Newspace parameters
| Level: | \( N \) | \(=\) | \( 1120 = 2^{5} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1120.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(66.0821392064\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 89x^{3} + 129x^{2} + 900x - 180 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 89x^{3} + 129x^{2} + 900x - 180 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 66\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + \nu - 36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 85\nu^{2} + 60\nu + 612 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta _1 + 36 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 3\beta_{2} + 67\beta _1 - 36 \)
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| \(\nu^{4}\) | \(=\) |
\( 6\beta_{4} + 85\beta_{3} - 145\beta _1 + 2448 \)
|
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 | −10.3110 | 0 | 5.00000 | 0 | −7.00000 | 0 | 79.3177 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −6.38145 | 0 | 5.00000 | 0 | −7.00000 | 0 | 13.7229 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.19527 | 0 | 5.00000 | 0 | −7.00000 | 0 | −22.1808 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.779096 | 0 | 5.00000 | 0 | −7.00000 | 0 | −26.3930 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 7.10867 | 0 | 5.00000 | 0 | −7.00000 | 0 | 23.5332 | 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 | 1120.4.a.q | ✓ | 5 |
| 4.b | odd | 2 | 1 | 1120.4.a.t | yes | 5 | |
| 8.b | even | 2 | 1 | 2240.4.a.cp | 5 | ||
| 8.d | odd | 2 | 1 | 2240.4.a.cm | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1120.4.a.q | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1120.4.a.t | yes | 5 | 4.b | odd | 2 | 1 | |
| 2240.4.a.cm | 5 | 8.d | odd | 2 | 1 | ||
| 2240.4.a.cp | 5 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1120))\):
|
\( T_{3}^{5} + 11T_{3}^{4} - 41T_{3}^{3} - 559T_{3}^{2} - 572T_{3} + 800 \)
|
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\( T_{11}^{5} - 9T_{11}^{4} - 4417T_{11}^{3} + 24829T_{11}^{2} + 4088916T_{11} + 8379680 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 11 T^{4} + \cdots + 800 \)
|
| $5$ |
\( (T - 5)^{5} \)
|
| $7$ |
\( (T + 7)^{5} \)
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| $11$ |
\( T^{5} - 9 T^{4} + \cdots + 8379680 \)
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| $13$ |
\( T^{5} - 91 T^{4} + \cdots - 180761220 \)
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| $17$ |
\( T^{5} - 161 T^{4} + \cdots - 181193300 \)
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| $19$ |
\( T^{5} + 84 T^{4} + \cdots - 128450560 \)
|
| $23$ |
\( T^{5} + \cdots - 6025936896 \)
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| $29$ |
\( T^{5} - 217 T^{4} + \cdots - 207299340 \)
|
| $31$ |
\( T^{5} + \cdots - 12695500800 \)
|
| $37$ |
\( T^{5} + \cdots - 2179440000 \)
|
| $41$ |
\( T^{5} + \cdots + 2073300000 \)
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| $43$ |
\( T^{5} + \cdots - 69329887232 \)
|
| $47$ |
\( T^{5} + \cdots - 5322104866240 \)
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| $53$ |
\( T^{5} + \cdots - 1226009088000 \)
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| $59$ |
\( T^{5} + \cdots + 188636774400 \)
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| $61$ |
\( T^{5} + \cdots + 2106472952800 \)
|
| $67$ |
\( T^{5} + \cdots + 16353263616 \)
|
| $71$ |
\( T^{5} + \cdots - 3434615377920 \)
|
| $73$ |
\( T^{5} + \cdots + 186391636000 \)
|
| $79$ |
\( T^{5} + \cdots + 2207702242880 \)
|
| $83$ |
\( T^{5} + \cdots - 38585932800000 \)
|
| $89$ |
\( T^{5} + \cdots - 55089402520608 \)
|
| $97$ |
\( T^{5} + \cdots - 615019097092500 \)
|
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