Newspace parameters
| Level: | \( N \) | \(=\) | \( 1166 = 2 \cdot 11 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1166.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(9.31055687568\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.67348.1 |
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|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 10x^{2} + 10x + 24 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 10x^{2} + 10x + 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 6 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 6\nu + 6 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 6 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 2\beta_{2} + 8\beta _1 + 6 \)
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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 |
|
1.00000 | −2.47808 | 1.00000 | 1.50599 | −2.47808 | 4.61899 | 1.00000 | 3.14090 | 1.50599 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.18683 | 1.00000 | −3.11376 | −1.18683 | −1.40462 | 1.00000 | −1.59145 | −3.11376 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 2.38249 | 1.00000 | 3.91504 | 2.38249 | −0.706244 | 1.00000 | 2.67624 | 3.91504 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 3.28242 | 1.00000 | −1.30728 | 3.28242 | 3.49188 | 1.00000 | 7.77430 | −1.30728 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(11\) | \( +1 \) |
| \(53\) | \( +1 \) |
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 | 1166.2.a.j | ✓ | 4 |
| 4.b | odd | 2 | 1 | 9328.2.a.y | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1166.2.a.j | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 9328.2.a.y | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1166))\):
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\( T_{3}^{4} - 2T_{3}^{3} - 10T_{3}^{2} + 12T_{3} + 23 \)
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\( T_{7}^{4} - 6T_{7}^{3} + 26T_{7} + 16 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
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| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 23 \)
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| $5$ |
\( T^{4} - T^{3} + \cdots + 24 \)
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| $7$ |
\( T^{4} - 6 T^{3} + \cdots + 16 \)
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| $11$ |
\( (T + 1)^{4} \)
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| $13$ |
\( T^{4} - 46 T^{2} + \cdots + 421 \)
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| $17$ |
\( T^{4} + 11 T^{3} + \cdots - 336 \)
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| $19$ |
\( T^{4} + 5 T^{3} + \cdots + 8 \)
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| $23$ |
\( T^{4} - 17 T^{3} + \cdots - 504 \)
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| $29$ |
\( T^{4} - 7 T^{3} + \cdots - 6 \)
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| $31$ |
\( T^{4} - 8 T^{3} + \cdots - 16 \)
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| $37$ |
\( T^{4} + 3 T^{3} + \cdots + 52 \)
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| $41$ |
\( T^{4} + 17 T^{3} + \cdots + 966 \)
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| $43$ |
\( T^{4} + 10 T^{3} + \cdots + 8 \)
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| $47$ |
\( T^{4} + 3 T^{3} + \cdots + 234 \)
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| $53$ |
\( (T + 1)^{4} \)
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| $59$ |
\( T^{4} - 26 T^{3} + \cdots + 1248 \)
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| $61$ |
\( T^{4} - 28 T^{3} + \cdots + 448 \)
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| $67$ |
\( T^{4} - 5 T^{3} + \cdots - 128 \)
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| $71$ |
\( T^{4} + 9 T^{3} + \cdots - 4008 \)
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| $73$ |
\( T^{4} - 16 T^{3} + \cdots - 1264 \)
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| $79$ |
\( T^{4} - 14 T^{3} + \cdots - 3511 \)
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| $83$ |
\( T^{4} - T^{3} + \cdots + 28734 \)
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| $89$ |
\( T^{4} + 8 T^{3} + \cdots + 936 \)
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| $97$ |
\( T^{4} - 17 T^{3} + \cdots - 1186 \)
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