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 \(x^{3} - x^{2} - 26 x - 22\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \( \nu^{2} - 2 \nu - 17 \) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{2} + 2 \beta_{1} + 17\) |
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
|
| \(3\) |
\(-1\) |
| \(5\) |
\(1\) |
| \(11\) |
\(-1\) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 2 T_{2}^{2} - 25 T_{2} - 48 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(165))\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( -48 - 25 T + 2 T^{2} + T^{3} \) |
| $3$ |
\( ( -3 + T )^{3} \) |
| $5$ |
\( ( 5 + T )^{3} \) |
| $7$ |
\( 3448 - 604 T - 10 T^{2} + T^{3} \) |
| $11$ |
\( ( -11 + T )^{3} \) |
| $13$ |
\( -37216 + 3712 T - 114 T^{2} + T^{3} \) |
| $17$ |
\( 8448 + 2792 T + 104 T^{2} + T^{3} \) |
| $19$ |
\( 65520 - 11496 T + 58 T^{2} + T^{3} \) |
| $23$ |
\( 148224 - 10736 T - 120 T^{2} + T^{3} \) |
| $29$ |
\( -629760 - 14308 T + 220 T^{2} + T^{3} \) |
| $31$ |
\( 9589248 - 38592 T - 248 T^{2} + T^{3} \) |
| $37$ |
\( -18607336 + 223548 T - 838 T^{2} + T^{3} \) |
| $41$ |
\( -3013632 - 106596 T - 156 T^{2} + T^{3} \) |
| $43$ |
\( 1445400 - 44940 T - 122 T^{2} + T^{3} \) |
| $47$ |
\( -4372224 + 82192 T - 504 T^{2} + T^{3} \) |
| $53$ |
\( -3654264 - 222068 T - 282 T^{2} + T^{3} \) |
| $59$ |
\( -1206720 + 57584 T - 548 T^{2} + T^{3} \) |
| $61$ |
\( 342344792 - 681332 T - 414 T^{2} + T^{3} \) |
| $67$ |
\( -8135552 - 206752 T + 428 T^{2} + T^{3} \) |
| $71$ |
\( -2867712 + 97888 T + 912 T^{2} + T^{3} \) |
| $73$ |
\( 26458592 - 100544 T - 618 T^{2} + T^{3} \) |
| $79$ |
\( -88503440 - 161224 T + 542 T^{2} + T^{3} \) |
| $83$ |
\( -434328048 - 1091340 T + T^{3} \) |
| $89$ |
\( 1941629400 - 2118532 T - 790 T^{2} + T^{3} \) |
| $97$ |
\( 98075336 + 967212 T - 2074 T^{2} + T^{3} \) |
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