Properties

Label 4012.2.b.b.237.38
Level 4012
Weight 2
Character 4012.237
Analytic conductor 32.036
Analytic rank 0
Dimension 46
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(46\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 237.38
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.9

$q$-expansion

\(f(q)\) \(=\) \(q+2.27623i q^{3} -2.55864i q^{5} +1.00518i q^{7} -2.18122 q^{9} +O(q^{10})\) \(q+2.27623i q^{3} -2.55864i q^{5} +1.00518i q^{7} -2.18122 q^{9} -4.20954i q^{11} -6.65655 q^{13} +5.82405 q^{15} +(2.10815 + 3.54340i) q^{17} +1.96359 q^{19} -2.28803 q^{21} +7.06438i q^{23} -1.54664 q^{25} +1.86373i q^{27} -2.50313i q^{29} -7.19153i q^{31} +9.58187 q^{33} +2.57190 q^{35} -5.03511i q^{37} -15.1518i q^{39} +11.6991i q^{41} +2.35785 q^{43} +5.58095i q^{45} -9.91253 q^{47} +5.98961 q^{49} +(-8.06560 + 4.79863i) q^{51} +0.260490 q^{53} -10.7707 q^{55} +4.46958i q^{57} +1.00000 q^{59} +10.4088i q^{61} -2.19252i q^{63} +17.0317i q^{65} +3.87736 q^{67} -16.0801 q^{69} -5.64239i q^{71} +8.82479i q^{73} -3.52050i q^{75} +4.23135 q^{77} +13.6738i q^{79} -10.7859 q^{81} +2.15593 q^{83} +(9.06629 - 5.39399i) q^{85} +5.69770 q^{87} +4.88983 q^{89} -6.69104i q^{91} +16.3696 q^{93} -5.02412i q^{95} +12.2137i q^{97} +9.18192i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times\).

\(n\) \(2007\) \(3129\) \(3777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 2.27623i 1.31418i 0.753811 + 0.657091i \(0.228213\pi\)
−0.753811 + 0.657091i \(0.771787\pi\)
\(4\) 0 0
\(5\) 2.55864i 1.14426i −0.820163 0.572129i \(-0.806118\pi\)
0.820163 0.572129i \(-0.193882\pi\)
\(6\) 0 0
\(7\) 1.00518i 0.379923i 0.981792 + 0.189962i \(0.0608363\pi\)
−0.981792 + 0.189962i \(0.939164\pi\)
\(8\) 0 0
\(9\) −2.18122 −0.727073
\(10\) 0 0
\(11\) 4.20954i 1.26922i −0.772831 0.634612i \(-0.781160\pi\)
0.772831 0.634612i \(-0.218840\pi\)
\(12\) 0 0
\(13\) −6.65655 −1.84619 −0.923097 0.384568i \(-0.874350\pi\)
−0.923097 + 0.384568i \(0.874350\pi\)
\(14\) 0 0
\(15\) 5.82405 1.50376
\(16\) 0 0
\(17\) 2.10815 + 3.54340i 0.511301 + 0.859402i
\(18\) 0 0
\(19\) 1.96359 0.450478 0.225239 0.974304i \(-0.427684\pi\)
0.225239 + 0.974304i \(0.427684\pi\)
\(20\) 0 0
\(21\) −2.28803 −0.499288
\(22\) 0 0
\(23\) 7.06438i 1.47302i 0.676424 + 0.736512i \(0.263529\pi\)
−0.676424 + 0.736512i \(0.736471\pi\)
\(24\) 0 0
\(25\) −1.54664 −0.309328
\(26\) 0 0
\(27\) 1.86373i 0.358675i
\(28\) 0 0
\(29\) 2.50313i 0.464820i −0.972618 0.232410i \(-0.925339\pi\)
0.972618 0.232410i \(-0.0746611\pi\)
\(30\) 0 0
\(31\) 7.19153i 1.29164i −0.763491 0.645818i \(-0.776516\pi\)
0.763491 0.645818i \(-0.223484\pi\)
\(32\) 0 0
\(33\) 9.58187 1.66799
\(34\) 0 0
\(35\) 2.57190 0.434730
\(36\) 0 0
\(37\) 5.03511i 0.827767i −0.910330 0.413884i \(-0.864172\pi\)
0.910330 0.413884i \(-0.135828\pi\)
\(38\) 0 0
\(39\) 15.1518i 2.42623i
\(40\) 0 0
\(41\) 11.6991i 1.82709i 0.406736 + 0.913546i \(0.366667\pi\)
−0.406736 + 0.913546i \(0.633333\pi\)
\(42\) 0 0
\(43\) 2.35785 0.359569 0.179785 0.983706i \(-0.442460\pi\)
0.179785 + 0.983706i \(0.442460\pi\)
\(44\) 0 0
\(45\) 5.58095i 0.831960i
\(46\) 0 0
\(47\) −9.91253 −1.44589 −0.722946 0.690905i \(-0.757212\pi\)
−0.722946 + 0.690905i \(0.757212\pi\)
\(48\) 0 0
\(49\) 5.98961 0.855658
\(50\) 0 0
\(51\) −8.06560 + 4.79863i −1.12941 + 0.671943i
\(52\) 0 0
\(53\) 0.260490 0.0357811 0.0178905 0.999840i \(-0.494305\pi\)
0.0178905 + 0.999840i \(0.494305\pi\)
\(54\) 0 0
\(55\) −10.7707 −1.45232
\(56\) 0 0
\(57\) 4.46958i 0.592010i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 10.4088i 1.33271i 0.745634 + 0.666355i \(0.232147\pi\)
−0.745634 + 0.666355i \(0.767853\pi\)
\(62\) 0 0
\(63\) 2.19252i 0.276232i
\(64\) 0 0
\(65\) 17.0317i 2.11252i
\(66\) 0 0
\(67\) 3.87736 0.473695 0.236848 0.971547i \(-0.423886\pi\)
0.236848 + 0.971547i \(0.423886\pi\)
\(68\) 0 0
\(69\) −16.0801 −1.93582
\(70\) 0 0
\(71\) 5.64239i 0.669629i −0.942284 0.334814i \(-0.891326\pi\)
0.942284 0.334814i \(-0.108674\pi\)
\(72\) 0 0
\(73\) 8.82479i 1.03286i 0.856328 + 0.516432i \(0.172740\pi\)
−0.856328 + 0.516432i \(0.827260\pi\)
\(74\) 0 0
\(75\) 3.52050i 0.406513i
\(76\) 0 0
\(77\) 4.23135 0.482207
\(78\) 0 0
\(79\) 13.6738i 1.53842i 0.638994 + 0.769212i \(0.279351\pi\)
−0.638994 + 0.769212i \(0.720649\pi\)
\(80\) 0 0
\(81\) −10.7859 −1.19844
\(82\) 0 0
\(83\) 2.15593 0.236644 0.118322 0.992975i \(-0.462248\pi\)
0.118322 + 0.992975i \(0.462248\pi\)
\(84\) 0 0
\(85\) 9.06629 5.39399i 0.983378 0.585061i
\(86\) 0 0
\(87\) 5.69770 0.610857
\(88\) 0 0
\(89\) 4.88983 0.518321 0.259161 0.965834i \(-0.416554\pi\)
0.259161 + 0.965834i \(0.416554\pi\)
\(90\) 0 0
\(91\) 6.69104i 0.701412i
\(92\) 0 0
\(93\) 16.3696 1.69744
\(94\) 0 0
\(95\) 5.02412i 0.515464i
\(96\) 0 0
\(97\) 12.2137i 1.24012i 0.784556 + 0.620058i \(0.212891\pi\)
−0.784556 + 0.620058i \(0.787109\pi\)
\(98\) 0 0
\(99\) 9.18192i 0.922818i
\(100\) 0 0
\(101\) 4.57122 0.454853 0.227427 0.973795i \(-0.426969\pi\)
0.227427 + 0.973795i \(0.426969\pi\)
\(102\) 0 0
\(103\) 6.37882 0.628524 0.314262 0.949336i \(-0.398243\pi\)
0.314262 + 0.949336i \(0.398243\pi\)
\(104\) 0 0
\(105\) 5.85423i 0.571315i
\(106\) 0 0
\(107\) 13.2575i 1.28165i 0.767685 + 0.640827i \(0.221408\pi\)
−0.767685 + 0.640827i \(0.778592\pi\)
\(108\) 0 0
\(109\) 17.6707i 1.69255i 0.532748 + 0.846274i \(0.321159\pi\)
−0.532748 + 0.846274i \(0.678841\pi\)
\(110\) 0 0
\(111\) 11.4611 1.08784
\(112\) 0 0
\(113\) 11.8665i 1.11630i 0.829739 + 0.558152i \(0.188489\pi\)
−0.829739 + 0.558152i \(0.811511\pi\)
\(114\) 0 0
\(115\) 18.0752 1.68552
\(116\) 0 0
\(117\) 14.5194 1.34232
\(118\) 0 0
\(119\) −3.56177 + 2.11907i −0.326507 + 0.194255i
\(120\) 0 0
\(121\) −6.72020 −0.610927
\(122\) 0 0
\(123\) −26.6298 −2.40113
\(124\) 0 0
\(125\) 8.83591i 0.790308i
\(126\) 0 0
\(127\) −12.7130 −1.12810 −0.564048 0.825742i \(-0.690757\pi\)
−0.564048 + 0.825742i \(0.690757\pi\)
\(128\) 0 0
\(129\) 5.36701i 0.472539i
\(130\) 0 0
\(131\) 0.992715i 0.0867339i −0.999059 0.0433670i \(-0.986192\pi\)
0.999059 0.0433670i \(-0.0138085\pi\)
\(132\) 0 0
\(133\) 1.97377i 0.171147i
\(134\) 0 0
\(135\) 4.76862 0.410417
\(136\) 0 0
\(137\) 9.80553 0.837743 0.418871 0.908046i \(-0.362426\pi\)
0.418871 + 0.908046i \(0.362426\pi\)
\(138\) 0 0
\(139\) 14.2279i 1.20680i −0.797440 0.603398i \(-0.793813\pi\)
0.797440 0.603398i \(-0.206187\pi\)
\(140\) 0 0
\(141\) 22.5632i 1.90016i
\(142\) 0 0
\(143\) 28.0210i 2.34323i
\(144\) 0 0
\(145\) −6.40461 −0.531874
\(146\) 0 0
\(147\) 13.6337i 1.12449i
\(148\) 0 0
\(149\) −19.7986 −1.62197 −0.810983 0.585070i \(-0.801067\pi\)
−0.810983 + 0.585070i \(0.801067\pi\)
\(150\) 0 0
\(151\) −18.6733 −1.51961 −0.759804 0.650152i \(-0.774705\pi\)
−0.759804 + 0.650152i \(0.774705\pi\)
\(152\) 0 0
\(153\) −4.59833 7.72894i −0.371753 0.624848i
\(154\) 0 0
\(155\) −18.4005 −1.47797
\(156\) 0 0
\(157\) −0.00844362 −0.000673874 −0.000336937 1.00000i \(-0.500107\pi\)
−0.000336937 1.00000i \(0.500107\pi\)
\(158\) 0 0
\(159\) 0.592936i 0.0470229i
\(160\) 0 0
\(161\) −7.10099 −0.559636
\(162\) 0 0
\(163\) 2.81580i 0.220550i −0.993901 0.110275i \(-0.964827\pi\)
0.993901 0.110275i \(-0.0351732\pi\)
\(164\) 0 0
\(165\) 24.5166i 1.90861i
\(166\) 0 0
\(167\) 15.8605i 1.22732i 0.789569 + 0.613661i \(0.210304\pi\)
−0.789569 + 0.613661i \(0.789696\pi\)
\(168\) 0 0
\(169\) 31.3096 2.40843
\(170\) 0 0
\(171\) −4.28302 −0.327531
\(172\) 0 0
\(173\) 15.4777i 1.17675i 0.808589 + 0.588374i \(0.200232\pi\)
−0.808589 + 0.588374i \(0.799768\pi\)
\(174\) 0 0
\(175\) 1.55465i 0.117521i
\(176\) 0 0
\(177\) 2.27623i 0.171092i
\(178\) 0 0
\(179\) 9.06073 0.677231 0.338616 0.940925i \(-0.390041\pi\)
0.338616 + 0.940925i \(0.390041\pi\)
\(180\) 0 0
\(181\) 2.64116i 0.196316i 0.995171 + 0.0981579i \(0.0312950\pi\)
−0.995171 + 0.0981579i \(0.968705\pi\)
\(182\) 0 0
\(183\) −23.6928 −1.75142
\(184\) 0 0
\(185\) −12.8830 −0.947180
\(186\) 0 0
\(187\) 14.9161 8.87433i 1.09077 0.648955i
\(188\) 0 0
\(189\) −1.87339 −0.136269
\(190\) 0 0
\(191\) −13.8146 −0.999592 −0.499796 0.866143i \(-0.666592\pi\)
−0.499796 + 0.866143i \(0.666592\pi\)
\(192\) 0 0
\(193\) 5.45032i 0.392322i 0.980572 + 0.196161i \(0.0628476\pi\)
−0.980572 + 0.196161i \(0.937152\pi\)
\(194\) 0 0
\(195\) −38.7681 −2.77624
\(196\) 0 0
\(197\) 22.1005i 1.57459i −0.616574 0.787297i \(-0.711480\pi\)
0.616574 0.787297i \(-0.288520\pi\)
\(198\) 0 0
\(199\) 11.8021i 0.836629i 0.908302 + 0.418315i \(0.137379\pi\)
−0.908302 + 0.418315i \(0.862621\pi\)
\(200\) 0 0
\(201\) 8.82576i 0.622521i
\(202\) 0 0
\(203\) 2.51610 0.176596
\(204\) 0 0
\(205\) 29.9338 2.09066
\(206\) 0 0
\(207\) 15.4090i 1.07100i
\(208\) 0 0
\(209\) 8.26580i 0.571758i
\(210\) 0 0
\(211\) 28.7529i 1.97943i 0.143060 + 0.989714i \(0.454306\pi\)
−0.143060 + 0.989714i \(0.545694\pi\)
\(212\) 0 0
\(213\) 12.8434 0.880014
\(214\) 0 0
\(215\) 6.03289i 0.411440i
\(216\) 0 0
\(217\) 7.22879 0.490722
\(218\) 0 0
\(219\) −20.0872 −1.35737
\(220\) 0 0
\(221\) −14.0330 23.5868i −0.943961 1.58662i
\(222\) 0 0
\(223\) 19.4175 1.30029 0.650147 0.759808i \(-0.274707\pi\)
0.650147 + 0.759808i \(0.274707\pi\)
\(224\) 0 0
\(225\) 3.37356 0.224904
\(226\) 0 0
\(227\) 21.4500i 1.42368i −0.702339 0.711842i \(-0.747861\pi\)
0.702339 0.711842i \(-0.252139\pi\)
\(228\) 0 0
\(229\) 10.6539 0.704032 0.352016 0.935994i \(-0.385496\pi\)
0.352016 + 0.935994i \(0.385496\pi\)
\(230\) 0 0
\(231\) 9.63153i 0.633708i
\(232\) 0 0
\(233\) 24.7340i 1.62038i −0.586166 0.810191i \(-0.699364\pi\)
0.586166 0.810191i \(-0.300636\pi\)
\(234\) 0 0
\(235\) 25.3626i 1.65447i
\(236\) 0 0
\(237\) −31.1247 −2.02177
\(238\) 0 0
\(239\) −23.6967 −1.53282 −0.766408 0.642355i \(-0.777958\pi\)
−0.766408 + 0.642355i \(0.777958\pi\)
\(240\) 0 0
\(241\) 6.04958i 0.389688i −0.980834 0.194844i \(-0.937580\pi\)
0.980834 0.194844i \(-0.0624200\pi\)
\(242\) 0 0
\(243\) 18.9601i 1.21629i
\(244\) 0 0
\(245\) 15.3253i 0.979094i
\(246\) 0 0
\(247\) −13.0707 −0.831670
\(248\) 0 0
\(249\) 4.90739i 0.310994i
\(250\) 0 0
\(251\) −4.14423 −0.261581 −0.130791 0.991410i \(-0.541752\pi\)
−0.130791 + 0.991410i \(0.541752\pi\)
\(252\) 0 0
\(253\) 29.7378 1.86960
\(254\) 0 0
\(255\) 12.2780 + 20.6370i 0.768876 + 1.29234i
\(256\) 0 0
\(257\) 29.4226 1.83533 0.917664 0.397357i \(-0.130072\pi\)
0.917664 + 0.397357i \(0.130072\pi\)
\(258\) 0 0
\(259\) 5.06120 0.314488
\(260\) 0 0
\(261\) 5.45988i 0.337958i
\(262\) 0 0
\(263\) −7.77247 −0.479271 −0.239636 0.970863i \(-0.577028\pi\)
−0.239636 + 0.970863i \(0.577028\pi\)
\(264\) 0 0
\(265\) 0.666501i 0.0409428i
\(266\) 0 0
\(267\) 11.1304i 0.681168i
\(268\) 0 0
\(269\) 3.48426i 0.212439i −0.994343 0.106220i \(-0.966125\pi\)
0.994343 0.106220i \(-0.0338747\pi\)
\(270\) 0 0
\(271\) 32.3356 1.96425 0.982124 0.188233i \(-0.0602760\pi\)
0.982124 + 0.188233i \(0.0602760\pi\)
\(272\) 0 0
\(273\) 15.2303 0.921782
\(274\) 0 0
\(275\) 6.51063i 0.392606i
\(276\) 0 0
\(277\) 28.1946i 1.69405i 0.531553 + 0.847025i \(0.321609\pi\)
−0.531553 + 0.847025i \(0.678391\pi\)
\(278\) 0 0
\(279\) 15.6863i 0.939114i
\(280\) 0 0
\(281\) −12.6074 −0.752092 −0.376046 0.926601i \(-0.622717\pi\)
−0.376046 + 0.926601i \(0.622717\pi\)
\(282\) 0 0
\(283\) 31.0152i 1.84366i −0.387594 0.921830i \(-0.626694\pi\)
0.387594 0.921830i \(-0.373306\pi\)
\(284\) 0 0
\(285\) 11.4360 0.677413
\(286\) 0 0
\(287\) −11.7597 −0.694154
\(288\) 0 0
\(289\) −8.11142 + 14.9400i −0.477142 + 0.878826i
\(290\) 0 0
\(291\) −27.8013 −1.62974
\(292\) 0 0
\(293\) −3.10723 −0.181527 −0.0907633 0.995872i \(-0.528931\pi\)
−0.0907633 + 0.995872i \(0.528931\pi\)
\(294\) 0 0
\(295\) 2.55864i 0.148970i
\(296\) 0 0
\(297\) 7.84545 0.455239
\(298\) 0 0
\(299\) 47.0243i 2.71949i
\(300\) 0 0
\(301\) 2.37007i 0.136609i
\(302\) 0 0
\(303\) 10.4051i 0.597760i
\(304\) 0 0
\(305\) 26.6324 1.52497
\(306\) 0 0
\(307\) −28.4347 −1.62285 −0.811426 0.584455i \(-0.801308\pi\)
−0.811426 + 0.584455i \(0.801308\pi\)
\(308\) 0 0
\(309\) 14.5197i 0.825995i
\(310\) 0 0
\(311\) 33.6607i 1.90873i 0.298651 + 0.954363i \(0.403463\pi\)
−0.298651 + 0.954363i \(0.596537\pi\)
\(312\) 0 0
\(313\) 12.3042i 0.695474i −0.937592 0.347737i \(-0.886950\pi\)
0.937592 0.347737i \(-0.113050\pi\)
\(314\) 0 0
\(315\) −5.60988 −0.316081
\(316\) 0 0
\(317\) 30.4950i 1.71277i 0.516339 + 0.856384i \(0.327294\pi\)
−0.516339 + 0.856384i \(0.672706\pi\)
\(318\) 0 0
\(319\) −10.5370 −0.589960
\(320\) 0 0
\(321\) −30.1772 −1.68433
\(322\) 0 0
\(323\) 4.13954 + 6.95779i 0.230330 + 0.387142i
\(324\) 0 0
\(325\) 10.2953 0.571079
\(326\) 0 0
\(327\) −40.2226 −2.22431
\(328\) 0 0
\(329\) 9.96390i 0.549328i
\(330\) 0 0
\(331\) −2.55529 −0.140451 −0.0702256 0.997531i \(-0.522372\pi\)
−0.0702256 + 0.997531i \(0.522372\pi\)
\(332\) 0 0
\(333\) 10.9827i 0.601847i
\(334\) 0 0
\(335\) 9.92077i 0.542030i
\(336\) 0 0
\(337\) 10.8364i 0.590297i 0.955451 + 0.295149i \(0.0953692\pi\)
−0.955451 + 0.295149i \(0.904631\pi\)
\(338\) 0 0
\(339\) −27.0108 −1.46703
\(340\) 0 0
\(341\) −30.2730 −1.63937
\(342\) 0 0
\(343\) 13.0569i 0.705008i
\(344\) 0 0
\(345\) 41.1433i 2.21508i
\(346\) 0 0
\(347\) 0.510915i 0.0274273i −0.999906 0.0137137i \(-0.995635\pi\)
0.999906 0.0137137i \(-0.00436533\pi\)
\(348\) 0 0
\(349\) −17.3955 −0.931157 −0.465579 0.885007i \(-0.654154\pi\)
−0.465579 + 0.885007i \(0.654154\pi\)
\(350\) 0 0
\(351\) 12.4060i 0.662184i
\(352\) 0 0
\(353\) 29.8410 1.58827 0.794137 0.607739i \(-0.207923\pi\)
0.794137 + 0.607739i \(0.207923\pi\)
\(354\) 0 0
\(355\) −14.4369 −0.766229
\(356\) 0 0
\(357\) −4.82350 8.10740i −0.255287 0.429089i
\(358\) 0 0
\(359\) −29.4112 −1.55226 −0.776131 0.630571i \(-0.782821\pi\)
−0.776131 + 0.630571i \(0.782821\pi\)
\(360\) 0 0
\(361\) −15.1443 −0.797069
\(362\) 0 0
\(363\) 15.2967i 0.802870i
\(364\) 0 0
\(365\) 22.5794 1.18186
\(366\) 0 0
\(367\) 16.6869i 0.871049i 0.900177 + 0.435525i \(0.143437\pi\)
−0.900177 + 0.435525i \(0.856563\pi\)
\(368\) 0 0
\(369\) 25.5183i 1.32843i
\(370\) 0 0
\(371\) 0.261840i 0.0135941i
\(372\) 0 0
\(373\) 8.46715 0.438413 0.219206 0.975679i \(-0.429653\pi\)
0.219206 + 0.975679i \(0.429653\pi\)
\(374\) 0 0
\(375\) 20.1126 1.03861
\(376\) 0 0
\(377\) 16.6622i 0.858147i
\(378\) 0 0
\(379\) 25.6790i 1.31904i −0.751686 0.659521i \(-0.770759\pi\)
0.751686 0.659521i \(-0.229241\pi\)
\(380\) 0 0
\(381\) 28.9377i 1.48252i
\(382\) 0 0
\(383\) 18.0159 0.920569 0.460284 0.887771i \(-0.347747\pi\)
0.460284 + 0.887771i \(0.347747\pi\)
\(384\) 0 0
\(385\) 10.8265i 0.551770i
\(386\) 0 0
\(387\) −5.14299 −0.261433
\(388\) 0 0
\(389\) −16.5210 −0.837649 −0.418825 0.908067i \(-0.637558\pi\)
−0.418825 + 0.908067i \(0.637558\pi\)
\(390\) 0 0
\(391\) −25.0319 + 14.8928i −1.26592 + 0.753159i
\(392\) 0 0
\(393\) 2.25965 0.113984
\(394\) 0 0
\(395\) 34.9864 1.76035
\(396\) 0 0
\(397\) 7.10777i 0.356729i −0.983965 0.178364i \(-0.942919\pi\)
0.983965 0.178364i \(-0.0570806\pi\)
\(398\) 0 0
\(399\) −4.49274 −0.224918
\(400\) 0 0
\(401\) 23.7556i 1.18630i −0.805093 0.593148i \(-0.797885\pi\)
0.805093 0.593148i \(-0.202115\pi\)
\(402\) 0 0
\(403\) 47.8707i 2.38461i
\(404\) 0 0
\(405\) 27.5973i 1.37132i
\(406\) 0 0
\(407\) −21.1955 −1.05062
\(408\) 0 0
\(409\) −35.3537 −1.74813 −0.874063 0.485813i \(-0.838524\pi\)
−0.874063 + 0.485813i \(0.838524\pi\)
\(410\) 0 0
\(411\) 22.3196i 1.10095i
\(412\) 0 0
\(413\) 1.00518i 0.0494618i
\(414\) 0 0
\(415\) 5.51625i 0.270782i
\(416\) 0 0
\(417\) 32.3860 1.58595
\(418\) 0 0
\(419\) 1.19866i 0.0585585i −0.999571 0.0292792i \(-0.990679\pi\)
0.999571 0.0292792i \(-0.00932120\pi\)
\(420\) 0 0
\(421\) 10.8949 0.530983 0.265492 0.964113i \(-0.414466\pi\)
0.265492 + 0.964113i \(0.414466\pi\)
\(422\) 0 0
\(423\) 21.6214 1.05127
\(424\) 0 0
\(425\) −3.26054 5.48036i −0.158160 0.265837i
\(426\) 0 0
\(427\) −10.4627 −0.506328
\(428\) 0 0
\(429\) −63.7822 −3.07943
\(430\) 0 0
\(431\) 0.595134i 0.0286666i −0.999897 0.0143333i \(-0.995437\pi\)
0.999897 0.0143333i \(-0.00456259\pi\)
\(432\) 0 0
\(433\) 20.6870 0.994154 0.497077 0.867706i \(-0.334407\pi\)
0.497077 + 0.867706i \(0.334407\pi\)
\(434\) 0 0
\(435\) 14.5784i 0.698979i
\(436\) 0 0
\(437\) 13.8715i 0.663566i
\(438\) 0 0
\(439\) 8.10626i 0.386891i 0.981111 + 0.193445i \(0.0619662\pi\)
−0.981111 + 0.193445i \(0.938034\pi\)
\(440\) 0 0
\(441\) −13.0647 −0.622126
\(442\) 0 0
\(443\) 6.07515 0.288639 0.144320 0.989531i \(-0.453901\pi\)
0.144320 + 0.989531i \(0.453901\pi\)
\(444\) 0 0
\(445\) 12.5113i 0.593094i
\(446\) 0 0
\(447\) 45.0662i 2.13156i
\(448\) 0 0
\(449\) 23.9914i 1.13222i −0.824329 0.566111i \(-0.808447\pi\)
0.824329 0.566111i \(-0.191553\pi\)
\(450\) 0 0
\(451\) 49.2478 2.31899
\(452\) 0 0
\(453\) 42.5046i 1.99704i
\(454\) 0 0
\(455\) −17.1200 −0.802596
\(456\) 0 0
\(457\) 3.45563 0.161647 0.0808237 0.996728i \(-0.474245\pi\)
0.0808237 + 0.996728i \(0.474245\pi\)
\(458\) 0 0
\(459\) −6.60396 + 3.92903i −0.308246 + 0.183391i
\(460\) 0 0
\(461\) 41.5005 1.93287 0.966436 0.256906i \(-0.0827032\pi\)
0.966436 + 0.256906i \(0.0827032\pi\)
\(462\) 0 0
\(463\) 0.955366 0.0443996 0.0221998 0.999754i \(-0.492933\pi\)
0.0221998 + 0.999754i \(0.492933\pi\)
\(464\) 0 0
\(465\) 41.8838i 1.94232i
\(466\) 0 0
\(467\) −28.3197 −1.31048 −0.655240 0.755421i \(-0.727432\pi\)
−0.655240 + 0.755421i \(0.727432\pi\)
\(468\) 0 0
\(469\) 3.89746i 0.179968i
\(470\) 0 0
\(471\) 0.0192196i 0.000885593i
\(472\) 0 0
\(473\) 9.92547i 0.456373i
\(474\) 0 0
\(475\) −3.03696 −0.139345
\(476\) 0 0
\(477\) −0.568186 −0.0260155
\(478\) 0 0
\(479\) 18.0796i 0.826079i −0.910713 0.413039i \(-0.864467\pi\)
0.910713 0.413039i \(-0.135533\pi\)
\(480\) 0 0
\(481\) 33.5164i 1.52822i
\(482\) 0 0
\(483\) 16.1635i 0.735463i
\(484\) 0 0
\(485\) 31.2505 1.41901
\(486\) 0 0
\(487\) 20.7402i 0.939828i −0.882712 0.469914i \(-0.844285\pi\)
0.882712 0.469914i \(-0.155715\pi\)
\(488\) 0 0
\(489\) 6.40940 0.289843
\(490\) 0 0
\(491\) 9.46942 0.427349 0.213674 0.976905i \(-0.431457\pi\)
0.213674 + 0.976905i \(0.431457\pi\)
\(492\) 0 0
\(493\) 8.86960 5.27697i 0.399467 0.237663i
\(494\) 0 0
\(495\) 23.4932 1.05594
\(496\) 0 0
\(497\) 5.67163 0.254408
\(498\) 0 0
\(499\) 30.8318i 1.38022i 0.723705 + 0.690110i \(0.242438\pi\)
−0.723705 + 0.690110i \(0.757562\pi\)
\(500\) 0 0
\(501\) −36.1021 −1.61292
\(502\) 0 0
\(503\) 21.0282i 0.937603i 0.883303 + 0.468802i \(0.155314\pi\)
−0.883303 + 0.468802i \(0.844686\pi\)
\(504\) 0 0
\(505\) 11.6961i 0.520470i
\(506\) 0 0
\(507\) 71.2678i 3.16511i
\(508\) 0 0
\(509\) −7.87582 −0.349090 −0.174545 0.984649i \(-0.555845\pi\)
−0.174545 + 0.984649i \(0.555845\pi\)
\(510\) 0 0
\(511\) −8.87052 −0.392409
\(512\) 0 0
\(513\) 3.65961i 0.161576i
\(514\) 0 0
\(515\) 16.3211i 0.719194i
\(516\) 0 0
\(517\) 41.7272i 1.83516i
\(518\) 0 0
\(519\) −35.2308 −1.54646
\(520\) 0 0
\(521\) 4.28579i 0.187764i 0.995583 + 0.0938820i \(0.0299276\pi\)
−0.995583 + 0.0938820i \(0.970072\pi\)
\(522\) 0 0
\(523\) 9.03467 0.395058 0.197529 0.980297i \(-0.436708\pi\)
0.197529 + 0.980297i \(0.436708\pi\)
\(524\) 0 0
\(525\) 3.53875 0.154444
\(526\) 0 0
\(527\) 25.4825 15.1608i 1.11003 0.660415i
\(528\) 0 0
\(529\) −26.9054 −1.16980
\(530\) 0 0
\(531\) −2.18122 −0.0946569
\(532\) 0 0
\(533\) 77.8755i 3.37316i
\(534\) 0 0
\(535\) 33.9212 1.46654
\(536\) 0 0
\(537\) 20.6243i 0.890005i
\(538\) 0 0
\(539\) 25.2135i 1.08602i
\(540\) 0 0
\(541\) 27.7720i 1.19401i −0.802237 0.597006i \(-0.796357\pi\)
0.802237 0.597006i \(-0.203643\pi\)
\(542\) 0 0
\(543\) −6.01189 −0.257995
\(544\) 0 0
\(545\) 45.2130 1.93671
\(546\) 0 0
\(547\) 12.2621i 0.524289i −0.965029 0.262144i \(-0.915570\pi\)
0.965029 0.262144i \(-0.0844297\pi\)
\(548\) 0 0
\(549\) 22.7039i 0.968978i
\(550\) 0 0
\(551\) 4.91512i 0.209391i
\(552\) 0 0
\(553\) −13.7447 −0.584483
\(554\) 0 0
\(555\) 29.3247i 1.24477i
\(556\) 0 0
\(557\) −7.41588 −0.314221 −0.157111 0.987581i \(-0.550218\pi\)
−0.157111 + 0.987581i \(0.550218\pi\)
\(558\) 0 0
\(559\) −15.6951 −0.663834
\(560\) 0 0
\(561\) 20.2000 + 33.9524i 0.852845 + 1.43347i
\(562\) 0 0
\(563\) −2.04178 −0.0860509 −0.0430255 0.999074i \(-0.513700\pi\)
−0.0430255 + 0.999074i \(0.513700\pi\)
\(564\) 0 0
\(565\) 30.3620 1.27734
\(566\) 0 0
\(567\) 10.8418i 0.455314i
\(568\) 0 0
\(569\) −29.6366 −1.24243 −0.621216 0.783639i \(-0.713361\pi\)
−0.621216 + 0.783639i \(0.713361\pi\)
\(570\) 0 0
\(571\) 29.3519i 1.22834i −0.789174 0.614169i \(-0.789491\pi\)
0.789174 0.614169i \(-0.210509\pi\)
\(572\) 0 0
\(573\) 31.4453i 1.31364i
\(574\) 0 0
\(575\) 10.9260i 0.455647i
\(576\) 0 0
\(577\) −3.38873 −0.141075 −0.0705374 0.997509i \(-0.522471\pi\)
−0.0705374 + 0.997509i \(0.522471\pi\)
\(578\) 0 0
\(579\) −12.4062 −0.515583
\(580\) 0 0
\(581\) 2.16710i 0.0899066i
\(582\) 0 0
\(583\) 1.09654i 0.0454142i
\(584\) 0 0
\(585\) 37.1499i 1.53596i
\(586\) 0 0
\(587\) −30.9740 −1.27844 −0.639218 0.769026i \(-0.720742\pi\)
−0.639218 + 0.769026i \(0.720742\pi\)
\(588\) 0 0
\(589\) 14.1212i 0.581854i
\(590\) 0 0
\(591\) 50.3057 2.06930
\(592\) 0 0
\(593\) 22.5290 0.925155 0.462577 0.886579i \(-0.346925\pi\)
0.462577 + 0.886579i \(0.346925\pi\)
\(594\) 0 0
\(595\) 5.42195 + 9.11328i 0.222278 + 0.373608i
\(596\) 0 0
\(597\) −26.8643 −1.09948
\(598\) 0 0
\(599\) 36.2890 1.48273 0.741364 0.671103i \(-0.234179\pi\)
0.741364 + 0.671103i \(0.234179\pi\)
\(600\) 0 0
\(601\) 5.37020i 0.219055i −0.993984 0.109527i \(-0.965066\pi\)
0.993984 0.109527i \(-0.0349338\pi\)
\(602\) 0 0
\(603\) −8.45738 −0.344411
\(604\) 0 0
\(605\) 17.1946i 0.699059i
\(606\) 0 0
\(607\) 29.1625i 1.18367i −0.806059 0.591835i \(-0.798404\pi\)
0.806059 0.591835i \(-0.201596\pi\)
\(608\) 0 0
\(609\) 5.72722i 0.232079i
\(610\) 0 0
\(611\) 65.9832 2.66940
\(612\) 0 0
\(613\) −29.7291 −1.20075 −0.600374 0.799719i \(-0.704982\pi\)
−0.600374 + 0.799719i \(0.704982\pi\)
\(614\) 0 0
\(615\) 68.1361i 2.74751i
\(616\) 0 0
\(617\) 0.474131i 0.0190878i −0.999954 0.00954391i \(-0.996962\pi\)
0.999954 0.00954391i \(-0.00303797\pi\)
\(618\) 0 0
\(619\) 41.5114i 1.66848i 0.551399 + 0.834242i \(0.314094\pi\)
−0.551399 + 0.834242i \(0.685906\pi\)
\(620\) 0 0
\(621\) −13.1661 −0.528338
\(622\) 0 0
\(623\) 4.91517i 0.196922i
\(624\) 0 0
\(625\) −30.3411 −1.21364
\(626\) 0 0
\(627\) 18.8149 0.751393
\(628\) 0 0
\(629\) 17.8414 10.6148i 0.711384 0.423238i
\(630\) 0 0
\(631\) −3.42025 −0.136158 −0.0680791 0.997680i \(-0.521687\pi\)
−0.0680791 + 0.997680i \(0.521687\pi\)
\(632\) 0 0
\(633\) −65.4481 −2.60133
\(634\) 0 0
\(635\) 32.5280i 1.29083i
\(636\) 0 0
\(637\) −39.8701 −1.57971
\(638\) 0 0
\(639\) 12.3073i 0.486869i
\(640\) 0 0
\(641\) 1.25740i 0.0496643i 0.999692 + 0.0248322i \(0.00790514\pi\)
−0.999692 + 0.0248322i \(0.992095\pi\)
\(642\) 0 0
\(643\) 43.0990i 1.69966i 0.527059 + 0.849829i \(0.323295\pi\)
−0.527059 + 0.849829i \(0.676705\pi\)
\(644\) 0 0
\(645\) 13.7323 0.540707
\(646\) 0 0
\(647\) 6.97098 0.274058 0.137029 0.990567i \(-0.456245\pi\)
0.137029 + 0.990567i \(0.456245\pi\)
\(648\) 0 0
\(649\) 4.20954i 0.165239i
\(650\) 0 0
\(651\) 16.4544i 0.644898i
\(652\) 0 0
\(653\) 29.9941i 1.17376i 0.809674 + 0.586880i \(0.199644\pi\)
−0.809674 + 0.586880i \(0.800356\pi\)
\(654\) 0 0
\(655\) −2.54000 −0.0992460
\(656\) 0 0
\(657\) 19.2488i 0.750967i
\(658\) 0 0
\(659\) 34.2812 1.33541 0.667703 0.744428i \(-0.267278\pi\)
0.667703 + 0.744428i \(0.267278\pi\)
\(660\) 0 0
\(661\) 31.6709 1.23186 0.615928 0.787803i \(-0.288781\pi\)
0.615928 + 0.787803i \(0.288781\pi\)
\(662\) 0 0
\(663\) 53.6890 31.9423i 2.08511 1.24054i
\(664\) 0 0
\(665\) 5.05015 0.195837
\(666\) 0 0
\(667\) 17.6831 0.684691
\(668\) 0 0
\(669\) 44.1988i 1.70882i
\(670\) 0 0
\(671\) 43.8162 1.69151
\(672\) 0 0
\(673\) 29.9400i 1.15410i 0.816708 + 0.577051i \(0.195797\pi\)
−0.816708 + 0.577051i \(0.804203\pi\)
\(674\) 0 0
\(675\) 2.88252i 0.110948i
\(676\) 0 0
\(677\) 3.31186i 0.127285i 0.997973 + 0.0636426i \(0.0202718\pi\)
−0.997973 + 0.0636426i \(0.979728\pi\)
\(678\) 0 0
\(679\) −12.2770 −0.471149
\(680\) 0 0
\(681\) 48.8250 1.87098
\(682\) 0 0
\(683\) 4.80915i 0.184017i 0.995758 + 0.0920085i \(0.0293287\pi\)
−0.995758 + 0.0920085i \(0.970671\pi\)
\(684\) 0 0
\(685\) 25.0888i 0.958594i
\(686\) 0 0
\(687\) 24.2508i 0.925226i
\(688\) 0 0
\(689\) −1.73397 −0.0660588
\(690\) 0 0
\(691\) 14.1889i 0.539772i −0.962892 0.269886i \(-0.913014\pi\)
0.962892 0.269886i \(-0.0869861\pi\)
\(692\) 0 0
\(693\) −9.22951 −0.350600
\(694\) 0 0
\(695\) −36.4041 −1.38089
\(696\) 0 0
\(697\) −41.4546 + 24.6634i −1.57021 + 0.934194i
\(698\) 0 0
\(699\) 56.3004 2.12948
\(700\) 0 0
\(701\) −2.33659 −0.0882517 −0.0441258 0.999026i \(-0.514050\pi\)
−0.0441258 + 0.999026i \(0.514050\pi\)
\(702\) 0 0
\(703\) 9.88689i 0.372891i
\(704\) 0 0
\(705\) −57.7311 −2.17428
\(706\) 0 0
\(707\) 4.59491i 0.172809i
\(708\) 0 0
\(709\) 5.65401i 0.212341i 0.994348 + 0.106170i \(0.0338589\pi\)
−0.994348 + 0.106170i \(0.966141\pi\)
\(710\) 0 0
\(711\) 29.8256i 1.11855i
\(712\) 0 0
\(713\) 50.8036 1.90261
\(714\) 0 0
\(715\) 71.6956 2.68126
\(716\) 0 0
\(717\) 53.9392i 2.01440i
\(718\) 0 0
\(719\) 27.2950i 1.01793i −0.860787 0.508966i \(-0.830028\pi\)
0.860787 0.508966i \(-0.169972\pi\)
\(720\) 0 0
\(721\) 6.41188i 0.238791i
\(722\) 0 0
\(723\) 13.7702 0.512120
\(724\) 0 0
\(725\) 3.87144i 0.143782i
\(726\) 0 0
\(727\) 7.29246 0.270462 0.135231 0.990814i \(-0.456822\pi\)
0.135231 + 0.990814i \(0.456822\pi\)
\(728\) 0 0
\(729\) 10.7997 0.399987
\(730\) 0 0
\(731\) 4.97070 + 8.35482i 0.183848 + 0.309014i
\(732\) 0 0
\(733\) 44.4584 1.64211 0.821055 0.570850i \(-0.193386\pi\)
0.821055 + 0.570850i \(0.193386\pi\)
\(734\) 0 0
\(735\) 34.8838 1.28671
\(736\) 0 0
\(737\) 16.3219i 0.601225i
\(738\) 0 0
\(739\) 39.3561 1.44774 0.723868 0.689939i \(-0.242362\pi\)
0.723868 + 0.689939i \(0.242362\pi\)
\(740\) 0 0
\(741\) 29.7520i 1.09297i
\(742\) 0 0
\(743\) 35.7207i 1.31047i −0.755426 0.655234i \(-0.772570\pi\)
0.755426 0.655234i \(-0.227430\pi\)
\(744\) 0 0
\(745\) 50.6575i 1.85595i
\(746\) 0 0
\(747\) −4.70256 −0.172058
\(748\) 0 0
\(749\) −13.3262 −0.486930
\(750\) 0 0
\(751\) 37.7059i 1.37591i −0.725754 0.687954i \(-0.758509\pi\)
0.725754 0.687954i \(-0.241491\pi\)
\(752\) 0 0
\(753\) 9.43321i 0.343765i
\(754\) 0 0
\(755\) 47.7782i 1.73883i
\(756\) 0 0
\(757\) 3.21403 0.116816 0.0584080 0.998293i \(-0.481398\pi\)
0.0584080 + 0.998293i \(0.481398\pi\)
\(758\) 0 0
\(759\) 67.6900i 2.45699i
\(760\) 0 0
\(761\) −20.2104 −0.732627 −0.366313 0.930492i \(-0.619380\pi\)
−0.366313 + 0.930492i \(0.619380\pi\)
\(762\) 0 0
\(763\) −17.7623 −0.643038
\(764\) 0 0
\(765\) −19.7756 + 11.7655i −0.714987 + 0.425382i
\(766\) 0 0
\(767\) −6.65655 −0.240354
\(768\) 0 0
\(769\) 8.75506 0.315716 0.157858 0.987462i \(-0.449541\pi\)
0.157858 + 0.987462i \(0.449541\pi\)
\(770\) 0 0
\(771\) 66.9725i 2.41195i
\(772\) 0 0
\(773\) 48.8172 1.75583 0.877916 0.478814i \(-0.158933\pi\)
0.877916 + 0.478814i \(0.158933\pi\)
\(774\) 0 0
\(775\) 11.1227i 0.399539i
\(776\) 0 0
\(777\) 11.5205i 0.413294i
\(778\) 0 0
\(779\) 22.9722i 0.823065i
\(780\) 0 0
\(781\) −23.7519 −0.849909
\(782\) 0 0
\(783\) 4.66517 0.166719
\(784\) 0 0
\(785\) 0.0216042i 0.000771086i
\(786\) 0 0
\(787\) 20.5500i 0.732527i −0.930511 0.366264i \(-0.880637\pi\)
0.930511 0.366264i \(-0.119363\pi\)
\(788\) 0 0
\(789\) 17.6919i 0.629849i
\(790\) 0 0
\(791\) −11.9280 −0.424110
\(792\) 0 0
\(793\) 69.2867i 2.46044i