Newspace parameters
| Level: | \( N \) | \(=\) | \( 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5712.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(45.6105496346\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.961.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 10x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2856) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 10x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + \nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} - \beta _1 + 8 \)
|
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 | 1.00000 | 0 | −2.29707 | 0 | −1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | 0.213198 | 0 | −1.00000 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 1.00000 | 0 | 4.08387 | 0 | −1.00000 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(17\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5712.2.a.bv | 3 | |
| 4.b | odd | 2 | 1 | 2856.2.a.r | ✓ | 3 | |
| 12.b | even | 2 | 1 | 8568.2.a.y | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2856.2.a.r | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 5712.2.a.bv | 3 | 1.a | even | 1 | 1 | trivial | |
| 8568.2.a.y | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5712))\):
|
\( T_{5}^{3} - 2T_{5}^{2} - 9T_{5} + 2 \)
|
|
\( T_{11}^{3} + 4T_{11}^{2} - 5T_{11} - 16 \)
|
|
\( T_{13}^{3} - 31T_{13} - 62 \)
|
|
\( T_{19}^{3} - T_{19}^{2} - 10T_{19} + 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T - 1)^{3} \)
|
| $5$ |
\( T^{3} - 2 T^{2} + \cdots + 2 \)
|
| $7$ |
\( (T + 1)^{3} \)
|
| $11$ |
\( T^{3} + 4 T^{2} + \cdots - 16 \)
|
| $13$ |
\( T^{3} - 31T - 62 \)
|
| $17$ |
\( (T + 1)^{3} \)
|
| $19$ |
\( T^{3} - T^{2} - 10T + 8 \)
|
| $23$ |
\( T^{3} + T^{2} - 10T - 8 \)
|
| $29$ |
\( (T - 6)^{3} \)
|
| $31$ |
\( T^{3} + 2 T^{2} + \cdots - 64 \)
|
| $37$ |
\( T^{3} - T^{2} - 10T + 8 \)
|
| $41$ |
\( T^{3} - 7 T^{2} + \cdots + 16 \)
|
| $43$ |
\( T^{3} + 6 T^{2} + \cdots - 116 \)
|
| $47$ |
\( T^{3} - 2 T^{2} + \cdots + 64 \)
|
| $53$ |
\( T^{3} - 13 T^{2} + \cdots + 92 \)
|
| $59$ |
\( T^{3} - 6 T^{2} + \cdots + 736 \)
|
| $61$ |
\( T^{3} - 124T - 496 \)
|
| $67$ |
\( T^{3} - 7 T^{2} + \cdots + 16 \)
|
| $71$ |
\( (T + 4)^{3} \)
|
| $73$ |
\( T^{3} - 15 T^{2} + \cdots + 92 \)
|
| $79$ |
\( T^{3} - 7 T^{2} + \cdots + 16 \)
|
| $83$ |
\( T^{3} - 9 T^{2} + \cdots + 376 \)
|
| $89$ |
\( T^{3} - 7 T^{2} + \cdots - 232 \)
|
| $97$ |
\( T^{3} - 9 T^{2} + \cdots + 872 \)
|
| show more | |
| show less |