Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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 \(x^{3} - x^{2} - 13 x + 1\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( \nu^{2} + 4 \nu - 11 \)\()/2\) |
| \(\beta_{2}\) | \(=\) | \((\)\( \nu^{2} - 2 \nu - 9 \)\()/2\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(-\beta_{2} + \beta_{1} + 1\)\()/3\) |
| \(\nu^{2}\) | \(=\) | \((\)\(4 \beta_{2} + 2 \beta_{1} + 29\)\()/3\) |
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 \( T_{2}^{3} + T_{2}^{2} - 17 T_{2} - 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( -3 - 17 T + T^{2} + T^{3} \) |
| $3$ |
\( T^{3} \) |
| $5$ |
\( ( -5 + T )^{3} \) |
| $7$ |
\( -5337 + 195 T + 43 T^{2} + T^{3} \) |
| $11$ |
\( -43548 - 2816 T - 14 T^{2} + T^{3} \) |
| $13$ |
\( 75364 - 2452 T - 40 T^{2} + T^{3} \) |
| $17$ |
\( 156324 + 8920 T + 166 T^{2} + T^{3} \) |
| $19$ |
\( 57316 + 7292 T + 164 T^{2} + T^{3} \) |
| $23$ |
\( 61209 - 4833 T - 171 T^{2} + T^{3} \) |
| $29$ |
\( -107067 + 27331 T + 335 T^{2} + T^{3} \) |
| $31$ |
\( -9860940 - 13932 T + 352 T^{2} + T^{3} \) |
| $37$ |
\( 3335284 + 24708 T - 402 T^{2} + T^{3} \) |
| $41$ |
\( 7228275 - 44597 T - 187 T^{2} + T^{3} \) |
| $43$ |
\( -15444524 + 9956 T + 602 T^{2} + T^{3} \) |
| $47$ |
\( -2487483 + 95281 T - 665 T^{2} + T^{3} \) |
| $53$ |
\( 3250536 + 106300 T + 730 T^{2} + T^{3} \) |
| $59$ |
\( -127375896 - 354644 T + 298 T^{2} + T^{3} \) |
| $61$ |
\( 55613497 + 589307 T + 1439 T^{2} + T^{3} \) |
| $67$ |
\( 208776159 + 1093677 T + 1849 T^{2} + T^{3} \) |
| $71$ |
\( 223775052 - 685460 T - 70 T^{2} + T^{3} \) |
| $73$ |
\( -134927744 - 372928 T + 368 T^{2} + T^{3} \) |
| $79$ |
\( -138322584 - 387924 T + 382 T^{2} + T^{3} \) |
| $83$ |
\( -955843821 - 1160973 T + 831 T^{2} + T^{3} \) |
| $89$ |
\( 125506395 + 238491 T - 1719 T^{2} + T^{3} \) |
| $97$ |
\( -16898264 - 67668 T + 282 T^{2} + T^{3} \) |
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