Restriction method for S68

S₆⁸

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


ell  m11m22m33m44m55m66m12m13m23m14m24m34m15m25m35m45m16m26m36m46m56   cusp:term   terms  dimM₁⁴⁸(ell)  #rels

  3    2  2  2  2  2  2  0  0  1  1  0  1  1  1  1  1  1  1  1  1  1   1/0: q^8/1      27      17         9
                                                                       1/1: q^17/3  

  6    2  2  2  2  2  4  0  0  0  0  0  0  0  0  0  0  1  1  1  1  1   1/0: q^10/1     61      49         9
                                                                       1/1: q^18/6  
                                                                       1/2: q^19/3  
                                                                       1/3: q^10/2  

 12    2  2  2  2  4  4  0  1  0  0  1  0  0  0 -1  1  1  1  0  1  2   1/0: q^11/1    123      97        18
                                                                       1/1: q^38/12 
                                                                       1/2: q^17/3  
                                                                       1/3: q^17/4  
                                                                       1/4: q^26/3  
                                                                       1/6: q^8/1   

NUMBER OF RESTRICTION MAPS:   3                         TOTAL RELS:   36

Total relations: 36.
Independent relations 26.
Because the net set of coefficients has size 26, and these 26 relations are independent, this implies that dim S₆⁸ = 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 M₁⁴⁸(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.