₄(4,4,4,4,4,11,12,19))
     C₆₇ = Grit(THBK₄(5,5,5,5,5,6,16,17))
     C₆₈ = Grit(THBK₄(5,6,7,7,7,7,7,20))
     C₆₉ = Grit(THBK₄(5,9,10,10,10,10,10,10))

Weight 2 "Tweak" Theta Blocks that yield Gritsenko lifts with Characters

     D₁ = Grit(THBK₂(1,8,12,12))
     D₂ = Grit(THBK₂(2,3,4,18))
     D₃ = Grit(THBK₂(2,3,12,14))
     D₄ = Grit(THBK₂(2,6,12,13))
     D₅ = Grit(THBK₂(3,10,10,12))
     D₆ = Grit(THBK₂(4,7,12,12))
     D₇ = Grit(THBK₂(5,6,6,16))
     D₈ = Grit(THBK₂(8,8,9,12))

The set A₄ of 4x4 matrices used in theta tracing. Here |A₄|=13.

     {{12,1,2,2},{1,20,1,8},{2,1,24,11},{2,8,11,30}}
     {{18,2,-5,-1},{2,18,4,5},{-5,4,18,6},{-1,5,6,28}}
     {{12,1,-1,1},{1,18,4,-8},{-1,4,24,3},{1,-8,3,30}}
     {{12,1,1,-2},{1,18,8,2},{1,8,20,9},{-2,2,9,40}}
     {{10,0,-1,-3},{0,16,6,-3},{-1,6,20,8},{-3,-3,8,50}}
     {{14,3,-2,-4},{3,16,0,7},{-2,0,20,9},{-4,7,9,38}}
     {{10,1,-1,-2},{1,24,10,1},{-1,10,28,14},{-2,1,14,30}}
     {{18,2,-9,-5},{2,18,0,1},{-9,0,20,7},{-5,1,7,28}}
     {{14,3,-2,3},{3,16,0,-7},{-2,0,20,1},{3,-7,1,34}}
     {{14,2,-4,-1},{2,16,3,3},{-4,3,18,1},{-1,3,1,36}}
     {{14,3,4,5},{3,16,6,3},{4,6,16,7},{5,3,7,48}}
     {{10,1,-2,-1},{1,12,0,-2},{-2,0,36,15},{-1,-2,15,36}}
     {{10,1,-1,-2},{1,12,1,-2},{-1,1,18,3},{-2,-2,3,60}}