Properties

Label 1216.3.e.g.1025.1
Level $1216$
Weight $3$
Character 1216.1025
Analytic conductor $33.134$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1216.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.1336001462\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-13}) \)
Defining polynomial: \(x^{2} + 13\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.1
Root \(-3.60555i\) of defining polynomial
Character \(\chi\) \(=\) 1216.1025
Dual form 1216.3.e.g.1025.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.60555i q^{3} -4.00000 q^{5} -5.00000 q^{7} -4.00000 q^{9} +O(q^{10})\) \(q-3.60555i q^{3} -4.00000 q^{5} -5.00000 q^{7} -4.00000 q^{9} +10.0000 q^{11} +3.60555i q^{13} +14.4222i q^{15} +15.0000 q^{17} +(6.00000 + 18.0278i) q^{19} +18.0278i q^{21} +35.0000 q^{23} -9.00000 q^{25} -18.0278i q^{27} +18.0278i q^{29} +36.0555i q^{31} -36.0555i q^{33} +20.0000 q^{35} -21.6333i q^{37} +13.0000 q^{39} -36.0555i q^{41} +20.0000 q^{43} +16.0000 q^{45} +10.0000 q^{47} -24.0000 q^{49} -54.0833i q^{51} -75.7166i q^{53} -40.0000 q^{55} +(65.0000 - 21.6333i) q^{57} +18.0278i q^{59} +40.0000 q^{61} +20.0000 q^{63} -14.4222i q^{65} +39.6611i q^{67} -126.194i q^{69} -108.167i q^{71} +105.000 q^{73} +32.4500i q^{75} -50.0000 q^{77} +36.0555i q^{79} -101.000 q^{81} +40.0000 q^{83} -60.0000 q^{85} +65.0000 q^{87} -18.0278i q^{91} +130.000 q^{93} +(-24.0000 - 72.1110i) q^{95} -122.589i q^{97} -40.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{5} - 10q^{7} - 8q^{9} + O(q^{10}) \) \( 2q - 8q^{5} - 10q^{7} - 8q^{9} + 20q^{11} + 30q^{17} + 12q^{19} + 70q^{23} - 18q^{25} + 40q^{35} + 26q^{39} + 40q^{43} + 32q^{45} + 20q^{47} - 48q^{49} - 80q^{55} + 130q^{57} + 80q^{61} + 40q^{63} + 210q^{73} - 100q^{77} - 202q^{81} + 80q^{83} - 120q^{85} + 130q^{87} + 260q^{93} - 48q^{95} - 80q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 3.60555i 1.20185i −0.799305 0.600925i \(-0.794799\pi\)
0.799305 0.600925i \(-0.205201\pi\)
\(4\) 0 0
\(5\) −4.00000 −0.800000 −0.400000 0.916515i \(-0.630990\pi\)
−0.400000 + 0.916515i \(0.630990\pi\)
\(6\) 0 0
\(7\) −5.00000 −0.714286 −0.357143 0.934050i \(-0.616249\pi\)
−0.357143 + 0.934050i \(0.616249\pi\)
\(8\) 0 0
\(9\) −4.00000 −0.444444
\(10\) 0 0
\(11\) 10.0000 0.909091 0.454545 0.890724i \(-0.349802\pi\)
0.454545 + 0.890724i \(0.349802\pi\)
\(12\) 0 0
\(13\) 3.60555i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) 14.4222i 0.961480i
\(16\) 0 0
\(17\) 15.0000 0.882353 0.441176 0.897420i \(-0.354561\pi\)
0.441176 + 0.897420i \(0.354561\pi\)
\(18\) 0 0
\(19\) 6.00000 + 18.0278i 0.315789 + 0.948829i
\(20\) 0 0
\(21\) 18.0278i 0.858465i
\(22\) 0 0
\(23\) 35.0000 1.52174 0.760870 0.648905i \(-0.224773\pi\)
0.760870 + 0.648905i \(0.224773\pi\)
\(24\) 0 0
\(25\) −9.00000 −0.360000
\(26\) 0 0
\(27\) 18.0278i 0.667695i
\(28\) 0 0
\(29\) 18.0278i 0.621647i 0.950468 + 0.310823i \(0.100605\pi\)
−0.950468 + 0.310823i \(0.899395\pi\)
\(30\) 0 0
\(31\) 36.0555i 1.16308i 0.813517 + 0.581541i \(0.197550\pi\)
−0.813517 + 0.581541i \(0.802450\pi\)
\(32\) 0 0
\(33\) 36.0555i 1.09259i
\(34\) 0 0
\(35\) 20.0000 0.571429
\(36\) 0 0
\(37\) 21.6333i 0.584684i −0.956314 0.292342i \(-0.905565\pi\)
0.956314 0.292342i \(-0.0944346\pi\)
\(38\) 0 0
\(39\) 13.0000 0.333333
\(40\) 0 0
\(41\) 36.0555i 0.879403i −0.898144 0.439701i \(-0.855084\pi\)
0.898144 0.439701i \(-0.144916\pi\)
\(42\) 0 0
\(43\) 20.0000 0.465116 0.232558 0.972582i \(-0.425290\pi\)
0.232558 + 0.972582i \(0.425290\pi\)
\(44\) 0 0
\(45\) 16.0000 0.355556
\(46\) 0 0
\(47\) 10.0000 0.212766 0.106383 0.994325i \(-0.466073\pi\)
0.106383 + 0.994325i \(0.466073\pi\)
\(48\) 0 0
\(49\) −24.0000 −0.489796
\(50\) 0 0
\(51\) 54.0833i 1.06046i
\(52\) 0 0
\(53\) 75.7166i 1.42861i −0.699832 0.714307i \(-0.746742\pi\)
0.699832 0.714307i \(-0.253258\pi\)
\(54\) 0 0
\(55\) −40.0000 −0.727273
\(56\) 0 0
\(57\) 65.0000 21.6333i 1.14035 0.379532i
\(58\) 0 0
\(59\) 18.0278i 0.305555i 0.988261 + 0.152778i \(0.0488218\pi\)
−0.988261 + 0.152778i \(0.951178\pi\)
\(60\) 0 0
\(61\) 40.0000 0.655738 0.327869 0.944723i \(-0.393670\pi\)
0.327869 + 0.944723i \(0.393670\pi\)
\(62\) 0 0
\(63\) 20.0000 0.317460
\(64\) 0 0
\(65\) 14.4222i 0.221880i
\(66\) 0 0
\(67\) 39.6611i 0.591956i 0.955195 + 0.295978i \(0.0956455\pi\)
−0.955195 + 0.295978i \(0.904354\pi\)
\(68\) 0 0
\(69\) 126.194i 1.82890i
\(70\) 0 0
\(71\) 108.167i 1.52347i −0.647887 0.761736i \(-0.724347\pi\)
0.647887 0.761736i \(-0.275653\pi\)
\(72\) 0 0
\(73\) 105.000 1.43836 0.719178 0.694826i \(-0.244519\pi\)
0.719178 + 0.694826i \(0.244519\pi\)
\(74\) 0 0
\(75\) 32.4500i 0.432666i
\(76\) 0 0
\(77\) −50.0000 −0.649351
\(78\) 0 0
\(79\) 36.0555i 0.456399i 0.973614 + 0.228199i \(0.0732838\pi\)
−0.973614 + 0.228199i \(0.926716\pi\)
\(80\) 0 0
\(81\) −101.000 −1.24691
\(82\) 0 0
\(83\) 40.0000 0.481928 0.240964 0.970534i \(-0.422536\pi\)
0.240964 + 0.970534i \(0.422536\pi\)
\(84\) 0 0
\(85\) −60.0000 −0.705882
\(86\) 0 0
\(87\) 65.0000 0.747126
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 18.0278i 0.198107i
\(92\) 0 0
\(93\) 130.000 1.39785
\(94\) 0 0
\(95\) −24.0000 72.1110i −0.252632 0.759063i
\(96\) 0 0
\(97\) 122.589i 1.26380i −0.775049 0.631901i \(-0.782275\pi\)
0.775049 0.631901i \(-0.217725\pi\)
\(98\) 0 0
\(99\) −40.0000 −0.404040
\(100\) 0 0
\(101\) 50.0000 0.495050 0.247525 0.968882i \(-0.420383\pi\)
0.247525 + 0.968882i \(0.420383\pi\)
\(102\) 0 0
\(103\) 57.6888i 0.560086i −0.959988 0.280043i \(-0.909651\pi\)
0.959988 0.280043i \(-0.0903487\pi\)
\(104\) 0 0
\(105\) 72.1110i 0.686772i
\(106\) 0 0
\(107\) 75.7166i 0.707632i 0.935315 + 0.353816i \(0.115116\pi\)
−0.935315 + 0.353816i \(0.884884\pi\)
\(108\) 0 0
\(109\) 198.305i 1.81931i −0.415359 0.909657i \(-0.636344\pi\)
0.415359 0.909657i \(-0.363656\pi\)
\(110\) 0 0
\(111\) −78.0000 −0.702703
\(112\) 0 0
\(113\) 122.589i 1.08486i 0.840102 + 0.542428i \(0.182495\pi\)
−0.840102 + 0.542428i \(0.817505\pi\)
\(114\) 0 0
\(115\) −140.000 −1.21739
\(116\) 0 0
\(117\) 14.4222i 0.123267i
\(118\) 0 0
\(119\) −75.0000 −0.630252
\(120\) 0 0
\(121\) −21.0000 −0.173554
\(122\) 0 0
\(123\) −130.000 −1.05691
\(124\) 0 0
\(125\) 136.000 1.08800
\(126\) 0 0
\(127\) 129.800i 1.02205i 0.859567 + 0.511023i \(0.170733\pi\)
−0.859567 + 0.511023i \(0.829267\pi\)
\(128\) 0 0
\(129\) 72.1110i 0.559000i
\(130\) 0 0
\(131\) −112.000 −0.854962 −0.427481 0.904024i \(-0.640599\pi\)
−0.427481 + 0.904024i \(0.640599\pi\)
\(132\) 0 0
\(133\) −30.0000 90.1388i −0.225564 0.677735i
\(134\) 0 0
\(135\) 72.1110i 0.534156i
\(136\) 0 0
\(137\) 125.000 0.912409 0.456204 0.889875i \(-0.349209\pi\)
0.456204 + 0.889875i \(0.349209\pi\)
\(138\) 0 0
\(139\) −50.0000 −0.359712 −0.179856 0.983693i \(-0.557563\pi\)
−0.179856 + 0.983693i \(0.557563\pi\)
\(140\) 0 0
\(141\) 36.0555i 0.255713i
\(142\) 0 0
\(143\) 36.0555i 0.252136i
\(144\) 0 0
\(145\) 72.1110i 0.497317i
\(146\) 0 0
\(147\) 86.5332i 0.588661i
\(148\) 0 0
\(149\) −70.0000 −0.469799 −0.234899 0.972020i \(-0.575476\pi\)
−0.234899 + 0.972020i \(0.575476\pi\)
\(150\) 0 0
\(151\) 36.0555i 0.238778i −0.992848 0.119389i \(-0.961906\pi\)
0.992848 0.119389i \(-0.0380936\pi\)
\(152\) 0 0
\(153\) −60.0000 −0.392157
\(154\) 0 0
\(155\) 144.222i 0.930465i
\(156\) 0 0
\(157\) −10.0000 −0.0636943 −0.0318471 0.999493i \(-0.510139\pi\)
−0.0318471 + 0.999493i \(0.510139\pi\)
\(158\) 0 0
\(159\) −273.000 −1.71698
\(160\) 0 0
\(161\) −175.000 −1.08696
\(162\) 0 0
\(163\) 270.000 1.65644 0.828221 0.560402i \(-0.189353\pi\)
0.828221 + 0.560402i \(0.189353\pi\)
\(164\) 0 0
\(165\) 144.222i 0.874073i
\(166\) 0 0
\(167\) 122.589i 0.734064i −0.930208 0.367032i \(-0.880374\pi\)
0.930208 0.367032i \(-0.119626\pi\)
\(168\) 0 0
\(169\) 156.000 0.923077
\(170\) 0 0
\(171\) −24.0000 72.1110i −0.140351 0.421702i
\(172\) 0 0
\(173\) 122.589i 0.708605i −0.935131 0.354303i \(-0.884718\pi\)
0.935131 0.354303i \(-0.115282\pi\)
\(174\) 0 0
\(175\) 45.0000 0.257143
\(176\) 0 0
\(177\) 65.0000 0.367232
\(178\) 0 0
\(179\) 36.0555i 0.201427i 0.994915 + 0.100714i \(0.0321126\pi\)
−0.994915 + 0.100714i \(0.967887\pi\)
\(180\) 0 0
\(181\) 108.167i 0.597605i 0.954315 + 0.298803i \(0.0965872\pi\)
−0.954315 + 0.298803i \(0.903413\pi\)
\(182\) 0 0
\(183\) 144.222i 0.788099i
\(184\) 0 0
\(185\) 86.5332i 0.467747i
\(186\) 0 0
\(187\) 150.000 0.802139
\(188\) 0 0
\(189\) 90.1388i 0.476925i
\(190\) 0 0
\(191\) 193.000 1.01047 0.505236 0.862981i \(-0.331406\pi\)
0.505236 + 0.862981i \(0.331406\pi\)
\(192\) 0 0
\(193\) 266.811i 1.38244i −0.722645 0.691220i \(-0.757074\pi\)
0.722645 0.691220i \(-0.242926\pi\)
\(194\) 0 0
\(195\) −52.0000 −0.266667
\(196\) 0 0
\(197\) −90.0000 −0.456853 −0.228426 0.973561i \(-0.573358\pi\)
−0.228426 + 0.973561i \(0.573358\pi\)
\(198\) 0 0
\(199\) 123.000 0.618090 0.309045 0.951047i \(-0.399991\pi\)
0.309045 + 0.951047i \(0.399991\pi\)
\(200\) 0 0
\(201\) 143.000 0.711443
\(202\) 0 0
\(203\) 90.1388i 0.444033i
\(204\) 0 0
\(205\) 144.222i 0.703522i
\(206\) 0 0
\(207\) −140.000 −0.676329
\(208\) 0 0
\(209\) 60.0000 + 180.278i 0.287081 + 0.862572i
\(210\) 0 0
\(211\) 234.361i 1.11071i −0.831612 0.555357i \(-0.812581\pi\)
0.831612 0.555357i \(-0.187419\pi\)
\(212\) 0 0
\(213\) −390.000 −1.83099
\(214\) 0 0
\(215\) −80.0000 −0.372093
\(216\) 0 0
\(217\) 180.278i 0.830772i
\(218\) 0 0
\(219\) 378.583i 1.72869i
\(220\) 0 0
\(221\) 54.0833i 0.244721i
\(222\) 0 0
\(223\) 201.911i 0.905430i 0.891655 + 0.452715i \(0.149544\pi\)
−0.891655 + 0.452715i \(0.850456\pi\)
\(224\) 0 0
\(225\) 36.0000 0.160000
\(226\) 0 0
\(227\) 255.994i 1.12773i 0.825868 + 0.563864i \(0.190686\pi\)
−0.825868 + 0.563864i \(0.809314\pi\)
\(228\) 0 0
\(229\) 160.000 0.698690 0.349345 0.936994i \(-0.386404\pi\)
0.349345 + 0.936994i \(0.386404\pi\)
\(230\) 0 0
\(231\) 180.278i 0.780422i
\(232\) 0 0
\(233\) −270.000 −1.15880 −0.579399 0.815044i \(-0.696713\pi\)
−0.579399 + 0.815044i \(0.696713\pi\)
\(234\) 0 0
\(235\) −40.0000 −0.170213
\(236\) 0 0
\(237\) 130.000 0.548523
\(238\) 0 0
\(239\) 197.000 0.824268 0.412134 0.911123i \(-0.364784\pi\)
0.412134 + 0.911123i \(0.364784\pi\)
\(240\) 0 0
\(241\) 396.611i 1.64569i −0.568268 0.822844i \(-0.692386\pi\)
0.568268 0.822844i \(-0.307614\pi\)
\(242\) 0 0
\(243\) 201.911i 0.830909i
\(244\) 0 0
\(245\) 96.0000 0.391837
\(246\) 0 0
\(247\) −65.0000 + 21.6333i −0.263158 + 0.0875842i
\(248\) 0 0
\(249\) 144.222i 0.579205i
\(250\) 0 0
\(251\) 402.000 1.60159 0.800797 0.598936i \(-0.204410\pi\)
0.800797 + 0.598936i \(0.204410\pi\)
\(252\) 0 0
\(253\) 350.000 1.38340
\(254\) 0 0
\(255\) 216.333i 0.848365i
\(256\) 0 0
\(257\) 418.244i 1.62741i 0.581279 + 0.813704i \(0.302552\pi\)
−0.581279 + 0.813704i \(0.697448\pi\)
\(258\) 0 0
\(259\) 108.167i 0.417631i
\(260\) 0 0
\(261\) 72.1110i 0.276287i
\(262\) 0 0
\(263\) 310.000 1.17871 0.589354 0.807875i \(-0.299382\pi\)
0.589354 + 0.807875i \(0.299382\pi\)
\(264\) 0 0
\(265\) 302.866i 1.14289i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 108.167i 0.402106i 0.979580 + 0.201053i \(0.0644364\pi\)
−0.979580 + 0.201053i \(0.935564\pi\)
\(270\) 0 0
\(271\) 105.000 0.387454 0.193727 0.981055i \(-0.437942\pi\)
0.193727 + 0.981055i \(0.437942\pi\)
\(272\) 0 0
\(273\) −65.0000 −0.238095
\(274\) 0 0
\(275\) −90.0000 −0.327273
\(276\) 0 0
\(277\) 50.0000 0.180505 0.0902527 0.995919i \(-0.471233\pi\)
0.0902527 + 0.995919i \(0.471233\pi\)
\(278\) 0 0
\(279\) 144.222i 0.516925i
\(280\) 0 0
\(281\) 288.444i 1.02649i 0.858242 + 0.513246i \(0.171557\pi\)
−0.858242 + 0.513246i \(0.828443\pi\)
\(282\) 0 0
\(283\) 320.000 1.13074 0.565371 0.824837i \(-0.308733\pi\)
0.565371 + 0.824837i \(0.308733\pi\)
\(284\) 0 0
\(285\) −260.000 + 86.5332i −0.912281 + 0.303625i
\(286\) 0 0
\(287\) 180.278i 0.628145i
\(288\) 0 0
\(289\) −64.0000 −0.221453
\(290\) 0 0
\(291\) −442.000 −1.51890
\(292\) 0 0
\(293\) 219.939i 0.750644i −0.926895 0.375322i \(-0.877532\pi\)
0.926895 0.375322i \(-0.122468\pi\)
\(294\) 0 0
\(295\) 72.1110i 0.244444i
\(296\) 0 0
\(297\) 180.278i 0.606995i
\(298\) 0 0
\(299\) 126.194i 0.422054i
\(300\) 0 0
\(301\) −100.000 −0.332226
\(302\) 0 0
\(303\) 180.278i 0.594975i
\(304\) 0 0
\(305\) −160.000 −0.524590
\(306\) 0 0
\(307\) 237.966i 0.775135i 0.921841 + 0.387567i \(0.126685\pi\)
−0.921841 + 0.387567i \(0.873315\pi\)
\(308\) 0 0
\(309\) −208.000 −0.673139
\(310\) 0 0
\(311\) 395.000 1.27010 0.635048 0.772472i \(-0.280980\pi\)
0.635048 + 0.772472i \(0.280980\pi\)
\(312\) 0 0
\(313\) 125.000 0.399361 0.199681 0.979861i \(-0.436010\pi\)
0.199681 + 0.979861i \(0.436010\pi\)
\(314\) 0 0
\(315\) −80.0000 −0.253968
\(316\) 0 0
\(317\) 3.60555i 0.0113740i 0.999984 + 0.00568699i \(0.00181023\pi\)
−0.999984 + 0.00568699i \(0.998190\pi\)
\(318\) 0 0
\(319\) 180.278i 0.565133i
\(320\) 0 0
\(321\) 273.000 0.850467
\(322\) 0 0
\(323\) 90.0000 + 270.416i 0.278638 + 0.837202i
\(324\) 0 0
\(325\) 32.4500i 0.0998460i
\(326\) 0 0
\(327\) −715.000 −2.18654
\(328\) 0 0
\(329\) −50.0000 −0.151976
\(330\) 0 0
\(331\) 198.305i 0.599110i −0.954079 0.299555i \(-0.903162\pi\)
0.954079 0.299555i \(-0.0968382\pi\)
\(332\) 0 0
\(333\) 86.5332i 0.259860i
\(334\) 0 0
\(335\) 158.644i 0.473565i
\(336\) 0 0
\(337\) 57.6888i 0.171183i −0.996330 0.0855917i \(-0.972722\pi\)
0.996330 0.0855917i \(-0.0272781\pi\)
\(338\) 0 0
\(339\) 442.000 1.30383
\(340\) 0 0
\(341\) 360.555i 1.05735i
\(342\) 0 0
\(343\) 365.000 1.06414
\(344\) 0 0
\(345\) 504.777i 1.46312i
\(346\) 0 0
\(347\) 40.0000 0.115274 0.0576369 0.998338i \(-0.481643\pi\)
0.0576369 + 0.998338i \(0.481643\pi\)
\(348\) 0 0
\(349\) −98.0000 −0.280802 −0.140401 0.990095i \(-0.544839\pi\)
−0.140401 + 0.990095i \(0.544839\pi\)
\(350\) 0 0
\(351\) 65.0000 0.185185
\(352\) 0 0
\(353\) −185.000 −0.524079 −0.262040 0.965057i \(-0.584395\pi\)
−0.262040 + 0.965057i \(0.584395\pi\)
\(354\) 0 0
\(355\) 432.666i 1.21878i
\(356\) 0 0
\(357\) 270.416i 0.757469i
\(358\) 0 0
\(359\) −225.000 −0.626741 −0.313370 0.949631i \(-0.601458\pi\)
−0.313370 + 0.949631i \(0.601458\pi\)
\(360\) 0 0
\(361\) −289.000 + 216.333i −0.800554 + 0.599261i
\(362\) 0 0
\(363\) 75.7166i 0.208586i
\(364\) 0 0
\(365\) −420.000 −1.15068
\(366\) 0 0
\(367\) 50.0000 0.136240 0.0681199 0.997677i \(-0.478300\pi\)
0.0681199 + 0.997677i \(0.478300\pi\)
\(368\) 0 0
\(369\) 144.222i 0.390846i
\(370\) 0 0
\(371\) 378.583i 1.02044i
\(372\) 0 0
\(373\) 436.272i 1.16963i −0.811167 0.584815i \(-0.801167\pi\)
0.811167 0.584815i \(-0.198833\pi\)
\(374\) 0 0
\(375\) 490.355i 1.30761i
\(376\) 0 0
\(377\) −65.0000 −0.172414
\(378\) 0 0
\(379\) 486.749i 1.28430i −0.766579 0.642150i \(-0.778043\pi\)
0.766579 0.642150i \(-0.221957\pi\)
\(380\) 0 0
\(381\) 468.000 1.22835
\(382\) 0 0
\(383\) 201.911i 0.527182i −0.964634 0.263591i \(-0.915093\pi\)
0.964634 0.263591i \(-0.0849070\pi\)
\(384\) 0 0
\(385\) 200.000 0.519481
\(386\) 0 0
\(387\) −80.0000 −0.206718
\(388\) 0 0
\(389\) 478.000 1.22879 0.614396 0.788998i \(-0.289400\pi\)
0.614396 + 0.788998i \(0.289400\pi\)
\(390\) 0 0
\(391\) 525.000 1.34271
\(392\) 0 0
\(393\) 403.822i 1.02754i
\(394\) 0 0
\(395\) 144.222i 0.365119i
\(396\) 0 0
\(397\) −750.000 −1.88917 −0.944584 0.328269i \(-0.893535\pi\)
−0.944584 + 0.328269i \(0.893535\pi\)
\(398\) 0 0
\(399\) −325.000 + 108.167i −0.814536 + 0.271094i
\(400\) 0 0
\(401\) 288.444i 0.719312i 0.933085 + 0.359656i \(0.117106\pi\)
−0.933085 + 0.359656i \(0.882894\pi\)
\(402\) 0 0
\(403\) −130.000 −0.322581
\(404\) 0 0
\(405\) 404.000 0.997531
\(406\) 0 0
\(407\) 216.333i 0.531531i
\(408\) 0 0
\(409\) 36.0555i 0.0881553i 0.999028 + 0.0440776i \(0.0140349\pi\)
−0.999028 + 0.0440776i \(0.985965\pi\)
\(410\) 0 0
\(411\) 450.694i 1.09658i
\(412\) 0 0
\(413\) 90.1388i 0.218254i
\(414\) 0 0
\(415\) −160.000 −0.385542
\(416\) 0 0
\(417\) 180.278i 0.432320i
\(418\) 0 0
\(419\) −112.000 −0.267303 −0.133652 0.991028i \(-0.542670\pi\)
−0.133652 + 0.991028i \(0.542670\pi\)
\(420\) 0 0
\(421\) 630.971i 1.49874i 0.662149 + 0.749372i \(0.269645\pi\)
−0.662149 + 0.749372i \(0.730355\pi\)
\(422\) 0 0
\(423\) −40.0000 −0.0945626
\(424\) 0 0
\(425\) −135.000 −0.317647
\(426\) 0 0
\(427\) −200.000 −0.468384
\(428\) 0 0
\(429\) 130.000 0.303030
\(430\) 0 0
\(431\) 432.666i 1.00387i 0.864907 + 0.501933i \(0.167378\pi\)
−0.864907 + 0.501933i \(0.832622\pi\)
\(432\) 0 0
\(433\) 735.532i 1.69869i 0.527839 + 0.849345i \(0.323003\pi\)
−0.527839 + 0.849345i \(0.676997\pi\)
\(434\) 0 0
\(435\) −260.000 −0.597701
\(436\) 0 0
\(437\) 210.000 + 630.971i 0.480549 + 1.44387i
\(438\) 0 0
\(439\) 793.221i 1.80688i −0.428712 0.903441i \(-0.641033\pi\)
0.428712 0.903441i \(-0.358967\pi\)
\(440\) 0 0
\(441\) 96.0000 0.217687
\(442\) 0 0
\(443\) −670.000 −1.51242 −0.756208 0.654332i \(-0.772950\pi\)
−0.756208 + 0.654332i \(0.772950\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 252.389i 0.564628i
\(448\) 0 0
\(449\) 36.0555i 0.0803018i 0.999194 + 0.0401509i \(0.0127839\pi\)
−0.999194 + 0.0401509i \(0.987216\pi\)
\(450\) 0 0
\(451\) 360.555i 0.799457i
\(452\) 0 0
\(453\) −130.000 −0.286976
\(454\) 0 0
\(455\) 72.1110i 0.158486i
\(456\) 0 0
\(457\) −755.000 −1.65208 −0.826039 0.563612i \(-0.809411\pi\)
−0.826039 + 0.563612i \(0.809411\pi\)
\(458\) 0 0
\(459\) 270.416i 0.589142i
\(460\) 0 0
\(461\) −772.000 −1.67462 −0.837310 0.546728i \(-0.815873\pi\)
−0.837310 + 0.546728i \(0.815873\pi\)
\(462\) 0 0
\(463\) −350.000 −0.755940 −0.377970 0.925818i \(-0.623378\pi\)
−0.377970 + 0.925818i \(0.623378\pi\)
\(464\) 0 0
\(465\) −520.000 −1.11828
\(466\) 0 0
\(467\) −70.0000 −0.149893 −0.0749465 0.997188i \(-0.523879\pi\)
−0.0749465 + 0.997188i \(0.523879\pi\)
\(468\) 0 0
\(469\) 198.305i 0.422826i
\(470\) 0 0
\(471\) 36.0555i 0.0765510i
\(472\) 0 0
\(473\) 200.000 0.422833
\(474\) 0 0
\(475\) −54.0000 162.250i −0.113684 0.341579i
\(476\) 0 0
\(477\) 302.866i 0.634940i
\(478\) 0 0
\(479\) −370.000 −0.772443 −0.386221 0.922406i \(-0.626220\pi\)
−0.386221 + 0.922406i \(0.626220\pi\)
\(480\) 0 0
\(481\) 78.0000 0.162162
\(482\) 0 0
\(483\) 630.971i 1.30636i
\(484\) 0 0
\(485\) 490.355i 1.01104i
\(486\) 0 0
\(487\) 519.199i 1.06612i 0.846078 + 0.533059i \(0.178958\pi\)
−0.846078 + 0.533059i \(0.821042\pi\)
\(488\) 0 0
\(489\) 973.499i 1.99080i
\(490\) 0 0
\(491\) 632.000 1.28717 0.643585 0.765375i \(-0.277446\pi\)
0.643585 + 0.765375i \(0.277446\pi\)
\(492\) 0 0
\(493\) 270.416i 0.548512i
\(494\) 0 0
\(495\) 160.000 0.323232
\(496\) 0 0
\(497\) 540.833i 1.08819i
\(498\) 0 0
\(499\) −380.000 −0.761523 −0.380762 0.924673i \(-0.624338\pi\)
−0.380762 + 0.924673i \(0.624338\pi\)
\(500\) 0 0
\(501\) −442.000 −0.882236
\(502\) 0 0
\(503\) −45.0000 −0.0894632 −0.0447316 0.998999i \(-0.514243\pi\)
−0.0447316 + 0.998999i \(0.514243\pi\)
\(504\) 0 0
\(505\) −200.000 −0.396040
\(506\) 0 0
\(507\) 562.466i 1.10940i
\(508\) 0 0
\(509\) 829.277i 1.62923i 0.580004 + 0.814614i \(0.303051\pi\)
−0.580004 + 0.814614i \(0.696949\pi\)
\(510\) 0 0
\(511\) −525.000 −1.02740
\(512\) 0 0
\(513\) 325.000 108.167i 0.633528 0.210851i
\(514\) 0 0
\(515\) 230.755i 0.448069i
\(516\) 0 0
\(517\) 100.000 0.193424
\(518\) 0 0
\(519\) −442.000 −0.851638
\(520\) 0 0
\(521\) 612.944i 1.17648i −0.808688 0.588238i \(-0.799822\pi\)
0.808688 0.588238i \(-0.200178\pi\)
\(522\) 0 0
\(523\) 465.116i 0.889323i 0.895699 + 0.444662i \(0.146676\pi\)
−0.895699 + 0.444662i \(0.853324\pi\)
\(524\) 0 0
\(525\) 162.250i 0.309047i
\(526\) 0 0
\(527\) 540.833i 1.02625i
\(528\) 0 0
\(529\) 696.000 1.31569
\(530\) 0 0
\(531\) 72.1110i 0.135802i
\(532\) 0 0
\(533\) 130.000 0.243902
\(534\) 0 0
\(535\) 302.866i 0.566105i
\(536\) 0 0
\(537\) 130.000 0.242086
\(538\) 0 0
\(539\) −240.000 −0.445269
\(540\) 0 0
\(541\) 600.000 1.10906 0.554529 0.832165i \(-0.312899\pi\)
0.554529 + 0.832165i \(0.312899\pi\)
\(542\) 0 0
\(543\) 390.000 0.718232
\(544\) 0 0
\(545\) 793.221i 1.45545i
\(546\) 0 0
\(547\) 598.522i 1.09419i −0.837071 0.547095i \(-0.815734\pi\)
0.837071 0.547095i \(-0.184266\pi\)
\(548\) 0 0
\(549\) −160.000 −0.291439
\(550\) 0 0
\(551\) −325.000 + 108.167i −0.589837 + 0.196310i
\(552\) 0 0
\(553\) 180.278i 0.325999i
\(554\) 0 0
\(555\) 312.000 0.562162
\(556\) 0 0
\(557\) −380.000 −0.682226 −0.341113 0.940022i \(-0.610804\pi\)
−0.341113 + 0.940022i \(0.610804\pi\)
\(558\) 0 0
\(559\) 72.1110i 0.129000i
\(560\) 0 0
\(561\) 540.833i 0.964051i
\(562\) 0 0
\(563\) 122.589i 0.217742i 0.994056 + 0.108871i \(0.0347235\pi\)
−0.994056 + 0.108871i \(0.965276\pi\)
\(564\) 0 0
\(565\) 490.355i 0.867885i
\(566\) 0 0
\(567\) 505.000 0.890653
\(568\) 0 0
\(569\) 36.0555i 0.0633665i −0.999498 0.0316832i \(-0.989913\pi\)
0.999498 0.0316832i \(-0.0100868\pi\)
\(570\) 0 0
\(571\) 790.000 1.38354 0.691769 0.722119i \(-0.256832\pi\)
0.691769 + 0.722119i \(0.256832\pi\)
\(572\) 0 0
\(573\) 695.871i 1.21444i
\(574\) 0 0
\(575\) −315.000 −0.547826
\(576\) 0 0
\(577\) 675.000 1.16984 0.584922 0.811090i \(-0.301125\pi\)
0.584922 + 0.811090i \(0.301125\pi\)
\(578\) 0 0
\(579\) −962.000 −1.66149
\(580\) 0 0
\(581\) −200.000 −0.344234
\(582\) 0 0
\(583\) 757.166i 1.29874i
\(584\) 0 0
\(585\) 57.6888i 0.0986134i
\(586\) 0 0
\(587\) 280.000 0.477002 0.238501 0.971142i \(-0.423344\pi\)
0.238501 + 0.971142i \(0.423344\pi\)
\(588\) 0 0
\(589\) −650.000 + 216.333i −1.10357 + 0.367289i
\(590\) 0 0
\(591\) 324.500i 0.549069i
\(592\) 0 0
\(593\) 750.000 1.26476 0.632378 0.774660i \(-0.282079\pi\)
0.632378 + 0.774660i \(0.282079\pi\)
\(594\) 0 0
\(595\) 300.000 0.504202
\(596\) 0 0
\(597\) 443.483i 0.742852i
\(598\) 0 0
\(599\) 504.777i 0.842700i −0.906898 0.421350i \(-0.861556\pi\)
0.906898 0.421350i \(-0.138444\pi\)
\(600\) 0 0
\(601\) 612.944i 1.01987i −0.860212 0.509937i \(-0.829669\pi\)
0.860212 0.509937i \(-0.170331\pi\)
\(602\) 0 0
\(603\) 158.644i 0.263092i
\(604\) 0 0
\(605\) 84.0000 0.138843
\(606\) 0 0
\(607\) 987.921i 1.62755i 0.581182 + 0.813774i \(0.302590\pi\)
−0.581182 + 0.813774i \(0.697410\pi\)
\(608\) 0 0
\(609\) −325.000 −0.533662
\(610\) 0 0
\(611\) 36.0555i 0.0590107i
\(612\) 0 0
\(613\) 1200.00 1.95759 0.978793 0.204853i \(-0.0656715\pi\)
0.978793 + 0.204853i \(0.0656715\pi\)
\(614\) 0 0
\(615\) 520.000 0.845528
\(616\) 0 0
\(617\) −350.000 −0.567261 −0.283630 0.958934i \(-0.591539\pi\)
−0.283630 + 0.958934i \(0.591539\pi\)
\(618\) 0 0
\(619\) −560.000 −0.904685 −0.452342 0.891844i \(-0.649412\pi\)
−0.452342 + 0.891844i \(0.649412\pi\)
\(620\) 0 0
\(621\) 630.971i 1.01606i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −319.000 −0.510400
\(626\) 0 0
\(627\) 650.000 216.333i 1.03668 0.345029i
\(628\) 0 0
\(629\) 324.500i 0.515898i
\(630\) 0 0
\(631\) −1050.00 −1.66403 −0.832013 0.554757i \(-0.812811\pi\)
−0.832013 + 0.554757i \(0.812811\pi\)
\(632\) 0 0
\(633\) −845.000 −1.33491
\(634\) 0 0
\(635\) 519.199i 0.817637i
\(636\) 0 0
\(637\) 86.5332i 0.135845i
\(638\) 0 0
\(639\) 432.666i 0.677099i
\(640\) 0 0
\(641\) 1225.89i 1.91246i −0.292615 0.956230i \(-0.594525\pi\)
0.292615 0.956230i \(-0.405475\pi\)
\(642\) 0 0
\(643\) 1030.00 1.60187 0.800933 0.598754i \(-0.204337\pi\)
0.800933 + 0.598754i \(0.204337\pi\)
\(644\) 0 0
\(645\) 288.444i 0.447200i
\(646\) 0 0
\(647\) 555.000 0.857805 0.428903 0.903351i \(-0.358900\pi\)
0.428903 + 0.903351i \(0.358900\pi\)
\(648\) 0 0
\(649\) 180.278i 0.277777i
\(650\) 0 0
\(651\) −650.000 −0.998464
\(652\) 0 0
\(653\) −50.0000 −0.0765697 −0.0382848 0.999267i \(-0.512189\pi\)
−0.0382848 + 0.999267i \(0.512189\pi\)
\(654\) 0 0
\(655\) 448.000 0.683969
\(656\) 0 0
\(657\) −420.000 −0.639269
\(658\) 0 0
\(659\) 198.305i 0.300919i 0.988616 + 0.150459i \(0.0480752\pi\)
−0.988616 + 0.150459i \(0.951925\pi\)
\(660\) 0 0
\(661\) 198.305i 0.300008i 0.988685 + 0.150004i \(0.0479287\pi\)
−0.988685 + 0.150004i \(0.952071\pi\)
\(662\) 0 0
\(663\) 195.000 0.294118
\(664\) 0 0
\(665\) 120.000 + 360.555i 0.180451 + 0.542188i
\(666\) 0 0
\(667\) 630.971i 0.945984i
\(668\) 0 0
\(669\) 728.000 1.08819
\(670\) 0 0
\(671\) 400.000 0.596125
\(672\) 0 0
\(673\) 598.522i 0.889334i −0.895696 0.444667i \(-0.853322\pi\)
0.895696 0.444667i \(-0.146678\pi\)
\(674\) 0 0
\(675\) 162.250i 0.240370i
\(676\) 0 0
\(677\) 68.5055i 0.101190i 0.998719 + 0.0505949i \(0.0161117\pi\)
−0.998719 + 0.0505949i \(0.983888\pi\)
\(678\) 0 0
\(679\) 612.944i 0.902715i
\(680\) 0 0
\(681\) 923.000 1.35536
\(682\) 0 0
\(683\) 237.966i 0.348413i −0.984709 0.174207i \(-0.944264\pi\)
0.984709 0.174207i \(-0.0557361\pi\)
\(684\) 0 0
\(685\) −500.000 −0.729927
\(686\) 0 0
\(687\) 576.888i 0.839721i
\(688\) 0 0
\(689\) 273.000 0.396226
\(690\) 0 0
\(691\) 820.000 1.18669 0.593343 0.804950i \(-0.297808\pi\)
0.593343 + 0.804950i \(0.297808\pi\)
\(692\) 0 0
\(693\) 200.000 0.288600
\(694\) 0 0
\(695\) 200.000 0.287770
\(696\) 0 0
\(697\) 540.833i 0.775944i
\(698\) 0 0
\(699\) 973.499i 1.39270i
\(700\) 0 0
\(701\) 540.000 0.770328 0.385164 0.922848i \(-0.374145\pi\)
0.385164 + 0.922848i \(0.374145\pi\)
\(702\) 0 0
\(703\) 390.000 129.800i 0.554765 0.184637i
\(704\) 0 0
\(705\) 144.222i 0.204570i
\(706\) 0 0
\(707\) −250.000 −0.353607
\(708\) 0 0
\(709\) −268.000 −0.377997 −0.188999 0.981977i \(-0.560524\pi\)
−0.188999 + 0.981977i \(0.560524\pi\)
\(710\) 0 0
\(711\) 144.222i 0.202844i
\(712\) 0 0
\(713\) 1261.94i 1.76991i
\(714\) 0 0
\(715\) 144.222i 0.201709i
\(716\) 0 0
\(717\) 710.294i 0.990647i
\(718\) 0 0
\(719\) 105.000 0.146036 0.0730181 0.997331i \(-0.476737\pi\)
0.0730181 + 0.997331i \(0.476737\pi\)
\(720\) 0 0
\(721\) 288.444i 0.400061i
\(722\) 0 0
\(723\) −1430.00 −1.97787
\(724\) 0 0
\(725\) 162.250i 0.223793i
\(726\) 0 0
\(727\) 695.000 0.955983 0.477992 0.878364i \(-0.341365\pi\)
0.477992 + 0.878364i \(0.341365\pi\)
\(728\) 0 0
\(729\) −181.000 −0.248285
\(730\) 0 0
\(731\) 300.000 0.410397
\(732\) 0 0
\(733\) −160.000 −0.218281 −0.109141 0.994026i \(-0.534810\pi\)
−0.109141 + 0.994026i \(0.534810\pi\)
\(734\) 0 0
\(735\) 346.133i 0.470929i
\(736\) 0 0
\(737\) 396.611i 0.538142i
\(738\) 0 0
\(739\) −1028.00 −1.39107 −0.695535 0.718493i \(-0.744832\pi\)
−0.695535 + 0.718493i \(0.744832\pi\)
\(740\) 0 0
\(741\) 78.0000 + 234.361i 0.105263 + 0.316276i
\(742\) 0 0
\(743\) 526.410i 0.708493i 0.935152 + 0.354247i \(0.115263\pi\)
−0.935152 + 0.354247i \(0.884737\pi\)
\(744\) 0 0
\(745\) 280.000 0.375839
\(746\) 0 0
\(747\) −160.000 −0.214190
\(748\) 0 0
\(749\) 378.583i 0.505451i
\(750\) 0 0
\(751\) 36.0555i 0.0480100i −0.999712 0.0240050i \(-0.992358\pi\)
0.999712 0.0240050i \(-0.00764176\pi\)
\(752\) 0 0
\(753\) 1449.43i 1.92488i
\(754\) 0 0
\(755\) 144.222i 0.191023i
\(756\) 0 0
\(757\) −60.0000 −0.0792602 −0.0396301 0.999214i \(-0.512618\pi\)
−0.0396301 + 0.999214i \(0.512618\pi\)
\(758\) 0 0
\(759\) 1261.94i 1.66264i
\(760\) 0 0
\(761\) −655.000 −0.860710 −0.430355 0.902660i \(-0.641612\pi\)
−0.430355 + 0.902660i \(0.641612\pi\)
\(762\) 0 0
\(763\) 991.527i 1.29951i
\(764\) 0 0
\(765\) 240.000 0.313725
\(766\) 0 0
\(767\) −65.0000 −0.0847458
\(768\) 0 0
\(769\) −185.000 −0.240572 −0.120286 0.992739i \(-0.538381\pi\)
−0.120286 + 0.992739i \(0.538381\pi\)
\(770\) 0 0
\(771\) 1508.00 1.95590
\(772\) 0 0
\(773\) 320.894i 0.415128i −0.978221 0.207564i \(-0.933446\pi\)
0.978221 0.207564i \(-0.0665536\pi\)
\(774\) 0 0
\(775\) 324.500i 0.418709i
\(776\) 0 0
\(777\) 390.000 0.501931
\(778\) 0 0
\(779\) 650.000 216.333i 0.834403 0.277706i
\(780\) 0 0
\(781\) 1081.67i 1.38497i
\(782\) 0 0
\(783\) 325.000 0.415070
\(784\) 0 0
\(785\) 40.0000 0.0509554
\(786\) 0 0
\(787\) 68.5055i 0.0870463i 0.999052 + 0.0435232i \(0.0138582\pi\)
−0.999052 + 0.0435232i \(0.986142\pi\)
\(788\) 0 0
\(789\) 1117.72i 1.41663i
\(790\) 0 0
\(791\) 612.944i 0.774897i
\(792\) 0 0
\(793\) 144.222i 0.181869i
\(794\) 0 0
\(795\) 1092.00 1.37358
\(796\) 0 0
\(797\) 1258.34i 1.57884i 0.613852 + 0.789421i \(0.289619\pi\)
−0.613852 + 0.789421i \(0.710381\pi\)
\(798\) 0 0
\(799\) 150.000 0.187735
\(800\) 0 0
\(801\) 0 0