Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below.
We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 14 x^{6} + 61 x^{4} + 84 x^{2} + 4\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( \nu^{5} + 9 \nu^{3} + 16 \nu \)\()/2\) |
| \(\beta_{2}\) | \(=\) | \((\)\( \nu^{6} + 11 \nu^{4} + 30 \nu^{2} + 4 \nu + 12 \)\()/4\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -\nu^{6} - 9 \nu^{4} - 16 \nu^{2} + 2 \)\()/2\) |
| \(\beta_{4}\) | \(=\) | \((\)\( -\nu^{6} - 11 \nu^{4} - 30 \nu^{2} + 4 \nu - 12 \)\()/4\) |
| \(\beta_{5}\) | \(=\) | \((\)\( \nu^{5} + 11 \nu^{3} + 26 \nu \)\()/2\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -\nu^{6} - 11 \nu^{4} - 26 \nu^{2} + 4 \)\()/2\) |
| \(\beta_{7}\) | \(=\) | \((\)\( \nu^{7} + 13 \nu^{5} + 50 \nu^{3} + 54 \nu \)\()/2\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(\beta_{4} + \beta_{2}\)\()/2\) |
| \(\nu^{2}\) | \(=\) | \((\)\(\beta_{6} - \beta_{4} + \beta_{2} - 8\)\()/2\) |
| \(\nu^{3}\) | \(=\) | \((\)\(2 \beta_{5} - 5 \beta_{4} - 5 \beta_{2} - 2 \beta_{1}\)\()/2\) |
| \(\nu^{4}\) | \(=\) | \((\)\(-7 \beta_{6} + 5 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} + 42\)\()/2\) |
| \(\nu^{5}\) | \(=\) | \((\)\(-18 \beta_{5} + 29 \beta_{4} + 29 \beta_{2} + 22 \beta_{1}\)\()/2\) |
| \(\nu^{6}\) | \(=\) | \((\)\(47 \beta_{6} - 29 \beta_{4} - 22 \beta_{3} + 29 \beta_{2} - 246\)\()/2\) |
| \(\nu^{7}\) | \(=\) | \((\)\(4 \beta_{7} + 134 \beta_{5} - 181 \beta_{4} - 181 \beta_{2} - 186 \beta_{1}\)\()/2\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).
| \(n\) |
\(127\) |
\(421\) |
\(449\) |
\(577\) |
| \(\chi(n)\) |
\(-1\) |
\(1\) |
\(1\) |
\(-1\) |
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
This newform subspace can be constructed as the kernel of the linear operator \( T_{19}^{4} - 4 T_{19}^{3} - 44 T_{19}^{2} + 96 T_{19} + 64 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\).
Hecke Characteristic Polynomials
| $p$ |
$F_p(T)$ |
| $2$
| \( \)
|
| $3$
| \( ( 1 - T )^{8} \)
|
| $5$
| \( 1 - 12 T^{2} + 56 T^{4} + 28 T^{6} - 1074 T^{8} + 700 T^{10} + 35000 T^{12} - 187500 T^{14} + 390625 T^{16} \)
|
| $7$
| \( 1 - 4 T + 8 T^{2} - 20 T^{3} + 46 T^{4} - 140 T^{5} + 392 T^{6} - 1372 T^{7} + 2401 T^{8} \)
|
| $11$
| \( 1 - 20 T^{2} + 296 T^{4} - 2268 T^{6} + 20718 T^{8} - 274428 T^{10} + 4333736 T^{12} - 35431220 T^{14} + 214358881 T^{16} \)
|
| $13$
| \( ( 1 - 18 T^{2} + 169 T^{4} )^{4} \)
|
| $17$
| \( 1 - 108 T^{2} + 5432 T^{4} - 166628 T^{6} + 3420078 T^{8} - 48155492 T^{10} + 453686072 T^{12} - 2606857452 T^{14} + 6975757441 T^{16} \)
|
| $19$
| \( ( 1 - 4 T + 32 T^{2} - 132 T^{3} + 558 T^{4} - 2508 T^{5} + 11552 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2} \)
|
| $23$
| \( 1 - 100 T^{2} + 5000 T^{4} - 174764 T^{6} + 4622926 T^{8} - 92450156 T^{10} + 1399205000 T^{12} - 14803588900 T^{14} + 78310985281 T^{16} \)
|
| $29$
| \( ( 1 + 36 T^{2} + 96 T^{3} + 390 T^{4} + 2784 T^{5} + 30276 T^{6} + 707281 T^{8} )^{2} \)
|
| $31$
| \( ( 1 + 8 T + 100 T^{2} + 616 T^{3} + 4534 T^{4} + 19096 T^{5} + 96100 T^{6} + 238328 T^{7} + 923521 T^{8} )^{2} \)
|
| $37$
| \( ( 1 - 4 T + 40 T^{2} + 292 T^{3} - 866 T^{4} + 10804 T^{5} + 54760 T^{6} - 202612 T^{7} + 1874161 T^{8} )^{2} \)
|
| $41$
| \( 1 - 204 T^{2} + 21176 T^{4} - 1439172 T^{6} + 69360622 T^{8} - 2419248132 T^{10} + 59838314936 T^{12} - 969021265164 T^{14} + 7984925229121 T^{16} \)
|
| $43$
| \( 1 - 192 T^{2} + 18396 T^{4} - 1162560 T^{6} + 55940774 T^{8} - 2149573440 T^{10} + 62892263196 T^{12} - 1213701705408 T^{14} + 11688200277601 T^{16} \)
|
| $47$
| \( ( 1 - 8 T + 108 T^{2} - 872 T^{3} + 6758 T^{4} - 40984 T^{5} + 238572 T^{6} - 830584 T^{7} + 4879681 T^{8} )^{2} \)
|
| $53$
| \( ( 1 - 8 T + 132 T^{2} - 632 T^{3} + 7718 T^{4} - 33496 T^{5} + 370788 T^{6} - 1191016 T^{7} + 7890481 T^{8} )^{2} \)
|
| $59$
| \( ( 1 - 8 T + 102 T^{2} - 472 T^{3} + 3481 T^{4} )^{4} \)
|
| $61$
| \( 1 - 184 T^{2} + 16284 T^{4} - 814280 T^{6} + 38728934 T^{8} - 3029935880 T^{10} + 225465674844 T^{12} - 9479748882424 T^{14} + 191707312997281 T^{16} \)
|
| $67$
| \( 1 - 240 T^{2} + 37436 T^{4} - 3815056 T^{6} + 299829030 T^{8} - 17125786384 T^{10} + 754377365756 T^{12} - 21710011720560 T^{14} + 406067677556641 T^{16} \)
|
| $71$
| \( 1 - 484 T^{2} + 107912 T^{4} - 14382508 T^{6} + 1250123214 T^{8} - 72502222828 T^{10} + 2742225320072 T^{12} - 62000537417764 T^{14} + 645753531245761 T^{16} \)
|
| $73$
| \( 1 - 408 T^{2} + 80380 T^{4} - 10037032 T^{6} + 869620166 T^{8} - 53487343528 T^{10} + 2282650611580 T^{12} - 61744364325912 T^{14} + 806460091894081 T^{16} \)
|
| $79$
| \( 1 - 368 T^{2} + 73436 T^{4} - 9593744 T^{6} + 892827078 T^{8} - 59874556304 T^{10} + 2860338148316 T^{12} - 89456183631728 T^{14} + 1517108809906561 T^{16} \)
|
| $83$
| \( ( 1 - 8 T + 124 T^{2} - 1480 T^{3} + 7318 T^{4} - 122840 T^{5} + 854236 T^{6} - 4574296 T^{7} + 47458321 T^{8} )^{2} \)
|
| $89$
| \( 1 + 52 T^{2} - 328 T^{4} + 280764 T^{6} + 120770670 T^{8} + 2223931644 T^{10} - 20579455048 T^{12} + 25843027129972 T^{14} + 3936588805702081 T^{16} \)
|
| $97$
| \( 1 - 344 T^{2} + 58172 T^{4} - 6404840 T^{6} + 615495942 T^{8} - 60263139560 T^{10} + 5149925334332 T^{12} - 286542369695576 T^{14} + 7837433594376961 T^{16} \)
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