Properties

Label 4012.2.b.b.237.20
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 237.20
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.27

$q$-expansion

\(f(q)\) \(=\) \(q-0.405081i q^{3} +1.20499i q^{5} -0.122460i q^{7} +2.83591 q^{9} +O(q^{10})\) \(q-0.405081i q^{3} +1.20499i q^{5} -0.122460i q^{7} +2.83591 q^{9} +1.65371i q^{11} -1.21607 q^{13} +0.488118 q^{15} +(-4.09888 + 0.446336i) q^{17} +5.49834 q^{19} -0.0496062 q^{21} -1.29315i q^{23} +3.54801 q^{25} -2.36402i q^{27} -9.31434i q^{29} -6.06362i q^{31} +0.669888 q^{33} +0.147562 q^{35} +4.07227i q^{37} +0.492607i q^{39} +1.98151i q^{41} +7.44725 q^{43} +3.41723i q^{45} -10.1355 q^{47} +6.98500 q^{49} +(0.180802 + 1.66038i) q^{51} +4.71660 q^{53} -1.99270 q^{55} -2.22728i q^{57} +1.00000 q^{59} +3.86956i q^{61} -0.347285i q^{63} -1.46535i q^{65} +2.82254 q^{67} -0.523832 q^{69} +12.5130i q^{71} +14.7913i q^{73} -1.43723i q^{75} +0.202513 q^{77} -0.917389i q^{79} +7.55011 q^{81} +12.7272 q^{83} +(-0.537829 - 4.93909i) q^{85} -3.77307 q^{87} -2.24107 q^{89} +0.148920i q^{91} -2.45626 q^{93} +6.62543i q^{95} -7.92714i q^{97} +4.68978i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

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
\(3\) 0.405081i 0.233874i −0.993139 0.116937i \(-0.962692\pi\)
0.993139 0.116937i \(-0.0373075\pi\)
\(4\) 0 0
\(5\) 1.20499i 0.538887i 0.963016 + 0.269443i \(0.0868397\pi\)
−0.963016 + 0.269443i \(0.913160\pi\)
\(6\) 0 0
\(7\) 0.122460i 0.0462854i −0.999732 0.0231427i \(-0.992633\pi\)
0.999732 0.0231427i \(-0.00736722\pi\)
\(8\) 0 0
\(9\) 2.83591 0.945303
\(10\) 0 0
\(11\) 1.65371i 0.498613i 0.968425 + 0.249306i \(0.0802026\pi\)
−0.968425 + 0.249306i \(0.919797\pi\)
\(12\) 0 0
\(13\) −1.21607 −0.337277 −0.168638 0.985678i \(-0.553937\pi\)
−0.168638 + 0.985678i \(0.553937\pi\)
\(14\) 0 0
\(15\) 0.488118 0.126032
\(16\) 0 0
\(17\) −4.09888 + 0.446336i −0.994123 + 0.108252i
\(18\) 0 0
\(19\) 5.49834 1.26141 0.630703 0.776025i \(-0.282767\pi\)
0.630703 + 0.776025i \(0.282767\pi\)
\(20\) 0 0
\(21\) −0.0496062 −0.0108250
\(22\) 0 0
\(23\) 1.29315i 0.269641i −0.990870 0.134820i \(-0.956954\pi\)
0.990870 0.134820i \(-0.0430458\pi\)
\(24\) 0 0
\(25\) 3.54801 0.709601
\(26\) 0 0
\(27\) 2.36402i 0.454956i
\(28\) 0 0
\(29\) 9.31434i 1.72963i −0.502091 0.864815i \(-0.667436\pi\)
0.502091 0.864815i \(-0.332564\pi\)
\(30\) 0 0
\(31\) 6.06362i 1.08906i −0.838742 0.544530i \(-0.816708\pi\)
0.838742 0.544530i \(-0.183292\pi\)
\(32\) 0 0
\(33\) 0.669888 0.116613
\(34\) 0 0
\(35\) 0.147562 0.0249426
\(36\) 0 0
\(37\) 4.07227i 0.669478i 0.942311 + 0.334739i \(0.108648\pi\)
−0.942311 + 0.334739i \(0.891352\pi\)
\(38\) 0 0
\(39\) 0.492607i 0.0788803i
\(40\) 0 0
\(41\) 1.98151i 0.309460i 0.987957 + 0.154730i \(0.0494508\pi\)
−0.987957 + 0.154730i \(0.950549\pi\)
\(42\) 0 0
\(43\) 7.44725 1.13570 0.567848 0.823134i \(-0.307776\pi\)
0.567848 + 0.823134i \(0.307776\pi\)
\(44\) 0 0
\(45\) 3.41723i 0.509411i
\(46\) 0 0
\(47\) −10.1355 −1.47841 −0.739207 0.673478i \(-0.764800\pi\)
−0.739207 + 0.673478i \(0.764800\pi\)
\(48\) 0 0
\(49\) 6.98500 0.997858
\(50\) 0 0
\(51\) 0.180802 + 1.66038i 0.0253174 + 0.232500i
\(52\) 0 0
\(53\) 4.71660 0.647874 0.323937 0.946079i \(-0.394993\pi\)
0.323937 + 0.946079i \(0.394993\pi\)
\(54\) 0 0
\(55\) −1.99270 −0.268696
\(56\) 0 0
\(57\) 2.22728i 0.295010i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 3.86956i 0.495447i 0.968831 + 0.247723i \(0.0796824\pi\)
−0.968831 + 0.247723i \(0.920318\pi\)
\(62\) 0 0
\(63\) 0.347285i 0.0437538i
\(64\) 0 0
\(65\) 1.46535i 0.181754i
\(66\) 0 0
\(67\) 2.82254 0.344828 0.172414 0.985025i \(-0.444843\pi\)
0.172414 + 0.985025i \(0.444843\pi\)
\(68\) 0 0
\(69\) −0.523832 −0.0630620
\(70\) 0 0
\(71\) 12.5130i 1.48502i 0.669834 + 0.742511i \(0.266365\pi\)
−0.669834 + 0.742511i \(0.733635\pi\)
\(72\) 0 0
\(73\) 14.7913i 1.73119i 0.500740 + 0.865597i \(0.333061\pi\)
−0.500740 + 0.865597i \(0.666939\pi\)
\(74\) 0 0
\(75\) 1.43723i 0.165957i
\(76\) 0 0
\(77\) 0.202513 0.0230785
\(78\) 0 0
\(79\) 0.917389i 0.103214i −0.998667 0.0516072i \(-0.983566\pi\)
0.998667 0.0516072i \(-0.0164344\pi\)
\(80\) 0 0
\(81\) 7.55011 0.838901
\(82\) 0 0
\(83\) 12.7272 1.39699 0.698494 0.715616i \(-0.253854\pi\)
0.698494 + 0.715616i \(0.253854\pi\)
\(84\) 0 0
\(85\) −0.537829 4.93909i −0.0583357 0.535720i
\(86\) 0 0
\(87\) −3.77307 −0.404515
\(88\) 0 0
\(89\) −2.24107 −0.237553 −0.118776 0.992921i \(-0.537897\pi\)
−0.118776 + 0.992921i \(0.537897\pi\)
\(90\) 0 0
\(91\) 0.148920i 0.0156110i
\(92\) 0 0
\(93\) −2.45626 −0.254702
\(94\) 0 0
\(95\) 6.62543i 0.679755i
\(96\) 0 0
\(97\) 7.92714i 0.804879i −0.915446 0.402440i \(-0.868162\pi\)
0.915446 0.402440i \(-0.131838\pi\)
\(98\) 0 0
\(99\) 4.68978i 0.471340i
\(100\) 0 0
\(101\) −4.46389 −0.444174 −0.222087 0.975027i \(-0.571287\pi\)
−0.222087 + 0.975027i \(0.571287\pi\)
\(102\) 0 0
\(103\) 8.70518 0.857746 0.428873 0.903365i \(-0.358911\pi\)
0.428873 + 0.903365i \(0.358911\pi\)
\(104\) 0 0
\(105\) 0.0597748i 0.00583343i
\(106\) 0 0
\(107\) 0.613356i 0.0592953i 0.999560 + 0.0296477i \(0.00943853\pi\)
−0.999560 + 0.0296477i \(0.990561\pi\)
\(108\) 0 0
\(109\) 9.44960i 0.905108i 0.891737 + 0.452554i \(0.149487\pi\)
−0.891737 + 0.452554i \(0.850513\pi\)
\(110\) 0 0
\(111\) 1.64960 0.156573
\(112\) 0 0
\(113\) 15.9263i 1.49822i −0.662447 0.749109i \(-0.730482\pi\)
0.662447 0.749109i \(-0.269518\pi\)
\(114\) 0 0
\(115\) 1.55823 0.145306
\(116\) 0 0
\(117\) −3.44866 −0.318829
\(118\) 0 0
\(119\) 0.0546582 + 0.501947i 0.00501051 + 0.0460134i
\(120\) 0 0
\(121\) 8.26524 0.751385
\(122\) 0 0
\(123\) 0.802673 0.0723746
\(124\) 0 0
\(125\) 10.3002i 0.921281i
\(126\) 0 0
\(127\) 0.0633634 0.00562259 0.00281130 0.999996i \(-0.499105\pi\)
0.00281130 + 0.999996i \(0.499105\pi\)
\(128\) 0 0
\(129\) 3.01675i 0.265610i
\(130\) 0 0
\(131\) 7.72225i 0.674697i −0.941380 0.337348i \(-0.890470\pi\)
0.941380 0.337348i \(-0.109530\pi\)
\(132\) 0 0
\(133\) 0.673325i 0.0583847i
\(134\) 0 0
\(135\) 2.84861 0.245170
\(136\) 0 0
\(137\) −5.92432 −0.506149 −0.253074 0.967447i \(-0.581442\pi\)
−0.253074 + 0.967447i \(0.581442\pi\)
\(138\) 0 0
\(139\) 16.4843i 1.39818i −0.715034 0.699090i \(-0.753589\pi\)
0.715034 0.699090i \(-0.246411\pi\)
\(140\) 0 0
\(141\) 4.10570i 0.345763i
\(142\) 0 0
\(143\) 2.01103i 0.168171i
\(144\) 0 0
\(145\) 11.2237 0.932074
\(146\) 0 0
\(147\) 2.82950i 0.233373i
\(148\) 0 0
\(149\) −0.397789 −0.0325881 −0.0162941 0.999867i \(-0.505187\pi\)
−0.0162941 + 0.999867i \(0.505187\pi\)
\(150\) 0 0
\(151\) 19.2318 1.56506 0.782532 0.622611i \(-0.213928\pi\)
0.782532 + 0.622611i \(0.213928\pi\)
\(152\) 0 0
\(153\) −11.6240 + 1.26577i −0.939748 + 0.102331i
\(154\) 0 0
\(155\) 7.30659 0.586879
\(156\) 0 0
\(157\) −7.26619 −0.579905 −0.289953 0.957041i \(-0.593640\pi\)
−0.289953 + 0.957041i \(0.593640\pi\)
\(158\) 0 0
\(159\) 1.91061i 0.151521i
\(160\) 0 0
\(161\) −0.158359 −0.0124805
\(162\) 0 0
\(163\) 10.3754i 0.812662i 0.913726 + 0.406331i \(0.133192\pi\)
−0.913726 + 0.406331i \(0.866808\pi\)
\(164\) 0 0
\(165\) 0.807207i 0.0628409i
\(166\) 0 0
\(167\) 14.1828i 1.09750i −0.835987 0.548750i \(-0.815104\pi\)
0.835987 0.548750i \(-0.184896\pi\)
\(168\) 0 0
\(169\) −11.5212 −0.886244
\(170\) 0 0
\(171\) 15.5928 1.19241
\(172\) 0 0
\(173\) 6.69164i 0.508756i −0.967105 0.254378i \(-0.918129\pi\)
0.967105 0.254378i \(-0.0818708\pi\)
\(174\) 0 0
\(175\) 0.434488i 0.0328442i
\(176\) 0 0
\(177\) 0.405081i 0.0304478i
\(178\) 0 0
\(179\) 11.8396 0.884931 0.442465 0.896786i \(-0.354104\pi\)
0.442465 + 0.896786i \(0.354104\pi\)
\(180\) 0 0
\(181\) 15.6430i 1.16274i 0.813641 + 0.581368i \(0.197482\pi\)
−0.813641 + 0.581368i \(0.802518\pi\)
\(182\) 0 0
\(183\) 1.56749 0.115872
\(184\) 0 0
\(185\) −4.90704 −0.360773
\(186\) 0 0
\(187\) −0.738111 6.77836i −0.0539760 0.495683i
\(188\) 0 0
\(189\) −0.289497 −0.0210578
\(190\) 0 0
\(191\) 8.26989 0.598388 0.299194 0.954192i \(-0.403282\pi\)
0.299194 + 0.954192i \(0.403282\pi\)
\(192\) 0 0
\(193\) 16.7443i 1.20528i 0.798012 + 0.602642i \(0.205885\pi\)
−0.798012 + 0.602642i \(0.794115\pi\)
\(194\) 0 0
\(195\) −0.593585 −0.0425075
\(196\) 0 0
\(197\) 8.46212i 0.602901i −0.953482 0.301450i \(-0.902529\pi\)
0.953482 0.301450i \(-0.0974708\pi\)
\(198\) 0 0
\(199\) 27.1755i 1.92642i 0.268756 + 0.963208i \(0.413388\pi\)
−0.268756 + 0.963208i \(0.586612\pi\)
\(200\) 0 0
\(201\) 1.14336i 0.0806463i
\(202\) 0 0
\(203\) −1.14063 −0.0800567
\(204\) 0 0
\(205\) −2.38770 −0.166764
\(206\) 0 0
\(207\) 3.66726i 0.254892i
\(208\) 0 0
\(209\) 9.09267i 0.628953i
\(210\) 0 0
\(211\) 8.91458i 0.613705i 0.951757 + 0.306853i \(0.0992759\pi\)
−0.951757 + 0.306853i \(0.900724\pi\)
\(212\) 0 0
\(213\) 5.06879 0.347308
\(214\) 0 0
\(215\) 8.97385i 0.612011i
\(216\) 0 0
\(217\) −0.742550 −0.0504076
\(218\) 0 0
\(219\) 5.99170 0.404881
\(220\) 0 0
\(221\) 4.98452 0.542775i 0.335295 0.0365110i
\(222\) 0 0
\(223\) 11.9201 0.798232 0.399116 0.916900i \(-0.369317\pi\)
0.399116 + 0.916900i \(0.369317\pi\)
\(224\) 0 0
\(225\) 10.0618 0.670788
\(226\) 0 0
\(227\) 13.2247i 0.877751i 0.898548 + 0.438876i \(0.144623\pi\)
−0.898548 + 0.438876i \(0.855377\pi\)
\(228\) 0 0
\(229\) −8.47499 −0.560043 −0.280022 0.959994i \(-0.590342\pi\)
−0.280022 + 0.959994i \(0.590342\pi\)
\(230\) 0 0
\(231\) 0.0820343i 0.00539746i
\(232\) 0 0
\(233\) 4.37200i 0.286419i 0.989692 + 0.143210i \(0.0457423\pi\)
−0.989692 + 0.143210i \(0.954258\pi\)
\(234\) 0 0
\(235\) 12.2132i 0.796698i
\(236\) 0 0
\(237\) −0.371617 −0.0241391
\(238\) 0 0
\(239\) 29.9809 1.93930 0.969651 0.244491i \(-0.0786210\pi\)
0.969651 + 0.244491i \(0.0786210\pi\)
\(240\) 0 0
\(241\) 20.5142i 1.32144i 0.750634 + 0.660718i \(0.229748\pi\)
−0.750634 + 0.660718i \(0.770252\pi\)
\(242\) 0 0
\(243\) 10.1505i 0.651153i
\(244\) 0 0
\(245\) 8.41684i 0.537732i
\(246\) 0 0
\(247\) −6.68636 −0.425443
\(248\) 0 0
\(249\) 5.15554i 0.326719i
\(250\) 0 0
\(251\) 13.4172 0.846888 0.423444 0.905922i \(-0.360821\pi\)
0.423444 + 0.905922i \(0.360821\pi\)
\(252\) 0 0
\(253\) 2.13850 0.134446
\(254\) 0 0
\(255\) −2.00074 + 0.217865i −0.125291 + 0.0136432i
\(256\) 0 0
\(257\) 15.9305 0.993717 0.496859 0.867831i \(-0.334487\pi\)
0.496859 + 0.867831i \(0.334487\pi\)
\(258\) 0 0
\(259\) 0.498690 0.0309871
\(260\) 0 0
\(261\) 26.4146i 1.63502i
\(262\) 0 0
\(263\) 3.30320 0.203684 0.101842 0.994801i \(-0.467526\pi\)
0.101842 + 0.994801i \(0.467526\pi\)
\(264\) 0 0
\(265\) 5.68344i 0.349131i
\(266\) 0 0
\(267\) 0.907815i 0.0555574i
\(268\) 0 0
\(269\) 18.0068i 1.09790i −0.835856 0.548948i \(-0.815028\pi\)
0.835856 0.548948i \(-0.184972\pi\)
\(270\) 0 0
\(271\) 8.49255 0.515886 0.257943 0.966160i \(-0.416955\pi\)
0.257943 + 0.966160i \(0.416955\pi\)
\(272\) 0 0
\(273\) 0.0603246 0.00365101
\(274\) 0 0
\(275\) 5.86738i 0.353816i
\(276\) 0 0
\(277\) 13.1893i 0.792467i −0.918150 0.396234i \(-0.870317\pi\)
0.918150 0.396234i \(-0.129683\pi\)
\(278\) 0 0
\(279\) 17.1959i 1.02949i
\(280\) 0 0
\(281\) 9.65973 0.576251 0.288126 0.957593i \(-0.406968\pi\)
0.288126 + 0.957593i \(0.406968\pi\)
\(282\) 0 0
\(283\) 11.4726i 0.681975i −0.940068 0.340987i \(-0.889239\pi\)
0.940068 0.340987i \(-0.110761\pi\)
\(284\) 0 0
\(285\) 2.68384 0.158977
\(286\) 0 0
\(287\) 0.242655 0.0143235
\(288\) 0 0
\(289\) 16.6016 3.65895i 0.976563 0.215232i
\(290\) 0 0
\(291\) −3.21114 −0.188240
\(292\) 0 0
\(293\) −2.06902 −0.120873 −0.0604366 0.998172i \(-0.519249\pi\)
−0.0604366 + 0.998172i \(0.519249\pi\)
\(294\) 0 0
\(295\) 1.20499i 0.0701571i
\(296\) 0 0
\(297\) 3.90941 0.226847
\(298\) 0 0
\(299\) 1.57256i 0.0909437i
\(300\) 0 0
\(301\) 0.911989i 0.0525662i
\(302\) 0 0
\(303\) 1.80824i 0.103881i
\(304\) 0 0
\(305\) −4.66278 −0.266990
\(306\) 0 0
\(307\) −19.5503 −1.11580 −0.557898 0.829909i \(-0.688392\pi\)
−0.557898 + 0.829909i \(0.688392\pi\)
\(308\) 0 0
\(309\) 3.52631i 0.200605i
\(310\) 0 0
\(311\) 23.0757i 1.30850i −0.756277 0.654252i \(-0.772984\pi\)
0.756277 0.654252i \(-0.227016\pi\)
\(312\) 0 0
\(313\) 14.2773i 0.807000i −0.914980 0.403500i \(-0.867793\pi\)
0.914980 0.403500i \(-0.132207\pi\)
\(314\) 0 0
\(315\) 0.418474 0.0235783
\(316\) 0 0
\(317\) 17.8592i 1.00307i 0.865137 + 0.501536i \(0.167231\pi\)
−0.865137 + 0.501536i \(0.832769\pi\)
\(318\) 0 0
\(319\) 15.4032 0.862415
\(320\) 0 0
\(321\) 0.248459 0.0138676
\(322\) 0 0
\(323\) −22.5370 + 2.45411i −1.25399 + 0.136550i
\(324\) 0 0
\(325\) −4.31462 −0.239332
\(326\) 0 0
\(327\) 3.82786 0.211681
\(328\) 0 0
\(329\) 1.24119i 0.0684291i
\(330\) 0 0
\(331\) 22.6188 1.24324 0.621621 0.783318i \(-0.286474\pi\)
0.621621 + 0.783318i \(0.286474\pi\)
\(332\) 0 0
\(333\) 11.5486i 0.632859i
\(334\) 0 0
\(335\) 3.40112i 0.185823i
\(336\) 0 0
\(337\) 22.3964i 1.22001i 0.792398 + 0.610004i \(0.208832\pi\)
−0.792398 + 0.610004i \(0.791168\pi\)
\(338\) 0 0
\(339\) −6.45144 −0.350394
\(340\) 0 0
\(341\) 10.0275 0.543019
\(342\) 0 0
\(343\) 1.71260i 0.0924717i
\(344\) 0 0
\(345\) 0.631211i 0.0339833i
\(346\) 0 0
\(347\) 17.9296i 0.962511i −0.876580 0.481256i \(-0.840181\pi\)
0.876580 0.481256i \(-0.159819\pi\)
\(348\) 0 0
\(349\) −15.7956 −0.845517 −0.422758 0.906242i \(-0.638938\pi\)
−0.422758 + 0.906242i \(0.638938\pi\)
\(350\) 0 0
\(351\) 2.87481i 0.153446i
\(352\) 0 0
\(353\) −14.9174 −0.793971 −0.396986 0.917825i \(-0.629944\pi\)
−0.396986 + 0.917825i \(0.629944\pi\)
\(354\) 0 0
\(355\) −15.0780 −0.800259
\(356\) 0 0
\(357\) 0.203330 0.0221410i 0.0107613 0.00117183i
\(358\) 0 0
\(359\) 8.98954 0.474450 0.237225 0.971455i \(-0.423762\pi\)
0.237225 + 0.971455i \(0.423762\pi\)
\(360\) 0 0
\(361\) 11.2317 0.591144
\(362\) 0 0
\(363\) 3.34809i 0.175729i
\(364\) 0 0
\(365\) −17.8234 −0.932918
\(366\) 0 0
\(367\) 4.92123i 0.256886i −0.991717 0.128443i \(-0.959002\pi\)
0.991717 0.128443i \(-0.0409979\pi\)
\(368\) 0 0
\(369\) 5.61939i 0.292534i
\(370\) 0 0
\(371\) 0.577593i 0.0299872i
\(372\) 0 0
\(373\) 2.22961 0.115445 0.0577225 0.998333i \(-0.481616\pi\)
0.0577225 + 0.998333i \(0.481616\pi\)
\(374\) 0 0
\(375\) 4.17244 0.215464
\(376\) 0 0
\(377\) 11.3269i 0.583364i
\(378\) 0 0
\(379\) 27.0371i 1.38880i 0.719588 + 0.694401i \(0.244331\pi\)
−0.719588 + 0.694401i \(0.755669\pi\)
\(380\) 0 0
\(381\) 0.0256673i 0.00131498i
\(382\) 0 0
\(383\) −9.31266 −0.475855 −0.237927 0.971283i \(-0.576468\pi\)
−0.237927 + 0.971283i \(0.576468\pi\)
\(384\) 0 0
\(385\) 0.244026i 0.0124367i
\(386\) 0 0
\(387\) 21.1197 1.07358
\(388\) 0 0
\(389\) −12.7814 −0.648045 −0.324023 0.946049i \(-0.605035\pi\)
−0.324023 + 0.946049i \(0.605035\pi\)
\(390\) 0 0
\(391\) 0.577180 + 5.30047i 0.0291893 + 0.268056i
\(392\) 0 0
\(393\) −3.12814 −0.157794
\(394\) 0 0
\(395\) 1.10544 0.0556208
\(396\) 0 0
\(397\) 20.4352i 1.02561i −0.858505 0.512806i \(-0.828606\pi\)
0.858505 0.512806i \(-0.171394\pi\)
\(398\) 0 0
\(399\) −0.272752 −0.0136547
\(400\) 0 0
\(401\) 22.5597i 1.12658i −0.826259 0.563290i \(-0.809535\pi\)
0.826259 0.563290i \(-0.190465\pi\)
\(402\) 0 0
\(403\) 7.37379i 0.367315i
\(404\) 0 0
\(405\) 9.09778i 0.452072i
\(406\) 0 0
\(407\) −6.73437 −0.333810
\(408\) 0 0
\(409\) 16.2860 0.805291 0.402646 0.915356i \(-0.368091\pi\)
0.402646 + 0.915356i \(0.368091\pi\)
\(410\) 0 0
\(411\) 2.39983i 0.118375i
\(412\) 0 0
\(413\) 0.122460i 0.00602585i
\(414\) 0 0
\(415\) 15.3361i 0.752818i
\(416\) 0 0
\(417\) −6.67748 −0.326998
\(418\) 0 0
\(419\) 3.56522i 0.174172i −0.996201 0.0870862i \(-0.972244\pi\)
0.996201 0.0870862i \(-0.0277556\pi\)
\(420\) 0 0
\(421\) −36.8154 −1.79427 −0.897136 0.441755i \(-0.854356\pi\)
−0.897136 + 0.441755i \(0.854356\pi\)
\(422\) 0 0
\(423\) −28.7434 −1.39755
\(424\) 0 0
\(425\) −14.5428 + 1.58360i −0.705431 + 0.0768160i
\(426\) 0 0
\(427\) 0.473866 0.0229320
\(428\) 0 0
\(429\) −0.814630 −0.0393307
\(430\) 0 0
\(431\) 7.27173i 0.350267i 0.984545 + 0.175134i \(0.0560357\pi\)
−0.984545 + 0.175134i \(0.943964\pi\)
\(432\) 0 0
\(433\) −13.9835 −0.672004 −0.336002 0.941861i \(-0.609075\pi\)
−0.336002 + 0.941861i \(0.609075\pi\)
\(434\) 0 0
\(435\) 4.54650i 0.217988i
\(436\) 0 0
\(437\) 7.11019i 0.340127i
\(438\) 0 0
\(439\) 20.9580i 1.00027i −0.865947 0.500136i \(-0.833283\pi\)
0.865947 0.500136i \(-0.166717\pi\)
\(440\) 0 0
\(441\) 19.8088 0.943278
\(442\) 0 0
\(443\) −10.8826 −0.517046 −0.258523 0.966005i \(-0.583236\pi\)
−0.258523 + 0.966005i \(0.583236\pi\)
\(444\) 0 0
\(445\) 2.70046i 0.128014i
\(446\) 0 0
\(447\) 0.161137i 0.00762151i
\(448\) 0 0
\(449\) 21.3267i 1.00647i −0.864150 0.503235i \(-0.832143\pi\)
0.864150 0.503235i \(-0.167857\pi\)
\(450\) 0 0
\(451\) −3.27685 −0.154301
\(452\) 0 0
\(453\) 7.79045i 0.366027i
\(454\) 0 0
\(455\) −0.179446 −0.00841257
\(456\) 0 0
\(457\) 3.67720 0.172012 0.0860061 0.996295i \(-0.472590\pi\)
0.0860061 + 0.996295i \(0.472590\pi\)
\(458\) 0 0
\(459\) 1.05515 + 9.68982i 0.0492500 + 0.452282i
\(460\) 0 0
\(461\) 1.43905 0.0670233 0.0335116 0.999438i \(-0.489331\pi\)
0.0335116 + 0.999438i \(0.489331\pi\)
\(462\) 0 0
\(463\) −3.24480 −0.150799 −0.0753994 0.997153i \(-0.524023\pi\)
−0.0753994 + 0.997153i \(0.524023\pi\)
\(464\) 0 0
\(465\) 2.95976i 0.137256i
\(466\) 0 0
\(467\) −23.9357 −1.10761 −0.553805 0.832646i \(-0.686825\pi\)
−0.553805 + 0.832646i \(0.686825\pi\)
\(468\) 0 0
\(469\) 0.345647i 0.0159605i
\(470\) 0 0
\(471\) 2.94340i 0.135625i
\(472\) 0 0
\(473\) 12.3156i 0.566272i
\(474\) 0 0
\(475\) 19.5081 0.895095
\(476\) 0 0
\(477\) 13.3758 0.612438
\(478\) 0 0
\(479\) 2.84658i 0.130063i 0.997883 + 0.0650317i \(0.0207149\pi\)
−0.997883 + 0.0650317i \(0.979285\pi\)
\(480\) 0 0
\(481\) 4.95217i 0.225799i
\(482\) 0 0
\(483\) 0.0641484i 0.00291885i
\(484\) 0 0
\(485\) 9.55211 0.433739
\(486\) 0 0
\(487\) 21.1362i 0.957773i −0.877877 0.478887i \(-0.841041\pi\)
0.877877 0.478887i \(-0.158959\pi\)
\(488\) 0 0
\(489\) 4.20287 0.190061
\(490\) 0 0
\(491\) −7.61991 −0.343882 −0.171941 0.985107i \(-0.555004\pi\)
−0.171941 + 0.985107i \(0.555004\pi\)
\(492\) 0 0
\(493\) 4.15732 + 38.1783i 0.187236 + 1.71947i
\(494\) 0 0
\(495\) −5.65112 −0.253999
\(496\) 0 0
\(497\) 1.53234 0.0687349
\(498\) 0 0
\(499\) 12.5751i 0.562938i 0.959570 + 0.281469i \(0.0908217\pi\)
−0.959570 + 0.281469i \(0.909178\pi\)
\(500\) 0 0
\(501\) −5.74520 −0.256676
\(502\) 0 0
\(503\) 26.3701i 1.17579i −0.808939 0.587893i \(-0.799958\pi\)
0.808939 0.587893i \(-0.200042\pi\)
\(504\) 0 0
\(505\) 5.37893i 0.239359i
\(506\) 0 0
\(507\) 4.66701i 0.207269i
\(508\) 0 0
\(509\) −5.57527 −0.247119 −0.123560 0.992337i \(-0.539431\pi\)
−0.123560 + 0.992337i \(0.539431\pi\)
\(510\) 0 0
\(511\) 1.81134 0.0801291
\(512\) 0 0
\(513\) 12.9982i 0.573884i
\(514\) 0 0
\(515\) 10.4896i 0.462228i
\(516\) 0 0
\(517\) 16.7612i 0.737157i
\(518\) 0 0
\(519\) −2.71066 −0.118985
\(520\) 0 0
\(521\) 8.87437i 0.388793i 0.980923 + 0.194397i \(0.0622749\pi\)
−0.980923 + 0.194397i \(0.937725\pi\)
\(522\) 0 0
\(523\) 4.70502 0.205736 0.102868 0.994695i \(-0.467198\pi\)
0.102868 + 0.994695i \(0.467198\pi\)
\(524\) 0 0
\(525\) −0.176003 −0.00768140
\(526\) 0 0
\(527\) 2.70641 + 24.8540i 0.117893 + 1.08266i
\(528\) 0 0
\(529\) 21.3278 0.927294
\(530\) 0 0
\(531\) 2.83591 0.123068
\(532\) 0 0
\(533\) 2.40966i 0.104374i
\(534\) 0 0
\(535\) −0.739086 −0.0319535
\(536\) 0 0
\(537\) 4.79599i 0.206962i
\(538\) 0 0
\(539\) 11.5512i 0.497545i
\(540\) 0 0
\(541\) 35.0127i 1.50531i −0.658413 0.752657i \(-0.728772\pi\)
0.658413 0.752657i \(-0.271228\pi\)
\(542\) 0 0
\(543\) 6.33669 0.271934
\(544\) 0 0
\(545\) −11.3867 −0.487751
\(546\) 0 0
\(547\) 33.3916i 1.42772i −0.700287 0.713862i \(-0.746944\pi\)
0.700287 0.713862i \(-0.253056\pi\)
\(548\) 0 0
\(549\) 10.9737i 0.468347i
\(550\) 0 0
\(551\) 51.2134i 2.18176i
\(552\) 0 0
\(553\) −0.112343 −0.00477732
\(554\) 0 0
\(555\) 1.98775i 0.0843753i
\(556\) 0 0
\(557\) −15.5185 −0.657540 −0.328770 0.944410i \(-0.606634\pi\)
−0.328770 + 0.944410i \(0.606634\pi\)
\(558\) 0 0
\(559\) −9.05638 −0.383044
\(560\) 0 0
\(561\) −2.74579 + 0.298995i −0.115927 + 0.0126236i
\(562\) 0 0
\(563\) 9.09863 0.383462 0.191731 0.981448i \(-0.438590\pi\)
0.191731 + 0.981448i \(0.438590\pi\)
\(564\) 0 0
\(565\) 19.1910 0.807370
\(566\) 0 0
\(567\) 0.924584i 0.0388289i
\(568\) 0 0
\(569\) −19.1267 −0.801834 −0.400917 0.916114i \(-0.631308\pi\)
−0.400917 + 0.916114i \(0.631308\pi\)
\(570\) 0 0
\(571\) 3.78922i 0.158574i 0.996852 + 0.0792869i \(0.0252643\pi\)
−0.996852 + 0.0792869i \(0.974736\pi\)
\(572\) 0 0
\(573\) 3.34998i 0.139947i
\(574\) 0 0
\(575\) 4.58811i 0.191337i
\(576\) 0 0
\(577\) −24.9530 −1.03881 −0.519403 0.854529i \(-0.673846\pi\)
−0.519403 + 0.854529i \(0.673846\pi\)
\(578\) 0 0
\(579\) 6.78282 0.281885
\(580\) 0 0
\(581\) 1.55857i 0.0646602i
\(582\) 0 0
\(583\) 7.79989i 0.323039i
\(584\) 0 0
\(585\) 4.15559i 0.171813i
\(586\) 0 0
\(587\) 5.06343 0.208990 0.104495 0.994525i \(-0.466677\pi\)
0.104495 + 0.994525i \(0.466677\pi\)
\(588\) 0 0
\(589\) 33.3399i 1.37375i
\(590\) 0 0
\(591\) −3.42785 −0.141003
\(592\) 0 0
\(593\) 11.2442 0.461746 0.230873 0.972984i \(-0.425842\pi\)
0.230873 + 0.972984i \(0.425842\pi\)
\(594\) 0 0
\(595\) −0.604840 + 0.0658624i −0.0247960 + 0.00270010i
\(596\) 0 0
\(597\) 11.0083 0.450539
\(598\) 0 0
\(599\) −14.4981 −0.592378 −0.296189 0.955129i \(-0.595716\pi\)
−0.296189 + 0.955129i \(0.595716\pi\)
\(600\) 0 0
\(601\) 12.1474i 0.495503i 0.968824 + 0.247751i \(0.0796916\pi\)
−0.968824 + 0.247751i \(0.920308\pi\)
\(602\) 0 0
\(603\) 8.00446 0.325967
\(604\) 0 0
\(605\) 9.95951i 0.404912i
\(606\) 0 0
\(607\) 45.1298i 1.83176i 0.401451 + 0.915881i \(0.368506\pi\)
−0.401451 + 0.915881i \(0.631494\pi\)
\(608\) 0 0
\(609\) 0.462049i 0.0187232i
\(610\) 0 0
\(611\) 12.3255 0.498635
\(612\) 0 0
\(613\) −42.8452 −1.73050 −0.865250 0.501340i \(-0.832841\pi\)
−0.865250 + 0.501340i \(0.832841\pi\)
\(614\) 0 0
\(615\) 0.967211i 0.0390017i
\(616\) 0 0
\(617\) 18.9274i 0.761988i 0.924577 + 0.380994i \(0.124418\pi\)
−0.924577 + 0.380994i \(0.875582\pi\)
\(618\) 0 0
\(619\) 18.8350i 0.757041i −0.925593 0.378520i \(-0.876433\pi\)
0.925593 0.378520i \(-0.123567\pi\)
\(620\) 0 0
\(621\) −3.05704 −0.122675
\(622\) 0 0
\(623\) 0.274441i 0.0109952i
\(624\) 0 0
\(625\) 5.32837 0.213135
\(626\) 0 0
\(627\) 3.68327 0.147096
\(628\) 0 0
\(629\) −1.81760 16.6917i −0.0724725 0.665543i
\(630\) 0 0
\(631\) 13.5185 0.538164 0.269082 0.963117i \(-0.413280\pi\)
0.269082 + 0.963117i \(0.413280\pi\)
\(632\) 0 0
\(633\) 3.61113 0.143530
\(634\) 0 0
\(635\) 0.0763521i 0.00302994i
\(636\) 0 0
\(637\) −8.49425 −0.336554
\(638\) 0 0
\(639\) 35.4858i 1.40380i
\(640\) 0 0
\(641\) 20.1887i 0.797407i 0.917080 + 0.398704i \(0.130540\pi\)
−0.917080 + 0.398704i \(0.869460\pi\)
\(642\) 0 0
\(643\) 13.4682i 0.531135i −0.964092 0.265567i \(-0.914441\pi\)
0.964092 0.265567i \(-0.0855592\pi\)
\(644\) 0 0
\(645\) 3.63514 0.143133
\(646\) 0 0
\(647\) 6.54797 0.257427 0.128714 0.991682i \(-0.458915\pi\)
0.128714 + 0.991682i \(0.458915\pi\)
\(648\) 0 0
\(649\) 1.65371i 0.0649139i
\(650\) 0 0
\(651\) 0.300793i 0.0117890i
\(652\) 0 0
\(653\) 27.3249i 1.06931i 0.845071 + 0.534654i \(0.179558\pi\)
−0.845071 + 0.534654i \(0.820442\pi\)
\(654\) 0 0
\(655\) 9.30522 0.363585
\(656\) 0 0
\(657\) 41.9469i 1.63650i
\(658\) 0 0
\(659\) −47.4256 −1.84744 −0.923719 0.383070i \(-0.874867\pi\)
−0.923719 + 0.383070i \(0.874867\pi\)
\(660\) 0 0
\(661\) −21.1368 −0.822127 −0.411064 0.911607i \(-0.634843\pi\)
−0.411064 + 0.911607i \(0.634843\pi\)
\(662\) 0 0
\(663\) −0.219868 2.01914i −0.00853897 0.0784167i
\(664\) 0 0
\(665\) 0.811349 0.0314627
\(666\) 0 0
\(667\) −12.0449 −0.466379
\(668\) 0 0
\(669\) 4.82863i 0.186686i
\(670\) 0 0
\(671\) −6.39914 −0.247036
\(672\) 0 0
\(673\) 16.7911i 0.647250i 0.946185 + 0.323625i \(0.104902\pi\)
−0.946185 + 0.323625i \(0.895098\pi\)
\(674\) 0 0
\(675\) 8.38755i 0.322837i
\(676\) 0 0
\(677\) 36.3838i 1.39834i 0.714953 + 0.699172i \(0.246448\pi\)
−0.714953 + 0.699172i \(0.753552\pi\)
\(678\) 0 0
\(679\) −0.970756 −0.0372542
\(680\) 0 0
\(681\) 5.35706 0.205283
\(682\) 0 0
\(683\) 11.1766i 0.427662i 0.976871 + 0.213831i \(0.0685941\pi\)
−0.976871 + 0.213831i \(0.931406\pi\)
\(684\) 0 0
\(685\) 7.13873i 0.272757i
\(686\) 0 0
\(687\) 3.43306i 0.130980i
\(688\) 0 0
\(689\) −5.73571 −0.218513
\(690\) 0 0
\(691\) 43.1149i 1.64017i 0.572244 + 0.820083i \(0.306073\pi\)
−0.572244 + 0.820083i \(0.693927\pi\)
\(692\) 0 0
\(693\) 0.574309 0.0218162
\(694\) 0 0
\(695\) 19.8634 0.753460
\(696\) 0 0
\(697\) −0.884419 8.12197i −0.0334998 0.307641i
\(698\) 0 0
\(699\) 1.77102 0.0669860
\(700\) 0 0
\(701\) 9.08849 0.343267 0.171634 0.985161i \(-0.445095\pi\)
0.171634 + 0.985161i \(0.445095\pi\)
\(702\) 0 0
\(703\) 22.3907i 0.844483i
\(704\) 0 0
\(705\) −4.94732 −0.186327
\(706\) 0 0
\(707\) 0.546647i 0.0205588i
\(708\) 0 0
\(709\) 27.8580i 1.04623i −0.852262 0.523115i \(-0.824770\pi\)
0.852262 0.523115i \(-0.175230\pi\)
\(710\) 0 0
\(711\) 2.60163i 0.0975688i
\(712\) 0 0
\(713\) −7.84119 −0.293655
\(714\) 0 0
\(715\) 2.42326 0.0906249
\(716\) 0 0
\(717\) 12.1447i 0.453552i
\(718\) 0 0
\(719\) 35.4791i 1.32315i −0.749880 0.661573i \(-0.769889\pi\)
0.749880 0.661573i \(-0.230111\pi\)
\(720\) 0 0
\(721\) 1.06603i 0.0397012i
\(722\) 0 0
\(723\) 8.30993 0.309050
\(724\) 0 0
\(725\) 33.0473i 1.22735i
\(726\) 0 0
\(727\) −15.9770 −0.592554 −0.296277 0.955102i \(-0.595745\pi\)
−0.296277 + 0.955102i \(0.595745\pi\)
\(728\) 0 0
\(729\) 18.5386 0.686613
\(730\) 0 0
\(731\) −30.5254 + 3.32398i −1.12902 + 0.122942i
\(732\) 0 0
\(733\) 6.93413 0.256118 0.128059 0.991767i \(-0.459125\pi\)
0.128059 + 0.991767i \(0.459125\pi\)
\(734\) 0 0
\(735\) 3.40951 0.125762
\(736\) 0 0
\(737\) 4.66766i 0.171936i
\(738\) 0 0
\(739\) −0.389535 −0.0143293 −0.00716464 0.999974i \(-0.502281\pi\)
−0.00716464 + 0.999974i \(0.502281\pi\)
\(740\) 0 0
\(741\) 2.70852i 0.0995000i
\(742\) 0 0
\(743\) 9.32540i 0.342116i 0.985261 + 0.171058i \(0.0547185\pi\)
−0.985261 + 0.171058i \(0.945281\pi\)
\(744\) 0 0
\(745\) 0.479331i 0.0175613i
\(746\) 0 0
\(747\) 36.0931 1.32058
\(748\) 0 0
\(749\) 0.0751114 0.00274451
\(750\) 0 0
\(751\) 28.4223i 1.03714i 0.855034 + 0.518572i \(0.173536\pi\)
−0.855034 + 0.518572i \(0.826464\pi\)
\(752\) 0 0
\(753\) 5.43507i 0.198065i
\(754\) 0 0
\(755\) 23.1741i 0.843392i
\(756\) 0 0
\(757\) −46.6449 −1.69534 −0.847668 0.530527i \(-0.821994\pi\)
−0.847668 + 0.530527i \(0.821994\pi\)
\(758\) 0 0
\(759\) 0.866267i 0.0314435i
\(760\) 0 0
\(761\) −14.8358 −0.537797 −0.268899 0.963168i \(-0.586660\pi\)
−0.268899 + 0.963168i \(0.586660\pi\)
\(762\) 0 0
\(763\) 1.15720 0.0418933
\(764\) 0 0
\(765\) −1.52523 14.0068i −0.0551449 0.506418i
\(766\) 0 0
\(767\) −1.21607 −0.0439097
\(768\) 0 0
\(769\) −3.96099 −0.142837 −0.0714185 0.997446i \(-0.522753\pi\)
−0.0714185 + 0.997446i \(0.522753\pi\)
\(770\) 0 0
\(771\) 6.45315i 0.232405i
\(772\) 0 0
\(773\) −45.9340 −1.65213 −0.826065 0.563575i \(-0.809426\pi\)
−0.826065 + 0.563575i \(0.809426\pi\)
\(774\) 0 0
\(775\) 21.5138i 0.772797i
\(776\) 0 0
\(777\) 0.202010i 0.00724707i
\(778\) 0 0
\(779\) 10.8950i 0.390355i
\(780\) 0 0
\(781\) −20.6929 −0.740451
\(782\) 0 0
\(783\) −22.0193 −0.786905
\(784\) 0 0
\(785\) 8.75567i 0.312503i
\(786\) 0 0
\(787\) 44.4809i 1.58557i 0.609500 + 0.792786i \(0.291370\pi\)
−0.609500 + 0.792786i \(0.708630\pi\)
\(788\) 0 0
\(789\) 1.33807i 0.0476364i
\(790\) 0 0
\(791\) −1.95033 −0.0693457
\(792\) 0 0
\(793\) 4.70566i 0.167103i