Properties

Label 684.3.y.g.145.1
Level $684$
Weight $3$
Character 684.145
Analytic conductor $18.638$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.y (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 34 x^{6} + 921 x^{4} - 7990 x^{2} + 55225\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.1
Root \(-4.27333 - 2.46721i\) of defining polynomial
Character \(\chi\) \(=\) 684.145
Dual form 684.3.y.g.217.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-3.48916 - 6.04340i) q^{5} -3.44949 q^{7} +O(q^{10})\) \(q+(-3.48916 - 6.04340i) q^{5} -3.44949 q^{7} -10.1150 q^{11} +(-0.151531 - 0.0874863i) q^{13} +(-1.56834 - 2.71645i) q^{17} +(11.3485 + 15.2385i) q^{19} +(-3.48916 + 6.04340i) q^{23} +(-11.8485 + 20.5222i) q^{25} +(4.70502 + 2.71645i) q^{29} +36.8017i q^{31} +(12.0358 + 20.8467i) q^{35} +27.1879i q^{37} +(51.2800 - 29.6065i) q^{41} +(10.9722 + 19.0044i) q^{43} +(-6.27337 + 10.8658i) q^{47} -37.1010 q^{49} +(36.1075 + 20.8467i) q^{53} +(35.2929 + 61.1290i) q^{55} +(-21.9924 + 12.6973i) q^{59} +(-26.1413 + 45.2781i) q^{61} +1.22102i q^{65} +(41.3105 + 23.8506i) q^{67} +(-14.1151 + 8.14934i) q^{71} +(14.9495 + 25.8933i) q^{73} +34.8916 q^{77} +(-53.1742 + 30.7002i) q^{79} -148.976 q^{83} +(-10.9444 + 18.9562i) q^{85} +(-54.9276 - 31.7124i) q^{89} +(0.522704 + 0.301783i) q^{91} +(52.4958 - 121.753i) q^{95} +(27.0000 - 15.5885i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7} + O(q^{10}) \) \( 8 q - 8 q^{7} - 60 q^{13} + 32 q^{19} - 36 q^{25} - 20 q^{43} - 336 q^{49} + 8 q^{55} + 124 q^{61} - 228 q^{67} + 100 q^{73} - 396 q^{79} + 128 q^{85} - 84 q^{91} + 216 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −3.48916 6.04340i −0.697832 1.20868i −0.969217 0.246209i \(-0.920815\pi\)
0.271385 0.962471i \(-0.412518\pi\)
\(6\) 0 0
\(7\) −3.44949 −0.492784 −0.246392 0.969170i \(-0.579245\pi\)
−0.246392 + 0.969170i \(0.579245\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −10.1150 −0.919546 −0.459773 0.888037i \(-0.652069\pi\)
−0.459773 + 0.888037i \(0.652069\pi\)
\(12\) 0 0
\(13\) −0.151531 0.0874863i −0.0116562 0.00672972i 0.494161 0.869371i \(-0.335475\pi\)
−0.505817 + 0.862641i \(0.668809\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.56834 2.71645i −0.0922554 0.159791i 0.816204 0.577763i \(-0.196074\pi\)
−0.908460 + 0.417972i \(0.862741\pi\)
\(18\) 0 0
\(19\) 11.3485 + 15.2385i 0.597288 + 0.802027i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.48916 + 6.04340i −0.151703 + 0.262757i −0.931853 0.362835i \(-0.881809\pi\)
0.780151 + 0.625591i \(0.215142\pi\)
\(24\) 0 0
\(25\) −11.8485 + 20.5222i −0.473939 + 0.820886i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.70502 + 2.71645i 0.162242 + 0.0936706i 0.578923 0.815382i \(-0.303473\pi\)
−0.416681 + 0.909053i \(0.636807\pi\)
\(30\) 0 0
\(31\) 36.8017i 1.18715i 0.804779 + 0.593575i \(0.202284\pi\)
−0.804779 + 0.593575i \(0.797716\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.0358 + 20.8467i 0.343881 + 0.595619i
\(36\) 0 0
\(37\) 27.1879i 0.734808i 0.930061 + 0.367404i \(0.119753\pi\)
−0.930061 + 0.367404i \(0.880247\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 51.2800 29.6065i 1.25073 0.722110i 0.279476 0.960153i \(-0.409839\pi\)
0.971255 + 0.238043i \(0.0765059\pi\)
\(42\) 0 0
\(43\) 10.9722 + 19.0044i 0.255167 + 0.441963i 0.964941 0.262467i \(-0.0845361\pi\)
−0.709774 + 0.704430i \(0.751203\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.27337 + 10.8658i −0.133476 + 0.231187i −0.925014 0.379933i \(-0.875947\pi\)
0.791538 + 0.611120i \(0.209281\pi\)
\(48\) 0 0
\(49\) −37.1010 −0.757164
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 36.1075 + 20.8467i 0.681273 + 0.393333i 0.800334 0.599554i \(-0.204655\pi\)
−0.119062 + 0.992887i \(0.537989\pi\)
\(54\) 0 0
\(55\) 35.2929 + 61.1290i 0.641688 + 1.11144i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −21.9924 + 12.6973i −0.372752 + 0.215209i −0.674660 0.738128i \(-0.735710\pi\)
0.301908 + 0.953337i \(0.402376\pi\)
\(60\) 0 0
\(61\) −26.1413 + 45.2781i −0.428546 + 0.742264i −0.996744 0.0806279i \(-0.974307\pi\)
0.568198 + 0.822892i \(0.307641\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.22102i 0.0187848i
\(66\) 0 0
\(67\) 41.3105 + 23.8506i 0.616574 + 0.355979i 0.775534 0.631306i \(-0.217481\pi\)
−0.158960 + 0.987285i \(0.550814\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.1151 + 8.14934i −0.198804 + 0.114779i −0.596097 0.802912i \(-0.703283\pi\)
0.397294 + 0.917692i \(0.369949\pi\)
\(72\) 0 0
\(73\) 14.9495 + 25.8933i 0.204788 + 0.354702i 0.950065 0.312052i \(-0.101016\pi\)
−0.745277 + 0.666754i \(0.767683\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 34.8916 0.453138
\(78\) 0 0
\(79\) −53.1742 + 30.7002i −0.673092 + 0.388610i −0.797247 0.603653i \(-0.793711\pi\)
0.124155 + 0.992263i \(0.460378\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −148.976 −1.79490 −0.897448 0.441119i \(-0.854581\pi\)
−0.897448 + 0.441119i \(0.854581\pi\)
\(84\) 0 0
\(85\) −10.9444 + 18.9562i −0.128757 + 0.223015i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −54.9276 31.7124i −0.617164 0.356320i 0.158600 0.987343i \(-0.449302\pi\)
−0.775764 + 0.631023i \(0.782635\pi\)
\(90\) 0 0
\(91\) 0.522704 + 0.301783i 0.00574400 + 0.00331630i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 52.4958 121.753i 0.552588 1.28161i
\(96\) 0 0
\(97\) 27.0000 15.5885i 0.278351 0.160706i −0.354326 0.935122i \(-0.615290\pi\)
0.632676 + 0.774416i \(0.281956\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −49.0067 + 84.8820i −0.485215 + 0.840416i −0.999856 0.0169894i \(-0.994592\pi\)
0.514641 + 0.857406i \(0.327925\pi\)
\(102\) 0 0
\(103\) 100.713i 0.977792i 0.872342 + 0.488896i \(0.162600\pi\)
−0.872342 + 0.488896i \(0.837400\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.43289i 0.0507747i 0.999678 + 0.0253874i \(0.00808191\pi\)
−0.999678 + 0.0253874i \(0.991918\pi\)
\(108\) 0 0
\(109\) −59.6969 + 34.4660i −0.547678 + 0.316202i −0.748185 0.663490i \(-0.769074\pi\)
0.200507 + 0.979692i \(0.435741\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 200.043i 1.77029i −0.465315 0.885145i \(-0.654059\pi\)
0.465315 0.885145i \(-0.345941\pi\)
\(114\) 0 0
\(115\) 48.6969 0.423452
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.40998 + 9.37036i 0.0454620 + 0.0787425i
\(120\) 0 0
\(121\) −18.6867 −0.154436
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.09318 −0.0727454
\(126\) 0 0
\(127\) 99.5755 + 57.4899i 0.784059 + 0.452677i 0.837867 0.545875i \(-0.183802\pi\)
−0.0538078 + 0.998551i \(0.517136\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 51.8265 + 89.7661i 0.395622 + 0.685237i 0.993180 0.116588i \(-0.0371957\pi\)
−0.597558 + 0.801825i \(0.703862\pi\)
\(132\) 0 0
\(133\) −39.1464 52.5651i −0.294334 0.395226i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −64.3732 + 111.498i −0.469877 + 0.813852i −0.999407 0.0344400i \(-0.989035\pi\)
0.529529 + 0.848292i \(0.322369\pi\)
\(138\) 0 0
\(139\) −9.17423 + 15.8902i −0.0660017 + 0.114318i −0.897138 0.441751i \(-0.854358\pi\)
0.831136 + 0.556069i \(0.187691\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.53273 + 0.884924i 0.0107184 + 0.00618828i
\(144\) 0 0
\(145\) 37.9125i 0.261465i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 114.437 + 198.211i 0.768036 + 1.33028i 0.938627 + 0.344935i \(0.112099\pi\)
−0.170591 + 0.985342i \(0.554568\pi\)
\(150\) 0 0
\(151\) 161.063i 1.06664i −0.845913 0.533321i \(-0.820944\pi\)
0.845913 0.533321i \(-0.179056\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 222.407 128.407i 1.43489 0.828431i
\(156\) 0 0
\(157\) −131.030 226.951i −0.834587 1.44555i −0.894367 0.447335i \(-0.852373\pi\)
0.0597799 0.998212i \(-0.480960\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0358 20.8467i 0.0747566 0.129482i
\(162\) 0 0
\(163\) 109.833 0.673823 0.336912 0.941536i \(-0.390618\pi\)
0.336912 + 0.941536i \(0.390618\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 198.882 + 114.825i 1.19091 + 0.687573i 0.958513 0.285048i \(-0.0920096\pi\)
0.232398 + 0.972621i \(0.425343\pi\)
\(168\) 0 0
\(169\) −84.4847 146.332i −0.499909 0.865869i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −244.400 + 141.104i −1.41271 + 0.815631i −0.995643 0.0932428i \(-0.970277\pi\)
−0.417071 + 0.908874i \(0.636943\pi\)
\(174\) 0 0
\(175\) 40.8712 70.7909i 0.233550 0.404520i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 22.9526i 0.128227i 0.997943 + 0.0641134i \(0.0204219\pi\)
−0.997943 + 0.0641134i \(0.979578\pi\)
\(180\) 0 0
\(181\) 27.4699 + 15.8598i 0.151767 + 0.0876230i 0.573961 0.818883i \(-0.305406\pi\)
−0.422193 + 0.906506i \(0.638740\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 164.307 94.8629i 0.888148 0.512772i
\(186\) 0 0
\(187\) 15.8638 + 27.4769i 0.0848330 + 0.146935i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 96.6746 0.506150 0.253075 0.967447i \(-0.418558\pi\)
0.253075 + 0.967447i \(0.418558\pi\)
\(192\) 0 0
\(193\) 92.5454 53.4311i 0.479510 0.276845i −0.240702 0.970599i \(-0.577378\pi\)
0.720212 + 0.693754i \(0.244044\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −253.651 −1.28757 −0.643785 0.765207i \(-0.722637\pi\)
−0.643785 + 0.765207i \(0.722637\pi\)
\(198\) 0 0
\(199\) 89.4115 154.865i 0.449304 0.778217i −0.549037 0.835798i \(-0.685005\pi\)
0.998341 + 0.0575809i \(0.0183387\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −16.2299 9.37036i −0.0799504 0.0461594i
\(204\) 0 0
\(205\) −357.848 206.604i −1.74560 1.00782i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −114.790 154.138i −0.549233 0.737500i
\(210\) 0 0
\(211\) 211.098 121.878i 1.00047 0.577619i 0.0920795 0.995752i \(-0.470649\pi\)
0.908386 + 0.418133i \(0.137315\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 76.5675 132.619i 0.356128 0.616831i
\(216\) 0 0
\(217\) 126.947i 0.585009i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.548834i 0.00248341i
\(222\) 0 0
\(223\) 12.3559 7.13366i 0.0554075 0.0319895i −0.472040 0.881577i \(-0.656482\pi\)
0.527448 + 0.849587i \(0.323149\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 259.256i 1.14210i −0.820917 0.571048i \(-0.806537\pi\)
0.820917 0.571048i \(-0.193463\pi\)
\(228\) 0 0
\(229\) 56.4143 0.246351 0.123175 0.992385i \(-0.460692\pi\)
0.123175 + 0.992385i \(0.460692\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −54.5751 94.5268i −0.234228 0.405694i 0.724820 0.688938i \(-0.241923\pi\)
−0.959048 + 0.283244i \(0.908589\pi\)
\(234\) 0 0
\(235\) 87.5551 0.372575
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −405.764 −1.69776 −0.848879 0.528587i \(-0.822722\pi\)
−0.848879 + 0.528587i \(0.822722\pi\)
\(240\) 0 0
\(241\) −245.787 141.905i −1.01986 0.588819i −0.105799 0.994387i \(-0.533740\pi\)
−0.914065 + 0.405569i \(0.867074\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 129.451 + 224.216i 0.528373 + 0.915169i
\(246\) 0 0
\(247\) −0.386481 3.30194i −0.00156470 0.0133682i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −139.566 + 241.736i −0.556041 + 0.963092i 0.441780 + 0.897123i \(0.354347\pi\)
−0.997822 + 0.0659686i \(0.978986\pi\)
\(252\) 0 0
\(253\) 35.2929 61.1290i 0.139497 0.241617i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −278.868 161.004i −1.08509 0.626476i −0.152823 0.988254i \(-0.548837\pi\)
−0.932264 + 0.361778i \(0.882170\pi\)
\(258\) 0 0
\(259\) 93.7844i 0.362102i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 44.1432 + 76.4583i 0.167845 + 0.290716i 0.937662 0.347549i \(-0.112986\pi\)
−0.769817 + 0.638265i \(0.779653\pi\)
\(264\) 0 0
\(265\) 290.949i 1.09792i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 198.407 114.550i 0.737572 0.425837i −0.0836140 0.996498i \(-0.526646\pi\)
0.821186 + 0.570661i \(0.193313\pi\)
\(270\) 0 0
\(271\) 25.0908 + 43.4586i 0.0925860 + 0.160364i 0.908599 0.417670i \(-0.137153\pi\)
−0.816013 + 0.578034i \(0.803820\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 119.847 207.582i 0.435808 0.754842i
\(276\) 0 0
\(277\) −323.918 −1.16938 −0.584690 0.811257i \(-0.698784\pi\)
−0.584690 + 0.811257i \(0.698784\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.05743 + 0.610508i 0.00376310 + 0.00217262i 0.501880 0.864937i \(-0.332642\pi\)
−0.498117 + 0.867110i \(0.665975\pi\)
\(282\) 0 0
\(283\) 203.247 + 352.035i 0.718189 + 1.24394i 0.961717 + 0.274046i \(0.0883620\pi\)
−0.243528 + 0.969894i \(0.578305\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −176.890 + 102.127i −0.616340 + 0.355844i
\(288\) 0 0
\(289\) 139.581 241.761i 0.482978 0.836542i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 362.604i 1.23756i 0.785566 + 0.618778i \(0.212372\pi\)
−0.785566 + 0.618778i \(0.787628\pi\)
\(294\) 0 0
\(295\) 153.470 + 88.6059i 0.520237 + 0.300359i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.05743 0.610508i 0.00353656 0.00204183i
\(300\) 0 0
\(301\) −37.8485 65.5555i −0.125742 0.217792i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 364.845 1.19621
\(306\) 0 0
\(307\) −54.3031 + 31.3519i −0.176883 + 0.102123i −0.585827 0.810436i \(-0.699230\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −248.471 −0.798942 −0.399471 0.916746i \(-0.630806\pi\)
−0.399471 + 0.916746i \(0.630806\pi\)
\(312\) 0 0
\(313\) 164.000 284.056i 0.523962 0.907528i −0.475649 0.879635i \(-0.657787\pi\)
0.999611 0.0278932i \(-0.00887983\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 292.983 + 169.154i 0.924235 + 0.533607i 0.884984 0.465622i \(-0.154169\pi\)
0.0392515 + 0.999229i \(0.487503\pi\)
\(318\) 0 0
\(319\) −47.5913 27.4769i −0.149189 0.0861344i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 23.5963 54.7267i 0.0730537 0.169433i
\(324\) 0 0
\(325\) 3.59082 2.07316i 0.0110487 0.00637895i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 21.6399 37.4814i 0.0657748 0.113925i
\(330\) 0 0
\(331\) 197.707i 0.597303i −0.954362 0.298652i \(-0.903463\pi\)
0.954362 0.298652i \(-0.0965369\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 332.874i 0.993654i
\(336\) 0 0
\(337\) −288.591 + 166.618i −0.856353 + 0.494415i −0.862789 0.505564i \(-0.831284\pi\)
0.00643660 + 0.999979i \(0.497951\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 372.249i 1.09164i
\(342\) 0 0
\(343\) 297.005 0.865903
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 204.221 + 353.721i 0.588533 + 1.01937i 0.994425 + 0.105448i \(0.0336275\pi\)
−0.405892 + 0.913921i \(0.633039\pi\)
\(348\) 0 0
\(349\) −378.444 −1.08437 −0.542183 0.840260i \(-0.682402\pi\)
−0.542183 + 0.840260i \(0.682402\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 356.528 1.00999 0.504997 0.863121i \(-0.331494\pi\)
0.504997 + 0.863121i \(0.331494\pi\)
\(354\) 0 0
\(355\) 98.4995 + 56.8687i 0.277463 + 0.160194i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 305.248 + 528.705i 0.850273 + 1.47272i 0.880962 + 0.473187i \(0.156897\pi\)
−0.0306887 + 0.999529i \(0.509770\pi\)
\(360\) 0 0
\(361\) −103.424 + 345.868i −0.286494 + 0.958082i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 104.322 180.692i 0.285815 0.495045i
\(366\) 0 0
\(367\) −12.7497 + 22.0832i −0.0347404 + 0.0601722i −0.882873 0.469612i \(-0.844394\pi\)
0.848132 + 0.529784i \(0.177727\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −124.552 71.9103i −0.335721 0.193828i
\(372\) 0 0
\(373\) 443.527i 1.18908i 0.804066 + 0.594540i \(0.202666\pi\)
−0.804066 + 0.594540i \(0.797334\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.475304 0.823251i −0.00126075 0.00218369i
\(378\) 0 0
\(379\) 146.375i 0.386214i 0.981178 + 0.193107i \(0.0618564\pi\)
−0.981178 + 0.193107i \(0.938144\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −434.822 + 251.045i −1.13531 + 0.655469i −0.945264 0.326307i \(-0.894196\pi\)
−0.190042 + 0.981776i \(0.560862\pi\)
\(384\) 0 0
\(385\) −121.742 210.864i −0.316214 0.547699i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 38.8561 67.3007i 0.0998870 0.173009i −0.811751 0.584004i \(-0.801485\pi\)
0.911638 + 0.410995i \(0.134819\pi\)
\(390\) 0 0
\(391\) 21.8888 0.0559815
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 371.067 + 214.236i 0.939410 + 0.542368i
\(396\) 0 0
\(397\) 99.3332 + 172.050i 0.250209 + 0.433376i 0.963583 0.267408i \(-0.0861672\pi\)
−0.713374 + 0.700784i \(0.752834\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −329.197 + 190.062i −0.820940 + 0.473970i −0.850740 0.525586i \(-0.823846\pi\)
0.0298006 + 0.999556i \(0.490513\pi\)
\(402\) 0 0
\(403\) 3.21964 5.57658i 0.00798919 0.0138377i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 275.006i 0.675689i
\(408\) 0 0
\(409\) 384.909 + 222.227i 0.941098 + 0.543343i 0.890304 0.455366i \(-0.150492\pi\)
0.0507938 + 0.998709i \(0.483825\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 75.8625 43.7992i 0.183686 0.106051i
\(414\) 0 0
\(415\) 519.803 + 900.324i 1.25254 + 2.16946i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −492.182 −1.17466 −0.587329 0.809348i \(-0.699820\pi\)
−0.587329 + 0.809348i \(0.699820\pi\)
\(420\) 0 0
\(421\) −435.681 + 251.541i −1.03487 + 0.597484i −0.918377 0.395708i \(-0.870499\pi\)
−0.116495 + 0.993191i \(0.537166\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 74.3298 0.174894
\(426\) 0 0
\(427\) 90.1742 156.186i 0.211181 0.365776i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −79.5101 45.9052i −0.184478 0.106509i 0.404917 0.914354i \(-0.367300\pi\)
−0.589395 + 0.807845i \(0.700634\pi\)
\(432\) 0 0
\(433\) 268.847 + 155.219i 0.620895 + 0.358474i 0.777217 0.629232i \(-0.216631\pi\)
−0.156323 + 0.987706i \(0.549964\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −131.689 + 15.4138i −0.301348 + 0.0352718i
\(438\) 0 0
\(439\) −195.901 + 113.103i −0.446243 + 0.257639i −0.706242 0.707970i \(-0.749611\pi\)
0.259999 + 0.965609i \(0.416278\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 49.3235 85.4309i 0.111340 0.192846i −0.804971 0.593314i \(-0.797819\pi\)
0.916311 + 0.400468i \(0.131152\pi\)
\(444\) 0 0
\(445\) 442.599i 0.994605i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 258.035i 0.574688i −0.957828 0.287344i \(-0.907228\pi\)
0.957828 0.287344i \(-0.0927722\pi\)
\(450\) 0 0
\(451\) −518.697 + 299.470i −1.15010 + 0.664013i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.21188i 0.00925688i
\(456\) 0 0
\(457\) 102.212 0.223659 0.111830 0.993727i \(-0.464329\pi\)
0.111830 + 0.993727i \(0.464329\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −53.9057 93.3675i −0.116932 0.202532i 0.801618 0.597836i \(-0.203973\pi\)
−0.918550 + 0.395304i \(0.870639\pi\)
\(462\) 0 0
\(463\) −271.692 −0.586809 −0.293404 0.955988i \(-0.594788\pi\)
−0.293404 + 0.955988i \(0.594788\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −617.403 −1.32206 −0.661031 0.750358i \(-0.729881\pi\)
−0.661031 + 0.750358i \(0.729881\pi\)
\(468\) 0 0
\(469\) −142.500 82.2724i −0.303838 0.175421i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −110.984 192.230i −0.234638 0.406405i
\(474\) 0 0
\(475\) −447.189 + 52.3420i −0.941451 + 0.110194i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −53.2364 + 92.2081i −0.111141 + 0.192501i −0.916230 0.400652i \(-0.868784\pi\)
0.805090 + 0.593153i \(0.202117\pi\)
\(480\) 0 0
\(481\) 2.37857 4.11980i 0.00494505 0.00856508i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −188.415 108.781i −0.388484 0.224291i
\(486\) 0 0
\(487\) 719.920i 1.47827i 0.673555 + 0.739137i \(0.264767\pi\)
−0.673555 + 0.739137i \(0.735233\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −43.8263 75.9094i −0.0892593 0.154602i 0.817939 0.575305i \(-0.195117\pi\)
−0.907198 + 0.420703i \(0.861783\pi\)
\(492\) 0 0
\(493\) 17.0413i 0.0345665i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 48.6898 28.1111i 0.0979674 0.0565615i
\(498\) 0 0
\(499\) 78.7293 + 136.363i 0.157774 + 0.273273i 0.934066 0.357101i \(-0.116235\pi\)
−0.776292 + 0.630374i \(0.782902\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 141.523 245.125i 0.281357 0.487325i −0.690362 0.723464i \(-0.742549\pi\)
0.971719 + 0.236139i \(0.0758820\pi\)
\(504\) 0 0
\(505\) 683.968 1.35439
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −747.314 431.462i −1.46820 0.847666i −0.468835 0.883286i \(-0.655326\pi\)
−0.999365 + 0.0356201i \(0.988659\pi\)
\(510\) 0 0
\(511\) −51.5681 89.3186i −0.100916 0.174792i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 608.647 351.402i 1.18184 0.682334i
\(516\) 0 0
\(517\) 63.4551 109.907i 0.122737 0.212587i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 426.578i 0.818767i 0.912362 + 0.409384i \(0.134256\pi\)
−0.912362 + 0.409384i \(0.865744\pi\)
\(522\) 0 0
\(523\) 703.309 + 406.056i 1.34476 + 0.776397i 0.987502 0.157608i \(-0.0503783\pi\)
0.357258 + 0.934006i \(0.383712\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 99.9698 57.7176i 0.189696 0.109521i
\(528\) 0 0
\(529\) 240.152 + 415.955i 0.453973 + 0.786304i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −10.3607 −0.0194384
\(534\) 0 0
\(535\) 32.8332 18.9562i 0.0613704 0.0354322i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 375.277 0.696247
\(540\) 0 0
\(541\) 49.7429 86.1572i 0.0919461 0.159255i −0.816384 0.577510i \(-0.804025\pi\)
0.908330 + 0.418254i \(0.137358\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 416.584 + 240.515i 0.764375 + 0.441312i
\(546\) 0 0
\(547\) −552.901 319.217i −1.01079 0.583578i −0.0993654 0.995051i \(-0.531681\pi\)
−0.911422 + 0.411473i \(0.865015\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0002 + 102.525i 0.0217790 + 0.186071i
\(552\) 0 0
\(553\) 183.424 105.900i 0.331689 0.191501i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.18040 15.9009i 0.0164819 0.0285474i −0.857667 0.514206i \(-0.828087\pi\)
0.874149 + 0.485658i \(0.161420\pi\)
\(558\) 0 0
\(559\) 3.83967i 0.00686882i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 579.618i 1.02952i 0.857335 + 0.514758i \(0.172118\pi\)
−0.857335 + 0.514758i \(0.827882\pi\)
\(564\) 0 0
\(565\) −1208.94 + 697.981i −2.13971 + 1.23536i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 785.519i 1.38053i 0.723559 + 0.690263i \(0.242505\pi\)
−0.723559 + 0.690263i \(0.757495\pi\)
\(570\) 0 0
\(571\) 37.6219 0.0658878 0.0329439 0.999457i \(-0.489512\pi\)
0.0329439 + 0.999457i \(0.489512\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −82.6824 143.210i −0.143795 0.249061i
\(576\) 0 0
\(577\) 720.504 1.24871 0.624354 0.781142i \(-0.285362\pi\)
0.624354 + 0.781142i \(0.285362\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 513.893 0.884497
\(582\) 0 0
\(583\) −365.227 210.864i −0.626461 0.361688i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −100.869 174.710i −0.171838 0.297632i 0.767225 0.641378i \(-0.221637\pi\)
−0.939062 + 0.343747i \(0.888304\pi\)
\(588\) 0 0
\(589\) −560.803 + 417.643i −0.952127 + 0.709070i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 70.3653 121.876i 0.118660 0.205525i −0.800577 0.599230i \(-0.795474\pi\)
0.919237 + 0.393705i \(0.128807\pi\)
\(594\) 0 0
\(595\) 37.7526 65.3893i 0.0634497 0.109898i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 841.997 + 486.127i 1.40567 + 0.811564i 0.994967 0.100205i \(-0.0319498\pi\)
0.410703 + 0.911769i \(0.365283\pi\)
\(600\) 0 0
\(601\) 486.915i 0.810174i −0.914278 0.405087i \(-0.867241\pi\)
0.914278 0.405087i \(-0.132759\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 65.2010 + 112.931i 0.107770 + 0.186664i
\(606\) 0 0
\(607\) 819.811i 1.35059i −0.737546 0.675297i \(-0.764015\pi\)
0.737546 0.675297i \(-0.235985\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.90122 1.09767i 0.00311165 0.00179651i
\(612\) 0 0
\(613\) 262.388 + 454.470i 0.428040 + 0.741386i 0.996699 0.0811869i \(-0.0258711\pi\)
−0.568659 + 0.822573i \(0.692538\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −420.462 + 728.261i −0.681461 + 1.18033i 0.293074 + 0.956090i \(0.405322\pi\)
−0.974535 + 0.224236i \(0.928012\pi\)
\(618\) 0 0
\(619\) −420.196 −0.678831 −0.339416 0.940637i \(-0.610229\pi\)
−0.339416 + 0.940637i \(0.610229\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 189.472 + 109.392i 0.304128 + 0.175589i
\(624\) 0 0
\(625\) 327.939 + 568.008i 0.524703 + 0.908812i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 73.8545 42.6399i 0.117416 0.0677900i
\(630\) 0 0
\(631\) 472.805 818.923i 0.749295 1.29782i −0.198866 0.980027i \(-0.563726\pi\)
0.948161 0.317791i \(-0.102941\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 802.366i 1.26357i
\(636\) 0 0
\(637\) 5.62195 + 3.24583i 0.00882566 + 0.00509550i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −244.875 + 141.379i −0.382020 + 0.220559i −0.678697 0.734419i \(-0.737455\pi\)
0.296677 + 0.954978i \(0.404122\pi\)
\(642\) 0 0
\(643\) 575.275 + 996.406i 0.894674 + 1.54962i 0.834208 + 0.551450i \(0.185925\pi\)
0.0604660 + 0.998170i \(0.480741\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 366.501 0.566461 0.283231 0.959052i \(-0.408594\pi\)
0.283231 + 0.959052i \(0.408594\pi\)
\(648\) 0 0
\(649\) 222.453 128.433i 0.342763 0.197894i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 712.248 1.09073 0.545366 0.838198i \(-0.316391\pi\)
0.545366 + 0.838198i \(0.316391\pi\)
\(654\) 0 0
\(655\) 361.662 626.417i 0.552155 0.956361i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −804.356 464.395i −1.22057 0.704697i −0.255531 0.966801i \(-0.582250\pi\)
−0.965040 + 0.262104i \(0.915584\pi\)
\(660\) 0 0
\(661\) −788.697 455.354i −1.19319 0.688887i −0.234159 0.972198i \(-0.575234\pi\)
−0.959028 + 0.283311i \(0.908567\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −181.084 + 419.986i −0.272306 + 0.631557i
\(666\) 0 0
\(667\) −32.8332 + 18.9562i −0.0492251 + 0.0284201i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 264.420 457.988i 0.394068 0.682546i
\(672\) 0 0
\(673\) 197.173i 0.292976i −0.989212 0.146488i \(-0.953203\pi\)
0.989212 0.146488i \(-0.0467970\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 38.5791i 0.0569854i 0.999594 + 0.0284927i \(0.00907073\pi\)
−0.999594 + 0.0284927i \(0.990929\pi\)
\(678\) 0 0
\(679\) −93.1362 + 53.7722i −0.137167 + 0.0791933i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1073.83i 1.57223i 0.618081 + 0.786114i \(0.287910\pi\)
−0.618081 + 0.786114i \(0.712090\pi\)
\(684\) 0 0
\(685\) 898.434 1.31158
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −3.64759 6.31782i −0.00529404 0.00916955i
\(690\) 0 0
\(691\) −966.786 −1.39911 −0.699555 0.714578i \(-0.746619\pi\)
−0.699555 + 0.714578i \(0.746619\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 128.041 0.184232
\(696\) 0 0
\(697\) −160.849 92.8662i −0.230773 0.133237i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −691.064 1196.96i −0.985825 1.70750i −0.638209 0.769863i \(-0.720325\pi\)
−0.347616 0.937637i \(-0.613009\pi\)
\(702\) 0 0
\(703\) −414.303 + 308.541i −0.589336 + 0.438892i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 169.048 292.800i 0.239106 0.414144i
\(708\) 0 0
\(709\) 183.197 317.306i 0.258388 0.447541i −0.707422 0.706791i \(-0.750142\pi\)
0.965810 + 0.259250i \(0.0834754\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −222.407 128.407i −0.311932 0.180094i
\(714\) 0 0
\(715\) 12.3506i 0.0172735i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −674.358 1168.02i −0.937912 1.62451i −0.769358 0.638818i \(-0.779424\pi\)
−0.168554 0.985692i \(-0.553910\pi\)
\(720\) 0 0
\(721\) 347.407i 0.481840i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −111.495 + 64.3715i −0.153786 + 0.0887882i
\(726\) 0 0
\(727\) −594.082 1028.98i −0.817170 1.41538i −0.907759 0.419491i \(-0.862209\pi\)
0.0905897 0.995888i \(-0.471125\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 34.4163 59.6108i 0.0470811 0.0815469i
\(732\) 0 0
\(733\) 47.5143 0.0648217 0.0324108 0.999475i \(-0.489682\pi\)
0.0324108 + 0.999475i \(0.489682\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −417.855 241.249i −0.566968 0.327339i
\(738\) 0 0
\(739\) 25.3717 + 43.9451i 0.0343325 + 0.0594656i 0.882681 0.469972i \(-0.155736\pi\)
−0.848349 + 0.529438i \(0.822403\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 712.157 411.164i 0.958489 0.553384i 0.0627811 0.998027i \(-0.480003\pi\)
0.895708 + 0.444644i \(0.146670\pi\)
\(744\) 0 0
\(745\) 798.580 1383.18i 1.07192 1.85662i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 18.7407i 0.0250210i
\(750\) 0 0
\(751\) 33.3115 + 19.2324i 0.0443562 + 0.0256091i 0.522014 0.852937i \(-0.325181\pi\)
−0.477658 + 0.878546i \(0.658514\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −973.369 + 561.975i −1.28923 + 0.744337i
\(756\) 0 0
\(757\) 155.046 + 268.547i 0.204816 + 0.354752i 0.950074 0.312024i \(-0.101007\pi\)
−0.745258 + 0.666776i \(0.767674\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −923.570 −1.21363 −0.606813 0.794844i \(-0.707552\pi\)
−0.606813 + 0.794844i \(0.707552\pi\)
\(762\) 0 0
\(763\) 205.924 118.890i 0.269887 0.155819i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.44336 0.00579317
\(768\) 0 0
\(769\) 749.307 1297.84i 0.974392 1.68770i 0.292463 0.956277i \(-0.405525\pi\)
0.681928 0.731419i \(-0.261142\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −588.080 339.528i −0.760776 0.439234i 0.0687981 0.997631i \(-0.478084\pi\)
−0.829574 + 0.558396i \(0.811417\pi\)
\(774\) 0 0
\(775\) −755.249 436.043i −0.974515 0.562637i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1033.11 + 445.442i 1.32620 + 0.571812i
\(780\) 0 0
\(781\) 142.774 82.4306i 0.182809 0.105545i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −914.370 + 1583.74i −1.16480 + 2.01750i
\(786\) 0 0
\(787\) 1166.31i 1.48197i −0.671524 0.740983i \(-0.734360\pi\)
0.671524 0.740983i \(-0.265640\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 690.045i 0.872371i
\(792\) 0 0
\(793\) 7.92243 4.57402i 0.00999045 0.00576799i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 771.416i 0.967899i −0.875096 0.483950i \(-0.839202\pi\)
0.875096 0.483950i \(-0.160798\pi\)
\(798\) 0 0
\(799\) 39.3551 0.0492555
\(800\) 0 0
\(801\) 0 0