Properties

Label 4012.2.b.b.237.14
Level 4012
Weight 2
Character 4012.237
Analytic conductor 32.036
Analytic rank 0
Dimension 46
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(46\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 237.14
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.33

$q$-expansion

\(f(q)\) \(=\) \(q-1.50352i q^{3} +4.31120i q^{5} -4.93216i q^{7} +0.739426 q^{9} +O(q^{10})\) \(q-1.50352i q^{3} +4.31120i q^{5} -4.93216i q^{7} +0.739426 q^{9} +1.58263i q^{11} -0.143694 q^{13} +6.48198 q^{15} +(-0.180560 + 4.11915i) q^{17} +0.296737 q^{19} -7.41561 q^{21} +6.24916i q^{23} -13.5864 q^{25} -5.62230i q^{27} +6.17923i q^{29} -4.07702i q^{31} +2.37952 q^{33} +21.2635 q^{35} +7.76874i q^{37} +0.216046i q^{39} +1.59168i q^{41} -6.67696 q^{43} +3.18781i q^{45} -11.8958 q^{47} -17.3262 q^{49} +(6.19323 + 0.271476i) q^{51} +6.38790 q^{53} -6.82305 q^{55} -0.446151i q^{57} +1.00000 q^{59} -0.241773i q^{61} -3.64697i q^{63} -0.619492i q^{65} +12.4554 q^{67} +9.39574 q^{69} +1.01095i q^{71} +0.169624i q^{73} +20.4275i q^{75} +7.80581 q^{77} -11.2445i q^{79} -6.23497 q^{81} -5.97834 q^{83} +(-17.7585 - 0.778431i) q^{85} +9.29059 q^{87} +3.73469 q^{89} +0.708721i q^{91} -6.12989 q^{93} +1.27929i q^{95} +4.04072i q^{97} +1.17024i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times\).

\(n\) \(2007\) \(3129\) \(3777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 1.50352i 0.868058i −0.900899 0.434029i \(-0.857092\pi\)
0.900899 0.434029i \(-0.142908\pi\)
\(4\) 0 0
\(5\) 4.31120i 1.92803i 0.265854 + 0.964013i \(0.414346\pi\)
−0.265854 + 0.964013i \(0.585654\pi\)
\(6\) 0 0
\(7\) 4.93216i 1.86418i −0.362223 0.932091i \(-0.617982\pi\)
0.362223 0.932091i \(-0.382018\pi\)
\(8\) 0 0
\(9\) 0.739426 0.246475
\(10\) 0 0
\(11\) 1.58263i 0.477182i 0.971120 + 0.238591i \(0.0766856\pi\)
−0.971120 + 0.238591i \(0.923314\pi\)
\(12\) 0 0
\(13\) −0.143694 −0.0398535 −0.0199267 0.999801i \(-0.506343\pi\)
−0.0199267 + 0.999801i \(0.506343\pi\)
\(14\) 0 0
\(15\) 6.48198 1.67364
\(16\) 0 0
\(17\) −0.180560 + 4.11915i −0.0437923 + 0.999041i
\(18\) 0 0
\(19\) 0.296737 0.0680762 0.0340381 0.999421i \(-0.489163\pi\)
0.0340381 + 0.999421i \(0.489163\pi\)
\(20\) 0 0
\(21\) −7.41561 −1.61822
\(22\) 0 0
\(23\) 6.24916i 1.30304i 0.758631 + 0.651520i \(0.225868\pi\)
−0.758631 + 0.651520i \(0.774132\pi\)
\(24\) 0 0
\(25\) −13.5864 −2.71729
\(26\) 0 0
\(27\) 5.62230i 1.08201i
\(28\) 0 0
\(29\) 6.17923i 1.14745i 0.819047 + 0.573727i \(0.194503\pi\)
−0.819047 + 0.573727i \(0.805497\pi\)
\(30\) 0 0
\(31\) 4.07702i 0.732255i −0.930565 0.366128i \(-0.880683\pi\)
0.930565 0.366128i \(-0.119317\pi\)
\(32\) 0 0
\(33\) 2.37952 0.414222
\(34\) 0 0
\(35\) 21.2635 3.59419
\(36\) 0 0
\(37\) 7.76874i 1.27717i 0.769550 + 0.638586i \(0.220480\pi\)
−0.769550 + 0.638586i \(0.779520\pi\)
\(38\) 0 0
\(39\) 0.216046i 0.0345951i
\(40\) 0 0
\(41\) 1.59168i 0.248579i 0.992246 + 0.124290i \(0.0396652\pi\)
−0.992246 + 0.124290i \(0.960335\pi\)
\(42\) 0 0
\(43\) −6.67696 −1.01823 −0.509113 0.860700i \(-0.670027\pi\)
−0.509113 + 0.860700i \(0.670027\pi\)
\(44\) 0 0
\(45\) 3.18781i 0.475211i
\(46\) 0 0
\(47\) −11.8958 −1.73518 −0.867592 0.497276i \(-0.834334\pi\)
−0.867592 + 0.497276i \(0.834334\pi\)
\(48\) 0 0
\(49\) −17.3262 −2.47518
\(50\) 0 0
\(51\) 6.19323 + 0.271476i 0.867225 + 0.0380142i
\(52\) 0 0
\(53\) 6.38790 0.877445 0.438722 0.898623i \(-0.355431\pi\)
0.438722 + 0.898623i \(0.355431\pi\)
\(54\) 0 0
\(55\) −6.82305 −0.920020
\(56\) 0 0
\(57\) 0.446151i 0.0590941i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 0.241773i 0.0309558i −0.999880 0.0154779i \(-0.995073\pi\)
0.999880 0.0154779i \(-0.00492697\pi\)
\(62\) 0 0
\(63\) 3.64697i 0.459475i
\(64\) 0 0
\(65\) 0.619492i 0.0768386i
\(66\) 0 0
\(67\) 12.4554 1.52167 0.760833 0.648948i \(-0.224791\pi\)
0.760833 + 0.648948i \(0.224791\pi\)
\(68\) 0 0
\(69\) 9.39574 1.13111
\(70\) 0 0
\(71\) 1.01095i 0.119977i 0.998199 + 0.0599886i \(0.0191064\pi\)
−0.998199 + 0.0599886i \(0.980894\pi\)
\(72\) 0 0
\(73\) 0.169624i 0.0198530i 0.999951 + 0.00992651i \(0.00315976\pi\)
−0.999951 + 0.00992651i \(0.996840\pi\)
\(74\) 0 0
\(75\) 20.4275i 2.35876i
\(76\) 0 0
\(77\) 7.80581 0.889555
\(78\) 0 0
\(79\) 11.2445i 1.26510i −0.774518 0.632551i \(-0.782008\pi\)
0.774518 0.632551i \(-0.217992\pi\)
\(80\) 0 0
\(81\) −6.23497 −0.692774
\(82\) 0 0
\(83\) −5.97834 −0.656208 −0.328104 0.944642i \(-0.606410\pi\)
−0.328104 + 0.944642i \(0.606410\pi\)
\(84\) 0 0
\(85\) −17.7585 0.778431i −1.92618 0.0844327i
\(86\) 0 0
\(87\) 9.29059 0.996056
\(88\) 0 0
\(89\) 3.73469 0.395877 0.197938 0.980214i \(-0.436575\pi\)
0.197938 + 0.980214i \(0.436575\pi\)
\(90\) 0 0
\(91\) 0.708721i 0.0742942i
\(92\) 0 0
\(93\) −6.12989 −0.635640
\(94\) 0 0
\(95\) 1.27929i 0.131253i
\(96\) 0 0
\(97\) 4.04072i 0.410273i 0.978733 + 0.205136i \(0.0657638\pi\)
−0.978733 + 0.205136i \(0.934236\pi\)
\(98\) 0 0
\(99\) 1.17024i 0.117614i
\(100\) 0 0
\(101\) −7.71033 −0.767206 −0.383603 0.923498i \(-0.625317\pi\)
−0.383603 + 0.923498i \(0.625317\pi\)
\(102\) 0 0
\(103\) 15.0201 1.47998 0.739988 0.672620i \(-0.234831\pi\)
0.739988 + 0.672620i \(0.234831\pi\)
\(104\) 0 0
\(105\) 31.9702i 3.11997i
\(106\) 0 0
\(107\) 9.34094i 0.903023i 0.892265 + 0.451512i \(0.149115\pi\)
−0.892265 + 0.451512i \(0.850885\pi\)
\(108\) 0 0
\(109\) 1.42532i 0.136521i 0.997668 + 0.0682605i \(0.0217449\pi\)
−0.997668 + 0.0682605i \(0.978255\pi\)
\(110\) 0 0
\(111\) 11.6805 1.10866
\(112\) 0 0
\(113\) 9.73563i 0.915851i 0.888991 + 0.457925i \(0.151407\pi\)
−0.888991 + 0.457925i \(0.848593\pi\)
\(114\) 0 0
\(115\) −26.9414 −2.51230
\(116\) 0 0
\(117\) −0.106251 −0.00982290
\(118\) 0 0
\(119\) 20.3163 + 0.890552i 1.86239 + 0.0816368i
\(120\) 0 0
\(121\) 8.49527 0.772297
\(122\) 0 0
\(123\) 2.39313 0.215781
\(124\) 0 0
\(125\) 37.0178i 3.31098i
\(126\) 0 0
\(127\) −10.8719 −0.964721 −0.482361 0.875973i \(-0.660221\pi\)
−0.482361 + 0.875973i \(0.660221\pi\)
\(128\) 0 0
\(129\) 10.0389i 0.883879i
\(130\) 0 0
\(131\) 20.7875i 1.81621i 0.418742 + 0.908105i \(0.362471\pi\)
−0.418742 + 0.908105i \(0.637529\pi\)
\(132\) 0 0
\(133\) 1.46356i 0.126907i
\(134\) 0 0
\(135\) 24.2389 2.08615
\(136\) 0 0
\(137\) −16.5099 −1.41054 −0.705268 0.708940i \(-0.749173\pi\)
−0.705268 + 0.708940i \(0.749173\pi\)
\(138\) 0 0
\(139\) 1.84066i 0.156122i 0.996949 + 0.0780612i \(0.0248729\pi\)
−0.996949 + 0.0780612i \(0.975127\pi\)
\(140\) 0 0
\(141\) 17.8856i 1.50624i
\(142\) 0 0
\(143\) 0.227415i 0.0190174i
\(144\) 0 0
\(145\) −26.6399 −2.21232
\(146\) 0 0
\(147\) 26.0504i 2.14860i
\(148\) 0 0
\(149\) 18.1016 1.48294 0.741471 0.670985i \(-0.234129\pi\)
0.741471 + 0.670985i \(0.234129\pi\)
\(150\) 0 0
\(151\) −5.77597 −0.470042 −0.235021 0.971990i \(-0.575516\pi\)
−0.235021 + 0.971990i \(0.575516\pi\)
\(152\) 0 0
\(153\) −0.133511 + 3.04581i −0.0107937 + 0.246239i
\(154\) 0 0
\(155\) 17.5769 1.41181
\(156\) 0 0
\(157\) −4.35499 −0.347566 −0.173783 0.984784i \(-0.555599\pi\)
−0.173783 + 0.984784i \(0.555599\pi\)
\(158\) 0 0
\(159\) 9.60433i 0.761673i
\(160\) 0 0
\(161\) 30.8219 2.42911
\(162\) 0 0
\(163\) 20.3997i 1.59783i 0.601446 + 0.798914i \(0.294592\pi\)
−0.601446 + 0.798914i \(0.705408\pi\)
\(164\) 0 0
\(165\) 10.2586i 0.798631i
\(166\) 0 0
\(167\) 18.6690i 1.44465i 0.691555 + 0.722324i \(0.256926\pi\)
−0.691555 + 0.722324i \(0.743074\pi\)
\(168\) 0 0
\(169\) −12.9794 −0.998412
\(170\) 0 0
\(171\) 0.219415 0.0167791
\(172\) 0 0
\(173\) 15.2485i 1.15932i 0.814858 + 0.579661i \(0.196815\pi\)
−0.814858 + 0.579661i \(0.803185\pi\)
\(174\) 0 0
\(175\) 67.0105i 5.06552i
\(176\) 0 0
\(177\) 1.50352i 0.113012i
\(178\) 0 0
\(179\) −2.07366 −0.154992 −0.0774962 0.996993i \(-0.524693\pi\)
−0.0774962 + 0.996993i \(0.524693\pi\)
\(180\) 0 0
\(181\) 0.645075i 0.0479481i −0.999713 0.0239740i \(-0.992368\pi\)
0.999713 0.0239740i \(-0.00763190\pi\)
\(182\) 0 0
\(183\) −0.363510 −0.0268715
\(184\) 0 0
\(185\) −33.4926 −2.46242
\(186\) 0 0
\(187\) −6.51911 0.285761i −0.476725 0.0208969i
\(188\) 0 0
\(189\) −27.7301 −2.01707
\(190\) 0 0
\(191\) 3.92528 0.284023 0.142012 0.989865i \(-0.454643\pi\)
0.142012 + 0.989865i \(0.454643\pi\)
\(192\) 0 0
\(193\) 21.9594i 1.58067i 0.612672 + 0.790337i \(0.290095\pi\)
−0.612672 + 0.790337i \(0.709905\pi\)
\(194\) 0 0
\(195\) −0.931419 −0.0667003
\(196\) 0 0
\(197\) 4.36989i 0.311342i −0.987809 0.155671i \(-0.950246\pi\)
0.987809 0.155671i \(-0.0497540\pi\)
\(198\) 0 0
\(199\) 26.5589i 1.88271i 0.337413 + 0.941357i \(0.390448\pi\)
−0.337413 + 0.941357i \(0.609552\pi\)
\(200\) 0 0
\(201\) 18.7269i 1.32089i
\(202\) 0 0
\(203\) 30.4770 2.13906
\(204\) 0 0
\(205\) −6.86206 −0.479267
\(206\) 0 0
\(207\) 4.62080i 0.321168i
\(208\) 0 0
\(209\) 0.469627i 0.0324848i
\(210\) 0 0
\(211\) 13.3998i 0.922479i −0.887276 0.461239i \(-0.847405\pi\)
0.887276 0.461239i \(-0.152595\pi\)
\(212\) 0 0
\(213\) 1.51998 0.104147
\(214\) 0 0
\(215\) 28.7857i 1.96317i
\(216\) 0 0
\(217\) −20.1086 −1.36506
\(218\) 0 0
\(219\) 0.255033 0.0172336
\(220\) 0 0
\(221\) 0.0259454 0.591896i 0.00174527 0.0398152i
\(222\) 0 0
\(223\) −6.03682 −0.404255 −0.202128 0.979359i \(-0.564786\pi\)
−0.202128 + 0.979359i \(0.564786\pi\)
\(224\) 0 0
\(225\) −10.0462 −0.669745
\(226\) 0 0
\(227\) 17.6954i 1.17448i −0.809412 0.587241i \(-0.800214\pi\)
0.809412 0.587241i \(-0.199786\pi\)
\(228\) 0 0
\(229\) 16.1121 1.06472 0.532359 0.846519i \(-0.321306\pi\)
0.532359 + 0.846519i \(0.321306\pi\)
\(230\) 0 0
\(231\) 11.7362i 0.772185i
\(232\) 0 0
\(233\) 17.3934i 1.13948i −0.821825 0.569739i \(-0.807044\pi\)
0.821825 0.569739i \(-0.192956\pi\)
\(234\) 0 0
\(235\) 51.2853i 3.34548i
\(236\) 0 0
\(237\) −16.9063 −1.09818
\(238\) 0 0
\(239\) −15.3331 −0.991819 −0.495909 0.868374i \(-0.665165\pi\)
−0.495909 + 0.868374i \(0.665165\pi\)
\(240\) 0 0
\(241\) 14.1882i 0.913943i 0.889481 + 0.456972i \(0.151066\pi\)
−0.889481 + 0.456972i \(0.848934\pi\)
\(242\) 0 0
\(243\) 7.49251i 0.480645i
\(244\) 0 0
\(245\) 74.6969i 4.77221i
\(246\) 0 0
\(247\) −0.0426393 −0.00271307
\(248\) 0 0
\(249\) 8.98856i 0.569627i
\(250\) 0 0
\(251\) −23.6445 −1.49243 −0.746215 0.665705i \(-0.768131\pi\)
−0.746215 + 0.665705i \(0.768131\pi\)
\(252\) 0 0
\(253\) −9.89014 −0.621788
\(254\) 0 0
\(255\) −1.17039 + 26.7002i −0.0732924 + 1.67203i
\(256\) 0 0
\(257\) 10.0533 0.627105 0.313553 0.949571i \(-0.398481\pi\)
0.313553 + 0.949571i \(0.398481\pi\)
\(258\) 0 0
\(259\) 38.3167 2.38088
\(260\) 0 0
\(261\) 4.56908i 0.282819i
\(262\) 0 0
\(263\) 15.8706 0.978623 0.489312 0.872109i \(-0.337248\pi\)
0.489312 + 0.872109i \(0.337248\pi\)
\(264\) 0 0
\(265\) 27.5395i 1.69174i
\(266\) 0 0
\(267\) 5.61519i 0.343644i
\(268\) 0 0
\(269\) 7.47473i 0.455742i 0.973691 + 0.227871i \(0.0731765\pi\)
−0.973691 + 0.227871i \(0.926824\pi\)
\(270\) 0 0
\(271\) −8.42094 −0.511536 −0.255768 0.966738i \(-0.582328\pi\)
−0.255768 + 0.966738i \(0.582328\pi\)
\(272\) 0 0
\(273\) 1.06558 0.0644916
\(274\) 0 0
\(275\) 21.5024i 1.29664i
\(276\) 0 0
\(277\) 11.6769i 0.701600i −0.936450 0.350800i \(-0.885910\pi\)
0.936450 0.350800i \(-0.114090\pi\)
\(278\) 0 0
\(279\) 3.01466i 0.180483i
\(280\) 0 0
\(281\) 17.5929 1.04950 0.524751 0.851256i \(-0.324158\pi\)
0.524751 + 0.851256i \(0.324158\pi\)
\(282\) 0 0
\(283\) 0.765834i 0.0455241i 0.999741 + 0.0227620i \(0.00724601\pi\)
−0.999741 + 0.0227620i \(0.992754\pi\)
\(284\) 0 0
\(285\) 1.92344 0.113935
\(286\) 0 0
\(287\) 7.85044 0.463397
\(288\) 0 0
\(289\) −16.9348 1.48751i −0.996164 0.0875005i
\(290\) 0 0
\(291\) 6.07530 0.356140
\(292\) 0 0
\(293\) −9.55957 −0.558476 −0.279238 0.960222i \(-0.590082\pi\)
−0.279238 + 0.960222i \(0.590082\pi\)
\(294\) 0 0
\(295\) 4.31120i 0.251008i
\(296\) 0 0
\(297\) 8.89805 0.516317
\(298\) 0 0
\(299\) 0.897965i 0.0519307i
\(300\) 0 0
\(301\) 32.9318i 1.89816i
\(302\) 0 0
\(303\) 11.5926i 0.665979i
\(304\) 0 0
\(305\) 1.04233 0.0596837
\(306\) 0 0
\(307\) −6.30200 −0.359674 −0.179837 0.983696i \(-0.557557\pi\)
−0.179837 + 0.983696i \(0.557557\pi\)
\(308\) 0 0
\(309\) 22.5831i 1.28470i
\(310\) 0 0
\(311\) 2.13077i 0.120825i 0.998174 + 0.0604124i \(0.0192416\pi\)
−0.998174 + 0.0604124i \(0.980758\pi\)
\(312\) 0 0
\(313\) 20.7160i 1.17094i −0.810695 0.585469i \(-0.800911\pi\)
0.810695 0.585469i \(-0.199089\pi\)
\(314\) 0 0
\(315\) 15.7228 0.885881
\(316\) 0 0
\(317\) 21.1861i 1.18993i 0.803751 + 0.594966i \(0.202834\pi\)
−0.803751 + 0.594966i \(0.797166\pi\)
\(318\) 0 0
\(319\) −9.77946 −0.547544
\(320\) 0 0
\(321\) 14.0443 0.783876
\(322\) 0 0
\(323\) −0.0535789 + 1.22231i −0.00298121 + 0.0680109i
\(324\) 0 0
\(325\) 1.95229 0.108293
\(326\) 0 0
\(327\) 2.14300 0.118508
\(328\) 0 0
\(329\) 58.6722i 3.23470i
\(330\) 0 0
\(331\) 11.8922 0.653656 0.326828 0.945084i \(-0.394020\pi\)
0.326828 + 0.945084i \(0.394020\pi\)
\(332\) 0 0
\(333\) 5.74441i 0.314792i
\(334\) 0 0
\(335\) 53.6976i 2.93381i
\(336\) 0 0
\(337\) 14.7656i 0.804336i −0.915566 0.402168i \(-0.868257\pi\)
0.915566 0.402168i \(-0.131743\pi\)
\(338\) 0 0
\(339\) 14.6377 0.795011
\(340\) 0 0
\(341\) 6.45244 0.349419
\(342\) 0 0
\(343\) 50.9307i 2.75000i
\(344\) 0 0
\(345\) 40.5069i 2.18082i
\(346\) 0 0
\(347\) 2.55012i 0.136897i −0.997655 0.0684487i \(-0.978195\pi\)
0.997655 0.0684487i \(-0.0218050\pi\)
\(348\) 0 0
\(349\) 17.6916 0.947011 0.473506 0.880791i \(-0.342988\pi\)
0.473506 + 0.880791i \(0.342988\pi\)
\(350\) 0 0
\(351\) 0.807890i 0.0431220i
\(352\) 0 0
\(353\) −16.1137 −0.857646 −0.428823 0.903389i \(-0.641072\pi\)
−0.428823 + 0.903389i \(0.641072\pi\)
\(354\) 0 0
\(355\) −4.35839 −0.231319
\(356\) 0 0
\(357\) 1.33896 30.5460i 0.0708655 1.61667i
\(358\) 0 0
\(359\) 0.424780 0.0224190 0.0112095 0.999937i \(-0.496432\pi\)
0.0112095 + 0.999937i \(0.496432\pi\)
\(360\) 0 0
\(361\) −18.9119 −0.995366
\(362\) 0 0
\(363\) 12.7728i 0.670399i
\(364\) 0 0
\(365\) −0.731284 −0.0382771
\(366\) 0 0
\(367\) 25.9082i 1.35240i 0.736719 + 0.676199i \(0.236374\pi\)
−0.736719 + 0.676199i \(0.763626\pi\)
\(368\) 0 0
\(369\) 1.17693i 0.0612686i
\(370\) 0 0
\(371\) 31.5061i 1.63572i
\(372\) 0 0
\(373\) 12.8854 0.667178 0.333589 0.942719i \(-0.391740\pi\)
0.333589 + 0.942719i \(0.391740\pi\)
\(374\) 0 0
\(375\) −55.6571 −2.87412
\(376\) 0 0
\(377\) 0.887916i 0.0457300i
\(378\) 0 0
\(379\) 22.9050i 1.17655i −0.808661 0.588274i \(-0.799808\pi\)
0.808661 0.588274i \(-0.200192\pi\)
\(380\) 0 0
\(381\) 16.3461i 0.837434i
\(382\) 0 0
\(383\) 36.2399 1.85177 0.925885 0.377804i \(-0.123321\pi\)
0.925885 + 0.377804i \(0.123321\pi\)
\(384\) 0 0
\(385\) 33.6524i 1.71509i
\(386\) 0 0
\(387\) −4.93712 −0.250968
\(388\) 0 0
\(389\) 16.1675 0.819725 0.409863 0.912147i \(-0.365577\pi\)
0.409863 + 0.912147i \(0.365577\pi\)
\(390\) 0 0
\(391\) −25.7412 1.12835i −1.30179 0.0570631i
\(392\) 0 0
\(393\) 31.2544 1.57658
\(394\) 0 0
\(395\) 48.4772 2.43915
\(396\) 0 0
\(397\) 26.4855i 1.32927i −0.747168 0.664635i \(-0.768587\pi\)
0.747168 0.664635i \(-0.231413\pi\)
\(398\) 0 0
\(399\) −2.20049 −0.110162
\(400\) 0 0
\(401\) 29.0998i 1.45318i 0.687074 + 0.726588i \(0.258895\pi\)
−0.687074 + 0.726588i \(0.741105\pi\)
\(402\) 0 0
\(403\) 0.585843i 0.0291829i
\(404\) 0 0
\(405\) 26.8802i 1.33569i
\(406\) 0 0
\(407\) −12.2951 −0.609444
\(408\) 0 0
\(409\) 40.4080 1.99805 0.999023 0.0441909i \(-0.0140710\pi\)
0.999023 + 0.0441909i \(0.0140710\pi\)
\(410\) 0 0
\(411\) 24.8230i 1.22443i
\(412\) 0 0
\(413\) 4.93216i 0.242696i
\(414\) 0 0
\(415\) 25.7738i 1.26519i
\(416\) 0 0
\(417\) 2.76746 0.135523
\(418\) 0 0
\(419\) 14.1646i 0.691986i 0.938237 + 0.345993i \(0.112458\pi\)
−0.938237 + 0.345993i \(0.887542\pi\)
\(420\) 0 0
\(421\) 32.0232 1.56072 0.780358 0.625333i \(-0.215037\pi\)
0.780358 + 0.625333i \(0.215037\pi\)
\(422\) 0 0
\(423\) −8.79609 −0.427680
\(424\) 0 0
\(425\) 2.45317 55.9646i 0.118996 2.71468i
\(426\) 0 0
\(427\) −1.19246 −0.0577074
\(428\) 0 0
\(429\) −0.341923 −0.0165082
\(430\) 0 0
\(431\) 18.1644i 0.874949i −0.899231 0.437475i \(-0.855873\pi\)
0.899231 0.437475i \(-0.144127\pi\)
\(432\) 0 0
\(433\) 2.10107 0.100971 0.0504854 0.998725i \(-0.483923\pi\)
0.0504854 + 0.998725i \(0.483923\pi\)
\(434\) 0 0
\(435\) 40.0536i 1.92042i
\(436\) 0 0
\(437\) 1.85436i 0.0887061i
\(438\) 0 0
\(439\) 5.43081i 0.259199i 0.991566 + 0.129599i \(0.0413691\pi\)
−0.991566 + 0.129599i \(0.958631\pi\)
\(440\) 0 0
\(441\) −12.8115 −0.610071
\(442\) 0 0
\(443\) −37.1788 −1.76642 −0.883209 0.468979i \(-0.844622\pi\)
−0.883209 + 0.468979i \(0.844622\pi\)
\(444\) 0 0
\(445\) 16.1010i 0.763261i
\(446\) 0 0
\(447\) 27.2161i 1.28728i
\(448\) 0 0
\(449\) 13.5969i 0.641676i 0.947134 + 0.320838i \(0.103964\pi\)
−0.947134 + 0.320838i \(0.896036\pi\)
\(450\) 0 0
\(451\) −2.51905 −0.118618
\(452\) 0 0
\(453\) 8.68429i 0.408024i
\(454\) 0 0
\(455\) −3.05544 −0.143241
\(456\) 0 0
\(457\) 25.6533 1.20001 0.600005 0.799996i \(-0.295165\pi\)
0.600005 + 0.799996i \(0.295165\pi\)
\(458\) 0 0
\(459\) 23.1591 + 1.01516i 1.08097 + 0.0473838i
\(460\) 0 0
\(461\) 21.6479 1.00824 0.504121 0.863633i \(-0.331817\pi\)
0.504121 + 0.863633i \(0.331817\pi\)
\(462\) 0 0
\(463\) 8.07378 0.375220 0.187610 0.982244i \(-0.439926\pi\)
0.187610 + 0.982244i \(0.439926\pi\)
\(464\) 0 0
\(465\) 26.4272i 1.22553i
\(466\) 0 0
\(467\) −24.5973 −1.13823 −0.569114 0.822259i \(-0.692714\pi\)
−0.569114 + 0.822259i \(0.692714\pi\)
\(468\) 0 0
\(469\) 61.4319i 2.83666i
\(470\) 0 0
\(471\) 6.54782i 0.301708i
\(472\) 0 0
\(473\) 10.5672i 0.485880i
\(474\) 0 0
\(475\) −4.03160 −0.184983
\(476\) 0 0
\(477\) 4.72338 0.216269
\(478\) 0 0
\(479\) 19.0113i 0.868647i −0.900757 0.434323i \(-0.856988\pi\)
0.900757 0.434323i \(-0.143012\pi\)
\(480\) 0 0
\(481\) 1.11632i 0.0508997i
\(482\) 0 0
\(483\) 46.3413i 2.10860i
\(484\) 0 0
\(485\) −17.4203 −0.791017
\(486\) 0 0
\(487\) 2.22891i 0.101001i −0.998724 0.0505007i \(-0.983918\pi\)
0.998724 0.0505007i \(-0.0160817\pi\)
\(488\) 0 0
\(489\) 30.6713 1.38701
\(490\) 0 0
\(491\) −1.68008 −0.0758208 −0.0379104 0.999281i \(-0.512070\pi\)
−0.0379104 + 0.999281i \(0.512070\pi\)
\(492\) 0 0
\(493\) −25.4532 1.11572i −1.14635 0.0502496i
\(494\) 0 0
\(495\) −5.04515 −0.226762
\(496\) 0 0
\(497\) 4.98615 0.223659
\(498\) 0 0
\(499\) 14.7992i 0.662502i −0.943543 0.331251i \(-0.892529\pi\)
0.943543 0.331251i \(-0.107471\pi\)
\(500\) 0 0
\(501\) 28.0692 1.25404
\(502\) 0 0
\(503\) 35.9433i 1.60263i −0.598240 0.801317i \(-0.704133\pi\)
0.598240 0.801317i \(-0.295867\pi\)
\(504\) 0 0
\(505\) 33.2408i 1.47919i
\(506\) 0 0
\(507\) 19.5147i 0.866679i
\(508\) 0 0
\(509\) 9.44784 0.418768 0.209384 0.977833i \(-0.432854\pi\)
0.209384 + 0.977833i \(0.432854\pi\)
\(510\) 0 0
\(511\) 0.836615 0.0370096
\(512\) 0 0
\(513\) 1.66835i 0.0736594i
\(514\) 0 0
\(515\) 64.7547i 2.85343i
\(516\) 0 0
\(517\) 18.8267i 0.827999i
\(518\) 0 0
\(519\) 22.9264 1.00636
\(520\) 0 0
\(521\) 7.49939i 0.328554i −0.986414 0.164277i \(-0.947471\pi\)
0.986414 0.164277i \(-0.0525291\pi\)
\(522\) 0 0
\(523\) 31.8814 1.39408 0.697038 0.717034i \(-0.254501\pi\)
0.697038 + 0.717034i \(0.254501\pi\)
\(524\) 0 0
\(525\) 100.752 4.39716
\(526\) 0 0
\(527\) 16.7939 + 0.736148i 0.731553 + 0.0320671i
\(528\) 0 0
\(529\) −16.0520 −0.697914
\(530\) 0 0
\(531\) 0.739426 0.0320884
\(532\) 0 0
\(533\) 0.228715i 0.00990674i
\(534\) 0 0
\(535\) −40.2707 −1.74105
\(536\) 0 0
\(537\) 3.11779i 0.134542i
\(538\) 0 0
\(539\) 27.4211i 1.18111i
\(540\) 0 0
\(541\) 29.9009i 1.28554i 0.766058 + 0.642771i \(0.222215\pi\)
−0.766058 + 0.642771i \(0.777785\pi\)
\(542\) 0 0
\(543\) −0.969884 −0.0416217
\(544\) 0 0
\(545\) −6.14485 −0.263216
\(546\) 0 0
\(547\) 11.5398i 0.493404i 0.969091 + 0.246702i \(0.0793469\pi\)
−0.969091 + 0.246702i \(0.920653\pi\)
\(548\) 0 0
\(549\) 0.178773i 0.00762986i
\(550\) 0 0
\(551\) 1.83361i 0.0781143i
\(552\) 0 0
\(553\) −55.4596 −2.35838
\(554\) 0 0
\(555\) 50.3568i 2.13753i
\(556\) 0 0
\(557\) 29.6636 1.25689 0.628444 0.777855i \(-0.283692\pi\)
0.628444 + 0.777855i \(0.283692\pi\)
\(558\) 0 0
\(559\) 0.959437 0.0405799
\(560\) 0 0
\(561\) −0.429647 + 9.80161i −0.0181397 + 0.413824i
\(562\) 0 0
\(563\) −8.39522 −0.353816 −0.176908 0.984227i \(-0.556610\pi\)
−0.176908 + 0.984227i \(0.556610\pi\)
\(564\) 0 0
\(565\) −41.9722 −1.76578
\(566\) 0 0
\(567\) 30.7519i 1.29146i
\(568\) 0 0
\(569\) 12.7721 0.535436 0.267718 0.963497i \(-0.413730\pi\)
0.267718 + 0.963497i \(0.413730\pi\)
\(570\) 0 0
\(571\) 18.3722i 0.768853i −0.923156 0.384426i \(-0.874399\pi\)
0.923156 0.384426i \(-0.125601\pi\)
\(572\) 0 0
\(573\) 5.90174i 0.246549i
\(574\) 0 0
\(575\) 84.9038i 3.54074i
\(576\) 0 0
\(577\) −15.8464 −0.659695 −0.329848 0.944034i \(-0.606997\pi\)
−0.329848 + 0.944034i \(0.606997\pi\)
\(578\) 0 0
\(579\) 33.0165 1.37212
\(580\) 0 0
\(581\) 29.4862i 1.22329i
\(582\) 0 0
\(583\) 10.1097i 0.418701i
\(584\) 0 0
\(585\) 0.458069i 0.0189388i
\(586\) 0 0
\(587\) −43.0686 −1.77763 −0.888816 0.458264i \(-0.848471\pi\)
−0.888816 + 0.458264i \(0.848471\pi\)
\(588\) 0 0
\(589\) 1.20981i 0.0498492i
\(590\) 0 0
\(591\) −6.57022 −0.270263
\(592\) 0 0
\(593\) −44.5851 −1.83089 −0.915446 0.402440i \(-0.868162\pi\)
−0.915446 + 0.402440i \(0.868162\pi\)
\(594\) 0 0
\(595\) −3.83935 + 87.5877i −0.157398 + 3.59075i
\(596\) 0 0
\(597\) 39.9319 1.63430
\(598\) 0 0
\(599\) 28.1693 1.15097 0.575483 0.817814i \(-0.304814\pi\)
0.575483 + 0.817814i \(0.304814\pi\)
\(600\) 0 0
\(601\) 40.3029i 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(602\) 0 0
\(603\) 9.20983 0.375053
\(604\) 0 0
\(605\) 36.6248i 1.48901i
\(606\) 0 0
\(607\) 17.9363i 0.728013i −0.931396 0.364006i \(-0.881409\pi\)
0.931396 0.364006i \(-0.118591\pi\)
\(608\) 0 0
\(609\) 45.8227i 1.85683i
\(610\) 0 0
\(611\) 1.70936 0.0691531
\(612\) 0 0
\(613\) 16.4913 0.666077 0.333039 0.942913i \(-0.391926\pi\)
0.333039 + 0.942913i \(0.391926\pi\)
\(614\) 0 0
\(615\) 10.3172i 0.416032i
\(616\) 0 0
\(617\) 6.31424i 0.254202i −0.991890 0.127101i \(-0.959433\pi\)
0.991890 0.127101i \(-0.0405672\pi\)
\(618\) 0 0
\(619\) 42.7568i 1.71854i −0.511520 0.859271i \(-0.670917\pi\)
0.511520 0.859271i \(-0.329083\pi\)
\(620\) 0 0
\(621\) 35.1347 1.40991
\(622\) 0 0
\(623\) 18.4201i 0.737987i
\(624\) 0 0
\(625\) 91.6590 3.66636
\(626\) 0 0
\(627\) 0.706094 0.0281987
\(628\) 0 0
\(629\) −32.0006 1.40272i −1.27595 0.0559303i
\(630\) 0 0
\(631\) −28.5154 −1.13518 −0.567591 0.823311i \(-0.692124\pi\)
−0.567591 + 0.823311i \(0.692124\pi\)
\(632\) 0 0
\(633\) −20.1468 −0.800765
\(634\) 0 0
\(635\) 46.8707i 1.86001i
\(636\) 0 0
\(637\) 2.48967 0.0986444
\(638\) 0 0
\(639\) 0.747520i 0.0295714i
\(640\) 0 0
\(641\) 12.3593i 0.488162i 0.969755 + 0.244081i \(0.0784862\pi\)
−0.969755 + 0.244081i \(0.921514\pi\)
\(642\) 0 0
\(643\) 15.3362i 0.604800i −0.953181 0.302400i \(-0.902212\pi\)
0.953181 0.302400i \(-0.0977879\pi\)
\(644\) 0 0
\(645\) −43.2799 −1.70414
\(646\) 0 0
\(647\) −46.9121 −1.84430 −0.922152 0.386828i \(-0.873571\pi\)
−0.922152 + 0.386828i \(0.873571\pi\)
\(648\) 0 0
\(649\) 1.58263i 0.0621238i
\(650\) 0 0
\(651\) 30.2336i 1.18495i
\(652\) 0 0
\(653\) 20.0020i 0.782740i −0.920233 0.391370i \(-0.872001\pi\)
0.920233 0.391370i \(-0.127999\pi\)
\(654\) 0 0
\(655\) −89.6190 −3.50170
\(656\) 0 0
\(657\) 0.125425i 0.00489328i
\(658\) 0 0
\(659\) −33.4677 −1.30372 −0.651858 0.758341i \(-0.726010\pi\)
−0.651858 + 0.758341i \(0.726010\pi\)
\(660\) 0 0
\(661\) 18.9379 0.736601 0.368300 0.929707i \(-0.379940\pi\)
0.368300 + 0.929707i \(0.379940\pi\)
\(662\) 0 0
\(663\) −0.889928 0.0390094i −0.0345619 0.00151500i
\(664\) 0 0
\(665\) 6.30969 0.244679
\(666\) 0 0
\(667\) −38.6150 −1.49518
\(668\) 0 0
\(669\) 9.07648i 0.350917i
\(670\) 0 0
\(671\) 0.382638 0.0147716
\(672\) 0 0
\(673\) 28.1073i 1.08346i −0.840553 0.541729i \(-0.817770\pi\)
0.840553 0.541729i \(-0.182230\pi\)
\(674\) 0 0
\(675\) 76.3871i 2.94014i
\(676\) 0 0
\(677\) 30.2847i 1.16394i −0.813212 0.581968i \(-0.802283\pi\)
0.813212 0.581968i \(-0.197717\pi\)
\(678\) 0 0
\(679\) 19.9295 0.764823
\(680\) 0 0
\(681\) −26.6053 −1.01952
\(682\) 0 0
\(683\) 20.4526i 0.782599i 0.920263 + 0.391299i \(0.127974\pi\)
−0.920263 + 0.391299i \(0.872026\pi\)
\(684\) 0 0
\(685\) 71.1775i 2.71955i
\(686\) 0 0
\(687\) 24.2249i 0.924237i
\(688\) 0 0
\(689\) −0.917900 −0.0349692
\(690\) 0 0
\(691\) 23.3610i 0.888696i −0.895854 0.444348i \(-0.853435\pi\)
0.895854 0.444348i \(-0.146565\pi\)
\(692\) 0 0
\(693\) 5.77183 0.219254
\(694\) 0 0
\(695\) −7.93543 −0.301008
\(696\) 0 0
\(697\) −6.55638 0.287394i −0.248341 0.0108858i
\(698\) 0 0
\(699\) −26.1513 −0.989134
\(700\) 0 0
\(701\) −0.457933 −0.0172959 −0.00864794 0.999963i \(-0.502753\pi\)
−0.00864794 + 0.999963i \(0.502753\pi\)
\(702\) 0 0
\(703\) 2.30527i 0.0869451i
\(704\) 0 0
\(705\) −77.1084 −2.90407
\(706\) 0 0
\(707\) 38.0286i 1.43021i
\(708\) 0 0
\(709\) 7.24043i 0.271920i 0.990714 + 0.135960i \(0.0434119\pi\)
−0.990714 + 0.135960i \(0.956588\pi\)
\(710\) 0 0
\(711\) 8.31446i 0.311817i
\(712\) 0 0
\(713\) 25.4780 0.954158
\(714\) 0 0
\(715\) 0.980430 0.0366660
\(716\) 0 0
\(717\) 23.0537i 0.860956i
\(718\) 0 0
\(719\) 42.6305i 1.58985i 0.606707 + 0.794925i \(0.292490\pi\)
−0.606707 + 0.794925i \(0.707510\pi\)
\(720\) 0 0
\(721\) 74.0817i 2.75895i
\(722\) 0 0
\(723\) 21.3323 0.793356
\(724\) 0 0
\(725\) 83.9536i 3.11796i
\(726\) 0 0
\(727\) 4.88783 0.181280 0.0906398 0.995884i \(-0.471109\pi\)
0.0906398 + 0.995884i \(0.471109\pi\)
\(728\) 0 0
\(729\) −29.9700 −1.11000
\(730\) 0 0
\(731\) 1.20559 27.5034i 0.0445904 1.01725i
\(732\) 0 0
\(733\) −26.6680 −0.985007 −0.492503 0.870311i \(-0.663918\pi\)
−0.492503 + 0.870311i \(0.663918\pi\)
\(734\) 0 0
\(735\) −112.308 −4.14255
\(736\) 0 0
\(737\) 19.7123i 0.726112i
\(738\) 0 0
\(739\) 8.74907 0.321840 0.160920 0.986967i \(-0.448554\pi\)
0.160920 + 0.986967i \(0.448554\pi\)
\(740\) 0 0
\(741\) 0.0641091i 0.00235510i
\(742\) 0 0
\(743\) 2.87111i 0.105331i −0.998612 0.0526655i \(-0.983228\pi\)
0.998612 0.0526655i \(-0.0167717\pi\)
\(744\) 0 0
\(745\) 78.0397i 2.85915i
\(746\) 0 0
\(747\) −4.42054 −0.161739
\(748\) 0 0
\(749\) 46.0711 1.68340
\(750\) 0 0
\(751\) 13.0671i 0.476825i −0.971164 0.238413i \(-0.923373\pi\)
0.971164 0.238413i \(-0.0766271\pi\)
\(752\) 0 0
\(753\) 35.5500i 1.29552i
\(754\) 0 0
\(755\) 24.9014i 0.906254i
\(756\) 0 0
\(757\) −17.3683 −0.631262 −0.315631 0.948882i \(-0.602216\pi\)
−0.315631 + 0.948882i \(0.602216\pi\)
\(758\) 0 0
\(759\) 14.8700i 0.539748i
\(760\) 0 0
\(761\) 18.3288 0.664419 0.332209 0.943206i \(-0.392206\pi\)
0.332209 + 0.943206i \(0.392206\pi\)
\(762\) 0 0
\(763\) 7.02992 0.254500
\(764\) 0 0
\(765\) −13.1311 0.575592i −0.474755 0.0208106i
\(766\) 0 0
\(767\) −0.143694 −0.00518848
\(768\) 0 0
\(769\) 6.99518 0.252253 0.126126 0.992014i \(-0.459746\pi\)
0.126126 + 0.992014i \(0.459746\pi\)
\(770\) 0 0
\(771\) 15.1153i 0.544364i
\(772\) 0 0
\(773\) −49.6691 −1.78647 −0.893237 0.449587i \(-0.851571\pi\)
−0.893237 + 0.449587i \(0.851571\pi\)
\(774\) 0 0
\(775\) 55.3922i 1.98975i
\(776\) 0 0
\(777\) 57.6099i 2.06674i
\(778\) 0 0
\(779\) 0.472312i 0.0169223i
\(780\) 0 0
\(781\) −1.59996 −0.0572510
\(782\) 0 0
\(783\) 34.7415 1.24156
\(784\) 0 0
\(785\) 18.7752i 0.670117i
\(786\) 0 0
\(787\) 21.5635i 0.768657i 0.923196 + 0.384328i \(0.125567\pi\)
−0.923196 + 0.384328i \(0.874433\pi\)
\(788\) 0 0
\(789\) 23.8618i 0.849501i
\(790\) 0 0
\(791\) 48.0177 1.70731
\(792\) 0 0
\(793\) 0.0347412i 0.00123370i