Newspace parameters
| Level: | \( N \) | \(=\) | \( 8410 = 2 \cdot 5 \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8410.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(67.1541880999\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.6120149.1 |
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|
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| Defining polynomial: |
\( x^{6} - x^{5} - 12x^{4} + 10x^{3} + 33x^{2} - 15x - 29 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 290) |
| 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
\( x^{6} - x^{5} - 12x^{4} + 10x^{3} + 33x^{2} - 15x - 29 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( 2\nu^{4} - \nu^{3} - 21\nu^{2} + 8\nu + 32 \)
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| \(\beta_{3}\) | \(=\) |
\( 2\nu^{5} - \nu^{4} - 21\nu^{3} + 8\nu^{2} + 33\nu - 1 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 9\nu^{3} + 30\nu^{2} + 5\nu - 40 \)
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| \(\beta_{5}\) | \(=\) |
\( -2\nu^{5} + 3\nu^{4} + 20\nu^{3} - 28\nu^{2} - 25\nu + 29 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{3} - \beta_{2} + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 4\beta_{4} - \beta_{3} + 4\beta_{2} + 6\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( 11\beta_{5} + 2\beta_{4} + 10\beta_{3} - 8\beta_{2} - \beta _1 + 27 \)
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| \(\nu^{5}\) | \(=\) |
\( 12\beta_{5} + 43\beta_{4} - 9\beta_{3} + 42\beta_{2} + 46\beta _1 + 19 \)
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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 |
|
−1.00000 | −2.90180 | 1.00000 | 1.00000 | 2.90180 | −3.12900 | −1.00000 | 5.42042 | −1.00000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.16223 | 1.00000 | 1.00000 | 1.16223 | 1.19995 | −1.00000 | −1.64922 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.12006 | 1.00000 | 1.00000 | 1.12006 | 4.76374 | −1.00000 | −1.74546 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.56511 | 1.00000 | 1.00000 | −1.56511 | −1.26978 | −1.00000 | −0.550442 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.65482 | 1.00000 | 1.00000 | −1.65482 | 0.525122 | −1.00000 | −0.261584 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.96417 | 1.00000 | 1.00000 | −2.96417 | −1.09003 | −1.00000 | 5.78629 | −1.00000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(29\) | \( +1 \) |
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 | 8410.2.a.be | 6 | |
| 29.b | even | 2 | 1 | 8410.2.a.bl | 6 | ||
| 29.e | even | 14 | 2 | 290.2.k.a | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 290.2.k.a | ✓ | 12 | 29.e | even | 14 | 2 | |
| 8410.2.a.be | 6 | 1.a | even | 1 | 1 | trivial | |
| 8410.2.a.bl | 6 | 29.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8410))\):
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\( T_{3}^{6} - T_{3}^{5} - 12T_{3}^{4} + 10T_{3}^{3} + 33T_{3}^{2} - 15T_{3} - 29 \)
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\( T_{7}^{6} - T_{7}^{5} - 18T_{7}^{4} - 7T_{7}^{3} + 33T_{7}^{2} + 12T_{7} - 13 \)
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\( T_{11}^{6} - 9T_{11}^{5} + 5T_{11}^{4} + 70T_{11}^{3} + 52T_{11}^{2} - 15T_{11} - 13 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
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| $3$ |
\( T^{6} - T^{5} + \cdots - 29 \)
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| $5$ |
\( (T - 1)^{6} \)
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| $7$ |
\( T^{6} - T^{5} + \cdots - 13 \)
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| $11$ |
\( T^{6} - 9 T^{5} + \cdots - 13 \)
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| $13$ |
\( T^{6} + 6 T^{5} + \cdots - 1967 \)
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| $17$ |
\( T^{6} - 27 T^{5} + \cdots - 8021 \)
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| $19$ |
\( T^{6} - 12 T^{5} + \cdots - 71 \)
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| $23$ |
\( T^{6} - T^{5} + \cdots - 463 \)
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| $29$ |
\( T^{6} \)
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| $31$ |
\( T^{6} - 13 T^{5} + \cdots - 2561 \)
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| $37$ |
\( T^{6} - 35 T^{5} + \cdots - 85457 \)
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| $41$ |
\( T^{6} - 4 T^{5} + \cdots - 31843 \)
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| $43$ |
\( T^{6} + 9 T^{5} + \cdots - 13 \)
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| $47$ |
\( (T^{3} - 3 T^{2} + \cdots + 293)^{2} \)
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| $53$ |
\( T^{6} + 13 T^{5} + \cdots + 301 \)
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| $59$ |
\( T^{6} + 9 T^{5} + \cdots - 22939 \)
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| $61$ |
\( T^{6} - 17 T^{5} + \cdots - 287 \)
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| $67$ |
\( T^{6} - 11 T^{5} + \cdots - 1594489 \)
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| $71$ |
\( T^{6} - 3 T^{5} + \cdots + 125 \)
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| $73$ |
\( T^{6} + 14 T^{5} + \cdots + 4843 \)
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| $79$ |
\( T^{6} - 10 T^{5} + \cdots - 88423 \)
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| $83$ |
\( T^{6} - 34 T^{5} + \cdots - 643069 \)
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| $89$ |
\( T^{6} - T^{5} + \cdots - 35447 \)
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| $97$ |
\( T^{6} - 245 T^{4} + \cdots + 13769 \)
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