Properties

Label 273.2.bw.a.254.1
Level $273$
Weight $2$
Character 273.254
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 254.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 273.254
Dual form 273.2.bw.a.158.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{3} +(-1.73205 - 1.00000i) q^{4} +(0.866025 + 2.50000i) q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} +(-1.73205 - 1.00000i) q^{4} +(0.866025 + 2.50000i) q^{7} +3.00000 q^{9} +(3.00000 + 1.73205i) q^{12} +(3.46410 + 1.00000i) q^{13} +(2.00000 + 3.46410i) q^{16} +(4.83013 + 4.83013i) q^{19} +(-1.50000 - 4.33013i) q^{21} +(4.33013 - 2.50000i) q^{25} -5.19615 q^{27} +(1.00000 - 5.19615i) q^{28} +(-2.86603 + 10.6962i) q^{31} +(-5.19615 - 3.00000i) q^{36} +(-11.0622 - 2.96410i) q^{37} +(-6.00000 - 1.73205i) q^{39} +(9.00000 - 5.19615i) q^{43} +(-3.46410 - 6.00000i) q^{48} +(-5.50000 + 4.33013i) q^{49} +(-5.00000 - 5.19615i) q^{52} +(-8.36603 - 8.36603i) q^{57} +15.5885 q^{61} +(2.59808 + 7.50000i) q^{63} -8.00000i q^{64} +(0.562178 + 0.562178i) q^{67} +(2.63397 + 0.705771i) q^{73} +(-7.50000 + 4.33013i) q^{75} +(-3.53590 - 13.1962i) q^{76} +(-6.06218 + 10.5000i) q^{79} +9.00000 q^{81} +(-1.73205 + 9.00000i) q^{84} +(0.500000 + 9.52628i) q^{91} +(4.96410 - 18.5263i) q^{93} +(-4.42820 + 16.5263i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} + 12q^{12} + 8q^{16} + 2q^{19} - 6q^{21} + 4q^{28} - 8q^{31} - 20q^{37} - 24q^{39} + 36q^{43} - 22q^{49} - 20q^{52} - 30q^{57} - 22q^{67} + 14q^{73} - 30q^{75} - 28q^{76} + 36q^{81} + 2q^{91} + 6q^{93} + 10q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\)

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 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(3\) −1.73205 −1.00000
\(4\) −1.73205 1.00000i −0.866025 0.500000i
\(5\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(6\) 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 \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) 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
\(19\) 4.83013 + 4.83013i 1.10811 + 1.10811i 0.993399 + 0.114708i \(0.0365932\pi\)
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
\(31\) −2.86603 + 10.6962i −0.514753 + 1.92109i −0.155543 + 0.987829i \(0.549713\pi\)
−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
\(37\) −11.0622 2.96410i −1.81861 0.487295i −0.821995 0.569495i \(-0.807139\pi\)
−0.996616 + 0.0821995i \(0.973806\pi\)
\(38\) 0 0
\(39\) −6.00000 1.73205i −0.960769 0.277350i
\(40\) 0 0
\(41\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(42\) 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 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(48\) −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
\(57\) −8.36603 8.36603i −1.10811 1.10811i
\(58\) 0 0
\(59\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(60\) 0 0
\(61\) 15.5885 1.99590 0.997949 0.0640184i \(-0.0203916\pi\)
0.997949 + 0.0640184i \(0.0203916\pi\)
\(62\) 0 0
\(63\) 2.59808 + 7.50000i 0.327327 + 0.944911i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.562178 + 0.562178i 0.0686810 + 0.0686810i 0.740613 0.671932i \(-0.234535\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(72\) 0 0
\(73\) 2.63397 + 0.705771i 0.308283 + 0.0826043i 0.409644 0.912245i \(-0.365653\pi\)
−0.101361 + 0.994850i \(0.532320\pi\)
\(74\) 0 0
\(75\) −7.50000 + 4.33013i −0.866025 + 0.500000i
\(76\) −3.53590 13.1962i −0.405595 1.51370i
\(77\) 0 0
\(78\) 0 0
\(79\) −6.06218 + 10.5000i −0.682048 + 1.18134i 0.292306 + 0.956325i \(0.405577\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) −1.73205 + 9.00000i −0.188982 + 0.981981i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(90\) 0 0
\(91\) 0.500000 + 9.52628i 0.0524142 + 0.998625i
\(92\) 0 0
\(93\) 4.96410 18.5263i 0.514753 1.92109i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.42820 + 16.5263i −0.449616 + 1.67799i 0.253837 + 0.967247i \(0.418307\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 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
\(109\) 3.43782 12.8301i 0.329284 1.22890i −0.580651 0.814152i \(-0.697202\pi\)
0.909935 0.414751i \(-0.136131\pi\)
\(110\) 0 0
\(111\) 19.1603 + 5.13397i 1.81861 + 0.487295i
\(112\) −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
\(124\) 15.6603 15.6603i 1.40633 1.40633i
\(125\) 0 0
\(126\) 0 0
\(127\) −17.3205 10.0000i −1.53695 0.887357i −0.999015 0.0443678i \(-0.985873\pi\)
−0.537931 0.842989i \(-0.680794\pi\)
\(128\) 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
\(133\) −7.89230 + 16.2583i −0.684350 + 1.40978i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(138\) 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
\(148\) 16.1962 + 16.1962i 1.33132 + 1.33132i
\(149\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(150\) 0 0
\(151\) 19.2942 + 5.16987i 1.57014 + 0.420718i 0.935857 0.352381i \(-0.114628\pi\)
0.634285 + 0.773099i \(0.281294\pi\)
\(152\) 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
\(163\) −1.02628 + 1.02628i −0.0803844 + 0.0803844i −0.746156 0.665771i \(-0.768103\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(168\) 0 0
\(169\) 11.0000 + 6.92820i 0.846154 + 0.532939i
\(170\) 0 0
\(171\) 14.4904 + 14.4904i 1.10811 + 1.10811i
\(172\) −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
\(193\) 12.8564 12.8564i 0.925424 0.925424i −0.0719816 0.997406i \(-0.522932\pi\)
0.997406 + 0.0719816i \(0.0229323\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 13.8564 2.00000i 0.989743 0.142857i
\(197\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(198\) 0 0
\(199\) −9.52628 5.50000i −0.675300 0.389885i 0.122782 0.992434i \(-0.460818\pi\)
−0.798082 + 0.602549i \(0.794152\pi\)
\(200\) 0 0
\(201\) −0.973721 0.973721i −0.0686810 0.0686810i
\(202\) 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 −0.894295 0.447478i \(-0.852322\pi\)
0.834675 + 0.550743i \(0.185655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −29.2224 + 2.09808i −1.98375 + 0.142427i
\(218\) 0 0
\(219\) −4.56218 1.22243i −0.308283 0.0826043i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −12.0263 + 3.22243i −0.805339 + 0.215790i −0.637927 0.770097i \(-0.720208\pi\)
−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 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(228\) 6.12436 + 22.8564i 0.405595 + 1.51370i
\(229\) −3.72243 13.8923i −0.245985 0.918029i −0.972886 0.231287i \(-0.925707\pi\)
0.726900 0.686743i \(-0.240960\pi\)
\(230\) 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 \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(240\) 0 0
\(241\) −2.50962 9.36603i −0.161659 0.603319i −0.998443 0.0557856i \(-0.982234\pi\)
0.836784 0.547533i \(-0.184433\pi\)
\(242\) 0 0
\(243\) −15.5885 −1.00000
\(244\) −27.0000 15.5885i −1.72850 0.997949i
\(245\) 0 0
\(246\) 0 0
\(247\) 11.9019 + 21.5622i 0.757301 + 1.37197i
\(248\) 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
\(259\) −2.16987 30.2224i −0.134829 1.87793i
\(260\) 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
\(268\) −0.411543 1.53590i −0.0251390 0.0938199i
\(269\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(270\) 0 0
\(271\) −21.7942 5.83975i −1.32391 0.354739i −0.473466 0.880812i \(-0.656997\pi\)
−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
\(279\) −8.59808 + 32.0885i −0.514753 + 1.92109i
\(280\) 0 0
\(281\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(282\) 0 0
\(283\) 32.0000i 1.90220i −0.308879 0.951101i \(-0.599954\pi\)
0.308879 0.951101i \(-0.400046\pi\)
\(284\) 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
\(291\) 7.66987 28.6244i 0.449616 1.67799i
\(292\) −3.85641 3.85641i −0.225679 0.225679i
\(293\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(294\) 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
\(304\) −7.07180 + 26.3923i −0.405595 + 1.51370i
\(305\) 0 0
\(306\) 0 0
\(307\) 16.6340 16.6340i 0.949351 0.949351i −0.0494267 0.998778i \(-0.515739\pi\)
0.998778 + 0.0494267i \(0.0157394\pi\)
\(308\) 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 \(0.453086\pi\)
−0.930062 + 0.367402i \(0.880247\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 21.0000 12.1244i 1.18134 0.682048i
\(317\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(318\) 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
\(327\) −5.95448 + 22.2224i −0.329284 + 1.22890i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −7.33975 7.33975i −0.403429 0.403429i 0.476011 0.879440i \(-0.342082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) −33.1865 8.89230i −1.81861 0.487295i
\(334\) 0 0
\(335\) 0 0
\(336\) 12.0000 13.8564i 0.654654 0.755929i
\(337\) 34.0000i 1.85210i 0.377403 + 0.926049i \(0.376817\pi\)
−0.377403 + 0.926049i \(0.623183\pi\)
\(338\) 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
\(349\) 7.72243 + 28.8205i 0.413372 + 1.54273i 0.788074 + 0.615581i \(0.211079\pi\)
−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 \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(360\) 0 0
\(361\) 27.6603i 1.45580i
\(362\) 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
\(372\) −27.1244 + 27.1244i −1.40633 + 1.40633i
\(373\) 36.3731 1.88333 0.941663 0.336557i \(-0.109263\pi\)
0.941663 + 0.336557i \(0.109263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 16.9904 4.55256i 0.872737 0.233849i 0.205466 0.978664i \(-0.434129\pi\)
0.667271 + 0.744815i \(0.267462\pi\)
\(380\) 0 0
\(381\) 30.0000 + 17.3205i 1.53695 + 0.887357i
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 27.0000 15.5885i 1.37249 0.792406i
\(388\) 24.1962 24.1962i 1.22837 1.22837i
\(389\) 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
\(397\) 6.60770 6.60770i 0.331631 0.331631i −0.521575 0.853206i \(-0.674655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 0 0
\(399\) 13.6699 28.1603i 0.684350 1.40978i
\(400\) 17.3205 + 10.0000i 0.866025 + 0.500000i
\(401\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(402\) 0 0
\(403\) −20.6244 + 34.1865i −1.02737 + 1.70295i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 35.4186 9.49038i 1.75134 0.469269i 0.766426 0.642333i \(-0.222033\pi\)
0.984911 + 0.173064i \(0.0553667\pi\)
\(410\) 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
\(421\) 27.6865 + 27.6865i 1.34936 + 1.34936i 0.886357 + 0.463002i \(0.153228\pi\)
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 \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(432\) −10.3923 18.0000i −0.500000 0.866025i
\(433\) −30.3109 + 17.5000i −1.45665 + 0.840996i −0.998845 0.0480569i \(-0.984697\pi\)
−0.457804 + 0.889053i \(0.651364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −18.7846 + 18.7846i −0.899620 + 0.899620i
\(437\) 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 \(0.666667\pi\)
\(444\) −28.0526 28.0526i −1.33132 1.33132i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 20.0000 6.92820i 0.944911 0.327327i
\(449\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −33.4186 8.95448i −1.57014 0.420718i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9.72243 36.2846i 0.454796 1.69732i −0.233890 0.972263i \(-0.575146\pi\)
0.688686 0.725059i \(-0.258188\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(462\) 0 0
\(463\) 29.0526 29.0526i 1.35019 1.35019i 0.464739 0.885448i \(-0.346148\pi\)
0.885448 0.464739i \(-0.153852\pi\)
\(464\) 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
\(469\) −0.918584 + 1.89230i −0.0424163 + 0.0873785i
\(470\) 0 0
\(471\) 21.6506 + 37.5000i 0.997609 + 1.72791i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 32.9904 + 8.83975i 1.51370 + 0.405595i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(480\) 0 0
\(481\) −35.3564 21.3301i −1.61211 0.972570i
\(482\) 0 0
\(483\) 0 0
\(484\) −11.0000 + 19.0526i −0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) −14.2321 + 3.81347i −0.644916 + 0.172805i −0.566429 0.824110i \(-0.691675\pi\)
−0.0784867 + 0.996915i \(0.525009\pi\)
\(488\) 0 0
\(489\) 1.77757 1.77757i 0.0803844 0.0803844i
\(490\) 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
\(496\) −42.7846 + 11.4641i −1.92109 + 0.514753i
\(497\) 0 0
\(498\) 0 0
\(499\) −37.0885 + 9.93782i −1.66031 + 0.444878i −0.962472 0.271380i \(-0.912520\pi\)
−0.697835 + 0.716258i \(0.745853\pi\)
\(500\) 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 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(510\) 0 0
\(511\) 0.516660 + 7.19615i 0.0228557 + 0.318339i
\(512\) 0 0
\(513\) −25.0981 25.0981i −1.10811 1.10811i
\(514\) 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
\(532\) 29.9282 20.2679i 1.29755 0.878727i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.15064 + 26.6865i 0.307430 + 1.14734i 0.930834 + 0.365444i \(0.119083\pi\)
−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 \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 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 \(-0.224668\pi\)
0.942293 0.334790i \(-0.108665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 24.0000i 1.00000i
\(577\) −6.54552 + 24.4282i −0.272493 + 1.01696i 0.685009 + 0.728535i \(0.259798\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) 0 0
\(579\) −22.2679 + 22.2679i −0.925424 + 0.925424i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(588\) −24.0000 + 3.46410i −0.989743 + 0.142857i
\(589\) −65.5070 + 37.8205i −2.69917 + 1.55837i
\(590\) 0 0
\(591\) 0 0
\(592\) −11.8564 44.2487i −0.487295 1.81861i
\(593\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(594\) 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 \(-0.488753\pi\)
0.847822 0.530281i \(-0.177914\pi\)
\(602\) 0 0
\(603\) 1.68653 + 1.68653i 0.0686810 + 0.0686810i
\(604\) −28.2487 28.2487i −1.14942 1.14942i
\(605\) 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
\(613\) 15.7058 + 15.7058i 0.634350 + 0.634350i 0.949156 0.314806i \(-0.101939\pi\)
−0.314806 + 0.949156i \(0.601939\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(618\) 0 0
\(619\) −4.16987 1.11731i −0.167601 0.0449086i 0.174042 0.984738i \(-0.444317\pi\)
−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
\(631\) −47.1147 12.6244i −1.87561 0.502568i −0.999802 0.0199047i \(-0.993664\pi\)
−0.875806 0.482663i \(-0.839670\pi\)
\(632\) 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
\(643\) 13.0263 48.6147i 0.513706 1.91718i 0.138027 0.990429i \(-0.455924\pi\)
0.375680 0.926750i \(-0.377409\pi\)
\(644\) 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
\(651\) 50.6147 3.63397i 1.98375 0.142427i
\(652\) 2.80385 0.751289i 0.109807 0.0294227i
\(653\) 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
\(657\) 7.90192 + 2.11731i 0.308283 + 0.0826043i
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) −32.2942 + 32.2942i −1.25610 + 1.25610i −0.303160 + 0.952940i \(0.598042\pi\)
−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
\(669\) 20.8301 5.58142i 0.805339 0.215790i
\(670\) 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
\(679\) −45.1506 + 3.24167i −1.73272 + 0.124404i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(684\) −10.6077 39.5885i −0.405595 1.51370i
\(685\) 0 0
\(686\) 0 0
\(687\) 6.44744 + 24.0622i 0.245985 + 0.918029i
\(688\) 36.0000 + 20.7846i 1.37249 + 0.792406i
\(689\) 0 0
\(690\) 0 0
\(691\) 10.9737 40.9545i 0.417460 1.55798i −0.362397 0.932024i \(-0.618041\pi\)
0.779857 0.625958i \(-0.215292\pi\)
\(692\) 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
\(703\) −39.1147 67.7487i −1.47524 2.55519i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −37.1506 + 37.1506i −1.39522 + 1.39522i −0.582115 + 0.813107i \(0.697775\pi\)
−0.813107 + 0.582115i \(0.802225\pi\)
\(710\) 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
\(723\) 4.34679 + 16.2224i 0.161659 + 0.603319i
\(724\) −19.0526 + 33.0000i −0.708083 + 1.22644i
\(725\) 0 0
\(726\) 0 0
\(727\) 49.0000i 1.81731i 0.417548 + 0.908655i \(0.362889\pi\)
−0.417548 + 0.908655i \(0.637111\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 46.7654 + 27.0000i 1.72850 + 0.997949i
\(733\) 41.4545 11.1077i 1.53116 0.410272i 0.607760 0.794121i \(-0.292068\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.41858 + 1.41858i −0.0521835 + 0.0521835i −0.732717 0.680534i \(-0.761748\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) −20.6147 37.3468i −0.757301 1.37197i
\(742\) 0 0
\(743\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(744\) 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 \(-0.489979\pi\)
0.849858 0.527011i \(-0.176688\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(762\) 0 0
\(763\) 35.0526 2.51666i 1.26899 0.0911092i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 13.8564 24.0000i 0.500000 0.866025i
\(769\) −52.2128 + 13.9904i −1.88284 + 0.504506i −0.883493 + 0.468445i \(0.844814\pi\)
−0.999350 + 0.0360609i \(0.988519\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −35.1244 + 9.41154i −1.26415 + 0.338729i
\(773\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(774\) 0 0
\(775\) 14.3301 + 53.4808i 0.514753 + 1.92109i
\(776\) 0 0
\(777\) 3.75833 + 52.3468i 0.134829 + 1.87793i
\(778\) 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
\(787\) 2.88526 + 10.7679i 0.102849 + 0.383836i 0.998092 0.0617409i \(-0.0196653\pi\)
−0.895244 + 0.445577i \(0.852999\pi\)
\(788\) 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