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: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.9248.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 4232) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 5x^{2} + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( 2\nu^{3} - 8\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 4\beta_1 ) / 2 \)
|
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 |
|
0 | −1.56155 | 0 | −1.32431 | 0 | 4.71659 | 0 | −0.561553 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.56155 | 0 | 1.32431 | 0 | −4.71659 | 0 | −0.561553 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 2.56155 | 0 | −4.27156 | 0 | −2.39871 | 0 | 3.56155 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.56155 | 0 | 4.27156 | 0 | 2.39871 | 0 | 3.56155 | 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 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 8464.2.a.bo | 4 | |
| 4.b | odd | 2 | 1 | 4232.2.a.r | ✓ | 4 | |
| 23.b | odd | 2 | 1 | inner | 8464.2.a.bo | 4 | |
| 92.b | even | 2 | 1 | 4232.2.a.r | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4232.2.a.r | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 4232.2.a.r | ✓ | 4 | 92.b | even | 2 | 1 | |
| 8464.2.a.bo | 4 | 1.a | even | 1 | 1 | trivial | |
| 8464.2.a.bo | 4 | 23.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8464))\):
|
\( T_{3}^{2} - T_{3} - 4 \)
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\( T_{5}^{4} - 20T_{5}^{2} + 32 \)
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\( T_{7}^{4} - 28T_{7}^{2} + 128 \)
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\( T_{13}^{2} - 5T_{13} + 2 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( (T^{2} - T - 4)^{2} \)
|
| $5$ |
\( T^{4} - 20T^{2} + 32 \)
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| $7$ |
\( T^{4} - 28T^{2} + 128 \)
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| $11$ |
\( T^{4} - 20T^{2} + 32 \)
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| $13$ |
\( (T^{2} - 5 T + 2)^{2} \)
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| $17$ |
\( T^{4} - 28T^{2} + 128 \)
|
| $19$ |
\( T^{4} - 40T^{2} + 128 \)
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| $23$ |
\( T^{4} \)
|
| $29$ |
\( (T^{2} + 5 T + 2)^{2} \)
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| $31$ |
\( (T^{2} + 7 T + 8)^{2} \)
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| $37$ |
\( T^{4} - 148T^{2} + 5408 \)
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| $41$ |
\( (T^{2} - 9 T - 18)^{2} \)
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| $43$ |
\( T^{4} - 184T^{2} + 8192 \)
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| $47$ |
\( (T^{2} + 3 T - 104)^{2} \)
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| $53$ |
\( T^{4} - 40T^{2} + 128 \)
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| $59$ |
\( (T + 4)^{4} \)
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| $61$ |
\( T^{4} - 56T^{2} + 512 \)
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| $67$ |
\( T^{4} - 20T^{2} + 32 \)
|
| $71$ |
\( (T^{2} - 9 T + 16)^{2} \)
|
| $73$ |
\( (T^{2} + 13 T + 38)^{2} \)
|
| $79$ |
\( T^{4} - 80T^{2} + 512 \)
|
| $83$ |
\( T^{4} - 244 T^{2} + 11552 \)
|
| $89$ |
\( T^{4} - 368 T^{2} + 32768 \)
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| $97$ |
\( T^{4} - 252 T^{2} + 10368 \)
|
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