Newspace parameters
| Level: | \( N \) | \(=\) | \( 171 = 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 171.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(88.0711279840\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{8} - x^{7} - 3356 x^{6} - 1330 x^{5} + 3186388 x^{4} - 1801192 x^{3} - 758043152 x^{2} + \cdots - 16080668672 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{29}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 19) |
| 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
\( x^{8} - x^{7} - 3356 x^{6} - 1330 x^{5} + 3186388 x^{4} - 1801192 x^{3} - 758043152 x^{2} + \cdots - 16080668672 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 839 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 155435 \nu^{7} + 155494 \nu^{6} + 500071971 \nu^{5} + 754417652 \nu^{4} + \cdots - 393975956593088 ) / 22333949760 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2442607 \nu^{7} + 89863001 \nu^{6} + 5717983110 \nu^{5} - 218132022350 \nu^{4} + \cdots + 27\!\cdots\!12 ) / 89335799040 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 420885 \nu^{7} + 10482859 \nu^{6} + 1092447946 \nu^{5} - 24198670538 \nu^{4} + \cdots + 138044266497152 ) / 14889299840 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2957621 \nu^{7} - 103632415 \nu^{6} - 7003343598 \nu^{5} + 249515611234 \nu^{4} + \cdots - 34\!\cdots\!12 ) / 89335799040 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 5088661 \nu^{7} + 120563999 \nu^{6} + 13443010254 \nu^{5} - 277519681442 \nu^{4} + \cdots + 78261663501824 ) / 89335799040 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 839 \)
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| \(\nu^{3}\) | \(=\) |
\( -5\beta_{7} - \beta_{6} + 5\beta_{5} + 2\beta_{4} + 8\beta_{3} + 4\beta_{2} + 1460\beta _1 + 1577 \)
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| \(\nu^{4}\) | \(=\) |
\( 93\beta_{7} - 301\beta_{6} - 257\beta_{5} - 292\beta_{4} - 2\beta_{3} + 1880\beta_{2} + 10430\beta _1 + 1223861 \)
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| \(\nu^{5}\) | \(=\) |
\( - 11883 \beta_{7} - 7181 \beta_{6} + 8443 \beta_{5} + 1596 \beta_{4} + 22534 \beta_{3} + 8806 \beta_{2} + \cdots + 8673147 \)
|
| \(\nu^{6}\) | \(=\) |
\( 235743 \beta_{7} - 911307 \beta_{6} - 751431 \beta_{5} - 840920 \beta_{4} + 91222 \beta_{3} + \cdots + 2014079189 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 23484551 \beta_{7} - 22663853 \beta_{6} + 11105367 \beta_{5} - 2747256 \beta_{4} + 49941242 \beta_{3} + \cdots + 23417657479 \)
|
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 |
|
−43.3971 | 0 | 1371.31 | −1236.46 | 0 | −10692.2 | −37291.4 | 0 | 53658.9 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −30.7538 | 0 | 433.799 | −18.5839 | 0 | 6967.83 | 2404.99 | 0 | 571.527 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −30.7385 | 0 | 432.858 | −2199.99 | 0 | 7318.17 | 2432.70 | 0 | 67624.5 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 3.24681 | 0 | −501.458 | 574.796 | 0 | 7689.68 | −3290.50 | 0 | 1866.25 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 4.59102 | 0 | −490.923 | −2263.73 | 0 | −2177.39 | −4604.44 | 0 | −10392.8 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 6.83232 | 0 | −465.319 | 1115.83 | 0 | −8602.80 | −6677.36 | 0 | 7623.74 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 32.1433 | 0 | 521.193 | 1240.15 | 0 | −8874.26 | 295.489 | 0 | 39862.5 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 43.0760 | 0 | 1343.54 | −1106.01 | 0 | 1237.92 | 35819.6 | 0 | −47642.6 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(19\) | \( -1 \) |
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 | 171.10.a.f | 8 | |
| 3.b | odd | 2 | 1 | 19.10.a.b | ✓ | 8 | |
| 12.b | even | 2 | 1 | 304.10.a.i | 8 | ||
| 57.d | even | 2 | 1 | 361.10.a.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.10.a.b | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 171.10.a.f | 8 | 1.a | even | 1 | 1 | trivial | |
| 304.10.a.i | 8 | 12.b | even | 2 | 1 | ||
| 361.10.a.c | 8 | 57.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 15 T_{2}^{7} - 3258 T_{2}^{6} - 41238 T_{2}^{5} + 2972568 T_{2}^{4} + 23100984 T_{2}^{3} + \cdots - 5784998400 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(171))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots - 5784998400 \)
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| $3$ |
\( T^{8} \)
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| $5$ |
\( T^{8} + \cdots - 10\!\cdots\!00 \)
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| $7$ |
\( T^{8} + \cdots + 86\!\cdots\!12 \)
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| $11$ |
\( T^{8} + \cdots - 14\!\cdots\!96 \)
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| $13$ |
\( T^{8} + \cdots - 47\!\cdots\!80 \)
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| $17$ |
\( T^{8} + \cdots + 41\!\cdots\!34 \)
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| $19$ |
\( (T - 130321)^{8} \)
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| $23$ |
\( T^{8} + \cdots + 52\!\cdots\!00 \)
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| $29$ |
\( T^{8} + \cdots - 95\!\cdots\!60 \)
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| $31$ |
\( T^{8} + \cdots + 47\!\cdots\!00 \)
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| $37$ |
\( T^{8} + \cdots + 13\!\cdots\!40 \)
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| $41$ |
\( T^{8} + \cdots - 30\!\cdots\!00 \)
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| $43$ |
\( T^{8} + \cdots + 25\!\cdots\!36 \)
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| $47$ |
\( T^{8} + \cdots - 55\!\cdots\!52 \)
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| $53$ |
\( T^{8} + \cdots + 15\!\cdots\!20 \)
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| $59$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
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| $61$ |
\( T^{8} + \cdots - 16\!\cdots\!24 \)
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| $67$ |
\( T^{8} + \cdots - 56\!\cdots\!04 \)
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| $71$ |
\( T^{8} + \cdots + 27\!\cdots\!88 \)
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| $73$ |
\( T^{8} + \cdots - 39\!\cdots\!66 \)
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| $79$ |
\( T^{8} + \cdots + 19\!\cdots\!40 \)
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| $83$ |
\( T^{8} + \cdots - 50\!\cdots\!40 \)
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| $89$ |
\( T^{8} + \cdots - 47\!\cdots\!00 \)
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| $97$ |
\( T^{8} + \cdots + 27\!\cdots\!44 \)
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