Properties

Label 4004.2.m.c.2157.23
Level 4004
Weight 2
Character 4004.2157
Analytic conductor 31.972
Analytic rank 0
Dimension 36
CM No

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Newspace parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.m (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(36\)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2157.23
Character \(\chi\) = 4004.2157
Dual form 4004.2.m.c.2157.24

$q$-expansion

\(f(q)\) \(=\) \(q\)\(+0.810205 q^{3}\) \(+3.82551i q^{5}\) \(-1.00000i q^{7}\) \(-2.34357 q^{9}\) \(+O(q^{10})\) \(q\)\(+0.810205 q^{3}\) \(+3.82551i q^{5}\) \(-1.00000i q^{7}\) \(-2.34357 q^{9}\) \(+1.00000i q^{11}\) \(+(3.12587 - 1.79692i) q^{13}\) \(+3.09945i q^{15}\) \(+1.91054 q^{17}\) \(-2.75048i q^{19}\) \(-0.810205i q^{21}\) \(-7.88826 q^{23}\) \(-9.63452 q^{25}\) \(-4.32939 q^{27}\) \(-8.14739 q^{29}\) \(+6.00616i q^{31}\) \(+0.810205i q^{33}\) \(+3.82551 q^{35}\) \(+5.83504i q^{37}\) \(+(2.53260 - 1.45587i) q^{39}\) \(-5.43959i q^{41}\) \(-5.59610 q^{43}\) \(-8.96534i q^{45}\) \(-12.1911i q^{47}\) \(-1.00000 q^{49}\) \(+1.54793 q^{51}\) \(-7.25361 q^{53}\) \(-3.82551 q^{55}\) \(-2.22845i q^{57}\) \(+10.5590i q^{59}\) \(+10.1797 q^{61}\) \(+2.34357i q^{63}\) \(+(6.87414 + 11.9581i) q^{65}\) \(+1.53230i q^{67}\) \(-6.39111 q^{69}\) \(+10.9857i q^{71}\) \(-9.21307i q^{73}\) \(-7.80594 q^{75}\) \(+1.00000 q^{77}\) \(-6.37576 q^{79}\) \(+3.52301 q^{81}\) \(+4.00995i q^{83}\) \(+7.30879i q^{85}\) \(-6.60106 q^{87}\) \(-8.61636i q^{89}\) \(+(-1.79692 - 3.12587i) q^{91}\) \(+4.86622i q^{93}\) \(+10.5220 q^{95}\) \(+1.03847i q^{97}\) \(-2.34357i q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 80q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 80q^{69} \) \(\mathstrut -\mathstrut 36q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 132q^{81} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(925\) \(2003\) \(3433\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 0.810205 0.467772 0.233886 0.972264i \(-0.424856\pi\)
0.233886 + 0.972264i \(0.424856\pi\)
\(4\) 0 0
\(5\) 3.82551i 1.71082i 0.517952 + 0.855410i \(0.326695\pi\)
−0.517952 + 0.855410i \(0.673305\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −2.34357 −0.781189
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) 3.12587 1.79692i 0.866961 0.498376i
\(14\) 0 0
\(15\) 3.09945i 0.800274i
\(16\) 0 0
\(17\) 1.91054 0.463374 0.231687 0.972790i \(-0.425575\pi\)
0.231687 + 0.972790i \(0.425575\pi\)
\(18\) 0 0
\(19\) 2.75048i 0.631003i −0.948925 0.315502i \(-0.897827\pi\)
0.948925 0.315502i \(-0.102173\pi\)
\(20\) 0 0
\(21\) 0.810205i 0.176801i
\(22\) 0 0
\(23\) −7.88826 −1.64482 −0.822408 0.568898i \(-0.807370\pi\)
−0.822408 + 0.568898i \(0.807370\pi\)
\(24\) 0 0
\(25\) −9.63452 −1.92690
\(26\) 0 0
\(27\) −4.32939 −0.833191
\(28\) 0 0
\(29\) −8.14739 −1.51293 −0.756467 0.654032i \(-0.773076\pi\)
−0.756467 + 0.654032i \(0.773076\pi\)
\(30\) 0 0
\(31\) 6.00616i 1.07874i 0.842070 + 0.539369i \(0.181337\pi\)
−0.842070 + 0.539369i \(0.818663\pi\)
\(32\) 0 0
\(33\) 0.810205i 0.141039i
\(34\) 0 0
\(35\) 3.82551 0.646629
\(36\) 0 0
\(37\) 5.83504i 0.959274i 0.877467 + 0.479637i \(0.159232\pi\)
−0.877467 + 0.479637i \(0.840768\pi\)
\(38\) 0 0
\(39\) 2.53260 1.45587i 0.405540 0.233126i
\(40\) 0 0
\(41\) 5.43959i 0.849521i −0.905306 0.424761i \(-0.860358\pi\)
0.905306 0.424761i \(-0.139642\pi\)
\(42\) 0 0
\(43\) −5.59610 −0.853397 −0.426699 0.904394i \(-0.640324\pi\)
−0.426699 + 0.904394i \(0.640324\pi\)
\(44\) 0 0
\(45\) 8.96534i 1.33647i
\(46\) 0 0
\(47\) 12.1911i 1.77825i −0.457661 0.889127i \(-0.651313\pi\)
0.457661 0.889127i \(-0.348687\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 1.54793 0.216754
\(52\) 0 0
\(53\) −7.25361 −0.996361 −0.498180 0.867073i \(-0.665998\pi\)
−0.498180 + 0.867073i \(0.665998\pi\)
\(54\) 0 0
\(55\) −3.82551 −0.515832
\(56\) 0 0
\(57\) 2.22845i 0.295166i
\(58\) 0 0
\(59\) 10.5590i 1.37466i 0.726345 + 0.687330i \(0.241217\pi\)
−0.726345 + 0.687330i \(0.758783\pi\)
\(60\) 0 0
\(61\) 10.1797 1.30338 0.651691 0.758484i \(-0.274060\pi\)
0.651691 + 0.758484i \(0.274060\pi\)
\(62\) 0 0
\(63\) 2.34357i 0.295262i
\(64\) 0 0
\(65\) 6.87414 + 11.9581i 0.852632 + 1.48321i
\(66\) 0 0
\(67\) 1.53230i 0.187200i 0.995610 + 0.0936000i \(0.0298375\pi\)
−0.995610 + 0.0936000i \(0.970163\pi\)
\(68\) 0 0
\(69\) −6.39111 −0.769399
\(70\) 0 0
\(71\) 10.9857i 1.30377i 0.758320 + 0.651883i \(0.226021\pi\)
−0.758320 + 0.651883i \(0.773979\pi\)
\(72\) 0 0
\(73\) 9.21307i 1.07831i −0.842207 0.539154i \(-0.818744\pi\)
0.842207 0.539154i \(-0.181256\pi\)
\(74\) 0 0
\(75\) −7.80594 −0.901352
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −6.37576 −0.717329 −0.358664 0.933467i \(-0.616768\pi\)
−0.358664 + 0.933467i \(0.616768\pi\)
\(80\) 0 0
\(81\) 3.52301 0.391446
\(82\) 0 0
\(83\) 4.00995i 0.440149i 0.975483 + 0.220075i \(0.0706301\pi\)
−0.975483 + 0.220075i \(0.929370\pi\)
\(84\) 0 0
\(85\) 7.30879i 0.792750i
\(86\) 0 0
\(87\) −6.60106 −0.707708
\(88\) 0 0
\(89\) 8.61636i 0.913332i −0.889638 0.456666i \(-0.849043\pi\)
0.889638 0.456666i \(-0.150957\pi\)
\(90\) 0 0
\(91\) −1.79692 3.12587i −0.188368 0.327680i
\(92\) 0 0
\(93\) 4.86622i 0.504604i
\(94\) 0 0
\(95\) 10.5220 1.07953
\(96\) 0 0
\(97\) 1.03847i 0.105441i 0.998609 + 0.0527203i \(0.0167892\pi\)
−0.998609 + 0.0527203i \(0.983211\pi\)
\(98\) 0 0
\(99\) 2.34357i 0.235537i
\(100\) 0 0
\(101\) 16.3268 1.62458 0.812290 0.583254i \(-0.198221\pi\)
0.812290 + 0.583254i \(0.198221\pi\)
\(102\) 0 0
\(103\) −17.4748 −1.72184 −0.860921 0.508738i \(-0.830112\pi\)
−0.860921 + 0.508738i \(0.830112\pi\)
\(104\) 0 0
\(105\) 3.09945 0.302475
\(106\) 0 0
\(107\) 8.44728 0.816629 0.408314 0.912841i \(-0.366117\pi\)
0.408314 + 0.912841i \(0.366117\pi\)
\(108\) 0 0
\(109\) 12.7130i 1.21768i −0.793291 0.608842i \(-0.791634\pi\)
0.793291 0.608842i \(-0.208366\pi\)
\(110\) 0 0
\(111\) 4.72758i 0.448722i
\(112\) 0 0
\(113\) −17.9640 −1.68991 −0.844955 0.534837i \(-0.820373\pi\)
−0.844955 + 0.534837i \(0.820373\pi\)
\(114\) 0 0
\(115\) 30.1766i 2.81398i
\(116\) 0 0
\(117\) −7.32569 + 4.21120i −0.677261 + 0.389326i
\(118\) 0 0
\(119\) 1.91054i 0.175139i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 4.40718i 0.397382i
\(124\) 0 0
\(125\) 17.7294i 1.58577i
\(126\) 0 0
\(127\) −10.2539 −0.909887 −0.454944 0.890520i \(-0.650341\pi\)
−0.454944 + 0.890520i \(0.650341\pi\)
\(128\) 0 0
\(129\) −4.53399 −0.399196
\(130\) 0 0
\(131\) −14.2031 −1.24093 −0.620467 0.784232i \(-0.713057\pi\)
−0.620467 + 0.784232i \(0.713057\pi\)
\(132\) 0 0
\(133\) −2.75048 −0.238497
\(134\) 0 0
\(135\) 16.5621i 1.42544i
\(136\) 0 0
\(137\) 3.55443i 0.303676i 0.988405 + 0.151838i \(0.0485192\pi\)
−0.988405 + 0.151838i \(0.951481\pi\)
\(138\) 0 0
\(139\) 6.82549 0.578930 0.289465 0.957189i \(-0.406523\pi\)
0.289465 + 0.957189i \(0.406523\pi\)
\(140\) 0 0
\(141\) 9.87728i 0.831817i
\(142\) 0 0
\(143\) 1.79692 + 3.12587i 0.150266 + 0.261399i
\(144\) 0 0
\(145\) 31.1679i 2.58836i
\(146\) 0 0
\(147\) −0.810205 −0.0668246
\(148\) 0 0
\(149\) 0.609809i 0.0499575i −0.999688 0.0249788i \(-0.992048\pi\)
0.999688 0.0249788i \(-0.00795181\pi\)
\(150\) 0 0
\(151\) 16.5888i 1.34998i −0.737826 0.674990i \(-0.764148\pi\)
0.737826 0.674990i \(-0.235852\pi\)
\(152\) 0 0
\(153\) −4.47748 −0.361983
\(154\) 0 0
\(155\) −22.9766 −1.84553
\(156\) 0 0
\(157\) 20.0683 1.60162 0.800811 0.598918i \(-0.204402\pi\)
0.800811 + 0.598918i \(0.204402\pi\)
\(158\) 0 0
\(159\) −5.87692 −0.466070
\(160\) 0 0
\(161\) 7.88826i 0.621682i
\(162\) 0 0
\(163\) 3.36355i 0.263454i 0.991286 + 0.131727i \(0.0420522\pi\)
−0.991286 + 0.131727i \(0.957948\pi\)
\(164\) 0 0
\(165\) −3.09945 −0.241292
\(166\) 0 0
\(167\) 18.6361i 1.44210i 0.692882 + 0.721051i \(0.256341\pi\)
−0.692882 + 0.721051i \(0.743659\pi\)
\(168\) 0 0
\(169\) 6.54215 11.2339i 0.503243 0.864145i
\(170\) 0 0
\(171\) 6.44593i 0.492933i
\(172\) 0 0
\(173\) −17.8784 −1.35927 −0.679636 0.733549i \(-0.737862\pi\)
−0.679636 + 0.733549i \(0.737862\pi\)
\(174\) 0 0
\(175\) 9.63452i 0.728301i
\(176\) 0 0
\(177\) 8.55493i 0.643028i
\(178\) 0 0
\(179\) −1.99525 −0.149132 −0.0745661 0.997216i \(-0.523757\pi\)
−0.0745661 + 0.997216i \(0.523757\pi\)
\(180\) 0 0
\(181\) −22.9006 −1.70219 −0.851093 0.525015i \(-0.824060\pi\)
−0.851093 + 0.525015i \(0.824060\pi\)
\(182\) 0 0
\(183\) 8.24768 0.609686
\(184\) 0 0
\(185\) −22.3220 −1.64114
\(186\) 0 0
\(187\) 1.91054i 0.139713i
\(188\) 0 0
\(189\) 4.32939i 0.314917i
\(190\) 0 0
\(191\) −20.5446 −1.48656 −0.743279 0.668982i \(-0.766730\pi\)
−0.743279 + 0.668982i \(0.766730\pi\)
\(192\) 0 0
\(193\) 19.1022i 1.37501i −0.726181 0.687503i \(-0.758707\pi\)
0.726181 0.687503i \(-0.241293\pi\)
\(194\) 0 0
\(195\) 5.56946 + 9.68848i 0.398837 + 0.693806i
\(196\) 0 0
\(197\) 12.1514i 0.865748i −0.901454 0.432874i \(-0.857499\pi\)
0.901454 0.432874i \(-0.142501\pi\)
\(198\) 0 0
\(199\) −7.12897 −0.505359 −0.252680 0.967550i \(-0.581312\pi\)
−0.252680 + 0.967550i \(0.581312\pi\)
\(200\) 0 0
\(201\) 1.24148i 0.0875669i
\(202\) 0 0
\(203\) 8.14739i 0.571835i
\(204\) 0 0
\(205\) 20.8092 1.45338
\(206\) 0 0
\(207\) 18.4867 1.28491
\(208\) 0 0
\(209\) 2.75048 0.190255
\(210\) 0 0
\(211\) −2.27714 −0.156765 −0.0783824 0.996923i \(-0.524976\pi\)
−0.0783824 + 0.996923i \(0.524976\pi\)
\(212\) 0 0
\(213\) 8.90069i 0.609865i
\(214\) 0 0
\(215\) 21.4079i 1.46001i
\(216\) 0 0
\(217\) 6.00616 0.407725
\(218\) 0 0
\(219\) 7.46448i 0.504403i
\(220\) 0 0
\(221\) 5.97211 3.43309i 0.401727 0.230935i
\(222\) 0 0
\(223\) 14.9117i 0.998563i 0.866440 + 0.499282i \(0.166403\pi\)
−0.866440 + 0.499282i \(0.833597\pi\)
\(224\) 0 0
\(225\) 22.5791 1.50528
\(226\) 0 0
\(227\) 22.1082i 1.46737i 0.679488 + 0.733687i \(0.262202\pi\)
−0.679488 + 0.733687i \(0.737798\pi\)
\(228\) 0 0
\(229\) 0.280435i 0.0185316i −0.999957 0.00926582i \(-0.997051\pi\)
0.999957 0.00926582i \(-0.00294944\pi\)
\(230\) 0 0
\(231\) 0.810205 0.0533076
\(232\) 0 0
\(233\) 16.6953 1.09374 0.546872 0.837216i \(-0.315818\pi\)
0.546872 + 0.837216i \(0.315818\pi\)
\(234\) 0 0
\(235\) 46.6371 3.04227
\(236\) 0 0
\(237\) −5.16567 −0.335546
\(238\) 0 0
\(239\) 24.2526i 1.56877i 0.620275 + 0.784384i \(0.287021\pi\)
−0.620275 + 0.784384i \(0.712979\pi\)
\(240\) 0 0
\(241\) 9.63690i 0.620767i −0.950611 0.310384i \(-0.899542\pi\)
0.950611 0.310384i \(-0.100458\pi\)
\(242\) 0 0
\(243\) 15.8425 1.01630
\(244\) 0 0
\(245\) 3.82551i 0.244403i
\(246\) 0 0
\(247\) −4.94239 8.59765i −0.314477 0.547055i
\(248\) 0 0
\(249\) 3.24888i 0.205890i
\(250\) 0 0
\(251\) 3.41128 0.215318 0.107659 0.994188i \(-0.465664\pi\)
0.107659 + 0.994188i \(0.465664\pi\)
\(252\) 0 0
\(253\) 7.88826i 0.495931i
\(254\) 0 0
\(255\) 5.92162i 0.370826i
\(256\) 0 0
\(257\) 18.0645 1.12684 0.563418 0.826172i \(-0.309486\pi\)
0.563418 + 0.826172i \(0.309486\pi\)
\(258\) 0 0
\(259\) 5.83504 0.362571
\(260\) 0 0
\(261\) 19.0940 1.18189
\(262\) 0 0
\(263\) 1.54515 0.0952779 0.0476389 0.998865i \(-0.484830\pi\)
0.0476389 + 0.998865i \(0.484830\pi\)
\(264\) 0 0
\(265\) 27.7488i 1.70459i
\(266\) 0 0
\(267\) 6.98102i 0.427231i
\(268\) 0 0
\(269\) 15.5203 0.946287 0.473143 0.880985i \(-0.343119\pi\)
0.473143 + 0.880985i \(0.343119\pi\)
\(270\) 0 0
\(271\) 5.86316i 0.356162i −0.984016 0.178081i \(-0.943011\pi\)
0.984016 0.178081i \(-0.0569889\pi\)
\(272\) 0 0
\(273\) −1.45587 2.53260i −0.0881135 0.153280i
\(274\) 0 0
\(275\) 9.63452i 0.580983i
\(276\) 0 0
\(277\) −18.3570 −1.10296 −0.551481 0.834187i \(-0.685937\pi\)
−0.551481 + 0.834187i \(0.685937\pi\)
\(278\) 0 0
\(279\) 14.0758i 0.842698i
\(280\) 0 0
\(281\) 15.8478i 0.945398i −0.881224 0.472699i \(-0.843280\pi\)
0.881224 0.472699i \(-0.156720\pi\)
\(282\) 0 0
\(283\) 27.2037 1.61709 0.808545 0.588434i \(-0.200255\pi\)
0.808545 + 0.588434i \(0.200255\pi\)
\(284\) 0 0
\(285\) 8.52496 0.504975
\(286\) 0 0
\(287\) −5.43959 −0.321089
\(288\) 0 0
\(289\) −13.3498 −0.785284
\(290\) 0 0
\(291\) 0.841373i 0.0493222i
\(292\) 0 0
\(293\) 10.9766i 0.641262i 0.947204 + 0.320631i \(0.103895\pi\)
−0.947204 + 0.320631i \(0.896105\pi\)
\(294\) 0 0
\(295\) −40.3934 −2.35179
\(296\) 0 0
\(297\) 4.32939i 0.251216i
\(298\) 0 0
\(299\) −24.6577 + 14.1746i −1.42599 + 0.819737i
\(300\) 0 0
\(301\) 5.59610i 0.322554i
\(302\) 0 0
\(303\) 13.2281 0.759933
\(304\) 0 0
\(305\) 38.9427i 2.22985i
\(306\) 0 0
\(307\) 29.6137i 1.69015i −0.534652 0.845073i \(-0.679557\pi\)
0.534652 0.845073i \(-0.320443\pi\)
\(308\) 0 0
\(309\) −14.1582 −0.805430
\(310\) 0 0
\(311\) −19.4926 −1.10532 −0.552662 0.833405i \(-0.686388\pi\)
−0.552662 + 0.833405i \(0.686388\pi\)
\(312\) 0 0
\(313\) −15.1618 −0.856994 −0.428497 0.903543i \(-0.640957\pi\)
−0.428497 + 0.903543i \(0.640957\pi\)
\(314\) 0 0
\(315\) −8.96534 −0.505140
\(316\) 0 0
\(317\) 26.2137i 1.47231i 0.676814 + 0.736154i \(0.263360\pi\)
−0.676814 + 0.736154i \(0.736640\pi\)
\(318\) 0 0
\(319\) 8.14739i 0.456166i
\(320\) 0 0
\(321\) 6.84403 0.381996
\(322\) 0 0
\(323\) 5.25490i 0.292391i
\(324\) 0 0
\(325\) −30.1163 + 17.3125i −1.67055 + 0.960323i
\(326\) 0 0
\(327\) 10.3001i 0.569599i
\(328\) 0 0
\(329\) −12.1911 −0.672117
\(330\) 0 0
\(331\) 8.31485i 0.457026i 0.973541 + 0.228513i \(0.0733863\pi\)
−0.973541 + 0.228513i \(0.926614\pi\)
\(332\) 0 0
\(333\) 13.6748i 0.749374i
\(334\) 0 0
\(335\) −5.86182 −0.320265
\(336\) 0 0
\(337\) 1.87403 0.102085 0.0510426 0.998696i \(-0.483746\pi\)
0.0510426 + 0.998696i \(0.483746\pi\)
\(338\) 0 0
\(339\) −14.5545 −0.790493
\(340\) 0 0
\(341\) −6.00616 −0.325252
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 24.4493i 1.31630i
\(346\) 0 0
\(347\) −16.3694 −0.878755 −0.439378 0.898302i \(-0.644801\pi\)
−0.439378 + 0.898302i \(0.644801\pi\)
\(348\) 0 0
\(349\) 14.7403i 0.789031i 0.918889 + 0.394515i \(0.129087\pi\)
−0.918889 + 0.394515i \(0.870913\pi\)
\(350\) 0 0
\(351\) −13.5331 + 7.77956i −0.722344 + 0.415242i
\(352\) 0 0
\(353\) 6.90003i 0.367252i 0.982996 + 0.183626i \(0.0587835\pi\)
−0.982996 + 0.183626i \(0.941217\pi\)
\(354\) 0 0
\(355\) −42.0260 −2.23051
\(356\) 0 0
\(357\) 1.54793i 0.0819251i
\(358\) 0 0
\(359\) 27.7588i 1.46505i 0.680740 + 0.732526i \(0.261659\pi\)
−0.680740 + 0.732526i \(0.738341\pi\)
\(360\) 0 0
\(361\) 11.4349 0.601835
\(362\) 0 0
\(363\) −0.810205 −0.0425247
\(364\) 0 0
\(365\) 35.2447 1.84479
\(366\) 0 0
\(367\) 32.5352 1.69833 0.849163 0.528131i \(-0.177107\pi\)
0.849163 + 0.528131i \(0.177107\pi\)
\(368\) 0 0
\(369\) 12.7480i 0.663637i
\(370\) 0 0
\(371\) 7.25361i 0.376589i
\(372\) 0 0
\(373\) −13.4309 −0.695424 −0.347712 0.937601i \(-0.613041\pi\)
−0.347712 + 0.937601i \(0.613041\pi\)
\(374\) 0 0
\(375\) 14.3645i 0.741777i
\(376\) 0 0
\(377\) −25.4677 + 14.6402i −1.31165 + 0.754010i
\(378\) 0 0
\(379\) 20.4132i 1.04856i −0.851547 0.524279i \(-0.824335\pi\)
0.851547 0.524279i \(-0.175665\pi\)
\(380\) 0 0
\(381\) −8.30777 −0.425620
\(382\) 0 0
\(383\) 17.8138i 0.910242i 0.890430 + 0.455121i \(0.150404\pi\)
−0.890430 + 0.455121i \(0.849596\pi\)
\(384\) 0 0
\(385\) 3.82551i 0.194966i
\(386\) 0 0
\(387\) 13.1148 0.666665
\(388\) 0 0
\(389\) 33.1676 1.68166 0.840831 0.541298i \(-0.182067\pi\)
0.840831 + 0.541298i \(0.182067\pi\)
\(390\) 0 0
\(391\) −15.0708 −0.762165
\(392\) 0 0
\(393\) −11.5075 −0.580475
\(394\) 0 0
\(395\) 24.3905i 1.22722i
\(396\) 0 0
\(397\) 7.90138i 0.396559i −0.980146 0.198279i \(-0.936465\pi\)
0.980146 0.198279i \(-0.0635353\pi\)
\(398\) 0 0
\(399\) −2.22845 −0.111562
\(400\) 0 0
\(401\) 37.1332i 1.85435i −0.374634 0.927173i \(-0.622232\pi\)
0.374634 0.927173i \(-0.377768\pi\)
\(402\) 0 0
\(403\) 10.7926 + 18.7745i 0.537617 + 0.935224i
\(404\) 0 0
\(405\) 13.4773i 0.669693i
\(406\) 0 0
\(407\) −5.83504 −0.289232
\(408\) 0 0
\(409\) 7.93769i 0.392493i 0.980555 + 0.196247i \(0.0628754\pi\)
−0.980555 + 0.196247i \(0.937125\pi\)
\(410\) 0 0
\(411\) 2.87982i 0.142051i
\(412\) 0 0
\(413\) 10.5590 0.519573
\(414\) 0 0
\(415\) −15.3401 −0.753016
\(416\) 0 0
\(417\) 5.53005 0.270808
\(418\) 0 0
\(419\) 2.23501 0.109187 0.0545937 0.998509i \(-0.482614\pi\)
0.0545937 + 0.998509i \(0.482614\pi\)
\(420\) 0 0
\(421\) 33.9151i 1.65292i 0.562994 + 0.826461i \(0.309649\pi\)
−0.562994 + 0.826461i \(0.690351\pi\)
\(422\) 0 0
\(423\) 28.5706i 1.38915i
\(424\) 0 0
\(425\) −18.4071 −0.892877
\(426\) 0 0
\(427\) 10.1797i 0.492632i
\(428\) 0 0
\(429\) 1.45587 + 2.53260i 0.0702903 + 0.122275i
\(430\) 0 0
\(431\) 15.9604i 0.768784i 0.923170 + 0.384392i \(0.125589\pi\)
−0.923170 + 0.384392i \(0.874411\pi\)
\(432\) 0 0
\(433\) −3.87431 −0.186188 −0.0930938 0.995657i \(-0.529676\pi\)
−0.0930938 + 0.995657i \(0.529676\pi\)
\(434\) 0 0
\(435\) 25.2524i 1.21076i
\(436\) 0 0
\(437\) 21.6965i 1.03788i
\(438\) 0 0
\(439\) −21.4656 −1.02450 −0.512248 0.858838i \(-0.671187\pi\)
−0.512248 + 0.858838i \(0.671187\pi\)
\(440\) 0 0
\(441\) 2.34357 0.111598
\(442\) 0 0
\(443\) −8.03686 −0.381843 −0.190921 0.981605i \(-0.561148\pi\)
−0.190921 + 0.981605i \(0.561148\pi\)
\(444\) 0 0
\(445\) 32.9620 1.56255
\(446\) 0 0
\(447\) 0.494070i 0.0233687i
\(448\) 0 0
\(449\) 33.5666i 1.58411i 0.610452 + 0.792054i \(0.290988\pi\)
−0.610452 + 0.792054i \(0.709012\pi\)
\(450\) 0 0
\(451\) 5.43959 0.256140
\(452\) 0 0
\(453\) 13.4404i 0.631484i
\(454\) 0 0
\(455\) 11.9581 6.87414i 0.560602 0.322264i
\(456\) 0 0
\(457\) 22.5771i 1.05611i 0.849209 + 0.528057i \(0.177079\pi\)
−0.849209 + 0.528057i \(0.822921\pi\)
\(458\) 0 0
\(459\) −8.27147 −0.386079
\(460\) 0 0
\(461\) 38.8120i 1.80765i −0.427897 0.903827i \(-0.640745\pi\)
0.427897 0.903827i \(-0.359255\pi\)
\(462\) 0 0
\(463\) 22.0060i 1.02270i −0.859371 0.511352i \(-0.829145\pi\)
0.859371 0.511352i \(-0.170855\pi\)
\(464\) 0 0
\(465\) −18.6158 −0.863286
\(466\) 0 0
\(467\) 25.5139 1.18064 0.590322 0.807168i \(-0.299001\pi\)
0.590322 + 0.807168i \(0.299001\pi\)
\(468\) 0 0
\(469\) 1.53230 0.0707549
\(470\) 0 0
\(471\) 16.2594 0.749194
\(472\) 0 0
\(473\) 5.59610i 0.257309i
\(474\) 0 0
\(475\) 26.4995i 1.21588i
\(476\) 0 0
\(477\) 16.9993 0.778346
\(478\) 0 0
\(479\) 12.6343i 0.577274i −0.957439 0.288637i \(-0.906798\pi\)
0.957439 0.288637i \(-0.0932021\pi\)
\(480\) 0 0
\(481\) 10.4851 + 18.2396i 0.478079 + 0.831653i
\(482\) 0 0
\(483\) 6.39111i 0.290806i
\(484\) 0 0
\(485\) −3.97267 −0.180390
\(486\) 0 0
\(487\) 11.8074i 0.535043i −0.963552 0.267522i \(-0.913795\pi\)
0.963552 0.267522i \(-0.0862047\pi\)
\(488\) 0 0
\(489\) 2.72517i 0.123236i
\(490\) 0 0
\(491\) −14.1537 −0.638747 −0.319374 0.947629i \(-0.603472\pi\)
−0.319374 + 0.947629i \(0.603472\pi\)
\(492\) 0 0
\(493\) −15.5659 −0.701054
\(494\) 0 0
\(495\) 8.96534 0.402962
\(496\) 0 0
\(497\) 10.9857 0.492777
\(498\) 0 0
\(499\) 17.3630i 0.777276i 0.921390 + 0.388638i \(0.127054\pi\)
−0.921390 + 0.388638i \(0.872946\pi\)
\(500\) 0 0
\(501\) 15.0990i 0.674576i
\(502\) 0 0
\(503\) −26.0761 −1.16267 −0.581337 0.813663i \(-0.697470\pi\)
−0.581337 + 0.813663i \(0.697470\pi\)
\(504\) 0 0
\(505\) 62.4584i 2.77936i
\(506\) 0 0
\(507\) 5.30049 9.10175i 0.235403 0.404223i
\(508\) 0 0
\(509\) 5.29781i 0.234821i −0.993083 0.117411i \(-0.962541\pi\)
0.993083 0.117411i \(-0.0374593\pi\)
\(510\) 0 0
\(511\) −9.21307 −0.407562
\(512\) 0 0
\(513\) 11.9079i 0.525746i
\(514\) 0 0
\(515\) 66.8500i 2.94576i
\(516\) 0 0
\(517\) 12.1911 0.536164
\(518\) 0 0
\(519\) −14.4852 −0.635830
\(520\) 0 0
\(521\) 6.39819 0.280310 0.140155 0.990130i \(-0.455240\pi\)
0.140155 + 0.990130i \(0.455240\pi\)
\(522\) 0 0
\(523\) −6.20893 −0.271498 −0.135749 0.990743i \(-0.543344\pi\)
−0.135749 + 0.990743i \(0.543344\pi\)
\(524\) 0 0
\(525\) 7.80594i 0.340679i
\(526\) 0 0
\(527\) 11.4750i 0.499859i
\(528\) 0 0
\(529\) 39.2247 1.70542
\(530\) 0 0
\(531\) 24.7456i 1.07387i
\(532\) 0 0
\(533\) −9.77451 17.0035i −0.423381 0.736502i
\(534\) 0 0
\(535\) 32.3151i 1.39710i
\(536\) 0 0
\(537\) −1.61656 −0.0697599
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 26.3385i 1.13238i 0.824275 + 0.566190i \(0.191583\pi\)
−0.824275 + 0.566190i \(0.808417\pi\)
\(542\) 0 0
\(543\) −18.5542 −0.796235
\(544\) 0 0
\(545\) 48.6337 2.08324
\(546\) 0 0
\(547\) −20.0106 −0.855593 −0.427796 0.903875i \(-0.640710\pi\)
−0.427796 + 0.903875i \(0.640710\pi\)
\(548\) 0 0
\(549\) −23.8569 −1.01819
\(550\) 0 0
\(551\) 22.4092i 0.954665i
\(552\) 0 0
\(553\) 6.37576i 0.271125i
\(554\) 0 0
\(555\) −18.0854 −0.767682
\(556\) 0 0
\(557\) 29.5912i 1.25382i 0.779093 + 0.626909i \(0.215680\pi\)
−0.779093 + 0.626909i \(0.784320\pi\)
\(558\) 0 0
\(559\) −17.4927 + 10.0557i −0.739862 + 0.425313i
\(560\) 0 0
\(561\) 1.54793i 0.0653537i
\(562\) 0 0
\(563\) −17.3869 −0.732768 −0.366384 0.930464i \(-0.619404\pi\)
−0.366384 + 0.930464i \(0.619404\pi\)
\(564\) 0 0
\(565\) 68.7214i 2.89113i
\(566\) 0 0
\(567\) 3.52301i 0.147953i
\(568\) 0 0
\(569\) 19.9401 0.835931 0.417965 0.908463i \(-0.362743\pi\)
0.417965 + 0.908463i \(0.362743\pi\)
\(570\) 0 0
\(571\) −14.9167 −0.624243 −0.312122 0.950042i \(-0.601040\pi\)
−0.312122 + 0.950042i \(0.601040\pi\)
\(572\) 0 0
\(573\) −16.6454 −0.695370
\(574\) 0 0
\(575\) 75.9996 3.16940
\(576\) 0 0
\(577\) 34.6442i 1.44226i 0.692801 + 0.721128i \(0.256376\pi\)
−0.692801 + 0.721128i \(0.743624\pi\)
\(578\) 0 0
\(579\) 15.4767i 0.643190i
\(580\) 0 0
\(581\) 4.00995 0.166361
\(582\) 0 0
\(583\) 7.25361i 0.300414i
\(584\) 0 0
\(585\) −16.1100 28.0245i −0.666066 1.15867i
\(586\) 0 0
\(587\) 25.7114i 1.06122i −0.847616 0.530611i \(-0.821963\pi\)
0.847616 0.530611i \(-0.178037\pi\)
\(588\) 0 0
\(589\) 16.5198 0.680687
\(590\) 0 0
\(591\) 9.84509i 0.404973i
\(592\) 0 0
\(593\) 30.6866i 1.26015i 0.776535 + 0.630074i \(0.216975\pi\)
−0.776535 + 0.630074i \(0.783025\pi\)
\(594\) 0 0
\(595\) 7.30879 0.299631
\(596\) 0 0
\(597\) −5.77593 −0.236393
\(598\) 0 0
\(599\) 35.4819 1.44975 0.724875 0.688880i \(-0.241897\pi\)
0.724875 + 0.688880i \(0.241897\pi\)
\(600\) 0 0
\(601\) −8.62650 −0.351882 −0.175941 0.984401i \(-0.556297\pi\)
−0.175941 + 0.984401i \(0.556297\pi\)
\(602\) 0 0
\(603\) 3.59104i 0.146239i
\(604\) 0 0
\(605\) 3.82551i 0.155529i
\(606\) 0 0
\(607\) 15.6481 0.635136 0.317568 0.948235i \(-0.397134\pi\)
0.317568 + 0.948235i \(0.397134\pi\)
\(608\) 0 0
\(609\) 6.60106i 0.267488i
\(610\) 0 0
\(611\) −21.9064 38.1078i −0.886239 1.54168i
\(612\) 0 0
\(613\) 24.6208i 0.994426i −0.867628 0.497213i \(-0.834357\pi\)
0.867628 0.497213i \(-0.165643\pi\)
\(614\) 0 0
\(615\) 16.8597 0.679850
\(616\) 0 0
\(617\) 44.8982i 1.80754i 0.428024 + 0.903768i \(0.359210\pi\)
−0.428024 + 0.903768i \(0.640790\pi\)
\(618\) 0 0
\(619\) 10.9600i 0.440520i 0.975441 + 0.220260i \(0.0706906\pi\)
−0.975441 + 0.220260i \(0.929309\pi\)
\(620\) 0 0
\(621\) 34.1513 1.37045
\(622\) 0 0
\(623\) −8.61636 −0.345207
\(624\) 0 0
\(625\) 19.6514 0.786055
\(626\) 0 0
\(627\) 2.22845 0.0889958
\(628\) 0 0
\(629\) 11.1481i 0.444503i
\(630\) 0 0
\(631\) 4.96234i 0.197548i 0.995110 + 0.0987738i \(0.0314920\pi\)
−0.995110 + 0.0987738i \(0.968508\pi\)
\(632\) 0 0
\(633\) −1.84495 −0.0733302
\(634\) 0 0
\(635\) 39.2264i 1.55665i
\(636\) 0 0
\(637\) −3.12587 + 1.79692i −0.123852 + 0.0711966i
\(638\) 0 0
\(639\) 25.7458i 1.01849i
\(640\) 0 0
\(641\) −48.0510 −1.89790 −0.948951 0.315423i \(-0.897854\pi\)
−0.948951 + 0.315423i \(0.897854\pi\)
\(642\) 0 0
\(643\) 13.3346i 0.525864i 0.964814 + 0.262932i \(0.0846895\pi\)
−0.964814 + 0.262932i \(0.915310\pi\)
\(644\) 0 0
\(645\) 17.3448i 0.682952i
\(646\) 0 0
\(647\) −24.5437 −0.964912 −0.482456 0.875920i \(-0.660255\pi\)
−0.482456 + 0.875920i \(0.660255\pi\)
\(648\) 0 0
\(649\) −10.5590 −0.414475
\(650\) 0 0
\(651\) 4.86622 0.190722
\(652\) 0 0
\(653\) 5.18967 0.203088 0.101544 0.994831i \(-0.467622\pi\)
0.101544 + 0.994831i \(0.467622\pi\)
\(654\) 0 0
\(655\) 54.3343i 2.12302i
\(656\) 0 0
\(657\) 21.5915i 0.842363i
\(658\) 0 0
\(659\) −25.2362 −0.983063 −0.491532 0.870860i \(-0.663563\pi\)
−0.491532 + 0.870860i \(0.663563\pi\)
\(660\) 0 0
\(661\) 4.79114i 0.186354i −0.995650 0.0931770i \(-0.970298\pi\)
0.995650 0.0931770i \(-0.0297022\pi\)
\(662\) 0 0
\(663\) 4.83863 2.78151i 0.187917 0.108025i
\(664\) 0 0
\(665\) 10.5220i 0.408025i
\(666\) 0 0
\(667\) 64.2688 2.48850
\(668\) 0 0
\(669\) 12.0816i 0.467100i
\(670\) 0 0
\(671\) 10.1797i 0.392985i
\(672\) 0 0
\(673\) −33.1840 −1.27915 −0.639575 0.768729i \(-0.720890\pi\)
−0.639575 + 0.768729i \(0.720890\pi\)
\(674\) 0 0
\(675\) 41.7116 1.60548
\(676\) 0 0
\(677\) 13.5797 0.521911 0.260955 0.965351i \(-0.415962\pi\)
0.260955 + 0.965351i \(0.415962\pi\)
\(678\) 0 0
\(679\) 1.03847 0.0398528
\(680\) 0 0
\(681\) 17.9122i 0.686396i
\(682\) 0 0
\(683\) 44.7327i 1.71165i 0.517267 + 0.855824i \(0.326949\pi\)
−0.517267 + 0.855824i \(0.673051\pi\)
\(684\) 0 0
\(685\) −13.5975 −0.519534
\(686\) 0 0
\(687\) 0.227210i 0.00866859i
\(688\) 0 0
\(689\) −22.6739 + 13.0342i −0.863806 + 0.496562i
\(690\) 0 0
\(691\) 10.3633i 0.394238i −0.980380 0.197119i \(-0.936841\pi\)
0.980380 0.197119i \(-0.0631585\pi\)
\(692\) 0 0
\(693\) −2.34357 −0.0890248
\(694\) 0 0
\(695\) 26.1110i 0.990445i
\(696\) 0 0
\(697\) 10.3926i 0.393646i
\(698\) 0 0
\(699\) 13.5266 0.511623
\(700\) 0 0
\(701\) 35.6871 1.34788 0.673941 0.738785i \(-0.264600\pi\)
0.673941 + 0.738785i \(0.264600\pi\)
\(702\) 0 0
\(703\) 16.0491 0.605305
\(704\) 0 0
\(705\) 37.7856 1.42309
\(706\) 0 0
\(707\) 16.3268i 0.614033i
\(708\) 0 0
\(709\) 24.8188i 0.932089i −0.884761 0.466045i \(-0.845679\pi\)
0.884761 0.466045i \(-0.154321\pi\)
\(710\) 0 0
\(711\) 14.9420 0.560369
\(712\) 0 0
\(713\) 47.3782i 1.77433i
\(714\) 0 0
\(715\) −11.9581 + 6.87414i −0.447206 + 0.257078i
\(716\) 0 0
\(717\) 19.6496i 0.733826i
\(718\) 0 0
\(719\) 5.03720 0.187856 0.0939279 0.995579i \(-0.470058\pi\)
0.0939279 + 0.995579i \(0.470058\pi\)
\(720\) 0 0
\(721\) 17.4748i 0.650795i
\(722\) 0 0
\(723\) 7.80787i 0.290378i
\(724\) 0 0
\(725\) 78.4962 2.91528
\(726\) 0 0
\(727\) 7.85553 0.291345 0.145673 0.989333i \(-0.453465\pi\)
0.145673 + 0.989333i \(0.453465\pi\)
\(728\) 0 0
\(729\) 2.26666 0.0839504
\(730\) 0 0
\(731\) −10.6916 −0.395442
\(732\) 0 0
\(733\) 50.9180i 1.88070i 0.340208 + 0.940350i \(0.389503\pi\)
−0.340208 + 0.940350i \(0.610497\pi\)
\(734\) 0 0
\(735\) 3.09945i 0.114325i
\(736\) 0 0
\(737\) −1.53230 −0.0564429
\(738\) 0 0
\(739\) 39.2431i 1.44358i −0.692111 0.721791i \(-0.743319\pi\)
0.692111 0.721791i \(-0.256681\pi\)
\(740\) 0 0
\(741\) −4.00435 6.96586i −0.147104 0.255897i
\(742\) 0 0
\(743\) 11.6893i 0.428839i 0.976742 + 0.214419i \(0.0687859\pi\)
−0.976742 + 0.214419i \(0.931214\pi\)
\(744\) 0 0
\(745\) 2.33283 0.0854683
\(746\) 0 0
\(747\) 9.39759i 0.343840i
\(748\) 0 0
\(749\) 8.44728i 0.308657i
\(750\) 0 0
\(751\) −30.9287 −1.12860 −0.564302 0.825568i \(-0.690855\pi\)
−0.564302 + 0.825568i \(0.690855\pi\)
\(752\) 0 0
\(753\) 2.76384 0.100720
\(754\) 0 0
\(755\) 63.4608 2.30957
\(756\) 0 0
\(757\) −8.11518 −0.294951 −0.147476 0.989066i \(-0.547115\pi\)
−0.147476 + 0.989066i \(0.547115\pi\)
\(758\) 0 0
\(759\) 6.39111i 0.231983i
\(760\) 0 0
\(761\) 8.76604i 0.317769i −0.987297 0.158884i \(-0.949210\pi\)
0.987297 0.158884i \(-0.0507897\pi\)
\(762\) 0 0
\(763\) −12.7130 −0.460241
\(764\) 0 0
\(765\) 17.1286i 0.619287i
\(766\) 0 0
\(767\) 18.9736 + 33.0060i 0.685097 + 1.19178i
\(768\) 0 0
\(769\) 1.81454i 0.0654342i −0.999465 0.0327171i \(-0.989584\pi\)
0.999465 0.0327171i \(-0.0104160\pi\)
\(770\) 0 0
\(771\) 14.6360 0.527102
\(772\) 0 0
\(773\) 18.8071i 0.676445i 0.941066 + 0.338223i \(0.109826\pi\)
−0.941066 + 0.338223i \(0.890174\pi\)
\(774\) 0 0
\(775\) 57.8665i 2.07862i
\(776\) 0 0
\(777\) 4.72758 0.169601
\(778\) 0 0
\(779\) −14.9615 −0.536050
\(780\) 0 0
\(781\) −10.9857 −0.393100
\(782\) 0 0
\(783\) 35.2732 1.26056
\(784\) 0 0
\(785\) 76.7713i 2.74008i
\(786\) 0 0
\(787\) 9.49371i 0.338414i 0.985581 + 0.169207i \(0.0541207\pi\)
−0.985581 + 0.169207i \(0.945879\pi\)
\(788\) 0 0
\(789\) 1.25189 0.0445683
\(790\) 0 0
\(791\) 17.9640i 0.638726i
\(792\) 0 0
\(793\) 31.8206 18.2922i 1.12998 0.649575i
\(794\) 0