Properties

Label 4004.2.m.c.2157.6
Level 4004
Weight 2
Character 4004.2157
Analytic conductor 31.972
Analytic rank 0
Dimension 36
CM No

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Newspace parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.m (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(36\)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2157.6
Character \(\chi\) = 4004.2157
Dual form 4004.2.m.c.2157.5

$q$-expansion

\(f(q)\) \(=\) \(q\)\(-2.75761 q^{3}\) \(-1.98236i q^{5}\) \(-1.00000i q^{7}\) \(+4.60441 q^{9}\) \(+O(q^{10})\) \(q\)\(-2.75761 q^{3}\) \(-1.98236i q^{5}\) \(-1.00000i q^{7}\) \(+4.60441 q^{9}\) \(+1.00000i q^{11}\) \(+(-2.54138 - 2.55761i) q^{13}\) \(+5.46656i q^{15}\) \(-7.39189 q^{17}\) \(+4.19788i q^{19}\) \(+2.75761i q^{21}\) \(+4.61924 q^{23}\) \(+1.07027 q^{25}\) \(-4.42434 q^{27}\) \(+4.07057 q^{29}\) \(-3.60497i q^{31}\) \(-2.75761i q^{33}\) \(-1.98236 q^{35}\) \(+9.67276i q^{37}\) \(+(7.00813 + 7.05290i) q^{39}\) \(+5.43161i q^{41}\) \(+12.3904 q^{43}\) \(-9.12758i q^{45}\) \(+0.284033i q^{47}\) \(-1.00000 q^{49}\) \(+20.3839 q^{51}\) \(-4.46945 q^{53}\) \(+1.98236 q^{55}\) \(-11.5761i q^{57}\) \(+1.96978i q^{59}\) \(-2.69145 q^{61}\) \(-4.60441i q^{63}\) \(+(-5.07010 + 5.03792i) q^{65}\) \(-12.8334i q^{67}\) \(-12.7381 q^{69}\) \(+8.00846i q^{71}\) \(+0.542368i q^{73}\) \(-2.95137 q^{75}\) \(+1.00000 q^{77}\) \(-13.5068 q^{79}\) \(-1.61263 q^{81}\) \(-3.32443i q^{83}\) \(+14.6533i q^{85}\) \(-11.2251 q^{87}\) \(+14.4386i q^{89}\) \(+(-2.55761 + 2.54138i) q^{91}\) \(+9.94111i q^{93}\) \(+8.32168 q^{95}\) \(+15.3264i q^{97}\) \(+4.60441i q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 80q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 80q^{69} \) \(\mathstrut -\mathstrut 36q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 132q^{81} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(925\) \(2003\) \(3433\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) −2.75761 −1.59211 −0.796053 0.605226i \(-0.793083\pi\)
−0.796053 + 0.605226i \(0.793083\pi\)
\(4\) 0 0
\(5\) 1.98236i 0.886536i −0.896389 0.443268i \(-0.853819\pi\)
0.896389 0.443268i \(-0.146181\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 4.60441 1.53480
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) −2.54138 2.55761i −0.704852 0.709355i
\(14\) 0 0
\(15\) 5.46656i 1.41146i
\(16\) 0 0
\(17\) −7.39189 −1.79280 −0.896398 0.443250i \(-0.853825\pi\)
−0.896398 + 0.443250i \(0.853825\pi\)
\(18\) 0 0
\(19\) 4.19788i 0.963059i 0.876430 + 0.481529i \(0.159919\pi\)
−0.876430 + 0.481529i \(0.840081\pi\)
\(20\) 0 0
\(21\) 2.75761i 0.601760i
\(22\) 0 0
\(23\) 4.61924 0.963178 0.481589 0.876397i \(-0.340060\pi\)
0.481589 + 0.876397i \(0.340060\pi\)
\(24\) 0 0
\(25\) 1.07027 0.214053
\(26\) 0 0
\(27\) −4.42434 −0.851465
\(28\) 0 0
\(29\) 4.07057 0.755887 0.377943 0.925829i \(-0.376631\pi\)
0.377943 + 0.925829i \(0.376631\pi\)
\(30\) 0 0
\(31\) 3.60497i 0.647472i −0.946147 0.323736i \(-0.895061\pi\)
0.946147 0.323736i \(-0.104939\pi\)
\(32\) 0 0
\(33\) 2.75761i 0.480038i
\(34\) 0 0
\(35\) −1.98236 −0.335079
\(36\) 0 0
\(37\) 9.67276i 1.59019i 0.606484 + 0.795096i \(0.292580\pi\)
−0.606484 + 0.795096i \(0.707420\pi\)
\(38\) 0 0
\(39\) 7.00813 + 7.05290i 1.12220 + 1.12937i
\(40\) 0 0
\(41\) 5.43161i 0.848275i 0.905598 + 0.424138i \(0.139423\pi\)
−0.905598 + 0.424138i \(0.860577\pi\)
\(42\) 0 0
\(43\) 12.3904 1.88953 0.944763 0.327755i \(-0.106292\pi\)
0.944763 + 0.327755i \(0.106292\pi\)
\(44\) 0 0
\(45\) 9.12758i 1.36066i
\(46\) 0 0
\(47\) 0.284033i 0.0414305i 0.999785 + 0.0207153i \(0.00659435\pi\)
−0.999785 + 0.0207153i \(0.993406\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 20.3839 2.85432
\(52\) 0 0
\(53\) −4.46945 −0.613926 −0.306963 0.951722i \(-0.599313\pi\)
−0.306963 + 0.951722i \(0.599313\pi\)
\(54\) 0 0
\(55\) 1.98236 0.267301
\(56\) 0 0
\(57\) 11.5761i 1.53329i
\(58\) 0 0
\(59\) 1.96978i 0.256444i 0.991745 + 0.128222i \(0.0409270\pi\)
−0.991745 + 0.128222i \(0.959073\pi\)
\(60\) 0 0
\(61\) −2.69145 −0.344605 −0.172303 0.985044i \(-0.555121\pi\)
−0.172303 + 0.985044i \(0.555121\pi\)
\(62\) 0 0
\(63\) 4.60441i 0.580101i
\(64\) 0 0
\(65\) −5.07010 + 5.03792i −0.628869 + 0.624877i
\(66\) 0 0
\(67\) 12.8334i 1.56784i −0.620860 0.783922i \(-0.713216\pi\)
0.620860 0.783922i \(-0.286784\pi\)
\(68\) 0 0
\(69\) −12.7381 −1.53348
\(70\) 0 0
\(71\) 8.00846i 0.950429i 0.879870 + 0.475214i \(0.157630\pi\)
−0.879870 + 0.475214i \(0.842370\pi\)
\(72\) 0 0
\(73\) 0.542368i 0.0634793i 0.999496 + 0.0317397i \(0.0101047\pi\)
−0.999496 + 0.0317397i \(0.989895\pi\)
\(74\) 0 0
\(75\) −2.95137 −0.340795
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −13.5068 −1.51964 −0.759819 0.650134i \(-0.774713\pi\)
−0.759819 + 0.650134i \(0.774713\pi\)
\(80\) 0 0
\(81\) −1.61263 −0.179181
\(82\) 0 0
\(83\) 3.32443i 0.364903i −0.983215 0.182452i \(-0.941597\pi\)
0.983215 0.182452i \(-0.0584033\pi\)
\(84\) 0 0
\(85\) 14.6533i 1.58938i
\(86\) 0 0
\(87\) −11.2251 −1.20345
\(88\) 0 0
\(89\) 14.4386i 1.53048i 0.643742 + 0.765242i \(0.277381\pi\)
−0.643742 + 0.765242i \(0.722619\pi\)
\(90\) 0 0
\(91\) −2.55761 + 2.54138i −0.268111 + 0.266409i
\(92\) 0 0
\(93\) 9.94111i 1.03084i
\(94\) 0 0
\(95\) 8.32168 0.853787
\(96\) 0 0
\(97\) 15.3264i 1.55616i 0.628164 + 0.778081i \(0.283807\pi\)
−0.628164 + 0.778081i \(0.716193\pi\)
\(98\) 0 0
\(99\) 4.60441i 0.462761i
\(100\) 0 0
\(101\) −6.90146 −0.686720 −0.343360 0.939204i \(-0.611565\pi\)
−0.343360 + 0.939204i \(0.611565\pi\)
\(102\) 0 0
\(103\) 13.1336 1.29409 0.647046 0.762451i \(-0.276004\pi\)
0.647046 + 0.762451i \(0.276004\pi\)
\(104\) 0 0
\(105\) 5.46656 0.533482
\(106\) 0 0
\(107\) 16.8985 1.63364 0.816822 0.576890i \(-0.195734\pi\)
0.816822 + 0.576890i \(0.195734\pi\)
\(108\) 0 0
\(109\) 12.6498i 1.21164i −0.795604 0.605818i \(-0.792846\pi\)
0.795604 0.605818i \(-0.207154\pi\)
\(110\) 0 0
\(111\) 26.6737i 2.53176i
\(112\) 0 0
\(113\) 5.99276 0.563752 0.281876 0.959451i \(-0.409043\pi\)
0.281876 + 0.959451i \(0.409043\pi\)
\(114\) 0 0
\(115\) 9.15698i 0.853892i
\(116\) 0 0
\(117\) −11.7016 11.7763i −1.08181 1.08872i
\(118\) 0 0
\(119\) 7.39189i 0.677613i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 14.9783i 1.35054i
\(124\) 0 0
\(125\) 12.0334i 1.07630i
\(126\) 0 0
\(127\) −4.60540 −0.408664 −0.204332 0.978902i \(-0.565502\pi\)
−0.204332 + 0.978902i \(0.565502\pi\)
\(128\) 0 0
\(129\) −34.1680 −3.00833
\(130\) 0 0
\(131\) −11.5436 −1.00857 −0.504283 0.863538i \(-0.668243\pi\)
−0.504283 + 0.863538i \(0.668243\pi\)
\(132\) 0 0
\(133\) 4.19788 0.364002
\(134\) 0 0
\(135\) 8.77062i 0.754855i
\(136\) 0 0
\(137\) 7.68246i 0.656357i −0.944616 0.328178i \(-0.893565\pi\)
0.944616 0.328178i \(-0.106435\pi\)
\(138\) 0 0
\(139\) 1.55906 0.132237 0.0661187 0.997812i \(-0.478938\pi\)
0.0661187 + 0.997812i \(0.478938\pi\)
\(140\) 0 0
\(141\) 0.783253i 0.0659618i
\(142\) 0 0
\(143\) 2.55761 2.54138i 0.213878 0.212521i
\(144\) 0 0
\(145\) 8.06932i 0.670121i
\(146\) 0 0
\(147\) 2.75761 0.227444
\(148\) 0 0
\(149\) 3.41910i 0.280103i 0.990144 + 0.140052i \(0.0447269\pi\)
−0.990144 + 0.140052i \(0.955273\pi\)
\(150\) 0 0
\(151\) 10.0152i 0.815024i −0.913200 0.407512i \(-0.866397\pi\)
0.913200 0.407512i \(-0.133603\pi\)
\(152\) 0 0
\(153\) −34.0353 −2.75159
\(154\) 0 0
\(155\) −7.14634 −0.574008
\(156\) 0 0
\(157\) 19.3142 1.54144 0.770719 0.637176i \(-0.219897\pi\)
0.770719 + 0.637176i \(0.219897\pi\)
\(158\) 0 0
\(159\) 12.3250 0.977435
\(160\) 0 0
\(161\) 4.61924i 0.364047i
\(162\) 0 0
\(163\) 1.51917i 0.118990i −0.998229 0.0594952i \(-0.981051\pi\)
0.998229 0.0594952i \(-0.0189491\pi\)
\(164\) 0 0
\(165\) −5.46656 −0.425571
\(166\) 0 0
\(167\) 4.56617i 0.353340i −0.984270 0.176670i \(-0.943467\pi\)
0.984270 0.176670i \(-0.0565326\pi\)
\(168\) 0 0
\(169\) −0.0827857 + 12.9997i −0.00636813 + 0.999980i
\(170\) 0 0
\(171\) 19.3288i 1.47811i
\(172\) 0 0
\(173\) 21.0406 1.59969 0.799845 0.600207i \(-0.204915\pi\)
0.799845 + 0.600207i \(0.204915\pi\)
\(174\) 0 0
\(175\) 1.07027i 0.0809045i
\(176\) 0 0
\(177\) 5.43190i 0.408286i
\(178\) 0 0
\(179\) −5.13541 −0.383839 −0.191919 0.981411i \(-0.561471\pi\)
−0.191919 + 0.981411i \(0.561471\pi\)
\(180\) 0 0
\(181\) 0.227231 0.0168899 0.00844497 0.999964i \(-0.497312\pi\)
0.00844497 + 0.999964i \(0.497312\pi\)
\(182\) 0 0
\(183\) 7.42198 0.548649
\(184\) 0 0
\(185\) 19.1748 1.40976
\(186\) 0 0
\(187\) 7.39189i 0.540548i
\(188\) 0 0
\(189\) 4.42434i 0.321824i
\(190\) 0 0
\(191\) 15.5458 1.12486 0.562428 0.826846i \(-0.309867\pi\)
0.562428 + 0.826846i \(0.309867\pi\)
\(192\) 0 0
\(193\) 24.9751i 1.79775i −0.438207 0.898874i \(-0.644386\pi\)
0.438207 0.898874i \(-0.355614\pi\)
\(194\) 0 0
\(195\) 13.9814 13.8926i 1.00123 0.994871i
\(196\) 0 0
\(197\) 20.4952i 1.46022i −0.683329 0.730110i \(-0.739469\pi\)
0.683329 0.730110i \(-0.260531\pi\)
\(198\) 0 0
\(199\) 2.18473 0.154872 0.0774358 0.996997i \(-0.475327\pi\)
0.0774358 + 0.996997i \(0.475327\pi\)
\(200\) 0 0
\(201\) 35.3894i 2.49617i
\(202\) 0 0
\(203\) 4.07057i 0.285698i
\(204\) 0 0
\(205\) 10.7674 0.752027
\(206\) 0 0
\(207\) 21.2689 1.47829
\(208\) 0 0
\(209\) −4.19788 −0.290373
\(210\) 0 0
\(211\) 5.26708 0.362601 0.181300 0.983428i \(-0.441969\pi\)
0.181300 + 0.983428i \(0.441969\pi\)
\(212\) 0 0
\(213\) 22.0842i 1.51318i
\(214\) 0 0
\(215\) 24.5623i 1.67513i
\(216\) 0 0
\(217\) −3.60497 −0.244721
\(218\) 0 0
\(219\) 1.49564i 0.101066i
\(220\) 0 0
\(221\) 18.7856 + 18.9056i 1.26366 + 1.27173i
\(222\) 0 0
\(223\) 10.2976i 0.689578i −0.938680 0.344789i \(-0.887950\pi\)
0.938680 0.344789i \(-0.112050\pi\)
\(224\) 0 0
\(225\) 4.92794 0.328530
\(226\) 0 0
\(227\) 27.0317i 1.79415i −0.441874 0.897077i \(-0.645686\pi\)
0.441874 0.897077i \(-0.354314\pi\)
\(228\) 0 0
\(229\) 2.21512i 0.146379i 0.997318 + 0.0731896i \(0.0233178\pi\)
−0.997318 + 0.0731896i \(0.976682\pi\)
\(230\) 0 0
\(231\) −2.75761 −0.181437
\(232\) 0 0
\(233\) −21.8938 −1.43431 −0.717154 0.696915i \(-0.754556\pi\)
−0.717154 + 0.696915i \(0.754556\pi\)
\(234\) 0 0
\(235\) 0.563055 0.0367297
\(236\) 0 0
\(237\) 37.2466 2.41943
\(238\) 0 0
\(239\) 4.09702i 0.265014i 0.991182 + 0.132507i \(0.0423027\pi\)
−0.991182 + 0.132507i \(0.957697\pi\)
\(240\) 0 0
\(241\) 2.77298i 0.178623i 0.996004 + 0.0893117i \(0.0284667\pi\)
−0.996004 + 0.0893117i \(0.971533\pi\)
\(242\) 0 0
\(243\) 17.7200 1.13674
\(244\) 0 0
\(245\) 1.98236i 0.126648i
\(246\) 0 0
\(247\) 10.7365 10.6684i 0.683150 0.678814i
\(248\) 0 0
\(249\) 9.16747i 0.580965i
\(250\) 0 0
\(251\) 20.7097 1.30718 0.653592 0.756847i \(-0.273261\pi\)
0.653592 + 0.756847i \(0.273261\pi\)
\(252\) 0 0
\(253\) 4.61924i 0.290409i
\(254\) 0 0
\(255\) 40.4082i 2.53046i
\(256\) 0 0
\(257\) 1.54096 0.0961227 0.0480613 0.998844i \(-0.484696\pi\)
0.0480613 + 0.998844i \(0.484696\pi\)
\(258\) 0 0
\(259\) 9.67276 0.601036
\(260\) 0 0
\(261\) 18.7426 1.16014
\(262\) 0 0
\(263\) 7.95053 0.490250 0.245125 0.969491i \(-0.421171\pi\)
0.245125 + 0.969491i \(0.421171\pi\)
\(264\) 0 0
\(265\) 8.86003i 0.544267i
\(266\) 0 0
\(267\) 39.8159i 2.43670i
\(268\) 0 0
\(269\) 8.18925 0.499307 0.249654 0.968335i \(-0.419683\pi\)
0.249654 + 0.968335i \(0.419683\pi\)
\(270\) 0 0
\(271\) 11.8033i 0.716999i 0.933530 + 0.358500i \(0.116712\pi\)
−0.933530 + 0.358500i \(0.883288\pi\)
\(272\) 0 0
\(273\) 7.05290 7.00813i 0.426861 0.424151i
\(274\) 0 0
\(275\) 1.07027i 0.0645394i
\(276\) 0 0
\(277\) −9.29260 −0.558338 −0.279169 0.960242i \(-0.590059\pi\)
−0.279169 + 0.960242i \(0.590059\pi\)
\(278\) 0 0
\(279\) 16.5988i 0.993743i
\(280\) 0 0
\(281\) 27.3565i 1.63195i 0.578084 + 0.815977i \(0.303801\pi\)
−0.578084 + 0.815977i \(0.696199\pi\)
\(282\) 0 0
\(283\) 19.0094 1.12999 0.564995 0.825094i \(-0.308878\pi\)
0.564995 + 0.825094i \(0.308878\pi\)
\(284\) 0 0
\(285\) −22.9480 −1.35932
\(286\) 0 0
\(287\) 5.43161 0.320618
\(288\) 0 0
\(289\) 37.6400 2.21412
\(290\) 0 0
\(291\) 42.2643i 2.47758i
\(292\) 0 0
\(293\) 9.11806i 0.532683i −0.963879 0.266341i \(-0.914185\pi\)
0.963879 0.266341i \(-0.0858148\pi\)
\(294\) 0 0
\(295\) 3.90481 0.227347
\(296\) 0 0
\(297\) 4.42434i 0.256726i
\(298\) 0 0
\(299\) −11.7392 11.8142i −0.678898 0.683235i
\(300\) 0 0
\(301\) 12.3904i 0.714173i
\(302\) 0 0
\(303\) 19.0315 1.09333
\(304\) 0 0
\(305\) 5.33542i 0.305505i
\(306\) 0 0
\(307\) 17.3379i 0.989524i −0.869028 0.494762i \(-0.835255\pi\)
0.869028 0.494762i \(-0.164745\pi\)
\(308\) 0 0
\(309\) −36.2173 −2.06033
\(310\) 0 0
\(311\) 2.81808 0.159799 0.0798995 0.996803i \(-0.474540\pi\)
0.0798995 + 0.996803i \(0.474540\pi\)
\(312\) 0 0
\(313\) −26.4075 −1.49264 −0.746319 0.665588i \(-0.768181\pi\)
−0.746319 + 0.665588i \(0.768181\pi\)
\(314\) 0 0
\(315\) −9.12758 −0.514281
\(316\) 0 0
\(317\) 3.99221i 0.224225i −0.993696 0.112113i \(-0.964238\pi\)
0.993696 0.112113i \(-0.0357617\pi\)
\(318\) 0 0
\(319\) 4.07057i 0.227908i
\(320\) 0 0
\(321\) −46.5996 −2.60094
\(322\) 0 0
\(323\) 31.0302i 1.72657i
\(324\) 0 0
\(325\) −2.71995 2.73733i −0.150876 0.151840i
\(326\) 0 0
\(327\) 34.8833i 1.92905i
\(328\) 0 0
\(329\) 0.284033 0.0156593
\(330\) 0 0
\(331\) 21.9021i 1.20385i −0.798554 0.601923i \(-0.794401\pi\)
0.798554 0.601923i \(-0.205599\pi\)
\(332\) 0 0
\(333\) 44.5374i 2.44063i
\(334\) 0 0
\(335\) −25.4403 −1.38995
\(336\) 0 0
\(337\) 25.2354 1.37466 0.687331 0.726344i \(-0.258782\pi\)
0.687331 + 0.726344i \(0.258782\pi\)
\(338\) 0 0
\(339\) −16.5257 −0.897553
\(340\) 0 0
\(341\) 3.60497 0.195220
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 25.2514i 1.35949i
\(346\) 0 0
\(347\) −3.98169 −0.213748 −0.106874 0.994273i \(-0.534084\pi\)
−0.106874 + 0.994273i \(0.534084\pi\)
\(348\) 0 0
\(349\) 1.38283i 0.0740214i 0.999315 + 0.0370107i \(0.0117836\pi\)
−0.999315 + 0.0370107i \(0.988216\pi\)
\(350\) 0 0
\(351\) 11.2439 + 11.3158i 0.600157 + 0.603991i
\(352\) 0 0
\(353\) 2.50659i 0.133412i −0.997773 0.0667060i \(-0.978751\pi\)
0.997773 0.0667060i \(-0.0212490\pi\)
\(354\) 0 0
\(355\) 15.8756 0.842590
\(356\) 0 0
\(357\) 20.3839i 1.07883i
\(358\) 0 0
\(359\) 15.6288i 0.824859i 0.910989 + 0.412430i \(0.135320\pi\)
−0.910989 + 0.412430i \(0.864680\pi\)
\(360\) 0 0
\(361\) 1.37783 0.0725176
\(362\) 0 0
\(363\) 2.75761 0.144737
\(364\) 0 0
\(365\) 1.07517 0.0562768
\(366\) 0 0
\(367\) 19.7136 1.02904 0.514522 0.857477i \(-0.327970\pi\)
0.514522 + 0.857477i \(0.327970\pi\)
\(368\) 0 0
\(369\) 25.0094i 1.30194i
\(370\) 0 0
\(371\) 4.46945i 0.232042i
\(372\) 0 0
\(373\) 11.5753 0.599346 0.299673 0.954042i \(-0.403122\pi\)
0.299673 + 0.954042i \(0.403122\pi\)
\(374\) 0 0
\(375\) 33.1835i 1.71359i
\(376\) 0 0
\(377\) −10.3449 10.4110i −0.532788 0.536192i
\(378\) 0 0
\(379\) 12.2568i 0.629588i −0.949160 0.314794i \(-0.898065\pi\)
0.949160 0.314794i \(-0.101935\pi\)
\(380\) 0 0
\(381\) 12.6999 0.650636
\(382\) 0 0
\(383\) 32.5093i 1.66115i 0.556909 + 0.830574i \(0.311987\pi\)
−0.556909 + 0.830574i \(0.688013\pi\)
\(384\) 0 0
\(385\) 1.98236i 0.101030i
\(386\) 0 0
\(387\) 57.0507 2.90005
\(388\) 0 0
\(389\) −7.79700 −0.395324 −0.197662 0.980270i \(-0.563335\pi\)
−0.197662 + 0.980270i \(0.563335\pi\)
\(390\) 0 0
\(391\) −34.1449 −1.72678
\(392\) 0 0
\(393\) 31.8327 1.60575
\(394\) 0 0
\(395\) 26.7754i 1.34722i
\(396\) 0 0
\(397\) 19.6911i 0.988269i −0.869386 0.494134i \(-0.835485\pi\)
0.869386 0.494134i \(-0.164515\pi\)
\(398\) 0 0
\(399\) −11.5761 −0.579530
\(400\) 0 0
\(401\) 0.958076i 0.0478440i 0.999714 + 0.0239220i \(0.00761534\pi\)
−0.999714 + 0.0239220i \(0.992385\pi\)
\(402\) 0 0
\(403\) −9.22013 + 9.16160i −0.459287 + 0.456372i
\(404\) 0 0
\(405\) 3.19680i 0.158850i
\(406\) 0 0
\(407\) −9.67276 −0.479461
\(408\) 0 0
\(409\) 35.2821i 1.74459i −0.488981 0.872294i \(-0.662631\pi\)
0.488981 0.872294i \(-0.337369\pi\)
\(410\) 0 0
\(411\) 21.1852i 1.04499i
\(412\) 0 0
\(413\) 1.96978 0.0969268
\(414\) 0 0
\(415\) −6.59019 −0.323500
\(416\) 0 0
\(417\) −4.29927 −0.210536
\(418\) 0 0
\(419\) 10.3266 0.504485 0.252243 0.967664i \(-0.418832\pi\)
0.252243 + 0.967664i \(0.418832\pi\)
\(420\) 0 0
\(421\) 13.1500i 0.640892i 0.947267 + 0.320446i \(0.103833\pi\)
−0.947267 + 0.320446i \(0.896167\pi\)
\(422\) 0 0
\(423\) 1.30781i 0.0635878i
\(424\) 0 0
\(425\) −7.91128 −0.383753
\(426\) 0 0
\(427\) 2.69145i 0.130249i
\(428\) 0 0
\(429\) −7.05290 + 7.00813i −0.340517 + 0.338356i
\(430\) 0 0
\(431\) 5.88317i 0.283382i −0.989911 0.141691i \(-0.954746\pi\)
0.989911 0.141691i \(-0.0452540\pi\)
\(432\) 0 0
\(433\) −36.6669 −1.76210 −0.881049 0.473025i \(-0.843162\pi\)
−0.881049 + 0.473025i \(0.843162\pi\)
\(434\) 0 0
\(435\) 22.2520i 1.06690i
\(436\) 0 0
\(437\) 19.3910i 0.927597i
\(438\) 0 0
\(439\) 27.3167 1.30376 0.651878 0.758324i \(-0.273981\pi\)
0.651878 + 0.758324i \(0.273981\pi\)
\(440\) 0 0
\(441\) −4.60441 −0.219258
\(442\) 0 0
\(443\) −33.3749 −1.58569 −0.792844 0.609424i \(-0.791401\pi\)
−0.792844 + 0.609424i \(0.791401\pi\)
\(444\) 0 0
\(445\) 28.6224 1.35683
\(446\) 0 0
\(447\) 9.42853i 0.445954i
\(448\) 0 0
\(449\) 17.2049i 0.811952i −0.913884 0.405976i \(-0.866932\pi\)
0.913884 0.405976i \(-0.133068\pi\)
\(450\) 0 0
\(451\) −5.43161 −0.255765
\(452\) 0 0
\(453\) 27.6179i 1.29760i
\(454\) 0 0
\(455\) 5.03792 + 5.07010i 0.236181 + 0.237690i
\(456\) 0 0
\(457\) 21.5943i 1.01014i −0.863079 0.505068i \(-0.831467\pi\)
0.863079 0.505068i \(-0.168533\pi\)
\(458\) 0 0
\(459\) 32.7042 1.52650
\(460\) 0 0
\(461\) 28.9024i 1.34612i −0.739588 0.673060i \(-0.764980\pi\)
0.739588 0.673060i \(-0.235020\pi\)
\(462\) 0 0
\(463\) 26.3410i 1.22417i 0.790792 + 0.612085i \(0.209669\pi\)
−0.790792 + 0.612085i \(0.790331\pi\)
\(464\) 0 0
\(465\) 19.7068 0.913881
\(466\) 0 0
\(467\) −28.4092 −1.31462 −0.657311 0.753619i \(-0.728306\pi\)
−0.657311 + 0.753619i \(0.728306\pi\)
\(468\) 0 0
\(469\) −12.8334 −0.592589
\(470\) 0 0
\(471\) −53.2609 −2.45413
\(472\) 0 0
\(473\) 12.3904i 0.569713i
\(474\) 0 0
\(475\) 4.49284i 0.206146i
\(476\) 0 0
\(477\) −20.5792 −0.942256
\(478\) 0 0
\(479\) 25.4893i 1.16464i 0.812961 + 0.582318i \(0.197854\pi\)
−0.812961 + 0.582318i \(0.802146\pi\)
\(480\) 0 0
\(481\) 24.7392 24.5821i 1.12801 1.12085i
\(482\) 0 0
\(483\) 12.7381i 0.579602i
\(484\) 0 0
\(485\) 30.3824 1.37959
\(486\) 0 0
\(487\) 33.5089i 1.51843i 0.650839 + 0.759216i \(0.274417\pi\)
−0.650839 + 0.759216i \(0.725583\pi\)
\(488\) 0 0
\(489\) 4.18927i 0.189445i
\(490\) 0 0
\(491\) 3.87148 0.174717 0.0873586 0.996177i \(-0.472157\pi\)
0.0873586 + 0.996177i \(0.472157\pi\)
\(492\) 0 0
\(493\) −30.0892 −1.35515
\(494\) 0 0
\(495\) 9.12758 0.410254
\(496\) 0 0
\(497\) 8.00846 0.359228
\(498\) 0 0
\(499\) 16.7283i 0.748862i −0.927255 0.374431i \(-0.877838\pi\)
0.927255 0.374431i \(-0.122162\pi\)
\(500\) 0 0
\(501\) 12.5917i 0.562556i
\(502\) 0 0
\(503\) 43.5572 1.94212 0.971059 0.238839i \(-0.0767669\pi\)
0.971059 + 0.238839i \(0.0767669\pi\)
\(504\) 0 0
\(505\) 13.6811i 0.608803i
\(506\) 0 0
\(507\) 0.228291 35.8482i 0.0101387 1.59207i
\(508\) 0 0
\(509\) 6.52062i 0.289021i 0.989503 + 0.144511i \(0.0461608\pi\)
−0.989503 + 0.144511i \(0.953839\pi\)
\(510\) 0 0
\(511\) 0.542368 0.0239929
\(512\) 0 0
\(513\) 18.5728i 0.820011i
\(514\) 0 0
\(515\) 26.0354i 1.14726i
\(516\) 0 0
\(517\) −0.284033 −0.0124918
\(518\) 0 0
\(519\) −58.0218 −2.54688
\(520\) 0 0
\(521\) −3.38853 −0.148454 −0.0742270 0.997241i \(-0.523649\pi\)
−0.0742270 + 0.997241i \(0.523649\pi\)
\(522\) 0 0
\(523\) 36.5564 1.59850 0.799249 0.601000i \(-0.205231\pi\)
0.799249 + 0.601000i \(0.205231\pi\)
\(524\) 0 0
\(525\) 2.95137i 0.128809i
\(526\) 0 0
\(527\) 26.6475i 1.16079i
\(528\) 0 0
\(529\) −1.66263 −0.0722881
\(530\) 0 0
\(531\) 9.06970i 0.393591i
\(532\) 0 0
\(533\) 13.8920 13.8038i 0.601728 0.597908i
\(534\) 0 0
\(535\) 33.4989i 1.44829i
\(536\) 0 0
\(537\) 14.1615 0.611112
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 46.0690i 1.98066i −0.138734 0.990330i \(-0.544303\pi\)
0.138734 0.990330i \(-0.455697\pi\)
\(542\) 0 0
\(543\) −0.626614 −0.0268906
\(544\) 0 0
\(545\) −25.0765 −1.07416
\(546\) 0 0
\(547\) −11.2186 −0.479674 −0.239837 0.970813i \(-0.577094\pi\)
−0.239837 + 0.970813i \(0.577094\pi\)
\(548\) 0 0
\(549\) −12.3926 −0.528902
\(550\) 0 0
\(551\) 17.0878i 0.727963i
\(552\) 0 0
\(553\) 13.5068i 0.574370i
\(554\) 0 0
\(555\) −52.8768 −2.24449
\(556\) 0 0
\(557\) 15.3169i 0.648998i −0.945886 0.324499i \(-0.894804\pi\)
0.945886 0.324499i \(-0.105196\pi\)
\(558\) 0 0
\(559\) −31.4888 31.6900i −1.33184 1.34034i
\(560\) 0 0
\(561\) 20.3839i 0.860611i
\(562\) 0 0
\(563\) 32.2582 1.35952 0.679760 0.733435i \(-0.262084\pi\)
0.679760 + 0.733435i \(0.262084\pi\)
\(564\) 0 0
\(565\) 11.8798i 0.499786i
\(566\) 0 0
\(567\) 1.61263i 0.0677239i
\(568\) 0 0
\(569\) 18.1639 0.761471 0.380735 0.924684i \(-0.375671\pi\)
0.380735 + 0.924684i \(0.375671\pi\)
\(570\) 0 0
\(571\) −26.6686 −1.11605 −0.558023 0.829825i \(-0.688440\pi\)
−0.558023 + 0.829825i \(0.688440\pi\)
\(572\) 0 0
\(573\) −42.8693 −1.79089
\(574\) 0 0
\(575\) 4.94381 0.206171
\(576\) 0 0
\(577\) 11.9219i 0.496315i 0.968720 + 0.248157i \(0.0798250\pi\)
−0.968720 + 0.248157i \(0.920175\pi\)
\(578\) 0 0
\(579\) 68.8716i 2.86221i
\(580\) 0 0
\(581\) −3.32443 −0.137920
\(582\) 0 0
\(583\) 4.46945i 0.185106i
\(584\) 0 0
\(585\) −23.3448 + 23.1966i −0.965190 + 0.959063i
\(586\) 0 0
\(587\) 2.09246i 0.0863651i 0.999067 + 0.0431826i \(0.0137497\pi\)
−0.999067 + 0.0431826i \(0.986250\pi\)
\(588\) 0 0
\(589\) 15.1332 0.623554
\(590\) 0 0
\(591\) 56.5177i 2.32483i
\(592\) 0 0
\(593\) 17.1115i 0.702685i −0.936247 0.351343i \(-0.885725\pi\)
0.936247 0.351343i \(-0.114275\pi\)
\(594\) 0 0
\(595\) 14.6533 0.600729
\(596\) 0 0
\(597\) −6.02464 −0.246572
\(598\) 0 0
\(599\) 43.5056 1.77759 0.888795 0.458304i \(-0.151543\pi\)
0.888795 + 0.458304i \(0.151543\pi\)
\(600\) 0 0
\(601\) −12.7825 −0.521411 −0.260705 0.965418i \(-0.583955\pi\)
−0.260705 + 0.965418i \(0.583955\pi\)
\(602\) 0 0
\(603\) 59.0900i 2.40633i
\(604\) 0 0
\(605\) 1.98236i 0.0805942i
\(606\) 0 0
\(607\) 23.6720 0.960815 0.480408 0.877045i \(-0.340489\pi\)
0.480408 + 0.877045i \(0.340489\pi\)
\(608\) 0 0
\(609\) 11.2251i 0.454862i
\(610\) 0 0
\(611\) 0.726448 0.721837i 0.0293889 0.0292024i
\(612\) 0 0
\(613\) 18.2292i 0.736271i 0.929772 + 0.368135i \(0.120004\pi\)
−0.929772 + 0.368135i \(0.879996\pi\)
\(614\) 0 0
\(615\) −29.6922 −1.19731
\(616\) 0 0
\(617\) 38.0001i 1.52983i 0.644134 + 0.764913i \(0.277218\pi\)
−0.644134 + 0.764913i \(0.722782\pi\)
\(618\) 0 0
\(619\) 14.2348i 0.572147i −0.958208 0.286073i \(-0.907650\pi\)
0.958208 0.286073i \(-0.0923501\pi\)
\(620\) 0 0
\(621\) −20.4371 −0.820112
\(622\) 0 0
\(623\) 14.4386 0.578469
\(624\) 0 0
\(625\) −18.5032 −0.740128
\(626\) 0 0
\(627\) 11.5761 0.462305
\(628\) 0 0
\(629\) 71.4999i 2.85089i
\(630\) 0 0
\(631\) 4.63927i 0.184687i −0.995727 0.0923433i \(-0.970564\pi\)
0.995727 0.0923433i \(-0.0294357\pi\)
\(632\) 0 0
\(633\) −14.5246 −0.577299
\(634\) 0 0
\(635\) 9.12955i 0.362295i
\(636\) 0 0
\(637\) 2.54138 + 2.55761i 0.100693 + 0.101336i
\(638\) 0 0
\(639\) 36.8742i 1.45872i
\(640\) 0 0
\(641\) 7.60473 0.300369 0.150184 0.988658i \(-0.452013\pi\)
0.150184 + 0.988658i \(0.452013\pi\)
\(642\) 0 0
\(643\) 25.1261i 0.990875i −0.868643 0.495438i \(-0.835008\pi\)
0.868643 0.495438i \(-0.164992\pi\)
\(644\) 0 0
\(645\) 67.7332i 2.66699i
\(646\) 0 0
\(647\) −29.1613 −1.14645 −0.573225 0.819398i \(-0.694308\pi\)
−0.573225 + 0.819398i \(0.694308\pi\)
\(648\) 0 0
\(649\) −1.96978 −0.0773208
\(650\) 0 0
\(651\) 9.94111 0.389623
\(652\) 0 0
\(653\) −21.4154 −0.838048 −0.419024 0.907975i \(-0.637628\pi\)
−0.419024 + 0.907975i \(0.637628\pi\)
\(654\) 0 0
\(655\) 22.8835i 0.894131i
\(656\) 0 0
\(657\) 2.49728i 0.0974284i
\(658\) 0 0
\(659\) 1.27587 0.0497008 0.0248504 0.999691i \(-0.492089\pi\)
0.0248504 + 0.999691i \(0.492089\pi\)
\(660\) 0 0
\(661\) 17.8826i 0.695554i 0.937577 + 0.347777i \(0.113063\pi\)
−0.937577 + 0.347777i \(0.886937\pi\)
\(662\) 0 0
\(663\) −51.8033 52.1343i −2.01187 2.02473i
\(664\) 0 0
\(665\) 8.32168i 0.322701i
\(666\) 0 0
\(667\) 18.8030 0.728053
\(668\) 0 0
\(669\) 28.3968i 1.09788i
\(670\) 0 0
\(671\) 2.69145i 0.103902i
\(672\) 0 0
\(673\) −10.0259 −0.386470 −0.193235 0.981152i \(-0.561898\pi\)
−0.193235 + 0.981152i \(0.561898\pi\)
\(674\) 0 0
\(675\) −4.73522 −0.182259
\(676\) 0 0
\(677\) 22.6036 0.868728 0.434364 0.900737i \(-0.356973\pi\)
0.434364 + 0.900737i \(0.356973\pi\)
\(678\) 0 0
\(679\) 15.3264 0.588174
\(680\) 0 0
\(681\) 74.5428i 2.85648i
\(682\) 0 0
\(683\) 27.3688i 1.04724i 0.851952 + 0.523619i \(0.175419\pi\)
−0.851952 + 0.523619i \(0.824581\pi\)
\(684\) 0 0
\(685\) −15.2294 −0.581884
\(686\) 0 0
\(687\) 6.10844i 0.233051i
\(688\) 0 0
\(689\) 11.3586 + 11.4311i 0.432727 + 0.435491i
\(690\) 0 0
\(691\) 22.4407i 0.853683i 0.904326 + 0.426842i \(0.140374\pi\)
−0.904326 + 0.426842i \(0.859626\pi\)
\(692\) 0 0
\(693\) 4.60441 0.174907
\(694\) 0 0
\(695\) 3.09060i 0.117233i
\(696\) 0 0
\(697\) 40.1498i 1.52078i
\(698\) 0 0
\(699\) 60.3744 2.28357
\(700\) 0 0
\(701\) −37.6066 −1.42038 −0.710191 0.704009i \(-0.751391\pi\)
−0.710191 + 0.704009i \(0.751391\pi\)
\(702\) 0 0
\(703\) −40.6050 −1.53145
\(704\) 0 0
\(705\) −1.55269 −0.0584776
\(706\) 0 0
\(707\) 6.90146i 0.259556i
\(708\) 0 0
\(709\) 42.6546i 1.60193i 0.598714 + 0.800963i \(0.295679\pi\)
−0.598714 + 0.800963i \(0.704321\pi\)
\(710\) 0 0
\(711\) −62.1911 −2.33235
\(712\) 0 0
\(713\) 16.6522i 0.623631i
\(714\) 0 0
\(715\) −5.03792 5.07010i −0.188407 0.189611i
\(716\) 0 0
\(717\) 11.2980i 0.421931i
\(718\) 0 0
\(719\) 42.2236 1.57467 0.787337 0.616523i \(-0.211459\pi\)
0.787337 + 0.616523i \(0.211459\pi\)
\(720\) 0 0
\(721\) 13.1336i 0.489120i
\(722\) 0 0
\(723\) 7.64680i 0.284388i
\(724\) 0 0
\(725\) 4.35659 0.161800
\(726\) 0 0
\(727\) 5.24000 0.194341 0.0971705 0.995268i \(-0.469021\pi\)
0.0971705 + 0.995268i \(0.469021\pi\)
\(728\) 0 0
\(729\) −44.0270 −1.63063
\(730\) 0 0
\(731\) −91.5888 −3.38753
\(732\) 0 0
\(733\) 24.2788i 0.896757i −0.893844 0.448378i \(-0.852002\pi\)
0.893844 0.448378i \(-0.147998\pi\)
\(734\) 0 0
\(735\) 5.46656i 0.201637i
\(736\) 0 0
\(737\) 12.8334 0.472723
\(738\) 0 0
\(739\) 5.37005i 0.197541i −0.995110 0.0987703i \(-0.968509\pi\)
0.995110 0.0987703i \(-0.0314909\pi\)
\(740\) 0 0
\(741\) −29.6072 + 29.4193i −1.08765 + 1.08074i
\(742\) 0 0
\(743\) 11.4181i 0.418890i 0.977820 + 0.209445i \(0.0671658\pi\)
−0.977820 + 0.209445i \(0.932834\pi\)
\(744\) 0 0
\(745\) 6.77787 0.248322
\(746\) 0 0
\(747\) 15.3070i 0.560055i
\(748\) 0 0
\(749\) 16.8985i 0.617459i
\(750\) 0 0
\(751\) 48.4887 1.76938 0.884689 0.466181i \(-0.154371\pi\)
0.884689 + 0.466181i \(0.154371\pi\)
\(752\) 0 0
\(753\) −57.1092 −2.08118
\(754\) 0 0
\(755\) −19.8536 −0.722548
\(756\) 0 0
\(757\) 1.79543 0.0652559 0.0326280 0.999468i \(-0.489612\pi\)
0.0326280 + 0.999468i \(0.489612\pi\)
\(758\) 0 0
\(759\) 12.7381i 0.462362i
\(760\) 0 0
\(761\) 8.79075i 0.318664i 0.987225 + 0.159332i \(0.0509341\pi\)
−0.987225 + 0.159332i \(0.949066\pi\)
\(762\) 0 0
\(763\) −12.6498 −0.457955
\(764\) 0 0
\(765\) 67.4701i 2.43939i
\(766\) 0 0
\(767\) 5.03795 5.00597i 0.181910 0.180755i
\(768\) 0 0
\(769\) 44.8648i 1.61787i −0.587901 0.808933i \(-0.700045\pi\)
0.587901 0.808933i \(-0.299955\pi\)
\(770\) 0 0
\(771\) −4.24938 −0.153038
\(772\) 0 0
\(773\) 3.23482i 0.116348i −0.998306 0.0581742i \(-0.981472\pi\)
0.998306 0.0581742i \(-0.0185279\pi\)
\(774\) 0 0
\(775\) 3.85828i 0.138593i
\(776\) 0 0
\(777\) −26.6737 −0.956913
\(778\) 0 0
\(779\) −22.8012 −0.816939
\(780\) 0 0
\(781\) −8.00846 −0.286565
\(782\) 0 0
\(783\) −18.0096 −0.643611
\(784\) 0 0
\(785\) 38.2875i 1.36654i
\(786\) 0 0
\(787\) 52.0427i 1.85512i −0.373673 0.927560i \(-0.621902\pi\)
0.373673 0.927560i \(-0.378098\pi\)
\(788\) 0 0
\(789\) −21.9245 −0.780531
\(790\) 0 0
\(791\) 5.99276i 0.213078i
\(792\) 0 0
\(793\) 6.84001 + 6.88370i 0.242896 + 0.244447i
\(794\) 0 0