Newspace parameters
| Level: | \( N \) | \(=\) | \( 8954 = 2 \cdot 11^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8954.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(71.4980499699\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 2x^{6} - 11x^{5} + 27x^{4} - 3x^{3} - 21x^{2} + 4x + 4 \)
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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
\( x^{7} - 2x^{6} - 11x^{5} + 27x^{4} - 3x^{3} - 21x^{2} + 4x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{6} + 37\nu^{4} - 11\nu^{3} - 57\nu^{2} + 5\nu + 10 ) / 4 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{6} + 4\nu^{5} + 33\nu^{4} - 55\nu^{3} + 3\nu^{2} + 13\nu - 6 ) / 4 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{5} + 11\nu^{4} - 27\nu^{3} + 3\nu^{2} + 19\nu - 2 ) / 2 \)
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| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 12\nu^{4} - 15\nu^{3} - 12\nu^{2} + 9\nu + 5 \)
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| \(\beta_{6}\) | \(=\) |
\( 2\nu^{6} - 2\nu^{5} - 23\nu^{4} + 30\nu^{3} + 13\nu^{2} - 14\nu - 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} + 8\beta _1 - 3 \)
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| \(\nu^{4}\) | \(=\) |
\( 12\beta_{6} + 2\beta_{5} + 11\beta_{4} + 11\beta_{3} + 11\beta_{2} - 4\beta _1 + 26 \)
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| \(\nu^{5}\) | \(=\) |
\( -3\beta_{6} + 13\beta_{5} - 26\beta_{4} + 8\beta_{3} - 16\beta_{2} + 82\beta _1 - 48 \)
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| \(\nu^{6}\) | \(=\) |
\( 129\beta_{6} + 21\beta_{5} + 124\beta_{4} + 113\beta_{3} + 119\beta_{2} - 77\beta _1 + 278 \)
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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 | −3.27585 | 1.00000 | −2.36104 | 3.27585 | −1.12156 | −1.00000 | 7.73117 | 2.36104 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −0.713098 | 1.00000 | 0.108716 | 0.713098 | −1.52087 | −1.00000 | −2.49149 | −0.108716 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −0.587361 | 1.00000 | −2.29467 | 0.587361 | 3.28545 | −1.00000 | −2.65501 | 2.29467 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −0.225918 | 1.00000 | 2.86053 | 0.225918 | 2.66826 | −1.00000 | −2.94896 | −2.86053 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.494236 | 1.00000 | 0.0992536 | −0.494236 | −4.61415 | −1.00000 | −2.75573 | −0.0992536 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 1.93544 | 1.00000 | 1.83896 | −1.93544 | 4.49319 | −1.00000 | 0.745917 | −1.83896 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 3.37255 | 1.00000 | −3.25175 | −3.37255 | −0.190313 | −1.00000 | 8.37410 | 3.25175 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(11\) | \( -1 \) |
| \(37\) | \( +1 \) |
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 | 8954.2.a.bh | ✓ | 7 |
| 11.b | odd | 2 | 1 | 8954.2.a.bj | yes | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8954.2.a.bh | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 8954.2.a.bj | yes | 7 | 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8954))\):
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\( T_{3}^{7} - T_{3}^{6} - 13T_{3}^{5} + 10T_{3}^{4} + 23T_{3}^{3} + 2T_{3}^{2} - 5T_{3} - 1 \)
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\( T_{5}^{7} + 3T_{5}^{6} - 12T_{5}^{5} - 35T_{5}^{4} + 33T_{5}^{3} + 87T_{5}^{2} - 19T_{5} + 1 \)
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\( T_{7}^{7} - 3T_{7}^{6} - 27T_{7}^{5} + 76T_{7}^{4} + 141T_{7}^{3} - 244T_{7}^{2} - 361T_{7} - 59 \)
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\( T_{17}^{7} - 3T_{17}^{6} - 74T_{17}^{5} + 267T_{17}^{4} + 1274T_{17}^{3} - 4905T_{17}^{2} - 2961T_{17} + 11637 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{7} \)
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| $3$ |
\( T^{7} - T^{6} - 13 T^{5} + \cdots - 1 \)
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| $5$ |
\( T^{7} + 3 T^{6} + \cdots + 1 \)
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| $7$ |
\( T^{7} - 3 T^{6} + \cdots - 59 \)
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| $11$ |
\( T^{7} \)
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| $13$ |
\( T^{7} - 2 T^{6} + \cdots - 6607 \)
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| $17$ |
\( T^{7} - 3 T^{6} + \cdots + 11637 \)
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| $19$ |
\( T^{7} - 10 T^{6} + \cdots + 1727 \)
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| $23$ |
\( T^{7} + 18 T^{6} + \cdots - 6124 \)
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| $29$ |
\( T^{7} - 14 T^{6} + \cdots + 2453 \)
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| $31$ |
\( T^{7} + 5 T^{6} + \cdots + 34363 \)
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| $37$ |
\( (T + 1)^{7} \)
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| $41$ |
\( T^{7} + 10 T^{6} + \cdots + 727 \)
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| $43$ |
\( T^{7} - 11 T^{6} + \cdots + 452861 \)
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| $47$ |
\( T^{7} + 16 T^{6} + \cdots + 3587 \)
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| $53$ |
\( T^{7} + 21 T^{6} + \cdots - 26207 \)
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| $59$ |
\( T^{7} - 29 T^{6} + \cdots - 439177 \)
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| $61$ |
\( T^{7} - 37 T^{6} + \cdots - 16 \)
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| $67$ |
\( T^{7} + 13 T^{6} + \cdots - 279221 \)
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| $71$ |
\( T^{7} + 15 T^{6} + \cdots + 1684 \)
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| $73$ |
\( T^{7} - 12 T^{6} + \cdots + 472049 \)
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| $79$ |
\( T^{7} + 9 T^{6} + \cdots + 251372 \)
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| $83$ |
\( T^{7} + 9 T^{6} + \cdots - 196421 \)
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| $89$ |
\( T^{7} + 17 T^{6} + \cdots + 451332 \)
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| $97$ |
\( T^{7} + 31 T^{6} + \cdots - 106807 \)
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