Properties

Label 4012.2.b.b.237.17
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 237.17
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.30

$q$-expansion

\(f(q)\) \(=\) \(q-1.38906i q^{3} -2.35139i q^{5} -0.244801i q^{7} +1.07052 q^{9} +O(q^{10})\) \(q-1.38906i q^{3} -2.35139i q^{5} -0.244801i q^{7} +1.07052 q^{9} -6.06593i q^{11} +0.379019 q^{13} -3.26622 q^{15} +(3.68677 - 1.84601i) q^{17} +1.83644 q^{19} -0.340043 q^{21} -3.35137i q^{23} -0.529047 q^{25} -5.65419i q^{27} +0.959029i q^{29} +7.69476i q^{31} -8.42593 q^{33} -0.575623 q^{35} -5.21503i q^{37} -0.526480i q^{39} -9.98774i q^{41} +3.23514 q^{43} -2.51721i q^{45} +2.89151 q^{47} +6.94007 q^{49} +(-2.56421 - 5.12114i) q^{51} +9.23452 q^{53} -14.2634 q^{55} -2.55092i q^{57} +1.00000 q^{59} +5.34838i q^{61} -0.262063i q^{63} -0.891223i q^{65} -3.05101 q^{67} -4.65525 q^{69} +12.1190i q^{71} -4.32377i q^{73} +0.734877i q^{75} -1.48494 q^{77} +9.09865i q^{79} -4.64244 q^{81} -14.4197 q^{83} +(-4.34069 - 8.66904i) q^{85} +1.33215 q^{87} +12.1589 q^{89} -0.0927842i q^{91} +10.6885 q^{93} -4.31819i q^{95} +17.9932i q^{97} -6.49368i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

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
\(3\) 1.38906i 0.801973i −0.916084 0.400987i \(-0.868667\pi\)
0.916084 0.400987i \(-0.131333\pi\)
\(4\) 0 0
\(5\) 2.35139i 1.05157i −0.850616 0.525787i \(-0.823771\pi\)
0.850616 0.525787i \(-0.176229\pi\)
\(6\) 0 0
\(7\) 0.244801i 0.0925260i −0.998929 0.0462630i \(-0.985269\pi\)
0.998929 0.0462630i \(-0.0147312\pi\)
\(8\) 0 0
\(9\) 1.07052 0.356839
\(10\) 0 0
\(11\) 6.06593i 1.82895i −0.404647 0.914473i \(-0.632606\pi\)
0.404647 0.914473i \(-0.367394\pi\)
\(12\) 0 0
\(13\) 0.379019 0.105121 0.0525605 0.998618i \(-0.483262\pi\)
0.0525605 + 0.998618i \(0.483262\pi\)
\(14\) 0 0
\(15\) −3.26622 −0.843335
\(16\) 0 0
\(17\) 3.68677 1.84601i 0.894173 0.447722i
\(18\) 0 0
\(19\) 1.83644 0.421308 0.210654 0.977561i \(-0.432441\pi\)
0.210654 + 0.977561i \(0.432441\pi\)
\(20\) 0 0
\(21\) −0.340043 −0.0742034
\(22\) 0 0
\(23\) 3.35137i 0.698809i −0.936972 0.349404i \(-0.886384\pi\)
0.936972 0.349404i \(-0.113616\pi\)
\(24\) 0 0
\(25\) −0.529047 −0.105809
\(26\) 0 0
\(27\) 5.65419i 1.08815i
\(28\) 0 0
\(29\) 0.959029i 0.178087i 0.996028 + 0.0890436i \(0.0283810\pi\)
−0.996028 + 0.0890436i \(0.971619\pi\)
\(30\) 0 0
\(31\) 7.69476i 1.38202i 0.722845 + 0.691010i \(0.242834\pi\)
−0.722845 + 0.691010i \(0.757166\pi\)
\(32\) 0 0
\(33\) −8.42593 −1.46677
\(34\) 0 0
\(35\) −0.575623 −0.0972980
\(36\) 0 0
\(37\) 5.21503i 0.857346i −0.903460 0.428673i \(-0.858981\pi\)
0.903460 0.428673i \(-0.141019\pi\)
\(38\) 0 0
\(39\) 0.526480i 0.0843042i
\(40\) 0 0
\(41\) 9.98774i 1.55982i −0.625889 0.779912i \(-0.715264\pi\)
0.625889 0.779912i \(-0.284736\pi\)
\(42\) 0 0
\(43\) 3.23514 0.493354 0.246677 0.969098i \(-0.420661\pi\)
0.246677 + 0.969098i \(0.420661\pi\)
\(44\) 0 0
\(45\) 2.51721i 0.375243i
\(46\) 0 0
\(47\) 2.89151 0.421770 0.210885 0.977511i \(-0.432365\pi\)
0.210885 + 0.977511i \(0.432365\pi\)
\(48\) 0 0
\(49\) 6.94007 0.991439
\(50\) 0 0
\(51\) −2.56421 5.12114i −0.359061 0.717103i
\(52\) 0 0
\(53\) 9.23452 1.26846 0.634229 0.773145i \(-0.281317\pi\)
0.634229 + 0.773145i \(0.281317\pi\)
\(54\) 0 0
\(55\) −14.2634 −1.92327
\(56\) 0 0
\(57\) 2.55092i 0.337878i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 5.34838i 0.684789i 0.939556 + 0.342395i \(0.111238\pi\)
−0.939556 + 0.342395i \(0.888762\pi\)
\(62\) 0 0
\(63\) 0.262063i 0.0330169i
\(64\) 0 0
\(65\) 0.891223i 0.110543i
\(66\) 0 0
\(67\) −3.05101 −0.372741 −0.186370 0.982480i \(-0.559672\pi\)
−0.186370 + 0.982480i \(0.559672\pi\)
\(68\) 0 0
\(69\) −4.65525 −0.560426
\(70\) 0 0
\(71\) 12.1190i 1.43826i 0.694874 + 0.719131i \(0.255460\pi\)
−0.694874 + 0.719131i \(0.744540\pi\)
\(72\) 0 0
\(73\) 4.32377i 0.506059i −0.967459 0.253029i \(-0.918573\pi\)
0.967459 0.253029i \(-0.0814269\pi\)
\(74\) 0 0
\(75\) 0.734877i 0.0848563i
\(76\) 0 0
\(77\) −1.48494 −0.169225
\(78\) 0 0
\(79\) 9.09865i 1.02368i 0.859081 + 0.511839i \(0.171036\pi\)
−0.859081 + 0.511839i \(0.828964\pi\)
\(80\) 0 0
\(81\) −4.64244 −0.515827
\(82\) 0 0
\(83\) −14.4197 −1.58277 −0.791386 0.611317i \(-0.790640\pi\)
−0.791386 + 0.611317i \(0.790640\pi\)
\(84\) 0 0
\(85\) −4.34069 8.66904i −0.470813 0.940289i
\(86\) 0 0
\(87\) 1.33215 0.142821
\(88\) 0 0
\(89\) 12.1589 1.28884 0.644418 0.764673i \(-0.277100\pi\)
0.644418 + 0.764673i \(0.277100\pi\)
\(90\) 0 0
\(91\) 0.0927842i 0.00972643i
\(92\) 0 0
\(93\) 10.6885 1.10834
\(94\) 0 0
\(95\) 4.31819i 0.443037i
\(96\) 0 0
\(97\) 17.9932i 1.82694i 0.406910 + 0.913468i \(0.366606\pi\)
−0.406910 + 0.913468i \(0.633394\pi\)
\(98\) 0 0
\(99\) 6.49368i 0.652639i
\(100\) 0 0
\(101\) −5.98935 −0.595963 −0.297982 0.954572i \(-0.596313\pi\)
−0.297982 + 0.954572i \(0.596313\pi\)
\(102\) 0 0
\(103\) 6.69831 0.660004 0.330002 0.943980i \(-0.392951\pi\)
0.330002 + 0.943980i \(0.392951\pi\)
\(104\) 0 0
\(105\) 0.799574i 0.0780304i
\(106\) 0 0
\(107\) 10.7325i 1.03755i −0.854912 0.518773i \(-0.826389\pi\)
0.854912 0.518773i \(-0.173611\pi\)
\(108\) 0 0
\(109\) 5.19178i 0.497282i 0.968596 + 0.248641i \(0.0799839\pi\)
−0.968596 + 0.248641i \(0.920016\pi\)
\(110\) 0 0
\(111\) −7.24399 −0.687569
\(112\) 0 0
\(113\) 4.91246i 0.462125i −0.972939 0.231063i \(-0.925780\pi\)
0.972939 0.231063i \(-0.0742203\pi\)
\(114\) 0 0
\(115\) −7.88039 −0.734850
\(116\) 0 0
\(117\) 0.405746 0.0375113
\(118\) 0 0
\(119\) −0.451904 0.902524i −0.0414260 0.0827343i
\(120\) 0 0
\(121\) −25.7955 −2.34504
\(122\) 0 0
\(123\) −13.8736 −1.25094
\(124\) 0 0
\(125\) 10.5130i 0.940308i
\(126\) 0 0
\(127\) −14.5330 −1.28959 −0.644796 0.764355i \(-0.723057\pi\)
−0.644796 + 0.764355i \(0.723057\pi\)
\(128\) 0 0
\(129\) 4.49380i 0.395657i
\(130\) 0 0
\(131\) 7.06962i 0.617676i −0.951115 0.308838i \(-0.900060\pi\)
0.951115 0.308838i \(-0.0999400\pi\)
\(132\) 0 0
\(133\) 0.449562i 0.0389820i
\(134\) 0 0
\(135\) −13.2952 −1.14427
\(136\) 0 0
\(137\) 11.3645 0.970936 0.485468 0.874255i \(-0.338649\pi\)
0.485468 + 0.874255i \(0.338649\pi\)
\(138\) 0 0
\(139\) 3.60281i 0.305586i −0.988258 0.152793i \(-0.951173\pi\)
0.988258 0.152793i \(-0.0488268\pi\)
\(140\) 0 0
\(141\) 4.01648i 0.338248i
\(142\) 0 0
\(143\) 2.29910i 0.192261i
\(144\) 0 0
\(145\) 2.25505 0.187272
\(146\) 0 0
\(147\) 9.64017i 0.795107i
\(148\) 0 0
\(149\) −5.97875 −0.489798 −0.244899 0.969549i \(-0.578755\pi\)
−0.244899 + 0.969549i \(0.578755\pi\)
\(150\) 0 0
\(151\) 10.2843 0.836925 0.418462 0.908234i \(-0.362569\pi\)
0.418462 + 0.908234i \(0.362569\pi\)
\(152\) 0 0
\(153\) 3.94675 1.97618i 0.319076 0.159765i
\(154\) 0 0
\(155\) 18.0934 1.45330
\(156\) 0 0
\(157\) 16.1612 1.28981 0.644903 0.764265i \(-0.276898\pi\)
0.644903 + 0.764265i \(0.276898\pi\)
\(158\) 0 0
\(159\) 12.8273i 1.01727i
\(160\) 0 0
\(161\) −0.820418 −0.0646580
\(162\) 0 0
\(163\) 17.9742i 1.40785i 0.710274 + 0.703925i \(0.248571\pi\)
−0.710274 + 0.703925i \(0.751429\pi\)
\(164\) 0 0
\(165\) 19.8127i 1.54241i
\(166\) 0 0
\(167\) 18.9921i 1.46965i 0.678256 + 0.734826i \(0.262736\pi\)
−0.678256 + 0.734826i \(0.737264\pi\)
\(168\) 0 0
\(169\) −12.8563 −0.988950
\(170\) 0 0
\(171\) 1.96594 0.150339
\(172\) 0 0
\(173\) 18.4506i 1.40277i 0.712782 + 0.701385i \(0.247435\pi\)
−0.712782 + 0.701385i \(0.752565\pi\)
\(174\) 0 0
\(175\) 0.129511i 0.00979013i
\(176\) 0 0
\(177\) 1.38906i 0.104408i
\(178\) 0 0
\(179\) −11.0649 −0.827031 −0.413515 0.910497i \(-0.635699\pi\)
−0.413515 + 0.910497i \(0.635699\pi\)
\(180\) 0 0
\(181\) 19.9796i 1.48507i 0.669806 + 0.742536i \(0.266377\pi\)
−0.669806 + 0.742536i \(0.733623\pi\)
\(182\) 0 0
\(183\) 7.42921 0.549183
\(184\) 0 0
\(185\) −12.2626 −0.901564
\(186\) 0 0
\(187\) −11.1977 22.3637i −0.818860 1.63539i
\(188\) 0 0
\(189\) −1.38415 −0.100682
\(190\) 0 0
\(191\) −8.50151 −0.615147 −0.307574 0.951524i \(-0.599517\pi\)
−0.307574 + 0.951524i \(0.599517\pi\)
\(192\) 0 0
\(193\) 3.59449i 0.258737i 0.991597 + 0.129369i \(0.0412950\pi\)
−0.991597 + 0.129369i \(0.958705\pi\)
\(194\) 0 0
\(195\) −1.23796 −0.0886522
\(196\) 0 0
\(197\) 25.0789i 1.78680i −0.449262 0.893400i \(-0.648313\pi\)
0.449262 0.893400i \(-0.351687\pi\)
\(198\) 0 0
\(199\) 9.89913i 0.701731i 0.936426 + 0.350865i \(0.114113\pi\)
−0.936426 + 0.350865i \(0.885887\pi\)
\(200\) 0 0
\(201\) 4.23804i 0.298928i
\(202\) 0 0
\(203\) 0.234771 0.0164777
\(204\) 0 0
\(205\) −23.4851 −1.64027
\(206\) 0 0
\(207\) 3.58770i 0.249362i
\(208\) 0 0
\(209\) 11.1397i 0.770550i
\(210\) 0 0
\(211\) 2.26006i 0.155589i 0.996969 + 0.0777944i \(0.0247878\pi\)
−0.996969 + 0.0777944i \(0.975212\pi\)
\(212\) 0 0
\(213\) 16.8340 1.15345
\(214\) 0 0
\(215\) 7.60709i 0.518799i
\(216\) 0 0
\(217\) 1.88368 0.127873
\(218\) 0 0
\(219\) −6.00596 −0.405845
\(220\) 0 0
\(221\) 1.39736 0.699672i 0.0939963 0.0470650i
\(222\) 0 0
\(223\) −0.818467 −0.0548086 −0.0274043 0.999624i \(-0.508724\pi\)
−0.0274043 + 0.999624i \(0.508724\pi\)
\(224\) 0 0
\(225\) −0.566354 −0.0377569
\(226\) 0 0
\(227\) 18.9195i 1.25573i 0.778322 + 0.627865i \(0.216071\pi\)
−0.778322 + 0.627865i \(0.783929\pi\)
\(228\) 0 0
\(229\) −15.8931 −1.05025 −0.525123 0.851026i \(-0.675981\pi\)
−0.525123 + 0.851026i \(0.675981\pi\)
\(230\) 0 0
\(231\) 2.06267i 0.135714i
\(232\) 0 0
\(233\) 6.37035i 0.417336i 0.977987 + 0.208668i \(0.0669127\pi\)
−0.977987 + 0.208668i \(0.933087\pi\)
\(234\) 0 0
\(235\) 6.79908i 0.443523i
\(236\) 0 0
\(237\) 12.6386 0.820963
\(238\) 0 0
\(239\) 8.98990 0.581508 0.290754 0.956798i \(-0.406094\pi\)
0.290754 + 0.956798i \(0.406094\pi\)
\(240\) 0 0
\(241\) 8.75918i 0.564229i 0.959381 + 0.282114i \(0.0910358\pi\)
−0.959381 + 0.282114i \(0.908964\pi\)
\(242\) 0 0
\(243\) 10.5139i 0.674469i
\(244\) 0 0
\(245\) 16.3188i 1.04257i
\(246\) 0 0
\(247\) 0.696046 0.0442883
\(248\) 0 0
\(249\) 20.0299i 1.26934i
\(250\) 0 0
\(251\) 5.10871 0.322459 0.161230 0.986917i \(-0.448454\pi\)
0.161230 + 0.986917i \(0.448454\pi\)
\(252\) 0 0
\(253\) −20.3292 −1.27808
\(254\) 0 0
\(255\) −12.0418 + 6.02947i −0.754087 + 0.377580i
\(256\) 0 0
\(257\) −17.0717 −1.06490 −0.532452 0.846460i \(-0.678729\pi\)
−0.532452 + 0.846460i \(0.678729\pi\)
\(258\) 0 0
\(259\) −1.27665 −0.0793269
\(260\) 0 0
\(261\) 1.02666i 0.0635484i
\(262\) 0 0
\(263\) −9.49782 −0.585661 −0.292830 0.956164i \(-0.594597\pi\)
−0.292830 + 0.956164i \(0.594597\pi\)
\(264\) 0 0
\(265\) 21.7140i 1.33388i
\(266\) 0 0
\(267\) 16.8894i 1.03361i
\(268\) 0 0
\(269\) 9.19982i 0.560923i 0.959865 + 0.280461i \(0.0904875\pi\)
−0.959865 + 0.280461i \(0.909513\pi\)
\(270\) 0 0
\(271\) 3.41812 0.207636 0.103818 0.994596i \(-0.466894\pi\)
0.103818 + 0.994596i \(0.466894\pi\)
\(272\) 0 0
\(273\) −0.128883 −0.00780034
\(274\) 0 0
\(275\) 3.20916i 0.193520i
\(276\) 0 0
\(277\) 24.2213i 1.45532i −0.685940 0.727658i \(-0.740609\pi\)
0.685940 0.727658i \(-0.259391\pi\)
\(278\) 0 0
\(279\) 8.23737i 0.493159i
\(280\) 0 0
\(281\) −26.9643 −1.60856 −0.804278 0.594254i \(-0.797448\pi\)
−0.804278 + 0.594254i \(0.797448\pi\)
\(282\) 0 0
\(283\) 4.65439i 0.276675i −0.990385 0.138337i \(-0.955824\pi\)
0.990385 0.138337i \(-0.0441758\pi\)
\(284\) 0 0
\(285\) −5.99822 −0.355304
\(286\) 0 0
\(287\) −2.44501 −0.144324
\(288\) 0 0
\(289\) 10.1845 13.6116i 0.599090 0.800682i
\(290\) 0 0
\(291\) 24.9937 1.46515
\(292\) 0 0
\(293\) −4.85974 −0.283909 −0.141955 0.989873i \(-0.545339\pi\)
−0.141955 + 0.989873i \(0.545339\pi\)
\(294\) 0 0
\(295\) 2.35139i 0.136903i
\(296\) 0 0
\(297\) −34.2979 −1.99016
\(298\) 0 0
\(299\) 1.27023i 0.0734595i
\(300\) 0 0
\(301\) 0.791965i 0.0456481i
\(302\) 0 0
\(303\) 8.31956i 0.477946i
\(304\) 0 0
\(305\) 12.5761 0.720107
\(306\) 0 0
\(307\) −22.0681 −1.25949 −0.629747 0.776800i \(-0.716841\pi\)
−0.629747 + 0.776800i \(0.716841\pi\)
\(308\) 0 0
\(309\) 9.30434i 0.529306i
\(310\) 0 0
\(311\) 3.19178i 0.180989i −0.995897 0.0904946i \(-0.971155\pi\)
0.995897 0.0904946i \(-0.0288448\pi\)
\(312\) 0 0
\(313\) 28.3216i 1.60083i 0.599446 + 0.800415i \(0.295388\pi\)
−0.599446 + 0.800415i \(0.704612\pi\)
\(314\) 0 0
\(315\) −0.616214 −0.0347197
\(316\) 0 0
\(317\) 22.0626i 1.23916i 0.784935 + 0.619579i \(0.212696\pi\)
−0.784935 + 0.619579i \(0.787304\pi\)
\(318\) 0 0
\(319\) 5.81740 0.325712
\(320\) 0 0
\(321\) −14.9080 −0.832085
\(322\) 0 0
\(323\) 6.77053 3.39008i 0.376722 0.188629i
\(324\) 0 0
\(325\) −0.200519 −0.0111228
\(326\) 0 0
\(327\) 7.21168 0.398807
\(328\) 0 0
\(329\) 0.707845i 0.0390247i
\(330\) 0 0
\(331\) −24.4525 −1.34403 −0.672014 0.740538i \(-0.734571\pi\)
−0.672014 + 0.740538i \(0.734571\pi\)
\(332\) 0 0
\(333\) 5.58278i 0.305935i
\(334\) 0 0
\(335\) 7.17413i 0.391965i
\(336\) 0 0
\(337\) 29.4804i 1.60590i 0.596047 + 0.802950i \(0.296737\pi\)
−0.596047 + 0.802950i \(0.703263\pi\)
\(338\) 0 0
\(339\) −6.82369 −0.370612
\(340\) 0 0
\(341\) 46.6759 2.52764
\(342\) 0 0
\(343\) 3.41254i 0.184260i
\(344\) 0 0
\(345\) 10.9463i 0.589330i
\(346\) 0 0
\(347\) 7.50267i 0.402764i −0.979513 0.201382i \(-0.935457\pi\)
0.979513 0.201382i \(-0.0645433\pi\)
\(348\) 0 0
\(349\) 15.6923 0.839987 0.419993 0.907527i \(-0.362032\pi\)
0.419993 + 0.907527i \(0.362032\pi\)
\(350\) 0 0
\(351\) 2.14305i 0.114387i
\(352\) 0 0
\(353\) 6.21440 0.330759 0.165380 0.986230i \(-0.447115\pi\)
0.165380 + 0.986230i \(0.447115\pi\)
\(354\) 0 0
\(355\) 28.4966 1.51244
\(356\) 0 0
\(357\) −1.25366 + 0.627721i −0.0663507 + 0.0332225i
\(358\) 0 0
\(359\) −21.6016 −1.14009 −0.570045 0.821613i \(-0.693074\pi\)
−0.570045 + 0.821613i \(0.693074\pi\)
\(360\) 0 0
\(361\) −15.6275 −0.822499
\(362\) 0 0
\(363\) 35.8314i 1.88066i
\(364\) 0 0
\(365\) −10.1669 −0.532158
\(366\) 0 0
\(367\) 6.27674i 0.327643i 0.986490 + 0.163821i \(0.0523821\pi\)
−0.986490 + 0.163821i \(0.947618\pi\)
\(368\) 0 0
\(369\) 10.6920i 0.556606i
\(370\) 0 0
\(371\) 2.26062i 0.117365i
\(372\) 0 0
\(373\) 25.6148 1.32628 0.663141 0.748494i \(-0.269223\pi\)
0.663141 + 0.748494i \(0.269223\pi\)
\(374\) 0 0
\(375\) −14.6031 −0.754102
\(376\) 0 0
\(377\) 0.363490i 0.0187207i
\(378\) 0 0
\(379\) 19.5785i 1.00568i −0.864379 0.502841i \(-0.832288\pi\)
0.864379 0.502841i \(-0.167712\pi\)
\(380\) 0 0
\(381\) 20.1871i 1.03422i
\(382\) 0 0
\(383\) −23.3826 −1.19480 −0.597399 0.801944i \(-0.703799\pi\)
−0.597399 + 0.801944i \(0.703799\pi\)
\(384\) 0 0
\(385\) 3.49169i 0.177953i
\(386\) 0 0
\(387\) 3.46327 0.176048
\(388\) 0 0
\(389\) 19.5733 0.992406 0.496203 0.868206i \(-0.334727\pi\)
0.496203 + 0.868206i \(0.334727\pi\)
\(390\) 0 0
\(391\) −6.18665 12.3557i −0.312872 0.624856i
\(392\) 0 0
\(393\) −9.82011 −0.495359
\(394\) 0 0
\(395\) 21.3945 1.07647
\(396\) 0 0
\(397\) 12.5936i 0.632053i −0.948750 0.316026i \(-0.897651\pi\)
0.948750 0.316026i \(-0.102349\pi\)
\(398\) 0 0
\(399\) −0.624468 −0.0312625
\(400\) 0 0
\(401\) 5.51985i 0.275648i 0.990457 + 0.137824i \(0.0440109\pi\)
−0.990457 + 0.137824i \(0.955989\pi\)
\(402\) 0 0
\(403\) 2.91646i 0.145279i
\(404\) 0 0
\(405\) 10.9162i 0.542431i
\(406\) 0 0
\(407\) −31.6340 −1.56804
\(408\) 0 0
\(409\) 8.49200 0.419902 0.209951 0.977712i \(-0.432670\pi\)
0.209951 + 0.977712i \(0.432670\pi\)
\(410\) 0 0
\(411\) 15.7860i 0.778664i
\(412\) 0 0
\(413\) 0.244801i 0.0120459i
\(414\) 0 0
\(415\) 33.9065i 1.66440i
\(416\) 0 0
\(417\) −5.00451 −0.245072
\(418\) 0 0
\(419\) 20.7166i 1.01207i 0.862512 + 0.506036i \(0.168890\pi\)
−0.862512 + 0.506036i \(0.831110\pi\)
\(420\) 0 0
\(421\) 19.6449 0.957434 0.478717 0.877969i \(-0.341102\pi\)
0.478717 + 0.877969i \(0.341102\pi\)
\(422\) 0 0
\(423\) 3.09541 0.150504
\(424\) 0 0
\(425\) −1.95047 + 0.976624i −0.0946119 + 0.0473732i
\(426\) 0 0
\(427\) 1.30929 0.0633608
\(428\) 0 0
\(429\) −3.19359 −0.154188
\(430\) 0 0
\(431\) 15.2837i 0.736190i 0.929788 + 0.368095i \(0.119990\pi\)
−0.929788 + 0.368095i \(0.880010\pi\)
\(432\) 0 0
\(433\) −9.63445 −0.463002 −0.231501 0.972835i \(-0.574364\pi\)
−0.231501 + 0.972835i \(0.574364\pi\)
\(434\) 0 0
\(435\) 3.13240i 0.150187i
\(436\) 0 0
\(437\) 6.15459i 0.294414i
\(438\) 0 0
\(439\) 18.0746i 0.862653i −0.902196 0.431326i \(-0.858046\pi\)
0.902196 0.431326i \(-0.141954\pi\)
\(440\) 0 0
\(441\) 7.42946 0.353784
\(442\) 0 0
\(443\) 6.17615 0.293438 0.146719 0.989178i \(-0.453129\pi\)
0.146719 + 0.989178i \(0.453129\pi\)
\(444\) 0 0
\(445\) 28.5902i 1.35531i
\(446\) 0 0
\(447\) 8.30483i 0.392805i
\(448\) 0 0
\(449\) 9.40253i 0.443733i −0.975077 0.221866i \(-0.928785\pi\)
0.975077 0.221866i \(-0.0712149\pi\)
\(450\) 0 0
\(451\) −60.5849 −2.85283
\(452\) 0 0
\(453\) 14.2855i 0.671191i
\(454\) 0 0
\(455\) −0.218172 −0.0102281
\(456\) 0 0
\(457\) 36.4982 1.70731 0.853657 0.520835i \(-0.174380\pi\)
0.853657 + 0.520835i \(0.174380\pi\)
\(458\) 0 0
\(459\) −10.4377 20.8457i −0.487188 0.972993i
\(460\) 0 0
\(461\) 29.4996 1.37393 0.686967 0.726688i \(-0.258942\pi\)
0.686967 + 0.726688i \(0.258942\pi\)
\(462\) 0 0
\(463\) 7.94349 0.369165 0.184583 0.982817i \(-0.440907\pi\)
0.184583 + 0.982817i \(0.440907\pi\)
\(464\) 0 0
\(465\) 25.1328i 1.16551i
\(466\) 0 0
\(467\) 25.3103 1.17122 0.585611 0.810593i \(-0.300855\pi\)
0.585611 + 0.810593i \(0.300855\pi\)
\(468\) 0 0
\(469\) 0.746891i 0.0344882i
\(470\) 0 0
\(471\) 22.4489i 1.03439i
\(472\) 0 0
\(473\) 19.6241i 0.902319i
\(474\) 0 0
\(475\) −0.971563 −0.0445784
\(476\) 0 0
\(477\) 9.88571 0.452635
\(478\) 0 0
\(479\) 26.5313i 1.21225i −0.795371 0.606124i \(-0.792724\pi\)
0.795371 0.606124i \(-0.207276\pi\)
\(480\) 0 0
\(481\) 1.97660i 0.0901251i
\(482\) 0 0
\(483\) 1.13961i 0.0518540i
\(484\) 0 0
\(485\) 42.3092 1.92116
\(486\) 0 0
\(487\) 26.6480i 1.20754i 0.797160 + 0.603768i \(0.206335\pi\)
−0.797160 + 0.603768i \(0.793665\pi\)
\(488\) 0 0
\(489\) 24.9673 1.12906
\(490\) 0 0
\(491\) −15.8809 −0.716697 −0.358349 0.933588i \(-0.616660\pi\)
−0.358349 + 0.933588i \(0.616660\pi\)
\(492\) 0 0
\(493\) 1.77037 + 3.53572i 0.0797336 + 0.159241i
\(494\) 0 0
\(495\) −15.2692 −0.686299
\(496\) 0 0
\(497\) 2.96675 0.133077
\(498\) 0 0
\(499\) 3.65268i 0.163516i 0.996652 + 0.0817581i \(0.0260535\pi\)
−0.996652 + 0.0817581i \(0.973947\pi\)
\(500\) 0 0
\(501\) 26.3811 1.17862
\(502\) 0 0
\(503\) 30.2692i 1.34964i −0.737985 0.674818i \(-0.764222\pi\)
0.737985 0.674818i \(-0.235778\pi\)
\(504\) 0 0
\(505\) 14.0833i 0.626700i
\(506\) 0 0
\(507\) 17.8582i 0.793111i
\(508\) 0 0
\(509\) 5.49284 0.243466 0.121733 0.992563i \(-0.461155\pi\)
0.121733 + 0.992563i \(0.461155\pi\)
\(510\) 0 0
\(511\) −1.05846 −0.0468236
\(512\) 0 0
\(513\) 10.3836i 0.458446i
\(514\) 0 0
\(515\) 15.7504i 0.694044i
\(516\) 0 0
\(517\) 17.5397i 0.771395i
\(518\) 0 0
\(519\) 25.6289 1.12498
\(520\) 0 0
\(521\) 25.4861i 1.11657i −0.829651 0.558283i \(-0.811461\pi\)
0.829651 0.558283i \(-0.188539\pi\)
\(522\) 0 0
\(523\) 9.75471 0.426544 0.213272 0.976993i \(-0.431588\pi\)
0.213272 + 0.976993i \(0.431588\pi\)
\(524\) 0 0
\(525\) 0.179899 0.00785142
\(526\) 0 0
\(527\) 14.2046 + 28.3688i 0.618761 + 1.23576i
\(528\) 0 0
\(529\) 11.7683 0.511666
\(530\) 0 0
\(531\) 1.07052 0.0464565
\(532\) 0 0
\(533\) 3.78555i 0.163970i
\(534\) 0 0
\(535\) −25.2362 −1.09106
\(536\) 0 0
\(537\) 15.3698i 0.663257i
\(538\) 0 0
\(539\) 42.0980i 1.81329i
\(540\) 0 0
\(541\) 32.3643i 1.39145i 0.718309 + 0.695724i \(0.244916\pi\)
−0.718309 + 0.695724i \(0.755084\pi\)
\(542\) 0 0
\(543\) 27.7528 1.19099
\(544\) 0 0
\(545\) 12.2079 0.522929
\(546\) 0 0
\(547\) 25.2865i 1.08117i −0.841289 0.540586i \(-0.818203\pi\)
0.841289 0.540586i \(-0.181797\pi\)
\(548\) 0 0
\(549\) 5.72553i 0.244359i
\(550\) 0 0
\(551\) 1.76120i 0.0750296i
\(552\) 0 0
\(553\) 2.22736 0.0947169
\(554\) 0 0
\(555\) 17.0335i 0.723030i
\(556\) 0 0
\(557\) −12.4140 −0.525998 −0.262999 0.964796i \(-0.584712\pi\)
−0.262999 + 0.964796i \(0.584712\pi\)
\(558\) 0 0
\(559\) 1.22618 0.0518619
\(560\) 0 0
\(561\) −31.0644 + 15.5543i −1.31154 + 0.656704i
\(562\) 0 0
\(563\) 2.45827 0.103604 0.0518019 0.998657i \(-0.483504\pi\)
0.0518019 + 0.998657i \(0.483504\pi\)
\(564\) 0 0
\(565\) −11.5511 −0.485959
\(566\) 0 0
\(567\) 1.13647i 0.0477274i
\(568\) 0 0
\(569\) 39.5479 1.65794 0.828968 0.559296i \(-0.188929\pi\)
0.828968 + 0.559296i \(0.188929\pi\)
\(570\) 0 0
\(571\) 0.210260i 0.00879912i −0.999990 0.00439956i \(-0.998600\pi\)
0.999990 0.00439956i \(-0.00140043\pi\)
\(572\) 0 0
\(573\) 11.8091i 0.493332i
\(574\) 0 0
\(575\) 1.77303i 0.0739405i
\(576\) 0 0
\(577\) −15.4221 −0.642032 −0.321016 0.947074i \(-0.604024\pi\)
−0.321016 + 0.947074i \(0.604024\pi\)
\(578\) 0 0
\(579\) 4.99296 0.207500
\(580\) 0 0
\(581\) 3.52996i 0.146448i
\(582\) 0 0
\(583\) 56.0159i 2.31994i
\(584\) 0 0
\(585\) 0.954069i 0.0394459i
\(586\) 0 0
\(587\) 25.0617 1.03441 0.517203 0.855863i \(-0.326973\pi\)
0.517203 + 0.855863i \(0.326973\pi\)
\(588\) 0 0
\(589\) 14.1310i 0.582256i
\(590\) 0 0
\(591\) −34.8361 −1.43297
\(592\) 0 0
\(593\) 29.0880 1.19450 0.597251 0.802054i \(-0.296259\pi\)
0.597251 + 0.802054i \(0.296259\pi\)
\(594\) 0 0
\(595\) −2.12219 + 1.06260i −0.0870013 + 0.0435625i
\(596\) 0 0
\(597\) 13.7505 0.562769
\(598\) 0 0
\(599\) −23.0386 −0.941331 −0.470666 0.882312i \(-0.655986\pi\)
−0.470666 + 0.882312i \(0.655986\pi\)
\(600\) 0 0
\(601\) 26.3979i 1.07679i −0.842692 0.538396i \(-0.819031\pi\)
0.842692 0.538396i \(-0.180969\pi\)
\(602\) 0 0
\(603\) −3.26616 −0.133008
\(604\) 0 0
\(605\) 60.6553i 2.46599i
\(606\) 0 0
\(607\) 22.5729i 0.916204i −0.888900 0.458102i \(-0.848529\pi\)
0.888900 0.458102i \(-0.151471\pi\)
\(608\) 0 0
\(609\) 0.326111i 0.0132147i
\(610\) 0 0
\(611\) 1.09594 0.0443369
\(612\) 0 0
\(613\) 3.12681 0.126291 0.0631454 0.998004i \(-0.479887\pi\)
0.0631454 + 0.998004i \(0.479887\pi\)
\(614\) 0 0
\(615\) 32.6222i 1.31545i
\(616\) 0 0
\(617\) 16.7263i 0.673377i 0.941616 + 0.336688i \(0.109307\pi\)
−0.941616 + 0.336688i \(0.890693\pi\)
\(618\) 0 0
\(619\) 32.9319i 1.32364i −0.749661 0.661822i \(-0.769783\pi\)
0.749661 0.661822i \(-0.230217\pi\)
\(620\) 0 0
\(621\) −18.9493 −0.760408
\(622\) 0 0
\(623\) 2.97650i 0.119251i
\(624\) 0 0
\(625\) −27.3653 −1.09461
\(626\) 0 0
\(627\) −15.4737 −0.617960
\(628\) 0 0
\(629\) −9.62699 19.2266i −0.383853 0.766616i
\(630\) 0 0
\(631\) 31.8339 1.26729 0.633645 0.773624i \(-0.281558\pi\)
0.633645 + 0.773624i \(0.281558\pi\)
\(632\) 0 0
\(633\) 3.13935 0.124778
\(634\) 0 0
\(635\) 34.1727i 1.35610i
\(636\) 0 0
\(637\) 2.63042 0.104221
\(638\) 0 0
\(639\) 12.9736i 0.513228i
\(640\) 0 0
\(641\) 50.2859i 1.98617i 0.117382 + 0.993087i \(0.462550\pi\)
−0.117382 + 0.993087i \(0.537450\pi\)
\(642\) 0 0
\(643\) 31.7570i 1.25238i −0.779672 0.626188i \(-0.784614\pi\)
0.779672 0.626188i \(-0.215386\pi\)
\(644\) 0 0
\(645\) −10.5667 −0.416063
\(646\) 0 0
\(647\) 23.9490 0.941533 0.470767 0.882258i \(-0.343977\pi\)
0.470767 + 0.882258i \(0.343977\pi\)
\(648\) 0 0
\(649\) 6.06593i 0.238108i
\(650\) 0 0
\(651\) 2.61655i 0.102551i
\(652\) 0 0
\(653\) 10.7631i 0.421191i −0.977573 0.210596i \(-0.932460\pi\)
0.977573 0.210596i \(-0.0675403\pi\)
\(654\) 0 0
\(655\) −16.6235 −0.649532
\(656\) 0 0
\(657\) 4.62866i 0.180581i
\(658\) 0 0
\(659\) −21.0210 −0.818862 −0.409431 0.912341i \(-0.634273\pi\)
−0.409431 + 0.912341i \(0.634273\pi\)
\(660\) 0 0
\(661\) −31.8955 −1.24059 −0.620295 0.784369i \(-0.712987\pi\)
−0.620295 + 0.784369i \(0.712987\pi\)
\(662\) 0 0
\(663\) −0.971885 1.94101i −0.0377449 0.0753826i
\(664\) 0 0
\(665\) −1.05710 −0.0409925
\(666\) 0 0
\(667\) 3.21406 0.124449
\(668\) 0 0
\(669\) 1.13690i 0.0439550i
\(670\) 0 0
\(671\) 32.4429 1.25244
\(672\) 0 0
\(673\) 17.1993i 0.662984i 0.943458 + 0.331492i \(0.107552\pi\)
−0.943458 + 0.331492i \(0.892448\pi\)
\(674\) 0 0
\(675\) 2.99133i 0.115136i
\(676\) 0 0
\(677\) 7.57787i 0.291241i 0.989341 + 0.145621i \(0.0465179\pi\)
−0.989341 + 0.145621i \(0.953482\pi\)
\(678\) 0 0
\(679\) 4.40476 0.169039
\(680\) 0 0
\(681\) 26.2803 1.00706
\(682\) 0 0
\(683\) 35.2582i 1.34912i 0.738221 + 0.674558i \(0.235666\pi\)
−0.738221 + 0.674558i \(0.764334\pi\)
\(684\) 0 0
\(685\) 26.7224i 1.02101i
\(686\) 0 0
\(687\) 22.0764i 0.842269i
\(688\) 0 0
\(689\) 3.50006 0.133342
\(690\) 0 0
\(691\) 17.3942i 0.661706i 0.943682 + 0.330853i \(0.107336\pi\)
−0.943682 + 0.330853i \(0.892664\pi\)
\(692\) 0 0
\(693\) −1.58966 −0.0603861
\(694\) 0 0
\(695\) −8.47161 −0.321347
\(696\) 0 0
\(697\) −18.4374 36.8225i −0.698368 1.39475i
\(698\) 0 0
\(699\) 8.84879 0.334692
\(700\) 0 0
\(701\) 32.0840 1.21180 0.605898 0.795542i \(-0.292814\pi\)
0.605898 + 0.795542i \(0.292814\pi\)
\(702\) 0 0
\(703\) 9.57710i 0.361207i
\(704\) 0 0
\(705\) −9.44432 −0.355694
\(706\) 0 0
\(707\) 1.46620i 0.0551421i
\(708\) 0 0
\(709\) 14.3393i 0.538524i −0.963067 0.269262i \(-0.913220\pi\)
0.963067 0.269262i \(-0.0867798\pi\)
\(710\) 0 0
\(711\) 9.74026i 0.365288i
\(712\) 0 0
\(713\) 25.7880 0.965768
\(714\) 0 0
\(715\) −5.40609 −0.202176
\(716\) 0 0
\(717\) 12.4875i 0.466354i
\(718\) 0 0
\(719\) 8.06288i 0.300695i 0.988633 + 0.150347i \(0.0480392\pi\)
−0.988633 + 0.150347i \(0.951961\pi\)
\(720\) 0 0
\(721\) 1.63975i 0.0610676i
\(722\) 0 0
\(723\) 12.1670 0.452496
\(724\) 0 0
\(725\) 0.507371i 0.0188433i
\(726\) 0 0
\(727\) 6.25393 0.231946 0.115973 0.993252i \(-0.463001\pi\)
0.115973 + 0.993252i \(0.463001\pi\)
\(728\) 0 0
\(729\) −28.5318 −1.05673
\(730\) 0 0
\(731\) 11.9272 5.97209i 0.441144 0.220886i
\(732\) 0 0
\(733\) 23.9489 0.884574 0.442287 0.896874i \(-0.354167\pi\)
0.442287 + 0.896874i \(0.354167\pi\)
\(734\) 0 0
\(735\) −22.6678 −0.836115
\(736\) 0 0
\(737\) 18.5072i 0.681722i
\(738\) 0 0
\(739\) 21.6470 0.796299 0.398149 0.917321i \(-0.369653\pi\)
0.398149 + 0.917321i \(0.369653\pi\)
\(740\) 0 0
\(741\) 0.966849i 0.0355181i
\(742\) 0 0
\(743\) 21.1026i 0.774178i 0.922042 + 0.387089i \(0.126519\pi\)
−0.922042 + 0.387089i \(0.873481\pi\)
\(744\) 0 0
\(745\) 14.0584i 0.515059i
\(746\) 0 0
\(747\) −15.4366 −0.564794
\(748\) 0 0
\(749\) −2.62732 −0.0960001
\(750\) 0 0
\(751\) 23.6334i 0.862394i 0.902258 + 0.431197i \(0.141909\pi\)
−0.902258 + 0.431197i \(0.858091\pi\)
\(752\) 0 0
\(753\) 7.09630i 0.258604i
\(754\) 0 0
\(755\) 24.1824i 0.880089i
\(756\) 0 0
\(757\) 48.8655 1.77605 0.888024 0.459797i \(-0.152078\pi\)
0.888024 + 0.459797i \(0.152078\pi\)
\(758\) 0 0
\(759\) 28.2384i 1.02499i
\(760\) 0 0
\(761\) 24.2939 0.880651 0.440326 0.897838i \(-0.354863\pi\)
0.440326 + 0.897838i \(0.354863\pi\)
\(762\) 0 0
\(763\) 1.27095 0.0460115
\(764\) 0 0
\(765\) −4.64678 9.28035i −0.168005 0.335532i
\(766\) 0 0
\(767\) 0.379019 0.0136856
\(768\) 0 0
\(769\) −38.2086 −1.37784 −0.688920 0.724838i \(-0.741915\pi\)
−0.688920 + 0.724838i \(0.741915\pi\)
\(770\) 0 0
\(771\) 23.7136i 0.854024i
\(772\) 0 0
\(773\) −39.0683 −1.40519 −0.702594 0.711591i \(-0.747975\pi\)
−0.702594 + 0.711591i \(0.747975\pi\)
\(774\) 0 0
\(775\) 4.07089i 0.146231i
\(776\) 0 0
\(777\) 1.77333i 0.0636180i
\(778\) 0 0
\(779\) 18.3419i 0.657166i
\(780\) 0 0
\(781\) 73.5131 2.63050
\(782\) 0 0
\(783\) 5.42253 0.193785
\(784\) 0 0
\(785\) 38.0014i 1.35633i
\(786\) 0 0
\(787\) 30.5306i 1.08830i 0.838989 + 0.544149i \(0.183147\pi\)
−0.838989 + 0.544149i \(0.816853\pi\)
\(788\) 0 0
\(789\) 13.1930i 0.469684i
\(790\) 0 0
\(791\) −1.20257 −0.0427586
\(792\) 0 0
\(793\) 2.02714i 0.0719858i