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: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{8} - 3 x^{7} - 135356 x^{6} - 24398 x^{5} + 5213582205 x^{4} + 598076469 x^{3} + \cdots + 15\!\cdots\!64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{25}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} - 3 x^{7} - 135356 x^{6} - 24398 x^{5} + 5213582205 x^{4} + 598076469 x^{3} + \cdots + 15\!\cdots\!64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4\nu - 33839 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 253915866823 \nu^{7} - 39044211581394 \nu^{6} + \cdots + 25\!\cdots\!04 ) / 65\!\cdots\!00 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 128047192432 \nu^{7} + 18240898422141 \nu^{6} + \cdots + 16\!\cdots\!24 ) / 25\!\cdots\!00 \)
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| \(\beta_{5}\) | \(=\) |
\( ( - 20167518840415 \nu^{7} + 193965141915267 \nu^{6} + \cdots + 30\!\cdots\!64 ) / 10\!\cdots\!00 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 8045120983423 \nu^{7} - 294583135275621 \nu^{6} + \cdots + 21\!\cdots\!46 ) / 40\!\cdots\!50 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 64477563979993 \nu^{7} + \cdots - 30\!\cdots\!88 ) / 32\!\cdots\!00 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4\beta _1 + 33839 \)
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| \(\nu^{3}\) | \(=\) |
\( 38\beta_{7} - \beta_{6} - 52\beta_{5} + 98\beta_{4} + 626\beta_{3} + 14\beta_{2} + 58466\beta _1 + 139767 \)
|
| \(\nu^{4}\) | \(=\) |
\( 259 \beta_{7} - 8 \beta_{6} - 261 \beta_{5} + 50474 \beta_{4} + 65058 \beta_{3} + 68702 \beta_{2} + \cdots + 1973949600 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2524897 \beta_{7} - 805649 \beta_{6} - 4357988 \beta_{5} + 13555057 \beta_{4} + 56382289 \beta_{3} + \cdots + 24916443019 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 8383266 \beta_{7} - 176233428 \beta_{6} - 163821986 \beta_{5} + 5408963204 \beta_{4} + \cdots + 123939846558045 \)
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| \(\nu^{7}\) | \(=\) |
\( 141881037808 \beta_{7} - 103564344341 \beta_{6} - 321123017632 \beta_{5} + 1343743782168 \beta_{4} + \cdots + 29\!\cdots\!63 \)
|
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 | −266.659 | 0 | −2715.88 | 0 | 3411.26 | 0 | 51423.9 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −263.871 | 0 | 2083.78 | 0 | −3949.77 | 0 | 49944.8 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −127.049 | 0 | 393.414 | 0 | −6601.13 | 0 | −3541.59 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −23.3087 | 0 | −760.329 | 0 | 8557.89 | 0 | −19139.7 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −11.3170 | 0 | 2354.21 | 0 | 4597.91 | 0 | −19554.9 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 88.9864 | 0 | −1188.35 | 0 | −3819.67 | 0 | −11764.4 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 209.704 | 0 | 488.996 | 0 | −11010.2 | 0 | 24292.6 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 252.514 | 0 | 1395.15 | 0 | 11230.7 | 0 | 44080.4 | 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.m | 8 | |
| 4.b | odd | 2 | 1 | 104.10.a.d | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.10.a.d | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 208.10.a.m | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 141 T_{3}^{7} - 126662 T_{3}^{6} - 14336666 T_{3}^{5} + 4560292185 T_{3}^{4} + \cdots - 11\!\cdots\!92 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(208))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots - 11\!\cdots\!92 \)
|
| $5$ |
\( T^{8} + \cdots - 32\!\cdots\!00 \)
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| $7$ |
\( T^{8} + \cdots + 16\!\cdots\!56 \)
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| $11$ |
\( T^{8} + \cdots - 13\!\cdots\!24 \)
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| $13$ |
\( (T - 28561)^{8} \)
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| $17$ |
\( T^{8} + \cdots + 24\!\cdots\!76 \)
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| $19$ |
\( T^{8} + \cdots + 53\!\cdots\!68 \)
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| $23$ |
\( T^{8} + \cdots + 54\!\cdots\!36 \)
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| $29$ |
\( T^{8} + \cdots + 52\!\cdots\!68 \)
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| $31$ |
\( T^{8} + \cdots - 12\!\cdots\!68 \)
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| $37$ |
\( T^{8} + \cdots - 83\!\cdots\!00 \)
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| $41$ |
\( T^{8} + \cdots + 38\!\cdots\!44 \)
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| $43$ |
\( T^{8} + \cdots - 21\!\cdots\!36 \)
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| $47$ |
\( T^{8} + \cdots - 24\!\cdots\!72 \)
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| $53$ |
\( T^{8} + \cdots + 45\!\cdots\!24 \)
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| $59$ |
\( T^{8} + \cdots + 15\!\cdots\!36 \)
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| $61$ |
\( T^{8} + \cdots + 24\!\cdots\!16 \)
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| $67$ |
\( T^{8} + \cdots - 10\!\cdots\!36 \)
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| $71$ |
\( T^{8} + \cdots - 52\!\cdots\!08 \)
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| $73$ |
\( T^{8} + \cdots + 14\!\cdots\!24 \)
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| $79$ |
\( T^{8} + \cdots - 43\!\cdots\!64 \)
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| $83$ |
\( T^{8} + \cdots - 10\!\cdots\!32 \)
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| $89$ |
\( T^{8} + \cdots - 27\!\cdots\!00 \)
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| $97$ |
\( T^{8} + \cdots - 13\!\cdots\!56 \)
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