Newspace parameters
| Level: | \( N \) | \(=\) | \( 9065 = 5 \cdot 7^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9065.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(72.3843894323\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.126032.1 |
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| Defining polynomial: |
\( x^{5} - 6x^{3} + 6x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1295) |
| 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
\( x^{5} - 6x^{3} + 6x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} - \nu + 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + \nu^{3} - 6\nu^{2} - 5\nu + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + 4\beta _1 - 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 5\beta_{2} + \beta _1 + 8 \)
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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 |
|
−2.23025 | 0.161179 | 2.97402 | 1.00000 | −0.359470 | 0 | −2.17231 | −2.97402 | −2.23025 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.815403 | 2.08209 | −1.33512 | 1.00000 | −1.69775 | 0 | 2.71947 | 1.33512 | −0.815403 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.392048 | −2.20143 | −1.84630 | 1.00000 | 0.863067 | 0 | 1.50793 | 1.84630 | −0.392048 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.33253 | −1.79565 | −0.224374 | 1.00000 | −2.39276 | 0 | −2.96404 | 0.224374 | 1.33253 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.10518 | 0.753811 | 2.43177 | 1.00000 | 1.58691 | 0 | 0.908949 | −2.43177 | 2.10518 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(37\) | \( +1 \) |
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 | 9065.2.a.i | 5 | |
| 7.b | odd | 2 | 1 | 1295.2.a.f | ✓ | 5 | |
| 35.c | odd | 2 | 1 | 6475.2.a.p | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1295.2.a.f | ✓ | 5 | 7.b | odd | 2 | 1 | |
| 6475.2.a.p | 5 | 35.c | odd | 2 | 1 | ||
| 9065.2.a.i | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9065))\):
|
\( T_{2}^{5} - 6T_{2}^{3} + 6T_{2} + 2 \)
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\( T_{3}^{5} + T_{3}^{4} - 6T_{3}^{3} - 4T_{3}^{2} + 7T_{3} - 1 \)
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\( T_{11}^{5} + 7T_{11}^{4} + 10T_{11}^{3} - 6T_{11}^{2} - 11T_{11} + 3 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 6 T^{3} + \cdots + 2 \)
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| $3$ |
\( T^{5} + T^{4} - 6 T^{3} + \cdots - 1 \)
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| $5$ |
\( (T - 1)^{5} \)
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| $7$ |
\( T^{5} \)
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| $11$ |
\( T^{5} + 7 T^{4} + \cdots + 3 \)
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| $13$ |
\( T^{5} - 3 T^{4} + \cdots - 1 \)
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| $17$ |
\( T^{5} + 5 T^{4} + \cdots + 1 \)
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| $19$ |
\( T^{5} - 32 T^{3} + \cdots - 2 \)
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| $23$ |
\( T^{5} - 4 T^{4} + \cdots + 2 \)
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| $29$ |
\( T^{5} + 17 T^{4} + \cdots - 2287 \)
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| $31$ |
\( T^{5} - 8 T^{4} + \cdots - 192 \)
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| $37$ |
\( (T + 1)^{5} \)
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| $41$ |
\( T^{5} - 10 T^{4} + \cdots - 32 \)
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| $43$ |
\( T^{5} + 6 T^{4} + \cdots + 7286 \)
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| $47$ |
\( T^{5} + 11 T^{4} + \cdots - 25 \)
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| $53$ |
\( T^{5} - 14 T^{4} + \cdots + 450 \)
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| $59$ |
\( T^{5} - 104 T^{3} + \cdots + 5046 \)
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| $61$ |
\( T^{5} - 4 T^{4} + \cdots - 8 \)
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| $67$ |
\( T^{5} + 28 T^{4} + \cdots + 1282 \)
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| $71$ |
\( T^{5} - 236 T^{3} + \cdots + 12 \)
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| $73$ |
\( T^{5} - 26 T^{4} + \cdots - 15584 \)
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| $79$ |
\( T^{5} + 17 T^{4} + \cdots - 4923 \)
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| $83$ |
\( T^{5} + 4 T^{4} + \cdots + 192 \)
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| $89$ |
\( T^{5} - 2 T^{4} + \cdots - 1376 \)
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| $97$ |
\( T^{5} - 35 T^{4} + \cdots + 161657 \)
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