Properties

Label 4004.2.m.c.2157.27
Level 4004
Weight 2
Character 4004.2157
Analytic conductor 31.972
Analytic rank 0
Dimension 36
CM No

Related objects

Downloads

Learn more about

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 2157.27
Character \(\chi\) = 4004.2157
Dual form 4004.2.m.c.2157.28

$q$-expansion

\(f(q)\) \(=\) \(q\)\(+1.47001 q^{3}\) \(-1.22404i q^{5}\) \(+1.00000i q^{7}\) \(-0.839061 q^{9}\) \(+O(q^{10})\) \(q\)\(+1.47001 q^{3}\) \(-1.22404i q^{5}\) \(+1.00000i q^{7}\) \(-0.839061 q^{9}\) \(-1.00000i q^{11}\) \(+(-3.29392 - 1.46633i) q^{13}\) \(-1.79935i q^{15}\) \(+5.50110 q^{17}\) \(+4.73877i q^{19}\) \(+1.47001i q^{21}\) \(+8.72579 q^{23}\) \(+3.50174 q^{25}\) \(-5.64347 q^{27}\) \(-1.63307 q^{29}\) \(+5.96177i q^{31}\) \(-1.47001i q^{33}\) \(+1.22404 q^{35}\) \(-9.25792i q^{37}\) \(+(-4.84210 - 2.15552i) q^{39}\) \(-6.84298i q^{41}\) \(+5.93685 q^{43}\) \(+1.02704i q^{45}\) \(+10.8026i q^{47}\) \(-1.00000 q^{49}\) \(+8.08669 q^{51}\) \(+3.59354 q^{53}\) \(-1.22404 q^{55}\) \(+6.96605i q^{57}\) \(+2.97310i q^{59}\) \(+3.03907 q^{61}\) \(-0.839061i q^{63}\) \(+(-1.79484 + 4.03187i) q^{65}\) \(-12.8053i q^{67}\) \(+12.8270 q^{69}\) \(-12.0729i q^{71}\) \(+1.51173i q^{73}\) \(+5.14760 q^{75}\) \(+1.00000 q^{77}\) \(+6.58964 q^{79}\) \(-5.77879 q^{81}\) \(-2.25080i q^{83}\) \(-6.73355i q^{85}\) \(-2.40063 q^{87}\) \(-9.64863i q^{89}\) \(+(1.46633 - 3.29392i) q^{91}\) \(+8.76388i q^{93}\) \(+5.80042 q^{95}\) \(-5.06930i q^{97}\) \(+0.839061i q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 80q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 80q^{69} \) \(\mathstrut -\mathstrut 36q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 132q^{81} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
\(3\) 1.47001 0.848713 0.424356 0.905495i \(-0.360500\pi\)
0.424356 + 0.905495i \(0.360500\pi\)
\(4\) 0 0
\(5\) 1.22404i 0.547406i −0.961814 0.273703i \(-0.911752\pi\)
0.961814 0.273703i \(-0.0882485\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −0.839061 −0.279687
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) −3.29392 1.46633i −0.913568 0.406686i
\(14\) 0 0
\(15\) 1.79935i 0.464590i
\(16\) 0 0
\(17\) 5.50110 1.33421 0.667107 0.744962i \(-0.267533\pi\)
0.667107 + 0.744962i \(0.267533\pi\)
\(18\) 0 0
\(19\) 4.73877i 1.08715i 0.839361 + 0.543574i \(0.182929\pi\)
−0.839361 + 0.543574i \(0.817071\pi\)
\(20\) 0 0
\(21\) 1.47001i 0.320783i
\(22\) 0 0
\(23\) 8.72579 1.81945 0.909726 0.415209i \(-0.136292\pi\)
0.909726 + 0.415209i \(0.136292\pi\)
\(24\) 0 0
\(25\) 3.50174 0.700347
\(26\) 0 0
\(27\) −5.64347 −1.08609
\(28\) 0 0
\(29\) −1.63307 −0.303253 −0.151627 0.988438i \(-0.548451\pi\)
−0.151627 + 0.988438i \(0.548451\pi\)
\(30\) 0 0
\(31\) 5.96177i 1.07077i 0.844610 + 0.535383i \(0.179833\pi\)
−0.844610 + 0.535383i \(0.820167\pi\)
\(32\) 0 0
\(33\) 1.47001i 0.255896i
\(34\) 0 0
\(35\) 1.22404 0.206900
\(36\) 0 0
\(37\) 9.25792i 1.52199i −0.648757 0.760996i \(-0.724711\pi\)
0.648757 0.760996i \(-0.275289\pi\)
\(38\) 0 0
\(39\) −4.84210 2.15552i −0.775357 0.345159i
\(40\) 0 0
\(41\) 6.84298i 1.06869i −0.845265 0.534347i \(-0.820558\pi\)
0.845265 0.534347i \(-0.179442\pi\)
\(42\) 0 0
\(43\) 5.93685 0.905362 0.452681 0.891673i \(-0.350468\pi\)
0.452681 + 0.891673i \(0.350468\pi\)
\(44\) 0 0
\(45\) 1.02704i 0.153102i
\(46\) 0 0
\(47\) 10.8026i 1.57572i 0.615852 + 0.787862i \(0.288812\pi\)
−0.615852 + 0.787862i \(0.711188\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 8.08669 1.13236
\(52\) 0 0
\(53\) 3.59354 0.493611 0.246805 0.969065i \(-0.420619\pi\)
0.246805 + 0.969065i \(0.420619\pi\)
\(54\) 0 0
\(55\) −1.22404 −0.165049
\(56\) 0 0
\(57\) 6.96605i 0.922676i
\(58\) 0 0
\(59\) 2.97310i 0.387065i 0.981094 + 0.193532i \(0.0619945\pi\)
−0.981094 + 0.193532i \(0.938006\pi\)
\(60\) 0 0
\(61\) 3.03907 0.389113 0.194556 0.980891i \(-0.437673\pi\)
0.194556 + 0.980891i \(0.437673\pi\)
\(62\) 0 0
\(63\) 0.839061i 0.105712i
\(64\) 0 0
\(65\) −1.79484 + 4.03187i −0.222622 + 0.500092i
\(66\) 0 0
\(67\) 12.8053i 1.56442i −0.623016 0.782209i \(-0.714093\pi\)
0.623016 0.782209i \(-0.285907\pi\)
\(68\) 0 0
\(69\) 12.8270 1.54419
\(70\) 0 0
\(71\) 12.0729i 1.43279i −0.697693 0.716397i \(-0.745790\pi\)
0.697693 0.716397i \(-0.254210\pi\)
\(72\) 0 0
\(73\) 1.51173i 0.176934i 0.996079 + 0.0884671i \(0.0281968\pi\)
−0.996079 + 0.0884671i \(0.971803\pi\)
\(74\) 0 0
\(75\) 5.14760 0.594393
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 6.58964 0.741393 0.370696 0.928754i \(-0.379119\pi\)
0.370696 + 0.928754i \(0.379119\pi\)
\(80\) 0 0
\(81\) −5.77879 −0.642088
\(82\) 0 0
\(83\) 2.25080i 0.247057i −0.992341 0.123529i \(-0.960579\pi\)
0.992341 0.123529i \(-0.0394210\pi\)
\(84\) 0 0
\(85\) 6.73355i 0.730356i
\(86\) 0 0
\(87\) −2.40063 −0.257375
\(88\) 0 0
\(89\) 9.64863i 1.02275i −0.859357 0.511376i \(-0.829136\pi\)
0.859357 0.511376i \(-0.170864\pi\)
\(90\) 0 0
\(91\) 1.46633 3.29392i 0.153713 0.345296i
\(92\) 0 0
\(93\) 8.76388i 0.908772i
\(94\) 0 0
\(95\) 5.80042 0.595111
\(96\) 0 0
\(97\) 5.06930i 0.514709i −0.966317 0.257354i \(-0.917149\pi\)
0.966317 0.257354i \(-0.0828508\pi\)
\(98\) 0 0
\(99\) 0.839061i 0.0843288i
\(100\) 0 0
\(101\) 19.9511 1.98520 0.992602 0.121413i \(-0.0387425\pi\)
0.992602 + 0.121413i \(0.0387425\pi\)
\(102\) 0 0
\(103\) 6.64635 0.654884 0.327442 0.944871i \(-0.393813\pi\)
0.327442 + 0.944871i \(0.393813\pi\)
\(104\) 0 0
\(105\) 1.79935 0.175599
\(106\) 0 0
\(107\) 1.27199 0.122968 0.0614840 0.998108i \(-0.480417\pi\)
0.0614840 + 0.998108i \(0.480417\pi\)
\(108\) 0 0
\(109\) 10.4180i 0.997862i −0.866642 0.498931i \(-0.833726\pi\)
0.866642 0.498931i \(-0.166274\pi\)
\(110\) 0 0
\(111\) 13.6093i 1.29173i
\(112\) 0 0
\(113\) 17.5094 1.64715 0.823573 0.567210i \(-0.191977\pi\)
0.823573 + 0.567210i \(0.191977\pi\)
\(114\) 0 0
\(115\) 10.6807i 0.995978i
\(116\) 0 0
\(117\) 2.76380 + 1.23034i 0.255513 + 0.113745i
\(118\) 0 0
\(119\) 5.50110i 0.504285i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 10.0593i 0.907014i
\(124\) 0 0
\(125\) 10.4064i 0.930780i
\(126\) 0 0
\(127\) −19.8793 −1.76400 −0.882001 0.471247i \(-0.843804\pi\)
−0.882001 + 0.471247i \(0.843804\pi\)
\(128\) 0 0
\(129\) 8.72725 0.768392
\(130\) 0 0
\(131\) 14.1726 1.23826 0.619132 0.785287i \(-0.287484\pi\)
0.619132 + 0.785287i \(0.287484\pi\)
\(132\) 0 0
\(133\) −4.73877 −0.410903
\(134\) 0 0
\(135\) 6.90781i 0.594530i
\(136\) 0 0
\(137\) 7.90227i 0.675136i −0.941301 0.337568i \(-0.890396\pi\)
0.941301 0.337568i \(-0.109604\pi\)
\(138\) 0 0
\(139\) −5.73294 −0.486261 −0.243131 0.969994i \(-0.578174\pi\)
−0.243131 + 0.969994i \(0.578174\pi\)
\(140\) 0 0
\(141\) 15.8800i 1.33734i
\(142\) 0 0
\(143\) −1.46633 + 3.29392i −0.122620 + 0.275451i
\(144\) 0 0
\(145\) 1.99894i 0.166003i
\(146\) 0 0
\(147\) −1.47001 −0.121245
\(148\) 0 0
\(149\) 11.6548i 0.954802i 0.878686 + 0.477401i \(0.158421\pi\)
−0.878686 + 0.477401i \(0.841579\pi\)
\(150\) 0 0
\(151\) 17.0042i 1.38378i 0.722002 + 0.691891i \(0.243222\pi\)
−0.722002 + 0.691891i \(0.756778\pi\)
\(152\) 0 0
\(153\) −4.61576 −0.373162
\(154\) 0 0
\(155\) 7.29742 0.586143
\(156\) 0 0
\(157\) −3.28773 −0.262389 −0.131195 0.991357i \(-0.541881\pi\)
−0.131195 + 0.991357i \(0.541881\pi\)
\(158\) 0 0
\(159\) 5.28255 0.418934
\(160\) 0 0
\(161\) 8.72579i 0.687688i
\(162\) 0 0
\(163\) 6.27858i 0.491776i 0.969298 + 0.245888i \(0.0790796\pi\)
−0.969298 + 0.245888i \(0.920920\pi\)
\(164\) 0 0
\(165\) −1.79935 −0.140079
\(166\) 0 0
\(167\) 10.7574i 0.832431i 0.909266 + 0.416216i \(0.136644\pi\)
−0.909266 + 0.416216i \(0.863356\pi\)
\(168\) 0 0
\(169\) 8.69977 + 9.65992i 0.669213 + 0.743071i
\(170\) 0 0
\(171\) 3.97612i 0.304061i
\(172\) 0 0
\(173\) −6.08041 −0.462285 −0.231143 0.972920i \(-0.574246\pi\)
−0.231143 + 0.972920i \(0.574246\pi\)
\(174\) 0 0
\(175\) 3.50174i 0.264706i
\(176\) 0 0
\(177\) 4.37050i 0.328507i
\(178\) 0 0
\(179\) 19.8805 1.48594 0.742970 0.669325i \(-0.233416\pi\)
0.742970 + 0.669325i \(0.233416\pi\)
\(180\) 0 0
\(181\) 4.71326 0.350334 0.175167 0.984539i \(-0.443954\pi\)
0.175167 + 0.984539i \(0.443954\pi\)
\(182\) 0 0
\(183\) 4.46747 0.330245
\(184\) 0 0
\(185\) −11.3320 −0.833147
\(186\) 0 0
\(187\) 5.50110i 0.402280i
\(188\) 0 0
\(189\) 5.64347i 0.410502i
\(190\) 0 0
\(191\) −12.2115 −0.883590 −0.441795 0.897116i \(-0.645658\pi\)
−0.441795 + 0.897116i \(0.645658\pi\)
\(192\) 0 0
\(193\) 13.8500i 0.996944i −0.866906 0.498472i \(-0.833895\pi\)
0.866906 0.498472i \(-0.166105\pi\)
\(194\) 0 0
\(195\) −2.63843 + 5.92691i −0.188942 + 0.424435i
\(196\) 0 0
\(197\) 8.42331i 0.600136i 0.953918 + 0.300068i \(0.0970094\pi\)
−0.953918 + 0.300068i \(0.902991\pi\)
\(198\) 0 0
\(199\) −23.6650 −1.67756 −0.838782 0.544467i \(-0.816732\pi\)
−0.838782 + 0.544467i \(0.816732\pi\)
\(200\) 0 0
\(201\) 18.8240i 1.32774i
\(202\) 0 0
\(203\) 1.63307i 0.114619i
\(204\) 0 0
\(205\) −8.37605 −0.585009
\(206\) 0 0
\(207\) −7.32147 −0.508877
\(208\) 0 0
\(209\) 4.73877 0.327787
\(210\) 0 0
\(211\) 15.1796 1.04501 0.522503 0.852638i \(-0.324998\pi\)
0.522503 + 0.852638i \(0.324998\pi\)
\(212\) 0 0
\(213\) 17.7474i 1.21603i
\(214\) 0 0
\(215\) 7.26692i 0.495600i
\(216\) 0 0
\(217\) −5.96177 −0.404711
\(218\) 0 0
\(219\) 2.22226i 0.150166i
\(220\) 0 0
\(221\) −18.1202 8.06641i −1.21889 0.542606i
\(222\) 0 0
\(223\) 23.3712i 1.56505i 0.622620 + 0.782524i \(0.286068\pi\)
−0.622620 + 0.782524i \(0.713932\pi\)
\(224\) 0 0
\(225\) −2.93817 −0.195878
\(226\) 0 0
\(227\) 12.8633i 0.853771i 0.904306 + 0.426885i \(0.140389\pi\)
−0.904306 + 0.426885i \(0.859611\pi\)
\(228\) 0 0
\(229\) 21.1385i 1.39687i −0.715673 0.698435i \(-0.753880\pi\)
0.715673 0.698435i \(-0.246120\pi\)
\(230\) 0 0
\(231\) 1.47001 0.0967198
\(232\) 0 0
\(233\) −25.8076 −1.69071 −0.845355 0.534206i \(-0.820611\pi\)
−0.845355 + 0.534206i \(0.820611\pi\)
\(234\) 0 0
\(235\) 13.2228 0.862560
\(236\) 0 0
\(237\) 9.68686 0.629229
\(238\) 0 0
\(239\) 5.78204i 0.374009i −0.982359 0.187004i \(-0.940122\pi\)
0.982359 0.187004i \(-0.0598779\pi\)
\(240\) 0 0
\(241\) 3.45665i 0.222662i −0.993783 0.111331i \(-0.964489\pi\)
0.993783 0.111331i \(-0.0355114\pi\)
\(242\) 0 0
\(243\) 8.43551 0.541138
\(244\) 0 0
\(245\) 1.22404i 0.0782008i
\(246\) 0 0
\(247\) 6.94858 15.6091i 0.442128 0.993183i
\(248\) 0 0
\(249\) 3.30870i 0.209680i
\(250\) 0 0
\(251\) 18.2778 1.15368 0.576842 0.816856i \(-0.304285\pi\)
0.576842 + 0.816856i \(0.304285\pi\)
\(252\) 0 0
\(253\) 8.72579i 0.548585i
\(254\) 0 0
\(255\) 9.89840i 0.619862i
\(256\) 0 0
\(257\) −7.98557 −0.498126 −0.249063 0.968487i \(-0.580123\pi\)
−0.249063 + 0.968487i \(0.580123\pi\)
\(258\) 0 0
\(259\) 9.25792 0.575259
\(260\) 0 0
\(261\) 1.37024 0.0848160
\(262\) 0 0
\(263\) −26.7116 −1.64711 −0.823553 0.567239i \(-0.808011\pi\)
−0.823553 + 0.567239i \(0.808011\pi\)
\(264\) 0 0
\(265\) 4.39862i 0.270205i
\(266\) 0 0
\(267\) 14.1836i 0.868023i
\(268\) 0 0
\(269\) 24.2396 1.47792 0.738959 0.673751i \(-0.235318\pi\)
0.738959 + 0.673751i \(0.235318\pi\)
\(270\) 0 0
\(271\) 26.7339i 1.62397i 0.583680 + 0.811984i \(0.301612\pi\)
−0.583680 + 0.811984i \(0.698388\pi\)
\(272\) 0 0
\(273\) 2.15552 4.84210i 0.130458 0.293057i
\(274\) 0 0
\(275\) 3.50174i 0.211163i
\(276\) 0 0
\(277\) 3.97508 0.238839 0.119420 0.992844i \(-0.461897\pi\)
0.119420 + 0.992844i \(0.461897\pi\)
\(278\) 0 0
\(279\) 5.00229i 0.299479i
\(280\) 0 0
\(281\) 22.9799i 1.37086i 0.728137 + 0.685432i \(0.240386\pi\)
−0.728137 + 0.685432i \(0.759614\pi\)
\(282\) 0 0
\(283\) 14.7483 0.876693 0.438347 0.898806i \(-0.355564\pi\)
0.438347 + 0.898806i \(0.355564\pi\)
\(284\) 0 0
\(285\) 8.52670 0.505078
\(286\) 0 0
\(287\) 6.84298 0.403928
\(288\) 0 0
\(289\) 13.2621 0.780125
\(290\) 0 0
\(291\) 7.45193i 0.436840i
\(292\) 0 0
\(293\) 9.61531i 0.561733i 0.959747 + 0.280866i \(0.0906217\pi\)
−0.959747 + 0.280866i \(0.909378\pi\)
\(294\) 0 0
\(295\) 3.63918 0.211881
\(296\) 0 0
\(297\) 5.64347i 0.327467i
\(298\) 0 0
\(299\) −28.7420 12.7949i −1.66219 0.739946i
\(300\) 0 0
\(301\) 5.93685i 0.342195i
\(302\) 0 0
\(303\) 29.3283 1.68487
\(304\) 0 0
\(305\) 3.71993i 0.213003i
\(306\) 0 0
\(307\) 28.7531i 1.64103i −0.571627 0.820514i \(-0.693687\pi\)
0.571627 0.820514i \(-0.306313\pi\)
\(308\) 0 0
\(309\) 9.77022 0.555808
\(310\) 0 0
\(311\) −13.8294 −0.784196 −0.392098 0.919923i \(-0.628251\pi\)
−0.392098 + 0.919923i \(0.628251\pi\)
\(312\) 0 0
\(313\) −0.104190 −0.00588916 −0.00294458 0.999996i \(-0.500937\pi\)
−0.00294458 + 0.999996i \(0.500937\pi\)
\(314\) 0 0
\(315\) −1.02704 −0.0578672
\(316\) 0 0
\(317\) 1.44379i 0.0810915i −0.999178 0.0405458i \(-0.987090\pi\)
0.999178 0.0405458i \(-0.0129097\pi\)
\(318\) 0 0
\(319\) 1.63307i 0.0914343i
\(320\) 0 0
\(321\) 1.86984 0.104364
\(322\) 0 0
\(323\) 26.0684i 1.45049i
\(324\) 0 0
\(325\) −11.5344 5.13469i −0.639815 0.284821i
\(326\) 0 0
\(327\) 15.3146i 0.846898i
\(328\) 0 0
\(329\) −10.8026 −0.595567
\(330\) 0 0
\(331\) 12.1057i 0.665392i 0.943034 + 0.332696i \(0.107958\pi\)
−0.943034 + 0.332696i \(0.892042\pi\)
\(332\) 0 0
\(333\) 7.76796i 0.425681i
\(334\) 0 0
\(335\) −15.6742 −0.856371
\(336\) 0 0
\(337\) −11.4835 −0.625544 −0.312772 0.949828i \(-0.601258\pi\)
−0.312772 + 0.949828i \(0.601258\pi\)
\(338\) 0 0
\(339\) 25.7391 1.39795
\(340\) 0 0
\(341\) 5.96177 0.322848
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 15.7007i 0.845299i
\(346\) 0 0
\(347\) −29.1793 −1.56643 −0.783213 0.621754i \(-0.786420\pi\)
−0.783213 + 0.621754i \(0.786420\pi\)
\(348\) 0 0
\(349\) 15.1279i 0.809779i 0.914366 + 0.404890i \(0.132690\pi\)
−0.914366 + 0.404890i \(0.867310\pi\)
\(350\) 0 0
\(351\) 18.5891 + 8.27517i 0.992214 + 0.441696i
\(352\) 0 0
\(353\) 1.06327i 0.0565920i 0.999600 + 0.0282960i \(0.00900810\pi\)
−0.999600 + 0.0282960i \(0.990992\pi\)
\(354\) 0 0
\(355\) −14.7777 −0.784319
\(356\) 0 0
\(357\) 8.08669i 0.427993i
\(358\) 0 0
\(359\) 10.4559i 0.551842i 0.961180 + 0.275921i \(0.0889829\pi\)
−0.961180 + 0.275921i \(0.911017\pi\)
\(360\) 0 0
\(361\) −3.45591 −0.181890
\(362\) 0 0
\(363\) −1.47001 −0.0771557
\(364\) 0 0
\(365\) 1.85041 0.0968548
\(366\) 0 0
\(367\) 3.62200 0.189067 0.0945334 0.995522i \(-0.469864\pi\)
0.0945334 + 0.995522i \(0.469864\pi\)
\(368\) 0 0
\(369\) 5.74168i 0.298900i
\(370\) 0 0
\(371\) 3.59354i 0.186567i
\(372\) 0 0
\(373\) −6.87493 −0.355971 −0.177985 0.984033i \(-0.556958\pi\)
−0.177985 + 0.984033i \(0.556958\pi\)
\(374\) 0 0
\(375\) 15.2976i 0.789964i
\(376\) 0 0
\(377\) 5.37919 + 2.39461i 0.277043 + 0.123329i
\(378\) 0 0
\(379\) 11.9962i 0.616201i −0.951354 0.308101i \(-0.900307\pi\)
0.951354 0.308101i \(-0.0996934\pi\)
\(380\) 0 0
\(381\) −29.2228 −1.49713
\(382\) 0 0
\(383\) 37.3087i 1.90639i −0.302359 0.953194i \(-0.597774\pi\)
0.302359 0.953194i \(-0.402226\pi\)
\(384\) 0 0
\(385\) 1.22404i 0.0623827i
\(386\) 0 0
\(387\) −4.98138 −0.253218
\(388\) 0 0
\(389\) −26.4583 −1.34149 −0.670745 0.741688i \(-0.734026\pi\)
−0.670745 + 0.741688i \(0.734026\pi\)
\(390\) 0 0
\(391\) 48.0014 2.42754
\(392\) 0 0
\(393\) 20.8339 1.05093
\(394\) 0 0
\(395\) 8.06596i 0.405843i
\(396\) 0 0
\(397\) 16.8355i 0.844951i 0.906374 + 0.422475i \(0.138839\pi\)
−0.906374 + 0.422475i \(0.861161\pi\)
\(398\) 0 0
\(399\) −6.96605 −0.348739
\(400\) 0 0
\(401\) 7.98670i 0.398837i 0.979914 + 0.199418i \(0.0639053\pi\)
−0.979914 + 0.199418i \(0.936095\pi\)
\(402\) 0 0
\(403\) 8.74190 19.6376i 0.435465 0.978217i
\(404\) 0 0
\(405\) 7.07345i 0.351483i
\(406\) 0 0
\(407\) −9.25792 −0.458898
\(408\) 0 0
\(409\) 22.8324i 1.12899i 0.825436 + 0.564495i \(0.190929\pi\)
−0.825436 + 0.564495i \(0.809071\pi\)
\(410\) 0 0
\(411\) 11.6164i 0.572996i
\(412\) 0 0
\(413\) −2.97310 −0.146297
\(414\) 0 0
\(415\) −2.75506 −0.135240
\(416\) 0 0
\(417\) −8.42749 −0.412696
\(418\) 0 0
\(419\) −7.30919 −0.357077 −0.178539 0.983933i \(-0.557137\pi\)
−0.178539 + 0.983933i \(0.557137\pi\)
\(420\) 0 0
\(421\) 3.92648i 0.191365i −0.995412 0.0956824i \(-0.969497\pi\)
0.995412 0.0956824i \(-0.0305033\pi\)
\(422\) 0 0
\(423\) 9.06405i 0.440709i
\(424\) 0 0
\(425\) 19.2634 0.934412
\(426\) 0 0
\(427\) 3.03907i 0.147071i
\(428\) 0 0
\(429\) −2.15552 + 4.84210i −0.104069 + 0.233779i
\(430\) 0 0
\(431\) 6.73121i 0.324231i 0.986772 + 0.162116i \(0.0518317\pi\)
−0.986772 + 0.162116i \(0.948168\pi\)
\(432\) 0 0
\(433\) −19.2354 −0.924395 −0.462198 0.886777i \(-0.652939\pi\)
−0.462198 + 0.886777i \(0.652939\pi\)
\(434\) 0 0
\(435\) 2.93846i 0.140888i
\(436\) 0 0
\(437\) 41.3495i 1.97801i
\(438\) 0 0
\(439\) 33.7190 1.60932 0.804660 0.593735i \(-0.202347\pi\)
0.804660 + 0.593735i \(0.202347\pi\)
\(440\) 0 0
\(441\) 0.839061 0.0399553
\(442\) 0 0
\(443\) 12.0171 0.570948 0.285474 0.958387i \(-0.407849\pi\)
0.285474 + 0.958387i \(0.407849\pi\)
\(444\) 0 0
\(445\) −11.8103 −0.559861
\(446\) 0 0
\(447\) 17.1328i 0.810352i
\(448\) 0 0
\(449\) 19.5615i 0.923162i 0.887098 + 0.461581i \(0.152718\pi\)
−0.887098 + 0.461581i \(0.847282\pi\)
\(450\) 0 0
\(451\) −6.84298 −0.322223
\(452\) 0 0
\(453\) 24.9964i 1.17443i
\(454\) 0 0
\(455\) −4.03187 1.79484i −0.189017 0.0841433i
\(456\) 0 0
\(457\) 33.4151i 1.56309i −0.623847 0.781547i \(-0.714431\pi\)
0.623847 0.781547i \(-0.285569\pi\)
\(458\) 0 0
\(459\) −31.0453 −1.44907
\(460\) 0 0
\(461\) 16.6872i 0.777201i −0.921406 0.388601i \(-0.872959\pi\)
0.921406 0.388601i \(-0.127041\pi\)
\(462\) 0 0
\(463\) 21.9389i 1.01959i 0.860297 + 0.509793i \(0.170278\pi\)
−0.860297 + 0.509793i \(0.829722\pi\)
\(464\) 0 0
\(465\) 10.7273 0.497467
\(466\) 0 0
\(467\) −9.82381 −0.454591 −0.227296 0.973826i \(-0.572988\pi\)
−0.227296 + 0.973826i \(0.572988\pi\)
\(468\) 0 0
\(469\) 12.8053 0.591294
\(470\) 0 0
\(471\) −4.83300 −0.222693
\(472\) 0 0
\(473\) 5.93685i 0.272977i
\(474\) 0 0
\(475\) 16.5939i 0.761381i
\(476\) 0 0
\(477\) −3.01520 −0.138057
\(478\) 0 0
\(479\) 15.6626i 0.715641i 0.933790 + 0.357820i \(0.116480\pi\)
−0.933790 + 0.357820i \(0.883520\pi\)
\(480\) 0 0
\(481\) −13.5751 + 30.4948i −0.618973 + 1.39044i
\(482\) 0 0
\(483\) 12.8270i 0.583650i
\(484\) 0 0
\(485\) −6.20500 −0.281755
\(486\) 0 0
\(487\) 5.04650i 0.228679i 0.993442 + 0.114339i \(0.0364751\pi\)
−0.993442 + 0.114339i \(0.963525\pi\)
\(488\) 0 0
\(489\) 9.22959i 0.417377i
\(490\) 0 0
\(491\) 0.766239 0.0345799 0.0172899 0.999851i \(-0.494496\pi\)
0.0172899 + 0.999851i \(0.494496\pi\)
\(492\) 0 0
\(493\) −8.98368 −0.404605
\(494\) 0 0
\(495\) 1.02704 0.0461621
\(496\) 0 0
\(497\) 12.0729 0.541545
\(498\) 0 0
\(499\) 15.0008i 0.671526i 0.941946 + 0.335763i \(0.108994\pi\)
−0.941946 + 0.335763i \(0.891006\pi\)
\(500\) 0 0
\(501\) 15.8135i 0.706495i
\(502\) 0 0
\(503\) 3.97436 0.177208 0.0886039 0.996067i \(-0.471759\pi\)
0.0886039 + 0.996067i \(0.471759\pi\)
\(504\) 0 0
\(505\) 24.4208i 1.08671i
\(506\) 0 0
\(507\) 12.7888 + 14.2002i 0.567970 + 0.630653i
\(508\) 0 0
\(509\) 14.8919i 0.660070i −0.943969 0.330035i \(-0.892939\pi\)
0.943969 0.330035i \(-0.107061\pi\)
\(510\) 0 0
\(511\) −1.51173 −0.0668748
\(512\) 0 0
\(513\) 26.7431i 1.18074i
\(514\) 0 0
\(515\) 8.13537i 0.358487i
\(516\) 0 0
\(517\) 10.8026 0.475098
\(518\) 0 0
\(519\) −8.93828 −0.392347
\(520\) 0 0
\(521\) −36.7258 −1.60898 −0.804492 0.593963i \(-0.797562\pi\)
−0.804492 + 0.593963i \(0.797562\pi\)
\(522\) 0 0
\(523\) 19.8183 0.866595 0.433297 0.901251i \(-0.357350\pi\)
0.433297 + 0.901251i \(0.357350\pi\)
\(524\) 0 0
\(525\) 5.14760i 0.224660i
\(526\) 0 0
\(527\) 32.7963i 1.42863i
\(528\) 0 0
\(529\) 53.1393 2.31041
\(530\) 0 0
\(531\) 2.49461i 0.108257i
\(532\) 0 0
\(533\) −10.0340 + 22.5402i −0.434623 + 0.976325i
\(534\) 0 0
\(535\) 1.55696i 0.0673134i
\(536\) 0 0
\(537\) 29.2246 1.26114
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 5.15187i 0.221496i −0.993849 0.110748i \(-0.964675\pi\)
0.993849 0.110748i \(-0.0353247\pi\)
\(542\) 0 0
\(543\) 6.92855 0.297333
\(544\) 0 0
\(545\) −12.7520 −0.546235
\(546\) 0 0
\(547\) 24.8419 1.06216 0.531082 0.847321i \(-0.321786\pi\)
0.531082 + 0.847321i \(0.321786\pi\)
\(548\) 0 0
\(549\) −2.54996 −0.108830
\(550\) 0 0
\(551\) 7.73873i 0.329681i
\(552\) 0 0
\(553\) 6.58964i 0.280220i
\(554\) 0 0
\(555\) −16.6582 −0.707102
\(556\) 0 0
\(557\) 20.5865i 0.872279i −0.899879 0.436140i \(-0.856345\pi\)
0.899879 0.436140i \(-0.143655\pi\)
\(558\) 0 0
\(559\) −19.5555 8.70537i −0.827109 0.368198i
\(560\) 0 0
\(561\) 8.08669i 0.341420i
\(562\) 0 0
\(563\) −25.8005 −1.08736 −0.543681 0.839292i \(-0.682970\pi\)
−0.543681 + 0.839292i \(0.682970\pi\)
\(564\) 0 0
\(565\) 21.4321i 0.901657i
\(566\) 0 0
\(567\) 5.77879i 0.242686i
\(568\) 0 0
\(569\) 14.4320 0.605019 0.302510 0.953146i \(-0.402176\pi\)
0.302510 + 0.953146i \(0.402176\pi\)
\(570\) 0 0
\(571\) −4.42797 −0.185305 −0.0926525 0.995699i \(-0.529535\pi\)
−0.0926525 + 0.995699i \(0.529535\pi\)
\(572\) 0 0
\(573\) −17.9510 −0.749914
\(574\) 0 0
\(575\) 30.5554 1.27425
\(576\) 0 0
\(577\) 9.57811i 0.398742i −0.979924 0.199371i \(-0.936110\pi\)
0.979924 0.199371i \(-0.0638899\pi\)
\(578\) 0 0
\(579\) 20.3597i 0.846119i
\(580\) 0 0
\(581\) 2.25080 0.0933788
\(582\) 0 0
\(583\) 3.59354i 0.148829i
\(584\) 0 0
\(585\) 1.50598 3.38299i 0.0622645 0.139869i
\(586\) 0 0
\(587\) 3.66683i 0.151346i 0.997133 + 0.0756730i \(0.0241105\pi\)
−0.997133 + 0.0756730i \(0.975889\pi\)
\(588\) 0 0
\(589\) −28.2514 −1.16408
\(590\) 0 0
\(591\) 12.3824i 0.509343i
\(592\) 0 0
\(593\) 24.4602i 1.00446i 0.864734 + 0.502229i \(0.167487\pi\)
−0.864734 + 0.502229i \(0.832513\pi\)
\(594\) 0 0
\(595\) 6.73355 0.276049
\(596\) 0 0
\(597\) −34.7878 −1.42377
\(598\) 0 0
\(599\) 13.5442 0.553400 0.276700 0.960956i \(-0.410759\pi\)
0.276700 + 0.960956i \(0.410759\pi\)
\(600\) 0 0
\(601\) 29.5453 1.20518 0.602589 0.798051i \(-0.294136\pi\)
0.602589 + 0.798051i \(0.294136\pi\)
\(602\) 0 0
\(603\) 10.7444i 0.437547i
\(604\) 0 0
\(605\) 1.22404i 0.0497641i
\(606\) 0 0
\(607\) 18.4428 0.748571 0.374286 0.927313i \(-0.377888\pi\)
0.374286 + 0.927313i \(0.377888\pi\)
\(608\) 0 0
\(609\) 2.40063i 0.0972786i
\(610\) 0 0
\(611\) 15.8402 35.5829i 0.640824 1.43953i
\(612\) 0 0
\(613\) 0.177819i 0.00718204i −0.999994 0.00359102i \(-0.998857\pi\)
0.999994 0.00359102i \(-0.00114306\pi\)
\(614\) 0 0
\(615\) −12.3129 −0.496504
\(616\) 0 0
\(617\) 22.4751i 0.904815i 0.891811 + 0.452407i \(0.149435\pi\)
−0.891811 + 0.452407i \(0.850565\pi\)
\(618\) 0 0
\(619\) 21.2262i 0.853155i −0.904451 0.426577i \(-0.859719\pi\)
0.904451 0.426577i \(-0.140281\pi\)
\(620\) 0 0
\(621\) −49.2437 −1.97608
\(622\) 0 0
\(623\) 9.64863 0.386564
\(624\) 0 0
\(625\) 4.77083 0.190833
\(626\) 0 0
\(627\) 6.96605 0.278197
\(628\) 0 0
\(629\) 50.9287i 2.03066i
\(630\) 0 0
\(631\) 44.3687i 1.76629i −0.469102 0.883144i \(-0.655422\pi\)
0.469102 0.883144i \(-0.344578\pi\)
\(632\) 0 0
\(633\) 22.3142 0.886909
\(634\) 0 0
\(635\) 24.3330i 0.965625i
\(636\) 0 0
\(637\) 3.29392 + 1.46633i 0.130510 + 0.0580980i
\(638\) 0 0
\(639\) 10.1299i 0.400734i
\(640\) 0 0
\(641\) −41.6766 −1.64613 −0.823064 0.567949i \(-0.807737\pi\)
−0.823064 + 0.567949i \(0.807737\pi\)
\(642\) 0 0
\(643\) 4.67552i 0.184385i −0.995741 0.0921923i \(-0.970613\pi\)
0.995741 0.0921923i \(-0.0293875\pi\)
\(644\) 0 0
\(645\) 10.6825i 0.420622i
\(646\) 0 0
\(647\) 10.0175 0.393829 0.196915 0.980421i \(-0.436908\pi\)
0.196915 + 0.980421i \(0.436908\pi\)
\(648\) 0 0
\(649\) 2.97310 0.116704
\(650\) 0 0
\(651\) −8.76388 −0.343483
\(652\) 0 0
\(653\) −27.6735 −1.08295 −0.541473 0.840718i \(-0.682133\pi\)
−0.541473 + 0.840718i \(0.682133\pi\)
\(654\) 0 0
\(655\) 17.3478i 0.677833i
\(656\) 0 0
\(657\) 1.26843i 0.0494862i
\(658\) 0 0
\(659\) −2.69784 −0.105093 −0.0525465 0.998618i \(-0.516734\pi\)
−0.0525465 + 0.998618i \(0.516734\pi\)
\(660\) 0 0
\(661\) 18.0359i 0.701515i 0.936466 + 0.350758i \(0.114076\pi\)
−0.936466 + 0.350758i \(0.885924\pi\)
\(662\) 0 0
\(663\) −26.6369 11.8577i −1.03449 0.460516i
\(664\) 0 0
\(665\) 5.80042i 0.224931i
\(666\) 0 0
\(667\) −14.2498 −0.551755
\(668\) 0 0
\(669\) 34.3559i 1.32828i
\(670\) 0 0
\(671\) 3.03907i 0.117322i
\(672\) 0 0
\(673\) 12.0042 0.462728 0.231364 0.972867i \(-0.425681\pi\)
0.231364 + 0.972867i \(0.425681\pi\)
\(674\) 0 0
\(675\) −19.7619 −0.760637
\(676\) 0 0
\(677\) −6.77083 −0.260224 −0.130112 0.991499i \(-0.541534\pi\)
−0.130112 + 0.991499i \(0.541534\pi\)
\(678\) 0 0
\(679\) 5.06930 0.194542
\(680\) 0 0
\(681\) 18.9093i 0.724606i
\(682\) 0 0
\(683\) 19.3666i 0.741042i −0.928824 0.370521i \(-0.879179\pi\)
0.928824 0.370521i \(-0.120821\pi\)
\(684\) 0 0
\(685\) −9.67266 −0.369573
\(686\) 0 0
\(687\) 31.0739i 1.18554i
\(688\) 0 0
\(689\) −11.8368 5.26931i −0.450947 0.200745i
\(690\) 0 0
\(691\) 21.5895i 0.821304i −0.911792 0.410652i \(-0.865301\pi\)
0.911792 0.410652i \(-0.134699\pi\)
\(692\) 0 0
\(693\) −0.839061 −0.0318733
\(694\) 0 0
\(695\) 7.01732i 0.266182i
\(696\) 0 0
\(697\) 37.6439i 1.42587i
\(698\) 0 0
\(699\) −37.9374 −1.43493
\(700\) 0 0
\(701\) −28.5044 −1.07660 −0.538299 0.842754i \(-0.680933\pi\)
−0.538299 + 0.842754i \(0.680933\pi\)
\(702\) 0 0
\(703\) 43.8711 1.65463
\(704\) 0 0
\(705\) 19.4377 0.732065
\(706\) 0 0
\(707\) 19.9511i 0.750337i
\(708\) 0 0
\(709\) 28.9901i 1.08875i −0.838843 0.544373i \(-0.816767\pi\)
0.838843 0.544373i \(-0.183233\pi\)
\(710\) 0 0
\(711\) −5.52911 −0.207358
\(712\) 0 0
\(713\) 52.0211i 1.94821i
\(714\) 0 0
\(715\) 4.03187 + 1.79484i 0.150783 + 0.0671231i
\(716\) 0 0
\(717\) 8.49967i 0.317426i
\(718\) 0 0
\(719\) −36.9074 −1.37641 −0.688206 0.725515i \(-0.741602\pi\)
−0.688206 + 0.725515i \(0.741602\pi\)
\(720\) 0 0
\(721\) 6.64635i 0.247523i
\(722\) 0 0
\(723\) 5.08132i 0.188976i
\(724\) 0 0
\(725\) −5.71858 −0.212383
\(726\) 0 0
\(727\) −1.96710 −0.0729555 −0.0364778 0.999334i \(-0.511614\pi\)
−0.0364778 + 0.999334i \(0.511614\pi\)
\(728\) 0 0
\(729\) 29.7367 1.10136
\(730\) 0 0
\(731\) 32.6592 1.20795
\(732\) 0 0
\(733\) 52.4302i 1.93655i 0.249883 + 0.968276i \(0.419608\pi\)
−0.249883 + 0.968276i \(0.580392\pi\)
\(734\) 0 0
\(735\) 1.79935i 0.0663700i
\(736\) 0 0
\(737\) −12.8053 −0.471690
\(738\) 0 0
\(739\) 47.7388i 1.75610i −0.478569 0.878050i \(-0.658844\pi\)
0.478569 0.878050i \(-0.341156\pi\)
\(740\) 0 0
\(741\) 10.2145 22.9456i 0.375239 0.842927i
\(742\) 0 0
\(743\) 14.1915i 0.520635i 0.965523 + 0.260317i \(0.0838272\pi\)
−0.965523 + 0.260317i \(0.916173\pi\)
\(744\) 0 0
\(745\) 14.2659 0.522664
\(746\) 0 0
\(747\) 1.88856i 0.0690986i
\(748\) 0 0
\(749\) 1.27199i 0.0464775i
\(750\) 0 0
\(751\) −25.1886 −0.919144 −0.459572 0.888141i \(-0.651997\pi\)
−0.459572 + 0.888141i \(0.651997\pi\)
\(752\) 0 0
\(753\) 26.8686 0.979146
\(754\) 0 0
\(755\) 20.8138 0.757491
\(756\) 0 0
\(757\) −45.9404 −1.66973 −0.834867 0.550452i \(-0.814455\pi\)
−0.834867 + 0.550452i \(0.814455\pi\)
\(758\) 0 0
\(759\) 12.8270i 0.465591i
\(760\) 0 0
\(761\) 16.6725i 0.604379i 0.953248 + 0.302189i \(0.0977175\pi\)
−0.953248 + 0.302189i \(0.902282\pi\)
\(762\) 0 0
\(763\) 10.4180 0.377156
\(764\) 0 0
\(765\) 5.64986i 0.204271i
\(766\) 0 0
\(767\) 4.35954 9.79315i 0.157414 0.353610i
\(768\) 0 0
\(769\) 1.35079i 0.0487108i 0.999703 + 0.0243554i \(0.00775334\pi\)
−0.999703 + 0.0243554i \(0.992247\pi\)
\(770\) 0 0
\(771\) −11.7389 −0.422766
\(772\) 0 0
\(773\) 37.8255i 1.36049i 0.732986 + 0.680243i \(0.238126\pi\)
−0.732986 + 0.680243i \(0.761874\pi\)
\(774\) 0 0
\(775\) 20.8765i 0.749907i
\(776\) 0 0
\(777\) 13.6093 0.488229
\(778\) 0 0
\(779\) 32.4273 1.16183
\(780\) 0 0
\(781\) −12.0729 −0.432004
\(782\) 0 0
\(783\) 9.21618 0.329359
\(784\) 0 0
\(785\) 4.02430i 0.143633i
\(786\) 0 0
\(787\) 39.9390i 1.42367i −0.702345 0.711837i \(-0.747864\pi\)
0.702345 0.711837i \(-0.252136\pi\)
\(788\) 0 0
\(789\) −39.2664 −1.39792
\(790\) 0 0
\(791\) 17.5094i 0.622563i
\(792\) 0 0
\(793\) −10.0104 4.45627i −0.355481 0.158247i
\(794\) 0