Newspace parameters
| Level: | \( N \) | \(=\) | \( 171 = 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 171.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(10.0893266110\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 45x^{4} + 516x^{2} - 436 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 45x^{4} + 516x^{2} - 436 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 23\nu^{2} + 16 ) / 6 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 15 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{3} - 22\nu \)
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| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 35\nu^{3} + 262\nu ) / 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 15 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 22\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( 23\beta_{3} + 6\beta_{2} + 329 \)
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| \(\nu^{5}\) | \(=\) |
\( 12\beta_{5} + 35\beta_{4} + 508\beta_1 \)
|
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 |
|
−5.02421 | 0 | 17.2427 | −16.6214 | 0 | −9.96112 | −46.4372 | 0 | 83.5095 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −4.34057 | 0 | 10.8405 | 6.67469 | 0 | 28.6293 | −12.3296 | 0 | −28.9719 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.957478 | 0 | −7.08324 | −20.3268 | 0 | −8.66820 | 14.4419 | 0 | 19.4624 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.957478 | 0 | −7.08324 | 20.3268 | 0 | −8.66820 | −14.4419 | 0 | 19.4624 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 4.34057 | 0 | 10.8405 | −6.67469 | 0 | 28.6293 | 12.3296 | 0 | −28.9719 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 5.02421 | 0 | 17.2427 | 16.6214 | 0 | −9.96112 | 46.4372 | 0 | 83.5095 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(19\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 171.4.a.i | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 171.4.a.i | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 171.4.a.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 171.4.a.i | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(171))\):
|
\( T_{2}^{6} - 45T_{2}^{4} + 516T_{2}^{2} - 436 \)
|
|
\( T_{5}^{6} - 734T_{5}^{4} + 144865T_{5}^{2} - 5085504 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 45 T^{4} + \cdots - 436 \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} - 734 T^{4} + \cdots - 5085504 \)
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| $7$ |
\( (T^{3} - 10 T^{2} + \cdots - 2472)^{2} \)
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| $11$ |
\( T^{6} + \cdots - 1279384884 \)
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| $13$ |
\( (T^{3} - 74 T^{2} + \cdots + 281832)^{2} \)
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| $17$ |
\( T^{6} + \cdots - 29761640784 \)
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| $19$ |
\( (T + 19)^{6} \)
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| $23$ |
\( T^{6} - 42444 T^{4} + \cdots - 119715136 \)
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| $29$ |
\( T^{6} + \cdots - 588012348736 \)
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| $31$ |
\( (T^{3} - 256 T^{2} + \cdots + 1132464)^{2} \)
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| $37$ |
\( (T^{3} - 250 T^{2} + \cdots - 196128)^{2} \)
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| $41$ |
\( T^{6} + \cdots - 4383750657024 \)
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| $43$ |
\( (T^{3} - 546 T^{2} + \cdots - 3535796)^{2} \)
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| $47$ |
\( T^{6} + \cdots - 613194157553076 \)
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| $53$ |
\( T^{6} + \cdots - 13951775658816 \)
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| $59$ |
\( T^{6} + \cdots - 634487493574656 \)
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| $61$ |
\( (T^{3} - 196 T^{2} + \cdots + 68426854)^{2} \)
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| $67$ |
\( (T^{3} - 324 T^{2} + \cdots + 234470848)^{2} \)
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| $71$ |
\( T^{6} + \cdots - 79\!\cdots\!36 \)
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| $73$ |
\( (T^{3} + 1452 T^{2} + \cdots - 129001834)^{2} \)
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| $79$ |
\( (T^{3} - 328 T^{2} + \cdots - 304622208)^{2} \)
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| $83$ |
\( T^{6} + \cdots - 15\!\cdots\!64 \)
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| $89$ |
\( T^{6} + \cdots - 69\!\cdots\!96 \)
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| $97$ |
\( (T^{3} - 4066 T^{2} + \cdots - 2008115192)^{2} \)
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