Newspace parameters
| Level: | \( N \) | \(=\) | \( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8379.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(66.9066518536\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.39110656.1 |
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|
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| Defining polynomial: |
\( x^{6} - 10x^{4} - 4x^{3} + 20x^{2} + 16x + 2 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2793) |
| 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
\( x^{6} - 10x^{4} - 4x^{3} + 20x^{2} + 16x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{5} + 3\nu^{4} + 18\nu^{3} - 14\nu^{2} - 34\nu - 11 ) / 5 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{5} + 3\nu^{4} + 18\nu^{3} - 19\nu^{2} - 29\nu + 9 ) / 5 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} + 2\nu^{4} + 27\nu^{3} - 6\nu^{2} - 46\nu - 14 ) / 5 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{5} + \nu^{4} + 41\nu^{3} + 2\nu^{2} - 88\nu - 27 ) / 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 2\beta_{4} + \beta_{2} + 6\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( -2\beta_{4} - 6\beta_{3} + 9\beta_{2} + 8\beta _1 + 25 \)
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| \(\nu^{5}\) | \(=\) |
\( 9\beta_{5} - 21\beta_{4} - 2\beta_{3} + 13\beta_{2} + 42\beta _1 + 22 \)
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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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−2.73812 | 0 | 5.49729 | 3.91150 | 0 | 0 | −9.57599 | 0 | −10.7102 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.59905 | 0 | 4.75505 | 0.340834 | 0 | 0 | −7.16050 | 0 | −0.885843 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.67222 | 0 | 0.796312 | −3.61790 | 0 | 0 | 2.01283 | 0 | 6.04992 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.05939 | 0 | −0.877693 | −2.46348 | 0 | 0 | −3.04860 | 0 | −2.60979 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.85705 | 0 | 1.44864 | −2.96557 | 0 | 0 | −1.02390 | 0 | −5.50722 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.09294 | 0 | 2.38040 | 0.794617 | 0 | 0 | 0.796162 | 0 | 1.66309 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(19\) | \( +1 \) |
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 | 8379.2.a.cf | 6 | |
| 3.b | odd | 2 | 1 | 2793.2.a.bl | yes | 6 | |
| 7.b | odd | 2 | 1 | 8379.2.a.cg | 6 | ||
| 21.c | even | 2 | 1 | 2793.2.a.bk | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2793.2.a.bk | ✓ | 6 | 21.c | even | 2 | 1 | |
| 2793.2.a.bl | yes | 6 | 3.b | odd | 2 | 1 | |
| 8379.2.a.cf | 6 | 1.a | even | 1 | 1 | trivial | |
| 8379.2.a.cg | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8379))\):
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\( T_{2}^{6} + 2T_{2}^{5} - 11T_{2}^{4} - 16T_{2}^{3} + 41T_{2}^{2} + 30T_{2} - 49 \)
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\( T_{5}^{6} + 4T_{5}^{5} - 14T_{5}^{4} - 68T_{5}^{3} - 16T_{5}^{2} + 96T_{5} - 28 \)
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\( T_{11}^{6} + 8T_{11}^{5} + 2T_{11}^{4} - 88T_{11}^{3} - 84T_{11}^{2} + 176T_{11} - 56 \)
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\( T_{13}^{6} - 4T_{13}^{5} - 12T_{13}^{4} + 40T_{13}^{3} + 48T_{13}^{2} - 64T_{13} - 16 \)
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\( T_{17}^{6} + 8T_{17}^{5} - 34T_{17}^{4} - 332T_{17}^{3} - 304T_{17}^{2} + 224T_{17} + 196 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 2 T^{5} + \cdots - 49 \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} + 4 T^{5} + \cdots - 28 \)
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| $7$ |
\( T^{6} \)
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| $11$ |
\( T^{6} + 8 T^{5} + \cdots - 56 \)
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| $13$ |
\( T^{6} - 4 T^{5} + \cdots - 16 \)
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| $17$ |
\( T^{6} + 8 T^{5} + \cdots + 196 \)
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| $19$ |
\( (T + 1)^{6} \)
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| $23$ |
\( T^{6} + 20 T^{5} + \cdots - 56 \)
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| $29$ |
\( T^{6} + 4 T^{5} + \cdots + 5596 \)
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| $31$ |
\( T^{6} - 16 T^{5} + \cdots + 1008 \)
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| $37$ |
\( T^{6} - 12 T^{5} + \cdots - 7648 \)
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| $41$ |
\( T^{6} - 16 T^{5} + \cdots - 56 \)
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| $43$ |
\( T^{6} - 4 T^{5} + \cdots - 2032 \)
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| $47$ |
\( T^{6} - 12 T^{5} + \cdots + 164 \)
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| $53$ |
\( T^{6} + 4 T^{5} + \cdots - 10628 \)
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| $59$ |
\( T^{6} + 4 T^{5} + \cdots - 76832 \)
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| $61$ |
\( T^{6} - 24 T^{5} + \cdots - 104776 \)
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| $67$ |
\( T^{6} + 8 T^{5} + \cdots - 29192 \)
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| $71$ |
\( T^{6} - 338 T^{4} + \cdots + 126748 \)
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| $73$ |
\( T^{6} - 24 T^{5} + \cdots - 12872 \)
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| $79$ |
\( T^{6} + 16 T^{5} + \cdots - 584 \)
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| $83$ |
\( T^{6} + 12 T^{5} + \cdots + 12676 \)
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| $89$ |
\( T^{6} - 44 T^{5} + \cdots - 158648 \)
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| $97$ |
\( T^{6} + 8 T^{5} + \cdots - 105984 \)
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