LEVEL 557

GRITSENKO LIFTS

Number of wt 2 Gritsenko lifts: 19

WEIGHT 4 CUSP FORMS

The weight 4 space of cusp forms (and the plus and minus parts) have dimensions:
     dim S⁴(K(557)) = 598
     dim S⁴(K(557))⁺ = 460
     dim S⁴(K(557))^- = 138
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 19 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 190 wt 4 plus forms.
We compute the Hecke operators T₂, T₃, T₄ on the above products of wt 2 Gritsenko lifts. This gives us an additional 570 wt 4 plus forms.

We computed the initial 1496 Fourier coefficients of these forms (which is up through determinant 679/4). (This of course required much longer expansions of the Fourier series before computing any Hecke operators - actually 96349 Fourier coefficients were computed.)
and the rank of these 760 truncated series was computed mod 19 to be 460.
So these forms span at least 460 linearly independent plus forms. Hence dim S⁴(K(557))⁺ ≥ 460.

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 190 combinations of P,Q∈A₄.
We truncated the Fourier series to the first 799 coefficients (which corresponded to going up to determinant 464/4),
and the rank of these truncated series was computed mod 19 to be 138,
and this proved dim S⁴(K(557))^- ≥ 138.

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(557)) = 598.

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(557))⁺, we could only compute that
     dim H(2)''⁺ ≤ 3
which did not imply anything about the nonexistence of nonlifts. Instead, we computed H(2)'⁺ which had dimension at most 115 and we used the "g1, g2" method by taking
     g1 = G₁₈
     g2 = G₁₉
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 the spanning set above for S⁴(K(557))^-, we could only compute that
     dim H(2)^- ≤ 21
which did not imply anything about the nonexistence of nonlifts. We used the "g1, g2" method by taking
     g1 = G₅
     g2 = G₁₈
and showed that there were no nonzero solutions to h2*g1=h1*g2 for h1,h2 ∈ H(2)^-. This proves that there are no minus wt 2 nonlifts.

Dimensions of subspaces of S⁴(K(557))⁺ 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(557))⁺	dimension
W	190
Above along with T₂(W)	380
Above along with T₃(W)	459
Above along with T₄(W)	460

Weight 2 Theta Blocks

     G₁ = Grit(THBK₂(4,4,5,6,8,9,11,13,15,19))
     G₂ = Grit(THBK₂(3,5,5,6,8,9,11,14,14,19))
     G₃ = Grit(THBK₂(3,4,6,6,9,9,10,13,15,19))
     G₄ = Grit(THBK₂(3,4,5,7,8,10,11,12,15,19))
     G₅ = Grit(THBK₂(3,4,5,7,8,9,11,13,16,18))
     G₆ = Grit(THBK₂(3,4,5,7,7,10,11,14,15,18))
     G₇ = Grit(THBK₂(3,4,5,6,9,9,11,14,15,18))
     G₈ = Grit(THBK₂(3,3,6,7,8,10,11,13,14,19))
     G₉ = Grit(THBK₂(3,3,6,7,7,10,12,13,15,18))
     G₁₀ = Grit(THBK₂(3,3,6,6,9,9,12,13,15,18))
     G₁₁ = Grit(THBK₂(3,3,6,6,7,9,10,13,15,20))
     G₁₂ = Grit(THBK₂(3,3,5,8,8,11,11,12,14,19))
     G₁₃ = Grit(THBK₂(3,3,4,7,7,10,10,13,17,18))
     G₁₄ = Grit(THBK₂(3,3,4,6,7,10,11,14,17,17))
     G₁₅ = Grit(THBK₂(2,5,6,7,8,9,10,13,15,19))
     G₁₆ = Grit(THBK₂(2,5,6,7,7,9,12,13,14,19))
     G₁₇ = Grit(THBK₂(2,5,6,7,7,9,11,13,16,18))
     G₁₈ = Grit(THBK₂(2,4,7,7,8,9,11,12,15,19))
     G₁₉ = Grit(THBK₂(2,3,6,8,8,10,11,14,14,18))

The set A₄ of 4x4 matrices used in theta tracing. Here |A₄|=19.

     {{18,2,1,5},{2,20,1,8},{1,1,22,10},{5,8,10,48}}
     {{16,1,2,-4},{1,18,3,-3},{2,3,24,1},{-4,-3,1,48}}
     {{16,0,-4,-3},{0,18,1,-4},{-4,1,32,16},{-3,-4,16,44}}
     {{16,1,2,-3},{1,18,3,4},{2,3,24,10},{-3,4,10,52}}
     {{16,1,-3,-4},{1,18,2,-3},{-3,2,28,14},{-4,-3,14,48}}
     {{20,1,-8,-5},{1,24,5,8},{-8,5,30,12},{-5,8,12,32}}
     {{20,1,1,-1},{1,22,9,3},{1,9,24,5},{-1,3,5,36}}
     {{20,3,1,-8},{3,22,4,5},{1,4,24,2},{-8,5,2,36}}
     {{16,3,2,0},{3,20,7,6},{2,7,24,11},{0,6,11,52}}
     {{22,1,-10,-9},{1,24,1,8},{-10,1,30,12},{-9,8,12,32}}
     {{20,7,5,-7},{7,22,4,-7},{5,4,30,2},{-7,-7,2,32}}
     {{16,3,2,-7},{3,20,7,-4},{2,7,24,0},{-7,-4,0,50}}
     {{16,3,4,-5},{3,18,5,-4},{4,5,22,0},{-5,-4,0,58}}
     {{14,1,-1,0},{1,20,0,9},{-1,0,40,18},{0,9,18,40}}
     {{14,0,-1,1},{0,22,7,-8},{-1,7,24,1},{1,-8,1,50}}
     {{20,7,9,8},{7,24,7,9},{9,7,28,6},{8,9,6,36}}
     {{16,3,2,-1},{3,20,4,-10},{2,4,26,7},{-1,-10,7,48}}
     {{14,0,-1,-6},{0,22,7,3},{-1,7,24,5},{-6,3,5,50}}
     {{22,10,6,3},{10,24,7,4},{6,7,26,4},{3,4,4,32}}