Newspace parameters
| Level: | \( N \) | \(=\) | \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 966.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(56.9958450655\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.12197.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 15x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{3} - x^{2} - 15x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 10 \)
|
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 |
|
2.00000 | −3.00000 | 4.00000 | −9.55073 | −6.00000 | 7.00000 | 8.00000 | 9.00000 | −19.1015 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | −3.00000 | 4.00000 | 4.81897 | −6.00000 | 7.00000 | 8.00000 | 9.00000 | 9.63795 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −3.00000 | 4.00000 | 13.7318 | −6.00000 | 7.00000 | 8.00000 | 9.00000 | 27.4635 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 966.4.a.f | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 966.4.a.f | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 9T_{5}^{2} - 111T_{5} + 632 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(966))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( T^{3} - 9 T^{2} + \cdots + 632 \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( T^{3} - 53 T^{2} + \cdots + 71001 \)
|
| $13$ |
\( T^{3} - 61 T^{2} + \cdots - 1822 \)
|
| $17$ |
\( T^{3} + 2 T^{2} + \cdots - 16024 \)
|
| $19$ |
\( T^{3} - 63 T^{2} + \cdots + 47899 \)
|
| $23$ |
\( (T + 23)^{3} \)
|
| $29$ |
\( T^{3} - 312 T^{2} + \cdots + 13186 \)
|
| $31$ |
\( T^{3} - 220 T^{2} + \cdots + 731490 \)
|
| $37$ |
\( T^{3} + 44 T^{2} + \cdots - 13366222 \)
|
| $41$ |
\( T^{3} - 171 T^{2} + \cdots + 31352967 \)
|
| $43$ |
\( T^{3} + 627 T^{2} + \cdots - 49544756 \)
|
| $47$ |
\( T^{3} - 353 T^{2} + \cdots + 84406692 \)
|
| $53$ |
\( T^{3} + 504 T^{2} + \cdots - 86011099 \)
|
| $59$ |
\( T^{3} - 538 T^{2} + \cdots + 43304511 \)
|
| $61$ |
\( T^{3} - 256 T^{2} + \cdots - 6141519 \)
|
| $67$ |
\( T^{3} - 1113 T^{2} + \cdots + 55713572 \)
|
| $71$ |
\( T^{3} - 53 T^{2} + \cdots - 93489564 \)
|
| $73$ |
\( T^{3} - 360 T^{2} + \cdots + 197977404 \)
|
| $79$ |
\( T^{3} - 134 T^{2} + \cdots - 45775266 \)
|
| $83$ |
\( T^{3} - 1506 T^{2} + \cdots - 107382958 \)
|
| $89$ |
\( T^{3} - 2381 T^{2} + \cdots + 55340850 \)
|
| $97$ |
\( T^{3} + 88 T^{2} + \cdots - 22908546 \)
|
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