Newspace parameters
| Level: | \( N \) | \(=\) | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1014.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(59.8279367458\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
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|
|
| Defining polynomial: |
\( x^{6} + 566x^{4} + 80089x^{2} + 736164 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 78) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} + 566x^{4} + 80089x^{2} + 736164 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 283\nu ) / 858 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} - 283\nu^{2} ) / 858 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{4} + 2273\nu^{2} + 163020 ) / 3432 \)
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| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 473\nu^{3} + 49480\nu ) / 3432 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 4\beta_{4} + 5\beta_{3} - 190 \)
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| \(\nu^{3}\) | \(=\) |
\( 858\beta_{2} - 283\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -1132\beta_{4} - 2273\beta_{3} + 53770 \)
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| \(\nu^{5}\) | \(=\) |
\( 3432\beta_{5} - 405834\beta_{2} + 84379\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).
| \(n\) | \(677\) | \(847\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
− | 2.00000i | 3.00000 | −4.00000 | − | 19.0305i | − | 6.00000i | 33.7672i | 8.00000i | 9.00000 | −38.0610 | |||||||||||||||||||||||||||||||||
| 337.2 | − | 2.00000i | 3.00000 | −4.00000 | − | 7.14134i | − | 6.00000i | − | 35.1063i | 8.00000i | 9.00000 | −14.2827 | |||||||||||||||||||||||||||||||||
| 337.3 | − | 2.00000i | 3.00000 | −4.00000 | 14.1718i | − | 6.00000i | 18.3392i | 8.00000i | 9.00000 | 28.3437 | |||||||||||||||||||||||||||||||||||
| 337.4 | 2.00000i | 3.00000 | −4.00000 | − | 14.1718i | 6.00000i | − | 18.3392i | − | 8.00000i | 9.00000 | 28.3437 | ||||||||||||||||||||||||||||||||||
| 337.5 | 2.00000i | 3.00000 | −4.00000 | 7.14134i | 6.00000i | 35.1063i | − | 8.00000i | 9.00000 | −14.2827 | ||||||||||||||||||||||||||||||||||||
| 337.6 | 2.00000i | 3.00000 | −4.00000 | 19.0305i | 6.00000i | − | 33.7672i | − | 8.00000i | 9.00000 | −38.0610 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1014.4.b.m | 6 | |
| 13.b | even | 2 | 1 | inner | 1014.4.b.m | 6 | |
| 13.d | odd | 4 | 1 | 1014.4.a.u | 3 | ||
| 13.d | odd | 4 | 1 | 1014.4.a.y | 3 | ||
| 13.f | odd | 12 | 2 | 78.4.e.d | ✓ | 6 | |
| 39.k | even | 12 | 2 | 234.4.h.j | 6 | ||
| 52.l | even | 12 | 2 | 624.4.q.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.4.e.d | ✓ | 6 | 13.f | odd | 12 | 2 | |
| 234.4.h.j | 6 | 39.k | even | 12 | 2 | ||
| 624.4.q.h | 6 | 52.l | even | 12 | 2 | ||
| 1014.4.a.u | 3 | 13.d | odd | 4 | 1 | ||
| 1014.4.a.y | 3 | 13.d | odd | 4 | 1 | ||
| 1014.4.b.m | 6 | 1.a | even | 1 | 1 | trivial | |
| 1014.4.b.m | 6 | 13.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1014, [\chi])\):
|
\( T_{5}^{6} + 614T_{5}^{4} + 101449T_{5}^{2} + 3709476 \)
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\( T_{7}^{6} + 2709T_{7}^{4} + 2203260T_{7}^{2} + 472627600 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{3} \)
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| $3$ |
\( (T - 3)^{6} \)
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| $5$ |
\( T^{6} + 614 T^{4} + \cdots + 3709476 \)
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| $7$ |
\( T^{6} + 2709 T^{4} + \cdots + 472627600 \)
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| $11$ |
\( T^{6} + \cdots + 21778675776 \)
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| $13$ |
\( T^{6} \)
|
| $17$ |
\( (T^{3} + 16 T^{2} + \cdots - 175392)^{2} \)
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| $19$ |
\( T^{6} + \cdots + 39394310400 \)
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| $23$ |
\( (T^{3} + 296 T^{2} + \cdots + 440280)^{2} \)
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| $29$ |
\( (T^{3} + 256 T^{2} + \cdots - 713970)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 1248636395776 \)
|
| $37$ |
\( T^{6} + \cdots + 12134674712196 \)
|
| $41$ |
\( T^{6} + \cdots + 30479011808400 \)
|
| $43$ |
\( (T^{3} + 55 T^{2} + \cdots - 3268)^{2} \)
|
| $47$ |
\( T^{6} + 14624 T^{4} + \cdots + 78428736 \)
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| $53$ |
\( (T^{3} + 1106 T^{2} + \cdots + 41277168)^{2} \)
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| $59$ |
\( T^{6} + \cdots + 343510935266304 \)
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| $61$ |
\( (T^{3} - 487 T^{2} + \cdots + 103143115)^{2} \)
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| $67$ |
\( T^{6} + \cdots + 947501727258256 \)
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| $71$ |
\( T^{6} + \cdots + 92\!\cdots\!96 \)
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| $73$ |
\( T^{6} + \cdots + 13\!\cdots\!25 \)
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| $79$ |
\( (T^{3} + 1107 T^{2} + \cdots - 16936400)^{2} \)
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| $83$ |
\( T^{6} + \cdots + 37641464349696 \)
|
| $89$ |
\( T^{6} + \cdots + 29\!\cdots\!64 \)
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| $97$ |
\( T^{6} + \cdots + 54\!\cdots\!24 \)
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