LEVEL 521


GRITSENKO LIFTS
Number of wt 2 Gritsenko lifts: 14


WEIGHT 4 CUSP FORMS
The weight 4 space of cusp forms (and the plus and minus parts) have dimensions:
     dim S4(K(521)) = 527
     dim S4(K(521))+ = 426
     dim S4(K(521))- = 101
We use the following wt 4 plus forms in an attempt to fill out the wt 4 plus space of cusp forms. We computed the initial 1325 Fourier coefficients of these forms (which is up through determinant 635/4). (This of course required much longer expansions of the Fourier series before computing any Hecke operators - actually 170763 Fourier coefficients were computed.)
and the rank of these 525 truncated series was computed mod 19 to be 426.
So these forms span at least 426 linearly independent plus forms. Hence dim S4(K(521))+ ≥ 426.

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(ϑPϑQ) for all 136 combinations of P,Q∈A4.
We truncated the Fourier series to the first 684 coefficients (which corresponded to going up to determinant 415/4),
and the rank of these truncated series was computed mod 19 to be 101,
and this proved dim S4(K(521))- ≥ 101.

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(521)) = 527.


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(521))+, we compute that
     dim H(2)''+ = 0
which implies that there are no plus wt 2 nonlifts.
Using a spanning set for S4(K(521))- computed above, we calculated that
     dim H(2)- ≤ 3.
Then dim H(2)- < (number of wt 2 Gritsenko lifts = 14) implies there are no wt 2 minus form, and in particular no wt 2 minus nonlift.


Dimensions of subspaces of S4(K(521))+ 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(521))+ dimension
W105
Above along with T2(W)210
Above along with T3(W)315
Above along with T4(W)420
Above along with T5(W)426


The set A4 of 4x4 matrices used in theta tracing. Here |A4|=16.
     {{12,1,-1,1},{1,18,3,-1},{-1,3,20,7},{1,-1,7,68}}
     {{12,1,0,-2},{1,20,2,8},{0,2,22,3},{-2,8,3,56}}
     {{12,1,1,-2},{1,20,6,-6},{1,6,28,9},{-2,-6,9,50}}
     {{14,0,-5,1},{0,16,4,-1},{-5,4,40,18},{1,-1,18,42}}
     {{14,1,-2,-1},{1,18,4,-6},{-2,4,22,5},{-1,-6,5,56}}
     {{16,2,3,0},{2,24,1,11},{3,1,28,14},{0,11,14,38}}
     {{16,4,3,5},{4,18,4,-1},{3,4,28,12},{5,-1,12,44}}
     {{16,4,4,7},{4,24,11,10},{4,11,28,11},{7,10,11,40}}
     {{16,7,7,2},{7,24,8,8},{7,8,26,1},{2,8,1,40}}
     {{18,2,-5,-7},{2,20,1,-10},{-5,1,22,10},{-7,-10,10,48}}
     {{18,2,-5,-1},{2,20,4,-1},{-5,4,20,3},{-1,-1,3,44}}
     {{18,2,-5,0},{2,20,4,9},{-5,4,20,5},{0,9,5,48}}
     {{18,4,-6,-5},{4,18,1,-6},{-6,1,26,1},{-5,-6,1,40}}
     {{18,4,5,0},{4,20,4,-7},{5,4,20,5},{0,-7,5,48}}
     {{18,4,5,2},{4,20,4,3},{5,4,20,7},{2,3,7,46}}
     {{20,1,-10,-7},{1,22,0,-3},{-10,0,24,7},{-7,-3,7,36}}