Newspace parameters
| Level: | \( N \) | \(=\) | \( 886 = 2 \cdot 443 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 886.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(7.07474561909\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{22 +2 \sqrt{5}})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 3x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - x^{3} - 3x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 1 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 2\nu + 1 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 1 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3\beta_1 \)
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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 |
|
1.00000 | −2.91300 | 1.00000 | −0.859226 | −2.91300 | −1.02811 | 1.00000 | 5.48555 | −0.859226 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.57549 | 1.00000 | −1.88039 | −1.57549 | 3.00509 | 1.00000 | −0.517822 | −1.88039 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.32307 | 1.00000 | 1.71333 | −1.32307 | −4.20796 | 1.00000 | −1.24948 | 1.71333 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.81156 | 1.00000 | −3.97371 | 1.81156 | −3.76902 | 1.00000 | 0.281754 | −3.97371 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(443\) | \( -1 \) |
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 | 886.2.a.g | ✓ | 4 |
| 3.b | odd | 2 | 1 | 7974.2.a.v | 4 | ||
| 4.b | odd | 2 | 1 | 7088.2.a.p | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 886.2.a.g | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 7088.2.a.p | 4 | 4.b | odd | 2 | 1 | ||
| 7974.2.a.v | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(886))\):
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\( T_{3}^{4} + 4T_{3}^{3} - 13T_{3} - 11 \)
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\( T_{5}^{4} + 5T_{5}^{3} + T_{5}^{2} - 15T_{5} - 11 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
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| $3$ |
\( T^{4} + 4 T^{3} + \cdots - 11 \)
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| $5$ |
\( T^{4} + 5 T^{3} + \cdots - 11 \)
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| $7$ |
\( T^{4} + 6 T^{3} + \cdots - 49 \)
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| $11$ |
\( T^{4} + 10 T^{3} + \cdots - 251 \)
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| $13$ |
\( T^{4} - 7 T^{3} + \cdots + 1 \)
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| $17$ |
\( T^{4} + 11 T^{3} + \cdots + 31 \)
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| $19$ |
\( T^{4} + 14 T^{3} + \cdots - 379 \)
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| $23$ |
\( T^{4} + 6 T^{3} + \cdots + 169 \)
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| $29$ |
\( (T^{2} - 3 T - 9)^{2} \)
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| $31$ |
\( T^{4} + 22 T^{3} + \cdots - 2719 \)
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| $37$ |
\( T^{4} - 7 T^{3} + \cdots + 139 \)
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| $41$ |
\( T^{4} + 10 T^{3} + \cdots - 821 \)
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| $43$ |
\( T^{4} + 14 T^{3} + \cdots - 451 \)
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| $47$ |
\( T^{4} + 15 T^{3} + \cdots - 1441 \)
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| $53$ |
\( T^{4} + 16 T^{3} + \cdots - 1091 \)
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| $59$ |
\( T^{4} + 21 T^{3} + \cdots - 109 \)
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| $61$ |
\( T^{4} + T^{3} + \cdots - 1279 \)
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| $67$ |
\( T^{4} - 19 T^{3} + \cdots - 3389 \)
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| $71$ |
\( T^{4} + 6 T^{3} + \cdots - 431 \)
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| $73$ |
\( T^{4} - 17 T^{3} + \cdots - 491 \)
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| $79$ |
\( T^{4} - 13 T^{3} + \cdots - 641 \)
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| $83$ |
\( T^{4} + 14 T^{3} + \cdots - 11111 \)
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| $89$ |
\( T^{4} + 8 T^{3} + \cdots + 256 \)
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| $97$ |
\( T^{4} - 16 T^{3} + \cdots + 539 \)
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