Newspace parameters
| Level: | \( N \) | \(=\) | \( 5488 = 2^{4} \cdot 7^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5488.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(43.8219006293\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1229312.1 |
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| Defining polynomial: |
\( x^{6} - 10x^{4} + 24x^{2} - 8 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1372) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 10x^{4} + 24x^{2} - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 4 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 6\nu ) / 2 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 6\nu^{2} + 4 ) / 4 \)
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| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 8\nu^{3} + 12\nu ) / 4 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 6\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} + 12\beta_{2} + 20 \)
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| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} + 16\beta_{3} + 36\beta_1 \)
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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 |
|
0 | −2.54832 | 0 | 0.349282 | 0 | 0 | 0 | 3.49396 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.76350 | 0 | 2.04360 | 0 | 0 | 0 | 0.109916 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.629384 | 0 | −3.96254 | 0 | 0 | 0 | −2.60388 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.629384 | 0 | 3.96254 | 0 | 0 | 0 | −2.60388 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.76350 | 0 | −2.04360 | 0 | 0 | 0 | 0.109916 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.54832 | 0 | −0.349282 | 0 | 0 | 0 | 3.49396 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5488.2.a.j | 6 | |
| 4.b | odd | 2 | 1 | 1372.2.a.c | ✓ | 6 | |
| 7.b | odd | 2 | 1 | inner | 5488.2.a.j | 6 | |
| 28.d | even | 2 | 1 | 1372.2.a.c | ✓ | 6 | |
| 28.f | even | 6 | 2 | 1372.2.e.b | 12 | ||
| 28.g | odd | 6 | 2 | 1372.2.e.b | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1372.2.a.c | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 1372.2.a.c | ✓ | 6 | 28.d | even | 2 | 1 | |
| 1372.2.e.b | 12 | 28.f | even | 6 | 2 | ||
| 1372.2.e.b | 12 | 28.g | odd | 6 | 2 | ||
| 5488.2.a.j | 6 | 1.a | even | 1 | 1 | trivial | |
| 5488.2.a.j | 6 | 7.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5488))\):
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\( T_{3}^{6} - 10T_{3}^{4} + 24T_{3}^{2} - 8 \)
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\( T_{5}^{6} - 20T_{5}^{4} + 68T_{5}^{2} - 8 \)
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\( T_{11}^{3} + 5T_{11}^{2} + 6T_{11} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} - 10 T^{4} + \cdots - 8 \)
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| $5$ |
\( T^{6} - 20 T^{4} + \cdots - 8 \)
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| $7$ |
\( T^{6} \)
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| $11$ |
\( (T^{3} + 5 T^{2} + 6 T + 1)^{2} \)
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| $13$ |
\( T^{6} - 12 T^{4} + \cdots - 8 \)
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| $17$ |
\( T^{6} - 34 T^{4} + \cdots - 8 \)
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| $19$ |
\( T^{6} - 52 T^{4} + \cdots - 1352 \)
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| $23$ |
\( (T^{3} + 9 T^{2} + 20 T - 1)^{2} \)
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| $29$ |
\( (T^{3} - 3 T^{2} - 46 T + 97)^{2} \)
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| $31$ |
\( T^{6} - 84 T^{4} + \cdots - 392 \)
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| $37$ |
\( (T^{3} - 11 T^{2} + \cdots + 29)^{2} \)
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| $41$ |
\( T^{6} - 178 T^{4} + \cdots - 129032 \)
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| $43$ |
\( (T^{3} + 9 T^{2} - 22 T - 71)^{2} \)
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| $47$ |
\( T^{6} - 180 T^{4} + \cdots - 129032 \)
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| $53$ |
\( (T^{3} - 9 T^{2} + \cdots + 169)^{2} \)
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| $59$ |
\( T^{6} - 250 T^{4} + \cdots - 228488 \)
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| $61$ |
\( (T^{2} - 162)^{3} \)
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| $67$ |
\( (T^{3} + 21 T^{2} + \cdots + 189)^{2} \)
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| $71$ |
\( (T^{3} + 35 T^{2} + \cdots + 1379)^{2} \)
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| $73$ |
\( T^{6} - 598 T^{4} + \cdots - 6237512 \)
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| $79$ |
\( (T^{3} + 3 T^{2} - 46 T - 97)^{2} \)
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| $83$ |
\( T^{6} - 404 T^{4} + \cdots - 631688 \)
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| $89$ |
\( T^{6} - 262 T^{4} + \cdots - 40328 \)
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| $97$ |
\( T^{6} - 426 T^{4} + \cdots - 262088 \)
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