Newspace parameters
| Level: | \( N \) | \(=\) | \( 9660 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9660.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(77.1354883526\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1101.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 9x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{3} - x^{2} - 9x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 7 \)
|
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 | −1.00000 | 0 | 1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | −1.00000 | 0 | 1.00000 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 1.00000 | 0 | −1.00000 | 0 | 1.00000 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 9660.2.a.n | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9660.2.a.n | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{3} + 7T_{11}^{2} + 7T_{11} - 18 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9660))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T - 1)^{3} \)
|
| $5$ |
\( (T + 1)^{3} \)
|
| $7$ |
\( (T - 1)^{3} \)
|
| $11$ |
\( T^{3} + 7 T^{2} + \cdots - 18 \)
|
| $13$ |
\( T^{3} + 5 T^{2} + \cdots - 8 \)
|
| $17$ |
\( T^{3} + 3 T^{2} + \cdots + 18 \)
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| $19$ |
\( T^{3} - 9 T^{2} + \cdots + 202 \)
|
| $23$ |
\( (T + 1)^{3} \)
|
| $29$ |
\( T^{3} - 4 T^{2} + \cdots - 24 \)
|
| $31$ |
\( (T - 2)^{3} \)
|
| $37$ |
\( T^{3} - 13 T^{2} + \cdots - 2 \)
|
| $41$ |
\( T^{3} + 7 T^{2} + \cdots - 18 \)
|
| $43$ |
\( T^{3} - 7 T^{2} + \cdots + 292 \)
|
| $47$ |
\( T^{3} + 8 T^{2} + \cdots - 96 \)
|
| $53$ |
\( T^{3} \)
|
| $59$ |
\( T^{3} - 3 T^{2} + \cdots + 324 \)
|
| $61$ |
\( T^{3} + 9 T^{2} + \cdots - 86 \)
|
| $67$ |
\( T^{3} + 5 T^{2} + \cdots + 124 \)
|
| $71$ |
\( T^{3} + 2 T^{2} + \cdots + 288 \)
|
| $73$ |
\( T^{3} - 27 T^{2} + \cdots - 248 \)
|
| $79$ |
\( T^{3} - 10 T^{2} + \cdots + 16 \)
|
| $83$ |
\( T^{3} - 3 T^{2} + \cdots + 324 \)
|
| $89$ |
\( T^{3} - 2 T^{2} + \cdots + 72 \)
|
| $97$ |
\( T^{3} + 6 T^{2} + \cdots - 512 \)
|
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