Newspace parameters
| Level: | \( N \) | \(=\) | \( 1218 = 2 \cdot 3 \cdot 7 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1218.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(9.72577896619\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.568.1 |
|
|
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| Defining polynomial: |
\( x^{3} - x^{2} - 6x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{3} - x^{2} - 6x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
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| \(\beta_{2}\) | \(=\) |
\( -\nu^{2} + 3\nu + 3 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 9 ) / 2 \)
|
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 | −1.00000 | 1.00000 | −2.64002 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | 2.64002 | |||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.00000 | 1.00000 | −0.864641 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | 0.864641 | ||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.00000 | 1.00000 | 3.50466 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | −3.50466 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(29\) | \( -1 \) |
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 | 1218.2.a.o | ✓ | 3 |
| 3.b | odd | 2 | 1 | 3654.2.a.bf | 3 | ||
| 4.b | odd | 2 | 1 | 9744.2.a.bm | 3 | ||
| 7.b | odd | 2 | 1 | 8526.2.a.bt | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1218.2.a.o | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 3654.2.a.bf | 3 | 3.b | odd | 2 | 1 | ||
| 8526.2.a.bt | 3 | 7.b | odd | 2 | 1 | ||
| 9744.2.a.bm | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1218))\):
|
\( T_{5}^{3} - 10T_{5} - 8 \)
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\( T_{11}^{3} + 4T_{11}^{2} - 20T_{11} - 64 \)
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\( T_{13}^{3} + 2T_{13}^{2} - 18T_{13} - 44 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{3} \)
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| $3$ |
\( (T + 1)^{3} \)
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| $5$ |
\( T^{3} - 10T - 8 \)
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| $7$ |
\( (T - 1)^{3} \)
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| $11$ |
\( T^{3} + 4 T^{2} + \cdots - 64 \)
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| $13$ |
\( T^{3} + 2 T^{2} + \cdots - 44 \)
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| $17$ |
\( T^{3} + 14 T^{2} + \cdots - 20 \)
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| $19$ |
\( T^{3} - 40T + 64 \)
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| $23$ |
\( T^{3} + 6 T^{2} + \cdots - 8 \)
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| $29$ |
\( (T - 1)^{3} \)
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| $31$ |
\( T^{3} + 4 T^{2} + \cdots - 200 \)
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| $37$ |
\( T^{3} + 2 T^{2} + \cdots - 32 \)
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| $41$ |
\( T^{3} - 2 T^{2} + \cdots - 100 \)
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| $43$ |
\( T^{3} - 2 T^{2} + \cdots + 32 \)
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| $47$ |
\( T^{3} + 8 T^{2} + \cdots - 64 \)
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| $53$ |
\( T^{3} + 10 T^{2} + \cdots - 472 \)
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| $59$ |
\( T^{3} + 20 T^{2} + \cdots + 88 \)
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| $61$ |
\( T^{3} - 8 T^{2} + \cdots + 80 \)
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| $67$ |
\( T^{3} - 4 T^{2} + \cdots + 352 \)
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| $71$ |
\( T^{3} + 18 T^{2} + \cdots - 88 \)
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| $73$ |
\( T^{3} + 24 T^{2} + \cdots - 424 \)
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| $79$ |
\( T^{3} + 2 T^{2} + \cdots + 160 \)
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| $83$ |
\( T^{3} + 20 T^{2} + \cdots + 88 \)
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| $89$ |
\( T^{3} + 2 T^{2} + \cdots + 916 \)
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| $97$ |
\( T^{3} + 8 T^{2} + \cdots - 3256 \)
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