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: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 2x^{6} - 221x^{5} - 596x^{4} + 5896x^{3} + 11344x^{2} - 34530x - 61956 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 221x^{5} - 596x^{4} + 5896x^{3} + 11344x^{2} - 34530x - 61956 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 143\nu^{6} - 1223\nu^{5} - 27474\nu^{4} + 91730\nu^{3} + 1101946\nu^{2} - 767670\nu - 6746724 ) / 211248 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -143\nu^{6} + 1223\nu^{5} + 27474\nu^{4} - 91730\nu^{3} - 1101946\nu^{2} + 1612662\nu + 6535476 ) / 211248 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 355\nu^{6} - 1723\nu^{5} - 74442\nu^{4} + 3178\nu^{3} + 2230706\nu^{2} - 1572222\nu - 8392500 ) / 211248 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -1433\nu^{6} + 9137\nu^{5} + 319470\nu^{4} - 620654\nu^{3} - 14432230\nu^{2} + 15021978\nu + 97071228 ) / 844992 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 799\nu^{6} - 5575\nu^{5} - 161442\nu^{4} + 359938\nu^{3} + 5669258\nu^{2} - 13092438\nu - 31595652 ) / 422496 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 581\nu^{6} - 829\nu^{5} - 124118\nu^{4} - 423914\nu^{3} + 2195502\nu^{2} + 4061358\nu - 4072140 ) / 281664 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 4 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{6} + 5\beta_{5} - 3\beta_{4} - 16\beta_{3} + 7\beta_{2} + 10\beta _1 + 258 ) / 4 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 33\beta_{6} + 69\beta_{5} - 15\beta_{4} - 120\beta_{3} + 229\beta_{2} + 196\beta _1 + 1768 ) / 4 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 523\beta_{6} + 667\beta_{5} - 237\beta_{4} - 1697\beta_{3} + 1325\beta_{2} + 1487\beta _1 + 23748 ) / 2 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 6267\beta_{6} + 10029\beta_{5} - 2985\beta_{4} - 21075\beta_{3} + 27092\beta_{2} + 24818\beta _1 + 300713 ) / 2 \)
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| \(\nu^{6}\) | \(=\) |
\( ( 124231 \beta_{6} + 172525 \beta_{5} - 54693 \beta_{4} - 406145 \beta_{3} + 388535 \beta_{2} + \cdots + 5670360 ) / 2 \)
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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 | −20.0067 | 0 | −6.48266 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −6.86553 | 0 | −32.9189 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −4.53773 | 0 | 26.4165 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | −1.09729 | 0 | 12.1118 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 1.13930 | 0 | 12.7208 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0 | 0 | 10.2927 | 0 | −3.45245 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0 | 0 | 17.0752 | 0 | −10.3951 | 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.r | ✓ | 7 |
| 3.b | odd | 2 | 1 | 1656.4.a.s | yes | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1656.4.a.r | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1656.4.a.s | yes | 7 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{7} + 4T_{5}^{6} - 426T_{5}^{5} - 940T_{5}^{4} + 29084T_{5}^{3} + 109536T_{5}^{2} - 40248T_{5} - 136944 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1656))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
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| $3$ |
\( T^{7} \)
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| $5$ |
\( T^{7} + 4 T^{6} + \cdots - 136944 \)
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| $7$ |
\( T^{7} + 2 T^{6} + \cdots - 31171104 \)
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| $11$ |
\( T^{7} - 46 T^{6} + \cdots - 454242816 \)
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| $13$ |
\( T^{7} + 18 T^{6} + \cdots + 208104448 \)
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| $17$ |
\( T^{7} + \cdots + 32425992768 \)
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| $19$ |
\( T^{7} + \cdots + 1289780651088 \)
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| $23$ |
\( (T + 23)^{7} \)
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| $29$ |
\( T^{7} + \cdots - 196690466061312 \)
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| $31$ |
\( T^{7} + \cdots + 10899964354560 \)
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| $37$ |
\( T^{7} + \cdots + 195216275543232 \)
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| $41$ |
\( T^{7} + \cdots - 36\!\cdots\!12 \)
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| $43$ |
\( T^{7} + \cdots - 25\!\cdots\!00 \)
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| $47$ |
\( T^{7} + \cdots + 32\!\cdots\!44 \)
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| $53$ |
\( T^{7} + \cdots - 31\!\cdots\!36 \)
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| $59$ |
\( T^{7} + \cdots + 15\!\cdots\!96 \)
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| $61$ |
\( T^{7} + \cdots - 37\!\cdots\!36 \)
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| $67$ |
\( T^{7} + \cdots + 48\!\cdots\!72 \)
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| $71$ |
\( T^{7} + \cdots - 35\!\cdots\!84 \)
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| $73$ |
\( T^{7} + \cdots + 49\!\cdots\!88 \)
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| $79$ |
\( T^{7} + \cdots + 40\!\cdots\!76 \)
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| $83$ |
\( T^{7} + \cdots + 46\!\cdots\!92 \)
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| $89$ |
\( T^{7} + \cdots + 65\!\cdots\!80 \)
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| $97$ |
\( T^{7} + \cdots - 35\!\cdots\!16 \)
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