Newspace parameters
| Level: | \( N \) | \(=\) | \( 9152 = 2^{6} \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9152.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(73.0790879299\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.106740016.1 |
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| Defining polynomial: |
\( x^{6} - x^{5} - 9x^{4} + 7x^{3} + 15x^{2} + x - 2 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 4576) |
| 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
\( x^{6} - x^{5} - 9x^{4} + 7x^{3} + 15x^{2} + x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + \nu^{2} - 6\nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 7\nu^{2} + 9\nu + 1 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 8\nu^{2} + 8\nu - 2 \)
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| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 8\nu^{3} - 14\nu^{2} - 7\nu + 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta_{2} + 5\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 7\beta_{4} - 6\beta_{3} + 5\beta _1 + 17 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 8\beta_{4} + 10\beta_{3} + 8\beta_{2} + 29\beta _1 - 5 \)
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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 | −2.63744 | 0 | 1.69975 | 0 | 3.25636 | 0 | 3.95610 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.822033 | 0 | −3.86135 | 0 | 2.53709 | 0 | −2.32426 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.531122 | 0 | 1.77141 | 0 | −3.48932 | 0 | −2.71791 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.320976 | 0 | −0.406820 | 0 | −1.49015 | 0 | −2.89697 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.13437 | 0 | −0.721145 | 0 | 3.27666 | 0 | 1.55552 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.53526 | 0 | 3.51816 | 0 | 0.909368 | 0 | 3.42753 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(11\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 9152.2.a.cr | 6 | |
| 4.b | odd | 2 | 1 | 9152.2.a.cp | 6 | ||
| 8.b | even | 2 | 1 | 4576.2.a.p | ✓ | 6 | |
| 8.d | odd | 2 | 1 | 4576.2.a.q | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4576.2.a.p | ✓ | 6 | 8.b | even | 2 | 1 | |
| 4576.2.a.q | yes | 6 | 8.d | odd | 2 | 1 | |
| 9152.2.a.cp | 6 | 4.b | odd | 2 | 1 | ||
| 9152.2.a.cr | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9152))\):
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\( T_{3}^{6} - T_{3}^{5} - 9T_{3}^{4} + 7T_{3}^{3} + 15T_{3}^{2} + T_{3} - 2 \)
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\( T_{5}^{6} - 2T_{5}^{5} - 15T_{5}^{4} + 34T_{5}^{3} + 10T_{5}^{2} - 32T_{5} - 12 \)
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\( T_{7}^{6} - 5T_{7}^{5} - 9T_{7}^{4} + 73T_{7}^{3} - 49T_{7}^{2} - 147T_{7} + 128 \)
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\( T_{17}^{6} - 6T_{17}^{5} - 10T_{17}^{4} + 88T_{17}^{3} - 60T_{17}^{2} - 120T_{17} + 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} - T^{5} - 9 T^{4} + \cdots - 2 \)
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| $5$ |
\( T^{6} - 2 T^{5} + \cdots - 12 \)
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| $7$ |
\( T^{6} - 5 T^{5} + \cdots + 128 \)
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| $11$ |
\( (T - 1)^{6} \)
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| $13$ |
\( (T - 1)^{6} \)
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| $17$ |
\( T^{6} - 6 T^{5} + \cdots + 32 \)
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| $19$ |
\( T^{6} - 2 T^{5} + \cdots - 32 \)
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| $23$ |
\( T^{6} - 20 T^{5} + \cdots - 73 \)
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| $29$ |
\( T^{6} - 3 T^{5} + \cdots - 144 \)
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| $31$ |
\( T^{6} - 11 T^{5} + \cdots - 6696 \)
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| $37$ |
\( T^{6} - 3 T^{5} + \cdots - 128 \)
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| $41$ |
\( T^{6} + T^{5} + \cdots - 4154 \)
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| $43$ |
\( T^{6} - 9 T^{5} + \cdots - 12184 \)
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| $47$ |
\( T^{6} + 8 T^{5} + \cdots - 63552 \)
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| $53$ |
\( T^{6} - 2 T^{5} + \cdots - 3204 \)
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| $59$ |
\( T^{6} + 6 T^{5} + \cdots + 13136 \)
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| $61$ |
\( T^{6} - 11 T^{5} + \cdots + 31744 \)
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| $67$ |
\( T^{6} + 14 T^{5} + \cdots + 51364 \)
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| $71$ |
\( T^{6} - 19 T^{5} + \cdots + 3312 \)
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| $73$ |
\( T^{6} + T^{5} + \cdots - 3746 \)
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| $79$ |
\( T^{6} - 36 T^{5} + \cdots + 17152 \)
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| $83$ |
\( T^{6} - 2 T^{5} + \cdots + 496 \)
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| $89$ |
\( T^{6} + 5 T^{5} + \cdots - 2541536 \)
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| $97$ |
\( T^{6} + T^{5} + \cdots - 14176 \)
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