Newspace parameters
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(146.270043669\) |
| 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: |
\( x^{4} - 2x^{3} - 431x^{2} - 916x + 26148 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 456) |
| 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
\( x^{4} - 2x^{3} - 431x^{2} - 916x + 26148 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 5\nu - 214 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 11\nu^{2} - 262\nu + 1172 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + 5\beta _1 + 214 \)
|
| \(\nu^{3}\) | \(=\) |
\( 10\beta_{3} + 22\beta_{2} + 317\beta _1 + 1182 \)
|
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 |
|
0 | 9.00000 | 0 | −104.744 | 0 | 21.6551 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | −15.6754 | 0 | 62.1515 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 9.00000 | 0 | 36.1596 | 0 | −186.762 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.00000 | 0 | 64.2602 | 0 | 32.9558 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 912.6.a.q | 4 | |
| 4.b | odd | 2 | 1 | 456.6.a.a | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 456.6.a.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 912.6.a.q | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 20T_{5}^{3} - 8127T_{5}^{2} + 114930T_{5} + 3815176 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(912))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 9)^{4} \)
|
| $5$ |
\( T^{4} + 20 T^{3} + \cdots + 3815176 \)
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| $7$ |
\( T^{4} + 70 T^{3} + \cdots - 8283840 \)
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| $11$ |
\( T^{4} + 164 T^{3} + \cdots + 750470640 \)
|
| $13$ |
\( T^{4} + \cdots - 25329362368 \)
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| $17$ |
\( T^{4} + \cdots - 153126627452 \)
|
| $19$ |
\( (T - 361)^{4} \)
|
| $23$ |
\( T^{4} + \cdots - 4653295992000 \)
|
| $29$ |
\( T^{4} + \cdots - 217238809282016 \)
|
| $31$ |
\( T^{4} + \cdots - 142841723952960 \)
|
| $37$ |
\( T^{4} + \cdots + 27582136294400 \)
|
| $41$ |
\( T^{4} + \cdots - 8054186228736 \)
|
| $43$ |
\( T^{4} + \cdots - 49\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 50\!\cdots\!56 \)
|
| $53$ |
\( T^{4} + \cdots + 52\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots - 34\!\cdots\!68 \)
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| $61$ |
\( T^{4} + \cdots - 279016297885124 \)
|
| $67$ |
\( T^{4} + \cdots + 95\!\cdots\!08 \)
|
| $71$ |
\( T^{4} + \cdots - 39\!\cdots\!44 \)
|
| $73$ |
\( T^{4} + \cdots - 99\!\cdots\!84 \)
|
| $79$ |
\( T^{4} + \cdots - 27\!\cdots\!60 \)
|
| $83$ |
\( T^{4} + \cdots + 13\!\cdots\!64 \)
|
| $89$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 21\!\cdots\!00 \)
|
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