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: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.16371248.1 |
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 7x^{4} + 8x^{3} + 15x^{2} - 6x - 8 \)
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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
\( x^{6} - 2x^{5} - 7x^{4} + 8x^{3} + 15x^{2} - 6x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 4\nu^{2} + 5\nu + 2 ) / 2 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 4\nu^{4} - \nu^{3} + 14\nu^{2} - 3\nu - 10 ) / 2 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + 7\nu^{3} - 8\nu^{2} - 13\nu + 4 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{3} + 2\beta_{2} + 6\beta _1 + 3 \)
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| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 3\beta_{3} + 9\beta_{2} + 13\beta _1 + 15 \)
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| \(\nu^{5}\) | \(=\) |
\( 9\beta_{5} - 2\beta_{4} + 13\beta_{3} + 24\beta_{2} + 47\beta _1 + 31 \)
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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 | −3.02951 | 0 | −1.51059 | 0 | −0.303583 | 0 | 6.17793 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.83986 | 0 | 3.05721 | 0 | −4.32043 | 0 | 0.385071 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.570243 | 0 | −2.97371 | 0 | 0.659060 | 0 | −2.67482 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.478300 | 0 | 1.68758 | 0 | 3.58359 | 0 | −2.77123 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.557301 | 0 | 1.63387 | 0 | −1.48700 | 0 | −2.68942 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.36061 | 0 | 0.105635 | 0 | 0.868366 | 0 | 2.57247 | 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.cn | 6 | |
| 4.b | odd | 2 | 1 | 9152.2.a.ct | 6 | ||
| 8.b | even | 2 | 1 | 4576.2.a.r | yes | 6 | |
| 8.d | odd | 2 | 1 | 4576.2.a.o | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4576.2.a.o | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 4576.2.a.r | yes | 6 | 8.b | even | 2 | 1 | |
| 9152.2.a.cn | 6 | 1.a | even | 1 | 1 | trivial | |
| 9152.2.a.ct | 6 | 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9152))\):
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\( T_{3}^{6} + 3T_{3}^{5} - 5T_{3}^{4} - 17T_{3}^{3} - 5T_{3}^{2} + 5T_{3} + 2 \)
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\( T_{5}^{6} - 2T_{5}^{5} - 11T_{5}^{4} + 22T_{5}^{3} + 18T_{5}^{2} - 40T_{5} + 4 \)
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\( T_{7}^{6} + T_{7}^{5} - 17T_{7}^{4} - 5T_{7}^{3} + 27T_{7}^{2} - 5T_{7} - 4 \)
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\( T_{17}^{6} + 6T_{17}^{5} - 26T_{17}^{4} - 88T_{17}^{3} + 148T_{17}^{2} + 328T_{17} - 128 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} + 3 T^{5} + \cdots + 2 \)
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| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 4 \)
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| $7$ |
\( T^{6} + T^{5} - 17 T^{4} + \cdots - 4 \)
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| $11$ |
\( (T - 1)^{6} \)
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| $13$ |
\( (T + 1)^{6} \)
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| $17$ |
\( T^{6} + 6 T^{5} + \cdots - 128 \)
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| $19$ |
\( T^{6} + 10 T^{5} + \cdots + 344 \)
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| $23$ |
\( T^{6} - 8 T^{5} + \cdots + 523 \)
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| $29$ |
\( T^{6} + 3 T^{5} + \cdots - 5744 \)
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| $31$ |
\( T^{6} - 3 T^{5} + \cdots - 8 \)
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| $37$ |
\( T^{6} - 3 T^{5} + \cdots + 5504 \)
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| $41$ |
\( T^{6} - 9 T^{5} + \cdots + 6406 \)
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| $43$ |
\( T^{6} + 13 T^{5} + \cdots - 8 \)
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| $47$ |
\( T^{6} - 4 T^{5} + \cdots - 64 \)
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| $53$ |
\( T^{6} + 10 T^{5} + \cdots + 244 \)
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| $59$ |
\( T^{6} + 18 T^{5} + \cdots - 37936 \)
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| $61$ |
\( T^{6} - T^{5} + \cdots + 5696 \)
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| $67$ |
\( T^{6} + 14 T^{5} + \cdots - 7468 \)
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| $71$ |
\( T^{6} - 23 T^{5} + \cdots + 2597792 \)
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| $73$ |
\( T^{6} - 9 T^{5} + \cdots - 20738 \)
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| $79$ |
\( T^{6} + 12 T^{5} + \cdots - 32192 \)
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| $83$ |
\( T^{6} + 38 T^{5} + \cdots + 5584 \)
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| $89$ |
\( T^{6} - 15 T^{5} + \cdots - 74912 \)
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| $97$ |
\( T^{6} - 11 T^{5} + \cdots + 2864 \)
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