Properties

Label 4004.2.m.c.2157.20
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 2157.20
Character \(\chi\) = 4004.2157
Dual form 4004.2.m.c.2157.19

$q$-expansion

\(f(q)\) \(=\) \(q\)\(-0.189991 q^{3}\) \(+1.01340i q^{5}\) \(+1.00000i q^{7}\) \(-2.96390 q^{9}\) \(+O(q^{10})\) \(q\)\(-0.189991 q^{3}\) \(+1.01340i q^{5}\) \(+1.00000i q^{7}\) \(-2.96390 q^{9}\) \(-1.00000i q^{11}\) \(+(1.24743 + 3.38289i) q^{13}\) \(-0.192538i q^{15}\) \(+3.81146 q^{17}\) \(+0.826026i q^{19}\) \(-0.189991i q^{21}\) \(+5.31508 q^{23}\) \(+3.97302 q^{25}\) \(+1.13309 q^{27}\) \(-4.80230 q^{29}\) \(+2.20033i q^{31}\) \(+0.189991i q^{33}\) \(-1.01340 q^{35}\) \(+0.485470i q^{37}\) \(+(-0.237001 - 0.642719i) q^{39}\) \(-5.22731i q^{41}\) \(+1.56315 q^{43}\) \(-3.00362i q^{45}\) \(-6.95964i q^{47}\) \(-1.00000 q^{49}\) \(-0.724145 q^{51}\) \(-11.3254 q^{53}\) \(+1.01340 q^{55}\) \(-0.156938i q^{57}\) \(+0.884512i q^{59}\) \(+1.77019 q^{61}\) \(-2.96390i q^{63}\) \(+(-3.42822 + 1.26415i) q^{65}\) \(+14.9710i q^{67}\) \(-1.00982 q^{69}\) \(+9.34192i q^{71}\) \(+4.59317i q^{73}\) \(-0.754839 q^{75}\) \(+1.00000 q^{77}\) \(+3.37613 q^{79}\) \(+8.67643 q^{81}\) \(+13.0825i q^{83}\) \(+3.86254i q^{85}\) \(+0.912394 q^{87}\) \(+12.4177i q^{89}\) \(+(-3.38289 + 1.24743i) q^{91}\) \(-0.418043i q^{93}\) \(-0.837096 q^{95}\) \(-9.93635i q^{97}\) \(+2.96390i q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 80q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 80q^{69} \) \(\mathstrut -\mathstrut 36q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 132q^{81} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
\(3\) −0.189991 −0.109692 −0.0548458 0.998495i \(-0.517467\pi\)
−0.0548458 + 0.998495i \(0.517467\pi\)
\(4\) 0 0
\(5\) 1.01340i 0.453207i 0.973987 + 0.226604i \(0.0727622\pi\)
−0.973987 + 0.226604i \(0.927238\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −2.96390 −0.987968
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) 1.24743 + 3.38289i 0.345976 + 0.938244i
\(14\) 0 0
\(15\) 0.192538i 0.0497130i
\(16\) 0 0
\(17\) 3.81146 0.924416 0.462208 0.886772i \(-0.347057\pi\)
0.462208 + 0.886772i \(0.347057\pi\)
\(18\) 0 0
\(19\) 0.826026i 0.189503i 0.995501 + 0.0947517i \(0.0302057\pi\)
−0.995501 + 0.0947517i \(0.969794\pi\)
\(20\) 0 0
\(21\) 0.189991i 0.0414595i
\(22\) 0 0
\(23\) 5.31508 1.10827 0.554136 0.832426i \(-0.313049\pi\)
0.554136 + 0.832426i \(0.313049\pi\)
\(24\) 0 0
\(25\) 3.97302 0.794603
\(26\) 0 0
\(27\) 1.13309 0.218063
\(28\) 0 0
\(29\) −4.80230 −0.891764 −0.445882 0.895092i \(-0.647110\pi\)
−0.445882 + 0.895092i \(0.647110\pi\)
\(30\) 0 0
\(31\) 2.20033i 0.395190i 0.980284 + 0.197595i \(0.0633132\pi\)
−0.980284 + 0.197595i \(0.936687\pi\)
\(32\) 0 0
\(33\) 0.189991i 0.0330732i
\(34\) 0 0
\(35\) −1.01340 −0.171296
\(36\) 0 0
\(37\) 0.485470i 0.0798108i 0.999203 + 0.0399054i \(0.0127057\pi\)
−0.999203 + 0.0399054i \(0.987294\pi\)
\(38\) 0 0
\(39\) −0.237001 0.642719i −0.0379506 0.102917i
\(40\) 0 0
\(41\) 5.22731i 0.816369i −0.912900 0.408184i \(-0.866162\pi\)
0.912900 0.408184i \(-0.133838\pi\)
\(42\) 0 0
\(43\) 1.56315 0.238378 0.119189 0.992872i \(-0.461971\pi\)
0.119189 + 0.992872i \(0.461971\pi\)
\(44\) 0 0
\(45\) 3.00362i 0.447754i
\(46\) 0 0
\(47\) 6.95964i 1.01517i −0.861603 0.507584i \(-0.830539\pi\)
0.861603 0.507584i \(-0.169461\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −0.724145 −0.101401
\(52\) 0 0
\(53\) −11.3254 −1.55566 −0.777832 0.628473i \(-0.783680\pi\)
−0.777832 + 0.628473i \(0.783680\pi\)
\(54\) 0 0
\(55\) 1.01340 0.136647
\(56\) 0 0
\(57\) 0.156938i 0.0207869i
\(58\) 0 0
\(59\) 0.884512i 0.115154i 0.998341 + 0.0575769i \(0.0183374\pi\)
−0.998341 + 0.0575769i \(0.981663\pi\)
\(60\) 0 0
\(61\) 1.77019 0.226650 0.113325 0.993558i \(-0.463850\pi\)
0.113325 + 0.993558i \(0.463850\pi\)
\(62\) 0 0
\(63\) 2.96390i 0.373417i
\(64\) 0 0
\(65\) −3.42822 + 1.26415i −0.425219 + 0.156799i
\(66\) 0 0
\(67\) 14.9710i 1.82900i 0.404587 + 0.914500i \(0.367415\pi\)
−0.404587 + 0.914500i \(0.632585\pi\)
\(68\) 0 0
\(69\) −1.00982 −0.121568
\(70\) 0 0
\(71\) 9.34192i 1.10868i 0.832289 + 0.554341i \(0.187030\pi\)
−0.832289 + 0.554341i \(0.812970\pi\)
\(72\) 0 0
\(73\) 4.59317i 0.537590i 0.963197 + 0.268795i \(0.0866255\pi\)
−0.963197 + 0.268795i \(0.913375\pi\)
\(74\) 0 0
\(75\) −0.754839 −0.0871613
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 3.37613 0.379844 0.189922 0.981799i \(-0.439176\pi\)
0.189922 + 0.981799i \(0.439176\pi\)
\(80\) 0 0
\(81\) 8.67643 0.964048
\(82\) 0 0
\(83\) 13.0825i 1.43600i 0.696046 + 0.717998i \(0.254941\pi\)
−0.696046 + 0.717998i \(0.745059\pi\)
\(84\) 0 0
\(85\) 3.86254i 0.418952i
\(86\) 0 0
\(87\) 0.912394 0.0978190
\(88\) 0 0
\(89\) 12.4177i 1.31627i 0.752900 + 0.658135i \(0.228654\pi\)
−0.752900 + 0.658135i \(0.771346\pi\)
\(90\) 0 0
\(91\) −3.38289 + 1.24743i −0.354623 + 0.130766i
\(92\) 0 0
\(93\) 0.418043i 0.0433490i
\(94\) 0 0
\(95\) −0.837096 −0.0858842
\(96\) 0 0
\(97\) 9.93635i 1.00888i −0.863446 0.504442i \(-0.831698\pi\)
0.863446 0.504442i \(-0.168302\pi\)
\(98\) 0 0
\(99\) 2.96390i 0.297883i
\(100\) 0 0
\(101\) 1.85017 0.184098 0.0920492 0.995754i \(-0.470658\pi\)
0.0920492 + 0.995754i \(0.470658\pi\)
\(102\) 0 0
\(103\) −4.51317 −0.444696 −0.222348 0.974967i \(-0.571372\pi\)
−0.222348 + 0.974967i \(0.571372\pi\)
\(104\) 0 0
\(105\) 0.192538 0.0187897
\(106\) 0 0
\(107\) −19.4364 −1.87899 −0.939494 0.342565i \(-0.888704\pi\)
−0.939494 + 0.342565i \(0.888704\pi\)
\(108\) 0 0
\(109\) 8.82552i 0.845331i 0.906286 + 0.422666i \(0.138906\pi\)
−0.906286 + 0.422666i \(0.861094\pi\)
\(110\) 0 0
\(111\) 0.0922351i 0.00875457i
\(112\) 0 0
\(113\) −16.2667 −1.53024 −0.765119 0.643889i \(-0.777320\pi\)
−0.765119 + 0.643889i \(0.777320\pi\)
\(114\) 0 0
\(115\) 5.38632i 0.502276i
\(116\) 0 0
\(117\) −3.69727 10.0265i −0.341813 0.926954i
\(118\) 0 0
\(119\) 3.81146i 0.349396i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 0.993143i 0.0895487i
\(124\) 0 0
\(125\) 9.09327i 0.813327i
\(126\) 0 0
\(127\) 10.6870 0.948321 0.474161 0.880438i \(-0.342752\pi\)
0.474161 + 0.880438i \(0.342752\pi\)
\(128\) 0 0
\(129\) −0.296984 −0.0261480
\(130\) 0 0
\(131\) 3.68425 0.321894 0.160947 0.986963i \(-0.448545\pi\)
0.160947 + 0.986963i \(0.448545\pi\)
\(132\) 0 0
\(133\) −0.826026 −0.0716255
\(134\) 0 0
\(135\) 1.14828i 0.0988278i
\(136\) 0 0
\(137\) 6.00305i 0.512875i 0.966561 + 0.256437i \(0.0825488\pi\)
−0.966561 + 0.256437i \(0.917451\pi\)
\(138\) 0 0
\(139\) −17.5839 −1.49144 −0.745722 0.666257i \(-0.767895\pi\)
−0.745722 + 0.666257i \(0.767895\pi\)
\(140\) 0 0
\(141\) 1.32227i 0.111355i
\(142\) 0 0
\(143\) 3.38289 1.24743i 0.282891 0.104316i
\(144\) 0 0
\(145\) 4.86666i 0.404154i
\(146\) 0 0
\(147\) 0.189991 0.0156702
\(148\) 0 0
\(149\) 11.3455i 0.929462i 0.885452 + 0.464731i \(0.153849\pi\)
−0.885452 + 0.464731i \(0.846151\pi\)
\(150\) 0 0
\(151\) 16.2793i 1.32479i 0.749153 + 0.662397i \(0.230461\pi\)
−0.749153 + 0.662397i \(0.769539\pi\)
\(152\) 0 0
\(153\) −11.2968 −0.913293
\(154\) 0 0
\(155\) −2.22982 −0.179103
\(156\) 0 0
\(157\) −0.684650 −0.0546410 −0.0273205 0.999627i \(-0.508697\pi\)
−0.0273205 + 0.999627i \(0.508697\pi\)
\(158\) 0 0
\(159\) 2.15173 0.170643
\(160\) 0 0
\(161\) 5.31508i 0.418887i
\(162\) 0 0
\(163\) 3.92170i 0.307171i −0.988135 0.153585i \(-0.950918\pi\)
0.988135 0.153585i \(-0.0490820\pi\)
\(164\) 0 0
\(165\) −0.192538 −0.0149890
\(166\) 0 0
\(167\) 9.28829i 0.718750i −0.933193 0.359375i \(-0.882990\pi\)
0.933193 0.359375i \(-0.117010\pi\)
\(168\) 0 0
\(169\) −9.88782 + 8.43984i −0.760602 + 0.649219i
\(170\) 0 0
\(171\) 2.44826i 0.187223i
\(172\) 0 0
\(173\) 23.5700 1.79199 0.895997 0.444060i \(-0.146462\pi\)
0.895997 + 0.444060i \(0.146462\pi\)
\(174\) 0 0
\(175\) 3.97302i 0.300332i
\(176\) 0 0
\(177\) 0.168050i 0.0126314i
\(178\) 0 0
\(179\) 22.5364 1.68445 0.842225 0.539126i \(-0.181245\pi\)
0.842225 + 0.539126i \(0.181245\pi\)
\(180\) 0 0
\(181\) 1.38826 0.103188 0.0515941 0.998668i \(-0.483570\pi\)
0.0515941 + 0.998668i \(0.483570\pi\)
\(182\) 0 0
\(183\) −0.336321 −0.0248616
\(184\) 0 0
\(185\) −0.491976 −0.0361708
\(186\) 0 0
\(187\) 3.81146i 0.278722i
\(188\) 0 0
\(189\) 1.13309i 0.0824201i
\(190\) 0 0
\(191\) −7.88487 −0.570529 −0.285265 0.958449i \(-0.592081\pi\)
−0.285265 + 0.958449i \(0.592081\pi\)
\(192\) 0 0
\(193\) 12.7152i 0.915263i −0.889142 0.457632i \(-0.848698\pi\)
0.889142 0.457632i \(-0.151302\pi\)
\(194\) 0 0
\(195\) 0.651332 0.240178i 0.0466429 0.0171995i
\(196\) 0 0
\(197\) 17.3188i 1.23391i −0.786997 0.616956i \(-0.788365\pi\)
0.786997 0.616956i \(-0.211635\pi\)
\(198\) 0 0
\(199\) 3.38792 0.240164 0.120082 0.992764i \(-0.461684\pi\)
0.120082 + 0.992764i \(0.461684\pi\)
\(200\) 0 0
\(201\) 2.84436i 0.200626i
\(202\) 0 0
\(203\) 4.80230i 0.337055i
\(204\) 0 0
\(205\) 5.29736 0.369984
\(206\) 0 0
\(207\) −15.7534 −1.09494
\(208\) 0 0
\(209\) 0.826026 0.0571374
\(210\) 0 0
\(211\) −4.25509 −0.292932 −0.146466 0.989216i \(-0.546790\pi\)
−0.146466 + 0.989216i \(0.546790\pi\)
\(212\) 0 0
\(213\) 1.77488i 0.121613i
\(214\) 0 0
\(215\) 1.58410i 0.108034i
\(216\) 0 0
\(217\) −2.20033 −0.149368
\(218\) 0 0
\(219\) 0.872663i 0.0589691i
\(220\) 0 0
\(221\) 4.75455 + 12.8937i 0.319825 + 0.867327i
\(222\) 0 0
\(223\) 15.5061i 1.03836i 0.854664 + 0.519181i \(0.173763\pi\)
−0.854664 + 0.519181i \(0.826237\pi\)
\(224\) 0 0
\(225\) −11.7756 −0.785043
\(226\) 0 0
\(227\) 0.0576844i 0.00382865i −0.999998 0.00191432i \(-0.999391\pi\)
0.999998 0.00191432i \(-0.000609349\pi\)
\(228\) 0 0
\(229\) 7.20446i 0.476084i 0.971255 + 0.238042i \(0.0765056\pi\)
−0.971255 + 0.238042i \(0.923494\pi\)
\(230\) 0 0
\(231\) −0.189991 −0.0125005
\(232\) 0 0
\(233\) −9.14158 −0.598885 −0.299442 0.954114i \(-0.596801\pi\)
−0.299442 + 0.954114i \(0.596801\pi\)
\(234\) 0 0
\(235\) 7.05291 0.460081
\(236\) 0 0
\(237\) −0.641435 −0.0416657
\(238\) 0 0
\(239\) 12.1085i 0.783231i 0.920129 + 0.391616i \(0.128084\pi\)
−0.920129 + 0.391616i \(0.871916\pi\)
\(240\) 0 0
\(241\) 6.56170i 0.422676i 0.977413 + 0.211338i \(0.0677821\pi\)
−0.977413 + 0.211338i \(0.932218\pi\)
\(242\) 0 0
\(243\) −5.04772 −0.323811
\(244\) 0 0
\(245\) 1.01340i 0.0647439i
\(246\) 0 0
\(247\) −2.79435 + 1.03041i −0.177800 + 0.0655635i
\(248\) 0 0
\(249\) 2.48557i 0.157516i
\(250\) 0 0
\(251\) −29.4185 −1.85688 −0.928441 0.371480i \(-0.878851\pi\)
−0.928441 + 0.371480i \(0.878851\pi\)
\(252\) 0 0
\(253\) 5.31508i 0.334156i
\(254\) 0 0
\(255\) 0.733850i 0.0459555i
\(256\) 0 0
\(257\) 28.6795 1.78898 0.894490 0.447088i \(-0.147539\pi\)
0.894490 + 0.447088i \(0.147539\pi\)
\(258\) 0 0
\(259\) −0.485470 −0.0301656
\(260\) 0 0
\(261\) 14.2335 0.881034
\(262\) 0 0
\(263\) 12.4611 0.768382 0.384191 0.923254i \(-0.374480\pi\)
0.384191 + 0.923254i \(0.374480\pi\)
\(264\) 0 0
\(265\) 11.4772i 0.705038i
\(266\) 0 0
\(267\) 2.35925i 0.144384i
\(268\) 0 0
\(269\) −29.0250 −1.76969 −0.884843 0.465889i \(-0.845735\pi\)
−0.884843 + 0.465889i \(0.845735\pi\)
\(270\) 0 0
\(271\) 29.0682i 1.76577i 0.469594 + 0.882883i \(0.344400\pi\)
−0.469594 + 0.882883i \(0.655600\pi\)
\(272\) 0 0
\(273\) 0.642719 0.237001i 0.0388991 0.0143440i
\(274\) 0 0
\(275\) 3.97302i 0.239582i
\(276\) 0 0
\(277\) 0.638994 0.0383934 0.0191967 0.999816i \(-0.493889\pi\)
0.0191967 + 0.999816i \(0.493889\pi\)
\(278\) 0 0
\(279\) 6.52156i 0.390435i
\(280\) 0 0
\(281\) 18.5484i 1.10650i −0.833014 0.553252i \(-0.813387\pi\)
0.833014 0.553252i \(-0.186613\pi\)
\(282\) 0 0
\(283\) −28.5214 −1.69542 −0.847710 0.530460i \(-0.822019\pi\)
−0.847710 + 0.530460i \(0.822019\pi\)
\(284\) 0 0
\(285\) 0.159041 0.00942077
\(286\) 0 0
\(287\) 5.22731 0.308558
\(288\) 0 0
\(289\) −2.47274 −0.145455
\(290\) 0 0
\(291\) 1.88782i 0.110666i
\(292\) 0 0
\(293\) 16.6740i 0.974105i 0.873373 + 0.487053i \(0.161928\pi\)
−0.873373 + 0.487053i \(0.838072\pi\)
\(294\) 0 0
\(295\) −0.896366 −0.0521885
\(296\) 0 0
\(297\) 1.13309i 0.0657485i
\(298\) 0 0
\(299\) 6.63021 + 17.9803i 0.383435 + 1.03983i
\(300\) 0 0
\(301\) 1.56315i 0.0900983i
\(302\) 0 0
\(303\) −0.351515 −0.0201940
\(304\) 0 0
\(305\) 1.79392i 0.102719i
\(306\) 0 0
\(307\) 5.03711i 0.287483i −0.989615 0.143741i \(-0.954087\pi\)
0.989615 0.143741i \(-0.0459134\pi\)
\(308\) 0 0
\(309\) 0.857463 0.0487793
\(310\) 0 0
\(311\) −4.38097 −0.248422 −0.124211 0.992256i \(-0.539640\pi\)
−0.124211 + 0.992256i \(0.539640\pi\)
\(312\) 0 0
\(313\) 9.18764 0.519316 0.259658 0.965701i \(-0.416390\pi\)
0.259658 + 0.965701i \(0.416390\pi\)
\(314\) 0 0
\(315\) 3.00362 0.169235
\(316\) 0 0
\(317\) 4.05702i 0.227865i 0.993489 + 0.113932i \(0.0363447\pi\)
−0.993489 + 0.113932i \(0.963655\pi\)
\(318\) 0 0
\(319\) 4.80230i 0.268877i
\(320\) 0 0
\(321\) 3.69275 0.206109
\(322\) 0 0
\(323\) 3.14837i 0.175180i
\(324\) 0 0
\(325\) 4.95607 + 13.4403i 0.274913 + 0.745531i
\(326\) 0 0
\(327\) 1.67677i 0.0927257i
\(328\) 0 0
\(329\) 6.95964 0.383697
\(330\) 0 0
\(331\) 25.5449i 1.40408i 0.712139 + 0.702039i \(0.247727\pi\)
−0.712139 + 0.702039i \(0.752273\pi\)
\(332\) 0 0
\(333\) 1.43889i 0.0788505i
\(334\) 0 0
\(335\) −15.1716 −0.828915
\(336\) 0 0
\(337\) −5.71536 −0.311336 −0.155668 0.987809i \(-0.549753\pi\)
−0.155668 + 0.987809i \(0.549753\pi\)
\(338\) 0 0
\(339\) 3.09052 0.167854
\(340\) 0 0
\(341\) 2.20033 0.119154
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 1.02335i 0.0550955i
\(346\) 0 0
\(347\) 11.6658 0.626254 0.313127 0.949711i \(-0.398623\pi\)
0.313127 + 0.949711i \(0.398623\pi\)
\(348\) 0 0
\(349\) 6.69248i 0.358240i −0.983827 0.179120i \(-0.942675\pi\)
0.983827 0.179120i \(-0.0573251\pi\)
\(350\) 0 0
\(351\) 1.41345 + 3.83311i 0.0754445 + 0.204596i
\(352\) 0 0
\(353\) 10.2069i 0.543258i −0.962402 0.271629i \(-0.912438\pi\)
0.962402 0.271629i \(-0.0875623\pi\)
\(354\) 0 0
\(355\) −9.46712 −0.502463
\(356\) 0 0
\(357\) 0.724145i 0.0383258i
\(358\) 0 0
\(359\) 8.62716i 0.455324i −0.973740 0.227662i \(-0.926892\pi\)
0.973740 0.227662i \(-0.0731081\pi\)
\(360\) 0 0
\(361\) 18.3177 0.964088
\(362\) 0 0
\(363\) 0.189991 0.00997196
\(364\) 0 0
\(365\) −4.65473 −0.243640
\(366\) 0 0
\(367\) 12.9414 0.675536 0.337768 0.941229i \(-0.390328\pi\)
0.337768 + 0.941229i \(0.390328\pi\)
\(368\) 0 0
\(369\) 15.4932i 0.806546i
\(370\) 0 0
\(371\) 11.3254i 0.587985i
\(372\) 0 0
\(373\) 24.4202 1.26443 0.632214 0.774793i \(-0.282146\pi\)
0.632214 + 0.774793i \(0.282146\pi\)
\(374\) 0 0
\(375\) 1.72764i 0.0892151i
\(376\) 0 0
\(377\) −5.99054 16.2456i −0.308529 0.836692i
\(378\) 0 0
\(379\) 2.11317i 0.108546i −0.998526 0.0542732i \(-0.982716\pi\)
0.998526 0.0542732i \(-0.0172842\pi\)
\(380\) 0 0
\(381\) −2.03044 −0.104023
\(382\) 0 0
\(383\) 7.92826i 0.405115i 0.979270 + 0.202558i \(0.0649254\pi\)
−0.979270 + 0.202558i \(0.935075\pi\)
\(384\) 0 0
\(385\) 1.01340i 0.0516477i
\(386\) 0 0
\(387\) −4.63302 −0.235509
\(388\) 0 0
\(389\) 6.70958 0.340189 0.170095 0.985428i \(-0.445593\pi\)
0.170095 + 0.985428i \(0.445593\pi\)
\(390\) 0 0
\(391\) 20.2583 1.02450
\(392\) 0 0
\(393\) −0.699975 −0.0353091
\(394\) 0 0
\(395\) 3.42137i 0.172148i
\(396\) 0 0
\(397\) 30.7601i 1.54380i −0.635741 0.771902i \(-0.719305\pi\)
0.635741 0.771902i \(-0.280695\pi\)
\(398\) 0 0
\(399\) 0.156938 0.00785671
\(400\) 0 0
\(401\) 1.24795i 0.0623198i 0.999514 + 0.0311599i \(0.00992011\pi\)
−0.999514 + 0.0311599i \(0.990080\pi\)
\(402\) 0 0
\(403\) −7.44345 + 2.74476i −0.370785 + 0.136726i
\(404\) 0 0
\(405\) 8.79271i 0.436913i
\(406\) 0 0
\(407\) 0.485470 0.0240639
\(408\) 0 0
\(409\) 21.2318i 1.04984i 0.851150 + 0.524922i \(0.175906\pi\)
−0.851150 + 0.524922i \(0.824094\pi\)
\(410\) 0 0
\(411\) 1.14053i 0.0562580i
\(412\) 0 0
\(413\) −0.884512 −0.0435240
\(414\) 0 0
\(415\) −13.2579 −0.650803
\(416\) 0 0
\(417\) 3.34078 0.163599
\(418\) 0 0
\(419\) −3.95797 −0.193360 −0.0966798 0.995316i \(-0.530822\pi\)
−0.0966798 + 0.995316i \(0.530822\pi\)
\(420\) 0 0
\(421\) 8.47820i 0.413202i 0.978425 + 0.206601i \(0.0662402\pi\)
−0.978425 + 0.206601i \(0.933760\pi\)
\(422\) 0 0
\(423\) 20.6277i 1.00295i
\(424\) 0 0
\(425\) 15.1430 0.734544
\(426\) 0 0
\(427\) 1.77019i 0.0856656i
\(428\) 0 0
\(429\) −0.642719 + 0.237001i −0.0310308 + 0.0114425i
\(430\) 0 0
\(431\) 11.9312i 0.574708i 0.957825 + 0.287354i \(0.0927755\pi\)
−0.957825 + 0.287354i \(0.907224\pi\)
\(432\) 0 0
\(433\) −19.6003 −0.941931 −0.470966 0.882152i \(-0.656094\pi\)
−0.470966 + 0.882152i \(0.656094\pi\)
\(434\) 0 0
\(435\) 0.924622i 0.0443322i
\(436\) 0 0
\(437\) 4.39040i 0.210021i
\(438\) 0 0
\(439\) −14.6490 −0.699159 −0.349579 0.936907i \(-0.613675\pi\)
−0.349579 + 0.936907i \(0.613675\pi\)
\(440\) 0 0
\(441\) 2.96390 0.141138
\(442\) 0 0
\(443\) 18.5646 0.882030 0.441015 0.897500i \(-0.354619\pi\)
0.441015 + 0.897500i \(0.354619\pi\)
\(444\) 0 0
\(445\) −12.5841 −0.596542
\(446\) 0 0
\(447\) 2.15555i 0.101954i
\(448\) 0 0
\(449\) 36.3052i 1.71335i −0.515860 0.856673i \(-0.672527\pi\)
0.515860 0.856673i \(-0.327473\pi\)
\(450\) 0 0
\(451\) −5.22731 −0.246144
\(452\) 0 0
\(453\) 3.09293i 0.145319i
\(454\) 0 0
\(455\) −1.26415 3.42822i −0.0592643 0.160718i
\(456\) 0 0
\(457\) 24.9697i 1.16803i 0.811741 + 0.584017i \(0.198520\pi\)
−0.811741 + 0.584017i \(0.801480\pi\)
\(458\) 0 0
\(459\) 4.31873 0.201581
\(460\) 0 0
\(461\) 33.4175i 1.55641i −0.628011 0.778205i \(-0.716131\pi\)
0.628011 0.778205i \(-0.283869\pi\)
\(462\) 0 0
\(463\) 39.3266i 1.82766i 0.406095 + 0.913831i \(0.366890\pi\)
−0.406095 + 0.913831i \(0.633110\pi\)
\(464\) 0 0
\(465\) 0.423645 0.0196461
\(466\) 0 0
\(467\) −0.479382 −0.0221831 −0.0110916 0.999938i \(-0.503531\pi\)
−0.0110916 + 0.999938i \(0.503531\pi\)
\(468\) 0 0
\(469\) −14.9710 −0.691297
\(470\) 0 0
\(471\) 0.130078 0.00599366
\(472\) 0 0
\(473\) 1.56315i 0.0718736i
\(474\) 0 0
\(475\) 3.28181i 0.150580i
\(476\) 0 0
\(477\) 33.5674 1.53695
\(478\) 0 0
\(479\) 4.66956i 0.213357i 0.994294 + 0.106679i \(0.0340216\pi\)
−0.994294 + 0.106679i \(0.965978\pi\)
\(480\) 0 0
\(481\) −1.64229 + 0.605591i −0.0748819 + 0.0276126i
\(482\) 0 0
\(483\) 1.00982i 0.0459484i
\(484\) 0 0
\(485\) 10.0695 0.457233
\(486\) 0 0
\(487\) 1.14144i 0.0517237i −0.999666 0.0258619i \(-0.991767\pi\)
0.999666 0.0258619i \(-0.00823300\pi\)
\(488\) 0 0
\(489\) 0.745088i 0.0336940i
\(490\) 0 0
\(491\) 28.8880 1.30370 0.651849 0.758349i \(-0.273993\pi\)
0.651849 + 0.758349i \(0.273993\pi\)
\(492\) 0 0
\(493\) −18.3038 −0.824361
\(494\) 0 0
\(495\) −3.00362 −0.135003
\(496\) 0 0
\(497\) −9.34192 −0.419043
\(498\) 0 0
\(499\) 42.9111i 1.92096i 0.278340 + 0.960482i \(0.410216\pi\)
−0.278340 + 0.960482i \(0.589784\pi\)
\(500\) 0 0
\(501\) 1.76469i 0.0788407i
\(502\) 0 0
\(503\) −9.00138 −0.401352 −0.200676 0.979658i \(-0.564314\pi\)
−0.200676 + 0.979658i \(0.564314\pi\)
\(504\) 0 0
\(505\) 1.87496i 0.0834347i
\(506\) 0 0
\(507\) 1.87860 1.60350i 0.0834316 0.0712138i
\(508\) 0 0
\(509\) 25.7572i 1.14167i −0.821066 0.570833i \(-0.806620\pi\)
0.821066 0.570833i \(-0.193380\pi\)
\(510\) 0 0
\(511\) −4.59317 −0.203190
\(512\) 0 0
\(513\) 0.935961i 0.0413237i
\(514\) 0 0
\(515\) 4.57365i 0.201539i
\(516\) 0 0
\(517\) −6.95964 −0.306084
\(518\) 0 0
\(519\) −4.47810 −0.196567
\(520\) 0 0
\(521\) −19.6501 −0.860885 −0.430442 0.902618i \(-0.641642\pi\)
−0.430442 + 0.902618i \(0.641642\pi\)
\(522\) 0 0
\(523\) −4.70868 −0.205896 −0.102948 0.994687i \(-0.532828\pi\)
−0.102948 + 0.994687i \(0.532828\pi\)
\(524\) 0 0
\(525\) 0.754839i 0.0329439i
\(526\) 0 0
\(527\) 8.38647i 0.365320i
\(528\) 0 0
\(529\) 5.25012 0.228266
\(530\) 0 0
\(531\) 2.62161i 0.113768i
\(532\) 0 0
\(533\) 17.6834 6.52072i 0.765953 0.282444i
\(534\) 0 0
\(535\) 19.6969i 0.851571i
\(536\) 0 0
\(537\) −4.28172 −0.184770
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 40.9304i 1.75974i −0.475218 0.879868i \(-0.657631\pi\)
0.475218 0.879868i \(-0.342369\pi\)
\(542\) 0 0
\(543\) −0.263756 −0.0113189
\(544\) 0 0
\(545\) −8.94379 −0.383110
\(546\) 0 0
\(547\) 8.47268 0.362265 0.181133 0.983459i \(-0.442024\pi\)
0.181133 + 0.983459i \(0.442024\pi\)
\(548\) 0 0
\(549\) −5.24668 −0.223923
\(550\) 0 0
\(551\) 3.96682i 0.168992i
\(552\) 0 0
\(553\) 3.37613i 0.143568i
\(554\) 0 0
\(555\) 0.0934712 0.00396763
\(556\) 0 0
\(557\) 14.8845i 0.630677i −0.948979 0.315338i \(-0.897882\pi\)
0.948979 0.315338i \(-0.102118\pi\)
\(558\) 0 0
\(559\) 1.94992 + 5.28795i 0.0824729 + 0.223656i
\(560\) 0 0
\(561\) 0.724145i 0.0305734i
\(562\) 0 0
\(563\) −35.5493 −1.49822 −0.749112 0.662443i \(-0.769520\pi\)
−0.749112 + 0.662443i \(0.769520\pi\)
\(564\) 0 0
\(565\) 16.4847i 0.693515i
\(566\) 0 0
\(567\) 8.67643i 0.364376i
\(568\) 0 0
\(569\) 17.5267 0.734758 0.367379 0.930071i \(-0.380255\pi\)
0.367379 + 0.930071i \(0.380255\pi\)
\(570\) 0 0
\(571\) 10.0904 0.422271 0.211136 0.977457i \(-0.432284\pi\)
0.211136 + 0.977457i \(0.432284\pi\)
\(572\) 0 0
\(573\) 1.49806 0.0625822
\(574\) 0 0
\(575\) 21.1169 0.880636
\(576\) 0 0
\(577\) 35.2321i 1.46673i −0.679834 0.733366i \(-0.737948\pi\)
0.679834 0.733366i \(-0.262052\pi\)
\(578\) 0 0
\(579\) 2.41579i 0.100397i
\(580\) 0 0
\(581\) −13.0825 −0.542755
\(582\) 0 0
\(583\) 11.3254i 0.469050i
\(584\) 0 0
\(585\) 10.1609 3.74682i 0.420102 0.154912i
\(586\) 0 0
\(587\) 14.5996i 0.602591i 0.953531 + 0.301296i \(0.0974191\pi\)
−0.953531 + 0.301296i \(0.902581\pi\)
\(588\) 0 0
\(589\) −1.81753 −0.0748899
\(590\) 0 0
\(591\) 3.29042i 0.135350i
\(592\) 0 0
\(593\) 11.2006i 0.459952i −0.973196 0.229976i \(-0.926135\pi\)
0.973196 0.229976i \(-0.0738648\pi\)
\(594\) 0 0
\(595\) −3.86254 −0.158349
\(596\) 0 0
\(597\) −0.643676 −0.0263439
\(598\) 0 0
\(599\) 34.2433 1.39914 0.699571 0.714563i \(-0.253374\pi\)
0.699571 + 0.714563i \(0.253374\pi\)
\(600\) 0 0
\(601\) 30.7394 1.25389 0.626943 0.779065i \(-0.284306\pi\)
0.626943 + 0.779065i \(0.284306\pi\)
\(602\) 0 0
\(603\) 44.3726i 1.80699i
\(604\) 0 0
\(605\) 1.01340i 0.0412006i
\(606\) 0 0
\(607\) −42.1938 −1.71260 −0.856298 0.516483i \(-0.827241\pi\)
−0.856298 + 0.516483i \(0.827241\pi\)
\(608\) 0 0
\(609\) 0.912394i 0.0369721i
\(610\) 0 0
\(611\) 23.5436 8.68168i 0.952474 0.351223i
\(612\) 0 0
\(613\) 20.7355i 0.837500i −0.908102 0.418750i \(-0.862468\pi\)
0.908102 0.418750i \(-0.137532\pi\)
\(614\) 0 0
\(615\) −1.00645 −0.0405841
\(616\) 0 0
\(617\) 4.00510i 0.161239i −0.996745 0.0806197i \(-0.974310\pi\)
0.996745 0.0806197i \(-0.0256899\pi\)
\(618\) 0 0
\(619\) 13.6865i 0.550106i −0.961429 0.275053i \(-0.911305\pi\)
0.961429 0.275053i \(-0.0886954\pi\)
\(620\) 0 0
\(621\) 6.02247 0.241673
\(622\) 0 0
\(623\) −12.4177 −0.497503
\(624\) 0 0
\(625\) 10.6499 0.425998
\(626\) 0 0
\(627\) −0.156938 −0.00626749
\(628\) 0 0
\(629\) 1.85035i 0.0737784i
\(630\) 0 0
\(631\) 9.19621i 0.366095i −0.983104 0.183048i \(-0.941404\pi\)
0.983104 0.183048i \(-0.0585962\pi\)
\(632\) 0 0
\(633\) 0.808429 0.0321322
\(634\) 0 0
\(635\) 10.8303i 0.429786i
\(636\) 0 0
\(637\) −1.24743 3.38289i −0.0494251 0.134035i
\(638\) 0 0
\(639\) 27.6886i 1.09534i
\(640\) 0 0
\(641\) 17.7227 0.700005 0.350002 0.936749i \(-0.386181\pi\)
0.350002 + 0.936749i \(0.386181\pi\)
\(642\) 0 0
\(643\) 28.1531i 1.11025i −0.831767 0.555125i \(-0.812671\pi\)
0.831767 0.555125i \(-0.187329\pi\)
\(644\) 0 0
\(645\) 0.300964i 0.0118505i
\(646\) 0 0
\(647\) 6.66981 0.262217 0.131109 0.991368i \(-0.458146\pi\)
0.131109 + 0.991368i \(0.458146\pi\)
\(648\) 0 0
\(649\) 0.884512 0.0347202
\(650\) 0 0
\(651\) 0.418043 0.0163844
\(652\) 0 0
\(653\) 35.7081 1.39737 0.698683 0.715432i \(-0.253770\pi\)
0.698683 + 0.715432i \(0.253770\pi\)
\(654\) 0 0
\(655\) 3.73362i 0.145885i
\(656\) 0 0
\(657\) 13.6137i 0.531122i
\(658\) 0 0
\(659\) −32.4747 −1.26503 −0.632517 0.774546i \(-0.717978\pi\)
−0.632517 + 0.774546i \(0.717978\pi\)
\(660\) 0 0
\(661\) 47.7693i 1.85801i −0.370068 0.929005i \(-0.620666\pi\)
0.370068 0.929005i \(-0.379334\pi\)
\(662\) 0 0
\(663\) −0.903322 2.44970i −0.0350821 0.0951384i
\(664\) 0 0
\(665\) 0.837096i 0.0324612i
\(666\) 0 0
\(667\) −25.5246 −0.988317
\(668\) 0 0
\(669\) 2.94602i 0.113900i
\(670\) 0 0
\(671\) 1.77019i 0.0683375i
\(672\) 0 0
\(673\) −31.9932 −1.23325 −0.616625 0.787257i \(-0.711500\pi\)
−0.616625 + 0.787257i \(0.711500\pi\)
\(674\) 0 0
\(675\) 4.50178 0.173274
\(676\) 0 0
\(677\) −0.974841 −0.0374662 −0.0187331 0.999825i \(-0.505963\pi\)
−0.0187331 + 0.999825i \(0.505963\pi\)
\(678\) 0 0
\(679\) 9.93635 0.381322
\(680\) 0 0
\(681\) 0.0109595i 0.000419970i
\(682\) 0 0
\(683\) 25.5297i 0.976865i 0.872602 + 0.488433i \(0.162431\pi\)
−0.872602 + 0.488433i \(0.837569\pi\)
\(684\) 0 0
\(685\) −6.08350 −0.232439
\(686\) 0 0
\(687\) 1.36878i 0.0522224i
\(688\) 0 0
\(689\) −14.1277 38.3125i −0.538221 1.45959i
\(690\) 0 0
\(691\) 23.7563i 0.903733i 0.892086 + 0.451866i \(0.149242\pi\)
−0.892086 + 0.451866i \(0.850758\pi\)
\(692\) 0 0
\(693\) −2.96390 −0.112589
\(694\) 0 0
\(695\) 17.8195i 0.675933i
\(696\) 0 0
\(697\) 19.9237i 0.754664i
\(698\) 0 0
\(699\) 1.73682 0.0656926
\(700\) 0 0
\(701\) −10.9239 −0.412589 −0.206294 0.978490i \(-0.566140\pi\)
−0.206294 + 0.978490i \(0.566140\pi\)
\(702\) 0 0
\(703\) −0.401011 −0.0151244
\(704\) 0 0
\(705\) −1.33999 −0.0504670
\(706\) 0 0
\(707\) 1.85017i 0.0695826i
\(708\) 0 0
\(709\) 35.2624i 1.32431i 0.749367 + 0.662154i \(0.230358\pi\)
−0.749367 + 0.662154i \(0.769642\pi\)
\(710\) 0 0
\(711\) −10.0065 −0.375274
\(712\) 0 0
\(713\) 11.6949i 0.437978i
\(714\) 0 0
\(715\) 1.26415 + 3.42822i 0.0472765 + 0.128208i
\(716\) 0 0
\(717\) 2.30050i 0.0859138i
\(718\) 0 0
\(719\) 48.8172 1.82057 0.910287 0.413978i \(-0.135861\pi\)
0.910287 + 0.413978i \(0.135861\pi\)
\(720\) 0 0
\(721\) 4.51317i 0.168079i
\(722\) 0 0
\(723\) 1.24667i 0.0463640i
\(724\) 0 0
\(725\) −19.0796 −0.708599
\(726\) 0 0
\(727\) −19.1217 −0.709185 −0.354592 0.935021i \(-0.615380\pi\)
−0.354592 + 0.935021i \(0.615380\pi\)
\(728\) 0 0
\(729\) −25.0703 −0.928529
\(730\) 0 0
\(731\) 5.95788 0.220360
\(732\) 0 0
\(733\) 36.0576i 1.33182i 0.746033 + 0.665909i \(0.231956\pi\)
−0.746033 + 0.665909i \(0.768044\pi\)
\(734\) 0 0
\(735\) 0.192538i 0.00710185i
\(736\) 0 0
\(737\) 14.9710 0.551464
\(738\) 0 0
\(739\) 15.3941i 0.566280i −0.959079 0.283140i \(-0.908624\pi\)
0.959079 0.283140i \(-0.0913760\pi\)
\(740\) 0 0
\(741\) 0.530902 0.195769i 0.0195032 0.00719176i
\(742\) 0 0
\(743\) 27.1836i 0.997271i −0.866812 0.498635i \(-0.833835\pi\)
0.866812 0.498635i \(-0.166165\pi\)
\(744\) 0 0
\(745\) −11.4976 −0.421239
\(746\) 0 0
\(747\) 38.7754i 1.41872i
\(748\) 0 0
\(749\) 19.4364i 0.710191i
\(750\) 0 0
\(751\) 2.48372 0.0906322 0.0453161 0.998973i \(-0.485571\pi\)
0.0453161 + 0.998973i \(0.485571\pi\)
\(752\) 0 0
\(753\) 5.58927 0.203684
\(754\) 0 0
\(755\) −16.4975 −0.600406
\(756\) 0 0
\(757\) 26.8686 0.976555 0.488278 0.872688i \(-0.337625\pi\)
0.488278 + 0.872688i \(0.337625\pi\)
\(758\) 0 0
\(759\) 1.00982i 0.0366541i
\(760\) 0 0
\(761\) 15.6260i 0.566442i −0.959055 0.283221i \(-0.908597\pi\)
0.959055 0.283221i \(-0.0914030\pi\)
\(762\) 0 0
\(763\) −8.82552 −0.319505
\(764\) 0 0
\(765\) 11.4482i 0.413911i
\(766\) 0 0
\(767\) −2.99220 + 1.10337i −0.108042 + 0.0398404i
\(768\) 0 0
\(769\) 28.8620i 1.04079i −0.853926 0.520395i \(-0.825785\pi\)
0.853926 0.520395i \(-0.174215\pi\)
\(770\) 0 0
\(771\) −5.44886 −0.196236
\(772\) 0 0
\(773\) 31.9189i 1.14804i 0.818840 + 0.574021i \(0.194617\pi\)
−0.818840 + 0.574021i \(0.805383\pi\)
\(774\) 0 0
\(775\) 8.74194i 0.314020i
\(776\) 0 0
\(777\) 0.0922351 0.00330892
\(778\) 0 0
\(779\) 4.31789 0.154705
\(780\) 0 0
\(781\) 9.34192 0.334280
\(782\) 0 0
\(783\) −5.44143 −0.194461
\(784\) 0 0
\(785\) 0.693826i 0.0247637i
\(786\) 0 0
\(787\) 34.1458i 1.21717i 0.793490 + 0.608584i \(0.208262\pi\)
−0.793490 + 0.608584i \(0.791738\pi\)
\(788\) 0 0
\(789\) −2.36749 −0.0842850
\(790\) 0 0
\(791\) 16.2667i 0.578376i
\(792\) 0 0
\(793\) 2.20820 + 5.98836i 0.0784153 + 0.212653i
\(794\)