Newspace parameters
| Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 462.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(144.321881774\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 2357x - 36078 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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
\( x^{3} - x^{2} - 2357x - 36078 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 23\nu - 1563 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6\beta_{2} + 23\beta _1 + 3149 ) / 2 \)
|
Embeddings
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−8.00000 | 27.0000 | 64.0000 | −328.406 | −216.000 | 343.000 | −512.000 | 729.000 | 2627.25 | |||||||||||||||||||||||||||
| 1.2 | −8.00000 | 27.0000 | 64.0000 | −87.8043 | −216.000 | 343.000 | −512.000 | 729.000 | 702.434 | ||||||||||||||||||||||||||||
| 1.3 | −8.00000 | 27.0000 | 64.0000 | 318.210 | −216.000 | 343.000 | −512.000 | 729.000 | −2545.68 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 462.8.a.c | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.8.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 98T_{5}^{2} - 103607T_{5} - 9175740 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(462))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{3} \)
|
| $3$ |
\( (T - 27)^{3} \)
|
| $5$ |
\( T^{3} + 98 T^{2} + \cdots - 9175740 \)
|
| $7$ |
\( (T - 343)^{3} \)
|
| $11$ |
\( (T - 1331)^{3} \)
|
| $13$ |
\( T^{3} + \cdots - 38007001446 \)
|
| $17$ |
\( T^{3} + \cdots - 2539109628000 \)
|
| $19$ |
\( T^{3} + \cdots - 5755052138374 \)
|
| $23$ |
\( T^{3} + \cdots - 4001875923696 \)
|
| $29$ |
\( T^{3} + \cdots - 21\!\cdots\!38 \)
|
| $31$ |
\( T^{3} + \cdots - 167490775353240 \)
|
| $37$ |
\( T^{3} + \cdots + 13\!\cdots\!82 \)
|
| $41$ |
\( T^{3} + \cdots + 11\!\cdots\!60 \)
|
| $43$ |
\( T^{3} + \cdots - 51\!\cdots\!56 \)
|
| $47$ |
\( T^{3} + \cdots + 33\!\cdots\!42 \)
|
| $53$ |
\( T^{3} + \cdots - 24\!\cdots\!04 \)
|
| $59$ |
\( T^{3} + \cdots + 45\!\cdots\!12 \)
|
| $61$ |
\( T^{3} + \cdots + 93\!\cdots\!88 \)
|
| $67$ |
\( T^{3} + \cdots - 74\!\cdots\!44 \)
|
| $71$ |
\( T^{3} + \cdots - 51\!\cdots\!28 \)
|
| $73$ |
\( T^{3} + \cdots - 15\!\cdots\!92 \)
|
| $79$ |
\( T^{3} + \cdots - 58\!\cdots\!56 \)
|
| $83$ |
\( T^{3} + \cdots - 36\!\cdots\!28 \)
|
| $89$ |
\( T^{3} + \cdots - 88\!\cdots\!24 \)
|
| $97$ |
\( T^{3} + \cdots - 21\!\cdots\!28 \)
|
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