Properties

Label 6422.2.a.n.1.3
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.2799281781\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) 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 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 6422.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +3.80194 q^{5} -1.24698 q^{6} +1.10992 q^{7} -1.00000 q^{8} -1.44504 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} +3.80194 q^{5} -1.24698 q^{6} +1.10992 q^{7} -1.00000 q^{8} -1.44504 q^{9} -3.80194 q^{10} -0.890084 q^{11} +1.24698 q^{12} -1.10992 q^{14} +4.74094 q^{15} +1.00000 q^{16} +2.66487 q^{17} +1.44504 q^{18} -1.00000 q^{19} +3.80194 q^{20} +1.38404 q^{21} +0.890084 q^{22} +6.31767 q^{23} -1.24698 q^{24} +9.45473 q^{25} -5.54288 q^{27} +1.10992 q^{28} +8.09783 q^{29} -4.74094 q^{30} -4.85086 q^{31} -1.00000 q^{32} -1.10992 q^{33} -2.66487 q^{34} +4.21983 q^{35} -1.44504 q^{36} +11.4819 q^{37} +1.00000 q^{38} -3.80194 q^{40} +12.0978 q^{41} -1.38404 q^{42} -4.98792 q^{43} -0.890084 q^{44} -5.49396 q^{45} -6.31767 q^{46} +6.98792 q^{47} +1.24698 q^{48} -5.76809 q^{49} -9.45473 q^{50} +3.32304 q^{51} -11.6039 q^{53} +5.54288 q^{54} -3.38404 q^{55} -1.10992 q^{56} -1.24698 q^{57} -8.09783 q^{58} -5.67994 q^{59} +4.74094 q^{60} -1.97823 q^{61} +4.85086 q^{62} -1.60388 q^{63} +1.00000 q^{64} +1.10992 q^{66} -7.74094 q^{67} +2.66487 q^{68} +7.87800 q^{69} -4.21983 q^{70} -2.84117 q^{71} +1.44504 q^{72} +2.66487 q^{73} -11.4819 q^{74} +11.7899 q^{75} -1.00000 q^{76} -0.987918 q^{77} -8.36658 q^{79} +3.80194 q^{80} -2.57673 q^{81} -12.0978 q^{82} +6.49396 q^{83} +1.38404 q^{84} +10.1317 q^{85} +4.98792 q^{86} +10.0978 q^{87} +0.890084 q^{88} +15.4276 q^{89} +5.49396 q^{90} +6.31767 q^{92} -6.04892 q^{93} -6.98792 q^{94} -3.80194 q^{95} -1.24698 q^{96} -7.92154 q^{97} +5.76809 q^{98} +1.28621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 7 q^{5} + q^{6} + 4 q^{7} - 3 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 7 q^{5} + q^{6} + 4 q^{7} - 3 q^{8} - 4 q^{9} - 7 q^{10} - 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 9 q^{17} + 4 q^{18} - 3 q^{19} + 7 q^{20} - 6 q^{21} + 2 q^{22} + 2 q^{23} + q^{24} + 6 q^{25} + 2 q^{27} + 4 q^{28} + 6 q^{29} - q^{31} - 3 q^{32} - 4 q^{33} - 9 q^{34} + 14 q^{35} - 4 q^{36} + 6 q^{37} + 3 q^{38} - 7 q^{40} + 18 q^{41} + 6 q^{42} + 4 q^{43} - 2 q^{44} - 7 q^{45} - 2 q^{46} + 2 q^{47} - q^{48} + 3 q^{49} - 6 q^{50} - 10 q^{51} - 26 q^{53} - 2 q^{54} - 4 q^{56} + q^{57} - 6 q^{58} + 7 q^{59} - 9 q^{61} + q^{62} + 4 q^{63} + 3 q^{64} + 4 q^{66} - 9 q^{67} + 9 q^{68} + 4 q^{69} - 14 q^{70} - 17 q^{71} + 4 q^{72} + 9 q^{73} - 6 q^{74} + 12 q^{75} - 3 q^{76} + 16 q^{77} + q^{79} + 7 q^{80} - 5 q^{81} - 18 q^{82} + 10 q^{83} - 6 q^{84} + 28 q^{85} - 4 q^{86} + 12 q^{87} + 2 q^{88} + 30 q^{89} + 7 q^{90} + 2 q^{92} - 9 q^{93} - 2 q^{94} - 7 q^{95} + q^{96} + 2 q^{97} - 3 q^{98} + 12 q^{99}+O(q^{100}) \) 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
\(3\) 1.24698 0.719944 0.359972 0.932963i \(-0.382786\pi\)
0.359972 + 0.932963i \(0.382786\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.80194 1.70028 0.850139 0.526558i \(-0.176518\pi\)
0.850139 + 0.526558i \(0.176518\pi\)
\(6\) −1.24698 −0.509077
\(7\) 1.10992 0.419509 0.209754 0.977754i \(-0.432734\pi\)
0.209754 + 0.977754i \(0.432734\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.44504 −0.481681
\(10\) −3.80194 −1.20228
\(11\) −0.890084 −0.268370 −0.134185 0.990956i \(-0.542842\pi\)
−0.134185 + 0.990956i \(0.542842\pi\)
\(12\) 1.24698 0.359972
\(13\) 0 0
\(14\) −1.10992 −0.296638
\(15\) 4.74094 1.22411
\(16\) 1.00000 0.250000
\(17\) 2.66487 0.646327 0.323163 0.946343i \(-0.395254\pi\)
0.323163 + 0.946343i \(0.395254\pi\)
\(18\) 1.44504 0.340600
\(19\) −1.00000 −0.229416
\(20\) 3.80194 0.850139
\(21\) 1.38404 0.302023
\(22\) 0.890084 0.189766
\(23\) 6.31767 1.31732 0.658662 0.752439i \(-0.271123\pi\)
0.658662 + 0.752439i \(0.271123\pi\)
\(24\) −1.24698 −0.254539
\(25\) 9.45473 1.89095
\(26\) 0 0
\(27\) −5.54288 −1.06673
\(28\) 1.10992 0.209754
\(29\) 8.09783 1.50373 0.751865 0.659317i \(-0.229154\pi\)
0.751865 + 0.659317i \(0.229154\pi\)
\(30\) −4.74094 −0.865573
\(31\) −4.85086 −0.871239 −0.435620 0.900131i \(-0.643471\pi\)
−0.435620 + 0.900131i \(0.643471\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.10992 −0.193212
\(34\) −2.66487 −0.457022
\(35\) 4.21983 0.713282
\(36\) −1.44504 −0.240840
\(37\) 11.4819 1.88761 0.943805 0.330504i \(-0.107219\pi\)
0.943805 + 0.330504i \(0.107219\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −3.80194 −0.601139
\(41\) 12.0978 1.88936 0.944682 0.327987i \(-0.106370\pi\)
0.944682 + 0.327987i \(0.106370\pi\)
\(42\) −1.38404 −0.213562
\(43\) −4.98792 −0.760650 −0.380325 0.924853i \(-0.624188\pi\)
−0.380325 + 0.924853i \(0.624188\pi\)
\(44\) −0.890084 −0.134185
\(45\) −5.49396 −0.818991
\(46\) −6.31767 −0.931489
\(47\) 6.98792 1.01929 0.509646 0.860384i \(-0.329776\pi\)
0.509646 + 0.860384i \(0.329776\pi\)
\(48\) 1.24698 0.179986
\(49\) −5.76809 −0.824012
\(50\) −9.45473 −1.33710
\(51\) 3.32304 0.465319
\(52\) 0 0
\(53\) −11.6039 −1.59391 −0.796957 0.604035i \(-0.793559\pi\)
−0.796957 + 0.604035i \(0.793559\pi\)
\(54\) 5.54288 0.754290
\(55\) −3.38404 −0.456304
\(56\) −1.10992 −0.148319
\(57\) −1.24698 −0.165166
\(58\) −8.09783 −1.06330
\(59\) −5.67994 −0.739465 −0.369733 0.929138i \(-0.620551\pi\)
−0.369733 + 0.929138i \(0.620551\pi\)
\(60\) 4.74094 0.612053
\(61\) −1.97823 −0.253286 −0.126643 0.991948i \(-0.540420\pi\)
−0.126643 + 0.991948i \(0.540420\pi\)
\(62\) 4.85086 0.616059
\(63\) −1.60388 −0.202069
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.10992 0.136621
\(67\) −7.74094 −0.945706 −0.472853 0.881141i \(-0.656776\pi\)
−0.472853 + 0.881141i \(0.656776\pi\)
\(68\) 2.66487 0.323163
\(69\) 7.87800 0.948400
\(70\) −4.21983 −0.504366
\(71\) −2.84117 −0.337184 −0.168592 0.985686i \(-0.553922\pi\)
−0.168592 + 0.985686i \(0.553922\pi\)
\(72\) 1.44504 0.170300
\(73\) 2.66487 0.311900 0.155950 0.987765i \(-0.450156\pi\)
0.155950 + 0.987765i \(0.450156\pi\)
\(74\) −11.4819 −1.33474
\(75\) 11.7899 1.36138
\(76\) −1.00000 −0.114708
\(77\) −0.987918 −0.112584
\(78\) 0 0
\(79\) −8.36658 −0.941314 −0.470657 0.882316i \(-0.655983\pi\)
−0.470657 + 0.882316i \(0.655983\pi\)
\(80\) 3.80194 0.425070
\(81\) −2.57673 −0.286303
\(82\) −12.0978 −1.33598
\(83\) 6.49396 0.712805 0.356402 0.934333i \(-0.384003\pi\)
0.356402 + 0.934333i \(0.384003\pi\)
\(84\) 1.38404 0.151011
\(85\) 10.1317 1.09894
\(86\) 4.98792 0.537861
\(87\) 10.0978 1.08260
\(88\) 0.890084 0.0948832
\(89\) 15.4276 1.63532 0.817660 0.575701i \(-0.195271\pi\)
0.817660 + 0.575701i \(0.195271\pi\)
\(90\) 5.49396 0.579114
\(91\) 0 0
\(92\) 6.31767 0.658662
\(93\) −6.04892 −0.627244
\(94\) −6.98792 −0.720749
\(95\) −3.80194 −0.390071
\(96\) −1.24698 −0.127269
\(97\) −7.92154 −0.804311 −0.402155 0.915571i \(-0.631739\pi\)
−0.402155 + 0.915571i \(0.631739\pi\)
\(98\) 5.76809 0.582665
\(99\) 1.28621 0.129269
\(100\) 9.45473 0.945473
\(101\) 0.637727 0.0634562 0.0317281 0.999497i \(-0.489899\pi\)
0.0317281 + 0.999497i \(0.489899\pi\)
\(102\) −3.32304 −0.329030
\(103\) 8.47219 0.834790 0.417395 0.908725i \(-0.362943\pi\)
0.417395 + 0.908725i \(0.362943\pi\)
\(104\) 0 0
\(105\) 5.26205 0.513523
\(106\) 11.6039 1.12707
\(107\) −9.65279 −0.933171 −0.466585 0.884476i \(-0.654516\pi\)
−0.466585 + 0.884476i \(0.654516\pi\)
\(108\) −5.54288 −0.533364
\(109\) 4.79225 0.459014 0.229507 0.973307i \(-0.426289\pi\)
0.229507 + 0.973307i \(0.426289\pi\)
\(110\) 3.38404 0.322656
\(111\) 14.3177 1.35897
\(112\) 1.10992 0.104877
\(113\) 1.50604 0.141676 0.0708382 0.997488i \(-0.477433\pi\)
0.0708382 + 0.997488i \(0.477433\pi\)
\(114\) 1.24698 0.116790
\(115\) 24.0194 2.23982
\(116\) 8.09783 0.751865
\(117\) 0 0
\(118\) 5.67994 0.522881
\(119\) 2.95779 0.271140
\(120\) −4.74094 −0.432787
\(121\) −10.2078 −0.927977
\(122\) 1.97823 0.179101
\(123\) 15.0858 1.36024
\(124\) −4.85086 −0.435620
\(125\) 16.9366 1.51486
\(126\) 1.60388 0.142885
\(127\) −18.3569 −1.62891 −0.814456 0.580226i \(-0.802964\pi\)
−0.814456 + 0.580226i \(0.802964\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.21983 −0.547626
\(130\) 0 0
\(131\) −5.10992 −0.446455 −0.223228 0.974766i \(-0.571659\pi\)
−0.223228 + 0.974766i \(0.571659\pi\)
\(132\) −1.10992 −0.0966058
\(133\) −1.10992 −0.0962419
\(134\) 7.74094 0.668715
\(135\) −21.0737 −1.81373
\(136\) −2.66487 −0.228511
\(137\) 0.570024 0.0487004 0.0243502 0.999703i \(-0.492248\pi\)
0.0243502 + 0.999703i \(0.492248\pi\)
\(138\) −7.87800 −0.670620
\(139\) 6.14138 0.520905 0.260452 0.965487i \(-0.416128\pi\)
0.260452 + 0.965487i \(0.416128\pi\)
\(140\) 4.21983 0.356641
\(141\) 8.71379 0.733834
\(142\) 2.84117 0.238425
\(143\) 0 0
\(144\) −1.44504 −0.120420
\(145\) 30.7875 2.55676
\(146\) −2.66487 −0.220547
\(147\) −7.19269 −0.593243
\(148\) 11.4819 0.943805
\(149\) 4.40150 0.360585 0.180293 0.983613i \(-0.442296\pi\)
0.180293 + 0.983613i \(0.442296\pi\)
\(150\) −11.7899 −0.962638
\(151\) 11.0368 0.898165 0.449082 0.893490i \(-0.351751\pi\)
0.449082 + 0.893490i \(0.351751\pi\)
\(152\) 1.00000 0.0811107
\(153\) −3.85086 −0.311323
\(154\) 0.987918 0.0796087
\(155\) −18.4426 −1.48135
\(156\) 0 0
\(157\) 22.0248 1.75777 0.878883 0.477037i \(-0.158289\pi\)
0.878883 + 0.477037i \(0.158289\pi\)
\(158\) 8.36658 0.665610
\(159\) −14.4698 −1.14753
\(160\) −3.80194 −0.300570
\(161\) 7.01208 0.552629
\(162\) 2.57673 0.202447
\(163\) 10.3177 0.808142 0.404071 0.914728i \(-0.367595\pi\)
0.404071 + 0.914728i \(0.367595\pi\)
\(164\) 12.0978 0.944682
\(165\) −4.21983 −0.328514
\(166\) −6.49396 −0.504029
\(167\) 16.7724 1.29789 0.648944 0.760837i \(-0.275211\pi\)
0.648944 + 0.760837i \(0.275211\pi\)
\(168\) −1.38404 −0.106781
\(169\) 0 0
\(170\) −10.1317 −0.777065
\(171\) 1.44504 0.110505
\(172\) −4.98792 −0.380325
\(173\) −9.35988 −0.711618 −0.355809 0.934559i \(-0.615795\pi\)
−0.355809 + 0.934559i \(0.615795\pi\)
\(174\) −10.0978 −0.765515
\(175\) 10.4940 0.793269
\(176\) −0.890084 −0.0670926
\(177\) −7.08277 −0.532374
\(178\) −15.4276 −1.15635
\(179\) −1.89679 −0.141773 −0.0708863 0.997484i \(-0.522583\pi\)
−0.0708863 + 0.997484i \(0.522583\pi\)
\(180\) −5.49396 −0.409496
\(181\) 6.57242 0.488524 0.244262 0.969709i \(-0.421454\pi\)
0.244262 + 0.969709i \(0.421454\pi\)
\(182\) 0 0
\(183\) −2.46681 −0.182352
\(184\) −6.31767 −0.465745
\(185\) 43.6534 3.20946
\(186\) 6.04892 0.443528
\(187\) −2.37196 −0.173455
\(188\) 6.98792 0.509646
\(189\) −6.15213 −0.447502
\(190\) 3.80194 0.275822
\(191\) −7.32975 −0.530362 −0.265181 0.964199i \(-0.585432\pi\)
−0.265181 + 0.964199i \(0.585432\pi\)
\(192\) 1.24698 0.0899930
\(193\) 14.8224 1.06694 0.533469 0.845820i \(-0.320888\pi\)
0.533469 + 0.845820i \(0.320888\pi\)
\(194\) 7.92154 0.568734
\(195\) 0 0
\(196\) −5.76809 −0.412006
\(197\) −6.70841 −0.477955 −0.238977 0.971025i \(-0.576812\pi\)
−0.238977 + 0.971025i \(0.576812\pi\)
\(198\) −1.28621 −0.0914068
\(199\) 22.5676 1.59978 0.799888 0.600149i \(-0.204892\pi\)
0.799888 + 0.600149i \(0.204892\pi\)
\(200\) −9.45473 −0.668550
\(201\) −9.65279 −0.680856
\(202\) −0.637727 −0.0448703
\(203\) 8.98792 0.630828
\(204\) 3.32304 0.232660
\(205\) 45.9952 3.21245
\(206\) −8.47219 −0.590285
\(207\) −9.12929 −0.634530
\(208\) 0 0
\(209\) 0.890084 0.0615684
\(210\) −5.26205 −0.363116
\(211\) −7.30127 −0.502640 −0.251320 0.967904i \(-0.580865\pi\)
−0.251320 + 0.967904i \(0.580865\pi\)
\(212\) −11.6039 −0.796957
\(213\) −3.54288 −0.242754
\(214\) 9.65279 0.659851
\(215\) −18.9638 −1.29332
\(216\) 5.54288 0.377145
\(217\) −5.38404 −0.365493
\(218\) −4.79225 −0.324572
\(219\) 3.32304 0.224551
\(220\) −3.38404 −0.228152
\(221\) 0 0
\(222\) −14.3177 −0.960939
\(223\) −23.4282 −1.56887 −0.784433 0.620213i \(-0.787046\pi\)
−0.784433 + 0.620213i \(0.787046\pi\)
\(224\) −1.10992 −0.0741594
\(225\) −13.6625 −0.910832
\(226\) −1.50604 −0.100180
\(227\) 13.2470 0.879233 0.439616 0.898186i \(-0.355114\pi\)
0.439616 + 0.898186i \(0.355114\pi\)
\(228\) −1.24698 −0.0825832
\(229\) 6.41789 0.424106 0.212053 0.977258i \(-0.431985\pi\)
0.212053 + 0.977258i \(0.431985\pi\)
\(230\) −24.0194 −1.58379
\(231\) −1.23191 −0.0810540
\(232\) −8.09783 −0.531649
\(233\) 9.16182 0.600211 0.300105 0.953906i \(-0.402978\pi\)
0.300105 + 0.953906i \(0.402978\pi\)
\(234\) 0 0
\(235\) 26.5676 1.73308
\(236\) −5.67994 −0.369733
\(237\) −10.4330 −0.677694
\(238\) −2.95779 −0.191725
\(239\) −10.7922 −0.698093 −0.349046 0.937105i \(-0.613494\pi\)
−0.349046 + 0.937105i \(0.613494\pi\)
\(240\) 4.74094 0.306026
\(241\) 24.8418 1.60020 0.800099 0.599868i \(-0.204780\pi\)
0.800099 + 0.599868i \(0.204780\pi\)
\(242\) 10.2078 0.656179
\(243\) 13.4155 0.860605
\(244\) −1.97823 −0.126643
\(245\) −21.9299 −1.40105
\(246\) −15.0858 −0.961832
\(247\) 0 0
\(248\) 4.85086 0.308030
\(249\) 8.09783 0.513179
\(250\) −16.9366 −1.07117
\(251\) 26.9879 1.70346 0.851731 0.523979i \(-0.175553\pi\)
0.851731 + 0.523979i \(0.175553\pi\)
\(252\) −1.60388 −0.101035
\(253\) −5.62325 −0.353531
\(254\) 18.3569 1.15181
\(255\) 12.6340 0.791172
\(256\) 1.00000 0.0625000
\(257\) −13.7259 −0.856196 −0.428098 0.903732i \(-0.640816\pi\)
−0.428098 + 0.903732i \(0.640816\pi\)
\(258\) 6.21983 0.387230
\(259\) 12.7439 0.791869
\(260\) 0 0
\(261\) −11.7017 −0.724318
\(262\) 5.10992 0.315692
\(263\) 14.5676 0.898279 0.449139 0.893462i \(-0.351731\pi\)
0.449139 + 0.893462i \(0.351731\pi\)
\(264\) 1.10992 0.0683106
\(265\) −44.1172 −2.71010
\(266\) 1.10992 0.0680533
\(267\) 19.2379 1.17734
\(268\) −7.74094 −0.472853
\(269\) −12.1521 −0.740928 −0.370464 0.928847i \(-0.620801\pi\)
−0.370464 + 0.928847i \(0.620801\pi\)
\(270\) 21.0737 1.28250
\(271\) −17.6474 −1.07200 −0.536002 0.844217i \(-0.680066\pi\)
−0.536002 + 0.844217i \(0.680066\pi\)
\(272\) 2.66487 0.161582
\(273\) 0 0
\(274\) −0.570024 −0.0344364
\(275\) −8.41550 −0.507474
\(276\) 7.87800 0.474200
\(277\) −28.9855 −1.74157 −0.870786 0.491663i \(-0.836389\pi\)
−0.870786 + 0.491663i \(0.836389\pi\)
\(278\) −6.14138 −0.368335
\(279\) 7.00969 0.419659
\(280\) −4.21983 −0.252183
\(281\) −5.18359 −0.309227 −0.154613 0.987975i \(-0.549413\pi\)
−0.154613 + 0.987975i \(0.549413\pi\)
\(282\) −8.71379 −0.518899
\(283\) 23.3250 1.38653 0.693263 0.720685i \(-0.256173\pi\)
0.693263 + 0.720685i \(0.256173\pi\)
\(284\) −2.84117 −0.168592
\(285\) −4.74094 −0.280829
\(286\) 0 0
\(287\) 13.4276 0.792605
\(288\) 1.44504 0.0851499
\(289\) −9.89844 −0.582261
\(290\) −30.7875 −1.80790
\(291\) −9.87800 −0.579059
\(292\) 2.66487 0.155950
\(293\) −4.81163 −0.281098 −0.140549 0.990074i \(-0.544887\pi\)
−0.140549 + 0.990074i \(0.544887\pi\)
\(294\) 7.19269 0.419486
\(295\) −21.5948 −1.25730
\(296\) −11.4819 −0.667371
\(297\) 4.93362 0.286278
\(298\) −4.40150 −0.254972
\(299\) 0 0
\(300\) 11.7899 0.680688
\(301\) −5.53617 −0.319100
\(302\) −11.0368 −0.635099
\(303\) 0.795233 0.0456849
\(304\) −1.00000 −0.0573539
\(305\) −7.52111 −0.430657
\(306\) 3.85086 0.220139
\(307\) 9.88231 0.564013 0.282007 0.959412i \(-0.409000\pi\)
0.282007 + 0.959412i \(0.409000\pi\)
\(308\) −0.987918 −0.0562919
\(309\) 10.5646 0.601002
\(310\) 18.4426 1.04747
\(311\) 15.4383 0.875428 0.437714 0.899114i \(-0.355788\pi\)
0.437714 + 0.899114i \(0.355788\pi\)
\(312\) 0 0
\(313\) −14.9855 −0.847032 −0.423516 0.905889i \(-0.639204\pi\)
−0.423516 + 0.905889i \(0.639204\pi\)
\(314\) −22.0248 −1.24293
\(315\) −6.09783 −0.343574
\(316\) −8.36658 −0.470657
\(317\) 2.26337 0.127124 0.0635618 0.997978i \(-0.479754\pi\)
0.0635618 + 0.997978i \(0.479754\pi\)
\(318\) 14.4698 0.811426
\(319\) −7.20775 −0.403557
\(320\) 3.80194 0.212535
\(321\) −12.0368 −0.671831
\(322\) −7.01208 −0.390768
\(323\) −2.66487 −0.148278
\(324\) −2.57673 −0.143152
\(325\) 0 0
\(326\) −10.3177 −0.571443
\(327\) 5.97584 0.330465
\(328\) −12.0978 −0.667991
\(329\) 7.75600 0.427602
\(330\) 4.21983 0.232294
\(331\) 1.44265 0.0792952 0.0396476 0.999214i \(-0.487376\pi\)
0.0396476 + 0.999214i \(0.487376\pi\)
\(332\) 6.49396 0.356402
\(333\) −16.5918 −0.909225
\(334\) −16.7724 −0.917745
\(335\) −29.4306 −1.60796
\(336\) 1.38404 0.0755057
\(337\) −30.5133 −1.66217 −0.831084 0.556147i \(-0.812279\pi\)
−0.831084 + 0.556147i \(0.812279\pi\)
\(338\) 0 0
\(339\) 1.87800 0.101999
\(340\) 10.1317 0.549468
\(341\) 4.31767 0.233815
\(342\) −1.44504 −0.0781389
\(343\) −14.1715 −0.765189
\(344\) 4.98792 0.268931
\(345\) 29.9517 1.61254
\(346\) 9.35988 0.503190
\(347\) −20.6461 −1.10834 −0.554170 0.832403i \(-0.686964\pi\)
−0.554170 + 0.832403i \(0.686964\pi\)
\(348\) 10.0978 0.541301
\(349\) 10.6310 0.569066 0.284533 0.958666i \(-0.408162\pi\)
0.284533 + 0.958666i \(0.408162\pi\)
\(350\) −10.4940 −0.560926
\(351\) 0 0
\(352\) 0.890084 0.0474416
\(353\) 0.0935228 0.00497772 0.00248886 0.999997i \(-0.499208\pi\)
0.00248886 + 0.999997i \(0.499208\pi\)
\(354\) 7.08277 0.376445
\(355\) −10.8019 −0.573307
\(356\) 15.4276 0.817660
\(357\) 3.68830 0.195206
\(358\) 1.89679 0.100248
\(359\) −28.6703 −1.51316 −0.756579 0.653902i \(-0.773131\pi\)
−0.756579 + 0.653902i \(0.773131\pi\)
\(360\) 5.49396 0.289557
\(361\) 1.00000 0.0526316
\(362\) −6.57242 −0.345439
\(363\) −12.7289 −0.668092
\(364\) 0 0
\(365\) 10.1317 0.530317
\(366\) 2.46681 0.128942
\(367\) 14.4940 0.756579 0.378289 0.925687i \(-0.376512\pi\)
0.378289 + 0.925687i \(0.376512\pi\)
\(368\) 6.31767 0.329331
\(369\) −17.4819 −0.910070
\(370\) −43.6534 −2.26943
\(371\) −12.8793 −0.668662
\(372\) −6.04892 −0.313622
\(373\) −20.5241 −1.06270 −0.531349 0.847153i \(-0.678315\pi\)
−0.531349 + 0.847153i \(0.678315\pi\)
\(374\) 2.37196 0.122651
\(375\) 21.1196 1.09061
\(376\) −6.98792 −0.360374
\(377\) 0 0
\(378\) 6.15213 0.316431
\(379\) 1.34050 0.0688570 0.0344285 0.999407i \(-0.489039\pi\)
0.0344285 + 0.999407i \(0.489039\pi\)
\(380\) −3.80194 −0.195035
\(381\) −22.8907 −1.17272
\(382\) 7.32975 0.375023
\(383\) −0.667858 −0.0341260 −0.0170630 0.999854i \(-0.505432\pi\)
−0.0170630 + 0.999854i \(0.505432\pi\)
\(384\) −1.24698 −0.0636347
\(385\) −3.75600 −0.191424
\(386\) −14.8224 −0.754439
\(387\) 7.20775 0.366391
\(388\) −7.92154 −0.402155
\(389\) 34.0388 1.72583 0.862917 0.505346i \(-0.168635\pi\)
0.862917 + 0.505346i \(0.168635\pi\)
\(390\) 0 0
\(391\) 16.8358 0.851422
\(392\) 5.76809 0.291332
\(393\) −6.37196 −0.321423
\(394\) 6.70841 0.337965
\(395\) −31.8092 −1.60050
\(396\) 1.28621 0.0646344
\(397\) −11.6063 −0.582502 −0.291251 0.956647i \(-0.594071\pi\)
−0.291251 + 0.956647i \(0.594071\pi\)
\(398\) −22.5676 −1.13121
\(399\) −1.38404 −0.0692888
\(400\) 9.45473 0.472737
\(401\) 11.2948 0.564037 0.282018 0.959409i \(-0.408996\pi\)
0.282018 + 0.959409i \(0.408996\pi\)
\(402\) 9.65279 0.481438
\(403\) 0 0
\(404\) 0.637727 0.0317281
\(405\) −9.79656 −0.486795
\(406\) −8.98792 −0.446063
\(407\) −10.2198 −0.506578
\(408\) −3.32304 −0.164515
\(409\) 6.07846 0.300560 0.150280 0.988643i \(-0.451982\pi\)
0.150280 + 0.988643i \(0.451982\pi\)
\(410\) −45.9952 −2.27154
\(411\) 0.710808 0.0350616
\(412\) 8.47219 0.417395
\(413\) −6.30426 −0.310212
\(414\) 9.12929 0.448680
\(415\) 24.6896 1.21197
\(416\) 0 0
\(417\) 7.65817 0.375022
\(418\) −0.890084 −0.0435354
\(419\) 18.9202 0.924313 0.462156 0.886798i \(-0.347076\pi\)
0.462156 + 0.886798i \(0.347076\pi\)
\(420\) 5.26205 0.256762
\(421\) 9.23191 0.449936 0.224968 0.974366i \(-0.427772\pi\)
0.224968 + 0.974366i \(0.427772\pi\)
\(422\) 7.30127 0.355420
\(423\) −10.0978 −0.490974
\(424\) 11.6039 0.563534
\(425\) 25.1957 1.22217
\(426\) 3.54288 0.171653
\(427\) −2.19567 −0.106256
\(428\) −9.65279 −0.466585
\(429\) 0 0
\(430\) 18.9638 0.914513
\(431\) −25.5646 −1.23141 −0.615703 0.787978i \(-0.711128\pi\)
−0.615703 + 0.787978i \(0.711128\pi\)
\(432\) −5.54288 −0.266682
\(433\) −30.3129 −1.45674 −0.728372 0.685182i \(-0.759723\pi\)
−0.728372 + 0.685182i \(0.759723\pi\)
\(434\) 5.38404 0.258442
\(435\) 38.3913 1.84072
\(436\) 4.79225 0.229507
\(437\) −6.31767 −0.302215
\(438\) −3.32304 −0.158781
\(439\) −0.576728 −0.0275257 −0.0137629 0.999905i \(-0.504381\pi\)
−0.0137629 + 0.999905i \(0.504381\pi\)
\(440\) 3.38404 0.161328
\(441\) 8.33513 0.396911
\(442\) 0 0
\(443\) −39.3250 −1.86839 −0.934193 0.356769i \(-0.883878\pi\)
−0.934193 + 0.356769i \(0.883878\pi\)
\(444\) 14.3177 0.679486
\(445\) 58.6547 2.78050
\(446\) 23.4282 1.10936
\(447\) 5.48858 0.259601
\(448\) 1.10992 0.0524386
\(449\) −5.92154 −0.279455 −0.139727 0.990190i \(-0.544623\pi\)
−0.139727 + 0.990190i \(0.544623\pi\)
\(450\) 13.6625 0.644056
\(451\) −10.7681 −0.507049
\(452\) 1.50604 0.0708382
\(453\) 13.7627 0.646628
\(454\) −13.2470 −0.621712
\(455\) 0 0
\(456\) 1.24698 0.0583952
\(457\) 27.6256 1.29227 0.646137 0.763222i \(-0.276384\pi\)
0.646137 + 0.763222i \(0.276384\pi\)
\(458\) −6.41789 −0.299889
\(459\) −14.7711 −0.689454
\(460\) 24.0194 1.11991
\(461\) 33.2325 1.54779 0.773896 0.633313i \(-0.218305\pi\)
0.773896 + 0.633313i \(0.218305\pi\)
\(462\) 1.23191 0.0573138
\(463\) 42.4784 1.97414 0.987070 0.160291i \(-0.0512433\pi\)
0.987070 + 0.160291i \(0.0512433\pi\)
\(464\) 8.09783 0.375933
\(465\) −22.9976 −1.06649
\(466\) −9.16182 −0.424413
\(467\) 35.7318 1.65347 0.826736 0.562590i \(-0.190195\pi\)
0.826736 + 0.562590i \(0.190195\pi\)
\(468\) 0 0
\(469\) −8.59179 −0.396732
\(470\) −26.5676 −1.22547
\(471\) 27.4644 1.26549
\(472\) 5.67994 0.261440
\(473\) 4.43967 0.204136
\(474\) 10.4330 0.479202
\(475\) −9.45473 −0.433813
\(476\) 2.95779 0.135570
\(477\) 16.7681 0.767758
\(478\) 10.7922 0.493626
\(479\) −8.43967 −0.385618 −0.192809 0.981236i \(-0.561760\pi\)
−0.192809 + 0.981236i \(0.561760\pi\)
\(480\) −4.74094 −0.216393
\(481\) 0 0
\(482\) −24.8418 −1.13151
\(483\) 8.74392 0.397862
\(484\) −10.2078 −0.463989
\(485\) −30.1172 −1.36755
\(486\) −13.4155 −0.608540
\(487\) −25.8689 −1.17223 −0.586116 0.810227i \(-0.699344\pi\)
−0.586116 + 0.810227i \(0.699344\pi\)
\(488\) 1.97823 0.0895503
\(489\) 12.8659 0.581817
\(490\) 21.9299 0.990692
\(491\) 13.9323 0.628756 0.314378 0.949298i \(-0.398204\pi\)
0.314378 + 0.949298i \(0.398204\pi\)
\(492\) 15.0858 0.680118
\(493\) 21.5797 0.971901
\(494\) 0 0
\(495\) 4.89008 0.219793
\(496\) −4.85086 −0.217810
\(497\) −3.15346 −0.141452
\(498\) −8.09783 −0.362873
\(499\) −32.2500 −1.44371 −0.721853 0.692046i \(-0.756709\pi\)
−0.721853 + 0.692046i \(0.756709\pi\)
\(500\) 16.9366 0.757428
\(501\) 20.9148 0.934406
\(502\) −26.9879 −1.20453
\(503\) −9.41417 −0.419757 −0.209879 0.977727i \(-0.567307\pi\)
−0.209879 + 0.977727i \(0.567307\pi\)
\(504\) 1.60388 0.0714423
\(505\) 2.42460 0.107893
\(506\) 5.62325 0.249984
\(507\) 0 0
\(508\) −18.3569 −0.814456
\(509\) −27.6039 −1.22352 −0.611760 0.791043i \(-0.709538\pi\)
−0.611760 + 0.791043i \(0.709538\pi\)
\(510\) −12.6340 −0.559443
\(511\) 2.95779 0.130845
\(512\) −1.00000 −0.0441942
\(513\) 5.54288 0.244724
\(514\) 13.7259 0.605422
\(515\) 32.2107 1.41937
\(516\) −6.21983 −0.273813
\(517\) −6.21983 −0.273548
\(518\) −12.7439 −0.559936
\(519\) −11.6716 −0.512325
\(520\) 0 0
\(521\) 20.6461 0.904522 0.452261 0.891886i \(-0.350618\pi\)
0.452261 + 0.891886i \(0.350618\pi\)
\(522\) 11.7017 0.512170
\(523\) 7.20775 0.315173 0.157586 0.987505i \(-0.449629\pi\)
0.157586 + 0.987505i \(0.449629\pi\)
\(524\) −5.10992 −0.223228
\(525\) 13.0858 0.571109
\(526\) −14.5676 −0.635179
\(527\) −12.9269 −0.563105
\(528\) −1.10992 −0.0483029
\(529\) 16.9129 0.735344
\(530\) 44.1172 1.91633
\(531\) 8.20775 0.356186
\(532\) −1.10992 −0.0481210
\(533\) 0 0
\(534\) −19.2379 −0.832505
\(535\) −36.6993 −1.58665
\(536\) 7.74094 0.334358
\(537\) −2.36526 −0.102068
\(538\) 12.1521 0.523915
\(539\) 5.13408 0.221140
\(540\) −21.0737 −0.906866
\(541\) −22.5827 −0.970906 −0.485453 0.874263i \(-0.661345\pi\)
−0.485453 + 0.874263i \(0.661345\pi\)
\(542\) 17.6474 0.758021
\(543\) 8.19567 0.351710
\(544\) −2.66487 −0.114256
\(545\) 18.2198 0.780452
\(546\) 0 0
\(547\) −11.3351 −0.484655 −0.242327 0.970195i \(-0.577911\pi\)
−0.242327 + 0.970195i \(0.577911\pi\)
\(548\) 0.570024 0.0243502
\(549\) 2.85862 0.122003
\(550\) 8.41550 0.358838
\(551\) −8.09783 −0.344979
\(552\) −7.87800 −0.335310
\(553\) −9.28621 −0.394890
\(554\) 28.9855 1.23148
\(555\) 54.4349 2.31063
\(556\) 6.14138 0.260452
\(557\) −17.9734 −0.761559 −0.380780 0.924666i \(-0.624344\pi\)
−0.380780 + 0.924666i \(0.624344\pi\)
\(558\) −7.00969 −0.296744
\(559\) 0 0
\(560\) 4.21983 0.178320
\(561\) −2.95779 −0.124878
\(562\) 5.18359 0.218656
\(563\) 31.3013 1.31919 0.659596 0.751621i \(-0.270728\pi\)
0.659596 + 0.751621i \(0.270728\pi\)
\(564\) 8.71379 0.366917
\(565\) 5.72587 0.240889
\(566\) −23.3250 −0.980421
\(567\) −2.85995 −0.120107
\(568\) 2.84117 0.119213
\(569\) −0.396125 −0.0166064 −0.00830320 0.999966i \(-0.502643\pi\)
−0.00830320 + 0.999966i \(0.502643\pi\)
\(570\) 4.74094 0.198576
\(571\) 46.1232 1.93020 0.965098 0.261891i \(-0.0843460\pi\)
0.965098 + 0.261891i \(0.0843460\pi\)
\(572\) 0 0
\(573\) −9.14005 −0.381831
\(574\) −13.4276 −0.560457
\(575\) 59.7318 2.49099
\(576\) −1.44504 −0.0602101
\(577\) 12.6987 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(578\) 9.89844 0.411721
\(579\) 18.4832 0.768136
\(580\) 30.7875 1.27838
\(581\) 7.20775 0.299028
\(582\) 9.87800 0.409456
\(583\) 10.3284 0.427759
\(584\) −2.66487 −0.110273
\(585\) 0 0
\(586\) 4.81163 0.198766
\(587\) −15.1051 −0.623455 −0.311728 0.950171i \(-0.600908\pi\)
−0.311728 + 0.950171i \(0.600908\pi\)
\(588\) −7.19269 −0.296621
\(589\) 4.85086 0.199876
\(590\) 21.5948 0.889043
\(591\) −8.36526 −0.344101
\(592\) 11.4819 0.471902
\(593\) −8.06292 −0.331104 −0.165552 0.986201i \(-0.552941\pi\)
−0.165552 + 0.986201i \(0.552941\pi\)
\(594\) −4.93362 −0.202429
\(595\) 11.2453 0.461013
\(596\) 4.40150 0.180293
\(597\) 28.1414 1.15175
\(598\) 0 0
\(599\) −30.5526 −1.24834 −0.624172 0.781287i \(-0.714564\pi\)
−0.624172 + 0.781287i \(0.714564\pi\)
\(600\) −11.7899 −0.481319
\(601\) −28.9396 −1.18047 −0.590235 0.807231i \(-0.700965\pi\)
−0.590235 + 0.807231i \(0.700965\pi\)
\(602\) 5.53617 0.225638
\(603\) 11.1860 0.455528
\(604\) 11.0368 0.449082
\(605\) −38.8092 −1.57782
\(606\) −0.795233 −0.0323041
\(607\) −30.1825 −1.22507 −0.612535 0.790443i \(-0.709850\pi\)
−0.612535 + 0.790443i \(0.709850\pi\)
\(608\) 1.00000 0.0405554
\(609\) 11.2078 0.454161
\(610\) 7.52111 0.304521
\(611\) 0 0
\(612\) −3.85086 −0.155662
\(613\) 37.9259 1.53181 0.765905 0.642953i \(-0.222291\pi\)
0.765905 + 0.642953i \(0.222291\pi\)
\(614\) −9.88231 −0.398818
\(615\) 57.3551 2.31278
\(616\) 0.987918 0.0398044
\(617\) −0.190293 −0.00766089 −0.00383044 0.999993i \(-0.501219\pi\)
−0.00383044 + 0.999993i \(0.501219\pi\)
\(618\) −10.5646 −0.424972
\(619\) −40.8310 −1.64114 −0.820568 0.571548i \(-0.806343\pi\)
−0.820568 + 0.571548i \(0.806343\pi\)
\(620\) −18.4426 −0.740675
\(621\) −35.0180 −1.40523
\(622\) −15.4383 −0.619021
\(623\) 17.1233 0.686032
\(624\) 0 0
\(625\) 17.1183 0.684731
\(626\) 14.9855 0.598942
\(627\) 1.10992 0.0443258
\(628\) 22.0248 0.878883
\(629\) 30.5978 1.22001
\(630\) 6.09783 0.242944
\(631\) 1.43237 0.0570217 0.0285109 0.999593i \(-0.490923\pi\)
0.0285109 + 0.999593i \(0.490923\pi\)
\(632\) 8.36658 0.332805
\(633\) −9.10454 −0.361873
\(634\) −2.26337 −0.0898900
\(635\) −69.7918 −2.76960
\(636\) −14.4698 −0.573765
\(637\) 0 0
\(638\) 7.20775 0.285358
\(639\) 4.10560 0.162415
\(640\) −3.80194 −0.150285
\(641\) −27.2814 −1.07755 −0.538776 0.842449i \(-0.681113\pi\)
−0.538776 + 0.842449i \(0.681113\pi\)
\(642\) 12.0368 0.475056
\(643\) −15.5555 −0.613451 −0.306725 0.951798i \(-0.599233\pi\)
−0.306725 + 0.951798i \(0.599233\pi\)
\(644\) 7.01208 0.276315
\(645\) −23.6474 −0.931116
\(646\) 2.66487 0.104848
\(647\) −45.8625 −1.80304 −0.901520 0.432738i \(-0.857548\pi\)
−0.901520 + 0.432738i \(0.857548\pi\)
\(648\) 2.57673 0.101223
\(649\) 5.05562 0.198451
\(650\) 0 0
\(651\) −6.71379 −0.263134
\(652\) 10.3177 0.404071
\(653\) −16.6950 −0.653326 −0.326663 0.945141i \(-0.605924\pi\)
−0.326663 + 0.945141i \(0.605924\pi\)
\(654\) −5.97584 −0.233674
\(655\) −19.4276 −0.759099
\(656\) 12.0978 0.472341
\(657\) −3.85086 −0.150236
\(658\) −7.75600 −0.302361
\(659\) −26.7162 −1.04071 −0.520357 0.853949i \(-0.674201\pi\)
−0.520357 + 0.853949i \(0.674201\pi\)
\(660\) −4.21983 −0.164257
\(661\) 7.31037 0.284340 0.142170 0.989842i \(-0.454592\pi\)
0.142170 + 0.989842i \(0.454592\pi\)
\(662\) −1.44265 −0.0560701
\(663\) 0 0
\(664\) −6.49396 −0.252014
\(665\) −4.21983 −0.163638
\(666\) 16.5918 0.642919
\(667\) 51.1594 1.98090
\(668\) 16.7724 0.648944
\(669\) −29.2145 −1.12950
\(670\) 29.4306 1.13700
\(671\) 1.76079 0.0679745
\(672\) −1.38404 −0.0533906
\(673\) −17.2513 −0.664988 −0.332494 0.943105i \(-0.607890\pi\)
−0.332494 + 0.943105i \(0.607890\pi\)
\(674\) 30.5133 1.17533
\(675\) −52.4064 −2.01712
\(676\) 0 0
\(677\) 15.9022 0.611170 0.305585 0.952165i \(-0.401148\pi\)
0.305585 + 0.952165i \(0.401148\pi\)
\(678\) −1.87800 −0.0721242
\(679\) −8.79225 −0.337416
\(680\) −10.1317 −0.388532
\(681\) 16.5187 0.632998
\(682\) −4.31767 −0.165332
\(683\) −45.5881 −1.74438 −0.872190 0.489168i \(-0.837300\pi\)
−0.872190 + 0.489168i \(0.837300\pi\)
\(684\) 1.44504 0.0552526
\(685\) 2.16719 0.0828042
\(686\) 14.1715 0.541071
\(687\) 8.00298 0.305333
\(688\) −4.98792 −0.190163
\(689\) 0 0
\(690\) −29.9517 −1.14024
\(691\) 45.5555 1.73301 0.866507 0.499164i \(-0.166360\pi\)
0.866507 + 0.499164i \(0.166360\pi\)
\(692\) −9.35988 −0.355809
\(693\) 1.42758 0.0542294
\(694\) 20.6461 0.783715
\(695\) 23.3491 0.885683
\(696\) −10.0978 −0.382757
\(697\) 32.2392 1.22115
\(698\) −10.6310 −0.402390
\(699\) 11.4246 0.432118
\(700\) 10.4940 0.396634
\(701\) −19.3448 −0.730644 −0.365322 0.930881i \(-0.619041\pi\)
−0.365322 + 0.930881i \(0.619041\pi\)
\(702\) 0 0
\(703\) −11.4819 −0.433047
\(704\) −0.890084 −0.0335463
\(705\) 33.1293 1.24772
\(706\) −0.0935228 −0.00351978
\(707\) 0.707824 0.0266205
\(708\) −7.08277 −0.266187
\(709\) −5.53558 −0.207893 −0.103947 0.994583i \(-0.533147\pi\)
−0.103947 + 0.994583i \(0.533147\pi\)
\(710\) 10.8019 0.405389
\(711\) 12.0901 0.453413
\(712\) −15.4276 −0.578173
\(713\) −30.6461 −1.14771
\(714\) −3.68830 −0.138031
\(715\) 0 0
\(716\) −1.89679 −0.0708863
\(717\) −13.4577 −0.502588
\(718\) 28.6703 1.06996
\(719\) 1.53617 0.0572895 0.0286448 0.999590i \(-0.490881\pi\)
0.0286448 + 0.999590i \(0.490881\pi\)
\(720\) −5.49396 −0.204748
\(721\) 9.40342 0.350202
\(722\) −1.00000 −0.0372161
\(723\) 30.9772 1.15205
\(724\) 6.57242 0.244262
\(725\) 76.5628 2.84347
\(726\) 12.7289 0.472412
\(727\) 17.5496 0.650878 0.325439 0.945563i \(-0.394488\pi\)
0.325439 + 0.945563i \(0.394488\pi\)
\(728\) 0 0
\(729\) 24.4590 0.905890
\(730\) −10.1317 −0.374991
\(731\) −13.2922 −0.491629
\(732\) −2.46681 −0.0911760
\(733\) 14.0140 0.517619 0.258809 0.965928i \(-0.416670\pi\)
0.258809 + 0.965928i \(0.416670\pi\)
\(734\) −14.4940 −0.534982
\(735\) −27.3461 −1.00868
\(736\) −6.31767 −0.232872
\(737\) 6.89008 0.253799
\(738\) 17.4819 0.643517
\(739\) 17.6125 0.647886 0.323943 0.946077i \(-0.394991\pi\)
0.323943 + 0.946077i \(0.394991\pi\)
\(740\) 43.6534 1.60473
\(741\) 0 0
\(742\) 12.8793 0.472815
\(743\) 34.5827 1.26872 0.634358 0.773039i \(-0.281265\pi\)
0.634358 + 0.773039i \(0.281265\pi\)
\(744\) 6.04892 0.221764
\(745\) 16.7342 0.613095
\(746\) 20.5241 0.751440
\(747\) −9.38404 −0.343344
\(748\) −2.37196 −0.0867275
\(749\) −10.7138 −0.391473
\(750\) −21.1196 −0.771179
\(751\) −24.0388 −0.877187 −0.438593 0.898686i \(-0.644523\pi\)
−0.438593 + 0.898686i \(0.644523\pi\)
\(752\) 6.98792 0.254823
\(753\) 33.6534 1.22640
\(754\) 0 0
\(755\) 41.9614 1.52713
\(756\) −6.15213 −0.223751
\(757\) −7.49289 −0.272334 −0.136167 0.990686i \(-0.543478\pi\)
−0.136167 + 0.990686i \(0.543478\pi\)
\(758\) −1.34050 −0.0486892
\(759\) −7.01208 −0.254522
\(760\) 3.80194 0.137911
\(761\) 34.0200 1.23322 0.616611 0.787268i \(-0.288505\pi\)
0.616611 + 0.787268i \(0.288505\pi\)
\(762\) 22.8907 0.829242
\(763\) 5.31900 0.192561
\(764\) −7.32975 −0.265181
\(765\) −14.6407 −0.529336
\(766\) 0.667858 0.0241307
\(767\) 0 0
\(768\) 1.24698 0.0449965
\(769\) 35.7342 1.28861 0.644305 0.764769i \(-0.277147\pi\)
0.644305 + 0.764769i \(0.277147\pi\)
\(770\) 3.75600 0.135357
\(771\) −17.1159 −0.616414
\(772\) 14.8224 0.533469
\(773\) −17.5797 −0.632298 −0.316149 0.948709i \(-0.602390\pi\)
−0.316149 + 0.948709i \(0.602390\pi\)
\(774\) −7.20775 −0.259077
\(775\) −45.8635 −1.64747
\(776\) 7.92154 0.284367
\(777\) 15.8914 0.570101
\(778\) −34.0388 −1.22035
\(779\) −12.0978 −0.433450
\(780\) 0 0
\(781\) 2.52888 0.0904903
\(782\) −16.8358 −0.602047
\(783\) −44.8853 −1.60407
\(784\) −5.76809 −0.206003
\(785\) 83.7367 2.98869
\(786\) 6.37196 0.227280
\(787\) 35.5687 1.26789 0.633944 0.773379i \(-0.281435\pi\)
0.633944 + 0.773379i \(0.281435\pi\)
\(788\) −6.70841 −0.238977
\(789\) 18.1655 0.646710
\(790\) 31.8092 1.13172
\(791\) 1.67158 0.0594345
\(792\) −1.28621 −0.0457034
\(793\) 0 0
\(794\) 11.6063 0.411891
\(795\) −55.0133 −1.95112
\(796\) 22.5676 0.799888
\(797\) 50.8805 1.80228 0.901140 0.433528i \(-0.142731\pi\)
0.901140 + 0.433528i \(0.142731\pi\)
\(798\) 1.38404 0.0489946
\(799\) 18.6219 0.658796
\(800\) −9.45473 −0.334275
\(801\) −22.2935 −0.787702
\(802\) −11.2948 −0.398834
\(803\) −2.37196 −0.0837047
\(804\) −9.65279 −0.340428
\(805\) 26.6595 0.939624
\(806\) 0 0
\(807\) −15.1535 −0.533427
\(808\) −0.637727 −0.0224352
\(809\) −6.20477 −0.218148 −0.109074 0.994034i \(-0.534789\pi\)
−0.109074 + 0.994034i \(0.534789\pi\)
\(810\) 9.79656 0.344216
\(811\) 39.3032 1.38012 0.690061 0.723751i \(-0.257584\pi\)
0.690061 + 0.723751i \(0.257584\pi\)
\(812\) 8.98792 0.315414
\(813\) −22.0060 −0.771783
\(814\) 10.2198 0.358205
\(815\) 39.2271 1.37407
\(816\) 3.32304 0.116330
\(817\) 4.98792 0.174505
\(818\) −6.07846 −0.212528
\(819\) 0 0
\(820\) 45.9952 1.60622
\(821\) 13.3817 0.467023 0.233511 0.972354i \(-0.424978\pi\)
0.233511 + 0.972354i \(0.424978\pi\)
\(822\) −0.710808 −0.0247923
\(823\) −47.0180 −1.63895 −0.819473 0.573118i \(-0.805734\pi\)
−0.819473 + 0.573118i \(0.805734\pi\)
\(824\) −8.47219 −0.295143
\(825\) −10.4940 −0.365353
\(826\) 6.30426 0.219353
\(827\) −18.5321 −0.644425 −0.322213 0.946667i \(-0.604427\pi\)
−0.322213 + 0.946667i \(0.604427\pi\)
\(828\) −9.12929 −0.317265
\(829\) −24.3236 −0.844795 −0.422397 0.906411i \(-0.638811\pi\)
−0.422397 + 0.906411i \(0.638811\pi\)
\(830\) −24.6896 −0.856990
\(831\) −36.1444 −1.25383
\(832\) 0 0
\(833\) −15.3712 −0.532581
\(834\) −7.65817 −0.265181
\(835\) 63.7676 2.20677
\(836\) 0.890084 0.0307842
\(837\) 26.8877 0.929375
\(838\) −18.9202 −0.653588
\(839\) 1.39506 0.0481628 0.0240814 0.999710i \(-0.492334\pi\)
0.0240814 + 0.999710i \(0.492334\pi\)
\(840\) −5.26205 −0.181558
\(841\) 36.5749 1.26120
\(842\) −9.23191 −0.318153
\(843\) −6.46383 −0.222626
\(844\) −7.30127 −0.251320
\(845\) 0 0
\(846\) 10.0978 0.347171
\(847\) −11.3297 −0.389295
\(848\) −11.6039 −0.398479
\(849\) 29.0858 0.998220
\(850\) −25.1957 −0.864204
\(851\) 72.5387 2.48659
\(852\) −3.54288 −0.121377
\(853\) −55.2006 −1.89003 −0.945016 0.327025i \(-0.893954\pi\)
−0.945016 + 0.327025i \(0.893954\pi\)
\(854\) 2.19567 0.0751343
\(855\) 5.49396 0.187889
\(856\) 9.65279 0.329926
\(857\) 33.5797 1.14706 0.573531 0.819184i \(-0.305573\pi\)
0.573531 + 0.819184i \(0.305573\pi\)
\(858\) 0 0
\(859\) −36.0581 −1.23029 −0.615144 0.788415i \(-0.710902\pi\)
−0.615144 + 0.788415i \(0.710902\pi\)
\(860\) −18.9638 −0.646659
\(861\) 16.7439 0.570631
\(862\) 25.5646 0.870735
\(863\) −51.9517 −1.76846 −0.884228 0.467056i \(-0.845315\pi\)
−0.884228 + 0.467056i \(0.845315\pi\)
\(864\) 5.54288 0.188572
\(865\) −35.5857 −1.20995
\(866\) 30.3129 1.03007
\(867\) −12.3432 −0.419196
\(868\) −5.38404 −0.182746
\(869\) 7.44696 0.252621
\(870\) −38.3913 −1.30159
\(871\) 0 0
\(872\) −4.79225 −0.162286
\(873\) 11.4470 0.387421
\(874\) 6.31767 0.213698
\(875\) 18.7982 0.635496
\(876\) 3.32304 0.112275
\(877\) 23.7017 0.800350 0.400175 0.916439i \(-0.368949\pi\)
0.400175 + 0.916439i \(0.368949\pi\)
\(878\) 0.576728 0.0194636
\(879\) −6.00000 −0.202375
\(880\) −3.38404 −0.114076
\(881\) −18.9573 −0.638688 −0.319344 0.947639i \(-0.603463\pi\)
−0.319344 + 0.947639i \(0.603463\pi\)
\(882\) −8.33513 −0.280658
\(883\) −26.6160 −0.895698 −0.447849 0.894109i \(-0.647810\pi\)
−0.447849 + 0.894109i \(0.647810\pi\)
\(884\) 0 0
\(885\) −26.9282 −0.905183
\(886\) 39.3250 1.32115
\(887\) 6.19998 0.208175 0.104087 0.994568i \(-0.466808\pi\)
0.104087 + 0.994568i \(0.466808\pi\)
\(888\) −14.3177 −0.480469
\(889\) −20.3746 −0.683343
\(890\) −58.6547 −1.96611
\(891\) 2.29350 0.0768353
\(892\) −23.4282 −0.784433
\(893\) −6.98792 −0.233842
\(894\) −5.48858 −0.183566
\(895\) −7.21147 −0.241053
\(896\) −1.10992 −0.0370797
\(897\) 0 0
\(898\) 5.92154 0.197604
\(899\) −39.2814 −1.31011
\(900\) −13.6625 −0.455416
\(901\) −30.9229 −1.03019
\(902\) 10.7681 0.358538
\(903\) −6.90349 −0.229734
\(904\) −1.50604 −0.0500902
\(905\) 24.9879 0.830627
\(906\) −13.7627 −0.457235
\(907\) 28.1086 0.933330 0.466665 0.884434i \(-0.345455\pi\)
0.466665 + 0.884434i \(0.345455\pi\)
\(908\) 13.2470 0.439616
\(909\) −0.921543 −0.0305656
\(910\) 0 0
\(911\) −6.49635 −0.215234 −0.107617 0.994192i \(-0.534322\pi\)
−0.107617 + 0.994192i \(0.534322\pi\)
\(912\) −1.24698 −0.0412916
\(913\) −5.78017 −0.191296
\(914\) −27.6256 −0.913775
\(915\) −9.37867 −0.310049
\(916\) 6.41789 0.212053
\(917\) −5.67158 −0.187292
\(918\) 14.7711 0.487518
\(919\) 27.7017 0.913795 0.456898 0.889519i \(-0.348961\pi\)
0.456898 + 0.889519i \(0.348961\pi\)
\(920\) −24.0194 −0.791895
\(921\) 12.3230 0.406058
\(922\) −33.2325 −1.09445
\(923\) 0 0
\(924\) −1.23191 −0.0405270
\(925\) 108.558 3.56937
\(926\) −42.4784 −1.39593
\(927\) −12.2427 −0.402102
\(928\) −8.09783 −0.265824
\(929\) −51.2529 −1.68155 −0.840777 0.541381i \(-0.817902\pi\)
−0.840777 + 0.541381i \(0.817902\pi\)
\(930\) 22.9976 0.754121
\(931\) 5.76809 0.189041
\(932\) 9.16182 0.300105
\(933\) 19.2513 0.630259
\(934\) −35.7318 −1.16918
\(935\) −9.01805 −0.294922
\(936\) 0 0
\(937\) −12.8461 −0.419663 −0.209831 0.977738i \(-0.567292\pi\)
−0.209831 + 0.977738i \(0.567292\pi\)
\(938\) 8.59179 0.280532
\(939\) −18.6866 −0.609816
\(940\) 26.5676 0.866541
\(941\) −46.0844 −1.50231 −0.751155 0.660126i \(-0.770503\pi\)
−0.751155 + 0.660126i \(0.770503\pi\)
\(942\) −27.4644 −0.894839
\(943\) 76.4301 2.48891
\(944\) −5.67994 −0.184866
\(945\) −23.3900 −0.760877
\(946\) −4.43967 −0.144346
\(947\) 3.09054 0.100429 0.0502145 0.998738i \(-0.484010\pi\)
0.0502145 + 0.998738i \(0.484010\pi\)
\(948\) −10.4330 −0.338847
\(949\) 0 0
\(950\) 9.45473 0.306752
\(951\) 2.82238 0.0915219
\(952\) −2.95779 −0.0958624
\(953\) 57.7512 1.87075 0.935373 0.353663i \(-0.115064\pi\)
0.935373 + 0.353663i \(0.115064\pi\)
\(954\) −16.7681 −0.542887
\(955\) −27.8672 −0.901763
\(956\) −10.7922 −0.349046
\(957\) −8.98792 −0.290538
\(958\) 8.43967 0.272673
\(959\) 0.632678 0.0204303
\(960\) 4.74094 0.153013
\(961\) −7.46921 −0.240942
\(962\) 0 0
\(963\) 13.9487 0.449490
\(964\) 24.8418 0.800099
\(965\) 56.3538 1.81409
\(966\) −8.74392 −0.281331
\(967\) −22.3177 −0.717688 −0.358844 0.933398i \(-0.616829\pi\)
−0.358844 + 0.933398i \(0.616829\pi\)
\(968\) 10.2078 0.328090
\(969\) −3.32304 −0.106752
\(970\) 30.1172 0.967005
\(971\) −36.4064 −1.16834 −0.584169 0.811632i \(-0.698579\pi\)
−0.584169 + 0.811632i \(0.698579\pi\)
\(972\) 13.4155 0.430302
\(973\) 6.81641 0.218524
\(974\) 25.8689 0.828893
\(975\) 0 0
\(976\) −1.97823 −0.0633216
\(977\) −39.2728 −1.25645 −0.628224 0.778032i \(-0.716218\pi\)
−0.628224 + 0.778032i \(0.716218\pi\)
\(978\) −12.8659 −0.411407
\(979\) −13.7318 −0.438872
\(980\) −21.9299 −0.700525
\(981\) −6.92500 −0.221098
\(982\) −13.9323 −0.444597
\(983\) −27.4174 −0.874480 −0.437240 0.899345i \(-0.644044\pi\)
−0.437240 + 0.899345i \(0.644044\pi\)
\(984\) −15.0858 −0.480916
\(985\) −25.5050 −0.812656
\(986\) −21.5797 −0.687238
\(987\) 9.67158 0.307850
\(988\) 0 0
\(989\) −31.5120 −1.00202
\(990\) −4.89008 −0.155417
\(991\) −59.2838 −1.88321 −0.941606 0.336716i \(-0.890684\pi\)
−0.941606 + 0.336716i \(0.890684\pi\)
\(992\) 4.85086 0.154015
\(993\) 1.79895 0.0570881
\(994\) 3.15346 0.100022
\(995\) 85.8007 2.72007
\(996\) 8.09783 0.256590
\(997\) 44.0103 1.39382 0.696910 0.717159i \(-0.254558\pi\)
0.696910 + 0.717159i \(0.254558\pi\)
\(998\) 32.2500 1.02085
\(999\) −63.6426 −2.01356
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 6422.2.a.n.1.3 3
13.12 even 2 6422.2.a.v.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6422.2.a.n.1.3 3 1.1 even 1 trivial
6422.2.a.v.1.3 yes 3 13.12 even 2