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: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 72131x^{4} + 310880x^{3} + 1323158708x^{2} - 12926602192x - 1707532200192 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{2} \) |
| 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
\( x^{6} - 2x^{5} - 72131x^{4} + 310880x^{3} + 1323158708x^{2} - 12926602192x - 1707532200192 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -76\nu^{5} - 917\nu^{4} + 4180218\nu^{3} + 25058752\nu^{2} - 48836424220\nu + 432797607042 ) / 1017191925 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 149\nu^{5} - 45937\nu^{4} - 198012\nu^{3} + 1836763552\nu^{2} - 183849555830\nu - 4766327538378 ) / 1130213250 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 79 \nu^{5} + 230377 \nu^{4} + 52528002 \nu^{3} - 13199429942 \nu^{2} + \cdots + 125958387049188 ) / 10171919250 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5801 \nu^{5} - 1020563 \nu^{4} - 223300938 \nu^{3} + 35086904548 \nu^{2} + \cdots + 15509439425328 ) / 10171919250 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} - \beta_{4} + 2\beta_{3} - 4\beta_{2} - 4\beta _1 + 24044 \)
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| \(\nu^{3}\) | \(=\) |
\( -47\beta_{5} + 70\beta_{4} + 157\beta_{3} - 89\beta_{2} + 35472\beta _1 - 95177 \)
|
| \(\nu^{4}\) | \(=\) |
\( -47382\beta_{5} - 27780\beta_{4} + 81642\beta_{3} - 214719\beta_{2} - 72356\beta _1 + 851902512 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2343154\beta_{5} + 3855668\beta_{4} + 8309816\beta_{3} - 17007487\beta_{2} + 1308031370\beta _1 - 1891381210 \)
|
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 | −182.617 | 0 | −149.732 | 0 | −2294.25 | 0 | 13666.0 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −174.799 | 0 | −722.350 | 0 | 6843.47 | 0 | 10871.6 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −21.2698 | 0 | 1834.98 | 0 | 4227.07 | 0 | −19230.6 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 54.4051 | 0 | −1319.68 | 0 | −6657.51 | 0 | −16723.1 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 176.417 | 0 | 1801.15 | 0 | 387.425 | 0 | 11439.9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 215.864 | 0 | −1620.37 | 0 | 7157.79 | 0 | 26914.2 | 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.k | 6 | |
| 4.b | odd | 2 | 1 | 104.10.a.a | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.10.a.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 208.10.a.k | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 68T_{3}^{5} - 70206T_{3}^{4} + 3455604T_{3}^{3} + 1260778797T_{3}^{2} - 41540331600T_{3} - 1406705130000 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(208))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 1406705130000 \)
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| $5$ |
\( T^{6} + \cdots + 76\!\cdots\!00 \)
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| $7$ |
\( T^{6} + \cdots + 12\!\cdots\!20 \)
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| $11$ |
\( T^{6} + \cdots + 12\!\cdots\!04 \)
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| $13$ |
\( (T - 28561)^{6} \)
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| $17$ |
\( T^{6} + \cdots - 37\!\cdots\!52 \)
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| $19$ |
\( T^{6} + \cdots - 60\!\cdots\!16 \)
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| $23$ |
\( T^{6} + \cdots - 23\!\cdots\!56 \)
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| $29$ |
\( T^{6} + \cdots - 39\!\cdots\!32 \)
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| $31$ |
\( T^{6} + \cdots + 16\!\cdots\!44 \)
|
| $37$ |
\( T^{6} + \cdots - 21\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots - 19\!\cdots\!00 \)
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| $43$ |
\( T^{6} + \cdots - 13\!\cdots\!60 \)
|
| $47$ |
\( T^{6} + \cdots + 31\!\cdots\!60 \)
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| $53$ |
\( T^{6} + \cdots - 40\!\cdots\!56 \)
|
| $59$ |
\( T^{6} + \cdots - 13\!\cdots\!04 \)
|
| $61$ |
\( T^{6} + \cdots + 11\!\cdots\!76 \)
|
| $67$ |
\( T^{6} + \cdots + 86\!\cdots\!80 \)
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| $71$ |
\( T^{6} + \cdots - 12\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots - 18\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots + 49\!\cdots\!60 \)
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| $83$ |
\( T^{6} + \cdots + 18\!\cdots\!04 \)
|
| $89$ |
\( T^{6} + \cdots + 27\!\cdots\!88 \)
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| $97$ |
\( T^{6} + \cdots - 14\!\cdots\!00 \)
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