Newspace parameters
Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 342.m (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.31882504112\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 80 x^{14} + 4582 x^{12} - 116048 x^{10} + 2090635 x^{8} - 20507888 x^{6} + 145417654 x^{4} - 570337064 x^{2} + 1506138481 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \( x^{16} - 80 x^{14} + 4582 x^{12} - 116048 x^{10} + 2090635 x^{8} - 20507888 x^{6} + 145417654 x^{4} - 570337064 x^{2} + 1506138481 \) :
\(\beta_{1}\) | \(=\) | \( ( - 163455856112 \nu^{14} + 12380288973470 \nu^{12} - 699476967224640 \nu^{10} + \cdots + 29\!\cdots\!60 ) / 46\!\cdots\!13 \) |
\(\beta_{2}\) | \(=\) | \( ( - 625586030420 \nu^{14} + 49550082836104 \nu^{12} + \cdots + 65\!\cdots\!92 ) / 46\!\cdots\!13 \) |
\(\beta_{3}\) | \(=\) | \( ( - 1947351576330 \nu^{14} + 148577931152752 \nu^{12} + \cdots + 62\!\cdots\!96 ) / 46\!\cdots\!13 \) |
\(\beta_{4}\) | \(=\) | \( ( 16507449391907 \nu^{14} + \cdots - 18\!\cdots\!07 ) / 18\!\cdots\!52 \) |
\(\beta_{5}\) | \(=\) | \( ( - 14098651228589 \nu^{14} + 891087694938548 \nu^{12} + \cdots - 40\!\cdots\!16 ) / 93\!\cdots\!26 \) |
\(\beta_{6}\) | \(=\) | \( ( 6279628042228 \nu^{14} - 537478001105760 \nu^{12} + \cdots - 23\!\cdots\!50 ) / 31\!\cdots\!42 \) |
\(\beta_{7}\) | \(=\) | \( ( 1505410050 \nu^{14} - 108378542575 \nu^{12} + 5956393343625 \nu^{10} - 121511691139200 \nu^{8} + \cdots + 99\!\cdots\!77 ) / 47\!\cdots\!28 \) |
\(\beta_{8}\) | \(=\) | \( ( 285440682588860 \nu^{15} + \cdots - 39\!\cdots\!67 \nu ) / 54\!\cdots\!66 \) |
\(\beta_{9}\) | \(=\) | \( ( - 3737417132 \nu^{15} + 295169480981 \nu^{13} - 16676852520672 \nu^{11} + 405931558747333 \nu^{9} + \cdots + 71\!\cdots\!38 \nu ) / 11\!\cdots\!94 \) |
\(\beta_{10}\) | \(=\) | \( ( - 203457785384199 \nu^{15} + \cdots - 63\!\cdots\!48 \nu ) / 52\!\cdots\!92 \) |
\(\beta_{11}\) | \(=\) | \( ( - 19\!\cdots\!33 \nu^{15} + \cdots - 94\!\cdots\!40 \nu ) / 36\!\cdots\!44 \) |
\(\beta_{12}\) | \(=\) | \( ( - 108973994552152 \nu^{15} + \cdots + 39\!\cdots\!11 \nu ) / 19\!\cdots\!76 \) |
\(\beta_{13}\) | \(=\) | \( ( - 96\!\cdots\!55 \nu^{15} + \cdots + 41\!\cdots\!28 \nu ) / 10\!\cdots\!32 \) |
\(\beta_{14}\) | \(=\) | \( ( - 33\!\cdots\!83 \nu^{15} + \cdots + 60\!\cdots\!22 \nu ) / 36\!\cdots\!44 \) |
\(\beta_{15}\) | \(=\) | \( ( 39\!\cdots\!84 \nu^{15} + \cdots - 13\!\cdots\!83 \nu ) / 36\!\cdots\!44 \) |
\(\nu\) | \(=\) | \( ( \beta_{15} + \beta_{12} - 2\beta_{11} + 2\beta_{10} + 3\beta_{9} + 3\beta_{8} ) / 6 \) |
\(\nu^{2}\) | \(=\) | \( 2\beta_{3} - \beta_{2} - 20\beta_1 \) |
\(\nu^{3}\) | \(=\) | \( ( 82 \beta_{15} + 18 \beta_{14} + 18 \beta_{13} - 14 \beta_{12} - 41 \beta_{11} - 7 \beta_{10} + 285 \beta_{9} + 18 \beta_{8} ) / 6 \) |
\(\nu^{4}\) | \(=\) | \( -6\beta_{6} - 6\beta_{5} + 12\beta_{4} + 58\beta_{3} + 58\beta_{2} - 697\beta _1 - 697 \) |
\(\nu^{5}\) | \(=\) | \( ( 1879\beta_{15} + 792\beta_{13} - 905\beta_{12} + 1879\beta_{11} + 113\beta_{10} - 15321\beta_{8} ) / 6 \) |
\(\nu^{6}\) | \(=\) | \( -864\beta_{7} + 450\beta_{6} - 900\beta_{5} - 2925\beta_{3} + 5850\beta_{2} - 30374 \) |
\(\nu^{7}\) | \(=\) | \( ( - 89099 \beta_{15} - 34182 \beta_{14} + 51085 \beta_{12} + 178198 \beta_{11} + 67988 \beta_{10} - 821499 \beta_{9} - 821499 \beta_{8} ) / 6 \) |
\(\nu^{8}\) | \(=\) | \( - 47304 \beta_{7} + 50184 \beta_{6} - 25092 \beta_{5} - 47304 \beta_{4} - 286504 \beta_{3} + 143252 \beta_{2} + 1417885 \beta_1 \) |
\(\nu^{9}\) | \(=\) | \( ( - 8551250 \beta_{15} - 1559232 \beta_{14} - 1559232 \beta_{13} + 5166382 \beta_{12} + 4275625 \beta_{11} + 2583191 \beta_{10} - 40498419 \beta_{9} - 1559232 \beta_{8} ) / 6 \) |
\(\nu^{10}\) | \(=\) | \( 1277436 \beta_{6} + 1277436 \beta_{5} - 2389920 \beta_{4} - 6956069 \beta_{3} - 6956069 \beta_{2} + 67677800 \beta _1 + 67677800 \) |
\(\nu^{11}\) | \(=\) | \( ( - 206090633 \beta_{15} - 73536354 \beta_{13} + 126810967 \beta_{12} - 206090633 \beta_{11} - 53274613 \beta_{10} + 1898588067 \beta_{8} ) / 6 \) |
\(\nu^{12}\) | \(=\) | \( 117426564 \beta_{7} - 62957970 \beta_{6} + 125915940 \beta_{5} + 336788262 \beta_{3} - 673576524 \beta_{2} + 3255835501 \) |
\(\nu^{13}\) | \(=\) | \( ( 9950641183 \beta_{15} + 3519826488 \beta_{14} - 6164259137 \beta_{12} - 19901282366 \beta_{11} - 8808691786 \beta_{10} + 95591247825 \beta_{9} + 95591247825 \beta_{8} ) / 6 \) |
\(\nu^{14}\) | \(=\) | \( 5710899168 \beta_{7} - 6131093868 \beta_{6} + 3065546934 \beta_{5} + 5710899168 \beta_{4} + 32577285170 \beta_{3} - 16288642585 \beta_{2} - 157087053074 \beta_1 \) |
\(\nu^{15}\) | \(=\) | \( ( 961510531510 \beta_{15} + 169484931702 \beta_{14} + 169484931702 \beta_{13} - 597156820250 \beta_{12} - 480755265755 \beta_{11} - 298578410125 \beta_{10} + \cdots + 169484931702 \beta_{8} ) / 6 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(325\) |
\(\chi(n)\) | \(1\) | \(1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
145.1 |
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−1.22474 | + | 0.707107i | 0 | 1.00000 | − | 1.73205i | −4.33044 | − | 7.50054i | 0 | 2.96184 | 2.82843i | 0 | 10.6074 | + | 6.12417i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.2 | −1.22474 | + | 0.707107i | 0 | 1.00000 | − | 1.73205i | −0.0575331 | − | 0.0996502i | 0 | −10.7734 | 2.82843i | 0 | 0.140927 | + | 0.0813641i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.3 | −1.22474 | + | 0.707107i | 0 | 1.00000 | − | 1.73205i | 0.375080 | + | 0.649658i | 0 | 0.433375 | 2.82843i | 0 | −0.918755 | − | 0.530444i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.4 | −1.22474 | + | 0.707107i | 0 | 1.00000 | − | 1.73205i | 4.01289 | + | 6.95053i | 0 | 13.3781 | 2.82843i | 0 | −9.82954 | − | 5.67509i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.5 | 1.22474 | − | 0.707107i | 0 | 1.00000 | − | 1.73205i | −4.01289 | − | 6.95053i | 0 | 13.3781 | − | 2.82843i | 0 | −9.82954 | − | 5.67509i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.6 | 1.22474 | − | 0.707107i | 0 | 1.00000 | − | 1.73205i | −0.375080 | − | 0.649658i | 0 | 0.433375 | − | 2.82843i | 0 | −0.918755 | − | 0.530444i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.7 | 1.22474 | − | 0.707107i | 0 | 1.00000 | − | 1.73205i | 0.0575331 | + | 0.0996502i | 0 | −10.7734 | − | 2.82843i | 0 | 0.140927 | + | 0.0813641i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.8 | 1.22474 | − | 0.707107i | 0 | 1.00000 | − | 1.73205i | 4.33044 | + | 7.50054i | 0 | 2.96184 | − | 2.82843i | 0 | 10.6074 | + | 6.12417i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
217.1 | −1.22474 | − | 0.707107i | 0 | 1.00000 | + | 1.73205i | −4.33044 | + | 7.50054i | 0 | 2.96184 | − | 2.82843i | 0 | 10.6074 | − | 6.12417i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
217.2 | −1.22474 | − | 0.707107i | 0 | 1.00000 | + | 1.73205i | −0.0575331 | + | 0.0996502i | 0 | −10.7734 | − | 2.82843i | 0 | 0.140927 | − | 0.0813641i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
217.3 | −1.22474 | − | 0.707107i | 0 | 1.00000 | + | 1.73205i | 0.375080 | − | 0.649658i | 0 | 0.433375 | − | 2.82843i | 0 | −0.918755 | + | 0.530444i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
217.4 | −1.22474 | − | 0.707107i | 0 | 1.00000 | + | 1.73205i | 4.01289 | − | 6.95053i | 0 | 13.3781 | − | 2.82843i | 0 | −9.82954 | + | 5.67509i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
217.5 | 1.22474 | + | 0.707107i | 0 | 1.00000 | + | 1.73205i | −4.01289 | + | 6.95053i | 0 | 13.3781 | 2.82843i | 0 | −9.82954 | + | 5.67509i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
217.6 | 1.22474 | + | 0.707107i | 0 | 1.00000 | + | 1.73205i | −0.375080 | + | 0.649658i | 0 | 0.433375 | 2.82843i | 0 | −0.918755 | + | 0.530444i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
217.7 | 1.22474 | + | 0.707107i | 0 | 1.00000 | + | 1.73205i | 0.0575331 | − | 0.0996502i | 0 | −10.7734 | 2.82843i | 0 | 0.140927 | − | 0.0813641i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
217.8 | 1.22474 | + | 0.707107i | 0 | 1.00000 | + | 1.73205i | 4.33044 | − | 7.50054i | 0 | 2.96184 | 2.82843i | 0 | 10.6074 | − | 6.12417i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
19.d | odd | 6 | 1 | inner |
57.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 342.3.m.d | ✓ | 16 |
3.b | odd | 2 | 1 | inner | 342.3.m.d | ✓ | 16 |
19.d | odd | 6 | 1 | inner | 342.3.m.d | ✓ | 16 |
57.f | even | 6 | 1 | inner | 342.3.m.d | ✓ | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
342.3.m.d | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
342.3.m.d | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
342.3.m.d | ✓ | 16 | 19.d | odd | 6 | 1 | inner |
342.3.m.d | ✓ | 16 | 57.f | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} + 140 T_{5}^{14} + 14688 T_{5}^{12} + 682112 T_{5}^{10} + 23737948 T_{5}^{8} + 13664928 T_{5}^{6} + 7573824 T_{5}^{4} + 100224 T_{5}^{2} + 1296 \)
acting on \(S_{3}^{\mathrm{new}}(342, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - 2 T^{2} + 4)^{4} \)
$3$
\( T^{16} \)
$5$
\( T^{16} + 140 T^{14} + 14688 T^{12} + \cdots + 1296 \)
$7$
\( (T^{4} - 6 T^{3} - 134 T^{2} + 486 T - 185)^{4} \)
$11$
\( (T^{8} - 812 T^{6} + 228184 T^{4} + \cdots + 738426276)^{2} \)
$13$
\( (T^{8} - 234 T^{6} + 47763 T^{4} + \cdots + 48902049)^{2} \)
$17$
\( T^{16} + 2288 T^{14} + \cdots + 24\!\cdots\!16 \)
$19$
\( (T^{8} - 196 T^{6} + \cdots + 16983563041)^{2} \)
$23$
\( T^{16} + 2876 T^{14} + \cdots + 22\!\cdots\!16 \)
$29$
\( T^{16} - 1680 T^{14} + \cdots + 557256278016 \)
$31$
\( (T^{8} + 3912 T^{6} + \cdots + 3913628481)^{2} \)
$37$
\( (T^{8} + 9900 T^{6} + \cdots + 372970368369)^{2} \)
$41$
\( T^{16} - 7056 T^{14} + \cdots + 74\!\cdots\!96 \)
$43$
\( (T^{8} - 106 T^{7} + 9238 T^{6} + \cdots + 784056001)^{2} \)
$47$
\( T^{16} + 8816 T^{14} + \cdots + 10\!\cdots\!96 \)
$53$
\( T^{16} - 7620 T^{14} + \cdots + 97\!\cdots\!56 \)
$59$
\( T^{16} - 23460 T^{14} + \cdots + 12\!\cdots\!56 \)
$61$
\( (T^{8} + 96 T^{7} + \cdots + 572481390625)^{2} \)
$67$
\( (T^{8} - 54 T^{7} + \cdots + 15671836160289)^{2} \)
$71$
\( T^{16} - 13248 T^{14} + \cdots + 20\!\cdots\!56 \)
$73$
\( (T^{8} + 20 T^{7} + \cdots + 20616240141121)^{2} \)
$79$
\( (T^{8} + 366 T^{7} + \cdots + 34109606357649)^{2} \)
$83$
\( (T^{8} - 21968 T^{6} + \cdots + 728195406129216)^{2} \)
$89$
\( T^{16} - 27924 T^{14} + \cdots + 22\!\cdots\!16 \)
$97$
\( (T^{8} - 180 T^{7} + \cdots + 33546476292624)^{2} \)
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