Newspace parameters
| Level: | \( N \) | \(=\) | \( 880 = 2^{4} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 880.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(51.9216808051\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 47x^{2} - 34x + 120 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 440) |
| 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
\( x^{4} - 47x^{2} - 34x + 120 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 24 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 41\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 24 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 42\beta _1 + 26 \)
|
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 | −6.02910 | 0 | −5.00000 | 0 | −26.4083 | 0 | 9.35008 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.294743 | 0 | −5.00000 | 0 | 21.3236 | 0 | −26.9131 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 3.12819 | 0 | −5.00000 | 0 | 18.4708 | 0 | −17.2144 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 7.19566 | 0 | −5.00000 | 0 | −15.3862 | 0 | 24.7775 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 880.4.a.bb | 4 | |
| 4.b | odd | 2 | 1 | 440.4.a.h | ✓ | 4 | |
| 20.d | odd | 2 | 1 | 2200.4.a.q | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 440.4.a.h | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 880.4.a.bb | 4 | 1.a | even | 1 | 1 | trivial | |
| 2200.4.a.q | 4 | 20.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(880))\):
|
\( T_{3}^{4} - 4T_{3}^{3} - 41T_{3}^{2} + 124T_{3} + 40 \)
|
|
\( T_{7}^{4} + 2T_{7}^{3} - 863T_{7}^{2} + 292T_{7} + 160036 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} - 4 T^{3} + \cdots + 40 \)
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| $5$ |
\( (T + 5)^{4} \)
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| $7$ |
\( T^{4} + 2 T^{3} + \cdots + 160036 \)
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| $11$ |
\( (T - 11)^{4} \)
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| $13$ |
\( T^{4} - 5604 T^{2} + \cdots + 929760 \)
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| $17$ |
\( T^{4} - 20 T^{3} + \cdots + 98241352 \)
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| $19$ |
\( T^{4} + 58 T^{3} + \cdots - 16790720 \)
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| $23$ |
\( T^{4} + 72 T^{3} + \cdots + 6769632 \)
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| $29$ |
\( T^{4} - 8 T^{3} + \cdots + 710034300 \)
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| $31$ |
\( T^{4} + 324 T^{3} + \cdots + 349986240 \)
|
| $37$ |
\( T^{4} - 354 T^{3} + \cdots - 371712860 \)
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| $41$ |
\( T^{4} - 276 T^{3} + \cdots + 53298480 \)
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| $43$ |
\( T^{4} + 692 T^{3} + \cdots - 180637440 \)
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| $47$ |
\( T^{4} + \cdots + 3539898528 \)
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| $53$ |
\( T^{4} + 182 T^{3} + \cdots + 139207940 \)
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| $59$ |
\( T^{4} + \cdots + 2890722112 \)
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| $61$ |
\( T^{4} + 528 T^{3} + \cdots + 544668268 \)
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| $67$ |
\( T^{4} + \cdots + 5598912000 \)
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| $71$ |
\( T^{4} + \cdots + 12436252800 \)
|
| $73$ |
\( T^{4} - 124 T^{3} + \cdots - 332187360 \)
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| $79$ |
\( T^{4} + \cdots - 95753291328 \)
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| $83$ |
\( T^{4} + \cdots + 37196118768 \)
|
| $89$ |
\( T^{4} + \cdots + 40186945060 \)
|
| $97$ |
\( T^{4} + \cdots - 30252643968 \)
|
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