Properties

Label 5520.2.be.b.1471.13
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.be (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} - 45408 x^{7} + 62624 x^{6} - 18048 x^{5} + 2160 x^{4} - 1664 x^{3} + 6272 x^{2} - 896 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.13
Root \(-2.64546 + 2.64546i\) of defining polynomial
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.b.1471.5

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{5} +1.23448 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{5} +1.23448 q^{7} -1.00000 q^{9} +6.11271 q^{11} -0.591966 q^{13} -1.00000 q^{15} -4.95937i q^{17} +7.62424 q^{19} +1.23448i q^{21} +(-4.72490 + 0.821792i) q^{23} -1.00000 q^{25} -1.00000i q^{27} +4.66359 q^{29} -7.50734i q^{31} +6.11271i q^{33} +1.23448i q^{35} +2.86288i q^{37} -0.591966i q^{39} +4.23178 q^{41} +4.79315 q^{43} -1.00000i q^{45} +5.19442i q^{47} -5.47607 q^{49} +4.95937 q^{51} -7.65960i q^{53} +6.11271i q^{55} +7.62424i q^{57} -11.5061i q^{59} -11.5924i q^{61} -1.23448 q^{63} -0.591966i q^{65} -2.90488 q^{67} +(-0.821792 - 4.72490i) q^{69} -15.7877i q^{71} +15.1738 q^{73} -1.00000i q^{75} +7.54599 q^{77} -15.6998 q^{79} +1.00000 q^{81} +3.91936 q^{83} +4.95937 q^{85} +4.66359i q^{87} -0.663092i q^{89} -0.730767 q^{91} +7.50734 q^{93} +7.62424i q^{95} -11.6244i q^{97} -6.11271 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{7} - 16q^{9} + O(q^{10}) \) \( 16q + 8q^{7} - 16q^{9} - 8q^{11} + 8q^{13} - 16q^{15} - 12q^{23} - 16q^{25} - 4q^{29} + 4q^{41} + 20q^{49} + 4q^{51} - 8q^{63} + 16q^{67} + 40q^{73} + 24q^{77} - 32q^{79} + 16q^{81} + 4q^{85} + 48q^{91} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1841\) \(4417\) \(4831\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 1.23448 0.466588 0.233294 0.972406i \(-0.425050\pi\)
0.233294 + 0.972406i \(0.425050\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 6.11271 1.84305 0.921526 0.388317i \(-0.126943\pi\)
0.921526 + 0.388317i \(0.126943\pi\)
\(12\) 0 0
\(13\) −0.591966 −0.164182 −0.0820909 0.996625i \(-0.526160\pi\)
−0.0820909 + 0.996625i \(0.526160\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 4.95937i 1.20282i −0.798939 0.601412i \(-0.794605\pi\)
0.798939 0.601412i \(-0.205395\pi\)
\(18\) 0 0
\(19\) 7.62424 1.74912 0.874560 0.484917i \(-0.161150\pi\)
0.874560 + 0.484917i \(0.161150\pi\)
\(20\) 0 0
\(21\) 1.23448i 0.269385i
\(22\) 0 0
\(23\) −4.72490 + 0.821792i −0.985209 + 0.171356i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 4.66359 0.866007 0.433004 0.901392i \(-0.357454\pi\)
0.433004 + 0.901392i \(0.357454\pi\)
\(30\) 0 0
\(31\) 7.50734i 1.34836i −0.738568 0.674179i \(-0.764498\pi\)
0.738568 0.674179i \(-0.235502\pi\)
\(32\) 0 0
\(33\) 6.11271i 1.06409i
\(34\) 0 0
\(35\) 1.23448i 0.208664i
\(36\) 0 0
\(37\) 2.86288i 0.470654i 0.971916 + 0.235327i \(0.0756162\pi\)
−0.971916 + 0.235327i \(0.924384\pi\)
\(38\) 0 0
\(39\) 0.591966i 0.0947904i
\(40\) 0 0
\(41\) 4.23178 0.660892 0.330446 0.943825i \(-0.392801\pi\)
0.330446 + 0.943825i \(0.392801\pi\)
\(42\) 0 0
\(43\) 4.79315 0.730948 0.365474 0.930821i \(-0.380907\pi\)
0.365474 + 0.930821i \(0.380907\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 5.19442i 0.757685i 0.925461 + 0.378842i \(0.123678\pi\)
−0.925461 + 0.378842i \(0.876322\pi\)
\(48\) 0 0
\(49\) −5.47607 −0.782296
\(50\) 0 0
\(51\) 4.95937 0.694451
\(52\) 0 0
\(53\) 7.65960i 1.05213i −0.850445 0.526064i \(-0.823667\pi\)
0.850445 0.526064i \(-0.176333\pi\)
\(54\) 0 0
\(55\) 6.11271i 0.824238i
\(56\) 0 0
\(57\) 7.62424i 1.00986i
\(58\) 0 0
\(59\) 11.5061i 1.49796i −0.662592 0.748981i \(-0.730543\pi\)
0.662592 0.748981i \(-0.269457\pi\)
\(60\) 0 0
\(61\) 11.5924i 1.48426i −0.670257 0.742129i \(-0.733816\pi\)
0.670257 0.742129i \(-0.266184\pi\)
\(62\) 0 0
\(63\) −1.23448 −0.155529
\(64\) 0 0
\(65\) 0.591966i 0.0734243i
\(66\) 0 0
\(67\) −2.90488 −0.354888 −0.177444 0.984131i \(-0.556783\pi\)
−0.177444 + 0.984131i \(0.556783\pi\)
\(68\) 0 0
\(69\) −0.821792 4.72490i −0.0989322 0.568811i
\(70\) 0 0
\(71\) 15.7877i 1.87365i −0.349796 0.936826i \(-0.613749\pi\)
0.349796 0.936826i \(-0.386251\pi\)
\(72\) 0 0
\(73\) 15.1738 1.77596 0.887980 0.459883i \(-0.152109\pi\)
0.887980 + 0.459883i \(0.152109\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 7.54599 0.859945
\(78\) 0 0
\(79\) −15.6998 −1.76637 −0.883183 0.469028i \(-0.844604\pi\)
−0.883183 + 0.469028i \(0.844604\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.91936 0.430206 0.215103 0.976591i \(-0.430991\pi\)
0.215103 + 0.976591i \(0.430991\pi\)
\(84\) 0 0
\(85\) 4.95937 0.537920
\(86\) 0 0
\(87\) 4.66359i 0.499989i
\(88\) 0 0
\(89\) 0.663092i 0.0702876i −0.999382 0.0351438i \(-0.988811\pi\)
0.999382 0.0351438i \(-0.0111889\pi\)
\(90\) 0 0
\(91\) −0.730767 −0.0766052
\(92\) 0 0
\(93\) 7.50734 0.778475
\(94\) 0 0
\(95\) 7.62424i 0.782230i
\(96\) 0 0
\(97\) 11.6244i 1.18028i −0.807301 0.590140i \(-0.799072\pi\)
0.807301 0.590140i \(-0.200928\pi\)
\(98\) 0 0
\(99\) −6.11271 −0.614351
\(100\) 0 0
\(101\) 1.20872 0.120272 0.0601361 0.998190i \(-0.480847\pi\)
0.0601361 + 0.998190i \(0.480847\pi\)
\(102\) 0 0
\(103\) −10.5686 −1.04136 −0.520680 0.853752i \(-0.674321\pi\)
−0.520680 + 0.853752i \(0.674321\pi\)
\(104\) 0 0
\(105\) −1.23448 −0.120472
\(106\) 0 0
\(107\) −8.55738 −0.827274 −0.413637 0.910442i \(-0.635742\pi\)
−0.413637 + 0.910442i \(0.635742\pi\)
\(108\) 0 0
\(109\) 2.98941i 0.286334i −0.989699 0.143167i \(-0.954271\pi\)
0.989699 0.143167i \(-0.0457286\pi\)
\(110\) 0 0
\(111\) −2.86288 −0.271732
\(112\) 0 0
\(113\) 19.9276i 1.87463i 0.348480 + 0.937316i \(0.386698\pi\)
−0.348480 + 0.937316i \(0.613302\pi\)
\(114\) 0 0
\(115\) −0.821792 4.72490i −0.0766325 0.440599i
\(116\) 0 0
\(117\) 0.591966 0.0547272
\(118\) 0 0
\(119\) 6.12222i 0.561223i
\(120\) 0 0
\(121\) 26.3652 2.39684
\(122\) 0 0
\(123\) 4.23178i 0.381566i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 6.95560i 0.617210i 0.951190 + 0.308605i \(0.0998621\pi\)
−0.951190 + 0.308605i \(0.900138\pi\)
\(128\) 0 0
\(129\) 4.79315i 0.422013i
\(130\) 0 0
\(131\) 8.92419i 0.779710i −0.920876 0.389855i \(-0.872525\pi\)
0.920876 0.389855i \(-0.127475\pi\)
\(132\) 0 0
\(133\) 9.41193 0.816118
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 2.86465i 0.244744i 0.992484 + 0.122372i \(0.0390501\pi\)
−0.992484 + 0.122372i \(0.960950\pi\)
\(138\) 0 0
\(139\) 14.7787i 1.25352i 0.779214 + 0.626758i \(0.215619\pi\)
−0.779214 + 0.626758i \(0.784381\pi\)
\(140\) 0 0
\(141\) −5.19442 −0.437449
\(142\) 0 0
\(143\) −3.61852 −0.302595
\(144\) 0 0
\(145\) 4.66359i 0.387290i
\(146\) 0 0
\(147\) 5.47607i 0.451659i
\(148\) 0 0
\(149\) 9.09665i 0.745227i 0.927987 + 0.372613i \(0.121538\pi\)
−0.927987 + 0.372613i \(0.878462\pi\)
\(150\) 0 0
\(151\) 11.9012i 0.968508i −0.874928 0.484254i \(-0.839091\pi\)
0.874928 0.484254i \(-0.160909\pi\)
\(152\) 0 0
\(153\) 4.95937i 0.400942i
\(154\) 0 0
\(155\) 7.50734 0.603004
\(156\) 0 0
\(157\) 3.87561i 0.309307i 0.987969 + 0.154654i \(0.0494262\pi\)
−0.987969 + 0.154654i \(0.950574\pi\)
\(158\) 0 0
\(159\) 7.65960 0.607446
\(160\) 0 0
\(161\) −5.83277 + 1.01448i −0.459687 + 0.0799524i
\(162\) 0 0
\(163\) 4.19414i 0.328510i 0.986418 + 0.164255i \(0.0525221\pi\)
−0.986418 + 0.164255i \(0.947478\pi\)
\(164\) 0 0
\(165\) −6.11271 −0.475874
\(166\) 0 0
\(167\) 11.9811i 0.927128i −0.886064 0.463564i \(-0.846570\pi\)
0.886064 0.463564i \(-0.153430\pi\)
\(168\) 0 0
\(169\) −12.6496 −0.973044
\(170\) 0 0
\(171\) −7.62424 −0.583040
\(172\) 0 0
\(173\) −11.9589 −0.909221 −0.454610 0.890690i \(-0.650221\pi\)
−0.454610 + 0.890690i \(0.650221\pi\)
\(174\) 0 0
\(175\) −1.23448 −0.0933176
\(176\) 0 0
\(177\) 11.5061 0.864849
\(178\) 0 0
\(179\) 6.19461i 0.463007i −0.972834 0.231504i \(-0.925635\pi\)
0.972834 0.231504i \(-0.0743645\pi\)
\(180\) 0 0
\(181\) 15.7081i 1.16758i 0.811906 + 0.583788i \(0.198430\pi\)
−0.811906 + 0.583788i \(0.801570\pi\)
\(182\) 0 0
\(183\) 11.5924 0.856937
\(184\) 0 0
\(185\) −2.86288 −0.210483
\(186\) 0 0
\(187\) 30.3152i 2.21687i
\(188\) 0 0
\(189\) 1.23448i 0.0897949i
\(190\) 0 0
\(191\) 22.6853 1.64145 0.820724 0.571325i \(-0.193571\pi\)
0.820724 + 0.571325i \(0.193571\pi\)
\(192\) 0 0
\(193\) 17.5627 1.26419 0.632095 0.774891i \(-0.282195\pi\)
0.632095 + 0.774891i \(0.282195\pi\)
\(194\) 0 0
\(195\) 0.591966 0.0423915
\(196\) 0 0
\(197\) 14.4444 1.02912 0.514559 0.857455i \(-0.327956\pi\)
0.514559 + 0.857455i \(0.327956\pi\)
\(198\) 0 0
\(199\) −2.16919 −0.153770 −0.0768850 0.997040i \(-0.524497\pi\)
−0.0768850 + 0.997040i \(0.524497\pi\)
\(200\) 0 0
\(201\) 2.90488i 0.204894i
\(202\) 0 0
\(203\) 5.75709 0.404068
\(204\) 0 0
\(205\) 4.23178i 0.295560i
\(206\) 0 0
\(207\) 4.72490 0.821792i 0.328403 0.0571185i
\(208\) 0 0
\(209\) 46.6048 3.22372
\(210\) 0 0
\(211\) 17.8897i 1.23158i 0.787910 + 0.615790i \(0.211163\pi\)
−0.787910 + 0.615790i \(0.788837\pi\)
\(212\) 0 0
\(213\) 15.7877 1.08175
\(214\) 0 0
\(215\) 4.79315i 0.326890i
\(216\) 0 0
\(217\) 9.26762i 0.629127i
\(218\) 0 0
\(219\) 15.1738i 1.02535i
\(220\) 0 0
\(221\) 2.93578i 0.197482i
\(222\) 0 0
\(223\) 19.1120i 1.27983i 0.768445 + 0.639916i \(0.221031\pi\)
−0.768445 + 0.639916i \(0.778969\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −12.7159 −0.843984 −0.421992 0.906600i \(-0.638669\pi\)
−0.421992 + 0.906600i \(0.638669\pi\)
\(228\) 0 0
\(229\) 22.2900i 1.47296i 0.676458 + 0.736481i \(0.263514\pi\)
−0.676458 + 0.736481i \(0.736486\pi\)
\(230\) 0 0
\(231\) 7.54599i 0.496490i
\(232\) 0 0
\(233\) −20.2452 −1.32631 −0.663153 0.748484i \(-0.730782\pi\)
−0.663153 + 0.748484i \(0.730782\pi\)
\(234\) 0 0
\(235\) −5.19442 −0.338847
\(236\) 0 0
\(237\) 15.6998i 1.01981i
\(238\) 0 0
\(239\) 6.08011i 0.393289i 0.980475 + 0.196645i \(0.0630045\pi\)
−0.980475 + 0.196645i \(0.936995\pi\)
\(240\) 0 0
\(241\) 14.7552i 0.950465i −0.879860 0.475232i \(-0.842364\pi\)
0.879860 0.475232i \(-0.157636\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 5.47607i 0.349853i
\(246\) 0 0
\(247\) −4.51329 −0.287174
\(248\) 0 0
\(249\) 3.91936i 0.248379i
\(250\) 0 0
\(251\) −4.97907 −0.314276 −0.157138 0.987577i \(-0.550227\pi\)
−0.157138 + 0.987577i \(0.550227\pi\)
\(252\) 0 0
\(253\) −28.8819 + 5.02338i −1.81579 + 0.315817i
\(254\) 0 0
\(255\) 4.95937i 0.310568i
\(256\) 0 0
\(257\) 24.0339 1.49919 0.749597 0.661895i \(-0.230247\pi\)
0.749597 + 0.661895i \(0.230247\pi\)
\(258\) 0 0
\(259\) 3.53415i 0.219602i
\(260\) 0 0
\(261\) −4.66359 −0.288669
\(262\) 0 0
\(263\) 13.3344 0.822232 0.411116 0.911583i \(-0.365139\pi\)
0.411116 + 0.911583i \(0.365139\pi\)
\(264\) 0 0
\(265\) 7.65960 0.470526
\(266\) 0 0
\(267\) 0.663092 0.0405805
\(268\) 0 0
\(269\) 29.3882 1.79183 0.895916 0.444222i \(-0.146520\pi\)
0.895916 + 0.444222i \(0.146520\pi\)
\(270\) 0 0
\(271\) 4.93152i 0.299568i −0.988719 0.149784i \(-0.952142\pi\)
0.988719 0.149784i \(-0.0478579\pi\)
\(272\) 0 0
\(273\) 0.730767i 0.0442280i
\(274\) 0 0
\(275\) −6.11271 −0.368610
\(276\) 0 0
\(277\) 1.67570 0.100683 0.0503415 0.998732i \(-0.483969\pi\)
0.0503415 + 0.998732i \(0.483969\pi\)
\(278\) 0 0
\(279\) 7.50734i 0.449453i
\(280\) 0 0
\(281\) 1.09883i 0.0655510i −0.999463 0.0327755i \(-0.989565\pi\)
0.999463 0.0327755i \(-0.0104346\pi\)
\(282\) 0 0
\(283\) −22.0056 −1.30810 −0.654049 0.756452i \(-0.726931\pi\)
−0.654049 + 0.756452i \(0.726931\pi\)
\(284\) 0 0
\(285\) −7.62424 −0.451621
\(286\) 0 0
\(287\) 5.22402 0.308364
\(288\) 0 0
\(289\) −7.59538 −0.446787
\(290\) 0 0
\(291\) 11.6244 0.681435
\(292\) 0 0
\(293\) 21.5662i 1.25991i 0.776632 + 0.629955i \(0.216927\pi\)
−0.776632 + 0.629955i \(0.783073\pi\)
\(294\) 0 0
\(295\) 11.5061 0.669909
\(296\) 0 0
\(297\) 6.11271i 0.354695i
\(298\) 0 0
\(299\) 2.79698 0.486473i 0.161753 0.0281334i
\(300\) 0 0
\(301\) 5.91702 0.341052
\(302\) 0 0
\(303\) 1.20872i 0.0694392i
\(304\) 0 0
\(305\) 11.5924 0.663780
\(306\) 0 0
\(307\) 2.70481i 0.154372i 0.997017 + 0.0771859i \(0.0245935\pi\)
−0.997017 + 0.0771859i \(0.975407\pi\)
\(308\) 0 0
\(309\) 10.5686i 0.601229i
\(310\) 0 0
\(311\) 14.5678i 0.826062i 0.910717 + 0.413031i \(0.135530\pi\)
−0.910717 + 0.413031i \(0.864470\pi\)
\(312\) 0 0
\(313\) 18.2833i 1.03343i 0.856156 + 0.516717i \(0.172846\pi\)
−0.856156 + 0.516717i \(0.827154\pi\)
\(314\) 0 0
\(315\) 1.23448i 0.0695548i
\(316\) 0 0
\(317\) −13.2025 −0.741526 −0.370763 0.928728i \(-0.620904\pi\)
−0.370763 + 0.928728i \(0.620904\pi\)
\(318\) 0 0
\(319\) 28.5072 1.59610
\(320\) 0 0
\(321\) 8.55738i 0.477627i
\(322\) 0 0
\(323\) 37.8114i 2.10388i
\(324\) 0 0
\(325\) 0.591966 0.0328363
\(326\) 0 0
\(327\) 2.98941 0.165315
\(328\) 0 0
\(329\) 6.41239i 0.353526i
\(330\) 0 0
\(331\) 20.3479i 1.11842i 0.829025 + 0.559212i \(0.188896\pi\)
−0.829025 + 0.559212i \(0.811104\pi\)
\(332\) 0 0
\(333\) 2.86288i 0.156885i
\(334\) 0 0
\(335\) 2.90488i 0.158711i
\(336\) 0 0
\(337\) 10.8261i 0.589736i 0.955538 + 0.294868i \(0.0952756\pi\)
−0.955538 + 0.294868i \(0.904724\pi\)
\(338\) 0 0
\(339\) −19.9276 −1.08232
\(340\) 0 0
\(341\) 45.8902i 2.48509i
\(342\) 0 0
\(343\) −15.4014 −0.831597
\(344\) 0 0
\(345\) 4.72490 0.821792i 0.254380 0.0442438i
\(346\) 0 0
\(347\) 20.8259i 1.11799i −0.829170 0.558996i \(-0.811187\pi\)
0.829170 0.558996i \(-0.188813\pi\)
\(348\) 0 0
\(349\) 17.0430 0.912289 0.456145 0.889906i \(-0.349230\pi\)
0.456145 + 0.889906i \(0.349230\pi\)
\(350\) 0 0
\(351\) 0.591966i 0.0315968i
\(352\) 0 0
\(353\) −11.1232 −0.592030 −0.296015 0.955183i \(-0.595658\pi\)
−0.296015 + 0.955183i \(0.595658\pi\)
\(354\) 0 0
\(355\) 15.7877 0.837922
\(356\) 0 0
\(357\) 6.12222 0.324022
\(358\) 0 0
\(359\) −3.33824 −0.176185 −0.0880927 0.996112i \(-0.528077\pi\)
−0.0880927 + 0.996112i \(0.528077\pi\)
\(360\) 0 0
\(361\) 39.1290 2.05942
\(362\) 0 0
\(363\) 26.3652i 1.38382i
\(364\) 0 0
\(365\) 15.1738i 0.794233i
\(366\) 0 0
\(367\) −15.5263 −0.810468 −0.405234 0.914213i \(-0.632810\pi\)
−0.405234 + 0.914213i \(0.632810\pi\)
\(368\) 0 0
\(369\) −4.23178 −0.220297
\(370\) 0 0
\(371\) 9.45559i 0.490910i
\(372\) 0 0
\(373\) 28.8438i 1.49347i 0.665120 + 0.746737i \(0.268381\pi\)
−0.665120 + 0.746737i \(0.731619\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −2.76069 −0.142183
\(378\) 0 0
\(379\) −5.77317 −0.296548 −0.148274 0.988946i \(-0.547372\pi\)
−0.148274 + 0.988946i \(0.547372\pi\)
\(380\) 0 0
\(381\) −6.95560 −0.356346
\(382\) 0 0
\(383\) 28.4701 1.45475 0.727376 0.686239i \(-0.240740\pi\)
0.727376 + 0.686239i \(0.240740\pi\)
\(384\) 0 0
\(385\) 7.54599i 0.384579i
\(386\) 0 0
\(387\) −4.79315 −0.243649
\(388\) 0 0
\(389\) 28.4547i 1.44271i −0.692564 0.721356i \(-0.743519\pi\)
0.692564 0.721356i \(-0.256481\pi\)
\(390\) 0 0
\(391\) 4.07557 + 23.4325i 0.206111 + 1.18503i
\(392\) 0 0
\(393\) 8.92419 0.450166
\(394\) 0 0
\(395\) 15.6998i 0.789943i
\(396\) 0 0
\(397\) 19.4303 0.975181 0.487590 0.873073i \(-0.337876\pi\)
0.487590 + 0.873073i \(0.337876\pi\)
\(398\) 0 0
\(399\) 9.41193i 0.471186i
\(400\) 0 0
\(401\) 1.04759i 0.0523141i 0.999658 + 0.0261571i \(0.00832700\pi\)
−0.999658 + 0.0261571i \(0.991673\pi\)
\(402\) 0 0
\(403\) 4.44409i 0.221376i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 17.4999i 0.867440i
\(408\) 0 0
\(409\) −22.6624 −1.12058 −0.560292 0.828295i \(-0.689311\pi\)
−0.560292 + 0.828295i \(0.689311\pi\)
\(410\) 0 0
\(411\) −2.86465 −0.141303
\(412\) 0 0
\(413\) 14.2040i 0.698931i
\(414\) 0 0
\(415\) 3.91936i 0.192394i
\(416\) 0 0
\(417\) −14.7787 −0.723718
\(418\) 0 0
\(419\) −11.2006 −0.547184 −0.273592 0.961846i \(-0.588212\pi\)
−0.273592 + 0.961846i \(0.588212\pi\)
\(420\) 0 0
\(421\) 11.6665i 0.568591i 0.958737 + 0.284295i \(0.0917596\pi\)
−0.958737 + 0.284295i \(0.908240\pi\)
\(422\) 0 0
\(423\) 5.19442i 0.252562i
\(424\) 0 0
\(425\) 4.95937i 0.240565i
\(426\) 0 0
\(427\) 14.3106i 0.692537i
\(428\) 0 0
\(429\) 3.61852i 0.174704i
\(430\) 0 0
\(431\) −14.6855 −0.707374 −0.353687 0.935364i \(-0.615072\pi\)
−0.353687 + 0.935364i \(0.615072\pi\)
\(432\) 0 0
\(433\) 29.7196i 1.42823i 0.700027 + 0.714116i \(0.253171\pi\)
−0.700027 + 0.714116i \(0.746829\pi\)
\(434\) 0 0
\(435\) −4.66359 −0.223602
\(436\) 0 0
\(437\) −36.0237 + 6.26554i −1.72325 + 0.299721i
\(438\) 0 0
\(439\) 16.5496i 0.789870i 0.918709 + 0.394935i \(0.129233\pi\)
−0.918709 + 0.394935i \(0.870767\pi\)
\(440\) 0 0
\(441\) 5.47607 0.260765
\(442\) 0 0
\(443\) 28.9462i 1.37528i −0.726054 0.687638i \(-0.758648\pi\)
0.726054 0.687638i \(-0.241352\pi\)
\(444\) 0 0
\(445\) 0.663092 0.0314336
\(446\) 0 0
\(447\) −9.09665 −0.430257
\(448\) 0 0
\(449\) 38.6399 1.82353 0.911765 0.410713i \(-0.134720\pi\)
0.911765 + 0.410713i \(0.134720\pi\)
\(450\) 0 0
\(451\) 25.8676 1.21806
\(452\) 0 0
\(453\) 11.9012 0.559168
\(454\) 0 0
\(455\) 0.730767i 0.0342589i
\(456\) 0 0
\(457\) 21.7846i 1.01904i −0.860458 0.509521i \(-0.829823\pi\)
0.860458 0.509521i \(-0.170177\pi\)
\(458\) 0 0
\(459\) −4.95937 −0.231484
\(460\) 0 0
\(461\) −3.04092 −0.141630 −0.0708150 0.997489i \(-0.522560\pi\)
−0.0708150 + 0.997489i \(0.522560\pi\)
\(462\) 0 0
\(463\) 9.10105i 0.422962i −0.977382 0.211481i \(-0.932171\pi\)
0.977382 0.211481i \(-0.0678286\pi\)
\(464\) 0 0
\(465\) 7.50734i 0.348144i
\(466\) 0 0
\(467\) 41.3771 1.91470 0.957352 0.288925i \(-0.0932980\pi\)
0.957352 + 0.288925i \(0.0932980\pi\)
\(468\) 0 0
\(469\) −3.58600 −0.165586
\(470\) 0 0
\(471\) −3.87561 −0.178579
\(472\) 0 0
\(473\) 29.2991 1.34718
\(474\) 0 0
\(475\) −7.62424 −0.349824
\(476\) 0 0
\(477\) 7.65960i 0.350709i
\(478\) 0 0
\(479\) −7.89200 −0.360595 −0.180297 0.983612i \(-0.557706\pi\)
−0.180297 + 0.983612i \(0.557706\pi\)
\(480\) 0 0
\(481\) 1.69473i 0.0772728i
\(482\) 0 0
\(483\) −1.01448 5.83277i −0.0461605 0.265400i
\(484\) 0 0
\(485\) 11.6244 0.527837
\(486\) 0 0
\(487\) 36.3469i 1.64704i 0.567291 + 0.823518i \(0.307992\pi\)
−0.567291 + 0.823518i \(0.692008\pi\)
\(488\) 0 0
\(489\) −4.19414 −0.189666
\(490\) 0 0
\(491\) 39.3024i 1.77369i 0.462065 + 0.886846i \(0.347109\pi\)
−0.462065 + 0.886846i \(0.652891\pi\)
\(492\) 0 0
\(493\) 23.1285i 1.04165i
\(494\) 0 0
\(495\) 6.11271i 0.274746i
\(496\) 0 0
\(497\) 19.4895i 0.874223i
\(498\) 0 0
\(499\) 15.2335i 0.681946i 0.940073 + 0.340973i \(0.110757\pi\)
−0.940073 + 0.340973i \(0.889243\pi\)
\(500\) 0 0
\(501\) 11.9811 0.535277
\(502\) 0 0
\(503\) −16.2999 −0.726776 −0.363388 0.931638i \(-0.618380\pi\)
−0.363388 + 0.931638i \(0.618380\pi\)
\(504\) 0 0
\(505\) 1.20872i 0.0537874i
\(506\) 0 0
\(507\) 12.6496i 0.561787i
\(508\) 0 0
\(509\) −31.6751 −1.40397 −0.701987 0.712189i \(-0.747704\pi\)
−0.701987 + 0.712189i \(0.747704\pi\)
\(510\) 0 0
\(511\) 18.7317 0.828641
\(512\) 0 0
\(513\) 7.62424i 0.336618i
\(514\) 0 0
\(515\) 10.5686i 0.465710i
\(516\) 0 0
\(517\) 31.7520i 1.39645i
\(518\) 0 0
\(519\) 11.9589i 0.524939i
\(520\) 0 0
\(521\) 24.8736i 1.08973i −0.838524 0.544865i \(-0.816581\pi\)
0.838524 0.544865i \(-0.183419\pi\)
\(522\) 0 0
\(523\) −5.37517 −0.235040 −0.117520 0.993071i \(-0.537494\pi\)
−0.117520 + 0.993071i \(0.537494\pi\)
\(524\) 0 0
\(525\) 1.23448i 0.0538769i
\(526\) 0 0
\(527\) −37.2317 −1.62184
\(528\) 0 0
\(529\) 21.6493 7.76577i 0.941275 0.337642i
\(530\) 0 0
\(531\) 11.5061i 0.499321i
\(532\) 0 0
\(533\) −2.50507 −0.108506
\(534\) 0 0
\(535\) 8.55738i 0.369968i
\(536\) 0 0
\(537\) 6.19461 0.267317
\(538\) 0 0
\(539\) −33.4736 −1.44181
\(540\) 0 0
\(541\) −30.7344 −1.32137 −0.660687 0.750662i \(-0.729735\pi\)
−0.660687 + 0.750662i \(0.729735\pi\)
\(542\) 0 0
\(543\) −15.7081 −0.674100
\(544\) 0 0
\(545\) 2.98941 0.128052
\(546\) 0 0
\(547\) 34.5025i 1.47522i 0.675226 + 0.737611i \(0.264046\pi\)
−0.675226 + 0.737611i \(0.735954\pi\)
\(548\) 0 0
\(549\) 11.5924i 0.494753i
\(550\) 0 0
\(551\) 35.5563 1.51475
\(552\) 0 0
\(553\) −19.3810 −0.824165
\(554\) 0 0
\(555\) 2.86288i 0.121522i
\(556\) 0 0
\(557\) 12.2831i 0.520452i −0.965548 0.260226i \(-0.916203\pi\)
0.965548 0.260226i \(-0.0837971\pi\)
\(558\) 0 0
\(559\) −2.83738 −0.120008
\(560\) 0 0
\(561\) 30.3152 1.27991
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 0.651545i \(-0.774121\pi\)
−0.758610 + 0.651545i \(0.774121\pi\)
\(564\) 0 0
\(565\) −19.9276 −0.838361
\(566\) 0 0
\(567\) 1.23448 0.0518431
\(568\) 0 0
\(569\) 36.4118i 1.52646i 0.646126 + 0.763231i \(0.276388\pi\)
−0.646126 + 0.763231i \(0.723612\pi\)
\(570\) 0 0
\(571\) −1.29464 −0.0541789 −0.0270895 0.999633i \(-0.508624\pi\)
−0.0270895 + 0.999633i \(0.508624\pi\)
\(572\) 0 0
\(573\) 22.6853i 0.947690i
\(574\) 0 0
\(575\) 4.72490 0.821792i 0.197042 0.0342711i
\(576\) 0 0
\(577\) 32.3549 1.34695 0.673476 0.739209i \(-0.264800\pi\)
0.673476 + 0.739209i \(0.264800\pi\)
\(578\) 0 0
\(579\) 17.5627i 0.729880i
\(580\) 0 0
\(581\) 4.83836 0.200729
\(582\) 0 0
\(583\) 46.8209i 1.93912i
\(584\) 0 0
\(585\) 0.591966i 0.0244748i
\(586\) 0 0
\(587\) 24.4859i 1.01064i −0.862932 0.505320i \(-0.831374\pi\)
0.862932 0.505320i \(-0.168626\pi\)
\(588\) 0 0
\(589\) 57.2377i 2.35844i
\(590\) 0 0
\(591\) 14.4444i 0.594161i
\(592\) 0 0
\(593\) −12.5612 −0.515826 −0.257913 0.966168i \(-0.583035\pi\)
−0.257913 + 0.966168i \(0.583035\pi\)
\(594\) 0 0
\(595\) 6.12222 0.250987
\(596\) 0 0
\(597\) 2.16919i 0.0887792i
\(598\) 0 0
\(599\) 45.7320i 1.86856i −0.356541 0.934280i \(-0.616044\pi\)
0.356541 0.934280i \(-0.383956\pi\)
\(600\) 0 0
\(601\) −0.449630 −0.0183408 −0.00917039 0.999958i \(-0.502919\pi\)
−0.00917039 + 0.999958i \(0.502919\pi\)
\(602\) 0 0
\(603\) 2.90488 0.118296
\(604\) 0 0
\(605\) 26.3652i 1.07190i
\(606\) 0 0
\(607\) 25.2249i 1.02385i 0.859031 + 0.511924i \(0.171067\pi\)
−0.859031 + 0.511924i \(0.828933\pi\)
\(608\) 0 0
\(609\) 5.75709i 0.233289i
\(610\) 0 0
\(611\) 3.07492i 0.124398i
\(612\) 0 0
\(613\) 23.6265i 0.954264i 0.878832 + 0.477132i \(0.158324\pi\)
−0.878832 + 0.477132i \(0.841676\pi\)
\(614\) 0 0
\(615\) −4.23178 −0.170642
\(616\) 0 0
\(617\) 31.3914i 1.26377i 0.775063 + 0.631884i \(0.217718\pi\)
−0.775063 + 0.631884i \(0.782282\pi\)
\(618\) 0 0
\(619\) 15.1598 0.609322 0.304661 0.952461i \(-0.401457\pi\)
0.304661 + 0.952461i \(0.401457\pi\)
\(620\) 0 0
\(621\) 0.821792 + 4.72490i 0.0329774 + 0.189604i
\(622\) 0 0
\(623\) 0.818570i 0.0327953i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 46.6048i 1.86122i
\(628\) 0 0
\(629\) 14.1981 0.566115
\(630\) 0 0
\(631\) −33.7377 −1.34308 −0.671538 0.740970i \(-0.734366\pi\)
−0.671538 + 0.740970i \(0.734366\pi\)
\(632\) 0 0
\(633\) −17.8897 −0.711053
\(634\) 0 0
\(635\) −6.95560 −0.276025
\(636\) 0 0
\(637\) 3.24165 0.128439
\(638\) 0 0
\(639\) 15.7877i 0.624551i
\(640\) 0 0
\(641\) 20.8579i 0.823837i 0.911221 + 0.411918i \(0.135141\pi\)
−0.911221 + 0.411918i \(0.864859\pi\)
\(642\) 0 0
\(643\) 40.3538 1.59140 0.795700 0.605692i \(-0.207103\pi\)
0.795700 + 0.605692i \(0.207103\pi\)
\(644\) 0 0
\(645\) −4.79315 −0.188730
\(646\) 0 0
\(647\) 11.2696i 0.443054i 0.975154 + 0.221527i \(0.0711041\pi\)
−0.975154 + 0.221527i \(0.928896\pi\)
\(648\) 0 0
\(649\) 70.3332i 2.76082i
\(650\) 0 0
\(651\) 9.26762 0.363227
\(652\) 0 0
\(653\) −13.7439 −0.537842 −0.268921 0.963162i \(-0.586667\pi\)
−0.268921 + 0.963162i \(0.586667\pi\)
\(654\) 0 0
\(655\) 8.92419 0.348697
\(656\) 0 0
\(657\) −15.1738 −0.591986
\(658\) 0 0
\(659\) 10.1309 0.394645 0.197322 0.980339i \(-0.436775\pi\)
0.197322 + 0.980339i \(0.436775\pi\)
\(660\) 0 0
\(661\) 36.1223i 1.40499i −0.711686 0.702497i \(-0.752068\pi\)
0.711686 0.702497i \(-0.247932\pi\)
\(662\) 0 0
\(663\) −2.93578 −0.114016
\(664\) 0 0
\(665\) 9.41193i 0.364979i
\(666\) 0 0
\(667\) −22.0350 + 3.83250i −0.853198 + 0.148395i
\(668\) 0 0
\(669\) −19.1120 −0.738912
\(670\) 0 0
\(671\) 70.8611i 2.73556i
\(672\) 0 0
\(673\) 30.2950 1.16779 0.583893 0.811831i \(-0.301529\pi\)
0.583893 + 0.811831i \(0.301529\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 46.2931i 1.77919i −0.456751 0.889594i \(-0.650987\pi\)
0.456751 0.889594i \(-0.349013\pi\)
\(678\) 0 0
\(679\) 14.3501i 0.550704i
\(680\) 0 0
\(681\) 12.7159i 0.487274i
\(682\) 0 0
\(683\) 13.7794i 0.527256i −0.964624 0.263628i \(-0.915081\pi\)
0.964624 0.263628i \(-0.0849191\pi\)
\(684\) 0 0
\(685\) −2.86465 −0.109453
\(686\) 0 0
\(687\) −22.2900 −0.850415
\(688\) 0 0
\(689\) 4.53422i 0.172740i
\(690\) 0 0
\(691\) 28.6475i 1.08980i 0.838500 + 0.544902i \(0.183433\pi\)
−0.838500 + 0.544902i \(0.816567\pi\)
\(692\) 0 0
\(693\) −7.54599 −0.286648
\(694\) 0 0
\(695\) −14.7787 −0.560589
\(696\) 0 0
\(697\) 20.9870i 0.794938i
\(698\) 0 0
\(699\) 20.2452i 0.765743i
\(700\) 0 0
\(701\) 17.6434i 0.666381i −0.942859 0.333191i \(-0.891875\pi\)
0.942859 0.333191i \(-0.108125\pi\)
\(702\) 0 0
\(703\) 21.8273i 0.823231i
\(704\) 0 0
\(705\) 5.19442i 0.195633i
\(706\) 0 0
\(707\) 1.49214 0.0561176
\(708\) 0 0
\(709\) 14.2902i 0.536679i 0.963324 + 0.268339i \(0.0864749\pi\)
−0.963324 + 0.268339i \(0.913525\pi\)
\(710\) 0 0
\(711\) 15.6998 0.588789
\(712\) 0 0
\(713\) 6.16947 + 35.4714i 0.231049 + 1.32841i
\(714\) 0 0
\(715\) 3.61852i 0.135325i
\(716\) 0 0
\(717\) −6.08011 −0.227066
\(718\) 0 0
\(719\) 20.3841i 0.760200i −0.924945 0.380100i \(-0.875890\pi\)
0.924945 0.380100i \(-0.124110\pi\)
\(720\) 0 0
\(721\) −13.0467 −0.485886
\(722\) 0 0
\(723\) 14.7552 0.548751
\(724\) 0 0
\(725\) −4.66359 −0.173201
\(726\) 0 0
\(727\) −28.5609 −1.05926 −0.529632 0.848228i \(-0.677670\pi\)
−0.529632 + 0.848228i \(0.677670\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 23.7710i 0.879203i
\(732\) 0 0
\(733\) 1.55848i 0.0575639i −0.999586 0.0287819i \(-0.990837\pi\)
0.999586 0.0287819i \(-0.00916284\pi\)
\(734\) 0 0
\(735\) 5.47607 0.201988
\(736\) 0 0
\(737\) −17.7567 −0.654076
\(738\) 0 0
\(739\) 7.69465i 0.283052i −0.989935 0.141526i \(-0.954799\pi\)
0.989935 0.141526i \(-0.0452009\pi\)
\(740\) 0 0
\(741\) 4.51329i 0.165800i
\(742\) 0 0
\(743\) 0.862609 0.0316461 0.0158230 0.999875i \(-0.494963\pi\)
0.0158230 + 0.999875i \(0.494963\pi\)
\(744\) 0 0
\(745\) −9.09665 −0.333276
\(746\) 0 0
\(747\) −3.91936 −0.143402
\(748\) 0 0
\(749\) −10.5639 −0.385996
\(750\) 0 0
\(751\) −3.83821 −0.140058 −0.0700292 0.997545i \(-0.522309\pi\)
−0.0700292 + 0.997545i \(0.522309\pi\)
\(752\) 0 0
\(753\) 4.97907i 0.181447i
\(754\) 0 0
\(755\) 11.9012 0.433130
\(756\) 0 0
\(757\) 25.5793i 0.929696i −0.885391 0.464848i \(-0.846109\pi\)
0.885391 0.464848i \(-0.153891\pi\)
\(758\) 0 0
\(759\) −5.02338 28.8819i −0.182337 1.04835i
\(760\) 0 0
\(761\) −19.4410 −0.704735 −0.352367 0.935862i \(-0.614623\pi\)
−0.352367 + 0.935862i \(0.614623\pi\)
\(762\) 0 0
\(763\) 3.69036i 0.133600i
\(764\) 0 0
\(765\) −4.95937 −0.179307
\(766\) 0 0
\(767\) 6.81119i 0.245938i
\(768\) 0 0
\(769\) 36.0564i 1.30023i 0.759838 + 0.650113i \(0.225278\pi\)
−0.759838 + 0.650113i \(0.774722\pi\)
\(770\) 0 0
\(771\) 24.0339i 0.865560i
\(772\) 0 0
\(773\) 16.6369i 0.598389i 0.954192 + 0.299194i \(0.0967179\pi\)
−0.954192 + 0.299194i \(0.903282\pi\)
\(774\) 0 0
\(775\) 7.50734i 0.269672i
\(776\) 0 0
\(777\) −3.53415 −0.126787
\(778\) 0 0
\(779\) 32.2641 1.15598
\(780\) 0 0
\(781\) 96.5055i 3.45324i
\(782\) 0 0
\(783\) 4.66359i 0.166663i
\(784\) 0 0
\(785\) −3.87561 −0.138327
\(786\) 0 0
\(787\) −13.5394 −0.482628 −0.241314 0.970447i \(-0.577578\pi\)
−0.241314 + 0.970447i \(0.577578\pi\)
\(788\) 0 0
\(789\) 13.3344i 0.474716i
\(790\) 0 0
\(791\) 24.6001i 0.874681i
\(792\) 0 0
\(793\) 6.86232i 0.243688i
\(794\) 0 0
\(795\) 7.65960i 0.271658i
\(796\) 0 0
\(797\) 24.8048i 0.878633i 0.898332 + 0.439316i \(0.144779\pi\)
−0.898332 + 0.439316i \(0.855221\pi\)
\(798\) 0 0
\(799\) 25.7611 0.911362
\(800\) 0 0
\(801\) 0.663092i 0.0234292i
\(802\) 0 0
\(803\) 92.7531 3.27318
\(804\) 0 0
\(805\) −1.014