Newspace parameters
| Level: | \( N \) | \(=\) | \( 5766 = 2 \cdot 3 \cdot 31^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5766.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(46.0417418055\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{32})^+\) |
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| Defining polynomial: |
\( x^{8} - 8x^{6} + 20x^{4} - 16x^{2} + 2 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{32} + \zeta_{32}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 5\nu \)
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| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 \)
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| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 7\nu^{5} + 14\nu^{3} - 7\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} + 10\beta_1 \)
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| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 6\beta_{4} + 15\beta_{2} + 20 \)
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| \(\nu^{7}\) | \(=\) |
\( \beta_{7} + 7\beta_{5} + 21\beta_{3} + 35\beta_1 \)
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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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−1.00000 | −1.00000 | 1.00000 | −2.07715 | 1.00000 | −0.677595 | −1.00000 | 1.00000 | 2.07715 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.00000 | 1.00000 | −1.52535 | 1.00000 | −1.59903 | −1.00000 | 1.00000 | 1.52535 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.00000 | 1.00000 | 0.452643 | 1.00000 | 1.03903 | −1.00000 | 1.00000 | −0.452643 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.00000 | 1.00000 | 0.696927 | 1.00000 | −4.92491 | −1.00000 | 1.00000 | −0.696927 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −1.00000 | 1.00000 | 1.24873 | 1.00000 | 1.54469 | −1.00000 | 1.00000 | −1.24873 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −1.00000 | 1.00000 | 2.02403 | 1.00000 | 0.218010 | −1.00000 | 1.00000 | −2.02403 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | −1.00000 | 1.00000 | 2.80439 | 1.00000 | 4.14115 | −1.00000 | 1.00000 | −2.80439 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | −1.00000 | 1.00000 | 4.37578 | 1.00000 | 0.258666 | −1.00000 | 1.00000 | −4.37578 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(31\) | \( +1 \) |
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 | 5766.2.a.bl | ✓ | 8 |
| 31.b | odd | 2 | 1 | 5766.2.a.bn | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5766.2.a.bl | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 5766.2.a.bn | yes | 8 | 31.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5766))\):
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\( T_{5}^{8} - 8T_{5}^{7} + 12T_{5}^{6} + 40T_{5}^{5} - 110T_{5}^{4} + 8T_{5}^{3} + 164T_{5}^{2} - 136T_{5} + 31 \)
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\( T_{7}^{8} - 24T_{7}^{6} + 16T_{7}^{5} + 64T_{7}^{4} - 48T_{7}^{3} - 24T_{7}^{2} + 16T_{7} - 2 \)
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\( T_{11}^{8} + 8T_{11}^{7} + 4T_{11}^{6} - 56T_{11}^{5} - 34T_{11}^{4} + 120T_{11}^{3} + 52T_{11}^{2} - 72T_{11} - 31 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
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| $3$ |
\( (T + 1)^{8} \)
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| $5$ |
\( T^{8} - 8 T^{7} + \cdots + 31 \)
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| $7$ |
\( T^{8} - 24 T^{6} + \cdots - 2 \)
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| $11$ |
\( T^{8} + 8 T^{7} + \cdots - 31 \)
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| $13$ |
\( T^{8} + 8 T^{7} + \cdots - 257 \)
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| $17$ |
\( T^{8} - 8 T^{7} + \cdots + 1697 \)
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| $19$ |
\( T^{8} + 8 T^{7} + \cdots - 12479 \)
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| $23$ |
\( T^{8} - 64 T^{6} + \cdots + 2878 \)
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| $29$ |
\( T^{8} + 16 T^{7} + \cdots - 940162 \)
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| $31$ |
\( T^{8} \)
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| $37$ |
\( T^{8} - 120 T^{6} + \cdots - 23428 \)
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| $41$ |
\( T^{8} - 16 T^{7} + \cdots - 440446 \)
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| $43$ |
\( T^{8} - 200 T^{6} + \cdots - 210878 \)
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| $47$ |
\( T^{8} - 24 T^{7} + \cdots + 6484607 \)
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| $53$ |
\( T^{8} + 16 T^{7} + \cdots + 805438 \)
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| $59$ |
\( T^{8} - 16 T^{7} + \cdots + 2687752 \)
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| $61$ |
\( T^{8} + 56 T^{7} + \cdots - 23279233 \)
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| $67$ |
\( T^{8} + 8 T^{7} + \cdots + 115937 \)
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| $71$ |
\( T^{8} - 8 T^{7} + \cdots - 5921 \)
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| $73$ |
\( T^{8} + 32 T^{7} + \cdots - 458366 \)
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| $79$ |
\( T^{8} - 8 T^{7} + \cdots - 14561 \)
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| $83$ |
\( T^{8} - 8 T^{7} + \cdots - 1898527 \)
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| $89$ |
\( T^{8} + 48 T^{7} + \cdots + 4085252 \)
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| $97$ |
\( T^{8} + 8 T^{7} + \cdots + 1011169 \)
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