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: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{7} - 2x^{6} - 174x^{5} + 280x^{4} + 8473x^{3} - 8454x^{2} - 81204x - 73800 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 174x^{5} + 280x^{4} + 8473x^{3} - 8454x^{2} - 81204x - 73800 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{6} + 7\nu^{5} + 453\nu^{4} - 951\nu^{3} - 17462\nu^{2} + 31356\nu + 75240 ) / 1920 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{6} - 7\nu^{5} - 453\nu^{4} + 951\nu^{3} + 19382\nu^{2} - 31356\nu - 173160 ) / 1920 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 5\nu^{5} + 159\nu^{4} - 757\nu^{3} - 6970\nu^{2} + 27828\nu + 53400 ) / 384 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} + 7\nu^{5} + 513\nu^{4} - 1311\nu^{3} - 23642\nu^{2} + 59676\nu + 156120 ) / 480 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 5\nu^{5} + 183\nu^{4} - 709\nu^{3} - 9058\nu^{2} + 23988\nu + 64824 ) / 192 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 51 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} - 2\beta_{4} - 2\beta_{3} + 2\beta_{2} + 79\beta _1 + 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{6} + 2\beta_{5} - 12\beta_{4} + 91\beta_{3} + 83\beta_{2} + 2\beta _1 + 3947 \)
|
| \(\nu^{5}\) | \(=\) |
\( 147\beta_{6} - 171\beta_{5} - 150\beta_{4} - 172\beta_{3} + 272\beta_{2} + 6513\beta _1 + 675 \)
|
| \(\nu^{6}\) | \(=\) |
\( 932\beta_{6} + 220\beta_{5} - 1528\beta_{4} + 8153\beta_{3} + 6073\beta_{2} + 908\beta _1 + 323579 \)
|
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 |
|
−8.38854 | −5.79088 | 38.3676 | −25.0000 | 48.5771 | −15.7775 | −53.4152 | −209.466 | 209.714 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.36057 | 29.6023 | 22.1780 | −25.0000 | −217.890 | 7.97017 | 72.2952 | 633.293 | 184.014 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.884207 | −14.7420 | −31.2182 | −25.0000 | 13.0350 | −152.021 | 55.8980 | −25.6739 | 22.1052 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.286458 | 7.80723 | −31.9179 | −25.0000 | −2.23644 | 62.8709 | 18.3098 | −182.047 | 7.16145 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 5.67039 | −26.2371 | 0.153344 | −25.0000 | −148.775 | 97.5249 | −180.583 | 445.388 | −141.760 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 9.66742 | 25.4107 | 61.4589 | −25.0000 | 245.656 | −170.362 | 284.792 | 402.704 | −241.685 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 10.5820 | 1.94982 | 79.9781 | −25.0000 | 20.6330 | 219.795 | 507.703 | −239.198 | −264.549 | ||||||||||||||||||||||||||||||||||||||||
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.b | ✓ | 7 |
| 3.b | odd | 2 | 1 | 765.6.a.h | 7 | ||
| 5.b | even | 2 | 1 | 425.6.a.e | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.6.a.b | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 425.6.a.e | 7 | 5.b | even | 2 | 1 | ||
| 765.6.a.h | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 9T_{2}^{6} - 141T_{2}^{5} + 1085T_{2}^{4} + 5688T_{2}^{3} - 30504T_{2}^{2} - 40848T_{2} - 9072 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(85))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 9 T^{6} + \cdots - 9072 \)
|
| $3$ |
\( T^{7} - 18 T^{6} + \cdots + 25647840 \)
|
| $5$ |
\( (T + 25)^{7} \)
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| $7$ |
\( T^{7} + \cdots + 4389022964864 \)
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| $11$ |
\( T^{7} + \cdots + 26\!\cdots\!20 \)
|
| $13$ |
\( T^{7} + \cdots + 24\!\cdots\!64 \)
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| $17$ |
\( (T - 289)^{7} \)
|
| $19$ |
\( T^{7} + \cdots - 18\!\cdots\!00 \)
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| $23$ |
\( T^{7} + \cdots - 58\!\cdots\!40 \)
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| $29$ |
\( T^{7} + \cdots - 43\!\cdots\!00 \)
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| $31$ |
\( T^{7} + \cdots - 29\!\cdots\!60 \)
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| $37$ |
\( T^{7} + \cdots - 42\!\cdots\!68 \)
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| $41$ |
\( T^{7} + \cdots - 21\!\cdots\!00 \)
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| $43$ |
\( T^{7} + \cdots + 79\!\cdots\!76 \)
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| $47$ |
\( T^{7} + \cdots + 37\!\cdots\!40 \)
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| $53$ |
\( T^{7} + \cdots - 70\!\cdots\!80 \)
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| $59$ |
\( T^{7} + \cdots + 93\!\cdots\!60 \)
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| $61$ |
\( T^{7} + \cdots - 92\!\cdots\!00 \)
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| $67$ |
\( T^{7} + \cdots + 80\!\cdots\!08 \)
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| $71$ |
\( T^{7} + \cdots - 18\!\cdots\!52 \)
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| $73$ |
\( T^{7} + \cdots + 41\!\cdots\!16 \)
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| $79$ |
\( T^{7} + \cdots + 43\!\cdots\!20 \)
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| $83$ |
\( T^{7} + \cdots + 24\!\cdots\!04 \)
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| $89$ |
\( T^{7} + \cdots - 14\!\cdots\!00 \)
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| $97$ |
\( T^{7} + \cdots - 78\!\cdots\!00 \)
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