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: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{8} - 3x^{7} - 216x^{6} + 384x^{5} + 14645x^{4} - 11755x^{3} - 306418x^{2} + 90238x + 1047460 \)
|
| 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
\( x^{8} - 3x^{7} - 216x^{6} + 384x^{5} + 14645x^{4} - 11755x^{3} - 306418x^{2} + 90238x + 1047460 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 55 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 281 \nu^{7} - 2322 \nu^{6} - 41078 \nu^{5} + 271814 \nu^{4} + 1754983 \nu^{3} - 7164016 \nu^{2} + \cdots + 25840748 ) / 1173696 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 78 \nu^{7} - 1667 \nu^{6} - 721 \nu^{5} + 188421 \nu^{4} - 542555 \nu^{3} - 4739090 \nu^{2} + \cdots + 15351244 ) / 293424 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 421 \nu^{7} + 6568 \nu^{6} + 44880 \nu^{5} - 892144 \nu^{4} - 520345 \nu^{3} + 30218388 \nu^{2} + \cdots - 154011108 ) / 1173696 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 367 \nu^{7} + 3533 \nu^{6} + 40619 \nu^{5} - 338491 \nu^{4} - 700344 \nu^{3} + 3941314 \nu^{2} + \cdots + 29842168 ) / 586848 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 529 \nu^{7} + 6525 \nu^{6} + 59515 \nu^{5} - 734059 \nu^{4} - 1670258 \nu^{3} + 17057786 \nu^{2} + \cdots - 44928112 ) / 586848 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 55 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{3} + 2\beta_{2} + 84\beta _1 + 72 \)
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| \(\nu^{4}\) | \(=\) |
\( 10\beta_{7} + 4\beta_{6} - 2\beta_{5} + 10\beta_{4} + 34\beta_{3} + 103\beta_{2} + 221\beta _1 + 4693 \)
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| \(\nu^{5}\) | \(=\) |
\( 92\beta_{7} + 101\beta_{6} + 121\beta_{5} + 213\beta_{4} + 555\beta_{3} + 360\beta_{2} + 8020\beta _1 + 14222 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2066 \beta_{7} + 490 \beta_{6} + 36 \beta_{5} + 2036 \beta_{4} + 6852 \beta_{3} + 10437 \beta_{2} + \cdots + 454635 \)
|
| \(\nu^{7}\) | \(=\) |
\( 20848 \beta_{7} + 8699 \beta_{6} + 13675 \beta_{5} + 32043 \beta_{4} + 90305 \beta_{3} + 52242 \beta_{2} + \cdots + 2156840 \)
|
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 |
|
−10.1813 | −19.2338 | 71.6598 | 25.0000 | 195.825 | −52.5302 | −403.790 | 126.937 | −254.534 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.42452 | 16.2153 | 38.9726 | 25.0000 | −136.606 | 216.971 | −58.7405 | 19.9355 | −210.613 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.45691 | −3.73602 | −12.1359 | 25.0000 | 16.6511 | −106.632 | 196.710 | −229.042 | −111.423 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.17089 | 28.2424 | −30.6290 | 25.0000 | −33.0688 | −72.8315 | 73.3318 | 554.632 | −29.2723 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.92830 | −24.4464 | −23.4251 | 25.0000 | −71.5865 | −192.001 | −162.301 | 354.628 | 73.2075 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 6.47558 | 20.5829 | 9.93308 | 25.0000 | 133.286 | 103.024 | −142.896 | 180.655 | 161.889 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 9.83081 | −16.3074 | 64.6448 | 25.0000 | −160.315 | 73.2314 | 320.925 | 22.9310 | 245.770 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 9.99899 | 16.6830 | 67.9798 | 25.0000 | 166.813 | −15.2319 | 359.761 | 35.3236 | 249.975 | |||||||||||||||||||||||||||||||||||||||||||
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.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 765.6.a.j | 8 | ||
| 5.b | even | 2 | 1 | 425.6.a.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.6.a.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 425.6.a.f | 8 | 5.b | even | 2 | 1 | ||
| 765.6.a.j | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 5T_{2}^{7} - 209T_{2}^{6} + 919T_{2}^{5} + 13290T_{2}^{4} - 46296T_{2}^{3} - 253248T_{2}^{2} + 498672T_{2} + 834336 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(85))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 5 T^{7} + \cdots + 834336 \)
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| $3$ |
\( T^{8} + \cdots + 4504847400 \)
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| $5$ |
\( (T - 25)^{8} \)
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| $7$ |
\( T^{8} + \cdots - 19\!\cdots\!16 \)
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| $11$ |
\( T^{8} + \cdots - 66\!\cdots\!80 \)
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| $13$ |
\( T^{8} + \cdots - 60\!\cdots\!88 \)
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| $17$ |
\( (T + 289)^{8} \)
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| $19$ |
\( T^{8} + \cdots - 48\!\cdots\!60 \)
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| $23$ |
\( T^{8} + \cdots - 38\!\cdots\!00 \)
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| $29$ |
\( T^{8} + \cdots + 55\!\cdots\!00 \)
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| $31$ |
\( T^{8} + \cdots - 23\!\cdots\!36 \)
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| $37$ |
\( T^{8} + \cdots + 35\!\cdots\!24 \)
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| $41$ |
\( T^{8} + \cdots - 25\!\cdots\!92 \)
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| $43$ |
\( T^{8} + \cdots + 43\!\cdots\!96 \)
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| $47$ |
\( T^{8} + \cdots - 31\!\cdots\!80 \)
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| $53$ |
\( T^{8} + \cdots - 10\!\cdots\!56 \)
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| $59$ |
\( T^{8} + \cdots - 27\!\cdots\!80 \)
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| $61$ |
\( T^{8} + \cdots - 77\!\cdots\!00 \)
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| $67$ |
\( T^{8} + \cdots + 11\!\cdots\!00 \)
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| $71$ |
\( T^{8} + \cdots - 40\!\cdots\!44 \)
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| $73$ |
\( T^{8} + \cdots + 15\!\cdots\!48 \)
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| $79$ |
\( T^{8} + \cdots + 63\!\cdots\!60 \)
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| $83$ |
\( T^{8} + \cdots + 15\!\cdots\!12 \)
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| $89$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
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| $97$ |
\( T^{8} + \cdots - 40\!\cdots\!00 \)
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