Newspace parameters
| Level: | \( N \) | \(=\) | \( 560 = 2^{4} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 560.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(174.935614271\) |
| 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: |
\( x^{4} - x^{3} - 7039x^{2} + 20029x + 3886530 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 140) |
| 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
\( x^{4} - x^{3} - 7039x^{2} + 20029x + 3886530 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 42\nu^{2} - 5779\nu - 136662 ) / 78 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 36\nu^{2} - 6013\nu + 137976 ) / 78 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} - 3\beta _1 + 3521 \)
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| \(\nu^{3}\) | \(=\) |
\( 42\beta_{3} + 36\beta_{2} + 5905\beta _1 - 11220 \)
|
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 | −93.0053 | 0 | −125.000 | 0 | −343.000 | 0 | 6462.98 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −40.3169 | 0 | −125.000 | 0 | −343.000 | 0 | −561.546 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 8.98037 | 0 | −125.000 | 0 | −343.000 | 0 | −2106.35 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 67.3418 | 0 | −125.000 | 0 | −343.000 | 0 | 2347.92 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(7\) | \(1\) |
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 | 560.8.a.m | 4 | |
| 4.b | odd | 2 | 1 | 140.8.a.d | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.8.a.d | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 560.8.a.m | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 57T_{3}^{3} - 5821T_{3}^{2} - 205557T_{3} + 2267640 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(560))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 57 T^{3} + \cdots + 2267640 \)
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| $5$ |
\( (T + 125)^{4} \)
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| $7$ |
\( (T + 343)^{4} \)
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| $11$ |
\( T^{4} + \cdots + 116447880320100 \)
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| $13$ |
\( T^{4} + \cdots - 18\!\cdots\!62 \)
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| $17$ |
\( T^{4} + \cdots + 61\!\cdots\!30 \)
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| $19$ |
\( T^{4} + \cdots - 21\!\cdots\!24 \)
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| $23$ |
\( T^{4} + \cdots + 26\!\cdots\!56 \)
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| $29$ |
\( T^{4} + \cdots + 53\!\cdots\!58 \)
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| $31$ |
\( T^{4} + \cdots + 50\!\cdots\!08 \)
|
| $37$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots + 49\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 19\!\cdots\!80 \)
|
| $47$ |
\( T^{4} + \cdots - 24\!\cdots\!44 \)
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| $53$ |
\( T^{4} + \cdots - 55\!\cdots\!00 \)
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| $59$ |
\( T^{4} + \cdots - 27\!\cdots\!60 \)
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| $61$ |
\( T^{4} + \cdots + 75\!\cdots\!40 \)
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| $67$ |
\( T^{4} + \cdots - 55\!\cdots\!60 \)
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| $71$ |
\( T^{4} + \cdots + 27\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 74\!\cdots\!00 \)
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| $79$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
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| $83$ |
\( T^{4} + \cdots - 31\!\cdots\!40 \)
|
| $89$ |
\( T^{4} + \cdots + 38\!\cdots\!40 \)
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| $97$ |
\( T^{4} + \cdots + 58\!\cdots\!34 \)
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