Properties

Label 3120.2.l.m.1249.4
Level $3120$
Weight $2$
Character 3120.1249
Analytic conductor $24.913$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.9133254306\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.4
Root \(-1.75233 - 1.75233i\) of defining polynomial
Character \(\chi\) \(=\) 3120.1249
Dual form 3120.2.l.m.1249.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +(-1.75233 + 1.38900i) q^{5} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(-1.75233 + 1.38900i) q^{5} -1.00000 q^{9} -1.50466 q^{11} -1.00000i q^{13} +(-1.38900 - 1.75233i) q^{15} -2.72666i q^{17} +0.726656 q^{19} -4.72666i q^{23} +(1.14134 - 4.86799i) q^{25} -1.00000i q^{27} +7.55602 q^{29} +3.00933 q^{31} -1.50466i q^{33} -5.00933i q^{37} +1.00000 q^{39} +5.78734 q^{41} +2.72666i q^{43} +(1.75233 - 1.38900i) q^{45} +10.2313i q^{47} +7.00000 q^{49} +2.72666 q^{51} +7.55602i q^{53} +(2.63667 - 2.08998i) q^{55} +0.726656i q^{57} -12.5140 q^{59} +6.28267 q^{61} +(1.38900 + 1.75233i) q^{65} +12.5653i q^{67} +4.72666 q^{69} -4.77801 q^{71} -12.0187i q^{73} +(4.86799 + 1.14134i) q^{75} +5.27334 q^{79} +1.00000 q^{81} +7.78734i q^{83} +(3.78734 + 4.77801i) q^{85} +7.55602i q^{87} -1.78734 q^{89} +3.00933i q^{93} +(-1.27334 + 1.00933i) q^{95} -6.00000i q^{97} +1.50466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} + 12 q^{11} - 2 q^{15} - 4 q^{19} - 10 q^{25} + 20 q^{29} - 24 q^{31} + 6 q^{39} - 20 q^{41} + 42 q^{49} + 8 q^{51} + 20 q^{55} - 12 q^{59} + 4 q^{61} + 2 q^{65} + 20 q^{69} - 16 q^{71} + 4 q^{75} + 40 q^{79} + 6 q^{81} - 32 q^{85} + 44 q^{89} - 16 q^{95} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times\).

\(n\) \(1951\) \(2081\) \(2341\) \(2497\) \(2641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −1.75233 + 1.38900i −0.783667 + 0.621181i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.50466 −0.453673 −0.226837 0.973933i \(-0.572838\pi\)
−0.226837 + 0.973933i \(0.572838\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) −1.38900 1.75233i −0.358639 0.452450i
\(16\) 0 0
\(17\) 2.72666i 0.661311i −0.943751 0.330656i \(-0.892730\pi\)
0.943751 0.330656i \(-0.107270\pi\)
\(18\) 0 0
\(19\) 0.726656 0.166706 0.0833532 0.996520i \(-0.473437\pi\)
0.0833532 + 0.996520i \(0.473437\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.72666i 0.985576i −0.870149 0.492788i \(-0.835978\pi\)
0.870149 0.492788i \(-0.164022\pi\)
\(24\) 0 0
\(25\) 1.14134 4.86799i 0.228267 0.973599i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 7.55602 1.40312 0.701558 0.712612i \(-0.252488\pi\)
0.701558 + 0.712612i \(0.252488\pi\)
\(30\) 0 0
\(31\) 3.00933 0.540491 0.270246 0.962791i \(-0.412895\pi\)
0.270246 + 0.962791i \(0.412895\pi\)
\(32\) 0 0
\(33\) 1.50466i 0.261928i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00933i 0.823529i −0.911290 0.411764i \(-0.864913\pi\)
0.911290 0.411764i \(-0.135087\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 5.78734 0.903830 0.451915 0.892061i \(-0.350741\pi\)
0.451915 + 0.892061i \(0.350741\pi\)
\(42\) 0 0
\(43\) 2.72666i 0.415811i 0.978149 + 0.207906i \(0.0666647\pi\)
−0.978149 + 0.207906i \(0.933335\pi\)
\(44\) 0 0
\(45\) 1.75233 1.38900i 0.261222 0.207060i
\(46\) 0 0
\(47\) 10.2313i 1.49239i 0.665727 + 0.746196i \(0.268122\pi\)
−0.665727 + 0.746196i \(0.731878\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 2.72666 0.381808
\(52\) 0 0
\(53\) 7.55602i 1.03790i 0.854805 + 0.518949i \(0.173677\pi\)
−0.854805 + 0.518949i \(0.826323\pi\)
\(54\) 0 0
\(55\) 2.63667 2.08998i 0.355529 0.281813i
\(56\) 0 0
\(57\) 0.726656i 0.0962480i
\(58\) 0 0
\(59\) −12.5140 −1.62918 −0.814592 0.580035i \(-0.803039\pi\)
−0.814592 + 0.580035i \(0.803039\pi\)
\(60\) 0 0
\(61\) 6.28267 0.804414 0.402207 0.915549i \(-0.368243\pi\)
0.402207 + 0.915549i \(0.368243\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.38900 + 1.75233i 0.172285 + 0.217350i
\(66\) 0 0
\(67\) 12.5653i 1.53510i 0.640988 + 0.767551i \(0.278525\pi\)
−0.640988 + 0.767551i \(0.721475\pi\)
\(68\) 0 0
\(69\) 4.72666 0.569023
\(70\) 0 0
\(71\) −4.77801 −0.567045 −0.283523 0.958966i \(-0.591503\pi\)
−0.283523 + 0.958966i \(0.591503\pi\)
\(72\) 0 0
\(73\) 12.0187i 1.40668i −0.710855 0.703339i \(-0.751692\pi\)
0.710855 0.703339i \(-0.248308\pi\)
\(74\) 0 0
\(75\) 4.86799 + 1.14134i 0.562107 + 0.131790i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.27334 0.593297 0.296649 0.954987i \(-0.404131\pi\)
0.296649 + 0.954987i \(0.404131\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.78734i 0.854771i 0.904069 + 0.427386i \(0.140565\pi\)
−0.904069 + 0.427386i \(0.859435\pi\)
\(84\) 0 0
\(85\) 3.78734 + 4.77801i 0.410794 + 0.518248i
\(86\) 0 0
\(87\) 7.55602i 0.810090i
\(88\) 0 0
\(89\) −1.78734 −0.189457 −0.0947286 0.995503i \(-0.530198\pi\)
−0.0947286 + 0.995503i \(0.530198\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.00933i 0.312053i
\(94\) 0 0
\(95\) −1.27334 + 1.00933i −0.130642 + 0.103555i
\(96\) 0 0
\(97\) 6.00000i 0.609208i −0.952479 0.304604i \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) 0 0
\(99\) 1.50466 0.151224
\(100\) 0 0
\(101\) −2.99067 −0.297583 −0.148791 0.988869i \(-0.547538\pi\)
−0.148791 + 0.988869i \(0.547538\pi\)
\(102\) 0 0
\(103\) 0.443984i 0.0437471i −0.999761 0.0218735i \(-0.993037\pi\)
0.999761 0.0218735i \(-0.00696312\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.00933i 0.677617i −0.940855 0.338809i \(-0.889976\pi\)
0.940855 0.338809i \(-0.110024\pi\)
\(108\) 0 0
\(109\) 13.8387 1.32551 0.662753 0.748838i \(-0.269388\pi\)
0.662753 + 0.748838i \(0.269388\pi\)
\(110\) 0 0
\(111\) 5.00933 0.475464
\(112\) 0 0
\(113\) 4.28267i 0.402880i 0.979501 + 0.201440i \(0.0645621\pi\)
−0.979501 + 0.201440i \(0.935438\pi\)
\(114\) 0 0
\(115\) 6.56534 + 8.28267i 0.612222 + 0.772363i
\(116\) 0 0
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.73599 −0.794180
\(122\) 0 0
\(123\) 5.78734i 0.521827i
\(124\) 0 0
\(125\) 4.76166 + 10.1157i 0.425896 + 0.904772i
\(126\) 0 0
\(127\) 5.71733i 0.507331i 0.967292 + 0.253665i \(0.0816362\pi\)
−0.967292 + 0.253665i \(0.918364\pi\)
\(128\) 0 0
\(129\) −2.72666 −0.240069
\(130\) 0 0
\(131\) −5.55602 −0.485431 −0.242716 0.970097i \(-0.578038\pi\)
−0.242716 + 0.970097i \(0.578038\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.38900 + 1.75233i 0.119546 + 0.150817i
\(136\) 0 0
\(137\) 6.67531i 0.570310i −0.958481 0.285155i \(-0.907955\pi\)
0.958481 0.285155i \(-0.0920450\pi\)
\(138\) 0 0
\(139\) 19.4720 1.65159 0.825795 0.563970i \(-0.190727\pi\)
0.825795 + 0.563970i \(0.190727\pi\)
\(140\) 0 0
\(141\) −10.2313 −0.861633
\(142\) 0 0
\(143\) 1.50466i 0.125826i
\(144\) 0 0
\(145\) −13.2406 + 10.4953i −1.09958 + 0.871590i
\(146\) 0 0
\(147\) 7.00000i 0.577350i
\(148\) 0 0
\(149\) 14.5140 1.18903 0.594516 0.804084i \(-0.297344\pi\)
0.594516 + 0.804084i \(0.297344\pi\)
\(150\) 0 0
\(151\) 4.46264 0.363165 0.181582 0.983376i \(-0.441878\pi\)
0.181582 + 0.983376i \(0.441878\pi\)
\(152\) 0 0
\(153\) 2.72666i 0.220437i
\(154\) 0 0
\(155\) −5.27334 + 4.17997i −0.423565 + 0.335743i
\(156\) 0 0
\(157\) 8.30133i 0.662518i 0.943540 + 0.331259i \(0.107473\pi\)
−0.943540 + 0.331259i \(0.892527\pi\)
\(158\) 0 0
\(159\) −7.55602 −0.599231
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 11.0093i 0.862317i −0.902276 0.431159i \(-0.858105\pi\)
0.902276 0.431159i \(-0.141895\pi\)
\(164\) 0 0
\(165\) 2.08998 + 2.63667i 0.162705 + 0.205265i
\(166\) 0 0
\(167\) 5.76868i 0.446394i 0.974773 + 0.223197i \(0.0716493\pi\)
−0.974773 + 0.223197i \(0.928351\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −0.726656 −0.0555688
\(172\) 0 0
\(173\) 4.90663i 0.373044i −0.982451 0.186522i \(-0.940278\pi\)
0.982451 0.186522i \(-0.0597215\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.5140i 0.940609i
\(178\) 0 0
\(179\) 9.45331 0.706574 0.353287 0.935515i \(-0.385064\pi\)
0.353287 + 0.935515i \(0.385064\pi\)
\(180\) 0 0
\(181\) 17.4720 1.29868 0.649341 0.760498i \(-0.275045\pi\)
0.649341 + 0.760498i \(0.275045\pi\)
\(182\) 0 0
\(183\) 6.28267i 0.464428i
\(184\) 0 0
\(185\) 6.95798 + 8.77801i 0.511561 + 0.645372i
\(186\) 0 0
\(187\) 4.10270i 0.300019i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.1120 0.804038 0.402019 0.915631i \(-0.368309\pi\)
0.402019 + 0.915631i \(0.368309\pi\)
\(192\) 0 0
\(193\) 6.10270i 0.439282i −0.975581 0.219641i \(-0.929511\pi\)
0.975581 0.219641i \(-0.0704886\pi\)
\(194\) 0 0
\(195\) −1.75233 + 1.38900i −0.125487 + 0.0994686i
\(196\) 0 0
\(197\) 21.3620i 1.52198i 0.648764 + 0.760990i \(0.275286\pi\)
−0.648764 + 0.760990i \(0.724714\pi\)
\(198\) 0 0
\(199\) 8.38538 0.594423 0.297212 0.954812i \(-0.403943\pi\)
0.297212 + 0.954812i \(0.403943\pi\)
\(200\) 0 0
\(201\) −12.5653 −0.886291
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −10.1413 + 8.03863i −0.708302 + 0.561443i
\(206\) 0 0
\(207\) 4.72666i 0.328525i
\(208\) 0 0
\(209\) −1.09337 −0.0756303
\(210\) 0 0
\(211\) 1.27334 0.0876606 0.0438303 0.999039i \(-0.486044\pi\)
0.0438303 + 0.999039i \(0.486044\pi\)
\(212\) 0 0
\(213\) 4.77801i 0.327384i
\(214\) 0 0
\(215\) −3.78734 4.77801i −0.258294 0.325857i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 12.0187 0.812146
\(220\) 0 0
\(221\) −2.72666 −0.183415
\(222\) 0 0
\(223\) 16.4626i 1.10242i −0.834367 0.551210i \(-0.814166\pi\)
0.834367 0.551210i \(-0.185834\pi\)
\(224\) 0 0
\(225\) −1.14134 + 4.86799i −0.0760891 + 0.324533i
\(226\) 0 0
\(227\) 12.2500i 0.813060i 0.913638 + 0.406530i \(0.133261\pi\)
−0.913638 + 0.406530i \(0.866739\pi\)
\(228\) 0 0
\(229\) 1.27334 0.0841449 0.0420725 0.999115i \(-0.486604\pi\)
0.0420725 + 0.999115i \(0.486604\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.3013i 1.19896i 0.800390 + 0.599480i \(0.204626\pi\)
−0.800390 + 0.599480i \(0.795374\pi\)
\(234\) 0 0
\(235\) −14.2113 17.9287i −0.927046 1.16954i
\(236\) 0 0
\(237\) 5.27334i 0.342540i
\(238\) 0 0
\(239\) 6.23132 0.403071 0.201535 0.979481i \(-0.435407\pi\)
0.201535 + 0.979481i \(0.435407\pi\)
\(240\) 0 0
\(241\) 19.5560 1.25971 0.629857 0.776711i \(-0.283114\pi\)
0.629857 + 0.776711i \(0.283114\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) −12.2663 + 9.72303i −0.783667 + 0.621181i
\(246\) 0 0
\(247\) 0.726656i 0.0462360i
\(248\) 0 0
\(249\) −7.78734 −0.493502
\(250\) 0 0
\(251\) −26.4813 −1.67148 −0.835742 0.549122i \(-0.814962\pi\)
−0.835742 + 0.549122i \(0.814962\pi\)
\(252\) 0 0
\(253\) 7.11203i 0.447130i
\(254\) 0 0
\(255\) −4.77801 + 3.78734i −0.299210 + 0.237172i
\(256\) 0 0
\(257\) 14.8294i 0.925030i −0.886611 0.462515i \(-0.846947\pi\)
0.886611 0.462515i \(-0.153053\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −7.55602 −0.467706
\(262\) 0 0
\(263\) 22.9507i 1.41520i −0.706612 0.707601i \(-0.749777\pi\)
0.706612 0.707601i \(-0.250223\pi\)
\(264\) 0 0
\(265\) −10.4953 13.2406i −0.644723 0.813367i
\(266\) 0 0
\(267\) 1.78734i 0.109383i
\(268\) 0 0
\(269\) 27.9160 1.70207 0.851033 0.525112i \(-0.175977\pi\)
0.851033 + 0.525112i \(0.175977\pi\)
\(270\) 0 0
\(271\) 5.65872 0.343743 0.171871 0.985119i \(-0.445019\pi\)
0.171871 + 0.985119i \(0.445019\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.71733 + 7.32469i −0.103559 + 0.441696i
\(276\) 0 0
\(277\) 7.73599i 0.464810i −0.972619 0.232405i \(-0.925340\pi\)
0.972619 0.232405i \(-0.0746595\pi\)
\(278\) 0 0
\(279\) −3.00933 −0.180164
\(280\) 0 0
\(281\) 5.11929 0.305391 0.152696 0.988273i \(-0.451205\pi\)
0.152696 + 0.988273i \(0.451205\pi\)
\(282\) 0 0
\(283\) 6.90663i 0.410556i −0.978704 0.205278i \(-0.934190\pi\)
0.978704 0.205278i \(-0.0658099\pi\)
\(284\) 0 0
\(285\) −1.00933 1.27334i −0.0597875 0.0754264i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 9.56534 0.562667
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 0 0
\(293\) 26.3527i 1.53954i 0.638321 + 0.769770i \(0.279629\pi\)
−0.638321 + 0.769770i \(0.720371\pi\)
\(294\) 0 0
\(295\) 21.9287 17.3820i 1.27674 1.01202i
\(296\) 0 0
\(297\) 1.50466i 0.0873095i
\(298\) 0 0
\(299\) −4.72666 −0.273350
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 2.99067i 0.171810i
\(304\) 0 0
\(305\) −11.0093 + 8.72666i −0.630392 + 0.499687i
\(306\) 0 0
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 0 0
\(309\) 0.443984 0.0252574
\(310\) 0 0
\(311\) 11.8973 0.674634 0.337317 0.941391i \(-0.390481\pi\)
0.337317 + 0.941391i \(0.390481\pi\)
\(312\) 0 0
\(313\) 17.7360i 1.00250i 0.865303 + 0.501249i \(0.167126\pi\)
−0.865303 + 0.501249i \(0.832874\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.32469i 0.0744023i 0.999308 + 0.0372011i \(0.0118442\pi\)
−0.999308 + 0.0372011i \(0.988156\pi\)
\(318\) 0 0
\(319\) −11.3693 −0.636557
\(320\) 0 0
\(321\) 7.00933 0.391223
\(322\) 0 0
\(323\) 1.98134i 0.110245i
\(324\) 0 0
\(325\) −4.86799 1.14134i −0.270028 0.0633099i
\(326\) 0 0
\(327\) 13.8387i 0.765281i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3.73599 0.205348 0.102674 0.994715i \(-0.467260\pi\)
0.102674 + 0.994715i \(0.467260\pi\)
\(332\) 0 0
\(333\) 5.00933i 0.274510i
\(334\) 0 0
\(335\) −17.4533 22.0187i −0.953576 1.20301i
\(336\) 0 0
\(337\) 23.3947i 1.27439i −0.770702 0.637195i \(-0.780094\pi\)
0.770702 0.637195i \(-0.219906\pi\)
\(338\) 0 0
\(339\) −4.28267 −0.232603
\(340\) 0 0
\(341\) −4.52803 −0.245207
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −8.28267 + 6.56534i −0.445924 + 0.353466i
\(346\) 0 0
\(347\) 30.5840i 1.64184i −0.571047 0.820918i \(-0.693462\pi\)
0.571047 0.820918i \(-0.306538\pi\)
\(348\) 0 0
\(349\) −1.37605 −0.0736581 −0.0368290 0.999322i \(-0.511726\pi\)
−0.0368290 + 0.999322i \(0.511726\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 19.9087i 1.05963i −0.848112 0.529817i \(-0.822261\pi\)
0.848112 0.529817i \(-0.177739\pi\)
\(354\) 0 0
\(355\) 8.37266 6.63667i 0.444374 0.352238i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.6940 1.40885 0.704427 0.709777i \(-0.251204\pi\)
0.704427 + 0.709777i \(0.251204\pi\)
\(360\) 0 0
\(361\) −18.4720 −0.972209
\(362\) 0 0
\(363\) 8.73599i 0.458520i
\(364\) 0 0
\(365\) 16.6940 + 21.0607i 0.873802 + 1.10237i
\(366\) 0 0
\(367\) 8.12136i 0.423932i −0.977277 0.211966i \(-0.932013\pi\)
0.977277 0.211966i \(-0.0679865\pi\)
\(368\) 0 0
\(369\) −5.78734 −0.301277
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 9.47197i 0.490440i −0.969467 0.245220i \(-0.921140\pi\)
0.969467 0.245220i \(-0.0788602\pi\)
\(374\) 0 0
\(375\) −10.1157 + 4.76166i −0.522370 + 0.245891i
\(376\) 0 0
\(377\) 7.55602i 0.389155i
\(378\) 0 0
\(379\) 16.7267 0.859191 0.429595 0.903022i \(-0.358656\pi\)
0.429595 + 0.903022i \(0.358656\pi\)
\(380\) 0 0
\(381\) −5.71733 −0.292908
\(382\) 0 0
\(383\) 19.6846i 1.00584i 0.864334 + 0.502919i \(0.167741\pi\)
−0.864334 + 0.502919i \(0.832259\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.72666i 0.138604i
\(388\) 0 0
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) −12.8880 −0.651773
\(392\) 0 0
\(393\) 5.55602i 0.280264i
\(394\) 0 0
\(395\) −9.24065 + 7.32469i −0.464948 + 0.368545i
\(396\) 0 0
\(397\) 3.35061i 0.168162i −0.996459 0.0840812i \(-0.973205\pi\)
0.996459 0.0840812i \(-0.0267955\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −36.3713 −1.81630 −0.908149 0.418647i \(-0.862504\pi\)
−0.908149 + 0.418647i \(0.862504\pi\)
\(402\) 0 0
\(403\) 3.00933i 0.149905i
\(404\) 0 0
\(405\) −1.75233 + 1.38900i −0.0870741 + 0.0690202i
\(406\) 0 0
\(407\) 7.53736i 0.373613i
\(408\) 0 0
\(409\) −35.5933 −1.75998 −0.879988 0.474995i \(-0.842450\pi\)
−0.879988 + 0.474995i \(0.842450\pi\)
\(410\) 0 0
\(411\) 6.67531 0.329269
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −10.8166 13.6460i −0.530968 0.669856i
\(416\) 0 0
\(417\) 19.4720i 0.953546i
\(418\) 0 0
\(419\) −13.9160 −0.679839 −0.339919 0.940455i \(-0.610400\pi\)
−0.339919 + 0.940455i \(0.610400\pi\)
\(420\) 0 0
\(421\) 4.70800 0.229454 0.114727 0.993397i \(-0.463401\pi\)
0.114727 + 0.993397i \(0.463401\pi\)
\(422\) 0 0
\(423\) 10.2313i 0.497464i
\(424\) 0 0
\(425\) −13.2733 3.11203i −0.643852 0.150956i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.50466 −0.0726459
\(430\) 0 0
\(431\) −7.89004 −0.380050 −0.190025 0.981779i \(-0.560857\pi\)
−0.190025 + 0.981779i \(0.560857\pi\)
\(432\) 0 0
\(433\) 2.30133i 0.110595i 0.998470 + 0.0552974i \(0.0176107\pi\)
−0.998470 + 0.0552974i \(0.982389\pi\)
\(434\) 0 0
\(435\) −10.4953 13.2406i −0.503213 0.634841i
\(436\) 0 0
\(437\) 3.43466i 0.164302i
\(438\) 0 0
\(439\) 41.1307 1.96306 0.981530 0.191307i \(-0.0612725\pi\)
0.981530 + 0.191307i \(0.0612725\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 29.5933i 1.40602i −0.711179 0.703011i \(-0.751839\pi\)
0.711179 0.703011i \(-0.248161\pi\)
\(444\) 0 0
\(445\) 3.13201 2.48262i 0.148471 0.117687i
\(446\) 0 0
\(447\) 14.5140i 0.686488i
\(448\) 0 0
\(449\) 11.4461 0.540173 0.270086 0.962836i \(-0.412948\pi\)
0.270086 + 0.962836i \(0.412948\pi\)
\(450\) 0 0
\(451\) −8.70800 −0.410044
\(452\) 0 0
\(453\) 4.46264i 0.209673i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16.5467i 0.774021i −0.922075 0.387011i \(-0.873508\pi\)
0.922075 0.387011i \(-0.126492\pi\)
\(458\) 0 0
\(459\) −2.72666 −0.127269
\(460\) 0 0
\(461\) 3.50466 0.163228 0.0816142 0.996664i \(-0.473992\pi\)
0.0816142 + 0.996664i \(0.473992\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) 0 0
\(465\) −4.17997 5.27334i −0.193841 0.244545i
\(466\) 0 0
\(467\) 6.44398i 0.298192i 0.988823 + 0.149096i \(0.0476363\pi\)
−0.988823 + 0.149096i \(0.952364\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −8.30133 −0.382505
\(472\) 0 0
\(473\) 4.10270i 0.188642i
\(474\) 0 0
\(475\) 0.829359 3.53736i 0.0380536 0.162305i
\(476\) 0 0
\(477\) 7.55602i 0.345966i
\(478\) 0 0
\(479\) 22.2313 1.01577 0.507887 0.861423i \(-0.330427\pi\)
0.507887 + 0.861423i \(0.330427\pi\)
\(480\) 0 0
\(481\) −5.00933 −0.228406
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.33402 + 10.5140i 0.378429 + 0.477416i
\(486\) 0 0
\(487\) 13.9160i 0.630592i −0.948993 0.315296i \(-0.897896\pi\)
0.948993 0.315296i \(-0.102104\pi\)
\(488\) 0 0
\(489\) 11.0093 0.497859
\(490\) 0 0
\(491\) 28.1400 1.26994 0.634971 0.772536i \(-0.281012\pi\)
0.634971 + 0.772536i \(0.281012\pi\)
\(492\) 0 0
\(493\) 20.6027i 0.927897i
\(494\) 0 0
\(495\) −2.63667 + 2.08998i −0.118510 + 0.0939378i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 38.8480 1.73908 0.869538 0.493866i \(-0.164417\pi\)
0.869538 + 0.493866i \(0.164417\pi\)
\(500\) 0 0
\(501\) −5.76868 −0.257726
\(502\) 0 0
\(503\) 8.19863i 0.365559i −0.983154 0.182779i \(-0.941491\pi\)
0.983154 0.182779i \(-0.0585094\pi\)
\(504\) 0 0
\(505\) 5.24065 4.15405i 0.233206 0.184853i
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) −16.0700 −0.712291 −0.356145 0.934431i \(-0.615909\pi\)
−0.356145 + 0.934431i \(0.615909\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0.726656i 0.0320827i
\(514\) 0 0
\(515\) 0.616696 + 0.778008i 0.0271749 + 0.0342831i
\(516\) 0 0
\(517\) 15.3947i 0.677058i
\(518\) 0 0
\(519\) 4.90663 0.215377
\(520\) 0 0
\(521\) −23.0280 −1.00887 −0.504437 0.863448i \(-0.668300\pi\)
−0.504437 + 0.863448i \(0.668300\pi\)
\(522\) 0 0
\(523\) 5.47875i 0.239569i −0.992800 0.119784i \(-0.961780\pi\)
0.992800 0.119784i \(-0.0382204\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.20541i 0.357433i
\(528\) 0 0
\(529\) 0.658719 0.0286399
\(530\) 0 0
\(531\) 12.5140 0.543061
\(532\) 0 0
\(533\) 5.78734i 0.250677i
\(534\) 0 0
\(535\) 9.73599 + 12.2827i 0.420923 + 0.531026i
\(536\) 0 0
\(537\) 9.45331i 0.407941i
\(538\) 0 0
\(539\) −10.5327 −0.453673
\(540\) 0 0
\(541\) 17.8387 0.766945 0.383473 0.923552i \(-0.374728\pi\)
0.383473 + 0.923552i \(0.374728\pi\)
\(542\) 0 0
\(543\) 17.4720i 0.749794i
\(544\) 0 0
\(545\) −24.2500 + 19.2220i −1.03875 + 0.823380i
\(546\) 0 0
\(547\) 29.1053i 1.24445i 0.782838 + 0.622225i \(0.213771\pi\)
−0.782838 + 0.622225i \(0.786229\pi\)
\(548\) 0 0
\(549\) −6.28267 −0.268138
\(550\) 0 0
\(551\) 5.49063 0.233909
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −8.77801 + 6.95798i −0.372606 + 0.295350i
\(556\) 0 0
\(557\) 44.0114i 1.86482i −0.361399 0.932411i \(-0.617701\pi\)
0.361399 0.932411i \(-0.382299\pi\)
\(558\) 0 0
\(559\) 2.72666 0.115325
\(560\) 0 0
\(561\) −4.10270 −0.173216
\(562\) 0 0
\(563\) 7.43466i 0.313333i −0.987652 0.156667i \(-0.949925\pi\)
0.987652 0.156667i \(-0.0500748\pi\)
\(564\) 0 0
\(565\) −5.94865 7.50466i −0.250262 0.315724i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8.48130 −0.355555 −0.177777 0.984071i \(-0.556891\pi\)
−0.177777 + 0.984071i \(0.556891\pi\)
\(570\) 0 0
\(571\) −40.7826 −1.70670 −0.853350 0.521339i \(-0.825433\pi\)
−0.853350 + 0.521339i \(0.825433\pi\)
\(572\) 0 0
\(573\) 11.1120i 0.464212i
\(574\) 0 0
\(575\) −23.0093 5.39470i −0.959555 0.224975i
\(576\) 0 0
\(577\) 1.57467i 0.0655545i −0.999463 0.0327773i \(-0.989565\pi\)
0.999463 0.0327773i \(-0.0104352\pi\)
\(578\) 0 0
\(579\) 6.10270 0.253620
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 11.3693i 0.470867i
\(584\) 0 0
\(585\) −1.38900 1.75233i −0.0574282 0.0724500i
\(586\) 0 0
\(587\) 27.8247i 1.14845i 0.818699 + 0.574223i \(0.194696\pi\)
−0.818699 + 0.574223i \(0.805304\pi\)
\(588\) 0 0
\(589\) 2.18675 0.0901034
\(590\) 0 0
\(591\) −21.3620 −0.878716
\(592\) 0 0
\(593\) 32.6940i 1.34258i 0.741195 + 0.671290i \(0.234260\pi\)
−0.741195 + 0.671290i \(0.765740\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.38538i 0.343191i
\(598\) 0 0
\(599\) −32.1400 −1.31321 −0.656603 0.754237i \(-0.728007\pi\)
−0.656603 + 0.754237i \(0.728007\pi\)
\(600\) 0 0
\(601\) 40.8667 1.66699 0.833493 0.552530i \(-0.186337\pi\)
0.833493 + 0.552530i \(0.186337\pi\)
\(602\) 0 0
\(603\) 12.5653i 0.511700i
\(604\) 0 0
\(605\) 15.3083 12.1343i 0.622373 0.493330i
\(606\) 0 0
\(607\) 14.9907i 0.608453i 0.952600 + 0.304226i \(0.0983979\pi\)
−0.952600 + 0.304226i \(0.901602\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 10.2313 0.413915
\(612\) 0 0
\(613\) 13.5747i 0.548276i 0.961690 + 0.274138i \(0.0883925\pi\)
−0.961690 + 0.274138i \(0.911608\pi\)
\(614\) 0 0
\(615\) −8.03863 10.1413i −0.324149 0.408938i
\(616\) 0 0
\(617\) 4.69396i 0.188972i 0.995526 + 0.0944859i \(0.0301207\pi\)
−0.995526 + 0.0944859i \(0.969879\pi\)
\(618\) 0 0
\(619\) −20.8667 −0.838702 −0.419351 0.907824i \(-0.637742\pi\)
−0.419351 + 0.907824i \(0.637742\pi\)
\(620\) 0 0
\(621\) −4.72666 −0.189674
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −22.3947 11.1120i −0.895788 0.444481i
\(626\) 0 0
\(627\) 1.09337i 0.0436652i
\(628\) 0 0
\(629\) −13.6587 −0.544609
\(630\) 0 0
\(631\) −45.9533 −1.82937 −0.914685 0.404167i \(-0.867562\pi\)
−0.914685 + 0.404167i \(0.867562\pi\)
\(632\) 0 0
\(633\) 1.27334i 0.0506109i
\(634\) 0 0
\(635\) −7.94139 10.0187i −0.315144 0.397578i
\(636\) 0 0
\(637\) 7.00000i 0.277350i
\(638\) 0 0
\(639\) 4.77801 0.189015
\(640\) 0 0
\(641\) 20.0187 0.790689 0.395345 0.918533i \(-0.370625\pi\)
0.395345 + 0.918533i \(0.370625\pi\)
\(642\) 0 0
\(643\) 26.5840i 1.04837i 0.851604 + 0.524185i \(0.175630\pi\)
−0.851604 + 0.524185i \(0.824370\pi\)
\(644\) 0 0
\(645\) 4.77801 3.78734i 0.188134 0.149126i
\(646\) 0 0
\(647\) 9.39470i 0.369344i 0.982800 + 0.184672i \(0.0591223\pi\)
−0.982800 + 0.184672i \(0.940878\pi\)
\(648\) 0 0
\(649\) 18.8294 0.739117
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.00000i 0.0782660i 0.999234 + 0.0391330i \(0.0124596\pi\)
−0.999234 + 0.0391330i \(0.987540\pi\)
\(654\) 0 0
\(655\) 9.73599 7.71733i 0.380416 0.301541i
\(656\) 0 0
\(657\) 12.0187i 0.468892i
\(658\) 0 0
\(659\) 47.6774 1.85725 0.928623 0.371024i \(-0.120993\pi\)
0.928623 + 0.371024i \(0.120993\pi\)
\(660\) 0 0
\(661\) −22.0959 −0.859432 −0.429716 0.902964i \(-0.641386\pi\)
−0.429716 + 0.902964i \(0.641386\pi\)
\(662\) 0 0
\(663\) 2.72666i 0.105895i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 35.7147i 1.38288i
\(668\) 0 0
\(669\) 16.4626 0.636482
\(670\) 0 0
\(671\) −9.45331 −0.364941
\(672\) 0 0
\(673\) 48.6027i 1.87349i −0.350006 0.936747i \(-0.613820\pi\)
0.350006 0.936747i \(-0.386180\pi\)
\(674\) 0 0
\(675\) −4.86799 1.14134i −0.187369 0.0439300i
\(676\) 0 0
\(677\) 10.4253i 0.400678i −0.979727 0.200339i \(-0.935796\pi\)
0.979727 0.200339i \(-0.0642043\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −12.2500 −0.469420
\(682\) 0 0
\(683\) 9.13795i 0.349654i 0.984599 + 0.174827i \(0.0559366\pi\)
−0.984599 + 0.174827i \(0.944063\pi\)
\(684\) 0 0
\(685\) 9.27203 + 11.6974i 0.354266 + 0.446933i
\(686\) 0 0
\(687\) 1.27334i 0.0485811i
\(688\) 0 0
\(689\) 7.55602 0.287861
\(690\) 0 0
\(691\) −33.7173 −1.28267 −0.641334 0.767262i \(-0.721619\pi\)
−0.641334 + 0.767262i \(0.721619\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −34.1214 + 27.0466i −1.29430 + 1.02594i
\(696\) 0 0
\(697\) 15.7801i 0.597713i
\(698\) 0 0
\(699\) −18.3013 −0.692220
\(700\) 0 0
\(701\) −19.9813 −0.754685 −0.377342 0.926074i \(-0.623162\pi\)
−0.377342 + 0.926074i \(0.623162\pi\)
\(702\) 0 0
\(703\) 3.64006i 0.137288i
\(704\) 0 0
\(705\) 17.9287 14.2113i 0.675233 0.535230i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 43.3293 1.62727 0.813633 0.581378i \(-0.197486\pi\)
0.813633 + 0.581378i \(0.197486\pi\)
\(710\) 0 0
\(711\) −5.27334 −0.197766
\(712\) 0 0
\(713\) 14.2241i 0.532695i
\(714\) 0 0
\(715\) −2.08998 2.63667i −0.0781610 0.0986059i
\(716\) 0 0
\(717\) 6.23132i 0.232713i
\(718\) 0 0
\(719\) −34.6867 −1.29360 −0.646798 0.762661i \(-0.723892\pi\)
−0.646798 + 0.762661i \(0.723892\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 19.5560i 0.727296i
\(724\) 0 0
\(725\) 8.62395 36.7826i 0.320286 1.36607i
\(726\) 0 0
\(727\) 5.39470i 0.200078i −0.994983 0.100039i \(-0.968103\pi\)
0.994983 0.100039i \(-0.0318968\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 7.43466 0.274981
\(732\) 0 0
\(733\) 9.11203i 0.336561i −0.985739 0.168280i \(-0.946179\pi\)
0.985739 0.168280i \(-0.0538214\pi\)
\(734\) 0 0
\(735\) −9.72303 12.2663i −0.358639 0.452450i
\(736\) 0 0
\(737\) 18.9066i 0.696435i
\(738\) 0 0
\(739\) −4.82936 −0.177651 −0.0888254 0.996047i \(-0.528311\pi\)
−0.0888254 + 0.996047i \(0.528311\pi\)
\(740\) 0 0
\(741\) 0.726656 0.0266944
\(742\) 0 0
\(743\) 50.4087i 1.84931i 0.380801 + 0.924657i \(0.375648\pi\)
−0.380801 + 0.924657i \(0.624352\pi\)
\(744\) 0 0
\(745\) −25.4333 + 20.1600i −0.931805 + 0.738605i
\(746\) 0 0
\(747\) 7.78734i 0.284924i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −4.88797 −0.178365 −0.0891823 0.996015i \(-0.528425\pi\)
−0.0891823 + 0.996015i \(0.528425\pi\)
\(752\) 0 0
\(753\) 26.4813i 0.965032i
\(754\) 0 0
\(755\) −7.82003 + 6.19863i −0.284600 + 0.225591i
\(756\) 0 0
\(757\) 32.2241i 1.17120i 0.810599 + 0.585602i \(0.199142\pi\)
−0.810599 + 0.585602i \(0.800858\pi\)
\(758\) 0 0
\(759\) −7.11203 −0.258150
\(760\) 0 0
\(761\) 36.4740 1.32218 0.661091 0.750305i \(-0.270094\pi\)
0.661091 + 0.750305i \(0.270094\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −3.78734 4.77801i −0.136931 0.172749i
\(766\) 0 0
\(767\) 12.5140i 0.451854i
\(768\) 0 0
\(769\) −13.1120 −0.472832 −0.236416 0.971652i \(-0.575973\pi\)
−0.236416 + 0.971652i \(0.575973\pi\)
\(770\) 0 0
\(771\) 14.8294 0.534066
\(772\) 0 0
\(773\) 12.5913i 0.452876i 0.974026 + 0.226438i \(0.0727081\pi\)
−0.974026 + 0.226438i \(0.927292\pi\)
\(774\) 0 0
\(775\) 3.43466 14.6494i 0.123376 0.526222i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.20541 0.150674
\(780\) 0 0
\(781\) 7.18930 0.257253
\(782\) 0 0
\(783\) 7.55602i 0.270030i
\(784\) 0 0
\(785\) −11.5306 14.5467i −0.411544 0.519194i
\(786\) 0 0
\(787\) 36.9253i 1.31624i −0.752911 0.658122i \(-0.771351\pi\)
0.752911 0.658122i \(-0.228649\pi\)
\(788\) 0 0
\(789\) 22.9507 0.817067
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6.28267i 0.223104i
\(794\) 0 0
\(795\) 13.2406 10.4953i 0.469597 0.372231i
\(796\) 0 0
\(797\) 25.0466i 0.887198i −0.896225 0.443599i \(-0.853702\pi\)
0.896225 0.443599i \(-0.146298\pi\)
\(798\) 0 0
\(799\) 27.8973 0.986935
\(800\) 0 0
\(801\) 1.78734 0.0631524
\(802\) 0 0