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: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.11344.1 |
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| Defining polynomial: |
\( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 4640) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 2 \)
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| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{3} - 3\beta_{2} + 7\beta _1 + 11 ) / 2 \)
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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 |
|
0 | −2.87834 | 0 | 1.00000 | 0 | −2.87834 | 0 | 5.28487 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0.296842 | 0 | 1.00000 | 0 | 0.296842 | 0 | −2.91188 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.53844 | 0 | 1.00000 | 0 | 1.53844 | 0 | −0.633188 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.04306 | 0 | 1.00000 | 0 | 3.04306 | 0 | 6.26020 | 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.cf | 4 | |
| 4.b | odd | 2 | 1 | 9280.2.a.bz | 4 | ||
| 8.b | even | 2 | 1 | 4640.2.a.l | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 4640.2.a.r | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4640.2.a.l | ✓ | 4 | 8.b | even | 2 | 1 | |
| 4640.2.a.r | yes | 4 | 8.d | odd | 2 | 1 | |
| 9280.2.a.bz | 4 | 4.b | odd | 2 | 1 | ||
| 9280.2.a.cf | 4 | 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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\( T_{3}^{4} - 2T_{3}^{3} - 8T_{3}^{2} + 16T_{3} - 4 \)
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\( T_{7}^{4} - 2T_{7}^{3} - 8T_{7}^{2} + 16T_{7} - 4 \)
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\( T_{11}^{4} + 2T_{11}^{3} - 24T_{11}^{2} - 32T_{11} + 20 \)
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\( T_{13}^{4} + 4T_{13}^{3} - 24T_{13}^{2} - 48T_{13} + 160 \)
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\( T_{19}^{4} - 6T_{19}^{3} - 28T_{19}^{2} + 160T_{19} + 36 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} - 2 T^{3} + \cdots - 4 \)
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| $5$ |
\( (T - 1)^{4} \)
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| $7$ |
\( T^{4} - 2 T^{3} + \cdots - 4 \)
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| $11$ |
\( T^{4} + 2 T^{3} + \cdots + 20 \)
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| $13$ |
\( T^{4} + 4 T^{3} + \cdots + 160 \)
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| $17$ |
\( T^{4} + 4 T^{3} + \cdots + 300 \)
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| $19$ |
\( T^{4} - 6 T^{3} + \cdots + 36 \)
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| $23$ |
\( T^{4} + 14 T^{3} + \cdots - 772 \)
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| $29$ |
\( (T + 1)^{4} \)
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| $31$ |
\( T^{4} + 10 T^{3} + \cdots + 100 \)
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| $37$ |
\( T^{4} - 16 T^{3} + \cdots - 596 \)
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| $41$ |
\( T^{4} - 72 T^{2} + \cdots + 480 \)
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| $43$ |
\( T^{4} - 14 T^{3} + \cdots + 60 \)
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| $47$ |
\( T^{4} - 6 T^{3} + \cdots - 372 \)
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| $53$ |
\( T^{4} + 8 T^{3} + \cdots + 352 \)
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| $59$ |
\( T^{4} + 12 T^{3} + \cdots - 720 \)
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| $61$ |
\( T^{4} + 16 T^{3} + \cdots - 4704 \)
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| $67$ |
\( T^{4} - 6 T^{3} + \cdots + 8524 \)
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| $71$ |
\( T^{4} - 4 T^{3} + \cdots + 48 \)
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| $73$ |
\( T^{4} - 12 T^{3} + \cdots - 2820 \)
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| $79$ |
\( T^{4} + 2 T^{3} + \cdots - 1004 \)
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| $83$ |
\( T^{4} - 22 T^{3} + \cdots - 1524 \)
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| $89$ |
\( T^{4} - 4 T^{3} + \cdots - 16 \)
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| $97$ |
\( T^{4} + 8 T^{3} + \cdots - 9092 \)
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