Newspace parameters
| Level: | \( N \) | \(=\) | \( 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8330.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(66.5153848837\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.12694016.1 |
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| Defining polynomial: |
\( x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 1 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu - 1 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} - \nu + 3 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 4\nu^{2} + 12\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} + \beta _1 + 12 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 7\beta_{3} + \beta_{2} + 17\beta _1 + 7 \)
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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 |
|
−1.00000 | −2.98922 | 1.00000 | 1.00000 | 2.98922 | 0 | −1.00000 | 5.93543 | −1.00000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.53308 | 1.00000 | 1.00000 | 2.53308 | 0 | −1.00000 | 3.41648 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −0.538903 | 1.00000 | 1.00000 | 0.538903 | 0 | −1.00000 | −2.70958 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.787153 | 1.00000 | 1.00000 | −0.787153 | 0 | −1.00000 | −2.38039 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.20207 | 1.00000 | 1.00000 | −1.20207 | 0 | −1.00000 | −1.55504 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.07198 | 1.00000 | 1.00000 | −2.07198 | 0 | −1.00000 | 1.29310 | −1.00000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(17\) | \( -1 \) |
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 | 8330.2.a.cn | ✓ | 6 |
| 7.b | odd | 2 | 1 | 8330.2.a.cq | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8330.2.a.cn | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 8330.2.a.cq | yes | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8330))\):
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\( T_{3}^{6} + 2T_{3}^{5} - 9T_{3}^{4} - 10T_{3}^{3} + 25T_{3}^{2} - 8 \)
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\( T_{11}^{6} + 6T_{11}^{5} + 7T_{11}^{4} - 10T_{11}^{3} - 11T_{11}^{2} + 4T_{11} + 2 \)
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\( T_{13}^{6} - 22T_{13}^{4} - 16T_{13}^{3} + 100T_{13}^{2} + 160T_{13} + 64 \)
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\( T_{19}^{6} - 6T_{19}^{5} - 9T_{19}^{4} + 82T_{19}^{3} - 131T_{19}^{2} + 68T_{19} - 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
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| $3$ |
\( T^{6} + 2 T^{5} + \cdots - 8 \)
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| $5$ |
\( (T - 1)^{6} \)
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| $7$ |
\( T^{6} \)
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| $11$ |
\( T^{6} + 6 T^{5} + \cdots + 2 \)
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| $13$ |
\( T^{6} - 22 T^{4} + \cdots + 64 \)
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| $17$ |
\( (T - 1)^{6} \)
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| $19$ |
\( T^{6} - 6 T^{5} + \cdots - 4 \)
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| $23$ |
\( T^{6} + 2 T^{5} + \cdots - 1784 \)
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| $29$ |
\( T^{6} + 2 T^{5} + \cdots - 8 \)
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| $31$ |
\( T^{6} + 2 T^{5} + \cdots + 4604 \)
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| $37$ |
\( T^{6} + 12 T^{5} + \cdots + 3592 \)
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| $41$ |
\( T^{6} - 6 T^{5} + \cdots + 508 \)
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| $43$ |
\( T^{6} + 26 T^{5} + \cdots - 47332 \)
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| $47$ |
\( T^{6} + 30 T^{5} + \cdots - 13550 \)
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| $53$ |
\( T^{6} - 14 T^{5} + \cdots + 73636 \)
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| $59$ |
\( T^{6} + 6 T^{5} + \cdots - 51100 \)
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| $61$ |
\( T^{6} - 12 T^{5} + \cdots + 246032 \)
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| $67$ |
\( T^{6} + 14 T^{5} + \cdots + 438848 \)
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| $71$ |
\( T^{6} - 12 T^{5} + \cdots + 56 \)
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| $73$ |
\( T^{6} + 12 T^{5} + \cdots + 64 \)
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| $79$ |
\( T^{6} + 36 T^{5} + \cdots - 7112 \)
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| $83$ |
\( T^{6} + 32 T^{5} + \cdots + 425336 \)
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| $89$ |
\( T^{6} - 8 T^{5} + \cdots - 43904 \)
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| $97$ |
\( T^{6} - 16 T^{5} + \cdots - 146944 \)
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