Properties

Label 4003.2.a.c.1.16
Level 4003
Weight 2
Character 4003.1
Self dual yes
Analytic conductor 31.964
Analytic rank 0
Dimension 179
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9641159291\)
Analytic rank: \(0\)
Dimension: \(179\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) = 4003.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.38397 q^{2} +1.62509 q^{3} +3.68332 q^{4} -0.985677 q^{5} -3.87416 q^{6} -1.38154 q^{7} -4.01299 q^{8} -0.359094 q^{9} +O(q^{10})\) \(q-2.38397 q^{2} +1.62509 q^{3} +3.68332 q^{4} -0.985677 q^{5} -3.87416 q^{6} -1.38154 q^{7} -4.01299 q^{8} -0.359094 q^{9} +2.34983 q^{10} +3.88920 q^{11} +5.98572 q^{12} +1.26448 q^{13} +3.29355 q^{14} -1.60181 q^{15} +2.20022 q^{16} +1.19457 q^{17} +0.856070 q^{18} +6.37297 q^{19} -3.63057 q^{20} -2.24512 q^{21} -9.27175 q^{22} +1.09885 q^{23} -6.52146 q^{24} -4.02844 q^{25} -3.01449 q^{26} -5.45882 q^{27} -5.08866 q^{28} +3.93133 q^{29} +3.81867 q^{30} +4.35012 q^{31} +2.78073 q^{32} +6.32029 q^{33} -2.84781 q^{34} +1.36175 q^{35} -1.32266 q^{36} +5.82343 q^{37} -15.1930 q^{38} +2.05490 q^{39} +3.95552 q^{40} -1.19420 q^{41} +5.35231 q^{42} +4.92877 q^{43} +14.3252 q^{44} +0.353951 q^{45} -2.61964 q^{46} -2.09466 q^{47} +3.57555 q^{48} -5.09135 q^{49} +9.60369 q^{50} +1.94127 q^{51} +4.65750 q^{52} +1.02747 q^{53} +13.0137 q^{54} -3.83350 q^{55} +5.54411 q^{56} +10.3566 q^{57} -9.37219 q^{58} -3.71709 q^{59} -5.89998 q^{60} -8.30371 q^{61} -10.3706 q^{62} +0.496103 q^{63} -11.0296 q^{64} -1.24637 q^{65} -15.0674 q^{66} -3.61329 q^{67} +4.39997 q^{68} +1.78573 q^{69} -3.24638 q^{70} +6.21892 q^{71} +1.44104 q^{72} -12.0245 q^{73} -13.8829 q^{74} -6.54656 q^{75} +23.4737 q^{76} -5.37309 q^{77} -4.89881 q^{78} -12.5180 q^{79} -2.16871 q^{80} -7.79377 q^{81} +2.84695 q^{82} +14.5960 q^{83} -8.26951 q^{84} -1.17746 q^{85} -11.7501 q^{86} +6.38876 q^{87} -15.6074 q^{88} -11.3971 q^{89} -0.843809 q^{90} -1.74693 q^{91} +4.04744 q^{92} +7.06932 q^{93} +4.99361 q^{94} -6.28169 q^{95} +4.51892 q^{96} +12.5169 q^{97} +12.1376 q^{98} -1.39659 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 179q + 22q^{2} + 16q^{3} + 196q^{4} + 61q^{5} + 7q^{6} + 21q^{7} + 60q^{8} + 221q^{9} + O(q^{10}) \) \( 179q + 22q^{2} + 16q^{3} + 196q^{4} + 61q^{5} + 7q^{6} + 21q^{7} + 60q^{8} + 221q^{9} + 9q^{10} + 46q^{11} + 33q^{12} + 47q^{13} + 22q^{14} + 36q^{15} + 222q^{16} + 103q^{17} + 43q^{18} + 12q^{19} + 102q^{20} + 50q^{21} + 39q^{22} + 121q^{23} - 3q^{24} + 246q^{25} + 52q^{26} + 49q^{27} + 41q^{28} + 138q^{29} + 28q^{30} + 5q^{31} + 137q^{32} + 63q^{33} + 2q^{34} + 72q^{35} + 279q^{36} + 118q^{37} + 123q^{38} + q^{39} + 9q^{40} + 50q^{41} + 48q^{42} + 48q^{43} + 108q^{44} + 158q^{45} + 13q^{46} + 85q^{47} + 50q^{48} + 230q^{49} + 78q^{50} + 15q^{51} + 41q^{52} + 399q^{53} - 5q^{54} + 24q^{55} + 53q^{56} + 45q^{57} + 27q^{58} + 48q^{59} + 66q^{60} + 46q^{61} + 81q^{62} + 78q^{63} + 252q^{64} + 153q^{65} + 6q^{66} + 70q^{67} + 240q^{68} + 120q^{69} - 31q^{70} + 86q^{71} + 89q^{72} + 45q^{73} + 68q^{74} + 17q^{75} - 13q^{76} + 362q^{77} + 69q^{78} + 31q^{79} + 169q^{80} + 303q^{81} + 25q^{82} + 106q^{83} + 13q^{84} + 115q^{85} + 95q^{86} + 32q^{87} + 83q^{88} + 105q^{89} - 38q^{90} + 3q^{91} + 310q^{92} + 298q^{93} - 17q^{94} + 102q^{95} - 82q^{96} + 34q^{97} + 81q^{98} + 58q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.38397 −1.68572 −0.842861 0.538131i \(-0.819131\pi\)
−0.842861 + 0.538131i \(0.819131\pi\)
\(3\) 1.62509 0.938244 0.469122 0.883133i \(-0.344570\pi\)
0.469122 + 0.883133i \(0.344570\pi\)
\(4\) 3.68332 1.84166
\(5\) −0.985677 −0.440808 −0.220404 0.975409i \(-0.570738\pi\)
−0.220404 + 0.975409i \(0.570738\pi\)
\(6\) −3.87416 −1.58162
\(7\) −1.38154 −0.522173 −0.261086 0.965315i \(-0.584081\pi\)
−0.261086 + 0.965315i \(0.584081\pi\)
\(8\) −4.01299 −1.41881
\(9\) −0.359094 −0.119698
\(10\) 2.34983 0.743080
\(11\) 3.88920 1.17264 0.586320 0.810080i \(-0.300576\pi\)
0.586320 + 0.810080i \(0.300576\pi\)
\(12\) 5.98572 1.72793
\(13\) 1.26448 0.350705 0.175352 0.984506i \(-0.443894\pi\)
0.175352 + 0.984506i \(0.443894\pi\)
\(14\) 3.29355 0.880239
\(15\) −1.60181 −0.413586
\(16\) 2.20022 0.550055
\(17\) 1.19457 0.289725 0.144862 0.989452i \(-0.453726\pi\)
0.144862 + 0.989452i \(0.453726\pi\)
\(18\) 0.856070 0.201778
\(19\) 6.37297 1.46206 0.731030 0.682345i \(-0.239040\pi\)
0.731030 + 0.682345i \(0.239040\pi\)
\(20\) −3.63057 −0.811819
\(21\) −2.24512 −0.489926
\(22\) −9.27175 −1.97674
\(23\) 1.09885 0.229127 0.114564 0.993416i \(-0.463453\pi\)
0.114564 + 0.993416i \(0.463453\pi\)
\(24\) −6.52146 −1.33119
\(25\) −4.02844 −0.805688
\(26\) −3.01449 −0.591191
\(27\) −5.45882 −1.05055
\(28\) −5.08866 −0.961666
\(29\) 3.93133 0.730030 0.365015 0.931002i \(-0.381064\pi\)
0.365015 + 0.931002i \(0.381064\pi\)
\(30\) 3.81867 0.697191
\(31\) 4.35012 0.781304 0.390652 0.920538i \(-0.372250\pi\)
0.390652 + 0.920538i \(0.372250\pi\)
\(32\) 2.78073 0.491568
\(33\) 6.32029 1.10022
\(34\) −2.84781 −0.488396
\(35\) 1.36175 0.230178
\(36\) −1.32266 −0.220443
\(37\) 5.82343 0.957366 0.478683 0.877988i \(-0.341114\pi\)
0.478683 + 0.877988i \(0.341114\pi\)
\(38\) −15.1930 −2.46463
\(39\) 2.05490 0.329047
\(40\) 3.95552 0.625422
\(41\) −1.19420 −0.186503 −0.0932516 0.995643i \(-0.529726\pi\)
−0.0932516 + 0.995643i \(0.529726\pi\)
\(42\) 5.35231 0.825879
\(43\) 4.92877 0.751631 0.375815 0.926695i \(-0.377363\pi\)
0.375815 + 0.926695i \(0.377363\pi\)
\(44\) 14.3252 2.15960
\(45\) 0.353951 0.0527639
\(46\) −2.61964 −0.386245
\(47\) −2.09466 −0.305537 −0.152769 0.988262i \(-0.548819\pi\)
−0.152769 + 0.988262i \(0.548819\pi\)
\(48\) 3.57555 0.516086
\(49\) −5.09135 −0.727335
\(50\) 9.60369 1.35817
\(51\) 1.94127 0.271833
\(52\) 4.65750 0.645879
\(53\) 1.02747 0.141133 0.0705667 0.997507i \(-0.477519\pi\)
0.0705667 + 0.997507i \(0.477519\pi\)
\(54\) 13.0137 1.77094
\(55\) −3.83350 −0.516909
\(56\) 5.54411 0.740863
\(57\) 10.3566 1.37177
\(58\) −9.37219 −1.23063
\(59\) −3.71709 −0.483924 −0.241962 0.970286i \(-0.577791\pi\)
−0.241962 + 0.970286i \(0.577791\pi\)
\(60\) −5.89998 −0.761685
\(61\) −8.30371 −1.06318 −0.531591 0.847001i \(-0.678406\pi\)
−0.531591 + 0.847001i \(0.678406\pi\)
\(62\) −10.3706 −1.31706
\(63\) 0.496103 0.0625031
\(64\) −11.0296 −1.37870
\(65\) −1.24637 −0.154594
\(66\) −15.0674 −1.85467
\(67\) −3.61329 −0.441434 −0.220717 0.975338i \(-0.570840\pi\)
−0.220717 + 0.975338i \(0.570840\pi\)
\(68\) 4.39997 0.533575
\(69\) 1.78573 0.214977
\(70\) −3.24638 −0.388016
\(71\) 6.21892 0.738050 0.369025 0.929419i \(-0.379692\pi\)
0.369025 + 0.929419i \(0.379692\pi\)
\(72\) 1.44104 0.169829
\(73\) −12.0245 −1.40736 −0.703679 0.710518i \(-0.748460\pi\)
−0.703679 + 0.710518i \(0.748460\pi\)
\(74\) −13.8829 −1.61385
\(75\) −6.54656 −0.755932
\(76\) 23.4737 2.69262
\(77\) −5.37309 −0.612320
\(78\) −4.89881 −0.554682
\(79\) −12.5180 −1.40839 −0.704193 0.710008i \(-0.748691\pi\)
−0.704193 + 0.710008i \(0.748691\pi\)
\(80\) −2.16871 −0.242469
\(81\) −7.79377 −0.865974
\(82\) 2.84695 0.314393
\(83\) 14.5960 1.60212 0.801062 0.598582i \(-0.204269\pi\)
0.801062 + 0.598582i \(0.204269\pi\)
\(84\) −8.26951 −0.902277
\(85\) −1.17746 −0.127713
\(86\) −11.7501 −1.26704
\(87\) 6.38876 0.684947
\(88\) −15.6074 −1.66375
\(89\) −11.3971 −1.20809 −0.604047 0.796949i \(-0.706446\pi\)
−0.604047 + 0.796949i \(0.706446\pi\)
\(90\) −0.843809 −0.0889453
\(91\) −1.74693 −0.183129
\(92\) 4.04744 0.421974
\(93\) 7.06932 0.733054
\(94\) 4.99361 0.515051
\(95\) −6.28169 −0.644488
\(96\) 4.51892 0.461210
\(97\) 12.5169 1.27090 0.635451 0.772141i \(-0.280814\pi\)
0.635451 + 0.772141i \(0.280814\pi\)
\(98\) 12.1376 1.22609
\(99\) −1.39659 −0.140363
\(100\) −14.8380 −1.48380
\(101\) 11.4870 1.14300 0.571498 0.820603i \(-0.306362\pi\)
0.571498 + 0.820603i \(0.306362\pi\)
\(102\) −4.62794 −0.458235
\(103\) 4.56537 0.449839 0.224920 0.974377i \(-0.427788\pi\)
0.224920 + 0.974377i \(0.427788\pi\)
\(104\) −5.07437 −0.497583
\(105\) 2.21296 0.215963
\(106\) −2.44945 −0.237912
\(107\) 15.1394 1.46358 0.731792 0.681528i \(-0.238684\pi\)
0.731792 + 0.681528i \(0.238684\pi\)
\(108\) −20.1066 −1.93476
\(109\) 3.64049 0.348696 0.174348 0.984684i \(-0.444218\pi\)
0.174348 + 0.984684i \(0.444218\pi\)
\(110\) 9.13895 0.871365
\(111\) 9.46358 0.898243
\(112\) −3.03969 −0.287224
\(113\) 4.66060 0.438432 0.219216 0.975676i \(-0.429650\pi\)
0.219216 + 0.975676i \(0.429650\pi\)
\(114\) −24.6899 −2.31242
\(115\) −1.08312 −0.101001
\(116\) 14.4804 1.34447
\(117\) −0.454069 −0.0419787
\(118\) 8.86144 0.815761
\(119\) −1.65034 −0.151286
\(120\) 6.42805 0.586798
\(121\) 4.12591 0.375082
\(122\) 19.7958 1.79223
\(123\) −1.94068 −0.174985
\(124\) 16.0229 1.43890
\(125\) 8.89913 0.795962
\(126\) −1.18270 −0.105363
\(127\) 4.97369 0.441344 0.220672 0.975348i \(-0.429175\pi\)
0.220672 + 0.975348i \(0.429175\pi\)
\(128\) 20.7328 1.83254
\(129\) 8.00968 0.705213
\(130\) 2.97132 0.260602
\(131\) 3.88753 0.339655 0.169828 0.985474i \(-0.445679\pi\)
0.169828 + 0.985474i \(0.445679\pi\)
\(132\) 23.2797 2.02624
\(133\) −8.80452 −0.763448
\(134\) 8.61398 0.744135
\(135\) 5.38063 0.463091
\(136\) −4.79379 −0.411064
\(137\) 2.11556 0.180745 0.0903724 0.995908i \(-0.471194\pi\)
0.0903724 + 0.995908i \(0.471194\pi\)
\(138\) −4.25714 −0.362392
\(139\) −9.42170 −0.799138 −0.399569 0.916703i \(-0.630840\pi\)
−0.399569 + 0.916703i \(0.630840\pi\)
\(140\) 5.01577 0.423910
\(141\) −3.40400 −0.286669
\(142\) −14.8257 −1.24415
\(143\) 4.91784 0.411250
\(144\) −0.790086 −0.0658405
\(145\) −3.87502 −0.321803
\(146\) 28.6660 2.37241
\(147\) −8.27388 −0.682418
\(148\) 21.4496 1.76314
\(149\) −7.43957 −0.609473 −0.304737 0.952437i \(-0.598568\pi\)
−0.304737 + 0.952437i \(0.598568\pi\)
\(150\) 15.6068 1.27429
\(151\) 17.2817 1.40636 0.703180 0.711012i \(-0.251763\pi\)
0.703180 + 0.711012i \(0.251763\pi\)
\(152\) −25.5747 −2.07438
\(153\) −0.428962 −0.0346795
\(154\) 12.8093 1.03220
\(155\) −4.28781 −0.344405
\(156\) 7.56884 0.605993
\(157\) 5.99724 0.478632 0.239316 0.970942i \(-0.423077\pi\)
0.239316 + 0.970942i \(0.423077\pi\)
\(158\) 29.8426 2.37415
\(159\) 1.66972 0.132417
\(160\) −2.74090 −0.216687
\(161\) −1.51811 −0.119644
\(162\) 18.5801 1.45979
\(163\) −20.1634 −1.57932 −0.789658 0.613547i \(-0.789742\pi\)
−0.789658 + 0.613547i \(0.789742\pi\)
\(164\) −4.39863 −0.343476
\(165\) −6.22977 −0.484987
\(166\) −34.7965 −2.70074
\(167\) 20.8663 1.61468 0.807342 0.590083i \(-0.200905\pi\)
0.807342 + 0.590083i \(0.200905\pi\)
\(168\) 9.00966 0.695110
\(169\) −11.4011 −0.877006
\(170\) 2.80702 0.215289
\(171\) −2.28850 −0.175006
\(172\) 18.1543 1.38425
\(173\) −5.97414 −0.454205 −0.227103 0.973871i \(-0.572925\pi\)
−0.227103 + 0.973871i \(0.572925\pi\)
\(174\) −15.2306 −1.15463
\(175\) 5.56545 0.420709
\(176\) 8.55710 0.645016
\(177\) −6.04059 −0.454039
\(178\) 27.1704 2.03651
\(179\) 18.1058 1.35329 0.676647 0.736308i \(-0.263432\pi\)
0.676647 + 0.736308i \(0.263432\pi\)
\(180\) 1.30372 0.0971732
\(181\) 19.8416 1.47481 0.737406 0.675450i \(-0.236051\pi\)
0.737406 + 0.675450i \(0.236051\pi\)
\(182\) 4.16464 0.308704
\(183\) −13.4943 −0.997524
\(184\) −4.40970 −0.325087
\(185\) −5.74002 −0.422015
\(186\) −16.8531 −1.23573
\(187\) 4.64591 0.339743
\(188\) −7.71531 −0.562697
\(189\) 7.54157 0.548569
\(190\) 14.9754 1.08643
\(191\) −23.4582 −1.69738 −0.848688 0.528894i \(-0.822607\pi\)
−0.848688 + 0.528894i \(0.822607\pi\)
\(192\) −17.9241 −1.29356
\(193\) 26.7484 1.92539 0.962697 0.270581i \(-0.0872157\pi\)
0.962697 + 0.270581i \(0.0872157\pi\)
\(194\) −29.8400 −2.14239
\(195\) −2.02546 −0.145046
\(196\) −18.7531 −1.33951
\(197\) −21.9533 −1.56411 −0.782055 0.623209i \(-0.785828\pi\)
−0.782055 + 0.623209i \(0.785828\pi\)
\(198\) 3.32943 0.236612
\(199\) 21.4122 1.51787 0.758935 0.651167i \(-0.225720\pi\)
0.758935 + 0.651167i \(0.225720\pi\)
\(200\) 16.1661 1.14312
\(201\) −5.87191 −0.414172
\(202\) −27.3846 −1.92678
\(203\) −5.43129 −0.381202
\(204\) 7.15034 0.500624
\(205\) 1.17710 0.0822121
\(206\) −10.8837 −0.758305
\(207\) −0.394592 −0.0274261
\(208\) 2.78214 0.192907
\(209\) 24.7858 1.71447
\(210\) −5.27564 −0.364054
\(211\) 14.8915 1.02517 0.512587 0.858635i \(-0.328687\pi\)
0.512587 + 0.858635i \(0.328687\pi\)
\(212\) 3.78449 0.259920
\(213\) 10.1063 0.692471
\(214\) −36.0920 −2.46720
\(215\) −4.85818 −0.331325
\(216\) 21.9062 1.49053
\(217\) −6.00986 −0.407976
\(218\) −8.67883 −0.587804
\(219\) −19.5408 −1.32044
\(220\) −14.1200 −0.951971
\(221\) 1.51051 0.101608
\(222\) −22.5609 −1.51419
\(223\) 16.6924 1.11781 0.558903 0.829233i \(-0.311222\pi\)
0.558903 + 0.829233i \(0.311222\pi\)
\(224\) −3.84168 −0.256683
\(225\) 1.44659 0.0964393
\(226\) −11.1107 −0.739076
\(227\) 27.5760 1.83029 0.915143 0.403130i \(-0.132078\pi\)
0.915143 + 0.403130i \(0.132078\pi\)
\(228\) 38.1468 2.52633
\(229\) −15.7633 −1.04167 −0.520836 0.853657i \(-0.674380\pi\)
−0.520836 + 0.853657i \(0.674380\pi\)
\(230\) 2.58212 0.170260
\(231\) −8.73173 −0.574506
\(232\) −15.7764 −1.03577
\(233\) 28.5806 1.87238 0.936188 0.351500i \(-0.114328\pi\)
0.936188 + 0.351500i \(0.114328\pi\)
\(234\) 1.08249 0.0707644
\(235\) 2.06466 0.134683
\(236\) −13.6912 −0.891224
\(237\) −20.3429 −1.32141
\(238\) 3.93437 0.255027
\(239\) 0.441817 0.0285788 0.0142894 0.999898i \(-0.495451\pi\)
0.0142894 + 0.999898i \(0.495451\pi\)
\(240\) −3.52433 −0.227495
\(241\) −7.51546 −0.484113 −0.242057 0.970262i \(-0.577822\pi\)
−0.242057 + 0.970262i \(0.577822\pi\)
\(242\) −9.83605 −0.632285
\(243\) 3.71091 0.238055
\(244\) −30.5853 −1.95802
\(245\) 5.01842 0.320615
\(246\) 4.62653 0.294977
\(247\) 8.05852 0.512752
\(248\) −17.4570 −1.10852
\(249\) 23.7198 1.50318
\(250\) −21.2153 −1.34177
\(251\) 28.0907 1.77307 0.886534 0.462664i \(-0.153106\pi\)
0.886534 + 0.462664i \(0.153106\pi\)
\(252\) 1.82731 0.115110
\(253\) 4.27367 0.268683
\(254\) −11.8571 −0.743984
\(255\) −1.91347 −0.119826
\(256\) −27.3673 −1.71045
\(257\) 30.0704 1.87574 0.937870 0.346987i \(-0.112795\pi\)
0.937870 + 0.346987i \(0.112795\pi\)
\(258\) −19.0949 −1.18879
\(259\) −8.04530 −0.499911
\(260\) −4.59079 −0.284709
\(261\) −1.41172 −0.0873832
\(262\) −9.26777 −0.572565
\(263\) −10.6895 −0.659141 −0.329571 0.944131i \(-0.606904\pi\)
−0.329571 + 0.944131i \(0.606904\pi\)
\(264\) −25.3633 −1.56100
\(265\) −1.01275 −0.0622127
\(266\) 20.9897 1.28696
\(267\) −18.5213 −1.13349
\(268\) −13.3089 −0.812971
\(269\) −17.1571 −1.04609 −0.523043 0.852306i \(-0.675203\pi\)
−0.523043 + 0.852306i \(0.675203\pi\)
\(270\) −12.8273 −0.780643
\(271\) −2.87675 −0.174750 −0.0873750 0.996175i \(-0.527848\pi\)
−0.0873750 + 0.996175i \(0.527848\pi\)
\(272\) 2.62831 0.159365
\(273\) −2.83892 −0.171819
\(274\) −5.04344 −0.304686
\(275\) −15.6674 −0.944781
\(276\) 6.57743 0.395915
\(277\) −20.5033 −1.23192 −0.615962 0.787776i \(-0.711233\pi\)
−0.615962 + 0.787776i \(0.711233\pi\)
\(278\) 22.4611 1.34713
\(279\) −1.56210 −0.0935206
\(280\) −5.46470 −0.326578
\(281\) −18.5394 −1.10597 −0.552984 0.833192i \(-0.686511\pi\)
−0.552984 + 0.833192i \(0.686511\pi\)
\(282\) 8.11505 0.483244
\(283\) −6.45245 −0.383558 −0.191779 0.981438i \(-0.561426\pi\)
−0.191779 + 0.981438i \(0.561426\pi\)
\(284\) 22.9063 1.35924
\(285\) −10.2083 −0.604687
\(286\) −11.7240 −0.693254
\(287\) 1.64984 0.0973869
\(288\) −0.998542 −0.0588397
\(289\) −15.5730 −0.916059
\(290\) 9.23795 0.542471
\(291\) 20.3411 1.19242
\(292\) −44.2900 −2.59188
\(293\) −2.30664 −0.134756 −0.0673778 0.997728i \(-0.521463\pi\)
−0.0673778 + 0.997728i \(0.521463\pi\)
\(294\) 19.7247 1.15037
\(295\) 3.66385 0.213318
\(296\) −23.3694 −1.35832
\(297\) −21.2305 −1.23192
\(298\) 17.7357 1.02740
\(299\) 1.38948 0.0803560
\(300\) −24.1131 −1.39217
\(301\) −6.80930 −0.392481
\(302\) −41.1990 −2.37073
\(303\) 18.6673 1.07241
\(304\) 14.0219 0.804213
\(305\) 8.18478 0.468659
\(306\) 1.02263 0.0584600
\(307\) −32.9828 −1.88242 −0.941212 0.337815i \(-0.890312\pi\)
−0.941212 + 0.337815i \(0.890312\pi\)
\(308\) −19.7908 −1.12769
\(309\) 7.41912 0.422059
\(310\) 10.2220 0.580572
\(311\) 5.31422 0.301342 0.150671 0.988584i \(-0.451857\pi\)
0.150671 + 0.988584i \(0.451857\pi\)
\(312\) −8.24628 −0.466854
\(313\) −5.65641 −0.319719 −0.159860 0.987140i \(-0.551104\pi\)
−0.159860 + 0.987140i \(0.551104\pi\)
\(314\) −14.2973 −0.806841
\(315\) −0.488997 −0.0275519
\(316\) −46.1079 −2.59377
\(317\) 3.34255 0.187736 0.0938681 0.995585i \(-0.470077\pi\)
0.0938681 + 0.995585i \(0.470077\pi\)
\(318\) −3.98057 −0.223219
\(319\) 15.2898 0.856062
\(320\) 10.8716 0.607743
\(321\) 24.6029 1.37320
\(322\) 3.61913 0.201686
\(323\) 7.61294 0.423595
\(324\) −28.7070 −1.59483
\(325\) −5.09390 −0.282559
\(326\) 48.0689 2.66229
\(327\) 5.91611 0.327162
\(328\) 4.79233 0.264612
\(329\) 2.89385 0.159543
\(330\) 14.8516 0.817553
\(331\) −8.30956 −0.456735 −0.228367 0.973575i \(-0.573339\pi\)
−0.228367 + 0.973575i \(0.573339\pi\)
\(332\) 53.7619 2.95057
\(333\) −2.09116 −0.114595
\(334\) −49.7447 −2.72191
\(335\) 3.56154 0.194587
\(336\) −4.93976 −0.269486
\(337\) 4.26862 0.232526 0.116263 0.993218i \(-0.462908\pi\)
0.116263 + 0.993218i \(0.462908\pi\)
\(338\) 27.1799 1.47839
\(339\) 7.57388 0.411357
\(340\) −4.33695 −0.235204
\(341\) 16.9185 0.916188
\(342\) 5.45571 0.295011
\(343\) 16.7047 0.901968
\(344\) −19.7791 −1.06642
\(345\) −1.76016 −0.0947636
\(346\) 14.2422 0.765664
\(347\) 23.7556 1.27527 0.637634 0.770340i \(-0.279913\pi\)
0.637634 + 0.770340i \(0.279913\pi\)
\(348\) 23.5319 1.26144
\(349\) −31.4886 −1.68555 −0.842774 0.538268i \(-0.819079\pi\)
−0.842774 + 0.538268i \(0.819079\pi\)
\(350\) −13.2679 −0.709198
\(351\) −6.90259 −0.368433
\(352\) 10.8148 0.576431
\(353\) −0.538111 −0.0286408 −0.0143204 0.999897i \(-0.504558\pi\)
−0.0143204 + 0.999897i \(0.504558\pi\)
\(354\) 14.4006 0.765383
\(355\) −6.12985 −0.325338
\(356\) −41.9793 −2.22490
\(357\) −2.68195 −0.141944
\(358\) −43.1638 −2.28128
\(359\) −7.98560 −0.421464 −0.210732 0.977544i \(-0.567585\pi\)
−0.210732 + 0.977544i \(0.567585\pi\)
\(360\) −1.42040 −0.0748618
\(361\) 21.6148 1.13762
\(362\) −47.3017 −2.48612
\(363\) 6.70496 0.351919
\(364\) −6.43452 −0.337261
\(365\) 11.8522 0.620375
\(366\) 32.1699 1.68155
\(367\) 24.6772 1.28814 0.644070 0.764967i \(-0.277245\pi\)
0.644070 + 0.764967i \(0.277245\pi\)
\(368\) 2.41772 0.126032
\(369\) 0.428831 0.0223241
\(370\) 13.6841 0.711400
\(371\) −1.41949 −0.0736960
\(372\) 26.0386 1.35004
\(373\) −20.8355 −1.07882 −0.539410 0.842043i \(-0.681353\pi\)
−0.539410 + 0.842043i \(0.681353\pi\)
\(374\) −11.0757 −0.572712
\(375\) 14.4618 0.746807
\(376\) 8.40585 0.433499
\(377\) 4.97111 0.256025
\(378\) −17.9789 −0.924735
\(379\) 10.8518 0.557417 0.278709 0.960376i \(-0.410094\pi\)
0.278709 + 0.960376i \(0.410094\pi\)
\(380\) −23.1375 −1.18693
\(381\) 8.08268 0.414088
\(382\) 55.9237 2.86130
\(383\) −0.999478 −0.0510710 −0.0255355 0.999674i \(-0.508129\pi\)
−0.0255355 + 0.999674i \(0.508129\pi\)
\(384\) 33.6926 1.71937
\(385\) 5.29613 0.269916
\(386\) −63.7675 −3.24568
\(387\) −1.76989 −0.0899688
\(388\) 46.1039 2.34057
\(389\) 18.5799 0.942038 0.471019 0.882123i \(-0.343886\pi\)
0.471019 + 0.882123i \(0.343886\pi\)
\(390\) 4.82865 0.244508
\(391\) 1.31265 0.0663838
\(392\) 20.4315 1.03195
\(393\) 6.31758 0.318680
\(394\) 52.3361 2.63666
\(395\) 12.3387 0.620828
\(396\) −5.14409 −0.258500
\(397\) 30.3144 1.52143 0.760717 0.649084i \(-0.224848\pi\)
0.760717 + 0.649084i \(0.224848\pi\)
\(398\) −51.0460 −2.55871
\(399\) −14.3081 −0.716301
\(400\) −8.86345 −0.443173
\(401\) 37.3576 1.86555 0.932775 0.360459i \(-0.117380\pi\)
0.932775 + 0.360459i \(0.117380\pi\)
\(402\) 13.9985 0.698180
\(403\) 5.50066 0.274007
\(404\) 42.3102 2.10501
\(405\) 7.68214 0.381729
\(406\) 12.9481 0.642601
\(407\) 22.6485 1.12265
\(408\) −7.79032 −0.385678
\(409\) −27.7895 −1.37410 −0.687050 0.726610i \(-0.741095\pi\)
−0.687050 + 0.726610i \(0.741095\pi\)
\(410\) −2.80617 −0.138587
\(411\) 3.43797 0.169583
\(412\) 16.8157 0.828452
\(413\) 5.13531 0.252692
\(414\) 0.940697 0.0462327
\(415\) −14.3870 −0.706229
\(416\) 3.51618 0.172395
\(417\) −15.3111 −0.749787
\(418\) −59.0886 −2.89012
\(419\) 7.73759 0.378006 0.189003 0.981977i \(-0.439474\pi\)
0.189003 + 0.981977i \(0.439474\pi\)
\(420\) 8.15106 0.397731
\(421\) 30.2169 1.47268 0.736341 0.676610i \(-0.236552\pi\)
0.736341 + 0.676610i \(0.236552\pi\)
\(422\) −35.5010 −1.72816
\(423\) 0.752180 0.0365722
\(424\) −4.12321 −0.200241
\(425\) −4.81224 −0.233428
\(426\) −24.0931 −1.16731
\(427\) 11.4719 0.555165
\(428\) 55.7634 2.69543
\(429\) 7.99191 0.385853
\(430\) 11.5818 0.558522
\(431\) 34.2524 1.64988 0.824939 0.565222i \(-0.191209\pi\)
0.824939 + 0.565222i \(0.191209\pi\)
\(432\) −12.0106 −0.577860
\(433\) −36.0183 −1.73093 −0.865464 0.500971i \(-0.832976\pi\)
−0.865464 + 0.500971i \(0.832976\pi\)
\(434\) 14.3273 0.687734
\(435\) −6.29725 −0.301930
\(436\) 13.4091 0.642180
\(437\) 7.00297 0.334998
\(438\) 46.5847 2.22590
\(439\) 20.2425 0.966123 0.483062 0.875586i \(-0.339525\pi\)
0.483062 + 0.875586i \(0.339525\pi\)
\(440\) 15.3838 0.733394
\(441\) 1.82827 0.0870606
\(442\) −3.60101 −0.171283
\(443\) −28.4706 −1.35268 −0.676340 0.736590i \(-0.736435\pi\)
−0.676340 + 0.736590i \(0.736435\pi\)
\(444\) 34.8574 1.65426
\(445\) 11.2339 0.532537
\(446\) −39.7942 −1.88431
\(447\) −12.0899 −0.571834
\(448\) 15.2378 0.719920
\(449\) 21.4256 1.01114 0.505568 0.862787i \(-0.331283\pi\)
0.505568 + 0.862787i \(0.331283\pi\)
\(450\) −3.44863 −0.162570
\(451\) −4.64450 −0.218701
\(452\) 17.1665 0.807444
\(453\) 28.0842 1.31951
\(454\) −65.7405 −3.08535
\(455\) 1.72191 0.0807245
\(456\) −41.5611 −1.94628
\(457\) −2.93875 −0.137469 −0.0687345 0.997635i \(-0.521896\pi\)
−0.0687345 + 0.997635i \(0.521896\pi\)
\(458\) 37.5794 1.75597
\(459\) −6.52092 −0.304371
\(460\) −3.98946 −0.186010
\(461\) 15.7378 0.732982 0.366491 0.930422i \(-0.380559\pi\)
0.366491 + 0.930422i \(0.380559\pi\)
\(462\) 20.8162 0.968458
\(463\) 19.7845 0.919466 0.459733 0.888057i \(-0.347945\pi\)
0.459733 + 0.888057i \(0.347945\pi\)
\(464\) 8.64980 0.401557
\(465\) −6.96807 −0.323136
\(466\) −68.1353 −3.15631
\(467\) 8.34339 0.386086 0.193043 0.981190i \(-0.438164\pi\)
0.193043 + 0.981190i \(0.438164\pi\)
\(468\) −1.67248 −0.0773105
\(469\) 4.99190 0.230505
\(470\) −4.92209 −0.227039
\(471\) 9.74603 0.449074
\(472\) 14.9167 0.686595
\(473\) 19.1690 0.881392
\(474\) 48.4968 2.22753
\(475\) −25.6732 −1.17796
\(476\) −6.07874 −0.278618
\(477\) −0.368957 −0.0168934
\(478\) −1.05328 −0.0481759
\(479\) −29.5028 −1.34802 −0.674009 0.738723i \(-0.735429\pi\)
−0.674009 + 0.738723i \(0.735429\pi\)
\(480\) −4.45419 −0.203305
\(481\) 7.36364 0.335753
\(482\) 17.9166 0.816080
\(483\) −2.46706 −0.112255
\(484\) 15.1970 0.690775
\(485\) −12.3376 −0.560224
\(486\) −8.84670 −0.401294
\(487\) 6.58140 0.298232 0.149116 0.988820i \(-0.452357\pi\)
0.149116 + 0.988820i \(0.452357\pi\)
\(488\) 33.3228 1.50845
\(489\) −32.7672 −1.48178
\(490\) −11.9638 −0.540469
\(491\) 3.50101 0.157998 0.0789991 0.996875i \(-0.474828\pi\)
0.0789991 + 0.996875i \(0.474828\pi\)
\(492\) −7.14816 −0.322264
\(493\) 4.69624 0.211508
\(494\) −19.2113 −0.864357
\(495\) 1.37659 0.0618730
\(496\) 9.57122 0.429760
\(497\) −8.59168 −0.385390
\(498\) −56.5474 −2.53395
\(499\) 8.20477 0.367296 0.183648 0.982992i \(-0.441209\pi\)
0.183648 + 0.982992i \(0.441209\pi\)
\(500\) 32.7784 1.46589
\(501\) 33.9096 1.51497
\(502\) −66.9674 −2.98890
\(503\) 44.4308 1.98107 0.990536 0.137251i \(-0.0438268\pi\)
0.990536 + 0.137251i \(0.0438268\pi\)
\(504\) −1.99086 −0.0886798
\(505\) −11.3224 −0.503842
\(506\) −10.1883 −0.452926
\(507\) −18.5277 −0.822846
\(508\) 18.3197 0.812806
\(509\) 9.31101 0.412703 0.206352 0.978478i \(-0.433841\pi\)
0.206352 + 0.978478i \(0.433841\pi\)
\(510\) 4.56166 0.201993
\(511\) 16.6123 0.734884
\(512\) 23.7771 1.05081
\(513\) −34.7889 −1.53597
\(514\) −71.6870 −3.16198
\(515\) −4.49998 −0.198293
\(516\) 29.5022 1.29876
\(517\) −8.14656 −0.358285
\(518\) 19.1798 0.842711
\(519\) −9.70849 −0.426155
\(520\) 5.00169 0.219338
\(521\) −22.8124 −0.999431 −0.499716 0.866190i \(-0.666562\pi\)
−0.499716 + 0.866190i \(0.666562\pi\)
\(522\) 3.36550 0.147304
\(523\) −32.9871 −1.44242 −0.721212 0.692715i \(-0.756415\pi\)
−0.721212 + 0.692715i \(0.756415\pi\)
\(524\) 14.3190 0.625530
\(525\) 9.04434 0.394727
\(526\) 25.4834 1.11113
\(527\) 5.19651 0.226363
\(528\) 13.9060 0.605182
\(529\) −21.7925 −0.947501
\(530\) 2.41437 0.104873
\(531\) 1.33479 0.0579247
\(532\) −32.4299 −1.40601
\(533\) −1.51005 −0.0654075
\(534\) 44.1543 1.91074
\(535\) −14.9226 −0.645160
\(536\) 14.5001 0.626309
\(537\) 29.4235 1.26972
\(538\) 40.9020 1.76341
\(539\) −19.8013 −0.852902
\(540\) 19.8186 0.852857
\(541\) 29.7693 1.27988 0.639942 0.768423i \(-0.278958\pi\)
0.639942 + 0.768423i \(0.278958\pi\)
\(542\) 6.85809 0.294580
\(543\) 32.2442 1.38373
\(544\) 3.32176 0.142419
\(545\) −3.58835 −0.153708
\(546\) 6.76791 0.289640
\(547\) 29.7546 1.27222 0.636108 0.771600i \(-0.280543\pi\)
0.636108 + 0.771600i \(0.280543\pi\)
\(548\) 7.79230 0.332871
\(549\) 2.98182 0.127261
\(550\) 37.3507 1.59264
\(551\) 25.0543 1.06735
\(552\) −7.16614 −0.305011
\(553\) 17.2941 0.735421
\(554\) 48.8793 2.07668
\(555\) −9.32803 −0.395953
\(556\) −34.7032 −1.47174
\(557\) 16.6704 0.706349 0.353174 0.935558i \(-0.385102\pi\)
0.353174 + 0.935558i \(0.385102\pi\)
\(558\) 3.72401 0.157650
\(559\) 6.23236 0.263601
\(560\) 2.99615 0.126611
\(561\) 7.55001 0.318762
\(562\) 44.1974 1.86435
\(563\) 9.27289 0.390806 0.195403 0.980723i \(-0.437399\pi\)
0.195403 + 0.980723i \(0.437399\pi\)
\(564\) −12.5380 −0.527947
\(565\) −4.59385 −0.193265
\(566\) 15.3825 0.646573
\(567\) 10.7674 0.452188
\(568\) −24.9565 −1.04715
\(569\) −31.5143 −1.32115 −0.660573 0.750762i \(-0.729687\pi\)
−0.660573 + 0.750762i \(0.729687\pi\)
\(570\) 24.3363 1.01933
\(571\) 17.4098 0.728579 0.364290 0.931286i \(-0.381312\pi\)
0.364290 + 0.931286i \(0.381312\pi\)
\(572\) 18.1140 0.757383
\(573\) −38.1216 −1.59255
\(574\) −3.93317 −0.164167
\(575\) −4.42667 −0.184605
\(576\) 3.96067 0.165028
\(577\) 1.37427 0.0572115 0.0286058 0.999591i \(-0.490893\pi\)
0.0286058 + 0.999591i \(0.490893\pi\)
\(578\) 37.1256 1.54422
\(579\) 43.4685 1.80649
\(580\) −14.2730 −0.592653
\(581\) −20.1650 −0.836585
\(582\) −48.4926 −2.01008
\(583\) 3.99602 0.165498
\(584\) 48.2541 1.99677
\(585\) 0.447565 0.0185045
\(586\) 5.49898 0.227161
\(587\) 0.883022 0.0364462 0.0182231 0.999834i \(-0.494199\pi\)
0.0182231 + 0.999834i \(0.494199\pi\)
\(588\) −30.4754 −1.25678
\(589\) 27.7232 1.14231
\(590\) −8.73451 −0.359594
\(591\) −35.6761 −1.46752
\(592\) 12.8128 0.526604
\(593\) −25.1310 −1.03201 −0.516003 0.856587i \(-0.672581\pi\)
−0.516003 + 0.856587i \(0.672581\pi\)
\(594\) 50.6128 2.07667
\(595\) 1.62670 0.0666883
\(596\) −27.4023 −1.12244
\(597\) 34.7966 1.42413
\(598\) −3.31249 −0.135458
\(599\) 32.2298 1.31687 0.658437 0.752636i \(-0.271218\pi\)
0.658437 + 0.752636i \(0.271218\pi\)
\(600\) 26.2713 1.07252
\(601\) 17.4491 0.711763 0.355881 0.934531i \(-0.384181\pi\)
0.355881 + 0.934531i \(0.384181\pi\)
\(602\) 16.2332 0.661615
\(603\) 1.29751 0.0528387
\(604\) 63.6539 2.59004
\(605\) −4.06681 −0.165339
\(606\) −44.5024 −1.80779
\(607\) 19.0974 0.775139 0.387569 0.921841i \(-0.373315\pi\)
0.387569 + 0.921841i \(0.373315\pi\)
\(608\) 17.7215 0.718701
\(609\) −8.82632 −0.357661
\(610\) −19.5123 −0.790029
\(611\) −2.64866 −0.107153
\(612\) −1.58000 −0.0638679
\(613\) −9.19146 −0.371240 −0.185620 0.982622i \(-0.559429\pi\)
−0.185620 + 0.982622i \(0.559429\pi\)
\(614\) 78.6300 3.17325
\(615\) 1.91289 0.0771350
\(616\) 21.5622 0.868765
\(617\) −11.0770 −0.445942 −0.222971 0.974825i \(-0.571576\pi\)
−0.222971 + 0.974825i \(0.571576\pi\)
\(618\) −17.6870 −0.711475
\(619\) −27.6053 −1.10955 −0.554775 0.832001i \(-0.687195\pi\)
−0.554775 + 0.832001i \(0.687195\pi\)
\(620\) −15.7934 −0.634278
\(621\) −5.99845 −0.240709
\(622\) −12.6690 −0.507979
\(623\) 15.7456 0.630834
\(624\) 4.52122 0.180994
\(625\) 11.3705 0.454822
\(626\) 13.4847 0.538958
\(627\) 40.2791 1.60859
\(628\) 22.0898 0.881478
\(629\) 6.95648 0.277373
\(630\) 1.16576 0.0464448
\(631\) −13.3080 −0.529784 −0.264892 0.964278i \(-0.585336\pi\)
−0.264892 + 0.964278i \(0.585336\pi\)
\(632\) 50.2347 1.99823
\(633\) 24.2000 0.961864
\(634\) −7.96854 −0.316471
\(635\) −4.90245 −0.194548
\(636\) 6.15012 0.243868
\(637\) −6.43793 −0.255080
\(638\) −36.4504 −1.44308
\(639\) −2.23318 −0.0883431
\(640\) −20.4359 −0.807799
\(641\) 20.3014 0.801858 0.400929 0.916109i \(-0.368687\pi\)
0.400929 + 0.916109i \(0.368687\pi\)
\(642\) −58.6526 −2.31483
\(643\) −29.2420 −1.15319 −0.576597 0.817029i \(-0.695620\pi\)
−0.576597 + 0.817029i \(0.695620\pi\)
\(644\) −5.59169 −0.220344
\(645\) −7.89496 −0.310864
\(646\) −18.1490 −0.714064
\(647\) −1.00324 −0.0394416 −0.0197208 0.999806i \(-0.506278\pi\)
−0.0197208 + 0.999806i \(0.506278\pi\)
\(648\) 31.2763 1.22865
\(649\) −14.4565 −0.567468
\(650\) 12.1437 0.476316
\(651\) −9.76655 −0.382781
\(652\) −74.2682 −2.90857
\(653\) 9.10509 0.356310 0.178155 0.984002i \(-0.442987\pi\)
0.178155 + 0.984002i \(0.442987\pi\)
\(654\) −14.1038 −0.551504
\(655\) −3.83185 −0.149723
\(656\) −2.62751 −0.102587
\(657\) 4.31792 0.168458
\(658\) −6.89887 −0.268946
\(659\) −9.61728 −0.374636 −0.187318 0.982299i \(-0.559980\pi\)
−0.187318 + 0.982299i \(0.559980\pi\)
\(660\) −22.9462 −0.893181
\(661\) −19.9268 −0.775064 −0.387532 0.921856i \(-0.626672\pi\)
−0.387532 + 0.921856i \(0.626672\pi\)
\(662\) 19.8098 0.769928
\(663\) 2.45471 0.0953330
\(664\) −58.5738 −2.27310
\(665\) 8.67841 0.336534
\(666\) 4.98527 0.193175
\(667\) 4.31996 0.167270
\(668\) 76.8574 2.97370
\(669\) 27.1266 1.04877
\(670\) −8.49060 −0.328021
\(671\) −32.2948 −1.24673
\(672\) −6.24307 −0.240832
\(673\) −2.85519 −0.110060 −0.0550298 0.998485i \(-0.517525\pi\)
−0.0550298 + 0.998485i \(0.517525\pi\)
\(674\) −10.1763 −0.391975
\(675\) 21.9905 0.846416
\(676\) −41.9939 −1.61515
\(677\) 7.85158 0.301761 0.150880 0.988552i \(-0.451789\pi\)
0.150880 + 0.988552i \(0.451789\pi\)
\(678\) −18.0559 −0.693433
\(679\) −17.2926 −0.663630
\(680\) 4.72513 0.181200
\(681\) 44.8134 1.71725
\(682\) −40.3332 −1.54444
\(683\) −4.40699 −0.168629 −0.0843145 0.996439i \(-0.526870\pi\)
−0.0843145 + 0.996439i \(0.526870\pi\)
\(684\) −8.42928 −0.322301
\(685\) −2.08526 −0.0796738
\(686\) −39.8235 −1.52047
\(687\) −25.6168 −0.977342
\(688\) 10.8444 0.413438
\(689\) 1.29921 0.0494961
\(690\) 4.19616 0.159745
\(691\) −3.20181 −0.121803 −0.0609013 0.998144i \(-0.519398\pi\)
−0.0609013 + 0.998144i \(0.519398\pi\)
\(692\) −22.0047 −0.836492
\(693\) 1.92945 0.0732936
\(694\) −56.6327 −2.14975
\(695\) 9.28675 0.352267
\(696\) −25.6380 −0.971807
\(697\) −1.42655 −0.0540346
\(698\) 75.0680 2.84137
\(699\) 46.4459 1.75675
\(700\) 20.4993 0.774803
\(701\) 14.0982 0.532483 0.266242 0.963906i \(-0.414218\pi\)
0.266242 + 0.963906i \(0.414218\pi\)
\(702\) 16.4556 0.621076
\(703\) 37.1126 1.39973
\(704\) −42.8964 −1.61672
\(705\) 3.35525 0.126366
\(706\) 1.28284 0.0482804
\(707\) −15.8697 −0.596842
\(708\) −22.2494 −0.836185
\(709\) 47.5619 1.78622 0.893112 0.449835i \(-0.148517\pi\)
0.893112 + 0.449835i \(0.148517\pi\)
\(710\) 14.6134 0.548430
\(711\) 4.49515 0.168581
\(712\) 45.7366 1.71405
\(713\) 4.78015 0.179018
\(714\) 6.39368 0.239278
\(715\) −4.84740 −0.181282
\(716\) 66.6896 2.49231
\(717\) 0.717991 0.0268139
\(718\) 19.0374 0.710471
\(719\) 31.9606 1.19193 0.595965 0.803011i \(-0.296770\pi\)
0.595965 + 0.803011i \(0.296770\pi\)
\(720\) 0.778770 0.0290230
\(721\) −6.30724 −0.234894
\(722\) −51.5291 −1.91771
\(723\) −12.2133 −0.454216
\(724\) 73.0829 2.71610
\(725\) −15.8371 −0.588177
\(726\) −15.9844 −0.593238
\(727\) 7.63297 0.283091 0.141546 0.989932i \(-0.454793\pi\)
0.141546 + 0.989932i \(0.454793\pi\)
\(728\) 7.01044 0.259824
\(729\) 29.4119 1.08933
\(730\) −28.2554 −1.04578
\(731\) 5.88775 0.217766
\(732\) −49.7037 −1.83710
\(733\) −46.4847 −1.71695 −0.858475 0.512855i \(-0.828588\pi\)
−0.858475 + 0.512855i \(0.828588\pi\)
\(734\) −58.8298 −2.17145
\(735\) 8.15537 0.300815
\(736\) 3.05561 0.112631
\(737\) −14.0528 −0.517642
\(738\) −1.02232 −0.0376322
\(739\) −34.3813 −1.26474 −0.632369 0.774667i \(-0.717917\pi\)
−0.632369 + 0.774667i \(0.717917\pi\)
\(740\) −21.1424 −0.777208
\(741\) 13.0958 0.481086
\(742\) 3.38401 0.124231
\(743\) −32.4758 −1.19142 −0.595711 0.803199i \(-0.703130\pi\)
−0.595711 + 0.803199i \(0.703130\pi\)
\(744\) −28.3691 −1.04006
\(745\) 7.33301 0.268661
\(746\) 49.6712 1.81859
\(747\) −5.24135 −0.191771
\(748\) 17.1124 0.625691
\(749\) −20.9157 −0.764244
\(750\) −34.4766 −1.25891
\(751\) −2.00830 −0.0732840 −0.0366420 0.999328i \(-0.511666\pi\)
−0.0366420 + 0.999328i \(0.511666\pi\)
\(752\) −4.60871 −0.168062
\(753\) 45.6498 1.66357
\(754\) −11.8510 −0.431587
\(755\) −17.0341 −0.619935
\(756\) 27.7780 1.01028
\(757\) −10.1703 −0.369644 −0.184822 0.982772i \(-0.559171\pi\)
−0.184822 + 0.982772i \(0.559171\pi\)
\(758\) −25.8703 −0.939651
\(759\) 6.94508 0.252091
\(760\) 25.2084 0.914405
\(761\) −16.3519 −0.592756 −0.296378 0.955071i \(-0.595779\pi\)
−0.296378 + 0.955071i \(0.595779\pi\)
\(762\) −19.2689 −0.698038
\(763\) −5.02948 −0.182080
\(764\) −86.4041 −3.12599
\(765\) 0.422818 0.0152870
\(766\) 2.38273 0.0860915
\(767\) −4.70020 −0.169714
\(768\) −44.4742 −1.60482
\(769\) −9.09299 −0.327902 −0.163951 0.986469i \(-0.552424\pi\)
−0.163951 + 0.986469i \(0.552424\pi\)
\(770\) −12.6258 −0.455003
\(771\) 48.8670 1.75990
\(772\) 98.5231 3.54592
\(773\) 40.6120 1.46071 0.730357 0.683066i \(-0.239354\pi\)
0.730357 + 0.683066i \(0.239354\pi\)
\(774\) 4.21938 0.151662
\(775\) −17.5242 −0.629488
\(776\) −50.2304 −1.80316
\(777\) −13.0743 −0.469038
\(778\) −44.2939 −1.58801
\(779\) −7.61062 −0.272679
\(780\) −7.46044 −0.267126
\(781\) 24.1866 0.865466
\(782\) −3.12933 −0.111905
\(783\) −21.4604 −0.766933
\(784\) −11.2021 −0.400074
\(785\) −5.91134 −0.210985
\(786\) −15.0609 −0.537205
\(787\) −25.1104 −0.895090 −0.447545 0.894262i \(-0.647702\pi\)
−0.447545 + 0.894262i \(0.647702\pi\)
\(788\) −80.8612 −2.88056
\(789\) −17.3713 −0.618435
\(790\) −29.4152 −1.04654
\(791\) −6.43881 −0.228938
\(792\) 5.60451 0.199148
\(793\) −10.4999 −0.372863
\(794\) −72.2686 −2.56472
\(795\) −1.64581 −0.0583707
\(796\) 78.8680 2.79540
\(797\) −33.6657 −1.19250 −0.596250 0.802799i \(-0.703343\pi\)
−0.596250 + 0.802799i \(0.703343\pi\)
\(798\) 34.1101 1.20748
\(799\) −2.50221 −0.0885218
\(800\) −11.2020 −0.396050
\(801\) 4.09264 0.144606
\(802\) −89.0595 −3.14480
\(803\) −46.7656 −1.65032
\(804\) −21.6281 −0.762765
\(805\) 1.49637 0.0527400
\(806\) −13.1134 −0.461900
\(807\) −27.8818 −0.981485
\(808\) −46.0972 −1.62169
\(809\) −24.3264 −0.855269 −0.427635 0.903952i \(-0.640653\pi\)
−0.427635 + 0.903952i \(0.640653\pi\)
\(810\) −18.3140 −0.643488
\(811\) −30.2095 −1.06080 −0.530400 0.847747i \(-0.677958\pi\)
−0.530400 + 0.847747i \(0.677958\pi\)
\(812\) −20.0052 −0.702045
\(813\) −4.67496 −0.163958
\(814\) −53.9934 −1.89247
\(815\) 19.8746 0.696176
\(816\) 4.27123 0.149523
\(817\) 31.4109 1.09893
\(818\) 66.2493 2.31635
\(819\) 0.627314 0.0219201
\(820\) 4.33563 0.151407
\(821\) −48.5910 −1.69584 −0.847918 0.530127i \(-0.822144\pi\)
−0.847918 + 0.530127i \(0.822144\pi\)
\(822\) −8.19603 −0.285869
\(823\) 1.21203 0.0422486 0.0211243 0.999777i \(-0.493275\pi\)
0.0211243 + 0.999777i \(0.493275\pi\)
\(824\) −18.3208 −0.638235
\(825\) −25.4609 −0.886436
\(826\) −12.2424 −0.425969
\(827\) 47.2210 1.64203 0.821017 0.570904i \(-0.193407\pi\)
0.821017 + 0.570904i \(0.193407\pi\)
\(828\) −1.45341 −0.0505095
\(829\) −23.7155 −0.823673 −0.411836 0.911258i \(-0.635112\pi\)
−0.411836 + 0.911258i \(0.635112\pi\)
\(830\) 34.2981 1.19051
\(831\) −33.3196 −1.15585
\(832\) −13.9468 −0.483517
\(833\) −6.08195 −0.210727
\(834\) 36.5012 1.26393
\(835\) −20.5675 −0.711766
\(836\) 91.2941 3.15747
\(837\) −23.7465 −0.820799
\(838\) −18.4462 −0.637213
\(839\) −49.4138 −1.70595 −0.852977 0.521948i \(-0.825206\pi\)
−0.852977 + 0.521948i \(0.825206\pi\)
\(840\) −8.88061 −0.306410
\(841\) −13.5446 −0.467056
\(842\) −72.0363 −2.48253
\(843\) −30.1281 −1.03767
\(844\) 54.8503 1.88802
\(845\) 11.2378 0.386591
\(846\) −1.79318 −0.0616507
\(847\) −5.70010 −0.195858
\(848\) 2.26065 0.0776311
\(849\) −10.4858 −0.359871
\(850\) 11.4722 0.393495
\(851\) 6.39911 0.219359
\(852\) 37.2247 1.27530
\(853\) 34.1047 1.16772 0.583861 0.811854i \(-0.301541\pi\)
0.583861 + 0.811854i \(0.301541\pi\)
\(854\) −27.3487 −0.935854
\(855\) 2.25572 0.0771440
\(856\) −60.7545 −2.07655
\(857\) 45.9077 1.56818 0.784089 0.620649i \(-0.213131\pi\)
0.784089 + 0.620649i \(0.213131\pi\)
\(858\) −19.0525 −0.650441
\(859\) 30.0054 1.02377 0.511886 0.859053i \(-0.328947\pi\)
0.511886 + 0.859053i \(0.328947\pi\)
\(860\) −17.8942 −0.610188
\(861\) 2.68113 0.0913727
\(862\) −81.6567 −2.78124
\(863\) 33.6886 1.14677 0.573386 0.819285i \(-0.305630\pi\)
0.573386 + 0.819285i \(0.305630\pi\)
\(864\) −15.1795 −0.516416
\(865\) 5.88857 0.200217
\(866\) 85.8666 2.91787
\(867\) −25.3075 −0.859487
\(868\) −22.1363 −0.751354
\(869\) −48.6851 −1.65153
\(870\) 15.0125 0.508970
\(871\) −4.56895 −0.154813
\(872\) −14.6093 −0.494732
\(873\) −4.49476 −0.152124
\(874\) −16.6949 −0.564713
\(875\) −12.2945 −0.415630
\(876\) −71.9750 −2.43181
\(877\) 22.0177 0.743485 0.371743 0.928336i \(-0.378760\pi\)
0.371743 + 0.928336i \(0.378760\pi\)
\(878\) −48.2576 −1.62862
\(879\) −3.74850 −0.126434
\(880\) −8.43454 −0.284328
\(881\) 51.8498 1.74686 0.873432 0.486947i \(-0.161889\pi\)
0.873432 + 0.486947i \(0.161889\pi\)
\(882\) −4.35855 −0.146760
\(883\) −20.5448 −0.691386 −0.345693 0.938348i \(-0.612356\pi\)
−0.345693 + 0.938348i \(0.612356\pi\)
\(884\) 5.56370 0.187127
\(885\) 5.95407 0.200144
\(886\) 67.8731 2.28024
\(887\) −23.7931 −0.798893 −0.399447 0.916756i \(-0.630798\pi\)
−0.399447 + 0.916756i \(0.630798\pi\)
\(888\) −37.9773 −1.27443
\(889\) −6.87135 −0.230458
\(890\) −26.7813 −0.897711
\(891\) −30.3116 −1.01548
\(892\) 61.4835 2.05862
\(893\) −13.3492 −0.446714
\(894\) 28.8221 0.963954
\(895\) −17.8465 −0.596543
\(896\) −28.6432 −0.956903
\(897\) 2.25803 0.0753935
\(898\) −51.0780 −1.70449
\(899\) 17.1018 0.570376
\(900\) 5.32826 0.177609
\(901\) 1.22738 0.0408898
\(902\) 11.0724 0.368669
\(903\) −11.0657 −0.368243
\(904\) −18.7030 −0.622051
\(905\) −19.5574 −0.650109
\(906\) −66.9519 −2.22433
\(907\) 30.1290 1.00042 0.500208 0.865905i \(-0.333257\pi\)
0.500208 + 0.865905i \(0.333257\pi\)
\(908\) 101.571 3.37077
\(909\) −4.12491 −0.136814
\(910\) −4.10499 −0.136079
\(911\) −37.0846 −1.22867 −0.614334 0.789046i \(-0.710575\pi\)
−0.614334 + 0.789046i \(0.710575\pi\)
\(912\) 22.7869 0.754549
\(913\) 56.7670 1.87871
\(914\) 7.00590 0.231735
\(915\) 13.3010 0.439717
\(916\) −58.0615 −1.91841
\(917\) −5.37078 −0.177359
\(918\) 15.5457 0.513084
\(919\) 28.0924 0.926682 0.463341 0.886180i \(-0.346651\pi\)
0.463341 + 0.886180i \(0.346651\pi\)
\(920\) 4.34654 0.143301
\(921\) −53.5998 −1.76617
\(922\) −37.5184 −1.23560
\(923\) 7.86372 0.258838
\(924\) −32.1618 −1.05805
\(925\) −23.4594 −0.771339
\(926\) −47.1658 −1.54996
\(927\) −1.63940 −0.0538449
\(928\) 10.9320 0.358859
\(929\) −14.4713 −0.474789 −0.237394 0.971413i \(-0.576293\pi\)
−0.237394 + 0.971413i \(0.576293\pi\)
\(930\) 16.6117 0.544718
\(931\) −32.4470 −1.06341
\(932\) 105.271 3.44828
\(933\) 8.63607 0.282732
\(934\) −19.8904 −0.650834
\(935\) −4.57937 −0.149761
\(936\) 1.82218 0.0595597
\(937\) −23.5798 −0.770320 −0.385160 0.922850i \(-0.625854\pi\)
−0.385160 + 0.922850i \(0.625854\pi\)
\(938\) −11.9006 −0.388567
\(939\) −9.19215 −0.299975
\(940\) 7.60480 0.248041
\(941\) −14.3974 −0.469342 −0.234671 0.972075i \(-0.575401\pi\)
−0.234671 + 0.972075i \(0.575401\pi\)
\(942\) −23.2343 −0.757013
\(943\) −1.31226 −0.0427329
\(944\) −8.17841 −0.266185
\(945\) −7.43356 −0.241814
\(946\) −45.6984 −1.48578
\(947\) −32.4809 −1.05549 −0.527744 0.849403i \(-0.676962\pi\)
−0.527744 + 0.849403i \(0.676962\pi\)
\(948\) −74.9293 −2.43359
\(949\) −15.2047 −0.493567
\(950\) 61.2041 1.98572
\(951\) 5.43193 0.176142
\(952\) 6.62281 0.214646
\(953\) −27.6339 −0.895150 −0.447575 0.894246i \(-0.647712\pi\)
−0.447575 + 0.894246i \(0.647712\pi\)
\(954\) 0.879583 0.0284776
\(955\) 23.1222 0.748217
\(956\) 1.62736 0.0526324
\(957\) 24.8472 0.803195
\(958\) 70.3339 2.27239
\(959\) −2.92274 −0.0943800
\(960\) 17.6673 0.570211
\(961\) −12.0765 −0.389563
\(962\) −17.5547 −0.565986
\(963\) −5.43648 −0.175188
\(964\) −27.6818 −0.891572
\(965\) −26.3653 −0.848730
\(966\) 5.88141 0.189231
\(967\) 30.0821 0.967374 0.483687 0.875241i \(-0.339297\pi\)
0.483687 + 0.875241i \(0.339297\pi\)
\(968\) −16.5572 −0.532170
\(969\) 12.3717 0.397436
\(970\) 29.4126 0.944382
\(971\) −0.390165 −0.0125210 −0.00626050 0.999980i \(-0.501993\pi\)
−0.00626050 + 0.999980i \(0.501993\pi\)
\(972\) 13.6685 0.438416
\(973\) 13.0165 0.417288
\(974\) −15.6899 −0.502736
\(975\) −8.27803 −0.265109
\(976\) −18.2700 −0.584808
\(977\) −11.5655 −0.370012 −0.185006 0.982737i \(-0.559230\pi\)
−0.185006 + 0.982737i \(0.559230\pi\)
\(978\) 78.1161 2.49788
\(979\) −44.3258 −1.41666
\(980\) 18.4845 0.590465
\(981\) −1.30728 −0.0417382
\(982\) −8.34630 −0.266341
\(983\) 7.44288 0.237391 0.118696 0.992931i \(-0.462129\pi\)
0.118696 + 0.992931i \(0.462129\pi\)
\(984\) 7.78795 0.248271
\(985\) 21.6389 0.689472
\(986\) −11.1957 −0.356544
\(987\) 4.70276 0.149691
\(988\) 29.6821 0.944315
\(989\) 5.41601 0.172219
\(990\) −3.28174 −0.104301
\(991\) 13.0064 0.413162 0.206581 0.978429i \(-0.433766\pi\)
0.206581 + 0.978429i \(0.433766\pi\)
\(992\) 12.0965 0.384064
\(993\) −13.5038 −0.428529
\(994\) 20.4823 0.649660
\(995\) −21.1055 −0.669089
\(996\) 87.3677 2.76835
\(997\) −9.39388 −0.297507 −0.148754 0.988874i \(-0.547526\pi\)
−0.148754 + 0.988874i \(0.547526\pi\)
\(998\) −19.5600 −0.619159
\(999\) −31.7891 −1.00576
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 4003.2.a.c.1.16 179
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4003.2.a.c.1.16 179 1.1 even 1 trivial