Newspace parameters
| Level: | \( N \) | \(=\) | \( 2576 = 2^{4} \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2576.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(20.5694635607\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.2147108.1 |
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| Defining polynomial: |
\( x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 161) |
| 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
\( x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu - 5 \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 8\nu^{2} + 5\nu + 11 ) / 2 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 10\nu^{2} - 5\nu + 19 ) / 2 \)
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| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 9 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} - 2\beta_{2} + 3\beta _1 + 3 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{4} + 2\beta_{3} + 9\beta_{2} + 9\beta _1 + 51 ) / 2 \)
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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 |
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0 | −2.68857 | 0 | −1.86253 | 0 | −1.00000 | 0 | 4.22838 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.46268 | 0 | 2.78847 | 0 | −1.00000 | 0 | 3.06481 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.269842 | 0 | −3.51109 | 0 | −1.00000 | 0 | −2.92719 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.84074 | 0 | 2.40920 | 0 | −1.00000 | 0 | 0.388311 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.04067 | 0 | −3.82405 | 0 | −1.00000 | 0 | 6.24568 | 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 | 2576.2.a.bd | 5 | |
| 4.b | odd | 2 | 1 | 161.2.a.d | ✓ | 5 | |
| 12.b | even | 2 | 1 | 1449.2.a.r | 5 | ||
| 20.d | odd | 2 | 1 | 4025.2.a.p | 5 | ||
| 28.d | even | 2 | 1 | 1127.2.a.h | 5 | ||
| 92.b | even | 2 | 1 | 3703.2.a.j | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 161.2.a.d | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 1127.2.a.h | 5 | 28.d | even | 2 | 1 | ||
| 1449.2.a.r | 5 | 12.b | even | 2 | 1 | ||
| 2576.2.a.bd | 5 | 1.a | even | 1 | 1 | trivial | |
| 3703.2.a.j | 5 | 92.b | even | 2 | 1 | ||
| 4025.2.a.p | 5 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2576))\):
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\( T_{3}^{5} - 13T_{3}^{3} + 38T_{3} - 10 \)
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\( T_{5}^{5} + 4T_{5}^{4} - 14T_{5}^{3} - 54T_{5}^{2} + 52T_{5} + 168 \)
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\( T_{11}^{5} - 4T_{11}^{4} - 28T_{11}^{3} + 148T_{11}^{2} - 160T_{11} + 48 \)
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\( T_{13}^{5} + 6T_{13}^{4} - 9T_{13}^{3} - 46T_{13}^{2} + 12T_{13} + 56 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
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| $3$ |
\( T^{5} - 13 T^{3} + \cdots - 10 \)
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| $5$ |
\( T^{5} + 4 T^{4} + \cdots + 168 \)
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| $7$ |
\( (T + 1)^{5} \)
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| $11$ |
\( T^{5} - 4 T^{4} + \cdots + 48 \)
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| $13$ |
\( T^{5} + 6 T^{4} + \cdots + 56 \)
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| $17$ |
\( T^{5} + 12 T^{4} + \cdots - 1536 \)
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| $19$ |
\( T^{5} + 6 T^{4} + \cdots - 128 \)
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| $23$ |
\( (T - 1)^{5} \)
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| $29$ |
\( T^{5} + 4 T^{4} + \cdots - 1452 \)
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| $31$ |
\( T^{5} + 30 T^{4} + \cdots + 5206 \)
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| $37$ |
\( T^{5} - 4 T^{4} + \cdots - 32 \)
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| $41$ |
\( T^{5} - 6 T^{4} + \cdots - 456 \)
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| $43$ |
\( T^{5} - 12 T^{4} + \cdots - 256 \)
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| $47$ |
\( T^{5} + 10 T^{4} + \cdots - 11142 \)
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| $53$ |
\( T^{5} - 16 T^{4} + \cdots - 480 \)
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| $59$ |
\( T^{5} + 22 T^{4} + \cdots - 1440 \)
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| $61$ |
\( T^{5} + 18 T^{4} + \cdots + 56 \)
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| $67$ |
\( T^{5} - 2 T^{4} + \cdots - 61936 \)
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| $71$ |
\( T^{5} + 4 T^{4} + \cdots + 5184 \)
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| $73$ |
\( T^{5} + 2 T^{4} + \cdots - 27656 \)
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| $79$ |
\( T^{5} + 30 T^{4} + \cdots - 1936 \)
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| $83$ |
\( T^{5} + 8 T^{4} + \cdots + 5376 \)
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| $89$ |
\( T^{5} + 20 T^{4} + \cdots - 4704 \)
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| $97$ |
\( T^{5} + 12 T^{4} + \cdots + 4120 \)
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