Properties

Label 5520.2.be.a.1471.1
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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 1471.1
Root \(0.0727486 - 0.0727486i\) of defining polynomial
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.a.1471.9

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.00000i q^{5} -4.59925 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +1.00000i q^{5} -4.59925 q^{7} -1.00000 q^{9} +4.88207 q^{11} +1.41451 q^{13} +1.00000 q^{15} -4.35085i q^{17} -1.34013 q^{19} +4.59925i q^{21} +(0.751600 + 4.73657i) q^{23} -1.00000 q^{25} +1.00000i q^{27} -8.47505 q^{29} +4.83021i q^{31} -4.88207i q^{33} -4.59925i q^{35} -2.55810i q^{37} -1.41451i q^{39} -1.94751 q^{41} +9.76592 q^{43} -1.00000i q^{45} +5.05444i q^{47} +14.1531 q^{49} -4.35085 q^{51} -11.0842i q^{53} +4.88207i q^{55} +1.34013i q^{57} +1.36588i q^{59} -9.11481i q^{61} +4.59925 q^{63} +1.41451i q^{65} +8.47015 q^{67} +(4.73657 - 0.751600i) q^{69} -0.721095i q^{71} +2.29450 q^{73} +1.00000i q^{75} -22.4538 q^{77} -15.2680 q^{79} +1.00000 q^{81} +13.3145 q^{83} +4.35085 q^{85} +8.47505i q^{87} +8.99269i q^{89} -6.50567 q^{91} +4.83021 q^{93} -1.34013i q^{95} +1.26729i q^{97} -4.88207 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 8q^{7} - 16q^{9} + O(q^{10}) \) \( 16q - 8q^{7} - 16q^{9} + 8q^{11} + 8q^{13} + 16q^{15} + 12q^{23} - 16q^{25} - 4q^{29} + 4q^{41} + 20q^{49} - 4q^{51} + 8q^{63} - 16q^{67} + 40q^{73} + 24q^{77} + 32q^{79} + 16q^{81} + 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

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
\(7\) −4.59925 −1.73835 −0.869176 0.494502i \(-0.835350\pi\)
−0.869176 + 0.494502i \(0.835350\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.88207 1.47200 0.735999 0.676982i \(-0.236713\pi\)
0.735999 + 0.676982i \(0.236713\pi\)
\(12\) 0 0
\(13\) 1.41451 0.392314 0.196157 0.980573i \(-0.437154\pi\)
0.196157 + 0.980573i \(0.437154\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 4.35085i 1.05524i −0.849482 0.527618i \(-0.823085\pi\)
0.849482 0.527618i \(-0.176915\pi\)
\(18\) 0 0
\(19\) −1.34013 −0.307448 −0.153724 0.988114i \(-0.549127\pi\)
−0.153724 + 0.988114i \(0.549127\pi\)
\(20\) 0 0
\(21\) 4.59925i 1.00364i
\(22\) 0 0
\(23\) 0.751600 + 4.73657i 0.156719 + 0.987643i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −8.47505 −1.57378 −0.786888 0.617095i \(-0.788309\pi\)
−0.786888 + 0.617095i \(0.788309\pi\)
\(30\) 0 0
\(31\) 4.83021i 0.867531i 0.901026 + 0.433766i \(0.142815\pi\)
−0.901026 + 0.433766i \(0.857185\pi\)
\(32\) 0 0
\(33\) 4.88207i 0.849859i
\(34\) 0 0
\(35\) 4.59925i 0.777415i
\(36\) 0 0
\(37\) 2.55810i 0.420549i −0.977642 0.210274i \(-0.932564\pi\)
0.977642 0.210274i \(-0.0674357\pi\)
\(38\) 0 0
\(39\) 1.41451i 0.226502i
\(40\) 0 0
\(41\) −1.94751 −0.304150 −0.152075 0.988369i \(-0.548596\pi\)
−0.152075 + 0.988369i \(0.548596\pi\)
\(42\) 0 0
\(43\) 9.76592 1.48929 0.744644 0.667461i \(-0.232619\pi\)
0.744644 + 0.667461i \(0.232619\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 5.05444i 0.737266i 0.929575 + 0.368633i \(0.120174\pi\)
−0.929575 + 0.368633i \(0.879826\pi\)
\(48\) 0 0
\(49\) 14.1531 2.02187
\(50\) 0 0
\(51\) −4.35085 −0.609241
\(52\) 0 0
\(53\) 11.0842i 1.52253i −0.648441 0.761265i \(-0.724578\pi\)
0.648441 0.761265i \(-0.275422\pi\)
\(54\) 0 0
\(55\) 4.88207i 0.658298i
\(56\) 0 0
\(57\) 1.34013i 0.177505i
\(58\) 0 0
\(59\) 1.36588i 0.177822i 0.996040 + 0.0889111i \(0.0283387\pi\)
−0.996040 + 0.0889111i \(0.971661\pi\)
\(60\) 0 0
\(61\) 9.11481i 1.16703i −0.812102 0.583516i \(-0.801676\pi\)
0.812102 0.583516i \(-0.198324\pi\)
\(62\) 0 0
\(63\) 4.59925 0.579451
\(64\) 0 0
\(65\) 1.41451i 0.175448i
\(66\) 0 0
\(67\) 8.47015 1.03479 0.517397 0.855746i \(-0.326901\pi\)
0.517397 + 0.855746i \(0.326901\pi\)
\(68\) 0 0
\(69\) 4.73657 0.751600i 0.570216 0.0904820i
\(70\) 0 0
\(71\) 0.721095i 0.0855783i −0.999084 0.0427891i \(-0.986376\pi\)
0.999084 0.0427891i \(-0.0136244\pi\)
\(72\) 0 0
\(73\) 2.29450 0.268551 0.134275 0.990944i \(-0.457129\pi\)
0.134275 + 0.990944i \(0.457129\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) −22.4538 −2.55885
\(78\) 0 0
\(79\) −15.2680 −1.71778 −0.858890 0.512160i \(-0.828846\pi\)
−0.858890 + 0.512160i \(0.828846\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.3145 1.46146 0.730729 0.682667i \(-0.239180\pi\)
0.730729 + 0.682667i \(0.239180\pi\)
\(84\) 0 0
\(85\) 4.35085 0.471916
\(86\) 0 0
\(87\) 8.47505i 0.908620i
\(88\) 0 0
\(89\) 8.99269i 0.953223i 0.879114 + 0.476612i \(0.158135\pi\)
−0.879114 + 0.476612i \(0.841865\pi\)
\(90\) 0 0
\(91\) −6.50567 −0.681980
\(92\) 0 0
\(93\) 4.83021 0.500869
\(94\) 0 0
\(95\) 1.34013i 0.137495i
\(96\) 0 0
\(97\) 1.26729i 0.128674i 0.997928 + 0.0643370i \(0.0204932\pi\)
−0.997928 + 0.0643370i \(0.979507\pi\)
\(98\) 0 0
\(99\) −4.88207 −0.490666
\(100\) 0 0
\(101\) −14.3919 −1.43205 −0.716025 0.698075i \(-0.754040\pi\)
−0.716025 + 0.698075i \(0.754040\pi\)
\(102\) 0 0
\(103\) 17.0569 1.68066 0.840331 0.542074i \(-0.182361\pi\)
0.840331 + 0.542074i \(0.182361\pi\)
\(104\) 0 0
\(105\) −4.59925 −0.448841
\(106\) 0 0
\(107\) −3.92835 −0.379768 −0.189884 0.981807i \(-0.560811\pi\)
−0.189884 + 0.981807i \(0.560811\pi\)
\(108\) 0 0
\(109\) 5.40581i 0.517783i 0.965906 + 0.258891i \(0.0833571\pi\)
−0.965906 + 0.258891i \(0.916643\pi\)
\(110\) 0 0
\(111\) −2.55810 −0.242804
\(112\) 0 0
\(113\) 13.5574i 1.27537i −0.770296 0.637686i \(-0.779892\pi\)
0.770296 0.637686i \(-0.220108\pi\)
\(114\) 0 0
\(115\) −4.73657 + 0.751600i −0.441687 + 0.0700870i
\(116\) 0 0
\(117\) −1.41451 −0.130771
\(118\) 0 0
\(119\) 20.0106i 1.83437i
\(120\) 0 0
\(121\) 12.8346 1.16678
\(122\) 0 0
\(123\) 1.94751i 0.175601i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 13.2910i 1.17938i −0.807628 0.589692i \(-0.799249\pi\)
0.807628 0.589692i \(-0.200751\pi\)
\(128\) 0 0
\(129\) 9.76592i 0.859841i
\(130\) 0 0
\(131\) 8.00063i 0.699019i −0.936933 0.349509i \(-0.886348\pi\)
0.936933 0.349509i \(-0.113652\pi\)
\(132\) 0 0
\(133\) 6.16361 0.534453
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 1.42388i 0.121650i 0.998148 + 0.0608251i \(0.0193732\pi\)
−0.998148 + 0.0608251i \(0.980627\pi\)
\(138\) 0 0
\(139\) 18.1054i 1.53568i −0.640641 0.767841i \(-0.721331\pi\)
0.640641 0.767841i \(-0.278669\pi\)
\(140\) 0 0
\(141\) 5.05444 0.425661
\(142\) 0 0
\(143\) 6.90572 0.577485
\(144\) 0 0
\(145\) 8.47505i 0.703814i
\(146\) 0 0
\(147\) 14.1531i 1.16733i
\(148\) 0 0
\(149\) 13.4277i 1.10004i −0.835151 0.550021i \(-0.814620\pi\)
0.835151 0.550021i \(-0.185380\pi\)
\(150\) 0 0
\(151\) 11.2003i 0.911466i 0.890116 + 0.455733i \(0.150623\pi\)
−0.890116 + 0.455733i \(0.849377\pi\)
\(152\) 0 0
\(153\) 4.35085i 0.351745i
\(154\) 0 0
\(155\) −4.83021 −0.387972
\(156\) 0 0
\(157\) 18.4353i 1.47129i −0.677365 0.735647i \(-0.736878\pi\)
0.677365 0.735647i \(-0.263122\pi\)
\(158\) 0 0
\(159\) −11.0842 −0.879033
\(160\) 0 0
\(161\) −3.45679 21.7847i −0.272434 1.71687i
\(162\) 0 0
\(163\) 12.1912i 0.954888i 0.878662 + 0.477444i \(0.158437\pi\)
−0.878662 + 0.477444i \(0.841563\pi\)
\(164\) 0 0
\(165\) 4.88207 0.380068
\(166\) 0 0
\(167\) 3.44145i 0.266307i 0.991095 + 0.133154i \(0.0425104\pi\)
−0.991095 + 0.133154i \(0.957490\pi\)
\(168\) 0 0
\(169\) −10.9992 −0.846090
\(170\) 0 0
\(171\) 1.34013 0.102483
\(172\) 0 0
\(173\) 21.8497 1.66120 0.830602 0.556867i \(-0.187997\pi\)
0.830602 + 0.556867i \(0.187997\pi\)
\(174\) 0 0
\(175\) 4.59925 0.347671
\(176\) 0 0
\(177\) 1.36588 0.102666
\(178\) 0 0
\(179\) 23.5630i 1.76118i −0.473876 0.880591i \(-0.657146\pi\)
0.473876 0.880591i \(-0.342854\pi\)
\(180\) 0 0
\(181\) 3.96386i 0.294631i −0.989090 0.147316i \(-0.952937\pi\)
0.989090 0.147316i \(-0.0470633\pi\)
\(182\) 0 0
\(183\) −9.11481 −0.673786
\(184\) 0 0
\(185\) 2.55810 0.188075
\(186\) 0 0
\(187\) 21.2411i 1.55331i
\(188\) 0 0
\(189\) 4.59925i 0.334546i
\(190\) 0 0
\(191\) 13.0157 0.941782 0.470891 0.882191i \(-0.343933\pi\)
0.470891 + 0.882191i \(0.343933\pi\)
\(192\) 0 0
\(193\) −7.98629 −0.574866 −0.287433 0.957801i \(-0.592802\pi\)
−0.287433 + 0.957801i \(0.592802\pi\)
\(194\) 0 0
\(195\) 1.41451 0.101295
\(196\) 0 0
\(197\) 22.2055 1.58208 0.791039 0.611765i \(-0.209540\pi\)
0.791039 + 0.611765i \(0.209540\pi\)
\(198\) 0 0
\(199\) 0.0306913 0.00217565 0.00108783 0.999999i \(-0.499654\pi\)
0.00108783 + 0.999999i \(0.499654\pi\)
\(200\) 0 0
\(201\) 8.47015i 0.597438i
\(202\) 0 0
\(203\) 38.9789 2.73578
\(204\) 0 0
\(205\) 1.94751i 0.136020i
\(206\) 0 0
\(207\) −0.751600 4.73657i −0.0522398 0.329214i
\(208\) 0 0
\(209\) −6.54263 −0.452563
\(210\) 0 0
\(211\) 10.9831i 0.756110i −0.925783 0.378055i \(-0.876593\pi\)
0.925783 0.378055i \(-0.123407\pi\)
\(212\) 0 0
\(213\) −0.721095 −0.0494086
\(214\) 0 0
\(215\) 9.76592i 0.666030i
\(216\) 0 0
\(217\) 22.2153i 1.50808i
\(218\) 0 0
\(219\) 2.29450i 0.155048i
\(220\) 0 0
\(221\) 6.15431i 0.413983i
\(222\) 0 0
\(223\) 27.7153i 1.85595i −0.372637 0.927977i \(-0.621546\pi\)
0.372637 0.927977i \(-0.378454\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 1.03113 0.0684385 0.0342192 0.999414i \(-0.489106\pi\)
0.0342192 + 0.999414i \(0.489106\pi\)
\(228\) 0 0
\(229\) 8.25485i 0.545496i −0.962086 0.272748i \(-0.912067\pi\)
0.962086 0.272748i \(-0.0879325\pi\)
\(230\) 0 0
\(231\) 22.4538i 1.47735i
\(232\) 0 0
\(233\) 9.59119 0.628340 0.314170 0.949367i \(-0.398274\pi\)
0.314170 + 0.949367i \(0.398274\pi\)
\(234\) 0 0
\(235\) −5.05444 −0.329716
\(236\) 0 0
\(237\) 15.2680i 0.991761i
\(238\) 0 0
\(239\) 21.5163i 1.39177i −0.718153 0.695885i \(-0.755012\pi\)
0.718153 0.695885i \(-0.244988\pi\)
\(240\) 0 0
\(241\) 8.52582i 0.549196i 0.961559 + 0.274598i \(0.0885448\pi\)
−0.961559 + 0.274598i \(0.911455\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 14.1531i 0.904208i
\(246\) 0 0
\(247\) −1.89563 −0.120616
\(248\) 0 0
\(249\) 13.3145i 0.843773i
\(250\) 0 0
\(251\) −11.6159 −0.733189 −0.366595 0.930381i \(-0.619476\pi\)
−0.366595 + 0.930381i \(0.619476\pi\)
\(252\) 0 0
\(253\) 3.66936 + 23.1243i 0.230691 + 1.45381i
\(254\) 0 0
\(255\) 4.35085i 0.272461i
\(256\) 0 0
\(257\) 5.71183 0.356294 0.178147 0.984004i \(-0.442990\pi\)
0.178147 + 0.984004i \(0.442990\pi\)
\(258\) 0 0
\(259\) 11.7653i 0.731062i
\(260\) 0 0
\(261\) 8.47505 0.524592
\(262\) 0 0
\(263\) −7.96311 −0.491027 −0.245513 0.969393i \(-0.578957\pi\)
−0.245513 + 0.969393i \(0.578957\pi\)
\(264\) 0 0
\(265\) 11.0842 0.680896
\(266\) 0 0
\(267\) 8.99269 0.550344
\(268\) 0 0
\(269\) 2.35108 0.143348 0.0716740 0.997428i \(-0.477166\pi\)
0.0716740 + 0.997428i \(0.477166\pi\)
\(270\) 0 0
\(271\) 25.2408i 1.53327i −0.642085 0.766634i \(-0.721930\pi\)
0.642085 0.766634i \(-0.278070\pi\)
\(272\) 0 0
\(273\) 6.50567i 0.393741i
\(274\) 0 0
\(275\) −4.88207 −0.294400
\(276\) 0 0
\(277\) 13.6182 0.818237 0.409118 0.912481i \(-0.365836\pi\)
0.409118 + 0.912481i \(0.365836\pi\)
\(278\) 0 0
\(279\) 4.83021i 0.289177i
\(280\) 0 0
\(281\) 28.3953i 1.69392i −0.531654 0.846962i \(-0.678429\pi\)
0.531654 0.846962i \(-0.321571\pi\)
\(282\) 0 0
\(283\) −8.70296 −0.517338 −0.258669 0.965966i \(-0.583284\pi\)
−0.258669 + 0.965966i \(0.583284\pi\)
\(284\) 0 0
\(285\) −1.34013 −0.0793827
\(286\) 0 0
\(287\) 8.95709 0.528720
\(288\) 0 0
\(289\) −1.92988 −0.113523
\(290\) 0 0
\(291\) 1.26729 0.0742899
\(292\) 0 0
\(293\) 13.6653i 0.798333i −0.916879 0.399166i \(-0.869300\pi\)
0.916879 0.399166i \(-0.130700\pi\)
\(294\) 0 0
\(295\) −1.36588 −0.0795245
\(296\) 0 0
\(297\) 4.88207i 0.283286i
\(298\) 0 0
\(299\) 1.06314 + 6.69991i 0.0614831 + 0.387466i
\(300\) 0 0
\(301\) −44.9159 −2.58891
\(302\) 0 0
\(303\) 14.3919i 0.826794i
\(304\) 0 0
\(305\) 9.11481 0.521913
\(306\) 0 0
\(307\) 6.87777i 0.392535i −0.980550 0.196268i \(-0.937118\pi\)
0.980550 0.196268i \(-0.0628821\pi\)
\(308\) 0 0
\(309\) 17.0569i 0.970331i
\(310\) 0 0
\(311\) 13.4738i 0.764028i 0.924157 + 0.382014i \(0.124769\pi\)
−0.924157 + 0.382014i \(0.875231\pi\)
\(312\) 0 0
\(313\) 2.61198i 0.147638i 0.997272 + 0.0738189i \(0.0235187\pi\)
−0.997272 + 0.0738189i \(0.976481\pi\)
\(314\) 0 0
\(315\) 4.59925i 0.259138i
\(316\) 0 0
\(317\) −15.2649 −0.857361 −0.428680 0.903456i \(-0.641021\pi\)
−0.428680 + 0.903456i \(0.641021\pi\)
\(318\) 0 0
\(319\) −41.3757 −2.31660
\(320\) 0 0
\(321\) 3.92835i 0.219259i
\(322\) 0 0
\(323\) 5.83072i 0.324430i
\(324\) 0 0
\(325\) −1.41451 −0.0784627
\(326\) 0 0
\(327\) 5.40581 0.298942
\(328\) 0 0
\(329\) 23.2466i 1.28163i
\(330\) 0 0
\(331\) 24.1660i 1.32829i 0.747606 + 0.664143i \(0.231203\pi\)
−0.747606 + 0.664143i \(0.768797\pi\)
\(332\) 0 0
\(333\) 2.55810i 0.140183i
\(334\) 0 0
\(335\) 8.47015i 0.462774i
\(336\) 0 0
\(337\) 28.9408i 1.57651i −0.615350 0.788254i \(-0.710985\pi\)
0.615350 0.788254i \(-0.289015\pi\)
\(338\) 0 0
\(339\) −13.5574 −0.736336
\(340\) 0 0
\(341\) 23.5814i 1.27700i
\(342\) 0 0
\(343\) −32.8989 −1.77637
\(344\) 0 0
\(345\) 0.751600 + 4.73657i 0.0404648 + 0.255008i
\(346\) 0 0
\(347\) 10.1722i 0.546071i −0.962004 0.273036i \(-0.911972\pi\)
0.962004 0.273036i \(-0.0880277\pi\)
\(348\) 0 0
\(349\) 2.12044 0.113505 0.0567524 0.998388i \(-0.481925\pi\)
0.0567524 + 0.998388i \(0.481925\pi\)
\(350\) 0 0
\(351\) 1.41451i 0.0755008i
\(352\) 0 0
\(353\) 7.05780 0.375649 0.187824 0.982203i \(-0.439856\pi\)
0.187824 + 0.982203i \(0.439856\pi\)
\(354\) 0 0
\(355\) 0.721095 0.0382718
\(356\) 0 0
\(357\) 20.0106 1.05908
\(358\) 0 0
\(359\) −19.6238 −1.03571 −0.517853 0.855470i \(-0.673268\pi\)
−0.517853 + 0.855470i \(0.673268\pi\)
\(360\) 0 0
\(361\) −17.2040 −0.905476
\(362\) 0 0
\(363\) 12.8346i 0.673641i
\(364\) 0 0
\(365\) 2.29450i 0.120100i
\(366\) 0 0
\(367\) −0.388755 −0.0202928 −0.0101464 0.999949i \(-0.503230\pi\)
−0.0101464 + 0.999949i \(0.503230\pi\)
\(368\) 0 0
\(369\) 1.94751 0.101383
\(370\) 0 0
\(371\) 50.9790i 2.64670i
\(372\) 0 0
\(373\) 6.08246i 0.314938i −0.987524 0.157469i \(-0.949667\pi\)
0.987524 0.157469i \(-0.0503334\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −11.9880 −0.617414
\(378\) 0 0
\(379\) 3.12546 0.160544 0.0802720 0.996773i \(-0.474421\pi\)
0.0802720 + 0.996773i \(0.474421\pi\)
\(380\) 0 0
\(381\) −13.2910 −0.680918
\(382\) 0 0
\(383\) 3.85924 0.197198 0.0985989 0.995127i \(-0.468564\pi\)
0.0985989 + 0.995127i \(0.468564\pi\)
\(384\) 0 0
\(385\) 22.4538i 1.14435i
\(386\) 0 0
\(387\) −9.76592 −0.496430
\(388\) 0 0
\(389\) 20.7792i 1.05355i 0.850006 + 0.526773i \(0.176598\pi\)
−0.850006 + 0.526773i \(0.823402\pi\)
\(390\) 0 0
\(391\) 20.6081 3.27010i 1.04220 0.165376i
\(392\) 0 0
\(393\) −8.00063 −0.403579
\(394\) 0 0
\(395\) 15.2680i 0.768215i
\(396\) 0 0
\(397\) −19.0440 −0.955791 −0.477896 0.878417i \(-0.658600\pi\)
−0.477896 + 0.878417i \(0.658600\pi\)
\(398\) 0 0
\(399\) 6.16361i 0.308566i
\(400\) 0 0
\(401\) 5.95750i 0.297503i 0.988875 + 0.148752i \(0.0475255\pi\)
−0.988875 + 0.148752i \(0.952475\pi\)
\(402\) 0 0
\(403\) 6.83237i 0.340344i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 12.4888i 0.619047i
\(408\) 0 0
\(409\) 15.8548 0.783968 0.391984 0.919972i \(-0.371789\pi\)
0.391984 + 0.919972i \(0.371789\pi\)
\(410\) 0 0
\(411\) 1.42388 0.0702348
\(412\) 0 0
\(413\) 6.28201i 0.309118i
\(414\) 0 0
\(415\) 13.3145i 0.653584i
\(416\) 0 0
\(417\) −18.1054 −0.886626
\(418\) 0 0
\(419\) 35.4203 1.73040 0.865198 0.501431i \(-0.167193\pi\)
0.865198 + 0.501431i \(0.167193\pi\)
\(420\) 0 0
\(421\) 10.6010i 0.516661i −0.966057 0.258331i \(-0.916828\pi\)
0.966057 0.258331i \(-0.0831724\pi\)
\(422\) 0 0
\(423\) 5.05444i 0.245755i
\(424\) 0 0
\(425\) 4.35085i 0.211047i
\(426\) 0 0
\(427\) 41.9213i 2.02871i
\(428\) 0 0
\(429\) 6.90572i 0.333411i
\(430\) 0 0
\(431\) 33.4942 1.61336 0.806679 0.590989i \(-0.201263\pi\)
0.806679 + 0.590989i \(0.201263\pi\)
\(432\) 0 0
\(433\) 11.1564i 0.536143i −0.963399 0.268071i \(-0.913614\pi\)
0.963399 0.268071i \(-0.0863863\pi\)
\(434\) 0 0
\(435\) −8.47505 −0.406347
\(436\) 0 0
\(437\) −1.00724 6.34764i −0.0481830 0.303649i
\(438\) 0 0
\(439\) 26.7286i 1.27568i 0.770167 + 0.637842i \(0.220173\pi\)
−0.770167 + 0.637842i \(0.779827\pi\)
\(440\) 0 0
\(441\) −14.1531 −0.673957
\(442\) 0 0
\(443\) 8.02805i 0.381424i 0.981646 + 0.190712i \(0.0610797\pi\)
−0.981646 + 0.190712i \(0.938920\pi\)
\(444\) 0 0
\(445\) −8.99269 −0.426294
\(446\) 0 0
\(447\) −13.4277 −0.635109
\(448\) 0 0
\(449\) 41.8897 1.97690 0.988449 0.151557i \(-0.0484287\pi\)
0.988449 + 0.151557i \(0.0484287\pi\)
\(450\) 0 0
\(451\) −9.50788 −0.447709
\(452\) 0 0
\(453\) 11.2003 0.526235
\(454\) 0 0
\(455\) 6.50567i 0.304991i
\(456\) 0 0
\(457\) 36.0975i 1.68857i −0.535894 0.844285i \(-0.680025\pi\)
0.535894 0.844285i \(-0.319975\pi\)
\(458\) 0 0
\(459\) 4.35085 0.203080
\(460\) 0 0
\(461\) 27.2086 1.26723 0.633616 0.773648i \(-0.281570\pi\)
0.633616 + 0.773648i \(0.281570\pi\)
\(462\) 0 0
\(463\) 21.3134i 0.990519i 0.868745 + 0.495259i \(0.164927\pi\)
−0.868745 + 0.495259i \(0.835073\pi\)
\(464\) 0 0
\(465\) 4.83021i 0.223996i
\(466\) 0 0
\(467\) −11.2590 −0.521004 −0.260502 0.965473i \(-0.583888\pi\)
−0.260502 + 0.965473i \(0.583888\pi\)
\(468\) 0 0
\(469\) −38.9563 −1.79884
\(470\) 0 0
\(471\) −18.4353 −0.849452
\(472\) 0 0
\(473\) 47.6779 2.19223
\(474\) 0 0
\(475\) 1.34013 0.0614896
\(476\) 0 0
\(477\) 11.0842i 0.507510i
\(478\) 0 0
\(479\) −22.5960 −1.03244 −0.516219 0.856457i \(-0.672661\pi\)
−0.516219 + 0.856457i \(0.672661\pi\)
\(480\) 0 0
\(481\) 3.61845i 0.164987i
\(482\) 0 0
\(483\) −21.7847 + 3.45679i −0.991237 + 0.157290i
\(484\) 0 0
\(485\) −1.26729 −0.0575447
\(486\) 0 0
\(487\) 14.2500i 0.645728i 0.946445 + 0.322864i \(0.104646\pi\)
−0.946445 + 0.322864i \(0.895354\pi\)
\(488\) 0 0
\(489\) 12.1912 0.551305
\(490\) 0 0
\(491\) 28.6030i 1.29084i 0.763830 + 0.645418i \(0.223317\pi\)
−0.763830 + 0.645418i \(0.776683\pi\)
\(492\) 0 0
\(493\) 36.8736i 1.66071i
\(494\) 0 0
\(495\) 4.88207i 0.219433i
\(496\) 0 0
\(497\) 3.31650i 0.148765i
\(498\) 0 0
\(499\) 17.5242i 0.784490i 0.919861 + 0.392245i \(0.128302\pi\)
−0.919861 + 0.392245i \(0.871698\pi\)
\(500\) 0 0
\(501\) 3.44145 0.153753
\(502\) 0 0
\(503\) −16.4284 −0.732508 −0.366254 0.930515i \(-0.619360\pi\)
−0.366254 + 0.930515i \(0.619360\pi\)
\(504\) 0 0
\(505\) 14.3919i 0.640432i
\(506\) 0 0
\(507\) 10.9992i 0.488490i
\(508\) 0 0
\(509\) 11.6812 0.517762 0.258881 0.965909i \(-0.416646\pi\)
0.258881 + 0.965909i \(0.416646\pi\)
\(510\) 0 0
\(511\) −10.5530 −0.466836
\(512\) 0 0
\(513\) 1.34013i 0.0591684i
\(514\) 0 0
\(515\) 17.0569i 0.751615i
\(516\) 0 0
\(517\) 24.6761i 1.08526i
\(518\) 0 0
\(519\) 21.8497i 0.959096i
\(520\) 0 0
\(521\) 22.9944i 1.00740i −0.863877 0.503702i \(-0.831971\pi\)
0.863877 0.503702i \(-0.168029\pi\)
\(522\) 0 0
\(523\) −16.7340 −0.731727 −0.365864 0.930668i \(-0.619226\pi\)
−0.365864 + 0.930668i \(0.619226\pi\)
\(524\) 0 0
\(525\) 4.59925i 0.200728i
\(526\) 0 0
\(527\) 21.0155 0.915450
\(528\) 0 0
\(529\) −21.8702 + 7.12001i −0.950878 + 0.309566i
\(530\) 0 0
\(531\) 1.36588i 0.0592741i
\(532\) 0 0
\(533\) −2.75477 −0.119322
\(534\) 0 0
\(535\) 3.92835i 0.169838i
\(536\) 0 0
\(537\) −23.5630 −1.01682
\(538\) 0 0
\(539\) 69.0964 2.97619
\(540\) 0 0
\(541\) 30.2431 1.30025 0.650126 0.759826i \(-0.274716\pi\)
0.650126 + 0.759826i \(0.274716\pi\)
\(542\) 0 0
\(543\) −3.96386 −0.170105
\(544\) 0 0
\(545\) −5.40581 −0.231559
\(546\) 0 0
\(547\) 39.8256i 1.70282i 0.524502 + 0.851409i \(0.324251\pi\)
−0.524502 + 0.851409i \(0.675749\pi\)
\(548\) 0 0
\(549\) 9.11481i 0.389011i
\(550\) 0 0
\(551\) 11.3577 0.483854
\(552\) 0 0
\(553\) 70.2212 2.98611
\(554\) 0 0
\(555\) 2.55810i 0.108585i
\(556\) 0 0
\(557\) 9.14289i 0.387397i 0.981061 + 0.193698i \(0.0620483\pi\)
−0.981061 + 0.193698i \(0.937952\pi\)
\(558\) 0 0
\(559\) 13.8140 0.584268
\(560\) 0 0
\(561\) −21.2411 −0.896801
\(562\) 0 0
\(563\) −5.43927 −0.229238 −0.114619 0.993410i \(-0.536565\pi\)
−0.114619 + 0.993410i \(0.536565\pi\)
\(564\) 0 0
\(565\) 13.5574 0.570364
\(566\) 0 0
\(567\) −4.59925 −0.193150
\(568\) 0 0
\(569\) 15.3131i 0.641960i 0.947086 + 0.320980i \(0.104012\pi\)
−0.947086 + 0.320980i \(0.895988\pi\)
\(570\) 0 0
\(571\) −28.7285 −1.20225 −0.601126 0.799154i \(-0.705281\pi\)
−0.601126 + 0.799154i \(0.705281\pi\)
\(572\) 0 0
\(573\) 13.0157i 0.543738i
\(574\) 0 0
\(575\) −0.751600 4.73657i −0.0313439 0.197529i
\(576\) 0 0
\(577\) −45.7627 −1.90513 −0.952563 0.304340i \(-0.901564\pi\)
−0.952563 + 0.304340i \(0.901564\pi\)
\(578\) 0 0
\(579\) 7.98629i 0.331899i
\(580\) 0 0
\(581\) −61.2368 −2.54053
\(582\) 0 0
\(583\) 54.1138i 2.24116i
\(584\) 0 0
\(585\) 1.41451i 0.0584827i
\(586\) 0 0
\(587\) 20.5254i 0.847176i −0.905855 0.423588i \(-0.860770\pi\)
0.905855 0.423588i \(-0.139230\pi\)
\(588\) 0 0
\(589\) 6.47313i 0.266721i
\(590\) 0 0
\(591\) 22.2055i 0.913413i
\(592\) 0 0
\(593\) 42.3259 1.73812 0.869058 0.494710i \(-0.164726\pi\)
0.869058 + 0.494710i \(0.164726\pi\)
\(594\) 0 0
\(595\) −20.0106 −0.820356
\(596\) 0 0
\(597\) 0.0306913i 0.00125611i
\(598\) 0 0
\(599\) 8.21773i 0.335767i 0.985807 + 0.167884i \(0.0536933\pi\)
−0.985807 + 0.167884i \(0.946307\pi\)
\(600\) 0 0
\(601\) −21.9012 −0.893370 −0.446685 0.894691i \(-0.647396\pi\)
−0.446685 + 0.894691i \(0.647396\pi\)
\(602\) 0 0
\(603\) −8.47015 −0.344931
\(604\) 0 0
\(605\) 12.8346i 0.521800i
\(606\) 0 0
\(607\) 41.6105i 1.68892i −0.535620 0.844459i \(-0.679922\pi\)
0.535620 0.844459i \(-0.320078\pi\)
\(608\) 0 0
\(609\) 38.9789i 1.57950i
\(610\) 0 0
\(611\) 7.14955i 0.289240i
\(612\) 0 0
\(613\) 35.6614i 1.44035i −0.693792 0.720175i \(-0.744061\pi\)
0.693792 0.720175i \(-0.255939\pi\)
\(614\) 0 0
\(615\) −1.94751 −0.0785312
\(616\) 0 0
\(617\) 33.4403i 1.34626i 0.739526 + 0.673128i \(0.235050\pi\)
−0.739526 + 0.673128i \(0.764950\pi\)
\(618\) 0 0
\(619\) 7.06018 0.283773 0.141886 0.989883i \(-0.454683\pi\)
0.141886 + 0.989883i \(0.454683\pi\)
\(620\) 0 0
\(621\) −4.73657 + 0.751600i −0.190072 + 0.0301607i
\(622\) 0 0
\(623\) 41.3596i 1.65704i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.54263i 0.261287i
\(628\) 0 0
\(629\) −11.1299 −0.443778
\(630\) 0 0
\(631\) 22.1241 0.880748 0.440374 0.897814i \(-0.354846\pi\)
0.440374 + 0.897814i \(0.354846\pi\)
\(632\) 0 0
\(633\) −10.9831 −0.436540
\(634\) 0 0
\(635\) 13.2910 0.527437
\(636\) 0 0
\(637\) 20.0196 0.793207
\(638\) 0 0
\(639\) 0.721095i 0.0285261i
\(640\) 0 0
\(641\) 28.6565i 1.13187i −0.824451 0.565933i \(-0.808516\pi\)
0.824451 0.565933i \(-0.191484\pi\)
\(642\) 0 0
\(643\) −35.1114 −1.38466 −0.692329 0.721582i \(-0.743415\pi\)
−0.692329 + 0.721582i \(0.743415\pi\)
\(644\) 0 0
\(645\) 9.76592 0.384533
\(646\) 0 0
\(647\) 27.4563i 1.07942i −0.841852 0.539709i \(-0.818534\pi\)
0.841852 0.539709i \(-0.181466\pi\)
\(648\) 0 0
\(649\) 6.66831i 0.261754i
\(650\) 0 0
\(651\) −22.2153 −0.870688
\(652\) 0 0
\(653\) −25.9667 −1.01615 −0.508077 0.861311i \(-0.669644\pi\)
−0.508077 + 0.861311i \(0.669644\pi\)
\(654\) 0 0
\(655\) 8.00063 0.312611
\(656\) 0 0
\(657\) −2.29450 −0.0895169
\(658\) 0 0
\(659\) 23.5211 0.916253 0.458127 0.888887i \(-0.348521\pi\)
0.458127 + 0.888887i \(0.348521\pi\)
\(660\) 0 0
\(661\) 41.0127i 1.59521i 0.603181 + 0.797605i \(0.293900\pi\)
−0.603181 + 0.797605i \(0.706100\pi\)
\(662\) 0 0
\(663\) −6.15431 −0.239013
\(664\) 0 0
\(665\) 6.16361i 0.239015i
\(666\) 0 0
\(667\) −6.36984 40.1427i −0.246641 1.55433i
\(668\) 0 0
\(669\) −27.7153 −1.07154
\(670\) 0 0
\(671\) 44.4991i 1.71787i
\(672\) 0 0
\(673\) 4.56660 0.176030 0.0880148 0.996119i \(-0.471948\pi\)
0.0880148 + 0.996119i \(0.471948\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 41.1466i 1.58139i −0.612210 0.790695i \(-0.709719\pi\)
0.612210 0.790695i \(-0.290281\pi\)
\(678\) 0 0
\(679\) 5.82859i 0.223681i
\(680\) 0 0
\(681\) 1.03113i 0.0395130i
\(682\) 0 0
\(683\) 35.5129i 1.35886i 0.733739 + 0.679432i \(0.237774\pi\)
−0.733739 + 0.679432i \(0.762226\pi\)
\(684\) 0 0
\(685\) −1.42388 −0.0544036
\(686\) 0 0
\(687\) −8.25485 −0.314942
\(688\) 0 0
\(689\) 15.6787i 0.597310i
\(690\) 0 0
\(691\) 31.3788i 1.19370i −0.802351 0.596852i \(-0.796418\pi\)
0.802351 0.596852i \(-0.203582\pi\)
\(692\) 0 0
\(693\) 22.4538 0.852951
\(694\) 0 0
\(695\) 18.1054 0.686777
\(696\) 0 0
\(697\) 8.47333i 0.320950i
\(698\) 0 0
\(699\) 9.59119i 0.362772i
\(700\) 0 0
\(701\) 23.9305i 0.903843i 0.892058 + 0.451921i \(0.149261\pi\)
−0.892058 + 0.451921i \(0.850739\pi\)
\(702\) 0 0
\(703\) 3.42819i 0.129297i
\(704\) 0 0
\(705\) 5.05444i 0.190361i
\(706\) 0 0
\(707\) 66.1920 2.48941
\(708\) 0 0
\(709\) 40.1421i 1.50757i 0.657122 + 0.753784i \(0.271774\pi\)
−0.657122 + 0.753784i \(0.728226\pi\)
\(710\) 0 0
\(711\) 15.2680 0.572593
\(712\) 0 0
\(713\) −22.8786 + 3.63038i −0.856811 + 0.135959i
\(714\) 0 0
\(715\) 6.90572i 0.258259i
\(716\) 0 0
\(717\) −21.5163 −0.803539
\(718\) 0 0
\(719\) 9.22449i 0.344015i −0.985096 0.172008i \(-0.944975\pi\)
0.985096 0.172008i \(-0.0550254\pi\)
\(720\) 0 0
\(721\) −78.4487 −2.92158
\(722\) 0 0
\(723\) 8.52582 0.317079
\(724\) 0 0
\(725\) 8.47505 0.314755
\(726\) 0 0
\(727\) 18.4434 0.684027 0.342014 0.939695i \(-0.388891\pi\)
0.342014 + 0.939695i \(0.388891\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 42.4900i 1.57155i
\(732\) 0 0
\(733\) 51.1562i 1.88950i 0.327797 + 0.944748i \(0.393694\pi\)
−0.327797 + 0.944748i \(0.606306\pi\)
\(734\) 0 0
\(735\) 14.1531 0.522045
\(736\) 0 0
\(737\) 41.3518 1.52321
\(738\) 0 0
\(739\) 17.9784i 0.661346i 0.943745 + 0.330673i \(0.107276\pi\)
−0.943745 + 0.330673i \(0.892724\pi\)
\(740\) 0 0
\(741\) 1.89563i 0.0696377i
\(742\) 0 0
\(743\) −45.8180 −1.68090 −0.840449 0.541890i \(-0.817709\pi\)
−0.840449 + 0.541890i \(0.817709\pi\)
\(744\) 0 0
\(745\) 13.4277 0.491954
\(746\) 0 0
\(747\) −13.3145 −0.487153
\(748\) 0 0
\(749\) 18.0675 0.660171
\(750\) 0 0
\(751\) −5.29283 −0.193138 −0.0965691 0.995326i \(-0.530787\pi\)
−0.0965691 + 0.995326i \(0.530787\pi\)
\(752\) 0 0
\(753\) 11.6159i 0.423307i
\(754\) 0 0
\(755\) −11.2003 −0.407620
\(756\) 0 0
\(757\) 24.6833i 0.897130i −0.893750 0.448565i \(-0.851935\pi\)
0.893750 0.448565i \(-0.148065\pi\)
\(758\) 0 0
\(759\) 23.1243 3.66936i 0.839357 0.133189i
\(760\) 0 0
\(761\) −19.9805 −0.724293 −0.362147 0.932121i \(-0.617956\pi\)
−0.362147 + 0.932121i \(0.617956\pi\)
\(762\) 0 0
\(763\) 24.8627i 0.900089i
\(764\) 0 0
\(765\) −4.35085 −0.157305
\(766\) 0 0
\(767\) 1.93204i 0.0697621i
\(768\) 0 0
\(769\) 4.23081i 0.152567i 0.997086 + 0.0762834i \(0.0243054\pi\)
−0.997086 + 0.0762834i \(0.975695\pi\)
\(770\) 0 0
\(771\) 5.71183i 0.205706i
\(772\) 0 0
\(773\) 4.37581i 0.157387i −0.996899 0.0786934i \(-0.974925\pi\)
0.996899 0.0786934i \(-0.0250748\pi\)
\(774\) 0 0
\(775\) 4.83021i 0.173506i
\(776\) 0 0
\(777\) 11.7653 0.422079
\(778\) 0 0
\(779\) 2.60993 0.0935103
\(780\) 0 0
\(781\) 3.52044i 0.125971i
\(782\) 0 0
\(783\) 8.47505i 0.302873i
\(784\) 0 0
\(785\) 18.4353 0.657983
\(786\) 0 0
\(787\) −5.64990 −0.201397 −0.100699 0.994917i \(-0.532108\pi\)
−0.100699 + 0.994917i \(0.532108\pi\)
\(788\) 0 0
\(789\) 7.96311i 0.283494i
\(790\) 0 0
\(791\) 62.3538i 2.21705i
\(792\) 0 0
\(793\) 12.8930i 0.457843i
\(794\) 0 0
\(795\) 11.0842i 0.393116i
\(796\) 0 0
\(797\) 33.8817i 1.20015i 0.799943 + 0.600076i \(0.204863\pi\)
−0.799943 + 0.600076i \(0.795137\pi\)
\(798\) 0 0
\(799\) 21.9911 0.777990
\(800\) 0 0
\(801\) 8.99269i 0.317741i
\(802\) 0 0
\(803\) 11.2019 0.395306
\(804\) 0 0