Newspace parameters
| Level: | \( N \) | \(=\) | \( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8664.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(69.1823883112\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.34025.1 |
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|
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| Defining polynomial: |
\( x^{4} - 2x^{3} - 19x^{2} + 20x + 80 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{4} - 2x^{3} - 19x^{2} + 20x + 80 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 12 ) / 4 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 12\nu ) / 4 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( 4\beta_{2} + \beta _1 + 12 \)
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| \(\nu^{3}\) | \(=\) |
\( 4\beta_{3} + 4\beta_{2} + 13\beta _1 + 12 \)
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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 | −1.00000 | 0 | −1.61803 | 0 | −4.69832 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −1.61803 | 0 | 3.08028 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | 0.618034 | 0 | −2.06234 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | 0.618034 | 0 | 2.68038 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(19\) | \( -1 \) |
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 | 8664.2.a.bb | ✓ | 4 |
| 19.b | odd | 2 | 1 | 8664.2.a.bd | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8664.2.a.bb | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 8664.2.a.bd | yes | 4 | 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8664))\):
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\( T_{5}^{2} + T_{5} - 1 \)
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\( T_{7}^{4} + T_{7}^{3} - 21T_{7}^{2} + 80 \)
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\( T_{13}^{4} - 23T_{13}^{2} - 20T_{13} + 61 \)
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\( T_{29}^{4} - 3T_{29}^{3} - 43T_{29}^{2} - 54T_{29} + 44 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( (T + 1)^{4} \)
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| $5$ |
\( (T^{2} + T - 1)^{2} \)
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| $7$ |
\( T^{4} + T^{3} + \cdots + 80 \)
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| $11$ |
\( T^{4} + 2 T^{3} + \cdots + 80 \)
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| $13$ |
\( T^{4} - 23 T^{2} + \cdots + 61 \)
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| $17$ |
\( T^{4} + T^{3} + \cdots + 25 \)
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| $19$ |
\( T^{4} \)
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| $23$ |
\( T^{4} - 2 T^{3} + \cdots + 80 \)
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| $29$ |
\( T^{4} - 3 T^{3} + \cdots + 44 \)
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| $31$ |
\( T^{4} - 17 T^{3} + \cdots + 64 \)
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| $37$ |
\( T^{4} - 5 T^{3} + \cdots - 244 \)
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| $41$ |
\( T^{4} + 16 T^{3} + \cdots + 289 \)
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| $43$ |
\( T^{4} + 12 T^{3} + \cdots - 1616 \)
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| $47$ |
\( T^{4} - T^{3} + \cdots + 80 \)
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| $53$ |
\( T^{4} - 7 T^{3} + \cdots + 55 \)
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| $59$ |
\( T^{4} - 20 T^{3} + \cdots - 64 \)
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| $61$ |
\( T^{4} + 15 T^{3} + \cdots - 964 \)
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| $67$ |
\( T^{4} - 11 T^{3} + \cdots - 1616 \)
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| $71$ |
\( T^{4} - T^{3} + \cdots + 80 \)
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| $73$ |
\( T^{4} + 33 T^{3} + \cdots + 449 \)
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| $79$ |
\( T^{4} + 24 T^{3} + \cdots - 3904 \)
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| $83$ |
\( T^{4} - 2 T^{3} + \cdots + 3904 \)
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| $89$ |
\( T^{4} - 25 T^{3} + \cdots - 944 \)
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| $97$ |
\( T^{4} + 21 T^{3} + \cdots + 5324 \)
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