Newspace parameters
| Level: | \( N \) | \(=\) | \( 5819 = 11 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5819.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(46.4649489362\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
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| Defining polynomial: |
\( x^{9} - x^{8} - 11x^{7} + 11x^{6} + 35x^{5} - 30x^{4} - 35x^{3} + 16x^{2} + 10x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - x^{8} - 11x^{7} + 11x^{6} + 35x^{5} - 30x^{4} - 35x^{3} + 16x^{2} + 10x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{7} - 10\nu^{5} + \nu^{4} + 26\nu^{3} - 3\nu^{2} - 13\nu \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{8} + \nu^{7} - 9\nu^{6} - 8\nu^{5} + 19\nu^{4} + 17\nu^{3} - 2\nu^{2} - 6\nu - 1 \)
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| \(\beta_{6}\) | \(=\) |
\( -\nu^{8} + 10\nu^{6} - \nu^{5} - 27\nu^{4} + 2\nu^{3} + 18\nu^{2} + 4\nu - 2 \)
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| \(\beta_{7}\) | \(=\) |
\( -\nu^{8} - \nu^{7} + 10\nu^{6} + 8\nu^{5} - 28\nu^{4} - 16\nu^{3} + 21\nu^{2} + 4\nu - 4 \)
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| \(\beta_{8}\) | \(=\) |
\( -\nu^{8} + \nu^{7} + 10\nu^{6} - 11\nu^{5} - 25\nu^{4} + 29\nu^{3} + 9\nu^{2} - 13\nu + 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - \beta_{6} - \beta_{4} - \beta_{3} + 6\beta_{2} + 14 \)
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| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + \beta_{6} - \beta_{4} + 8\beta_{3} + 19\beta _1 - 2 \)
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| \(\nu^{6}\) | \(=\) |
\( 9\beta_{8} + \beta_{7} - 9\beta_{6} + \beta_{5} - 9\beta_{4} - 10\beta_{3} + 35\beta_{2} - 2\beta _1 + 74 \)
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| \(\nu^{7}\) | \(=\) |
\( -\beta_{8} - 10\beta_{7} + 11\beta_{6} - 8\beta_{4} + 55\beta_{3} - 3\beta_{2} + 99\beta _1 - 25 \)
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| \(\nu^{8}\) | \(=\) |
\( 63\beta_{8} + 11\beta_{7} - 65\beta_{6} + 10\beta_{5} - 62\beta_{4} - 79\beta_{3} + 206\beta_{2} - 27\beta _1 + 416 \)
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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 |
|
−2.32903 | 2.21751 | 3.42439 | −4.23025 | −5.16466 | −4.78391 | −3.31746 | 1.91737 | 9.85239 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.79859 | −1.32341 | 1.23493 | 4.13566 | 2.38027 | 0.907060 | 1.37605 | −1.24859 | −7.43836 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.74483 | −3.07887 | 1.04444 | −1.70017 | 5.37211 | −1.49607 | 1.66729 | 6.47943 | 2.96652 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.750662 | 2.38297 | −1.43651 | −1.87437 | −1.78881 | 4.11774 | 2.57965 | 2.67857 | 1.40702 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.0898052 | 1.17282 | −1.99194 | 3.53340 | −0.105326 | −1.58572 | 0.358496 | −1.62448 | −0.317318 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.536892 | −2.60739 | −1.71175 | 0.983758 | −1.39989 | −0.697853 | −1.99281 | 3.79849 | 0.528172 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.822125 | 1.82868 | −1.32411 | −0.670767 | 1.50340 | −2.49149 | −2.73283 | 0.344072 | −0.551455 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.80127 | −0.356052 | 1.24457 | 1.18345 | −0.641345 | 4.77750 | −1.36074 | −2.87323 | 2.13170 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.55264 | −1.23627 | 4.51597 | −3.36070 | −3.15576 | 0.252760 | 6.42236 | −1.47163 | −8.57866 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 5819.2.a.h | ✓ | 9 |
| 23.b | odd | 2 | 1 | 5819.2.a.i | yes | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5819.2.a.h | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 5819.2.a.i | yes | 9 | 23.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5819))\):
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\( T_{2}^{9} + T_{2}^{8} - 11T_{2}^{7} - 11T_{2}^{6} + 35T_{2}^{5} + 30T_{2}^{4} - 35T_{2}^{3} - 16T_{2}^{2} + 10T_{2} + 1 \)
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\( T_{5}^{9} + 2T_{5}^{8} - 32T_{5}^{7} - 64T_{5}^{6} + 286T_{5}^{5} + 577T_{5}^{4} - 574T_{5}^{3} - 1113T_{5}^{2} + 393T_{5} + 517 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + T^{8} - 11 T^{7} + \cdots + 1 \)
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| $3$ |
\( T^{9} + T^{8} + \cdots + 53 \)
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| $5$ |
\( T^{9} + 2 T^{8} + \cdots + 517 \)
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| $7$ |
\( T^{9} + T^{8} + \cdots + 89 \)
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| $11$ |
\( (T - 1)^{9} \)
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| $13$ |
\( T^{9} - 8 T^{8} + \cdots - 539 \)
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| $17$ |
\( T^{9} + 13 T^{8} + \cdots - 29375 \)
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| $19$ |
\( T^{9} + 7 T^{8} + \cdots + 1399 \)
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| $23$ |
\( T^{9} \)
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| $29$ |
\( T^{9} - 3 T^{8} + \cdots + 831569 \)
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| $31$ |
\( T^{9} + 6 T^{8} + \cdots + 333467 \)
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| $37$ |
\( T^{9} + 27 T^{8} + \cdots + 1316935 \)
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| $41$ |
\( T^{9} - T^{8} + \cdots - 14441017 \)
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| $43$ |
\( T^{9} + 11 T^{8} + \cdots + 668115 \)
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| $47$ |
\( T^{9} + 8 T^{8} + \cdots + 202069 \)
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| $53$ |
\( T^{9} - 8 T^{8} + \cdots - 34337 \)
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| $59$ |
\( T^{9} + 16 T^{8} + \cdots - 614395 \)
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| $61$ |
\( T^{9} + 34 T^{8} + \cdots + 7555975 \)
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| $67$ |
\( T^{9} + 8 T^{8} + \cdots - 34027609 \)
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| $71$ |
\( T^{9} - 349 T^{7} + \cdots + 36881525 \)
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| $73$ |
\( T^{9} + 17 T^{8} + \cdots - 44575 \)
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| $79$ |
\( T^{9} + 7 T^{8} + \cdots + 252589 \)
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| $83$ |
\( T^{9} + 14 T^{8} + \cdots + 56735 \)
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| $89$ |
\( T^{9} - T^{8} + \cdots + 31475 \)
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| $97$ |
\( T^{9} + 2 T^{8} + \cdots + 558573841 \)
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