Restriction method for S510

S₅¹⁰

(An explanation of this table appears at the bottom of this page.)


ell  m11m22m33m44m55m12m13m23m14m24m34m15m25m35m45   cusp:term   terms  dimM₁⁵⁰(ell)  #rels

  4    2  2  2  2  2  0  0  0  1  0  0  1  1  1  0   1/0: q^10/1     38      26        12
                                                     1/1: q^20/4  
                                                     1/2: q^5/1   

  4    2  2  2  2  3  0  0  0  0  0  0  1  1  1  1   1/0: q^11/1     36      26        10
                                                     1/1: q^15/4  
                                                     1/2: q^7/1   

  4    2  2  4  4  4  0  0  0  0  0  0  1  1  2  2   1/0: q^13/1     33      26         7
                                                     1/1: q^12/4  
                                                     1/2: q^5/1   

  6    2  2  2  2  2  0  0  0  1  1  0  0  1  1  0   1/0: q^10/1     78      51        27
                                                     1/1: q^27/6  
                                                     1/2: q^21/3  
                                                     1/3: q^16/2  

  6    2  2  2  3  3  1  1  0  0  1  0  0  1  0  0   1/0: q^12/1     67      51        16
                                                     1/1: q^20/6  
                                                     1/2: q^13/3  
                                                     1/3: q^18/2  

  8    2  2  2  2  4  0  0  0  0  0  0  1  1  1  1   1/0: q^12/1     62      51        11
                                                     1/1: q^25/8  
                                                     1/2: q^11/2  
                                                     1/4: q^10/1  

  8    2  2  2  4  4  1  1  0  1  0  0  1  0  0  2   1/0: q^13/1     66      51        15
                                                     1/1: q^23/8  
                                                     1/2: q^16/2  
                                                     1/4: q^10/1  

  8    2  2  4  6  6  0  0  0  1  1  2  1  1 -2  0   1/0: q^17/1     56      51         5
                                                     1/1: q^17/8  
                                                     1/2: q^8/2   
                                                     1/4: q^10/1  

NUMBER OF RESTRICTION MAPS:   8                         TOTAL RELS:  103

Including 34 Witt-Erohkin relations, total relations: 137.
Total independent relations: 93.
The net set of coefficients has size 135, and the goal set has size 54. These relations eliminate down to 54 independent relations on the goal set. This implies the kernel of S₅¹⁰ under the Witt-Erohkin map has dimension at most 0. Because the image of the Witt-Erohkin map is 0-dimensional, this implies that dim S₅¹⁰ = 0.

ell: This is the level.
The next 15 numbers give the entries of the matrix.
The cusps (of G₀(ell)) are listed vertically in form a/c. To the right of each of these is the maximum term kept in the corresponding cusp expansion. For example, q^15/3 would indicate that the expansion is in powers of q^1/3, and that we keep the first 16 terms (q^0/3 through q^15/3).
The number under "terms" is the total number of terms kept over the various cusps.
The next entry is dim M₁⁵⁰(G₀(ell)).
The number under "rels" is the number of relations (on the net set) obtained from this particular restriction map. The number is usually the difference between the "terms" and "dim" unless the matrix has some "self-similarity" in the sense that the matrix box-operation with a cusp yields something that is a multiple of the original matrix.