Newspace parameters
| Level: | \( N \) | \(=\) | \( 85 = 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 85.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(13.6326246841\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 217x^{6} + 529x^{5} + 14270x^{4} - 26568x^{3} - 322656x^{2} + 436112x + 1913952 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 a root \(\nu\) of
\( x^{8} - 3x^{7} - 217x^{6} + 529x^{5} + 14270x^{4} - 26568x^{3} - 322656x^{2} + 436112x + 1913952 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 688 \nu^{7} - 1197 \nu^{6} + 144705 \nu^{5} + 321087 \nu^{4} - 8483271 \nu^{3} + \cdots + 270968496 ) / 674544 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 688 \nu^{7} + 1197 \nu^{6} - 144705 \nu^{5} - 321087 \nu^{4} + 8483271 \nu^{3} + 21983732 \nu^{2} + \cdots - 308068416 ) / 674544 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1197 \nu^{7} + 1000 \nu^{6} - 250924 \nu^{5} - 371228 \nu^{4} + 14812703 \nu^{3} + 30199984 \nu^{2} + \cdots - 453977232 ) / 674544 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 4195 \nu^{7} - 5113 \nu^{6} + 883813 \nu^{5} + 1493531 \nu^{4} - 52173800 \nu^{3} + \cdots + 1486073808 ) / 1349088 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 457 \nu^{7} - 473 \nu^{6} + 94157 \nu^{5} + 151891 \nu^{4} - 5363722 \nu^{3} - 11497048 \nu^{2} + \cdots + 169715952 ) / 103776 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 18671 \nu^{7} - 34629 \nu^{6} + 3931857 \nu^{5} + 8999055 \nu^{4} - 231803664 \nu^{3} + \cdots + 8030310192 ) / 2698176 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 55 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{7} + \beta_{5} + 2\beta_{4} - 10\beta_{3} + 4\beta_{2} + 85\beta _1 + 23 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{7} - 2\beta_{6} - 12\beta_{5} - 14\beta_{4} + 93\beta_{3} + 141\beta_{2} + 153\beta _1 + 4805 \)
|
| \(\nu^{5}\) | \(=\) |
\( -322\beta_{7} - 54\beta_{6} + 207\beta_{5} + 264\beta_{4} - 1624\beta_{3} + 622\beta_{2} + 8793\beta _1 + 4753 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 1040 \beta_{7} - 388 \beta_{6} - 1868 \beta_{5} - 2744 \beta_{4} + 8001 \beta_{3} + 17653 \beta_{2} + \cdots + 503499 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 43122 \beta_{7} - 11616 \beta_{6} + 28857 \beta_{5} + 29106 \beta_{4} - 219758 \beta_{3} + \cdots + 772915 \)
|
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 |
|
−11.2246 | 10.9613 | 93.9923 | −25.0000 | −123.037 | 11.7630 | −695.840 | −122.849 | 280.616 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.36478 | −26.0390 | 37.9695 | −25.0000 | 217.810 | −172.129 | −49.9338 | 435.029 | 209.119 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.66602 | −21.9907 | −10.2283 | −25.0000 | 102.609 | 220.039 | 197.038 | 240.592 | 116.650 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.56759 | 10.8158 | −11.1371 | −25.0000 | −49.4023 | 149.201 | 197.033 | −126.018 | 114.190 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.15916 | 20.5755 | −27.3380 | −25.0000 | 44.4258 | −142.667 | −128.120 | 180.352 | −53.9789 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 5.69415 | −2.00710 | 0.423381 | −25.0000 | −11.4287 | 89.3933 | −179.802 | −238.972 | −142.354 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 7.27269 | −2.60431 | 20.8920 | −25.0000 | −18.9403 | −120.946 | −80.7853 | −236.218 | −181.817 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10.6970 | −25.7116 | 82.4263 | −25.0000 | −275.037 | −218.656 | 539.411 | 418.084 | −267.426 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(17\) | \( +1 \) |
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 | 85.6.a.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 765.6.a.k | 8 | ||
| 5.b | even | 2 | 1 | 425.6.a.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.6.a.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 425.6.a.g | 8 | 5.b | even | 2 | 1 | ||
| 765.6.a.k | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 3T_{2}^{7} - 217T_{2}^{6} - 529T_{2}^{5} + 14270T_{2}^{4} + 26568T_{2}^{3} - 322656T_{2}^{2} - 436112T_{2} + 1913952 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(85))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 3 T^{7} + \cdots + 1913952 \)
|
| $3$ |
\( T^{8} + 36 T^{7} + \cdots - 187727400 \)
|
| $5$ |
\( (T + 25)^{8} \)
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| $7$ |
\( T^{8} + \cdots + 22\!\cdots\!76 \)
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| $11$ |
\( T^{8} + \cdots + 76\!\cdots\!80 \)
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| $13$ |
\( T^{8} + \cdots - 15\!\cdots\!56 \)
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| $17$ |
\( (T + 289)^{8} \)
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| $19$ |
\( T^{8} + \cdots + 13\!\cdots\!40 \)
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| $23$ |
\( T^{8} + \cdots - 31\!\cdots\!20 \)
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| $29$ |
\( T^{8} + \cdots - 25\!\cdots\!00 \)
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| $31$ |
\( T^{8} + \cdots + 11\!\cdots\!84 \)
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| $37$ |
\( T^{8} + \cdots + 18\!\cdots\!12 \)
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| $41$ |
\( T^{8} + \cdots - 30\!\cdots\!08 \)
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| $43$ |
\( T^{8} + \cdots + 15\!\cdots\!48 \)
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| $47$ |
\( T^{8} + \cdots + 33\!\cdots\!40 \)
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| $53$ |
\( T^{8} + \cdots - 23\!\cdots\!44 \)
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| $59$ |
\( T^{8} + \cdots - 10\!\cdots\!20 \)
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| $61$ |
\( T^{8} + \cdots - 35\!\cdots\!00 \)
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| $67$ |
\( T^{8} + \cdots + 95\!\cdots\!48 \)
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| $71$ |
\( T^{8} + \cdots + 45\!\cdots\!08 \)
|
| $73$ |
\( T^{8} + \cdots + 30\!\cdots\!96 \)
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| $79$ |
\( T^{8} + \cdots - 16\!\cdots\!20 \)
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| $83$ |
\( T^{8} + \cdots - 13\!\cdots\!44 \)
|
| $89$ |
\( T^{8} + \cdots - 83\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots - 80\!\cdots\!00 \)
|
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