Newspace parameters
| Level: | \( N \) | \(=\) | \( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1656.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(97.7071629695\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 394x^{4} + 1617x^{3} + 37348x^{2} - 237382x + 130496 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 552) |
| 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
\( x^{6} - 2x^{5} - 394x^{4} + 1617x^{3} + 37348x^{2} - 237382x + 130496 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 31\nu^{5} - 51\nu^{4} - 14296\nu^{3} + 28543\nu^{2} + 1576569\nu - 4566271 ) / 191943 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -356\nu^{5} + 15033\nu^{4} + 23828\nu^{3} - 3060392\nu^{2} + 15848157\nu - 28796764 ) / 2111373 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 16\nu^{5} + 203\nu^{4} - 5544\nu^{3} - 73328\nu^{2} + 373643\nu + 4289442 ) / 63981 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1754\nu^{5} + 12479\nu^{4} - 600652\nu^{3} - 2408254\nu^{2} + 49012445\nu - 23648146 ) / 703791 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 8\beta_{4} + \beta_{3} - 2\beta_{2} - 7\beta _1 + 529 ) / 4 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 11\beta_{5} - 25\beta_{4} - 22\beta_{3} - 154\beta_{2} + 405\beta _1 - 1513 ) / 4 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 148\beta_{5} - 878\beta_{4} + 280\beta_{3} - 632\beta_{2} - 1045\beta _1 + 51575 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4639\beta_{5} - 7052\beta_{4} - 10145\beta_{3} - 46490\beta_{2} + 88063\beta _1 - 527629 ) / 4 \)
|
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 | 0 | 0 | −16.6385 | 0 | 4.34238 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −12.9270 | 0 | −4.19260 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 0.263270 | 0 | 33.9571 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 6.18183 | 0 | 22.9284 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 9.37523 | 0 | −36.0711 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0 | 0 | 19.7452 | 0 | −18.9642 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 1656.4.a.q | 6 | |
| 3.b | odd | 2 | 1 | 552.4.a.k | ✓ | 6 | |
| 12.b | even | 2 | 1 | 1104.4.a.z | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 552.4.a.k | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 1104.4.a.z | 6 | 12.b | even | 2 | 1 | ||
| 1656.4.a.q | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 6T_{5}^{5} - 462T_{5}^{4} + 2180T_{5}^{3} + 44160T_{5}^{2} - 257904T_{5} + 64800 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1656))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} - 6 T^{5} + \cdots + 64800 \)
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| $7$ |
\( T^{6} - 2 T^{5} + \cdots - 9696384 \)
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| $11$ |
\( T^{6} + \cdots - 1787636736 \)
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| $13$ |
\( T^{6} + \cdots - 10630802944 \)
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| $17$ |
\( T^{6} + \cdots + 58696168704 \)
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| $19$ |
\( T^{6} + \cdots - 40399896960 \)
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| $23$ |
\( (T + 23)^{6} \)
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| $29$ |
\( T^{6} + \cdots - 824669780160 \)
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| $31$ |
\( T^{6} + \cdots + 24091485388800 \)
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| $37$ |
\( T^{6} + \cdots - 405386572032 \)
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| $41$ |
\( T^{6} + \cdots + 5609148528960 \)
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| $43$ |
\( T^{6} + \cdots - 13085434109952 \)
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| $47$ |
\( T^{6} + \cdots - 106348351586304 \)
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| $53$ |
\( T^{6} + \cdots + 18816672491040 \)
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| $59$ |
\( T^{6} + \cdots + 58\!\cdots\!08 \)
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| $61$ |
\( T^{6} + \cdots + 54\!\cdots\!24 \)
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| $67$ |
\( T^{6} + \cdots - 313120902765312 \)
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| $71$ |
\( T^{6} + \cdots + 34\!\cdots\!84 \)
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| $73$ |
\( T^{6} + \cdots + 535518639424320 \)
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| $79$ |
\( T^{6} + \cdots - 53\!\cdots\!28 \)
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| $83$ |
\( T^{6} + \cdots - 56\!\cdots\!16 \)
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| $89$ |
\( T^{6} + \cdots + 47\!\cdots\!00 \)
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| $97$ |
\( T^{6} + \cdots - 24\!\cdots\!72 \)
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