Properties

Label 4560.2.d.j.2431.10
Level $4560$
Weight $2$
Character 4560.2431
Analytic conductor $36.412$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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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 2431.10
Root \(-5.32019i\) of defining polynomial
Character \(\chi\) \(=\) 4560.2431
Dual form 4560.2.d.j.2431.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +3.21222i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +3.21222i q^{7} +1.00000 q^{9} -2.53893i q^{11} +1.54784i q^{13} -1.00000 q^{15} -2.67107 q^{17} +(-1.05331 - 4.22972i) q^{19} -3.21222i q^{21} -5.98643i q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.925176i q^{29} -3.27279 q^{31} +2.53893i q^{33} +3.21222i q^{35} +10.3117i q^{37} -1.54784i q^{39} +9.19865i q^{41} +11.9254i q^{43} +1.00000 q^{45} -4.08676i q^{47} -3.31834 q^{49} +2.67107 q^{51} +2.52233i q^{53} -2.53893i q^{55} +(1.05331 + 4.22972i) q^{57} -9.59113 q^{59} -3.68166 q^{61} +3.21222i q^{63} +1.54784i q^{65} +14.6323 q^{67} +5.98643i q^{69} -7.94385 q^{71} +7.83724 q^{73} -1.00000 q^{75} +8.15558 q^{77} -16.2951 q^{79} +1.00000 q^{81} -11.0509i q^{83} -2.67107 q^{85} -0.925176i q^{87} -2.71016i q^{89} -4.97199 q^{91} +3.27279 q^{93} +(-1.05331 - 4.22972i) q^{95} +8.46130i q^{97} -2.53893i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} + 12q^{5} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} + 12q^{5} + 12q^{9} - 12q^{15} + 4q^{17} + 12q^{25} - 12q^{27} + 12q^{31} + 12q^{45} - 28q^{49} - 4q^{51} - 52q^{59} - 56q^{61} + 32q^{67} - 8q^{71} + 32q^{73} - 12q^{75} + 24q^{77} + 28q^{79} + 12q^{81} + 4q^{85} - 32q^{91} - 12q^{93} + O(q^{100}) \)

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
\(7\) 3.21222i 1.21410i 0.794662 + 0.607052i \(0.207648\pi\)
−0.794662 + 0.607052i \(0.792352\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.53893i 0.765515i −0.923849 0.382757i \(-0.874975\pi\)
0.923849 0.382757i \(-0.125025\pi\)
\(12\) 0 0
\(13\) 1.54784i 0.429293i 0.976692 + 0.214646i \(0.0688599\pi\)
−0.976692 + 0.214646i \(0.931140\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −2.67107 −0.647829 −0.323914 0.946086i \(-0.604999\pi\)
−0.323914 + 0.946086i \(0.604999\pi\)
\(18\) 0 0
\(19\) −1.05331 4.22972i −0.241645 0.970365i
\(20\) 0 0
\(21\) 3.21222i 0.700963i
\(22\) 0 0
\(23\) 5.98643i 1.24826i −0.781322 0.624129i \(-0.785454\pi\)
0.781322 0.624129i \(-0.214546\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.925176i 0.171801i 0.996304 + 0.0859005i \(0.0273767\pi\)
−0.996304 + 0.0859005i \(0.972623\pi\)
\(30\) 0 0
\(31\) −3.27279 −0.587810 −0.293905 0.955835i \(-0.594955\pi\)
−0.293905 + 0.955835i \(0.594955\pi\)
\(32\) 0 0
\(33\) 2.53893i 0.441970i
\(34\) 0 0
\(35\) 3.21222i 0.542964i
\(36\) 0 0
\(37\) 10.3117i 1.69522i 0.530616 + 0.847612i \(0.321961\pi\)
−0.530616 + 0.847612i \(0.678039\pi\)
\(38\) 0 0
\(39\) 1.54784i 0.247852i
\(40\) 0 0
\(41\) 9.19865i 1.43659i 0.695740 + 0.718294i \(0.255077\pi\)
−0.695740 + 0.718294i \(0.744923\pi\)
\(42\) 0 0
\(43\) 11.9254i 1.81861i 0.416134 + 0.909303i \(0.363385\pi\)
−0.416134 + 0.909303i \(0.636615\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 4.08676i 0.596115i −0.954548 0.298058i \(-0.903661\pi\)
0.954548 0.298058i \(-0.0963388\pi\)
\(48\) 0 0
\(49\) −3.31834 −0.474048
\(50\) 0 0
\(51\) 2.67107 0.374024
\(52\) 0 0
\(53\) 2.52233i 0.346469i 0.984881 + 0.173234i \(0.0554218\pi\)
−0.984881 + 0.173234i \(0.944578\pi\)
\(54\) 0 0
\(55\) 2.53893i 0.342349i
\(56\) 0 0
\(57\) 1.05331 + 4.22972i 0.139514 + 0.560240i
\(58\) 0 0
\(59\) −9.59113 −1.24866 −0.624329 0.781161i \(-0.714628\pi\)
−0.624329 + 0.781161i \(0.714628\pi\)
\(60\) 0 0
\(61\) −3.68166 −0.471388 −0.235694 0.971827i \(-0.575736\pi\)
−0.235694 + 0.971827i \(0.575736\pi\)
\(62\) 0 0
\(63\) 3.21222i 0.404701i
\(64\) 0 0
\(65\) 1.54784i 0.191985i
\(66\) 0 0
\(67\) 14.6323 1.78762 0.893812 0.448441i \(-0.148021\pi\)
0.893812 + 0.448441i \(0.148021\pi\)
\(68\) 0 0
\(69\) 5.98643i 0.720682i
\(70\) 0 0
\(71\) −7.94385 −0.942762 −0.471381 0.881930i \(-0.656244\pi\)
−0.471381 + 0.881930i \(0.656244\pi\)
\(72\) 0 0
\(73\) 7.83724 0.917280 0.458640 0.888622i \(-0.348337\pi\)
0.458640 + 0.888622i \(0.348337\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 8.15558 0.929415
\(78\) 0 0
\(79\) −16.2951 −1.83334 −0.916672 0.399640i \(-0.869135\pi\)
−0.916672 + 0.399640i \(0.869135\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.0509i 1.21299i −0.795088 0.606494i \(-0.792575\pi\)
0.795088 0.606494i \(-0.207425\pi\)
\(84\) 0 0
\(85\) −2.67107 −0.289718
\(86\) 0 0
\(87\) 0.925176i 0.0991893i
\(88\) 0 0
\(89\) 2.71016i 0.287276i −0.989630 0.143638i \(-0.954120\pi\)
0.989630 0.143638i \(-0.0458801\pi\)
\(90\) 0 0
\(91\) −4.97199 −0.521206
\(92\) 0 0
\(93\) 3.27279 0.339372
\(94\) 0 0
\(95\) −1.05331 4.22972i −0.108067 0.433960i
\(96\) 0 0
\(97\) 8.46130i 0.859115i 0.903040 + 0.429557i \(0.141330\pi\)
−0.903040 + 0.429557i \(0.858670\pi\)
\(98\) 0 0
\(99\) 2.53893i 0.255172i
\(100\) 0 0
\(101\) −13.3407 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(102\) 0 0
\(103\) −14.6323 −1.44177 −0.720883 0.693056i \(-0.756264\pi\)
−0.720883 + 0.693056i \(0.756264\pi\)
\(104\) 0 0
\(105\) 3.21222i 0.313480i
\(106\) 0 0
\(107\) 14.5762 1.40913 0.704567 0.709638i \(-0.251141\pi\)
0.704567 + 0.709638i \(0.251141\pi\)
\(108\) 0 0
\(109\) 10.1423i 0.971454i −0.874111 0.485727i \(-0.838555\pi\)
0.874111 0.485727i \(-0.161445\pi\)
\(110\) 0 0
\(111\) 10.3117i 0.978739i
\(112\) 0 0
\(113\) 17.2216i 1.62007i −0.586383 0.810034i \(-0.699449\pi\)
0.586383 0.810034i \(-0.300551\pi\)
\(114\) 0 0
\(115\) 5.98643i 0.558238i
\(116\) 0 0
\(117\) 1.54784i 0.143098i
\(118\) 0 0
\(119\) 8.58004i 0.786531i
\(120\) 0 0
\(121\) 4.55386 0.413987
\(122\) 0 0
\(123\) 9.19865i 0.829414i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −20.4696 −1.81638 −0.908191 0.418557i \(-0.862536\pi\)
−0.908191 + 0.418557i \(0.862536\pi\)
\(128\) 0 0
\(129\) 11.9254i 1.04997i
\(130\) 0 0
\(131\) 19.6268i 1.71481i 0.514646 + 0.857403i \(0.327923\pi\)
−0.514646 + 0.857403i \(0.672077\pi\)
\(132\) 0 0
\(133\) 13.5868 3.38345i 1.17812 0.293382i
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −18.0819 −1.54484 −0.772420 0.635112i \(-0.780954\pi\)
−0.772420 + 0.635112i \(0.780954\pi\)
\(138\) 0 0
\(139\) 0.978154i 0.0829659i 0.999139 + 0.0414830i \(0.0132082\pi\)
−0.999139 + 0.0414830i \(0.986792\pi\)
\(140\) 0 0
\(141\) 4.08676i 0.344167i
\(142\) 0 0
\(143\) 3.92984 0.328630
\(144\) 0 0
\(145\) 0.925176i 0.0768317i
\(146\) 0 0
\(147\) 3.31834 0.273692
\(148\) 0 0
\(149\) 12.4894 1.02317 0.511587 0.859232i \(-0.329058\pi\)
0.511587 + 0.859232i \(0.329058\pi\)
\(150\) 0 0
\(151\) 7.14498 0.581451 0.290725 0.956807i \(-0.406103\pi\)
0.290725 + 0.956807i \(0.406103\pi\)
\(152\) 0 0
\(153\) −2.67107 −0.215943
\(154\) 0 0
\(155\) −3.27279 −0.262877
\(156\) 0 0
\(157\) −24.2739 −1.93727 −0.968635 0.248487i \(-0.920067\pi\)
−0.968635 + 0.248487i \(0.920067\pi\)
\(158\) 0 0
\(159\) 2.52233i 0.200034i
\(160\) 0 0
\(161\) 19.2297 1.51551
\(162\) 0 0
\(163\) 15.2689i 1.19595i 0.801514 + 0.597976i \(0.204028\pi\)
−0.801514 + 0.597976i \(0.795972\pi\)
\(164\) 0 0
\(165\) 2.53893i 0.197655i
\(166\) 0 0
\(167\) −14.4754 −1.12014 −0.560071 0.828445i \(-0.689226\pi\)
−0.560071 + 0.828445i \(0.689226\pi\)
\(168\) 0 0
\(169\) 10.6042 0.815708
\(170\) 0 0
\(171\) −1.05331 4.22972i −0.0805483 0.323455i
\(172\) 0 0
\(173\) 17.7746i 1.35138i 0.737186 + 0.675690i \(0.236154\pi\)
−0.737186 + 0.675690i \(0.763846\pi\)
\(174\) 0 0
\(175\) 3.21222i 0.242821i
\(176\) 0 0
\(177\) 9.59113 0.720913
\(178\) 0 0
\(179\) −25.6372 −1.91622 −0.958109 0.286404i \(-0.907540\pi\)
−0.958109 + 0.286404i \(0.907540\pi\)
\(180\) 0 0
\(181\) 2.28875i 0.170121i −0.996376 0.0850607i \(-0.972892\pi\)
0.996376 0.0850607i \(-0.0271084\pi\)
\(182\) 0 0
\(183\) 3.68166 0.272156
\(184\) 0 0
\(185\) 10.3117i 0.758128i
\(186\) 0 0
\(187\) 6.78164i 0.495922i
\(188\) 0 0
\(189\) 3.21222i 0.233654i
\(190\) 0 0
\(191\) 8.12456i 0.587872i −0.955825 0.293936i \(-0.905035\pi\)
0.955825 0.293936i \(-0.0949653\pi\)
\(192\) 0 0
\(193\) 7.21598i 0.519417i 0.965687 + 0.259709i \(0.0836266\pi\)
−0.965687 + 0.259709i \(0.916373\pi\)
\(194\) 0 0
\(195\) 1.54784i 0.110843i
\(196\) 0 0
\(197\) 0.319681 0.0227763 0.0113882 0.999935i \(-0.496375\pi\)
0.0113882 + 0.999935i \(0.496375\pi\)
\(198\) 0 0
\(199\) 18.2126i 1.29106i 0.763735 + 0.645530i \(0.223363\pi\)
−0.763735 + 0.645530i \(0.776637\pi\)
\(200\) 0 0
\(201\) −14.6323 −1.03209
\(202\) 0 0
\(203\) −2.97187 −0.208584
\(204\) 0 0
\(205\) 9.19865i 0.642461i
\(206\) 0 0
\(207\) 5.98643i 0.416086i
\(208\) 0 0
\(209\) −10.7389 + 2.67427i −0.742829 + 0.184983i
\(210\) 0 0
\(211\) −14.7936 −1.01844 −0.509218 0.860638i \(-0.670065\pi\)
−0.509218 + 0.860638i \(0.670065\pi\)
\(212\) 0 0
\(213\) 7.94385 0.544304
\(214\) 0 0
\(215\) 11.9254i 0.813306i
\(216\) 0 0
\(217\) 10.5129i 0.713662i
\(218\) 0 0
\(219\) −7.83724 −0.529592
\(220\) 0 0
\(221\) 4.13437i 0.278108i
\(222\) 0 0
\(223\) 21.7598 1.45714 0.728571 0.684970i \(-0.240185\pi\)
0.728571 + 0.684970i \(0.240185\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 1.41229 0.0937370 0.0468685 0.998901i \(-0.485076\pi\)
0.0468685 + 0.998901i \(0.485076\pi\)
\(228\) 0 0
\(229\) −16.9705 −1.12144 −0.560722 0.828004i \(-0.689476\pi\)
−0.560722 + 0.828004i \(0.689476\pi\)
\(230\) 0 0
\(231\) −8.15558 −0.536598
\(232\) 0 0
\(233\) −0.281071 −0.0184135 −0.00920677 0.999958i \(-0.502931\pi\)
−0.00920677 + 0.999958i \(0.502931\pi\)
\(234\) 0 0
\(235\) 4.08676i 0.266591i
\(236\) 0 0
\(237\) 16.2951 1.05848
\(238\) 0 0
\(239\) 6.10518i 0.394911i −0.980312 0.197455i \(-0.936732\pi\)
0.980312 0.197455i \(-0.0632678\pi\)
\(240\) 0 0
\(241\) 2.30218i 0.148296i −0.997247 0.0741481i \(-0.976376\pi\)
0.997247 0.0741481i \(-0.0236237\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −3.31834 −0.212001
\(246\) 0 0
\(247\) 6.54692 1.63034i 0.416570 0.103736i
\(248\) 0 0
\(249\) 11.0509i 0.700319i
\(250\) 0 0
\(251\) 13.8121i 0.871814i 0.899992 + 0.435907i \(0.143572\pi\)
−0.899992 + 0.435907i \(0.856428\pi\)
\(252\) 0 0
\(253\) −15.1991 −0.955559
\(254\) 0 0
\(255\) 2.67107 0.167269
\(256\) 0 0
\(257\) 2.47301i 0.154262i 0.997021 + 0.0771311i \(0.0245760\pi\)
−0.997021 + 0.0771311i \(0.975424\pi\)
\(258\) 0 0
\(259\) −33.1233 −2.05818
\(260\) 0 0
\(261\) 0.925176i 0.0572670i
\(262\) 0 0
\(263\) 24.2851i 1.49748i −0.662862 0.748742i \(-0.730658\pi\)
0.662862 0.748742i \(-0.269342\pi\)
\(264\) 0 0
\(265\) 2.52233i 0.154945i
\(266\) 0 0
\(267\) 2.71016i 0.165859i
\(268\) 0 0
\(269\) 17.7800i 1.08407i 0.840357 + 0.542033i \(0.182345\pi\)
−0.840357 + 0.542033i \(0.817655\pi\)
\(270\) 0 0
\(271\) 12.5298i 0.761129i −0.924754 0.380564i \(-0.875730\pi\)
0.924754 0.380564i \(-0.124270\pi\)
\(272\) 0 0
\(273\) 4.97199 0.300918
\(274\) 0 0
\(275\) 2.53893i 0.153103i
\(276\) 0 0
\(277\) 0.337218 0.0202615 0.0101307 0.999949i \(-0.496775\pi\)
0.0101307 + 0.999949i \(0.496775\pi\)
\(278\) 0 0
\(279\) −3.27279 −0.195937
\(280\) 0 0
\(281\) 24.9732i 1.48977i −0.667190 0.744887i \(-0.732503\pi\)
0.667190 0.744887i \(-0.267497\pi\)
\(282\) 0 0
\(283\) 30.2093i 1.79576i −0.440243 0.897879i \(-0.645108\pi\)
0.440243 0.897879i \(-0.354892\pi\)
\(284\) 0 0
\(285\) 1.05331 + 4.22972i 0.0623925 + 0.250547i
\(286\) 0 0
\(287\) −29.5481 −1.74417
\(288\) 0 0
\(289\) −9.86541 −0.580318
\(290\) 0 0
\(291\) 8.46130i 0.496010i
\(292\) 0 0
\(293\) 3.55233i 0.207529i −0.994602 0.103765i \(-0.966911\pi\)
0.994602 0.103765i \(-0.0330889\pi\)
\(294\) 0 0
\(295\) −9.59113 −0.558417
\(296\) 0 0
\(297\) 2.53893i 0.147323i
\(298\) 0 0
\(299\) 9.26601 0.535867
\(300\) 0 0
\(301\) −38.3070 −2.20798
\(302\) 0 0
\(303\) 13.3407 0.766401
\(304\) 0 0
\(305\) −3.68166 −0.210811
\(306\) 0 0
\(307\) −17.5146 −0.999611 −0.499806 0.866138i \(-0.666595\pi\)
−0.499806 + 0.866138i \(0.666595\pi\)
\(308\) 0 0
\(309\) 14.6323 0.832405
\(310\) 0 0
\(311\) 23.1622i 1.31341i 0.754148 + 0.656705i \(0.228050\pi\)
−0.754148 + 0.656705i \(0.771950\pi\)
\(312\) 0 0
\(313\) −28.3068 −1.60000 −0.799998 0.600003i \(-0.795166\pi\)
−0.799998 + 0.600003i \(0.795166\pi\)
\(314\) 0 0
\(315\) 3.21222i 0.180988i
\(316\) 0 0
\(317\) 30.4729i 1.71153i 0.517365 + 0.855765i \(0.326913\pi\)
−0.517365 + 0.855765i \(0.673087\pi\)
\(318\) 0 0
\(319\) 2.34895 0.131516
\(320\) 0 0
\(321\) −14.5762 −0.813564
\(322\) 0 0
\(323\) 2.81345 + 11.2979i 0.156544 + 0.628630i
\(324\) 0 0
\(325\) 1.54784i 0.0858585i
\(326\) 0 0
\(327\) 10.1423i 0.560869i
\(328\) 0 0
\(329\) 13.1276 0.723746
\(330\) 0 0
\(331\) 15.3547 0.843970 0.421985 0.906603i \(-0.361334\pi\)
0.421985 + 0.906603i \(0.361334\pi\)
\(332\) 0 0
\(333\) 10.3117i 0.565075i
\(334\) 0 0
\(335\) 14.6323 0.799450
\(336\) 0 0
\(337\) 19.1023i 1.04057i −0.853993 0.520285i \(-0.825826\pi\)
0.853993 0.520285i \(-0.174174\pi\)
\(338\) 0 0
\(339\) 17.2216i 0.935346i
\(340\) 0 0
\(341\) 8.30936i 0.449977i
\(342\) 0 0
\(343\) 11.8263i 0.638560i
\(344\) 0 0
\(345\) 5.98643i 0.322299i
\(346\) 0 0
\(347\) 4.94300i 0.265354i 0.991159 + 0.132677i \(0.0423573\pi\)
−0.991159 + 0.132677i \(0.957643\pi\)
\(348\) 0 0
\(349\) −24.1725 −1.29392 −0.646961 0.762523i \(-0.723960\pi\)
−0.646961 + 0.762523i \(0.723960\pi\)
\(350\) 0 0
\(351\) 1.54784i 0.0826174i
\(352\) 0 0
\(353\) 29.8335 1.58788 0.793939 0.607998i \(-0.208027\pi\)
0.793939 + 0.607998i \(0.208027\pi\)
\(354\) 0 0
\(355\) −7.94385 −0.421616
\(356\) 0 0
\(357\) 8.58004i 0.454104i
\(358\) 0 0
\(359\) 21.0003i 1.10835i −0.832399 0.554176i \(-0.813033\pi\)
0.832399 0.554176i \(-0.186967\pi\)
\(360\) 0 0
\(361\) −16.7811 + 8.91038i −0.883215 + 0.468967i
\(362\) 0 0
\(363\) −4.55386 −0.239016
\(364\) 0 0
\(365\) 7.83724 0.410220
\(366\) 0 0
\(367\) 3.70124i 0.193203i −0.995323 0.0966017i \(-0.969203\pi\)
0.995323 0.0966017i \(-0.0307973\pi\)
\(368\) 0 0
\(369\) 9.19865i 0.478863i
\(370\) 0 0
\(371\) −8.10227 −0.420649
\(372\) 0 0
\(373\) 17.7289i 0.917967i 0.888445 + 0.458983i \(0.151786\pi\)
−0.888445 + 0.458983i \(0.848214\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −1.43202 −0.0737528
\(378\) 0 0
\(379\) 6.78792 0.348672 0.174336 0.984686i \(-0.444222\pi\)
0.174336 + 0.984686i \(0.444222\pi\)
\(380\) 0 0
\(381\) 20.4696 1.04869
\(382\) 0 0
\(383\) 20.8106 1.06337 0.531686 0.846941i \(-0.321559\pi\)
0.531686 + 0.846941i \(0.321559\pi\)
\(384\) 0 0
\(385\) 8.15558 0.415647
\(386\) 0 0
\(387\) 11.9254i 0.606202i
\(388\) 0 0
\(389\) −7.55535 −0.383072 −0.191536 0.981486i \(-0.561347\pi\)
−0.191536 + 0.981486i \(0.561347\pi\)
\(390\) 0 0
\(391\) 15.9901i 0.808657i
\(392\) 0 0
\(393\) 19.6268i 0.990043i
\(394\) 0 0
\(395\) −16.2951 −0.819896
\(396\) 0 0
\(397\) 8.03143 0.403086 0.201543 0.979480i \(-0.435404\pi\)
0.201543 + 0.979480i \(0.435404\pi\)
\(398\) 0 0
\(399\) −13.5868 + 3.38345i −0.680190 + 0.169384i
\(400\) 0 0
\(401\) 28.1366i 1.40508i 0.711646 + 0.702538i \(0.247950\pi\)
−0.711646 + 0.702538i \(0.752050\pi\)
\(402\) 0 0
\(403\) 5.06574i 0.252342i
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 26.1805 1.29772
\(408\) 0 0
\(409\) 6.71758i 0.332163i 0.986112 + 0.166081i \(0.0531115\pi\)
−0.986112 + 0.166081i \(0.946889\pi\)
\(410\) 0 0
\(411\) 18.0819 0.891914
\(412\) 0 0
\(413\) 30.8088i 1.51600i
\(414\) 0 0
\(415\) 11.0509i 0.542465i
\(416\) 0 0
\(417\) 0.978154i 0.0479004i
\(418\) 0 0
\(419\) 22.6853i 1.10825i −0.832433 0.554125i \(-0.813053\pi\)
0.832433 0.554125i \(-0.186947\pi\)
\(420\) 0 0
\(421\) 9.23308i 0.449993i −0.974360 0.224996i \(-0.927763\pi\)
0.974360 0.224996i \(-0.0722370\pi\)
\(422\) 0 0
\(423\) 4.08676i 0.198705i
\(424\) 0 0
\(425\) −2.67107 −0.129566
\(426\) 0 0
\(427\) 11.8263i 0.572315i
\(428\) 0 0
\(429\) −3.92984 −0.189734
\(430\) 0 0
\(431\) −18.8019 −0.905657 −0.452829 0.891598i \(-0.649585\pi\)
−0.452829 + 0.891598i \(0.649585\pi\)
\(432\) 0 0
\(433\) 30.5862i 1.46988i −0.678134 0.734938i \(-0.737211\pi\)
0.678134 0.734938i \(-0.262789\pi\)
\(434\) 0 0
\(435\) 0.925176i 0.0443588i
\(436\) 0 0
\(437\) −25.3209 + 6.30554i −1.21126 + 0.301635i
\(438\) 0 0
\(439\) −15.7175 −0.750154 −0.375077 0.926994i \(-0.622384\pi\)
−0.375077 + 0.926994i \(0.622384\pi\)
\(440\) 0 0
\(441\) −3.31834 −0.158016
\(442\) 0 0
\(443\) 18.4828i 0.878146i 0.898451 + 0.439073i \(0.144693\pi\)
−0.898451 + 0.439073i \(0.855307\pi\)
\(444\) 0 0
\(445\) 2.71016i 0.128474i
\(446\) 0 0
\(447\) −12.4894 −0.590730
\(448\) 0 0
\(449\) 38.2554i 1.80538i 0.430287 + 0.902692i \(0.358413\pi\)
−0.430287 + 0.902692i \(0.641587\pi\)
\(450\) 0 0
\(451\) 23.3547 1.09973
\(452\) 0 0
\(453\) −7.14498 −0.335701
\(454\) 0 0
\(455\) −4.97199 −0.233090
\(456\) 0 0
\(457\) −20.4936 −0.958652 −0.479326 0.877637i \(-0.659119\pi\)
−0.479326 + 0.877637i \(0.659119\pi\)
\(458\) 0 0
\(459\) 2.67107 0.124675
\(460\) 0 0
\(461\) 32.4120 1.50958 0.754788 0.655969i \(-0.227740\pi\)
0.754788 + 0.655969i \(0.227740\pi\)
\(462\) 0 0
\(463\) 16.8208i 0.781729i 0.920448 + 0.390864i \(0.127824\pi\)
−0.920448 + 0.390864i \(0.872176\pi\)
\(464\) 0 0
\(465\) 3.27279 0.151772
\(466\) 0 0
\(467\) 6.42141i 0.297148i −0.988901 0.148574i \(-0.952532\pi\)
0.988901 0.148574i \(-0.0474682\pi\)
\(468\) 0 0
\(469\) 47.0022i 2.17036i
\(470\) 0 0
\(471\) 24.2739 1.11848
\(472\) 0 0
\(473\) 30.2777 1.39217
\(474\) 0 0
\(475\) −1.05331 4.22972i −0.0483290 0.194073i
\(476\) 0 0
\(477\) 2.52233i 0.115490i
\(478\) 0 0
\(479\) 20.3372i 0.929232i −0.885512 0.464616i \(-0.846192\pi\)
0.885512 0.464616i \(-0.153808\pi\)
\(480\) 0 0
\(481\) −15.9607 −0.727747
\(482\) 0 0
\(483\) −19.2297 −0.874982
\(484\) 0 0
\(485\) 8.46130i 0.384208i
\(486\) 0 0
\(487\) 34.8567 1.57951 0.789755 0.613423i \(-0.210208\pi\)
0.789755 + 0.613423i \(0.210208\pi\)
\(488\) 0 0
\(489\) 15.2689i 0.690484i
\(490\) 0 0
\(491\) 9.84278i 0.444198i 0.975024 + 0.222099i \(0.0712909\pi\)
−0.975024 + 0.222099i \(0.928709\pi\)
\(492\) 0 0
\(493\) 2.47121i 0.111298i
\(494\) 0 0
\(495\) 2.53893i 0.114116i
\(496\) 0 0
\(497\) 25.5174i 1.14461i
\(498\) 0 0
\(499\) 30.0869i 1.34687i 0.739245 + 0.673437i \(0.235183\pi\)
−0.739245 + 0.673437i \(0.764817\pi\)
\(500\) 0 0
\(501\) 14.4754 0.646714
\(502\) 0 0
\(503\) 3.61248i 0.161072i −0.996752 0.0805362i \(-0.974337\pi\)
0.996752 0.0805362i \(-0.0256632\pi\)
\(504\) 0 0
\(505\) −13.3407 −0.593652
\(506\) 0 0
\(507\) −10.6042 −0.470949
\(508\) 0 0
\(509\) 23.6635i 1.04886i 0.851452 + 0.524432i \(0.175722\pi\)
−0.851452 + 0.524432i \(0.824278\pi\)
\(510\) 0 0
\(511\) 25.1749i 1.11367i
\(512\) 0 0
\(513\) 1.05331 + 4.22972i 0.0465046 + 0.186747i
\(514\) 0 0
\(515\) −14.6323 −0.644778
\(516\) 0 0
\(517\) −10.3760 −0.456335
\(518\) 0 0
\(519\) 17.7746i 0.780220i
\(520\) 0 0
\(521\) 14.9835i 0.656438i −0.944602 0.328219i \(-0.893552\pi\)
0.944602 0.328219i \(-0.106448\pi\)
\(522\) 0 0
\(523\) −4.41765 −0.193170 −0.0965852 0.995325i \(-0.530792\pi\)
−0.0965852 + 0.995325i \(0.530792\pi\)
\(524\) 0 0
\(525\) 3.21222i 0.140193i
\(526\) 0 0
\(527\) 8.74183 0.380800
\(528\) 0 0
\(529\) −12.8374 −0.558146
\(530\) 0 0
\(531\) −9.59113 −0.416219
\(532\) 0 0
\(533\) −14.2380 −0.616716
\(534\) 0 0
\(535\) 14.5762 0.630184
\(536\) 0 0
\(537\) 25.6372 1.10633
\(538\) 0 0
\(539\) 8.42501i 0.362891i
\(540\) 0 0
\(541\) −18.1358 −0.779717 −0.389859 0.920875i \(-0.627476\pi\)
−0.389859 + 0.920875i \(0.627476\pi\)
\(542\) 0 0
\(543\) 2.28875i 0.0982196i
\(544\) 0 0
\(545\) 10.1423i 0.434447i
\(546\) 0 0
\(547\) −27.4794 −1.17493 −0.587466 0.809249i \(-0.699874\pi\)
−0.587466 + 0.809249i \(0.699874\pi\)
\(548\) 0 0
\(549\) −3.68166 −0.157129
\(550\) 0 0
\(551\) 3.91324 0.974494i 0.166710 0.0415148i
\(552\) 0 0
\(553\) 52.3435i 2.22587i
\(554\) 0 0
\(555\) 10.3117i 0.437705i
\(556\) 0 0
\(557\) 14.2719 0.604718 0.302359 0.953194i \(-0.402226\pi\)
0.302359 + 0.953194i \(0.402226\pi\)
\(558\) 0 0
\(559\) −18.4586 −0.780714
\(560\) 0 0
\(561\) 6.78164i 0.286321i
\(562\) 0 0
\(563\) −10.1877 −0.429358 −0.214679 0.976685i \(-0.568871\pi\)
−0.214679 + 0.976685i \(0.568871\pi\)
\(564\) 0 0
\(565\) 17.2216i 0.724516i
\(566\) 0 0
\(567\) 3.21222i 0.134900i
\(568\) 0 0
\(569\) 14.9875i 0.628308i −0.949372 0.314154i \(-0.898279\pi\)
0.949372 0.314154i \(-0.101721\pi\)
\(570\) 0 0
\(571\) 11.3338i 0.474303i 0.971473 + 0.237151i \(0.0762138\pi\)
−0.971473 + 0.237151i \(0.923786\pi\)
\(572\) 0 0
\(573\) 8.12456i 0.339408i
\(574\) 0 0
\(575\) 5.98643i 0.249651i
\(576\) 0 0
\(577\) −7.05725 −0.293797 −0.146899 0.989152i \(-0.546929\pi\)
−0.146899 + 0.989152i \(0.546929\pi\)
\(578\) 0 0
\(579\) 7.21598i 0.299886i
\(580\) 0 0
\(581\) 35.4977 1.47269
\(582\) 0 0
\(583\) 6.40401 0.265227
\(584\) 0 0
\(585\) 1.54784i 0.0639951i
\(586\) 0 0
\(587\) 6.81923i 0.281460i 0.990048 + 0.140730i \(0.0449449\pi\)
−0.990048 + 0.140730i \(0.955055\pi\)
\(588\) 0 0
\(589\) 3.44725 + 13.8430i 0.142041 + 0.570390i
\(590\) 0 0
\(591\) −0.319681 −0.0131499
\(592\) 0 0
\(593\) 34.6470 1.42278 0.711391 0.702796i \(-0.248065\pi\)
0.711391 + 0.702796i \(0.248065\pi\)
\(594\) 0 0
\(595\) 8.58004i 0.351747i
\(596\) 0 0
\(597\) 18.2126i 0.745394i
\(598\) 0 0
\(599\) 9.55670 0.390476 0.195238 0.980756i \(-0.437452\pi\)
0.195238 + 0.980756i \(0.437452\pi\)
\(600\) 0 0
\(601\) 13.3033i 0.542654i 0.962487 + 0.271327i \(0.0874624\pi\)
−0.962487 + 0.271327i \(0.912538\pi\)
\(602\) 0 0
\(603\) 14.6323 0.595875
\(604\) 0 0
\(605\) 4.55386 0.185141
\(606\) 0 0
\(607\) −32.2367 −1.30845 −0.654224 0.756301i \(-0.727004\pi\)
−0.654224 + 0.756301i \(0.727004\pi\)
\(608\) 0 0
\(609\) 2.97187 0.120426
\(610\) 0 0
\(611\) 6.32564 0.255908
\(612\) 0 0
\(613\) 21.8868 0.883998 0.441999 0.897015i \(-0.354269\pi\)
0.441999 + 0.897015i \(0.354269\pi\)
\(614\) 0 0
\(615\) 9.19865i 0.370925i
\(616\) 0 0
\(617\) −18.9822 −0.764196 −0.382098 0.924122i \(-0.624798\pi\)
−0.382098 + 0.924122i \(0.624798\pi\)
\(618\) 0 0
\(619\) 6.97762i 0.280454i −0.990119 0.140227i \(-0.955217\pi\)
0.990119 0.140227i \(-0.0447833\pi\)
\(620\) 0 0
\(621\) 5.98643i 0.240227i
\(622\) 0 0
\(623\) 8.70561 0.348783
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 10.7389 2.67427i 0.428872 0.106800i
\(628\) 0 0
\(629\) 27.5431i 1.09822i
\(630\) 0 0
\(631\) 29.7974i 1.18622i −0.805123 0.593109i \(-0.797900\pi\)
0.805123 0.593109i \(-0.202100\pi\)
\(632\) 0 0
\(633\) 14.7936 0.587994
\(634\) 0 0
\(635\) −20.4696 −0.812310
\(636\) 0 0
\(637\) 5.13624i 0.203505i
\(638\) 0 0
\(639\) −7.94385 −0.314254
\(640\) 0 0
\(641\) 36.0432i 1.42362i −0.702372 0.711810i \(-0.747876\pi\)
0.702372 0.711810i \(-0.252124\pi\)
\(642\) 0 0
\(643\) 1.79655i 0.0708491i 0.999372 + 0.0354245i \(0.0112783\pi\)
−0.999372 + 0.0354245i \(0.988722\pi\)
\(644\) 0 0
\(645\) 11.9254i 0.469562i
\(646\) 0 0
\(647\) 1.72893i 0.0679713i 0.999422 + 0.0339857i \(0.0108201\pi\)
−0.999422 + 0.0339857i \(0.989180\pi\)
\(648\) 0 0
\(649\) 24.3512i 0.955866i
\(650\) 0 0
\(651\) 10.5129i 0.412033i
\(652\) 0 0
\(653\) 15.4671 0.605274 0.302637 0.953106i \(-0.402133\pi\)
0.302637 + 0.953106i \(0.402133\pi\)
\(654\) 0 0
\(655\) 19.6268i 0.766884i
\(656\) 0 0
\(657\) 7.83724 0.305760
\(658\) 0 0
\(659\) −34.7171 −1.35239 −0.676193 0.736724i \(-0.736372\pi\)
−0.676193 + 0.736724i \(0.736372\pi\)
\(660\) 0 0
\(661\) 40.1807i 1.56285i −0.624000 0.781424i \(-0.714494\pi\)
0.624000 0.781424i \(-0.285506\pi\)
\(662\) 0 0
\(663\) 4.13437i 0.160566i
\(664\) 0 0
\(665\) 13.5868 3.38345i 0.526873 0.131204i
\(666\) 0 0
\(667\) 5.53850 0.214452
\(668\) 0 0
\(669\) −21.7598 −0.841282
\(670\) 0 0
\(671\) 9.34746i 0.360855i
\(672\) 0 0
\(673\) 8.94691i 0.344878i 0.985020 + 0.172439i \(0.0551648\pi\)
−0.985020 + 0.172439i \(0.944835\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 40.4341i 1.55401i −0.629495 0.777005i \(-0.716738\pi\)
0.629495 0.777005i \(-0.283262\pi\)
\(678\) 0 0
\(679\) −27.1795 −1.04305
\(680\) 0 0
\(681\) −1.41229 −0.0541191
\(682\) 0 0
\(683\) 42.3252 1.61953 0.809765 0.586755i \(-0.199595\pi\)
0.809765 + 0.586755i \(0.199595\pi\)
\(684\) 0 0
\(685\) −18.0819 −0.690874
\(686\) 0 0
\(687\) 16.9705 0.647466
\(688\) 0 0
\(689\) −3.90415 −0.148736
\(690\) 0 0
\(691\) 11.8439i 0.450562i −0.974294 0.225281i \(-0.927670\pi\)
0.974294 0.225281i \(-0.0723300\pi\)
\(692\) 0 0
\(693\) 8.15558 0.309805
\(694\) 0 0
\(695\) 0.978154i 0.0371035i
\(696\) 0 0
\(697\) 24.5702i 0.930662i
\(698\) 0 0
\(699\) 0.281071 0.0106311
\(700\) 0 0
\(701\) −7.57075 −0.285943 −0.142972 0.989727i \(-0.545666\pi\)
−0.142972 + 0.989727i \(0.545666\pi\)
\(702\) 0 0
\(703\) 43.6154 10.8613i 1.64499 0.409643i
\(704\) 0 0
\(705\) 4.08676i 0.153916i
\(706\) 0 0
\(707\) 42.8531i 1.61166i
\(708\) 0 0
\(709\) −15.7826 −0.592728 −0.296364 0.955075i \(-0.595774\pi\)
−0.296364 + 0.955075i \(0.595774\pi\)
\(710\) 0 0
\(711\) −16.2951 −0.611115
\(712\) 0 0
\(713\) 19.5923i 0.733738i
\(714\) 0 0
\(715\) 3.92984 0.146968
\(716\) 0 0
\(717\) 6.10518i 0.228002i
\(718\) 0 0
\(719\) 5.52963i 0.206220i −0.994670 0.103110i \(-0.967121\pi\)
0.994670 0.103110i \(-0.0328794\pi\)
\(720\) 0 0
\(721\) 47.0022i 1.75045i
\(722\) 0 0
\(723\) 2.30218i 0.0856188i
\(724\) 0 0
\(725\) 0.925176i 0.0343602i
\(726\) 0 0
\(727\) 49.6316i 1.84073i 0.391056 + 0.920367i \(0.372110\pi\)
−0.391056 + 0.920367i \(0.627890\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 31.8535i 1.17815i
\(732\) 0 0
\(733\) −42.1979 −1.55862 −0.779308 0.626641i \(-0.784429\pi\)
−0.779308 + 0.626641i \(0.784429\pi\)
\(734\) 0 0
\(735\) 3.31834 0.122399
\(736\) 0 0
\(737\) 37.1504i 1.36845i
\(738\) 0 0
\(739\) 11.8402i 0.435547i 0.975999 + 0.217774i \(0.0698794\pi\)
−0.975999 + 0.217774i \(0.930121\pi\)
\(740\) 0 0
\(741\) −6.54692 + 1.63034i −0.240507 + 0.0598922i
\(742\) 0 0
\(743\) 22.2273 0.815440 0.407720 0.913107i \(-0.366324\pi\)
0.407720 + 0.913107i \(0.366324\pi\)
\(744\) 0 0
\(745\) 12.4894 0.457577
\(746\) 0 0
\(747\) 11.0509i 0.404330i
\(748\) 0 0
\(749\) 46.8219i 1.71083i
\(750\) 0 0
\(751\) 14.2573 0.520257 0.260129 0.965574i \(-0.416235\pi\)
0.260129 + 0.965574i \(0.416235\pi\)
\(752\) 0 0
\(753\) 13.8121i 0.503342i
\(754\) 0 0
\(755\) 7.14498 0.260033
\(756\) 0 0
\(757\) 25.3860 0.922669 0.461335 0.887226i \(-0.347371\pi\)
0.461335 + 0.887226i \(0.347371\pi\)
\(758\) 0 0
\(759\) 15.1991 0.551692
\(760\) 0 0
\(761\) 20.7027 0.750470 0.375235 0.926930i \(-0.377562\pi\)
0.375235 + 0.926930i \(0.377562\pi\)
\(762\) 0 0
\(763\) 32.5792 1.17945
\(764\) 0 0
\(765\) −2.67107 −0.0965726
\(766\) 0 0
\(767\) 14.8455i 0.536040i
\(768\) 0 0
\(769\) −7.80505 −0.281457 −0.140729 0.990048i \(-0.544944\pi\)
−0.140729 + 0.990048i \(0.544944\pi\)
\(770\) 0 0
\(771\) 2.47301i 0.0890634i
\(772\) 0 0
\(773\) 8.05834i 0.289838i 0.989443 + 0.144919i \(0.0462922\pi\)
−0.989443 + 0.144919i \(0.953708\pi\)
\(774\) 0 0
\(775\) −3.27279 −0.117562
\(776\) 0 0
\(777\) 33.1233 1.18829
\(778\) 0 0
\(779\) 38.9077 9.68899i 1.39401 0.347144i
\(780\) 0 0
\(781\) 20.1688i 0.721698i
\(782\) 0 0
\(783\) 0.925176i 0.0330631i
\(784\) 0 0
\(785\) −24.2739 −0.866374
\(786\) 0 0
\(787\) 42.0475 1.49883 0.749417 0.662099i \(-0.230334\pi\)
0.749417 + 0.662099i \(0.230334\pi\)
\(788\) 0 0
\(789\) 24.2851i 0.864572i
\(790\) 0 0
\(791\) 55.3194 1.96693
\(792\) 0 0
\(793\) 5.69861i 0.202364i
\(794\) 0 0
\(795\) 2.52233i 0.0894578i
\(796\) 0 0
\(797\) 34.5098i 1.22240i 0.791477 + 0.611199i \(0.209313\pi\)
−0.791477 + 0.611199i \(0.790687\pi\)
\(798\) 0 0
\(799\) 10.9160i 0.386181i
\(800\) 0 0
\(801\) 2.71016i 0.0957586i
\(802\) 0 0
\(803\) 19.8982i 0.702191i
\(804\) 0