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: | 6.6.300125.1 |
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| Defining polynomial: |
\( x^{6} - x^{5} - 7x^{4} + 2x^{3} + 7x^{2} - 2x - 1 \)
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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
\( x^{6} - x^{5} - 7x^{4} + 2x^{3} + 7x^{2} - 2x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( -2\nu^{5} + \nu^{4} + 14\nu^{3} + 4\nu^{2} - 10\nu - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( -2\nu^{5} + 15\nu^{3} + 10\nu^{2} - 10\nu - 5 \)
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| \(\beta_{4}\) | \(=\) |
\( 3\nu^{5} - \nu^{4} - 21\nu^{3} - 9\nu^{2} + 12\nu + 4 \)
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| \(\beta_{5}\) | \(=\) |
\( -4\nu^{5} + \nu^{4} + 29\nu^{3} + 13\nu^{2} - 19\nu - 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{3} + \beta_{2} + \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + \beta_{3} + 2\beta_{2} + 6\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( -6\beta_{5} + 2\beta_{4} + 6\beta_{3} + 9\beta_{2} + 12\beta _1 + 10 \)
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| \(\nu^{5}\) | \(=\) |
\( -5\beta_{5} + 15\beta_{4} + 12\beta_{3} + 20\beta_{2} + 45\beta _1 + 15 \)
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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 | −2.36064 | 1.00000 | 1.00000 | 2.36064 | −2.42483 | −1.00000 | 2.57261 | −1.00000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.57261 | 1.00000 | 1.00000 | 1.57261 | −2.16424 | −1.00000 | −0.526887 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 1.21572 | 1.00000 | 1.00000 | −1.21572 | 3.65820 | −1.00000 | −1.52203 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.52689 | 1.00000 | 1.00000 | −1.52689 | −4.85141 | −1.00000 | −0.668617 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.66862 | 1.00000 | 1.00000 | −1.66862 | −2.17905 | −1.00000 | −0.215718 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.52203 | 1.00000 | 1.00000 | −2.52203 | 2.96132 | −1.00000 | 3.36064 | −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.bh | 6 | |
| 29.b | even | 2 | 1 | 8410.2.a.bi | 6 | ||
| 29.d | even | 7 | 2 | 290.2.k.b | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 290.2.k.b | ✓ | 12 | 29.d | even | 7 | 2 | |
| 8410.2.a.bh | 6 | 1.a | even | 1 | 1 | trivial | |
| 8410.2.a.bi | 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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\( T_{3}^{6} - 3T_{3}^{5} - 6T_{3}^{4} + 24T_{3}^{3} - 3T_{3}^{2} - 41T_{3} + 29 \)
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\( T_{7}^{6} + 5T_{7}^{5} - 18T_{7}^{4} - 107T_{7}^{3} + 11T_{7}^{2} + 558T_{7} + 601 \)
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\( T_{11}^{6} - 13T_{11}^{5} + 53T_{11}^{4} - 32T_{11}^{3} - 258T_{11}^{2} + 501T_{11} - 181 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
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| $3$ |
\( T^{6} - 3 T^{5} + \cdots + 29 \)
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| $5$ |
\( (T - 1)^{6} \)
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| $7$ |
\( T^{6} + 5 T^{5} + \cdots + 601 \)
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| $11$ |
\( T^{6} - 13 T^{5} + \cdots - 181 \)
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| $13$ |
\( T^{6} + 10 T^{5} + \cdots + 71 \)
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| $17$ |
\( T^{6} + 3 T^{5} + \cdots + 281 \)
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| $19$ |
\( T^{6} - 4 T^{5} + \cdots + 1189 \)
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| $23$ |
\( T^{6} + 9 T^{5} + \cdots + 71 \)
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| $29$ |
\( T^{6} \)
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| $31$ |
\( T^{6} - 9 T^{5} + \cdots + 12151 \)
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| $37$ |
\( T^{6} + 17 T^{5} + \cdots + 41 \)
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| $41$ |
\( T^{6} - 16 T^{5} + \cdots + 349 \)
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| $43$ |
\( T^{6} - 9 T^{5} + \cdots + 101569 \)
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| $47$ |
\( T^{6} - 2 T^{5} + \cdots + 71 \)
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| $53$ |
\( T^{6} + 27 T^{5} + \cdots - 24389 \)
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| $59$ |
\( T^{6} - 41 T^{5} + \cdots + 58841 \)
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| $61$ |
\( T^{6} - 19 T^{5} + \cdots + 618169 \)
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| $67$ |
\( T^{6} - 11 T^{5} + \cdots - 139 \)
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| $71$ |
\( T^{6} + 9 T^{5} + \cdots + 24389 \)
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| $73$ |
\( T^{6} - 18 T^{5} + \cdots - 9799 \)
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| $79$ |
\( T^{6} - 14 T^{5} + \cdots - 233659 \)
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| $83$ |
\( T^{6} + 18 T^{5} + \cdots - 123031 \)
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| $89$ |
\( T^{6} - 31 T^{5} + \cdots + 2419789 \)
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| $97$ |
\( T^{6} - 28 T^{5} + \cdots - 2292661 \)
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