Properties

Label 4012.2.b.b.237.4
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 237.4
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.43

$q$-expansion

\(f(q)\) \(=\) \(q-3.13247i q^{3} -3.91603i q^{5} +2.98385i q^{7} -6.81236 q^{9} +O(q^{10})\) \(q-3.13247i q^{3} -3.91603i q^{5} +2.98385i q^{7} -6.81236 q^{9} -5.81225i q^{11} +1.72232 q^{13} -12.2668 q^{15} +(-2.58285 + 3.21386i) q^{17} -0.658321 q^{19} +9.34682 q^{21} +9.22325i q^{23} -10.3353 q^{25} +11.9421i q^{27} +10.2249i q^{29} -5.64524i q^{31} -18.2067 q^{33} +11.6848 q^{35} -3.51339i q^{37} -5.39511i q^{39} +0.336674i q^{41} -2.85742 q^{43} +26.6774i q^{45} -4.71431 q^{47} -1.90336 q^{49} +(10.0673 + 8.09069i) q^{51} -3.06039 q^{53} -22.7609 q^{55} +2.06217i q^{57} +1.00000 q^{59} +4.36582i q^{61} -20.3271i q^{63} -6.74465i q^{65} +4.44979 q^{67} +28.8915 q^{69} +10.8686i q^{71} -3.72041i q^{73} +32.3750i q^{75} +17.3429 q^{77} +5.49978i q^{79} +16.9712 q^{81} -15.3146 q^{83} +(12.5856 + 10.1145i) q^{85} +32.0293 q^{87} -9.45135 q^{89} +5.13914i q^{91} -17.6835 q^{93} +2.57801i q^{95} -9.73000i q^{97} +39.5952i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

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
\(3\) 3.13247i 1.80853i −0.426970 0.904266i \(-0.640419\pi\)
0.426970 0.904266i \(-0.359581\pi\)
\(4\) 0 0
\(5\) 3.91603i 1.75130i −0.482944 0.875651i \(-0.660433\pi\)
0.482944 0.875651i \(-0.339567\pi\)
\(6\) 0 0
\(7\) 2.98385i 1.12779i 0.825847 + 0.563895i \(0.190698\pi\)
−0.825847 + 0.563895i \(0.809302\pi\)
\(8\) 0 0
\(9\) −6.81236 −2.27079
\(10\) 0 0
\(11\) 5.81225i 1.75246i −0.481894 0.876229i \(-0.660051\pi\)
0.481894 0.876229i \(-0.339949\pi\)
\(12\) 0 0
\(13\) 1.72232 0.477685 0.238842 0.971058i \(-0.423232\pi\)
0.238842 + 0.971058i \(0.423232\pi\)
\(14\) 0 0
\(15\) −12.2668 −3.16729
\(16\) 0 0
\(17\) −2.58285 + 3.21386i −0.626433 + 0.779475i
\(18\) 0 0
\(19\) −0.658321 −0.151029 −0.0755146 0.997145i \(-0.524060\pi\)
−0.0755146 + 0.997145i \(0.524060\pi\)
\(20\) 0 0
\(21\) 9.34682 2.03964
\(22\) 0 0
\(23\) 9.22325i 1.92318i 0.274491 + 0.961590i \(0.411491\pi\)
−0.274491 + 0.961590i \(0.588509\pi\)
\(24\) 0 0
\(25\) −10.3353 −2.06706
\(26\) 0 0
\(27\) 11.9421i 2.29826i
\(28\) 0 0
\(29\) 10.2249i 1.89872i 0.314187 + 0.949361i \(0.398268\pi\)
−0.314187 + 0.949361i \(0.601732\pi\)
\(30\) 0 0
\(31\) 5.64524i 1.01392i −0.861971 0.506958i \(-0.830770\pi\)
0.861971 0.506958i \(-0.169230\pi\)
\(32\) 0 0
\(33\) −18.2067 −3.16938
\(34\) 0 0
\(35\) 11.6848 1.97510
\(36\) 0 0
\(37\) 3.51339i 0.577598i −0.957390 0.288799i \(-0.906744\pi\)
0.957390 0.288799i \(-0.0932559\pi\)
\(38\) 0 0
\(39\) 5.39511i 0.863909i
\(40\) 0 0
\(41\) 0.336674i 0.0525797i 0.999654 + 0.0262899i \(0.00836928\pi\)
−0.999654 + 0.0262899i \(0.991631\pi\)
\(42\) 0 0
\(43\) −2.85742 −0.435752 −0.217876 0.975976i \(-0.569913\pi\)
−0.217876 + 0.975976i \(0.569913\pi\)
\(44\) 0 0
\(45\) 26.6774i 3.97684i
\(46\) 0 0
\(47\) −4.71431 −0.687652 −0.343826 0.939033i \(-0.611723\pi\)
−0.343826 + 0.939033i \(0.611723\pi\)
\(48\) 0 0
\(49\) −1.90336 −0.271909
\(50\) 0 0
\(51\) 10.0673 + 8.09069i 1.40971 + 1.13292i
\(52\) 0 0
\(53\) −3.06039 −0.420377 −0.210189 0.977661i \(-0.567408\pi\)
−0.210189 + 0.977661i \(0.567408\pi\)
\(54\) 0 0
\(55\) −22.7609 −3.06909
\(56\) 0 0
\(57\) 2.06217i 0.273141i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 4.36582i 0.558986i 0.960148 + 0.279493i \(0.0901664\pi\)
−0.960148 + 0.279493i \(0.909834\pi\)
\(62\) 0 0
\(63\) 20.3271i 2.56097i
\(64\) 0 0
\(65\) 6.74465i 0.836571i
\(66\) 0 0
\(67\) 4.44979 0.543628 0.271814 0.962350i \(-0.412376\pi\)
0.271814 + 0.962350i \(0.412376\pi\)
\(68\) 0 0
\(69\) 28.8915 3.47813
\(70\) 0 0
\(71\) 10.8686i 1.28987i 0.764238 + 0.644935i \(0.223115\pi\)
−0.764238 + 0.644935i \(0.776885\pi\)
\(72\) 0 0
\(73\) 3.72041i 0.435441i −0.976011 0.217721i \(-0.930138\pi\)
0.976011 0.217721i \(-0.0698622\pi\)
\(74\) 0 0
\(75\) 32.3750i 3.73834i
\(76\) 0 0
\(77\) 17.3429 1.97640
\(78\) 0 0
\(79\) 5.49978i 0.618773i 0.950936 + 0.309387i \(0.100124\pi\)
−0.950936 + 0.309387i \(0.899876\pi\)
\(80\) 0 0
\(81\) 16.9712 1.88569
\(82\) 0 0
\(83\) −15.3146 −1.68099 −0.840496 0.541817i \(-0.817736\pi\)
−0.840496 + 0.541817i \(0.817736\pi\)
\(84\) 0 0
\(85\) 12.5856 + 10.1145i 1.36510 + 1.09707i
\(86\) 0 0
\(87\) 32.0293 3.43390
\(88\) 0 0
\(89\) −9.45135 −1.00184 −0.500921 0.865493i \(-0.667005\pi\)
−0.500921 + 0.865493i \(0.667005\pi\)
\(90\) 0 0
\(91\) 5.13914i 0.538728i
\(92\) 0 0
\(93\) −17.6835 −1.83370
\(94\) 0 0
\(95\) 2.57801i 0.264498i
\(96\) 0 0
\(97\) 9.73000i 0.987932i −0.869481 0.493966i \(-0.835547\pi\)
0.869481 0.493966i \(-0.164453\pi\)
\(98\) 0 0
\(99\) 39.5952i 3.97946i
\(100\) 0 0
\(101\) −6.03437 −0.600442 −0.300221 0.953870i \(-0.597060\pi\)
−0.300221 + 0.953870i \(0.597060\pi\)
\(102\) 0 0
\(103\) −11.9289 −1.17539 −0.587696 0.809082i \(-0.699965\pi\)
−0.587696 + 0.809082i \(0.699965\pi\)
\(104\) 0 0
\(105\) 36.6024i 3.57203i
\(106\) 0 0
\(107\) 8.75201i 0.846088i −0.906109 0.423044i \(-0.860962\pi\)
0.906109 0.423044i \(-0.139038\pi\)
\(108\) 0 0
\(109\) 13.2858i 1.27255i −0.771463 0.636274i \(-0.780475\pi\)
0.771463 0.636274i \(-0.219525\pi\)
\(110\) 0 0
\(111\) −11.0056 −1.04460
\(112\) 0 0
\(113\) 0.860415i 0.0809410i 0.999181 + 0.0404705i \(0.0128857\pi\)
−0.999181 + 0.0404705i \(0.987114\pi\)
\(114\) 0 0
\(115\) 36.1185 3.36807
\(116\) 0 0
\(117\) −11.7331 −1.08472
\(118\) 0 0
\(119\) −9.58968 7.70683i −0.879084 0.706484i
\(120\) 0 0
\(121\) −22.7822 −2.07111
\(122\) 0 0
\(123\) 1.05462 0.0950921
\(124\) 0 0
\(125\) 20.8932i 1.86874i
\(126\) 0 0
\(127\) 1.07177 0.0951040 0.0475520 0.998869i \(-0.484858\pi\)
0.0475520 + 0.998869i \(0.484858\pi\)
\(128\) 0 0
\(129\) 8.95078i 0.788072i
\(130\) 0 0
\(131\) 10.9683i 0.958303i −0.877732 0.479152i \(-0.840944\pi\)
0.877732 0.479152i \(-0.159056\pi\)
\(132\) 0 0
\(133\) 1.96433i 0.170329i
\(134\) 0 0
\(135\) 46.7657 4.02495
\(136\) 0 0
\(137\) −16.2010 −1.38415 −0.692073 0.721828i \(-0.743302\pi\)
−0.692073 + 0.721828i \(0.743302\pi\)
\(138\) 0 0
\(139\) 17.2007i 1.45894i 0.684010 + 0.729472i \(0.260234\pi\)
−0.684010 + 0.729472i \(0.739766\pi\)
\(140\) 0 0
\(141\) 14.7674i 1.24364i
\(142\) 0 0
\(143\) 10.0105i 0.837123i
\(144\) 0 0
\(145\) 40.0411 3.32524
\(146\) 0 0
\(147\) 5.96223i 0.491756i
\(148\) 0 0
\(149\) 11.9746 0.981000 0.490500 0.871441i \(-0.336814\pi\)
0.490500 + 0.871441i \(0.336814\pi\)
\(150\) 0 0
\(151\) 2.12731 0.173118 0.0865590 0.996247i \(-0.472413\pi\)
0.0865590 + 0.996247i \(0.472413\pi\)
\(152\) 0 0
\(153\) 17.5953 21.8940i 1.42250 1.77002i
\(154\) 0 0
\(155\) −22.1069 −1.77567
\(156\) 0 0
\(157\) −13.8654 −1.10658 −0.553290 0.832989i \(-0.686628\pi\)
−0.553290 + 0.832989i \(0.686628\pi\)
\(158\) 0 0
\(159\) 9.58659i 0.760266i
\(160\) 0 0
\(161\) −27.5208 −2.16894
\(162\) 0 0
\(163\) 24.8869i 1.94929i −0.223759 0.974644i \(-0.571833\pi\)
0.223759 0.974644i \(-0.428167\pi\)
\(164\) 0 0
\(165\) 71.2980i 5.55054i
\(166\) 0 0
\(167\) 5.21241i 0.403349i 0.979453 + 0.201674i \(0.0646383\pi\)
−0.979453 + 0.201674i \(0.935362\pi\)
\(168\) 0 0
\(169\) −10.0336 −0.771817
\(170\) 0 0
\(171\) 4.48472 0.342955
\(172\) 0 0
\(173\) 9.30877i 0.707733i −0.935296 0.353866i \(-0.884867\pi\)
0.935296 0.353866i \(-0.115133\pi\)
\(174\) 0 0
\(175\) 30.8390i 2.33121i
\(176\) 0 0
\(177\) 3.13247i 0.235451i
\(178\) 0 0
\(179\) −24.2082 −1.80941 −0.904703 0.426042i \(-0.859908\pi\)
−0.904703 + 0.426042i \(0.859908\pi\)
\(180\) 0 0
\(181\) 9.13559i 0.679043i 0.940598 + 0.339522i \(0.110265\pi\)
−0.940598 + 0.339522i \(0.889735\pi\)
\(182\) 0 0
\(183\) 13.6758 1.01094
\(184\) 0 0
\(185\) −13.7585 −1.01155
\(186\) 0 0
\(187\) 18.6798 + 15.0122i 1.36600 + 1.09780i
\(188\) 0 0
\(189\) −35.6335 −2.59195
\(190\) 0 0
\(191\) 18.8385 1.36311 0.681554 0.731768i \(-0.261304\pi\)
0.681554 + 0.731768i \(0.261304\pi\)
\(192\) 0 0
\(193\) 13.3985i 0.964448i 0.876048 + 0.482224i \(0.160171\pi\)
−0.876048 + 0.482224i \(0.839829\pi\)
\(194\) 0 0
\(195\) −21.1274 −1.51296
\(196\) 0 0
\(197\) 8.45154i 0.602148i −0.953601 0.301074i \(-0.902655\pi\)
0.953601 0.301074i \(-0.0973450\pi\)
\(198\) 0 0
\(199\) 14.2354i 1.00912i 0.863376 + 0.504561i \(0.168345\pi\)
−0.863376 + 0.504561i \(0.831655\pi\)
\(200\) 0 0
\(201\) 13.9388i 0.983169i
\(202\) 0 0
\(203\) −30.5097 −2.14136
\(204\) 0 0
\(205\) 1.31843 0.0920829
\(206\) 0 0
\(207\) 62.8321i 4.36713i
\(208\) 0 0
\(209\) 3.82633i 0.264673i
\(210\) 0 0
\(211\) 17.0329i 1.17260i −0.810095 0.586298i \(-0.800585\pi\)
0.810095 0.586298i \(-0.199415\pi\)
\(212\) 0 0
\(213\) 34.0457 2.33277
\(214\) 0 0
\(215\) 11.1897i 0.763134i
\(216\) 0 0
\(217\) 16.8446 1.14348
\(218\) 0 0
\(219\) −11.6541 −0.787510
\(220\) 0 0
\(221\) −4.44849 + 5.53529i −0.299238 + 0.372344i
\(222\) 0 0
\(223\) 9.69697 0.649357 0.324678 0.945824i \(-0.394744\pi\)
0.324678 + 0.945824i \(0.394744\pi\)
\(224\) 0 0
\(225\) 70.4078 4.69385
\(226\) 0 0
\(227\) 17.6422i 1.17095i −0.810690 0.585476i \(-0.800908\pi\)
0.810690 0.585476i \(-0.199092\pi\)
\(228\) 0 0
\(229\) 13.6189 0.899960 0.449980 0.893039i \(-0.351431\pi\)
0.449980 + 0.893039i \(0.351431\pi\)
\(230\) 0 0
\(231\) 54.3260i 3.57439i
\(232\) 0 0
\(233\) 1.42634i 0.0934428i 0.998908 + 0.0467214i \(0.0148773\pi\)
−0.998908 + 0.0467214i \(0.985123\pi\)
\(234\) 0 0
\(235\) 18.4614i 1.20429i
\(236\) 0 0
\(237\) 17.2279 1.11907
\(238\) 0 0
\(239\) 10.1072 0.653783 0.326892 0.945062i \(-0.393999\pi\)
0.326892 + 0.945062i \(0.393999\pi\)
\(240\) 0 0
\(241\) 26.5467i 1.71003i −0.518607 0.855013i \(-0.673549\pi\)
0.518607 0.855013i \(-0.326451\pi\)
\(242\) 0 0
\(243\) 17.3355i 1.11207i
\(244\) 0 0
\(245\) 7.45363i 0.476195i
\(246\) 0 0
\(247\) −1.13384 −0.0721444
\(248\) 0 0
\(249\) 47.9724i 3.04013i
\(250\) 0 0
\(251\) 9.33762 0.589386 0.294693 0.955592i \(-0.404783\pi\)
0.294693 + 0.955592i \(0.404783\pi\)
\(252\) 0 0
\(253\) 53.6078 3.37029
\(254\) 0 0
\(255\) 31.6834 39.4239i 1.98409 2.46882i
\(256\) 0 0
\(257\) −10.8648 −0.677729 −0.338864 0.940835i \(-0.610043\pi\)
−0.338864 + 0.940835i \(0.610043\pi\)
\(258\) 0 0
\(259\) 10.4834 0.651409
\(260\) 0 0
\(261\) 69.6560i 4.31160i
\(262\) 0 0
\(263\) −1.53470 −0.0946337 −0.0473169 0.998880i \(-0.515067\pi\)
−0.0473169 + 0.998880i \(0.515067\pi\)
\(264\) 0 0
\(265\) 11.9846i 0.736208i
\(266\) 0 0
\(267\) 29.6061i 1.81186i
\(268\) 0 0
\(269\) 19.1124i 1.16530i 0.812722 + 0.582651i \(0.197985\pi\)
−0.812722 + 0.582651i \(0.802015\pi\)
\(270\) 0 0
\(271\) 27.0407 1.64261 0.821303 0.570493i \(-0.193248\pi\)
0.821303 + 0.570493i \(0.193248\pi\)
\(272\) 0 0
\(273\) 16.0982 0.974307
\(274\) 0 0
\(275\) 60.0713i 3.62244i
\(276\) 0 0
\(277\) 31.4152i 1.88756i −0.330578 0.943779i \(-0.607244\pi\)
0.330578 0.943779i \(-0.392756\pi\)
\(278\) 0 0
\(279\) 38.4574i 2.30239i
\(280\) 0 0
\(281\) −19.9424 −1.18966 −0.594831 0.803851i \(-0.702781\pi\)
−0.594831 + 0.803851i \(0.702781\pi\)
\(282\) 0 0
\(283\) 7.80395i 0.463897i −0.972728 0.231948i \(-0.925490\pi\)
0.972728 0.231948i \(-0.0745100\pi\)
\(284\) 0 0
\(285\) 8.07553 0.478353
\(286\) 0 0
\(287\) −1.00459 −0.0592988
\(288\) 0 0
\(289\) −3.65779 16.6018i −0.215164 0.976578i
\(290\) 0 0
\(291\) −30.4789 −1.78671
\(292\) 0 0
\(293\) −8.42796 −0.492366 −0.246183 0.969223i \(-0.579176\pi\)
−0.246183 + 0.969223i \(0.579176\pi\)
\(294\) 0 0
\(295\) 3.91603i 0.228000i
\(296\) 0 0
\(297\) 69.4105 4.02761
\(298\) 0 0
\(299\) 15.8854i 0.918674i
\(300\) 0 0
\(301\) 8.52611i 0.491437i
\(302\) 0 0
\(303\) 18.9025i 1.08592i
\(304\) 0 0
\(305\) 17.0967 0.978954
\(306\) 0 0
\(307\) 29.3463 1.67488 0.837440 0.546529i \(-0.184051\pi\)
0.837440 + 0.546529i \(0.184051\pi\)
\(308\) 0 0
\(309\) 37.3670i 2.12574i
\(310\) 0 0
\(311\) 0.880200i 0.0499116i −0.999689 0.0249558i \(-0.992056\pi\)
0.999689 0.0249558i \(-0.00794450\pi\)
\(312\) 0 0
\(313\) 3.59647i 0.203285i −0.994821 0.101642i \(-0.967590\pi\)
0.994821 0.101642i \(-0.0324097\pi\)
\(314\) 0 0
\(315\) −79.6015 −4.48503
\(316\) 0 0
\(317\) 10.6419i 0.597707i 0.954299 + 0.298853i \(0.0966042\pi\)
−0.954299 + 0.298853i \(0.903396\pi\)
\(318\) 0 0
\(319\) 59.4298 3.32743
\(320\) 0 0
\(321\) −27.4154 −1.53018
\(322\) 0 0
\(323\) 1.70034 2.11575i 0.0946097 0.117724i
\(324\) 0 0
\(325\) −17.8007 −0.987403
\(326\) 0 0
\(327\) −41.6174 −2.30145
\(328\) 0 0
\(329\) 14.0668i 0.775527i
\(330\) 0 0
\(331\) 19.5501 1.07457 0.537285 0.843401i \(-0.319450\pi\)
0.537285 + 0.843401i \(0.319450\pi\)
\(332\) 0 0
\(333\) 23.9345i 1.31160i
\(334\) 0 0
\(335\) 17.4255i 0.952057i
\(336\) 0 0
\(337\) 20.1471i 1.09748i 0.835992 + 0.548742i \(0.184893\pi\)
−0.835992 + 0.548742i \(0.815107\pi\)
\(338\) 0 0
\(339\) 2.69522 0.146384
\(340\) 0 0
\(341\) −32.8116 −1.77685
\(342\) 0 0
\(343\) 15.2076i 0.821133i
\(344\) 0 0
\(345\) 113.140i 6.09126i
\(346\) 0 0
\(347\) 36.3318i 1.95040i −0.221336 0.975198i \(-0.571042\pi\)
0.221336 0.975198i \(-0.428958\pi\)
\(348\) 0 0
\(349\) −30.0188 −1.60687 −0.803436 0.595392i \(-0.796997\pi\)
−0.803436 + 0.595392i \(0.796997\pi\)
\(350\) 0 0
\(351\) 20.5681i 1.09784i
\(352\) 0 0
\(353\) −3.99202 −0.212474 −0.106237 0.994341i \(-0.533880\pi\)
−0.106237 + 0.994341i \(0.533880\pi\)
\(354\) 0 0
\(355\) 42.5619 2.25895
\(356\) 0 0
\(357\) −24.1414 + 30.0394i −1.27770 + 1.58985i
\(358\) 0 0
\(359\) −5.14723 −0.271661 −0.135830 0.990732i \(-0.543370\pi\)
−0.135830 + 0.990732i \(0.543370\pi\)
\(360\) 0 0
\(361\) −18.5666 −0.977190
\(362\) 0 0
\(363\) 71.3647i 3.74567i
\(364\) 0 0
\(365\) −14.5693 −0.762590
\(366\) 0 0
\(367\) 5.00098i 0.261049i 0.991445 + 0.130524i \(0.0416661\pi\)
−0.991445 + 0.130524i \(0.958334\pi\)
\(368\) 0 0
\(369\) 2.29355i 0.119397i
\(370\) 0 0
\(371\) 9.13176i 0.474097i
\(372\) 0 0
\(373\) −19.2402 −0.996221 −0.498110 0.867114i \(-0.665973\pi\)
−0.498110 + 0.867114i \(0.665973\pi\)
\(374\) 0 0
\(375\) 65.4473 3.37968
\(376\) 0 0
\(377\) 17.6106i 0.906991i
\(378\) 0 0
\(379\) 13.5274i 0.694858i −0.937706 0.347429i \(-0.887055\pi\)
0.937706 0.347429i \(-0.112945\pi\)
\(380\) 0 0
\(381\) 3.35728i 0.171999i
\(382\) 0 0
\(383\) 31.6922 1.61940 0.809698 0.586847i \(-0.199631\pi\)
0.809698 + 0.586847i \(0.199631\pi\)
\(384\) 0 0
\(385\) 67.9153i 3.46128i
\(386\) 0 0
\(387\) 19.4658 0.989502
\(388\) 0 0
\(389\) −11.4870 −0.582412 −0.291206 0.956660i \(-0.594056\pi\)
−0.291206 + 0.956660i \(0.594056\pi\)
\(390\) 0 0
\(391\) −29.6422 23.8222i −1.49907 1.20474i
\(392\) 0 0
\(393\) −34.3578 −1.73312
\(394\) 0 0
\(395\) 21.5373 1.08366
\(396\) 0 0
\(397\) 6.04883i 0.303582i −0.988413 0.151791i \(-0.951496\pi\)
0.988413 0.151791i \(-0.0485040\pi\)
\(398\) 0 0
\(399\) −6.15321 −0.308046
\(400\) 0 0
\(401\) 14.2256i 0.710393i 0.934792 + 0.355197i \(0.115586\pi\)
−0.934792 + 0.355197i \(0.884414\pi\)
\(402\) 0 0
\(403\) 9.72290i 0.484332i
\(404\) 0 0
\(405\) 66.4598i 3.30241i
\(406\) 0 0
\(407\) −20.4207 −1.01222
\(408\) 0 0
\(409\) 30.8935 1.52759 0.763793 0.645461i \(-0.223335\pi\)
0.763793 + 0.645461i \(0.223335\pi\)
\(410\) 0 0
\(411\) 50.7491i 2.50327i
\(412\) 0 0
\(413\) 2.98385i 0.146826i
\(414\) 0 0
\(415\) 59.9723i 2.94393i
\(416\) 0 0
\(417\) 53.8807 2.63855
\(418\) 0 0
\(419\) 39.9490i 1.95164i 0.218582 + 0.975819i \(0.429857\pi\)
−0.218582 + 0.975819i \(0.570143\pi\)
\(420\) 0 0
\(421\) −0.500758 −0.0244054 −0.0122027 0.999926i \(-0.503884\pi\)
−0.0122027 + 0.999926i \(0.503884\pi\)
\(422\) 0 0
\(423\) 32.1156 1.56151
\(424\) 0 0
\(425\) 26.6945 33.2162i 1.29487 1.61122i
\(426\) 0 0
\(427\) −13.0270 −0.630419
\(428\) 0 0
\(429\) −31.3577 −1.51396
\(430\) 0 0
\(431\) 5.04039i 0.242787i −0.992604 0.121394i \(-0.961264\pi\)
0.992604 0.121394i \(-0.0387363\pi\)
\(432\) 0 0
\(433\) 4.55396 0.218850 0.109425 0.993995i \(-0.465099\pi\)
0.109425 + 0.993995i \(0.465099\pi\)
\(434\) 0 0
\(435\) 125.428i 6.01380i
\(436\) 0 0
\(437\) 6.07186i 0.290456i
\(438\) 0 0
\(439\) 6.28646i 0.300036i −0.988683 0.150018i \(-0.952067\pi\)
0.988683 0.150018i \(-0.0479332\pi\)
\(440\) 0 0
\(441\) 12.9664 0.617448
\(442\) 0 0
\(443\) −30.8639 −1.46639 −0.733195 0.680018i \(-0.761972\pi\)
−0.733195 + 0.680018i \(0.761972\pi\)
\(444\) 0 0
\(445\) 37.0118i 1.75453i
\(446\) 0 0
\(447\) 37.5102i 1.77417i
\(448\) 0 0
\(449\) 11.8093i 0.557316i −0.960390 0.278658i \(-0.910110\pi\)
0.960390 0.278658i \(-0.0898896\pi\)
\(450\) 0 0
\(451\) 1.95684 0.0921438
\(452\) 0 0
\(453\) 6.66373i 0.313089i
\(454\) 0 0
\(455\) 20.1250 0.943476
\(456\) 0 0
\(457\) −21.8513 −1.02216 −0.511079 0.859534i \(-0.670754\pi\)
−0.511079 + 0.859534i \(0.670754\pi\)
\(458\) 0 0
\(459\) −38.3803 30.8447i −1.79144 1.43971i
\(460\) 0 0
\(461\) −22.1769 −1.03288 −0.516441 0.856322i \(-0.672744\pi\)
−0.516441 + 0.856322i \(0.672744\pi\)
\(462\) 0 0
\(463\) −24.9244 −1.15834 −0.579168 0.815208i \(-0.696623\pi\)
−0.579168 + 0.815208i \(0.696623\pi\)
\(464\) 0 0
\(465\) 69.2493i 3.21136i
\(466\) 0 0
\(467\) −30.9741 −1.43331 −0.716656 0.697427i \(-0.754328\pi\)
−0.716656 + 0.697427i \(0.754328\pi\)
\(468\) 0 0
\(469\) 13.2775i 0.613098i
\(470\) 0 0
\(471\) 43.4330i 2.00129i
\(472\) 0 0
\(473\) 16.6080i 0.763638i
\(474\) 0 0
\(475\) 6.80395 0.312186
\(476\) 0 0
\(477\) 20.8485 0.954588
\(478\) 0 0
\(479\) 31.6766i 1.44734i 0.690146 + 0.723670i \(0.257546\pi\)
−0.690146 + 0.723670i \(0.742454\pi\)
\(480\) 0 0
\(481\) 6.05118i 0.275910i
\(482\) 0 0
\(483\) 86.2080i 3.92260i
\(484\) 0 0
\(485\) −38.1030 −1.73017
\(486\) 0 0
\(487\) 13.9104i 0.630341i −0.949035 0.315171i \(-0.897938\pi\)
0.949035 0.315171i \(-0.102062\pi\)
\(488\) 0 0
\(489\) −77.9573 −3.52535
\(490\) 0 0
\(491\) 36.8827 1.66449 0.832246 0.554406i \(-0.187055\pi\)
0.832246 + 0.554406i \(0.187055\pi\)
\(492\) 0 0
\(493\) −32.8615 26.4094i −1.48001 1.18942i
\(494\) 0 0
\(495\) 155.056 6.96924
\(496\) 0 0
\(497\) −32.4304 −1.45470
\(498\) 0 0
\(499\) 31.6469i 1.41671i 0.705856 + 0.708355i \(0.250563\pi\)
−0.705856 + 0.708355i \(0.749437\pi\)
\(500\) 0 0
\(501\) 16.3277 0.729469
\(502\) 0 0
\(503\) 18.2068i 0.811800i 0.913917 + 0.405900i \(0.133042\pi\)
−0.913917 + 0.405900i \(0.866958\pi\)
\(504\) 0 0
\(505\) 23.6308i 1.05156i
\(506\) 0 0
\(507\) 31.4300i 1.39586i
\(508\) 0 0
\(509\) −20.2981 −0.899696 −0.449848 0.893105i \(-0.648522\pi\)
−0.449848 + 0.893105i \(0.648522\pi\)
\(510\) 0 0
\(511\) 11.1012 0.491086
\(512\) 0 0
\(513\) 7.86175i 0.347105i
\(514\) 0 0
\(515\) 46.7141i 2.05847i
\(516\) 0 0
\(517\) 27.4007i 1.20508i
\(518\) 0 0
\(519\) −29.1594 −1.27996
\(520\) 0 0
\(521\) 12.0908i 0.529707i −0.964289 0.264854i \(-0.914676\pi\)
0.964289 0.264854i \(-0.0853236\pi\)
\(522\) 0 0
\(523\) 7.57044 0.331032 0.165516 0.986207i \(-0.447071\pi\)
0.165516 + 0.986207i \(0.447071\pi\)
\(524\) 0 0
\(525\) −96.6022 −4.21606
\(526\) 0 0
\(527\) 18.1430 + 14.5808i 0.790322 + 0.635150i
\(528\) 0 0
\(529\) −62.0683 −2.69862
\(530\) 0 0
\(531\) −6.81236 −0.295631
\(532\) 0 0
\(533\) 0.579860i 0.0251165i
\(534\) 0 0
\(535\) −34.2731 −1.48176
\(536\) 0 0
\(537\) 75.8315i 3.27237i
\(538\) 0 0
\(539\) 11.0628i 0.476509i
\(540\) 0 0
\(541\) 10.5176i 0.452185i −0.974106 0.226092i \(-0.927405\pi\)
0.974106 0.226092i \(-0.0725952\pi\)
\(542\) 0 0
\(543\) 28.6170 1.22807
\(544\) 0 0
\(545\) −52.0276 −2.22862
\(546\) 0 0
\(547\) 23.5505i 1.00695i −0.864010 0.503474i \(-0.832055\pi\)
0.864010 0.503474i \(-0.167945\pi\)
\(548\) 0 0
\(549\) 29.7416i 1.26934i
\(550\) 0 0
\(551\) 6.73129i 0.286763i
\(552\) 0 0
\(553\) −16.4105 −0.697846
\(554\) 0 0
\(555\) 43.0982i 1.82942i
\(556\) 0 0
\(557\) −17.4459 −0.739205 −0.369602 0.929190i \(-0.620506\pi\)
−0.369602 + 0.929190i \(0.620506\pi\)
\(558\) 0 0
\(559\) −4.92138 −0.208152
\(560\) 0 0
\(561\) 47.0251 58.5137i 1.98540 2.47045i
\(562\) 0 0
\(563\) 18.2805 0.770433 0.385217 0.922826i \(-0.374127\pi\)
0.385217 + 0.922826i \(0.374127\pi\)
\(564\) 0 0
\(565\) 3.36941 0.141752
\(566\) 0 0
\(567\) 50.6396i 2.12666i
\(568\) 0 0
\(569\) 20.6682 0.866454 0.433227 0.901285i \(-0.357375\pi\)
0.433227 + 0.901285i \(0.357375\pi\)
\(570\) 0 0
\(571\) 25.4837i 1.06646i −0.845970 0.533230i \(-0.820978\pi\)
0.845970 0.533230i \(-0.179022\pi\)
\(572\) 0 0
\(573\) 59.0111i 2.46522i
\(574\) 0 0
\(575\) 95.3250i 3.97533i
\(576\) 0 0
\(577\) −2.37458 −0.0988549 −0.0494274 0.998778i \(-0.515740\pi\)
−0.0494274 + 0.998778i \(0.515740\pi\)
\(578\) 0 0
\(579\) 41.9705 1.74424
\(580\) 0 0
\(581\) 45.6964i 1.89581i
\(582\) 0 0
\(583\) 17.7878i 0.736694i
\(584\) 0 0
\(585\) 45.9470i 1.89967i
\(586\) 0 0
\(587\) 24.8290 1.02480 0.512401 0.858746i \(-0.328756\pi\)
0.512401 + 0.858746i \(0.328756\pi\)
\(588\) 0 0
\(589\) 3.71638i 0.153131i
\(590\) 0 0
\(591\) −26.4742 −1.08900
\(592\) 0 0
\(593\) −12.7367 −0.523034 −0.261517 0.965199i \(-0.584223\pi\)
−0.261517 + 0.965199i \(0.584223\pi\)
\(594\) 0 0
\(595\) −30.1802 + 37.5535i −1.23727 + 1.53954i
\(596\) 0 0
\(597\) 44.5920 1.82503
\(598\) 0 0
\(599\) −25.4577 −1.04017 −0.520086 0.854114i \(-0.674100\pi\)
−0.520086 + 0.854114i \(0.674100\pi\)
\(600\) 0 0
\(601\) 33.2443i 1.35606i 0.735033 + 0.678032i \(0.237167\pi\)
−0.735033 + 0.678032i \(0.762833\pi\)
\(602\) 0 0
\(603\) −30.3136 −1.23446
\(604\) 0 0
\(605\) 89.2159i 3.62714i
\(606\) 0 0
\(607\) 44.2653i 1.79667i 0.439307 + 0.898337i \(0.355224\pi\)
−0.439307 + 0.898337i \(0.644776\pi\)
\(608\) 0 0
\(609\) 95.5706i 3.87272i
\(610\) 0 0
\(611\) −8.11953 −0.328481
\(612\) 0 0
\(613\) −10.1077 −0.408246 −0.204123 0.978945i \(-0.565434\pi\)
−0.204123 + 0.978945i \(0.565434\pi\)
\(614\) 0 0
\(615\) 4.12993i 0.166535i
\(616\) 0 0
\(617\) 18.8646i 0.759459i −0.925098 0.379730i \(-0.876017\pi\)
0.925098 0.379730i \(-0.123983\pi\)
\(618\) 0 0
\(619\) 12.9507i 0.520532i 0.965537 + 0.260266i \(0.0838102\pi\)
−0.965537 + 0.260266i \(0.916190\pi\)
\(620\) 0 0
\(621\) −110.145 −4.41997
\(622\) 0 0
\(623\) 28.2014i 1.12987i
\(624\) 0 0
\(625\) 30.1419 1.20567
\(626\) 0 0
\(627\) 11.9859 0.478669
\(628\) 0 0
\(629\) 11.2915 + 9.07456i 0.450223 + 0.361826i
\(630\) 0 0
\(631\) 27.1397 1.08041 0.540207 0.841532i \(-0.318346\pi\)
0.540207 + 0.841532i \(0.318346\pi\)
\(632\) 0 0
\(633\) −53.3552 −2.12068
\(634\) 0 0
\(635\) 4.19708i 0.166556i
\(636\) 0 0
\(637\) −3.27820 −0.129887
\(638\) 0 0
\(639\) 74.0411i 2.92902i
\(640\) 0 0
\(641\) 5.00559i 0.197709i −0.995102 0.0988545i \(-0.968482\pi\)
0.995102 0.0988545i \(-0.0315178\pi\)
\(642\) 0 0
\(643\) 29.5653i 1.16594i 0.812492 + 0.582972i \(0.198110\pi\)
−0.812492 + 0.582972i \(0.801890\pi\)
\(644\) 0 0
\(645\) 35.0515 1.38015
\(646\) 0 0
\(647\) 19.9091 0.782709 0.391354 0.920240i \(-0.372007\pi\)
0.391354 + 0.920240i \(0.372007\pi\)
\(648\) 0 0
\(649\) 5.81225i 0.228151i
\(650\) 0 0
\(651\) 52.7651i 2.06803i
\(652\) 0 0
\(653\) 11.8521i 0.463807i −0.972739 0.231904i \(-0.925505\pi\)
0.972739 0.231904i \(-0.0744954\pi\)
\(654\) 0 0
\(655\) −42.9521 −1.67828
\(656\) 0 0
\(657\) 25.3448i 0.988795i
\(658\) 0 0
\(659\) 29.3068 1.14163 0.570815 0.821079i \(-0.306627\pi\)
0.570815 + 0.821079i \(0.306627\pi\)
\(660\) 0 0
\(661\) 20.0032 0.778033 0.389017 0.921231i \(-0.372815\pi\)
0.389017 + 0.921231i \(0.372815\pi\)
\(662\) 0 0
\(663\) 17.3391 + 13.9347i 0.673395 + 0.541181i
\(664\) 0 0
\(665\) −7.69239 −0.298298
\(666\) 0 0
\(667\) −94.3070 −3.65158
\(668\) 0 0
\(669\) 30.3754i 1.17438i
\(670\) 0 0
\(671\) 25.3752 0.979601
\(672\) 0 0
\(673\) 35.9777i 1.38684i −0.720534 0.693420i \(-0.756103\pi\)
0.720534 0.693420i \(-0.243897\pi\)
\(674\) 0 0
\(675\) 123.425i 4.75064i
\(676\) 0 0
\(677\) 6.04713i 0.232410i 0.993225 + 0.116205i \(0.0370730\pi\)
−0.993225 + 0.116205i \(0.962927\pi\)
\(678\) 0 0
\(679\) 29.0329 1.11418
\(680\) 0 0
\(681\) −55.2635 −2.11770
\(682\) 0 0
\(683\) 8.85864i 0.338966i −0.985533 0.169483i \(-0.945790\pi\)
0.985533 0.169483i \(-0.0542098\pi\)
\(684\) 0 0
\(685\) 63.4436i 2.42406i
\(686\) 0 0
\(687\) 42.6607i 1.62761i
\(688\) 0 0
\(689\) −5.27097 −0.200808
\(690\) 0 0
\(691\) 1.00329i 0.0381668i 0.999818 + 0.0190834i \(0.00607481\pi\)
−0.999818 + 0.0190834i \(0.993925\pi\)
\(692\) 0 0
\(693\) −118.146 −4.48800
\(694\) 0 0
\(695\) 67.3585 2.55505
\(696\) 0 0
\(697\) −1.08202 0.869579i −0.0409846 0.0329376i
\(698\) 0 0
\(699\) 4.46797 0.168994
\(700\) 0 0
\(701\) −23.9352 −0.904020 −0.452010 0.892013i \(-0.649293\pi\)
−0.452010 + 0.892013i \(0.649293\pi\)
\(702\) 0 0
\(703\) 2.31294i 0.0872342i
\(704\) 0 0
\(705\) 57.8297 2.17799
\(706\) 0 0
\(707\) 18.0056i 0.677172i
\(708\) 0 0
\(709\) 10.7276i 0.402883i −0.979501 0.201442i \(-0.935437\pi\)
0.979501 0.201442i \(-0.0645626\pi\)
\(710\) 0 0
\(711\) 37.4665i 1.40510i
\(712\) 0 0
\(713\) 52.0675 1.94994
\(714\) 0 0
\(715\) −39.2016 −1.46606
\(716\) 0 0
\(717\) 31.6606i 1.18239i
\(718\) 0 0
\(719\) 20.6924i 0.771697i −0.922562 0.385848i \(-0.873909\pi\)
0.922562 0.385848i \(-0.126091\pi\)
\(720\) 0 0
\(721\) 35.5942i 1.32560i
\(722\) 0 0
\(723\) −83.1569 −3.09264
\(724\) 0 0
\(725\) 105.678i 3.92477i
\(726\) 0 0
\(727\) −49.3708 −1.83106 −0.915531 0.402248i \(-0.868229\pi\)
−0.915531 + 0.402248i \(0.868229\pi\)
\(728\) 0 0
\(729\) −3.38916 −0.125525
\(730\) 0 0
\(731\) 7.38028 9.18335i 0.272970 0.339658i
\(732\) 0 0
\(733\) 32.4054 1.19692 0.598460 0.801153i \(-0.295779\pi\)
0.598460 + 0.801153i \(0.295779\pi\)
\(734\) 0 0
\(735\) 23.3483 0.861214
\(736\) 0 0
\(737\) 25.8633i 0.952686i
\(738\) 0 0
\(739\) −10.1692 −0.374080 −0.187040 0.982352i \(-0.559889\pi\)
−0.187040 + 0.982352i \(0.559889\pi\)
\(740\) 0 0
\(741\) 3.55171i 0.130475i
\(742\) 0 0
\(743\) 46.6302i 1.71070i 0.518053 + 0.855348i \(0.326657\pi\)
−0.518053 + 0.855348i \(0.673343\pi\)
\(744\) 0 0
\(745\) 46.8930i 1.71803i
\(746\) 0 0
\(747\) 104.328 3.81718
\(748\) 0 0
\(749\) 26.1147 0.954210
\(750\) 0 0
\(751\) 40.3355i 1.47186i −0.677056 0.735932i \(-0.736744\pi\)
0.677056 0.735932i \(-0.263256\pi\)
\(752\) 0 0
\(753\) 29.2498i 1.06592i
\(754\) 0 0
\(755\) 8.33061i 0.303182i
\(756\) 0 0
\(757\) −34.3459 −1.24832 −0.624162 0.781295i \(-0.714560\pi\)
−0.624162 + 0.781295i \(0.714560\pi\)
\(758\) 0 0
\(759\) 167.925i 6.09528i
\(760\) 0 0
\(761\) −29.3644 −1.06446 −0.532229 0.846601i \(-0.678645\pi\)
−0.532229 + 0.846601i \(0.678645\pi\)
\(762\) 0 0
\(763\) 39.6428 1.43517
\(764\) 0 0
\(765\) −85.7375 68.9038i −3.09985 2.49122i
\(766\) 0 0
\(767\) 1.72232 0.0621893
\(768\) 0 0
\(769\) 11.4053 0.411287 0.205644 0.978627i \(-0.434071\pi\)
0.205644 + 0.978627i \(0.434071\pi\)
\(770\) 0 0
\(771\) 34.0337i 1.22569i
\(772\) 0 0
\(773\) 13.0912 0.470857 0.235428 0.971892i \(-0.424351\pi\)
0.235428 + 0.971892i \(0.424351\pi\)
\(774\) 0 0
\(775\) 58.3452i 2.09582i
\(776\) 0 0
\(777\) 32.8390i 1.17809i
\(778\) 0 0
\(779\) 0.221640i 0.00794107i
\(780\) 0 0
\(781\) 63.1712 2.26044
\(782\) 0 0
\(783\) −122.107 −4.36376
\(784\) 0 0
\(785\) 54.2974i 1.93796i
\(786\) 0 0
\(787\) 31.1097i 1.10894i 0.832203 + 0.554471i \(0.187079\pi\)
−0.832203 + 0.554471i \(0.812921\pi\)
\(788\) 0 0
\(789\) 4.80740i 0.171148i
\(790\) 0 0
\(791\) −2.56735 −0.0912844
\(792\) 0 0
\(793\) 7.51933i 0.267019i