Newspace parameters
| Level: | \( N \) | \(=\) | \( 1288 = 2^{3} \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1288.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(10.2847317803\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.3385684.1 |
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|
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| Defining polynomial: |
\( x^{5} - 2x^{4} - 12x^{3} + 22x^{2} + 20x - 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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
\( x^{5} - 2x^{4} - 12x^{3} + 22x^{2} + 20x - 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 8\nu^{2} + 2\nu - 4 ) / 4 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 12\nu^{2} + 2\nu + 20 ) / 4 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 4\nu^{3} - 8\nu^{2} - 34\nu ) / 4 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 6 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{2} + 9\beta _1 - 1 \)
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| \(\nu^{4}\) | \(=\) |
\( -8\beta_{3} + 12\beta_{2} - 2\beta _1 + 52 \)
|
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 | −3.37084 | 0 | 2.66293 | 0 | 1.00000 | 0 | 8.36257 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.40396 | 0 | −3.17666 | 0 | 1.00000 | 0 | 2.77901 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.768257 | 0 | 3.04812 | 0 | 1.00000 | 0 | −2.40978 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.191721 | 0 | −1.09901 | 0 | 1.00000 | 0 | −2.96324 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.35133 | 0 | 0.564619 | 0 | 1.00000 | 0 | 8.23144 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 1288.2.a.p | ✓ | 5 |
| 4.b | odd | 2 | 1 | 2576.2.a.be | 5 | ||
| 7.b | odd | 2 | 1 | 9016.2.a.bg | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1288.2.a.p | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 2576.2.a.be | 5 | 4.b | odd | 2 | 1 | ||
| 9016.2.a.bg | 5 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1288))\):
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\( T_{3}^{5} + 3T_{3}^{4} - 10T_{3}^{3} - 34T_{3}^{2} - 14T_{3} + 4 \)
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\( T_{5}^{5} - 2T_{5}^{4} - 12T_{5}^{3} + 22T_{5}^{2} + 20T_{5} - 16 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
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| $3$ |
\( T^{5} + 3 T^{4} + \cdots + 4 \)
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| $5$ |
\( T^{5} - 2 T^{4} + \cdots - 16 \)
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| $7$ |
\( (T - 1)^{5} \)
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| $11$ |
\( T^{5} + 2 T^{4} + \cdots + 128 \)
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| $13$ |
\( T^{5} - 9 T^{4} + \cdots + 128 \)
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| $17$ |
\( T^{5} - 12 T^{4} + \cdots + 64 \)
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| $19$ |
\( T^{5} + 12 T^{4} + \cdots + 2560 \)
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| $23$ |
\( (T + 1)^{5} \)
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| $29$ |
\( T^{5} - T^{4} + \cdots - 200 \)
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| $31$ |
\( T^{5} + 3 T^{4} + \cdots + 3592 \)
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| $37$ |
\( T^{5} - 2 T^{4} + \cdots - 12448 \)
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| $41$ |
\( T^{5} - 19 T^{4} + \cdots + 5584 \)
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| $43$ |
\( T^{5} - 136 T^{3} + \cdots + 2048 \)
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| $47$ |
\( T^{5} - 17 T^{4} + \cdots - 5240 \)
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| $53$ |
\( (T - 6)^{5} \)
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| $59$ |
\( T^{5} + 14 T^{4} + \cdots + 80128 \)
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| $61$ |
\( T^{5} - 2 T^{4} + \cdots - 5200 \)
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| $67$ |
\( T^{5} + 2 T^{4} + \cdots + 14368 \)
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| $71$ |
\( T^{5} - 43 T^{4} + \cdots + 102400 \)
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| $73$ |
\( T^{5} - 17 T^{4} + \cdots + 464 \)
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| $79$ |
\( T^{5} + 18 T^{4} + \cdots + 16960 \)
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| $83$ |
\( T^{5} - 2 T^{4} + \cdots - 128 \)
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| $89$ |
\( T^{5} - 16 T^{4} + \cdots - 5200 \)
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| $97$ |
\( T^{5} - 18 T^{4} + \cdots + 80704 \)
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