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: | \(5\) |
| Coefficient field: | 5.5.702521.1 |
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|
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| Defining polynomial: |
\( x^{5} - x^{4} - 7x^{3} + 9x^{2} + 4x - 5 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 7x^{3} + 9x^{2} + 4x - 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} + 3\nu + 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} + \nu^{3} - 6\nu^{2} - 2\nu + 4 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 2\nu + 6 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{2} - \beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 5\beta _1 - 2 \)
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| \(\nu^{4}\) | \(=\) |
\( -6\beta_{4} + 7\beta_{2} - 9\beta _1 + 22 \)
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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 | −2.91554 | 1.00000 | 2.44556 | 2.91554 | 4.66015 | −1.00000 | 5.50036 | −2.44556 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0.437894 | 1.00000 | 2.39312 | −0.437894 | −0.216918 | −1.00000 | −2.80825 | −2.39312 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0.724763 | 1.00000 | −3.93390 | −0.724763 | −2.35799 | −1.00000 | −2.47472 | 3.93390 | ||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.88265 | 1.00000 | −1.54950 | −1.88265 | 1.25031 | −1.00000 | 0.544362 | 1.54950 | ||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.87023 | 1.00000 | 0.644719 | −2.87023 | −0.335541 | −1.00000 | 5.23824 | −0.644719 | ||||||||||||||||||||||||||||||||||
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.z | ✓ | 5 |
| 11.b | odd | 2 | 1 | 8954.2.a.bd | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8954.2.a.z | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 8954.2.a.bd | yes | 5 | 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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\( T_{3}^{5} - 3T_{3}^{4} - 6T_{3}^{3} + 25T_{3}^{2} - 21T_{3} + 5 \)
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\( T_{5}^{5} - 15T_{5}^{3} + 12T_{5}^{2} + 34T_{5} - 23 \)
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\( T_{7}^{5} - 3T_{7}^{4} - 10T_{7}^{3} + 9T_{7}^{2} + 7T_{7} + 1 \)
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\( T_{17}^{5} + 7T_{17}^{4} + 9T_{17}^{3} - 10T_{17}^{2} - 3T_{17} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{5} \)
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| $3$ |
\( T^{5} - 3 T^{4} + \cdots + 5 \)
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| $5$ |
\( T^{5} - 15 T^{3} + \cdots - 23 \)
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| $7$ |
\( T^{5} - 3 T^{4} + \cdots + 1 \)
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| $11$ |
\( T^{5} \)
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| $13$ |
\( T^{5} + 11 T^{4} + \cdots - 955 \)
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| $17$ |
\( T^{5} + 7 T^{4} + \cdots + 1 \)
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| $19$ |
\( T^{5} + 2 T^{4} + \cdots - 135 \)
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| $23$ |
\( T^{5} - 16 T^{4} + \cdots + 1511 \)
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| $29$ |
\( T^{5} + 15 T^{4} + \cdots - 7705 \)
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| $31$ |
\( T^{5} - 11 T^{4} + \cdots + 5 \)
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| $37$ |
\( (T - 1)^{5} \)
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| $41$ |
\( T^{5} - 4 T^{4} + \cdots + 1255 \)
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| $43$ |
\( T^{5} + 4 T^{4} + \cdots + 225 \)
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| $47$ |
\( T^{5} - 7 T^{4} + \cdots - 18197 \)
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| $53$ |
\( T^{5} + 14 T^{4} + \cdots - 7349 \)
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| $59$ |
\( T^{5} - 3 T^{4} + \cdots + 1135 \)
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| $61$ |
\( T^{5} + 27 T^{4} + \cdots - 18125 \)
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| $67$ |
\( T^{5} - 12 T^{4} + \cdots + 8425 \)
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| $71$ |
\( T^{5} + 11 T^{4} + \cdots + 16839 \)
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| $73$ |
\( T^{5} + 15 T^{4} + \cdots + 21325 \)
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| $79$ |
\( T^{5} - 5 T^{4} + \cdots - 17531 \)
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| $83$ |
\( T^{5} + 8 T^{4} + \cdots - 27 \)
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| $89$ |
\( T^{5} - 27 T^{4} + \cdots + 2403 \)
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| $97$ |
\( T^{5} + 7 T^{4} + \cdots - 3379 \)
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