Properties

Label 4560.2.d.j.2431.5
Level $4560$
Weight $2$
Character 4560.2431
Analytic conductor $36.412$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 35 x^{10} + 202 x^{8} + 362 x^{6} + 245 x^{4} + 63 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.5
Root \(1.19434i\) of defining polynomial
Character \(\chi\) \(=\) 4560.2431
Dual form 4560.2.d.j.2431.8

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -1.25182i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -1.25182i q^{7} +1.00000 q^{9} -0.630063i q^{11} -3.96496i q^{13} -1.00000 q^{15} -6.90373 q^{17} +(2.11425 + 3.81182i) q^{19} +1.25182i q^{21} +6.49272i q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.83404i q^{29} +7.48802 q^{31} +0.630063i q^{33} -1.25182i q^{35} +9.57487i q^{37} +3.96496i q^{39} -7.74454i q^{41} +7.37089i q^{43} +1.00000 q^{45} +3.33490i q^{47} +5.43294 q^{49} +6.90373 q^{51} -9.95682i q^{53} -0.630063i q^{55} +(-2.11425 - 3.81182i) q^{57} +9.92095 q^{59} -12.4329 q^{61} -1.25182i q^{63} -3.96496i q^{65} +7.82681 q^{67} -6.49272i q^{69} -1.41571 q^{71} +7.64421 q^{73} -1.00000 q^{75} -0.788728 q^{77} +13.8485 q^{79} +1.00000 q^{81} -9.45397i q^{83} -6.90373 q^{85} -2.83404i q^{87} -4.52855i q^{89} -4.96343 q^{91} -7.48802 q^{93} +(2.11425 + 3.81182i) q^{95} +3.90679i q^{97} -0.630063i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} + 12q^{5} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} + 12q^{5} + 12q^{9} - 12q^{15} + 4q^{17} + 12q^{25} - 12q^{27} + 12q^{31} + 12q^{45} - 28q^{49} - 4q^{51} - 52q^{59} - 56q^{61} + 32q^{67} - 8q^{71} + 32q^{73} - 12q^{75} + 24q^{77} + 28q^{79} + 12q^{81} + 4q^{85} - 32q^{91} - 12q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(1711\) \(1921\) \(2737\) \(3041\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.25182i 0.473145i −0.971614 0.236573i \(-0.923976\pi\)
0.971614 0.236573i \(-0.0760241\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.630063i 0.189971i −0.995479 0.0949855i \(-0.969720\pi\)
0.995479 0.0949855i \(-0.0302805\pi\)
\(12\) 0 0
\(13\) 3.96496i 1.09968i −0.835269 0.549841i \(-0.814688\pi\)
0.835269 0.549841i \(-0.185312\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −6.90373 −1.67440 −0.837200 0.546896i \(-0.815809\pi\)
−0.837200 + 0.546896i \(0.815809\pi\)
\(18\) 0 0
\(19\) 2.11425 + 3.81182i 0.485041 + 0.874491i
\(20\) 0 0
\(21\) 1.25182i 0.273170i
\(22\) 0 0
\(23\) 6.49272i 1.35382i 0.736064 + 0.676912i \(0.236682\pi\)
−0.736064 + 0.676912i \(0.763318\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.83404i 0.526268i 0.964759 + 0.263134i \(0.0847561\pi\)
−0.964759 + 0.263134i \(0.915244\pi\)
\(30\) 0 0
\(31\) 7.48802 1.34489 0.672444 0.740148i \(-0.265245\pi\)
0.672444 + 0.740148i \(0.265245\pi\)
\(32\) 0 0
\(33\) 0.630063i 0.109680i
\(34\) 0 0
\(35\) 1.25182i 0.211597i
\(36\) 0 0
\(37\) 9.57487i 1.57410i 0.616890 + 0.787049i \(0.288392\pi\)
−0.616890 + 0.787049i \(0.711608\pi\)
\(38\) 0 0
\(39\) 3.96496i 0.634902i
\(40\) 0 0
\(41\) 7.74454i 1.20949i −0.796418 0.604747i \(-0.793274\pi\)
0.796418 0.604747i \(-0.206726\pi\)
\(42\) 0 0
\(43\) 7.37089i 1.12405i 0.827120 + 0.562025i \(0.189978\pi\)
−0.827120 + 0.562025i \(0.810022\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 3.33490i 0.486445i 0.969971 + 0.243222i \(0.0782045\pi\)
−0.969971 + 0.243222i \(0.921796\pi\)
\(48\) 0 0
\(49\) 5.43294 0.776134
\(50\) 0 0
\(51\) 6.90373 0.966716
\(52\) 0 0
\(53\) 9.95682i 1.36767i −0.729635 0.683837i \(-0.760310\pi\)
0.729635 0.683837i \(-0.239690\pi\)
\(54\) 0 0
\(55\) 0.630063i 0.0849576i
\(56\) 0 0
\(57\) −2.11425 3.81182i −0.280039 0.504888i
\(58\) 0 0
\(59\) 9.92095 1.29160 0.645799 0.763507i \(-0.276524\pi\)
0.645799 + 0.763507i \(0.276524\pi\)
\(60\) 0 0
\(61\) −12.4329 −1.59187 −0.795937 0.605379i \(-0.793021\pi\)
−0.795937 + 0.605379i \(0.793021\pi\)
\(62\) 0 0
\(63\) 1.25182i 0.157715i
\(64\) 0 0
\(65\) 3.96496i 0.491793i
\(66\) 0 0
\(67\) 7.82681 0.956197 0.478098 0.878306i \(-0.341326\pi\)
0.478098 + 0.878306i \(0.341326\pi\)
\(68\) 0 0
\(69\) 6.49272i 0.781631i
\(70\) 0 0
\(71\) −1.41571 −0.168014 −0.0840072 0.996465i \(-0.526772\pi\)
−0.0840072 + 0.996465i \(0.526772\pi\)
\(72\) 0 0
\(73\) 7.64421 0.894687 0.447343 0.894362i \(-0.352370\pi\)
0.447343 + 0.894362i \(0.352370\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −0.788728 −0.0898839
\(78\) 0 0
\(79\) 13.8485 1.55808 0.779040 0.626974i \(-0.215707\pi\)
0.779040 + 0.626974i \(0.215707\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.45397i 1.03771i −0.854863 0.518854i \(-0.826359\pi\)
0.854863 0.518854i \(-0.173641\pi\)
\(84\) 0 0
\(85\) −6.90373 −0.748815
\(86\) 0 0
\(87\) 2.83404i 0.303841i
\(88\) 0 0
\(89\) 4.52855i 0.480026i −0.970770 0.240013i \(-0.922848\pi\)
0.970770 0.240013i \(-0.0771516\pi\)
\(90\) 0 0
\(91\) −4.96343 −0.520309
\(92\) 0 0
\(93\) −7.48802 −0.776471
\(94\) 0 0
\(95\) 2.11425 + 3.81182i 0.216917 + 0.391084i
\(96\) 0 0
\(97\) 3.90679i 0.396674i 0.980134 + 0.198337i \(0.0635541\pi\)
−0.980134 + 0.198337i \(0.936446\pi\)
\(98\) 0 0
\(99\) 0.630063i 0.0633237i
\(100\) 0 0
\(101\) 14.7934 1.47200 0.736001 0.676981i \(-0.236712\pi\)
0.736001 + 0.676981i \(0.236712\pi\)
\(102\) 0 0
\(103\) −7.82681 −0.771198 −0.385599 0.922666i \(-0.626005\pi\)
−0.385599 + 0.922666i \(0.626005\pi\)
\(104\) 0 0
\(105\) 1.25182i 0.122166i
\(106\) 0 0
\(107\) 1.24252 0.120119 0.0600596 0.998195i \(-0.480871\pi\)
0.0600596 + 0.998195i \(0.480871\pi\)
\(108\) 0 0
\(109\) 17.2068i 1.64811i −0.566507 0.824057i \(-0.691706\pi\)
0.566507 0.824057i \(-0.308294\pi\)
\(110\) 0 0
\(111\) 9.57487i 0.908806i
\(112\) 0 0
\(113\) 9.29604i 0.874498i 0.899341 + 0.437249i \(0.144047\pi\)
−0.899341 + 0.437249i \(0.855953\pi\)
\(114\) 0 0
\(115\) 6.49272i 0.605449i
\(116\) 0 0
\(117\) 3.96496i 0.366561i
\(118\) 0 0
\(119\) 8.64226i 0.792235i
\(120\) 0 0
\(121\) 10.6030 0.963911
\(122\) 0 0
\(123\) 7.74454i 0.698302i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −13.4710 −1.19536 −0.597680 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(128\) 0 0
\(129\) 7.37089i 0.648971i
\(130\) 0 0
\(131\) 8.50842i 0.743385i 0.928356 + 0.371692i \(0.121222\pi\)
−0.928356 + 0.371692i \(0.878778\pi\)
\(132\) 0 0
\(133\) 4.77173 2.64667i 0.413761 0.229495i
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −0.608472 −0.0519853 −0.0259926 0.999662i \(-0.508275\pi\)
−0.0259926 + 0.999662i \(0.508275\pi\)
\(138\) 0 0
\(139\) 15.1578i 1.28567i −0.766006 0.642834i \(-0.777759\pi\)
0.766006 0.642834i \(-0.222241\pi\)
\(140\) 0 0
\(141\) 3.33490i 0.280849i
\(142\) 0 0
\(143\) −2.49817 −0.208908
\(144\) 0 0
\(145\) 2.83404i 0.235354i
\(146\) 0 0
\(147\) −5.43294 −0.448101
\(148\) 0 0
\(149\) −15.5603 −1.27475 −0.637376 0.770553i \(-0.719980\pi\)
−0.637376 + 0.770553i \(0.719980\pi\)
\(150\) 0 0
\(151\) −6.31793 −0.514146 −0.257073 0.966392i \(-0.582758\pi\)
−0.257073 + 0.966392i \(0.582758\pi\)
\(152\) 0 0
\(153\) −6.90373 −0.558134
\(154\) 0 0
\(155\) 7.48802 0.601452
\(156\) 0 0
\(157\) 14.9069 1.18970 0.594851 0.803836i \(-0.297211\pi\)
0.594851 + 0.803836i \(0.297211\pi\)
\(158\) 0 0
\(159\) 9.95682i 0.789627i
\(160\) 0 0
\(161\) 8.12774 0.640556
\(162\) 0 0
\(163\) 11.8536i 0.928447i 0.885718 + 0.464223i \(0.153667\pi\)
−0.885718 + 0.464223i \(0.846333\pi\)
\(164\) 0 0
\(165\) 0.630063i 0.0490503i
\(166\) 0 0
\(167\) 13.4742 1.04267 0.521333 0.853353i \(-0.325435\pi\)
0.521333 + 0.853353i \(0.325435\pi\)
\(168\) 0 0
\(169\) −2.72091 −0.209301
\(170\) 0 0
\(171\) 2.11425 + 3.81182i 0.161680 + 0.291497i
\(172\) 0 0
\(173\) 8.69008i 0.660695i −0.943859 0.330347i \(-0.892834\pi\)
0.943859 0.330347i \(-0.107166\pi\)
\(174\) 0 0
\(175\) 1.25182i 0.0946290i
\(176\) 0 0
\(177\) −9.92095 −0.745704
\(178\) 0 0
\(179\) −3.95895 −0.295906 −0.147953 0.988994i \(-0.547268\pi\)
−0.147953 + 0.988994i \(0.547268\pi\)
\(180\) 0 0
\(181\) 11.1264i 0.827016i −0.910500 0.413508i \(-0.864303\pi\)
0.910500 0.413508i \(-0.135697\pi\)
\(182\) 0 0
\(183\) 12.4329 0.919069
\(184\) 0 0
\(185\) 9.57487i 0.703958i
\(186\) 0 0
\(187\) 4.34978i 0.318088i
\(188\) 0 0
\(189\) 1.25182i 0.0910568i
\(190\) 0 0
\(191\) 9.75195i 0.705626i −0.935694 0.352813i \(-0.885225\pi\)
0.935694 0.352813i \(-0.114775\pi\)
\(192\) 0 0
\(193\) 17.5048i 1.26002i 0.776586 + 0.630011i \(0.216950\pi\)
−0.776586 + 0.630011i \(0.783050\pi\)
\(194\) 0 0
\(195\) 3.96496i 0.283937i
\(196\) 0 0
\(197\) 21.6568 1.54298 0.771492 0.636239i \(-0.219511\pi\)
0.771492 + 0.636239i \(0.219511\pi\)
\(198\) 0 0
\(199\) 4.70101i 0.333246i −0.986021 0.166623i \(-0.946714\pi\)
0.986021 0.166623i \(-0.0532863\pi\)
\(200\) 0 0
\(201\) −7.82681 −0.552061
\(202\) 0 0
\(203\) 3.54772 0.249001
\(204\) 0 0
\(205\) 7.74454i 0.540902i
\(206\) 0 0
\(207\) 6.49272i 0.451275i
\(208\) 0 0
\(209\) 2.40168 1.33211i 0.166128 0.0921438i
\(210\) 0 0
\(211\) 28.4183 1.95640 0.978198 0.207674i \(-0.0665893\pi\)
0.978198 + 0.207674i \(0.0665893\pi\)
\(212\) 0 0
\(213\) 1.41571 0.0970032
\(214\) 0 0
\(215\) 7.37089i 0.502691i
\(216\) 0 0
\(217\) 9.37368i 0.636327i
\(218\) 0 0
\(219\) −7.64421 −0.516548
\(220\) 0 0
\(221\) 27.3730i 1.84131i
\(222\) 0 0
\(223\) −0.509638 −0.0341279 −0.0170639 0.999854i \(-0.505432\pi\)
−0.0170639 + 0.999854i \(0.505432\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 16.3056 1.08224 0.541122 0.840944i \(-0.318000\pi\)
0.541122 + 0.840944i \(0.318000\pi\)
\(228\) 0 0
\(229\) 19.6375 1.29768 0.648840 0.760925i \(-0.275254\pi\)
0.648840 + 0.760925i \(0.275254\pi\)
\(230\) 0 0
\(231\) 0.788728 0.0518945
\(232\) 0 0
\(233\) −17.0910 −1.11967 −0.559836 0.828604i \(-0.689136\pi\)
−0.559836 + 0.828604i \(0.689136\pi\)
\(234\) 0 0
\(235\) 3.33490i 0.217545i
\(236\) 0 0
\(237\) −13.8485 −0.899558
\(238\) 0 0
\(239\) 19.0418i 1.23171i 0.787860 + 0.615855i \(0.211189\pi\)
−0.787860 + 0.615855i \(0.788811\pi\)
\(240\) 0 0
\(241\) 3.56019i 0.229332i 0.993404 + 0.114666i \(0.0365798\pi\)
−0.993404 + 0.114666i \(0.963420\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 5.43294 0.347098
\(246\) 0 0
\(247\) 15.1137 8.38290i 0.961662 0.533391i
\(248\) 0 0
\(249\) 9.45397i 0.599121i
\(250\) 0 0
\(251\) 15.5397i 0.980860i −0.871481 0.490430i \(-0.836840\pi\)
0.871481 0.490430i \(-0.163160\pi\)
\(252\) 0 0
\(253\) 4.09082 0.257187
\(254\) 0 0
\(255\) 6.90373 0.432328
\(256\) 0 0
\(257\) 1.13092i 0.0705449i −0.999378 0.0352725i \(-0.988770\pi\)
0.999378 0.0352725i \(-0.0112299\pi\)
\(258\) 0 0
\(259\) 11.9861 0.744777
\(260\) 0 0
\(261\) 2.83404i 0.175423i
\(262\) 0 0
\(263\) 4.33000i 0.266999i 0.991049 + 0.133500i \(0.0426215\pi\)
−0.991049 + 0.133500i \(0.957378\pi\)
\(264\) 0 0
\(265\) 9.95682i 0.611642i
\(266\) 0 0
\(267\) 4.52855i 0.277143i
\(268\) 0 0
\(269\) 2.64379i 0.161195i −0.996747 0.0805974i \(-0.974317\pi\)
0.996747 0.0805974i \(-0.0256828\pi\)
\(270\) 0 0
\(271\) 25.2272i 1.53244i 0.642576 + 0.766222i \(0.277866\pi\)
−0.642576 + 0.766222i \(0.722134\pi\)
\(272\) 0 0
\(273\) 4.96343 0.300401
\(274\) 0 0
\(275\) 0.630063i 0.0379942i
\(276\) 0 0
\(277\) 23.6753 1.42251 0.711256 0.702933i \(-0.248127\pi\)
0.711256 + 0.702933i \(0.248127\pi\)
\(278\) 0 0
\(279\) 7.48802 0.448296
\(280\) 0 0
\(281\) 25.0714i 1.49563i −0.663906 0.747816i \(-0.731103\pi\)
0.663906 0.747816i \(-0.268897\pi\)
\(282\) 0 0
\(283\) 10.4772i 0.622802i −0.950279 0.311401i \(-0.899202\pi\)
0.950279 0.311401i \(-0.100798\pi\)
\(284\) 0 0
\(285\) −2.11425 3.81182i −0.125237 0.225793i
\(286\) 0 0
\(287\) −9.69480 −0.572266
\(288\) 0 0
\(289\) 30.6615 1.80362
\(290\) 0 0
\(291\) 3.90679i 0.229020i
\(292\) 0 0
\(293\) 29.4541i 1.72073i −0.509681 0.860364i \(-0.670237\pi\)
0.509681 0.860364i \(-0.329763\pi\)
\(294\) 0 0
\(295\) 9.92095 0.577620
\(296\) 0 0
\(297\) 0.630063i 0.0365599i
\(298\) 0 0
\(299\) 25.7434 1.48878
\(300\) 0 0
\(301\) 9.22706 0.531839
\(302\) 0 0
\(303\) −14.7934 −0.849861
\(304\) 0 0
\(305\) −12.4329 −0.711908
\(306\) 0 0
\(307\) 30.3610 1.73279 0.866396 0.499358i \(-0.166431\pi\)
0.866396 + 0.499358i \(0.166431\pi\)
\(308\) 0 0
\(309\) 7.82681 0.445252
\(310\) 0 0
\(311\) 19.7798i 1.12161i 0.827948 + 0.560805i \(0.189508\pi\)
−0.827948 + 0.560805i \(0.810492\pi\)
\(312\) 0 0
\(313\) −21.1152 −1.19350 −0.596752 0.802426i \(-0.703542\pi\)
−0.596752 + 0.802426i \(0.703542\pi\)
\(314\) 0 0
\(315\) 1.25182i 0.0705323i
\(316\) 0 0
\(317\) 14.7057i 0.825955i −0.910741 0.412977i \(-0.864489\pi\)
0.910741 0.412977i \(-0.135511\pi\)
\(318\) 0 0
\(319\) 1.78562 0.0999756
\(320\) 0 0
\(321\) −1.24252 −0.0693509
\(322\) 0 0
\(323\) −14.5962 26.3158i −0.812154 1.46425i
\(324\) 0 0
\(325\) 3.96496i 0.219936i
\(326\) 0 0
\(327\) 17.2068i 0.951539i
\(328\) 0 0
\(329\) 4.17471 0.230159
\(330\) 0 0
\(331\) −12.8795 −0.707924 −0.353962 0.935260i \(-0.615166\pi\)
−0.353962 + 0.935260i \(0.615166\pi\)
\(332\) 0 0
\(333\) 9.57487i 0.524700i
\(334\) 0 0
\(335\) 7.82681 0.427624
\(336\) 0 0
\(337\) 6.25842i 0.340918i 0.985365 + 0.170459i \(0.0545250\pi\)
−0.985365 + 0.170459i \(0.945475\pi\)
\(338\) 0 0
\(339\) 9.29604i 0.504891i
\(340\) 0 0
\(341\) 4.71792i 0.255490i
\(342\) 0 0
\(343\) 15.5639i 0.840369i
\(344\) 0 0
\(345\) 6.49272i 0.349556i
\(346\) 0 0
\(347\) 31.2171i 1.67582i −0.545808 0.837910i \(-0.683777\pi\)
0.545808 0.837910i \(-0.316223\pi\)
\(348\) 0 0
\(349\) 11.8911 0.636515 0.318257 0.948004i \(-0.396902\pi\)
0.318257 + 0.948004i \(0.396902\pi\)
\(350\) 0 0
\(351\) 3.96496i 0.211634i
\(352\) 0 0
\(353\) −30.7603 −1.63720 −0.818602 0.574361i \(-0.805251\pi\)
−0.818602 + 0.574361i \(0.805251\pi\)
\(354\) 0 0
\(355\) −1.41571 −0.0751383
\(356\) 0 0
\(357\) 8.64226i 0.457397i
\(358\) 0 0
\(359\) 24.6285i 1.29984i 0.760002 + 0.649920i \(0.225198\pi\)
−0.760002 + 0.649920i \(0.774802\pi\)
\(360\) 0 0
\(361\) −10.0599 + 16.1182i −0.529470 + 0.848329i
\(362\) 0 0
\(363\) −10.6030 −0.556514
\(364\) 0 0
\(365\) 7.64421 0.400116
\(366\) 0 0
\(367\) 9.12358i 0.476247i −0.971235 0.238123i \(-0.923468\pi\)
0.971235 0.238123i \(-0.0765323\pi\)
\(368\) 0 0
\(369\) 7.74454i 0.403165i
\(370\) 0 0
\(371\) −12.4642 −0.647108
\(372\) 0 0
\(373\) 26.8785i 1.39171i 0.718180 + 0.695857i \(0.244975\pi\)
−0.718180 + 0.695857i \(0.755025\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 11.2369 0.578727
\(378\) 0 0
\(379\) −8.96192 −0.460343 −0.230171 0.973150i \(-0.573929\pi\)
−0.230171 + 0.973150i \(0.573929\pi\)
\(380\) 0 0
\(381\) 13.4710 0.690141
\(382\) 0 0
\(383\) 3.84395 0.196416 0.0982082 0.995166i \(-0.468689\pi\)
0.0982082 + 0.995166i \(0.468689\pi\)
\(384\) 0 0
\(385\) −0.788728 −0.0401973
\(386\) 0 0
\(387\) 7.37089i 0.374684i
\(388\) 0 0
\(389\) −3.35048 −0.169876 −0.0849380 0.996386i \(-0.527069\pi\)
−0.0849380 + 0.996386i \(0.527069\pi\)
\(390\) 0 0
\(391\) 44.8240i 2.26685i
\(392\) 0 0
\(393\) 8.50842i 0.429193i
\(394\) 0 0
\(395\) 13.8485 0.696794
\(396\) 0 0
\(397\) 3.42125 0.171708 0.0858538 0.996308i \(-0.472638\pi\)
0.0858538 + 0.996308i \(0.472638\pi\)
\(398\) 0 0
\(399\) −4.77173 + 2.64667i −0.238885 + 0.132499i
\(400\) 0 0
\(401\) 30.4328i 1.51974i 0.650073 + 0.759872i \(0.274738\pi\)
−0.650073 + 0.759872i \(0.725262\pi\)
\(402\) 0 0
\(403\) 29.6897i 1.47895i
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 6.03277 0.299033
\(408\) 0 0
\(409\) 14.1192i 0.698151i 0.937095 + 0.349075i \(0.113504\pi\)
−0.937095 + 0.349075i \(0.886496\pi\)
\(410\) 0 0
\(411\) 0.608472 0.0300137
\(412\) 0 0
\(413\) 12.4193i 0.611113i
\(414\) 0 0
\(415\) 9.45397i 0.464077i
\(416\) 0 0
\(417\) 15.1578i 0.742280i
\(418\) 0 0
\(419\) 19.0252i 0.929440i 0.885458 + 0.464720i \(0.153845\pi\)
−0.885458 + 0.464720i \(0.846155\pi\)
\(420\) 0 0
\(421\) 11.9451i 0.582168i 0.956698 + 0.291084i \(0.0940159\pi\)
−0.956698 + 0.291084i \(0.905984\pi\)
\(422\) 0 0
\(423\) 3.33490i 0.162148i
\(424\) 0 0
\(425\) −6.90373 −0.334880
\(426\) 0 0
\(427\) 15.5639i 0.753188i
\(428\) 0 0
\(429\) 2.49817 0.120613
\(430\) 0 0
\(431\) −2.95197 −0.142191 −0.0710957 0.997470i \(-0.522650\pi\)
−0.0710957 + 0.997470i \(0.522650\pi\)
\(432\) 0 0
\(433\) 29.6192i 1.42341i 0.702478 + 0.711705i \(0.252077\pi\)
−0.702478 + 0.711705i \(0.747923\pi\)
\(434\) 0 0
\(435\) 2.83404i 0.135882i
\(436\) 0 0
\(437\) −24.7491 + 13.7272i −1.18391 + 0.656661i
\(438\) 0 0
\(439\) 37.6919 1.79894 0.899469 0.436985i \(-0.143954\pi\)
0.899469 + 0.436985i \(0.143954\pi\)
\(440\) 0 0
\(441\) 5.43294 0.258711
\(442\) 0 0
\(443\) 25.8140i 1.22646i 0.789904 + 0.613231i \(0.210130\pi\)
−0.789904 + 0.613231i \(0.789870\pi\)
\(444\) 0 0
\(445\) 4.52855i 0.214674i
\(446\) 0 0
\(447\) 15.5603 0.735978
\(448\) 0 0
\(449\) 11.2815i 0.532408i −0.963917 0.266204i \(-0.914231\pi\)
0.963917 0.266204i \(-0.0857694\pi\)
\(450\) 0 0
\(451\) −4.87955 −0.229769
\(452\) 0 0
\(453\) 6.31793 0.296842
\(454\) 0 0
\(455\) −4.96343 −0.232689
\(456\) 0 0
\(457\) 24.7642 1.15842 0.579209 0.815179i \(-0.303361\pi\)
0.579209 + 0.815179i \(0.303361\pi\)
\(458\) 0 0
\(459\) 6.90373 0.322239
\(460\) 0 0
\(461\) −17.7142 −0.825031 −0.412515 0.910951i \(-0.635350\pi\)
−0.412515 + 0.910951i \(0.635350\pi\)
\(462\) 0 0
\(463\) 0.191937i 0.00892008i 0.999990 + 0.00446004i \(0.00141968\pi\)
−0.999990 + 0.00446004i \(0.998580\pi\)
\(464\) 0 0
\(465\) −7.48802 −0.347248
\(466\) 0 0
\(467\) 31.4589i 1.45574i 0.685713 + 0.727872i \(0.259490\pi\)
−0.685713 + 0.727872i \(0.740510\pi\)
\(468\) 0 0
\(469\) 9.79779i 0.452420i
\(470\) 0 0
\(471\) −14.9069 −0.686875
\(472\) 0 0
\(473\) 4.64412 0.213537
\(474\) 0 0
\(475\) 2.11425 + 3.81182i 0.0970083 + 0.174898i
\(476\) 0 0
\(477\) 9.95682i 0.455891i
\(478\) 0 0
\(479\) 23.7854i 1.08678i −0.839480 0.543391i \(-0.817140\pi\)
0.839480 0.543391i \(-0.182860\pi\)
\(480\) 0 0
\(481\) 37.9640 1.73101
\(482\) 0 0
\(483\) −8.12774 −0.369825
\(484\) 0 0
\(485\) 3.90679i 0.177398i
\(486\) 0 0
\(487\) −4.55349 −0.206338 −0.103169 0.994664i \(-0.532898\pi\)
−0.103169 + 0.994664i \(0.532898\pi\)
\(488\) 0 0
\(489\) 11.8536i 0.536039i
\(490\) 0 0
\(491\) 36.5290i 1.64853i 0.566204 + 0.824265i \(0.308412\pi\)
−0.566204 + 0.824265i \(0.691588\pi\)
\(492\) 0 0
\(493\) 19.5654i 0.881183i
\(494\) 0 0
\(495\) 0.630063i 0.0283192i
\(496\) 0 0
\(497\) 1.77223i 0.0794952i
\(498\) 0 0
\(499\) 0.0346536i 0.00155131i −1.00000 0.000775653i \(-0.999753\pi\)
1.00000 0.000775653i \(-0.000246898\pi\)
\(500\) 0 0
\(501\) −13.4742 −0.601983
\(502\) 0 0
\(503\) 14.6538i 0.653383i 0.945131 + 0.326691i \(0.105934\pi\)
−0.945131 + 0.326691i \(0.894066\pi\)
\(504\) 0 0
\(505\) 14.7934 0.658299
\(506\) 0 0
\(507\) 2.72091 0.120840
\(508\) 0 0
\(509\) 34.1830i 1.51513i −0.652758 0.757567i \(-0.726388\pi\)
0.652758 0.757567i \(-0.273612\pi\)
\(510\) 0 0
\(511\) 9.56921i 0.423317i
\(512\) 0 0
\(513\) −2.11425 3.81182i −0.0933463 0.168296i
\(514\) 0 0
\(515\) −7.82681 −0.344890
\(516\) 0 0
\(517\) 2.10119 0.0924104
\(518\) 0 0
\(519\) 8.69008i 0.381452i
\(520\) 0 0
\(521\) 21.6319i 0.947712i 0.880602 + 0.473856i \(0.157138\pi\)
−0.880602 + 0.473856i \(0.842862\pi\)
\(522\) 0 0
\(523\) 26.3171 1.15077 0.575383 0.817884i \(-0.304853\pi\)
0.575383 + 0.817884i \(0.304853\pi\)
\(524\) 0 0
\(525\) 1.25182i 0.0546341i
\(526\) 0 0
\(527\) −51.6953 −2.25188
\(528\) 0 0
\(529\) −19.1554 −0.832842
\(530\) 0 0
\(531\) 9.92095 0.430533
\(532\) 0 0
\(533\) −30.7068 −1.33006
\(534\) 0 0
\(535\) 1.24252 0.0537190
\(536\) 0 0
\(537\) 3.95895 0.170841
\(538\) 0 0
\(539\) 3.42309i 0.147443i
\(540\) 0 0
\(541\) 16.6108 0.714155 0.357078 0.934075i \(-0.383773\pi\)
0.357078 + 0.934075i \(0.383773\pi\)
\(542\) 0 0
\(543\) 11.1264i 0.477478i
\(544\) 0 0
\(545\) 17.2068i 0.737059i
\(546\) 0 0
\(547\) −37.7975 −1.61611 −0.808053 0.589110i \(-0.799478\pi\)
−0.808053 + 0.589110i \(0.799478\pi\)
\(548\) 0 0
\(549\) −12.4329 −0.530625
\(550\) 0 0
\(551\) −10.8028 + 5.99186i −0.460217 + 0.255262i
\(552\) 0 0
\(553\) 17.3359i 0.737198i
\(554\) 0 0
\(555\) 9.57487i 0.406431i
\(556\) 0 0
\(557\) 9.58935 0.406314 0.203157 0.979146i \(-0.434880\pi\)
0.203157 + 0.979146i \(0.434880\pi\)
\(558\) 0 0
\(559\) 29.2253 1.23610
\(560\) 0 0
\(561\) 4.34978i 0.183648i
\(562\) 0 0
\(563\) −46.0307 −1.93996 −0.969982 0.243177i \(-0.921810\pi\)
−0.969982 + 0.243177i \(0.921810\pi\)
\(564\) 0 0
\(565\) 9.29604i 0.391087i
\(566\) 0 0
\(567\) 1.25182i 0.0525717i
\(568\) 0 0
\(569\) 23.2020i 0.972679i 0.873770 + 0.486339i \(0.161668\pi\)
−0.873770 + 0.486339i \(0.838332\pi\)
\(570\) 0 0
\(571\) 20.4551i 0.856020i −0.903774 0.428010i \(-0.859215\pi\)
0.903774 0.428010i \(-0.140785\pi\)
\(572\) 0 0
\(573\) 9.75195i 0.407393i
\(574\) 0 0
\(575\) 6.49272i 0.270765i
\(576\) 0 0
\(577\) −32.0188 −1.33296 −0.666480 0.745522i \(-0.732200\pi\)
−0.666480 + 0.745522i \(0.732200\pi\)
\(578\) 0 0
\(579\) 17.5048i 0.727474i
\(580\) 0 0
\(581\) −11.8347 −0.490986
\(582\) 0 0
\(583\) −6.27342 −0.259818
\(584\) 0 0
\(585\) 3.96496i 0.163931i
\(586\) 0 0
\(587\) 13.9566i 0.576052i 0.957623 + 0.288026i \(0.0929990\pi\)
−0.957623 + 0.288026i \(0.907001\pi\)
\(588\) 0 0
\(589\) 15.8315 + 28.5430i 0.652326 + 1.17609i
\(590\) 0 0
\(591\) −21.6568 −0.890842
\(592\) 0 0
\(593\) 30.2855 1.24367 0.621837 0.783146i \(-0.286386\pi\)
0.621837 + 0.783146i \(0.286386\pi\)
\(594\) 0 0
\(595\) 8.64226i 0.354298i
\(596\) 0 0
\(597\) 4.70101i 0.192400i
\(598\) 0 0
\(599\) 35.4402 1.44805 0.724024 0.689775i \(-0.242290\pi\)
0.724024 + 0.689775i \(0.242290\pi\)
\(600\) 0 0
\(601\) 13.2505i 0.540497i 0.962791 + 0.270249i \(0.0871059\pi\)
−0.962791 + 0.270249i \(0.912894\pi\)
\(602\) 0 0
\(603\) 7.82681 0.318732
\(604\) 0 0
\(605\) 10.6030 0.431074
\(606\) 0 0
\(607\) 28.2364 1.14608 0.573040 0.819528i \(-0.305764\pi\)
0.573040 + 0.819528i \(0.305764\pi\)
\(608\) 0 0
\(609\) −3.54772 −0.143761
\(610\) 0 0
\(611\) 13.2227 0.534935
\(612\) 0 0
\(613\) 15.1176 0.610593 0.305297 0.952257i \(-0.401244\pi\)
0.305297 + 0.952257i \(0.401244\pi\)
\(614\) 0 0
\(615\) 7.74454i 0.312290i
\(616\) 0 0
\(617\) −5.32628 −0.214428 −0.107214 0.994236i \(-0.534193\pi\)
−0.107214 + 0.994236i \(0.534193\pi\)
\(618\) 0 0
\(619\) 37.8063i 1.51956i 0.650178 + 0.759782i \(0.274694\pi\)
−0.650178 + 0.759782i \(0.725306\pi\)
\(620\) 0 0
\(621\) 6.49272i 0.260544i
\(622\) 0 0
\(623\) −5.66895 −0.227122
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −2.40168 + 1.33211i −0.0959140 + 0.0531993i
\(628\) 0 0
\(629\) 66.1023i 2.63567i
\(630\) 0 0
\(631\) 6.40333i 0.254912i −0.991844 0.127456i \(-0.959319\pi\)
0.991844 0.127456i \(-0.0406812\pi\)
\(632\) 0 0
\(633\) −28.4183 −1.12953
\(634\) 0 0
\(635\) −13.4710 −0.534581
\(636\) 0 0
\(637\) 21.5414i 0.853500i
\(638\) 0 0
\(639\) −1.41571 −0.0560048
\(640\) 0 0
\(641\) 33.4809i 1.32242i −0.750203 0.661208i \(-0.770044\pi\)
0.750203 0.661208i \(-0.229956\pi\)
\(642\) 0 0
\(643\) 24.5225i 0.967072i −0.875325 0.483536i \(-0.839352\pi\)
0.875325 0.483536i \(-0.160648\pi\)
\(644\) 0 0
\(645\) 7.37089i 0.290229i
\(646\) 0 0
\(647\) 41.4957i 1.63136i −0.578501 0.815682i \(-0.696362\pi\)
0.578501 0.815682i \(-0.303638\pi\)
\(648\) 0 0
\(649\) 6.25082i 0.245366i
\(650\) 0 0
\(651\) 9.37368i 0.367384i
\(652\) 0 0
\(653\) 6.80018 0.266112 0.133056 0.991109i \(-0.457521\pi\)
0.133056 + 0.991109i \(0.457521\pi\)
\(654\) 0 0
\(655\) 8.50842i 0.332452i
\(656\) 0 0
\(657\) 7.64421 0.298229
\(658\) 0 0
\(659\) 36.8583 1.43580 0.717898 0.696149i \(-0.245105\pi\)
0.717898 + 0.696149i \(0.245105\pi\)
\(660\) 0 0
\(661\) 6.97211i 0.271183i 0.990765 + 0.135592i \(0.0432936\pi\)
−0.990765 + 0.135592i \(0.956706\pi\)
\(662\) 0 0
\(663\) 27.3730i 1.06308i
\(664\) 0 0
\(665\) 4.77173 2.64667i 0.185040 0.102633i
\(666\) 0 0
\(667\) −18.4006 −0.712474
\(668\) 0 0
\(669\) 0.509638 0.0197037
\(670\) 0 0
\(671\) 7.83353i 0.302410i
\(672\) 0 0
\(673\) 16.1423i 0.622239i −0.950371 0.311119i \(-0.899296\pi\)
0.950371 0.311119i \(-0.100704\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 34.9941i 1.34493i 0.740128 + 0.672466i \(0.234765\pi\)
−0.740128 + 0.672466i \(0.765235\pi\)
\(678\) 0 0
\(679\) 4.89062 0.187685
\(680\) 0 0
\(681\) −16.3056 −0.624833
\(682\) 0 0
\(683\) −22.5170 −0.861589 −0.430795 0.902450i \(-0.641767\pi\)
−0.430795 + 0.902450i \(0.641767\pi\)
\(684\) 0 0
\(685\) −0.608472 −0.0232485
\(686\) 0 0
\(687\) −19.6375 −0.749216
\(688\) 0 0
\(689\) −39.4784 −1.50401
\(690\) 0 0
\(691\) 39.5236i 1.50355i −0.659420 0.751775i \(-0.729198\pi\)
0.659420 0.751775i \(-0.270802\pi\)
\(692\) 0 0
\(693\) −0.788728 −0.0299613
\(694\) 0 0
\(695\) 15.1578i 0.574968i
\(696\) 0 0
\(697\) 53.4662i 2.02518i
\(698\) 0 0
\(699\) 17.0910 0.646442
\(700\) 0 0
\(701\) 33.7767 1.27573 0.637864 0.770149i \(-0.279818\pi\)
0.637864 + 0.770149i \(0.279818\pi\)
\(702\) 0 0
\(703\) −36.4977 + 20.2436i −1.37654 + 0.763503i
\(704\) 0 0
\(705\) 3.33490i 0.125600i
\(706\) 0 0
\(707\) 18.5188i 0.696470i
\(708\) 0 0
\(709\) 1.19262 0.0447898 0.0223949 0.999749i \(-0.492871\pi\)
0.0223949 + 0.999749i \(0.492871\pi\)
\(710\) 0 0
\(711\) 13.8485 0.519360
\(712\) 0 0
\(713\) 48.6176i 1.82074i
\(714\) 0 0
\(715\) −2.49817 −0.0934264
\(716\) 0 0
\(717\) 19.0418i 0.711128i
\(718\) 0 0
\(719\) 40.1949i 1.49902i 0.661994 + 0.749509i \(0.269710\pi\)
−0.661994 + 0.749509i \(0.730290\pi\)
\(720\) 0 0
\(721\) 9.79779i 0.364889i
\(722\) 0 0
\(723\) 3.56019i 0.132405i
\(724\) 0 0
\(725\) 2.83404i 0.105254i
\(726\) 0 0
\(727\) 2.40487i 0.0891918i −0.999005 0.0445959i \(-0.985800\pi\)
0.999005 0.0445959i \(-0.0142000\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 50.8867i 1.88211i
\(732\) 0 0
\(733\) −17.5401 −0.647859 −0.323930 0.946081i \(-0.605004\pi\)
−0.323930 + 0.946081i \(0.605004\pi\)
\(734\) 0 0
\(735\) −5.43294 −0.200397
\(736\) 0 0
\(737\) 4.93138i 0.181650i
\(738\) 0 0
\(739\) 16.4628i 0.605592i 0.953055 + 0.302796i \(0.0979201\pi\)
−0.953055 + 0.302796i \(0.902080\pi\)
\(740\) 0 0
\(741\) −15.1137 + 8.38290i −0.555216 + 0.307954i
\(742\) 0 0
\(743\) −37.3965 −1.37195 −0.685973 0.727627i \(-0.740623\pi\)
−0.685973 + 0.727627i \(0.740623\pi\)
\(744\) 0 0
\(745\) −15.5603 −0.570086
\(746\) 0 0
\(747\) 9.45397i 0.345903i
\(748\) 0 0
\(749\) 1.55542i 0.0568339i
\(750\) 0 0
\(751\) −33.2116 −1.21191 −0.605954 0.795500i \(-0.707208\pi\)
−0.605954 + 0.795500i \(0.707208\pi\)
\(752\) 0 0
\(753\) 15.5397i 0.566300i
\(754\) 0 0
\(755\) −6.31793 −0.229933
\(756\) 0 0
\(757\) −12.3936 −0.450452 −0.225226 0.974307i \(-0.572312\pi\)
−0.225226 + 0.974307i \(0.572312\pi\)
\(758\) 0 0
\(759\) −4.09082 −0.148487
\(760\) 0 0
\(761\) −20.0173 −0.725627 −0.362813 0.931862i \(-0.618184\pi\)
−0.362813 + 0.931862i \(0.618184\pi\)
\(762\) 0 0
\(763\) −21.5399 −0.779797
\(764\) 0 0
\(765\) −6.90373 −0.249605
\(766\) 0 0
\(767\) 39.3362i 1.42035i
\(768\) 0 0
\(769\) 43.6864 1.57537 0.787686 0.616077i \(-0.211279\pi\)
0.787686 + 0.616077i \(0.211279\pi\)
\(770\) 0 0
\(771\) 1.13092i 0.0407291i
\(772\) 0 0
\(773\) 32.6219i 1.17333i 0.809830 + 0.586664i \(0.199559\pi\)
−0.809830 + 0.586664i \(0.800441\pi\)
\(774\) 0 0
\(775\) 7.48802 0.268978
\(776\) 0 0
\(777\) −11.9861 −0.429997
\(778\) 0 0
\(779\) 29.5208 16.3739i 1.05769 0.586655i
\(780\) 0 0
\(781\) 0.891989i 0.0319179i
\(782\) 0 0
\(783\) 2.83404i 0.101280i
\(784\) 0 0
\(785\) 14.9069 0.532051
\(786\) 0 0
\(787\) −44.0146 −1.56895 −0.784475 0.620160i \(-0.787068\pi\)
−0.784475 + 0.620160i \(0.787068\pi\)
\(788\) 0 0
\(789\) 4.33000i 0.154152i
\(790\) 0 0
\(791\) 11.6370 0.413764
\(792\) 0 0
\(793\) 49.2961i 1.75056i
\(794\) 0 0
\(795\) 9.95682i 0.353132i
\(796\) 0 0
\(797\) 46.1194i 1.63363i −0.576896 0.816817i \(-0.695736\pi\)
0.576896 0.816817i \(-0.304264\pi\)
\(798\) 0 0
\(799\) 23.0232i 0.814504i
\(800\) 0 0
\(801\) 4.52855i 0.160009i
\(802\) 0 0
\(803\) 4.81633i 0.169965i
\(804\) 0