Newspace parameters
| Level: | \( N \) | \(=\) | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 570.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(178.059464526\) |
| 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: |
\( x^{4} - 2x^{3} - 4122x^{2} - 49773x + 620550 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3 \) |
| 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
\( x^{4} - 2x^{3} - 4122x^{2} - 49773x + 620550 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 2 \)
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| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{3} - 40\nu^{2} - 14770\nu - 84194 ) / 233 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -8\nu^{3} + 1012\nu^{2} + 8104\nu - 1742678 ) / 1631 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 4 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{3} + 2\beta_{2} + 23\beta _1 + 8248 ) / 4 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 140\beta_{3} + 506\beta_{2} + 7845\beta _1 + 348118 ) / 8 \)
|
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 |
|
8.00000 | −27.0000 | 64.0000 | 125.000 | −216.000 | −1165.98 | 512.000 | 729.000 | 1000.00 | ||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | −27.0000 | 64.0000 | 125.000 | −216.000 | −903.592 | 512.000 | 729.000 | 1000.00 | |||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | −27.0000 | 64.0000 | 125.000 | −216.000 | 206.862 | 512.000 | 729.000 | 1000.00 | |||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | −27.0000 | 64.0000 | 125.000 | −216.000 | 366.713 | 512.000 | 729.000 | 1000.00 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 570.8.a.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 570.8.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 1496T_{7}^{3} - 57624T_{7}^{2} - 447306872T_{7} + 79922833472 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(570))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{4} \)
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| $3$ |
\( (T + 27)^{4} \)
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| $5$ |
\( (T - 125)^{4} \)
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| $7$ |
\( T^{4} + \cdots + 79922833472 \)
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| $11$ |
\( T^{4} + \cdots + 293521583520 \)
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| $13$ |
\( T^{4} + \cdots - 703307956188096 \)
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| $17$ |
\( T^{4} + \cdots - 41\!\cdots\!00 \)
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| $19$ |
\( (T - 6859)^{4} \)
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| $23$ |
\( T^{4} + \cdots + 18\!\cdots\!20 \)
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| $29$ |
\( T^{4} + \cdots - 14\!\cdots\!32 \)
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| $31$ |
\( T^{4} + \cdots + 16\!\cdots\!08 \)
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| $37$ |
\( T^{4} + \cdots + 71\!\cdots\!44 \)
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| $41$ |
\( T^{4} + \cdots - 29\!\cdots\!00 \)
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| $43$ |
\( T^{4} + \cdots + 34\!\cdots\!60 \)
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| $47$ |
\( T^{4} + \cdots + 47\!\cdots\!20 \)
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| $53$ |
\( T^{4} + \cdots + 43\!\cdots\!56 \)
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| $59$ |
\( T^{4} + \cdots - 10\!\cdots\!92 \)
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| $61$ |
\( T^{4} + \cdots + 91\!\cdots\!56 \)
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| $67$ |
\( T^{4} + \cdots - 38\!\cdots\!00 \)
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| $71$ |
\( T^{4} + \cdots + 75\!\cdots\!00 \)
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| $73$ |
\( T^{4} + \cdots + 13\!\cdots\!44 \)
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| $79$ |
\( T^{4} + \cdots + 40\!\cdots\!32 \)
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| $83$ |
\( T^{4} + \cdots + 16\!\cdots\!40 \)
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| $89$ |
\( T^{4} + \cdots + 80\!\cdots\!40 \)
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| $97$ |
\( T^{4} + \cdots - 54\!\cdots\!60 \)
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