Restriction method for S412

S₄¹²

There are 8 relations from doing Witt-Erohkin method on 8 reducible forms (those marked by * in the table of coefficient forms).
(An explanation of this table appears at the bottom of this page.)


ell  m11m22m33m44m12m13m23m14m24m34   cusp:term   terms  dimM₁⁴⁸(ell)  #rels

  2    2  2  2  2  0  0  0  1  1  1   1/0: q^8/1      18      13         2
                                      1/1: q^8/2   

  5    2  2  2  2  1  0  0  1  0  1   1/0: q^8/1      27      25         2
                                      1/1: q^17/5  

  6    2  2  2  4  1  1  0  1  0  0   1/0: q^10/1     54      49         5
                                      1/1: q^14/6  
                                      1/2: q^17/3  
                                      1/3: q^9/2   

  8    3  3  3  3  1  1  1 -1  1  1   1/0: q^13/1     53      49         4
                                      1/1: q^18/8  
                                      1/2: q^11/2  
                                      1/4: q^7/1   

  9    4  4  4  4 -1  2  1 -2  2 -1   1/0: q^17/1     54      49         2
                                      1/1: q^17/9  
                                      1/3: q^8/1   
                                      2/3: q^8/1   

NUMBER OF RESTRICTION MAPS:   7                         TOTAL RELS:   15

Including 8 Witt-Erohkin relations, total relations: 23.
Total independent relations: 22.
Because the net set of coefficients has size 23, this implies the kernel of S₄¹² under the Witt-Erohkin map has dimension at most 23-22=1. Because the image of the Witt-Erohkin map is 1-dimensional, this proves that dim S₄¹² is at most 2. Because we can produce two linearly independent cusp forms in S₄¹² (for example, E₄ J₈ and ?), we conclude dim S₄¹² = 2.

ell: This is the level.
The next 10 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.