Properties

Label 1200.3.bg.h.193.2
Level $1200$
Weight $3$
Character 1200.193
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.bg (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 193.2
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1200.193
Dual form 1200.3.bg.h.1057.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.22474 + 1.22474i) q^{3} +(-2.44949 + 2.44949i) q^{7} +3.00000i q^{9} +O(q^{10})\) \(q+(1.22474 + 1.22474i) q^{3} +(-2.44949 + 2.44949i) q^{7} +3.00000i q^{9} -6.00000 q^{11} +(-12.2474 - 12.2474i) q^{13} +(14.6969 - 14.6969i) q^{17} -10.0000i q^{19} -6.00000 q^{21} +(29.3939 + 29.3939i) q^{23} +(-3.67423 + 3.67423i) q^{27} -48.0000i q^{29} +26.0000 q^{31} +(-7.34847 - 7.34847i) q^{33} +(31.8434 - 31.8434i) q^{37} -30.0000i q^{39} +30.0000 q^{41} +(29.3939 + 29.3939i) q^{43} +(14.6969 - 14.6969i) q^{47} +37.0000i q^{49} +36.0000 q^{51} +(14.6969 + 14.6969i) q^{53} +(12.2474 - 12.2474i) q^{57} +78.0000i q^{59} +2.00000 q^{61} +(-7.34847 - 7.34847i) q^{63} +(63.6867 - 63.6867i) q^{67} +72.0000i q^{69} -120.000 q^{71} +(83.2827 + 83.2827i) q^{73} +(14.6969 - 14.6969i) q^{77} -74.0000i q^{79} -9.00000 q^{81} +(-44.0908 - 44.0908i) q^{83} +(58.7878 - 58.7878i) q^{87} -150.000i q^{89} +60.0000 q^{91} +(31.8434 + 31.8434i) q^{93} +(4.89898 - 4.89898i) q^{97} -18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q - 24 q^{11} - 24 q^{21} + 104 q^{31} + 120 q^{41} + 144 q^{51} + 8 q^{61} - 480 q^{71} - 36 q^{81} + 240 q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

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
\(3\) 1.22474 + 1.22474i 0.408248 + 0.408248i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.44949 + 2.44949i −0.349927 + 0.349927i −0.860082 0.510155i \(-0.829588\pi\)
0.510155 + 0.860082i \(0.329588\pi\)
\(8\) 0 0
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) −6.00000 −0.545455 −0.272727 0.962091i \(-0.587926\pi\)
−0.272727 + 0.962091i \(0.587926\pi\)
\(12\) 0 0
\(13\) −12.2474 12.2474i −0.942111 0.942111i 0.0563023 0.998414i \(-0.482069\pi\)
−0.998414 + 0.0563023i \(0.982069\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 14.6969 14.6969i 0.864526 0.864526i −0.127334 0.991860i \(-0.540642\pi\)
0.991860 + 0.127334i \(0.0406420\pi\)
\(18\) 0 0
\(19\) 10.0000i 0.526316i −0.964753 0.263158i \(-0.915236\pi\)
0.964753 0.263158i \(-0.0847640\pi\)
\(20\) 0 0
\(21\) −6.00000 −0.285714
\(22\) 0 0
\(23\) 29.3939 + 29.3939i 1.27799 + 1.27799i 0.941788 + 0.336206i \(0.109144\pi\)
0.336206 + 0.941788i \(0.390856\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.67423 + 3.67423i −0.136083 + 0.136083i
\(28\) 0 0
\(29\) 48.0000i 1.65517i −0.561339 0.827586i \(-0.689713\pi\)
0.561339 0.827586i \(-0.310287\pi\)
\(30\) 0 0
\(31\) 26.0000 0.838710 0.419355 0.907822i \(-0.362256\pi\)
0.419355 + 0.907822i \(0.362256\pi\)
\(32\) 0 0
\(33\) −7.34847 7.34847i −0.222681 0.222681i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 31.8434 31.8434i 0.860632 0.860632i −0.130780 0.991411i \(-0.541748\pi\)
0.991411 + 0.130780i \(0.0417481\pi\)
\(38\) 0 0
\(39\) 30.0000i 0.769231i
\(40\) 0 0
\(41\) 30.0000 0.731707 0.365854 0.930672i \(-0.380777\pi\)
0.365854 + 0.930672i \(0.380777\pi\)
\(42\) 0 0
\(43\) 29.3939 + 29.3939i 0.683579 + 0.683579i 0.960805 0.277226i \(-0.0894151\pi\)
−0.277226 + 0.960805i \(0.589415\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 14.6969 14.6969i 0.312701 0.312701i −0.533254 0.845955i \(-0.679031\pi\)
0.845955 + 0.533254i \(0.179031\pi\)
\(48\) 0 0
\(49\) 37.0000i 0.755102i
\(50\) 0 0
\(51\) 36.0000 0.705882
\(52\) 0 0
\(53\) 14.6969 + 14.6969i 0.277301 + 0.277301i 0.832031 0.554730i \(-0.187178\pi\)
−0.554730 + 0.832031i \(0.687178\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 12.2474 12.2474i 0.214868 0.214868i
\(58\) 0 0
\(59\) 78.0000i 1.32203i 0.750371 + 0.661017i \(0.229875\pi\)
−0.750371 + 0.661017i \(0.770125\pi\)
\(60\) 0 0
\(61\) 2.00000 0.0327869 0.0163934 0.999866i \(-0.494782\pi\)
0.0163934 + 0.999866i \(0.494782\pi\)
\(62\) 0 0
\(63\) −7.34847 7.34847i −0.116642 0.116642i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 63.6867 63.6867i 0.950548 0.950548i −0.0482853 0.998834i \(-0.515376\pi\)
0.998834 + 0.0482853i \(0.0153757\pi\)
\(68\) 0 0
\(69\) 72.0000i 1.04348i
\(70\) 0 0
\(71\) −120.000 −1.69014 −0.845070 0.534655i \(-0.820442\pi\)
−0.845070 + 0.534655i \(0.820442\pi\)
\(72\) 0 0
\(73\) 83.2827 + 83.2827i 1.14086 + 1.14086i 0.988293 + 0.152565i \(0.0487533\pi\)
0.152565 + 0.988293i \(0.451247\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 14.6969 14.6969i 0.190869 0.190869i
\(78\) 0 0
\(79\) 74.0000i 0.936709i −0.883541 0.468354i \(-0.844847\pi\)
0.883541 0.468354i \(-0.155153\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) −44.0908 44.0908i −0.531215 0.531215i 0.389719 0.920934i \(-0.372572\pi\)
−0.920934 + 0.389719i \(0.872572\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 58.7878 58.7878i 0.675721 0.675721i
\(88\) 0 0
\(89\) 150.000i 1.68539i −0.538389 0.842697i \(-0.680967\pi\)
0.538389 0.842697i \(-0.319033\pi\)
\(90\) 0 0
\(91\) 60.0000 0.659341
\(92\) 0 0
\(93\) 31.8434 + 31.8434i 0.342402 + 0.342402i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.89898 4.89898i 0.0505049 0.0505049i −0.681403 0.731908i \(-0.738630\pi\)
0.731908 + 0.681403i \(0.238630\pi\)
\(98\) 0 0
\(99\) 18.0000i 0.181818i
\(100\) 0 0
\(101\) −12.0000 −0.118812 −0.0594059 0.998234i \(-0.518921\pi\)
−0.0594059 + 0.998234i \(0.518921\pi\)
\(102\) 0 0
\(103\) −41.6413 41.6413i −0.404285 0.404285i 0.475455 0.879740i \(-0.342283\pi\)
−0.879740 + 0.475455i \(0.842283\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 102.879 102.879i 0.961482 0.961482i −0.0378032 0.999285i \(-0.512036\pi\)
0.999285 + 0.0378032i \(0.0120360\pi\)
\(108\) 0 0
\(109\) 74.0000i 0.678899i 0.940624 + 0.339450i \(0.110241\pi\)
−0.940624 + 0.339450i \(0.889759\pi\)
\(110\) 0 0
\(111\) 78.0000 0.702703
\(112\) 0 0
\(113\) 132.272 + 132.272i 1.17055 + 1.17055i 0.982078 + 0.188475i \(0.0603543\pi\)
0.188475 + 0.982078i \(0.439646\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 36.7423 36.7423i 0.314037 0.314037i
\(118\) 0 0
\(119\) 72.0000i 0.605042i
\(120\) 0 0
\(121\) −85.0000 −0.702479
\(122\) 0 0
\(123\) 36.7423 + 36.7423i 0.298718 + 0.298718i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 95.5301 95.5301i 0.752206 0.752206i −0.222685 0.974890i \(-0.571482\pi\)
0.974890 + 0.222685i \(0.0714821\pi\)
\(128\) 0 0
\(129\) 72.0000i 0.558140i
\(130\) 0 0
\(131\) −102.000 −0.778626 −0.389313 0.921106i \(-0.627288\pi\)
−0.389313 + 0.921106i \(0.627288\pi\)
\(132\) 0 0
\(133\) 24.4949 + 24.4949i 0.184172 + 0.184172i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 44.0908 44.0908i 0.321831 0.321831i −0.527638 0.849469i \(-0.676922\pi\)
0.849469 + 0.527638i \(0.176922\pi\)
\(138\) 0 0
\(139\) 122.000i 0.877698i 0.898561 + 0.438849i \(0.144614\pi\)
−0.898561 + 0.438849i \(0.855386\pi\)
\(140\) 0 0
\(141\) 36.0000 0.255319
\(142\) 0 0
\(143\) 73.4847 + 73.4847i 0.513879 + 0.513879i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −45.3156 + 45.3156i −0.308269 + 0.308269i
\(148\) 0 0
\(149\) 36.0000i 0.241611i −0.992676 0.120805i \(-0.961452\pi\)
0.992676 0.120805i \(-0.0385477\pi\)
\(150\) 0 0
\(151\) −70.0000 −0.463576 −0.231788 0.972766i \(-0.574458\pi\)
−0.231788 + 0.972766i \(0.574458\pi\)
\(152\) 0 0
\(153\) 44.0908 + 44.0908i 0.288175 + 0.288175i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 139.621 139.621i 0.889305 0.889305i −0.105151 0.994456i \(-0.533533\pi\)
0.994456 + 0.105151i \(0.0335326\pi\)
\(158\) 0 0
\(159\) 36.0000i 0.226415i
\(160\) 0 0
\(161\) −144.000 −0.894410
\(162\) 0 0
\(163\) 97.9796 + 97.9796i 0.601102 + 0.601102i 0.940605 0.339503i \(-0.110259\pi\)
−0.339503 + 0.940605i \(0.610259\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 29.3939 29.3939i 0.176011 0.176011i −0.613603 0.789614i \(-0.710281\pi\)
0.789614 + 0.613603i \(0.210281\pi\)
\(168\) 0 0
\(169\) 131.000i 0.775148i
\(170\) 0 0
\(171\) 30.0000 0.175439
\(172\) 0 0
\(173\) −191.060 191.060i −1.10439 1.10439i −0.993874 0.110520i \(-0.964748\pi\)
−0.110520 0.993874i \(-0.535252\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −95.5301 + 95.5301i −0.539718 + 0.539718i
\(178\) 0 0
\(179\) 222.000i 1.24022i −0.784514 0.620112i \(-0.787087\pi\)
0.784514 0.620112i \(-0.212913\pi\)
\(180\) 0 0
\(181\) 190.000 1.04972 0.524862 0.851187i \(-0.324117\pi\)
0.524862 + 0.851187i \(0.324117\pi\)
\(182\) 0 0
\(183\) 2.44949 + 2.44949i 0.0133852 + 0.0133852i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −88.1816 + 88.1816i −0.471560 + 0.471560i
\(188\) 0 0
\(189\) 18.0000i 0.0952381i
\(190\) 0 0
\(191\) −204.000 −1.06806 −0.534031 0.845465i \(-0.679324\pi\)
−0.534031 + 0.845465i \(0.679324\pi\)
\(192\) 0 0
\(193\) −68.5857 68.5857i −0.355366 0.355366i 0.506735 0.862102i \(-0.330852\pi\)
−0.862102 + 0.506735i \(0.830852\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 205.757 205.757i 1.04445 1.04445i 0.0454876 0.998965i \(-0.485516\pi\)
0.998965 0.0454876i \(-0.0144841\pi\)
\(198\) 0 0
\(199\) 46.0000i 0.231156i 0.993298 + 0.115578i \(0.0368720\pi\)
−0.993298 + 0.115578i \(0.963128\pi\)
\(200\) 0 0
\(201\) 156.000 0.776119
\(202\) 0 0
\(203\) 117.576 + 117.576i 0.579190 + 0.579190i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −88.1816 + 88.1816i −0.425998 + 0.425998i
\(208\) 0 0
\(209\) 60.0000i 0.287081i
\(210\) 0 0
\(211\) 310.000 1.46919 0.734597 0.678504i \(-0.237371\pi\)
0.734597 + 0.678504i \(0.237371\pi\)
\(212\) 0 0
\(213\) −146.969 146.969i −0.689997 0.689997i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −63.6867 + 63.6867i −0.293487 + 0.293487i
\(218\) 0 0
\(219\) 204.000i 0.931507i
\(220\) 0 0
\(221\) −360.000 −1.62896
\(222\) 0 0
\(223\) 183.712 + 183.712i 0.823819 + 0.823819i 0.986653 0.162834i \(-0.0520635\pi\)
−0.162834 + 0.986653i \(0.552064\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −73.4847 + 73.4847i −0.323721 + 0.323721i −0.850193 0.526472i \(-0.823515\pi\)
0.526472 + 0.850193i \(0.323515\pi\)
\(228\) 0 0
\(229\) 242.000i 1.05677i −0.849005 0.528384i \(-0.822798\pi\)
0.849005 0.528384i \(-0.177202\pi\)
\(230\) 0 0
\(231\) 36.0000 0.155844
\(232\) 0 0
\(233\) 73.4847 + 73.4847i 0.315385 + 0.315385i 0.846991 0.531607i \(-0.178411\pi\)
−0.531607 + 0.846991i \(0.678411\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 90.6311 90.6311i 0.382410 0.382410i
\(238\) 0 0
\(239\) 324.000i 1.35565i −0.735224 0.677824i \(-0.762923\pi\)
0.735224 0.677824i \(-0.237077\pi\)
\(240\) 0 0
\(241\) 398.000 1.65145 0.825726 0.564071i \(-0.190766\pi\)
0.825726 + 0.564071i \(0.190766\pi\)
\(242\) 0 0
\(243\) −11.0227 11.0227i −0.0453609 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −122.474 + 122.474i −0.495848 + 0.495848i
\(248\) 0 0
\(249\) 108.000i 0.433735i
\(250\) 0 0
\(251\) 162.000 0.645418 0.322709 0.946498i \(-0.395406\pi\)
0.322709 + 0.946498i \(0.395406\pi\)
\(252\) 0 0
\(253\) −176.363 176.363i −0.697088 0.697088i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −102.879 + 102.879i −0.400306 + 0.400306i −0.878341 0.478035i \(-0.841349\pi\)
0.478035 + 0.878341i \(0.341349\pi\)
\(258\) 0 0
\(259\) 156.000i 0.602317i
\(260\) 0 0
\(261\) 144.000 0.551724
\(262\) 0 0
\(263\) −117.576 117.576i −0.447055 0.447055i 0.447319 0.894374i \(-0.352379\pi\)
−0.894374 + 0.447319i \(0.852379\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 183.712 183.712i 0.688059 0.688059i
\(268\) 0 0
\(269\) 48.0000i 0.178439i 0.996012 + 0.0892193i \(0.0284372\pi\)
−0.996012 + 0.0892193i \(0.971563\pi\)
\(270\) 0 0
\(271\) 46.0000 0.169742 0.0848708 0.996392i \(-0.472952\pi\)
0.0848708 + 0.996392i \(0.472952\pi\)
\(272\) 0 0
\(273\) 73.4847 + 73.4847i 0.269175 + 0.269175i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −80.8332 + 80.8332i −0.291816 + 0.291816i −0.837798 0.545981i \(-0.816157\pi\)
0.545981 + 0.837798i \(0.316157\pi\)
\(278\) 0 0
\(279\) 78.0000i 0.279570i
\(280\) 0 0
\(281\) −414.000 −1.47331 −0.736655 0.676269i \(-0.763596\pi\)
−0.736655 + 0.676269i \(0.763596\pi\)
\(282\) 0 0
\(283\) −279.242 279.242i −0.986720 0.986720i 0.0131927 0.999913i \(-0.495801\pi\)
−0.999913 + 0.0131927i \(0.995801\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −73.4847 + 73.4847i −0.256044 + 0.256044i
\(288\) 0 0
\(289\) 143.000i 0.494810i
\(290\) 0 0
\(291\) 12.0000 0.0412371
\(292\) 0 0
\(293\) −235.151 235.151i −0.802563 0.802563i 0.180932 0.983496i \(-0.442088\pi\)
−0.983496 + 0.180932i \(0.942088\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 22.0454 22.0454i 0.0742270 0.0742270i
\(298\) 0 0
\(299\) 720.000i 2.40803i
\(300\) 0 0
\(301\) −144.000 −0.478405
\(302\) 0 0
\(303\) −14.6969 14.6969i −0.0485047 0.0485047i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −44.0908 + 44.0908i −0.143618 + 0.143618i −0.775260 0.631642i \(-0.782381\pi\)
0.631642 + 0.775260i \(0.282381\pi\)
\(308\) 0 0
\(309\) 102.000i 0.330097i
\(310\) 0 0
\(311\) −204.000 −0.655949 −0.327974 0.944687i \(-0.606366\pi\)
−0.327974 + 0.944687i \(0.606366\pi\)
\(312\) 0 0
\(313\) 372.322 + 372.322i 1.18953 + 1.18953i 0.977198 + 0.212331i \(0.0681054\pi\)
0.212331 + 0.977198i \(0.431895\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −352.727 + 352.727i −1.11270 + 1.11270i −0.119918 + 0.992784i \(0.538263\pi\)
−0.992784 + 0.119918i \(0.961737\pi\)
\(318\) 0 0
\(319\) 288.000i 0.902821i
\(320\) 0 0
\(321\) 252.000 0.785047
\(322\) 0 0
\(323\) −146.969 146.969i −0.455014 0.455014i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −90.6311 + 90.6311i −0.277159 + 0.277159i
\(328\) 0 0
\(329\) 72.0000i 0.218845i
\(330\) 0 0
\(331\) −542.000 −1.63746 −0.818731 0.574177i \(-0.805322\pi\)
−0.818731 + 0.574177i \(0.805322\pi\)
\(332\) 0 0
\(333\) 95.5301 + 95.5301i 0.286877 + 0.286877i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −347.828 + 347.828i −1.03213 + 1.03213i −0.0326628 + 0.999466i \(0.510399\pi\)
−0.999466 + 0.0326628i \(0.989601\pi\)
\(338\) 0 0
\(339\) 324.000i 0.955752i
\(340\) 0 0
\(341\) −156.000 −0.457478
\(342\) 0 0
\(343\) −210.656 210.656i −0.614158 0.614158i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −191.060 + 191.060i −0.550606 + 0.550606i −0.926616 0.376010i \(-0.877296\pi\)
0.376010 + 0.926616i \(0.377296\pi\)
\(348\) 0 0
\(349\) 358.000i 1.02579i 0.858452 + 0.512894i \(0.171427\pi\)
−0.858452 + 0.512894i \(0.828573\pi\)
\(350\) 0 0
\(351\) 90.0000 0.256410
\(352\) 0 0
\(353\) 426.211 + 426.211i 1.20740 + 1.20740i 0.971867 + 0.235530i \(0.0756826\pi\)
0.235530 + 0.971867i \(0.424317\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −88.1816 + 88.1816i −0.247007 + 0.247007i
\(358\) 0 0
\(359\) 132.000i 0.367688i −0.982955 0.183844i \(-0.941146\pi\)
0.982955 0.183844i \(-0.0588541\pi\)
\(360\) 0 0
\(361\) 261.000 0.722992
\(362\) 0 0
\(363\) −104.103 104.103i −0.286786 0.286786i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 120.025 120.025i 0.327044 0.327044i −0.524418 0.851461i \(-0.675717\pi\)
0.851461 + 0.524418i \(0.175717\pi\)
\(368\) 0 0
\(369\) 90.0000i 0.243902i
\(370\) 0 0
\(371\) −72.0000 −0.194070
\(372\) 0 0
\(373\) −409.065 409.065i −1.09669 1.09669i −0.994795 0.101893i \(-0.967510\pi\)
−0.101893 0.994795i \(-0.532490\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −587.878 + 587.878i −1.55936 + 1.55936i
\(378\) 0 0
\(379\) 26.0000i 0.0686016i 0.999412 + 0.0343008i \(0.0109204\pi\)
−0.999412 + 0.0343008i \(0.989080\pi\)
\(380\) 0 0
\(381\) 234.000 0.614173
\(382\) 0 0
\(383\) −426.211 426.211i −1.11282 1.11282i −0.992767 0.120056i \(-0.961693\pi\)
−0.120056 0.992767i \(-0.538307\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −88.1816 + 88.1816i −0.227860 + 0.227860i
\(388\) 0 0
\(389\) 72.0000i 0.185090i −0.995709 0.0925450i \(-0.970500\pi\)
0.995709 0.0925450i \(-0.0295002\pi\)
\(390\) 0 0
\(391\) 864.000 2.20972
\(392\) 0 0
\(393\) −124.924 124.924i −0.317873 0.317873i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 227.803 227.803i 0.573810 0.573810i −0.359381 0.933191i \(-0.617012\pi\)
0.933191 + 0.359381i \(0.117012\pi\)
\(398\) 0 0
\(399\) 60.0000i 0.150376i
\(400\) 0 0
\(401\) −414.000 −1.03242 −0.516209 0.856462i \(-0.672657\pi\)
−0.516209 + 0.856462i \(0.672657\pi\)
\(402\) 0 0
\(403\) −318.434 318.434i −0.790158 0.790158i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −191.060 + 191.060i −0.469435 + 0.469435i
\(408\) 0 0
\(409\) 482.000i 1.17848i −0.807957 0.589242i \(-0.799426\pi\)
0.807957 0.589242i \(-0.200574\pi\)
\(410\) 0 0
\(411\) 108.000 0.262774
\(412\) 0 0
\(413\) −191.060 191.060i −0.462615 0.462615i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −149.419 + 149.419i −0.358319 + 0.358319i
\(418\) 0 0
\(419\) 126.000i 0.300716i −0.988632 0.150358i \(-0.951957\pi\)
0.988632 0.150358i \(-0.0480426\pi\)
\(420\) 0 0
\(421\) 430.000 1.02138 0.510689 0.859766i \(-0.329390\pi\)
0.510689 + 0.859766i \(0.329390\pi\)
\(422\) 0 0
\(423\) 44.0908 + 44.0908i 0.104234 + 0.104234i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4.89898 + 4.89898i −0.0114730 + 0.0114730i
\(428\) 0 0
\(429\) 180.000i 0.419580i
\(430\) 0 0
\(431\) −228.000 −0.529002 −0.264501 0.964385i \(-0.585207\pi\)
−0.264501 + 0.964385i \(0.585207\pi\)
\(432\) 0 0
\(433\) 249.848 + 249.848i 0.577016 + 0.577016i 0.934080 0.357064i \(-0.116222\pi\)
−0.357064 + 0.934080i \(0.616222\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 293.939 293.939i 0.672629 0.672629i
\(438\) 0 0
\(439\) 194.000i 0.441913i −0.975284 0.220957i \(-0.929082\pi\)
0.975284 0.220957i \(-0.0709180\pi\)
\(440\) 0 0
\(441\) −111.000 −0.251701
\(442\) 0 0
\(443\) 338.030 + 338.030i 0.763046 + 0.763046i 0.976872 0.213825i \(-0.0685924\pi\)
−0.213825 + 0.976872i \(0.568592\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 44.0908 44.0908i 0.0986372 0.0986372i
\(448\) 0 0
\(449\) 6.00000i 0.0133630i −0.999978 0.00668151i \(-0.997873\pi\)
0.999978 0.00668151i \(-0.00212681\pi\)
\(450\) 0 0
\(451\) −180.000 −0.399113
\(452\) 0 0
\(453\) −85.7321 85.7321i −0.189254 0.189254i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −367.423 + 367.423i −0.803990 + 0.803990i −0.983717 0.179727i \(-0.942479\pi\)
0.179727 + 0.983717i \(0.442479\pi\)
\(458\) 0 0
\(459\) 108.000i 0.235294i
\(460\) 0 0
\(461\) −204.000 −0.442516 −0.221258 0.975215i \(-0.571016\pi\)
−0.221258 + 0.975215i \(0.571016\pi\)
\(462\) 0 0
\(463\) 100.429 + 100.429i 0.216909 + 0.216909i 0.807195 0.590285i \(-0.200985\pi\)
−0.590285 + 0.807195i \(0.700985\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −191.060 + 191.060i −0.409122 + 0.409122i −0.881432 0.472310i \(-0.843420\pi\)
0.472310 + 0.881432i \(0.343420\pi\)
\(468\) 0 0
\(469\) 312.000i 0.665245i
\(470\) 0 0
\(471\) 342.000 0.726115
\(472\) 0 0
\(473\) −176.363 176.363i −0.372861 0.372861i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −44.0908 + 44.0908i −0.0924336 + 0.0924336i
\(478\) 0 0
\(479\) 888.000i 1.85386i 0.375232 + 0.926931i \(0.377563\pi\)
−0.375232 + 0.926931i \(0.622437\pi\)
\(480\) 0 0
\(481\) −780.000 −1.62162
\(482\) 0 0
\(483\) −176.363 176.363i −0.365141 0.365141i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −531.539 + 531.539i −1.09146 + 1.09146i −0.0960831 + 0.995373i \(0.530631\pi\)
−0.995373 + 0.0960831i \(0.969369\pi\)
\(488\) 0 0
\(489\) 240.000i 0.490798i
\(490\) 0 0
\(491\) −534.000 −1.08758 −0.543788 0.839223i \(-0.683011\pi\)
−0.543788 + 0.839223i \(0.683011\pi\)
\(492\) 0 0
\(493\) −705.453 705.453i −1.43094 1.43094i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 293.939 293.939i 0.591426 0.591426i
\(498\) 0 0
\(499\) 658.000i 1.31864i 0.751864 + 0.659319i \(0.229155\pi\)
−0.751864 + 0.659319i \(0.770845\pi\)
\(500\) 0 0
\(501\) 72.0000 0.143713
\(502\) 0 0
\(503\) 191.060 + 191.060i 0.379841 + 0.379841i 0.871045 0.491203i \(-0.163443\pi\)
−0.491203 + 0.871045i \(0.663443\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −160.442 + 160.442i −0.316453 + 0.316453i
\(508\) 0 0
\(509\) 324.000i 0.636542i −0.948000 0.318271i \(-0.896898\pi\)
0.948000 0.318271i \(-0.103102\pi\)
\(510\) 0 0
\(511\) −408.000 −0.798434
\(512\) 0 0
\(513\) 36.7423 + 36.7423i 0.0716225 + 0.0716225i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −88.1816 + 88.1816i −0.170564 + 0.170564i
\(518\) 0 0
\(519\) 468.000i 0.901734i
\(520\) 0 0
\(521\) 342.000 0.656430 0.328215 0.944603i \(-0.393553\pi\)
0.328215 + 0.944603i \(0.393553\pi\)
\(522\) 0 0
\(523\) 578.080 + 578.080i 1.10531 + 1.10531i 0.993758 + 0.111557i \(0.0355836\pi\)
0.111557 + 0.993758i \(0.464416\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 382.120 382.120i 0.725086 0.725086i
\(528\) 0 0
\(529\) 1199.00i 2.26654i
\(530\) 0 0
\(531\) −234.000 −0.440678
\(532\) 0 0
\(533\) −367.423 367.423i −0.689350 0.689350i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 271.893 271.893i 0.506319 0.506319i
\(538\) 0 0
\(539\) 222.000i 0.411874i
\(540\) 0 0
\(541\) 98.0000 0.181146 0.0905730 0.995890i \(-0.471130\pi\)
0.0905730 + 0.995890i \(0.471130\pi\)
\(542\) 0 0
\(543\) 232.702 + 232.702i 0.428548 + 0.428548i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −519.292 + 519.292i −0.949345 + 0.949345i −0.998777 0.0494323i \(-0.984259\pi\)
0.0494323 + 0.998777i \(0.484259\pi\)
\(548\) 0 0
\(549\) 6.00000i 0.0109290i
\(550\) 0 0
\(551\) −480.000 −0.871143
\(552\) 0 0
\(553\) 181.262 + 181.262i 0.327780 + 0.327780i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 396.817 396.817i 0.712419 0.712419i −0.254622 0.967041i \(-0.581951\pi\)
0.967041 + 0.254622i \(0.0819510\pi\)
\(558\) 0 0
\(559\) 720.000i 1.28801i
\(560\) 0 0
\(561\) −216.000 −0.385027
\(562\) 0 0
\(563\) −73.4847 73.4847i −0.130523 0.130523i 0.638827 0.769350i \(-0.279420\pi\)
−0.769350 + 0.638827i \(0.779420\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 22.0454 22.0454i 0.0388808 0.0388808i
\(568\) 0 0
\(569\) 762.000i 1.33919i −0.742726 0.669596i \(-0.766467\pi\)
0.742726 0.669596i \(-0.233533\pi\)
\(570\) 0 0
\(571\) 850.000 1.48862 0.744308 0.667836i \(-0.232779\pi\)
0.744308 + 0.667836i \(0.232779\pi\)
\(572\) 0 0
\(573\) −249.848 249.848i −0.436035 0.436035i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 4.89898 4.89898i 0.00849043 0.00849043i −0.702849 0.711339i \(-0.748089\pi\)
0.711339 + 0.702849i \(0.248089\pi\)
\(578\) 0 0
\(579\) 168.000i 0.290155i
\(580\) 0 0
\(581\) 216.000 0.371773
\(582\) 0 0
\(583\) −88.1816 88.1816i −0.151255 0.151255i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 132.272 132.272i 0.225336 0.225336i −0.585405 0.810741i \(-0.699064\pi\)
0.810741 + 0.585405i \(0.199064\pi\)
\(588\) 0 0
\(589\) 260.000i 0.441426i
\(590\) 0 0
\(591\) 504.000 0.852792
\(592\) 0 0
\(593\) −102.879 102.879i −0.173488 0.173488i 0.615022 0.788510i \(-0.289147\pi\)
−0.788510 + 0.615022i \(0.789147\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −56.3383 + 56.3383i −0.0943690 + 0.0943690i
\(598\) 0 0
\(599\) 732.000i 1.22204i 0.791616 + 0.611018i \(0.209240\pi\)
−0.791616 + 0.611018i \(0.790760\pi\)
\(600\) 0 0
\(601\) 778.000 1.29451 0.647255 0.762274i \(-0.275917\pi\)
0.647255 + 0.762274i \(0.275917\pi\)
\(602\) 0 0
\(603\) 191.060 + 191.060i 0.316849 + 0.316849i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −105.328 + 105.328i −0.173522 + 0.173522i −0.788525 0.615003i \(-0.789155\pi\)
0.615003 + 0.788525i \(0.289155\pi\)
\(608\) 0 0
\(609\) 288.000i 0.472906i
\(610\) 0 0
\(611\) −360.000 −0.589198
\(612\) 0 0
\(613\) 71.0352 + 71.0352i 0.115881 + 0.115881i 0.762670 0.646788i \(-0.223888\pi\)
−0.646788 + 0.762670i \(0.723888\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −455.605 + 455.605i −0.738420 + 0.738420i −0.972272 0.233852i \(-0.924867\pi\)
0.233852 + 0.972272i \(0.424867\pi\)
\(618\) 0 0
\(619\) 362.000i 0.584814i −0.956294 0.292407i \(-0.905544\pi\)
0.956294 0.292407i \(-0.0944562\pi\)
\(620\) 0 0
\(621\) −216.000 −0.347826
\(622\) 0 0
\(623\) 367.423 + 367.423i 0.589765 + 0.589765i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −73.4847 + 73.4847i −0.117200 + 0.117200i
\(628\) 0 0
\(629\) 936.000i 1.48808i
\(630\) 0 0
\(631\) 478.000 0.757528 0.378764 0.925493i \(-0.376349\pi\)
0.378764 + 0.925493i \(0.376349\pi\)
\(632\) 0 0
\(633\) 379.671 + 379.671i 0.599796 + 0.599796i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 453.156 453.156i 0.711390 0.711390i
\(638\) 0 0
\(639\) 360.000i 0.563380i
\(640\) 0 0
\(641\) 354.000 0.552262 0.276131 0.961120i \(-0.410948\pi\)
0.276131 + 0.961120i \(0.410948\pi\)
\(642\) 0 0
\(643\) 264.545 + 264.545i 0.411423 + 0.411423i 0.882234 0.470811i \(-0.156039\pi\)
−0.470811 + 0.882234i \(0.656039\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −484.999 + 484.999i −0.749612 + 0.749612i −0.974406 0.224794i \(-0.927829\pi\)
0.224794 + 0.974406i \(0.427829\pi\)
\(648\) 0 0
\(649\) 468.000i 0.721109i
\(650\) 0 0
\(651\) −156.000 −0.239631
\(652\) 0 0
\(653\) −29.3939 29.3939i −0.0450136 0.0450136i 0.684242 0.729255i \(-0.260133\pi\)
−0.729255 + 0.684242i \(0.760133\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −249.848 + 249.848i −0.380286 + 0.380286i
\(658\) 0 0
\(659\) 414.000i 0.628225i −0.949386 0.314112i \(-0.898293\pi\)
0.949386 0.314112i \(-0.101707\pi\)
\(660\) 0 0
\(661\) −1202.00 −1.81846 −0.909228 0.416298i \(-0.863327\pi\)
−0.909228 + 0.416298i \(0.863327\pi\)
\(662\) 0 0
\(663\) −440.908 440.908i −0.665020 0.665020i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1410.91 1410.91i 2.11530 2.11530i
\(668\) 0 0
\(669\) 450.000i 0.672646i
\(670\) 0 0
\(671\) −12.0000 −0.0178838
\(672\) 0 0
\(673\) 171.464 + 171.464i 0.254776 + 0.254776i 0.822925 0.568149i \(-0.192340\pi\)
−0.568149 + 0.822925i \(0.692340\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 235.151 235.151i 0.347343 0.347343i −0.511776 0.859119i \(-0.671012\pi\)
0.859119 + 0.511776i \(0.171012\pi\)
\(678\) 0 0
\(679\) 24.0000i 0.0353461i
\(680\) 0 0
\(681\) −180.000 −0.264317
\(682\) 0 0
\(683\) −690.756 690.756i −1.01136 1.01136i −0.999935 0.0114212i \(-0.996364\pi\)
−0.0114212 0.999935i \(-0.503636\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 296.388 296.388i 0.431424 0.431424i
\(688\) 0 0
\(689\) 360.000i 0.522496i
\(690\) 0 0
\(691\) −778.000 −1.12590 −0.562952 0.826489i \(-0.690335\pi\)
−0.562952 + 0.826489i \(0.690335\pi\)
\(692\) 0 0
\(693\) 44.0908 + 44.0908i 0.0636231 + 0.0636231i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 440.908 440.908i 0.632580 0.632580i
\(698\) 0 0
\(699\) 180.000i 0.257511i
\(700\) 0 0
\(701\) −84.0000 −0.119829 −0.0599144 0.998204i \(-0.519083\pi\)
−0.0599144 + 0.998204i \(0.519083\pi\)
\(702\) 0 0
\(703\) −318.434 318.434i −0.452964 0.452964i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 29.3939 29.3939i 0.0415755 0.0415755i
\(708\) 0 0
\(709\) 502.000i 0.708039i 0.935238 + 0.354020i \(0.115185\pi\)
−0.935238 + 0.354020i \(0.884815\pi\)
\(710\) 0 0
\(711\) 222.000 0.312236
\(712\) 0 0
\(713\) 764.241 + 764.241i 1.07187 + 1.07187i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 396.817 396.817i 0.553441 0.553441i
\(718\) 0 0
\(719\) 1356.00i 1.88595i 0.332860 + 0.942976i \(0.391986\pi\)
−0.332860 + 0.942976i \(0.608014\pi\)
\(720\) 0 0
\(721\) 204.000 0.282940
\(722\) 0 0
\(723\) 487.448 + 487.448i 0.674203 + 0.674203i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 815.680 815.680i 1.12198 1.12198i 0.130537 0.991443i \(-0.458330\pi\)
0.991443 0.130537i \(-0.0416703\pi\)
\(728\) 0 0
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) 864.000 1.18194
\(732\) 0 0
\(733\) 962.649 + 962.649i 1.31330 + 1.31330i 0.918968 + 0.394333i \(0.129024\pi\)
0.394333 + 0.918968i \(0.370976\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −382.120 + 382.120i −0.518481 + 0.518481i
\(738\) 0 0
\(739\) 610.000i 0.825440i −0.910858 0.412720i \(-0.864579\pi\)
0.910858 0.412720i \(-0.135421\pi\)
\(740\) 0 0
\(741\) −300.000 −0.404858
\(742\) 0 0
\(743\) 44.0908 + 44.0908i 0.0593416 + 0.0593416i 0.736155 0.676813i \(-0.236640\pi\)
−0.676813 + 0.736155i \(0.736640\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 132.272 132.272i 0.177072 0.177072i
\(748\) 0 0
\(749\) 504.000i 0.672897i
\(750\) 0 0
\(751\) 1058.00 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(752\) 0 0
\(753\) 198.409 + 198.409i 0.263491 + 0.263491i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −26.9444 + 26.9444i −0.0355936 + 0.0355936i −0.724680 0.689086i \(-0.758012\pi\)
0.689086 + 0.724680i \(0.258012\pi\)
\(758\) 0 0
\(759\) 432.000i 0.569170i
\(760\) 0 0
\(761\) 1122.00 1.47438 0.737188 0.675688i \(-0.236153\pi\)
0.737188 + 0.675688i \(0.236153\pi\)
\(762\) 0 0
\(763\) −181.262 181.262i −0.237565 0.237565i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 955.301 955.301i 1.24550 1.24550i
\(768\) 0 0
\(769\) 274.000i 0.356307i −0.984003 0.178153i \(-0.942988\pi\)
0.984003 0.178153i \(-0.0570123\pi\)
\(770\) 0 0
\(771\) −252.000 −0.326848
\(772\) 0 0
\(773\) 382.120 + 382.120i 0.494334 + 0.494334i 0.909669 0.415334i \(-0.136335\pi\)
−0.415334 + 0.909669i \(0.636335\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −191.060 + 191.060i −0.245895 + 0.245895i
\(778\) 0 0
\(779\) 300.000i 0.385109i
\(780\) 0 0
\(781\) 720.000 0.921895
\(782\) 0 0
\(783\) 176.363 + 176.363i 0.225240 + 0.225240i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −421.312 + 421.312i −0.535340 + 0.535340i −0.922157 0.386817i \(-0.873575\pi\)
0.386817 + 0.922157i \(0.373575\pi\)
\(788\) 0 0
\(789\) 288.000i 0.365019i
\(790\) 0 0
\(791\) −648.000 −0.819216
\(792\) 0 0
\(793\) −24.4949 24.4949i −0.0308889 0.0308889i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 396.817 396.817i 0.497889 0.497889i −0.412891 0.910780i \(-0.635481\pi\)
0.910780 + 0.412891i \(0.135481\pi\)
\(798\) 0 0
\(799\) 432.000i 0.540676i
\(800\) 0 0 </