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 \(x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\( 406 \nu^{7} - 714 \nu^{6} + 747 \nu^{5} - 1896 \nu^{4} + 2103 \nu^{3} + 4949 \nu^{2} + 1065 \nu - 7800 \)\()/1355\) |
| \(\beta_{3}\) | \(=\) | \((\)\( 420 \nu^{7} - 776 \nu^{6} + 698 \nu^{5} - 1924 \nu^{4} + 2297 \nu^{3} + 5129 \nu^{2} + 1055 \nu - 10265 \)\()/1355\) |
| \(\beta_{4}\) | \(=\) | \((\)\( 728 \nu^{7} - 1327 \nu^{6} + 1246 \nu^{5} - 3353 \nu^{4} + 3584 \nu^{3} + 8547 \nu^{2} + 2190 \nu - 15715 \)\()/1355\) |
| \(\beta_{5}\) | \(=\) | \((\)\( -857 \nu^{7} + 1666 \nu^{6} - 1743 \nu^{5} + 4424 \nu^{4} - 4907 \nu^{3} - 9470 \nu^{2} - 2485 \nu + 18200 \)\()/1355\) |
| \(\beta_{6}\) | \(=\) | \((\)\( 891 \nu^{7} - 1623 \nu^{6} + 1624 \nu^{5} - 4492 \nu^{4} + 4991 \nu^{3} + 9520 \nu^{2} + 3235 \nu - 18960 \)\()/1355\) |
| \(\beta_{7}\) | \(=\) | \((\)\( 955 \nu^{7} - 1829 \nu^{6} + 1942 \nu^{5} - 4891 \nu^{4} + 5723 \nu^{3} + 9646 \nu^{2} + 2415 \nu - 20550 \)\()/1355\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(-\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2}\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{7} + \beta_{5} - 3 \beta_{4} + 4 \beta_{3} + \beta_{2} + \beta_{1} + 3\) |
| \(\nu^{4}\) | \(=\) | \(5 \beta_{7} - 5 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 2\) |
| \(\nu^{5}\) | \(=\) | \(4 \beta_{7} - 6 \beta_{6} - 7 \beta_{3} + 11 \beta_{2} + 4 \beta_{1} - 13\) |
| \(\nu^{6}\) | \(=\) | \(7 \beta_{6} + 7 \beta_{5} - 8 \beta_{4} - 7 \beta_{3} + 21 \beta_{2} - 2 \beta_{1} - 21\) |
| \(\nu^{7}\) | \(=\) | \(23 \beta_{7} + 38 \beta_{5} + 23 \beta_{4} - 23 \beta_{3} + 12 \beta_{2}\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/125\mathbb{Z}\right)^\times\).
| \(n\) |
\(2\) |
| \(\chi(n)\) |
\(\beta_{2}\) |
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 \(T_{2}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(125, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( 1 + 10 T + 26 T^{2} - 10 T^{3} + T^{4} - 10 T^{5} + 11 T^{6} - 5 T^{7} + T^{8} \) |
| $3$ |
\( 16 - 80 T + 144 T^{2} - 110 T^{3} + 51 T^{4} - 15 T^{5} + 9 T^{6} - 5 T^{7} + T^{8} \) |
| $5$ |
\( T^{8} \) |
| $7$ |
\( 16 + 116 T^{2} + 121 T^{4} + 21 T^{6} + T^{8} \) |
| $11$ |
\( ( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} )^{2} \) |
| $13$ |
\( 1 + 5 T + 4 T^{2} - 5 T^{3} + 21 T^{4} + 5 T^{5} + 4 T^{6} - 5 T^{7} + T^{8} \) |
| $17$ |
\( 1936 - 880 T - 784 T^{2} + 550 T^{3} + 31 T^{4} - 125 T^{5} + 56 T^{6} - 10 T^{7} + T^{8} \) |
| $19$ |
\( 400 + 200 T + 400 T^{2} - 100 T^{3} - 15 T^{4} + 40 T^{5} + 30 T^{6} + 5 T^{7} + T^{8} \) |
| $23$ |
\( 256 + 320 T - 16 T^{2} + 60 T^{3} + 241 T^{4} + 15 T^{5} - T^{6} + 5 T^{7} + T^{8} \) |
| $29$ |
\( 483025 + 142475 T + 49350 T^{2} + 4525 T^{3} + 485 T^{4} - 5 T^{5} + 30 T^{6} + 5 T^{7} + T^{8} \) |
| $31$ |
\( 1936 + 23496 T + 110532 T^{2} + 31178 T^{3} + 6855 T^{4} + 917 T^{5} + 117 T^{6} + 9 T^{7} + T^{8} \) |
| $37$ |
\( 116281 + 192665 T + 141631 T^{2} + 61170 T^{3} + 17321 T^{4} + 3270 T^{5} + 406 T^{6} + 30 T^{7} + T^{8} \) |
| $41$ |
\( 13456 - 1624 T + 9492 T^{2} - 7622 T^{3} + 2655 T^{4} + 457 T^{5} + 52 T^{6} + 4 T^{7} + T^{8} \) |
| $43$ |
\( 246016 + 56784 T^{2} + 4421 T^{4} + 129 T^{6} + T^{8} \) |
| $47$ |
\( 65536 + 61440 T - 4864 T^{2} - 9840 T^{3} + 4101 T^{4} - 615 T^{5} + 16 T^{6} + T^{8} \) |
| $53$ |
\( 8755681 - 4157395 T + 722619 T^{2} - 29590 T^{3} - 8079 T^{4} + 1290 T^{5} - 6 T^{6} - 10 T^{7} + T^{8} \) |
| $59$ |
\( 4080400 + 1333200 T + 407000 T^{2} + 54150 T^{3} + 5635 T^{4} + 15 T^{5} + T^{8} \) |
| $61$ |
\( 116281 - 59334 T + 139032 T^{2} + 12978 T^{3} + 16405 T^{4} - 1068 T^{5} - 43 T^{6} + 9 T^{7} + T^{8} \) |
| $67$ |
\( 246016 - 198400 T + 74816 T^{2} - 6080 T^{3} - 2384 T^{4} - 80 T^{5} + 116 T^{6} + 20 T^{7} + T^{8} \) |
| $71$ |
\( 24245776 - 5889104 T + 966712 T^{2} - 53922 T^{3} + 3455 T^{4} + 297 T^{5} + 142 T^{6} - 6 T^{7} + T^{8} \) |
| $73$ |
\( 1 + 4 T^{2} - 30 T^{3} + 91 T^{4} - 120 T^{5} + 49 T^{6} + 15 T^{7} + T^{8} \) |
| $79$ |
\( 33408400 + 4913000 T + 1416100 T^{2} + 79050 T^{3} + 12185 T^{4} + 600 T^{5} + 100 T^{6} - 15 T^{7} + T^{8} \) |
| $83$ |
\( 99856 - 284400 T + 329344 T^{2} - 199950 T^{3} + 70651 T^{4} - 11175 T^{5} + 949 T^{6} - 45 T^{7} + T^{8} \) |
| $89$ |
\( 1392400 + 1640200 T + 888600 T^{2} + 258800 T^{3} + 47985 T^{4} + 5890 T^{5} + 520 T^{6} + 25 T^{7} + T^{8} \) |
| $97$ |
\( 301334881 - 63360350 T + 5529361 T^{2} - 971040 T^{3} + 213086 T^{4} - 24990 T^{5} + 1636 T^{6} - 60 T^{7} + T^{8} \) |
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