Newspace parameters
| Level: | \( N \) | \(=\) | \( 9280 = 2^{6} \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9280.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(74.1011730757\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.469.1 |
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|
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| Defining polynomial: |
\( x^{3} - x^{2} - 5x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 290) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 5x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
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 |
|
0 | −1.71871 | 0 | −1.00000 | 0 | −2.39138 | 0 | −0.0460370 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −0.683969 | 0 | −1.00000 | 0 | 2.16425 | 0 | −2.53219 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.40268 | 0 | −1.00000 | 0 | −0.772866 | 0 | 8.57822 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 9280.2.a.bp | 3 | |
| 4.b | odd | 2 | 1 | 9280.2.a.bn | 3 | ||
| 8.b | even | 2 | 1 | 290.2.a.d | ✓ | 3 | |
| 8.d | odd | 2 | 1 | 2320.2.a.q | 3 | ||
| 24.h | odd | 2 | 1 | 2610.2.a.w | 3 | ||
| 40.f | even | 2 | 1 | 1450.2.a.r | 3 | ||
| 40.i | odd | 4 | 2 | 1450.2.b.j | 6 | ||
| 232.g | even | 2 | 1 | 8410.2.a.w | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 290.2.a.d | ✓ | 3 | 8.b | even | 2 | 1 | |
| 1450.2.a.r | 3 | 40.f | even | 2 | 1 | ||
| 1450.2.b.j | 6 | 40.i | odd | 4 | 2 | ||
| 2320.2.a.q | 3 | 8.d | odd | 2 | 1 | ||
| 2610.2.a.w | 3 | 24.h | odd | 2 | 1 | ||
| 8410.2.a.w | 3 | 232.g | even | 2 | 1 | ||
| 9280.2.a.bn | 3 | 4.b | odd | 2 | 1 | ||
| 9280.2.a.bp | 3 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9280))\):
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\( T_{3}^{3} - T_{3}^{2} - 7T_{3} - 4 \)
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\( T_{7}^{3} + T_{7}^{2} - 5T_{7} - 4 \)
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\( T_{11}^{3} + 2T_{11}^{2} - 20T_{11} - 32 \)
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\( T_{13}^{3} + 5T_{13}^{2} - 21T_{13} - 98 \)
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\( T_{19}^{3} + 2T_{19}^{2} - 28T_{19} + 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
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| $3$ |
\( T^{3} - T^{2} - 7T - 4 \)
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| $5$ |
\( (T + 1)^{3} \)
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| $7$ |
\( T^{3} + T^{2} - 5T - 4 \)
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| $11$ |
\( T^{3} + 2 T^{2} + \cdots - 32 \)
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| $13$ |
\( T^{3} + 5 T^{2} + \cdots - 98 \)
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| $17$ |
\( T^{3} + 5 T^{2} + \cdots - 2 \)
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| $19$ |
\( T^{3} + 2 T^{2} + \cdots + 32 \)
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| $23$ |
\( T^{3} - T^{2} - 7T - 4 \)
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| $29$ |
\( (T + 1)^{3} \)
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| $31$ |
\( T^{3} + 5 T^{2} + \cdots - 268 \)
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| $37$ |
\( T^{3} - 4 T^{2} + \cdots + 8 \)
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| $41$ |
\( T^{3} - 8T^{2} + 56 \)
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| $43$ |
\( T^{3} + 5 T^{2} + \cdots - 500 \)
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| $47$ |
\( (T + 8)^{3} \)
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| $53$ |
\( T^{3} + 3 T^{2} + \cdots + 14 \)
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| $59$ |
\( T^{3} + T^{2} + \cdots - 196 \)
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| $61$ |
\( T^{3} + 5T^{2} + T - 14 \)
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| $67$ |
\( T^{3} + 4 T^{2} + \cdots + 256 \)
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| $71$ |
\( T^{3} + 4 T^{2} + \cdots + 256 \)
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| $73$ |
\( T^{3} - 11 T^{2} + \cdots + 862 \)
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| $79$ |
\( T^{3} - 9 T^{2} + \cdots + 2052 \)
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| $83$ |
\( T^{3} - 26 T^{2} + \cdots - 448 \)
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| $89$ |
\( T^{3} - 8 T^{2} + \cdots - 136 \)
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| $97$ |
\( T^{3} - 21 T^{2} + \cdots - 98 \)
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