Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) 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 \(x^{7} - 2 x^{6} - 44 x^{5} + 74 x^{4} + 479 x^{3} - 460 x^{2} - 1200 x + 288\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \( \nu^{2} - 13 \) |
| \(\beta_{3}\) | \(=\) | \( \nu^{3} - 23 \nu - 1 \) |
| \(\beta_{4}\) | \(=\) | \((\)\( \nu^{6} - \nu^{5} - 39 \nu^{4} + 35 \nu^{3} + 310 \nu^{2} - 114 \nu - 240 \)\()/12\) |
| \(\beta_{5}\) | \(=\) | \((\)\( -\nu^{6} + 4 \nu^{5} + 42 \nu^{4} - 140 \nu^{3} - 397 \nu^{2} + 756 \nu + 672 \)\()/24\) |
| \(\beta_{6}\) | \(=\) | \((\)\( \nu^{6} + 2 \nu^{5} - 48 \nu^{4} - 82 \nu^{3} + 595 \nu^{2} + 732 \nu - 1104 \)\()/24\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{2} + 13\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{3} + 23 \beta_{1} + 1\) |
| \(\nu^{4}\) | \(=\) | \(-2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 31 \beta_{2} - 6 \beta_{1} + 294\) |
| \(\nu^{5}\) | \(=\) | \(2 \beta_{6} + 6 \beta_{5} + 2 \beta_{4} + 36 \beta_{3} - 2 \beta_{2} + 597 \beta_{1} - 26\) |
| \(\nu^{6}\) | \(=\) | \(-76 \beta_{6} + 84 \beta_{5} + 92 \beta_{4} - 38 \beta_{3} + 897 \beta_{2} - 328 \beta_{1} + 7615\) |
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.
| \( p \) |
Sign
|
| \(3\) |
\(-1\) |
| \(5\) |
\(-1\) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 2 T_{2}^{6} - 44 T_{2}^{5} + 74 T_{2}^{4} + 479 T_{2}^{3} - 460 T_{2}^{2} - 1200 T_{2} + 288 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( 288 - 1200 T - 460 T^{2} + 479 T^{3} + 74 T^{4} - 44 T^{5} - 2 T^{6} + T^{7} \) |
| $3$ |
\( T^{7} \) |
| $5$ |
\( ( -5 + T )^{7} \) |
| $7$ |
\( 210450744 - 68052735 T - 4867848 T^{2} + 650337 T^{3} + 26142 T^{4} - 1571 T^{5} - 22 T^{6} + T^{7} \) |
| $11$ |
\( -20019444768 - 2234576400 T + 4473656 T^{2} + 5647316 T^{3} + 46184 T^{4} - 4166 T^{5} - 23 T^{6} + T^{7} \) |
| $13$ |
\( 159234392576 + 54441888 T - 558177424 T^{2} + 1597000 T^{3} + 436772 T^{4} - 3188 T^{5} - 96 T^{6} + T^{7} \) |
| $17$ |
\( 60588009792 + 7871650176 T - 2499369808 T^{2} - 27251020 T^{3} + 2125940 T^{4} - 9644 T^{5} - 161 T^{6} + T^{7} \) |
| $19$ |
\( -1377989598400 + 1417080000 T + 6594215888 T^{2} - 224819660 T^{3} + 1854320 T^{4} + 15442 T^{5} - 279 T^{6} + T^{7} \) |
| $23$ |
\( -4061042235738 + 319448690109 T - 7434227790 T^{2} + 899433 T^{3} + 1879776 T^{4} - 14439 T^{5} - 96 T^{6} + T^{7} \) |
| $29$ |
\( -12128150971026 - 2048542254411 T - 92897749316 T^{2} - 1753720489 T^{3} - 13795250 T^{4} - 16301 T^{5} + 296 T^{6} + T^{7} \) |
| $31$ |
\( 29481260805504 + 2691313729344 T - 21117961440 T^{2} - 1452653784 T^{3} + 22032132 T^{4} - 69584 T^{5} - 244 T^{6} + T^{7} \) |
| $37$ |
\( -83646911884544 - 8022049775936 T - 196405787520 T^{2} + 585288704 T^{3} + 34349204 T^{4} - 86580 T^{5} - 404 T^{6} + T^{7} \) |
| $41$ |
\( 1524782419390161 + 58383942911499 T + 692598630925 T^{2} + 1021803995 T^{3} - 31456877 T^{4} - 161351 T^{5} + 47 T^{6} + T^{7} \) |
| $43$ |
\( -1328908142680384 + 146770517416224 T - 1523916256048 T^{2} - 3470994956 T^{3} + 73393160 T^{4} - 94898 T^{5} - 525 T^{6} + T^{7} \) |
| $47$ |
\( 6505162038618372 + 292893869594349 T + 4217212042898 T^{2} + 20820619937 T^{3} - 8230474 T^{4} - 267791 T^{5} - 164 T^{6} + T^{7} \) |
| $53$ |
\( 111107669935704192 + 3313074698297664 T - 34008897116512 T^{2} + 39458567216 T^{3} + 279547832 T^{4} - 500516 T^{5} - 506 T^{6} + T^{7} \) |
| $59$ |
\( -595651050751670208 - 38178523175987136 T + 14741411894096 T^{2} + 375330469568 T^{3} - 71749996 T^{4} - 1106312 T^{5} + 85 T^{6} + T^{7} \) |
| $61$ |
\( 1223316036378446 - 192599375971827 T + 7109002299836 T^{2} - 97105034237 T^{3} + 471979010 T^{4} - 451073 T^{5} - 828 T^{6} + T^{7} \) |
| $67$ |
\( 29225224702431603 + 686210632919667 T - 11724435478647 T^{2} - 9567655617 T^{3} + 273237249 T^{4} - 88235 T^{5} - 1093 T^{6} + T^{7} \) |
| $71$ |
\( 329554038570193728 - 2740127166757248 T - 99927513780416 T^{2} + 322939201712 T^{3} + 436205092 T^{4} - 1229012 T^{5} - 328 T^{6} + T^{7} \) |
| $73$ |
\( 82873748372160512 + 2981349592535040 T + 20542100801024 T^{2} - 128439358016 T^{3} - 117820816 T^{4} + 1327444 T^{5} - 2085 T^{6} + T^{7} \) |
| $79$ |
\( 16163225966939188608 + 91608055797302208 T - 217501872207264 T^{2} - 941918607696 T^{3} + 2167211400 T^{4} + 15124 T^{5} - 2110 T^{6} + T^{7} \) |
| $83$ |
\( -6624473796314288172 + 68135962294721277 T - 175355495435376 T^{2} - 206497270215 T^{3} + 1234542222 T^{4} - 720135 T^{5} - 1290 T^{6} + T^{7} \) |
| $89$ |
\( \)\(32\!\cdots\!50\)\( + 277911272642123745 T - 3417793050944628 T^{2} - 8680332870945 T^{3} - 6568870986 T^{4} + 788319 T^{5} + 3048 T^{6} + T^{7} \) |
| $97$ |
\( \)\(47\!\cdots\!76\)\( + 331725638028814336 T - 3031047860438448 T^{2} + 320540186240 T^{3} + 4996509932 T^{4} - 2304468 T^{5} - 1787 T^{6} + T^{7} \) |
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