Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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 \(x^{12} - 5 x^{10} - 2 x^{9} + 15 x^{8} + 2 x^{7} - 30 x^{6} + 4 x^{5} + 60 x^{4} - 16 x^{3} - 80 x^{2} + 64\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\( \nu^{11} - 5 \nu^{9} - 2 \nu^{8} + 15 \nu^{7} + 2 \nu^{6} - 30 \nu^{5} + 4 \nu^{4} + 60 \nu^{3} - 16 \nu^{2} - 80 \nu \)\()/32\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 6 \nu^{8} - 13 \nu^{7} + 30 \nu^{6} - 6 \nu^{5} - 28 \nu^{4} - 4 \nu^{3} + 16 \nu^{2} - 48 \nu + 96 \)\()/32\) |
| \(\beta_{4}\) | \(=\) | \((\)\( 3 \nu^{11} + 2 \nu^{10} - 11 \nu^{9} - 8 \nu^{8} + 21 \nu^{7} + 4 \nu^{6} - 42 \nu^{5} + 84 \nu^{3} + 24 \nu^{2} - 64 \nu - 64 \)\()/32\) |
| \(\beta_{5}\) | \(=\) | \((\)\( \nu^{11} + 3 \nu^{10} - 3 \nu^{9} - 13 \nu^{8} + 7 \nu^{7} + 23 \nu^{6} - 26 \nu^{5} - 38 \nu^{4} + 60 \nu^{3} + 68 \nu^{2} - 56 \nu - 64 \)\()/16\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -5 \nu^{11} + 4 \nu^{10} + 13 \nu^{9} - 10 \nu^{8} - 23 \nu^{7} + 42 \nu^{6} + 10 \nu^{5} - 68 \nu^{4} - 20 \nu^{3} + 48 \nu^{2} - 32 \nu + 64 \)\()/32\) |
| \(\beta_{7}\) | \(=\) | \((\)\( -5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} - 68 \nu^{3} + 120 \nu^{2} + 16 \nu - 32 \)\()/32\) |
| \(\beta_{8}\) | \(=\) | \((\)\( -\nu^{11} + \nu^{10} + 5 \nu^{9} - 3 \nu^{8} - 13 \nu^{7} + 13 \nu^{6} + 20 \nu^{5} - 34 \nu^{4} - 28 \nu^{3} + 52 \nu^{2} + 24 \nu - 32 \)\()/8\) |
| \(\beta_{9}\) | \(=\) | \((\)\( -\nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + 8 \nu^{3} + 108 \nu^{2} - 16 \nu - 80 \)\()/16\) |
| \(\beta_{10}\) | \(=\) | \((\)\( -3 \nu^{11} + 15 \nu^{9} - 2 \nu^{8} - 37 \nu^{7} + 18 \nu^{6} + 66 \nu^{5} - 68 \nu^{4} - 92 \nu^{3} + 96 \nu^{2} + 96 \nu - 64 \)\()/16\) |
| \(\beta_{11}\) | \(=\) | \((\)\( -\nu^{11} + 4 \nu^{10} + 9 \nu^{9} - 18 \nu^{8} - 27 \nu^{7} + 50 \nu^{6} + 34 \nu^{5} - 116 \nu^{4} - 20 \nu^{3} + 192 \nu^{2} + 16 \nu - 160 \)\()/16\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(-\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 1\) |
| \(\nu^{3}\) | \(=\) | \(-\beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1}\) |
| \(\nu^{4}\) | \(=\) | \(-\beta_{11} + \beta_{9} + \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1\) |
| \(\nu^{5}\) | \(=\) | \(-2 \beta_{11} + 4 \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{1} + 1\) |
| \(\nu^{6}\) | \(=\) | \(2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} - 2\) |
| \(\nu^{7}\) | \(=\) | \(-5 \beta_{11} - 3 \beta_{10} + 5 \beta_{9} + 7 \beta_{8} - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 7 \beta_{4} - 3 \beta_{3} + \beta_{2} + 1\) |
| \(\nu^{8}\) | \(=\) | \(-3 \beta_{11} + 7 \beta_{8} + \beta_{7} - 5 \beta_{6} - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 10 \beta_{2} + 8 \beta_{1} - 3\) |
| \(\nu^{9}\) | \(=\) | \(-7 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - 4 \beta_{7} - 11 \beta_{6} + 12 \beta_{5} - 17 \beta_{4} + 7 \beta_{3} + 7 \beta_{2} + \beta_{1} - 8\) |
| \(\nu^{10}\) | \(=\) | \(-17 \beta_{11} + 4 \beta_{10} + 13 \beta_{9} + 18 \beta_{8} - 9 \beta_{7} - 8 \beta_{6} + 3 \beta_{5} + 7 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} - 6 \beta_{1} - 11\) |
| \(\nu^{11}\) | \(=\) | \(18 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} - 13 \beta_{8} - 21 \beta_{7} - 11 \beta_{6} - \beta_{5} - 26 \beta_{4} + 22 \beta_{3} + 14 \beta_{2} + 3 \beta_{1} - 7\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).
| \(n\) |
\(561\) |
\(911\) |
\(1093\) |
\(1249\) |
| \(\chi(n)\) |
\(1 - \beta_{9}\) |
\(1\) |
\(1\) |
\(1\) |
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 \(T_{3}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1456, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{12} \) |
| $3$ |
\( 16 + 80 T + 300 T^{2} + 516 T^{3} + 709 T^{4} + 390 T^{5} + 287 T^{6} + 42 T^{7} + 96 T^{8} + 4 T^{9} + 11 T^{10} + T^{12} \) |
| $5$ |
\( 3481 + 16148 T^{2} + 13040 T^{4} + 4146 T^{6} + 600 T^{8} + 40 T^{10} + T^{12} \) |
| $7$ |
\( ( 1 - T^{2} + T^{4} )^{3} \) |
| $11$ |
\( 256 - 768 T - 80 T^{2} + 2544 T^{3} + 2025 T^{4} - 1494 T^{5} - 771 T^{6} + 522 T^{7} + 248 T^{8} - 114 T^{9} - 7 T^{10} + 6 T^{11} + T^{12} \) |
| $13$ |
\( 4826809 - 1485172 T + 599781 T^{2} - 70304 T^{3} - 23998 T^{4} + 12012 T^{5} - 6587 T^{6} + 924 T^{7} - 142 T^{8} - 32 T^{9} + 21 T^{10} - 4 T^{11} + T^{12} \) |
| $17$ |
\( 241081 - 109984 T + 132173 T^{2} - 21512 T^{3} + 31018 T^{4} - 2576 T^{5} + 5229 T^{6} + 148 T^{7} + 514 T^{8} + 36 T^{9} + 37 T^{10} + 4 T^{11} + T^{12} \) |
| $19$ |
\( 55696 + 138768 T + 137196 T^{2} + 54684 T^{3} - 4499 T^{4} - 9486 T^{5} + 299 T^{6} + 2370 T^{7} + 748 T^{8} - 29 T^{10} + T^{12} \) |
| $23$ |
\( 38539264 - 9138176 T + 12198912 T^{2} - 5170176 T^{3} + 3630592 T^{4} - 1115904 T^{5} + 332096 T^{6} - 52416 T^{7} + 9312 T^{8} - 976 T^{9} + 164 T^{10} - 12 T^{11} + T^{12} \) |
| $29$ |
\( 10042561 - 1064784 T + 4587524 T^{2} - 3112876 T^{3} + 2323356 T^{4} - 854112 T^{5} + 261878 T^{6} - 47832 T^{7} + 7876 T^{8} - 780 T^{9} + 108 T^{10} - 8 T^{11} + T^{12} \) |
| $31$ |
\( 913936 + 1285560 T^{2} + 568225 T^{4} + 96896 T^{6} + 5854 T^{8} + 136 T^{10} + T^{12} \) |
| $37$ |
\( 1755945216 - 633588480 T - 260200512 T^{2} + 121383360 T^{3} + 65323584 T^{4} + 8644320 T^{5} - 396036 T^{6} - 159840 T^{7} + 8109 T^{8} + 5670 T^{9} + 723 T^{10} + 42 T^{11} + T^{12} \) |
| $41$ |
\( 884705536 - 421175040 T - 8238656 T^{2} + 35739840 T^{3} - 872384 T^{4} - 2742240 T^{5} + 452252 T^{6} + 35760 T^{7} - 11651 T^{8} - 450 T^{9} + 315 T^{10} - 30 T^{11} + T^{12} \) |
| $43$ |
\( 2408704 - 5860352 T + 10500784 T^{2} - 8862336 T^{3} + 5690569 T^{4} - 1037954 T^{5} + 282645 T^{6} - 3650 T^{7} + 9640 T^{8} - 38 T^{9} + 113 T^{10} + 2 T^{11} + T^{12} \) |
| $47$ |
\( 9461776 + 47561752 T^{2} + 10113609 T^{4} + 722232 T^{6} + 21782 T^{8} + 272 T^{10} + T^{12} \) |
| $53$ |
\( ( -2339 + 7302 T - 3353 T^{2} - 700 T^{3} + 91 T^{4} + 22 T^{5} + T^{6} )^{2} \) |
| $59$ |
\( 4571923456 + 3919023360 T + 1190986848 T^{2} + 61031880 T^{3} - 34227911 T^{4} - 3548430 T^{5} + 1035110 T^{6} + 185580 T^{7} - 2237 T^{8} - 1980 T^{9} - 2 T^{10} + 18 T^{11} + T^{12} \) |
| $61$ |
\( 5607424 - 3788800 T + 7030784 T^{2} - 3685376 T^{3} + 6036160 T^{4} - 2984960 T^{5} + 1823136 T^{6} - 177656 T^{7} + 29281 T^{8} - 1614 T^{9} + 283 T^{10} - 14 T^{11} + T^{12} \) |
| $67$ |
\( 613651984 + 1248211536 T + 790406444 T^{2} - 113725716 T^{3} - 37845559 T^{4} + 4849356 T^{5} + 1364502 T^{6} - 185988 T^{7} - 11949 T^{8} + 2352 T^{9} + 94 T^{10} - 24 T^{11} + T^{12} \) |
| $71$ |
\( 46895104 - 26296320 T - 3083264 T^{2} + 4485120 T^{3} + 367104 T^{4} - 620544 T^{5} + 82752 T^{6} + 17280 T^{7} - 3808 T^{8} - 480 T^{9} + 212 T^{10} - 24 T^{11} + T^{12} \) |
| $73$ |
\( 1386221824 + 513361280 T^{2} + 55965104 T^{4} + 2238456 T^{6} + 40473 T^{8} + 334 T^{10} + T^{12} \) |
| $79$ |
\( ( -512 - 1664 T - 1584 T^{2} - 192 T^{3} + 212 T^{4} - 28 T^{5} + T^{6} )^{2} \) |
| $83$ |
\( 141324544 + 454322976 T^{2} + 48190849 T^{4} + 1905008 T^{6} + 35086 T^{8} + 304 T^{10} + T^{12} \) |
| $89$ |
\( 1834580224 + 1271596416 T - 422744080 T^{2} - 496650552 T^{3} + 291555473 T^{4} - 41341974 T^{5} - 2811261 T^{6} + 792798 T^{7} + 66288 T^{8} - 3660 T^{9} - 257 T^{10} + 12 T^{11} + T^{12} \) |
| $97$ |
\( 53465344 + 190755456 T + 211235504 T^{2} - 55750056 T^{3} - 14421439 T^{4} + 4289964 T^{5} + 938031 T^{6} - 349092 T^{7} + 28032 T^{8} + 1110 T^{9} - 173 T^{10} - 6 T^{11} + T^{12} \) |
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