Properties

Label 2880.2.o.f.2879.12
Level $2880$
Weight $2$
Character 2880.2879
Analytic conductor $22.997$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.426337261060096.1
Defining polynomial: \(x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 1440)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2879.12
Root \(-0.0912546 - 1.41127i\) of defining polynomial
Character \(\chi\) \(=\) 2880.2879
Dual form 2880.2.o.f.2879.11

$q$-expansion

\(f(q)\) \(=\) \(q+(2.20963 + 0.342849i) q^{5} +2.64002 q^{7} +O(q^{10})\) \(q+(2.20963 + 0.342849i) q^{5} +2.64002 q^{7} +3.00504 q^{11} -0.640023i q^{13} -0.685698 q^{17} +5.28005i q^{19} +2.27653i q^{23} +(4.76491 + 1.51514i) q^{25} +8.15281i q^{29} -2.96972i q^{31} +(5.83347 + 0.905130i) q^{35} +1.60975i q^{37} -7.42430i q^{41} +11.2195 q^{43} +4.19982i q^{47} -0.0302761 q^{49} -9.60984 q^{53} +(6.64002 + 1.03028i) q^{55} +7.20487 q^{59} -10.4995 q^{61} +(0.219432 - 1.41421i) q^{65} +8.49954 q^{67} -13.1240 q^{71} -14.2498i q^{73} +7.93338 q^{77} -11.4693i q^{79} +13.1240i q^{83} +(-1.51514 - 0.235091i) q^{85} -10.2527i q^{89} -1.68968i q^{91} +(-1.81026 + 11.6669i) q^{95} +8.31032i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q - 8q^{25} + 64q^{43} - 4q^{49} + 48q^{55} + 8q^{61} - 32q^{67} - 20q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.20963 + 0.342849i 0.988176 + 0.153327i
\(6\) 0 0
\(7\) 2.64002 0.997835 0.498918 0.866649i \(-0.333731\pi\)
0.498918 + 0.866649i \(0.333731\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00504 0.906054 0.453027 0.891497i \(-0.350344\pi\)
0.453027 + 0.891497i \(0.350344\pi\)
\(12\) 0 0
\(13\) 0.640023i 0.177511i −0.996053 0.0887553i \(-0.971711\pi\)
0.996053 0.0887553i \(-0.0282889\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.685698 −0.166306 −0.0831531 0.996537i \(-0.526499\pi\)
−0.0831531 + 0.996537i \(0.526499\pi\)
\(18\) 0 0
\(19\) 5.28005i 1.21133i 0.795721 + 0.605663i \(0.207092\pi\)
−0.795721 + 0.605663i \(0.792908\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.27653i 0.474689i 0.971426 + 0.237344i \(0.0762770\pi\)
−0.971426 + 0.237344i \(0.923723\pi\)
\(24\) 0 0
\(25\) 4.76491 + 1.51514i 0.952982 + 0.303028i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.15281i 1.51394i 0.653450 + 0.756970i \(0.273321\pi\)
−0.653450 + 0.756970i \(0.726679\pi\)
\(30\) 0 0
\(31\) 2.96972i 0.533378i −0.963783 0.266689i \(-0.914070\pi\)
0.963783 0.266689i \(-0.0859297\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.83347 + 0.905130i 0.986036 + 0.152995i
\(36\) 0 0
\(37\) 1.60975i 0.264641i 0.991207 + 0.132320i \(0.0422428\pi\)
−0.991207 + 0.132320i \(0.957757\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.42430i 1.15948i −0.814801 0.579740i \(-0.803154\pi\)
0.814801 0.579740i \(-0.196846\pi\)
\(42\) 0 0
\(43\) 11.2195 1.71096 0.855478 0.517838i \(-0.173263\pi\)
0.855478 + 0.517838i \(0.173263\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.19982i 0.612607i 0.951934 + 0.306304i \(0.0990923\pi\)
−0.951934 + 0.306304i \(0.900908\pi\)
\(48\) 0 0
\(49\) −0.0302761 −0.00432516
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.60984 −1.32001 −0.660007 0.751260i \(-0.729447\pi\)
−0.660007 + 0.751260i \(0.729447\pi\)
\(54\) 0 0
\(55\) 6.64002 + 1.03028i 0.895341 + 0.138922i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.20487 0.937994 0.468997 0.883200i \(-0.344616\pi\)
0.468997 + 0.883200i \(0.344616\pi\)
\(60\) 0 0
\(61\) −10.4995 −1.34433 −0.672164 0.740402i \(-0.734635\pi\)
−0.672164 + 0.740402i \(0.734635\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.219432 1.41421i 0.0272171 0.175412i
\(66\) 0 0
\(67\) 8.49954 1.03838 0.519192 0.854658i \(-0.326233\pi\)
0.519192 + 0.854658i \(0.326233\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.1240 −1.55753 −0.778764 0.627317i \(-0.784153\pi\)
−0.778764 + 0.627317i \(0.784153\pi\)
\(72\) 0 0
\(73\) 14.2498i 1.66781i −0.551908 0.833905i \(-0.686100\pi\)
0.551908 0.833905i \(-0.313900\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.93338 0.904093
\(78\) 0 0
\(79\) 11.4693i 1.29039i −0.764017 0.645197i \(-0.776775\pi\)
0.764017 0.645197i \(-0.223225\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.1240i 1.44054i 0.693692 + 0.720271i \(0.255983\pi\)
−0.693692 + 0.720271i \(0.744017\pi\)
\(84\) 0 0
\(85\) −1.51514 0.235091i −0.164340 0.0254992i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.2527i 1.08679i −0.839478 0.543393i \(-0.817139\pi\)
0.839478 0.543393i \(-0.182861\pi\)
\(90\) 0 0
\(91\) 1.68968i 0.177126i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.81026 + 11.6669i −0.185729 + 1.19700i
\(96\) 0 0
\(97\) 8.31032i 0.843785i 0.906646 + 0.421893i \(0.138634\pi\)
−0.906646 + 0.421893i \(0.861366\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.51413i 0.349669i 0.984598 + 0.174834i \(0.0559390\pi\)
−0.984598 + 0.174834i \(0.944061\pi\)
\(102\) 0 0
\(103\) 3.13957 0.309351 0.154675 0.987965i \(-0.450567\pi\)
0.154675 + 0.987965i \(0.450567\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.46711i 0.721873i 0.932590 + 0.360937i \(0.117543\pi\)
−0.932590 + 0.360937i \(0.882457\pi\)
\(108\) 0 0
\(109\) −2.96972 −0.284448 −0.142224 0.989835i \(-0.545425\pi\)
−0.142224 + 0.989835i \(0.545425\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −5.67761 −0.534105 −0.267053 0.963682i \(-0.586050\pi\)
−0.267053 + 0.963682i \(0.586050\pi\)
\(114\) 0 0
\(115\) −0.780505 + 5.03028i −0.0727825 + 0.469076i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.81026 −0.165946
\(120\) 0 0
\(121\) −1.96972 −0.179066
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.0092 + 4.98154i 0.895251 + 0.445562i
\(126\) 0 0
\(127\) 19.1396 1.69836 0.849181 0.528102i \(-0.177096\pi\)
0.849181 + 0.528102i \(0.177096\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −11.5760 −1.01140 −0.505698 0.862711i \(-0.668765\pi\)
−0.505698 + 0.862711i \(0.668765\pi\)
\(132\) 0 0
\(133\) 13.9394i 1.20870i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.49596 0.213244 0.106622 0.994300i \(-0.465997\pi\)
0.106622 + 0.994300i \(0.465997\pi\)
\(138\) 0 0
\(139\) 7.46927i 0.633535i −0.948503 0.316767i \(-0.897403\pi\)
0.948503 0.316767i \(-0.102597\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.92330i 0.160834i
\(144\) 0 0
\(145\) −2.79518 + 18.0147i −0.232127 + 1.49604i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.60984i 0.787269i −0.919267 0.393634i \(-0.871218\pi\)
0.919267 0.393634i \(-0.128782\pi\)
\(150\) 0 0
\(151\) 19.4693i 1.58439i −0.610270 0.792193i \(-0.708939\pi\)
0.610270 0.792193i \(-0.291061\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.01817 6.56198i 0.0817812 0.527071i
\(156\) 0 0
\(157\) 12.8292i 1.02388i −0.859020 0.511942i \(-0.828926\pi\)
0.859020 0.511942i \(-0.171074\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.01008i 0.473661i
\(162\) 0 0
\(163\) 21.2800 1.66678 0.833391 0.552684i \(-0.186396\pi\)
0.833391 + 0.552684i \(0.186396\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.8676i 1.76955i 0.466020 + 0.884774i \(0.345688\pi\)
−0.466020 + 0.884774i \(0.654312\pi\)
\(168\) 0 0
\(169\) 12.5904 0.968490
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.98932 −0.455360 −0.227680 0.973736i \(-0.573114\pi\)
−0.227680 + 0.973736i \(0.573114\pi\)
\(174\) 0 0
\(175\) 12.5795 + 4.00000i 0.950919 + 0.302372i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.65181 0.198206 0.0991029 0.995077i \(-0.468403\pi\)
0.0991029 + 0.995077i \(0.468403\pi\)
\(180\) 0 0
\(181\) 11.4693 0.852504 0.426252 0.904605i \(-0.359834\pi\)
0.426252 + 0.904605i \(0.359834\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.551901 + 3.55694i −0.0405765 + 0.261512i
\(186\) 0 0
\(187\) −2.06055 −0.150682
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.30362 −0.383757 −0.191878 0.981419i \(-0.561458\pi\)
−0.191878 + 0.981419i \(0.561458\pi\)
\(192\) 0 0
\(193\) 10.0606i 0.724174i −0.932144 0.362087i \(-0.882064\pi\)
0.932144 0.362087i \(-0.117936\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.79065 −0.412567 −0.206283 0.978492i \(-0.566137\pi\)
−0.206283 + 0.978492i \(0.566137\pi\)
\(198\) 0 0
\(199\) 24.0899i 1.70769i 0.520529 + 0.853844i \(0.325735\pi\)
−0.520529 + 0.853844i \(0.674265\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 21.5236i 1.51066i
\(204\) 0 0
\(205\) 2.54541 16.4049i 0.177779 1.14577i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 15.8668i 1.09753i
\(210\) 0 0
\(211\) 23.4693i 1.61569i −0.589394 0.807845i \(-0.700634\pi\)
0.589394 0.807845i \(-0.299366\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 24.7909 + 3.84659i 1.69073 + 0.262336i
\(216\) 0 0
\(217\) 7.84014i 0.532223i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.438863i 0.0295211i
\(222\) 0 0
\(223\) −13.3600 −0.894650 −0.447325 0.894371i \(-0.647623\pi\)
−0.447325 + 0.894371i \(0.647623\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.57092i 0.568872i −0.958695 0.284436i \(-0.908194\pi\)
0.958695 0.284436i \(-0.0918063\pi\)
\(228\) 0 0
\(229\) 8.90917 0.588735 0.294367 0.955692i \(-0.404891\pi\)
0.294367 + 0.955692i \(0.404891\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.0196 1.57357 0.786787 0.617224i \(-0.211743\pi\)
0.786787 + 0.617224i \(0.211743\pi\)
\(234\) 0 0
\(235\) −1.43991 + 9.28005i −0.0939291 + 0.605364i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 14.4097 0.932088 0.466044 0.884762i \(-0.345679\pi\)
0.466044 + 0.884762i \(0.345679\pi\)
\(240\) 0 0
\(241\) 3.87890 0.249862 0.124931 0.992165i \(-0.460129\pi\)
0.124931 + 0.992165i \(0.460129\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.0668989 0.0103801i −0.00427401 0.000663162i
\(246\) 0 0
\(247\) 3.37935 0.215023
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −24.1754 −1.52594 −0.762970 0.646434i \(-0.776259\pi\)
−0.762970 + 0.646434i \(0.776259\pi\)
\(252\) 0 0
\(253\) 6.84106i 0.430094i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −12.4383 −0.775878 −0.387939 0.921685i \(-0.626813\pi\)
−0.387939 + 0.921685i \(0.626813\pi\)
\(258\) 0 0
\(259\) 4.24977i 0.264068i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.2471i 1.18683i −0.804898 0.593413i \(-0.797780\pi\)
0.804898 0.593413i \(-0.202220\pi\)
\(264\) 0 0
\(265\) −21.2342 3.29473i −1.30440 0.202393i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.8875i 0.968679i −0.874880 0.484340i \(-0.839060\pi\)
0.874880 0.484340i \(-0.160940\pi\)
\(270\) 0 0
\(271\) 6.02936i 0.366258i 0.983089 + 0.183129i \(0.0586225\pi\)
−0.983089 + 0.183129i \(0.941377\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14.3188 + 4.55305i 0.863453 + 0.274559i
\(276\) 0 0
\(277\) 12.3591i 0.742584i 0.928516 + 0.371292i \(0.121085\pi\)
−0.928516 + 0.371292i \(0.878915\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 17.7198i 1.05708i 0.848909 + 0.528538i \(0.177260\pi\)
−0.848909 + 0.528538i \(0.822740\pi\)
\(282\) 0 0
\(283\) 20.6206 1.22577 0.612885 0.790172i \(-0.290009\pi\)
0.612885 + 0.790172i \(0.290009\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 19.6003i 1.15697i
\(288\) 0 0
\(289\) −16.5298 −0.972342
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −21.6574 −1.26524 −0.632620 0.774463i \(-0.718020\pi\)
−0.632620 + 0.774463i \(0.718020\pi\)
\(294\) 0 0
\(295\) 15.9201 + 2.47018i 0.926902 + 0.143820i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.45703 0.0842622
\(300\) 0 0
\(301\) 29.6197 1.70725
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −23.2001 3.59976i −1.32843 0.206122i
\(306\) 0 0
\(307\) −3.87890 −0.221380 −0.110690 0.993855i \(-0.535306\pi\)
−0.110690 + 0.993855i \(0.535306\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.71654 0.380860 0.190430 0.981701i \(-0.439012\pi\)
0.190430 + 0.981701i \(0.439012\pi\)
\(312\) 0 0
\(313\) 21.9394i 1.24009i −0.784566 0.620045i \(-0.787114\pi\)
0.784566 0.620045i \(-0.212886\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.3616 0.806626 0.403313 0.915062i \(-0.367859\pi\)
0.403313 + 0.915062i \(0.367859\pi\)
\(318\) 0 0
\(319\) 24.4995i 1.37171i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.62052i 0.201451i
\(324\) 0 0
\(325\) 0.969724 3.04965i 0.0537906 0.169164i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.0876i 0.611281i
\(330\) 0 0
\(331\) 4.78051i 0.262760i −0.991332 0.131380i \(-0.958059\pi\)
0.991332 0.131380i \(-0.0419409\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 18.7808 + 2.91406i 1.02611 + 0.159212i
\(336\) 0 0
\(337\) 33.1883i 1.80788i 0.427657 + 0.903941i \(0.359339\pi\)
−0.427657 + 0.903941i \(0.640661\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.92414i 0.483270i
\(342\) 0 0
\(343\) −18.5601 −1.00215
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.8304i 0.742456i 0.928542 + 0.371228i \(0.121063\pi\)
−0.928542 + 0.371228i \(0.878937\pi\)
\(348\) 0 0
\(349\) −20.4390 −1.09407 −0.547037 0.837108i \(-0.684244\pi\)
−0.547037 + 0.837108i \(0.684244\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 20.8379 1.10909 0.554545 0.832154i \(-0.312892\pi\)
0.554545 + 0.832154i \(0.312892\pi\)
\(354\) 0 0
\(355\) −28.9991 4.49954i −1.53911 0.238811i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 22.0481 1.16365 0.581827 0.813312i \(-0.302338\pi\)
0.581827 + 0.813312i \(0.302338\pi\)
\(360\) 0 0
\(361\) −8.87890 −0.467310
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.88552 31.4867i 0.255720 1.64809i
\(366\) 0 0
\(367\) 4.70058 0.245368 0.122684 0.992446i \(-0.460850\pi\)
0.122684 + 0.992446i \(0.460850\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −25.3702 −1.31716
\(372\) 0 0
\(373\) 35.5104i 1.83866i 0.393486 + 0.919330i \(0.371269\pi\)
−0.393486 + 0.919330i \(0.628731\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.21799 0.268740
\(378\) 0 0
\(379\) 1.52982i 0.0785815i −0.999228 0.0392907i \(-0.987490\pi\)
0.999228 0.0392907i \(-0.0125098\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.16349i 0.110549i 0.998471 + 0.0552746i \(0.0176034\pi\)
−0.998471 + 0.0552746i \(0.982397\pi\)
\(384\) 0 0
\(385\) 17.5298 + 2.71995i 0.893402 + 0.138622i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0.0207603i 0.00105259i 1.00000 0.000526294i \(0.000167524\pi\)
−1.00000 0.000526294i \(0.999832\pi\)
\(390\) 0 0
\(391\) 1.56101i 0.0789437i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.93223 25.3428i 0.197852 1.27513i
\(396\) 0 0
\(397\) 36.6694i 1.84038i −0.391468 0.920192i \(-0.628033\pi\)
0.391468 0.920192i \(-0.371967\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.59556i 0.379304i −0.981851 0.189652i \(-0.939264\pi\)
0.981851 0.189652i \(-0.0607360\pi\)
\(402\) 0 0
\(403\) −1.90069 −0.0946803
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.83736i 0.239779i
\(408\) 0 0
\(409\) −26.5289 −1.31177 −0.655885 0.754861i \(-0.727704\pi\)
−0.655885 + 0.754861i \(0.727704\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 19.0210 0.935963
\(414\) 0 0
\(415\) −4.49954 + 28.9991i −0.220874 + 1.42351i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.841553 0.0411125 0.0205563 0.999789i \(-0.493456\pi\)
0.0205563 + 0.999789i \(0.493456\pi\)
\(420\) 0 0
\(421\) −26.1505 −1.27450 −0.637248 0.770659i \(-0.719927\pi\)
−0.637248 + 0.770659i \(0.719927\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.26729 1.03893i −0.158487 0.0503954i
\(426\) 0 0
\(427\) −27.7190 −1.34142
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −38.8474 −1.87121 −0.935607 0.353044i \(-0.885147\pi\)
−0.935607 + 0.353044i \(0.885147\pi\)
\(432\) 0 0
\(433\) 30.7787i 1.47913i 0.673086 + 0.739564i \(0.264968\pi\)
−0.673086 + 0.739564i \(0.735032\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.0202 −0.575003
\(438\) 0 0
\(439\) 26.1505i 1.24809i −0.781387 0.624047i \(-0.785487\pi\)
0.781387 0.624047i \(-0.214513\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 13.8304i 0.657103i −0.944486 0.328552i \(-0.893439\pi\)
0.944486 0.328552i \(-0.106561\pi\)
\(444\) 0 0
\(445\) 3.51514 22.6547i 0.166634 1.07394i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.07915i 0.0981212i 0.998796 + 0.0490606i \(0.0156228\pi\)
−0.998796 + 0.0490606i \(0.984377\pi\)
\(450\) 0 0
\(451\) 22.3103i 1.05055i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.579304 3.73356i 0.0271582 0.175032i
\(456\) 0 0
\(457\) 18.6282i 0.871391i −0.900094 0.435695i \(-0.856502\pi\)
0.900094 0.435695i \(-0.143498\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 23.0014i 1.07128i −0.844446 0.535641i \(-0.820070\pi\)
0.844446 0.535641i \(-0.179930\pi\)
\(462\) 0 0
\(463\) −25.9806 −1.20742 −0.603711 0.797203i \(-0.706312\pi\)
−0.603711 + 0.797203i \(0.706312\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 33.3177i 1.54176i −0.636981 0.770880i \(-0.719817\pi\)
0.636981 0.770880i \(-0.280183\pi\)
\(468\) 0 0
\(469\) 22.4390 1.03614
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 33.7151 1.55022
\(474\) 0 0
\(475\) −8.00000 + 25.1589i −0.367065 + 1.15437i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 16.0380 0.732796 0.366398 0.930458i \(-0.380591\pi\)
0.366398 + 0.930458i \(0.380591\pi\)
\(480\) 0 0
\(481\) 1.03028 0.0469765
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.84919 + 18.3627i −0.129375 + 0.833808i
\(486\) 0 0
\(487\) −19.1396 −0.867296 −0.433648 0.901082i \(-0.642774\pi\)
−0.433648 + 0.901082i \(0.642774\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −36.3669 −1.64121 −0.820607 0.571493i \(-0.806364\pi\)
−0.820607 + 0.571493i \(0.806364\pi\)
\(492\) 0 0
\(493\) 5.59037i 0.251778i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −34.6476 −1.55416
\(498\) 0 0
\(499\) 8.65940i 0.387648i 0.981036 + 0.193824i \(0.0620891\pi\)
−0.981036 + 0.193824i \(0.937911\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 31.1542i 1.38910i −0.719445 0.694549i \(-0.755604\pi\)
0.719445 0.694549i \(-0.244396\pi\)
\(504\) 0 0
\(505\) −1.20482 + 7.76491i −0.0536136 + 0.345534i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 28.3907i 1.25839i 0.777246 + 0.629197i \(0.216616\pi\)
−0.777246 + 0.629197i \(0.783384\pi\)
\(510\) 0 0
\(511\) 37.6197i 1.66420i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.93727 + 1.07640i 0.305693 + 0.0474317i
\(516\) 0 0
\(517\) 12.6206i 0.555055i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 16.4341i 0.719990i 0.932954 + 0.359995i \(0.117222\pi\)
−0.932954 + 0.359995i \(0.882778\pi\)
\(522\) 0 0
\(523\) −14.2791 −0.624383 −0.312191 0.950019i \(-0.601063\pi\)
−0.312191 + 0.950019i \(0.601063\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.03633i 0.0887041i
\(528\) 0 0
\(529\) 17.8174 0.774671
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.75172 −0.205820
\(534\) 0 0
\(535\) −2.56009 + 16.4995i −0.110683 + 0.713337i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.0909809 −0.00391883
\(540\) 0 0
\(541\) −1.40871 −0.0605653 −0.0302827 0.999541i \(-0.509641\pi\)
−0.0302827 + 0.999541i \(0.509641\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −6.56198 1.01817i −0.281085 0.0436135i
\(546\) 0 0
\(547\) −15.3406 −0.655917 −0.327958 0.944692i \(-0.606361\pi\)
−0.327958 + 0.944692i \(0.606361\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −43.0472 −1.83387
\(552\) 0 0
\(553\) 30.2791i 1.28760i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.53199 −0.319141 −0.159570 0.987187i \(-0.551011\pi\)
−0.159570 + 0.987187i \(0.551011\pi\)
\(558\) 0 0
\(559\) 7.18074i 0.303713i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 40.0784i 1.68910i 0.535475 + 0.844551i \(0.320132\pi\)
−0.535475 + 0.844551i \(0.679868\pi\)
\(564\) 0 0
\(565\) −12.5454 1.94657i −0.527790 0.0818926i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 28.6803i 1.20234i −0.799121 0.601171i \(-0.794701\pi\)
0.799121 0.601171i \(-0.205299\pi\)
\(570\) 0 0
\(571\) 33.9007i 1.41870i −0.704857 0.709350i \(-0.748989\pi\)
0.704857 0.709350i \(-0.251011\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.44925 + 10.8474i −0.143844 + 0.452370i
\(576\) 0 0
\(577\) 18.4002i 0.766012i −0.923746 0.383006i \(-0.874889\pi\)
0.923746 0.383006i \(-0.125111\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 34.6476i 1.43742i
\(582\) 0 0
\(583\) −28.8780 −1.19600
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 27.8869i 1.15102i −0.817796 0.575508i \(-0.804804\pi\)
0.817796 0.575508i \(-0.195196\pi\)
\(588\) 0 0
\(589\) 15.6803 0.646095
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −34.2295 −1.40564 −0.702818 0.711370i \(-0.748075\pi\)
−0.702818 + 0.711370i \(0.748075\pi\)
\(594\) 0 0
\(595\) −4.00000 0.620646i −0.163984 0.0254440i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −22.7546 −0.929727 −0.464863 0.885382i \(-0.653897\pi\)
−0.464863 + 0.885382i \(0.653897\pi\)
\(600\) 0 0
\(601\) 11.5298 0.470311 0.235156 0.971958i \(-0.424440\pi\)
0.235156 + 0.971958i \(0.424440\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.35236 0.675318i −0.176948 0.0274556i
\(606\) 0 0
\(607\) −17.0790 −0.693216 −0.346608 0.938010i \(-0.612667\pi\)
−0.346608 + 0.938010i \(0.612667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.68799 0.108744
\(612\) 0 0
\(613\) 43.1084i 1.74113i 0.492053 + 0.870565i \(0.336247\pi\)
−0.492053 + 0.870565i \(0.663753\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.7935 0.957890 0.478945 0.877845i \(-0.341019\pi\)
0.478945 + 0.877845i \(0.341019\pi\)
\(618\) 0 0
\(619\) 32.4683i 1.30501i 0.757783 + 0.652507i \(0.226283\pi\)
−0.757783 + 0.652507i \(0.773717\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 27.0674i 1.08443i
\(624\) 0 0
\(625\) 20.4087 + 14.4390i 0.816349 + 0.577560i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.10380i 0.0440114i
\(630\) 0 0
\(631\) 13.0303i 0.518727i −0.965780 0.259364i \(-0.916487\pi\)
0.965780 0.259364i \(-0.0835128\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 42.2913 + 6.56198i 1.67828 + 0.260404i
\(636\) 0 0
\(637\) 0.0193774i 0.000767761i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 38.1397i 1.50643i −0.657777 0.753213i \(-0.728503\pi\)
0.657777 0.753213i \(-0.271497\pi\)
\(642\) 0 0
\(643\) −17.1589 −0.676683 −0.338341 0.941023i \(-0.609866\pi\)
−0.338341 + 0.941023i \(0.609866\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30.4478i 1.19702i −0.801113 0.598512i \(-0.795759\pi\)
0.801113 0.598512i \(-0.204241\pi\)
\(648\) 0 0
\(649\) 21.6509 0.849873
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11.7151 0.458447 0.229224 0.973374i \(-0.426381\pi\)
0.229224 + 0.973374i \(0.426381\pi\)
\(654\) 0 0
\(655\) −25.5786 3.96881i −0.999437 0.155074i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 10.1189 0.394177 0.197089 0.980386i \(-0.436851\pi\)
0.197089 + 0.980386i \(0.436851\pi\)
\(660\) 0 0
\(661\) −11.1202 −0.432525 −0.216263 0.976335i \(-0.569387\pi\)
−0.216263 + 0.976335i \(0.569387\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.77913 + 30.8010i −0.185327 + 1.19441i
\(666\) 0 0
\(667\) −18.5601 −0.718650
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −31.5516 −1.21803
\(672\) 0 0
\(673\) 15.0984i 0.582000i −0.956723 0.291000i \(-0.906012\pi\)
0.956723 0.291000i \(-0.0939880\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.82699 0.300816 0.150408 0.988624i \(-0.451941\pi\)
0.150408 + 0.988624i \(0.451941\pi\)
\(678\) 0 0
\(679\) 21.9394i 0.841959i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 34.4763i 1.31920i −0.751617 0.659600i \(-0.770726\pi\)
0.751617 0.659600i \(-0.229274\pi\)
\(684\) 0 0
\(685\) 5.51514 + 0.855737i 0.210723 + 0.0326960i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.15052i 0.234316i
\(690\) 0 0
\(691\) 36.7181i 1.39682i 0.715696 + 0.698412i \(0.246109\pi\)
−0.715696 + 0.698412i \(0.753891\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.56083 16.5043i 0.0971379 0.626044i
\(696\) 0 0
\(697\) 5.09083i 0.192829i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.47779i 0.0558154i 0.999611 + 0.0279077i \(0.00888445\pi\)
−0.999611 + 0.0279077i \(0.991116\pi\)
\(702\) 0 0
\(703\) −8.49954 −0.320566
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.27737i 0.348912i
\(708\) 0 0
\(709\) −6.02936 −0.226437 −0.113219 0.993570i \(-0.536116\pi\)
−0.113219 + 0.993570i \(0.536116\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6.76066 0.253189
\(714\) 0 0
\(715\) 0.659401 4.24977i 0.0246602 0.158932i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −13.3059 −0.496227 −0.248114 0.968731i \(-0.579811\pi\)
−0.248114 + 0.968731i \(0.579811\pi\)
\(720\) 0 0
\(721\) 8.28853 0.308681
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −12.3526 + 38.8474i −0.458765 + 1.44276i
\(726\) 0 0
\(727\) −17.7384 −0.657881 −0.328941 0.944351i \(-0.606692\pi\)
−0.328941 + 0.944351i \(0.606692\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −7.69319 −0.284543
\(732\) 0 0
\(733\) 29.6391i 1.09475i −0.836889 0.547373i \(-0.815628\pi\)
0.836889 0.547373i \(-0.184372\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 25.5415 0.940832
\(738\) 0 0
\(739\) 19.2195i 0.707001i 0.935434 + 0.353500i \(0.115009\pi\)
−0.935434 + 0.353500i \(0.884991\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.8768i 0.802584i 0.915950 + 0.401292i \(0.131439\pi\)
−0.915950 + 0.401292i \(0.868561\pi\)
\(744\) 0 0
\(745\) 3.29473 21.2342i 0.120709 0.777960i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 19.7134i 0.720310i
\(750\) 0 0
\(751\) 14.5913i 0.532444i −0.963912 0.266222i \(-0.914225\pi\)
0.963912 0.266222i \(-0.0857754\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6.67502 43.0198i 0.242929 1.56565i
\(756\) 0 0
\(757\) 5.26067i 0.191202i −0.995420 0.0956011i \(-0.969523\pi\)
0.995420 0.0956011i \(-0.0304773\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.6261i 0.820196i −0.912041 0.410098i \(-0.865494\pi\)
0.912041 0.410098i \(-0.134506\pi\)
\(762\) 0 0
\(763\) −7.84014 −0.283832
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.61128i 0.166504i
\(768\) 0 0
\(769\) 0.150464 0.00542586 0.00271293 0.999996i \(-0.499136\pi\)
0.00271293 + 0.999996i \(0.499136\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.23760 0.0445133 0.0222567 0.999752i \(-0.492915\pi\)
0.0222567 + 0.999752i \(0.492915\pi\)
\(774\) 0 0
\(775\) 4.49954 14.1505i 0.161628 0.508300i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 39.2006 1.40451
\(780\) 0 0
\(781\) −39.4381 −1.41121
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.39850 28.3478i 0.156989 1.01178i
\(786\) 0 0
\(787\) 6.84106 0.243857 0.121929 0.992539i \(-0.461092\pi\)
0.121929 + 0.992539i \(0.461092\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −14.9890 −0.532949
\(792\) 0 0
\(793\) 6.71995i 0.238633i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 32.9437 1.16693 0.583463 0.812140i \(-0.301697\pi\)
0.583463 + 0.812140i \(0.301697\pi\)
\(798\) 0 0
\(799\) 2.87981i 0.101880i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 42.8212i 1.51113i
\(804\) 0 0
\(805\) −2.06055 + 13.2800i −0.0726249 + 0.468060i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 53.6532i 1.88635i