Newspace parameters
| Level: | \( N \) | \(=\) | \( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5328.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(42.5442941969\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.48389.1 |
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|
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| Defining polynomial: |
\( x^{4} - 2x^{3} - 8x^{2} + 3x + 10 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 296) |
| 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
\( x^{4} - 2x^{3} - 8x^{2} + 3x + 10 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu^{2} - 4\nu + 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + 10\beta _1 + 7 \)
|
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 | 0 | 0 | −3.65512 | 0 | −2.28805 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −2.87345 | 0 | 4.19623 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −0.687447 | 0 | 2.16737 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 2.21602 | 0 | −3.07555 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(37\) | \( -1 \) |
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 | 5328.2.a.bp | 4 | |
| 3.b | odd | 2 | 1 | 592.2.a.j | 4 | ||
| 4.b | odd | 2 | 1 | 2664.2.a.r | 4 | ||
| 12.b | even | 2 | 1 | 296.2.a.d | ✓ | 4 | |
| 24.f | even | 2 | 1 | 2368.2.a.bg | 4 | ||
| 24.h | odd | 2 | 1 | 2368.2.a.bh | 4 | ||
| 60.h | even | 2 | 1 | 7400.2.a.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 296.2.a.d | ✓ | 4 | 12.b | even | 2 | 1 | |
| 592.2.a.j | 4 | 3.b | odd | 2 | 1 | ||
| 2368.2.a.bg | 4 | 24.f | even | 2 | 1 | ||
| 2368.2.a.bh | 4 | 24.h | odd | 2 | 1 | ||
| 2664.2.a.r | 4 | 4.b | odd | 2 | 1 | ||
| 5328.2.a.bp | 4 | 1.a | even | 1 | 1 | trivial | |
| 7400.2.a.n | 4 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5328))\):
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\( T_{5}^{4} + 5T_{5}^{3} - T_{5}^{2} - 26T_{5} - 16 \)
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\( T_{7}^{4} - T_{7}^{3} - 18T_{7}^{2} + 4T_{7} + 64 \)
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\( T_{11}^{4} - 4T_{11}^{3} - 12T_{11}^{2} + 63T_{11} - 52 \)
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\( T_{13}^{4} - 5T_{13}^{3} - 17T_{13}^{2} + 122T_{13} - 160 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 5 T^{3} + \cdots - 16 \)
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| $7$ |
\( T^{4} - T^{3} + \cdots + 64 \)
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| $11$ |
\( T^{4} - 4 T^{3} + \cdots - 52 \)
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| $13$ |
\( T^{4} - 5 T^{3} + \cdots - 160 \)
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| $17$ |
\( (T + 2)^{4} \)
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| $19$ |
\( T^{4} + 2 T^{3} + \cdots + 640 \)
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| $23$ |
\( T^{4} + 9 T^{3} + \cdots - 472 \)
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| $29$ |
\( T^{4} + 7 T^{3} + \cdots + 4 \)
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| $31$ |
\( T^{4} - T^{3} + \cdots + 848 \)
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| $37$ |
\( (T - 1)^{4} \)
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| $41$ |
\( T^{4} + 2 T^{3} + \cdots - 422 \)
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| $43$ |
\( T^{4} + 6 T^{3} + \cdots + 128 \)
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| $47$ |
\( T^{4} + 29 T^{3} + \cdots - 2336 \)
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| $53$ |
\( T^{4} - 5 T^{3} + \cdots - 8 \)
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| $59$ |
\( T^{4} + 10 T^{3} + \cdots + 512 \)
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| $61$ |
\( T^{4} + T^{3} + \cdots + 664 \)
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| $67$ |
\( T^{4} - T^{3} + \cdots - 16 \)
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| $71$ |
\( T^{4} + 17 T^{3} + \cdots + 6976 \)
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| $73$ |
\( T^{4} - 8 T^{3} + \cdots + 2938 \)
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| $79$ |
\( T^{4} + 15 T^{3} + \cdots + 17320 \)
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| $83$ |
\( T^{4} - 15 T^{3} + \cdots + 4160 \)
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| $89$ |
\( T^{4} - 20 T^{3} + \cdots - 3392 \)
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| $97$ |
\( T^{4} - 2 T^{3} + \cdots - 160 \)
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