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: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 27x^{2} - 13x + 88 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 57) |
| 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
\( x^{4} - x^{3} - 27x^{2} - 13x + 88 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 19\nu + 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 6\nu^{2} + 11\nu - 56 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + \beta _1 + 28 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{3} + 12\beta_{2} + 23\beta _1 + 123 ) / 4 \)
|
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 | −16.8076 | 0 | −27.3697 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −10.4211 | 0 | −5.73511 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | 9.72745 | 0 | 21.1323 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −3.00000 | 0 | 11.5013 | 0 | −26.0275 | 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.t | 4 | |
| 4.b | odd | 2 | 1 | 57.4.a.d | ✓ | 4 | |
| 12.b | even | 2 | 1 | 171.4.a.h | 4 | ||
| 20.d | odd | 2 | 1 | 1425.4.a.k | 4 | ||
| 76.d | even | 2 | 1 | 1083.4.a.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.4.a.d | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 171.4.a.h | 4 | 12.b | even | 2 | 1 | ||
| 912.4.a.t | 4 | 1.a | even | 1 | 1 | trivial | |
| 1083.4.a.d | 4 | 76.d | even | 2 | 1 | ||
| 1425.4.a.k | 4 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(912))\):
|
\( T_{5}^{4} + 6T_{5}^{3} - 291T_{5}^{2} - 672T_{5} + 19596 \)
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\( T_{7}^{4} + 38T_{7}^{3} - 231T_{7}^{2} - 17440T_{7} - 86336 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( T^{4} + 6 T^{3} + \cdots + 19596 \)
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| $7$ |
\( T^{4} + 38 T^{3} + \cdots - 86336 \)
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| $11$ |
\( T^{4} - 3555 T^{2} + \cdots + 611496 \)
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| $13$ |
\( T^{4} - 68 T^{3} + \cdots + 1448560 \)
|
| $17$ |
\( T^{4} - 30 T^{3} + \cdots + 12247500 \)
|
| $19$ |
\( (T + 19)^{4} \)
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| $23$ |
\( T^{4} - 246 T^{3} + \cdots - 23258112 \)
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| $29$ |
\( T^{4} - 6 T^{3} + \cdots + 266996640 \)
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| $31$ |
\( T^{4} + 188 T^{3} + \cdots + 26116096 \)
|
| $37$ |
\( T^{4} + \cdots - 2908078592 \)
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| $41$ |
\( T^{4} + \cdots + 5830938624 \)
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| $43$ |
\( T^{4} + \cdots + 4295336224 \)
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| $47$ |
\( T^{4} + \cdots - 2545654992 \)
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| $53$ |
\( T^{4} + \cdots + 8557525344 \)
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| $59$ |
\( T^{4} + \cdots - 15363391488 \)
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| $61$ |
\( T^{4} + \cdots + 1866482620 \)
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| $67$ |
\( T^{4} + \cdots + 108714503680 \)
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| $71$ |
\( T^{4} + \cdots + 94181583360 \)
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| $73$ |
\( T^{4} + \cdots - 61797409892 \)
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| $79$ |
\( T^{4} + \cdots + 56420337664 \)
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| $83$ |
\( T^{4} + \cdots + 526934353536 \)
|
| $89$ |
\( T^{4} + \cdots - 181099849920 \)
|
| $97$ |
\( T^{4} + \cdots - 73056760016 \)
|
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