Newspace parameters
| Level: | \( N \) | \(=\) | \( 8118 = 2 \cdot 3^{2} \cdot 11 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8118.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(64.8225563609\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.148.1 |
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|
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| Defining polynomial: |
\( x^{3} - x^{2} - 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 2706) |
| 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
\( x^{3} - x^{2} - 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\beta_{2}\) | \(=\) |
\( -\nu^{2} + 2\nu + 2 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 2 \)
|
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 | 0 | 1.00000 | −1.52543 | 0 | 1.52543 | 1.00000 | 0 | −1.52543 | |||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | −0.630898 | 0 | 0.630898 | 1.00000 | 0 | −0.630898 | ||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | 4.15633 | 0 | −4.15633 | 1.00000 | 0 | 4.15633 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(11\) | \( +1 \) |
| \(41\) | \( +1 \) |
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 | 8118.2.a.bi | 3 | |
| 3.b | odd | 2 | 1 | 2706.2.a.x | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2706.2.a.x | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 8118.2.a.bi | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8118))\):
|
\( T_{5}^{3} - 2T_{5}^{2} - 8T_{5} - 4 \)
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\( T_{7}^{3} + 2T_{7}^{2} - 8T_{7} + 4 \)
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\( T_{13}^{3} + 2T_{13}^{2} - 4T_{13} - 4 \)
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\( T_{17}^{3} + 6T_{17}^{2} - 4T_{17} - 40 \)
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\( T_{19}^{3} + 8T_{19}^{2} - 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{3} \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} - 2 T^{2} + \cdots - 4 \)
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| $7$ |
\( T^{3} + 2 T^{2} + \cdots + 4 \)
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| $11$ |
\( (T + 1)^{3} \)
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| $13$ |
\( T^{3} + 2 T^{2} + \cdots - 4 \)
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| $17$ |
\( T^{3} + 6 T^{2} + \cdots - 40 \)
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| $19$ |
\( T^{3} + 8T^{2} - 32 \)
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| $23$ |
\( T^{3} - 6 T^{2} + \cdots - 4 \)
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| $29$ |
\( T^{3} - 16T - 16 \)
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| $31$ |
\( T^{3} + 16 T^{2} + \cdots + 32 \)
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| $37$ |
\( T^{3} - 2 T^{2} + \cdots + 8 \)
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| $41$ |
\( (T + 1)^{3} \)
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| $43$ |
\( T^{3} + 10 T^{2} + \cdots - 136 \)
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| $47$ |
\( T^{3} + 10T^{2} - 148 \)
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| $53$ |
\( T^{3} + 18 T^{2} + \cdots - 92 \)
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| $59$ |
\( T^{3} - 14 T^{2} + \cdots + 152 \)
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| $61$ |
\( T^{3} + 6 T^{2} + \cdots - 100 \)
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| $67$ |
\( T^{3} - 16T + 16 \)
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| $71$ |
\( T^{3} - 2 T^{2} + \cdots + 452 \)
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| $73$ |
\( T^{3} - 10 T^{2} + \cdots + 8 \)
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| $79$ |
\( T^{3} + 22 T^{2} + \cdots - 1828 \)
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| $83$ |
\( T^{3} + 8 T^{2} + \cdots - 128 \)
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| $89$ |
\( T^{3} - 16 T^{2} + \cdots - 80 \)
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| $97$ |
\( T^{3} + 18 T^{2} + \cdots - 1080 \)
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