Newspace parameters
| Level: | \( N \) | \(=\) | \( 1682 = 2 \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1682.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(13.4308376200\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.13716913.1 |
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 13x^{4} + 17x^{3} + 52x^{2} - 32x - 64 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 58) |
| 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
\( x^{6} - 2x^{5} - 13x^{4} + 17x^{3} + 52x^{2} - 32x - 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 13\nu^{3} + 17\nu^{2} + 36\nu - 16 ) / 16 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 9\nu^{2} + 9\nu + 16 ) / 4 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 9\nu^{2} + 16\nu + 4 ) / 4 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{5} - 10\nu^{4} - 49\nu^{3} + 69\nu^{2} + 100\nu - 96 ) / 16 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} - \beta_{2} + 6 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} - 6\beta_{2} + 5\beta _1 + 7 \)
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| \(\nu^{4}\) | \(=\) |
\( 13\beta_{5} - 11\beta_{4} + 4\beta_{3} - 21\beta_{2} + \beta _1 + 52 \)
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| \(\nu^{5}\) | \(=\) |
\( 35\beta_{5} - 18\beta_{4} + 8\beta_{3} - 87\beta_{2} + 31\beta _1 + 109 \)
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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 | −2.44077 | 1.00000 | −2.88581 | −2.44077 | −3.04359 | 1.00000 | 2.95734 | −2.88581 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.74168 | 1.00000 | −0.494698 | −1.74168 | 3.13840 | 1.00000 | 0.0334417 | −0.494698 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.17047 | 1.00000 | −2.97240 | −1.17047 | 0.520906 | 1.00000 | −1.63001 | −2.97240 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.63883 | 1.00000 | 1.19379 | 1.63883 | 2.04359 | 1.00000 | −0.314239 | 1.19379 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.29664 | 1.00000 | 3.54362 | 2.29664 | −4.13840 | 1.00000 | 2.27454 | 3.54362 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 3.41744 | 1.00000 | 1.61551 | 3.41744 | −1.52091 | 1.00000 | 8.67893 | 1.61551 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(29\) | \( +1 \) |
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 | 1682.2.a.t | 6 | |
| 29.b | even | 2 | 1 | 1682.2.a.q | 6 | ||
| 29.c | odd | 4 | 2 | 1682.2.b.i | 12 | ||
| 29.d | even | 7 | 2 | 58.2.d.b | ✓ | 12 | |
| 87.j | odd | 14 | 2 | 522.2.k.h | 12 | ||
| 116.j | odd | 14 | 2 | 464.2.u.h | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.2.d.b | ✓ | 12 | 29.d | even | 7 | 2 | |
| 464.2.u.h | 12 | 116.j | odd | 14 | 2 | ||
| 522.2.k.h | 12 | 87.j | odd | 14 | 2 | ||
| 1682.2.a.q | 6 | 29.b | even | 2 | 1 | ||
| 1682.2.a.t | 6 | 1.a | even | 1 | 1 | trivial | |
| 1682.2.b.i | 12 | 29.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1682))\):
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\( T_{3}^{6} - 2T_{3}^{5} - 13T_{3}^{4} + 17T_{3}^{3} + 52T_{3}^{2} - 32T_{3} - 64 \)
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\( T_{5}^{6} - 17T_{5}^{4} + 66T_{5}^{2} - 28T_{5} - 29 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
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| $3$ |
\( T^{6} - 2 T^{5} + \cdots - 64 \)
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| $5$ |
\( T^{6} - 17 T^{4} + \cdots - 29 \)
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| $7$ |
\( T^{6} + 3 T^{5} + \cdots - 64 \)
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| $11$ |
\( T^{6} + T^{5} + \cdots - 64 \)
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| $13$ |
\( T^{6} + 3 T^{5} + \cdots - 7 \)
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| $17$ |
\( T^{6} + 6 T^{5} + \cdots - 1259 \)
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| $19$ |
\( T^{6} - 18 T^{5} + \cdots - 3584 \)
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| $23$ |
\( T^{6} + 14 T^{5} + \cdots - 2752 \)
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| $29$ |
\( T^{6} \)
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| $31$ |
\( T^{6} - 17 T^{5} + \cdots + 1856 \)
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| $37$ |
\( T^{6} - 12 T^{5} + \cdots + 8273 \)
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| $41$ |
\( T^{6} + 15 T^{5} + \cdots - 9653 \)
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| $43$ |
\( T^{6} - T^{5} + \cdots - 64 \)
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| $47$ |
\( T^{6} - 8 T^{5} + \cdots + 2752 \)
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| $53$ |
\( T^{6} - 3 T^{5} + \cdots - 5977 \)
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| $59$ |
\( T^{6} - 19 T^{5} + \cdots - 3584 \)
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| $61$ |
\( T^{6} - 15 T^{5} + \cdots + 9857 \)
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| $67$ |
\( T^{6} - 21 T^{5} + \cdots - 64 \)
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| $71$ |
\( T^{6} + 9 T^{5} + \cdots + 60992 \)
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| $73$ |
\( T^{6} - 31 T^{5} + \cdots - 700517 \)
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| $79$ |
\( T^{6} + 2 T^{5} + \cdots - 337408 \)
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| $83$ |
\( T^{6} + 2 T^{5} + \cdots - 448 \)
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| $89$ |
\( T^{6} - 2 T^{5} + \cdots - 11243 \)
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| $97$ |
\( T^{6} - 43 T^{5} + \cdots - 190568 \)
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