Properties

Label 4012.2.b.b.237.18
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 237.18
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.29

$q$-expansion

\(f(q)\) \(=\) \(q-1.13278i q^{3} -2.09684i q^{5} +2.71039i q^{7} +1.71682 q^{9} +O(q^{10})\) \(q-1.13278i q^{3} -2.09684i q^{5} +2.71039i q^{7} +1.71682 q^{9} -3.86380i q^{11} -2.19264 q^{13} -2.37525 q^{15} +(1.04117 - 3.98948i) q^{17} +7.57941 q^{19} +3.07027 q^{21} +3.94101i q^{23} +0.603254 q^{25} -5.34310i q^{27} -2.64568i q^{29} -4.90386i q^{31} -4.37683 q^{33} +5.68326 q^{35} +11.3346i q^{37} +2.48378i q^{39} +4.94646i q^{41} -5.42674 q^{43} -3.59989i q^{45} +6.03939 q^{47} -0.346228 q^{49} +(-4.51919 - 1.17941i) q^{51} -1.40329 q^{53} -8.10178 q^{55} -8.58578i q^{57} +1.00000 q^{59} -13.3980i q^{61} +4.65324i q^{63} +4.59763i q^{65} +1.98465 q^{67} +4.46428 q^{69} -4.42275i q^{71} +1.70785i q^{73} -0.683352i q^{75} +10.4724 q^{77} -4.77934i q^{79} -0.902097 q^{81} +9.20975 q^{83} +(-8.36531 - 2.18317i) q^{85} -2.99696 q^{87} -4.58193 q^{89} -5.94293i q^{91} -5.55498 q^{93} -15.8928i q^{95} -14.2966i q^{97} -6.63343i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

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
\(3\) 1.13278i 0.654009i −0.945023 0.327005i \(-0.893961\pi\)
0.945023 0.327005i \(-0.106039\pi\)
\(4\) 0 0
\(5\) 2.09684i 0.937736i −0.883268 0.468868i \(-0.844662\pi\)
0.883268 0.468868i \(-0.155338\pi\)
\(6\) 0 0
\(7\) 2.71039i 1.02443i 0.858857 + 0.512216i \(0.171175\pi\)
−0.858857 + 0.512216i \(0.828825\pi\)
\(8\) 0 0
\(9\) 1.71682 0.572272
\(10\) 0 0
\(11\) 3.86380i 1.16498i −0.812838 0.582490i \(-0.802079\pi\)
0.812838 0.582490i \(-0.197921\pi\)
\(12\) 0 0
\(13\) −2.19264 −0.608130 −0.304065 0.952651i \(-0.598344\pi\)
−0.304065 + 0.952651i \(0.598344\pi\)
\(14\) 0 0
\(15\) −2.37525 −0.613288
\(16\) 0 0
\(17\) 1.04117 3.98948i 0.252521 0.967591i
\(18\) 0 0
\(19\) 7.57941 1.73884 0.869418 0.494078i \(-0.164494\pi\)
0.869418 + 0.494078i \(0.164494\pi\)
\(20\) 0 0
\(21\) 3.07027 0.669988
\(22\) 0 0
\(23\) 3.94101i 0.821757i 0.911690 + 0.410878i \(0.134778\pi\)
−0.911690 + 0.410878i \(0.865222\pi\)
\(24\) 0 0
\(25\) 0.603254 0.120651
\(26\) 0 0
\(27\) 5.34310i 1.02828i
\(28\) 0 0
\(29\) 2.64568i 0.491290i −0.969360 0.245645i \(-0.921000\pi\)
0.969360 0.245645i \(-0.0789998\pi\)
\(30\) 0 0
\(31\) 4.90386i 0.880759i −0.897812 0.440380i \(-0.854844\pi\)
0.897812 0.440380i \(-0.145156\pi\)
\(32\) 0 0
\(33\) −4.37683 −0.761907
\(34\) 0 0
\(35\) 5.68326 0.960647
\(36\) 0 0
\(37\) 11.3346i 1.86339i 0.363242 + 0.931695i \(0.381670\pi\)
−0.363242 + 0.931695i \(0.618330\pi\)
\(38\) 0 0
\(39\) 2.48378i 0.397723i
\(40\) 0 0
\(41\) 4.94646i 0.772508i 0.922393 + 0.386254i \(0.126231\pi\)
−0.922393 + 0.386254i \(0.873769\pi\)
\(42\) 0 0
\(43\) −5.42674 −0.827571 −0.413785 0.910374i \(-0.635794\pi\)
−0.413785 + 0.910374i \(0.635794\pi\)
\(44\) 0 0
\(45\) 3.59989i 0.536640i
\(46\) 0 0
\(47\) 6.03939 0.880936 0.440468 0.897768i \(-0.354812\pi\)
0.440468 + 0.897768i \(0.354812\pi\)
\(48\) 0 0
\(49\) −0.346228 −0.0494611
\(50\) 0 0
\(51\) −4.51919 1.17941i −0.632814 0.165151i
\(52\) 0 0
\(53\) −1.40329 −0.192756 −0.0963782 0.995345i \(-0.530726\pi\)
−0.0963782 + 0.995345i \(0.530726\pi\)
\(54\) 0 0
\(55\) −8.10178 −1.09244
\(56\) 0 0
\(57\) 8.58578i 1.13721i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 13.3980i 1.71544i −0.514120 0.857718i \(-0.671881\pi\)
0.514120 0.857718i \(-0.328119\pi\)
\(62\) 0 0
\(63\) 4.65324i 0.586254i
\(64\) 0 0
\(65\) 4.59763i 0.570266i
\(66\) 0 0
\(67\) 1.98465 0.242464 0.121232 0.992624i \(-0.461316\pi\)
0.121232 + 0.992624i \(0.461316\pi\)
\(68\) 0 0
\(69\) 4.46428 0.537436
\(70\) 0 0
\(71\) 4.42275i 0.524883i −0.964948 0.262442i \(-0.915472\pi\)
0.964948 0.262442i \(-0.0845277\pi\)
\(72\) 0 0
\(73\) 1.70785i 0.199889i 0.994993 + 0.0999445i \(0.0318665\pi\)
−0.994993 + 0.0999445i \(0.968133\pi\)
\(74\) 0 0
\(75\) 0.683352i 0.0789067i
\(76\) 0 0
\(77\) 10.4724 1.19344
\(78\) 0 0
\(79\) 4.77934i 0.537717i −0.963180 0.268859i \(-0.913354\pi\)
0.963180 0.268859i \(-0.0866465\pi\)
\(80\) 0 0
\(81\) −0.902097 −0.100233
\(82\) 0 0
\(83\) 9.20975 1.01090 0.505451 0.862856i \(-0.331326\pi\)
0.505451 + 0.862856i \(0.331326\pi\)
\(84\) 0 0
\(85\) −8.36531 2.18317i −0.907346 0.236798i
\(86\) 0 0
\(87\) −2.99696 −0.321308
\(88\) 0 0
\(89\) −4.58193 −0.485683 −0.242842 0.970066i \(-0.578080\pi\)
−0.242842 + 0.970066i \(0.578080\pi\)
\(90\) 0 0
\(91\) 5.94293i 0.622988i
\(92\) 0 0
\(93\) −5.55498 −0.576025
\(94\) 0 0
\(95\) 15.8928i 1.63057i
\(96\) 0 0
\(97\) 14.2966i 1.45160i −0.687906 0.725800i \(-0.741470\pi\)
0.687906 0.725800i \(-0.258530\pi\)
\(98\) 0 0
\(99\) 6.63343i 0.666685i
\(100\) 0 0
\(101\) 2.19359 0.218270 0.109135 0.994027i \(-0.465192\pi\)
0.109135 + 0.994027i \(0.465192\pi\)
\(102\) 0 0
\(103\) −6.86304 −0.676236 −0.338118 0.941104i \(-0.609790\pi\)
−0.338118 + 0.941104i \(0.609790\pi\)
\(104\) 0 0
\(105\) 6.43787i 0.628272i
\(106\) 0 0
\(107\) 7.37881i 0.713337i −0.934231 0.356668i \(-0.883913\pi\)
0.934231 0.356668i \(-0.116087\pi\)
\(108\) 0 0
\(109\) 8.73085i 0.836264i 0.908386 + 0.418132i \(0.137315\pi\)
−0.908386 + 0.418132i \(0.862685\pi\)
\(110\) 0 0
\(111\) 12.8395 1.21867
\(112\) 0 0
\(113\) 18.8264i 1.77103i 0.464607 + 0.885517i \(0.346196\pi\)
−0.464607 + 0.885517i \(0.653804\pi\)
\(114\) 0 0
\(115\) 8.26367 0.770591
\(116\) 0 0
\(117\) −3.76437 −0.348016
\(118\) 0 0
\(119\) 10.8131 + 2.82198i 0.991232 + 0.258691i
\(120\) 0 0
\(121\) −3.92895 −0.357178
\(122\) 0 0
\(123\) 5.60324 0.505227
\(124\) 0 0
\(125\) 11.7491i 1.05087i
\(126\) 0 0
\(127\) 0.105365 0.00934965 0.00467482 0.999989i \(-0.498512\pi\)
0.00467482 + 0.999989i \(0.498512\pi\)
\(128\) 0 0
\(129\) 6.14729i 0.541239i
\(130\) 0 0
\(131\) 11.2515i 0.983050i −0.870864 0.491525i \(-0.836440\pi\)
0.870864 0.491525i \(-0.163560\pi\)
\(132\) 0 0
\(133\) 20.5432i 1.78132i
\(134\) 0 0
\(135\) −11.2036 −0.964256
\(136\) 0 0
\(137\) 12.3796 1.05766 0.528830 0.848728i \(-0.322631\pi\)
0.528830 + 0.848728i \(0.322631\pi\)
\(138\) 0 0
\(139\) 11.3513i 0.962806i −0.876499 0.481403i \(-0.840127\pi\)
0.876499 0.481403i \(-0.159873\pi\)
\(140\) 0 0
\(141\) 6.84129i 0.576140i
\(142\) 0 0
\(143\) 8.47194i 0.708459i
\(144\) 0 0
\(145\) −5.54757 −0.460701
\(146\) 0 0
\(147\) 0.392199i 0.0323480i
\(148\) 0 0
\(149\) 5.29922 0.434129 0.217064 0.976157i \(-0.430352\pi\)
0.217064 + 0.976157i \(0.430352\pi\)
\(150\) 0 0
\(151\) −21.8368 −1.77705 −0.888527 0.458825i \(-0.848271\pi\)
−0.888527 + 0.458825i \(0.848271\pi\)
\(152\) 0 0
\(153\) 1.78750 6.84920i 0.144511 0.553725i
\(154\) 0 0
\(155\) −10.2826 −0.825920
\(156\) 0 0
\(157\) −24.9017 −1.98737 −0.993687 0.112188i \(-0.964214\pi\)
−0.993687 + 0.112188i \(0.964214\pi\)
\(158\) 0 0
\(159\) 1.58961i 0.126064i
\(160\) 0 0
\(161\) −10.6817 −0.841834
\(162\) 0 0
\(163\) 24.0233i 1.88165i −0.338893 0.940825i \(-0.610053\pi\)
0.338893 0.940825i \(-0.389947\pi\)
\(164\) 0 0
\(165\) 9.17751i 0.714468i
\(166\) 0 0
\(167\) 2.28288i 0.176654i −0.996092 0.0883272i \(-0.971848\pi\)
0.996092 0.0883272i \(-0.0281521\pi\)
\(168\) 0 0
\(169\) −8.19231 −0.630178
\(170\) 0 0
\(171\) 13.0124 0.995086
\(172\) 0 0
\(173\) 4.75068i 0.361187i −0.983558 0.180594i \(-0.942198\pi\)
0.983558 0.180594i \(-0.0578019\pi\)
\(174\) 0 0
\(175\) 1.63505i 0.123598i
\(176\) 0 0
\(177\) 1.13278i 0.0851448i
\(178\) 0 0
\(179\) 20.3267 1.51929 0.759644 0.650339i \(-0.225373\pi\)
0.759644 + 0.650339i \(0.225373\pi\)
\(180\) 0 0
\(181\) 2.46150i 0.182962i 0.995807 + 0.0914810i \(0.0291601\pi\)
−0.995807 + 0.0914810i \(0.970840\pi\)
\(182\) 0 0
\(183\) −15.1769 −1.12191
\(184\) 0 0
\(185\) 23.7668 1.74737
\(186\) 0 0
\(187\) −15.4146 4.02288i −1.12722 0.294182i
\(188\) 0 0
\(189\) 14.4819 1.05340
\(190\) 0 0
\(191\) −6.81181 −0.492885 −0.246443 0.969157i \(-0.579262\pi\)
−0.246443 + 0.969157i \(0.579262\pi\)
\(192\) 0 0
\(193\) 1.68256i 0.121114i 0.998165 + 0.0605568i \(0.0192876\pi\)
−0.998165 + 0.0605568i \(0.980712\pi\)
\(194\) 0 0
\(195\) 5.20809 0.372959
\(196\) 0 0
\(197\) 24.2376i 1.72686i 0.504471 + 0.863429i \(0.331688\pi\)
−0.504471 + 0.863429i \(0.668312\pi\)
\(198\) 0 0
\(199\) 6.44922i 0.457173i 0.973524 + 0.228586i \(0.0734103\pi\)
−0.973524 + 0.228586i \(0.926590\pi\)
\(200\) 0 0
\(201\) 2.24817i 0.158574i
\(202\) 0 0
\(203\) 7.17083 0.503293
\(204\) 0 0
\(205\) 10.3720 0.724409
\(206\) 0 0
\(207\) 6.76598i 0.470268i
\(208\) 0 0
\(209\) 29.2853i 2.02571i
\(210\) 0 0
\(211\) 0.132264i 0.00910545i 0.999990 + 0.00455273i \(0.00144918\pi\)
−0.999990 + 0.00455273i \(0.998551\pi\)
\(212\) 0 0
\(213\) −5.00999 −0.343279
\(214\) 0 0
\(215\) 11.3790i 0.776043i
\(216\) 0 0
\(217\) 13.2914 0.902278
\(218\) 0 0
\(219\) 1.93462 0.130729
\(220\) 0 0
\(221\) −2.28292 + 8.74752i −0.153566 + 0.588422i
\(222\) 0 0
\(223\) 13.9089 0.931406 0.465703 0.884941i \(-0.345801\pi\)
0.465703 + 0.884941i \(0.345801\pi\)
\(224\) 0 0
\(225\) 1.03568 0.0690450
\(226\) 0 0
\(227\) 9.66541i 0.641516i 0.947161 + 0.320758i \(0.103938\pi\)
−0.947161 + 0.320758i \(0.896062\pi\)
\(228\) 0 0
\(229\) −8.96547 −0.592455 −0.296228 0.955117i \(-0.595729\pi\)
−0.296228 + 0.955117i \(0.595729\pi\)
\(230\) 0 0
\(231\) 11.8629i 0.780522i
\(232\) 0 0
\(233\) 8.30445i 0.544042i −0.962291 0.272021i \(-0.912308\pi\)
0.962291 0.272021i \(-0.0876921\pi\)
\(234\) 0 0
\(235\) 12.6637i 0.826086i
\(236\) 0 0
\(237\) −5.41392 −0.351672
\(238\) 0 0
\(239\) −20.9404 −1.35452 −0.677260 0.735744i \(-0.736833\pi\)
−0.677260 + 0.735744i \(0.736833\pi\)
\(240\) 0 0
\(241\) 17.1231i 1.10300i −0.834175 0.551499i \(-0.814056\pi\)
0.834175 0.551499i \(-0.185944\pi\)
\(242\) 0 0
\(243\) 15.0074i 0.962727i
\(244\) 0 0
\(245\) 0.725985i 0.0463815i
\(246\) 0 0
\(247\) −16.6189 −1.05744
\(248\) 0 0
\(249\) 10.4326i 0.661139i
\(250\) 0 0
\(251\) 5.61892 0.354663 0.177332 0.984151i \(-0.443253\pi\)
0.177332 + 0.984151i \(0.443253\pi\)
\(252\) 0 0
\(253\) 15.2273 0.957330
\(254\) 0 0
\(255\) −2.47305 + 9.47604i −0.154868 + 0.593412i
\(256\) 0 0
\(257\) −25.3108 −1.57884 −0.789421 0.613852i \(-0.789619\pi\)
−0.789421 + 0.613852i \(0.789619\pi\)
\(258\) 0 0
\(259\) −30.7211 −1.90892
\(260\) 0 0
\(261\) 4.54214i 0.281152i
\(262\) 0 0
\(263\) −23.6724 −1.45970 −0.729851 0.683606i \(-0.760411\pi\)
−0.729851 + 0.683606i \(0.760411\pi\)
\(264\) 0 0
\(265\) 2.94247i 0.180755i
\(266\) 0 0
\(267\) 5.19030i 0.317641i
\(268\) 0 0
\(269\) 10.1941i 0.621547i −0.950484 0.310774i \(-0.899412\pi\)
0.950484 0.310774i \(-0.100588\pi\)
\(270\) 0 0
\(271\) −27.4239 −1.66588 −0.832940 0.553363i \(-0.813344\pi\)
−0.832940 + 0.553363i \(0.813344\pi\)
\(272\) 0 0
\(273\) −6.73201 −0.407440
\(274\) 0 0
\(275\) 2.33085i 0.140556i
\(276\) 0 0
\(277\) 11.8920i 0.714522i −0.934005 0.357261i \(-0.883711\pi\)
0.934005 0.357261i \(-0.116289\pi\)
\(278\) 0 0
\(279\) 8.41902i 0.504034i
\(280\) 0 0
\(281\) 10.7275 0.639947 0.319974 0.947426i \(-0.396326\pi\)
0.319974 + 0.947426i \(0.396326\pi\)
\(282\) 0 0
\(283\) 8.18183i 0.486360i 0.969981 + 0.243180i \(0.0781905\pi\)
−0.969981 + 0.243180i \(0.921809\pi\)
\(284\) 0 0
\(285\) −18.0030 −1.06641
\(286\) 0 0
\(287\) −13.4069 −0.791382
\(288\) 0 0
\(289\) −14.8319 8.30746i −0.872466 0.488674i
\(290\) 0 0
\(291\) −16.1949 −0.949360
\(292\) 0 0
\(293\) 30.6434 1.79021 0.895103 0.445860i \(-0.147102\pi\)
0.895103 + 0.445860i \(0.147102\pi\)
\(294\) 0 0
\(295\) 2.09684i 0.122083i
\(296\) 0 0
\(297\) −20.6447 −1.19793
\(298\) 0 0
\(299\) 8.64123i 0.499735i
\(300\) 0 0
\(301\) 14.7086i 0.847790i
\(302\) 0 0
\(303\) 2.48485i 0.142751i
\(304\) 0 0
\(305\) −28.0935 −1.60863
\(306\) 0 0
\(307\) 24.9866 1.42606 0.713030 0.701134i \(-0.247322\pi\)
0.713030 + 0.701134i \(0.247322\pi\)
\(308\) 0 0
\(309\) 7.77430i 0.442264i
\(310\) 0 0
\(311\) 34.7067i 1.96804i −0.178072 0.984018i \(-0.556986\pi\)
0.178072 0.984018i \(-0.443014\pi\)
\(312\) 0 0
\(313\) 20.8502i 1.17852i 0.807943 + 0.589261i \(0.200581\pi\)
−0.807943 + 0.589261i \(0.799419\pi\)
\(314\) 0 0
\(315\) 9.75712 0.549751
\(316\) 0 0
\(317\) 0.723762i 0.0406505i −0.999793 0.0203253i \(-0.993530\pi\)
0.999793 0.0203253i \(-0.00647018\pi\)
\(318\) 0 0
\(319\) −10.2224 −0.572343
\(320\) 0 0
\(321\) −8.35855 −0.466529
\(322\) 0 0
\(323\) 7.89145 30.2379i 0.439092 1.68248i
\(324\) 0 0
\(325\) −1.32272 −0.0733714
\(326\) 0 0
\(327\) 9.89011 0.546925
\(328\) 0 0
\(329\) 16.3691i 0.902459i
\(330\) 0 0
\(331\) −33.5150 −1.84215 −0.921076 0.389384i \(-0.872688\pi\)
−0.921076 + 0.389384i \(0.872688\pi\)
\(332\) 0 0
\(333\) 19.4593i 1.06637i
\(334\) 0 0
\(335\) 4.16150i 0.227367i
\(336\) 0 0
\(337\) 8.40561i 0.457883i 0.973440 + 0.228941i \(0.0735264\pi\)
−0.973440 + 0.228941i \(0.926474\pi\)
\(338\) 0 0
\(339\) 21.3261 1.15827
\(340\) 0 0
\(341\) −18.9475 −1.02607
\(342\) 0 0
\(343\) 18.0343i 0.973763i
\(344\) 0 0
\(345\) 9.36090i 0.503974i
\(346\) 0 0
\(347\) 1.71718i 0.0921828i 0.998937 + 0.0460914i \(0.0146765\pi\)
−0.998937 + 0.0460914i \(0.985323\pi\)
\(348\) 0 0
\(349\) 7.24134 0.387620 0.193810 0.981039i \(-0.437915\pi\)
0.193810 + 0.981039i \(0.437915\pi\)
\(350\) 0 0
\(351\) 11.7155i 0.625328i
\(352\) 0 0
\(353\) −4.15382 −0.221086 −0.110543 0.993871i \(-0.535259\pi\)
−0.110543 + 0.993871i \(0.535259\pi\)
\(354\) 0 0
\(355\) −9.27380 −0.492202
\(356\) 0 0
\(357\) 3.19668 12.2488i 0.169186 0.648275i
\(358\) 0 0
\(359\) 16.0839 0.848873 0.424437 0.905458i \(-0.360472\pi\)
0.424437 + 0.905458i \(0.360472\pi\)
\(360\) 0 0
\(361\) 38.4474 2.02355
\(362\) 0 0
\(363\) 4.45063i 0.233597i
\(364\) 0 0
\(365\) 3.58110 0.187443
\(366\) 0 0
\(367\) 11.1878i 0.583997i 0.956419 + 0.291998i \(0.0943202\pi\)
−0.956419 + 0.291998i \(0.905680\pi\)
\(368\) 0 0
\(369\) 8.49217i 0.442084i
\(370\) 0 0
\(371\) 3.80346i 0.197466i
\(372\) 0 0
\(373\) −16.9616 −0.878236 −0.439118 0.898429i \(-0.644709\pi\)
−0.439118 + 0.898429i \(0.644709\pi\)
\(374\) 0 0
\(375\) −13.3092 −0.687282
\(376\) 0 0
\(377\) 5.80103i 0.298768i
\(378\) 0 0
\(379\) 12.3024i 0.631930i −0.948771 0.315965i \(-0.897672\pi\)
0.948771 0.315965i \(-0.102328\pi\)
\(380\) 0 0
\(381\) 0.119355i 0.00611476i
\(382\) 0 0
\(383\) 3.44810 0.176190 0.0880948 0.996112i \(-0.471922\pi\)
0.0880948 + 0.996112i \(0.471922\pi\)
\(384\) 0 0
\(385\) 21.9590i 1.11913i
\(386\) 0 0
\(387\) −9.31672 −0.473595
\(388\) 0 0
\(389\) 33.5064 1.69884 0.849421 0.527716i \(-0.176951\pi\)
0.849421 + 0.527716i \(0.176951\pi\)
\(390\) 0 0
\(391\) 15.7226 + 4.10326i 0.795125 + 0.207511i
\(392\) 0 0
\(393\) −12.7455 −0.642924
\(394\) 0 0
\(395\) −10.0215 −0.504237
\(396\) 0 0
\(397\) 37.1344i 1.86372i 0.362816 + 0.931861i \(0.381815\pi\)
−0.362816 + 0.931861i \(0.618185\pi\)
\(398\) 0 0
\(399\) 23.2708 1.16500
\(400\) 0 0
\(401\) 7.63712i 0.381379i −0.981650 0.190690i \(-0.938928\pi\)
0.981650 0.190690i \(-0.0610724\pi\)
\(402\) 0 0
\(403\) 10.7524i 0.535616i
\(404\) 0 0
\(405\) 1.89155i 0.0939921i
\(406\) 0 0
\(407\) 43.7945 2.17081
\(408\) 0 0
\(409\) 1.03726 0.0512892 0.0256446 0.999671i \(-0.491836\pi\)
0.0256446 + 0.999671i \(0.491836\pi\)
\(410\) 0 0
\(411\) 14.0233i 0.691720i
\(412\) 0 0
\(413\) 2.71039i 0.133370i
\(414\) 0 0
\(415\) 19.3114i 0.947959i
\(416\) 0 0
\(417\) −12.8585 −0.629684
\(418\) 0 0
\(419\) 7.64135i 0.373304i 0.982426 + 0.186652i \(0.0597637\pi\)
−0.982426 + 0.186652i \(0.940236\pi\)
\(420\) 0 0
\(421\) 9.05833 0.441476 0.220738 0.975333i \(-0.429153\pi\)
0.220738 + 0.975333i \(0.429153\pi\)
\(422\) 0 0
\(423\) 10.3685 0.504135
\(424\) 0 0
\(425\) 0.628090 2.40667i 0.0304668 0.116741i
\(426\) 0 0
\(427\) 36.3138 1.75735
\(428\) 0 0
\(429\) 9.59682 0.463339
\(430\) 0 0
\(431\) 9.22394i 0.444302i 0.975012 + 0.222151i \(0.0713077\pi\)
−0.975012 + 0.222151i \(0.928692\pi\)
\(432\) 0 0
\(433\) 17.3283 0.832745 0.416373 0.909194i \(-0.363301\pi\)
0.416373 + 0.909194i \(0.363301\pi\)
\(434\) 0 0
\(435\) 6.28416i 0.301302i
\(436\) 0 0
\(437\) 29.8705i 1.42890i
\(438\) 0 0
\(439\) 10.5581i 0.503912i −0.967739 0.251956i \(-0.918926\pi\)
0.967739 0.251956i \(-0.0810738\pi\)
\(440\) 0 0
\(441\) −0.594409 −0.0283052
\(442\) 0 0
\(443\) 20.7429 0.985525 0.492762 0.870164i \(-0.335987\pi\)
0.492762 + 0.870164i \(0.335987\pi\)
\(444\) 0 0
\(445\) 9.60757i 0.455443i
\(446\) 0 0
\(447\) 6.00283i 0.283924i
\(448\) 0 0
\(449\) 20.7979i 0.981514i 0.871296 + 0.490757i \(0.163280\pi\)
−0.871296 + 0.490757i \(0.836720\pi\)
\(450\) 0 0
\(451\) 19.1121 0.899956
\(452\) 0 0
\(453\) 24.7362i 1.16221i
\(454\) 0 0
\(455\) −12.4614 −0.584199
\(456\) 0 0
\(457\) 22.8328 1.06807 0.534036 0.845462i \(-0.320675\pi\)
0.534036 + 0.845462i \(0.320675\pi\)
\(458\) 0 0
\(459\) −21.3162 5.56308i −0.994955 0.259662i
\(460\) 0 0
\(461\) 19.2864 0.898259 0.449129 0.893467i \(-0.351734\pi\)
0.449129 + 0.893467i \(0.351734\pi\)
\(462\) 0 0
\(463\) −36.8200 −1.71117 −0.855584 0.517664i \(-0.826802\pi\)
−0.855584 + 0.517664i \(0.826802\pi\)
\(464\) 0 0
\(465\) 11.6479i 0.540159i
\(466\) 0 0
\(467\) 35.6659 1.65042 0.825210 0.564826i \(-0.191057\pi\)
0.825210 + 0.564826i \(0.191057\pi\)
\(468\) 0 0
\(469\) 5.37918i 0.248388i
\(470\) 0 0
\(471\) 28.2081i 1.29976i
\(472\) 0 0
\(473\) 20.9679i 0.964103i
\(474\) 0 0
\(475\) 4.57231 0.209792
\(476\) 0 0
\(477\) −2.40919 −0.110309
\(478\) 0 0
\(479\) 12.6243i 0.576820i −0.957507 0.288410i \(-0.906873\pi\)
0.957507 0.288410i \(-0.0931266\pi\)
\(480\) 0 0
\(481\) 24.8527i 1.13318i
\(482\) 0 0
\(483\) 12.1000i 0.550567i
\(484\) 0 0
\(485\) −29.9777 −1.36122
\(486\) 0 0
\(487\) 4.63497i 0.210031i 0.994471 + 0.105015i \(0.0334892\pi\)
−0.994471 + 0.105015i \(0.966511\pi\)
\(488\) 0 0
\(489\) −27.2130 −1.23062
\(490\) 0 0
\(491\) −2.28619 −0.103174 −0.0515872 0.998668i \(-0.516428\pi\)
−0.0515872 + 0.998668i \(0.516428\pi\)
\(492\) 0 0
\(493\) −10.5549 2.75460i −0.475368 0.124061i
\(494\) 0 0
\(495\) −13.9093 −0.625175
\(496\) 0 0
\(497\) 11.9874 0.537707
\(498\) 0 0
\(499\) 11.2080i 0.501739i 0.968021 + 0.250869i \(0.0807165\pi\)
−0.968021 + 0.250869i \(0.919284\pi\)
\(500\) 0 0
\(501\) −2.58599 −0.115534
\(502\) 0 0
\(503\) 14.8013i 0.659959i −0.943988 0.329980i \(-0.892958\pi\)
0.943988 0.329980i \(-0.107042\pi\)
\(504\) 0 0
\(505\) 4.59961i 0.204680i
\(506\) 0 0
\(507\) 9.28006i 0.412142i
\(508\) 0 0
\(509\) 38.9964 1.72848 0.864242 0.503076i \(-0.167798\pi\)
0.864242 + 0.503076i \(0.167798\pi\)
\(510\) 0 0
\(511\) −4.62895 −0.204773
\(512\) 0 0
\(513\) 40.4975i 1.78801i
\(514\) 0 0
\(515\) 14.3907i 0.634131i
\(516\) 0 0
\(517\) 23.3350i 1.02627i
\(518\) 0 0
\(519\) −5.38146 −0.236220
\(520\) 0 0
\(521\) 40.1427i 1.75868i 0.476191 + 0.879342i \(0.342017\pi\)
−0.476191 + 0.879342i \(0.657983\pi\)
\(522\) 0 0
\(523\) −27.9817 −1.22355 −0.611777 0.791030i \(-0.709545\pi\)
−0.611777 + 0.791030i \(0.709545\pi\)
\(524\) 0 0
\(525\) 1.85215 0.0808346
\(526\) 0 0
\(527\) −19.5639 5.10575i −0.852215 0.222410i
\(528\) 0 0
\(529\) 7.46847 0.324716
\(530\) 0 0
\(531\) 1.71682 0.0745035
\(532\) 0 0
\(533\) 10.8458i 0.469785i
\(534\) 0 0
\(535\) −15.4722 −0.668922
\(536\) 0 0
\(537\) 23.0256i 0.993629i
\(538\) 0 0
\(539\) 1.33775i 0.0576212i
\(540\) 0 0
\(541\) 42.9837i 1.84801i 0.382376 + 0.924007i \(0.375106\pi\)
−0.382376 + 0.924007i \(0.624894\pi\)
\(542\) 0 0
\(543\) 2.78833 0.119659
\(544\) 0 0
\(545\) 18.3072 0.784195
\(546\) 0 0
\(547\) 35.7790i 1.52980i 0.644150 + 0.764899i \(0.277211\pi\)
−0.644150 + 0.764899i \(0.722789\pi\)
\(548\) 0 0
\(549\) 23.0019i 0.981696i
\(550\) 0 0
\(551\) 20.0527i 0.854273i
\(552\) 0 0
\(553\) 12.9539 0.550855
\(554\) 0 0
\(555\) 26.9225i 1.14279i
\(556\) 0 0
\(557\) 21.8747 0.926862 0.463431 0.886133i \(-0.346618\pi\)
0.463431 + 0.886133i \(0.346618\pi\)
\(558\) 0 0
\(559\) 11.8989 0.503271
\(560\) 0 0
\(561\) −4.55702 + 17.4613i −0.192398 + 0.737215i
\(562\) 0 0
\(563\) 38.8742 1.63835 0.819176 0.573542i \(-0.194431\pi\)
0.819176 + 0.573542i \(0.194431\pi\)
\(564\) 0 0
\(565\) 39.4759 1.66076
\(566\) 0 0
\(567\) 2.44504i 0.102682i
\(568\) 0 0
\(569\) −22.6908 −0.951246 −0.475623 0.879649i \(-0.657777\pi\)
−0.475623 + 0.879649i \(0.657777\pi\)
\(570\) 0 0
\(571\) 9.37266i 0.392233i −0.980581 0.196117i \(-0.937167\pi\)
0.980581 0.196117i \(-0.0628331\pi\)
\(572\) 0 0
\(573\) 7.71627i 0.322352i
\(574\) 0 0
\(575\) 2.37743i 0.0991456i
\(576\) 0 0
\(577\) −10.9027 −0.453886 −0.226943 0.973908i \(-0.572873\pi\)
−0.226943 + 0.973908i \(0.572873\pi\)
\(578\) 0 0
\(579\) 1.90597 0.0792094
\(580\) 0 0
\(581\) 24.9620i 1.03560i
\(582\) 0 0
\(583\) 5.42202i 0.224557i
\(584\) 0 0
\(585\) 7.89328i 0.326347i
\(586\) 0 0
\(587\) −31.6476 −1.30623 −0.653117 0.757257i \(-0.726539\pi\)
−0.653117 + 0.757257i \(0.726539\pi\)
\(588\) 0 0
\(589\) 37.1683i 1.53149i
\(590\) 0 0
\(591\) 27.4558 1.12938
\(592\) 0 0
\(593\) −16.2493 −0.667277 −0.333639 0.942701i \(-0.608276\pi\)
−0.333639 + 0.942701i \(0.608276\pi\)
\(594\) 0 0
\(595\) 5.91725 22.6733i 0.242584 0.929514i
\(596\) 0 0
\(597\) 7.30553 0.298995
\(598\) 0 0
\(599\) −31.1316 −1.27200 −0.636000 0.771689i \(-0.719412\pi\)
−0.636000 + 0.771689i \(0.719412\pi\)
\(600\) 0 0
\(601\) 9.38086i 0.382653i 0.981526 + 0.191327i \(0.0612790\pi\)
−0.981526 + 0.191327i \(0.938721\pi\)
\(602\) 0 0
\(603\) 3.40728 0.138755
\(604\) 0 0
\(605\) 8.23839i 0.334938i
\(606\) 0 0
\(607\) 19.2786i 0.782493i −0.920286 0.391246i \(-0.872044\pi\)
0.920286 0.391246i \(-0.127956\pi\)
\(608\) 0 0
\(609\) 8.12295i 0.329159i
\(610\) 0 0
\(611\) −13.2422 −0.535724
\(612\) 0 0
\(613\) 37.5758 1.51767 0.758836 0.651282i \(-0.225768\pi\)
0.758836 + 0.651282i \(0.225768\pi\)
\(614\) 0 0
\(615\) 11.7491i 0.473770i
\(616\) 0 0
\(617\) 41.3023i 1.66277i −0.555698 0.831384i \(-0.687549\pi\)
0.555698 0.831384i \(-0.312451\pi\)
\(618\) 0 0
\(619\) 7.57462i 0.304450i −0.988346 0.152225i \(-0.951356\pi\)
0.988346 0.152225i \(-0.0486438\pi\)
\(620\) 0 0
\(621\) 21.0572 0.844996
\(622\) 0 0
\(623\) 12.4188i 0.497549i
\(624\) 0 0
\(625\) −21.6198 −0.864793
\(626\) 0 0
\(627\) −33.1737 −1.32483
\(628\) 0 0
\(629\) 45.2190 + 11.8012i 1.80300 + 0.470545i
\(630\) 0 0
\(631\) 14.8708 0.591997 0.295998 0.955188i \(-0.404348\pi\)
0.295998 + 0.955188i \(0.404348\pi\)
\(632\) 0 0
\(633\) 0.149826 0.00595505
\(634\) 0 0
\(635\) 0.220934i 0.00876750i
\(636\) 0 0
\(637\) 0.759154 0.0300788
\(638\) 0 0
\(639\) 7.59304i 0.300376i
\(640\) 0 0
\(641\) 37.6789i 1.48823i 0.668053 + 0.744114i \(0.267128\pi\)
−0.668053 + 0.744114i \(0.732872\pi\)
\(642\) 0 0
\(643\) 0.295144i 0.0116393i −0.999983 0.00581966i \(-0.998148\pi\)
0.999983 0.00581966i \(-0.00185247\pi\)
\(644\) 0 0
\(645\) 12.8899 0.507539
\(646\) 0 0
\(647\) −29.3720 −1.15473 −0.577366 0.816486i \(-0.695919\pi\)
−0.577366 + 0.816486i \(0.695919\pi\)
\(648\) 0 0
\(649\) 3.86380i 0.151667i
\(650\) 0 0
\(651\) 15.0562i 0.590098i
\(652\) 0 0
\(653\) 16.1187i 0.630775i −0.948963 0.315387i \(-0.897866\pi\)
0.948963 0.315387i \(-0.102134\pi\)
\(654\) 0 0
\(655\) −23.5927 −0.921841
\(656\) 0 0
\(657\) 2.93207i 0.114391i
\(658\) 0 0
\(659\) 23.4823 0.914741 0.457370 0.889276i \(-0.348791\pi\)
0.457370 + 0.889276i \(0.348791\pi\)
\(660\) 0 0
\(661\) 8.39444 0.326506 0.163253 0.986584i \(-0.447801\pi\)
0.163253 + 0.986584i \(0.447801\pi\)
\(662\) 0 0
\(663\) 9.90899 + 2.58604i 0.384833 + 0.100433i
\(664\) 0 0
\(665\) 43.0758 1.67041
\(666\) 0 0
\(667\) 10.4266 0.403721
\(668\) 0 0
\(669\) 15.7556i 0.609148i
\(670\) 0 0
\(671\) −51.7671 −1.99845
\(672\) 0 0
\(673\) 11.4909i 0.442943i −0.975167 0.221472i \(-0.928914\pi\)
0.975167 0.221472i \(-0.0710860\pi\)
\(674\) 0 0
\(675\) 3.22325i 0.124063i
\(676\) 0 0
\(677\) 5.23834i 0.201326i 0.994921 + 0.100663i \(0.0320964\pi\)
−0.994921 + 0.100663i \(0.967904\pi\)
\(678\) 0 0
\(679\) 38.7494 1.48707
\(680\) 0 0
\(681\) 10.9488 0.419557
\(682\) 0 0
\(683\) 27.0797i 1.03618i 0.855327 + 0.518088i \(0.173356\pi\)
−0.855327 + 0.518088i \(0.826644\pi\)
\(684\) 0 0
\(685\) 25.9581i 0.991807i
\(686\) 0 0
\(687\) 10.1559i 0.387471i
\(688\) 0 0
\(689\) 3.07691 0.117221
\(690\) 0 0
\(691\) 4.40834i 0.167701i −0.996478 0.0838506i \(-0.973278\pi\)
0.996478 0.0838506i \(-0.0267218\pi\)
\(692\) 0 0
\(693\) 17.9792 0.682974
\(694\) 0 0
\(695\) −23.8019 −0.902858
\(696\) 0 0
\(697\) 19.7338 + 5.15011i 0.747472 + 0.195074i
\(698\) 0 0
\(699\) −9.40709 −0.355809
\(700\) 0 0
\(701\) 13.5986 0.513613 0.256806 0.966463i \(-0.417330\pi\)
0.256806 + 0.966463i \(0.417330\pi\)
\(702\) 0 0
\(703\) 85.9092i 3.24013i
\(704\) 0 0
\(705\) −14.3451 −0.540268
\(706\) 0 0
\(707\) 5.94549i 0.223603i
\(708\) 0 0
\(709\) 1.00228i 0.0376413i 0.999823 + 0.0188207i \(0.00599116\pi\)
−0.999823 + 0.0188207i \(0.994009\pi\)
\(710\) 0 0
\(711\) 8.20524i 0.307721i
\(712\) 0 0
\(713\) 19.3261 0.723770
\(714\) 0 0
\(715\) 17.7643 0.664348
\(716\) 0 0
\(717\) 23.7208i 0.885868i
\(718\) 0 0
\(719\) 30.6299i 1.14230i 0.820845 + 0.571151i \(0.193503\pi\)
−0.820845 + 0.571151i \(0.806497\pi\)
\(720\) 0 0
\(721\) 18.6015i 0.692758i
\(722\) 0 0
\(723\) −19.3967 −0.721371
\(724\) 0 0
\(725\) 1.59602i 0.0592745i
\(726\) 0 0
\(727\) 16.3145 0.605070 0.302535 0.953138i \(-0.402167\pi\)
0.302535 + 0.953138i \(0.402167\pi\)
\(728\) 0 0
\(729\) −19.7064 −0.729865
\(730\) 0 0
\(731\) −5.65017 + 21.6499i −0.208979 + 0.800750i
\(732\) 0 0
\(733\) 42.5135 1.57027 0.785136 0.619324i \(-0.212593\pi\)
0.785136 + 0.619324i \(0.212593\pi\)
\(734\) 0 0
\(735\) 0.822379 0.0303339
\(736\) 0 0
\(737\) 7.66830i 0.282465i
\(738\) 0 0
\(739\) 19.9685 0.734554 0.367277 0.930112i \(-0.380290\pi\)
0.367277 + 0.930112i \(0.380290\pi\)
\(740\) 0 0
\(741\) 18.8256i 0.691574i
\(742\) 0 0
\(743\) 33.9693i 1.24621i −0.782138 0.623106i \(-0.785871\pi\)
0.782138 0.623106i \(-0.214129\pi\)
\(744\) 0 0
\(745\) 11.1116i 0.407098i
\(746\) 0 0
\(747\) 15.8114 0.578510
\(748\) 0 0
\(749\) 19.9995 0.730765
\(750\) 0 0
\(751\) 46.5988i 1.70042i −0.526447 0.850208i \(-0.676476\pi\)
0.526447 0.850208i \(-0.323524\pi\)
\(752\) 0 0
\(753\) 6.36499i 0.231953i
\(754\) 0 0
\(755\) 45.7883i 1.66641i
\(756\) 0 0
\(757\) −52.1700 −1.89615 −0.948075 0.318047i \(-0.896973\pi\)
−0.948075 + 0.318047i \(0.896973\pi\)
\(758\) 0 0
\(759\) 17.2491i 0.626103i
\(760\) 0 0
\(761\) 0.909512 0.0329698 0.0164849 0.999864i \(-0.494752\pi\)
0.0164849 + 0.999864i \(0.494752\pi\)
\(762\) 0 0
\(763\) −23.6640 −0.856696
\(764\) 0 0
\(765\) −14.3617 3.74810i −0.519248 0.135513i
\(766\) 0 0
\(767\) −2.19264 −0.0791718
\(768\) 0 0
\(769\) −19.5541 −0.705137 −0.352569 0.935786i \(-0.614692\pi\)
−0.352569 + 0.935786i \(0.614692\pi\)
\(770\) 0 0
\(771\) 28.6715i 1.03258i
\(772\) 0 0
\(773\) 8.52160 0.306501 0.153250 0.988187i \(-0.451026\pi\)
0.153250 + 0.988187i \(0.451026\pi\)
\(774\) 0 0
\(775\) 2.95827i 0.106264i
\(776\) 0 0
\(777\) 34.8002i 1.24845i
\(778\) 0 0
\(779\) 37.4913i 1.34326i
\(780\) 0 0
\(781\) −17.0886 −0.611478
\(782\) 0 0
\(783\) −14.1361 −0.505184
\(784\) 0 0
\(785\) 52.2150i 1.86363i
\(786\) 0 0
\(787\) 1.83539i 0.0654248i −0.999465 0.0327124i \(-0.989585\pi\)
0.999465 0.0327124i \(-0.0104145\pi\)
\(788\) 0 0
\(789\) 26.8156i 0.954659i
\(790\) 0 0
\(791\) −51.0268 −1.81430
\(792\) 0 0
\(793\) 29.3770i 1.04321i