Newspace parameters
| Level: | \( N \) | \(=\) | \( 792 = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 792.m (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(6.32415184009\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-11})\) |
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| Defining polynomial: |
\( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu + 5 \)
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| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 3\nu^{2} + 19\nu - 9 ) / 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{3} - 6\nu^{2} + 44\nu - 21 ) / 3 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{3} - 2\beta_{2} + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 2\beta_{2} + 2\beta _1 - 9 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -8\beta_{3} + 19\beta_{2} + 3\beta _1 - 14 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/792\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(199\) | \(353\) | \(397\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 197.1 |
|
− | 1.41421i | 0 | −2.00000 | 0 | 0 | 0 | 2.82843i | 0 | 0 | |||||||||||||||||||||||||||||
| 197.2 | − | 1.41421i | 0 | −2.00000 | 0 | 0 | 0 | 2.82843i | 0 | 0 | ||||||||||||||||||||||||||||||
| 197.3 | 1.41421i | 0 | −2.00000 | 0 | 0 | 0 | − | 2.82843i | 0 | 0 | ||||||||||||||||||||||||||||||
| 197.4 | 1.41421i | 0 | −2.00000 | 0 | 0 | 0 | − | 2.82843i | 0 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 88.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-22}) \) |
| 3.b | odd | 2 | 1 | inner |
| 264.m | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 792.2.m.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 792.2.m.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 3168.2.m.a | 4 | ||
| 8.b | even | 2 | 1 | 792.2.m.b | yes | 4 | |
| 8.d | odd | 2 | 1 | 3168.2.m.b | 4 | ||
| 11.b | odd | 2 | 1 | 792.2.m.b | yes | 4 | |
| 12.b | even | 2 | 1 | 3168.2.m.a | 4 | ||
| 24.f | even | 2 | 1 | 3168.2.m.b | 4 | ||
| 24.h | odd | 2 | 1 | 792.2.m.b | yes | 4 | |
| 33.d | even | 2 | 1 | 792.2.m.b | yes | 4 | |
| 44.c | even | 2 | 1 | 3168.2.m.b | 4 | ||
| 88.b | odd | 2 | 1 | CM | 792.2.m.a | ✓ | 4 |
| 88.g | even | 2 | 1 | 3168.2.m.a | 4 | ||
| 132.d | odd | 2 | 1 | 3168.2.m.b | 4 | ||
| 264.m | even | 2 | 1 | inner | 792.2.m.a | ✓ | 4 |
| 264.p | odd | 2 | 1 | 3168.2.m.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 792.2.m.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 792.2.m.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 792.2.m.a | ✓ | 4 | 88.b | odd | 2 | 1 | CM |
| 792.2.m.a | ✓ | 4 | 264.m | even | 2 | 1 | inner |
| 792.2.m.b | yes | 4 | 8.b | even | 2 | 1 | |
| 792.2.m.b | yes | 4 | 11.b | odd | 2 | 1 | |
| 792.2.m.b | yes | 4 | 24.h | odd | 2 | 1 | |
| 792.2.m.b | yes | 4 | 33.d | even | 2 | 1 | |
| 3168.2.m.a | 4 | 4.b | odd | 2 | 1 | ||
| 3168.2.m.a | 4 | 12.b | even | 2 | 1 | ||
| 3168.2.m.a | 4 | 88.g | even | 2 | 1 | ||
| 3168.2.m.a | 4 | 264.p | odd | 2 | 1 | ||
| 3168.2.m.b | 4 | 8.d | odd | 2 | 1 | ||
| 3168.2.m.b | 4 | 24.f | even | 2 | 1 | ||
| 3168.2.m.b | 4 | 44.c | even | 2 | 1 | ||
| 3168.2.m.b | 4 | 132.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(792, [\chi])\):
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\( T_{5} \)
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\( T_{13}^{2} + 4T_{13} - 18 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{2} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( (T^{2} + 11)^{2} \)
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| $13$ |
\( (T^{2} + 4 T - 18)^{2} \)
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| $17$ |
\( T^{4} \)
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| $19$ |
\( (T^{2} - 8 T - 6)^{2} \)
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| $23$ |
\( T^{4} + 92T^{2} + 1764 \)
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| $29$ |
\( (T^{2} + 44)^{2} \)
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| $31$ |
\( (T^{2} - 88)^{2} \)
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| $37$ |
\( T^{4} \)
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| $41$ |
\( T^{4} \)
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| $43$ |
\( (T^{2} + 16 T + 42)^{2} \)
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| $47$ |
\( T^{4} + 188T^{2} + 36 \)
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| $53$ |
\( T^{4} \)
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| $59$ |
\( T^{4} \)
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| $61$ |
\( (T^{2} - 20 T + 78)^{2} \)
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| $67$ |
\( T^{4} \)
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| $71$ |
\( T^{4} + 284T^{2} + 2916 \)
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| $73$ |
\( T^{4} \)
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| $79$ |
\( T^{4} \)
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| $83$ |
\( (T^{2} + 44)^{2} \)
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| $89$ |
\( T^{4} + 356 T^{2} + 30276 \)
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| $97$ |
\( (T^{2} - 352)^{2} \)
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