Properties

Label 354.3.d.a
Level 354
Weight 3
Character orbit 354.d
Analytic conductor 9.646
Analytic rank 0
Dimension 20
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 354.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(9.64580135835\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} -\beta_{7} q^{3} -2 q^{4} + \beta_{5} q^{5} + \beta_{6} q^{6} + \beta_{9} q^{7} -2 \beta_{4} q^{8} + 3 q^{9} +O(q^{10})\) \( q + \beta_{4} q^{2} -\beta_{7} q^{3} -2 q^{4} + \beta_{5} q^{5} + \beta_{6} q^{6} + \beta_{9} q^{7} -2 \beta_{4} q^{8} + 3 q^{9} + \beta_{3} q^{10} + ( -\beta_{6} - \beta_{8} ) q^{11} + 2 \beta_{7} q^{12} + ( -\beta_{4} - 2 \beta_{6} + \beta_{16} + \beta_{18} ) q^{13} + \beta_{10} q^{14} + ( -1 + \beta_{1} ) q^{15} + 4 q^{16} + ( 1 + \beta_{1} + \beta_{5} - \beta_{9} - \beta_{11} + \beta_{14} - \beta_{15} ) q^{17} + 3 \beta_{4} q^{18} + ( 3 - \beta_{2} - \beta_{5} + \beta_{13} + \beta_{15} ) q^{19} -2 \beta_{5} q^{20} + ( \beta_{5} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{21} + ( 1 - 2 \beta_{7} + \beta_{13} ) q^{22} + ( -2 \beta_{6} - \beta_{12} + 2 \beta_{18} ) q^{23} -2 \beta_{6} q^{24} + ( 7 + \beta_{2} + \beta_{5} + 2 \beta_{7} + \beta_{9} + 2 \beta_{14} - \beta_{15} ) q^{25} + ( 3 + \beta_{1} - 4 \beta_{7} + \beta_{11} + \beta_{13} + \beta_{15} ) q^{26} -3 \beta_{7} q^{27} -2 \beta_{9} q^{28} + ( 1 - \beta_{1} - 2 \beta_{2} + 4 \beta_{7} + \beta_{9} + \beta_{11} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{29} + ( -\beta_{4} - \beta_{8} - \beta_{16} + \beta_{17} ) q^{30} + ( 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{12} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{31} + 4 \beta_{4} q^{32} + ( \beta_{3} - 2 \beta_{4} - \beta_{8} - \beta_{17} ) q^{33} + ( \beta_{8} - \beta_{10} - \beta_{12} + 2 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{34} + ( -4 + 3 \beta_{1} - \beta_{2} + 3 \beta_{5} - 6 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} - \beta_{13} + \beta_{15} ) q^{35} -6 q^{36} + ( -\beta_{3} + 6 \beta_{4} + \beta_{6} + 3 \beta_{10} - 3 \beta_{12} + 2 \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{37} + ( -\beta_{3} + 3 \beta_{4} + 2 \beta_{8} - \beta_{10} + \beta_{12} - \beta_{16} - \beta_{17} + \beta_{19} ) q^{38} + ( -5 \beta_{4} - \beta_{6} - \beta_{8} + \beta_{12} - \beta_{16} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{39} -2 \beta_{3} q^{40} + ( -3 + \beta_{1} + 2 \beta_{5} - 4 \beta_{7} + 3 \beta_{9} + \beta_{11} + \beta_{14} + 3 \beta_{15} ) q^{41} + ( -\beta_{4} + \beta_{10} - \beta_{12} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{42} + ( 3 \beta_{3} + 11 \beta_{4} - \beta_{6} - \beta_{8} + \beta_{12} - \beta_{16} + 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{43} + ( 2 \beta_{6} + 2 \beta_{8} ) q^{44} + 3 \beta_{5} q^{45} + ( 1 - \beta_{2} - \beta_{5} - 4 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} - \beta_{14} ) q^{46} + ( 4 \beta_{3} + 14 \beta_{4} - 3 \beta_{6} - \beta_{8} - 4 \beta_{10} - \beta_{12} - 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{19} ) q^{47} -4 \beta_{7} q^{48} + ( 5 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{5} - 2 \beta_{11} + 5 \beta_{13} - \beta_{14} + \beta_{15} ) q^{49} + ( -\beta_{3} + 4 \beta_{4} - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{10} - 3 \beta_{12} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{50} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{5} - 2 \beta_{7} + \beta_{9} + \beta_{11} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{51} + ( 2 \beta_{4} + 4 \beta_{6} - 2 \beta_{16} - 2 \beta_{18} ) q^{52} + ( -2 \beta_{1} + 3 \beta_{5} + 8 \beta_{7} - 2 \beta_{13} + 2 \beta_{15} ) q^{53} + 3 \beta_{6} q^{54} + ( -3 \beta_{3} - 7 \beta_{4} + 6 \beta_{6} + 4 \beta_{8} + 2 \beta_{10} + 3 \beta_{16} + \beta_{18} ) q^{55} -2 \beta_{10} q^{56} + ( -1 - \beta_{1} - \beta_{2} - 3 \beta_{5} - 2 \beta_{7} - 2 \beta_{9} - \beta_{11} - \beta_{14} - 2 \beta_{15} ) q^{57} + ( -\beta_{3} - 4 \beta_{6} + \beta_{8} + 3 \beta_{10} + \beta_{12} + 2 \beta_{16} - 2 \beta_{18} + 3 \beta_{19} ) q^{58} + ( 11 - \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 5 \beta_{7} - 3 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} + 3 \beta_{13} + \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{59} + ( 2 - 2 \beta_{1} ) q^{60} + ( \beta_{3} - \beta_{4} + 5 \beta_{6} + \beta_{10} - \beta_{12} + 3 \beta_{16} + \beta_{17} + 3 \beta_{18} - 2 \beta_{19} ) q^{61} + ( 1 + \beta_{2} - 3 \beta_{5} - 4 \beta_{7} - 3 \beta_{9} + \beta_{11} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{62} + 3 \beta_{9} q^{63} -8 q^{64} + ( -7 \beta_{3} + 3 \beta_{4} - 9 \beta_{6} + 5 \beta_{8} - 2 \beta_{10} - 3 \beta_{12} + 7 \beta_{16} + \beta_{17} + 6 \beta_{18} + \beta_{19} ) q^{65} + ( 6 + \beta_{1} - 2 \beta_{5} - \beta_{9} + \beta_{13} - \beta_{15} ) q^{66} + ( \beta_{3} - 11 \beta_{4} + 3 \beta_{6} + \beta_{8} + 6 \beta_{10} + \beta_{12} + \beta_{16} - 3 \beta_{18} + 3 \beta_{19} ) q^{67} + ( -2 - 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{14} + 2 \beta_{15} ) q^{68} + ( 2 \beta_{3} - 6 \beta_{4} - 3 \beta_{8} + \beta_{10} + 2 \beta_{12} - 3 \beta_{18} + \beta_{19} ) q^{69} + ( 3 \beta_{3} - 4 \beta_{4} + 6 \beta_{6} - 5 \beta_{8} - \beta_{10} + \beta_{12} - 4 \beta_{16} + 2 \beta_{17} - 4 \beta_{18} + \beta_{19} ) q^{70} + ( -29 + \beta_{1} - 2 \beta_{2} - 3 \beta_{5} + 10 \beta_{7} - 2 \beta_{9} - \beta_{11} - 5 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{71} -6 \beta_{4} q^{72} + ( -\beta_{3} - 9 \beta_{4} - \beta_{6} + \beta_{10} + 3 \beta_{12} - 3 \beta_{16} + 3 \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{73} + ( -14 + \beta_{1} - 3 \beta_{2} - \beta_{5} + 2 \beta_{7} - 5 \beta_{9} + 2 \beta_{11} + 2 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{74} + ( -5 + \beta_{2} - 2 \beta_{5} - 5 \beta_{7} + 3 \beta_{9} + \beta_{11} - 2 \beta_{15} ) q^{75} + ( -6 + 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{13} - 2 \beta_{15} ) q^{76} + ( 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} + 3 \beta_{8} - 5 \beta_{10} + 3 \beta_{12} - 2 \beta_{16} + 2 \beta_{18} - 2 \beta_{19} ) q^{77} + ( 12 + 2 \beta_{2} - 2 \beta_{7} - \beta_{9} - \beta_{11} + \beta_{13} - 2 \beta_{15} ) q^{78} + ( 33 + 4 \beta_{1} + 3 \beta_{2} + \beta_{5} + 8 \beta_{7} - 5 \beta_{9} - 2 \beta_{11} + 3 \beta_{13} - 2 \beta_{14} - 3 \beta_{15} ) q^{79} + 4 \beta_{5} q^{80} + 9 q^{81} + ( \beta_{3} - 2 \beta_{4} + 4 \beta_{6} - 3 \beta_{8} + \beta_{10} - \beta_{12} - 4 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{82} + ( -4 \beta_{3} - 6 \beta_{4} + 5 \beta_{6} - 5 \beta_{8} - 5 \beta_{12} - 6 \beta_{16} + 4 \beta_{17} - 2 \beta_{18} - 2 \beta_{19} ) q^{83} + ( -2 \beta_{5} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{84} + ( 32 - 6 \beta_{1} + 6 \beta_{2} + 2 \beta_{5} - 12 \beta_{7} + 2 \beta_{9} - 6 \beta_{11} - \beta_{13} + 3 \beta_{14} - 7 \beta_{15} ) q^{85} + ( -23 - 3 \beta_{1} - 4 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} - \beta_{11} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{86} + ( -12 - 2 \beta_{1} - 3 \beta_{5} + 5 \beta_{9} - 3 \beta_{11} - \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{87} + ( -2 + 4 \beta_{7} - 2 \beta_{13} ) q^{88} + ( -\beta_{3} + 10 \beta_{4} - 11 \beta_{6} + 5 \beta_{8} + 3 \beta_{10} + 4 \beta_{12} + \beta_{16} - 5 \beta_{17} - 4 \beta_{18} - 3 \beta_{19} ) q^{89} + 3 \beta_{3} q^{90} + ( -4 \beta_{3} - 5 \beta_{4} + 3 \beta_{6} - \beta_{8} - 6 \beta_{10} + 6 \beta_{12} - 5 \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{91} + ( 4 \beta_{6} + 2 \beta_{12} - 4 \beta_{18} ) q^{92} + ( \beta_{3} - 9 \beta_{4} - 3 \beta_{8} - \beta_{10} + \beta_{12} + 3 \beta_{17} + 2 \beta_{19} ) q^{93} + ( -26 - 3 \beta_{2} - 7 \beta_{5} - 6 \beta_{7} + 4 \beta_{9} - 3 \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{94} + ( -29 - 6 \beta_{1} - 5 \beta_{2} + 2 \beta_{5} + 14 \beta_{7} + 3 \beta_{11} - 4 \beta_{14} + 8 \beta_{15} ) q^{95} + 4 \beta_{6} q^{96} + ( 2 \beta_{3} + 15 \beta_{4} + 6 \beta_{6} - 2 \beta_{8} - 4 \beta_{10} + 6 \beta_{12} + \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{97} + ( -3 \beta_{3} + 4 \beta_{4} + 4 \beta_{8} - 3 \beta_{10} - \beta_{12} - 3 \beta_{16} + \beta_{17} + 4 \beta_{18} - 3 \beta_{19} ) q^{98} + ( -3 \beta_{6} - 3 \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 40q^{4} + 8q^{7} + 60q^{9} + O(q^{10}) \) \( 20q - 40q^{4} + 8q^{7} + 60q^{9} - 24q^{15} + 80q^{16} + 72q^{19} + 16q^{22} + 140q^{25} + 64q^{26} - 16q^{28} + 56q^{29} - 80q^{35} - 120q^{36} - 8q^{41} + 16q^{46} + 52q^{49} + 32q^{53} - 48q^{57} + 192q^{59} + 48q^{60} - 16q^{62} + 24q^{63} - 160q^{64} + 96q^{66} - 568q^{71} - 288q^{74} - 96q^{75} - 144q^{76} + 192q^{78} + 528q^{79} + 180q^{81} + 568q^{85} - 416q^{86} - 216q^{87} - 32q^{88} - 480q^{94} - 456q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 300 x^{18} - 88 x^{17} + 37952 x^{16} + 27000 x^{15} - 2574056 x^{14} - 3295208 x^{13} + 98138134 x^{12} + 195465720 x^{11} - 1935828112 x^{10} - 5590976152 x^{9} + 12661641204 x^{8} + 55962519336 x^{7} + 75365256136 x^{6} + 219349710792 x^{5} + 570681930321 x^{4} + 614778274152 x^{3} + 1559028888924 x^{2} + 534075251472 x + 2455573689828\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(11\!\cdots\!96\)\( \nu^{19} + \)\(15\!\cdots\!35\)\( \nu^{18} + \)\(20\!\cdots\!22\)\( \nu^{17} - \)\(36\!\cdots\!53\)\( \nu^{16} - \)\(48\!\cdots\!84\)\( \nu^{15} + \)\(30\!\cdots\!71\)\( \nu^{14} + \)\(48\!\cdots\!86\)\( \nu^{13} - \)\(88\!\cdots\!53\)\( \nu^{12} - \)\(24\!\cdots\!00\)\( \nu^{11} - \)\(18\!\cdots\!55\)\( \nu^{10} + \)\(55\!\cdots\!82\)\( \nu^{9} + \)\(15\!\cdots\!25\)\( \nu^{8} - \)\(21\!\cdots\!64\)\( \nu^{7} - \)\(18\!\cdots\!67\)\( \nu^{6} - \)\(76\!\cdots\!58\)\( \nu^{5} - \)\(20\!\cdots\!71\)\( \nu^{4} - \)\(36\!\cdots\!08\)\( \nu^{3} - \)\(57\!\cdots\!28\)\( \nu^{2} - \)\(46\!\cdots\!12\)\( \nu - \)\(49\!\cdots\!00\)\(\)\()/ \)\(78\!\cdots\!56\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(30\!\cdots\!89\)\( \nu^{19} + \)\(11\!\cdots\!07\)\( \nu^{18} - \)\(35\!\cdots\!01\)\( \nu^{17} - \)\(34\!\cdots\!95\)\( \nu^{16} - \)\(56\!\cdots\!71\)\( \nu^{15} + \)\(42\!\cdots\!11\)\( \nu^{14} + \)\(15\!\cdots\!91\)\( \nu^{13} - \)\(27\!\cdots\!75\)\( \nu^{12} - \)\(13\!\cdots\!13\)\( \nu^{11} + \)\(91\!\cdots\!53\)\( \nu^{10} + \)\(64\!\cdots\!93\)\( \nu^{9} - \)\(12\!\cdots\!41\)\( \nu^{8} - \)\(15\!\cdots\!25\)\( \nu^{7} - \)\(11\!\cdots\!71\)\( \nu^{6} + \)\(13\!\cdots\!01\)\( \nu^{5} + \)\(34\!\cdots\!15\)\( \nu^{4} + \)\(39\!\cdots\!44\)\( \nu^{3} + \)\(95\!\cdots\!92\)\( \nu^{2} + \)\(81\!\cdots\!00\)\( \nu + \)\(67\!\cdots\!28\)\(\)\()/ \)\(18\!\cdots\!64\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(10\!\cdots\!82\)\( \nu^{19} - \)\(39\!\cdots\!27\)\( \nu^{18} - \)\(31\!\cdots\!04\)\( \nu^{17} + \)\(10\!\cdots\!41\)\( \nu^{16} + \)\(38\!\cdots\!98\)\( \nu^{15} - \)\(10\!\cdots\!55\)\( \nu^{14} - \)\(25\!\cdots\!16\)\( \nu^{13} + \)\(54\!\cdots\!45\)\( \nu^{12} + \)\(99\!\cdots\!02\)\( \nu^{11} - \)\(11\!\cdots\!69\)\( \nu^{10} - \)\(21\!\cdots\!20\)\( \nu^{9} + \)\(26\!\cdots\!95\)\( \nu^{8} + \)\(20\!\cdots\!86\)\( \nu^{7} + \)\(16\!\cdots\!19\)\( \nu^{6} - \)\(14\!\cdots\!72\)\( \nu^{5} + \)\(67\!\cdots\!11\)\( \nu^{4} - \)\(24\!\cdots\!28\)\( \nu^{3} - \)\(13\!\cdots\!36\)\( \nu^{2} - \)\(25\!\cdots\!48\)\( \nu - \)\(22\!\cdots\!64\)\(\)\()/ \)\(31\!\cdots\!72\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(11\!\cdots\!45\)\( \nu^{19} + \)\(74\!\cdots\!74\)\( \nu^{18} + \)\(35\!\cdots\!11\)\( \nu^{17} - \)\(11\!\cdots\!68\)\( \nu^{16} - \)\(44\!\cdots\!53\)\( \nu^{15} - \)\(49\!\cdots\!14\)\( \nu^{14} + \)\(29\!\cdots\!35\)\( \nu^{13} + \)\(20\!\cdots\!48\)\( \nu^{12} - \)\(11\!\cdots\!15\)\( \nu^{11} - \)\(16\!\cdots\!86\)\( \nu^{10} + \)\(22\!\cdots\!57\)\( \nu^{9} + \)\(51\!\cdots\!00\)\( \nu^{8} - \)\(14\!\cdots\!15\)\( \nu^{7} - \)\(51\!\cdots\!18\)\( \nu^{6} - \)\(77\!\cdots\!87\)\( \nu^{5} - \)\(25\!\cdots\!44\)\( \nu^{4} - \)\(62\!\cdots\!68\)\( \nu^{3} - \)\(74\!\cdots\!36\)\( \nu^{2} - \)\(27\!\cdots\!32\)\( \nu - \)\(80\!\cdots\!76\)\(\)\()/ \)\(22\!\cdots\!04\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(11\!\cdots\!45\)\( \nu^{19} + \)\(74\!\cdots\!74\)\( \nu^{18} + \)\(35\!\cdots\!11\)\( \nu^{17} - \)\(11\!\cdots\!68\)\( \nu^{16} - \)\(44\!\cdots\!53\)\( \nu^{15} - \)\(49\!\cdots\!14\)\( \nu^{14} + \)\(29\!\cdots\!35\)\( \nu^{13} + \)\(20\!\cdots\!48\)\( \nu^{12} - \)\(11\!\cdots\!15\)\( \nu^{11} - \)\(16\!\cdots\!86\)\( \nu^{10} + \)\(22\!\cdots\!57\)\( \nu^{9} + \)\(51\!\cdots\!00\)\( \nu^{8} - \)\(14\!\cdots\!15\)\( \nu^{7} - \)\(51\!\cdots\!18\)\( \nu^{6} - \)\(77\!\cdots\!87\)\( \nu^{5} - \)\(25\!\cdots\!44\)\( \nu^{4} - \)\(62\!\cdots\!68\)\( \nu^{3} - \)\(74\!\cdots\!36\)\( \nu^{2} - \)\(54\!\cdots\!28\)\( \nu - \)\(80\!\cdots\!76\)\(\)\()/ \)\(22\!\cdots\!04\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(23\!\cdots\!82\)\( \nu^{19} + \)\(80\!\cdots\!67\)\( \nu^{18} - \)\(71\!\cdots\!88\)\( \nu^{17} - \)\(26\!\cdots\!65\)\( \nu^{16} + \)\(90\!\cdots\!30\)\( \nu^{15} + \)\(36\!\cdots\!87\)\( \nu^{14} - \)\(61\!\cdots\!16\)\( \nu^{13} - \)\(28\!\cdots\!57\)\( \nu^{12} + \)\(22\!\cdots\!38\)\( \nu^{11} + \)\(12\!\cdots\!05\)\( \nu^{10} - \)\(38\!\cdots\!72\)\( \nu^{9} - \)\(31\!\cdots\!63\)\( \nu^{8} - \)\(49\!\cdots\!26\)\( \nu^{7} + \)\(30\!\cdots\!53\)\( \nu^{6} + \)\(61\!\cdots\!96\)\( \nu^{5} + \)\(44\!\cdots\!05\)\( \nu^{4} + \)\(11\!\cdots\!12\)\( \nu^{3} + \)\(28\!\cdots\!80\)\( \nu^{2} + \)\(18\!\cdots\!12\)\( \nu + \)\(17\!\cdots\!28\)\(\)\()/ \)\(25\!\cdots\!48\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(30\!\cdots\!13\)\( \nu^{19} + \)\(30\!\cdots\!25\)\( \nu^{18} + \)\(93\!\cdots\!93\)\( \nu^{17} - \)\(73\!\cdots\!27\)\( \nu^{16} - \)\(11\!\cdots\!25\)\( \nu^{15} + \)\(53\!\cdots\!53\)\( \nu^{14} + \)\(83\!\cdots\!69\)\( \nu^{13} + \)\(25\!\cdots\!73\)\( \nu^{12} - \)\(33\!\cdots\!79\)\( \nu^{11} - \)\(21\!\cdots\!53\)\( \nu^{10} + \)\(77\!\cdots\!11\)\( \nu^{9} + \)\(10\!\cdots\!59\)\( \nu^{8} - \)\(86\!\cdots\!59\)\( \nu^{7} - \)\(16\!\cdots\!89\)\( \nu^{6} + \)\(24\!\cdots\!03\)\( \nu^{5} + \)\(43\!\cdots\!19\)\( \nu^{4} - \)\(32\!\cdots\!76\)\( \nu^{3} + \)\(11\!\cdots\!88\)\( \nu^{2} + \)\(12\!\cdots\!44\)\( \nu + \)\(27\!\cdots\!76\)\(\)\()/ \)\(32\!\cdots\!72\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(50\!\cdots\!80\)\( \nu^{19} + \)\(16\!\cdots\!84\)\( \nu^{18} + \)\(15\!\cdots\!45\)\( \nu^{17} - \)\(47\!\cdots\!87\)\( \nu^{16} - \)\(20\!\cdots\!78\)\( \nu^{15} + \)\(54\!\cdots\!02\)\( \nu^{14} + \)\(14\!\cdots\!19\)\( \nu^{13} - \)\(32\!\cdots\!61\)\( \nu^{12} - \)\(62\!\cdots\!92\)\( \nu^{11} + \)\(97\!\cdots\!48\)\( \nu^{10} + \)\(15\!\cdots\!43\)\( \nu^{9} - \)\(14\!\cdots\!85\)\( \nu^{8} - \)\(21\!\cdots\!74\)\( \nu^{7} + \)\(84\!\cdots\!14\)\( \nu^{6} + \)\(10\!\cdots\!61\)\( \nu^{5} - \)\(10\!\cdots\!59\)\( \nu^{4} + \)\(13\!\cdots\!56\)\( \nu^{3} + \)\(13\!\cdots\!88\)\( \nu^{2} - \)\(55\!\cdots\!08\)\( \nu + \)\(20\!\cdots\!36\)\(\)\()/ \)\(39\!\cdots\!28\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(17\!\cdots\!88\)\( \nu^{19} + \)\(63\!\cdots\!53\)\( \nu^{18} + \)\(54\!\cdots\!78\)\( \nu^{17} + \)\(19\!\cdots\!61\)\( \nu^{16} - \)\(70\!\cdots\!32\)\( \nu^{15} - \)\(58\!\cdots\!07\)\( \nu^{14} + \)\(49\!\cdots\!26\)\( \nu^{13} + \)\(69\!\cdots\!17\)\( \nu^{12} - \)\(19\!\cdots\!36\)\( \nu^{11} - \)\(41\!\cdots\!05\)\( \nu^{10} + \)\(40\!\cdots\!70\)\( \nu^{9} + \)\(12\!\cdots\!31\)\( \nu^{8} - \)\(30\!\cdots\!92\)\( \nu^{7} - \)\(12\!\cdots\!61\)\( \nu^{6} - \)\(99\!\cdots\!90\)\( \nu^{5} - \)\(34\!\cdots\!33\)\( \nu^{4} - \)\(13\!\cdots\!80\)\( \nu^{3} - \)\(92\!\cdots\!12\)\( \nu^{2} - \)\(10\!\cdots\!96\)\( \nu - \)\(26\!\cdots\!64\)\(\)\()/ \)\(54\!\cdots\!92\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(18\!\cdots\!37\)\( \nu^{19} - \)\(45\!\cdots\!45\)\( \nu^{18} - \)\(54\!\cdots\!43\)\( \nu^{17} + \)\(12\!\cdots\!15\)\( \nu^{16} + \)\(68\!\cdots\!73\)\( \nu^{15} - \)\(13\!\cdots\!57\)\( \nu^{14} - \)\(47\!\cdots\!63\)\( \nu^{13} + \)\(66\!\cdots\!27\)\( \nu^{12} + \)\(18\!\cdots\!99\)\( \nu^{11} - \)\(13\!\cdots\!75\)\( \nu^{10} - \)\(42\!\cdots\!61\)\( \nu^{9} - \)\(34\!\cdots\!23\)\( \nu^{8} + \)\(44\!\cdots\!47\)\( \nu^{7} + \)\(19\!\cdots\!77\)\( \nu^{6} - \)\(88\!\cdots\!37\)\( \nu^{5} + \)\(31\!\cdots\!05\)\( \nu^{4} + \)\(22\!\cdots\!00\)\( \nu^{3} - \)\(17\!\cdots\!76\)\( \nu^{2} + \)\(17\!\cdots\!24\)\( \nu - \)\(28\!\cdots\!04\)\(\)\()/ \)\(54\!\cdots\!92\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(60\!\cdots\!59\)\( \nu^{19} + \)\(25\!\cdots\!58\)\( \nu^{18} + \)\(18\!\cdots\!89\)\( \nu^{17} - \)\(68\!\cdots\!96\)\( \nu^{16} - \)\(24\!\cdots\!99\)\( \nu^{15} + \)\(73\!\cdots\!06\)\( \nu^{14} + \)\(18\!\cdots\!73\)\( \nu^{13} - \)\(37\!\cdots\!68\)\( \nu^{12} - \)\(80\!\cdots\!85\)\( \nu^{11} + \)\(78\!\cdots\!66\)\( \nu^{10} + \)\(20\!\cdots\!19\)\( \nu^{9} + \)\(31\!\cdots\!68\)\( \nu^{8} - \)\(28\!\cdots\!57\)\( \nu^{7} - \)\(34\!\cdots\!62\)\( \nu^{6} + \)\(12\!\cdots\!91\)\( \nu^{5} + \)\(28\!\cdots\!36\)\( \nu^{4} + \)\(24\!\cdots\!72\)\( \nu^{3} + \)\(76\!\cdots\!60\)\( \nu^{2} + \)\(82\!\cdots\!48\)\( \nu + \)\(17\!\cdots\!32\)\(\)\()/ \)\(14\!\cdots\!28\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(19\!\cdots\!79\)\( \nu^{19} - \)\(19\!\cdots\!70\)\( \nu^{18} + \)\(58\!\cdots\!49\)\( \nu^{17} + \)\(76\!\cdots\!98\)\( \nu^{16} - \)\(74\!\cdots\!75\)\( \nu^{15} - \)\(12\!\cdots\!22\)\( \nu^{14} + \)\(50\!\cdots\!97\)\( \nu^{13} + \)\(11\!\cdots\!06\)\( \nu^{12} - \)\(18\!\cdots\!93\)\( \nu^{11} - \)\(57\!\cdots\!58\)\( \nu^{10} + \)\(35\!\cdots\!51\)\( \nu^{9} + \)\(14\!\cdots\!62\)\( \nu^{8} - \)\(16\!\cdots\!85\)\( \nu^{7} - \)\(13\!\cdots\!10\)\( \nu^{6} - \)\(23\!\cdots\!45\)\( \nu^{5} - \)\(52\!\cdots\!22\)\( \nu^{4} - \)\(13\!\cdots\!24\)\( \nu^{3} - \)\(25\!\cdots\!92\)\( \nu^{2} - \)\(55\!\cdots\!00\)\( \nu - \)\(29\!\cdots\!56\)\(\)\()/ \)\(42\!\cdots\!84\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(26\!\cdots\!84\)\( \nu^{19} + \)\(22\!\cdots\!73\)\( \nu^{18} - \)\(79\!\cdots\!08\)\( \nu^{17} - \)\(92\!\cdots\!49\)\( \nu^{16} + \)\(10\!\cdots\!92\)\( \nu^{15} + \)\(16\!\cdots\!65\)\( \nu^{14} - \)\(70\!\cdots\!76\)\( \nu^{13} - \)\(14\!\cdots\!29\)\( \nu^{12} + \)\(27\!\cdots\!04\)\( \nu^{11} + \)\(77\!\cdots\!55\)\( \nu^{10} - \)\(56\!\cdots\!64\)\( \nu^{9} - \)\(20\!\cdots\!51\)\( \nu^{8} + \)\(41\!\cdots\!40\)\( \nu^{7} + \)\(21\!\cdots\!59\)\( \nu^{6} + \)\(13\!\cdots\!96\)\( \nu^{5} + \)\(43\!\cdots\!85\)\( \nu^{4} + \)\(22\!\cdots\!20\)\( \nu^{3} + \)\(10\!\cdots\!28\)\( \nu^{2} + \)\(17\!\cdots\!00\)\( \nu + \)\(11\!\cdots\!48\)\(\)\()/ \)\(54\!\cdots\!92\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(34\!\cdots\!88\)\( \nu^{19} - \)\(66\!\cdots\!51\)\( \nu^{18} - \)\(10\!\cdots\!84\)\( \nu^{17} + \)\(16\!\cdots\!71\)\( \nu^{16} + \)\(13\!\cdots\!72\)\( \nu^{15} - \)\(15\!\cdots\!75\)\( \nu^{14} - \)\(96\!\cdots\!68\)\( \nu^{13} + \)\(45\!\cdots\!87\)\( \nu^{12} + \)\(39\!\cdots\!00\)\( \nu^{11} + \)\(11\!\cdots\!79\)\( \nu^{10} - \)\(93\!\cdots\!52\)\( \nu^{9} - \)\(10\!\cdots\!63\)\( \nu^{8} + \)\(10\!\cdots\!84\)\( \nu^{7} + \)\(19\!\cdots\!27\)\( \nu^{6} - \)\(30\!\cdots\!48\)\( \nu^{5} - \)\(57\!\cdots\!83\)\( \nu^{4} + \)\(27\!\cdots\!76\)\( \nu^{3} - \)\(15\!\cdots\!76\)\( \nu^{2} - \)\(16\!\cdots\!68\)\( \nu - \)\(12\!\cdots\!08\)\(\)\()/ \)\(54\!\cdots\!92\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(85\!\cdots\!45\)\( \nu^{19} - \)\(18\!\cdots\!33\)\( \nu^{18} - \)\(26\!\cdots\!14\)\( \nu^{17} + \)\(46\!\cdots\!77\)\( \nu^{16} + \)\(33\!\cdots\!81\)\( \nu^{15} - \)\(44\!\cdots\!55\)\( \nu^{14} - \)\(23\!\cdots\!78\)\( \nu^{13} + \)\(16\!\cdots\!59\)\( \nu^{12} + \)\(96\!\cdots\!87\)\( \nu^{11} + \)\(38\!\cdots\!09\)\( \nu^{10} - \)\(22\!\cdots\!78\)\( \nu^{9} - \)\(19\!\cdots\!65\)\( \nu^{8} + \)\(24\!\cdots\!79\)\( \nu^{7} + \)\(39\!\cdots\!35\)\( \nu^{6} - \)\(62\!\cdots\!46\)\( \nu^{5} - \)\(10\!\cdots\!35\)\( \nu^{4} + \)\(27\!\cdots\!96\)\( \nu^{3} - \)\(24\!\cdots\!20\)\( \nu^{2} - \)\(39\!\cdots\!88\)\( \nu - \)\(18\!\cdots\!76\)\(\)\()/ \)\(13\!\cdots\!48\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(26\!\cdots\!13\)\( \nu^{19} - \)\(40\!\cdots\!24\)\( \nu^{18} + \)\(79\!\cdots\!77\)\( \nu^{17} + \)\(14\!\cdots\!96\)\( \nu^{16} - \)\(10\!\cdots\!09\)\( \nu^{15} - \)\(22\!\cdots\!32\)\( \nu^{14} + \)\(67\!\cdots\!57\)\( \nu^{13} + \)\(19\!\cdots\!84\)\( \nu^{12} - \)\(25\!\cdots\!27\)\( \nu^{11} - \)\(94\!\cdots\!44\)\( \nu^{10} + \)\(45\!\cdots\!23\)\( \nu^{9} + \)\(24\!\cdots\!56\)\( \nu^{8} - \)\(12\!\cdots\!79\)\( \nu^{7} - \)\(24\!\cdots\!72\)\( \nu^{6} - \)\(47\!\cdots\!01\)\( \nu^{5} - \)\(53\!\cdots\!16\)\( \nu^{4} - \)\(14\!\cdots\!92\)\( \nu^{3} - \)\(27\!\cdots\!96\)\( \nu^{2} - \)\(41\!\cdots\!08\)\( \nu - \)\(23\!\cdots\!68\)\(\)\()/ \)\(42\!\cdots\!84\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(37\!\cdots\!17\)\( \nu^{19} - \)\(29\!\cdots\!86\)\( \nu^{18} + \)\(11\!\cdots\!59\)\( \nu^{17} + \)\(12\!\cdots\!36\)\( \nu^{16} - \)\(14\!\cdots\!05\)\( \nu^{15} - \)\(21\!\cdots\!90\)\( \nu^{14} + \)\(98\!\cdots\!51\)\( \nu^{13} + \)\(20\!\cdots\!48\)\( \nu^{12} - \)\(37\!\cdots\!87\)\( \nu^{11} - \)\(10\!\cdots\!46\)\( \nu^{10} + \)\(72\!\cdots\!33\)\( \nu^{9} + \)\(27\!\cdots\!20\)\( \nu^{8} - \)\(40\!\cdots\!27\)\( \nu^{7} - \)\(27\!\cdots\!86\)\( \nu^{6} - \)\(42\!\cdots\!75\)\( \nu^{5} - \)\(80\!\cdots\!68\)\( \nu^{4} - \)\(16\!\cdots\!84\)\( \nu^{3} - \)\(18\!\cdots\!32\)\( \nu^{2} - \)\(56\!\cdots\!88\)\( \nu - \)\(69\!\cdots\!40\)\(\)\()/ \)\(54\!\cdots\!92\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(45\!\cdots\!07\)\( \nu^{19} + \)\(30\!\cdots\!32\)\( \nu^{18} - \)\(13\!\cdots\!91\)\( \nu^{17} - \)\(13\!\cdots\!04\)\( \nu^{16} + \)\(17\!\cdots\!95\)\( \nu^{15} + \)\(24\!\cdots\!92\)\( \nu^{14} - \)\(11\!\cdots\!71\)\( \nu^{13} - \)\(23\!\cdots\!60\)\( \nu^{12} + \)\(44\!\cdots\!81\)\( \nu^{11} + \)\(12\!\cdots\!96\)\( \nu^{10} - \)\(86\!\cdots\!93\)\( \nu^{9} - \)\(33\!\cdots\!08\)\( \nu^{8} + \)\(45\!\cdots\!21\)\( \nu^{7} + \)\(34\!\cdots\!44\)\( \nu^{6} + \)\(57\!\cdots\!07\)\( \nu^{5} + \)\(78\!\cdots\!80\)\( \nu^{4} + \)\(14\!\cdots\!48\)\( \nu^{3} + \)\(11\!\cdots\!12\)\( \nu^{2} + \)\(36\!\cdots\!80\)\( \nu - \)\(12\!\cdots\!84\)\(\)\()/ \)\(54\!\cdots\!92\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(32\!\cdots\!27\)\( \nu^{19} + \)\(16\!\cdots\!59\)\( \nu^{18} - \)\(97\!\cdots\!97\)\( \nu^{17} - \)\(80\!\cdots\!27\)\( \nu^{16} + \)\(12\!\cdots\!47\)\( \nu^{15} + \)\(15\!\cdots\!15\)\( \nu^{14} - \)\(84\!\cdots\!53\)\( \nu^{13} - \)\(15\!\cdots\!23\)\( \nu^{12} + \)\(31\!\cdots\!41\)\( \nu^{11} + \)\(85\!\cdots\!65\)\( \nu^{10} - \)\(61\!\cdots\!03\)\( \nu^{9} - \)\(23\!\cdots\!25\)\( \nu^{8} + \)\(33\!\cdots\!97\)\( \nu^{7} + \)\(24\!\cdots\!09\)\( \nu^{6} + \)\(39\!\cdots\!33\)\( \nu^{5} + \)\(56\!\cdots\!83\)\( \nu^{4} + \)\(10\!\cdots\!88\)\( \nu^{3} + \)\(84\!\cdots\!96\)\( \nu^{2} + \)\(29\!\cdots\!20\)\( \nu - \)\(75\!\cdots\!48\)\(\)\()/ \)\(26\!\cdots\!52\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{5} - \beta_{4}\)
\(\nu^{2}\)\(=\)\(-\beta_{15} + 2 \beta_{14} + \beta_{9} + 2 \beta_{7} + \beta_{5} - 2 \beta_{3} + \beta_{2} + 30\)
\(\nu^{3}\)\(=\)\(-3 \beta_{19} - 3 \beta_{17} - 3 \beta_{16} + 4 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + 9 \beta_{12} - 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} + 12 \beta_{7} + 6 \beta_{6} + 49 \beta_{5} - 85 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + 4 \beta_{1} + 16\)
\(\nu^{4}\)\(=\)\(24 \beta_{19} - 16 \beta_{18} + 32 \beta_{16} - 85 \beta_{15} + 108 \beta_{14} - 6 \beta_{13} + 8 \beta_{12} - 2 \beta_{11} + 32 \beta_{10} + 51 \beta_{9} + 40 \beta_{8} - 54 \beta_{7} + 48 \beta_{6} + 29 \beta_{5} - 88 \beta_{4} - 220 \beta_{3} + 51 \beta_{2} - 26 \beta_{1} + 1416\)
\(\nu^{5}\)\(=\)\(-325 \beta_{19} - 20 \beta_{18} - 335 \beta_{17} - 595 \beta_{16} + 238 \beta_{15} - 140 \beta_{14} + 64 \beta_{13} + 915 \beta_{12} - 164 \beta_{11} - 750 \beta_{10} - 242 \beta_{9} - 860 \beta_{8} + 1304 \beta_{7} - 190 \beta_{6} + 2271 \beta_{5} - 7299 \beta_{4} + 435 \beta_{3} + 194 \beta_{2} + 436 \beta_{1} + 90\)
\(\nu^{6}\)\(=\)\(2484 \beta_{19} - 2288 \beta_{18} - 1188 \beta_{17} + 4684 \beta_{16} - 4747 \beta_{15} + 4766 \beta_{14} - 296 \beta_{13} + 484 \beta_{12} - 272 \beta_{11} + 4504 \beta_{10} + 2995 \beta_{9} + 6080 \beta_{8} - 9306 \beta_{7} + 8784 \beta_{6} - 1141 \beta_{5} - 5168 \beta_{4} - 18930 \beta_{3} + 2435 \beta_{2} - 2616 \beta_{1} + 57892\)
\(\nu^{7}\)\(=\)\(-25641 \beta_{19} - 4368 \beta_{18} - 24521 \beta_{17} - 68425 \beta_{16} + 12404 \beta_{15} - 8622 \beta_{14} + 182 \beta_{13} + 76699 \beta_{12} - 7398 \beta_{11} - 73738 \beta_{10} - 19142 \beta_{9} - 88242 \beta_{8} + 79108 \beta_{7} - 70014 \beta_{6} + 81853 \beta_{5} - 553433 \beta_{4} + 53753 \beta_{3} + 4916 \beta_{2} + 27524 \beta_{1} - 65192\)
\(\nu^{8}\)\(=\)\(204624 \beta_{19} - 206176 \beta_{18} - 165312 \beta_{17} + 499072 \beta_{16} - 168489 \beta_{15} + 126656 \beta_{14} - 2250 \beta_{13} - 10384 \beta_{12} - 12294 \beta_{11} + 484480 \beta_{10} + 132523 \beta_{9} + 657776 \beta_{8} - 560270 \beta_{7} + 967968 \beta_{6} - 194335 \beta_{5} + 89520 \beta_{4} - 1464056 \beta_{3} + 69883 \beta_{2} - 147062 \beta_{1} + 1398812\)
\(\nu^{9}\)\(=\)\(-1832157 \beta_{19} - 436332 \beta_{18} - 1462971 \beta_{17} - 6287367 \beta_{16} + 421506 \beta_{15} - 364688 \beta_{14} - 128252 \beta_{13} + 5705571 \beta_{12} - 80000 \beta_{11} - 6345630 \beta_{10} - 784130 \beta_{9} - 7783344 \beta_{8} + 2424768 \beta_{7} - 8448030 \beta_{6} + 482815 \beta_{5} - 38678923 \beta_{4} + 5589063 \beta_{3} - 179122 \beta_{2} + 860028 \beta_{1} - 5685682\)
\(\nu^{10}\)\(=\)\(15307372 \beta_{19} - 15036544 \beta_{18} - 14836164 \beta_{17} + 44940012 \beta_{16} + 3076317 \beta_{15} - 5516278 \beta_{14} + 700692 \beta_{13} - 5823828 \beta_{12} + 325164 \beta_{11} + 44024872 \beta_{10} - 72145 \beta_{9} + 59528840 \beta_{8} - 8348554 \beta_{7} + 86383872 \beta_{6} - 10071725 \beta_{5} + 49743016 \beta_{4} - 105374518 \beta_{3} - 2867697 \beta_{2} - 1975028 \beta_{1} - 75783484\)
\(\nu^{11}\)\(=\)\(-125173037 \beta_{19} - 28031784 \beta_{18} - 70936833 \beta_{17} - 500549225 \beta_{16} - 22370312 \beta_{15} + 10500534 \beta_{14} - 8327838 \beta_{13} + 385536943 \beta_{12} + 26053702 \beta_{11} - 490560642 \beta_{10} + 30134994 \beta_{9} - 612046270 \beta_{8} - 168278084 \beta_{7} - 761581062 \beta_{6} - 321838299 \beta_{5} - 2518737145 \beta_{4} + 500334153 \beta_{3} - 33671344 \beta_{2} - 59193348 \beta_{1} - 112108372\)
\(\nu^{12}\)\(=\)\(1062670936 \beta_{19} - 942502256 \beta_{18} - 1051657104 \beta_{17} + 3557900432 \beta_{16} + 1375925959 \beta_{15} - 1357445676 \beta_{14} + 68660698 \beta_{13} - 795519640 \beta_{12} + 99742190 \beta_{11} + 3495987264 \beta_{10} - 903107305 \beta_{9} + 4695379512 \beta_{8} + 2778423378 \beta_{7} + 6706804944 \beta_{6} + 393564993 \beta_{5} + 6360219768 \beta_{4} - 7059358404 \beta_{3} - 720186313 \beta_{2} + 777702278 \beta_{1} - 16622556444\)
\(\nu^{13}\)\(=\)\(-8123893089 \beta_{19} - 1082448276 \beta_{18} - 2405988091 \beta_{17} - 35188337951 \beta_{16} - 7003180210 \beta_{15} + 4878316204 \beta_{14} - 130484072 \beta_{13} + 23561151367 \beta_{12} + 3735435140 \beta_{11} - 34045287094 \beta_{10} + 10352613110 \beta_{9} - 43029944932 \beta_{8} - 41520682456 \beta_{7} - 57464727814 \beta_{6} - 46105055481 \beta_{5} - 151119238347 \beta_{4} + 38999816063 \beta_{3} - 2484743454 \beta_{2} - 14479302604 \beta_{1} + 37434050762\)
\(\nu^{14}\)\(=\)\(67111931348 \beta_{19} - 50400346448 \beta_{18} - 61156315220 \beta_{17} + 247175087068 \beta_{16} + 178219432237 \beta_{15} - 158441331498 \beta_{14} + 4019914688 \beta_{13} - 75611435292 \beta_{12} + 8987440280 \beta_{11} + 242578280344 \beta_{10} - 137002913133 \beta_{9} + 323315978832 \beta_{8} + 478585788726 \beta_{7} + 456095497072 \beta_{6} + 155129182515 \beta_{5} + 571604318592 \beta_{4} - 429708452810 \beta_{3} - 81072791101 \beta_{2} + 134709421344 \beta_{1} - 1883906818388\)
\(\nu^{15}\)\(=\)\(-478148148233 \beta_{19} + 5843040736 \beta_{18} - 4148820705 \beta_{17} - 2138290857937 \beta_{16} - 982072045748 \beta_{15} + 730210382842 \beta_{14} + 30311048942 \beta_{13} + 1251128744555 \beta_{12} + 359819731890 \beta_{11} - 2059055043114 \beta_{10} + 1379993530746 \beta_{9} - 2631616237578 \beta_{8} - 5209399366668 \beta_{7} - 3662190581598 \beta_{6} - 4723058319875 \beta_{5} - 7990641491561 \beta_{4} + 2604957285729 \beta_{3} - 92230866020 \beta_{2} - 1788182253580 \beta_{1} + 7169280884976\)
\(\nu^{16}\)\(=\)\(3611012467360 \beta_{19} - 2086838653888 \beta_{18} - 2732907357952 \beta_{17} + 14399130551680 \beta_{16} + 17721292301927 \beta_{15} - 14989765340664 \beta_{14} + 132326104510 \beta_{13} - 5555083772960 \beta_{12} + 455680449218 \beta_{11} + 14078600855808 \beta_{10} - 14940285253805 \beta_{9} + 18600882290272 \beta_{8} + 54146776113490 \beta_{7} + 25979400195392 \beta_{6} + 22038723810737 \beta_{5} + 39981341159904 \beta_{4} - 22173074382960 \beta_{3} - 7244352072957 \beta_{2} + 15504106523778 \beta_{1} - 175295813267812\)
\(\nu^{17}\)\(=\)\(-22622149643317 \beta_{19} + 5160254586820 \beta_{18} + 7839490976085 \beta_{17} - 101513362710887 \beta_{16} - 107048356918598 \beta_{15} + 81901413941000 \beta_{14} + 4507501388780 \beta_{13} + 49715925451867 \beta_{12} + 29165151837768 \beta_{11} - 97684692915022 \beta_{10} + 141635314228462 \beta_{9} - 126468558449880 \beta_{8} - 522737688380016 \beta_{7} - 181737085173198 \beta_{6} - 417947329486353 \beta_{5} - 320554893267979 \beta_{4} + 137710913141415 \beta_{3} + 4040217694934 \beta_{2} - 176327636179188 \beta_{1} + 868159104771558\)
\(\nu^{18}\)\(=\)\(130657989007868 \beta_{19} - 33243943776992 \beta_{18} - 55766768971716 \beta_{17} + 584092890745836 \beta_{16} + 1535035139822973 \beta_{15} - 1262517147852094 \beta_{14} - 4225935236212 \beta_{13} - 295174377501156 \beta_{12} + 795706038468 \beta_{11} + 566900520486728 \beta_{10} - 1384491190844649 \beta_{9} + 738121140364216 \beta_{8} + 5088886578797046 \beta_{7} + 1018107038157600 \beta_{6} + 2354592927040595 \beta_{5} + 2007195062536568 \beta_{4} - 742315036548222 \beta_{3} - 569992058313257 \beta_{2} + 1483840998207188 \beta_{1} - 14602148259697708\)
\(\nu^{19}\)\(=\)\(-447769033388069 \beta_{19} + 492561905784744 \beta_{18} + 770621941544639 \beta_{17} - 1959381963676985 \beta_{16} - 10012300457668448 \beta_{15} + 7779402723366110 \beta_{14} + 424475348925562 \beta_{13} + 226232181731047 \beta_{12} + 2113808089162398 \beta_{11} - 1888761569117554 \beta_{10} + 12535713457468738 \beta_{9} - 2571229920050582 \beta_{8} - 45859720242426708 \beta_{7} - 4070921980391222 \beta_{6} - 33542732620996011 \beta_{5} - 1850210341337081 \beta_{4} + 3749442865882745 \beta_{3} + 1176591402274792 \beta_{2} - 15207549654619892 \beta_{1} + 85250807684934260\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/354\mathbb{Z}\right)^\times\).

\(n\) \(61\) \(119\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
−6.71849 + 1.41421i
−4.74052 + 1.41421i
0.383479 + 1.41421i
6.82349 + 1.41421i
7.71614 + 1.41421i
−8.79591 + 1.41421i
−2.72105 + 1.41421i
−0.662289 + 1.41421i
0.969662 + 1.41421i
7.74549 + 1.41421i
−6.71849 1.41421i
−4.74052 1.41421i
0.383479 1.41421i
6.82349 1.41421i
7.71614 1.41421i
−8.79591 1.41421i
−2.72105 1.41421i
−0.662289 1.41421i
0.969662 1.41421i
7.74549 1.41421i
1.41421i −1.73205 −2.00000 −6.71849 2.44949i −6.32195 2.82843i 3.00000 9.50137i
235.2 1.41421i −1.73205 −2.00000 −4.74052 2.44949i 12.4076 2.82843i 3.00000 6.70411i
235.3 1.41421i −1.73205 −2.00000 0.383479 2.44949i 1.10699 2.82843i 3.00000 0.542321i
235.4 1.41421i −1.73205 −2.00000 6.82349 2.44949i 2.37240 2.82843i 3.00000 9.64987i
235.5 1.41421i −1.73205 −2.00000 7.71614 2.44949i −7.56506 2.82843i 3.00000 10.9123i
235.6 1.41421i 1.73205 −2.00000 −8.79591 2.44949i 6.82395 2.82843i 3.00000 12.4393i
235.7 1.41421i 1.73205 −2.00000 −2.72105 2.44949i −3.73640 2.82843i 3.00000 3.84815i
235.8 1.41421i 1.73205 −2.00000 −0.662289 2.44949i −11.5463 2.82843i 3.00000 0.936618i
235.9 1.41421i 1.73205 −2.00000 0.969662 2.44949i 3.05407 2.82843i 3.00000 1.37131i
235.10 1.41421i 1.73205 −2.00000 7.74549 2.44949i 7.40466 2.82843i 3.00000 10.9538i
235.11 1.41421i −1.73205 −2.00000 −6.71849 2.44949i −6.32195 2.82843i 3.00000 9.50137i
235.12 1.41421i −1.73205 −2.00000 −4.74052 2.44949i 12.4076 2.82843i 3.00000 6.70411i
235.13 1.41421i −1.73205 −2.00000 0.383479 2.44949i 1.10699 2.82843i 3.00000 0.542321i
235.14 1.41421i −1.73205 −2.00000 6.82349 2.44949i 2.37240 2.82843i 3.00000 9.64987i
235.15 1.41421i −1.73205 −2.00000 7.71614 2.44949i −7.56506 2.82843i 3.00000 10.9123i
235.16 1.41421i 1.73205 −2.00000 −8.79591 2.44949i 6.82395 2.82843i 3.00000 12.4393i
235.17 1.41421i 1.73205 −2.00000 −2.72105 2.44949i −3.73640 2.82843i 3.00000 3.84815i
235.18 1.41421i 1.73205 −2.00000 −0.662289 2.44949i −11.5463 2.82843i 3.00000 0.936618i
235.19 1.41421i 1.73205 −2.00000 0.969662 2.44949i 3.05407 2.82843i 3.00000 1.37131i
235.20 1.41421i 1.73205 −2.00000 7.74549 2.44949i 7.40466 2.82843i 3.00000 10.9538i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.20
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
59.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(354, [\chi])\).