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: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{7} - x^{6} - 11x^{5} + 10x^{4} + 37x^{3} - 30x^{2} - 37x + 27 \)
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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
\( x^{7} - x^{6} - 11x^{5} + 10x^{4} + 37x^{3} - 30x^{2} - 37x + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 2\nu^{5} - 9\nu^{4} - 13\nu^{3} + 22\nu^{2} + 16\nu - 17 ) / 4 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{5} + 9\nu^{4} - 15\nu^{3} - 22\nu^{2} + 24\nu + 9 ) / 4 \)
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| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 9\nu^{4} - \nu^{3} + 22\nu^{2} + 4\nu - 13 ) / 2 \)
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| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} - 2\nu^{5} + 13\nu^{4} + 17\nu^{3} - 50\nu^{2} - 32\nu + 49 ) / 4 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} - \beta_{4} + \beta_{3} + 4\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + 7\beta_{2} + 13 \)
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| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} - 6\beta_{4} + 8\beta_{3} + 18\beta _1 + 2 \)
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| \(\nu^{6}\) | \(=\) |
\( 9\beta_{6} + 10\beta_{5} + 8\beta_{4} + \beta_{3} + 41\beta_{2} + 64 \)
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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 | −2.39288 | 1.00000 | 0.159905 | −2.39288 | −4.43030 | 1.00000 | 2.72586 | 0.159905 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.05670 | 1.00000 | 0.725757 | −2.05670 | 1.48708 | 1.00000 | 1.23001 | 0.725757 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.38741 | 1.00000 | −3.87174 | −1.38741 | 1.84420 | 1.00000 | −1.07508 | −3.87174 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.684252 | 1.00000 | 2.89682 | −0.684252 | −3.41001 | 1.00000 | −2.53180 | 2.89682 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.33735 | 1.00000 | 1.53172 | 1.33735 | 1.53228 | 1.00000 | −1.21150 | 1.53172 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.85738 | 1.00000 | −3.77878 | 1.85738 | 2.79762 | 1.00000 | 0.449865 | −3.77878 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.32651 | 1.00000 | −0.663682 | 2.32651 | −2.82086 | 1.00000 | 2.41265 | −0.663682 | ||||||||||||||||||||||||||||||||||||||||
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.bi | yes | 7 |
| 11.b | odd | 2 | 1 | 8954.2.a.bg | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8954.2.a.bg | ✓ | 7 | 11.b | odd | 2 | 1 | |
| 8954.2.a.bi | yes | 7 | 1.a | even | 1 | 1 | trivial |
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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\( T_{3}^{7} + T_{3}^{6} - 11T_{3}^{5} - 10T_{3}^{4} + 37T_{3}^{3} + 30T_{3}^{2} - 37T_{3} - 27 \)
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\( T_{5}^{7} + 3T_{5}^{6} - 16T_{5}^{5} - 29T_{5}^{4} + 79T_{5}^{3} - T_{5}^{2} - 33T_{5} + 5 \)
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\( T_{7}^{7} + 3T_{7}^{6} - 23T_{7}^{5} - 40T_{7}^{4} + 205T_{7}^{3} + 66T_{7}^{2} - 677T_{7} + 501 \)
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\( T_{17}^{7} + 3T_{17}^{6} - 46T_{17}^{5} - 97T_{17}^{4} + 494T_{17}^{3} + 887T_{17}^{2} - 745T_{17} - 437 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
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| $3$ |
\( T^{7} + T^{6} + \cdots - 27 \)
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| $5$ |
\( T^{7} + 3 T^{6} + \cdots + 5 \)
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| $7$ |
\( T^{7} + 3 T^{6} + \cdots + 501 \)
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| $11$ |
\( T^{7} \)
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| $13$ |
\( T^{7} + 10 T^{6} + \cdots - 575 \)
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| $17$ |
\( T^{7} + 3 T^{6} + \cdots - 437 \)
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| $19$ |
\( T^{7} + 10 T^{6} + \cdots - 2945 \)
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| $23$ |
\( T^{7} + 6 T^{6} + \cdots - 1356 \)
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| $29$ |
\( T^{7} + 2 T^{6} + \cdots + 30385 \)
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| $31$ |
\( T^{7} - T^{6} + \cdots + 800549 \)
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| $37$ |
\( (T + 1)^{7} \)
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| $41$ |
\( T^{7} - 6 T^{6} + \cdots + 162675 \)
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| $43$ |
\( T^{7} - T^{6} + \cdots + 81 \)
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| $47$ |
\( T^{7} + 10 T^{6} + \cdots + 101295 \)
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| $53$ |
\( T^{7} - 7 T^{6} + \cdots - 211985 \)
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| $59$ |
\( T^{7} + 19 T^{6} + \cdots + 79 \)
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| $61$ |
\( T^{7} + 19 T^{6} + \cdots - 11344 \)
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| $67$ |
\( T^{7} - T^{6} + \cdots - 469439 \)
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| $71$ |
\( T^{7} - 9 T^{6} + \cdots - 33804 \)
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| $73$ |
\( T^{7} + 16 T^{6} + \cdots + 1052417 \)
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| $79$ |
\( T^{7} + 47 T^{6} + \cdots - 203076 \)
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| $83$ |
\( T^{7} - 13 T^{6} + \cdots + 720877 \)
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| $89$ |
\( T^{7} - 3 T^{6} + \cdots + 79380 \)
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| $97$ |
\( T^{7} - T^{6} + \cdots - 291133 \)
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