Newspace parameters
| Level: | \( N \) | \(=\) | \( 172 = 2^{2} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 172.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(10.1483285210\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} - x^{5} - 117x^{4} + 138x^{3} + 3969x^{2} - 3321x - 38637 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{6} - x^{5} - 117x^{4} + 138x^{3} + 3969x^{2} - 3321x - 38637 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 2\nu^{4} + 75\nu^{3} + 123\nu^{2} - 1008\nu - 2619 ) / 108 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} - 8\nu^{4} + 99\nu^{3} + 699\nu^{2} - 2376\nu - 13365 ) / 162 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{5} + 22\nu^{4} - 423\nu^{3} - 1443\nu^{2} + 8100\nu + 21951 ) / 324 \)
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| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 8\nu^{4} - 72\nu^{3} - 564\nu^{2} + 1161\nu + 8505 ) / 81 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 39 \)
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| \(\nu^{3}\) | \(=\) |
\( 3\beta_{5} - 5\beta_{4} + \beta_{3} - 5\beta_{2} + 50\beta _1 - 15 \)
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| \(\nu^{4}\) | \(=\) |
\( 12\beta_{5} + 76\beta_{4} + 73\beta_{3} + 94\beta_{2} - 124\beta _1 + 1893 \)
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| \(\nu^{5}\) | \(=\) |
\( 201\beta_{5} - 404\beta_{4} + 52\beta_{3} - 548\beta_{2} + 2867\beta _1 - 2733 \)
|
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 |
|
0 | −7.44599 | 0 | −13.6757 | 0 | −0.436015 | 0 | 28.4427 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −4.74807 | 0 | −14.0296 | 0 | −15.8799 | 0 | −4.45584 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.45055 | 0 | 13.6689 | 0 | 9.15532 | 0 | −20.9948 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 5.85928 | 0 | 1.61930 | 0 | 33.8833 | 0 | 7.33121 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 7.68240 | 0 | −7.26278 | 0 | −4.11743 | 0 | 32.0193 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 8.10293 | 0 | 18.6799 | 0 | −2.60524 | 0 | 38.6574 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(43\) | \( -1 \) |
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 | 172.4.a.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1548.4.a.g | 6 | ||
| 4.b | odd | 2 | 1 | 688.4.a.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 172.4.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 688.4.a.h | 6 | 4.b | odd | 2 | 1 | ||
| 1548.4.a.g | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 7T_{3}^{5} - 97T_{3}^{4} + 576T_{3}^{3} + 2878T_{3}^{2} - 10388T_{3} - 31600 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(172))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} - 7 T^{5} + \cdots - 31600 \)
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| $5$ |
\( T^{6} + T^{5} + \cdots - 576144 \)
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| $7$ |
\( T^{6} - 20 T^{5} + \cdots + 23040 \)
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| $11$ |
\( T^{6} - 65 T^{5} + \cdots + 558581184 \)
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| $13$ |
\( T^{6} + \cdots + 1439453976 \)
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| $17$ |
\( T^{6} + \cdots + 20972543814 \)
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| $19$ |
\( T^{6} + \cdots - 744861085472 \)
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| $23$ |
\( T^{6} + \cdots + 744165651096 \)
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| $29$ |
\( T^{6} + \cdots + 20266243656 \)
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| $31$ |
\( T^{6} + \cdots - 2219564525136 \)
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| $37$ |
\( T^{6} + \cdots - 7937921826048 \)
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| $41$ |
\( T^{6} + \cdots + 173026112910102 \)
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| $43$ |
\( (T - 43)^{6} \)
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| $47$ |
\( T^{6} + \cdots + 2564037960192 \)
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| $53$ |
\( T^{6} + \cdots + 844228498783752 \)
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| $59$ |
\( T^{6} + \cdots - 33\!\cdots\!48 \)
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| $61$ |
\( T^{6} + \cdots - 63\!\cdots\!80 \)
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| $67$ |
\( T^{6} + \cdots + 295870788072688 \)
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| $71$ |
\( T^{6} + \cdots - 37\!\cdots\!64 \)
|
| $73$ |
\( T^{6} + \cdots + 13\!\cdots\!68 \)
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| $79$ |
\( T^{6} + \cdots + 70\!\cdots\!20 \)
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| $83$ |
\( T^{6} + \cdots - 22\!\cdots\!24 \)
|
| $89$ |
\( T^{6} + \cdots - 14\!\cdots\!16 \)
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| $97$ |
\( T^{6} + \cdots - 16\!\cdots\!94 \)
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