Newspace parameters
| Level: | \( N \) | \(=\) | \( 8464 = 2^{4} \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8464.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(67.5853802708\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.621.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 6x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 529) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $N(\mathrm{U}(1))$ |
$q$-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
\( x^{3} - 6x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + \beta _1 + 8 ) / 2 \)
|
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 |
|
0 | −3.20147 | 0 | 0 | 0 | 0 | 0 | 7.24943 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0.454904 | 0 | 0 | 0 | 0 | 0 | −2.79306 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 2.74657 | 0 | 0 | 0 | 0 | 0 | 4.54364 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(23\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 8464.2.a.bk | 3 | |
| 4.b | odd | 2 | 1 | 529.2.a.f | ✓ | 3 | |
| 12.b | even | 2 | 1 | 4761.2.a.ba | 3 | ||
| 23.b | odd | 2 | 1 | CM | 8464.2.a.bk | 3 | |
| 92.b | even | 2 | 1 | 529.2.a.f | ✓ | 3 | |
| 92.g | odd | 22 | 10 | 529.2.c.p | 30 | ||
| 92.h | even | 22 | 10 | 529.2.c.p | 30 | ||
| 276.h | odd | 2 | 1 | 4761.2.a.ba | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 529.2.a.f | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 529.2.a.f | ✓ | 3 | 92.b | even | 2 | 1 | |
| 529.2.c.p | 30 | 92.g | odd | 22 | 10 | ||
| 529.2.c.p | 30 | 92.h | even | 22 | 10 | ||
| 4761.2.a.ba | 3 | 12.b | even | 2 | 1 | ||
| 4761.2.a.ba | 3 | 276.h | odd | 2 | 1 | ||
| 8464.2.a.bk | 3 | 1.a | even | 1 | 1 | trivial | |
| 8464.2.a.bk | 3 | 23.b | odd | 2 | 1 | CM | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8464))\):
|
\( T_{3}^{3} - 9T_{3} + 4 \)
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\( T_{5} \)
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\( T_{7} \)
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\( T_{13}^{3} - 39T_{13} + 74 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 9T + 4 \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} \)
|
| $13$ |
\( T^{3} - 39T + 74 \)
|
| $17$ |
\( T^{3} \)
|
| $19$ |
\( T^{3} \)
|
| $23$ |
\( T^{3} \)
|
| $29$ |
\( T^{3} - 87T - 282 \)
|
| $31$ |
\( T^{3} - 93T - 344 \)
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| $37$ |
\( T^{3} \)
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| $41$ |
\( T^{3} - 123T - 426 \)
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| $43$ |
\( T^{3} \)
|
| $47$ |
\( T^{3} - 141T + 48 \)
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| $53$ |
\( T^{3} \)
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| $59$ |
\( (T + 12)^{3} \)
|
| $61$ |
\( T^{3} \)
|
| $67$ |
\( T^{3} \)
|
| $71$ |
\( T^{3} - 213T + 1176 \)
|
| $73$ |
\( T^{3} - 219T + 1226 \)
|
| $79$ |
\( T^{3} \)
|
| $83$ |
\( T^{3} \)
|
| $89$ |
\( T^{3} \)
|
| $97$ |
\( T^{3} \)
|
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