Properties

Label 4012.2.b.b.237.11
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 237.11
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.36

$q$-expansion

\(f(q)\) \(=\) \(q-1.79340i q^{3} +1.19915i q^{5} -1.20458i q^{7} -0.216279 q^{9} +O(q^{10})\) \(q-1.79340i q^{3} +1.19915i q^{5} -1.20458i q^{7} -0.216279 q^{9} -0.564620i q^{11} -6.78189 q^{13} +2.15056 q^{15} +(-2.99791 + 2.83064i) q^{17} -5.87932 q^{19} -2.16030 q^{21} -2.74169i q^{23} +3.56203 q^{25} -4.99232i q^{27} +8.94813i q^{29} -4.79978i q^{31} -1.01259 q^{33} +1.44448 q^{35} +5.32745i q^{37} +12.1626i q^{39} +6.25269i q^{41} +8.95816 q^{43} -0.259352i q^{45} +9.79572 q^{47} +5.54898 q^{49} +(5.07646 + 5.37645i) q^{51} +11.8332 q^{53} +0.677067 q^{55} +10.5440i q^{57} +1.00000 q^{59} -0.445356i q^{61} +0.260526i q^{63} -8.13254i q^{65} -13.0800 q^{67} -4.91694 q^{69} -9.22255i q^{71} +13.5216i q^{73} -6.38814i q^{75} -0.680133 q^{77} +15.6374i q^{79} -9.60206 q^{81} -6.77674 q^{83} +(-3.39437 - 3.59496i) q^{85} +16.0476 q^{87} -2.83173 q^{89} +8.16937i q^{91} -8.60792 q^{93} -7.05022i q^{95} +12.9863i q^{97} +0.122115i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

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
\(3\) 1.79340i 1.03542i −0.855556 0.517710i \(-0.826785\pi\)
0.855556 0.517710i \(-0.173215\pi\)
\(4\) 0 0
\(5\) 1.19915i 0.536278i 0.963380 + 0.268139i \(0.0864087\pi\)
−0.963380 + 0.268139i \(0.913591\pi\)
\(6\) 0 0
\(7\) 1.20458i 0.455290i −0.973744 0.227645i \(-0.926897\pi\)
0.973744 0.227645i \(-0.0731026\pi\)
\(8\) 0 0
\(9\) −0.216279 −0.0720929
\(10\) 0 0
\(11\) 0.564620i 0.170239i −0.996371 0.0851197i \(-0.972873\pi\)
0.996371 0.0851197i \(-0.0271273\pi\)
\(12\) 0 0
\(13\) −6.78189 −1.88096 −0.940480 0.339850i \(-0.889624\pi\)
−0.940480 + 0.339850i \(0.889624\pi\)
\(14\) 0 0
\(15\) 2.15056 0.555273
\(16\) 0 0
\(17\) −2.99791 + 2.83064i −0.727101 + 0.686531i
\(18\) 0 0
\(19\) −5.87932 −1.34881 −0.674405 0.738362i \(-0.735600\pi\)
−0.674405 + 0.738362i \(0.735600\pi\)
\(20\) 0 0
\(21\) −2.16030 −0.471416
\(22\) 0 0
\(23\) 2.74169i 0.571682i −0.958277 0.285841i \(-0.907727\pi\)
0.958277 0.285841i \(-0.0922728\pi\)
\(24\) 0 0
\(25\) 3.56203 0.712406
\(26\) 0 0
\(27\) 4.99232i 0.960773i
\(28\) 0 0
\(29\) 8.94813i 1.66163i 0.556552 + 0.830813i \(0.312124\pi\)
−0.556552 + 0.830813i \(0.687876\pi\)
\(30\) 0 0
\(31\) 4.79978i 0.862066i −0.902336 0.431033i \(-0.858149\pi\)
0.902336 0.431033i \(-0.141851\pi\)
\(32\) 0 0
\(33\) −1.01259 −0.176269
\(34\) 0 0
\(35\) 1.44448 0.244162
\(36\) 0 0
\(37\) 5.32745i 0.875828i 0.899017 + 0.437914i \(0.144283\pi\)
−0.899017 + 0.437914i \(0.855717\pi\)
\(38\) 0 0
\(39\) 12.1626i 1.94758i
\(40\) 0 0
\(41\) 6.25269i 0.976506i 0.872702 + 0.488253i \(0.162366\pi\)
−0.872702 + 0.488253i \(0.837634\pi\)
\(42\) 0 0
\(43\) 8.95816 1.36611 0.683053 0.730369i \(-0.260652\pi\)
0.683053 + 0.730369i \(0.260652\pi\)
\(44\) 0 0
\(45\) 0.259352i 0.0386619i
\(46\) 0 0
\(47\) 9.79572 1.42885 0.714426 0.699711i \(-0.246688\pi\)
0.714426 + 0.699711i \(0.246688\pi\)
\(48\) 0 0
\(49\) 5.54898 0.792711
\(50\) 0 0
\(51\) 5.07646 + 5.37645i 0.710847 + 0.752854i
\(52\) 0 0
\(53\) 11.8332 1.62541 0.812706 0.582674i \(-0.197994\pi\)
0.812706 + 0.582674i \(0.197994\pi\)
\(54\) 0 0
\(55\) 0.677067 0.0912956
\(56\) 0 0
\(57\) 10.5440i 1.39658i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 0.445356i 0.0570220i −0.999593 0.0285110i \(-0.990923\pi\)
0.999593 0.0285110i \(-0.00907656\pi\)
\(62\) 0 0
\(63\) 0.260526i 0.0328232i
\(64\) 0 0
\(65\) 8.13254i 1.00872i
\(66\) 0 0
\(67\) −13.0800 −1.59797 −0.798985 0.601350i \(-0.794630\pi\)
−0.798985 + 0.601350i \(0.794630\pi\)
\(68\) 0 0
\(69\) −4.91694 −0.591930
\(70\) 0 0
\(71\) 9.22255i 1.09451i −0.836964 0.547257i \(-0.815672\pi\)
0.836964 0.547257i \(-0.184328\pi\)
\(72\) 0 0
\(73\) 13.5216i 1.58258i 0.611440 + 0.791291i \(0.290591\pi\)
−0.611440 + 0.791291i \(0.709409\pi\)
\(74\) 0 0
\(75\) 6.38814i 0.737639i
\(76\) 0 0
\(77\) −0.680133 −0.0775083
\(78\) 0 0
\(79\) 15.6374i 1.75934i 0.475582 + 0.879671i \(0.342237\pi\)
−0.475582 + 0.879671i \(0.657763\pi\)
\(80\) 0 0
\(81\) −9.60206 −1.06690
\(82\) 0 0
\(83\) −6.77674 −0.743844 −0.371922 0.928264i \(-0.621301\pi\)
−0.371922 + 0.928264i \(0.621301\pi\)
\(84\) 0 0
\(85\) −3.39437 3.59496i −0.368171 0.389928i
\(86\) 0 0
\(87\) 16.0476 1.72048
\(88\) 0 0
\(89\) −2.83173 −0.300163 −0.150081 0.988674i \(-0.547954\pi\)
−0.150081 + 0.988674i \(0.547954\pi\)
\(90\) 0 0
\(91\) 8.16937i 0.856383i
\(92\) 0 0
\(93\) −8.60792 −0.892600
\(94\) 0 0
\(95\) 7.05022i 0.723337i
\(96\) 0 0
\(97\) 12.9863i 1.31855i 0.751900 + 0.659277i \(0.229138\pi\)
−0.751900 + 0.659277i \(0.770862\pi\)
\(98\) 0 0
\(99\) 0.122115i 0.0122731i
\(100\) 0 0
\(101\) 15.7177 1.56397 0.781983 0.623299i \(-0.214208\pi\)
0.781983 + 0.623299i \(0.214208\pi\)
\(102\) 0 0
\(103\) −18.8995 −1.86223 −0.931114 0.364729i \(-0.881162\pi\)
−0.931114 + 0.364729i \(0.881162\pi\)
\(104\) 0 0
\(105\) 2.59053i 0.252810i
\(106\) 0 0
\(107\) 9.21894i 0.891228i 0.895225 + 0.445614i \(0.147015\pi\)
−0.895225 + 0.445614i \(0.852985\pi\)
\(108\) 0 0
\(109\) 8.69650i 0.832973i −0.909142 0.416487i \(-0.863261\pi\)
0.909142 0.416487i \(-0.136739\pi\)
\(110\) 0 0
\(111\) 9.55425 0.906849
\(112\) 0 0
\(113\) 10.8045i 1.01640i 0.861240 + 0.508199i \(0.169689\pi\)
−0.861240 + 0.508199i \(0.830311\pi\)
\(114\) 0 0
\(115\) 3.28771 0.306580
\(116\) 0 0
\(117\) 1.46678 0.135604
\(118\) 0 0
\(119\) 3.40974 + 3.61124i 0.312571 + 0.331042i
\(120\) 0 0
\(121\) 10.6812 0.971019
\(122\) 0 0
\(123\) 11.2136 1.01109
\(124\) 0 0
\(125\) 10.2672i 0.918326i
\(126\) 0 0
\(127\) −2.90644 −0.257904 −0.128952 0.991651i \(-0.541161\pi\)
−0.128952 + 0.991651i \(0.541161\pi\)
\(128\) 0 0
\(129\) 16.0656i 1.41449i
\(130\) 0 0
\(131\) 13.8548i 1.21050i −0.796035 0.605251i \(-0.793073\pi\)
0.796035 0.605251i \(-0.206927\pi\)
\(132\) 0 0
\(133\) 7.08215i 0.614100i
\(134\) 0 0
\(135\) 5.98657 0.515242
\(136\) 0 0
\(137\) 5.94936 0.508288 0.254144 0.967166i \(-0.418206\pi\)
0.254144 + 0.967166i \(0.418206\pi\)
\(138\) 0 0
\(139\) 16.0659i 1.36269i 0.731962 + 0.681346i \(0.238605\pi\)
−0.731962 + 0.681346i \(0.761395\pi\)
\(140\) 0 0
\(141\) 17.5676i 1.47946i
\(142\) 0 0
\(143\) 3.82919i 0.320213i
\(144\) 0 0
\(145\) −10.7302 −0.891094
\(146\) 0 0
\(147\) 9.95152i 0.820788i
\(148\) 0 0
\(149\) −0.867969 −0.0711068 −0.0355534 0.999368i \(-0.511319\pi\)
−0.0355534 + 0.999368i \(0.511319\pi\)
\(150\) 0 0
\(151\) 7.69235 0.625995 0.312997 0.949754i \(-0.398667\pi\)
0.312997 + 0.949754i \(0.398667\pi\)
\(152\) 0 0
\(153\) 0.648385 0.612207i 0.0524188 0.0494940i
\(154\) 0 0
\(155\) 5.75568 0.462307
\(156\) 0 0
\(157\) −20.9266 −1.67012 −0.835062 0.550155i \(-0.814569\pi\)
−0.835062 + 0.550155i \(0.814569\pi\)
\(158\) 0 0
\(159\) 21.2216i 1.68298i
\(160\) 0 0
\(161\) −3.30260 −0.260281
\(162\) 0 0
\(163\) 4.63463i 0.363012i 0.983390 + 0.181506i \(0.0580972\pi\)
−0.983390 + 0.181506i \(0.941903\pi\)
\(164\) 0 0
\(165\) 1.21425i 0.0945293i
\(166\) 0 0
\(167\) 15.9930i 1.23758i 0.785557 + 0.618789i \(0.212377\pi\)
−0.785557 + 0.618789i \(0.787623\pi\)
\(168\) 0 0
\(169\) 32.9941 2.53801
\(170\) 0 0
\(171\) 1.27157 0.0972396
\(172\) 0 0
\(173\) 7.78024i 0.591521i 0.955262 + 0.295760i \(0.0955730\pi\)
−0.955262 + 0.295760i \(0.904427\pi\)
\(174\) 0 0
\(175\) 4.29077i 0.324351i
\(176\) 0 0
\(177\) 1.79340i 0.134800i
\(178\) 0 0
\(179\) −12.3247 −0.921192 −0.460596 0.887610i \(-0.652364\pi\)
−0.460596 + 0.887610i \(0.652364\pi\)
\(180\) 0 0
\(181\) 19.0861i 1.41866i −0.704879 0.709328i \(-0.748999\pi\)
0.704879 0.709328i \(-0.251001\pi\)
\(182\) 0 0
\(183\) −0.798701 −0.0590417
\(184\) 0 0
\(185\) −6.38844 −0.469687
\(186\) 0 0
\(187\) 1.59823 + 1.69268i 0.116875 + 0.123781i
\(188\) 0 0
\(189\) −6.01368 −0.437431
\(190\) 0 0
\(191\) 9.57292 0.692673 0.346336 0.938110i \(-0.387426\pi\)
0.346336 + 0.938110i \(0.387426\pi\)
\(192\) 0 0
\(193\) 3.91168i 0.281569i 0.990040 + 0.140785i \(0.0449625\pi\)
−0.990040 + 0.140785i \(0.955038\pi\)
\(194\) 0 0
\(195\) −14.5849 −1.04445
\(196\) 0 0
\(197\) 4.48496i 0.319540i 0.987154 + 0.159770i \(0.0510753\pi\)
−0.987154 + 0.159770i \(0.948925\pi\)
\(198\) 0 0
\(199\) 15.1850i 1.07644i 0.842805 + 0.538219i \(0.180903\pi\)
−0.842805 + 0.538219i \(0.819097\pi\)
\(200\) 0 0
\(201\) 23.4576i 1.65457i
\(202\) 0 0
\(203\) 10.7788 0.756522
\(204\) 0 0
\(205\) −7.49794 −0.523679
\(206\) 0 0
\(207\) 0.592969i 0.0412142i
\(208\) 0 0
\(209\) 3.31958i 0.229620i
\(210\) 0 0
\(211\) 8.61658i 0.593190i −0.955003 0.296595i \(-0.904149\pi\)
0.955003 0.296595i \(-0.0958511\pi\)
\(212\) 0 0
\(213\) −16.5397 −1.13328
\(214\) 0 0
\(215\) 10.7422i 0.732613i
\(216\) 0 0
\(217\) −5.78174 −0.392490
\(218\) 0 0
\(219\) 24.2496 1.63864
\(220\) 0 0
\(221\) 20.3315 19.1971i 1.36765 1.29134i
\(222\) 0 0
\(223\) 19.6359 1.31491 0.657457 0.753492i \(-0.271632\pi\)
0.657457 + 0.753492i \(0.271632\pi\)
\(224\) 0 0
\(225\) −0.770391 −0.0513594
\(226\) 0 0
\(227\) 17.2664i 1.14601i 0.819552 + 0.573005i \(0.194222\pi\)
−0.819552 + 0.573005i \(0.805778\pi\)
\(228\) 0 0
\(229\) −7.99676 −0.528441 −0.264220 0.964462i \(-0.585115\pi\)
−0.264220 + 0.964462i \(0.585115\pi\)
\(230\) 0 0
\(231\) 1.21975i 0.0802536i
\(232\) 0 0
\(233\) 24.3644i 1.59616i 0.602550 + 0.798081i \(0.294151\pi\)
−0.602550 + 0.798081i \(0.705849\pi\)
\(234\) 0 0
\(235\) 11.7466i 0.766262i
\(236\) 0 0
\(237\) 28.0441 1.82166
\(238\) 0 0
\(239\) −4.97573 −0.321853 −0.160927 0.986966i \(-0.551448\pi\)
−0.160927 + 0.986966i \(0.551448\pi\)
\(240\) 0 0
\(241\) 10.0453i 0.647073i 0.946216 + 0.323537i \(0.104872\pi\)
−0.946216 + 0.323537i \(0.895128\pi\)
\(242\) 0 0
\(243\) 2.24336i 0.143911i
\(244\) 0 0
\(245\) 6.65408i 0.425113i
\(246\) 0 0
\(247\) 39.8730 2.53706
\(248\) 0 0
\(249\) 12.1534i 0.770190i
\(250\) 0 0
\(251\) 8.24542 0.520446 0.260223 0.965548i \(-0.416204\pi\)
0.260223 + 0.965548i \(0.416204\pi\)
\(252\) 0 0
\(253\) −1.54801 −0.0973227
\(254\) 0 0
\(255\) −6.44720 + 6.08746i −0.403739 + 0.381212i
\(256\) 0 0
\(257\) −17.0017 −1.06054 −0.530269 0.847829i \(-0.677909\pi\)
−0.530269 + 0.847829i \(0.677909\pi\)
\(258\) 0 0
\(259\) 6.41737 0.398756
\(260\) 0 0
\(261\) 1.93529i 0.119791i
\(262\) 0 0
\(263\) −3.00933 −0.185563 −0.0927815 0.995686i \(-0.529576\pi\)
−0.0927815 + 0.995686i \(0.529576\pi\)
\(264\) 0 0
\(265\) 14.1898i 0.871673i
\(266\) 0 0
\(267\) 5.07842i 0.310794i
\(268\) 0 0
\(269\) 19.3968i 1.18264i −0.806435 0.591322i \(-0.798606\pi\)
0.806435 0.591322i \(-0.201394\pi\)
\(270\) 0 0
\(271\) 1.69220 0.102794 0.0513970 0.998678i \(-0.483633\pi\)
0.0513970 + 0.998678i \(0.483633\pi\)
\(272\) 0 0
\(273\) 14.6509 0.886715
\(274\) 0 0
\(275\) 2.01119i 0.121279i
\(276\) 0 0
\(277\) 19.0746i 1.14608i −0.819527 0.573040i \(-0.805764\pi\)
0.819527 0.573040i \(-0.194236\pi\)
\(278\) 0 0
\(279\) 1.03809i 0.0621489i
\(280\) 0 0
\(281\) −21.6400 −1.29093 −0.645467 0.763788i \(-0.723337\pi\)
−0.645467 + 0.763788i \(0.723337\pi\)
\(282\) 0 0
\(283\) 27.6229i 1.64201i −0.570921 0.821005i \(-0.693414\pi\)
0.570921 0.821005i \(-0.306586\pi\)
\(284\) 0 0
\(285\) −12.6439 −0.748957
\(286\) 0 0
\(287\) 7.53190 0.444594
\(288\) 0 0
\(289\) 0.974975 16.9720i 0.0573515 0.998354i
\(290\) 0 0
\(291\) 23.2895 1.36526
\(292\) 0 0
\(293\) −15.3433 −0.896367 −0.448183 0.893942i \(-0.647929\pi\)
−0.448183 + 0.893942i \(0.647929\pi\)
\(294\) 0 0
\(295\) 1.19915i 0.0698175i
\(296\) 0 0
\(297\) −2.81876 −0.163561
\(298\) 0 0
\(299\) 18.5938i 1.07531i
\(300\) 0 0
\(301\) 10.7909i 0.621975i
\(302\) 0 0
\(303\) 28.1880i 1.61936i
\(304\) 0 0
\(305\) 0.534051 0.0305797
\(306\) 0 0
\(307\) 3.03723 0.173344 0.0866719 0.996237i \(-0.472377\pi\)
0.0866719 + 0.996237i \(0.472377\pi\)
\(308\) 0 0
\(309\) 33.8944i 1.92819i
\(310\) 0 0
\(311\) 7.05485i 0.400044i 0.979791 + 0.200022i \(0.0641013\pi\)
−0.979791 + 0.200022i \(0.935899\pi\)
\(312\) 0 0
\(313\) 6.10368i 0.345000i −0.985010 0.172500i \(-0.944816\pi\)
0.985010 0.172500i \(-0.0551845\pi\)
\(314\) 0 0
\(315\) −0.312411 −0.0176024
\(316\) 0 0
\(317\) 5.36501i 0.301329i −0.988585 0.150664i \(-0.951859\pi\)
0.988585 0.150664i \(-0.0481413\pi\)
\(318\) 0 0
\(319\) 5.05229 0.282874
\(320\) 0 0
\(321\) 16.5332 0.922795
\(322\) 0 0
\(323\) 17.6257 16.6422i 0.980721 0.925999i
\(324\) 0 0
\(325\) −24.1573 −1.34001
\(326\) 0 0
\(327\) −15.5963 −0.862477
\(328\) 0 0
\(329\) 11.7998i 0.650543i
\(330\) 0 0
\(331\) −7.13788 −0.392333 −0.196167 0.980571i \(-0.562849\pi\)
−0.196167 + 0.980571i \(0.562849\pi\)
\(332\) 0 0
\(333\) 1.15222i 0.0631410i
\(334\) 0 0
\(335\) 15.6849i 0.856957i
\(336\) 0 0
\(337\) 3.72226i 0.202765i −0.994848 0.101382i \(-0.967673\pi\)
0.994848 0.101382i \(-0.0323265\pi\)
\(338\) 0 0
\(339\) 19.3767 1.05240
\(340\) 0 0
\(341\) −2.71005 −0.146758
\(342\) 0 0
\(343\) 15.1163i 0.816204i
\(344\) 0 0
\(345\) 5.89617i 0.317439i
\(346\) 0 0
\(347\) 9.93694i 0.533443i −0.963774 0.266721i \(-0.914060\pi\)
0.963774 0.266721i \(-0.0859404\pi\)
\(348\) 0 0
\(349\) 16.0432 0.858775 0.429387 0.903120i \(-0.358729\pi\)
0.429387 + 0.903120i \(0.358729\pi\)
\(350\) 0 0
\(351\) 33.8574i 1.80717i
\(352\) 0 0
\(353\) −0.866248 −0.0461057 −0.0230529 0.999734i \(-0.507339\pi\)
−0.0230529 + 0.999734i \(0.507339\pi\)
\(354\) 0 0
\(355\) 11.0593 0.586964
\(356\) 0 0
\(357\) 6.47640 6.11503i 0.342767 0.323642i
\(358\) 0 0
\(359\) 29.4865 1.55624 0.778119 0.628118i \(-0.216174\pi\)
0.778119 + 0.628118i \(0.216174\pi\)
\(360\) 0 0
\(361\) 15.5665 0.819287
\(362\) 0 0
\(363\) 19.1557i 1.00541i
\(364\) 0 0
\(365\) −16.2145 −0.848704
\(366\) 0 0
\(367\) 14.5988i 0.762051i 0.924565 + 0.381025i \(0.124429\pi\)
−0.924565 + 0.381025i \(0.875571\pi\)
\(368\) 0 0
\(369\) 1.35232i 0.0703992i
\(370\) 0 0
\(371\) 14.2541i 0.740034i
\(372\) 0 0
\(373\) −18.7235 −0.969465 −0.484733 0.874662i \(-0.661083\pi\)
−0.484733 + 0.874662i \(0.661083\pi\)
\(374\) 0 0
\(375\) 18.4132 0.950852
\(376\) 0 0
\(377\) 60.6853i 3.12545i
\(378\) 0 0
\(379\) 21.8231i 1.12098i 0.828163 + 0.560488i \(0.189386\pi\)
−0.828163 + 0.560488i \(0.810614\pi\)
\(380\) 0 0
\(381\) 5.21240i 0.267039i
\(382\) 0 0
\(383\) 17.6275 0.900722 0.450361 0.892847i \(-0.351295\pi\)
0.450361 + 0.892847i \(0.351295\pi\)
\(384\) 0 0
\(385\) 0.815584i 0.0415660i
\(386\) 0 0
\(387\) −1.93746 −0.0984866
\(388\) 0 0
\(389\) −11.7294 −0.594702 −0.297351 0.954768i \(-0.596103\pi\)
−0.297351 + 0.954768i \(0.596103\pi\)
\(390\) 0 0
\(391\) 7.76073 + 8.21935i 0.392477 + 0.415670i
\(392\) 0 0
\(393\) −24.8472 −1.25338
\(394\) 0 0
\(395\) −18.7516 −0.943497
\(396\) 0 0
\(397\) 32.8927i 1.65084i 0.564520 + 0.825420i \(0.309061\pi\)
−0.564520 + 0.825420i \(0.690939\pi\)
\(398\) 0 0
\(399\) 12.7011 0.635851
\(400\) 0 0
\(401\) 9.79236i 0.489007i 0.969648 + 0.244504i \(0.0786250\pi\)
−0.969648 + 0.244504i \(0.921375\pi\)
\(402\) 0 0
\(403\) 32.5516i 1.62151i
\(404\) 0 0
\(405\) 11.5144i 0.572153i
\(406\) 0 0
\(407\) 3.00799 0.149100
\(408\) 0 0
\(409\) −14.3786 −0.710977 −0.355488 0.934681i \(-0.615685\pi\)
−0.355488 + 0.934681i \(0.615685\pi\)
\(410\) 0 0
\(411\) 10.6696i 0.526292i
\(412\) 0 0
\(413\) 1.20458i 0.0592738i
\(414\) 0 0
\(415\) 8.12636i 0.398907i
\(416\) 0 0
\(417\) 28.8126 1.41096
\(418\) 0 0
\(419\) 1.17302i 0.0573057i 0.999589 + 0.0286529i \(0.00912174\pi\)
−0.999589 + 0.0286529i \(0.990878\pi\)
\(420\) 0 0
\(421\) 3.06357 0.149309 0.0746546 0.997209i \(-0.476215\pi\)
0.0746546 + 0.997209i \(0.476215\pi\)
\(422\) 0 0
\(423\) −2.11861 −0.103010
\(424\) 0 0
\(425\) −10.6787 + 10.0828i −0.517991 + 0.489088i
\(426\) 0 0
\(427\) −0.536469 −0.0259616
\(428\) 0 0
\(429\) 6.86727 0.331555
\(430\) 0 0
\(431\) 5.03954i 0.242746i 0.992607 + 0.121373i \(0.0387297\pi\)
−0.992607 + 0.121373i \(0.961270\pi\)
\(432\) 0 0
\(433\) −19.8369 −0.953300 −0.476650 0.879093i \(-0.658149\pi\)
−0.476650 + 0.879093i \(0.658149\pi\)
\(434\) 0 0
\(435\) 19.2435i 0.922656i
\(436\) 0 0
\(437\) 16.1193i 0.771089i
\(438\) 0 0
\(439\) 12.9200i 0.616636i −0.951283 0.308318i \(-0.900234\pi\)
0.951283 0.308318i \(-0.0997660\pi\)
\(440\) 0 0
\(441\) −1.20013 −0.0571488
\(442\) 0 0
\(443\) 9.80471 0.465836 0.232918 0.972496i \(-0.425173\pi\)
0.232918 + 0.972496i \(0.425173\pi\)
\(444\) 0 0
\(445\) 3.39568i 0.160971i
\(446\) 0 0
\(447\) 1.55661i 0.0736253i
\(448\) 0 0
\(449\) 41.2582i 1.94710i −0.228480 0.973549i \(-0.573376\pi\)
0.228480 0.973549i \(-0.426624\pi\)
\(450\) 0 0
\(451\) 3.53039 0.166240
\(452\) 0 0
\(453\) 13.7955i 0.648167i
\(454\) 0 0
\(455\) −9.79634 −0.459259
\(456\) 0 0
\(457\) 32.0203 1.49785 0.748923 0.662658i \(-0.230571\pi\)
0.748923 + 0.662658i \(0.230571\pi\)
\(458\) 0 0
\(459\) 14.1315 + 14.9666i 0.659600 + 0.698579i
\(460\) 0 0
\(461\) −12.8193 −0.597056 −0.298528 0.954401i \(-0.596496\pi\)
−0.298528 + 0.954401i \(0.596496\pi\)
\(462\) 0 0
\(463\) −9.65804 −0.448847 −0.224424 0.974492i \(-0.572050\pi\)
−0.224424 + 0.974492i \(0.572050\pi\)
\(464\) 0 0
\(465\) 10.3222i 0.478682i
\(466\) 0 0
\(467\) −20.1862 −0.934106 −0.467053 0.884229i \(-0.654684\pi\)
−0.467053 + 0.884229i \(0.654684\pi\)
\(468\) 0 0
\(469\) 15.7559i 0.727541i
\(470\) 0 0
\(471\) 37.5297i 1.72928i
\(472\) 0 0
\(473\) 5.05796i 0.232565i
\(474\) 0 0
\(475\) −20.9423 −0.960900
\(476\) 0 0
\(477\) −2.55927 −0.117181
\(478\) 0 0
\(479\) 2.78186i 0.127106i −0.997978 0.0635532i \(-0.979757\pi\)
0.997978 0.0635532i \(-0.0202432\pi\)
\(480\) 0 0
\(481\) 36.1302i 1.64740i
\(482\) 0 0
\(483\) 5.92287i 0.269500i
\(484\) 0 0
\(485\) −15.5725 −0.707112
\(486\) 0 0
\(487\) 35.1232i 1.59158i 0.605570 + 0.795792i \(0.292945\pi\)
−0.605570 + 0.795792i \(0.707055\pi\)
\(488\) 0 0
\(489\) 8.31174 0.375870
\(490\) 0 0
\(491\) −16.3223 −0.736613 −0.368307 0.929704i \(-0.620062\pi\)
−0.368307 + 0.929704i \(0.620062\pi\)
\(492\) 0 0
\(493\) −25.3289 26.8257i −1.14076 1.20817i
\(494\) 0 0
\(495\) −0.146435 −0.00658177
\(496\) 0 0
\(497\) −11.1093 −0.498322
\(498\) 0 0
\(499\) 4.76824i 0.213456i 0.994288 + 0.106728i \(0.0340373\pi\)
−0.994288 + 0.106728i \(0.965963\pi\)
\(500\) 0 0
\(501\) 28.6819 1.28141
\(502\) 0 0
\(503\) 27.8201i 1.24044i −0.784429 0.620218i \(-0.787044\pi\)
0.784429 0.620218i \(-0.212956\pi\)
\(504\) 0 0
\(505\) 18.8479i 0.838721i
\(506\) 0 0
\(507\) 59.1716i 2.62790i
\(508\) 0 0
\(509\) 31.8898 1.41349 0.706745 0.707468i \(-0.250163\pi\)
0.706745 + 0.707468i \(0.250163\pi\)
\(510\) 0 0
\(511\) 16.2879 0.720534
\(512\) 0 0
\(513\) 29.3515i 1.29590i
\(514\) 0 0
\(515\) 22.6635i 0.998672i
\(516\) 0 0
\(517\) 5.53086i 0.243247i
\(518\) 0 0
\(519\) 13.9531 0.612472
\(520\) 0 0
\(521\) 19.5058i 0.854566i 0.904118 + 0.427283i \(0.140529\pi\)
−0.904118 + 0.427283i \(0.859471\pi\)
\(522\) 0 0
\(523\) −25.7686 −1.12678 −0.563391 0.826191i \(-0.690503\pi\)
−0.563391 + 0.826191i \(0.690503\pi\)
\(524\) 0 0
\(525\) −7.69505 −0.335840
\(526\) 0 0
\(527\) 13.5864 + 14.3893i 0.591835 + 0.626809i
\(528\) 0 0
\(529\) 15.4831 0.673180
\(530\) 0 0
\(531\) −0.216279 −0.00938570
\(532\) 0 0
\(533\) 42.4051i 1.83677i
\(534\) 0 0
\(535\) −11.0549 −0.477946
\(536\) 0 0
\(537\) 22.1031i 0.953820i
\(538\) 0 0
\(539\) 3.13306i 0.134951i
\(540\) 0 0
\(541\) 12.9792i 0.558017i −0.960289 0.279009i \(-0.909994\pi\)
0.960289 0.279009i \(-0.0900058\pi\)
\(542\) 0 0
\(543\) −34.2289 −1.46890
\(544\) 0 0
\(545\) 10.4284 0.446705
\(546\) 0 0
\(547\) 7.67782i 0.328280i 0.986437 + 0.164140i \(0.0524849\pi\)
−0.986437 + 0.164140i \(0.947515\pi\)
\(548\) 0 0
\(549\) 0.0963211i 0.00411088i
\(550\) 0 0
\(551\) 52.6090i 2.24122i
\(552\) 0 0
\(553\) 18.8365 0.801012
\(554\) 0 0
\(555\) 11.4570i 0.486323i
\(556\) 0 0
\(557\) 17.0374 0.721899 0.360949 0.932585i \(-0.382453\pi\)
0.360949 + 0.932585i \(0.382453\pi\)
\(558\) 0 0
\(559\) −60.7533 −2.56959
\(560\) 0 0
\(561\) 3.03565 2.86627i 0.128165 0.121014i
\(562\) 0 0
\(563\) −35.0979 −1.47920 −0.739601 0.673046i \(-0.764986\pi\)
−0.739601 + 0.673046i \(0.764986\pi\)
\(564\) 0 0
\(565\) −12.9562 −0.545072
\(566\) 0 0
\(567\) 11.5665i 0.485747i
\(568\) 0 0
\(569\) 6.60353 0.276835 0.138417 0.990374i \(-0.455798\pi\)
0.138417 + 0.990374i \(0.455798\pi\)
\(570\) 0 0
\(571\) 26.0165i 1.08876i 0.838840 + 0.544378i \(0.183234\pi\)
−0.838840 + 0.544378i \(0.816766\pi\)
\(572\) 0 0
\(573\) 17.1681i 0.717206i
\(574\) 0 0
\(575\) 9.76597i 0.407269i
\(576\) 0 0
\(577\) 24.5374 1.02151 0.510753 0.859728i \(-0.329367\pi\)
0.510753 + 0.859728i \(0.329367\pi\)
\(578\) 0 0
\(579\) 7.01521 0.291542
\(580\) 0 0
\(581\) 8.16316i 0.338665i
\(582\) 0 0
\(583\) 6.68125i 0.276709i
\(584\) 0 0
\(585\) 1.75890i 0.0727214i
\(586\) 0 0
\(587\) −15.7405 −0.649682 −0.324841 0.945769i \(-0.605311\pi\)
−0.324841 + 0.945769i \(0.605311\pi\)
\(588\) 0 0
\(589\) 28.2195i 1.16276i
\(590\) 0 0
\(591\) 8.04332 0.330858
\(592\) 0 0
\(593\) −28.6638 −1.17708 −0.588540 0.808468i \(-0.700297\pi\)
−0.588540 + 0.808468i \(0.700297\pi\)
\(594\) 0 0
\(595\) −4.33044 + 4.08881i −0.177531 + 0.167625i
\(596\) 0 0
\(597\) 27.2328 1.11456
\(598\) 0 0
\(599\) −35.9389 −1.46842 −0.734211 0.678921i \(-0.762448\pi\)
−0.734211 + 0.678921i \(0.762448\pi\)
\(600\) 0 0
\(601\) 45.3256i 1.84887i 0.381342 + 0.924434i \(0.375462\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(602\) 0 0
\(603\) 2.82892 0.115202
\(604\) 0 0
\(605\) 12.8084i 0.520736i
\(606\) 0 0
\(607\) 36.8702i 1.49651i −0.663409 0.748257i \(-0.730891\pi\)
0.663409 0.748257i \(-0.269109\pi\)
\(608\) 0 0
\(609\) 19.3307i 0.783318i
\(610\) 0 0
\(611\) −66.4335 −2.68761
\(612\) 0 0
\(613\) 3.24520 0.131073 0.0655363 0.997850i \(-0.479124\pi\)
0.0655363 + 0.997850i \(0.479124\pi\)
\(614\) 0 0
\(615\) 13.4468i 0.542227i
\(616\) 0 0
\(617\) 8.05121i 0.324129i −0.986780 0.162065i \(-0.948185\pi\)
0.986780 0.162065i \(-0.0518153\pi\)
\(618\) 0 0
\(619\) 41.3037i 1.66013i 0.557663 + 0.830067i \(0.311698\pi\)
−0.557663 + 0.830067i \(0.688302\pi\)
\(620\) 0 0
\(621\) −13.6874 −0.549256
\(622\) 0 0
\(623\) 3.41106i 0.136661i
\(624\) 0 0
\(625\) 5.49819 0.219928
\(626\) 0 0
\(627\) 5.95334 0.237753
\(628\) 0 0
\(629\) −15.0801 15.9712i −0.601283 0.636815i
\(630\) 0 0
\(631\) −5.75140 −0.228960 −0.114480 0.993426i \(-0.536520\pi\)
−0.114480 + 0.993426i \(0.536520\pi\)
\(632\) 0 0
\(633\) −15.4530 −0.614200
\(634\) 0 0
\(635\) 3.48526i 0.138309i
\(636\) 0 0
\(637\) −37.6326 −1.49106
\(638\) 0 0
\(639\) 1.99464i 0.0789068i
\(640\) 0 0
\(641\) 5.01188i 0.197957i −0.995090 0.0989787i \(-0.968442\pi\)
0.995090 0.0989787i \(-0.0315576\pi\)
\(642\) 0 0
\(643\) 44.2604i 1.74546i 0.488203 + 0.872730i \(0.337653\pi\)
−0.488203 + 0.872730i \(0.662347\pi\)
\(644\) 0 0
\(645\) 19.2651 0.758562
\(646\) 0 0
\(647\) 1.68354 0.0661869 0.0330935 0.999452i \(-0.489464\pi\)
0.0330935 + 0.999452i \(0.489464\pi\)
\(648\) 0 0
\(649\) 0.564620i 0.0221633i
\(650\) 0 0
\(651\) 10.3690i 0.406392i
\(652\) 0 0
\(653\) 10.1938i 0.398913i −0.979907 0.199456i \(-0.936082\pi\)
0.979907 0.199456i \(-0.0639176\pi\)
\(654\) 0 0
\(655\) 16.6141 0.649166
\(656\) 0 0
\(657\) 2.92443i 0.114093i
\(658\) 0 0
\(659\) 8.86756 0.345431 0.172715 0.984972i \(-0.444746\pi\)
0.172715 + 0.984972i \(0.444746\pi\)
\(660\) 0 0
\(661\) −3.88174 −0.150982 −0.0754910 0.997146i \(-0.524052\pi\)
−0.0754910 + 0.997146i \(0.524052\pi\)
\(662\) 0 0
\(663\) −34.4280 36.4626i −1.33707 1.41609i
\(664\) 0 0
\(665\) −8.49259 −0.329328
\(666\) 0 0
\(667\) 24.5330 0.949921
\(668\) 0 0
\(669\) 35.2149i 1.36149i
\(670\) 0 0
\(671\) −0.251457 −0.00970739
\(672\) 0 0
\(673\) 9.49461i 0.365990i 0.983114 + 0.182995i \(0.0585792\pi\)
−0.983114 + 0.182995i \(0.941421\pi\)
\(674\) 0 0
\(675\) 17.7828i 0.684460i
\(676\) 0 0
\(677\) 23.6839i 0.910246i 0.890428 + 0.455123i \(0.150405\pi\)
−0.890428 + 0.455123i \(0.849595\pi\)
\(678\) 0 0
\(679\) 15.6431 0.600325
\(680\) 0 0
\(681\) 30.9655 1.18660
\(682\) 0 0
\(683\) 25.4349i 0.973240i 0.873614 + 0.486620i \(0.161770\pi\)
−0.873614 + 0.486620i \(0.838230\pi\)
\(684\) 0 0
\(685\) 7.13421i 0.272584i
\(686\) 0 0
\(687\) 14.3414i 0.547158i
\(688\) 0 0
\(689\) −80.2514 −3.05733
\(690\) 0 0
\(691\) 7.48165i 0.284616i −0.989822 0.142308i \(-0.954548\pi\)
0.989822 0.142308i \(-0.0454523\pi\)
\(692\) 0 0
\(693\) 0.147098 0.00558780
\(694\) 0 0
\(695\) −19.2655 −0.730782
\(696\) 0 0
\(697\) −17.6991 18.7450i −0.670401 0.710019i
\(698\) 0 0
\(699\) 43.6950 1.65270
\(700\) 0 0
\(701\) −37.4544 −1.41463 −0.707316 0.706898i \(-0.750094\pi\)
−0.707316 + 0.706898i \(0.750094\pi\)
\(702\) 0 0
\(703\) 31.3218i 1.18133i
\(704\) 0 0
\(705\) 21.0663 0.793403
\(706\) 0 0
\(707\) 18.9333i 0.712059i
\(708\) 0 0
\(709\) 19.5713i 0.735016i −0.930020 0.367508i \(-0.880211\pi\)
0.930020 0.367508i \(-0.119789\pi\)
\(710\) 0 0
\(711\) 3.38203i 0.126836i
\(712\) 0 0
\(713\) −13.1595 −0.492827
\(714\) 0 0
\(715\) −4.59179 −0.171723
\(716\) 0 0
\(717\) 8.92347i 0.333253i
\(718\) 0 0
\(719\) 1.37742i 0.0513689i −0.999670 0.0256845i \(-0.991823\pi\)
0.999670 0.0256845i \(-0.00817652\pi\)
\(720\) 0 0
\(721\) 22.7661i 0.847854i
\(722\) 0 0
\(723\) 18.0152 0.669992
\(724\) 0 0
\(725\) 31.8735i 1.18375i
\(726\) 0 0
\(727\) −12.4574 −0.462018 −0.231009 0.972952i \(-0.574203\pi\)
−0.231009 + 0.972952i \(0.574203\pi\)
\(728\) 0 0
\(729\) −24.7829 −0.917887
\(730\) 0 0
\(731\) −26.8558 + 25.3573i −0.993297 + 0.937874i
\(732\) 0 0
\(733\) 13.5840 0.501735 0.250868 0.968021i \(-0.419284\pi\)
0.250868 + 0.968021i \(0.419284\pi\)
\(734\) 0 0
\(735\) 11.9334 0.440171
\(736\) 0 0
\(737\) 7.38520i 0.272038i
\(738\) 0 0
\(739\) 37.9653 1.39658 0.698289 0.715816i \(-0.253945\pi\)
0.698289 + 0.715816i \(0.253945\pi\)
\(740\) 0 0
\(741\) 71.5081i 2.62692i
\(742\) 0 0
\(743\) 10.6464i 0.390579i 0.980746 + 0.195290i \(0.0625647\pi\)
−0.980746 + 0.195290i \(0.937435\pi\)
\(744\) 0 0
\(745\) 1.04083i 0.0381330i
\(746\) 0 0
\(747\) 1.46566 0.0536259
\(748\) 0 0
\(749\) 11.1050 0.405768
\(750\) 0 0
\(751\) 16.1764i 0.590287i 0.955453 + 0.295143i \(0.0953675\pi\)
−0.955453 + 0.295143i \(0.904633\pi\)
\(752\) 0 0
\(753\) 14.7873i 0.538880i
\(754\) 0 0
\(755\) 9.22432i 0.335707i
\(756\) 0 0
\(757\) 50.8517 1.84824 0.924118 0.382108i \(-0.124802\pi\)
0.924118 + 0.382108i \(0.124802\pi\)
\(758\) 0 0
\(759\) 2.77620i 0.100770i
\(760\) 0 0
\(761\) −0.0134063 −0.000485977 −0.000242989 1.00000i \(-0.500077\pi\)
−0.000242989 1.00000i \(0.500077\pi\)
\(762\) 0 0
\(763\) −10.4757 −0.379245
\(764\) 0 0
\(765\) 0.734131 + 0.777514i 0.0265426 + 0.0281111i
\(766\) 0 0
\(767\) −6.78189 −0.244880
\(768\) 0 0
\(769\) 29.0410 1.04725 0.523623 0.851950i \(-0.324580\pi\)
0.523623 + 0.851950i \(0.324580\pi\)
\(770\) 0 0
\(771\) 30.4909i 1.09810i
\(772\) 0 0
\(773\) 21.8675 0.786519 0.393260 0.919428i \(-0.371347\pi\)
0.393260 + 0.919428i \(0.371347\pi\)
\(774\) 0 0
\(775\) 17.0970i 0.614141i
\(776\) 0 0
\(777\) 11.5089i 0.412880i
\(778\) 0 0
\(779\) 36.7616i 1.31712i
\(780\) 0 0
\(781\) −5.20723 −0.186329
\(782\) 0 0
\(783\) 44.6719 1.59645
\(784\) 0 0
\(785\) 25.0942i 0.895652i
\(786\) 0 0
\(787\) 26.1478i 0.932068i −0.884767 0.466034i \(-0.845682\pi\)
0.884767 0.466034i \(-0.154318\pi\)
\(788\) 0 0
\(789\) 5.39692i 0.192136i
\(790\) 0 0
\(791\) 13.0149 0.462756
\(792\) 0 0
\(793\) 3.02036i 0.107256i