LEVEL 113


GRITSENKO LIFTS
Number of wt 2 Gritsenko lifts: 3


WEIGHT 4 CUSP FORMS
The weight 4 space of cusp forms (and the plus and minus parts) have dimensions:
     dim S4(K(113)) = 34
     dim S4(K(113))+ = 33
     dim S4(K(113))- = 1
Theta tracing was used to span the whole space of cusp forms, both plus and minus.
The quadratic forms used in Theta Tracing are listed at the end of this web page.
We computed Tr(ϑPϑQ) for all 78 combinations of P,Q∈A4.

We truncated the Fourier series to the first 68 coefficients (which corresponded to going up to determinant 95/4)
and this proved dim S4(K(113))+ ≥ 33 and dim S4(K(113))- ≥ 1 which sufficed to prove the wt 4 dimensions as claimed above because we know a priori that dim S4(K(113)) = 34.


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(113))+, we compute that
     dim H(2)''+ = 0
which implies that there are no plus wt 2 nonlifts.
Using a spanning set for S4(K(113))- computed above, we calculated that
     dim H(2)- = 0.
Then dim H(2)- < (number of wt 2 Gritsenko lifts = 3) implies there are no wt 2 minus form, and in particular no wt 2 minus nonlift.


The set A4 of 4x4 matrices used in theta tracing. Here |A4|=12.
     {{8,3,-3,2},{3,8,1,3},{-3,1,10,2},{2,3,2,30}}
     {{8,0,-3,-1},{0,10,2,-1},{-3,2,10,3},{-1,-1,3,20}}
     {{6,1,-1,0},{1,8,4,3},{-1,4,12,4},{0,3,4,30}}
     {{6,1,-3,-2},{1,6,0,-3},{-3,0,8,4},{-2,-3,4,60}}
     {{4,2,-1,-1},{2,6,1,2},{-1,1,12,1},{-1,2,1,58}}
     {{4,1,0,-2},{1,6,1,0},{0,1,10,5},{-2,0,5,60}}
     {{12,6,-3,-1},{6,12,1,-6},{-3,1,14,5},{-1,-6,5,16}}
     {{6,1,2,0},{1,12,3,1},{2,3,14,5},{0,1,5,16}}
     {{4,1,-1,0},{1,14,4,-7},{-1,4,18,4},{0,-7,4,20}}
     {{4,1,-2,0},{1,12,5,-3},{-2,5,18,1},{0,-3,1,20}}
     {{2,1,0,-1},{1,12,2,0},{0,2,20,5},{-1,0,5,30}}
     {{2,1,-1,-1},{1,18,1,-9},{-1,1,20,3},{-1,-9,3,24}}