Properties

Label 5520.2.be.c.1471.6
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.be (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.6
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.c.1471.5

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{5} +3.12483 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{5} +3.12483 q^{7} -1.00000 q^{9} -1.96967 q^{11} -2.52369 q^{13} -1.00000 q^{15} -3.66071i q^{17} -6.78607 q^{19} +3.12483i q^{21} +(2.20899 - 4.25680i) q^{23} -1.00000 q^{25} -1.00000i q^{27} +2.23487 q^{29} -0.377839i q^{31} -1.96967i q^{33} +3.12483i q^{35} -5.26726i q^{37} -2.52369i q^{39} +5.40401 q^{41} +12.5558 q^{43} -1.00000i q^{45} -1.35089i q^{47} +2.76459 q^{49} +3.66071 q^{51} +3.17126i q^{53} -1.96967i q^{55} -6.78607i q^{57} +11.9962i q^{59} -4.62698i q^{61} -3.12483 q^{63} -2.52369i q^{65} +10.4936 q^{67} +(4.25680 + 2.20899i) q^{69} -15.1992i q^{71} +13.4156 q^{73} -1.00000i q^{75} -6.15489 q^{77} +5.38677 q^{79} +1.00000 q^{81} +14.7579 q^{83} +3.66071 q^{85} +2.23487i q^{87} +0.338033i q^{89} -7.88610 q^{91} +0.377839 q^{93} -6.78607i q^{95} +11.0882i q^{97} +1.96967 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 8q^{7} - 32q^{9} + O(q^{10}) \) \( 32q - 8q^{7} - 32q^{9} + 8q^{11} - 8q^{13} - 32q^{15} - 32q^{25} + 4q^{29} + 20q^{41} + 52q^{49} - 4q^{51} + 8q^{63} + 32q^{67} - 40q^{73} - 24q^{77} + 32q^{79} + 32q^{81} - 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1841\) \(4417\) \(4831\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 3.12483 1.18108 0.590538 0.807010i \(-0.298916\pi\)
0.590538 + 0.807010i \(0.298916\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.96967 −0.593878 −0.296939 0.954896i \(-0.595966\pi\)
−0.296939 + 0.954896i \(0.595966\pi\)
\(12\) 0 0
\(13\) −2.52369 −0.699945 −0.349972 0.936760i \(-0.613809\pi\)
−0.349972 + 0.936760i \(0.613809\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 3.66071i 0.887854i −0.896063 0.443927i \(-0.853585\pi\)
0.896063 0.443927i \(-0.146415\pi\)
\(18\) 0 0
\(19\) −6.78607 −1.55683 −0.778415 0.627750i \(-0.783976\pi\)
−0.778415 + 0.627750i \(0.783976\pi\)
\(20\) 0 0
\(21\) 3.12483i 0.681895i
\(22\) 0 0
\(23\) 2.20899 4.25680i 0.460606 0.887605i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 2.23487 0.415005 0.207503 0.978234i \(-0.433466\pi\)
0.207503 + 0.978234i \(0.433466\pi\)
\(30\) 0 0
\(31\) 0.377839i 0.0678619i −0.999424 0.0339309i \(-0.989197\pi\)
0.999424 0.0339309i \(-0.0108026\pi\)
\(32\) 0 0
\(33\) 1.96967i 0.342875i
\(34\) 0 0
\(35\) 3.12483i 0.528193i
\(36\) 0 0
\(37\) 5.26726i 0.865933i −0.901410 0.432966i \(-0.857467\pi\)
0.901410 0.432966i \(-0.142533\pi\)
\(38\) 0 0
\(39\) 2.52369i 0.404113i
\(40\) 0 0
\(41\) 5.40401 0.843965 0.421982 0.906604i \(-0.361334\pi\)
0.421982 + 0.906604i \(0.361334\pi\)
\(42\) 0 0
\(43\) 12.5558 1.91475 0.957375 0.288849i \(-0.0932727\pi\)
0.957375 + 0.288849i \(0.0932727\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 1.35089i 0.197047i −0.995135 0.0985237i \(-0.968588\pi\)
0.995135 0.0985237i \(-0.0314120\pi\)
\(48\) 0 0
\(49\) 2.76459 0.394941
\(50\) 0 0
\(51\) 3.66071 0.512603
\(52\) 0 0
\(53\) 3.17126i 0.435606i 0.975993 + 0.217803i \(0.0698891\pi\)
−0.975993 + 0.217803i \(0.930111\pi\)
\(54\) 0 0
\(55\) 1.96967i 0.265590i
\(56\) 0 0
\(57\) 6.78607i 0.898837i
\(58\) 0 0
\(59\) 11.9962i 1.56177i 0.624675 + 0.780885i \(0.285231\pi\)
−0.624675 + 0.780885i \(0.714769\pi\)
\(60\) 0 0
\(61\) 4.62698i 0.592424i −0.955122 0.296212i \(-0.904276\pi\)
0.955122 0.296212i \(-0.0957235\pi\)
\(62\) 0 0
\(63\) −3.12483 −0.393692
\(64\) 0 0
\(65\) 2.52369i 0.313025i
\(66\) 0 0
\(67\) 10.4936 1.28200 0.640998 0.767543i \(-0.278521\pi\)
0.640998 + 0.767543i \(0.278521\pi\)
\(68\) 0 0
\(69\) 4.25680 + 2.20899i 0.512459 + 0.265931i
\(70\) 0 0
\(71\) 15.1992i 1.80381i −0.431935 0.901905i \(-0.642169\pi\)
0.431935 0.901905i \(-0.357831\pi\)
\(72\) 0 0
\(73\) 13.4156 1.57017 0.785087 0.619385i \(-0.212618\pi\)
0.785087 + 0.619385i \(0.212618\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) −6.15489 −0.701415
\(78\) 0 0
\(79\) 5.38677 0.606059 0.303029 0.952981i \(-0.402002\pi\)
0.303029 + 0.952981i \(0.402002\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.7579 1.61989 0.809943 0.586508i \(-0.199498\pi\)
0.809943 + 0.586508i \(0.199498\pi\)
\(84\) 0 0
\(85\) 3.66071 0.397060
\(86\) 0 0
\(87\) 2.23487i 0.239604i
\(88\) 0 0
\(89\) 0.338033i 0.0358315i 0.999840 + 0.0179157i \(0.00570306\pi\)
−0.999840 + 0.0179157i \(0.994297\pi\)
\(90\) 0 0
\(91\) −7.88610 −0.826688
\(92\) 0 0
\(93\) 0.377839 0.0391801
\(94\) 0 0
\(95\) 6.78607i 0.696236i
\(96\) 0 0
\(97\) 11.0882i 1.12584i 0.826512 + 0.562919i \(0.190322\pi\)
−0.826512 + 0.562919i \(0.809678\pi\)
\(98\) 0 0
\(99\) 1.96967 0.197959
\(100\) 0 0
\(101\) 6.64873 0.661573 0.330786 0.943706i \(-0.392686\pi\)
0.330786 + 0.943706i \(0.392686\pi\)
\(102\) 0 0
\(103\) 0.583656 0.0575093 0.0287547 0.999586i \(-0.490846\pi\)
0.0287547 + 0.999586i \(0.490846\pi\)
\(104\) 0 0
\(105\) −3.12483 −0.304953
\(106\) 0 0
\(107\) 2.04526 0.197723 0.0988615 0.995101i \(-0.468480\pi\)
0.0988615 + 0.995101i \(0.468480\pi\)
\(108\) 0 0
\(109\) 15.7438i 1.50798i 0.656884 + 0.753992i \(0.271874\pi\)
−0.656884 + 0.753992i \(0.728126\pi\)
\(110\) 0 0
\(111\) 5.26726 0.499946
\(112\) 0 0
\(113\) 12.6691i 1.19180i −0.803057 0.595902i \(-0.796795\pi\)
0.803057 0.595902i \(-0.203205\pi\)
\(114\) 0 0
\(115\) 4.25680 + 2.20899i 0.396949 + 0.205989i
\(116\) 0 0
\(117\) 2.52369 0.233315
\(118\) 0 0
\(119\) 11.4391i 1.04862i
\(120\) 0 0
\(121\) −7.12040 −0.647309
\(122\) 0 0
\(123\) 5.40401i 0.487263i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 21.2834i 1.88859i −0.329093 0.944297i \(-0.606743\pi\)
0.329093 0.944297i \(-0.393257\pi\)
\(128\) 0 0
\(129\) 12.5558i 1.10548i
\(130\) 0 0
\(131\) 13.9512i 1.21892i −0.792816 0.609461i \(-0.791386\pi\)
0.792816 0.609461i \(-0.208614\pi\)
\(132\) 0 0
\(133\) −21.2053 −1.83874
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 2.32226i 0.198404i 0.995067 + 0.0992021i \(0.0316290\pi\)
−0.995067 + 0.0992021i \(0.968371\pi\)
\(138\) 0 0
\(139\) 15.4464i 1.31014i 0.755567 + 0.655071i \(0.227361\pi\)
−0.755567 + 0.655071i \(0.772639\pi\)
\(140\) 0 0
\(141\) 1.35089 0.113765
\(142\) 0 0
\(143\) 4.97083 0.415682
\(144\) 0 0
\(145\) 2.23487i 0.185596i
\(146\) 0 0
\(147\) 2.76459i 0.228019i
\(148\) 0 0
\(149\) 4.84162i 0.396641i 0.980137 + 0.198320i \(0.0635487\pi\)
−0.980137 + 0.198320i \(0.936451\pi\)
\(150\) 0 0
\(151\) 19.8598i 1.61617i 0.589069 + 0.808083i \(0.299495\pi\)
−0.589069 + 0.808083i \(0.700505\pi\)
\(152\) 0 0
\(153\) 3.66071i 0.295951i
\(154\) 0 0
\(155\) 0.377839 0.0303488
\(156\) 0 0
\(157\) 16.2502i 1.29691i 0.761255 + 0.648453i \(0.224584\pi\)
−0.761255 + 0.648453i \(0.775416\pi\)
\(158\) 0 0
\(159\) −3.17126 −0.251497
\(160\) 0 0
\(161\) 6.90273 13.3018i 0.544011 1.04833i
\(162\) 0 0
\(163\) 21.5017i 1.68415i −0.539363 0.842073i \(-0.681335\pi\)
0.539363 0.842073i \(-0.318665\pi\)
\(164\) 0 0
\(165\) 1.96967 0.153339
\(166\) 0 0
\(167\) 13.7046i 1.06049i −0.847844 0.530245i \(-0.822100\pi\)
0.847844 0.530245i \(-0.177900\pi\)
\(168\) 0 0
\(169\) −6.63101 −0.510078
\(170\) 0 0
\(171\) 6.78607 0.518944
\(172\) 0 0
\(173\) −12.6695 −0.963244 −0.481622 0.876379i \(-0.659952\pi\)
−0.481622 + 0.876379i \(0.659952\pi\)
\(174\) 0 0
\(175\) −3.12483 −0.236215
\(176\) 0 0
\(177\) −11.9962 −0.901688
\(178\) 0 0
\(179\) 19.8164i 1.48115i −0.671974 0.740575i \(-0.734553\pi\)
0.671974 0.740575i \(-0.265447\pi\)
\(180\) 0 0
\(181\) 2.00943i 0.149360i 0.997208 + 0.0746801i \(0.0237936\pi\)
−0.997208 + 0.0746801i \(0.976206\pi\)
\(182\) 0 0
\(183\) 4.62698 0.342036
\(184\) 0 0
\(185\) 5.26726 0.387257
\(186\) 0 0
\(187\) 7.21040i 0.527277i
\(188\) 0 0
\(189\) 3.12483i 0.227298i
\(190\) 0 0
\(191\) 18.8616 1.36478 0.682388 0.730990i \(-0.260941\pi\)
0.682388 + 0.730990i \(0.260941\pi\)
\(192\) 0 0
\(193\) −25.8785 −1.86277 −0.931387 0.364030i \(-0.881401\pi\)
−0.931387 + 0.364030i \(0.881401\pi\)
\(194\) 0 0
\(195\) 2.52369 0.180725
\(196\) 0 0
\(197\) 11.3975 0.812040 0.406020 0.913864i \(-0.366916\pi\)
0.406020 + 0.913864i \(0.366916\pi\)
\(198\) 0 0
\(199\) 10.1484 0.719398 0.359699 0.933068i \(-0.382879\pi\)
0.359699 + 0.933068i \(0.382879\pi\)
\(200\) 0 0
\(201\) 10.4936i 0.740160i
\(202\) 0 0
\(203\) 6.98361 0.490153
\(204\) 0 0
\(205\) 5.40401i 0.377432i
\(206\) 0 0
\(207\) −2.20899 + 4.25680i −0.153535 + 0.295868i
\(208\) 0 0
\(209\) 13.3663 0.924567
\(210\) 0 0
\(211\) 4.20971i 0.289808i −0.989446 0.144904i \(-0.953713\pi\)
0.989446 0.144904i \(-0.0462874\pi\)
\(212\) 0 0
\(213\) 15.1992 1.04143
\(214\) 0 0
\(215\) 12.5558i 0.856302i
\(216\) 0 0
\(217\) 1.18068i 0.0801501i
\(218\) 0 0
\(219\) 13.4156i 0.906541i
\(220\) 0 0
\(221\) 9.23850i 0.621448i
\(222\) 0 0
\(223\) 20.2218i 1.35415i 0.735912 + 0.677077i \(0.236754\pi\)
−0.735912 + 0.677077i \(0.763246\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −29.8101 −1.97857 −0.989284 0.146005i \(-0.953359\pi\)
−0.989284 + 0.146005i \(0.953359\pi\)
\(228\) 0 0
\(229\) 12.3772i 0.817909i 0.912555 + 0.408955i \(0.134107\pi\)
−0.912555 + 0.408955i \(0.865893\pi\)
\(230\) 0 0
\(231\) 6.15489i 0.404962i
\(232\) 0 0
\(233\) −13.5820 −0.889789 −0.444895 0.895583i \(-0.646759\pi\)
−0.444895 + 0.895583i \(0.646759\pi\)
\(234\) 0 0
\(235\) 1.35089 0.0881222
\(236\) 0 0
\(237\) 5.38677i 0.349908i
\(238\) 0 0
\(239\) 8.77772i 0.567784i −0.958856 0.283892i \(-0.908374\pi\)
0.958856 0.283892i \(-0.0916257\pi\)
\(240\) 0 0
\(241\) 13.8466i 0.891935i −0.895049 0.445968i \(-0.852860\pi\)
0.895049 0.445968i \(-0.147140\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 2.76459i 0.176623i
\(246\) 0 0
\(247\) 17.1259 1.08970
\(248\) 0 0
\(249\) 14.7579i 0.935242i
\(250\) 0 0
\(251\) 26.2244 1.65527 0.827635 0.561267i \(-0.189686\pi\)
0.827635 + 0.561267i \(0.189686\pi\)
\(252\) 0 0
\(253\) −4.35098 + 8.38449i −0.273544 + 0.527129i
\(254\) 0 0
\(255\) 3.66071i 0.229243i
\(256\) 0 0
\(257\) −3.73811 −0.233177 −0.116588 0.993180i \(-0.537196\pi\)
−0.116588 + 0.993180i \(0.537196\pi\)
\(258\) 0 0
\(259\) 16.4593i 1.02273i
\(260\) 0 0
\(261\) −2.23487 −0.138335
\(262\) 0 0
\(263\) 19.3159 1.19107 0.595536 0.803328i \(-0.296940\pi\)
0.595536 + 0.803328i \(0.296940\pi\)
\(264\) 0 0
\(265\) −3.17126 −0.194809
\(266\) 0 0
\(267\) −0.338033 −0.0206873
\(268\) 0 0
\(269\) −32.5176 −1.98264 −0.991318 0.131489i \(-0.958024\pi\)
−0.991318 + 0.131489i \(0.958024\pi\)
\(270\) 0 0
\(271\) 19.1381i 1.16255i −0.813706 0.581277i \(-0.802553\pi\)
0.813706 0.581277i \(-0.197447\pi\)
\(272\) 0 0
\(273\) 7.88610i 0.477289i
\(274\) 0 0
\(275\) 1.96967 0.118776
\(276\) 0 0
\(277\) 9.32487 0.560277 0.280139 0.959960i \(-0.409620\pi\)
0.280139 + 0.959960i \(0.409620\pi\)
\(278\) 0 0
\(279\) 0.377839i 0.0226206i
\(280\) 0 0
\(281\) 12.4885i 0.745001i −0.928032 0.372500i \(-0.878501\pi\)
0.928032 0.372500i \(-0.121499\pi\)
\(282\) 0 0
\(283\) −10.1536 −0.603567 −0.301783 0.953377i \(-0.597582\pi\)
−0.301783 + 0.953377i \(0.597582\pi\)
\(284\) 0 0
\(285\) 6.78607 0.401972
\(286\) 0 0
\(287\) 16.8866 0.996787
\(288\) 0 0
\(289\) 3.59917 0.211716
\(290\) 0 0
\(291\) −11.0882 −0.650003
\(292\) 0 0
\(293\) 24.8954i 1.45440i −0.686424 0.727201i \(-0.740821\pi\)
0.686424 0.727201i \(-0.259179\pi\)
\(294\) 0 0
\(295\) −11.9962 −0.698444
\(296\) 0 0
\(297\) 1.96967i 0.114292i
\(298\) 0 0
\(299\) −5.57480 + 10.7428i −0.322399 + 0.621274i
\(300\) 0 0
\(301\) 39.2349 2.26146
\(302\) 0 0
\(303\) 6.64873i 0.381959i
\(304\) 0 0
\(305\) 4.62698 0.264940
\(306\) 0 0
\(307\) 14.7305i 0.840712i −0.907359 0.420356i \(-0.861905\pi\)
0.907359 0.420356i \(-0.138095\pi\)
\(308\) 0 0
\(309\) 0.583656i 0.0332030i
\(310\) 0 0
\(311\) 2.02347i 0.114740i 0.998353 + 0.0573702i \(0.0182715\pi\)
−0.998353 + 0.0573702i \(0.981728\pi\)
\(312\) 0 0
\(313\) 19.6539i 1.11091i 0.831548 + 0.555453i \(0.187455\pi\)
−0.831548 + 0.555453i \(0.812545\pi\)
\(314\) 0 0
\(315\) 3.12483i 0.176064i
\(316\) 0 0
\(317\) 0.0518500 0.00291219 0.00145609 0.999999i \(-0.499537\pi\)
0.00145609 + 0.999999i \(0.499537\pi\)
\(318\) 0 0
\(319\) −4.40196 −0.246463
\(320\) 0 0
\(321\) 2.04526i 0.114155i
\(322\) 0 0
\(323\) 24.8419i 1.38224i
\(324\) 0 0
\(325\) 2.52369 0.139989
\(326\) 0 0
\(327\) −15.7438 −0.870635
\(328\) 0 0
\(329\) 4.22130i 0.232728i
\(330\) 0 0
\(331\) 3.22601i 0.177317i −0.996062 0.0886586i \(-0.971742\pi\)
0.996062 0.0886586i \(-0.0282580\pi\)
\(332\) 0 0
\(333\) 5.26726i 0.288644i
\(334\) 0 0
\(335\) 10.4936i 0.573326i
\(336\) 0 0
\(337\) 32.7479i 1.78389i 0.452141 + 0.891946i \(0.350660\pi\)
−0.452141 + 0.891946i \(0.649340\pi\)
\(338\) 0 0
\(339\) 12.6691 0.688089
\(340\) 0 0
\(341\) 0.744218i 0.0403017i
\(342\) 0 0
\(343\) −13.2350 −0.714621
\(344\) 0 0
\(345\) −2.20899 + 4.25680i −0.118928 + 0.229179i
\(346\) 0 0
\(347\) 24.5999i 1.32059i −0.751006 0.660295i \(-0.770431\pi\)
0.751006 0.660295i \(-0.229569\pi\)
\(348\) 0 0
\(349\) 29.6353 1.58634 0.793171 0.608999i \(-0.208429\pi\)
0.793171 + 0.608999i \(0.208429\pi\)
\(350\) 0 0
\(351\) 2.52369i 0.134704i
\(352\) 0 0
\(353\) 31.6740 1.68583 0.842917 0.538043i \(-0.180836\pi\)
0.842917 + 0.538043i \(0.180836\pi\)
\(354\) 0 0
\(355\) 15.1992 0.806688
\(356\) 0 0
\(357\) 11.4391 0.605423
\(358\) 0 0
\(359\) 4.92427 0.259893 0.129947 0.991521i \(-0.458519\pi\)
0.129947 + 0.991521i \(0.458519\pi\)
\(360\) 0 0
\(361\) 27.0507 1.42372
\(362\) 0 0
\(363\) 7.12040i 0.373724i
\(364\) 0 0
\(365\) 13.4156i 0.702203i
\(366\) 0 0
\(367\) 19.7214 1.02945 0.514725 0.857355i \(-0.327894\pi\)
0.514725 + 0.857355i \(0.327894\pi\)
\(368\) 0 0
\(369\) −5.40401 −0.281322
\(370\) 0 0
\(371\) 9.90967i 0.514484i
\(372\) 0 0
\(373\) 6.38425i 0.330564i 0.986246 + 0.165282i \(0.0528534\pi\)
−0.986246 + 0.165282i \(0.947147\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −5.64012 −0.290481
\(378\) 0 0
\(379\) 2.70427 0.138909 0.0694546 0.997585i \(-0.477874\pi\)
0.0694546 + 0.997585i \(0.477874\pi\)
\(380\) 0 0
\(381\) 21.2834 1.09038
\(382\) 0 0
\(383\) 30.4327 1.55504 0.777519 0.628860i \(-0.216478\pi\)
0.777519 + 0.628860i \(0.216478\pi\)
\(384\) 0 0
\(385\) 6.15489i 0.313682i
\(386\) 0 0
\(387\) −12.5558 −0.638250
\(388\) 0 0
\(389\) 24.3346i 1.23381i −0.787037 0.616906i \(-0.788386\pi\)
0.787037 0.616906i \(-0.211614\pi\)
\(390\) 0 0
\(391\) −15.5829 8.08648i −0.788063 0.408951i
\(392\) 0 0
\(393\) 13.9512 0.703745
\(394\) 0 0
\(395\) 5.38677i 0.271038i
\(396\) 0 0
\(397\) 9.96983 0.500371 0.250186 0.968198i \(-0.419508\pi\)
0.250186 + 0.968198i \(0.419508\pi\)
\(398\) 0 0
\(399\) 21.2053i 1.06159i
\(400\) 0 0
\(401\) 14.6150i 0.729839i −0.931039 0.364920i \(-0.881096\pi\)
0.931039 0.364920i \(-0.118904\pi\)
\(402\) 0 0
\(403\) 0.953547i 0.0474996i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 10.3748i 0.514258i
\(408\) 0 0
\(409\) 5.77841 0.285724 0.142862 0.989743i \(-0.454369\pi\)
0.142862 + 0.989743i \(0.454369\pi\)
\(410\) 0 0
\(411\) −2.32226 −0.114549
\(412\) 0 0
\(413\) 37.4861i 1.84457i
\(414\) 0 0
\(415\) 14.7579i 0.724435i
\(416\) 0 0
\(417\) −15.4464 −0.756411
\(418\) 0 0
\(419\) 25.4826 1.24491 0.622453 0.782657i \(-0.286136\pi\)
0.622453 + 0.782657i \(0.286136\pi\)
\(420\) 0 0
\(421\) 8.35143i 0.407024i 0.979072 + 0.203512i \(0.0652356\pi\)
−0.979072 + 0.203512i \(0.934764\pi\)
\(422\) 0 0
\(423\) 1.35089i 0.0656824i
\(424\) 0 0
\(425\) 3.66071i 0.177571i
\(426\) 0 0
\(427\) 14.4585i 0.699698i
\(428\) 0 0
\(429\) 4.97083i 0.239994i
\(430\) 0 0
\(431\) −2.81003 −0.135354 −0.0676771 0.997707i \(-0.521559\pi\)
−0.0676771 + 0.997707i \(0.521559\pi\)
\(432\) 0 0
\(433\) 21.5271i 1.03453i −0.855826 0.517263i \(-0.826951\pi\)
0.855826 0.517263i \(-0.173049\pi\)
\(434\) 0 0
\(435\) −2.23487 −0.107154
\(436\) 0 0
\(437\) −14.9904 + 28.8870i −0.717086 + 1.38185i
\(438\) 0 0
\(439\) 14.5927i 0.696471i −0.937407 0.348235i \(-0.886781\pi\)
0.937407 0.348235i \(-0.113219\pi\)
\(440\) 0 0
\(441\) −2.76459 −0.131647
\(442\) 0 0
\(443\) 19.8921i 0.945101i 0.881304 + 0.472550i \(0.156667\pi\)
−0.881304 + 0.472550i \(0.843333\pi\)
\(444\) 0 0
\(445\) −0.338033 −0.0160243
\(446\) 0 0
\(447\) −4.84162 −0.229001
\(448\) 0 0
\(449\) −20.8817 −0.985470 −0.492735 0.870180i \(-0.664003\pi\)
−0.492735 + 0.870180i \(0.664003\pi\)
\(450\) 0 0
\(451\) −10.6441 −0.501212
\(452\) 0 0
\(453\) −19.8598 −0.933094
\(454\) 0 0
\(455\) 7.88610i 0.369706i
\(456\) 0 0
\(457\) 2.71008i 0.126772i −0.997989 0.0633861i \(-0.979810\pi\)
0.997989 0.0633861i \(-0.0201900\pi\)
\(458\) 0 0
\(459\) −3.66071 −0.170868
\(460\) 0 0
\(461\) −23.9568 −1.11578 −0.557889 0.829915i \(-0.688389\pi\)
−0.557889 + 0.829915i \(0.688389\pi\)
\(462\) 0 0
\(463\) 26.9255i 1.25133i 0.780090 + 0.625667i \(0.215173\pi\)
−0.780090 + 0.625667i \(0.784827\pi\)
\(464\) 0 0
\(465\) 0.377839i 0.0175219i
\(466\) 0 0
\(467\) −10.6351 −0.492134 −0.246067 0.969253i \(-0.579138\pi\)
−0.246067 + 0.969253i \(0.579138\pi\)
\(468\) 0 0
\(469\) 32.7907 1.51413
\(470\) 0 0
\(471\) −16.2502 −0.748769
\(472\) 0 0
\(473\) −24.7309 −1.13713
\(474\) 0 0
\(475\) 6.78607 0.311366
\(476\) 0 0
\(477\) 3.17126i 0.145202i
\(478\) 0 0
\(479\) 32.4647 1.48335 0.741676 0.670759i \(-0.234031\pi\)
0.741676 + 0.670759i \(0.234031\pi\)
\(480\) 0 0
\(481\) 13.2929i 0.606105i
\(482\) 0 0
\(483\) 13.3018 + 6.90273i 0.605253 + 0.314085i
\(484\) 0 0
\(485\) −11.0882 −0.503490
\(486\) 0 0
\(487\) 24.0618i 1.09034i −0.838324 0.545172i \(-0.816464\pi\)
0.838324 0.545172i \(-0.183536\pi\)
\(488\) 0 0
\(489\) 21.5017 0.972343
\(490\) 0 0
\(491\) 0.570628i 0.0257521i 0.999917 + 0.0128760i \(0.00409868\pi\)
−0.999917 + 0.0128760i \(0.995901\pi\)
\(492\) 0 0
\(493\) 8.18123i 0.368464i
\(494\) 0 0
\(495\) 1.96967i 0.0885301i
\(496\) 0 0
\(497\) 47.4949i 2.13044i
\(498\) 0 0
\(499\) 22.2429i 0.995727i 0.867255 + 0.497864i \(0.165882\pi\)
−0.867255 + 0.497864i \(0.834118\pi\)
\(500\) 0 0
\(501\) 13.7046 0.612275
\(502\) 0 0
\(503\) 32.1596 1.43392 0.716962 0.697112i \(-0.245532\pi\)
0.716962 + 0.697112i \(0.245532\pi\)
\(504\) 0 0
\(505\) 6.64873i 0.295864i
\(506\) 0 0
\(507\) 6.63101i 0.294493i
\(508\) 0 0
\(509\) 5.93120 0.262896 0.131448 0.991323i \(-0.458037\pi\)
0.131448 + 0.991323i \(0.458037\pi\)
\(510\) 0 0
\(511\) 41.9215 1.85450
\(512\) 0 0
\(513\) 6.78607i 0.299612i
\(514\) 0 0
\(515\) 0.583656i 0.0257189i
\(516\) 0 0
\(517\) 2.66080i 0.117022i
\(518\) 0 0
\(519\) 12.6695i 0.556129i
\(520\) 0 0
\(521\) 0.545006i 0.0238771i 0.999929 + 0.0119386i \(0.00380025\pi\)
−0.999929 + 0.0119386i \(0.996200\pi\)
\(522\) 0 0
\(523\) −32.0697 −1.40231 −0.701155 0.713009i \(-0.747332\pi\)
−0.701155 + 0.713009i \(0.747332\pi\)
\(524\) 0 0
\(525\) 3.12483i 0.136379i
\(526\) 0 0
\(527\) −1.38316 −0.0602514
\(528\) 0 0
\(529\) −13.2407 18.8065i −0.575684 0.817672i
\(530\) 0 0
\(531\) 11.9962i 0.520590i
\(532\) 0 0
\(533\) −13.6380 −0.590728
\(534\) 0 0
\(535\) 2.04526i 0.0884244i
\(536\) 0 0
\(537\) 19.8164 0.855142
\(538\) 0 0
\(539\) −5.44533 −0.234547
\(540\) 0 0
\(541\) −36.4749 −1.56818 −0.784089 0.620648i \(-0.786870\pi\)
−0.784089 + 0.620648i \(0.786870\pi\)
\(542\) 0 0
\(543\) −2.00943 −0.0862331
\(544\) 0 0
\(545\) −15.7438 −0.674391
\(546\) 0 0
\(547\) 0.286412i 0.0122461i 0.999981 + 0.00612304i \(0.00194904\pi\)
−0.999981 + 0.00612304i \(0.998051\pi\)
\(548\) 0 0
\(549\) 4.62698i 0.197475i
\(550\) 0 0
\(551\) −15.1660 −0.646093
\(552\) 0 0
\(553\) 16.8328 0.715802
\(554\) 0 0
\(555\) 5.26726i 0.223583i
\(556\) 0 0
\(557\) 7.77446i 0.329414i −0.986343 0.164707i \(-0.947332\pi\)
0.986343 0.164707i \(-0.0526679\pi\)
\(558\) 0 0
\(559\) −31.6870 −1.34022
\(560\) 0 0
\(561\) −7.21040 −0.304423
\(562\) 0 0
\(563\) −13.1915 −0.555955 −0.277977 0.960588i \(-0.589664\pi\)
−0.277977 + 0.960588i \(0.589664\pi\)
\(564\) 0 0
\(565\) 12.6691 0.532991
\(566\) 0 0
\(567\) 3.12483 0.131231
\(568\) 0 0
\(569\) 18.9198i 0.793159i 0.918000 + 0.396580i \(0.129803\pi\)
−0.918000 + 0.396580i \(0.870197\pi\)
\(570\) 0 0
\(571\) 2.35834 0.0986933 0.0493466 0.998782i \(-0.484286\pi\)
0.0493466 + 0.998782i \(0.484286\pi\)
\(572\) 0 0
\(573\) 18.8616i 0.787954i
\(574\) 0 0
\(575\) −2.20899 + 4.25680i −0.0921212 + 0.177521i
\(576\) 0 0
\(577\) 39.1515 1.62990 0.814949 0.579533i \(-0.196765\pi\)
0.814949 + 0.579533i \(0.196765\pi\)
\(578\) 0 0
\(579\) 25.8785i 1.07547i
\(580\) 0 0
\(581\) 46.1159 1.91321
\(582\) 0 0
\(583\) 6.24634i 0.258697i
\(584\) 0 0
\(585\) 2.52369i 0.104342i
\(586\) 0 0
\(587\) 28.8938i 1.19258i 0.802771 + 0.596288i \(0.203358\pi\)
−0.802771 + 0.596288i \(0.796642\pi\)
\(588\) 0 0
\(589\) 2.56404i 0.105649i
\(590\) 0 0
\(591\) 11.3975i 0.468832i
\(592\) 0 0
\(593\) 6.95439 0.285583 0.142791 0.989753i \(-0.454392\pi\)
0.142791 + 0.989753i \(0.454392\pi\)
\(594\) 0 0
\(595\) 11.4391 0.468958
\(596\) 0 0
\(597\) 10.1484i 0.415344i
\(598\) 0 0
\(599\) 39.5319i 1.61523i −0.589711 0.807614i \(-0.700758\pi\)
0.589711 0.807614i \(-0.299242\pi\)
\(600\) 0 0
\(601\) 23.3570 0.952751 0.476376 0.879242i \(-0.341950\pi\)
0.476376 + 0.879242i \(0.341950\pi\)
\(602\) 0 0
\(603\) −10.4936 −0.427332
\(604\) 0 0
\(605\) 7.12040i 0.289485i
\(606\) 0 0
\(607\) 6.94207i 0.281770i −0.990026 0.140885i \(-0.955005\pi\)
0.990026 0.140885i \(-0.0449947\pi\)
\(608\) 0 0
\(609\) 6.98361i 0.282990i
\(610\) 0 0
\(611\) 3.40922i 0.137922i
\(612\) 0 0
\(613\) 48.1058i 1.94298i 0.237089 + 0.971488i \(0.423807\pi\)
−0.237089 + 0.971488i \(0.576193\pi\)
\(614\) 0 0
\(615\) −5.40401 −0.217911
\(616\) 0 0
\(617\) 4.62309i 0.186119i 0.995661 + 0.0930593i \(0.0296646\pi\)
−0.995661 + 0.0930593i \(0.970335\pi\)
\(618\) 0 0
\(619\) −34.8837 −1.40210 −0.701048 0.713114i \(-0.747284\pi\)
−0.701048 + 0.713114i \(0.747284\pi\)
\(620\) 0 0
\(621\) −4.25680 2.20899i −0.170820 0.0886437i
\(622\) 0 0
\(623\) 1.05630i 0.0423197i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 13.3663i 0.533799i
\(628\) 0 0
\(629\) −19.2819 −0.768821
\(630\) 0 0
\(631\) −1.52455 −0.0606915 −0.0303458 0.999539i \(-0.509661\pi\)
−0.0303458 + 0.999539i \(0.509661\pi\)
\(632\) 0 0
\(633\) 4.20971 0.167321
\(634\) 0 0
\(635\) 21.2834 0.844605
\(636\) 0 0
\(637\) −6.97695 −0.276437
\(638\) 0 0
\(639\) 15.1992i 0.601270i
\(640\) 0 0
\(641\) 33.0371i 1.30489i −0.757837 0.652443i \(-0.773744\pi\)
0.757837 0.652443i \(-0.226256\pi\)
\(642\) 0 0
\(643\) 10.4205 0.410943 0.205472 0.978663i \(-0.434127\pi\)
0.205472 + 0.978663i \(0.434127\pi\)
\(644\) 0 0
\(645\) −12.5558 −0.494386
\(646\) 0 0
\(647\) 6.28523i 0.247098i 0.992338 + 0.123549i \(0.0394276\pi\)
−0.992338 + 0.123549i \(0.960572\pi\)
\(648\) 0 0
\(649\) 23.6285i 0.927500i
\(650\) 0 0
\(651\) 1.18068 0.0462747
\(652\) 0 0
\(653\) 21.7826 0.852419 0.426209 0.904625i \(-0.359849\pi\)
0.426209 + 0.904625i \(0.359849\pi\)
\(654\) 0 0
\(655\) 13.9512 0.545119
\(656\) 0 0
\(657\) −13.4156 −0.523391
\(658\) 0 0
\(659\) −18.2120 −0.709438 −0.354719 0.934973i \(-0.615424\pi\)
−0.354719 + 0.934973i \(0.615424\pi\)
\(660\) 0 0
\(661\) 17.9091i 0.696583i −0.937386 0.348291i \(-0.886762\pi\)
0.937386 0.348291i \(-0.113238\pi\)
\(662\) 0 0
\(663\) −9.23850 −0.358793
\(664\) 0 0
\(665\) 21.2053i 0.822308i
\(666\) 0 0
\(667\) 4.93681 9.51341i 0.191154 0.368361i
\(668\) 0 0
\(669\) −20.2218 −0.781821
\(670\) 0 0
\(671\) 9.11362i 0.351828i
\(672\) 0 0
\(673\) −40.4411 −1.55889 −0.779444 0.626471i \(-0.784499\pi\)
−0.779444 + 0.626471i \(0.784499\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 12.0827i 0.464375i −0.972671 0.232187i \(-0.925412\pi\)
0.972671 0.232187i \(-0.0745882\pi\)
\(678\) 0 0
\(679\) 34.6489i 1.32970i
\(680\) 0 0
\(681\) 29.8101i 1.14233i
\(682\) 0 0
\(683\) 44.2598i 1.69356i −0.531947 0.846778i \(-0.678539\pi\)
0.531947 0.846778i \(-0.321461\pi\)
\(684\) 0 0
\(685\) −2.32226 −0.0887290
\(686\) 0 0
\(687\) −12.3772 −0.472220
\(688\) 0 0
\(689\) 8.00327i 0.304900i
\(690\) 0 0
\(691\) 17.0262i 0.647705i −0.946108 0.323853i \(-0.895022\pi\)
0.946108 0.323853i \(-0.104978\pi\)
\(692\) 0 0
\(693\) 6.15489 0.233805
\(694\) 0 0
\(695\) −15.4464 −0.585914
\(696\) 0 0
\(697\) 19.7825i 0.749317i
\(698\) 0 0
\(699\) 13.5820i 0.513720i
\(700\) 0 0
\(701\) 15.2474i 0.575887i −0.957647 0.287943i \(-0.907029\pi\)
0.957647 0.287943i \(-0.0929715\pi\)
\(702\) 0 0
\(703\) 35.7440i 1.34811i
\(704\) 0 0
\(705\) 1.35089i 0.0508774i
\(706\) 0 0
\(707\) 20.7762 0.781368
\(708\) 0 0
\(709\) 17.8601i 0.670751i 0.942085 + 0.335375i \(0.108863\pi\)
−0.942085 + 0.335375i \(0.891137\pi\)
\(710\) 0 0
\(711\) −5.38677 −0.202020
\(712\) 0 0
\(713\) −1.60839 0.834642i −0.0602345 0.0312576i
\(714\) 0 0
\(715\) 4.97083i 0.185898i
\(716\) 0 0
\(717\) 8.77772 0.327810
\(718\) 0 0
\(719\) 11.5363i 0.430231i 0.976589 + 0.215116i \(0.0690129\pi\)
−0.976589 + 0.215116i \(0.930987\pi\)
\(720\) 0 0
\(721\) 1.82383 0.0679229
\(722\) 0 0
\(723\) 13.8466 0.514959
\(724\) 0 0
\(725\) −2.23487 −0.0830011
\(726\) 0 0
\(727\) 0.409166 0.0151751 0.00758756 0.999971i \(-0.497585\pi\)
0.00758756 + 0.999971i \(0.497585\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 45.9634i 1.70002i
\(732\) 0 0
\(733\) 28.6370i 1.05773i 0.848705 + 0.528866i \(0.177383\pi\)
−0.848705 + 0.528866i \(0.822617\pi\)
\(734\) 0 0
\(735\) −2.76459 −0.101973
\(736\) 0 0
\(737\) −20.6689 −0.761348
\(738\) 0 0
\(739\) 6.08323i 0.223775i 0.993721 + 0.111888i \(0.0356897\pi\)
−0.993721 + 0.111888i \(0.964310\pi\)
\(740\) 0 0
\(741\) 17.1259i 0.629136i
\(742\) 0 0
\(743\) −12.0715 −0.442860 −0.221430 0.975176i \(-0.571072\pi\)
−0.221430 + 0.975176i \(0.571072\pi\)
\(744\) 0 0
\(745\) −4.84162 −0.177383
\(746\) 0 0
\(747\) −14.7579 −0.539962
\(748\) 0 0
\(749\) 6.39111 0.233526
\(750\) 0 0
\(751\) −49.1905 −1.79499 −0.897493 0.441029i \(-0.854614\pi\)
−0.897493 + 0.441029i \(0.854614\pi\)
\(752\) 0 0
\(753\) 26.2244i 0.955670i
\(754\) 0 0
\(755\) −19.8598 −0.722771
\(756\) 0 0
\(757\) 21.0572i 0.765335i −0.923886 0.382668i \(-0.875005\pi\)
0.923886 0.382668i \(-0.124995\pi\)
\(758\) 0 0
\(759\) −8.38449 4.35098i −0.304338 0.157931i
\(760\) 0 0
\(761\) −44.4485 −1.61126 −0.805628 0.592422i \(-0.798172\pi\)
−0.805628 + 0.592422i \(0.798172\pi\)
\(762\) 0 0
\(763\) 49.1968i 1.78104i
\(764\) 0 0
\(765\) −3.66071 −0.132353
\(766\) 0 0
\(767\) 30.2746i 1.09315i
\(768\) 0 0
\(769\) 27.8869i 1.00563i −0.864395 0.502814i \(-0.832298\pi\)
0.864395 0.502814i \(-0.167702\pi\)
\(770\) 0 0
\(771\) 3.73811i 0.134625i
\(772\) 0 0
\(773\) 4.96861i 0.178709i 0.996000 + 0.0893543i \(0.0284803\pi\)
−0.996000 + 0.0893543i \(0.971520\pi\)
\(774\) 0 0
\(775\) 0.377839i 0.0135724i
\(776\) 0 0
\(777\) 16.4593 0.590475
\(778\) 0 0
\(779\) −36.6720 −1.31391
\(780\) 0 0
\(781\) 29.9373i 1.07124i
\(782\) 0 0
\(783\) 2.23487i 0.0798678i
\(784\) 0 0
\(785\) −16.2502 −0.579994
\(786\) 0 0
\(787\) −21.1512 −0.753957 −0.376979 0.926222i \(-0.623037\pi\)
−0.376979 + 0.926222i \(0.623037\pi\)
\(788\) 0 0
\(789\) 19.3159i 0.687666i
\(790\) 0 0
\(791\) 39.5887i 1.40761i
\(792\) 0 0
\(793\) 11.6770i 0.414664i
\(794\) 0 0
\(795\) 3.17126i 0.112473i
\(796\) 0 0
\(797\) 4.80963i 0.170366i −0.996365 0.0851830i \(-0.972853\pi\)
0.996365 0.0851830i \(-0.0271475\pi\)
\(798\) 0 0
\(799\) −4.94522 −0.174949
\(800\) 0 0
\(801\) 0.338033i 0.0119438i
\(802\) 0 0
\(803\) −26.4243 −0.932492
\(804\) 0 0
\(805\) 13.3018 + 6.90273i 0.468827 + 0.243289i
\(806\) 0 0