Newspace parameters
| Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 208.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(107.127453922\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} - 69754x^{4} - 2752492x^{3} + 1089377733x^{2} + 50183965132x - 2195812679340 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 104) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 69754x^{4} - 2752492x^{3} + 1089377733x^{2} + 50183965132x - 2195812679340 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 368 \nu^{5} - 45703 \nu^{4} - 19288734 \nu^{3} + 1079565658 \nu^{2} + 229184428576 \nu - 1111353084345 ) / 2334126375 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 4837 \nu^{5} + 674552 \nu^{4} + 199554531 \nu^{3} - 12219130022 \nu^{2} + \cdots - 23038541410395 ) / 11670631875 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 67849 \nu^{5} - 10621154 \nu^{4} - 3012685212 \nu^{3} + 282627831344 \nu^{2} + \cdots - 930979994747460 ) / 23341263750 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 9059 \nu^{5} + 1534489 \nu^{4} + 387694092 \nu^{3} - 41746828279 \nu^{2} + \cdots + 133358837451735 ) / 2334126375 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 2\beta_{3} - 7\beta_{2} + 61\beta _1 + 23252 \)
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| \(\nu^{3}\) | \(=\) |
\( 134\beta_{5} + 246\beta_{4} - 59\beta_{3} - 1392\beta_{2} + 38879\beta _1 + 1376603 \)
|
| \(\nu^{4}\) | \(=\) |
\( 71274\beta_{5} + 126464\beta_{4} + 61554\beta_{3} - 415291\beta_{2} + 5203232\beta _1 + 895673683 \)
|
| \(\nu^{5}\) | \(=\) |
\( 12941740 \beta_{5} + 22732830 \beta_{4} - 1315120 \beta_{3} - 97659990 \beta_{2} + 1882315401 \beta _1 + 118198901710 \)
|
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 | −172.332 | 0 | 107.467 | 0 | 615.682 | 0 | 10015.3 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −170.070 | 0 | 476.881 | 0 | 9274.85 | 0 | 9240.75 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −98.4022 | 0 | −1673.83 | 0 | −9266.18 | 0 | −10000.0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 18.3984 | 0 | 2458.39 | 0 | −6435.39 | 0 | −19344.5 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 127.372 | 0 | 811.825 | 0 | 899.641 | 0 | −3459.32 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 235.033 | 0 | −2004.73 | 0 | 1495.40 | 0 | 35557.7 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 208.10.a.j | 6 | |
| 4.b | odd | 2 | 1 | 104.10.a.b | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.10.a.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 208.10.a.j | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 60T_{3}^{5} - 68254T_{3}^{4} - 5522652T_{3}^{3} + 965100573T_{3}^{2} + 70867356192T_{3} - 1588484286720 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(208))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} + \cdots - 1588484286720 \)
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| $5$ |
\( T^{6} + \cdots + 34\!\cdots\!00 \)
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| $7$ |
\( T^{6} + \cdots + 45\!\cdots\!00 \)
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| $11$ |
\( T^{6} + \cdots + 99\!\cdots\!88 \)
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| $13$ |
\( (T + 28561)^{6} \)
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| $17$ |
\( T^{6} + \cdots + 33\!\cdots\!80 \)
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| $19$ |
\( T^{6} + \cdots - 43\!\cdots\!28 \)
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| $23$ |
\( T^{6} + \cdots - 79\!\cdots\!00 \)
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| $29$ |
\( T^{6} + \cdots + 27\!\cdots\!56 \)
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| $31$ |
\( T^{6} + \cdots + 85\!\cdots\!92 \)
|
| $37$ |
\( T^{6} + \cdots - 14\!\cdots\!20 \)
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| $41$ |
\( T^{6} + \cdots + 63\!\cdots\!32 \)
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| $43$ |
\( T^{6} + \cdots + 10\!\cdots\!20 \)
|
| $47$ |
\( T^{6} + \cdots + 22\!\cdots\!00 \)
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| $53$ |
\( T^{6} + \cdots + 19\!\cdots\!68 \)
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| $59$ |
\( T^{6} + \cdots - 12\!\cdots\!60 \)
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| $61$ |
\( T^{6} + \cdots - 54\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 85\!\cdots\!32 \)
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| $71$ |
\( T^{6} + \cdots - 68\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots + 86\!\cdots\!16 \)
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| $79$ |
\( T^{6} + \cdots + 44\!\cdots\!00 \)
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| $83$ |
\( T^{6} + \cdots + 51\!\cdots\!92 \)
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| $89$ |
\( T^{6} + \cdots + 14\!\cdots\!48 \)
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| $97$ |
\( T^{6} + \cdots + 16\!\cdots\!44 \)
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