Properties

Label 418.2.a.e.1.1
Level $418$
Weight $2$
Character 418.1
Self dual yes
Analytic conductor $3.338$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.33774680449\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 418.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.56155 q^{6} +3.56155 q^{7} -1.00000 q^{8} -0.561553 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.56155 q^{6} +3.56155 q^{7} -1.00000 q^{8} -0.561553 q^{9} -2.00000 q^{10} +1.00000 q^{11} -1.56155 q^{12} -3.56155 q^{13} -3.56155 q^{14} -3.12311 q^{15} +1.00000 q^{16} +3.56155 q^{17} +0.561553 q^{18} -1.00000 q^{19} +2.00000 q^{20} -5.56155 q^{21} -1.00000 q^{22} +5.56155 q^{23} +1.56155 q^{24} -1.00000 q^{25} +3.56155 q^{26} +5.56155 q^{27} +3.56155 q^{28} +6.68466 q^{29} +3.12311 q^{30} +2.00000 q^{31} -1.00000 q^{32} -1.56155 q^{33} -3.56155 q^{34} +7.12311 q^{35} -0.561553 q^{36} +3.12311 q^{37} +1.00000 q^{38} +5.56155 q^{39} -2.00000 q^{40} +2.00000 q^{41} +5.56155 q^{42} +1.00000 q^{44} -1.12311 q^{45} -5.56155 q^{46} +8.00000 q^{47} -1.56155 q^{48} +5.68466 q^{49} +1.00000 q^{50} -5.56155 q^{51} -3.56155 q^{52} -7.80776 q^{53} -5.56155 q^{54} +2.00000 q^{55} -3.56155 q^{56} +1.56155 q^{57} -6.68466 q^{58} +4.68466 q^{59} -3.12311 q^{60} -10.2462 q^{61} -2.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -7.12311 q^{65} +1.56155 q^{66} +4.68466 q^{67} +3.56155 q^{68} -8.68466 q^{69} -7.12311 q^{70} +6.00000 q^{71} +0.561553 q^{72} +2.68466 q^{73} -3.12311 q^{74} +1.56155 q^{75} -1.00000 q^{76} +3.56155 q^{77} -5.56155 q^{78} -4.00000 q^{79} +2.00000 q^{80} -7.00000 q^{81} -2.00000 q^{82} -2.24621 q^{83} -5.56155 q^{84} +7.12311 q^{85} -10.4384 q^{87} -1.00000 q^{88} -9.12311 q^{89} +1.12311 q^{90} -12.6847 q^{91} +5.56155 q^{92} -3.12311 q^{93} -8.00000 q^{94} -2.00000 q^{95} +1.56155 q^{96} +1.12311 q^{97} -5.68466 q^{98} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} - 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} - 2 q^{8} + 3 q^{9} - 4 q^{10} + 2 q^{11} + q^{12} - 3 q^{13} - 3 q^{14} + 2 q^{15} + 2 q^{16} + 3 q^{17} - 3 q^{18} - 2 q^{19} + 4 q^{20} - 7 q^{21} - 2 q^{22} + 7 q^{23} - q^{24} - 2 q^{25} + 3 q^{26} + 7 q^{27} + 3 q^{28} + q^{29} - 2 q^{30} + 4 q^{31} - 2 q^{32} + q^{33} - 3 q^{34} + 6 q^{35} + 3 q^{36} - 2 q^{37} + 2 q^{38} + 7 q^{39} - 4 q^{40} + 4 q^{41} + 7 q^{42} + 2 q^{44} + 6 q^{45} - 7 q^{46} + 16 q^{47} + q^{48} - q^{49} + 2 q^{50} - 7 q^{51} - 3 q^{52} + 5 q^{53} - 7 q^{54} + 4 q^{55} - 3 q^{56} - q^{57} - q^{58} - 3 q^{59} + 2 q^{60} - 4 q^{61} - 4 q^{62} - 4 q^{63} + 2 q^{64} - 6 q^{65} - q^{66} - 3 q^{67} + 3 q^{68} - 5 q^{69} - 6 q^{70} + 12 q^{71} - 3 q^{72} - 7 q^{73} + 2 q^{74} - q^{75} - 2 q^{76} + 3 q^{77} - 7 q^{78} - 8 q^{79} + 4 q^{80} - 14 q^{81} - 4 q^{82} + 12 q^{83} - 7 q^{84} + 6 q^{85} - 25 q^{87} - 2 q^{88} - 10 q^{89} - 6 q^{90} - 13 q^{91} + 7 q^{92} + 2 q^{93} - 16 q^{94} - 4 q^{95} - q^{96} - 6 q^{97} + q^{98} + 3 q^{99}+O(q^{100}) \) 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 \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 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 \(-0.664429\pi\)
−0.493899 + 0.869520i \(0.664429\pi\)
\(14\) −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
\(19\) −1.00000 −0.229416
\(20\) 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 \(-0.286867\pi\)
0.620655 + 0.784084i \(0.286867\pi\)
\(30\) 3.12311 0.570198
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −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 \(-0.417359\pi\)
0.256718 + 0.966486i \(0.417359\pi\)
\(38\) 1.00000 0.162221
\(39\) 5.56155 0.890561
\(40\) −2.00000 −0.316228
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 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 \(-0.680156\pi\)
−0.536239 + 0.844066i \(0.680156\pi\)
\(54\) −5.56155 −0.756831
\(55\) 2.00000 0.269680
\(56\) −3.56155 −0.475933
\(57\) 1.56155 0.206833
\(58\) −6.68466 −0.877739
\(59\) 4.68466 0.609891 0.304945 0.952370i \(-0.401362\pi\)
0.304945 + 0.952370i \(0.401362\pi\)
\(60\) −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 \(-0.407621\pi\)
0.286161 + 0.958182i \(0.407621\pi\)
\(68\) 3.56155 0.431902
\(69\) −8.68466 −1.04551
\(70\) −7.12311 −0.851374
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 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
\(76\) −1.00000 −0.114708
\(77\) 3.56155 0.405877
\(78\) −5.56155 −0.629722
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 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 \(-0.660643\pi\)
−0.483524 + 0.875331i \(0.660643\pi\)
\(90\) 1.12311 0.118386
\(91\) −12.6847 −1.32971
\(92\) 5.56155 0.579832
\(93\) −3.12311 −0.323851
\(94\) −8.00000 −0.825137
\(95\) −2.00000 −0.205196
\(96\) 1.56155 0.159375
\(97\) 1.12311 0.114034 0.0570170 0.998373i \(-0.481841\pi\)
0.0570170 + 0.998373i \(0.481841\pi\)
\(98\) −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 \(-0.310753\pi\)
0.560126 + 0.828407i \(0.310753\pi\)
\(104\) 3.56155 0.349239
\(105\) −11.1231 −1.08550
\(106\) 7.80776 0.758357
\(107\) 18.9309 1.83012 0.915058 0.403322i \(-0.132145\pi\)
0.915058 + 0.403322i \(0.132145\pi\)
\(108\) 5.56155 0.535161
\(109\) −18.6847 −1.78967 −0.894833 0.446401i \(-0.852705\pi\)
−0.894833 + 0.446401i \(0.852705\pi\)
\(110\) −2.00000 −0.190693
\(111\) −4.87689 −0.462894
\(112\) 3.56155 0.336535
\(113\) 5.12311 0.481941 0.240971 0.970532i \(-0.422534\pi\)
0.240971 + 0.970532i \(0.422534\pi\)
\(114\) −1.56155 −0.146253
\(115\) 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 \(-0.717796\pi\)
−0.632073 + 0.774909i \(0.717796\pi\)
\(128\) −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
\(133\) −3.56155 −0.308826
\(134\) −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 \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 1.00000 0.0811107
\(153\) −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 \(-0.938791\pi\)
−0.981568 + 0.191111i \(0.938791\pi\)
\(168\) 5.56155 0.429083
\(169\) −0.315342 −0.0242570
\(170\) −7.12311 −0.546317
\(171\) 0.561553 0.0429430
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 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 \(-0.187551\pi\)
0.831380 + 0.555704i \(0.187551\pi\)
\(180\) −1.12311 −0.0837114
\(181\) −19.1231 −1.42141 −0.710705 0.703491i \(-0.751624\pi\)
−0.710705 + 0.703491i \(0.751624\pi\)
\(182\) 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
\(190\) 2.00000 0.145095
\(191\) −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 \(-0.512870\pi\)
−0.0404215 + 0.999183i \(0.512870\pi\)
\(194\) −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
\(209\) −1.00000 −0.0691714
\(210\) 11.1231 0.767567
\(211\) −12.6847 −0.873248 −0.436624 0.899644i \(-0.643826\pi\)
−0.436624 + 0.899644i \(0.643826\pi\)
\(212\) −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 \(-0.810698\pi\)
−0.828311 + 0.560269i \(0.810698\pi\)
\(224\) −3.56155 −0.237966
\(225\) 0.561553 0.0374369
\(226\) −5.12311 −0.340784
\(227\) −17.1771 −1.14008 −0.570041 0.821616i \(-0.693073\pi\)
−0.570041 + 0.821616i \(0.693073\pi\)
\(228\) 1.56155 0.103416
\(229\) 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 \(-0.254877\pi\)
0.696189 + 0.717858i \(0.254877\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −5.75379 −0.369106
\(244\) −10.2462 −0.655946
\(245\) 11.3693 0.726359
\(246\) 3.12311 0.199122
\(247\) 3.56155 0.226616
\(248\) −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 \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 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
\(266\) 3.56155 0.218373
\(267\) 14.2462 0.871854
\(268\) 4.68466 0.286161
\(269\) 11.1231 0.678188 0.339094 0.940753i \(-0.389880\pi\)
0.339094 + 0.940753i \(0.389880\pi\)
\(270\) −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 \(-0.505988\pi\)
−0.0188117 + 0.999823i \(0.505988\pi\)
\(282\) 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
\(285\) 3.12311 0.184997
\(286\) 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 \(-0.759661\pi\)
−0.728238 + 0.685324i \(0.759661\pi\)
\(294\) 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
\(304\) −1.00000 −0.0573539
\(305\) −20.4924 −1.17339
\(306\) 2.00000 0.114332
\(307\) −5.75379 −0.328386 −0.164193 0.986428i \(-0.552502\pi\)
−0.164193 + 0.986428i \(0.552502\pi\)
\(308\) 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 \(-0.180195\pi\)
0.844000 + 0.536344i \(0.180195\pi\)
\(318\) −12.1922 −0.683707
\(319\) 6.68466 0.374269
\(320\) 2.00000 0.111803
\(321\) −29.5616 −1.64996
\(322\) −19.8078 −1.10384
\(323\) −3.56155 −0.198170
\(324\) −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 \(-0.592616\pi\)
−0.286874 + 0.957968i \(0.592616\pi\)
\(332\) −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 \(-0.475033\pi\)
0.0783572 + 0.996925i \(0.475033\pi\)
\(338\) 0.315342 0.0171523
\(339\) −8.00000 −0.434500
\(340\) 7.12311 0.386305
\(341\) 2.00000 0.108306
\(342\) −0.561553 −0.0303653
\(343\) −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
\(361\) 1.00000 0.0526316
\(362\) 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 \(-0.437282\pi\)
0.195761 + 0.980652i \(0.437282\pi\)
\(374\) −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 \(-0.480052\pi\)
0.0626273 + 0.998037i \(0.480052\pi\)
\(380\) −2.00000 −0.102598
\(381\) 22.2462 1.13971
\(382\) 20.6847 1.05832
\(383\) −2.00000 −0.102195 −0.0510976 0.998694i \(-0.516272\pi\)
−0.0510976 + 0.998694i \(0.516272\pi\)
\(384\) 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
\(399\) 5.56155 0.278426
\(400\) −1.00000 −0.0500000
\(401\) −24.2462 −1.21080 −0.605399 0.795922i \(-0.706986\pi\)
−0.605399 + 0.795922i \(0.706986\pi\)
\(402\) 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 \(-0.501938\pi\)
−0.00608718 + 0.999981i \(0.501938\pi\)
\(410\) −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
\(418\) 1.00000 0.0489116
\(419\) −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 \(-0.508294\pi\)
−0.0260531 + 0.999661i \(0.508294\pi\)
\(422\) 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 \(-0.714029\pi\)
−0.622858 + 0.782335i \(0.714029\pi\)
\(432\) 5.56155 0.267580
\(433\) 31.3693 1.50751 0.753757 0.657154i \(-0.228240\pi\)
0.753757 + 0.657154i \(0.228240\pi\)
\(434\) −7.12311 −0.341920
\(435\) −20.8769 −1.00097
\(436\) −18.6847 −0.894833
\(437\) −5.56155 −0.266045
\(438\) 4.19224 0.200313
\(439\) 31.6155 1.50893 0.754463 0.656342i \(-0.227897\pi\)
0.754463 + 0.656342i \(0.227897\pi\)
\(440\) −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 \(-0.636451\pi\)
−0.415664 + 0.909518i \(0.636451\pi\)
\(450\) −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
\(456\) −1.56155 −0.0731264
\(457\) 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
\(475\) 1.00000 0.0458831
\(476\) 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 \(-0.326556\pi\)
0.518326 + 0.855183i \(0.326556\pi\)
\(488\) 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
\(494\) −3.56155 −0.160242
\(495\) −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 \(-0.546946\pi\)
−0.146950 + 0.989144i \(0.546946\pi\)
\(510\) 11.1231 0.492539
\(511\) 9.56155 0.422978
\(512\) −1.00000 −0.0441942
\(513\) −5.56155 −0.245549
\(514\) 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 \(-0.492168\pi\)
0.0246021 + 0.999697i \(0.492168\pi\)
\(522\) 3.75379 0.164299
\(523\) −6.05398 −0.264722 −0.132361 0.991202i \(-0.542256\pi\)
−0.132361 + 0.991202i \(0.542256\pi\)
\(524\) −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
\(532\) −3.56155 −0.154413
\(533\) −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 \(-0.453984\pi\)
0.144062 + 0.989569i \(0.453984\pi\)
\(548\) 8.43845 0.360473
\(549\) 5.75379 0.245566
\(550\) 1.00000 0.0426401
\(551\) −6.68466 −0.284776
\(552\) 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 \(-0.454647\pi\)
0.142000 + 0.989867i \(0.454647\pi\)
\(564\) −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 \(-0.477501\pi\)
0.0706246 + 0.997503i \(0.477501\pi\)
\(570\) −3.12311 −0.130812
\(571\) 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
\(589\) −2.00000 −0.0824086
\(590\) −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 \(-0.197092\pi\)
0.814353 + 0.580370i \(0.197092\pi\)
\(600\) −1.56155 −0.0637501
\(601\) −3.36932 −0.137437 −0.0687187 0.997636i \(-0.521891\pi\)
−0.0687187 + 0.997636i \(0.521891\pi\)
\(602\) 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 \(-0.310632\pi\)
0.560440 + 0.828195i \(0.310632\pi\)
\(608\) 1.00000 0.0405554
\(609\) −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
\(627\) 1.56155 0.0623624
\(628\) 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 \(-0.560815\pi\)
−0.189895 + 0.981804i \(0.560815\pi\)
\(642\) 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
\(646\) 3.56155 0.140127
\(647\) 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 \(-0.125170\pi\)
0.923675 + 0.383178i \(0.125170\pi\)
\(660\) −3.12311 −0.121567
\(661\) −22.4384 −0.872754 −0.436377 0.899764i \(-0.643739\pi\)
−0.436377 + 0.899764i \(0.643739\pi\)
\(662\) 10.4384 0.405702
\(663\) 19.8078 0.769270
\(664\) 2.24621 0.0871699
\(665\) −7.12311 −0.276222
\(666\) 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 \(-0.618530\pi\)
−0.363825 + 0.931467i \(0.618530\pi\)
\(674\) −2.87689 −0.110814
\(675\) −5.56155 −0.214064
\(676\) −0.315342 −0.0121285
\(677\) −39.1771 −1.50570 −0.752849 0.658194i \(-0.771321\pi\)
−0.752849 + 0.658194i \(0.771321\pi\)
\(678\) 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 \(-0.527392\pi\)
−0.0859489 + 0.996300i \(0.527392\pi\)
\(684\) 0.561553 0.0214715
\(685\) 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
\(703\) −3.12311 −0.117790
\(704\) 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
\(722\) −1.00000 −0.0372161
\(723\) −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
\(741\) −5.56155 −0.204309
\(742\) 27.8078 1.02086
\(743\) −25.7538 −0.944815 −0.472407 0.881380i \(-0.656615\pi\)
−0.472407 + 0.881380i \(0.656615\pi\)
\(744\) 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 \(-0.771042\pi\)
−0.752271 + 0.658853i \(0.771042\pi\)
\(752\) 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
\(760\) 2.00000 0.0725476
\(761\) −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 \(-0.447852\pi\)
0.163097 + 0.986610i \(0.447852\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) −1.12311 −0.0403171
\(777\) −17.3693 −0.623121
\(778\) −18.0000 −0.645331
\(779\) −2.00000 −0.0716574
\(780\) 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 \(-0.481180\pi\)
0.0590896 + 0.998253i \(0.481180\pi\)
\(788\) 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 \(-0.718665\pi\)
−0.634188 + 0.773179i \(0.718665\pi\)
\(798\) −5.56155 −0.196877
\(799\) 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 \(-0.381695\pi\)
0.363168 + 0.931724i \(0.381695\pi\)
\(812\) 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 \(-0.380617\pi\)
0.366323 + 0.930488i \(0.380617\pi\)
\(828\) −3.12311 −0.108535
\(829\) −33.1771 −1.15229 −0.576144 0.817348i \(-0.695443\pi\)
−0.576144 + 0.817348i \(0.695443\pi\)
\(830\) 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
\(836\) −1.00000 −0.0345857
\(837\) 11.1231 0.384471
\(838\) 23.1231 0.798774
\(839\) 1.12311 0.0387739 0.0193870 0.999812i \(-0.493829\pi\)
0.0193870 + 0.999812i \(0.493829\pi\)
\(840\) 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
\(855\) 1.12311 0.0384094
\(856\) −18.9309 −0.647044
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) −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 \(-0.589179\pi\)
−0.276514 + 0.961010i \(0.589179\pi\)
\(864\) −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
\(874\) 5.56155 0.188122
\(875\) −42.7386 −1.44483
\(876\) −4.19224 −0.141643
\(877\) −40.4384 −1.36551 −0.682755 0.730648i \(-0.739218\pi\)
−0.682755 + 0.730648i \(0.739218\pi\)
\(878\) −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 \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 4.87689 0.163658
\(889\) −50.7386 −1.70172
\(890\) 18.2462 0.611614
\(891\) −7.00000 −0.234509
\(892\) −24.7386 −0.828311
\(893\) −8.00000 −0.267710
\(894\) −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 \(-0.108102\pi\)
0.942884 + 0.333122i \(0.108102\pi\)
\(908\) −17.1771 −0.570041
\(909\) 8.49242 0.281676
\(910\) 25.3693 0.840985
\(911\) −46.9848 −1.55668 −0.778339 0.627845i \(-0.783937\pi\)
−0.778339 + 0.627845i \(0.783937\pi\)
\(912\) 1.56155 0.0517082
\(913\) −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
\(931\) −5.68466 −0.186307
\(932\) 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 \(-0.391893\pi\)
0.333137 + 0.942878i \(0.391893\pi\)
\(942\) 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
\(950\) −1.00000 −0.0324443
\(951\) −46.9309 −1.52184
\(952\) −12.6847 −0.411112
\(953\) −9.12311 −0.295526 −0.147763 0.989023i \(-0.547207\pi\)
−0.147763 + 0.989023i \(0.547207\pi\)
\(954\) −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
\(969\) 5.56155 0.178663
\(970\) −2.24621 −0.0721215
\(971\) −14.2462 −0.457183 −0.228591 0.973522i \(-0.573412\pi\)
−0.228591 + 0.973522i \(0.573412\pi\)
\(972\) −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 \(-0.153672\pi\)
0.885711 + 0.464238i \(0.153672\pi\)
\(978\) −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 \(-0.779448\pi\)
−0.769408 + 0.638758i \(0.779448\pi\)
\(984\) 3.12311 0.0995610
\(985\) 3.50758 0.111761
\(986\) −23.8078 −0.758194
\(987\) −44.4924 −1.41621
\(988\) 3.56155 0.113308
\(989\) 0 0
\(990\) 1.12311 0.0356946
\(991\) 21.1231 0.670998 0.335499 0.942041i \(-0.391095\pi\)
0.335499 + 0.942041i \(0.391095\pi\)
\(992\) −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 418.2.a.e.1.1 2
3.2 odd 2 3762.2.a.y.1.2 2
4.3 odd 2 3344.2.a.k.1.2 2
11.10 odd 2 4598.2.a.bj.1.1 2
19.18 odd 2 7942.2.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.e.1.1 2 1.1 even 1 trivial
3344.2.a.k.1.2 2 4.3 odd 2
3762.2.a.y.1.2 2 3.2 odd 2
4598.2.a.bj.1.1 2 11.10 odd 2
7942.2.a.x.1.2 2 19.18 odd 2