LEVEL 587

GRITSENKO LIFTS

Number of wt 2 Gritsenko lifts: 18

WEIGHT 4 CUSP FORMS

The weight 4 space of cusp forms (and the plus and minus parts) have dimensions:
     dim S⁴(K(587)) = 649
     dim S⁴(K(587))⁺ = 480
     dim S⁴(K(587))^- = 169
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 18 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 171 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 684 wt 4 plus forms.

We computed the initial 2519 Fourier coefficients of these forms (which is up through determinant 999/4). (This of course required much longer expansions of the Fourier series before computing any Hecke operators - actually 337786 Fourier coefficients were computed.)
and the rank of these 855 truncated series was computed mod 19 to be 480.
So these forms span at least 480 linearly independent plus forms. Hence dim S⁴(K(587))⁺ ≥ 480.

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(ϑ_Pϑ_Q) for all 210 combinations of P,Q∈A₄.
We truncated the Fourier series to the first 852 coefficients (which corresponded to going up to determinant 479/4),
and the rank of these truncated series was computed mod 541 to be 169,
and this proved dim S⁴(K(587))^- ≥ 169.

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(587)) = 649.

PROVING UPPER BOUND ON NUMBER OF WEIGHT 2 NONLIFTS

We claim:
     At most number of wt 2 plus nonlifts: 1
     At most number of wt 2 minus nonlifts: 1
Using the spanning set above for S⁴(K(587))⁺, we compute that
     dim H(2)''⁺ ≤ 1
which implies that there is at most one wt 2 plus nonlifts.
Using the spanning set above for S⁴(K(587))^-, and using the "g1, g2" method by taking
     g1 = G₈
     g2 = G₁₅+G₁₈
and showed that there were at most 1 nonzero solutions to h2*g1=h1*g2 for h1,h2 ∈ H(2)^-. This proves that dim S²(K(587))^- ≤ 1.

Dimensions of subspaces of S⁴(K(587))⁺ 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(587))⁺	dimension
W	171
Above along with T₂(W)	342
Above along with T₃(W)	476
Above along with T₄(W)	479
Above along with T₅(W)	480

Weight 2 Theta Blocks

     G₁ = Grit(THBK₂(3,5,6,6,7,9,12,13,15,20))
     G₂ = Grit(THBK₂(3,5,5,7,8,8,12,13,15,20))
     G₃ = Grit(THBK₂(3,5,5,6,7,10,12,13,16,19))
     G₄ = Grit(THBK₂(3,4,5,7,8,9,12,13,16,19))
     G₅ = Grit(THBK₂(3,4,5,7,7,10,12,14,15,19))
     G₆ = Grit(THBK₂(3,4,5,6,8,11,11,14,15,19))
     G₇ = Grit(THBK₂(3,4,4,8,8,11,12,12,14,20))
     G₈ = Grit(THBK₂(3,4,4,7,8,11,12,13,15,19))
     G₉ = Grit(THBK₂(3,3,5,7,8,10,12,15,15,18))
     G₁₀ = Grit(THBK₂(3,3,5,6,8,9,12,13,14,21))
     G₁₁ = Grit(THBK₂(3,3,4,6,7,10,13,13,16,19))
     G₁₂ = Grit(THBK₂(3,3,4,5,7,8,12,13,17,20))
     G₁₃ = Grit(THBK₂(2,5,6,7,8,9,11,13,15,20))
     G₁₄ = Grit(THBK₂(2,5,6,7,7,9,12,13,16,19))
     G₁₅ = Grit(THBK₂(2,5,5,7,9,10,11,12,15,20))
     G₁₆ = Grit(THBK₂(2,5,5,7,8,9,12,13,17,18))
     G₁₇ = Grit(THBK₂(2,4,5,6,7,9,12,13,17,19))
     G₁₈ = Grit(THBK₂(2,4,5,5,6,9,11,13,16,21))

The set A₄ of 4x4 matrices used in theta tracing. Here |A₄|=20.

     {{20,4,3,7},{4,24,3,10},{3,3,26,7},{7,10,7,36}}
     {{20,1,3,1},{1,22,2,10},{3,2,22,5},{1,10,5,42}}
     {{18,4,5,3},{4,24,2,5},{5,2,24,1},{3,5,1,38}}
     {{16,3,2,2},{3,20,5,4},{2,5,24,3},{2,4,3,50}}
     {{16,1,4,2},{1,28,2,11},{4,2,30,13},{2,11,13,36}}
     {{16,5,3,-2},{5,24,5,-4},{3,5,34,17},{-2,-4,17,40}}
     {{18,7,0,2},{7,30,5,13},{0,5,32,13},{2,13,13,32}}
     {{16,3,3,-4},{3,20,5,5},{3,5,28,10},{-4,5,10,48}}
     {{16,3,2,7},{3,20,5,0},{2,5,24,10},{7,0,10,56}}
     {{24,10,2,-7},{10,24,3,-11},{2,3,30,1},{-7,-11,1,30}}
     {{20,9,-2,-7},{9,26,2,-11},{-2,2,30,1},{-7,-11,1,32}}
     {{20,7,5,-1},{7,24,4,-11},{5,4,26,0},{-1,-11,0,38}}
     {{20,3,5,-6},{3,22,3,10},{5,3,26,1},{-6,10,1,40}}
     {{20,1,6,1},{1,22,6,-5},{6,6,22,3},{1,-5,3,44}}
     {{18,1,-3,2},{1,20,5,-7},{-3,5,28,6},{2,-7,6,42}}
     {{14,1,4,-1},{1,20,8,6},{4,8,42,21},{-1,6,21,44}}
     {{16,3,6,7},{3,18,8,2},{6,8,26,5},{7,2,5,62}}
     {{18,1,5,8},{1,20,8,2},{5,8,24,11},{8,2,11,56}}
     {{18,5,-2,-3},{5,18,1,-4},{-2,1,20,2},{-3,-4,2,60}}
     {{18,5,7,-5},{5,24,9,7},{7,9,28,5},{-5,7,5,42}}