Properties

Label 7942.2.a.x.1.2
Level $7942$
Weight $2$
Character 7942.1
Self dual yes
Analytic conductor $63.417$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.4171892853\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 7942.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.56155 q^{6} +3.56155 q^{7} +1.00000 q^{8} -0.561553 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.56155 q^{6} +3.56155 q^{7} +1.00000 q^{8} -0.561553 q^{9} +2.00000 q^{10} +1.00000 q^{11} +1.56155 q^{12} +3.56155 q^{13} +3.56155 q^{14} +3.12311 q^{15} +1.00000 q^{16} +3.56155 q^{17} -0.561553 q^{18} +2.00000 q^{20} +5.56155 q^{21} +1.00000 q^{22} +5.56155 q^{23} +1.56155 q^{24} -1.00000 q^{25} +3.56155 q^{26} -5.56155 q^{27} +3.56155 q^{28} -6.68466 q^{29} +3.12311 q^{30} -2.00000 q^{31} +1.00000 q^{32} +1.56155 q^{33} +3.56155 q^{34} +7.12311 q^{35} -0.561553 q^{36} -3.12311 q^{37} +5.56155 q^{39} +2.00000 q^{40} -2.00000 q^{41} +5.56155 q^{42} +1.00000 q^{44} -1.12311 q^{45} +5.56155 q^{46} +8.00000 q^{47} +1.56155 q^{48} +5.68466 q^{49} -1.00000 q^{50} +5.56155 q^{51} +3.56155 q^{52} +7.80776 q^{53} -5.56155 q^{54} +2.00000 q^{55} +3.56155 q^{56} -6.68466 q^{58} -4.68466 q^{59} +3.12311 q^{60} -10.2462 q^{61} -2.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +7.12311 q^{65} +1.56155 q^{66} -4.68466 q^{67} +3.56155 q^{68} +8.68466 q^{69} +7.12311 q^{70} -6.00000 q^{71} -0.561553 q^{72} +2.68466 q^{73} -3.12311 q^{74} -1.56155 q^{75} +3.56155 q^{77} +5.56155 q^{78} +4.00000 q^{79} +2.00000 q^{80} -7.00000 q^{81} -2.00000 q^{82} -2.24621 q^{83} +5.56155 q^{84} +7.12311 q^{85} -10.4384 q^{87} +1.00000 q^{88} +9.12311 q^{89} -1.12311 q^{90} +12.6847 q^{91} +5.56155 q^{92} -3.12311 q^{93} +8.00000 q^{94} +1.56155 q^{96} -1.12311 q^{97} +5.68466 q^{98} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} + 2 q^{8} + 3 q^{9} + 4 q^{10} + 2 q^{11} - q^{12} + 3 q^{13} + 3 q^{14} - 2 q^{15} + 2 q^{16} + 3 q^{17} + 3 q^{18} + 4 q^{20} + 7 q^{21} + 2 q^{22} + 7 q^{23} - q^{24} - 2 q^{25} + 3 q^{26} - 7 q^{27} + 3 q^{28} - q^{29} - 2 q^{30} - 4 q^{31} + 2 q^{32} - q^{33} + 3 q^{34} + 6 q^{35} + 3 q^{36} + 2 q^{37} + 7 q^{39} + 4 q^{40} - 4 q^{41} + 7 q^{42} + 2 q^{44} + 6 q^{45} + 7 q^{46} + 16 q^{47} - q^{48} - q^{49} - 2 q^{50} + 7 q^{51} + 3 q^{52} - 5 q^{53} - 7 q^{54} + 4 q^{55} + 3 q^{56} - q^{58} + 3 q^{59} - 2 q^{60} - 4 q^{61} - 4 q^{62} - 4 q^{63} + 2 q^{64} + 6 q^{65} - q^{66} + 3 q^{67} + 3 q^{68} + 5 q^{69} + 6 q^{70} - 12 q^{71} + 3 q^{72} - 7 q^{73} + 2 q^{74} + q^{75} + 3 q^{77} + 7 q^{78} + 8 q^{79} + 4 q^{80} - 14 q^{81} - 4 q^{82} + 12 q^{83} + 7 q^{84} + 6 q^{85} - 25 q^{87} + 2 q^{88} + 10 q^{89} + 6 q^{90} + 13 q^{91} + 7 q^{92} + 2 q^{93} + 16 q^{94} - q^{96} + 6 q^{97} - q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.56155 0.901563 0.450781 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.56155 0.637501
\(7\) 3.56155 1.34614 0.673070 0.739579i \(-0.264975\pi\)
0.673070 + 0.739579i \(0.264975\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.561553 −0.187184
\(10\) 2.00000 0.632456
\(11\) 1.00000 0.301511
\(12\) 1.56155 0.450781
\(13\) 3.56155 0.987797 0.493899 0.869520i \(-0.335571\pi\)
0.493899 + 0.869520i \(0.335571\pi\)
\(14\) 3.56155 0.951865
\(15\) 3.12311 0.806382
\(16\) 1.00000 0.250000
\(17\) 3.56155 0.863803 0.431902 0.901921i \(-0.357843\pi\)
0.431902 + 0.901921i \(0.357843\pi\)
\(18\) −0.561553 −0.132359
\(19\) 0 0
\(20\) 2.00000 0.447214
\(21\) 5.56155 1.21363
\(22\) 1.00000 0.213201
\(23\) 5.56155 1.15966 0.579832 0.814736i \(-0.303118\pi\)
0.579832 + 0.814736i \(0.303118\pi\)
\(24\) 1.56155 0.318751
\(25\) −1.00000 −0.200000
\(26\) 3.56155 0.698478
\(27\) −5.56155 −1.07032
\(28\) 3.56155 0.673070
\(29\) −6.68466 −1.24131 −0.620655 0.784084i \(-0.713133\pi\)
−0.620655 + 0.784084i \(0.713133\pi\)
\(30\) 3.12311 0.570198
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.56155 0.271831
\(34\) 3.56155 0.610801
\(35\) 7.12311 1.20402
\(36\) −0.561553 −0.0935921
\(37\) −3.12311 −0.513435 −0.256718 0.966486i \(-0.582641\pi\)
−0.256718 + 0.966486i \(0.582641\pi\)
\(38\) 0 0
\(39\) 5.56155 0.890561
\(40\) 2.00000 0.316228
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 5.56155 0.858166
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.12311 −0.167423
\(46\) 5.56155 0.820006
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.56155 0.225391
\(49\) 5.68466 0.812094
\(50\) −1.00000 −0.141421
\(51\) 5.56155 0.778773
\(52\) 3.56155 0.493899
\(53\) 7.80776 1.07248 0.536239 0.844066i \(-0.319844\pi\)
0.536239 + 0.844066i \(0.319844\pi\)
\(54\) −5.56155 −0.756831
\(55\) 2.00000 0.269680
\(56\) 3.56155 0.475933
\(57\) 0 0
\(58\) −6.68466 −0.877739
\(59\) −4.68466 −0.609891 −0.304945 0.952370i \(-0.598638\pi\)
−0.304945 + 0.952370i \(0.598638\pi\)
\(60\) 3.12311 0.403191
\(61\) −10.2462 −1.31189 −0.655946 0.754807i \(-0.727730\pi\)
−0.655946 + 0.754807i \(0.727730\pi\)
\(62\) −2.00000 −0.254000
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 7.12311 0.883513
\(66\) 1.56155 0.192214
\(67\) −4.68466 −0.572322 −0.286161 0.958182i \(-0.592379\pi\)
−0.286161 + 0.958182i \(0.592379\pi\)
\(68\) 3.56155 0.431902
\(69\) 8.68466 1.04551
\(70\) 7.12311 0.851374
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −0.561553 −0.0661796
\(73\) 2.68466 0.314216 0.157108 0.987581i \(-0.449783\pi\)
0.157108 + 0.987581i \(0.449783\pi\)
\(74\) −3.12311 −0.363054
\(75\) −1.56155 −0.180313
\(76\) 0 0
\(77\) 3.56155 0.405877
\(78\) 5.56155 0.629722
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 2.00000 0.223607
\(81\) −7.00000 −0.777778
\(82\) −2.00000 −0.220863
\(83\) −2.24621 −0.246554 −0.123277 0.992372i \(-0.539340\pi\)
−0.123277 + 0.992372i \(0.539340\pi\)
\(84\) 5.56155 0.606815
\(85\) 7.12311 0.772609
\(86\) 0 0
\(87\) −10.4384 −1.11912
\(88\) 1.00000 0.106600
\(89\) 9.12311 0.967047 0.483524 0.875331i \(-0.339357\pi\)
0.483524 + 0.875331i \(0.339357\pi\)
\(90\) −1.12311 −0.118386
\(91\) 12.6847 1.32971
\(92\) 5.56155 0.579832
\(93\) −3.12311 −0.323851
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.56155 0.159375
\(97\) −1.12311 −0.114034 −0.0570170 0.998373i \(-0.518159\pi\)
−0.0570170 + 0.998373i \(0.518159\pi\)
\(98\) 5.68466 0.574237
\(99\) −0.561553 −0.0564382
\(100\) −1.00000 −0.100000
\(101\) −15.1231 −1.50481 −0.752403 0.658703i \(-0.771105\pi\)
−0.752403 + 0.658703i \(0.771105\pi\)
\(102\) 5.56155 0.550676
\(103\) −11.3693 −1.12025 −0.560126 0.828407i \(-0.689247\pi\)
−0.560126 + 0.828407i \(0.689247\pi\)
\(104\) 3.56155 0.349239
\(105\) 11.1231 1.08550
\(106\) 7.80776 0.758357
\(107\) −18.9309 −1.83012 −0.915058 0.403322i \(-0.867855\pi\)
−0.915058 + 0.403322i \(0.867855\pi\)
\(108\) −5.56155 −0.535161
\(109\) 18.6847 1.78967 0.894833 0.446401i \(-0.147295\pi\)
0.894833 + 0.446401i \(0.147295\pi\)
\(110\) 2.00000 0.190693
\(111\) −4.87689 −0.462894
\(112\) 3.56155 0.336535
\(113\) −5.12311 −0.481941 −0.240971 0.970532i \(-0.577466\pi\)
−0.240971 + 0.970532i \(0.577466\pi\)
\(114\) 0 0
\(115\) 11.1231 1.03723
\(116\) −6.68466 −0.620655
\(117\) −2.00000 −0.184900
\(118\) −4.68466 −0.431258
\(119\) 12.6847 1.16280
\(120\) 3.12311 0.285099
\(121\) 1.00000 0.0909091
\(122\) −10.2462 −0.927648
\(123\) −3.12311 −0.281601
\(124\) −2.00000 −0.179605
\(125\) −12.0000 −1.07331
\(126\) −2.00000 −0.178174
\(127\) 14.2462 1.26415 0.632073 0.774909i \(-0.282204\pi\)
0.632073 + 0.774909i \(0.282204\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 7.12311 0.624738
\(131\) −7.12311 −0.622349 −0.311174 0.950353i \(-0.600722\pi\)
−0.311174 + 0.950353i \(0.600722\pi\)
\(132\) 1.56155 0.135916
\(133\) 0 0
\(134\) −4.68466 −0.404693
\(135\) −11.1231 −0.957325
\(136\) 3.56155 0.305401
\(137\) 8.43845 0.720945 0.360473 0.932770i \(-0.382615\pi\)
0.360473 + 0.932770i \(0.382615\pi\)
\(138\) 8.68466 0.739287
\(139\) −17.3693 −1.47325 −0.736623 0.676303i \(-0.763581\pi\)
−0.736623 + 0.676303i \(0.763581\pi\)
\(140\) 7.12311 0.602012
\(141\) 12.4924 1.05205
\(142\) −6.00000 −0.503509
\(143\) 3.56155 0.297832
\(144\) −0.561553 −0.0467961
\(145\) −13.3693 −1.11026
\(146\) 2.68466 0.222184
\(147\) 8.87689 0.732154
\(148\) −3.12311 −0.256718
\(149\) −15.1231 −1.23893 −0.619467 0.785023i \(-0.712651\pi\)
−0.619467 + 0.785023i \(0.712651\pi\)
\(150\) −1.56155 −0.127500
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 3.56155 0.286998
\(155\) −4.00000 −0.321288
\(156\) 5.56155 0.445281
\(157\) 18.4924 1.47586 0.737928 0.674879i \(-0.235804\pi\)
0.737928 + 0.674879i \(0.235804\pi\)
\(158\) 4.00000 0.318223
\(159\) 12.1922 0.966907
\(160\) 2.00000 0.158114
\(161\) 19.8078 1.56107
\(162\) −7.00000 −0.549972
\(163\) −13.3693 −1.04717 −0.523583 0.851975i \(-0.675405\pi\)
−0.523583 + 0.851975i \(0.675405\pi\)
\(164\) −2.00000 −0.156174
\(165\) 3.12311 0.243133
\(166\) −2.24621 −0.174340
\(167\) 25.3693 1.96314 0.981568 0.191111i \(-0.0612092\pi\)
0.981568 + 0.191111i \(0.0612092\pi\)
\(168\) 5.56155 0.429083
\(169\) −0.315342 −0.0242570
\(170\) 7.12311 0.546317
\(171\) 0 0
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −10.4384 −0.791337
\(175\) −3.56155 −0.269228
\(176\) 1.00000 0.0753778
\(177\) −7.31534 −0.549855
\(178\) 9.12311 0.683806
\(179\) −22.2462 −1.66276 −0.831380 0.555704i \(-0.812449\pi\)
−0.831380 + 0.555704i \(0.812449\pi\)
\(180\) −1.12311 −0.0837114
\(181\) 19.1231 1.42141 0.710705 0.703491i \(-0.248376\pi\)
0.710705 + 0.703491i \(0.248376\pi\)
\(182\) 12.6847 0.940249
\(183\) −16.0000 −1.18275
\(184\) 5.56155 0.410003
\(185\) −6.24621 −0.459231
\(186\) −3.12311 −0.228997
\(187\) 3.56155 0.260447
\(188\) 8.00000 0.583460
\(189\) −19.8078 −1.44080
\(190\) 0 0
\(191\) −20.6847 −1.49669 −0.748345 0.663310i \(-0.769151\pi\)
−0.748345 + 0.663310i \(0.769151\pi\)
\(192\) 1.56155 0.112695
\(193\) 1.12311 0.0808429 0.0404215 0.999183i \(-0.487130\pi\)
0.0404215 + 0.999183i \(0.487130\pi\)
\(194\) −1.12311 −0.0806343
\(195\) 11.1231 0.796542
\(196\) 5.68466 0.406047
\(197\) 1.75379 0.124952 0.0624761 0.998046i \(-0.480100\pi\)
0.0624761 + 0.998046i \(0.480100\pi\)
\(198\) −0.561553 −0.0399078
\(199\) −0.684658 −0.0485341 −0.0242671 0.999706i \(-0.507725\pi\)
−0.0242671 + 0.999706i \(0.507725\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −7.31534 −0.515984
\(202\) −15.1231 −1.06406
\(203\) −23.8078 −1.67098
\(204\) 5.56155 0.389387
\(205\) −4.00000 −0.279372
\(206\) −11.3693 −0.792138
\(207\) −3.12311 −0.217071
\(208\) 3.56155 0.246949
\(209\) 0 0
\(210\) 11.1231 0.767567
\(211\) 12.6847 0.873248 0.436624 0.899644i \(-0.356174\pi\)
0.436624 + 0.899644i \(0.356174\pi\)
\(212\) 7.80776 0.536239
\(213\) −9.36932 −0.641975
\(214\) −18.9309 −1.29409
\(215\) 0 0
\(216\) −5.56155 −0.378416
\(217\) −7.12311 −0.483548
\(218\) 18.6847 1.26548
\(219\) 4.19224 0.283285
\(220\) 2.00000 0.134840
\(221\) 12.6847 0.853262
\(222\) −4.87689 −0.327316
\(223\) 24.7386 1.65662 0.828311 0.560269i \(-0.189302\pi\)
0.828311 + 0.560269i \(0.189302\pi\)
\(224\) 3.56155 0.237966
\(225\) 0.561553 0.0374369
\(226\) −5.12311 −0.340784
\(227\) 17.1771 1.14008 0.570041 0.821616i \(-0.306927\pi\)
0.570041 + 0.821616i \(0.306927\pi\)
\(228\) 0 0
\(229\) 21.1231 1.39585 0.697927 0.716169i \(-0.254106\pi\)
0.697927 + 0.716169i \(0.254106\pi\)
\(230\) 11.1231 0.733436
\(231\) 5.56155 0.365923
\(232\) −6.68466 −0.438869
\(233\) 20.7386 1.35863 0.679317 0.733845i \(-0.262276\pi\)
0.679317 + 0.733845i \(0.262276\pi\)
\(234\) −2.00000 −0.130744
\(235\) 16.0000 1.04372
\(236\) −4.68466 −0.304945
\(237\) 6.24621 0.405735
\(238\) 12.6847 0.822224
\(239\) −4.93087 −0.318951 −0.159476 0.987202i \(-0.550980\pi\)
−0.159476 + 0.987202i \(0.550980\pi\)
\(240\) 3.12311 0.201596
\(241\) −21.6155 −1.39238 −0.696189 0.717858i \(-0.745123\pi\)
−0.696189 + 0.717858i \(0.745123\pi\)
\(242\) 1.00000 0.0642824
\(243\) 5.75379 0.369106
\(244\) −10.2462 −0.655946
\(245\) 11.3693 0.726359
\(246\) −3.12311 −0.199122
\(247\) 0 0
\(248\) −2.00000 −0.127000
\(249\) −3.50758 −0.222284
\(250\) −12.0000 −0.758947
\(251\) 8.87689 0.560305 0.280152 0.959956i \(-0.409615\pi\)
0.280152 + 0.959956i \(0.409615\pi\)
\(252\) −2.00000 −0.125988
\(253\) 5.56155 0.349652
\(254\) 14.2462 0.893887
\(255\) 11.1231 0.696556
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) −11.1231 −0.691156
\(260\) 7.12311 0.441756
\(261\) 3.75379 0.232354
\(262\) −7.12311 −0.440067
\(263\) 17.1231 1.05586 0.527928 0.849289i \(-0.322969\pi\)
0.527928 + 0.849289i \(0.322969\pi\)
\(264\) 1.56155 0.0961069
\(265\) 15.6155 0.959254
\(266\) 0 0
\(267\) 14.2462 0.871854
\(268\) −4.68466 −0.286161
\(269\) −11.1231 −0.678188 −0.339094 0.940753i \(-0.610120\pi\)
−0.339094 + 0.940753i \(0.610120\pi\)
\(270\) −11.1231 −0.676931
\(271\) 13.3153 0.808849 0.404425 0.914571i \(-0.367472\pi\)
0.404425 + 0.914571i \(0.367472\pi\)
\(272\) 3.56155 0.215951
\(273\) 19.8078 1.19882
\(274\) 8.43845 0.509785
\(275\) −1.00000 −0.0603023
\(276\) 8.68466 0.522755
\(277\) −24.4924 −1.47161 −0.735804 0.677195i \(-0.763195\pi\)
−0.735804 + 0.677195i \(0.763195\pi\)
\(278\) −17.3693 −1.04174
\(279\) 1.12311 0.0672386
\(280\) 7.12311 0.425687
\(281\) 0.630683 0.0376234 0.0188117 0.999823i \(-0.494012\pi\)
0.0188117 + 0.999823i \(0.494012\pi\)
\(282\) 12.4924 0.743913
\(283\) −13.3693 −0.794723 −0.397362 0.917662i \(-0.630074\pi\)
−0.397362 + 0.917662i \(0.630074\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 3.56155 0.210599
\(287\) −7.12311 −0.420464
\(288\) −0.561553 −0.0330898
\(289\) −4.31534 −0.253844
\(290\) −13.3693 −0.785073
\(291\) −1.75379 −0.102809
\(292\) 2.68466 0.157108
\(293\) 24.9309 1.45648 0.728238 0.685324i \(-0.240339\pi\)
0.728238 + 0.685324i \(0.240339\pi\)
\(294\) 8.87689 0.517711
\(295\) −9.36932 −0.545503
\(296\) −3.12311 −0.181527
\(297\) −5.56155 −0.322714
\(298\) −15.1231 −0.876058
\(299\) 19.8078 1.14551
\(300\) −1.56155 −0.0901563
\(301\) 0 0
\(302\) −4.00000 −0.230174
\(303\) −23.6155 −1.35668
\(304\) 0 0
\(305\) −20.4924 −1.17339
\(306\) −2.00000 −0.114332
\(307\) 5.75379 0.328386 0.164193 0.986428i \(-0.447498\pi\)
0.164193 + 0.986428i \(0.447498\pi\)
\(308\) 3.56155 0.202938
\(309\) −17.7538 −1.00998
\(310\) −4.00000 −0.227185
\(311\) 15.8078 0.896376 0.448188 0.893939i \(-0.352070\pi\)
0.448188 + 0.893939i \(0.352070\pi\)
\(312\) 5.56155 0.314861
\(313\) 3.56155 0.201311 0.100655 0.994921i \(-0.467906\pi\)
0.100655 + 0.994921i \(0.467906\pi\)
\(314\) 18.4924 1.04359
\(315\) −4.00000 −0.225374
\(316\) 4.00000 0.225018
\(317\) −30.0540 −1.68800 −0.844000 0.536344i \(-0.819805\pi\)
−0.844000 + 0.536344i \(0.819805\pi\)
\(318\) 12.1922 0.683707
\(319\) −6.68466 −0.374269
\(320\) 2.00000 0.111803
\(321\) −29.5616 −1.64996
\(322\) 19.8078 1.10384
\(323\) 0 0
\(324\) −7.00000 −0.388889
\(325\) −3.56155 −0.197559
\(326\) −13.3693 −0.740458
\(327\) 29.1771 1.61350
\(328\) −2.00000 −0.110432
\(329\) 28.4924 1.57084
\(330\) 3.12311 0.171921
\(331\) 10.4384 0.573749 0.286874 0.957968i \(-0.407384\pi\)
0.286874 + 0.957968i \(0.407384\pi\)
\(332\) −2.24621 −0.123277
\(333\) 1.75379 0.0961070
\(334\) 25.3693 1.38815
\(335\) −9.36932 −0.511900
\(336\) 5.56155 0.303408
\(337\) −2.87689 −0.156714 −0.0783572 0.996925i \(-0.524967\pi\)
−0.0783572 + 0.996925i \(0.524967\pi\)
\(338\) −0.315342 −0.0171523
\(339\) −8.00000 −0.434500
\(340\) 7.12311 0.386305
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) −4.68466 −0.252948
\(344\) 0 0
\(345\) 17.3693 0.935133
\(346\) −18.0000 −0.967686
\(347\) −31.6155 −1.69721 −0.848605 0.529027i \(-0.822557\pi\)
−0.848605 + 0.529027i \(0.822557\pi\)
\(348\) −10.4384 −0.559560
\(349\) −19.6155 −1.05000 −0.524998 0.851104i \(-0.675934\pi\)
−0.524998 + 0.851104i \(0.675934\pi\)
\(350\) −3.56155 −0.190373
\(351\) −19.8078 −1.05726
\(352\) 1.00000 0.0533002
\(353\) −23.1771 −1.23359 −0.616796 0.787123i \(-0.711570\pi\)
−0.616796 + 0.787123i \(0.711570\pi\)
\(354\) −7.31534 −0.388806
\(355\) −12.0000 −0.636894
\(356\) 9.12311 0.483524
\(357\) 19.8078 1.04834
\(358\) −22.2462 −1.17575
\(359\) 12.9309 0.682465 0.341233 0.939979i \(-0.389156\pi\)
0.341233 + 0.939979i \(0.389156\pi\)
\(360\) −1.12311 −0.0591929
\(361\) 0 0
\(362\) 19.1231 1.00509
\(363\) 1.56155 0.0819603
\(364\) 12.6847 0.664857
\(365\) 5.36932 0.281043
\(366\) −16.0000 −0.836333
\(367\) 13.7538 0.717942 0.358971 0.933349i \(-0.383128\pi\)
0.358971 + 0.933349i \(0.383128\pi\)
\(368\) 5.56155 0.289916
\(369\) 1.12311 0.0584665
\(370\) −6.24621 −0.324725
\(371\) 27.8078 1.44371
\(372\) −3.12311 −0.161925
\(373\) −7.56155 −0.391522 −0.195761 0.980652i \(-0.562718\pi\)
−0.195761 + 0.980652i \(0.562718\pi\)
\(374\) 3.56155 0.184164
\(375\) −18.7386 −0.967659
\(376\) 8.00000 0.412568
\(377\) −23.8078 −1.22616
\(378\) −19.8078 −1.01880
\(379\) −2.43845 −0.125255 −0.0626273 0.998037i \(-0.519948\pi\)
−0.0626273 + 0.998037i \(0.519948\pi\)
\(380\) 0 0
\(381\) 22.2462 1.13971
\(382\) −20.6847 −1.05832
\(383\) 2.00000 0.102195 0.0510976 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) 1.56155 0.0796877
\(385\) 7.12311 0.363027
\(386\) 1.12311 0.0571646
\(387\) 0 0
\(388\) −1.12311 −0.0570170
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 11.1231 0.563240
\(391\) 19.8078 1.00172
\(392\) 5.68466 0.287119
\(393\) −11.1231 −0.561086
\(394\) 1.75379 0.0883546
\(395\) 8.00000 0.402524
\(396\) −0.561553 −0.0282191
\(397\) −26.9848 −1.35433 −0.677165 0.735831i \(-0.736792\pi\)
−0.677165 + 0.735831i \(0.736792\pi\)
\(398\) −0.684658 −0.0343188
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 24.2462 1.21080 0.605399 0.795922i \(-0.293014\pi\)
0.605399 + 0.795922i \(0.293014\pi\)
\(402\) −7.31534 −0.364856
\(403\) −7.12311 −0.354827
\(404\) −15.1231 −0.752403
\(405\) −14.0000 −0.695666
\(406\) −23.8078 −1.18156
\(407\) −3.12311 −0.154807
\(408\) 5.56155 0.275338
\(409\) 0.246211 0.0121744 0.00608718 0.999981i \(-0.498062\pi\)
0.00608718 + 0.999981i \(0.498062\pi\)
\(410\) −4.00000 −0.197546
\(411\) 13.1771 0.649977
\(412\) −11.3693 −0.560126
\(413\) −16.6847 −0.820998
\(414\) −3.12311 −0.153492
\(415\) −4.49242 −0.220524
\(416\) 3.56155 0.174619
\(417\) −27.1231 −1.32822
\(418\) 0 0
\(419\) −23.1231 −1.12964 −0.564819 0.825215i \(-0.691054\pi\)
−0.564819 + 0.825215i \(0.691054\pi\)
\(420\) 11.1231 0.542752
\(421\) 1.06913 0.0521062 0.0260531 0.999661i \(-0.491706\pi\)
0.0260531 + 0.999661i \(0.491706\pi\)
\(422\) 12.6847 0.617480
\(423\) −4.49242 −0.218429
\(424\) 7.80776 0.379179
\(425\) −3.56155 −0.172761
\(426\) −9.36932 −0.453945
\(427\) −36.4924 −1.76599
\(428\) −18.9309 −0.915058
\(429\) 5.56155 0.268514
\(430\) 0 0
\(431\) 25.8617 1.24572 0.622858 0.782335i \(-0.285971\pi\)
0.622858 + 0.782335i \(0.285971\pi\)
\(432\) −5.56155 −0.267580
\(433\) −31.3693 −1.50751 −0.753757 0.657154i \(-0.771760\pi\)
−0.753757 + 0.657154i \(0.771760\pi\)
\(434\) −7.12311 −0.341920
\(435\) −20.8769 −1.00097
\(436\) 18.6847 0.894833
\(437\) 0 0
\(438\) 4.19224 0.200313
\(439\) −31.6155 −1.50893 −0.754463 0.656342i \(-0.772103\pi\)
−0.754463 + 0.656342i \(0.772103\pi\)
\(440\) 2.00000 0.0953463
\(441\) −3.19224 −0.152011
\(442\) 12.6847 0.603348
\(443\) −17.8617 −0.848637 −0.424318 0.905513i \(-0.639486\pi\)
−0.424318 + 0.905513i \(0.639486\pi\)
\(444\) −4.87689 −0.231447
\(445\) 18.2462 0.864953
\(446\) 24.7386 1.17141
\(447\) −23.6155 −1.11698
\(448\) 3.56155 0.168268
\(449\) 17.6155 0.831328 0.415664 0.909518i \(-0.363549\pi\)
0.415664 + 0.909518i \(0.363549\pi\)
\(450\) 0.561553 0.0264719
\(451\) −2.00000 −0.0941763
\(452\) −5.12311 −0.240971
\(453\) −6.24621 −0.293473
\(454\) 17.1771 0.806160
\(455\) 25.3693 1.18933
\(456\) 0 0
\(457\) 32.4384 1.51741 0.758703 0.651436i \(-0.225833\pi\)
0.758703 + 0.651436i \(0.225833\pi\)
\(458\) 21.1231 0.987018
\(459\) −19.8078 −0.924547
\(460\) 11.1231 0.518617
\(461\) −22.7386 −1.05904 −0.529522 0.848296i \(-0.677629\pi\)
−0.529522 + 0.848296i \(0.677629\pi\)
\(462\) 5.56155 0.258747
\(463\) 36.4924 1.69595 0.847973 0.530039i \(-0.177823\pi\)
0.847973 + 0.530039i \(0.177823\pi\)
\(464\) −6.68466 −0.310327
\(465\) −6.24621 −0.289661
\(466\) 20.7386 0.960699
\(467\) 26.2462 1.21453 0.607265 0.794499i \(-0.292267\pi\)
0.607265 + 0.794499i \(0.292267\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −16.6847 −0.770426
\(470\) 16.0000 0.738025
\(471\) 28.8769 1.33058
\(472\) −4.68466 −0.215629
\(473\) 0 0
\(474\) 6.24621 0.286898
\(475\) 0 0
\(476\) 12.6847 0.581400
\(477\) −4.38447 −0.200751
\(478\) −4.93087 −0.225533
\(479\) 11.3693 0.519477 0.259739 0.965679i \(-0.416364\pi\)
0.259739 + 0.965679i \(0.416364\pi\)
\(480\) 3.12311 0.142550
\(481\) −11.1231 −0.507170
\(482\) −21.6155 −0.984560
\(483\) 30.9309 1.40740
\(484\) 1.00000 0.0454545
\(485\) −2.24621 −0.101995
\(486\) 5.75379 0.260997
\(487\) −22.8769 −1.03665 −0.518326 0.855183i \(-0.673444\pi\)
−0.518326 + 0.855183i \(0.673444\pi\)
\(488\) −10.2462 −0.463824
\(489\) −20.8769 −0.944086
\(490\) 11.3693 0.513613
\(491\) −14.6307 −0.660273 −0.330137 0.943933i \(-0.607095\pi\)
−0.330137 + 0.943933i \(0.607095\pi\)
\(492\) −3.12311 −0.140800
\(493\) −23.8078 −1.07225
\(494\) 0 0
\(495\) −1.12311 −0.0504798
\(496\) −2.00000 −0.0898027
\(497\) −21.3693 −0.958545
\(498\) −3.50758 −0.157178
\(499\) 26.2462 1.17494 0.587471 0.809245i \(-0.300124\pi\)
0.587471 + 0.809245i \(0.300124\pi\)
\(500\) −12.0000 −0.536656
\(501\) 39.6155 1.76989
\(502\) 8.87689 0.396195
\(503\) 9.31534 0.415351 0.207675 0.978198i \(-0.433410\pi\)
0.207675 + 0.978198i \(0.433410\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −30.2462 −1.34594
\(506\) 5.56155 0.247241
\(507\) −0.492423 −0.0218693
\(508\) 14.2462 0.632073
\(509\) 6.63068 0.293900 0.146950 0.989144i \(-0.453054\pi\)
0.146950 + 0.989144i \(0.453054\pi\)
\(510\) 11.1231 0.492539
\(511\) 9.56155 0.422978
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) −22.7386 −1.00198
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) −11.1231 −0.488721
\(519\) −28.1080 −1.23380
\(520\) 7.12311 0.312369
\(521\) −1.12311 −0.0492042 −0.0246021 0.999697i \(-0.507832\pi\)
−0.0246021 + 0.999697i \(0.507832\pi\)
\(522\) 3.75379 0.164299
\(523\) 6.05398 0.264722 0.132361 0.991202i \(-0.457744\pi\)
0.132361 + 0.991202i \(0.457744\pi\)
\(524\) −7.12311 −0.311174
\(525\) −5.56155 −0.242726
\(526\) 17.1231 0.746603
\(527\) −7.12311 −0.310287
\(528\) 1.56155 0.0679579
\(529\) 7.93087 0.344820
\(530\) 15.6155 0.678295
\(531\) 2.63068 0.114162
\(532\) 0 0
\(533\) −7.12311 −0.308536
\(534\) 14.2462 0.616494
\(535\) −37.8617 −1.63691
\(536\) −4.68466 −0.202346
\(537\) −34.7386 −1.49908
\(538\) −11.1231 −0.479551
\(539\) 5.68466 0.244856
\(540\) −11.1231 −0.478662
\(541\) 43.2311 1.85865 0.929324 0.369265i \(-0.120391\pi\)
0.929324 + 0.369265i \(0.120391\pi\)
\(542\) 13.3153 0.571943
\(543\) 29.8617 1.28149
\(544\) 3.56155 0.152700
\(545\) 37.3693 1.60073
\(546\) 19.8078 0.847694
\(547\) −6.73863 −0.288123 −0.144062 0.989569i \(-0.546016\pi\)
−0.144062 + 0.989569i \(0.546016\pi\)
\(548\) 8.43845 0.360473
\(549\) 5.75379 0.245566
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) 8.68466 0.369644
\(553\) 14.2462 0.605811
\(554\) −24.4924 −1.04058
\(555\) −9.75379 −0.414025
\(556\) −17.3693 −0.736623
\(557\) 8.87689 0.376126 0.188063 0.982157i \(-0.439779\pi\)
0.188063 + 0.982157i \(0.439779\pi\)
\(558\) 1.12311 0.0475449
\(559\) 0 0
\(560\) 7.12311 0.301006
\(561\) 5.56155 0.234809
\(562\) 0.630683 0.0266038
\(563\) −6.73863 −0.284000 −0.142000 0.989867i \(-0.545353\pi\)
−0.142000 + 0.989867i \(0.545353\pi\)
\(564\) 12.4924 0.526026
\(565\) −10.2462 −0.431061
\(566\) −13.3693 −0.561954
\(567\) −24.9309 −1.04700
\(568\) −6.00000 −0.251754
\(569\) −3.36932 −0.141249 −0.0706246 0.997503i \(-0.522499\pi\)
−0.0706246 + 0.997503i \(0.522499\pi\)
\(570\) 0 0
\(571\) 35.6155 1.49046 0.745232 0.666806i \(-0.232339\pi\)
0.745232 + 0.666806i \(0.232339\pi\)
\(572\) 3.56155 0.148916
\(573\) −32.3002 −1.34936
\(574\) −7.12311 −0.297313
\(575\) −5.56155 −0.231933
\(576\) −0.561553 −0.0233980
\(577\) 3.94602 0.164275 0.0821376 0.996621i \(-0.473825\pi\)
0.0821376 + 0.996621i \(0.473825\pi\)
\(578\) −4.31534 −0.179495
\(579\) 1.75379 0.0728850
\(580\) −13.3693 −0.555131
\(581\) −8.00000 −0.331896
\(582\) −1.75379 −0.0726969
\(583\) 7.80776 0.323365
\(584\) 2.68466 0.111092
\(585\) −4.00000 −0.165380
\(586\) 24.9309 1.02988
\(587\) −7.50758 −0.309871 −0.154935 0.987925i \(-0.549517\pi\)
−0.154935 + 0.987925i \(0.549517\pi\)
\(588\) 8.87689 0.366077
\(589\) 0 0
\(590\) −9.36932 −0.385729
\(591\) 2.73863 0.112652
\(592\) −3.12311 −0.128359
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) −5.56155 −0.228193
\(595\) 25.3693 1.04004
\(596\) −15.1231 −0.619467
\(597\) −1.06913 −0.0437566
\(598\) 19.8078 0.810000
\(599\) −39.8617 −1.62871 −0.814353 0.580370i \(-0.802908\pi\)
−0.814353 + 0.580370i \(0.802908\pi\)
\(600\) −1.56155 −0.0637501
\(601\) 3.36932 0.137437 0.0687187 0.997636i \(-0.478109\pi\)
0.0687187 + 0.997636i \(0.478109\pi\)
\(602\) 0 0
\(603\) 2.63068 0.107130
\(604\) −4.00000 −0.162758
\(605\) 2.00000 0.0813116
\(606\) −23.6155 −0.959315
\(607\) −27.6155 −1.12088 −0.560440 0.828195i \(-0.689368\pi\)
−0.560440 + 0.828195i \(0.689368\pi\)
\(608\) 0 0
\(609\) −37.1771 −1.50649
\(610\) −20.4924 −0.829714
\(611\) 28.4924 1.15268
\(612\) −2.00000 −0.0808452
\(613\) 1.75379 0.0708349 0.0354174 0.999373i \(-0.488724\pi\)
0.0354174 + 0.999373i \(0.488724\pi\)
\(614\) 5.75379 0.232204
\(615\) −6.24621 −0.251872
\(616\) 3.56155 0.143499
\(617\) −4.73863 −0.190770 −0.0953851 0.995440i \(-0.530408\pi\)
−0.0953851 + 0.995440i \(0.530408\pi\)
\(618\) −17.7538 −0.714162
\(619\) −26.2462 −1.05492 −0.527462 0.849579i \(-0.676856\pi\)
−0.527462 + 0.849579i \(0.676856\pi\)
\(620\) −4.00000 −0.160644
\(621\) −30.9309 −1.24121
\(622\) 15.8078 0.633834
\(623\) 32.4924 1.30178
\(624\) 5.56155 0.222640
\(625\) −19.0000 −0.760000
\(626\) 3.56155 0.142348
\(627\) 0 0
\(628\) 18.4924 0.737928
\(629\) −11.1231 −0.443507
\(630\) −4.00000 −0.159364
\(631\) 36.4924 1.45274 0.726370 0.687304i \(-0.241206\pi\)
0.726370 + 0.687304i \(0.241206\pi\)
\(632\) 4.00000 0.159111
\(633\) 19.8078 0.787288
\(634\) −30.0540 −1.19360
\(635\) 28.4924 1.13069
\(636\) 12.1922 0.483454
\(637\) 20.2462 0.802184
\(638\) −6.68466 −0.264648
\(639\) 3.36932 0.133288
\(640\) 2.00000 0.0790569
\(641\) 9.61553 0.379791 0.189895 0.981804i \(-0.439185\pi\)
0.189895 + 0.981804i \(0.439185\pi\)
\(642\) −29.5616 −1.16670
\(643\) −37.3693 −1.47370 −0.736851 0.676055i \(-0.763688\pi\)
−0.736851 + 0.676055i \(0.763688\pi\)
\(644\) 19.8078 0.780535
\(645\) 0 0
\(646\) 0 0
\(647\) 22.5464 0.886390 0.443195 0.896425i \(-0.353845\pi\)
0.443195 + 0.896425i \(0.353845\pi\)
\(648\) −7.00000 −0.274986
\(649\) −4.68466 −0.183889
\(650\) −3.56155 −0.139696
\(651\) −11.1231 −0.435949
\(652\) −13.3693 −0.523583
\(653\) 25.6155 1.00241 0.501207 0.865328i \(-0.332890\pi\)
0.501207 + 0.865328i \(0.332890\pi\)
\(654\) 29.1771 1.14091
\(655\) −14.2462 −0.556646
\(656\) −2.00000 −0.0780869
\(657\) −1.50758 −0.0588162
\(658\) 28.4924 1.11075
\(659\) −47.4233 −1.84735 −0.923675 0.383178i \(-0.874830\pi\)
−0.923675 + 0.383178i \(0.874830\pi\)
\(660\) 3.12311 0.121567
\(661\) 22.4384 0.872754 0.436377 0.899764i \(-0.356261\pi\)
0.436377 + 0.899764i \(0.356261\pi\)
\(662\) 10.4384 0.405702
\(663\) 19.8078 0.769270
\(664\) −2.24621 −0.0871699
\(665\) 0 0
\(666\) 1.75379 0.0679579
\(667\) −37.1771 −1.43950
\(668\) 25.3693 0.981568
\(669\) 38.6307 1.49355
\(670\) −9.36932 −0.361968
\(671\) −10.2462 −0.395551
\(672\) 5.56155 0.214542
\(673\) 18.8769 0.727651 0.363825 0.931467i \(-0.381470\pi\)
0.363825 + 0.931467i \(0.381470\pi\)
\(674\) −2.87689 −0.110814
\(675\) 5.56155 0.214064
\(676\) −0.315342 −0.0121285
\(677\) 39.1771 1.50570 0.752849 0.658194i \(-0.228679\pi\)
0.752849 + 0.658194i \(0.228679\pi\)
\(678\) −8.00000 −0.307238
\(679\) −4.00000 −0.153506
\(680\) 7.12311 0.273159
\(681\) 26.8229 1.02786
\(682\) −2.00000 −0.0765840
\(683\) 4.49242 0.171898 0.0859489 0.996300i \(-0.472608\pi\)
0.0859489 + 0.996300i \(0.472608\pi\)
\(684\) 0 0
\(685\) 16.8769 0.644833
\(686\) −4.68466 −0.178861
\(687\) 32.9848 1.25845
\(688\) 0 0
\(689\) 27.8078 1.05939
\(690\) 17.3693 0.661239
\(691\) −43.6155 −1.65921 −0.829606 0.558349i \(-0.811435\pi\)
−0.829606 + 0.558349i \(0.811435\pi\)
\(692\) −18.0000 −0.684257
\(693\) −2.00000 −0.0759737
\(694\) −31.6155 −1.20011
\(695\) −34.7386 −1.31771
\(696\) −10.4384 −0.395668
\(697\) −7.12311 −0.269807
\(698\) −19.6155 −0.742459
\(699\) 32.3845 1.22489
\(700\) −3.56155 −0.134614
\(701\) 24.9848 0.943665 0.471832 0.881688i \(-0.343593\pi\)
0.471832 + 0.881688i \(0.343593\pi\)
\(702\) −19.8078 −0.747596
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) 24.9848 0.940984
\(706\) −23.1771 −0.872281
\(707\) −53.8617 −2.02568
\(708\) −7.31534 −0.274927
\(709\) −9.50758 −0.357065 −0.178532 0.983934i \(-0.557135\pi\)
−0.178532 + 0.983934i \(0.557135\pi\)
\(710\) −12.0000 −0.450352
\(711\) −2.24621 −0.0842395
\(712\) 9.12311 0.341903
\(713\) −11.1231 −0.416564
\(714\) 19.8078 0.741287
\(715\) 7.12311 0.266389
\(716\) −22.2462 −0.831380
\(717\) −7.69981 −0.287555
\(718\) 12.9309 0.482576
\(719\) −6.43845 −0.240114 −0.120057 0.992767i \(-0.538308\pi\)
−0.120057 + 0.992767i \(0.538308\pi\)
\(720\) −1.12311 −0.0418557
\(721\) −40.4924 −1.50802
\(722\) 0 0
\(723\) −33.7538 −1.25532
\(724\) 19.1231 0.710705
\(725\) 6.68466 0.248262
\(726\) 1.56155 0.0579547
\(727\) −39.4233 −1.46213 −0.731064 0.682308i \(-0.760976\pi\)
−0.731064 + 0.682308i \(0.760976\pi\)
\(728\) 12.6847 0.470125
\(729\) 29.9848 1.11055
\(730\) 5.36932 0.198727
\(731\) 0 0
\(732\) −16.0000 −0.591377
\(733\) −48.1080 −1.77691 −0.888454 0.458966i \(-0.848220\pi\)
−0.888454 + 0.458966i \(0.848220\pi\)
\(734\) 13.7538 0.507662
\(735\) 17.7538 0.654858
\(736\) 5.56155 0.205002
\(737\) −4.68466 −0.172562
\(738\) 1.12311 0.0413421
\(739\) 20.9848 0.771940 0.385970 0.922511i \(-0.373867\pi\)
0.385970 + 0.922511i \(0.373867\pi\)
\(740\) −6.24621 −0.229615
\(741\) 0 0
\(742\) 27.8078 1.02086
\(743\) 25.7538 0.944815 0.472407 0.881380i \(-0.343385\pi\)
0.472407 + 0.881380i \(0.343385\pi\)
\(744\) −3.12311 −0.114499
\(745\) −30.2462 −1.10814
\(746\) −7.56155 −0.276848
\(747\) 1.26137 0.0461510
\(748\) 3.56155 0.130223
\(749\) −67.4233 −2.46359
\(750\) −18.7386 −0.684238
\(751\) 41.2311 1.50454 0.752271 0.658853i \(-0.228958\pi\)
0.752271 + 0.658853i \(0.228958\pi\)
\(752\) 8.00000 0.291730
\(753\) 13.8617 0.505150
\(754\) −23.8078 −0.867028
\(755\) −8.00000 −0.291150
\(756\) −19.8078 −0.720401
\(757\) −9.50758 −0.345559 −0.172779 0.984961i \(-0.555275\pi\)
−0.172779 + 0.984961i \(0.555275\pi\)
\(758\) −2.43845 −0.0885684
\(759\) 8.68466 0.315233
\(760\) 0 0
\(761\) −22.1922 −0.804468 −0.402234 0.915537i \(-0.631766\pi\)
−0.402234 + 0.915537i \(0.631766\pi\)
\(762\) 22.2462 0.805895
\(763\) 66.5464 2.40914
\(764\) −20.6847 −0.748345
\(765\) −4.00000 −0.144620
\(766\) 2.00000 0.0722629
\(767\) −16.6847 −0.602448
\(768\) 1.56155 0.0563477
\(769\) −1.31534 −0.0474324 −0.0237162 0.999719i \(-0.507550\pi\)
−0.0237162 + 0.999719i \(0.507550\pi\)
\(770\) 7.12311 0.256699
\(771\) 34.3542 1.23723
\(772\) 1.12311 0.0404215
\(773\) −9.06913 −0.326194 −0.163097 0.986610i \(-0.552148\pi\)
−0.163097 + 0.986610i \(0.552148\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) −1.12311 −0.0403171
\(777\) −17.3693 −0.623121
\(778\) 18.0000 0.645331
\(779\) 0 0
\(780\) 11.1231 0.398271
\(781\) −6.00000 −0.214697
\(782\) 19.8078 0.708324
\(783\) 37.1771 1.32860
\(784\) 5.68466 0.203024
\(785\) 36.9848 1.32005
\(786\) −11.1231 −0.396748
\(787\) −3.31534 −0.118179 −0.0590896 0.998253i \(-0.518820\pi\)
−0.0590896 + 0.998253i \(0.518820\pi\)
\(788\) 1.75379 0.0624761
\(789\) 26.7386 0.951921
\(790\) 8.00000 0.284627
\(791\) −18.2462 −0.648761
\(792\) −0.561553 −0.0199539
\(793\) −36.4924 −1.29588
\(794\) −26.9848 −0.957656
\(795\) 24.3845 0.864828
\(796\) −0.684658 −0.0242671
\(797\) 35.8078 1.26838 0.634188 0.773179i \(-0.281335\pi\)
0.634188 + 0.773179i \(0.281335\pi\)
\(798\) 0 0
\(799\) 28.4924 1.00799
\(800\) −1.00000 −0.0353553
\(801\) −5.12311 −0.181016
\(802\) 24.2462 0.856163
\(803\) 2.68466 0.0947395
\(804\) −7.31534 −0.257992
\(805\) 39.6155 1.39626
\(806\) −7.12311 −0.250901
\(807\) −17.3693 −0.611429
\(808\) −15.1231 −0.532029
\(809\) 36.9309 1.29842 0.649210 0.760609i \(-0.275100\pi\)
0.649210 + 0.760609i \(0.275100\pi\)
\(810\) −14.0000 −0.491910
\(811\) −20.6847 −0.726337 −0.363168 0.931724i \(-0.618305\pi\)
−0.363168 + 0.931724i \(0.618305\pi\)
\(812\) −23.8078 −0.835489
\(813\) 20.7926 0.729229
\(814\) −3.12311 −0.109465
\(815\) −26.7386 −0.936613
\(816\) 5.56155 0.194693
\(817\) 0 0
\(818\) 0.246211 0.00860857
\(819\) −7.12311 −0.248901
\(820\) −4.00000 −0.139686
\(821\) 44.9848 1.56998 0.784991 0.619507i \(-0.212668\pi\)
0.784991 + 0.619507i \(0.212668\pi\)
\(822\) 13.1771 0.459603
\(823\) 14.9309 0.520457 0.260229 0.965547i \(-0.416202\pi\)
0.260229 + 0.965547i \(0.416202\pi\)
\(824\) −11.3693 −0.396069
\(825\) −1.56155 −0.0543663
\(826\) −16.6847 −0.580534
\(827\) −21.0691 −0.732645 −0.366323 0.930488i \(-0.619383\pi\)
−0.366323 + 0.930488i \(0.619383\pi\)
\(828\) −3.12311 −0.108535
\(829\) 33.1771 1.15229 0.576144 0.817348i \(-0.304557\pi\)
0.576144 + 0.817348i \(0.304557\pi\)
\(830\) −4.49242 −0.155934
\(831\) −38.2462 −1.32675
\(832\) 3.56155 0.123475
\(833\) 20.2462 0.701490
\(834\) −27.1231 −0.939196
\(835\) 50.7386 1.75588
\(836\) 0 0
\(837\) 11.1231 0.384471
\(838\) −23.1231 −0.798774
\(839\) −1.12311 −0.0387739 −0.0193870 0.999812i \(-0.506171\pi\)
−0.0193870 + 0.999812i \(0.506171\pi\)
\(840\) 11.1231 0.383784
\(841\) 15.6847 0.540850
\(842\) 1.06913 0.0368447
\(843\) 0.984845 0.0339199
\(844\) 12.6847 0.436624
\(845\) −0.630683 −0.0216962
\(846\) −4.49242 −0.154453
\(847\) 3.56155 0.122376
\(848\) 7.80776 0.268120
\(849\) −20.8769 −0.716493
\(850\) −3.56155 −0.122160
\(851\) −17.3693 −0.595413
\(852\) −9.36932 −0.320988
\(853\) 38.3542 1.31322 0.656611 0.754230i \(-0.271989\pi\)
0.656611 + 0.754230i \(0.271989\pi\)
\(854\) −36.4924 −1.24874
\(855\) 0 0
\(856\) −18.9309 −0.647044
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 5.56155 0.189868
\(859\) −41.8617 −1.42830 −0.714152 0.699991i \(-0.753188\pi\)
−0.714152 + 0.699991i \(0.753188\pi\)
\(860\) 0 0
\(861\) −11.1231 −0.379074
\(862\) 25.8617 0.880854
\(863\) 16.2462 0.553027 0.276514 0.961010i \(-0.410821\pi\)
0.276514 + 0.961010i \(0.410821\pi\)
\(864\) −5.56155 −0.189208
\(865\) −36.0000 −1.22404
\(866\) −31.3693 −1.06597
\(867\) −6.73863 −0.228856
\(868\) −7.12311 −0.241774
\(869\) 4.00000 0.135691
\(870\) −20.8769 −0.707793
\(871\) −16.6847 −0.565338
\(872\) 18.6847 0.632742
\(873\) 0.630683 0.0213454
\(874\) 0 0
\(875\) −42.7386 −1.44483
\(876\) 4.19224 0.141643
\(877\) 40.4384 1.36551 0.682755 0.730648i \(-0.260782\pi\)
0.682755 + 0.730648i \(0.260782\pi\)
\(878\) −31.6155 −1.06697
\(879\) 38.9309 1.31311
\(880\) 2.00000 0.0674200
\(881\) −28.7386 −0.968229 −0.484115 0.875005i \(-0.660858\pi\)
−0.484115 + 0.875005i \(0.660858\pi\)
\(882\) −3.19224 −0.107488
\(883\) −54.7386 −1.84210 −0.921051 0.389442i \(-0.872668\pi\)
−0.921051 + 0.389442i \(0.872668\pi\)
\(884\) 12.6847 0.426631
\(885\) −14.6307 −0.491805
\(886\) −17.8617 −0.600077
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) −4.87689 −0.163658
\(889\) 50.7386 1.70172
\(890\) 18.2462 0.611614
\(891\) −7.00000 −0.234509
\(892\) 24.7386 0.828311
\(893\) 0 0
\(894\) −23.6155 −0.789821
\(895\) −44.4924 −1.48722
\(896\) 3.56155 0.118983
\(897\) 30.9309 1.03275
\(898\) 17.6155 0.587838
\(899\) 13.3693 0.445892
\(900\) 0.561553 0.0187184
\(901\) 27.8078 0.926411
\(902\) −2.00000 −0.0665927
\(903\) 0 0
\(904\) −5.12311 −0.170392
\(905\) 38.2462 1.27135
\(906\) −6.24621 −0.207516
\(907\) −56.7926 −1.88577 −0.942884 0.333122i \(-0.891898\pi\)
−0.942884 + 0.333122i \(0.891898\pi\)
\(908\) 17.1771 0.570041
\(909\) 8.49242 0.281676
\(910\) 25.3693 0.840985
\(911\) 46.9848 1.55668 0.778339 0.627845i \(-0.216063\pi\)
0.778339 + 0.627845i \(0.216063\pi\)
\(912\) 0 0
\(913\) −2.24621 −0.0743387
\(914\) 32.4384 1.07297
\(915\) −32.0000 −1.05789
\(916\) 21.1231 0.697927
\(917\) −25.3693 −0.837769
\(918\) −19.8078 −0.653754
\(919\) 28.4384 0.938098 0.469049 0.883172i \(-0.344597\pi\)
0.469049 + 0.883172i \(0.344597\pi\)
\(920\) 11.1231 0.366718
\(921\) 8.98485 0.296061
\(922\) −22.7386 −0.748857
\(923\) −21.3693 −0.703380
\(924\) 5.56155 0.182962
\(925\) 3.12311 0.102687
\(926\) 36.4924 1.19922
\(927\) 6.38447 0.209694
\(928\) −6.68466 −0.219435
\(929\) −16.9309 −0.555484 −0.277742 0.960656i \(-0.589586\pi\)
−0.277742 + 0.960656i \(0.589586\pi\)
\(930\) −6.24621 −0.204821
\(931\) 0 0
\(932\) 20.7386 0.679317
\(933\) 24.6847 0.808139
\(934\) 26.2462 0.858802
\(935\) 7.12311 0.232950
\(936\) −2.00000 −0.0653720
\(937\) −54.3002 −1.77391 −0.886955 0.461856i \(-0.847184\pi\)
−0.886955 + 0.461856i \(0.847184\pi\)
\(938\) −16.6847 −0.544773
\(939\) 5.56155 0.181494
\(940\) 16.0000 0.521862
\(941\) −20.4384 −0.666274 −0.333137 0.942878i \(-0.608107\pi\)
−0.333137 + 0.942878i \(0.608107\pi\)
\(942\) 28.8769 0.940860
\(943\) −11.1231 −0.362218
\(944\) −4.68466 −0.152473
\(945\) −39.6155 −1.28869
\(946\) 0 0
\(947\) 28.9848 0.941881 0.470940 0.882165i \(-0.343915\pi\)
0.470940 + 0.882165i \(0.343915\pi\)
\(948\) 6.24621 0.202868
\(949\) 9.56155 0.310381
\(950\) 0 0
\(951\) −46.9309 −1.52184
\(952\) 12.6847 0.411112
\(953\) 9.12311 0.295526 0.147763 0.989023i \(-0.452793\pi\)
0.147763 + 0.989023i \(0.452793\pi\)
\(954\) −4.38447 −0.141953
\(955\) −41.3693 −1.33868
\(956\) −4.93087 −0.159476
\(957\) −10.4384 −0.337427
\(958\) 11.3693 0.367326
\(959\) 30.0540 0.970493
\(960\) 3.12311 0.100798
\(961\) −27.0000 −0.870968
\(962\) −11.1231 −0.358623
\(963\) 10.6307 0.342569
\(964\) −21.6155 −0.696189
\(965\) 2.24621 0.0723081
\(966\) 30.9309 0.995184
\(967\) −3.86174 −0.124185 −0.0620926 0.998070i \(-0.519777\pi\)
−0.0620926 + 0.998070i \(0.519777\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −2.24621 −0.0721215
\(971\) 14.2462 0.457183 0.228591 0.973522i \(-0.426588\pi\)
0.228591 + 0.973522i \(0.426588\pi\)
\(972\) 5.75379 0.184553
\(973\) −61.8617 −1.98320
\(974\) −22.8769 −0.733023
\(975\) −5.56155 −0.178112
\(976\) −10.2462 −0.327973
\(977\) −55.3693 −1.77142 −0.885711 0.464238i \(-0.846328\pi\)
−0.885711 + 0.464238i \(0.846328\pi\)
\(978\) −20.8769 −0.667569
\(979\) 9.12311 0.291576
\(980\) 11.3693 0.363180
\(981\) −10.4924 −0.334997
\(982\) −14.6307 −0.466884
\(983\) 48.2462 1.53882 0.769408 0.638758i \(-0.220552\pi\)
0.769408 + 0.638758i \(0.220552\pi\)
\(984\) −3.12311 −0.0995610
\(985\) 3.50758 0.111761
\(986\) −23.8078 −0.758194
\(987\) 44.4924 1.41621
\(988\) 0 0
\(989\) 0 0
\(990\) −1.12311 −0.0356946
\(991\) −21.1231 −0.670998 −0.335499 0.942041i \(-0.608905\pi\)
−0.335499 + 0.942041i \(0.608905\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 16.3002 0.517271
\(994\) −21.3693 −0.677794
\(995\) −1.36932 −0.0434103
\(996\) −3.50758 −0.111142
\(997\) −53.3693 −1.69022 −0.845112 0.534590i \(-0.820466\pi\)
−0.845112 + 0.534590i \(0.820466\pi\)
\(998\) 26.2462 0.830809
\(999\) 17.3693 0.549541
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7942.2.a.x.1.2 2
19.18 odd 2 418.2.a.e.1.1 2
57.56 even 2 3762.2.a.y.1.2 2
76.75 even 2 3344.2.a.k.1.2 2
209.208 even 2 4598.2.a.bj.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.e.1.1 2 19.18 odd 2
3344.2.a.k.1.2 2 76.75 even 2
3762.2.a.y.1.2 2 57.56 even 2
4598.2.a.bj.1.1 2 209.208 even 2
7942.2.a.x.1.2 2 1.1 even 1 trivial