Newspace parameters
| Level: | \( N \) | \(=\) | \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8624.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(68.8629867032\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.352076.1 |
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| Defining polynomial: |
\( x^{5} - x^{4} - 8x^{3} + 3x^{2} + 8x - 4 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 616) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 8x^{3} + 3x^{2} + 8x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 8\nu^{2} + \nu + 6 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + 9\nu^{2} + 4\nu - 8 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{4} - \nu^{3} - 24\nu^{2} - 7\nu + 16 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 3\beta_{2} + 5\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( 9\beta_{4} + 8\beta_{3} - 9\beta_{2} + 4\beta _1 + 19 \)
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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 |
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0 | −2.95501 | 0 | 2.64159 | 0 | 0 | 0 | 5.73208 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.52551 | 0 | −2.88640 | 0 | 0 | 0 | −0.672805 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.39030 | 0 | 3.08458 | 0 | 0 | 0 | −1.06708 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.464071 | 0 | 1.08161 | 0 | 0 | 0 | −2.78464 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.40675 | 0 | 0.0786216 | 0 | 0 | 0 | 2.79244 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(11\) | \( +1 \) |
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 | 8624.2.a.da | 5 | |
| 4.b | odd | 2 | 1 | 4312.2.a.bh | 5 | ||
| 7.b | odd | 2 | 1 | 8624.2.a.dd | 5 | ||
| 7.c | even | 3 | 2 | 1232.2.q.p | 10 | ||
| 28.d | even | 2 | 1 | 4312.2.a.be | 5 | ||
| 28.g | odd | 6 | 2 | 616.2.q.e | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 616.2.q.e | ✓ | 10 | 28.g | odd | 6 | 2 | |
| 1232.2.q.p | 10 | 7.c | even | 3 | 2 | ||
| 4312.2.a.be | 5 | 28.d | even | 2 | 1 | ||
| 4312.2.a.bh | 5 | 4.b | odd | 2 | 1 | ||
| 8624.2.a.da | 5 | 1.a | even | 1 | 1 | trivial | |
| 8624.2.a.dd | 5 | 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8624))\):
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\( T_{3}^{5} + 3T_{3}^{4} - 5T_{3}^{3} - 18T_{3}^{2} - 6T_{3} + 7 \)
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\( T_{5}^{5} - 4T_{5}^{4} - 5T_{5}^{3} + 33T_{5}^{2} - 28T_{5} + 2 \)
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\( T_{13}^{5} - 7T_{13}^{4} - 29T_{13}^{3} + 272T_{13}^{2} - 410T_{13} + 141 \)
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\( T_{17}^{5} - 5T_{17}^{4} - 28T_{17}^{3} + 43T_{17}^{2} + 116T_{17} - 148 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
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| $3$ |
\( T^{5} + 3 T^{4} + \cdots + 7 \)
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| $5$ |
\( T^{5} - 4 T^{4} + \cdots + 2 \)
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| $7$ |
\( T^{5} \)
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| $11$ |
\( (T + 1)^{5} \)
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| $13$ |
\( T^{5} - 7 T^{4} + \cdots + 141 \)
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| $17$ |
\( T^{5} - 5 T^{4} + \cdots - 148 \)
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| $19$ |
\( T^{5} + 13 T^{4} + \cdots - 4606 \)
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| $23$ |
\( T^{5} + 8 T^{4} + \cdots + 914 \)
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| $29$ |
\( T^{5} - 7 T^{4} + \cdots + 1057 \)
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| $31$ |
\( T^{5} + 11 T^{4} + \cdots - 16 \)
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| $37$ |
\( T^{5} + 14 T^{4} + \cdots - 1392 \)
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| $41$ |
\( T^{5} + 7 T^{4} + \cdots + 134 \)
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| $43$ |
\( T^{5} + 12 T^{4} + \cdots + 14864 \)
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| $47$ |
\( T^{5} + 11 T^{4} + \cdots - 788 \)
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| $53$ |
\( T^{5} - 7 T^{4} + \cdots + 1402 \)
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| $59$ |
\( T^{5} - 114 T^{3} + \cdots + 201 \)
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| $61$ |
\( T^{5} - 18 T^{4} + \cdots + 13976 \)
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| $67$ |
\( T^{5} - 12 T^{4} + \cdots - 2088 \)
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| $71$ |
\( T^{5} + 19 T^{4} + \cdots - 954 \)
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| $73$ |
\( T^{5} + 12 T^{4} + \cdots + 11374 \)
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| $79$ |
\( T^{5} + 3 T^{4} + \cdots - 4363 \)
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| $83$ |
\( T^{5} + 23 T^{4} + \cdots - 636 \)
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| $89$ |
\( T^{5} - 5 T^{4} + \cdots - 88592 \)
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| $97$ |
\( T^{5} + 13 T^{4} + \cdots - 2693 \)
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