Newspace parameters
| Level: | \( N \) | \(=\) | \( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8712.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(69.5656702409\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1436.1 |
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|
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| Defining polynomial: |
\( x^{3} - 11x - 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{3} - 11x - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 7 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 7 \)
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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 |
|
0 | 0 | 0 | −3.76644 | 0 | 0.346835 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 1.28282 | 0 | 2.78872 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 2.48361 | 0 | −4.13555 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 8712.2.a.bw | yes | 3 |
| 3.b | odd | 2 | 1 | 8712.2.a.bv | ✓ | 3 | |
| 11.b | odd | 2 | 1 | 8712.2.a.bx | yes | 3 | |
| 33.d | even | 2 | 1 | 8712.2.a.by | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8712.2.a.bv | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 8712.2.a.bw | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 8712.2.a.bx | yes | 3 | 11.b | odd | 2 | 1 | |
| 8712.2.a.by | yes | 3 | 33.d | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8712))\):
|
\( T_{5}^{3} - 11T_{5} + 12 \)
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\( T_{7}^{3} + T_{7}^{2} - 12T_{7} + 4 \)
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\( T_{13}^{3} - 4T_{13}^{2} - 7T_{13} + 6 \)
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\( T_{17}^{3} - 8T_{17}^{2} - 23T_{17} + 192 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} - 11T + 12 \)
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| $7$ |
\( T^{3} + T^{2} - 12T + 4 \)
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| $11$ |
\( T^{3} \)
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| $13$ |
\( T^{3} - 4 T^{2} + \cdots + 6 \)
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| $17$ |
\( T^{3} - 8 T^{2} + \cdots + 192 \)
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| $19$ |
\( T^{3} + T^{2} + \cdots + 32 \)
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| $23$ |
\( T^{3} - 4 T^{2} + \cdots + 128 \)
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| $29$ |
\( T^{3} - 12 T^{2} + \cdots - 8 \)
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| $31$ |
\( T^{3} - T^{2} + \cdots - 16 \)
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| $37$ |
\( T^{3} + 3 T^{2} + \cdots - 139 \)
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| $41$ |
\( T^{3} - 4 T^{2} + \cdots + 356 \)
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| $43$ |
\( T^{3} - 44T + 96 \)
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| $47$ |
\( (T - 4)^{3} \)
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| $53$ |
\( T^{3} + 12 T^{2} + \cdots + 32 \)
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| $59$ |
\( T^{3} + 20 T^{2} + \cdots + 32 \)
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| $61$ |
\( T^{3} - 7 T^{2} + \cdots + 116 \)
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| $67$ |
\( T^{3} + 11 T^{2} + \cdots - 4 \)
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| $71$ |
\( T^{3} - 8 T^{2} + \cdots + 48 \)
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| $73$ |
\( T^{3} + 3 T^{2} + \cdots + 316 \)
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| $79$ |
\( T^{3} + 15 T^{2} + \cdots - 832 \)
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| $83$ |
\( T^{3} - 12 T^{2} + \cdots + 1408 \)
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| $89$ |
\( T^{3} - 28 T^{2} + \cdots - 492 \)
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| $97$ |
\( T^{3} + T^{2} + \cdots - 1229 \)
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