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: | \(0\) |
| 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: |
\( x^{3} - 9231x - 33826 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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
\( x^{3} - 9231x - 33826 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 5\nu - 6154 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + 5\beta _1 + 6154 \)
|
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 | −499.302 | 216.000 | 343.000 | −512.000 | 729.000 | 3994.42 | |||||||||||||||||||||||||||
| 1.2 | −8.00000 | −27.0000 | 64.0000 | 8.34873 | 216.000 | 343.000 | −512.000 | 729.000 | −66.7898 | ||||||||||||||||||||||||||||
| 1.3 | −8.00000 | −27.0000 | 64.0000 | 460.953 | 216.000 | 343.000 | −512.000 | 729.000 | −3687.63 | ||||||||||||||||||||||||||||
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.b | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.8.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 30T_{5}^{2} - 230475T_{5} + 1921500 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(462))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{3} \)
|
| $3$ |
\( (T + 27)^{3} \)
|
| $5$ |
\( T^{3} + 30 T^{2} + \cdots + 1921500 \)
|
| $7$ |
\( (T - 343)^{3} \)
|
| $11$ |
\( (T - 1331)^{3} \)
|
| $13$ |
\( T^{3} + \cdots - 4710518438 \)
|
| $17$ |
\( T^{3} + \cdots - 6661815655392 \)
|
| $19$ |
\( T^{3} + \cdots - 15901728389750 \)
|
| $23$ |
\( T^{3} + \cdots - 9223806216240 \)
|
| $29$ |
\( T^{3} + \cdots + 303067702345554 \)
|
| $31$ |
\( T^{3} + \cdots + 21\!\cdots\!08 \)
|
| $37$ |
\( T^{3} + \cdots - 320495136554 \)
|
| $41$ |
\( T^{3} + \cdots + 52\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots + 14\!\cdots\!56 \)
|
| $47$ |
\( T^{3} + \cdots - 47\!\cdots\!58 \)
|
| $53$ |
\( T^{3} + \cdots - 28\!\cdots\!48 \)
|
| $59$ |
\( T^{3} + \cdots - 15\!\cdots\!48 \)
|
| $61$ |
\( T^{3} + \cdots - 73\!\cdots\!44 \)
|
| $67$ |
\( T^{3} + \cdots + 18\!\cdots\!16 \)
|
| $71$ |
\( T^{3} + \cdots + 17\!\cdots\!88 \)
|
| $73$ |
\( T^{3} + \cdots + 70\!\cdots\!32 \)
|
| $79$ |
\( T^{3} + \cdots + 15\!\cdots\!44 \)
|
| $83$ |
\( T^{3} + \cdots - 37\!\cdots\!16 \)
|
| $89$ |
\( T^{3} + \cdots + 44\!\cdots\!48 \)
|
| $97$ |
\( T^{3} + \cdots + 25\!\cdots\!96 \)
|
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