Newspace parameters
| Level: | \( N \) | \(=\) | \( 1360 = 2^{4} \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1360.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(80.2425976078\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
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| Defining polynomial: |
\( x^{5} - 2x^{4} - 20x^{3} + 38x^{2} + 69x - 126 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 85) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 20x^{3} + 38x^{2} + 69x - 126 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 17\nu^{2} - 13\nu + 42 ) / 6 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} - \nu^{3} + 23\nu^{2} + 13\nu - 93 ) / 3 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{4} + \nu^{3} + 97\nu^{2} - \nu - 300 ) / 6 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 17 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{4} - 2\beta_{3} + 6\beta_{2} + 11\beta _1 + 7 ) / 2 \)
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| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{4} + 19\beta_{3} + 40\beta_{2} + 2\beta _1 + 211 ) / 2 \)
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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 | −6.08748 | 0 | 5.00000 | 0 | −10.8418 | 0 | 10.0574 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.57070 | 0 | 5.00000 | 0 | −25.2782 | 0 | −20.3915 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.820806 | 0 | 5.00000 | 0 | 22.0083 | 0 | −26.3263 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.13153 | 0 | 5.00000 | 0 | 26.5778 | 0 | −25.7196 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 7.70584 | 0 | 5.00000 | 0 | −22.4661 | 0 | 32.3800 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(17\) | \( -1 \) |
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 | 1360.4.a.w | 5 | |
| 4.b | odd | 2 | 1 | 85.4.a.g | ✓ | 5 | |
| 12.b | even | 2 | 1 | 765.4.a.m | 5 | ||
| 20.d | odd | 2 | 1 | 425.4.a.i | 5 | ||
| 20.e | even | 4 | 2 | 425.4.b.i | 10 | ||
| 68.d | odd | 2 | 1 | 1445.4.a.l | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.4.a.g | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 425.4.a.i | 5 | 20.d | odd | 2 | 1 | ||
| 425.4.b.i | 10 | 20.e | even | 4 | 2 | ||
| 765.4.a.m | 5 | 12.b | even | 2 | 1 | ||
| 1360.4.a.w | 5 | 1.a | even | 1 | 1 | trivial | |
| 1445.4.a.l | 5 | 68.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1360))\):
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\( T_{3}^{5} - T_{3}^{4} - 52T_{3}^{3} - 20T_{3}^{2} + 188T_{3} - 112 \)
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\( T_{7}^{5} + 10T_{7}^{4} - 1176T_{7}^{3} - 12316T_{7}^{2} + 335816T_{7} + 3601472 \)
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\( T_{11}^{5} + 126T_{11}^{4} + 4896T_{11}^{3} + 65756T_{11}^{2} + 296064T_{11} + 167488 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
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| $3$ |
\( T^{5} - T^{4} + \cdots - 112 \)
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| $5$ |
\( (T - 5)^{5} \)
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| $7$ |
\( T^{5} + 10 T^{4} + \cdots + 3601472 \)
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| $11$ |
\( T^{5} + 126 T^{4} + \cdots + 167488 \)
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| $13$ |
\( T^{5} - 83 T^{4} + \cdots - 7643696 \)
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| $17$ |
\( (T - 17)^{5} \)
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| $19$ |
\( T^{5} + \cdots + 2274428800 \)
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| $23$ |
\( T^{5} + \cdots - 2278533056 \)
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| $29$ |
\( T^{5} + \cdots - 2844978640 \)
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| $31$ |
\( T^{5} + \cdots + 73509549120 \)
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| $37$ |
\( T^{5} + \cdots - 694556688448 \)
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| $41$ |
\( T^{5} + \cdots - 478624610720 \)
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| $43$ |
\( T^{5} + \cdots - 1497205751872 \)
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| $47$ |
\( T^{5} + \cdots - 840515565952 \)
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| $53$ |
\( T^{5} + \cdots - 474378006640 \)
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| $59$ |
\( T^{5} + \cdots - 1194050514560 \)
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| $61$ |
\( T^{5} + \cdots - 37656238184848 \)
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| $67$ |
\( T^{5} + \cdots + 3947395137728 \)
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| $71$ |
\( T^{5} + \cdots - 26612610106464 \)
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| $73$ |
\( T^{5} + \cdots - 61959193488 \)
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| $79$ |
\( T^{5} + \cdots - 10071329239040 \)
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| $83$ |
\( T^{5} + \cdots + 968622815428096 \)
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| $89$ |
\( T^{5} + \cdots - 30709087651760 \)
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| $97$ |
\( T^{5} + \cdots + 17052960082000 \)
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