Newspace parameters
| Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 588.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(183.682394985\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 655x^{2} - 5704x - 84 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{3}\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 84) |
| 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
\( x^{4} - 2x^{3} - 655x^{2} - 5704x - 84 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 5\nu^{3} - 84\nu^{2} - 1805\nu + 2184 ) / 21 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 42\nu^{2} + 487\nu - 8778 ) / 42 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} - 21\nu^{2} - 1100\nu - 3073 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{3} + 6\beta_{2} + 3\beta _1 + 64 ) / 126 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{3} + 90\beta_{2} + 3\beta _1 + 20693 ) / 63 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -554\beta_{3} + 5190\beta_{2} + 1713\beta _1 + 663352 ) / 126 \)
|
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 | −27.0000 | 0 | −160.171 | 0 | 0 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −27.0000 | 0 | −129.570 | 0 | 0 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −27.0000 | 0 | 40.1550 | 0 | 0 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −27.0000 | 0 | 445.586 | 0 | 0 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 588.8.a.i | 4 | |
| 7.b | odd | 2 | 1 | 588.8.a.j | 4 | ||
| 7.c | even | 3 | 2 | 84.8.i.a | ✓ | 8 | |
| 7.d | odd | 6 | 2 | 588.8.i.o | 8 | ||
| 21.h | odd | 6 | 2 | 252.8.k.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.8.i.a | ✓ | 8 | 7.c | even | 3 | 2 | |
| 252.8.k.b | 8 | 21.h | odd | 6 | 2 | ||
| 588.8.a.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 588.8.a.j | 4 | 7.b | odd | 2 | 1 | ||
| 588.8.i.o | 8 | 7.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 196T_{5}^{3} - 102093T_{5}^{2} - 4896540T_{5} + 371328300 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(588))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 27)^{4} \)
|
| $5$ |
\( T^{4} - 196 T^{3} + \cdots + 371328300 \)
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| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 193939400174724 \)
|
| $13$ |
\( T^{4} + \cdots + 31670239466400 \)
|
| $17$ |
\( T^{4} + \cdots + 16\!\cdots\!08 \)
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| $19$ |
\( T^{4} + \cdots - 31\!\cdots\!84 \)
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| $23$ |
\( T^{4} + \cdots + 20\!\cdots\!04 \)
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| $29$ |
\( T^{4} + \cdots - 28\!\cdots\!32 \)
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| $31$ |
\( T^{4} + \cdots + 28\!\cdots\!71 \)
|
| $37$ |
\( T^{4} + \cdots + 26\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots + 74\!\cdots\!96 \)
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| $43$ |
\( T^{4} + \cdots + 33\!\cdots\!04 \)
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| $47$ |
\( T^{4} + \cdots + 98\!\cdots\!56 \)
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| $53$ |
\( T^{4} + \cdots + 41\!\cdots\!36 \)
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| $59$ |
\( T^{4} + \cdots - 52\!\cdots\!52 \)
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| $61$ |
\( T^{4} + \cdots + 23\!\cdots\!24 \)
|
| $67$ |
\( T^{4} + \cdots + 45\!\cdots\!68 \)
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| $71$ |
\( T^{4} + \cdots + 28\!\cdots\!08 \)
|
| $73$ |
\( T^{4} + \cdots + 29\!\cdots\!24 \)
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| $79$ |
\( T^{4} + \cdots - 53\!\cdots\!57 \)
|
| $83$ |
\( T^{4} + \cdots + 95\!\cdots\!92 \)
|
| $89$ |
\( T^{4} + \cdots - 62\!\cdots\!92 \)
|
| $97$ |
\( T^{4} + \cdots - 45\!\cdots\!56 \)
|
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