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: | \(5\) |
| Coefficient field: | 5.5.2186192.1 |
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| Defining polynomial: |
\( x^{5} - 12x^{3} - 2x^{2} + 30x + 8 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 814) |
| 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
\( x^{5} - 12x^{3} - 2x^{2} + 30x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 4\nu^{3} + 6\nu^{2} - 22\nu - 2 ) / 10 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{4} - 2\nu^{3} - 28\nu^{2} + 6\nu + 36 ) / 10 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 4\nu^{3} + 16\nu^{2} - 32\nu - 52 ) / 10 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{2} + \beta _1 + 5 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 2\beta_{2} + 7\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( 10\beta_{4} + 4\beta_{3} - 8\beta_{2} + 12\beta _1 + 36 \)
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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 | −2.79452 | 1.00000 | 2.39985 | −2.79452 | 5.27135 | 1.00000 | 4.80934 | 2.39985 | |||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.85130 | 1.00000 | −3.06790 | −1.85130 | 2.94902 | 1.00000 | 0.427321 | −3.06790 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.269612 | 1.00000 | 0.301993 | −0.269612 | −2.39898 | 1.00000 | −2.92731 | 0.301993 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.90549 | 1.00000 | 3.66992 | 1.90549 | 2.71086 | 1.00000 | 0.630907 | 3.66992 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 3.00994 | 1.00000 | 2.69614 | 3.00994 | −2.53225 | 1.00000 | 6.05974 | 2.69614 | ||||||||||||||||||||||||||||||||||
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.bb | 5 | |
| 11.b | odd | 2 | 1 | 814.2.a.j | ✓ | 5 | |
| 33.d | even | 2 | 1 | 7326.2.a.bs | 5 | ||
| 44.c | even | 2 | 1 | 6512.2.a.s | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 814.2.a.j | ✓ | 5 | 11.b | odd | 2 | 1 | |
| 6512.2.a.s | 5 | 44.c | even | 2 | 1 | ||
| 7326.2.a.bs | 5 | 33.d | even | 2 | 1 | ||
| 8954.2.a.bb | 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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\( T_{3}^{5} - 12T_{3}^{3} - 2T_{3}^{2} + 30T_{3} + 8 \)
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\( T_{5}^{5} - 6T_{5}^{4} + 54T_{5}^{2} - 89T_{5} + 22 \)
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\( T_{7}^{5} - 6T_{7}^{4} - 10T_{7}^{3} + 78T_{7}^{2} + 22T_{7} - 256 \)
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\( T_{17}^{5} + 8T_{17}^{4} - 12T_{17}^{3} - 170T_{17}^{2} - 21T_{17} + 842 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{5} \)
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| $3$ |
\( T^{5} - 12 T^{3} + \cdots + 8 \)
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| $5$ |
\( T^{5} - 6 T^{4} + \cdots + 22 \)
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| $7$ |
\( T^{5} - 6 T^{4} + \cdots - 256 \)
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| $11$ |
\( T^{5} \)
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| $13$ |
\( T^{5} - 2 T^{4} + \cdots + 508 \)
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| $17$ |
\( T^{5} + 8 T^{4} + \cdots + 842 \)
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| $19$ |
\( T^{5} - 8 T^{4} + \cdots - 216 \)
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| $23$ |
\( T^{5} - 4 T^{4} + \cdots + 4544 \)
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| $29$ |
\( T^{5} - 98 T^{3} + \cdots + 604 \)
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| $31$ |
\( T^{5} - 2 T^{4} + \cdots + 7136 \)
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| $37$ |
\( (T - 1)^{5} \)
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| $41$ |
\( T^{5} - 2 T^{4} + \cdots - 52 \)
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| $43$ |
\( T^{5} - 10 T^{4} + \cdots + 5688 \)
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| $47$ |
\( T^{5} - 10 T^{4} + \cdots + 13744 \)
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| $53$ |
\( T^{5} - 10 T^{4} + \cdots + 64 \)
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| $59$ |
\( T^{5} - 12 T^{4} + \cdots + 4864 \)
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| $61$ |
\( T^{5} + 6 T^{4} + \cdots + 14564 \)
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| $67$ |
\( T^{5} + 18 T^{4} + \cdots + 5176 \)
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| $71$ |
\( T^{5} - 34 T^{4} + \cdots + 206208 \)
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| $73$ |
\( T^{5} - 104 T^{3} + \cdots - 556 \)
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| $79$ |
\( T^{5} + 8 T^{4} + \cdots + 28088 \)
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| $83$ |
\( T^{5} + 10 T^{4} + \cdots + 118044 \)
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| $89$ |
\( T^{5} - 60 T^{4} + \cdots - 129168 \)
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| $97$ |
\( T^{5} - 20 T^{4} + \cdots + 4124 \)
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