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: | \(6\) |
| Coefficient field: | 6.6.390126348.1 |
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| Defining polynomial: |
\( x^{6} - x^{5} - 9x^{4} + 6x^{3} + 20x^{2} - 5x - 2 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 9x^{4} + 6x^{3} + 20x^{2} - 5x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} - 2\nu \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 5\nu^{2} + 15\nu - 2 \)
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| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{3} + 5\beta _1 + 2 ) / 2 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} + \beta _1 + 15 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{5} + 2\beta_{4} + 16\beta_{3} + 27\beta _1 + 20 ) / 2 \)
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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 | −3.16958 | 1.00000 | 2.79530 | 3.16958 | −0.904876 | −1.00000 | 7.04622 | −2.79530 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.86335 | 1.00000 | −1.59734 | 1.86335 | −4.28828 | −1.00000 | 0.472079 | 1.59734 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.29869 | 1.00000 | −2.13185 | 1.29869 | 4.53072 | −1.00000 | −1.31340 | 2.13185 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.0956587 | 1.00000 | 2.95103 | −0.0956587 | 0.442599 | −1.00000 | −2.99085 | −2.95103 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.42009 | 1.00000 | −3.35652 | −2.42009 | −5.04243 | −1.00000 | 2.85683 | 3.35652 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.81587 | 1.00000 | 0.339393 | −2.81587 | 3.26227 | −1.00000 | 4.92912 | −0.339393 | |||||||||||||||||||||||||||||||||||||
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.k | ✓ | 6 |
| 4.b | odd | 2 | 1 | 9328.2.a.be | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1166.2.a.k | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 9328.2.a.be | 6 | 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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\( T_{3}^{6} + T_{3}^{5} - 14T_{3}^{4} - 13T_{3}^{3} + 46T_{3}^{2} + 48T_{3} - 5 \)
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\( T_{7}^{6} + 2T_{7}^{5} - 36T_{7}^{4} - 48T_{7}^{3} + 320T_{7}^{2} + 160T_{7} - 128 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
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| $3$ |
\( T^{6} + T^{5} - 14 T^{4} + \cdots - 5 \)
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| $5$ |
\( T^{6} + T^{5} + \cdots - 32 \)
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| $7$ |
\( T^{6} + 2 T^{5} + \cdots - 128 \)
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| $11$ |
\( (T - 1)^{6} \)
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| $13$ |
\( T^{6} - 5 T^{5} + \cdots - 215 \)
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| $17$ |
\( T^{6} + 2 T^{5} + \cdots + 1312 \)
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| $19$ |
\( T^{6} - 20 T^{5} + \cdots - 11360 \)
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| $23$ |
\( T^{6} + 14 T^{5} + \cdots - 1280 \)
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| $29$ |
\( T^{6} - 13 T^{5} + \cdots - 1514 \)
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| $31$ |
\( T^{6} \)
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| $37$ |
\( T^{6} + 2 T^{5} + \cdots + 1312 \)
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| $41$ |
\( T^{6} - 13 T^{5} + \cdots + 1912 \)
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| $43$ |
\( T^{6} - 3 T^{5} + \cdots - 32 \)
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| $47$ |
\( T^{6} + 25 T^{5} + \cdots + 35848 \)
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| $53$ |
\( (T + 1)^{6} \)
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| $59$ |
\( T^{6} - 8 T^{5} + \cdots + 40192 \)
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| $61$ |
\( T^{6} - 34 T^{5} + \cdots - 2048 \)
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| $67$ |
\( T^{6} - 29 T^{5} + \cdots - 3728 \)
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| $71$ |
\( T^{6} - 8 T^{5} + \cdots - 363200 \)
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| $73$ |
\( T^{6} - 41 T^{5} + \cdots - 1539968 \)
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| $79$ |
\( T^{6} - 17 T^{5} + \cdots - 58265 \)
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| $83$ |
\( T^{6} + 2 T^{5} + \cdots + 15280 \)
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| $89$ |
\( T^{6} - 17 T^{5} + \cdots - 1208 \)
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| $97$ |
\( T^{6} - 3 T^{5} + \cdots + 70846 \)
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