Newspace parameters
| Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 462.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(74.0973247536\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 7217x^{2} - 11004x + 4670860 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 7217x^{2} - 11004x + 4670860 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{3} - 19\nu^{2} - 47324\nu + 355 ) / 4365 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 548\nu^{2} + 3733\nu - 1973570 ) / 8730 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 16\beta_{3} + \beta_{2} + 4\beta _1 + 3617 \)
|
| \(\nu^{3}\) | \(=\) |
\( 38\beta_{3} + 548\beta_{2} + 5925\beta _1 + 8546 \)
|
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 |
|
4.00000 | −9.00000 | 16.0000 | −90.6390 | −36.0000 | 49.0000 | 64.0000 | 81.0000 | −362.556 | ||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −9.00000 | 16.0000 | −38.7884 | −36.0000 | 49.0000 | 64.0000 | 81.0000 | −155.154 | |||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −9.00000 | 16.0000 | 14.8829 | −36.0000 | 49.0000 | 64.0000 | 81.0000 | 59.5316 | |||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | −9.00000 | 16.0000 | 70.5445 | −36.0000 | 49.0000 | 64.0000 | 81.0000 | 282.178 | |||||||||||||||||||||||||||||||
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 | 462.6.a.r | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.6.a.r | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 44T_{5}^{3} - 6491T_{5}^{2} - 164454T_{5} + 3691200 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(462))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{4} \)
|
| $3$ |
\( (T + 9)^{4} \)
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| $5$ |
\( T^{4} + 44 T^{3} + \cdots + 3691200 \)
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| $7$ |
\( (T - 49)^{4} \)
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| $11$ |
\( (T + 121)^{4} \)
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| $13$ |
\( T^{4} + \cdots + 12792848460 \)
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| $17$ |
\( T^{4} + \cdots + 3842514218304 \)
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| $19$ |
\( T^{4} + \cdots - 39448858580200 \)
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| $23$ |
\( T^{4} + \cdots + 24139786340352 \)
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| $29$ |
\( T^{4} + \cdots + 232598643231996 \)
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| $31$ |
\( T^{4} + \cdots + 78449516806080 \)
|
| $37$ |
\( T^{4} + \cdots - 36\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots - 23\!\cdots\!80 \)
|
| $43$ |
\( T^{4} + \cdots + 48\!\cdots\!12 \)
|
| $47$ |
\( T^{4} + \cdots + 30\!\cdots\!68 \)
|
| $53$ |
\( T^{4} + \cdots - 39\!\cdots\!80 \)
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| $59$ |
\( T^{4} + \cdots - 16\!\cdots\!00 \)
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| $61$ |
\( T^{4} + \cdots - 19\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots - 17\!\cdots\!92 \)
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| $71$ |
\( T^{4} + \cdots - 12\!\cdots\!80 \)
|
| $73$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 26\!\cdots\!48 \)
|
| $83$ |
\( T^{4} + \cdots + 77\!\cdots\!84 \)
|
| $89$ |
\( T^{4} + \cdots - 12\!\cdots\!32 \)
|
| $97$ |
\( T^{4} + \cdots - 19\!\cdots\!44 \)
|
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