S416
There are 527 relations from doing Witt-Erohkin method on 527 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 dimM164(ell) #rels
2 2 2 2 2 0 0 0 1 1 1 1/0: q14/1 30 17 6
1/1: q14/2
4 2 2 3 3 0 1 1 1 1 1 1/0: q18/1 47 33 7
1/1: q18/4
1/2: q8/1
5 2 2 2 2 1 0 0 1 0 1 1/0: q14/1 45 33 12
1/1: q29/5
5 2 2 4 4 1 1 0 1 1 2 1/0: q22/1 46 33 6
1/1: q22/5
6 2 2 2 4 1 1 0 1 0 0 1/0: q18/1 98 65 33
1/1: q31/6
1/2: q29/3
1/3: q16/2
6 2 2 4 4 0 1 1 1 1 1 1/0: q22/1 92 65 6
1/1: q22/6
1/2: q22/3
1/3: q22/2
6 2 3 3 4 -1 1 -1 0 1 1 1/0: q22/1 92 65 6
1/1: q22/6
1/2: q22/3
1/3: q22/2
7 3 3 3 3 1 0 1 -1 0 1 1/0: q24/1 50 43 3
1/1: q24/7
8 2 2 2 3 1 1 1 0 1 1 1/0: q16/1 88 65 23
1/1: q46/8
1/2: q13/2
1/4: q9/1
8 2 2 4 4 0 1 1 1 1 2 1/0: q22/1 82 65 17
1/1: q29/8
1/2: q13/2
1/4: q14/1
8 2 2 5 5 0 1 1 1 1 1 1/0: q24/1 74 65 4
1/1: q24/8
1/2: q11/2
1/4: q11/1
8 2 2 6 6 1 1 0 1 1 3 1/0: q25/1 82 65 8
1/1: q25/8
1/2: q14/2
1/4: q14/1
8 3 3 3 3 1 1 1 -1 1 1 1/0: q22/1 92 65 27
1/1: q31/8
1/2: q21/2
1/4: q14/1
8 3 3 4 4 1 1 -1 1 -1 2 1/0: q26/1 78 65 6
1/1: q26/8
1/2: q11/2
1/4: q11/1
9 2 2 2 3 1 1 1 1 1 1 1/0: q16/1 80 65 15
1/1: q46/9
1/3: q7/1
2/3: q7/1
9 2 3 3 3 1 1 1 1 1 0 1/0: q20/1 81 65 16
1/1: q35/9
1/3: q11/1
2/3: q11/1
9 2 3 4 6 1 0 1 1 1 2 1/0: q26/1 72 65 3
1/1: q26/9
1/3: q8/1
2/3: q8/1
9 4 4 4 4 -1 2 1 -2 2 -1 1/0: q29/1 90 65 12
1/1: q29/9
1/3: q14/1
2/3: q14/1
10 2 2 2 4 0 0 0 1 1 1 1/0: q18/1 127 97 30
1/1: q41/10
1/2: q44/5
1/5: q20/2
10 2 2 6 6 0 1 1 1 1 1 1/0: q26/1 108 97 2
1/1: q26/10
1/2: q26/5
1/5: q26/2
10 2 3 4 4 0 1 -1 1 1 0 1/0: q28/1 116 97 4
1/1: q28/10
1/2: q28/5
1/5: q28/2
10 2 4 4 6 1 0 2 0 1 2 1/0: q28/1 116 97 4
1/1: q28/10
1/2: q28/5
1/5: q28/2
11 2 2 2 3 0 0 1 1 1 1 1/0: q17/1 74 64 10
1/1: q55/11
12 2 2 4 5 0 0 0 1 1 2 1/0: q24/1 168 129 39
1/1: q41/12
1/2: q18/3
1/3: q24/4
1/4: q44/3
1/6: q11/1
12 2 4 4 4 1 0 2 1 2 2 1/0: q26/1 169 129 40
1/1: q43/12
1/2: q18/3
1/3: q24/4
1/4: q41/3
1/6: q11/1
12 2 4 4 8 1 1 1 0 2 2 1/0: q30/1 154 129 6
1/1: q30/12
1/2: q14/3
1/3: q30/4
1/4: q30/3
1/6: q14/1
12 2 4 6 6 0 1 2 1 2 3 1/0: q31/1 158 129 14
1/1: q31/12
1/2: q14/3
1/3: q31/4
1/4: q31/3
1/6: q14/1
12 3 3 3 3 1 1 1 1 1 1 1/0: q22/1 171 129 21
1/1: q37/12
1/2: q31/3
1/3: q22/4
1/4: q37/3
1/6: q16/1
14 2 2 2 5 0 0 0 1 1 1 1/0: q19/1 148 128 20
1/1: q53/14
1/2: q50/7
1/7: q22/2
14 2 2 3 4 0 1 1 0 0 1 1/0: q21/1 158 128 15
1/1: q56/14
1/2: q56/7
1/7: q21/2
14 2 2 4 4 -1 -1 1 -1 1 2 1/0: q21/1 156 128 28
1/1: q57/14
1/2: q54/7
1/7: q20/2
14 2 4 6 6 0 -1 1 -1 -1 2 1/0: q32/1 132 128 1
1/1: q32/14
1/2: q32/7
1/7: q32/2
14 4 4 6 6 2 2 0 2 2 3 1/0: q37/1 140 128 5
1/1: q37/14
1/2: q31/7
1/7: q31/2
15 2 2 3 3 1 1 1 1 1 0 1/0: q18/1 164 128 36
1/1: q73/15
1/3: q38/5
1/5: q31/3
15 2 2 4 5 0 1 1 1 0 2 1/0: q23/1 156 128 28
1/1: q52/15
1/3: q54/5
1/5: q23/3
15 3 3 4 4 0 1 1 1 1 -1 1/0: q26/1 146 128 18
1/1: q48/15
1/3: q27/5
1/5: q41/3
15 4 4 6 6 2 1 2 1 -1 -3 1/0: q37/1 152 128 5
1/1: q37/15
1/3: q37/5
1/5: q37/3
18 2 2 2 6 0 0 0 1 1 1 1/0: q20/1 227 193 34
1/1: q63/18
1/2: q56/9
1/3: q14/2
2/3: q14/2
1/6: q14/1
5/6: q14/1
1/9: q24/2
18 2 2 3 3 1 0 0 -1 -1 1 1/0: q19/1 239 193 46
1/1: q81/18
1/2: q56/9
1/3: q14/2
2/3: q14/2
1/6: q10/1
5/6: q10/1
1/9: q27/2
18 2 2 6 7 0 0 0 1 1 3 1/0: q27/1 229 193 36
1/1: q49/18
1/2: q47/9
1/3: q18/2
2/3: q18/2
1/6: q17/1
5/6: q17/1
1/9: q28/2
18 2 2 10 10 0 1 1 1 1 1 1/0: q34/1 200 193 1
1/1: q34/18
1/2: q34/9
1/3: q14/2
2/3: q14/2
1/6: q14/1
5/6: q14/1
1/9: q34/2
18 2 4 4 6 0 0 -2 -1 -1 2 1/0: q28/1 231 193 38
1/1: q50/18
1/2: q47/9
1/3: q17/2
2/3: q17/2
1/6: q18/1
5/6: q18/1
1/9: q28/2
18 2 4 6 10 0 1 2 0 2 1 1/0: q36/1 208 193 3
1/1: q36/18
1/2: q36/9
1/3: q14/2
2/3: q14/2
1/6: q14/1
5/6: q14/1
1/9: q36/2
18 2 6 6 8 0 0 0 1 3 3 1/0: q38/1 250 193 28
1/1: q39/18
1/2: q38/9
1/3: q22/2
2/3: q22/2
1/6: q22/1
5/6: q22/1
1/9: q39/2
18 3 3 3 3 0 0 0 1 1 1 1/0: q23/1 229 193 36
1/1: q59/18
1/2: q44/9
1/3: q17/2
2/3: q17/2
1/6: q13/1
5/6: q13/1
1/9: q35/2
30 2 2 2 9 0 0 0 1 1 1 1/0: q23/1 441 382 59
1/1: q89/30
1/2: q74/15
1/3: q60/10
1/5: q48/6
1/6: q62/5
1/10: q47/3
1/15: q30/2
30 2 2 3 5 -1 -1 1 -1 1 1 1/0: q21/1 448 382 66
1/1: q110/30
1/2: q81/15
1/3: q65/10
1/5: q49/6
1/6: q46/5
1/10: q37/3
1/15: q31/2
30 2 2 4 5 0 0 0 1 1 1 1/0: q24/1 447 382 65
1/1: q95/30
1/2: q91/15
1/3: q59/10
1/5: q45/6
1/6: q56/5
1/10: q44/3
1/15: q25/2
30 2 2 4 6 0 1 1 1 1 1 1/0: q24/1 436 382 54
1/1: q94/30
1/2: q96/15
1/3: q51/10
1/5: q41/6
1/6: q53/5
1/10: q45/3
1/15: q24/2
30 2 4 4 4 0 1 1 0 2 2 1/0: q26/1 450 382 68
1/1: q101/30
1/2: q96/15
1/3: q55/10
1/5: q43/6
1/6: q54/5
1/10: q43/3
1/15: q24/2
30 2 4 4 8 1 1 1 0 1 1 1/0: q30/1 420 382 38
1/1: q69/30
1/2: q74/15
1/3: q71/10
1/5: q32/6
1/6: q72/5
1/10: q31/3
1/15: q33/2
30 2 6 6 6 1 0 3 1 3 3 1/0: q34/1 425 382 43
1/1: q78/30
1/2: q77/15
1/3: q67/10
1/5: q34/6
1/6: q64/5
1/10: q30/3
1/15: q33/2
30 2 6 6 8 1 1 0 1 3 3 1/0: q37/1 406 382 24
1/1: q66/30
1/2: q66/15
1/3: q35/10
1/5: q57/6
1/6: q38/5
1/10: q62/3
1/15: q37/2
30 3 3 3 5 0 1 1 1 1 1 1/0: q27/1 431 382 49
1/1: q87/30
1/2: q63/15
1/3: q85/10
1/5: q38/6
1/6: q57/5
1/10: q27/3
1/15: q39/2
30 3 7 8 9 -1 1 3 0 0 3 1/0: q48/1 396 382 2
1/1: q49/30
1/2: q48/15
1/3: q49/10
1/5: q49/6
1/6: q48/5
1/10: q48/3
1/15: q49/2
30 4 4 6 6 2 1 2 1 2 1 1/0: q37/1 407 382 25
1/1: q67/30
1/2: q68/15
1/3: q33/10
1/5: q59/6
1/6: q36/5
1/10: q61/3
1/15: q38/2
30 4 6 8 8 0 1 3 1 3 3 1/0: q51/1 394 382 4
1/1: q49/30
1/2: q51/15
1/3: q48/10
1/5: q48/6
1/6: q45/5
1/10: q45/3
1/15: q49/2
30 6 6 8 8 0 3 3 3 3 3 1/0: q54/1 408 382 5
1/1: q54/30
1/2: q54/15
1/3: q46/10
1/5: q46/6
1/6: q46/5
1/10: q46/3
1/15: q54/2
NUMBER OF RESTRICTION MAPS: 58 TOTAL RELS: 1260
Including 527 Witt-Erohkin relations, total relations: 1787.
Total independent relations: 1685.
The net set of coefficients has size 2249,
and the goal set has size 148.
These relations eliminate down to 145 independent relations on the goal set.
This implies the kernel of S416 under the Witt-Erohkin map has dimension at most 148-145=3.
Because the image of the Witt-Erohkin map is 4-dimensional,
this proves that
dim S416 is at most 7.
Because we can produce 7 linearly independent cusp forms in S416
(see web page [yet to be made]),
we conclude
dim S416 = 7.
- 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, q15/3 would indicate that
the expansion is in powers of q1/3,
and that we keep the first 16 terms (q0/3 through q15/3).
- The number under "terms" is the total number of terms kept
over the various cusps.
- The next entry is dim M164(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.