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: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{5})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 6x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} - 6x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 4\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} + 4\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −2.23607 | 0 | −2.28825 | 0 | −3.70246 | 0 | 2.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.23607 | 0 | 2.28825 | 0 | 3.70246 | 0 | 2.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 2.23607 | 0 | −0.874032 | 0 | 0.540182 | 0 | 2.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.23607 | 0 | 0.874032 | 0 | −0.540182 | 0 | 2.00000 | 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.bm | 4 | |
| 4.b | odd | 2 | 1 | 4232.2.a.t | ✓ | 4 | |
| 23.b | odd | 2 | 1 | inner | 8464.2.a.bm | 4 | |
| 92.b | even | 2 | 1 | 4232.2.a.t | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4232.2.a.t | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 4232.2.a.t | ✓ | 4 | 92.b | even | 2 | 1 | |
| 8464.2.a.bm | 4 | 1.a | even | 1 | 1 | trivial | |
| 8464.2.a.bm | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8464))\):
|
\( T_{3}^{2} - 5 \)
|
|
\( T_{5}^{4} - 6T_{5}^{2} + 4 \)
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\( T_{7}^{4} - 14T_{7}^{2} + 4 \)
|
|
\( T_{13}^{2} + 8T_{13} + 11 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 5)^{2} \)
|
| $5$ |
\( T^{4} - 6T^{2} + 4 \)
|
| $7$ |
\( T^{4} - 14T^{2} + 4 \)
|
| $11$ |
\( T^{4} - 14T^{2} + 4 \)
|
| $13$ |
\( (T^{2} + 8 T + 11)^{2} \)
|
| $17$ |
\( T^{4} - 14T^{2} + 4 \)
|
| $19$ |
\( T^{4} - 36T^{2} + 4 \)
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| $23$ |
\( T^{4} \)
|
| $29$ |
\( (T^{2} + 4 T - 41)^{2} \)
|
| $31$ |
\( (T^{2} + 2 T - 79)^{2} \)
|
| $37$ |
\( T^{4} - 94T^{2} + 4 \)
|
| $41$ |
\( (T^{2} + 8 T - 29)^{2} \)
|
| $43$ |
\( (T^{2} - 72)^{2} \)
|
| $47$ |
\( (T^{2} - 10 T + 5)^{2} \)
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| $53$ |
\( T^{4} - 196T^{2} + 6724 \)
|
| $59$ |
\( (T^{2} + 14 T + 4)^{2} \)
|
| $61$ |
\( (T^{2} - 90)^{2} \)
|
| $67$ |
\( T^{4} - 174T^{2} + 1444 \)
|
| $71$ |
\( (T^{2} + 12 T + 31)^{2} \)
|
| $73$ |
\( (T^{2} + 12 T + 31)^{2} \)
|
| $79$ |
\( T^{4} - 180T^{2} + 100 \)
|
| $83$ |
\( T^{4} - 126T^{2} + 3844 \)
|
| $89$ |
\( T^{4} - 120T^{2} + 1600 \)
|
| $97$ |
\( T^{4} - 230 T^{2} + 12100 \)
|
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