Newspace parameters
| Level: | \( N \) | \(=\) | \( 1568 = 2^{5} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1568.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(92.5149948890\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{32 +2 \sqrt{67}})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 48x^{2} - 18x + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 224) |
| 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
\( x^{4} - 2x^{3} - 48x^{2} - 18x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 4\nu^{2} + 114\nu + 18 ) / 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 7\nu^{2} - 72\nu + 36 ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} - 2\nu - 24 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{3} + \beta_{2} + \beta _1 + 50 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 85\beta_{3} + 61\beta_{2} + 43\beta _1 + 350 ) / 4 \)
|
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 | −8.32891 | 0 | −19.3707 | 0 | 0 | 0 | 42.3707 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −6.05221 | 0 | 13.3707 | 0 | 0 | 0 | 9.62929 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 6.05221 | 0 | 13.3707 | 0 | 0 | 0 | 9.62929 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 8.32891 | 0 | −19.3707 | 0 | 0 | 0 | 42.3707 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1568.4.a.z | 4 | |
| 4.b | odd | 2 | 1 | inner | 1568.4.a.z | 4 | |
| 7.b | odd | 2 | 1 | 1568.4.a.bd | 4 | ||
| 7.d | odd | 6 | 2 | 224.4.i.b | ✓ | 8 | |
| 28.d | even | 2 | 1 | 1568.4.a.bd | 4 | ||
| 28.f | even | 6 | 2 | 224.4.i.b | ✓ | 8 | |
| 56.j | odd | 6 | 2 | 448.4.i.n | 8 | ||
| 56.m | even | 6 | 2 | 448.4.i.n | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.4.i.b | ✓ | 8 | 7.d | odd | 6 | 2 | |
| 224.4.i.b | ✓ | 8 | 28.f | even | 6 | 2 | |
| 448.4.i.n | 8 | 56.j | odd | 6 | 2 | ||
| 448.4.i.n | 8 | 56.m | even | 6 | 2 | ||
| 1568.4.a.z | 4 | 1.a | even | 1 | 1 | trivial | |
| 1568.4.a.z | 4 | 4.b | odd | 2 | 1 | inner | |
| 1568.4.a.bd | 4 | 7.b | odd | 2 | 1 | ||
| 1568.4.a.bd | 4 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1568))\):
|
\( T_{3}^{4} - 106T_{3}^{2} + 2541 \)
|
|
\( T_{5}^{2} + 6T_{5} - 259 \)
|
|
\( T_{11}^{4} - 2034T_{11}^{2} + 491589 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 106T^{2} + 2541 \)
|
| $5$ |
\( (T^{2} + 6 T - 259)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 2034 T^{2} + 491589 \)
|
| $13$ |
\( (T^{2} + 140 T + 4632)^{2} \)
|
| $17$ |
\( (T^{2} - 26 T - 2243)^{2} \)
|
| $19$ |
\( T^{4} - 7186 T^{2} + 6798981 \)
|
| $23$ |
\( T^{4} - 27066 T^{2} + 115873821 \)
|
| $29$ |
\( (T^{2} - 68 T - 44136)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 2303358309 \)
|
| $37$ |
\( (T^{2} + 242 T + 13569)^{2} \)
|
| $41$ |
\( (T^{2} - 116 T - 3336)^{2} \)
|
| $43$ |
\( T^{4} - 74272 T^{2} + 307082496 \)
|
| $47$ |
\( T^{4} + \cdots + 26988166821 \)
|
| $53$ |
\( (T^{2} + 946 T + 206577)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 199103344461 \)
|
| $61$ |
\( (T^{2} + 750 T + 139553)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 46727168061 \)
|
| $71$ |
\( T^{4} + \cdots + 408969974784 \)
|
| $73$ |
\( (T^{2} - 282 T - 66951)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 167280561021 \)
|
| $83$ |
\( T^{4} + \cdots + 368302927104 \)
|
| $89$ |
\( (T^{2} - 562 T - 75407)^{2} \)
|
| $97$ |
\( (T^{2} + 468 T - 352872)^{2} \)
|
| show more | |
| show less |