Newspace parameters
| Level: | \( N \) | \(=\) | \( 6069 = 3 \cdot 7 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6069.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(48.4612089867\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.2803712.1 |
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| Defining polynomial: |
\( x^{6} - 6x^{4} + 8x^{2} - 2 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 357) |
| 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
\( x^{6} - 6x^{4} + 8x^{2} - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 3 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 4\nu \)
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| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 6\nu^{3} - 7\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 3\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 5\beta_{2} + 7 \)
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| \(\nu^{5}\) | \(=\) |
\( 5\beta_{5} + 6\beta_{4} + 11\beta_1 \)
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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 |
|
−2.05288 | −1.00000 | 2.21432 | 2.57565 | 2.05288 | −1.00000 | −0.439973 | 1.00000 | −5.28751 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.20864 | −1.00000 | −0.539189 | −2.16204 | 1.20864 | −1.00000 | 3.06897 | 1.00000 | 2.61313 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.569973 | −1.00000 | −1.67513 | 0.169110 | 0.569973 | −1.00000 | 2.09473 | 1.00000 | −0.0963880 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.569973 | −1.00000 | −1.67513 | −1.51937 | −0.569973 | −1.00000 | −2.09473 | 1.00000 | −0.866001 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.20864 | −1.00000 | −0.539189 | 3.08366 | −1.20864 | −1.00000 | −3.06897 | 1.00000 | 3.72704 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.05288 | −1.00000 | 2.21432 | 3.85299 | −2.05288 | −1.00000 | 0.439973 | 1.00000 | 7.90972 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(17\) | \( +1 \) |
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 | 6069.2.a.x | 6 | |
| 17.b | even | 2 | 1 | 6069.2.a.y | 6 | ||
| 17.d | even | 8 | 2 | 357.2.k.a | ✓ | 12 | |
| 51.g | odd | 8 | 2 | 1071.2.n.a | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 357.2.k.a | ✓ | 12 | 17.d | even | 8 | 2 | |
| 1071.2.n.a | 12 | 51.g | odd | 8 | 2 | ||
| 6069.2.a.x | 6 | 1.a | even | 1 | 1 | trivial | |
| 6069.2.a.y | 6 | 17.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6069))\):
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\( T_{2}^{6} - 6T_{2}^{4} + 8T_{2}^{2} - 2 \)
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\( T_{5}^{6} - 6T_{5}^{5} - T_{5}^{4} + 48T_{5}^{3} - 23T_{5}^{2} - 98T_{5} + 17 \)
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\( T_{11}^{6} - 2T_{11}^{5} - 43T_{11}^{4} + 32T_{11}^{3} + 423T_{11}^{2} + 18T_{11} - 17 \)
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\( T_{23}^{6} + 10T_{23}^{5} - 7T_{23}^{4} - 124T_{23}^{3} + 119T_{23}^{2} + 234T_{23} - 241 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 6 T^{4} + \cdots - 2 \)
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| $3$ |
\( (T + 1)^{6} \)
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| $5$ |
\( T^{6} - 6 T^{5} + \cdots + 17 \)
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| $7$ |
\( (T + 1)^{6} \)
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| $11$ |
\( T^{6} - 2 T^{5} + \cdots - 17 \)
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| $13$ |
\( T^{6} + 2 T^{5} + \cdots + 167 \)
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| $17$ |
\( T^{6} \)
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| $19$ |
\( T^{6} + 10 T^{5} + \cdots + 151 \)
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| $23$ |
\( T^{6} + 10 T^{5} + \cdots - 241 \)
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| $29$ |
\( T^{6} + 4 T^{5} + \cdots + 10300 \)
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| $31$ |
\( T^{6} - 108 T^{4} + \cdots - 13346 \)
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| $37$ |
\( T^{6} - 16 T^{5} + \cdots + 26674 \)
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| $41$ |
\( T^{6} - 18 T^{5} + \cdots + 601 \)
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| $43$ |
\( T^{6} + 26 T^{5} + \cdots + 112681 \)
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| $47$ |
\( T^{6} + 16 T^{5} + \cdots - 2558 \)
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| $53$ |
\( T^{6} - 12 T^{5} + \cdots + 43166 \)
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| $59$ |
\( T^{6} + 12 T^{5} + \cdots - 765118 \)
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| $61$ |
\( T^{6} - 20 T^{5} + \cdots + 5134 \)
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| $67$ |
\( T^{6} + 28 T^{5} + \cdots - 412 \)
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| $71$ |
\( T^{6} - 20 T^{5} + \cdots + 22172 \)
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| $73$ |
\( T^{6} - 36 T^{5} + \cdots + 2046400 \)
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| $79$ |
\( T^{6} + 12 T^{5} + \cdots - 7022 \)
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| $83$ |
\( T^{6} + 12 T^{5} + \cdots + 6200 \)
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| $89$ |
\( T^{6} + 32 T^{5} + \cdots + 29824 \)
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| $97$ |
\( T^{6} - 36 T^{5} + \cdots - 1932808 \)
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