Newspace parameters
| Level: | \( N \) | \(=\) | \( 578 = 2 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 578.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(92.7018478519\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{16} - 4 x^{15} - 2234 x^{14} - 5644 x^{13} + 1696673 x^{12} + 12813520 x^{11} - 472386300 x^{10} + \cdots + 29\!\cdots\!28 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 17^{10} \) |
| Twist minimal: | no (minimal twist has level 34) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 4 x^{15} - 2234 x^{14} - 5644 x^{13} + 1696673 x^{12} + 12813520 x^{11} - 472386300 x^{10} + \cdots + 29\!\cdots\!28 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 14\!\cdots\!85 \nu^{15} + \cdots + 14\!\cdots\!08 ) / 52\!\cdots\!08 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 40\!\cdots\!66 \nu^{15} + \cdots + 16\!\cdots\!20 ) / 38\!\cdots\!28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 24\!\cdots\!92 \nu^{15} + \cdots - 12\!\cdots\!20 ) / 18\!\cdots\!44 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 70\!\cdots\!49 \nu^{15} + \cdots + 13\!\cdots\!40 ) / 38\!\cdots\!28 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 10\!\cdots\!18 \nu^{15} + \cdots - 11\!\cdots\!52 ) / 52\!\cdots\!08 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 10\!\cdots\!54 \nu^{15} + \cdots - 27\!\cdots\!84 ) / 38\!\cdots\!28 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 96\!\cdots\!79 \nu^{15} + \cdots - 17\!\cdots\!28 ) / 19\!\cdots\!64 \)
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| \(\beta_{8}\) | \(=\) |
\( ( - 60\!\cdots\!73 \nu^{15} + \cdots - 20\!\cdots\!48 ) / 52\!\cdots\!08 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 84\!\cdots\!54 \nu^{15} + \cdots + 49\!\cdots\!12 ) / 52\!\cdots\!08 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 92\!\cdots\!27 \nu^{15} + \cdots + 27\!\cdots\!48 ) / 52\!\cdots\!08 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 94\!\cdots\!30 \nu^{15} + \cdots + 46\!\cdots\!88 ) / 52\!\cdots\!08 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 11\!\cdots\!79 \nu^{15} + \cdots + 51\!\cdots\!68 ) / 52\!\cdots\!08 \)
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| \(\beta_{13}\) | \(=\) |
\( ( 13\!\cdots\!15 \nu^{15} + \cdots + 53\!\cdots\!76 ) / 52\!\cdots\!08 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 21\!\cdots\!18 \nu^{15} + \cdots + 91\!\cdots\!12 ) / 52\!\cdots\!08 \)
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| \(\beta_{15}\) | \(=\) |
\( ( 13\!\cdots\!24 \nu^{15} + \cdots - 38\!\cdots\!24 ) / 26\!\cdots\!04 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{12} - \beta_{8} + \beta_{7} - 2\beta_{6} + 2\beta _1 + 9 ) / 34 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3 \beta_{13} - 12 \beta_{12} - 16 \beta_{10} - 2 \beta_{8} + 42 \beta_{7} + 4 \beta_{6} - 238 \beta_{3} + \cdots + 9530 ) / 34 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 15 \beta_{15} - 51 \beta_{14} + 557 \beta_{13} - 1148 \beta_{12} + 42 \beta_{11} - 408 \beta_{10} + \cdots + 93314 ) / 34 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 140 \beta_{15} - 2060 \beta_{14} + 7293 \beta_{13} - 21472 \beta_{12} + 2520 \beta_{11} + \cdots + 7284942 ) / 34 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 21645 \beta_{15} - 110215 \beta_{14} + 602883 \beta_{13} - 1161832 \beta_{12} + 72510 \beta_{11} + \cdots + 140450832 ) / 34 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 440346 \beta_{15} - 4076846 \beta_{14} + 11360103 \beta_{13} - 28395806 \beta_{12} + \cdots + 7109315178 ) / 34 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 29526525 \beta_{15} - 173636771 \beta_{14} + 615908251 \beta_{13} - 1221534684 \beta_{12} + \cdots + 178983871778 ) / 34 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 794753336 \beta_{15} - 6338678376 \beta_{14} + 14883950425 \beta_{13} - 34484657576 \beta_{12} + \cdots + 7480147109286 ) / 34 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 39379649769 \beta_{15} - 248883822543 \beta_{14} + 648533347323 \beta_{13} - 1325126594278 \beta_{12} + \cdots + 215085326534586 ) / 34 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 1211774622406 \beta_{15} - 9030702157394 \beta_{14} + 18141894707937 \beta_{13} + \cdots + 81\!\cdots\!70 ) / 34 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 51731590026083 \beta_{15} - 340301287128121 \beta_{14} + 703655294932553 \beta_{13} + \cdots + 25\!\cdots\!58 ) / 34 \)
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| \(\nu^{12}\) | \(=\) |
\( ( - 17\!\cdots\!12 \beta_{15} + \cdots + 90\!\cdots\!34 ) / 34 \)
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| \(\nu^{13}\) | \(=\) |
\( ( - 67\!\cdots\!71 \beta_{15} + \cdots + 29\!\cdots\!30 ) / 34 \)
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| \(\nu^{14}\) | \(=\) |
\( ( - 22\!\cdots\!14 \beta_{15} + \cdots + 10\!\cdots\!46 ) / 34 \)
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| \(\nu^{15}\) | \(=\) |
\( ( - 86\!\cdots\!85 \beta_{15} + \cdots + 33\!\cdots\!02 ) / 34 \)
|
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 |
|
−4.00000 | −26.4096 | 16.0000 | −2.40235 | 105.638 | −84.9138 | −64.0000 | 454.467 | 9.60942 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | −25.6242 | 16.0000 | −95.2917 | 102.497 | 115.280 | −64.0000 | 413.601 | 381.167 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | −21.2324 | 16.0000 | −54.9040 | 84.9296 | −170.486 | −64.0000 | 207.815 | 219.616 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | −18.1990 | 16.0000 | 107.153 | 72.7958 | 12.5445 | −64.0000 | 88.2020 | −428.614 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −4.00000 | −14.2021 | 16.0000 | 71.5351 | 56.8084 | −19.2261 | −64.0000 | −41.3005 | −286.140 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −4.00000 | −9.35513 | 16.0000 | 38.1445 | 37.4205 | 229.803 | −64.0000 | −155.482 | −152.578 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −4.00000 | −6.25932 | 16.0000 | −36.2693 | 25.0373 | 146.885 | −64.0000 | −203.821 | 145.077 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −4.00000 | −1.87552 | 16.0000 | 37.9165 | 7.50206 | 155.651 | −64.0000 | −239.482 | −151.666 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −4.00000 | 1.87552 | 16.0000 | −37.9165 | −7.50206 | −155.651 | −64.0000 | −239.482 | 151.666 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −4.00000 | 6.25932 | 16.0000 | 36.2693 | −25.0373 | −146.885 | −64.0000 | −203.821 | −145.077 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −4.00000 | 9.35513 | 16.0000 | −38.1445 | −37.4205 | −229.803 | −64.0000 | −155.482 | 152.578 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | −4.00000 | 14.2021 | 16.0000 | −71.5351 | −56.8084 | 19.2261 | −64.0000 | −41.3005 | 286.140 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | −4.00000 | 18.1990 | 16.0000 | −107.153 | −72.7958 | −12.5445 | −64.0000 | 88.2020 | 428.614 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | −4.00000 | 21.2324 | 16.0000 | 54.9040 | −84.9296 | 170.486 | −64.0000 | 207.815 | −219.616 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | −4.00000 | 25.6242 | 16.0000 | 95.2917 | −102.497 | −115.280 | −64.0000 | 413.601 | −381.167 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | −4.00000 | 26.4096 | 16.0000 | 2.40235 | −105.638 | 84.9138 | −64.0000 | 454.467 | −9.60942 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(17\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 578.6.a.r | 16 | |
| 17.b | even | 2 | 1 | inner | 578.6.a.r | 16 | |
| 17.e | odd | 16 | 2 | 34.6.d.b | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 34.6.d.b | ✓ | 16 | 17.e | odd | 16 | 2 | |
| 578.6.a.r | 16 | 1.a | even | 1 | 1 | trivial | |
| 578.6.a.r | 16 | 17.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} - 2468 T_{3}^{14} + 2405310 T_{3}^{12} - 1178507904 T_{3}^{10} + 306266589212 T_{3}^{8} + \cdots + 16\!\cdots\!64 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(578))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{16} \)
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| $3$ |
\( T^{16} + \cdots + 16\!\cdots\!64 \)
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| $5$ |
\( T^{16} + \cdots + 25\!\cdots\!76 \)
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| $7$ |
\( T^{16} + \cdots + 44\!\cdots\!84 \)
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| $11$ |
\( T^{16} + \cdots + 17\!\cdots\!44 \)
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| $13$ |
\( (T^{8} + \cdots + 34\!\cdots\!24)^{2} \)
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| $17$ |
\( T^{16} \)
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| $19$ |
\( (T^{8} + \cdots - 13\!\cdots\!08)^{2} \)
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| $23$ |
\( T^{16} + \cdots + 29\!\cdots\!16 \)
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| $29$ |
\( T^{16} + \cdots + 37\!\cdots\!24 \)
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| $31$ |
\( T^{16} + \cdots + 31\!\cdots\!76 \)
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| $37$ |
\( T^{16} + \cdots + 14\!\cdots\!16 \)
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| $41$ |
\( T^{16} + \cdots + 19\!\cdots\!64 \)
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| $43$ |
\( (T^{8} + \cdots - 11\!\cdots\!64)^{2} \)
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| $47$ |
\( (T^{8} + \cdots + 59\!\cdots\!04)^{2} \)
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| $53$ |
\( (T^{8} + \cdots + 87\!\cdots\!64)^{2} \)
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| $59$ |
\( (T^{8} + \cdots + 80\!\cdots\!64)^{2} \)
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| $61$ |
\( T^{16} + \cdots + 11\!\cdots\!44 \)
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| $67$ |
\( (T^{8} + \cdots - 10\!\cdots\!36)^{2} \)
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| $71$ |
\( T^{16} + \cdots + 35\!\cdots\!16 \)
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| $73$ |
\( T^{16} + \cdots + 11\!\cdots\!16 \)
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| $79$ |
\( T^{16} + \cdots + 12\!\cdots\!16 \)
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| $83$ |
\( (T^{8} + \cdots - 35\!\cdots\!12)^{2} \)
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| $89$ |
\( (T^{8} + \cdots - 45\!\cdots\!64)^{2} \)
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| $97$ |
\( T^{16} + \cdots + 11\!\cdots\!96 \)
|
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