Newspace parameters
| Level: | \( N \) | \(=\) | \( 343 = 7^{3} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 343.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(20.2376551320\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
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| Defining polynomial: |
\( x^{18} - 173 x^{16} + 11193 x^{14} - 342063 x^{12} + 5260982 x^{10} - 43296548 x^{8} + 195166152 x^{6} + \cdots - 223058368 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 7^{11} \) |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 173 x^{16} + 11193 x^{14} - 342063 x^{12} + 5260982 x^{10} - 43296548 x^{8} + 195166152 x^{6} + \cdots - 223058368 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 64\!\cdots\!49 \nu^{16} + \cdots - 35\!\cdots\!88 ) / 79\!\cdots\!52 \)
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| \(\beta_{2}\) | \(=\) |
\( ( 12\!\cdots\!47 \nu^{16} + \cdots + 43\!\cdots\!96 ) / 79\!\cdots\!52 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 14\!\cdots\!59 \nu^{16} + \cdots - 59\!\cdots\!40 ) / 79\!\cdots\!52 \)
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| \(\beta_{4}\) | \(=\) |
\( ( - 17\!\cdots\!07 \nu^{16} + \cdots - 71\!\cdots\!16 ) / 79\!\cdots\!52 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 13\!\cdots\!55 \nu^{16} + \cdots + 62\!\cdots\!32 ) / 39\!\cdots\!76 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 30\!\cdots\!35 \nu^{16} + \cdots - 12\!\cdots\!44 ) / 79\!\cdots\!52 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 34\!\cdots\!23 \nu^{16} + \cdots - 13\!\cdots\!04 ) / 79\!\cdots\!52 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 19\!\cdots\!53 \nu^{16} + \cdots + 85\!\cdots\!76 ) / 39\!\cdots\!76 \)
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| \(\beta_{9}\) | \(=\) |
\( ( - 44\!\cdots\!83 \nu^{17} + \cdots + 39\!\cdots\!52 \nu ) / 51\!\cdots\!88 \)
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| \(\beta_{10}\) | \(=\) |
\( ( - 62\!\cdots\!34 \nu^{17} + \cdots + 91\!\cdots\!28 \nu ) / 51\!\cdots\!88 \)
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| \(\beta_{11}\) | \(=\) |
\( ( - 17\!\cdots\!87 \nu^{17} + \cdots - 63\!\cdots\!44 \nu ) / 10\!\cdots\!76 \)
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| \(\beta_{12}\) | \(=\) |
\( ( - 54\!\cdots\!24 \nu^{17} + \cdots - 21\!\cdots\!20 \nu ) / 25\!\cdots\!44 \)
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| \(\beta_{13}\) | \(=\) |
\( ( 10\!\cdots\!78 \nu^{17} + \cdots + 33\!\cdots\!76 \nu ) / 39\!\cdots\!76 \)
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| \(\beta_{14}\) | \(=\) |
\( ( - 55\!\cdots\!79 \nu^{17} + \cdots - 17\!\cdots\!76 \nu ) / 14\!\cdots\!68 \)
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| \(\beta_{15}\) | \(=\) |
\( ( 31\!\cdots\!96 \nu^{17} + \cdots + 11\!\cdots\!16 \nu ) / 51\!\cdots\!88 \)
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| \(\beta_{16}\) | \(=\) |
\( ( - 35\!\cdots\!79 \nu^{17} + \cdots - 10\!\cdots\!48 \nu ) / 51\!\cdots\!88 \)
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| \(\beta_{17}\) | \(=\) |
\( ( 59\!\cdots\!67 \nu^{17} + \cdots + 26\!\cdots\!16 \nu ) / 51\!\cdots\!88 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{16} + \beta_{15} - \beta_{9} ) / 7 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{8} + 12\beta_{7} - 3\beta_{6} + 5\beta_{5} + 11\beta_{4} - 6\beta_{3} + 22\beta_{2} + 4\beta _1 + 135 ) / 7 \)
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| \(\nu^{3}\) | \(=\) |
\( ( - 6 \beta_{17} + 35 \beta_{16} + 40 \beta_{15} + 3 \beta_{14} - 9 \beta_{13} - 51 \beta_{12} + \cdots - 46 \beta_{9} ) / 7 \)
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| \(\nu^{4}\) | \(=\) |
\( ( - 5 \beta_{8} + 668 \beta_{7} + 29 \beta_{6} + 483 \beta_{5} + 477 \beta_{4} - 516 \beta_{3} + \cdots + 5801 ) / 7 \)
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| \(\nu^{5}\) | \(=\) |
\( ( - 440 \beta_{17} + 1723 \beta_{16} + 1986 \beta_{15} - 219 \beta_{14} - 687 \beta_{13} + \cdots - 2198 \beta_{9} ) / 7 \)
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| \(\nu^{6}\) | \(=\) |
\( ( - 6101 \beta_{8} + 35610 \beta_{7} + 7453 \beta_{6} + 36801 \beta_{5} + 20717 \beta_{4} - 38162 \beta_{3} + \cdots + 295949 ) / 7 \)
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| \(\nu^{7}\) | \(=\) |
\( ( - 30160 \beta_{17} + 92301 \beta_{16} + 106470 \beta_{15} - 34139 \beta_{14} - 46713 \beta_{13} + \cdots - 111506 \beta_{9} ) / 7 \)
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| \(\nu^{8}\) | \(=\) |
\( ( - 659473 \beta_{8} + 1976512 \beta_{7} + 624857 \beta_{6} + 2596195 \beta_{5} + 918937 \beta_{4} + \cdots + 16080061 ) / 7 \)
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| \(\nu^{9}\) | \(=\) |
\( ( - 2001032 \beta_{17} + 5145195 \beta_{16} + 5969918 \beta_{15} - 3038307 \beta_{14} + \cdots - 5970786 \beta_{9} ) / 7 \)
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| \(\nu^{10}\) | \(=\) |
\( ( - 53479337 \beta_{8} + 114340286 \beta_{7} + 43739149 \beta_{6} + 175913049 \beta_{5} + \cdots + 910843425 ) / 7 \)
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| \(\nu^{11}\) | \(=\) |
\( ( - 130246420 \beta_{17} + 295449517 \beta_{16} + 345864902 \beta_{15} - 229815195 \beta_{14} + \cdots - 334521778 \beta_{9} ) / 7 \)
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| \(\nu^{12}\) | \(=\) |
\( ( - 3882469109 \beta_{8} + 6824899244 \beta_{7} + 2870269713 \beta_{6} + 11642533899 \beta_{5} + \cdots + 53247644497 ) / 7 \)
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| \(\nu^{13}\) | \(=\) |
\( ( - 8386010300 \beta_{17} + 17380187827 \beta_{16} + 20542343862 \beta_{15} + \cdots - 19423894362 \beta_{9} ) / 7 \)
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| \(\nu^{14}\) | \(=\) |
\( - 38140398267 \beta_{8} + 59464840062 \beta_{7} + 26184737643 \beta_{6} + 108504874847 \beta_{5} + \cdots + 455360052739 \)
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| \(\nu^{15}\) | \(=\) |
\( ( - 536638499088 \beta_{17} + 1042389720269 \beta_{16} + 1243129277774 \beta_{15} + \cdots - 1158431848266 \beta_{9} ) / 7 \)
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| \(\nu^{16}\) | \(=\) |
\( ( - 17818070235769 \beta_{8} + 25751016041448 \beta_{7} + 11572642421561 \beta_{6} + \cdots + 194150191926133 ) / 7 \)
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| \(\nu^{17}\) | \(=\) |
\( ( - 34221673050320 \beta_{17} + 63466415997227 \beta_{16} + 76271200105822 \beta_{15} + \cdots - 70432176002322 \beta_{9} ) / 7 \)
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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 |
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−4.19636 | −9.80952 | 9.60947 | −5.62939 | 41.1643 | 0 | −6.75390 | 69.2266 | 23.6230 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.19636 | 9.80952 | 9.60947 | 5.62939 | −41.1643 | 0 | −6.75390 | 69.2266 | −23.6230 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.65258 | −3.00261 | 5.34131 | −14.2215 | 10.9673 | 0 | 9.71106 | −17.9843 | 51.9452 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.65258 | 3.00261 | 5.34131 | 14.2215 | −10.9673 | 0 | 9.71106 | −17.9843 | −51.9452 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.73379 | −9.17648 | −0.526413 | 20.6226 | 25.0865 | 0 | 23.3094 | 57.2078 | −56.3779 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.73379 | 9.17648 | −0.526413 | −20.6226 | −25.0865 | 0 | 23.3094 | 57.2078 | 56.3779 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.494558 | −4.93674 | −7.75541 | −8.46346 | −2.44151 | 0 | −7.79196 | −2.62855 | −4.18567 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.494558 | 4.93674 | −7.75541 | 8.46346 | 2.44151 | 0 | −7.79196 | −2.62855 | 4.18567 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.928639 | −4.95012 | −7.13763 | 17.8089 | −4.59687 | 0 | −14.0574 | −2.49635 | 16.5381 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.928639 | 4.95012 | −7.13763 | −17.8089 | 4.59687 | 0 | −14.0574 | −2.49635 | −16.5381 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.68563 | −5.05609 | −0.787366 | −19.0790 | −13.5788 | 0 | −23.5997 | −1.43597 | −51.2391 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.68563 | 5.05609 | −0.787366 | 19.0790 | 13.5788 | 0 | −23.5997 | −1.43597 | 51.2391 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 3.47696 | −9.36076 | 4.08923 | −14.8148 | −32.5470 | 0 | −13.5976 | 60.6238 | −51.5104 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 3.47696 | 9.36076 | 4.08923 | 14.8148 | 32.5470 | 0 | −13.5976 | 60.6238 | 51.5104 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 4.68427 | −6.32100 | 13.9424 | −10.2573 | −29.6093 | 0 | 27.8357 | 12.9550 | −48.0479 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 4.68427 | 6.32100 | 13.9424 | 10.2573 | 29.6093 | 0 | 27.8357 | 12.9550 | 48.0479 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 5.31267 | −8.57507 | 20.2244 | 11.9281 | −45.5565 | 0 | 64.9443 | 46.5319 | 63.3702 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 5.31267 | 8.57507 | 20.2244 | −11.9281 | 45.5565 | 0 | 64.9443 | 46.5319 | −63.3702 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 343.4.a.f | ✓ | 18 |
| 7.b | odd | 2 | 1 | inner | 343.4.a.f | ✓ | 18 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 343.4.a.f | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
| 343.4.a.f | ✓ | 18 | 7.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(343))\):
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\( T_{2}^{9} - 7T_{2}^{8} - 30T_{2}^{7} + 267T_{2}^{6} + 146T_{2}^{5} - 3170T_{2}^{4} + 2127T_{2}^{3} + 11090T_{2}^{2} - 14680T_{2} + 4472 \)
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\( T_{3}^{18} - 465 T_{3}^{16} + 91632 T_{3}^{14} - 9983637 T_{3}^{12} + 658810758 T_{3}^{10} + \cdots - 287115717095872 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{9} - 7 T^{8} + \cdots + 4472)^{2} \)
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| $3$ |
\( T^{18} + \cdots - 287115717095872 \)
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| $5$ |
\( T^{18} + \cdots - 74\!\cdots\!88 \)
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| $7$ |
\( T^{18} \)
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| $11$ |
\( (T^{9} + \cdots - 1346524796968)^{2} \)
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| $13$ |
\( T^{18} + \cdots - 52\!\cdots\!72 \)
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| $17$ |
\( T^{18} + \cdots - 59\!\cdots\!72 \)
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| $19$ |
\( T^{18} + \cdots - 19\!\cdots\!12 \)
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| $23$ |
\( (T^{9} + \cdots + 86\!\cdots\!41)^{2} \)
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| $29$ |
\( (T^{9} + \cdots + 81\!\cdots\!88)^{2} \)
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| $31$ |
\( T^{18} + \cdots - 75\!\cdots\!52 \)
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| $37$ |
\( (T^{9} + \cdots + 14\!\cdots\!72)^{2} \)
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| $41$ |
\( T^{18} + \cdots - 57\!\cdots\!72 \)
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| $43$ |
\( (T^{9} + \cdots + 30\!\cdots\!92)^{2} \)
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| $47$ |
\( T^{18} + \cdots - 56\!\cdots\!68 \)
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| $53$ |
\( (T^{9} + \cdots - 66\!\cdots\!44)^{2} \)
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| $59$ |
\( T^{18} + \cdots - 78\!\cdots\!32 \)
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| $61$ |
\( T^{18} + \cdots - 43\!\cdots\!08 \)
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| $67$ |
\( (T^{9} + \cdots + 16\!\cdots\!76)^{2} \)
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| $71$ |
\( (T^{9} + \cdots + 38\!\cdots\!23)^{2} \)
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| $73$ |
\( T^{18} + \cdots - 37\!\cdots\!88 \)
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| $79$ |
\( (T^{9} + \cdots + 95\!\cdots\!83)^{2} \)
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| $83$ |
\( T^{18} + \cdots - 11\!\cdots\!28 \)
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| $89$ |
\( T^{18} + \cdots - 24\!\cdots\!52 \)
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| $97$ |
\( T^{18} + \cdots - 43\!\cdots\!52 \)
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