Restriction method for S414

S₄¹⁴

There are 69 relations from doing Witt-Erohkin method on 69 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  dimM₁⁵⁶(ell)  #rels

  2    2  2  2  2  0  0  0  1  1  1   1/0: q^12/1     26      15         5
                                      1/1: q^12/2  

  4    2  2  3  3  0  1  1  1  1  1   1/0: q^14/1     37      29         4
                                      1/1: q^14/4  
                                      1/2: q^6/1   

  5    2  2  2  2  1  0  0  1  0  1   1/0: q^12/1     39      29        10
                                      1/1: q^25/5  

  5    2  2  4  4  1  1  0  1  1  2   1/0: q^15/1     32      29         1
                                      1/1: q^15/5  

  6    2  2  2  4  1  1  0  1  0  0   1/0: q^14/1     75      57        18
                                      1/1: q^19/6  
                                      1/2: q^25/3  
                                      1/3: q^13/2  

  6    2  2  4  4  0  1  1  1  1  1   1/0: q^16/1     68      57         2
                                      1/1: q^16/6  
                                      1/2: q^16/3  
                                      1/3: q^16/2  

  6    2  3  3  4 -1  1 -1  0  1  1   1/0: q^15/1     64      57         1
                                      1/1: q^15/6  
                                      1/2: q^15/3  
                                      1/3: q^15/2  

  7    3  3  3  3  1  0  1 -1  0  1   1/0: q^19/1     40      37         1
                                      1/1: q^19/7  

  8    2  2  2  3  1  1  1  0  1  1   1/0: q^13/1     64      57         7
                                      1/1: q^31/8  
                                      1/2: q^9/2   
                                      1/4: q^7/1   

  8    2  2  4  4  0  1  1  1  1  2   1/0: q^15/1     63      57         6
                                      1/1: q^23/8  
                                      1/2: q^9/2   
                                      1/4: q^12/1  

  8    2  2  6  6  1  1  0  1  1  3   1/0: q^18/1     64      57         3
                                      1/1: q^18/8  
                                      1/2: q^12/2  
                                      1/4: q^12/1  

  8    3  3  3  3  1  1  1 -1  1  1   1/0: q^19/1     73      57        16
                                      1/1: q^26/8  
                                      1/2: q^15/2  
                                      1/4: q^9/1   

  8    3  3  4  4  1  1 -1  1 -1  2   1/0: q^20/1     60      57         1
                                      1/1: q^20/8  
                                      1/2: q^8/2   
                                      1/4: q^8/1   

  9    2  2  2  3  1  1  1  1  1  1   1/0: q^13/1     62      57         5
                                      1/1: q^33/9  
                                      1/3: q^6/1   
                                      2/3: q^6/1   

  9    2  3  3  3  1  1  1  1  1  0   1/0: q^15/1     61      57         4
                                      1/1: q^28/9  
                                      1/3: q^7/1   
                                      2/3: q^7/1   

  9    4  4  4  4 -1  2  1 -2  2 -1   1/0: q^25/1     78      57        10
                                      1/1: q^25/9  
                                      1/3: q^12/1  
                                      2/3: q^12/1  

 10    2  2  2  4  0  0  0  1  1  1   1/0: q^14/1     90      85         5
                                      1/1: q^31/10 
                                      1/2: q^25/5  
                                      1/5: q^16/2  

 11    2  2  2  3  0  0  1  1  1  1   1/0: q^13/1     57      56         1
                                      1/1: q^42/11 

 12    2  2  4  5  0  0  0  1  1  2   1/0: q^16/1    123     113        10
                                      1/1: q^31/12 
                                      1/2: q^12/3  
                                      1/3: q^20/4  
                                      1/4: q^30/3  
                                      1/6: q^8/1   

 12    2  4  4  4  1  0  2  1  2  2   1/0: q^16/1    120     113         7
                                      1/1: q^35/12 
                                      1/2: q^14/3  
                                      1/3: q^18/4  
                                      1/4: q^24/3  
                                      1/6: q^7/1   

 12    3  3  3  3  1  1  1  1  1  1   1/0: q^19/1    139     113        13
                                      1/1: q^32/12 
                                      1/2: q^18/3  
                                      1/3: q^19/4  
                                      1/4: q^32/3  
                                      1/6: q^13/1  

 15    2  2  3  3  1  1  1  1  1  0   1/0: q^14/1    120     112         8
                                      1/1: q^52/15 
                                      1/3: q^30/5  
                                      1/5: q^20/3  

 15    2  2  4  5  0  1  1  1  0  2   1/0: q^16/1    119     112         7
                                      1/1: q^35/15 
                                      1/3: q^45/5  
                                      1/5: q^19/3  

NUMBER OF RESTRICTION MAPS:  23                         TOTAL RELS:  145

Including 69 Witt-Erohkin relations, total relations: 214.
Total independent relations: 209.
The net set of coefficients has size 276, and the goal set has size 85. These relations eliminate down to 83 independent relations on the goal set. This implies the kernel of S₄¹⁴ under the Witt-Erohkin map has dimension at most 85-83=2. Because the image of the Witt-Erohkin map is 1-dimensional, this proves that dim S₄¹⁴ is at most 3. Because we can produce 3 linearly independent cusp forms in S₄¹⁴ (for example, E₄ G₁₀, E₆ J₈ and ?), we conclude dim S₄¹⁴ = 3.

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 M₁⁵⁶(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.