Newspace parameters
| Level: | \( N \) | \(=\) | \( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3168.h (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(25.2966073603\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
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|
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| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 264) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 4\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 2 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times\).
| \(n\) | \(353\) | \(991\) | \(1189\) | \(1729\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2287.1 |
|
0 | 0 | 0 | 0 | 0 | −2.44949 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 2287.2 | 0 | 0 | 0 | 0 | 0 | −2.44949 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 2287.3 | 0 | 0 | 0 | 0 | 0 | 2.44949 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 2287.4 | 0 | 0 | 0 | 0 | 0 | 2.44949 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 88.g | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3168.2.h.e | 4 | |
| 3.b | odd | 2 | 1 | 1056.2.h.c | 4 | ||
| 4.b | odd | 2 | 1 | 792.2.h.d | 4 | ||
| 8.b | even | 2 | 1 | 792.2.h.d | 4 | ||
| 8.d | odd | 2 | 1 | inner | 3168.2.h.e | 4 | |
| 11.b | odd | 2 | 1 | inner | 3168.2.h.e | 4 | |
| 12.b | even | 2 | 1 | 264.2.h.c | ✓ | 4 | |
| 24.f | even | 2 | 1 | 1056.2.h.c | 4 | ||
| 24.h | odd | 2 | 1 | 264.2.h.c | ✓ | 4 | |
| 33.d | even | 2 | 1 | 1056.2.h.c | 4 | ||
| 44.c | even | 2 | 1 | 792.2.h.d | 4 | ||
| 88.b | odd | 2 | 1 | 792.2.h.d | 4 | ||
| 88.g | even | 2 | 1 | inner | 3168.2.h.e | 4 | |
| 132.d | odd | 2 | 1 | 264.2.h.c | ✓ | 4 | |
| 264.m | even | 2 | 1 | 264.2.h.c | ✓ | 4 | |
| 264.p | odd | 2 | 1 | 1056.2.h.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 264.2.h.c | ✓ | 4 | 12.b | even | 2 | 1 | |
| 264.2.h.c | ✓ | 4 | 24.h | odd | 2 | 1 | |
| 264.2.h.c | ✓ | 4 | 132.d | odd | 2 | 1 | |
| 264.2.h.c | ✓ | 4 | 264.m | even | 2 | 1 | |
| 792.2.h.d | 4 | 4.b | odd | 2 | 1 | ||
| 792.2.h.d | 4 | 8.b | even | 2 | 1 | ||
| 792.2.h.d | 4 | 44.c | even | 2 | 1 | ||
| 792.2.h.d | 4 | 88.b | odd | 2 | 1 | ||
| 1056.2.h.c | 4 | 3.b | odd | 2 | 1 | ||
| 1056.2.h.c | 4 | 24.f | even | 2 | 1 | ||
| 1056.2.h.c | 4 | 33.d | even | 2 | 1 | ||
| 1056.2.h.c | 4 | 264.p | odd | 2 | 1 | ||
| 3168.2.h.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 3168.2.h.e | 4 | 8.d | odd | 2 | 1 | inner | |
| 3168.2.h.e | 4 | 11.b | odd | 2 | 1 | inner | |
| 3168.2.h.e | 4 | 88.g | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3168, [\chi])\):
|
\( T_{5} \)
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\( T_{7}^{2} - 6 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( (T^{2} - 6)^{2} \)
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| $11$ |
\( (T^{2} - 6 T + 11)^{2} \)
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| $13$ |
\( (T^{2} - 24)^{2} \)
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| $17$ |
\( (T^{2} + 50)^{2} \)
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| $19$ |
\( (T^{2} + 18)^{2} \)
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| $23$ |
\( (T^{2} + 48)^{2} \)
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| $29$ |
\( (T^{2} - 6)^{2} \)
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| $31$ |
\( (T^{2} + 108)^{2} \)
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| $37$ |
\( (T^{2} + 12)^{2} \)
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| $41$ |
\( (T^{2} + 50)^{2} \)
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| $43$ |
\( (T^{2} + 18)^{2} \)
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| $47$ |
\( (T^{2} + 108)^{2} \)
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| $53$ |
\( T^{4} \)
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| $59$ |
\( (T + 6)^{4} \)
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| $61$ |
\( (T^{2} - 96)^{2} \)
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| $67$ |
\( (T + 8)^{4} \)
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| $71$ |
\( (T^{2} + 12)^{2} \)
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| $73$ |
\( T^{4} \)
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| $79$ |
\( (T^{2} - 6)^{2} \)
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| $83$ |
\( (T^{2} + 32)^{2} \)
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| $89$ |
\( (T + 6)^{4} \)
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| $97$ |
\( (T + 8)^{4} \)
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