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: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.29901.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 39x + 102 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{3} - x^{2} - 39x + 102 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3\nu - 27 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3\beta _1 + 27 \)
|
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 | −8.85699 | −6.00000 | 7.00000 | −8.00000 | 9.00000 | 17.7140 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 3.00000 | 4.00000 | 1.18167 | −6.00000 | 7.00000 | −8.00000 | 9.00000 | −2.36334 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 3.00000 | 4.00000 | 2.67532 | −6.00000 | 7.00000 | −8.00000 | 9.00000 | −5.35065 | ||||||||||||||||||||||||||||
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.d | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 966.4.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 5T_{5}^{2} - 31T_{5} + 28 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(966))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( T^{3} + 5 T^{2} + \cdots + 28 \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( T^{3} + 3 T^{2} + \cdots + 3077 \)
|
| $13$ |
\( T^{3} + 79 T^{2} + \cdots - 27958 \)
|
| $17$ |
\( T^{3} + 26 T^{2} + \cdots - 48536 \)
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| $19$ |
\( T^{3} + 15 T^{2} + \cdots - 446023 \)
|
| $23$ |
\( (T - 23)^{3} \)
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| $29$ |
\( T^{3} + 74 T^{2} + \cdots - 275562 \)
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| $31$ |
\( T^{3} + 378 T^{2} + \cdots - 1080198 \)
|
| $37$ |
\( T^{3} + 260 T^{2} + \cdots - 3205706 \)
|
| $41$ |
\( T^{3} - 293 T^{2} + \cdots + 4110057 \)
|
| $43$ |
\( T^{3} + 625 T^{2} + \cdots - 110900584 \)
|
| $47$ |
\( T^{3} - 163 T^{2} + \cdots + 6557268 \)
|
| $53$ |
\( T^{3} - 178 T^{2} + \cdots + 3527949 \)
|
| $59$ |
\( T^{3} + 624 T^{2} + \cdots - 185492979 \)
|
| $61$ |
\( T^{3} + 502 T^{2} + \cdots - 177073753 \)
|
| $67$ |
\( T^{3} + 141 T^{2} + \cdots + 33105448 \)
|
| $71$ |
\( T^{3} - 29 T^{2} + \cdots - 115600548 \)
|
| $73$ |
\( T^{3} + 1198 T^{2} + \cdots - 85258908 \)
|
| $79$ |
\( T^{3} + 1654 T^{2} + \cdots - 562415718 \)
|
| $83$ |
\( T^{3} + 614 T^{2} + \cdots - 87895674 \)
|
| $89$ |
\( T^{3} + 549 T^{2} + \cdots - 82051758 \)
|
| $97$ |
\( T^{3} - 182 T^{2} + \cdots - 386199534 \)
|
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