Properties

Label 273.2.bw.a.149.1
Level $273$
Weight $2$
Character 273.149
Analytic conductor $2.180$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bw (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label 149.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 273.149
Dual form 273.2.bw.a.11.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205 q^{3} +(1.73205 + 1.00000i) q^{4} +(-0.866025 - 2.50000i) q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{3} +(1.73205 + 1.00000i) q^{4} +(-0.866025 - 2.50000i) q^{7} +3.00000 q^{9} +(3.00000 + 1.73205i) q^{12} +(-3.46410 - 1.00000i) q^{13} +(2.00000 + 3.46410i) q^{16} +(-3.83013 + 3.83013i) q^{19} +(-1.50000 - 4.33013i) q^{21} +(-4.33013 + 2.50000i) q^{25} +5.19615 q^{27} +(1.00000 - 5.19615i) q^{28} +(-1.13397 - 0.303848i) q^{31} +(5.19615 + 3.00000i) q^{36} +(1.06218 - 3.96410i) q^{37} +(-6.00000 - 1.73205i) q^{39} +(9.00000 - 5.19615i) q^{43} +(3.46410 + 6.00000i) q^{48} +(-5.50000 + 4.33013i) q^{49} +(-5.00000 - 5.19615i) q^{52} +(-6.63397 + 6.63397i) q^{57} -15.5885 q^{61} +(-2.59808 - 7.50000i) q^{63} +8.00000i q^{64} +(-11.5622 + 11.5622i) q^{67} +(4.36603 - 16.2942i) q^{73} +(-7.50000 + 4.33013i) q^{75} +(-10.4641 + 2.80385i) q^{76} +(6.06218 - 10.5000i) q^{79} +9.00000 q^{81} +(1.73205 - 9.00000i) q^{84} +(0.500000 + 9.52628i) q^{91} +(-1.96410 - 0.526279i) q^{93} +(9.42820 + 2.52628i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} + 12q^{12} + 8q^{16} + 2q^{19} - 6q^{21} + 4q^{28} - 8q^{31} - 20q^{37} - 24q^{39} + 36q^{43} - 22q^{49} - 20q^{52} - 30q^{57} - 22q^{67} + 14q^{73} - 30q^{75} - 28q^{76} + 36q^{81} + 2q^{91} + 6q^{93} + 10q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\)

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 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(3\) 1.73205 1.00000
\(4\) 1.73205 + 1.00000i 0.866025 + 0.500000i
\(5\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(6\) 0 0
\(7\) −0.866025 2.50000i −0.327327 0.944911i
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(12\) 3.00000 + 1.73205i 0.866025 + 0.500000i
\(13\) −3.46410 1.00000i −0.960769 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −3.83013 + 3.83013i −0.878691 + 0.878691i −0.993399 0.114708i \(-0.963407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) −1.50000 4.33013i −0.327327 0.944911i
\(22\) 0 0
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) −4.33013 + 2.50000i −0.866025 + 0.500000i
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 1.00000 5.19615i 0.188982 0.981981i
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) −1.13397 0.303848i −0.203668 0.0545726i 0.155543 0.987829i \(-0.450287\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 5.19615 + 3.00000i 0.866025 + 0.500000i
\(37\) 1.06218 3.96410i 0.174621 0.651694i −0.821995 0.569495i \(-0.807139\pi\)
0.996616 0.0821995i \(-0.0261945\pi\)
\(38\) 0 0
\(39\) −6.00000 1.73205i −0.960769 0.277350i
\(40\) 0 0
\(41\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(42\) 0 0
\(43\) 9.00000 5.19615i 1.37249 0.792406i 0.381246 0.924473i \(-0.375495\pi\)
0.991241 + 0.132068i \(0.0421616\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(48\) 3.46410 + 6.00000i 0.500000 + 0.866025i
\(49\) −5.50000 + 4.33013i −0.785714 + 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) −5.00000 5.19615i −0.693375 0.720577i
\(53\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.63397 + 6.63397i −0.878691 + 0.878691i
\(58\) 0 0
\(59\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(60\) 0 0
\(61\) −15.5885 −1.99590 −0.997949 0.0640184i \(-0.979608\pi\)
−0.997949 + 0.0640184i \(0.979608\pi\)
\(62\) 0 0
\(63\) −2.59808 7.50000i −0.327327 0.944911i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) −11.5622 + 11.5622i −1.41254 + 1.41254i −0.671932 + 0.740613i \(0.734535\pi\)
−0.740613 + 0.671932i \(0.765465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(72\) 0 0
\(73\) 4.36603 16.2942i 0.511005 1.90710i 0.101361 0.994850i \(-0.467680\pi\)
0.409644 0.912245i \(-0.365653\pi\)
\(74\) 0 0
\(75\) −7.50000 + 4.33013i −0.866025 + 0.500000i
\(76\) −10.4641 + 2.80385i −1.20031 + 0.321623i
\(77\) 0 0
\(78\) 0 0
\(79\) 6.06218 10.5000i 0.682048 1.18134i −0.292306 0.956325i \(-0.594423\pi\)
0.974355 0.225018i \(-0.0722440\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 1.73205 9.00000i 0.188982 0.981981i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(90\) 0 0
\(91\) 0.500000 + 9.52628i 0.0524142 + 0.998625i
\(92\) 0 0
\(93\) −1.96410 0.526279i −0.203668 0.0545726i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.42820 + 2.52628i 0.957289 + 0.256505i 0.703452 0.710742i \(-0.251641\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −10.0000 −1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −3.00000 1.73205i −0.295599 0.170664i 0.344865 0.938652i \(-0.387925\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) 9.00000 + 5.19615i 0.866025 + 0.500000i
\(109\) 15.5622 + 4.16987i 1.49059 + 0.399401i 0.909935 0.414751i \(-0.136131\pi\)
0.580651 + 0.814152i \(0.302798\pi\)
\(110\) 0 0
\(111\) 1.83975 6.86603i 0.174621 0.651694i
\(112\) 6.92820 8.00000i 0.654654 0.755929i
\(113\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −10.3923 3.00000i −0.960769 0.277350i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000i 1.00000i
\(122\) 0 0
\(123\) 0 0
\(124\) −1.66025 1.66025i −0.149095 0.149095i
\(125\) 0 0
\(126\) 0 0
\(127\) 17.3205 + 10.0000i 1.53695 + 0.887357i 0.999015 + 0.0443678i \(0.0141274\pi\)
0.537931 + 0.842989i \(0.319206\pi\)
\(128\) 0 0
\(129\) 15.5885 9.00000i 1.37249 0.792406i
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) 12.8923 + 6.25833i 1.11790 + 0.542666i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(138\) 0 0
\(139\) 3.50000 + 6.06218i 0.296866 + 0.514187i 0.975417 0.220366i \(-0.0707252\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 6.00000 + 10.3923i 0.500000 + 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) −9.52628 + 7.50000i −0.785714 + 0.618590i
\(148\) 5.80385 5.80385i 0.477073 0.477073i
\(149\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(150\) 0 0
\(151\) 3.70577 13.8301i 0.301571 1.12548i −0.634285 0.773099i \(-0.718706\pi\)
0.935857 0.352381i \(-0.114628\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −8.66025 9.00000i −0.693375 0.720577i
\(157\) −12.5000 21.6506i −0.997609 1.72791i −0.558661 0.829396i \(-0.688685\pi\)
−0.438948 0.898513i \(-0.644649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 18.0263 + 18.0263i 1.41193 + 1.41193i 0.746156 + 0.665771i \(0.231897\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(168\) 0 0
\(169\) 11.0000 + 6.92820i 0.846154 + 0.532939i
\(170\) 0 0
\(171\) −11.4904 + 11.4904i −0.878691 + 0.878691i
\(172\) 20.7846 1.58481
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 10.0000 + 8.66025i 0.755929 + 0.654654i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 19.0526i 1.41617i −0.706129 0.708083i \(-0.749560\pi\)
0.706129 0.708083i \(-0.250440\pi\)
\(182\) 0 0
\(183\) −27.0000 −1.99590
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −4.50000 12.9904i −0.327327 0.944911i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 13.8564i 1.00000i
\(193\) −14.8564 14.8564i −1.06939 1.06939i −0.997406 0.0719816i \(-0.977068\pi\)
−0.0719816 0.997406i \(-0.522932\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −13.8564 + 2.00000i −0.989743 + 0.142857i
\(197\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(198\) 0 0
\(199\) 9.52628 + 5.50000i 0.675300 + 0.389885i 0.798082 0.602549i \(-0.205848\pi\)
−0.122782 + 0.992434i \(0.539182\pi\)
\(200\) 0 0
\(201\) −20.0263 + 20.0263i −1.41254 + 1.41254i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −3.46410 14.0000i −0.240192 0.970725i
\(209\) 0 0
\(210\) 0 0
\(211\) 0.866025 1.50000i 0.0596196 0.103264i −0.834675 0.550743i \(-0.814345\pi\)
0.894295 + 0.447478i \(0.147678\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.222432 + 3.09808i 0.0150997 + 0.210311i
\(218\) 0 0
\(219\) 7.56218 28.2224i 0.511005 1.90710i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 7.02628 + 26.2224i 0.470514 + 1.75598i 0.637927 + 0.770097i \(0.279792\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) 0 0
\(225\) −12.9904 + 7.50000i −0.866025 + 0.500000i
\(226\) 0 0
\(227\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(228\) −18.1244 + 4.85641i −1.20031 + 0.321623i
\(229\) 25.7224 6.89230i 1.69979 0.455456i 0.726900 0.686743i \(-0.240960\pi\)
0.972886 + 0.231287i \(0.0742935\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 10.5000 18.1865i 0.682048 1.18134i
\(238\) 0 0
\(239\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(240\) 0 0
\(241\) −28.4904 + 7.63397i −1.83523 + 0.491748i −0.998443 0.0557856i \(-0.982234\pi\)
−0.836784 + 0.547533i \(0.815567\pi\)
\(242\) 0 0
\(243\) 15.5885 1.00000
\(244\) −27.0000 15.5885i −1.72850 0.997949i
\(245\) 0 0
\(246\) 0 0
\(247\) 17.0981 9.43782i 1.08792 0.600514i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(252\) 3.00000 15.5885i 0.188982 0.981981i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 + 13.8564i −0.500000 + 0.866025i
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) −10.8301 + 0.777568i −0.672951 + 0.0483157i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −31.5885 + 8.46410i −1.92957 + 0.517027i
\(269\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(270\) 0 0
\(271\) −6.20577 + 23.1603i −0.376974 + 1.40689i 0.473466 + 0.880812i \(0.343003\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) 0.866025 + 16.5000i 0.0524142 + 0.998625i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −28.5000 16.4545i −1.71240 0.988654i −0.931305 0.364241i \(-0.881328\pi\)
−0.781094 0.624413i \(-0.785338\pi\)
\(278\) 0 0
\(279\) −3.40192 0.911543i −0.203668 0.0545726i
\(280\) 0 0
\(281\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(282\) 0 0
\(283\) 32.0000i 1.90220i 0.308879 + 0.951101i \(0.400046\pi\)
−0.308879 + 0.951101i \(0.599954\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 + 14.7224i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) 16.3301 + 4.37564i 0.957289 + 0.256505i
\(292\) 23.8564 23.8564i 1.39609 1.39609i
\(293\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −17.3205 −1.00000
\(301\) −20.7846 18.0000i −1.19800 1.03750i
\(302\) 0 0
\(303\) 0 0
\(304\) −20.9282 5.60770i −1.20031 0.321623i
\(305\) 0 0
\(306\) 0 0
\(307\) 18.3660 + 18.3660i 1.04820 + 1.04820i 0.998778 + 0.0494267i \(0.0157394\pi\)
0.0494267 + 0.998778i \(0.484261\pi\)
\(308\) 0 0
\(309\) −5.19615 3.00000i −0.295599 0.170664i
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 13.8564 24.0000i 0.783210 1.35656i −0.146852 0.989158i \(-0.546914\pi\)
0.930062 0.367402i \(-0.119753\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 21.0000 12.1244i 1.18134 0.682048i
\(317\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 15.5885 + 9.00000i 0.866025 + 0.500000i
\(325\) 17.5000 4.33013i 0.970725 0.240192i
\(326\) 0 0
\(327\) 26.9545 + 7.22243i 1.49059 + 0.399401i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −24.6603 + 24.6603i −1.35545 + 1.35545i −0.476011 + 0.879440i \(0.657918\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) 3.18653 11.8923i 0.174621 0.651694i
\(334\) 0 0
\(335\) 0 0
\(336\) 12.0000 13.8564i 0.654654 0.755929i
\(337\) 34.0000i 1.85210i −0.377403 0.926049i \(-0.623183\pi\)
0.377403 0.926049i \(-0.376817\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 15.5885 + 10.0000i 0.841698 + 0.539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) −21.7224 + 5.82051i −1.16278 + 0.311565i −0.788074 0.615581i \(-0.788921\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −18.0000 5.19615i −0.960769 0.277350i
\(352\) 0 0
\(353\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(360\) 0 0
\(361\) 10.3397i 0.544197i
\(362\) 0 0
\(363\) 19.0526i 1.00000i
\(364\) −8.66025 + 17.0000i −0.453921 + 0.891042i
\(365\) 0 0
\(366\) 0 0
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −2.87564 2.87564i −0.149095 0.149095i
\(373\) −36.3731 −1.88333 −0.941663 0.336557i \(-0.890737\pi\)
−0.941663 + 0.336557i \(0.890737\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −8.99038 33.5526i −0.461805 1.72348i −0.667271 0.744815i \(-0.732538\pi\)
0.205466 0.978664i \(-0.434129\pi\)
\(380\) 0 0
\(381\) 30.0000 + 17.3205i 1.53695 + 0.887357i
\(382\) 0 0
\(383\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 27.0000 15.5885i 1.37249 0.792406i
\(388\) 13.8038 + 13.8038i 0.700784 + 0.700784i
\(389\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 27.3923 + 27.3923i 1.37478 + 1.37478i 0.853206 + 0.521575i \(0.174655\pi\)
0.521575 + 0.853206i \(0.325345\pi\)
\(398\) 0 0
\(399\) 22.3301 + 10.8397i 1.11790 + 0.542666i
\(400\) −17.3205 10.0000i −0.866025 0.500000i
\(401\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(402\) 0 0
\(403\) 3.62436 + 2.18653i 0.180542 + 0.108919i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −4.41858 16.4904i −0.218485 0.815397i −0.984911 0.173064i \(-0.944633\pi\)
0.766426 0.642333i \(-0.222033\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3.46410 6.00000i −0.170664 0.295599i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.06218 + 10.5000i 0.296866 + 0.514187i
\(418\) 0 0
\(419\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) −8.68653 + 8.68653i −0.423356 + 0.423356i −0.886357 0.463002i \(-0.846772\pi\)
0.463002 + 0.886357i \(0.346772\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 13.5000 + 38.9711i 0.653311 + 1.88595i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(432\) 10.3923 + 18.0000i 0.500000 + 0.866025i
\(433\) 30.3109 17.5000i 1.45665 0.840996i 0.457804 0.889053i \(-0.348636\pi\)
0.998845 + 0.0480569i \(0.0153029\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 22.7846 + 22.7846i 1.09118 + 1.09118i
\(437\) 0 0
\(438\) 0 0
\(439\) 7.50000 4.33013i 0.357955 0.206666i −0.310228 0.950662i \(-0.600405\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) −16.5000 + 12.9904i −0.785714 + 0.618590i
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 10.0526 10.0526i 0.477073 0.477073i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 20.0000 6.92820i 0.944911 0.327327i
\(449\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 6.41858 23.9545i 0.301571 1.12548i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.7224 5.28461i −0.922576 0.247204i −0.233890 0.972263i \(-0.575146\pi\)
−0.688686 + 0.725059i \(0.741812\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(462\) 0 0
\(463\) −9.05256 9.05256i −0.420708 0.420708i 0.464739 0.885448i \(-0.346148\pi\)
−0.885448 + 0.464739i \(0.846148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) −15.0000 15.5885i −0.693375 0.720577i
\(469\) 38.9186 + 18.8923i 1.79709 + 0.872366i
\(470\) 0 0
\(471\) −21.6506 37.5000i −0.997609 1.72791i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7.00962 26.1603i 0.321623 1.20031i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(480\) 0 0
\(481\) −7.64359 + 12.6699i −0.348518 + 0.577696i
\(482\) 0 0
\(483\) 0 0
\(484\) −11.0000 + 19.0526i −0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) −10.7679 40.1865i −0.487942 1.82103i −0.566429 0.824110i \(-0.691675\pi\)
0.0784867 0.996915i \(-0.474991\pi\)
\(488\) 0 0
\(489\) 31.2224 + 31.2224i 1.41193 + 1.41193i
\(490\) 0 0
\(491\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.21539 4.53590i −0.0545726 0.203668i
\(497\) 0 0
\(498\) 0 0
\(499\) −5.91154 22.0622i −0.264637 0.987639i −0.962472 0.271380i \(-0.912520\pi\)
0.697835 0.716258i \(-0.254147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 19.0526 + 12.0000i 0.846154 + 0.532939i
\(508\) 20.0000 + 34.6410i 0.887357 + 1.53695i
\(509\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(510\) 0 0
\(511\) −44.5167 + 3.19615i −1.96930 + 0.141389i
\(512\) 0 0
\(513\) −19.9019 + 19.9019i −0.878691 + 0.878691i
\(514\) 0 0
\(515\) 0 0
\(516\) 36.0000 1.58481
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 0 0
\(523\) −21.5000 37.2391i −0.940129 1.62835i −0.765222 0.643767i \(-0.777371\pi\)
−0.174908 0.984585i \(-0.555963\pi\)
\(524\) 0 0
\(525\) 17.3205 + 15.0000i 0.755929 + 0.654654i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 19.9186i 0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 16.0718 + 23.7321i 0.696801 + 1.02891i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −36.1506 + 9.68653i −1.55424 + 0.416457i −0.930834 0.365444i \(-0.880917\pi\)
−0.623404 + 0.781900i \(0.714251\pi\)
\(542\) 0 0
\(543\) 33.0000i 1.41617i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.00000 −0.0427569 −0.0213785 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) 0 0
\(549\) −46.7654 −1.99590
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −31.5000 6.06218i −1.33952 0.257790i
\(554\) 0 0
\(555\) 0 0
\(556\) 14.0000i 0.593732i
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) −36.3731 + 9.00000i −1.53842 + 0.380659i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −7.79423 22.5000i −0.327327 0.944911i
\(568\) 0 0
\(569\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(570\) 0 0
\(571\) −40.7032 + 23.5000i −1.70338 + 0.983444i −0.761083 + 0.648655i \(0.775332\pi\)
−0.942293 + 0.334790i \(0.891335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 24.0000i 1.00000i
\(577\) −39.4545 10.5718i −1.64251 0.440110i −0.685009 0.728535i \(-0.740202\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) 0 0
\(579\) −25.7321 25.7321i −1.06939 1.06939i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(588\) −24.0000 + 3.46410i −0.989743 + 0.142857i
\(589\) 5.50704 3.17949i 0.226914 0.131009i
\(590\) 0 0
\(591\) 0 0
\(592\) 15.8564 4.24871i 0.651694 0.174621i
\(593\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.5000 + 9.52628i 0.675300 + 0.389885i
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) −21.6506 + 37.5000i −0.883148 + 1.52966i −0.0353259 + 0.999376i \(0.511247\pi\)
−0.847822 + 0.530281i \(0.822086\pi\)
\(602\) 0 0
\(603\) −34.6865 + 34.6865i −1.41254 + 1.41254i
\(604\) 20.2487 20.2487i 0.823908 0.823908i
\(605\) 0 0
\(606\) 0 0
\(607\) −29.0000 −1.17707 −0.588537 0.808470i \(-0.700296\pi\)
−0.588537 + 0.808470i \(0.700296\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 31.2942 31.2942i 1.26396 1.26396i 0.314806 0.949156i \(-0.398061\pi\)
0.949156 0.314806i \(-0.101939\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(618\) 0 0
\(619\) −12.8301 + 47.8827i −0.515686 + 1.92457i −0.174042 + 0.984738i \(0.555683\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −6.00000 24.2487i −0.240192 0.970725i
\(625\) 12.5000 21.6506i 0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 50.0000i 1.99522i
\(629\) 0 0
\(630\) 0 0
\(631\) 3.11474 11.6244i 0.123996 0.462758i −0.875806 0.482663i \(-0.839670\pi\)
0.999802 + 0.0199047i \(0.00633628\pi\)
\(632\) 0 0
\(633\) 1.50000 2.59808i 0.0596196 0.103264i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 23.3827 9.50000i 0.926456 0.376404i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(642\) 0 0
\(643\) −6.02628 1.61474i −0.237653 0.0636790i 0.138027 0.990429i \(-0.455924\pi\)
−0.375680 + 0.926750i \(0.622591\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.385263 + 5.36603i 0.0150997 + 0.210311i
\(652\) 13.1962 + 49.2487i 0.516801 + 1.92873i
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 13.0981 48.8827i 0.511005 1.90710i
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) −16.7058 16.7058i −0.649779 0.649779i 0.303160 0.952940i \(-0.401958\pi\)
−0.952940 + 0.303160i \(0.901958\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 12.1699 + 45.4186i 0.470514 + 1.75598i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 12.0000 + 6.92820i 0.462566 + 0.267063i 0.713123 0.701039i \(-0.247280\pi\)
−0.250557 + 0.968102i \(0.580614\pi\)
\(674\) 0 0
\(675\) −22.5000 + 12.9904i −0.866025 + 0.500000i
\(676\) 12.1244 + 23.0000i 0.466321 + 0.884615i
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) −1.84936 25.7583i −0.0709721 0.988514i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(684\) −31.3923 + 8.41154i −1.20031 + 0.321623i
\(685\) 0 0
\(686\) 0 0
\(687\) 44.5526 11.9378i 1.69979 0.455456i
\(688\) 36.0000 + 20.7846i 1.37249 + 0.792406i
\(689\) 0 0
\(690\) 0 0
\(691\) 30.0263 + 8.04552i 1.14225 + 0.306066i 0.779857 0.625958i \(-0.215292\pi\)
0.362397 + 0.932024i \(0.381959\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 8.66025 + 25.0000i 0.327327 + 0.944911i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 11.1147 + 19.2513i 0.419200 + 0.726076i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.15064 + 6.15064i 0.230992 + 0.230992i 0.813107 0.582115i \(-0.197775\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) 0 0
\(711\) 18.1865 31.5000i 0.682048 1.18134i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −1.73205 + 9.00000i −0.0645049 + 0.335178i
\(722\) 0 0
\(723\) −49.3468 + 13.2224i −1.83523 + 0.491748i
\(724\) 19.0526 33.0000i 0.708083 1.22644i
\(725\) 0 0
\(726\) 0 0
\(727\) 49.0000i 1.81731i −0.417548 0.908655i \(-0.637111\pi\)
0.417548 0.908655i \(-0.362889\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −46.7654 27.0000i −1.72850 0.997949i
\(733\) 8.54552 + 31.8923i 0.315636 + 1.17797i 0.923396 + 0.383849i \(0.125402\pi\)
−0.607760 + 0.794121i \(0.707932\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 38.4186 + 38.4186i 1.41325 + 1.41325i 0.732717 + 0.680534i \(0.238252\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) 29.6147 16.3468i 1.08792 0.600514i
\(742\) 0 0
\(743\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 46.5000 26.8468i 1.69681 0.979653i 0.748056 0.663636i \(-0.230988\pi\)
0.948753 0.316017i \(-0.102346\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 5.19615 27.0000i 0.188982 0.981981i
\(757\) −24.2487 + 42.0000i −0.881334 + 1.52652i −0.0314762 + 0.999505i \(0.510021\pi\)
−0.849858 + 0.527011i \(0.823312\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(762\) 0 0
\(763\) −3.05256 42.5167i −0.110510 1.53921i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −13.8564 + 24.0000i −0.500000 + 0.866025i
\(769\) 3.21281 + 11.9904i 0.115857 + 0.432384i 0.999350 0.0360609i \(-0.0114810\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.8756 40.5885i −0.391423 1.46081i
\(773\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(774\) 0 0
\(775\) 5.66987 1.51924i 0.203668 0.0545726i
\(776\) 0 0
\(777\) −18.7583 + 1.34679i −0.672951 + 0.0483157i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −26.0000 10.3923i −0.928571 0.371154i
\(785\) 0 0
\(786\) 0 0
\(787\) 53.1147 14.2321i 1.89334 0.507318i 0.895244 0.445577i \(-0.147001\pi\)
0.998092 0.0617409i \(-0.0196653\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 54.0000 + 15.5885i 1.91760 + 0.553562i
\(794\) 0 0
\(795\) 0 0
\(796\) 11.0000 + 19.0526i 0.389885 + 0.675300i
\(797\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\)