Newspace parameters
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(62.8704573667\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
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| Defining polynomial: |
\( x^{5} - 167x^{3} - 387x^{2} + 1720x + 2340 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 56) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 167x^{3} - 387x^{2} + 1720x + 2340 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -133\nu^{4} + 490\nu^{3} + 21255\nu^{2} - 17495\nu - 229698 ) / 8068 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 91\nu^{4} + 514\nu^{3} - 21337\nu^{2} - 37287\nu + 416824 ) / 4034 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 74\nu^{4} + 152\nu^{3} - 12918\nu^{2} - 54082\nu + 104341 ) / 2017 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -1485\nu^{4} + 2074\nu^{3} + 227615\nu^{2} + 176881\nu - 1166786 ) / 8068 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} - 4\beta_{3} + 2\beta_{2} + 5\beta _1 - 2 ) / 56 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -17\beta_{4} - 19\beta_{3} - 15\beta_{2} + 127\beta _1 + 3690 ) / 56 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -97\beta_{4} - 192\beta_{3} + 131\beta_{2} + 835\beta _1 + 6141 ) / 28 \)
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| \(\nu^{4}\) | \(=\) |
\( ( -3300\beta_{4} - 3925\beta_{3} - 1695\beta_{2} + 22394\beta _1 + 538504 ) / 56 \)
|
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 | −18.5692 | 0 | 57.9194 | 0 | 0 | 0 | 101.814 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −14.5316 | 0 | −44.8226 | 0 | 0 | 0 | −31.8333 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 6.22560 | 0 | 9.40551 | 0 | 0 | 0 | −204.242 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 14.6817 | 0 | −72.1603 | 0 | 0 | 0 | −27.4480 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 25.1934 | 0 | 80.6580 | 0 | 0 | 0 | 391.710 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( +1 \) |
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 | 392.6.a.l | 5 | |
| 4.b | odd | 2 | 1 | 784.6.a.bj | 5 | ||
| 7.b | odd | 2 | 1 | 392.6.a.i | 5 | ||
| 7.c | even | 3 | 2 | 56.6.i.a | ✓ | 10 | |
| 7.d | odd | 6 | 2 | 392.6.i.p | 10 | ||
| 21.h | odd | 6 | 2 | 504.6.s.d | 10 | ||
| 28.d | even | 2 | 1 | 784.6.a.bm | 5 | ||
| 28.g | odd | 6 | 2 | 112.6.i.g | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.6.i.a | ✓ | 10 | 7.c | even | 3 | 2 | |
| 112.6.i.g | 10 | 28.g | odd | 6 | 2 | ||
| 392.6.a.i | 5 | 7.b | odd | 2 | 1 | ||
| 392.6.a.l | 5 | 1.a | even | 1 | 1 | trivial | |
| 392.6.i.p | 10 | 7.d | odd | 6 | 2 | ||
| 504.6.s.d | 10 | 21.h | odd | 6 | 2 | ||
| 784.6.a.bj | 5 | 4.b | odd | 2 | 1 | ||
| 784.6.a.bm | 5 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 13T_{3}^{4} - 638T_{3}^{3} + 5718T_{3}^{2} + 90573T_{3} - 621369 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(392))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 13 T^{4} + \cdots - 621369 \)
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| $5$ |
\( T^{5} - 31 T^{4} + \cdots - 142117891 \)
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| $7$ |
\( T^{5} \)
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| $11$ |
\( T^{5} + \cdots + 16452319223691 \)
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| $13$ |
\( T^{5} + \cdots + 173219419488 \)
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| $17$ |
\( T^{5} + \cdots + 49362505838685 \)
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| $19$ |
\( T^{5} + \cdots + 151493645339129 \)
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| $23$ |
\( T^{5} + \cdots + 18\!\cdots\!17 \)
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| $29$ |
\( T^{5} + \cdots - 36\!\cdots\!96 \)
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| $31$ |
\( T^{5} + \cdots + 12\!\cdots\!15 \)
|
| $37$ |
\( T^{5} + \cdots + 83\!\cdots\!09 \)
|
| $41$ |
\( T^{5} + \cdots - 92\!\cdots\!36 \)
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| $43$ |
\( T^{5} + \cdots + 27\!\cdots\!28 \)
|
| $47$ |
\( T^{5} + \cdots - 77\!\cdots\!15 \)
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| $53$ |
\( T^{5} + \cdots - 12\!\cdots\!71 \)
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| $59$ |
\( T^{5} + \cdots + 17\!\cdots\!75 \)
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| $61$ |
\( T^{5} + \cdots + 25\!\cdots\!45 \)
|
| $67$ |
\( T^{5} + \cdots - 16\!\cdots\!93 \)
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| $71$ |
\( T^{5} + \cdots - 95\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots + 73\!\cdots\!09 \)
|
| $79$ |
\( T^{5} + \cdots + 89\!\cdots\!37 \)
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| $83$ |
\( T^{5} + \cdots + 11\!\cdots\!16 \)
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| $89$ |
\( T^{5} + \cdots + 73\!\cdots\!33 \)
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| $97$ |
\( T^{5} + \cdots + 11\!\cdots\!52 \)
|
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