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: | \(5\) |
| Coefficient field: | 5.5.6083172.1 |
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| Defining polynomial: |
\( x^{5} - x^{4} - 13x^{3} + 10x^{2} + 40x - 28 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1160) |
| 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
\( x^{5} - x^{4} - 13x^{3} + 10x^{2} + 40x - 28 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 7\nu + 5 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 11\nu + 1 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 7\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 9\beta_{2} + 3\beta _1 + 34 \)
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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 | −2.57991 | 0 | −1.00000 | 0 | −2.42414 | 0 | 3.65592 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.36347 | 0 | −1.00000 | 0 | 2.16993 | 0 | 2.58601 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.689426 | 0 | −1.00000 | 0 | 4.55109 | 0 | −2.52469 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.13670 | 0 | −1.00000 | 0 | −4.33279 | 0 | 1.56547 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.11726 | 0 | −1.00000 | 0 | −0.964080 | 0 | 6.71730 | 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.cj | 5 | |
| 4.b | odd | 2 | 1 | 9280.2.a.ch | 5 | ||
| 8.b | even | 2 | 1 | 1160.2.a.i | ✓ | 5 | |
| 8.d | odd | 2 | 1 | 2320.2.a.w | 5 | ||
| 40.f | even | 2 | 1 | 5800.2.a.v | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1160.2.a.i | ✓ | 5 | 8.b | even | 2 | 1 | |
| 2320.2.a.w | 5 | 8.d | odd | 2 | 1 | ||
| 5800.2.a.v | 5 | 40.f | even | 2 | 1 | ||
| 9280.2.a.ch | 5 | 4.b | odd | 2 | 1 | ||
| 9280.2.a.cj | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9280))\):
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\( T_{3}^{5} - T_{3}^{4} - 13T_{3}^{3} + 10T_{3}^{2} + 40T_{3} - 28 \)
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\( T_{7}^{5} + T_{7}^{4} - 25T_{7}^{3} - 28T_{7}^{2} + 100T_{7} + 100 \)
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\( T_{11}^{5} - 4T_{11}^{4} - 16T_{11}^{3} + 28T_{11}^{2} + 40T_{11} - 48 \)
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\( T_{13}^{5} + 13T_{13}^{4} + 35T_{13}^{3} - 142T_{13}^{2} - 740T_{13} - 824 \)
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\( T_{19}^{5} - 76T_{19}^{3} + 12T_{19}^{2} + 1432T_{19} - 432 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
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| $3$ |
\( T^{5} - T^{4} + \cdots - 28 \)
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| $5$ |
\( (T + 1)^{5} \)
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| $7$ |
\( T^{5} + T^{4} + \cdots + 100 \)
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| $11$ |
\( T^{5} - 4 T^{4} + \cdots - 48 \)
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| $13$ |
\( T^{5} + 13 T^{4} + \cdots - 824 \)
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| $17$ |
\( T^{5} - 3 T^{4} + \cdots - 1964 \)
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| $19$ |
\( T^{5} - 76 T^{3} + \cdots - 432 \)
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| $23$ |
\( T^{5} + 7 T^{4} + \cdots - 900 \)
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| $29$ |
\( (T - 1)^{5} \)
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| $31$ |
\( T^{5} - 11 T^{4} + \cdots - 36 \)
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| $37$ |
\( T^{5} + 6 T^{4} + \cdots - 16 \)
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| $41$ |
\( T^{5} - 16 T^{4} + \cdots + 14816 \)
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| $43$ |
\( T^{5} + 9 T^{4} + \cdots + 1156 \)
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| $47$ |
\( T^{5} - 2 T^{4} + \cdots + 64 \)
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| $53$ |
\( T^{5} + 7 T^{4} + \cdots + 33336 \)
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| $59$ |
\( T^{5} + 5 T^{4} + \cdots + 16 \)
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| $61$ |
\( T^{5} + 7 T^{4} + \cdots + 11112 \)
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| $67$ |
\( T^{5} - 4 T^{4} + \cdots + 192 \)
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| $71$ |
\( T^{5} - 12 T^{4} + \cdots - 30464 \)
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| $73$ |
\( T^{5} - 7 T^{4} + \cdots + 1588 \)
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| $79$ |
\( T^{5} - 45 T^{4} + \cdots + 36204 \)
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| $83$ |
\( T^{5} - 22 T^{4} + \cdots - 25776 \)
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| $89$ |
\( T^{5} - 24 T^{4} + \cdots + 88672 \)
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| $97$ |
\( T^{5} - 17 T^{4} + \cdots - 196 \)
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