Newspace parameters
| Level: | \( N \) | \(=\) | \( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5808.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(46.3771134940\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{3}, \sqrt{11})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 7x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 363) |
| 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
\( x^{4} - 7x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 5\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} + 5\beta_1 \)
|
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 | −2.37228 | 0 | −0.792287 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −2.37228 | 0 | 0.792287 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | 3.37228 | 0 | −2.52434 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | 3.37228 | 0 | 2.52434 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(11\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5808.2.a.ck | 4 | |
| 4.b | odd | 2 | 1 | 363.2.a.j | ✓ | 4 | |
| 11.b | odd | 2 | 1 | inner | 5808.2.a.ck | 4 | |
| 12.b | even | 2 | 1 | 1089.2.a.u | 4 | ||
| 20.d | odd | 2 | 1 | 9075.2.a.cv | 4 | ||
| 44.c | even | 2 | 1 | 363.2.a.j | ✓ | 4 | |
| 44.g | even | 10 | 4 | 363.2.e.n | 16 | ||
| 44.h | odd | 10 | 4 | 363.2.e.n | 16 | ||
| 132.d | odd | 2 | 1 | 1089.2.a.u | 4 | ||
| 220.g | even | 2 | 1 | 9075.2.a.cv | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 363.2.a.j | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 363.2.a.j | ✓ | 4 | 44.c | even | 2 | 1 | |
| 363.2.e.n | 16 | 44.g | even | 10 | 4 | ||
| 363.2.e.n | 16 | 44.h | odd | 10 | 4 | ||
| 1089.2.a.u | 4 | 12.b | even | 2 | 1 | ||
| 1089.2.a.u | 4 | 132.d | odd | 2 | 1 | ||
| 5808.2.a.ck | 4 | 1.a | even | 1 | 1 | trivial | |
| 5808.2.a.ck | 4 | 11.b | odd | 2 | 1 | inner | |
| 9075.2.a.cv | 4 | 20.d | odd | 2 | 1 | ||
| 9075.2.a.cv | 4 | 220.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5808))\):
|
\( T_{5}^{2} - T_{5} - 8 \)
|
|
\( T_{7}^{4} - 7T_{7}^{2} + 4 \)
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\( T_{13}^{4} - 51T_{13}^{2} + 576 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 1)^{4} \)
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| $5$ |
\( (T^{2} - T - 8)^{2} \)
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| $7$ |
\( T^{4} - 7T^{2} + 4 \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} - 51T^{2} + 576 \)
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| $17$ |
\( T^{4} - 43T^{2} + 256 \)
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| $19$ |
\( T^{4} - 19T^{2} + 16 \)
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| $23$ |
\( (T + 2)^{4} \)
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| $29$ |
\( T^{4} - 7T^{2} + 4 \)
|
| $31$ |
\( (T^{2} + 9 T + 12)^{2} \)
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| $37$ |
\( (T - 5)^{4} \)
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| $41$ |
\( T^{4} - 151T^{2} + 3844 \)
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| $43$ |
\( (T^{2} - 44)^{2} \)
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| $47$ |
\( (T^{2} - 14 T + 16)^{2} \)
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| $53$ |
\( (T^{2} - 9 T - 54)^{2} \)
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| $59$ |
\( (T - 6)^{4} \)
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| $61$ |
\( T^{4} - 43T^{2} + 256 \)
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| $67$ |
\( (T^{2} + 15 T - 18)^{2} \)
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| $71$ |
\( (T^{2} - 10 T - 8)^{2} \)
|
| $73$ |
\( T^{4} - 139T^{2} + 4624 \)
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| $79$ |
\( T^{4} - 51T^{2} + 576 \)
|
| $83$ |
\( T^{4} - 76T^{2} + 256 \)
|
| $89$ |
\( (T^{2} + 7 T + 4)^{2} \)
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| $97$ |
\( (T^{2} + 2 T - 131)^{2} \)
|
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