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: | 5.5.303952.1 |
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|
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| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 2x^{2} + 5x + 1 \)
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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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 6x^{3} + 2x^{2} + 5x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 2\nu + 1 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 2\nu + 4 \)
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| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 5\nu^{2} - 7\nu - 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 5\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 5\beta_{3} + 7\beta_{2} + 3\beta _1 + 16 \)
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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 | −3.08717 | 1.00000 | −0.665485 | −3.08717 | 1.92435 | 1.00000 | 6.53062 | −0.665485 | |||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.04977 | 1.00000 | 2.68092 | −1.04977 | 1.35768 | 1.00000 | −1.89799 | 2.68092 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0.199967 | 1.00000 | −1.97156 | 0.199967 | −3.66431 | 1.00000 | −2.96001 | −1.97156 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.08036 | 1.00000 | −0.238066 | 1.08036 | 0.882095 | 1.00000 | −1.83282 | −0.238066 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.85661 | 1.00000 | 1.19418 | 2.85661 | −4.49982 | 1.00000 | 5.16021 | 1.19418 | ||||||||||||||||||||||||||||||||||
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.ba | yes | 5 |
| 11.b | odd | 2 | 1 | 8954.2.a.x | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8954.2.a.x | ✓ | 5 | 11.b | odd | 2 | 1 | |
| 8954.2.a.ba | yes | 5 | 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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\( T_{3}^{5} - 10T_{3}^{3} + 2T_{3}^{2} + 10T_{3} - 2 \)
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\( T_{5}^{5} - T_{5}^{4} - 6T_{5}^{3} + 2T_{5}^{2} + 5T_{5} + 1 \)
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\( T_{7}^{5} + 4T_{7}^{4} - 12T_{7}^{3} - 26T_{7}^{2} + 72T_{7} - 38 \)
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\( T_{17}^{5} - 7T_{17}^{4} - 10T_{17}^{3} + 114T_{17}^{2} - 63T_{17} - 43 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{5} \)
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| $3$ |
\( T^{5} - 10 T^{3} + \cdots - 2 \)
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| $5$ |
\( T^{5} - T^{4} - 6 T^{3} + \cdots + 1 \)
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| $7$ |
\( T^{5} + 4 T^{4} + \cdots - 38 \)
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| $11$ |
\( T^{5} \)
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| $13$ |
\( T^{5} + 6 T^{4} + \cdots + 2 \)
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| $17$ |
\( T^{5} - 7 T^{4} + \cdots - 43 \)
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| $19$ |
\( T^{5} + 10 T^{4} + \cdots - 874 \)
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| $23$ |
\( T^{5} + 12 T^{4} + \cdots - 128 \)
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| $29$ |
\( T^{5} - 2 T^{4} + \cdots + 1718 \)
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| $31$ |
\( T^{5} + 10 T^{4} + \cdots + 2648 \)
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| $37$ |
\( (T - 1)^{5} \)
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| $41$ |
\( T^{5} - 4 T^{4} + \cdots - 2 \)
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| $43$ |
\( T^{5} + 20 T^{4} + \cdots - 50 \)
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| $47$ |
\( T^{5} - 3 T^{4} + \cdots - 6361 \)
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| $53$ |
\( T^{5} + 16 T^{4} + \cdots + 1216 \)
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| $59$ |
\( T^{5} + 4 T^{4} + \cdots + 4 \)
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| $61$ |
\( T^{5} + 14 T^{4} + \cdots + 974 \)
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| $67$ |
\( T^{5} - 26 T^{4} + \cdots - 650 \)
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| $71$ |
\( T^{5} + 5 T^{4} + \cdots + 10861 \)
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| $73$ |
\( T^{5} + 28 T^{4} + \cdots - 2246 \)
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| $79$ |
\( T^{5} + 21 T^{4} + \cdots + 323 \)
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| $83$ |
\( T^{5} + 19 T^{4} + \cdots + 15173 \)
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| $89$ |
\( T^{5} + 10 T^{4} + \cdots - 1112 \)
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| $97$ |
\( T^{5} + 10 T^{4} + \cdots - 6346 \)
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