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: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 3x^{6} - 10x^{5} + 17x^{4} + 42x^{3} + 2x^{2} - 20x - 2 \)
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| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 10x^{5} + 17x^{4} + 42x^{3} + 2x^{2} - 20x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{6} - 10\nu^{5} - 3\nu^{4} + 52\nu^{3} - 5\nu^{2} - 46\nu + 13 ) / 3 \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{6} + 4\nu^{5} + 6\nu^{4} - 22\nu^{3} - 21\nu^{2} + 11\nu + 6 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 4\nu^{5} - 6\nu^{4} + 23\nu^{3} + 19\nu^{2} - 16\nu - 5 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{6} + 17\nu^{5} + 18\nu^{4} - 89\nu^{3} - 50\nu^{2} + 50\nu + 1 ) / 3 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 4\nu^{6} - 17\nu^{5} - 18\nu^{4} + 89\nu^{3} + 53\nu^{2} - 56\nu - 10 ) / 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 2\beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} + 9\beta _1 + 5 \)
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| \(\nu^{4}\) | \(=\) |
\( 11\beta_{6} + 9\beta_{5} + 2\beta_{4} + 4\beta_{3} - \beta_{2} + 28\beta _1 + 24 \)
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| \(\nu^{5}\) | \(=\) |
\( 34\beta_{6} + 25\beta_{5} + 13\beta_{4} + 21\beta_{3} - 6\beta_{2} + 103\beta _1 + 70 \)
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| \(\nu^{6}\) | \(=\) |
\( 137\beta_{6} + 89\beta_{5} + 42\beta_{4} + 85\beta_{3} - 30\beta_{2} + 351\beta _1 + 257 \)
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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 |
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1.00000 | −2.56081 | 1.00000 | 1.41575 | −2.56081 | 1.64447 | 1.00000 | 3.55774 | 1.41575 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.10691 | 1.00000 | −0.807854 | −2.10691 | −4.71309 | 1.00000 | 1.43907 | −0.807854 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0.329888 | 1.00000 | 4.05456 | 0.329888 | −1.01910 | 1.00000 | −2.89117 | 4.05456 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.10105 | 1.00000 | 2.48988 | 1.10105 | 4.69727 | 1.00000 | −1.78768 | 2.48988 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.15455 | 1.00000 | 0.590473 | 2.15455 | −0.748024 | 1.00000 | 1.64209 | 0.590473 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.29068 | 1.00000 | −4.03371 | 2.29068 | 1.82206 | 1.00000 | 2.24721 | −4.03371 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.79155 | 1.00000 | 0.290901 | 2.79155 | 0.316408 | 1.00000 | 4.79275 | 0.290901 | ||||||||||||||||||||||||||||||||||||||||
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.l | ✓ | 7 |
| 4.b | odd | 2 | 1 | 9328.2.a.bg | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1166.2.a.l | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 9328.2.a.bg | 7 | 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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\( T_{3}^{7} - 4T_{3}^{6} - 7T_{3}^{5} + 43T_{3}^{4} - 15T_{3}^{3} - 106T_{3}^{2} + 117T_{3} - 27 \)
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\( T_{7}^{7} - 2T_{7}^{6} - 24T_{7}^{5} + 48T_{7}^{4} + 42T_{7}^{3} - 76T_{7}^{2} - 32T_{7} + 16 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
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| $3$ |
\( T^{7} - 4 T^{6} + \cdots - 27 \)
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| $5$ |
\( T^{7} - 4 T^{6} + \cdots - 8 \)
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| $7$ |
\( T^{7} - 2 T^{6} + \cdots + 16 \)
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| $11$ |
\( (T - 1)^{7} \)
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| $13$ |
\( T^{7} - 6 T^{6} + \cdots - 717 \)
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| $17$ |
\( T^{7} + 4 T^{6} + \cdots - 24 \)
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| $19$ |
\( T^{7} - 18 T^{6} + \cdots + 6840 \)
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| $23$ |
\( T^{7} - 8 T^{6} + \cdots + 2496 \)
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| $29$ |
\( T^{7} + 11 T^{6} + \cdots - 15775 \)
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| $31$ |
\( T^{7} - 16 T^{6} + \cdots + 85248 \)
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| $37$ |
\( T^{7} + 8 T^{6} + \cdots - 714808 \)
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| $41$ |
\( T^{7} + 4 T^{6} + \cdots + 5242 \)
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| $43$ |
\( T^{7} - 7 T^{6} + \cdots - 846854 \)
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| $47$ |
\( T^{7} - 2 T^{6} + \cdots + 7734 \)
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| $53$ |
\( (T - 1)^{7} \)
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| $59$ |
\( T^{7} + 16 T^{6} + \cdots - 4560 \)
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| $61$ |
\( T^{7} - 26 T^{6} + \cdots - 37888 \)
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| $67$ |
\( T^{7} + 16 T^{6} + \cdots + 331136 \)
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| $71$ |
\( T^{7} - 2 T^{6} + \cdots - 1547528 \)
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| $73$ |
\( T^{7} + 3 T^{6} + \cdots + 2120936 \)
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| $79$ |
\( T^{7} - 26 T^{6} + \cdots + 536345 \)
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| $83$ |
\( T^{7} - 22 T^{6} + \cdots - 12568 \)
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| $89$ |
\( T^{7} - 3 T^{6} + \cdots + 66300 \)
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| $97$ |
\( T^{7} + 25 T^{6} + \cdots + 114623 \)
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