Properties

Label 668.2.a.b.1.1
Level $668$
Weight $2$
Character 668.1
Self dual yes
Analytic conductor $5.334$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 668 = 2^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 668.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.33400685502\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.826865.1
Defining polynomial: \(x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 6 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.75474\) of defining polynomial
Character \(\chi\) \(=\) 668.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.83385 q^{3} +2.00000 q^{5} +1.53681 q^{7} +0.363011 q^{9} +O(q^{10})\) \(q-1.83385 q^{3} +2.00000 q^{5} +1.53681 q^{7} +0.363011 q^{9} +6.05178 q^{11} -5.50948 q^{13} -3.66770 q^{15} +1.11576 q^{17} -2.87249 q^{19} -2.81828 q^{21} +6.59409 q^{23} -1.00000 q^{25} +4.83585 q^{27} +3.97267 q^{29} -1.23351 q^{31} -11.0981 q^{33} +3.07362 q^{35} +11.1772 q^{37} +10.1036 q^{39} +2.55195 q^{41} +11.0190 q^{43} +0.726021 q^{45} +8.14647 q^{47} -4.63822 q^{49} -2.04613 q^{51} -6.62523 q^{53} +12.1036 q^{55} +5.26772 q^{57} -12.3391 q^{59} -0.918459 q^{61} +0.557878 q^{63} -11.0190 q^{65} +12.9768 q^{67} -12.0926 q^{69} -11.3775 q^{71} +5.47833 q^{73} +1.83385 q^{75} +9.30043 q^{77} +10.1457 q^{79} -9.95726 q^{81} +14.5284 q^{83} +2.23151 q^{85} -7.28529 q^{87} +4.26971 q^{89} -8.46701 q^{91} +2.26207 q^{93} -5.74498 q^{95} -16.8392 q^{97} +2.19686 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 3q^{3} + 10q^{5} + 9q^{7} + 6q^{9} + O(q^{10}) \) \( 5q + 3q^{3} + 10q^{5} + 9q^{7} + 6q^{9} + 5q^{11} - 4q^{13} + 6q^{15} - 2q^{17} + 5q^{19} - 4q^{21} + 6q^{23} - 5q^{25} + 12q^{27} - 5q^{29} + 9q^{31} - 8q^{33} + 18q^{35} + 8q^{37} - 4q^{41} + 8q^{43} + 12q^{45} + 13q^{47} + 14q^{49} - 2q^{53} + 10q^{55} - 12q^{57} + 4q^{59} + 11q^{61} - q^{63} - 8q^{65} + 28q^{67} - 16q^{69} + 2q^{71} + 8q^{73} - 3q^{75} - 12q^{77} - 10q^{79} - 15q^{81} + 2q^{83} - 4q^{85} + 4q^{87} - 17q^{89} - 12q^{91} - 12q^{93} + 10q^{95} - 27q^{97} + 3q^{99} + O(q^{100}) \)

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
\(3\) −1.83385 −1.05877 −0.529387 0.848380i \(-0.677578\pi\)
−0.529387 + 0.848380i \(0.677578\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 1.53681 0.580859 0.290429 0.956896i \(-0.406202\pi\)
0.290429 + 0.956896i \(0.406202\pi\)
\(8\) 0 0
\(9\) 0.363011 0.121004
\(10\) 0 0
\(11\) 6.05178 1.82468 0.912341 0.409432i \(-0.134273\pi\)
0.912341 + 0.409432i \(0.134273\pi\)
\(12\) 0 0
\(13\) −5.50948 −1.52805 −0.764027 0.645184i \(-0.776781\pi\)
−0.764027 + 0.645184i \(0.776781\pi\)
\(14\) 0 0
\(15\) −3.66770 −0.946997
\(16\) 0 0
\(17\) 1.11576 0.270610 0.135305 0.990804i \(-0.456799\pi\)
0.135305 + 0.990804i \(0.456799\pi\)
\(18\) 0 0
\(19\) −2.87249 −0.658994 −0.329497 0.944157i \(-0.606879\pi\)
−0.329497 + 0.944157i \(0.606879\pi\)
\(20\) 0 0
\(21\) −2.81828 −0.614999
\(22\) 0 0
\(23\) 6.59409 1.37496 0.687481 0.726202i \(-0.258716\pi\)
0.687481 + 0.726202i \(0.258716\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 4.83585 0.930659
\(28\) 0 0
\(29\) 3.97267 0.737707 0.368853 0.929488i \(-0.379750\pi\)
0.368853 + 0.929488i \(0.379750\pi\)
\(30\) 0 0
\(31\) −1.23351 −0.221544 −0.110772 0.993846i \(-0.535332\pi\)
−0.110772 + 0.993846i \(0.535332\pi\)
\(32\) 0 0
\(33\) −11.0981 −1.93193
\(34\) 0 0
\(35\) 3.07362 0.519536
\(36\) 0 0
\(37\) 11.1772 1.83752 0.918759 0.394819i \(-0.129193\pi\)
0.918759 + 0.394819i \(0.129193\pi\)
\(38\) 0 0
\(39\) 10.1036 1.61787
\(40\) 0 0
\(41\) 2.55195 0.398547 0.199274 0.979944i \(-0.436142\pi\)
0.199274 + 0.979944i \(0.436142\pi\)
\(42\) 0 0
\(43\) 11.0190 1.68038 0.840188 0.542296i \(-0.182445\pi\)
0.840188 + 0.542296i \(0.182445\pi\)
\(44\) 0 0
\(45\) 0.726021 0.108229
\(46\) 0 0
\(47\) 8.14647 1.18828 0.594142 0.804360i \(-0.297492\pi\)
0.594142 + 0.804360i \(0.297492\pi\)
\(48\) 0 0
\(49\) −4.63822 −0.662603
\(50\) 0 0
\(51\) −2.04613 −0.286515
\(52\) 0 0
\(53\) −6.62523 −0.910046 −0.455023 0.890480i \(-0.650369\pi\)
−0.455023 + 0.890480i \(0.650369\pi\)
\(54\) 0 0
\(55\) 12.1036 1.63204
\(56\) 0 0
\(57\) 5.26772 0.697727
\(58\) 0 0
\(59\) −12.3391 −1.60641 −0.803205 0.595703i \(-0.796874\pi\)
−0.803205 + 0.595703i \(0.796874\pi\)
\(60\) 0 0
\(61\) −0.918459 −0.117597 −0.0587983 0.998270i \(-0.518727\pi\)
−0.0587983 + 0.998270i \(0.518727\pi\)
\(62\) 0 0
\(63\) 0.557878 0.0702860
\(64\) 0 0
\(65\) −11.0190 −1.36673
\(66\) 0 0
\(67\) 12.9768 1.58537 0.792685 0.609631i \(-0.208682\pi\)
0.792685 + 0.609631i \(0.208682\pi\)
\(68\) 0 0
\(69\) −12.0926 −1.45577
\(70\) 0 0
\(71\) −11.3775 −1.35027 −0.675133 0.737696i \(-0.735914\pi\)
−0.675133 + 0.737696i \(0.735914\pi\)
\(72\) 0 0
\(73\) 5.47833 0.641190 0.320595 0.947216i \(-0.396117\pi\)
0.320595 + 0.947216i \(0.396117\pi\)
\(74\) 0 0
\(75\) 1.83385 0.211755
\(76\) 0 0
\(77\) 9.30043 1.05988
\(78\) 0 0
\(79\) 10.1457 1.14148 0.570741 0.821130i \(-0.306656\pi\)
0.570741 + 0.821130i \(0.306656\pi\)
\(80\) 0 0
\(81\) −9.95726 −1.10636
\(82\) 0 0
\(83\) 14.5284 1.59470 0.797352 0.603515i \(-0.206234\pi\)
0.797352 + 0.603515i \(0.206234\pi\)
\(84\) 0 0
\(85\) 2.23151 0.242041
\(86\) 0 0
\(87\) −7.28529 −0.781065
\(88\) 0 0
\(89\) 4.26971 0.452589 0.226294 0.974059i \(-0.427339\pi\)
0.226294 + 0.974059i \(0.427339\pi\)
\(90\) 0 0
\(91\) −8.46701 −0.887584
\(92\) 0 0
\(93\) 2.26207 0.234565
\(94\) 0 0
\(95\) −5.74498 −0.589423
\(96\) 0 0
\(97\) −16.8392 −1.70976 −0.854882 0.518822i \(-0.826371\pi\)
−0.854882 + 0.518822i \(0.826371\pi\)
\(98\) 0 0
\(99\) 2.19686 0.220793
\(100\) 0 0
\(101\) −1.81063 −0.180164 −0.0900821 0.995934i \(-0.528713\pi\)
−0.0900821 + 0.995934i \(0.528713\pi\)
\(102\) 0 0
\(103\) −13.0031 −1.28123 −0.640617 0.767860i \(-0.721321\pi\)
−0.640617 + 0.767860i \(0.721321\pi\)
\(104\) 0 0
\(105\) −5.63655 −0.550071
\(106\) 0 0
\(107\) −4.69470 −0.453854 −0.226927 0.973912i \(-0.572868\pi\)
−0.226927 + 0.973912i \(0.572868\pi\)
\(108\) 0 0
\(109\) 4.03514 0.386496 0.193248 0.981150i \(-0.438098\pi\)
0.193248 + 0.981150i \(0.438098\pi\)
\(110\) 0 0
\(111\) −20.4973 −1.94552
\(112\) 0 0
\(113\) −10.0153 −0.942160 −0.471080 0.882091i \(-0.656136\pi\)
−0.471080 + 0.882091i \(0.656136\pi\)
\(114\) 0 0
\(115\) 13.1882 1.22980
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 1.71470 0.157186
\(120\) 0 0
\(121\) 25.6241 2.32946
\(122\) 0 0
\(123\) −4.67989 −0.421972
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −4.19901 −0.372602 −0.186301 0.982493i \(-0.559650\pi\)
−0.186301 + 0.982493i \(0.559650\pi\)
\(128\) 0 0
\(129\) −20.2071 −1.77914
\(130\) 0 0
\(131\) −10.7874 −0.942504 −0.471252 0.881999i \(-0.656198\pi\)
−0.471252 + 0.881999i \(0.656198\pi\)
\(132\) 0 0
\(133\) −4.41447 −0.382783
\(134\) 0 0
\(135\) 9.67169 0.832407
\(136\) 0 0
\(137\) 12.0711 1.03130 0.515651 0.856799i \(-0.327550\pi\)
0.515651 + 0.856799i \(0.327550\pi\)
\(138\) 0 0
\(139\) −13.4237 −1.13858 −0.569291 0.822136i \(-0.692782\pi\)
−0.569291 + 0.822136i \(0.692782\pi\)
\(140\) 0 0
\(141\) −14.9394 −1.25813
\(142\) 0 0
\(143\) −33.3422 −2.78821
\(144\) 0 0
\(145\) 7.94534 0.659825
\(146\) 0 0
\(147\) 8.50581 0.701547
\(148\) 0 0
\(149\) −12.7874 −1.04759 −0.523794 0.851845i \(-0.675484\pi\)
−0.523794 + 0.851845i \(0.675484\pi\)
\(150\) 0 0
\(151\) −21.2035 −1.72551 −0.862757 0.505619i \(-0.831264\pi\)
−0.862757 + 0.505619i \(0.831264\pi\)
\(152\) 0 0
\(153\) 0.405031 0.0327448
\(154\) 0 0
\(155\) −2.46701 −0.198155
\(156\) 0 0
\(157\) 7.71917 0.616057 0.308028 0.951377i \(-0.400331\pi\)
0.308028 + 0.951377i \(0.400331\pi\)
\(158\) 0 0
\(159\) 12.1497 0.963534
\(160\) 0 0
\(161\) 10.1338 0.798659
\(162\) 0 0
\(163\) 2.25901 0.176939 0.0884696 0.996079i \(-0.471802\pi\)
0.0884696 + 0.996079i \(0.471802\pi\)
\(164\) 0 0
\(165\) −22.1961 −1.72797
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 17.3544 1.33495
\(170\) 0 0
\(171\) −1.04274 −0.0797407
\(172\) 0 0
\(173\) −5.80099 −0.441041 −0.220520 0.975382i \(-0.570776\pi\)
−0.220520 + 0.975382i \(0.570776\pi\)
\(174\) 0 0
\(175\) −1.53681 −0.116172
\(176\) 0 0
\(177\) 22.6280 1.70083
\(178\) 0 0
\(179\) 2.31568 0.173082 0.0865412 0.996248i \(-0.472419\pi\)
0.0865412 + 0.996248i \(0.472419\pi\)
\(180\) 0 0
\(181\) 12.3820 0.920345 0.460172 0.887830i \(-0.347788\pi\)
0.460172 + 0.887830i \(0.347788\pi\)
\(182\) 0 0
\(183\) 1.68432 0.124508
\(184\) 0 0
\(185\) 22.3544 1.64353
\(186\) 0 0
\(187\) 6.75231 0.493778
\(188\) 0 0
\(189\) 7.43177 0.540582
\(190\) 0 0
\(191\) 16.1682 1.16989 0.584944 0.811073i \(-0.301116\pi\)
0.584944 + 0.811073i \(0.301116\pi\)
\(192\) 0 0
\(193\) −20.4552 −1.47239 −0.736197 0.676767i \(-0.763380\pi\)
−0.736197 + 0.676767i \(0.763380\pi\)
\(194\) 0 0
\(195\) 20.2071 1.44706
\(196\) 0 0
\(197\) 1.98415 0.141365 0.0706824 0.997499i \(-0.477482\pi\)
0.0706824 + 0.997499i \(0.477482\pi\)
\(198\) 0 0
\(199\) −2.07107 −0.146814 −0.0734071 0.997302i \(-0.523387\pi\)
−0.0734071 + 0.997302i \(0.523387\pi\)
\(200\) 0 0
\(201\) −23.7976 −1.67855
\(202\) 0 0
\(203\) 6.10523 0.428503
\(204\) 0 0
\(205\) 5.10389 0.356471
\(206\) 0 0
\(207\) 2.39372 0.166375
\(208\) 0 0
\(209\) −17.3837 −1.20245
\(210\) 0 0
\(211\) −10.3374 −0.711656 −0.355828 0.934551i \(-0.615801\pi\)
−0.355828 + 0.934551i \(0.615801\pi\)
\(212\) 0 0
\(213\) 20.8647 1.42963
\(214\) 0 0
\(215\) 22.0379 1.50297
\(216\) 0 0
\(217\) −1.89566 −0.128686
\(218\) 0 0
\(219\) −10.0464 −0.678876
\(220\) 0 0
\(221\) −6.14723 −0.413508
\(222\) 0 0
\(223\) 2.25043 0.150700 0.0753499 0.997157i \(-0.475993\pi\)
0.0753499 + 0.997157i \(0.475993\pi\)
\(224\) 0 0
\(225\) −0.363011 −0.0242007
\(226\) 0 0
\(227\) 20.4704 1.35867 0.679336 0.733828i \(-0.262268\pi\)
0.679336 + 0.733828i \(0.262268\pi\)
\(228\) 0 0
\(229\) −11.3081 −0.747259 −0.373629 0.927578i \(-0.621887\pi\)
−0.373629 + 0.927578i \(0.621887\pi\)
\(230\) 0 0
\(231\) −17.0556 −1.12218
\(232\) 0 0
\(233\) −13.1712 −0.862875 −0.431437 0.902143i \(-0.641993\pi\)
−0.431437 + 0.902143i \(0.641993\pi\)
\(234\) 0 0
\(235\) 16.2929 1.06283
\(236\) 0 0
\(237\) −18.6057 −1.20857
\(238\) 0 0
\(239\) −20.9212 −1.35328 −0.676641 0.736313i \(-0.736565\pi\)
−0.676641 + 0.736313i \(0.736565\pi\)
\(240\) 0 0
\(241\) 15.1265 0.974385 0.487192 0.873295i \(-0.338021\pi\)
0.487192 + 0.873295i \(0.338021\pi\)
\(242\) 0 0
\(243\) 3.75259 0.240729
\(244\) 0 0
\(245\) −9.27644 −0.592650
\(246\) 0 0
\(247\) 15.8259 1.00698
\(248\) 0 0
\(249\) −26.6430 −1.68843
\(250\) 0 0
\(251\) 13.4275 0.847536 0.423768 0.905771i \(-0.360707\pi\)
0.423768 + 0.905771i \(0.360707\pi\)
\(252\) 0 0
\(253\) 39.9060 2.50887
\(254\) 0 0
\(255\) −4.09226 −0.256267
\(256\) 0 0
\(257\) −18.8519 −1.17595 −0.587974 0.808880i \(-0.700074\pi\)
−0.587974 + 0.808880i \(0.700074\pi\)
\(258\) 0 0
\(259\) 17.1772 1.06734
\(260\) 0 0
\(261\) 1.44212 0.0892651
\(262\) 0 0
\(263\) −13.9726 −0.861585 −0.430792 0.902451i \(-0.641766\pi\)
−0.430792 + 0.902451i \(0.641766\pi\)
\(264\) 0 0
\(265\) −13.2505 −0.813970
\(266\) 0 0
\(267\) −7.83002 −0.479190
\(268\) 0 0
\(269\) 10.4209 0.635372 0.317686 0.948196i \(-0.397094\pi\)
0.317686 + 0.948196i \(0.397094\pi\)
\(270\) 0 0
\(271\) −5.01896 −0.304880 −0.152440 0.988313i \(-0.548713\pi\)
−0.152440 + 0.988313i \(0.548713\pi\)
\(272\) 0 0
\(273\) 15.5272 0.939751
\(274\) 0 0
\(275\) −6.05178 −0.364936
\(276\) 0 0
\(277\) −7.14603 −0.429364 −0.214682 0.976684i \(-0.568871\pi\)
−0.214682 + 0.976684i \(0.568871\pi\)
\(278\) 0 0
\(279\) −0.447776 −0.0268076
\(280\) 0 0
\(281\) −14.1005 −0.841165 −0.420583 0.907254i \(-0.638174\pi\)
−0.420583 + 0.907254i \(0.638174\pi\)
\(282\) 0 0
\(283\) 21.7546 1.29318 0.646589 0.762838i \(-0.276195\pi\)
0.646589 + 0.762838i \(0.276195\pi\)
\(284\) 0 0
\(285\) 10.5354 0.624066
\(286\) 0 0
\(287\) 3.92185 0.231500
\(288\) 0 0
\(289\) −15.7551 −0.926770
\(290\) 0 0
\(291\) 30.8806 1.81026
\(292\) 0 0
\(293\) 25.7571 1.50474 0.752372 0.658738i \(-0.228909\pi\)
0.752372 + 0.658738i \(0.228909\pi\)
\(294\) 0 0
\(295\) −24.6781 −1.43682
\(296\) 0 0
\(297\) 29.2655 1.69816
\(298\) 0 0
\(299\) −36.3300 −2.10102
\(300\) 0 0
\(301\) 16.9340 0.976061
\(302\) 0 0
\(303\) 3.32042 0.190753
\(304\) 0 0
\(305\) −1.83692 −0.105182
\(306\) 0 0
\(307\) −15.5551 −0.887774 −0.443887 0.896083i \(-0.646401\pi\)
−0.443887 + 0.896083i \(0.646401\pi\)
\(308\) 0 0
\(309\) 23.8458 1.35654
\(310\) 0 0
\(311\) 9.27245 0.525793 0.262896 0.964824i \(-0.415322\pi\)
0.262896 + 0.964824i \(0.415322\pi\)
\(312\) 0 0
\(313\) −27.3312 −1.54485 −0.772425 0.635106i \(-0.780956\pi\)
−0.772425 + 0.635106i \(0.780956\pi\)
\(314\) 0 0
\(315\) 1.11576 0.0628657
\(316\) 0 0
\(317\) 21.1414 1.18742 0.593711 0.804678i \(-0.297662\pi\)
0.593711 + 0.804678i \(0.297662\pi\)
\(318\) 0 0
\(319\) 24.0417 1.34608
\(320\) 0 0
\(321\) 8.60939 0.480529
\(322\) 0 0
\(323\) −3.20500 −0.178331
\(324\) 0 0
\(325\) 5.50948 0.305611
\(326\) 0 0
\(327\) −7.39984 −0.409212
\(328\) 0 0
\(329\) 12.5196 0.690226
\(330\) 0 0
\(331\) 29.1730 1.60349 0.801745 0.597666i \(-0.203905\pi\)
0.801745 + 0.597666i \(0.203905\pi\)
\(332\) 0 0
\(333\) 4.05744 0.222346
\(334\) 0 0
\(335\) 25.9536 1.41800
\(336\) 0 0
\(337\) −22.1811 −1.20828 −0.604139 0.796879i \(-0.706483\pi\)
−0.604139 + 0.796879i \(0.706483\pi\)
\(338\) 0 0
\(339\) 18.3666 0.997535
\(340\) 0 0
\(341\) −7.46491 −0.404248
\(342\) 0 0
\(343\) −17.8857 −0.965738
\(344\) 0 0
\(345\) −24.1851 −1.30208
\(346\) 0 0
\(347\) −26.7862 −1.43796 −0.718980 0.695030i \(-0.755391\pi\)
−0.718980 + 0.695030i \(0.755391\pi\)
\(348\) 0 0
\(349\) 6.53266 0.349685 0.174843 0.984596i \(-0.444058\pi\)
0.174843 + 0.984596i \(0.444058\pi\)
\(350\) 0 0
\(351\) −26.6430 −1.42210
\(352\) 0 0
\(353\) −0.267135 −0.0142182 −0.00710908 0.999975i \(-0.502263\pi\)
−0.00710908 + 0.999975i \(0.502263\pi\)
\(354\) 0 0
\(355\) −22.7551 −1.20771
\(356\) 0 0
\(357\) −3.14451 −0.166425
\(358\) 0 0
\(359\) −18.1019 −0.955382 −0.477691 0.878528i \(-0.658526\pi\)
−0.477691 + 0.878528i \(0.658526\pi\)
\(360\) 0 0
\(361\) −10.7488 −0.565726
\(362\) 0 0
\(363\) −46.9908 −2.46637
\(364\) 0 0
\(365\) 10.9567 0.573498
\(366\) 0 0
\(367\) −4.55517 −0.237778 −0.118889 0.992908i \(-0.537933\pi\)
−0.118889 + 0.992908i \(0.537933\pi\)
\(368\) 0 0
\(369\) 0.926384 0.0482256
\(370\) 0 0
\(371\) −10.1817 −0.528608
\(372\) 0 0
\(373\) 9.89643 0.512418 0.256209 0.966621i \(-0.417526\pi\)
0.256209 + 0.966621i \(0.417526\pi\)
\(374\) 0 0
\(375\) 22.0062 1.13640
\(376\) 0 0
\(377\) −21.8874 −1.12726
\(378\) 0 0
\(379\) 25.1072 1.28967 0.644836 0.764321i \(-0.276926\pi\)
0.644836 + 0.764321i \(0.276926\pi\)
\(380\) 0 0
\(381\) 7.70037 0.394502
\(382\) 0 0
\(383\) −20.1530 −1.02977 −0.514884 0.857260i \(-0.672165\pi\)
−0.514884 + 0.857260i \(0.672165\pi\)
\(384\) 0 0
\(385\) 18.6009 0.947987
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) 11.2040 0.568067 0.284033 0.958814i \(-0.408327\pi\)
0.284033 + 0.958814i \(0.408327\pi\)
\(390\) 0 0
\(391\) 7.35739 0.372079
\(392\) 0 0
\(393\) 19.7826 0.997899
\(394\) 0 0
\(395\) 20.2914 1.02097
\(396\) 0 0
\(397\) 12.5098 0.627847 0.313923 0.949448i \(-0.398357\pi\)
0.313923 + 0.949448i \(0.398357\pi\)
\(398\) 0 0
\(399\) 8.09547 0.405281
\(400\) 0 0
\(401\) 5.26912 0.263127 0.131564 0.991308i \(-0.458000\pi\)
0.131564 + 0.991308i \(0.458000\pi\)
\(402\) 0 0
\(403\) 6.79598 0.338532
\(404\) 0 0
\(405\) −19.9145 −0.989560
\(406\) 0 0
\(407\) 67.6419 3.35288
\(408\) 0 0
\(409\) 8.34625 0.412695 0.206348 0.978479i \(-0.433842\pi\)
0.206348 + 0.978479i \(0.433842\pi\)
\(410\) 0 0
\(411\) −22.1365 −1.09192
\(412\) 0 0
\(413\) −18.9628 −0.933097
\(414\) 0 0
\(415\) 29.0569 1.42635
\(416\) 0 0
\(417\) 24.6170 1.20550
\(418\) 0 0
\(419\) −12.8854 −0.629491 −0.314745 0.949176i \(-0.601919\pi\)
−0.314745 + 0.949176i \(0.601919\pi\)
\(420\) 0 0
\(421\) 3.27748 0.159735 0.0798674 0.996805i \(-0.474550\pi\)
0.0798674 + 0.996805i \(0.474550\pi\)
\(422\) 0 0
\(423\) 2.95726 0.143787
\(424\) 0 0
\(425\) −1.11576 −0.0541221
\(426\) 0 0
\(427\) −1.41149 −0.0683070
\(428\) 0 0
\(429\) 61.1446 2.95209
\(430\) 0 0
\(431\) −6.89917 −0.332321 −0.166161 0.986099i \(-0.553137\pi\)
−0.166161 + 0.986099i \(0.553137\pi\)
\(432\) 0 0
\(433\) −9.36148 −0.449884 −0.224942 0.974372i \(-0.572219\pi\)
−0.224942 + 0.974372i \(0.572219\pi\)
\(434\) 0 0
\(435\) −14.5706 −0.698606
\(436\) 0 0
\(437\) −18.9414 −0.906092
\(438\) 0 0
\(439\) 20.0639 0.957597 0.478799 0.877925i \(-0.341072\pi\)
0.478799 + 0.877925i \(0.341072\pi\)
\(440\) 0 0
\(441\) −1.68372 −0.0801773
\(442\) 0 0
\(443\) −39.1949 −1.86221 −0.931104 0.364754i \(-0.881153\pi\)
−0.931104 + 0.364754i \(0.881153\pi\)
\(444\) 0 0
\(445\) 8.53943 0.404808
\(446\) 0 0
\(447\) 23.4503 1.10916
\(448\) 0 0
\(449\) −30.4199 −1.43560 −0.717802 0.696248i \(-0.754852\pi\)
−0.717802 + 0.696248i \(0.754852\pi\)
\(450\) 0 0
\(451\) 15.4438 0.727222
\(452\) 0 0
\(453\) 38.8840 1.82693
\(454\) 0 0
\(455\) −16.9340 −0.793879
\(456\) 0 0
\(457\) −36.0801 −1.68775 −0.843877 0.536537i \(-0.819732\pi\)
−0.843877 + 0.536537i \(0.819732\pi\)
\(458\) 0 0
\(459\) 5.39562 0.251846
\(460\) 0 0
\(461\) −17.5701 −0.818323 −0.409162 0.912462i \(-0.634179\pi\)
−0.409162 + 0.912462i \(0.634179\pi\)
\(462\) 0 0
\(463\) 40.3895 1.87706 0.938530 0.345199i \(-0.112188\pi\)
0.938530 + 0.345199i \(0.112188\pi\)
\(464\) 0 0
\(465\) 4.52413 0.209802
\(466\) 0 0
\(467\) 18.3549 0.849361 0.424681 0.905343i \(-0.360386\pi\)
0.424681 + 0.905343i \(0.360386\pi\)
\(468\) 0 0
\(469\) 19.9429 0.920877
\(470\) 0 0
\(471\) −14.1558 −0.652265
\(472\) 0 0
\(473\) 66.6844 3.06615
\(474\) 0 0
\(475\) 2.87249 0.131799
\(476\) 0 0
\(477\) −2.40503 −0.110119
\(478\) 0 0
\(479\) 9.30912 0.425344 0.212672 0.977124i \(-0.431783\pi\)
0.212672 + 0.977124i \(0.431783\pi\)
\(480\) 0 0
\(481\) −61.5805 −2.80783
\(482\) 0 0
\(483\) −18.5840 −0.845600
\(484\) 0 0
\(485\) −33.6785 −1.52926
\(486\) 0 0
\(487\) 30.5893 1.38613 0.693067 0.720873i \(-0.256259\pi\)
0.693067 + 0.720873i \(0.256259\pi\)
\(488\) 0 0
\(489\) −4.14269 −0.187339
\(490\) 0 0
\(491\) 5.65120 0.255035 0.127517 0.991836i \(-0.459299\pi\)
0.127517 + 0.991836i \(0.459299\pi\)
\(492\) 0 0
\(493\) 4.43253 0.199631
\(494\) 0 0
\(495\) 4.39372 0.197483
\(496\) 0 0
\(497\) −17.4851 −0.784314
\(498\) 0 0
\(499\) −12.8497 −0.575234 −0.287617 0.957746i \(-0.592863\pi\)
−0.287617 + 0.957746i \(0.592863\pi\)
\(500\) 0 0
\(501\) 1.83385 0.0819304
\(502\) 0 0
\(503\) 22.0032 0.981072 0.490536 0.871421i \(-0.336801\pi\)
0.490536 + 0.871421i \(0.336801\pi\)
\(504\) 0 0
\(505\) −3.62126 −0.161144
\(506\) 0 0
\(507\) −31.8253 −1.41341
\(508\) 0 0
\(509\) −42.0580 −1.86419 −0.932094 0.362216i \(-0.882020\pi\)
−0.932094 + 0.362216i \(0.882020\pi\)
\(510\) 0 0
\(511\) 8.41914 0.372441
\(512\) 0 0
\(513\) −13.8909 −0.613299
\(514\) 0 0
\(515\) −26.0062 −1.14597
\(516\) 0 0
\(517\) 49.3007 2.16824
\(518\) 0 0
\(519\) 10.6381 0.466963
\(520\) 0 0
\(521\) −2.84758 −0.124755 −0.0623774 0.998053i \(-0.519868\pi\)
−0.0623774 + 0.998053i \(0.519868\pi\)
\(522\) 0 0
\(523\) 28.3973 1.24173 0.620864 0.783919i \(-0.286782\pi\)
0.620864 + 0.783919i \(0.286782\pi\)
\(524\) 0 0
\(525\) 2.81828 0.123000
\(526\) 0 0
\(527\) −1.37629 −0.0599522
\(528\) 0 0
\(529\) 20.4820 0.890521
\(530\) 0 0
\(531\) −4.47921 −0.194381
\(532\) 0 0
\(533\) −14.0599 −0.609002
\(534\) 0 0
\(535\) −9.38941 −0.405939
\(536\) 0 0
\(537\) −4.24662 −0.183255
\(538\) 0 0
\(539\) −28.0695 −1.20904
\(540\) 0 0
\(541\) −26.9524 −1.15878 −0.579388 0.815052i \(-0.696708\pi\)
−0.579388 + 0.815052i \(0.696708\pi\)
\(542\) 0 0
\(543\) −22.7067 −0.974437
\(544\) 0 0
\(545\) 8.07028 0.345693
\(546\) 0 0
\(547\) −9.65949 −0.413010 −0.206505 0.978446i \(-0.566209\pi\)
−0.206505 + 0.978446i \(0.566209\pi\)
\(548\) 0 0
\(549\) −0.333410 −0.0142296
\(550\) 0 0
\(551\) −11.4115 −0.486145
\(552\) 0 0
\(553\) 15.5920 0.663039
\(554\) 0 0
\(555\) −40.9946 −1.74012
\(556\) 0 0
\(557\) 26.3948 1.11839 0.559193 0.829038i \(-0.311111\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(558\) 0 0
\(559\) −60.7087 −2.56771
\(560\) 0 0
\(561\) −12.3827 −0.522799
\(562\) 0 0
\(563\) 3.99989 0.168575 0.0842877 0.996441i \(-0.473139\pi\)
0.0842877 + 0.996441i \(0.473139\pi\)
\(564\) 0 0
\(565\) −20.0306 −0.842693
\(566\) 0 0
\(567\) −15.3024 −0.642640
\(568\) 0 0
\(569\) 33.9714 1.42416 0.712078 0.702101i \(-0.247754\pi\)
0.712078 + 0.702101i \(0.247754\pi\)
\(570\) 0 0
\(571\) −4.18108 −0.174973 −0.0874863 0.996166i \(-0.527883\pi\)
−0.0874863 + 0.996166i \(0.527883\pi\)
\(572\) 0 0
\(573\) −29.6500 −1.23865
\(574\) 0 0
\(575\) −6.59409 −0.274992
\(576\) 0 0
\(577\) 4.85476 0.202106 0.101053 0.994881i \(-0.467779\pi\)
0.101053 + 0.994881i \(0.467779\pi\)
\(578\) 0 0
\(579\) 37.5117 1.55893
\(580\) 0 0
\(581\) 22.3274 0.926297
\(582\) 0 0
\(583\) −40.0945 −1.66054
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) −41.1977 −1.70041 −0.850206 0.526450i \(-0.823523\pi\)
−0.850206 + 0.526450i \(0.823523\pi\)
\(588\) 0 0
\(589\) 3.54323 0.145996
\(590\) 0 0
\(591\) −3.63863 −0.149673
\(592\) 0 0
\(593\) −37.4546 −1.53808 −0.769038 0.639203i \(-0.779264\pi\)
−0.769038 + 0.639203i \(0.779264\pi\)
\(594\) 0 0
\(595\) 3.42940 0.140592
\(596\) 0 0
\(597\) 3.79803 0.155443
\(598\) 0 0
\(599\) 24.8799 1.01656 0.508282 0.861191i \(-0.330281\pi\)
0.508282 + 0.861191i \(0.330281\pi\)
\(600\) 0 0
\(601\) −12.4360 −0.507276 −0.253638 0.967299i \(-0.581627\pi\)
−0.253638 + 0.967299i \(0.581627\pi\)
\(602\) 0 0
\(603\) 4.71072 0.191836
\(604\) 0 0
\(605\) 51.2482 2.08353
\(606\) 0 0
\(607\) −30.0532 −1.21982 −0.609911 0.792470i \(-0.708795\pi\)
−0.609911 + 0.792470i \(0.708795\pi\)
\(608\) 0 0
\(609\) −11.1961 −0.453688
\(610\) 0 0
\(611\) −44.8828 −1.81576
\(612\) 0 0
\(613\) 29.2271 1.18047 0.590236 0.807231i \(-0.299035\pi\)
0.590236 + 0.807231i \(0.299035\pi\)
\(614\) 0 0
\(615\) −9.35978 −0.377423
\(616\) 0 0
\(617\) 17.1003 0.688432 0.344216 0.938891i \(-0.388145\pi\)
0.344216 + 0.938891i \(0.388145\pi\)
\(618\) 0 0
\(619\) 1.58589 0.0637422 0.0318711 0.999492i \(-0.489853\pi\)
0.0318711 + 0.999492i \(0.489853\pi\)
\(620\) 0 0
\(621\) 31.8880 1.27962
\(622\) 0 0
\(623\) 6.56173 0.262890
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 31.8791 1.27313
\(628\) 0 0
\(629\) 12.4710 0.497251
\(630\) 0 0
\(631\) −18.7899 −0.748012 −0.374006 0.927426i \(-0.622016\pi\)
−0.374006 + 0.927426i \(0.622016\pi\)
\(632\) 0 0
\(633\) 18.9573 0.753483
\(634\) 0 0
\(635\) −8.39803 −0.333266
\(636\) 0 0
\(637\) 25.5542 1.01249
\(638\) 0 0
\(639\) −4.13017 −0.163387
\(640\) 0 0
\(641\) 12.9249 0.510501 0.255251 0.966875i \(-0.417842\pi\)
0.255251 + 0.966875i \(0.417842\pi\)
\(642\) 0 0
\(643\) 1.09799 0.0433007 0.0216503 0.999766i \(-0.493108\pi\)
0.0216503 + 0.999766i \(0.493108\pi\)
\(644\) 0 0
\(645\) −40.4143 −1.59131
\(646\) 0 0
\(647\) 22.9563 0.902506 0.451253 0.892396i \(-0.350977\pi\)
0.451253 + 0.892396i \(0.350977\pi\)
\(648\) 0 0
\(649\) −74.6734 −2.93119
\(650\) 0 0
\(651\) 3.47636 0.136249
\(652\) 0 0
\(653\) 15.4333 0.603952 0.301976 0.953315i \(-0.402354\pi\)
0.301976 + 0.953315i \(0.402354\pi\)
\(654\) 0 0
\(655\) −21.5749 −0.843001
\(656\) 0 0
\(657\) 1.98869 0.0775863
\(658\) 0 0
\(659\) −8.26179 −0.321834 −0.160917 0.986968i \(-0.551445\pi\)
−0.160917 + 0.986968i \(0.551445\pi\)
\(660\) 0 0
\(661\) −28.7197 −1.11707 −0.558534 0.829482i \(-0.688636\pi\)
−0.558534 + 0.829482i \(0.688636\pi\)
\(662\) 0 0
\(663\) 11.2731 0.437811
\(664\) 0 0
\(665\) −8.82893 −0.342371
\(666\) 0 0
\(667\) 26.1961 1.01432
\(668\) 0 0
\(669\) −4.12695 −0.159557
\(670\) 0 0
\(671\) −5.55831 −0.214576
\(672\) 0 0
\(673\) 35.9750 1.38674 0.693368 0.720584i \(-0.256126\pi\)
0.693368 + 0.720584i \(0.256126\pi\)
\(674\) 0 0
\(675\) −4.83585 −0.186132
\(676\) 0 0
\(677\) 20.6402 0.793268 0.396634 0.917977i \(-0.370178\pi\)
0.396634 + 0.917977i \(0.370178\pi\)
\(678\) 0 0
\(679\) −25.8787 −0.993132
\(680\) 0 0
\(681\) −37.5398 −1.43853
\(682\) 0 0
\(683\) −13.5046 −0.516740 −0.258370 0.966046i \(-0.583185\pi\)
−0.258370 + 0.966046i \(0.583185\pi\)
\(684\) 0 0
\(685\) 24.1421 0.922424
\(686\) 0 0
\(687\) 20.7373 0.791179
\(688\) 0 0
\(689\) 36.5016 1.39060
\(690\) 0 0
\(691\) −5.99213 −0.227951 −0.113976 0.993484i \(-0.536359\pi\)
−0.113976 + 0.993484i \(0.536359\pi\)
\(692\) 0 0
\(693\) 3.37615 0.128250
\(694\) 0 0
\(695\) −26.8473 −1.01838
\(696\) 0 0
\(697\) 2.84735 0.107851
\(698\) 0 0
\(699\) 24.1540 0.913590
\(700\) 0 0
\(701\) −37.3266 −1.40981 −0.704903 0.709304i \(-0.749009\pi\)
−0.704903 + 0.709304i \(0.749009\pi\)
\(702\) 0 0
\(703\) −32.1063 −1.21091
\(704\) 0 0
\(705\) −29.8788 −1.12530
\(706\) 0 0
\(707\) −2.78259 −0.104650
\(708\) 0 0
\(709\) −0.250714 −0.00941576 −0.00470788 0.999989i \(-0.501499\pi\)
−0.00470788 + 0.999989i \(0.501499\pi\)
\(710\) 0 0
\(711\) 3.68300 0.138123
\(712\) 0 0
\(713\) −8.13384 −0.304615
\(714\) 0 0
\(715\) −66.6844 −2.49385
\(716\) 0 0
\(717\) 38.3664 1.43282
\(718\) 0 0
\(719\) −16.7688 −0.625371 −0.312685 0.949857i \(-0.601229\pi\)
−0.312685 + 0.949857i \(0.601229\pi\)
\(720\) 0 0
\(721\) −19.9833 −0.744216
\(722\) 0 0
\(723\) −27.7398 −1.03165
\(724\) 0 0
\(725\) −3.97267 −0.147541
\(726\) 0 0
\(727\) 27.8678 1.03356 0.516780 0.856118i \(-0.327130\pi\)
0.516780 + 0.856118i \(0.327130\pi\)
\(728\) 0 0
\(729\) 22.9901 0.851484
\(730\) 0 0
\(731\) 12.2945 0.454727
\(732\) 0 0
\(733\) 5.46236 0.201757 0.100878 0.994899i \(-0.467835\pi\)
0.100878 + 0.994899i \(0.467835\pi\)
\(734\) 0 0
\(735\) 17.0116 0.627483
\(736\) 0 0
\(737\) 78.5329 2.89280
\(738\) 0 0
\(739\) −36.4399 −1.34046 −0.670232 0.742151i \(-0.733806\pi\)
−0.670232 + 0.742151i \(0.733806\pi\)
\(740\) 0 0
\(741\) −29.0224 −1.06616
\(742\) 0 0
\(743\) −13.8614 −0.508525 −0.254262 0.967135i \(-0.581833\pi\)
−0.254262 + 0.967135i \(0.581833\pi\)
\(744\) 0 0
\(745\) −25.5749 −0.936992
\(746\) 0 0
\(747\) 5.27398 0.192965
\(748\) 0 0
\(749\) −7.21486 −0.263625
\(750\) 0 0
\(751\) 32.9296 1.20162 0.600809 0.799393i \(-0.294845\pi\)
0.600809 + 0.799393i \(0.294845\pi\)
\(752\) 0 0
\(753\) −24.6240 −0.897349
\(754\) 0 0
\(755\) −42.4069 −1.54335
\(756\) 0 0
\(757\) −14.9978 −0.545103 −0.272552 0.962141i \(-0.587868\pi\)
−0.272552 + 0.962141i \(0.587868\pi\)
\(758\) 0 0
\(759\) −73.1816 −2.65633
\(760\) 0 0
\(761\) −13.4658 −0.488133 −0.244067 0.969758i \(-0.578482\pi\)
−0.244067 + 0.969758i \(0.578482\pi\)
\(762\) 0 0
\(763\) 6.20123 0.224500
\(764\) 0 0
\(765\) 0.810062 0.0292879
\(766\) 0 0
\(767\) 67.9818 2.45468
\(768\) 0 0
\(769\) 26.9048 0.970211 0.485106 0.874456i \(-0.338781\pi\)
0.485106 + 0.874456i \(0.338781\pi\)
\(770\) 0 0
\(771\) 34.5716 1.24506
\(772\) 0 0
\(773\) −18.2551 −0.656590 −0.328295 0.944575i \(-0.606474\pi\)
−0.328295 + 0.944575i \(0.606474\pi\)
\(774\) 0 0
\(775\) 1.23351 0.0443088
\(776\) 0 0
\(777\) −31.5004 −1.13007
\(778\) 0 0
\(779\) −7.33044 −0.262640
\(780\) 0 0
\(781\) −68.8544 −2.46381
\(782\) 0 0
\(783\) 19.2112 0.686553
\(784\) 0 0
\(785\) 15.4383 0.551018
\(786\) 0 0
\(787\) 16.4967 0.588042 0.294021 0.955799i \(-0.405006\pi\)
0.294021 + 0.955799i \(0.405006\pi\)
\(788\) 0 0
\(789\) 25.6236 0.912224
\(790\) 0 0
\(791\) −15.3916 −0.547262
\(792\) 0 0
\(793\) 5.06023 0.179694
\(794\) 0 0
\(795\) 24.2994 0.861811
\(796\) 0 0
\(797\) −11.1958 −0.396576 −0.198288 0.980144i \(-0.563538\pi\)
−0.198288 + 0.980144i \(0.563538\pi\)
\(798\) 0 0
\(799\) 9.08947 0.321562
\(800\) 0 0
\(801\) 1.54995 0.0547649
\(802\) 0 0
\(803\) 33.1537 1.16997
\(804\) 0 0
\(805\) 20.2677 0.714342
\(806\) 0 0
\(807\) −19.1103 −0.672716
\(808\) 0 0
\(809\) 49.7828 1.75027 0.875135 0.483878i \(-0.160772\pi\)
0.875135 + 0.483878i \(0.160772\pi\)
\(810\) 0 0
\(811\) 7.70131 0.270430 0.135215 0.990816i \(-0.456828\pi\)
0.135215 + 0.990816i \(0.456828\pi\)
\(812\) 0 0
\(813\) 9.20402 0.322799
\(814\) 0 0
\(815\) 4.51802 0.158259
\(816\) 0 0
\(817\) −31.6519 −1.10736
\(818\) 0 0
\(819\) −3.07362 −0.107401
\(820\) 0 0
\(821\) 13.9531 0.486966 0.243483 0.969905i \(-0.421710\pi\)
0.243483 + 0.969905i \(0.421710\pi\)
\(822\) 0 0
\(823\) 25.0452 0.873022 0.436511 0.899699i \(-0.356214\pi\)
0.436511 + 0.899699i \(0.356214\pi\)
\(824\) 0 0
\(825\) 11.0981 0.386385
\(826\) 0 0
\(827\) 12.6564 0.440106 0.220053 0.975488i \(-0.429377\pi\)
0.220053 + 0.975488i \(0.429377\pi\)
\(828\) 0 0
\(829\) 34.6964 1.20506 0.602529 0.798097i \(-0.294160\pi\)
0.602529 + 0.798097i \(0.294160\pi\)
\(830\) 0 0
\(831\) 13.1048 0.454599
\(832\) 0 0
\(833\) −5.17512 −0.179307
\(834\) 0 0
\(835\) −2.00000 −0.0692129
\(836\) 0 0
\(837\) −5.96504 −0.206182
\(838\) 0 0
\(839\) −23.1380 −0.798813 −0.399406 0.916774i \(-0.630784\pi\)
−0.399406 + 0.916774i \(0.630784\pi\)
\(840\) 0 0
\(841\) −13.2179 −0.455789
\(842\) 0 0
\(843\) 25.8582 0.890604
\(844\) 0 0
\(845\) 34.7087 1.19402
\(846\) 0 0
\(847\) 39.3793 1.35309
\(848\) 0 0
\(849\) −39.8947 −1.36918
\(850\) 0 0
\(851\) 73.7033 2.52652
\(852\) 0 0
\(853\) −29.6187 −1.01412 −0.507062 0.861909i \(-0.669269\pi\)
−0.507062 + 0.861909i \(0.669269\pi\)
\(854\) 0 0
\(855\) −2.08549 −0.0713222
\(856\) 0 0
\(857\) 21.7803 0.744002 0.372001 0.928232i \(-0.378672\pi\)
0.372001 + 0.928232i \(0.378672\pi\)
\(858\) 0 0
\(859\) 14.4256 0.492195 0.246097 0.969245i \(-0.420852\pi\)
0.246097 + 0.969245i \(0.420852\pi\)
\(860\) 0 0
\(861\) −7.19209 −0.245106
\(862\) 0 0
\(863\) 0.0664423 0.00226172 0.00113086 0.999999i \(-0.499640\pi\)
0.00113086 + 0.999999i \(0.499640\pi\)
\(864\) 0 0
\(865\) −11.6020 −0.394479
\(866\) 0 0
\(867\) 28.8925 0.981240
\(868\) 0 0
\(869\) 61.3996 2.08284
\(870\) 0 0
\(871\) −71.4955 −2.42253
\(872\) 0 0
\(873\) −6.11282 −0.206888
\(874\) 0 0
\(875\) −18.4417 −0.623443
\(876\) 0 0
\(877\) 2.91486 0.0984279 0.0492139 0.998788i \(-0.484328\pi\)
0.0492139 + 0.998788i \(0.484328\pi\)
\(878\) 0 0
\(879\) −47.2347 −1.59319
\(880\) 0 0
\(881\) −9.91063 −0.333898 −0.166949 0.985966i \(-0.553391\pi\)
−0.166949 + 0.985966i \(0.553391\pi\)
\(882\) 0 0
\(883\) −12.3842 −0.416760 −0.208380 0.978048i \(-0.566819\pi\)
−0.208380 + 0.978048i \(0.566819\pi\)
\(884\) 0 0
\(885\) 45.2560 1.52126
\(886\) 0 0
\(887\) −30.6773 −1.03004 −0.515021 0.857178i \(-0.672216\pi\)
−0.515021 + 0.857178i \(0.672216\pi\)
\(888\) 0 0
\(889\) −6.45308 −0.216429
\(890\) 0 0
\(891\) −60.2592 −2.01876
\(892\) 0 0
\(893\) −23.4007 −0.783073
\(894\) 0 0
\(895\) 4.63137 0.154810
\(896\) 0 0
\(897\) 66.6238 2.22450
\(898\) 0 0
\(899\) −4.90031 −0.163435
\(900\) 0 0
\(901\) −7.39214 −0.246268
\(902\) 0 0
\(903\) −31.0545 −1.03343
\(904\) 0 0
\(905\) 24.7639 0.823181
\(906\) 0 0
\(907\) 15.7014 0.521355 0.260678 0.965426i \(-0.416054\pi\)
0.260678 + 0.965426i \(0.416054\pi\)
\(908\) 0 0
\(909\) −0.657278 −0.0218005
\(910\) 0 0
\(911\) 10.6183 0.351801 0.175901 0.984408i \(-0.443716\pi\)
0.175901 + 0.984408i \(0.443716\pi\)
\(912\) 0 0
\(913\) 87.9230 2.90983
\(914\) 0 0
\(915\) 3.36863 0.111364
\(916\) 0 0
\(917\) −16.5782 −0.547462
\(918\) 0 0
\(919\) 32.4157 1.06929 0.534647 0.845075i \(-0.320444\pi\)
0.534647 + 0.845075i \(0.320444\pi\)
\(920\) 0 0
\(921\) 28.5257 0.939952
\(922\) 0 0
\(923\) 62.6844 2.06328
\(924\) 0 0
\(925\) −11.1772 −0.367503
\(926\) 0 0
\(927\) −4.72027 −0.155034
\(928\) 0 0
\(929\) 0.0892981 0.00292977 0.00146489 0.999999i \(-0.499534\pi\)
0.00146489 + 0.999999i \(0.499534\pi\)
\(930\) 0 0
\(931\) 13.3232 0.436652
\(932\) 0 0
\(933\) −17.0043 −0.556696
\(934\) 0 0
\(935\) 13.5046 0.441648
\(936\) 0 0
\(937\) −6.80760 −0.222395 −0.111197 0.993798i \(-0.535469\pi\)
−0.111197 + 0.993798i \(0.535469\pi\)
\(938\) 0 0
\(939\) 50.1213 1.63565
\(940\) 0 0
\(941\) −19.2599 −0.627856 −0.313928 0.949447i \(-0.601645\pi\)
−0.313928 + 0.949447i \(0.601645\pi\)
\(942\) 0 0
\(943\) 16.8278 0.547987
\(944\) 0 0
\(945\) 14.8635 0.483511
\(946\) 0 0
\(947\) −37.2451 −1.21030 −0.605151 0.796110i \(-0.706887\pi\)
−0.605151 + 0.796110i \(0.706887\pi\)
\(948\) 0 0
\(949\) −30.1828 −0.979774
\(950\) 0 0
\(951\) −38.7703 −1.25721
\(952\) 0 0
\(953\) 0.648190 0.0209969 0.0104985 0.999945i \(-0.496658\pi\)
0.0104985 + 0.999945i \(0.496658\pi\)
\(954\) 0 0
\(955\) 32.3364 1.04638
\(956\) 0 0
\(957\) −44.0890 −1.42519
\(958\) 0 0
\(959\) 18.5509 0.599040
\(960\) 0 0
\(961\) −29.4785 −0.950918
\(962\) 0 0
\(963\) −1.70423 −0.0549180
\(964\) 0 0
\(965\) −40.9103 −1.31695
\(966\) 0 0
\(967\) 35.5793 1.14415 0.572076 0.820201i \(-0.306138\pi\)
0.572076 + 0.820201i \(0.306138\pi\)
\(968\) 0 0
\(969\) 5.87749 0.188812
\(970\) 0 0
\(971\) 16.3660 0.525210 0.262605 0.964903i \(-0.415418\pi\)
0.262605 + 0.964903i \(0.415418\pi\)
\(972\) 0 0
\(973\) −20.6296 −0.661355
\(974\) 0 0
\(975\) −10.1036 −0.323573
\(976\) 0 0
\(977\) 21.8152 0.697929 0.348964 0.937136i \(-0.386533\pi\)
0.348964 + 0.937136i \(0.386533\pi\)
\(978\) 0 0
\(979\) 25.8394 0.825830
\(980\) 0 0
\(981\) 1.46480 0.0467674
\(982\) 0 0
\(983\) 47.7594 1.52329 0.761644 0.647996i \(-0.224393\pi\)
0.761644 + 0.647996i \(0.224393\pi\)
\(984\) 0 0
\(985\) 3.96830 0.126440
\(986\) 0 0
\(987\) −22.9590 −0.730793
\(988\) 0 0
\(989\) 72.6600 2.31045
\(990\) 0 0
\(991\) 19.3121 0.613470 0.306735 0.951795i \(-0.400763\pi\)
0.306735 + 0.951795i \(0.400763\pi\)
\(992\) 0 0
\(993\) −53.4989 −1.69774
\(994\) 0 0
\(995\) −4.14214 −0.131315
\(996\) 0 0
\(997\) 8.21884 0.260293 0.130147 0.991495i \(-0.458455\pi\)
0.130147 + 0.991495i \(0.458455\pi\)
\(998\) 0 0
\(999\) 54.0511 1.71010
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 668.2.a.b.1.1 5
3.2 odd 2 6012.2.a.d.1.3 5
4.3 odd 2 2672.2.a.j.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
668.2.a.b.1.1 5 1.1 even 1 trivial
2672.2.a.j.1.5 5 4.3 odd 2
6012.2.a.d.1.3 5 3.2 odd 2