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: | \(3\) |
| Coefficient field: | 3.3.257.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 4x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 4x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
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 |
|
−2.49086 | 1.00000 | 4.20440 | −0.286462 | −2.49086 | 1.00000 | −5.49086 | 1.00000 | 0.713538 | |||||||||||||||||||||||||||
| 1.2 | 0.656620 | 1.00000 | −1.56885 | −2.91223 | 0.656620 | 1.00000 | −2.34338 | 1.00000 | −1.91223 | ||||||||||||||||||||||||||||
| 1.3 | 1.83424 | 1.00000 | 1.36445 | 1.19869 | 1.83424 | 1.00000 | −1.16576 | 1.00000 | 2.19869 | ||||||||||||||||||||||||||||
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.q | yes | 3 |
| 17.b | even | 2 | 1 | 6069.2.a.o | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6069.2.a.o | ✓ | 3 | 17.b | even | 2 | 1 | |
| 6069.2.a.q | yes | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6069))\):
|
\( T_{2}^{3} - 5T_{2} + 3 \)
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\( T_{5}^{3} + 2T_{5}^{2} - 3T_{5} - 1 \)
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\( T_{11}^{3} + 2T_{11}^{2} - 23T_{11} + 21 \)
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\( T_{23}^{3} + 2T_{23}^{2} - 3T_{23} - 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 5T + 3 \)
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| $3$ |
\( (T - 1)^{3} \)
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| $5$ |
\( T^{3} + 2 T^{2} + \cdots - 1 \)
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| $7$ |
\( (T - 1)^{3} \)
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| $11$ |
\( T^{3} + 2 T^{2} + \cdots + 21 \)
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| $13$ |
\( T^{3} + 9 T^{2} + \cdots + 9 \)
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| $17$ |
\( T^{3} \)
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| $19$ |
\( T^{3} + T^{2} + \cdots + 105 \)
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| $23$ |
\( T^{3} + 2 T^{2} + \cdots - 1 \)
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| $29$ |
\( T^{3} - 7 T^{2} + \cdots + 183 \)
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| $31$ |
\( T^{3} + T^{2} - 17T + 7 \)
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| $37$ |
\( T^{3} - T^{2} + \cdots + 109 \)
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| $41$ |
\( T^{3} - 3 T^{2} + \cdots + 147 \)
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| $43$ |
\( T^{3} + 15 T^{2} + \cdots - 171 \)
|
| $47$ |
\( T^{3} + 8T^{2} - T - 75 \)
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| $53$ |
\( T^{3} - 3 T^{2} + \cdots - 5 \)
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| $59$ |
\( T^{3} - 3 T^{2} + \cdots + 125 \)
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| $61$ |
\( T^{3} - 13 T^{2} + \cdots + 441 \)
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| $67$ |
\( T^{3} - 5 T^{2} + \cdots - 123 \)
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| $71$ |
\( T^{3} + 13 T^{2} + \cdots - 513 \)
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| $73$ |
\( T^{3} + 9 T^{2} + \cdots - 405 \)
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| $79$ |
\( (T + 13)^{3} \)
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| $83$ |
\( T^{3} + 26 T^{2} + \cdots + 477 \)
|
| $89$ |
\( T^{3} - 33 T^{2} + \cdots - 1135 \)
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| $97$ |
\( T^{3} + 9 T^{2} + \cdots + 67 \)
|
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