Properties

Label 4012.2.b.b.237.1
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 237.1
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.46

$q$-expansion

\(f(q)\) \(=\) \(q-3.24132i q^{3} +0.276658i q^{5} -5.07344i q^{7} -7.50616 q^{9} +O(q^{10})\) \(q-3.24132i q^{3} +0.276658i q^{5} -5.07344i q^{7} -7.50616 q^{9} -3.78403i q^{11} +3.39697 q^{13} +0.896737 q^{15} +(0.332074 + 4.10971i) q^{17} -7.59929 q^{19} -16.4446 q^{21} -8.61298i q^{23} +4.92346 q^{25} +14.6059i q^{27} +8.08384i q^{29} +6.90939i q^{31} -12.2652 q^{33} +1.40361 q^{35} -5.47331i q^{37} -11.0107i q^{39} -5.51443i q^{41} -4.87514 q^{43} -2.07664i q^{45} -5.79509 q^{47} -18.7398 q^{49} +(13.3209 - 1.07636i) q^{51} -3.84973 q^{53} +1.04688 q^{55} +24.6317i q^{57} +1.00000 q^{59} -12.2085i q^{61} +38.0820i q^{63} +0.939798i q^{65} +8.67495 q^{67} -27.9174 q^{69} -0.288925i q^{71} -1.40640i q^{73} -15.9585i q^{75} -19.1980 q^{77} +11.0542i q^{79} +24.8239 q^{81} +17.0647 q^{83} +(-1.13698 + 0.0918708i) q^{85} +26.2023 q^{87} +3.59463 q^{89} -17.2343i q^{91} +22.3955 q^{93} -2.10240i q^{95} -0.104655i q^{97} +28.4035i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

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
\(3\) 3.24132i 1.87138i −0.352827 0.935689i \(-0.614780\pi\)
0.352827 0.935689i \(-0.385220\pi\)
\(4\) 0 0
\(5\) 0.276658i 0.123725i 0.998085 + 0.0618626i \(0.0197041\pi\)
−0.998085 + 0.0618626i \(0.980296\pi\)
\(6\) 0 0
\(7\) 5.07344i 1.91758i −0.284118 0.958789i \(-0.591701\pi\)
0.284118 0.958789i \(-0.408299\pi\)
\(8\) 0 0
\(9\) −7.50616 −2.50205
\(10\) 0 0
\(11\) 3.78403i 1.14093i −0.821323 0.570463i \(-0.806764\pi\)
0.821323 0.570463i \(-0.193236\pi\)
\(12\) 0 0
\(13\) 3.39697 0.942149 0.471075 0.882093i \(-0.343866\pi\)
0.471075 + 0.882093i \(0.343866\pi\)
\(14\) 0 0
\(15\) 0.896737 0.231536
\(16\) 0 0
\(17\) 0.332074 + 4.10971i 0.0805397 + 0.996751i
\(18\) 0 0
\(19\) −7.59929 −1.74340 −0.871699 0.490042i \(-0.836981\pi\)
−0.871699 + 0.490042i \(0.836981\pi\)
\(20\) 0 0
\(21\) −16.4446 −3.58851
\(22\) 0 0
\(23\) 8.61298i 1.79593i −0.440067 0.897965i \(-0.645045\pi\)
0.440067 0.897965i \(-0.354955\pi\)
\(24\) 0 0
\(25\) 4.92346 0.984692
\(26\) 0 0
\(27\) 14.6059i 2.81091i
\(28\) 0 0
\(29\) 8.08384i 1.50113i 0.660796 + 0.750565i \(0.270219\pi\)
−0.660796 + 0.750565i \(0.729781\pi\)
\(30\) 0 0
\(31\) 6.90939i 1.24096i 0.784221 + 0.620481i \(0.213063\pi\)
−0.784221 + 0.620481i \(0.786937\pi\)
\(32\) 0 0
\(33\) −12.2652 −2.13510
\(34\) 0 0
\(35\) 1.40361 0.237253
\(36\) 0 0
\(37\) 5.47331i 0.899806i −0.893077 0.449903i \(-0.851458\pi\)
0.893077 0.449903i \(-0.148542\pi\)
\(38\) 0 0
\(39\) 11.0107i 1.76312i
\(40\) 0 0
\(41\) 5.51443i 0.861209i −0.902541 0.430604i \(-0.858300\pi\)
0.902541 0.430604i \(-0.141700\pi\)
\(42\) 0 0
\(43\) −4.87514 −0.743451 −0.371726 0.928343i \(-0.621234\pi\)
−0.371726 + 0.928343i \(0.621234\pi\)
\(44\) 0 0
\(45\) 2.07664i 0.309567i
\(46\) 0 0
\(47\) −5.79509 −0.845301 −0.422650 0.906293i \(-0.638900\pi\)
−0.422650 + 0.906293i \(0.638900\pi\)
\(48\) 0 0
\(49\) −18.7398 −2.67711
\(50\) 0 0
\(51\) 13.3209 1.07636i 1.86530 0.150720i
\(52\) 0 0
\(53\) −3.84973 −0.528801 −0.264401 0.964413i \(-0.585174\pi\)
−0.264401 + 0.964413i \(0.585174\pi\)
\(54\) 0 0
\(55\) 1.04688 0.141161
\(56\) 0 0
\(57\) 24.6317i 3.26255i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 12.2085i 1.56314i −0.623815 0.781572i \(-0.714418\pi\)
0.623815 0.781572i \(-0.285582\pi\)
\(62\) 0 0
\(63\) 38.0820i 4.79788i
\(64\) 0 0
\(65\) 0.939798i 0.116568i
\(66\) 0 0
\(67\) 8.67495 1.05981 0.529907 0.848056i \(-0.322227\pi\)
0.529907 + 0.848056i \(0.322227\pi\)
\(68\) 0 0
\(69\) −27.9174 −3.36086
\(70\) 0 0
\(71\) 0.288925i 0.0342891i −0.999853 0.0171445i \(-0.994542\pi\)
0.999853 0.0171445i \(-0.00545754\pi\)
\(72\) 0 0
\(73\) 1.40640i 0.164606i −0.996607 0.0823031i \(-0.973772\pi\)
0.996607 0.0823031i \(-0.0262276\pi\)
\(74\) 0 0
\(75\) 15.9585i 1.84273i
\(76\) 0 0
\(77\) −19.1980 −2.18782
\(78\) 0 0
\(79\) 11.0542i 1.24369i 0.783141 + 0.621845i \(0.213616\pi\)
−0.783141 + 0.621845i \(0.786384\pi\)
\(80\) 0 0
\(81\) 24.8239 2.75822
\(82\) 0 0
\(83\) 17.0647 1.87309 0.936546 0.350544i \(-0.114003\pi\)
0.936546 + 0.350544i \(0.114003\pi\)
\(84\) 0 0
\(85\) −1.13698 + 0.0918708i −0.123323 + 0.00996479i
\(86\) 0 0
\(87\) 26.2023 2.80918
\(88\) 0 0
\(89\) 3.59463 0.381030 0.190515 0.981684i \(-0.438984\pi\)
0.190515 + 0.981684i \(0.438984\pi\)
\(90\) 0 0
\(91\) 17.2343i 1.80665i
\(92\) 0 0
\(93\) 22.3955 2.32231
\(94\) 0 0
\(95\) 2.10240i 0.215702i
\(96\) 0 0
\(97\) 0.104655i 0.0106261i −0.999986 0.00531304i \(-0.998309\pi\)
0.999986 0.00531304i \(-0.00169120\pi\)
\(98\) 0 0
\(99\) 28.4035i 2.85466i
\(100\) 0 0
\(101\) 4.92014 0.489572 0.244786 0.969577i \(-0.421282\pi\)
0.244786 + 0.969577i \(0.421282\pi\)
\(102\) 0 0
\(103\) −5.46641 −0.538622 −0.269311 0.963053i \(-0.586796\pi\)
−0.269311 + 0.963053i \(0.586796\pi\)
\(104\) 0 0
\(105\) 4.54954i 0.443989i
\(106\) 0 0
\(107\) 2.68445i 0.259515i 0.991546 + 0.129758i \(0.0414199\pi\)
−0.991546 + 0.129758i \(0.958580\pi\)
\(108\) 0 0
\(109\) 2.26847i 0.217280i 0.994081 + 0.108640i \(0.0346496\pi\)
−0.994081 + 0.108640i \(0.965350\pi\)
\(110\) 0 0
\(111\) −17.7407 −1.68388
\(112\) 0 0
\(113\) 5.23453i 0.492423i −0.969216 0.246212i \(-0.920814\pi\)
0.969216 0.246212i \(-0.0791858\pi\)
\(114\) 0 0
\(115\) 2.38285 0.222202
\(116\) 0 0
\(117\) −25.4982 −2.35731
\(118\) 0 0
\(119\) 20.8504 1.68476i 1.91135 0.154441i
\(120\) 0 0
\(121\) −3.31885 −0.301714
\(122\) 0 0
\(123\) −17.8740 −1.61165
\(124\) 0 0
\(125\) 2.74540i 0.245556i
\(126\) 0 0
\(127\) 12.9181 1.14629 0.573147 0.819453i \(-0.305722\pi\)
0.573147 + 0.819453i \(0.305722\pi\)
\(128\) 0 0
\(129\) 15.8019i 1.39128i
\(130\) 0 0
\(131\) 18.4940i 1.61583i −0.589301 0.807914i \(-0.700597\pi\)
0.589301 0.807914i \(-0.299403\pi\)
\(132\) 0 0
\(133\) 38.5545i 3.34310i
\(134\) 0 0
\(135\) −4.04084 −0.347780
\(136\) 0 0
\(137\) 0.913152 0.0780159 0.0390079 0.999239i \(-0.487580\pi\)
0.0390079 + 0.999239i \(0.487580\pi\)
\(138\) 0 0
\(139\) 16.0487i 1.36123i −0.732640 0.680616i \(-0.761712\pi\)
0.732640 0.680616i \(-0.238288\pi\)
\(140\) 0 0
\(141\) 18.7837i 1.58188i
\(142\) 0 0
\(143\) 12.8542i 1.07492i
\(144\) 0 0
\(145\) −2.23646 −0.185728
\(146\) 0 0
\(147\) 60.7415i 5.00988i
\(148\) 0 0
\(149\) −17.9063 −1.46694 −0.733471 0.679721i \(-0.762101\pi\)
−0.733471 + 0.679721i \(0.762101\pi\)
\(150\) 0 0
\(151\) −11.5846 −0.942740 −0.471370 0.881936i \(-0.656240\pi\)
−0.471370 + 0.881936i \(0.656240\pi\)
\(152\) 0 0
\(153\) −2.49260 30.8481i −0.201515 2.49392i
\(154\) 0 0
\(155\) −1.91154 −0.153538
\(156\) 0 0
\(157\) 1.15856 0.0924632 0.0462316 0.998931i \(-0.485279\pi\)
0.0462316 + 0.998931i \(0.485279\pi\)
\(158\) 0 0
\(159\) 12.4782i 0.989586i
\(160\) 0 0
\(161\) −43.6974 −3.44384
\(162\) 0 0
\(163\) 12.9501i 1.01433i 0.861850 + 0.507164i \(0.169306\pi\)
−0.861850 + 0.507164i \(0.830694\pi\)
\(164\) 0 0
\(165\) 3.39327i 0.264166i
\(166\) 0 0
\(167\) 12.4370i 0.962407i −0.876609 0.481204i \(-0.840200\pi\)
0.876609 0.481204i \(-0.159800\pi\)
\(168\) 0 0
\(169\) −1.46061 −0.112355
\(170\) 0 0
\(171\) 57.0415 4.36207
\(172\) 0 0
\(173\) 11.6216i 0.883571i 0.897121 + 0.441785i \(0.145655\pi\)
−0.897121 + 0.441785i \(0.854345\pi\)
\(174\) 0 0
\(175\) 24.9789i 1.88822i
\(176\) 0 0
\(177\) 3.24132i 0.243633i
\(178\) 0 0
\(179\) 2.42470 0.181231 0.0906154 0.995886i \(-0.471117\pi\)
0.0906154 + 0.995886i \(0.471117\pi\)
\(180\) 0 0
\(181\) 12.0773i 0.897699i 0.893607 + 0.448850i \(0.148166\pi\)
−0.893607 + 0.448850i \(0.851834\pi\)
\(182\) 0 0
\(183\) −39.5718 −2.92523
\(184\) 0 0
\(185\) 1.51423 0.111329
\(186\) 0 0
\(187\) 15.5513 1.25658i 1.13722 0.0918899i
\(188\) 0 0
\(189\) 74.1021 5.39014
\(190\) 0 0
\(191\) 17.1010 1.23739 0.618693 0.785633i \(-0.287663\pi\)
0.618693 + 0.785633i \(0.287663\pi\)
\(192\) 0 0
\(193\) 12.1921i 0.877608i 0.898583 + 0.438804i \(0.144598\pi\)
−0.898583 + 0.438804i \(0.855402\pi\)
\(194\) 0 0
\(195\) 3.04619 0.218142
\(196\) 0 0
\(197\) 4.68400i 0.333721i −0.985980 0.166861i \(-0.946637\pi\)
0.985980 0.166861i \(-0.0533630\pi\)
\(198\) 0 0
\(199\) 6.25454i 0.443373i −0.975118 0.221686i \(-0.928844\pi\)
0.975118 0.221686i \(-0.0711561\pi\)
\(200\) 0 0
\(201\) 28.1183i 1.98331i
\(202\) 0 0
\(203\) 41.0128 2.87854
\(204\) 0 0
\(205\) 1.52561 0.106553
\(206\) 0 0
\(207\) 64.6504i 4.49351i
\(208\) 0 0
\(209\) 28.7559i 1.98909i
\(210\) 0 0
\(211\) 10.1533i 0.698985i −0.936939 0.349493i \(-0.886354\pi\)
0.936939 0.349493i \(-0.113646\pi\)
\(212\) 0 0
\(213\) −0.936498 −0.0641678
\(214\) 0 0
\(215\) 1.34874i 0.0919836i
\(216\) 0 0
\(217\) 35.0543 2.37964
\(218\) 0 0
\(219\) −4.55858 −0.308040
\(220\) 0 0
\(221\) 1.12804 + 13.9606i 0.0758804 + 0.939089i
\(222\) 0 0
\(223\) −25.7556 −1.72472 −0.862362 0.506292i \(-0.831016\pi\)
−0.862362 + 0.506292i \(0.831016\pi\)
\(224\) 0 0
\(225\) −36.9563 −2.46375
\(226\) 0 0
\(227\) 19.4328i 1.28980i 0.764268 + 0.644899i \(0.223101\pi\)
−0.764268 + 0.644899i \(0.776899\pi\)
\(228\) 0 0
\(229\) −15.0991 −0.997774 −0.498887 0.866667i \(-0.666258\pi\)
−0.498887 + 0.866667i \(0.666258\pi\)
\(230\) 0 0
\(231\) 62.2269i 4.09423i
\(232\) 0 0
\(233\) 2.48681i 0.162917i −0.996677 0.0814583i \(-0.974042\pi\)
0.996677 0.0814583i \(-0.0259577\pi\)
\(234\) 0 0
\(235\) 1.60326i 0.104585i
\(236\) 0 0
\(237\) 35.8301 2.32741
\(238\) 0 0
\(239\) 16.6172 1.07488 0.537440 0.843302i \(-0.319392\pi\)
0.537440 + 0.843302i \(0.319392\pi\)
\(240\) 0 0
\(241\) 11.2170i 0.722554i −0.932459 0.361277i \(-0.882341\pi\)
0.932459 0.361277i \(-0.117659\pi\)
\(242\) 0 0
\(243\) 36.6446i 2.35075i
\(244\) 0 0
\(245\) 5.18450i 0.331226i
\(246\) 0 0
\(247\) −25.8145 −1.64254
\(248\) 0 0
\(249\) 55.3121i 3.50526i
\(250\) 0 0
\(251\) 12.8626 0.811883 0.405941 0.913899i \(-0.366944\pi\)
0.405941 + 0.913899i \(0.366944\pi\)
\(252\) 0 0
\(253\) −32.5917 −2.04902
\(254\) 0 0
\(255\) 0.297783 + 3.68533i 0.0186479 + 0.230784i
\(256\) 0 0
\(257\) 5.84511 0.364608 0.182304 0.983242i \(-0.441644\pi\)
0.182304 + 0.983242i \(0.441644\pi\)
\(258\) 0 0
\(259\) −27.7685 −1.72545
\(260\) 0 0
\(261\) 60.6786i 3.75591i
\(262\) 0 0
\(263\) 11.5038 0.709357 0.354678 0.934988i \(-0.384590\pi\)
0.354678 + 0.934988i \(0.384590\pi\)
\(264\) 0 0
\(265\) 1.06506i 0.0654260i
\(266\) 0 0
\(267\) 11.6513i 0.713050i
\(268\) 0 0
\(269\) 25.5810i 1.55970i −0.625967 0.779850i \(-0.715295\pi\)
0.625967 0.779850i \(-0.284705\pi\)
\(270\) 0 0
\(271\) 1.81347 0.110160 0.0550802 0.998482i \(-0.482459\pi\)
0.0550802 + 0.998482i \(0.482459\pi\)
\(272\) 0 0
\(273\) −55.8619 −3.38091
\(274\) 0 0
\(275\) 18.6305i 1.12346i
\(276\) 0 0
\(277\) 4.74191i 0.284914i 0.989801 + 0.142457i \(0.0455002\pi\)
−0.989801 + 0.142457i \(0.954500\pi\)
\(278\) 0 0
\(279\) 51.8630i 3.10495i
\(280\) 0 0
\(281\) 10.8429 0.646833 0.323417 0.946257i \(-0.395168\pi\)
0.323417 + 0.946257i \(0.395168\pi\)
\(282\) 0 0
\(283\) 26.3092i 1.56392i 0.623329 + 0.781959i \(0.285780\pi\)
−0.623329 + 0.781959i \(0.714220\pi\)
\(284\) 0 0
\(285\) −6.81456 −0.403660
\(286\) 0 0
\(287\) −27.9771 −1.65144
\(288\) 0 0
\(289\) −16.7795 + 2.72945i −0.987027 + 0.160556i
\(290\) 0 0
\(291\) −0.339219 −0.0198854
\(292\) 0 0
\(293\) −3.73918 −0.218445 −0.109223 0.994017i \(-0.534836\pi\)
−0.109223 + 0.994017i \(0.534836\pi\)
\(294\) 0 0
\(295\) 0.276658i 0.0161076i
\(296\) 0 0
\(297\) 55.2691 3.20704
\(298\) 0 0
\(299\) 29.2580i 1.69203i
\(300\) 0 0
\(301\) 24.7337i 1.42563i
\(302\) 0 0
\(303\) 15.9477i 0.916174i
\(304\) 0 0
\(305\) 3.37759 0.193400
\(306\) 0 0
\(307\) −11.1199 −0.634647 −0.317323 0.948317i \(-0.602784\pi\)
−0.317323 + 0.948317i \(0.602784\pi\)
\(308\) 0 0
\(309\) 17.7184i 1.00796i
\(310\) 0 0
\(311\) 11.1602i 0.632837i 0.948620 + 0.316418i \(0.102480\pi\)
−0.948620 + 0.316418i \(0.897520\pi\)
\(312\) 0 0
\(313\) 9.60169i 0.542720i 0.962478 + 0.271360i \(0.0874734\pi\)
−0.962478 + 0.271360i \(0.912527\pi\)
\(314\) 0 0
\(315\) −10.5357 −0.593619
\(316\) 0 0
\(317\) 17.8680i 1.00357i −0.864993 0.501784i \(-0.832677\pi\)
0.864993 0.501784i \(-0.167323\pi\)
\(318\) 0 0
\(319\) 30.5894 1.71268
\(320\) 0 0
\(321\) 8.70115 0.485651
\(322\) 0 0
\(323\) −2.52353 31.2309i −0.140413 1.73773i
\(324\) 0 0
\(325\) 16.7248 0.927727
\(326\) 0 0
\(327\) 7.35283 0.406613
\(328\) 0 0
\(329\) 29.4010i 1.62093i
\(330\) 0 0
\(331\) −8.92605 −0.490620 −0.245310 0.969445i \(-0.578890\pi\)
−0.245310 + 0.969445i \(0.578890\pi\)
\(332\) 0 0
\(333\) 41.0835i 2.25136i
\(334\) 0 0
\(335\) 2.39999i 0.131126i
\(336\) 0 0
\(337\) 14.8344i 0.808083i −0.914741 0.404041i \(-0.867605\pi\)
0.914741 0.404041i \(-0.132395\pi\)
\(338\) 0 0
\(339\) −16.9668 −0.921509
\(340\) 0 0
\(341\) 26.1453 1.41585
\(342\) 0 0
\(343\) 59.5609i 3.21599i
\(344\) 0 0
\(345\) 7.72358i 0.415823i
\(346\) 0 0
\(347\) 1.83432i 0.0984717i 0.998787 + 0.0492359i \(0.0156786\pi\)
−0.998787 + 0.0492359i \(0.984321\pi\)
\(348\) 0 0
\(349\) −3.42257 −0.183206 −0.0916030 0.995796i \(-0.529199\pi\)
−0.0916030 + 0.995796i \(0.529199\pi\)
\(350\) 0 0
\(351\) 49.6158i 2.64829i
\(352\) 0 0
\(353\) 2.13318 0.113538 0.0567689 0.998387i \(-0.481920\pi\)
0.0567689 + 0.998387i \(0.481920\pi\)
\(354\) 0 0
\(355\) 0.0799333 0.00424242
\(356\) 0 0
\(357\) −5.46083 67.5827i −0.289018 3.57686i
\(358\) 0 0
\(359\) −12.2739 −0.647793 −0.323897 0.946092i \(-0.604993\pi\)
−0.323897 + 0.946092i \(0.604993\pi\)
\(360\) 0 0
\(361\) 38.7492 2.03943
\(362\) 0 0
\(363\) 10.7575i 0.564620i
\(364\) 0 0
\(365\) 0.389091 0.0203659
\(366\) 0 0
\(367\) 20.8627i 1.08902i −0.838753 0.544511i \(-0.816715\pi\)
0.838753 0.544511i \(-0.183285\pi\)
\(368\) 0 0
\(369\) 41.3922i 2.15479i
\(370\) 0 0
\(371\) 19.5314i 1.01402i
\(372\) 0 0
\(373\) 10.8451 0.561540 0.280770 0.959775i \(-0.409410\pi\)
0.280770 + 0.959775i \(0.409410\pi\)
\(374\) 0 0
\(375\) 8.89873 0.459529
\(376\) 0 0
\(377\) 27.4605i 1.41429i
\(378\) 0 0
\(379\) 31.1103i 1.59803i 0.601310 + 0.799016i \(0.294646\pi\)
−0.601310 + 0.799016i \(0.705354\pi\)
\(380\) 0 0
\(381\) 41.8716i 2.14515i
\(382\) 0 0
\(383\) 10.5419 0.538665 0.269333 0.963047i \(-0.413197\pi\)
0.269333 + 0.963047i \(0.413197\pi\)
\(384\) 0 0
\(385\) 5.31128i 0.270688i
\(386\) 0 0
\(387\) 36.5936 1.86015
\(388\) 0 0
\(389\) 26.2390 1.33037 0.665185 0.746678i \(-0.268353\pi\)
0.665185 + 0.746678i \(0.268353\pi\)
\(390\) 0 0
\(391\) 35.3969 2.86014i 1.79010 0.144644i
\(392\) 0 0
\(393\) −59.9449 −3.02382
\(394\) 0 0
\(395\) −3.05822 −0.153876
\(396\) 0 0
\(397\) 5.97201i 0.299727i −0.988707 0.149863i \(-0.952117\pi\)
0.988707 0.149863i \(-0.0478834\pi\)
\(398\) 0 0
\(399\) 124.968 6.25620
\(400\) 0 0
\(401\) 7.75394i 0.387213i 0.981079 + 0.193607i \(0.0620185\pi\)
−0.981079 + 0.193607i \(0.937981\pi\)
\(402\) 0 0
\(403\) 23.4710i 1.16917i
\(404\) 0 0
\(405\) 6.86774i 0.341261i
\(406\) 0 0
\(407\) −20.7111 −1.02661
\(408\) 0 0
\(409\) 20.4178 1.00959 0.504797 0.863238i \(-0.331567\pi\)
0.504797 + 0.863238i \(0.331567\pi\)
\(410\) 0 0
\(411\) 2.95982i 0.145997i
\(412\) 0 0
\(413\) 5.07344i 0.249647i
\(414\) 0 0
\(415\) 4.72108i 0.231749i
\(416\) 0 0
\(417\) −52.0189 −2.54738
\(418\) 0 0
\(419\) 7.34885i 0.359015i −0.983757 0.179507i \(-0.942550\pi\)
0.983757 0.179507i \(-0.0574504\pi\)
\(420\) 0 0
\(421\) −6.95825 −0.339124 −0.169562 0.985519i \(-0.554235\pi\)
−0.169562 + 0.985519i \(0.554235\pi\)
\(422\) 0 0
\(423\) 43.4989 2.11499
\(424\) 0 0
\(425\) 1.63495 + 20.2340i 0.0793068 + 0.981493i
\(426\) 0 0
\(427\) −61.9393 −2.99745
\(428\) 0 0
\(429\) −41.6646 −2.01159
\(430\) 0 0
\(431\) 2.72846i 0.131425i 0.997839 + 0.0657126i \(0.0209320\pi\)
−0.997839 + 0.0657126i \(0.979068\pi\)
\(432\) 0 0
\(433\) 34.3330 1.64994 0.824970 0.565177i \(-0.191192\pi\)
0.824970 + 0.565177i \(0.191192\pi\)
\(434\) 0 0
\(435\) 7.24907i 0.347566i
\(436\) 0 0
\(437\) 65.4525i 3.13102i
\(438\) 0 0
\(439\) 0.862214i 0.0411512i 0.999788 + 0.0205756i \(0.00654988\pi\)
−0.999788 + 0.0205756i \(0.993450\pi\)
\(440\) 0 0
\(441\) 140.664 6.69826
\(442\) 0 0
\(443\) −18.3223 −0.870520 −0.435260 0.900305i \(-0.643344\pi\)
−0.435260 + 0.900305i \(0.643344\pi\)
\(444\) 0 0
\(445\) 0.994482i 0.0471429i
\(446\) 0 0
\(447\) 58.0401i 2.74520i
\(448\) 0 0
\(449\) 27.0182i 1.27507i −0.770422 0.637534i \(-0.779954\pi\)
0.770422 0.637534i \(-0.220046\pi\)
\(450\) 0 0
\(451\) −20.8667 −0.982576
\(452\) 0 0
\(453\) 37.5493i 1.76422i
\(454\) 0 0
\(455\) 4.76800 0.223527
\(456\) 0 0
\(457\) −31.6572 −1.48086 −0.740431 0.672133i \(-0.765378\pi\)
−0.740431 + 0.672133i \(0.765378\pi\)
\(458\) 0 0
\(459\) −60.0261 + 4.85024i −2.80178 + 0.226390i
\(460\) 0 0
\(461\) −5.48691 −0.255551 −0.127775 0.991803i \(-0.540784\pi\)
−0.127775 + 0.991803i \(0.540784\pi\)
\(462\) 0 0
\(463\) 20.4473 0.950266 0.475133 0.879914i \(-0.342400\pi\)
0.475133 + 0.879914i \(0.342400\pi\)
\(464\) 0 0
\(465\) 6.19590i 0.287328i
\(466\) 0 0
\(467\) −19.8991 −0.920819 −0.460410 0.887707i \(-0.652297\pi\)
−0.460410 + 0.887707i \(0.652297\pi\)
\(468\) 0 0
\(469\) 44.0118i 2.03228i
\(470\) 0 0
\(471\) 3.75527i 0.173034i
\(472\) 0 0
\(473\) 18.4476i 0.848224i
\(474\) 0 0
\(475\) −37.4148 −1.71671
\(476\) 0 0
\(477\) 28.8967 1.32309
\(478\) 0 0
\(479\) 13.2483i 0.605329i −0.953097 0.302664i \(-0.902124\pi\)
0.953097 0.302664i \(-0.0978761\pi\)
\(480\) 0 0
\(481\) 18.5927i 0.847752i
\(482\) 0 0
\(483\) 141.637i 6.44472i
\(484\) 0 0
\(485\) 0.0289535 0.00131471
\(486\) 0 0
\(487\) 24.3643i 1.10405i 0.833827 + 0.552026i \(0.186145\pi\)
−0.833827 + 0.552026i \(0.813855\pi\)
\(488\) 0 0
\(489\) 41.9753 1.89819
\(490\) 0 0
\(491\) 6.31961 0.285200 0.142600 0.989780i \(-0.454454\pi\)
0.142600 + 0.989780i \(0.454454\pi\)
\(492\) 0 0
\(493\) −33.2222 + 2.68443i −1.49625 + 0.120901i
\(494\) 0 0
\(495\) −7.85805 −0.353193
\(496\) 0 0
\(497\) −1.46584 −0.0657520
\(498\) 0 0
\(499\) 16.9416i 0.758409i −0.925313 0.379205i \(-0.876198\pi\)
0.925313 0.379205i \(-0.123802\pi\)
\(500\) 0 0
\(501\) −40.3124 −1.80103
\(502\) 0 0
\(503\) 9.18135i 0.409376i −0.978827 0.204688i \(-0.934382\pi\)
0.978827 0.204688i \(-0.0656180\pi\)
\(504\) 0 0
\(505\) 1.36120i 0.0605724i
\(506\) 0 0
\(507\) 4.73431i 0.210258i
\(508\) 0 0
\(509\) 17.5530 0.778022 0.389011 0.921233i \(-0.372817\pi\)
0.389011 + 0.921233i \(0.372817\pi\)
\(510\) 0 0
\(511\) −7.13526 −0.315645
\(512\) 0 0
\(513\) 110.995i 4.90053i
\(514\) 0 0
\(515\) 1.51233i 0.0666410i
\(516\) 0 0
\(517\) 21.9288i 0.964426i
\(518\) 0 0
\(519\) 37.6692 1.65349
\(520\) 0 0
\(521\) 2.93306i 0.128499i 0.997934 + 0.0642497i \(0.0204654\pi\)
−0.997934 + 0.0642497i \(0.979535\pi\)
\(522\) 0 0
\(523\) −37.8910 −1.65686 −0.828428 0.560095i \(-0.810764\pi\)
−0.828428 + 0.560095i \(0.810764\pi\)
\(524\) 0 0
\(525\) −80.9645 −3.53358
\(526\) 0 0
\(527\) −28.3956 + 2.29443i −1.23693 + 0.0999468i
\(528\) 0 0
\(529\) −51.1834 −2.22537
\(530\) 0 0
\(531\) −7.50616 −0.325740
\(532\) 0 0
\(533\) 18.7323i 0.811387i
\(534\) 0 0
\(535\) −0.742673 −0.0321086
\(536\) 0 0
\(537\) 7.85924i 0.339151i
\(538\) 0 0
\(539\) 70.9117i 3.05438i
\(540\) 0 0
\(541\) 5.49180i 0.236111i −0.993007 0.118055i \(-0.962334\pi\)
0.993007 0.118055i \(-0.0376661\pi\)
\(542\) 0 0
\(543\) 39.1464 1.67993
\(544\) 0 0
\(545\) −0.627590 −0.0268830
\(546\) 0 0
\(547\) 41.6097i 1.77910i −0.456835 0.889551i \(-0.651017\pi\)
0.456835 0.889551i \(-0.348983\pi\)
\(548\) 0 0
\(549\) 91.6393i 3.91107i
\(550\) 0 0
\(551\) 61.4314i 2.61707i
\(552\) 0 0
\(553\) 56.0825 2.38487
\(554\) 0 0
\(555\) 4.90812i 0.208338i
\(556\) 0 0
\(557\) −30.2076 −1.27994 −0.639969 0.768401i \(-0.721053\pi\)
−0.639969 + 0.768401i \(0.721053\pi\)
\(558\) 0 0
\(559\) −16.5607 −0.700442
\(560\) 0 0
\(561\) −4.07296 50.4066i −0.171961 2.12817i
\(562\) 0 0
\(563\) −18.2689 −0.769943 −0.384972 0.922928i \(-0.625789\pi\)
−0.384972 + 0.922928i \(0.625789\pi\)
\(564\) 0 0
\(565\) 1.44817 0.0609251
\(566\) 0 0
\(567\) 125.943i 5.28910i
\(568\) 0 0
\(569\) −43.7680 −1.83485 −0.917424 0.397910i \(-0.869736\pi\)
−0.917424 + 0.397910i \(0.869736\pi\)
\(570\) 0 0
\(571\) 14.1406i 0.591767i −0.955224 0.295884i \(-0.904386\pi\)
0.955224 0.295884i \(-0.0956140\pi\)
\(572\) 0 0
\(573\) 55.4299i 2.31562i
\(574\) 0 0
\(575\) 42.4057i 1.76844i
\(576\) 0 0
\(577\) −38.7490 −1.61314 −0.806572 0.591136i \(-0.798680\pi\)
−0.806572 + 0.591136i \(0.798680\pi\)
\(578\) 0 0
\(579\) 39.5186 1.64234
\(580\) 0 0
\(581\) 86.5766i 3.59180i
\(582\) 0 0
\(583\) 14.5675i 0.603323i
\(584\) 0 0
\(585\) 7.05427i 0.291658i
\(586\) 0 0
\(587\) 11.2133 0.462821 0.231411 0.972856i \(-0.425666\pi\)
0.231411 + 0.972856i \(0.425666\pi\)
\(588\) 0 0
\(589\) 52.5064i 2.16349i
\(590\) 0 0
\(591\) −15.1823 −0.624518
\(592\) 0 0
\(593\) −9.60251 −0.394328 −0.197164 0.980371i \(-0.563173\pi\)
−0.197164 + 0.980371i \(0.563173\pi\)
\(594\) 0 0
\(595\) 0.466101 + 5.76842i 0.0191083 + 0.236482i
\(596\) 0 0
\(597\) −20.2730 −0.829718
\(598\) 0 0
\(599\) −7.85745 −0.321047 −0.160523 0.987032i \(-0.551318\pi\)
−0.160523 + 0.987032i \(0.551318\pi\)
\(600\) 0 0
\(601\) 0.430072i 0.0175430i 0.999962 + 0.00877150i \(0.00279209\pi\)
−0.999962 + 0.00877150i \(0.997208\pi\)
\(602\) 0 0
\(603\) −65.1156 −2.65171
\(604\) 0 0
\(605\) 0.918186i 0.0373296i
\(606\) 0 0
\(607\) 29.1391i 1.18272i 0.806408 + 0.591360i \(0.201409\pi\)
−0.806408 + 0.591360i \(0.798591\pi\)
\(608\) 0 0
\(609\) 132.936i 5.38683i
\(610\) 0 0
\(611\) −19.6857 −0.796400
\(612\) 0 0
\(613\) −10.6321 −0.429428 −0.214714 0.976677i \(-0.568882\pi\)
−0.214714 + 0.976677i \(0.568882\pi\)
\(614\) 0 0
\(615\) 4.94499i 0.199401i
\(616\) 0 0
\(617\) 13.6550i 0.549729i 0.961483 + 0.274865i \(0.0886330\pi\)
−0.961483 + 0.274865i \(0.911367\pi\)
\(618\) 0 0
\(619\) 11.8643i 0.476866i 0.971159 + 0.238433i \(0.0766337\pi\)
−0.971159 + 0.238433i \(0.923366\pi\)
\(620\) 0 0
\(621\) 125.800 5.04819
\(622\) 0 0
\(623\) 18.2371i 0.730654i
\(624\) 0 0
\(625\) 23.8578 0.954311
\(626\) 0 0
\(627\) 93.2071 3.72233
\(628\) 0 0
\(629\) 22.4937 1.81754i 0.896883 0.0724702i
\(630\) 0 0
\(631\) −9.01753 −0.358982 −0.179491 0.983760i \(-0.557445\pi\)
−0.179491 + 0.983760i \(0.557445\pi\)
\(632\) 0 0
\(633\) −32.9102 −1.30806
\(634\) 0 0
\(635\) 3.57389i 0.141825i
\(636\) 0 0
\(637\) −63.6583 −2.52223
\(638\) 0 0
\(639\) 2.16871i 0.0857930i
\(640\) 0 0
\(641\) 2.31701i 0.0915166i −0.998953 0.0457583i \(-0.985430\pi\)
0.998953 0.0457583i \(-0.0145704\pi\)
\(642\) 0 0
\(643\) 0.784054i 0.0309201i −0.999880 0.0154600i \(-0.995079\pi\)
0.999880 0.0154600i \(-0.00492128\pi\)
\(644\) 0 0
\(645\) −4.37171 −0.172136
\(646\) 0 0
\(647\) 15.7454 0.619015 0.309508 0.950897i \(-0.399836\pi\)
0.309508 + 0.950897i \(0.399836\pi\)
\(648\) 0 0
\(649\) 3.78403i 0.148536i
\(650\) 0 0
\(651\) 113.622i 4.45321i
\(652\) 0 0
\(653\) 22.7524i 0.890372i −0.895438 0.445186i \(-0.853138\pi\)
0.895438 0.445186i \(-0.146862\pi\)
\(654\) 0 0
\(655\) 5.11651 0.199918
\(656\) 0 0
\(657\) 10.5566i 0.411854i
\(658\) 0 0
\(659\) −33.4013 −1.30113 −0.650564 0.759451i \(-0.725467\pi\)
−0.650564 + 0.759451i \(0.725467\pi\)
\(660\) 0 0
\(661\) −28.2313 −1.09807 −0.549035 0.835799i \(-0.685005\pi\)
−0.549035 + 0.835799i \(0.685005\pi\)
\(662\) 0 0
\(663\) 45.2506 3.65635i 1.75739 0.142001i
\(664\) 0 0
\(665\) −10.6664 −0.413626
\(666\) 0 0
\(667\) 69.6259 2.69593
\(668\) 0 0
\(669\) 83.4822i 3.22761i
\(670\) 0 0
\(671\) −46.1975 −1.78343
\(672\) 0 0
\(673\) 5.74164i 0.221324i −0.993858 0.110662i \(-0.964703\pi\)
0.993858 0.110662i \(-0.0352971\pi\)
\(674\) 0 0
\(675\) 71.9116i 2.76788i
\(676\) 0 0
\(677\) 42.8922i 1.64848i −0.566241 0.824240i \(-0.691603\pi\)
0.566241 0.824240i \(-0.308397\pi\)
\(678\) 0 0
\(679\) −0.530959 −0.0203763
\(680\) 0 0
\(681\) 62.9878 2.41370
\(682\) 0 0
\(683\) 14.8568i 0.568481i −0.958753 0.284241i \(-0.908259\pi\)
0.958753 0.284241i \(-0.0917415\pi\)
\(684\) 0 0
\(685\) 0.252631i 0.00965253i
\(686\) 0 0
\(687\) 48.9409i 1.86721i
\(688\) 0 0
\(689\) −13.0774 −0.498210
\(690\) 0 0
\(691\) 34.5431i 1.31408i −0.753854 0.657042i \(-0.771808\pi\)
0.753854 0.657042i \(-0.228192\pi\)
\(692\) 0 0
\(693\) 144.103 5.47403
\(694\) 0 0
\(695\) 4.43999 0.168419
\(696\) 0 0
\(697\) 22.6627 1.83120i 0.858411 0.0693615i
\(698\) 0 0
\(699\) −8.06056 −0.304878
\(700\) 0 0
\(701\) −14.1138 −0.533071 −0.266536 0.963825i \(-0.585879\pi\)
−0.266536 + 0.963825i \(0.585879\pi\)
\(702\) 0 0
\(703\) 41.5933i 1.56872i
\(704\) 0 0
\(705\) −5.19667 −0.195718
\(706\) 0 0
\(707\) 24.9620i 0.938793i
\(708\) 0 0
\(709\) 42.2180i 1.58553i 0.609527 + 0.792765i \(0.291359\pi\)
−0.609527 + 0.792765i \(0.708641\pi\)
\(710\) 0 0
\(711\) 82.9742i 3.11178i
\(712\) 0 0
\(713\) 59.5104 2.22868
\(714\) 0 0
\(715\) 3.55622 0.132995
\(716\) 0 0
\(717\) 53.8618i 2.01151i
\(718\) 0 0
\(719\) 29.5661i 1.10263i −0.834297 0.551315i \(-0.814126\pi\)
0.834297 0.551315i \(-0.185874\pi\)
\(720\) 0 0
\(721\) 27.7335i 1.03285i
\(722\) 0 0
\(723\) −36.3580 −1.35217
\(724\) 0 0
\(725\) 39.8004i 1.47815i
\(726\) 0 0
\(727\) −17.5612 −0.651309 −0.325654 0.945489i \(-0.605585\pi\)
−0.325654 + 0.945489i \(0.605585\pi\)
\(728\) 0 0
\(729\) −44.3052 −1.64093
\(730\) 0 0
\(731\) −1.61891 20.0354i −0.0598774 0.741036i
\(732\) 0 0
\(733\) 19.1176 0.706126 0.353063 0.935600i \(-0.385140\pi\)
0.353063 + 0.935600i \(0.385140\pi\)
\(734\) 0 0
\(735\) −16.8046 −0.619848
\(736\) 0 0
\(737\) 32.8263i 1.20917i
\(738\) 0 0
\(739\) −26.3055 −0.967663 −0.483832 0.875161i \(-0.660755\pi\)
−0.483832 + 0.875161i \(0.660755\pi\)
\(740\) 0 0
\(741\) 83.6732i 3.07381i
\(742\) 0 0
\(743\) 8.87849i 0.325720i 0.986649 + 0.162860i \(0.0520719\pi\)
−0.986649 + 0.162860i \(0.947928\pi\)
\(744\) 0 0
\(745\) 4.95392i 0.181498i
\(746\) 0 0
\(747\) −128.090 −4.68658
\(748\) 0 0
\(749\) 13.6194 0.497641
\(750\) 0 0
\(751\) 24.6864i 0.900820i −0.892822 0.450410i \(-0.851278\pi\)
0.892822 0.450410i \(-0.148722\pi\)
\(752\) 0 0
\(753\) 41.6919i 1.51934i
\(754\) 0 0
\(755\) 3.20497i 0.116641i
\(756\) 0 0
\(757\) −10.0533 −0.365395 −0.182698 0.983169i \(-0.558483\pi\)
−0.182698 + 0.983169i \(0.558483\pi\)
\(758\) 0 0
\(759\) 105.640i 3.83450i
\(760\) 0 0
\(761\) −4.71026 −0.170747 −0.0853734 0.996349i \(-0.527208\pi\)
−0.0853734 + 0.996349i \(0.527208\pi\)
\(762\) 0 0
\(763\) 11.5089 0.416651
\(764\) 0 0
\(765\) 8.53438 0.689597i 0.308561 0.0249324i
\(766\) 0 0
\(767\) 3.39697 0.122657
\(768\) 0 0
\(769\) 12.0516 0.434593 0.217296 0.976106i \(-0.430276\pi\)
0.217296 + 0.976106i \(0.430276\pi\)
\(770\) 0 0
\(771\) 18.9459i 0.682320i
\(772\) 0 0
\(773\) −12.4335 −0.447201 −0.223600 0.974681i \(-0.571781\pi\)
−0.223600 + 0.974681i \(0.571781\pi\)
\(774\) 0 0
\(775\) 34.0181i 1.22197i
\(776\) 0 0
\(777\) 90.0065i 3.22897i
\(778\) 0 0
\(779\) 41.9057i 1.50143i
\(780\) 0 0
\(781\) −1.09330 −0.0391213
\(782\) 0 0
\(783\) −118.072 −4.21954
\(784\) 0 0
\(785\) 0.320525i 0.0114400i
\(786\) 0 0
\(787\) 2.86175i 0.102010i −0.998698 0.0510052i \(-0.983757\pi\)
0.998698 0.0510052i \(-0.0162425\pi\)
\(788\) 0 0
\(789\) 37.2876i 1.32747i
\(790\) 0 0
\(791\) −26.5570 −0.944260
\(792\) 0 0
\(793\) 41.4720i