LEVEL 277
Existence discussion Integrality Congruence Weight 8


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.) 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 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: 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
W1495
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}}