Newspace parameters
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(53.8097419252\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 37x^{2} + 90x + 32 \)
|
| 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
\( x^{4} - x^{3} - 37x^{2} + 90x + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu - 19 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 3\nu^{2} - 25\nu - 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 5\nu - 20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 5\beta _1 + 75 ) / 4 \)
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| \(\nu^{3}\) | \(=\) |
\( 7\beta_{3} + \beta_{2} - 10\beta _1 - 40 \)
|
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 | 3.00000 | 0 | −16.7637 | 0 | −15.5196 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | −3.47753 | 0 | 20.0805 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 3.00000 | 0 | 4.02542 | 0 | −10.9228 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.00000 | 0 | 20.2158 | 0 | 20.3619 | 0 | 9.00000 | 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 | 912.4.a.v | 4 | |
| 4.b | odd | 2 | 1 | 456.4.a.d | ✓ | 4 | |
| 12.b | even | 2 | 1 | 1368.4.a.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 456.4.a.d | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 912.4.a.v | 4 | 1.a | even | 1 | 1 | trivial | |
| 1368.4.a.h | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(912))\):
|
\( T_{5}^{4} - 4T_{5}^{3} - 351T_{5}^{2} + 234T_{5} + 4744 \)
|
|
\( T_{7}^{4} - 14T_{7}^{3} - 491T_{7}^{2} + 3956T_{7} + 69312 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( (T - 3)^{4} \)
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| $5$ |
\( T^{4} - 4 T^{3} + \cdots + 4744 \)
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| $7$ |
\( T^{4} - 14 T^{3} + \cdots + 69312 \)
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| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 1427568 \)
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| $13$ |
\( T^{4} - 34 T^{3} + \cdots - 50368 \)
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| $17$ |
\( T^{4} - 6 T^{3} + \cdots + 3579172 \)
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| $19$ |
\( (T - 19)^{4} \)
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| $23$ |
\( T^{4} - 28 T^{3} + \cdots - 16129728 \)
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| $29$ |
\( T^{4} - 122 T^{3} + \cdots - 140199008 \)
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| $31$ |
\( T^{4} - 406 T^{3} + \cdots - 504207936 \)
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| $37$ |
\( T^{4} + 298 T^{3} + \cdots - 401007616 \)
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| $41$ |
\( T^{4} + 410 T^{3} + \cdots - 520350720 \)
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| $43$ |
\( T^{4} + \cdots - 19506717088 \)
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| $47$ |
\( T^{4} - 398 T^{3} + \cdots - 1107396 \)
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| $53$ |
\( T^{4} + 186 T^{3} + \cdots + 471097888 \)
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| $59$ |
\( T^{4} + \cdots - 22500689920 \)
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| $61$ |
\( T^{4} + \cdots + 5246148412 \)
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| $67$ |
\( T^{4} + \cdots + 38301585408 \)
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| $71$ |
\( T^{4} + \cdots - 15379571712 \)
|
| $73$ |
\( T^{4} + \cdots + 217739666388 \)
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| $79$ |
\( T^{4} + \cdots - 252780395008 \)
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| $83$ |
\( T^{4} + \cdots - 31109315968 \)
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| $89$ |
\( T^{4} + \cdots + 538261932672 \)
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| $97$ |
\( T^{4} + \cdots + 699514334000 \)
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