Properties

Label 840.2.t.e.169.6
Level $840$
Weight $2$
Character 840.169
Analytic conductor $6.707$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.t (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 169.6
Root \(0.403032 - 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 840.169
Dual form 840.2.t.e.169.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +(1.48119 - 1.67513i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(1.48119 - 1.67513i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +0.387873 q^{11} -2.96239i q^{13} +(1.67513 + 1.48119i) q^{15} -3.35026i q^{17} +2.96239 q^{19} +1.00000 q^{21} -0.962389i q^{23} +(-0.612127 - 4.96239i) q^{25} -1.00000i q^{27} -1.22425 q^{29} +2.96239 q^{31} +0.387873i q^{33} +(-1.67513 - 1.48119i) q^{35} -5.92478i q^{37} +2.96239 q^{39} +1.03761 q^{41} +10.7005i q^{43} +(-1.48119 + 1.67513i) q^{45} -3.22425i q^{47} -1.00000 q^{49} +3.35026 q^{51} -5.66291i q^{53} +(0.574515 - 0.649738i) q^{55} +2.96239i q^{57} -3.22425 q^{59} +14.6253 q^{61} +1.00000i q^{63} +(-4.96239 - 4.38787i) q^{65} +0.962389 q^{69} +5.53690 q^{71} +6.18664i q^{73} +(4.96239 - 0.612127i) q^{75} -0.387873i q^{77} +13.9248 q^{79} +1.00000 q^{81} +3.22425i q^{83} +(-5.61213 - 4.96239i) q^{85} -1.22425i q^{87} -3.73813 q^{89} -2.96239 q^{91} +2.96239i q^{93} +(4.38787 - 4.96239i) q^{95} +7.73813i q^{97} -0.387873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{5} - 6 q^{9} + 4 q^{11} - 4 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} - 4 q^{31} - 4 q^{39} + 28 q^{41} + 2 q^{45} - 6 q^{49} - 20 q^{55} - 16 q^{59} + 4 q^{61} - 8 q^{65} - 16 q^{69} - 12 q^{71} + 8 q^{75} + 40 q^{79} + 6 q^{81} - 32 q^{85} - 4 q^{89} + 4 q^{91} + 28 q^{95} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.48119 1.67513i 0.662410 0.749141i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0.387873 0.116948 0.0584741 0.998289i \(-0.481377\pi\)
0.0584741 + 0.998289i \(0.481377\pi\)
\(12\) 0 0
\(13\) 2.96239i 0.821619i −0.911721 0.410809i \(-0.865246\pi\)
0.911721 0.410809i \(-0.134754\pi\)
\(14\) 0 0
\(15\) 1.67513 + 1.48119i 0.432517 + 0.382443i
\(16\) 0 0
\(17\) 3.35026i 0.812558i −0.913749 0.406279i \(-0.866826\pi\)
0.913749 0.406279i \(-0.133174\pi\)
\(18\) 0 0
\(19\) 2.96239 0.679619 0.339809 0.940494i \(-0.389637\pi\)
0.339809 + 0.940494i \(0.389637\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 0.962389i 0.200672i −0.994954 0.100336i \(-0.968008\pi\)
0.994954 0.100336i \(-0.0319918\pi\)
\(24\) 0 0
\(25\) −0.612127 4.96239i −0.122425 0.992478i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −1.22425 −0.227338 −0.113669 0.993519i \(-0.536260\pi\)
−0.113669 + 0.993519i \(0.536260\pi\)
\(30\) 0 0
\(31\) 2.96239 0.532061 0.266030 0.963965i \(-0.414288\pi\)
0.266030 + 0.963965i \(0.414288\pi\)
\(32\) 0 0
\(33\) 0.387873i 0.0675200i
\(34\) 0 0
\(35\) −1.67513 1.48119i −0.283149 0.250368i
\(36\) 0 0
\(37\) 5.92478i 0.974027i −0.873394 0.487014i \(-0.838086\pi\)
0.873394 0.487014i \(-0.161914\pi\)
\(38\) 0 0
\(39\) 2.96239 0.474362
\(40\) 0 0
\(41\) 1.03761 0.162048 0.0810238 0.996712i \(-0.474181\pi\)
0.0810238 + 0.996712i \(0.474181\pi\)
\(42\) 0 0
\(43\) 10.7005i 1.63181i 0.578183 + 0.815907i \(0.303762\pi\)
−0.578183 + 0.815907i \(0.696238\pi\)
\(44\) 0 0
\(45\) −1.48119 + 1.67513i −0.220803 + 0.249714i
\(46\) 0 0
\(47\) 3.22425i 0.470306i −0.971958 0.235153i \(-0.924441\pi\)
0.971958 0.235153i \(-0.0755591\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 3.35026 0.469130
\(52\) 0 0
\(53\) 5.66291i 0.777861i −0.921267 0.388930i \(-0.872845\pi\)
0.921267 0.388930i \(-0.127155\pi\)
\(54\) 0 0
\(55\) 0.574515 0.649738i 0.0774677 0.0876107i
\(56\) 0 0
\(57\) 2.96239i 0.392378i
\(58\) 0 0
\(59\) −3.22425 −0.419762 −0.209881 0.977727i \(-0.567308\pi\)
−0.209881 + 0.977727i \(0.567308\pi\)
\(60\) 0 0
\(61\) 14.6253 1.87258 0.936289 0.351231i \(-0.114237\pi\)
0.936289 + 0.351231i \(0.114237\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) −4.96239 4.38787i −0.615509 0.544249i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0.962389 0.115858
\(70\) 0 0
\(71\) 5.53690 0.657110 0.328555 0.944485i \(-0.393438\pi\)
0.328555 + 0.944485i \(0.393438\pi\)
\(72\) 0 0
\(73\) 6.18664i 0.724092i 0.932160 + 0.362046i \(0.117922\pi\)
−0.932160 + 0.362046i \(0.882078\pi\)
\(74\) 0 0
\(75\) 4.96239 0.612127i 0.573007 0.0706823i
\(76\) 0 0
\(77\) 0.387873i 0.0442022i
\(78\) 0 0
\(79\) 13.9248 1.56666 0.783330 0.621606i \(-0.213520\pi\)
0.783330 + 0.621606i \(0.213520\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.22425i 0.353908i 0.984219 + 0.176954i \(0.0566244\pi\)
−0.984219 + 0.176954i \(0.943376\pi\)
\(84\) 0 0
\(85\) −5.61213 4.96239i −0.608721 0.538247i
\(86\) 0 0
\(87\) 1.22425i 0.131254i
\(88\) 0 0
\(89\) −3.73813 −0.396242 −0.198121 0.980178i \(-0.563484\pi\)
−0.198121 + 0.980178i \(0.563484\pi\)
\(90\) 0 0
\(91\) −2.96239 −0.310543
\(92\) 0 0
\(93\) 2.96239i 0.307185i
\(94\) 0 0
\(95\) 4.38787 4.96239i 0.450186 0.509130i
\(96\) 0 0
\(97\) 7.73813i 0.785689i 0.919605 + 0.392844i \(0.128509\pi\)
−0.919605 + 0.392844i \(0.871491\pi\)
\(98\) 0 0
\(99\) −0.387873 −0.0389827
\(100\) 0 0
\(101\) −7.58769 −0.755003 −0.377502 0.926009i \(-0.623217\pi\)
−0.377502 + 0.926009i \(0.623217\pi\)
\(102\) 0 0
\(103\) 14.7005i 1.44849i 0.689545 + 0.724243i \(0.257811\pi\)
−0.689545 + 0.724243i \(0.742189\pi\)
\(104\) 0 0
\(105\) 1.48119 1.67513i 0.144550 0.163476i
\(106\) 0 0
\(107\) 16.4387i 1.58919i −0.607143 0.794593i \(-0.707685\pi\)
0.607143 0.794593i \(-0.292315\pi\)
\(108\) 0 0
\(109\) 2.77575 0.265868 0.132934 0.991125i \(-0.457560\pi\)
0.132934 + 0.991125i \(0.457560\pi\)
\(110\) 0 0
\(111\) 5.92478 0.562355
\(112\) 0 0
\(113\) 4.26187i 0.400923i 0.979702 + 0.200461i \(0.0642441\pi\)
−0.979702 + 0.200461i \(0.935756\pi\)
\(114\) 0 0
\(115\) −1.61213 1.42548i −0.150332 0.132927i
\(116\) 0 0
\(117\) 2.96239i 0.273873i
\(118\) 0 0
\(119\) −3.35026 −0.307118
\(120\) 0 0
\(121\) −10.8496 −0.986323
\(122\) 0 0
\(123\) 1.03761i 0.0935583i
\(124\) 0 0
\(125\) −9.21933 6.32487i −0.824602 0.565713i
\(126\) 0 0
\(127\) 14.5501i 1.29111i 0.763714 + 0.645555i \(0.223374\pi\)
−0.763714 + 0.645555i \(0.776626\pi\)
\(128\) 0 0
\(129\) −10.7005 −0.942129
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 2.96239i 0.256872i
\(134\) 0 0
\(135\) −1.67513 1.48119i −0.144172 0.127481i
\(136\) 0 0
\(137\) 10.4387i 0.891835i −0.895074 0.445917i \(-0.852878\pi\)
0.895074 0.445917i \(-0.147122\pi\)
\(138\) 0 0
\(139\) −5.66291 −0.480322 −0.240161 0.970733i \(-0.577200\pi\)
−0.240161 + 0.970733i \(0.577200\pi\)
\(140\) 0 0
\(141\) 3.22425 0.271531
\(142\) 0 0
\(143\) 1.14903i 0.0960868i
\(144\) 0 0
\(145\) −1.81336 + 2.05079i −0.150591 + 0.170308i
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) −20.5501 −1.68353 −0.841764 0.539846i \(-0.818483\pi\)
−0.841764 + 0.539846i \(0.818483\pi\)
\(150\) 0 0
\(151\) −22.7005 −1.84734 −0.923671 0.383186i \(-0.874827\pi\)
−0.923671 + 0.383186i \(0.874827\pi\)
\(152\) 0 0
\(153\) 3.35026i 0.270853i
\(154\) 0 0
\(155\) 4.38787 4.96239i 0.352442 0.398589i
\(156\) 0 0
\(157\) 2.18664i 0.174513i 0.996186 + 0.0872565i \(0.0278100\pi\)
−0.996186 + 0.0872565i \(0.972190\pi\)
\(158\) 0 0
\(159\) 5.66291 0.449098
\(160\) 0 0
\(161\) −0.962389 −0.0758468
\(162\) 0 0
\(163\) 7.22425i 0.565847i 0.959142 + 0.282924i \(0.0913043\pi\)
−0.959142 + 0.282924i \(0.908696\pi\)
\(164\) 0 0
\(165\) 0.649738 + 0.574515i 0.0505821 + 0.0447260i
\(166\) 0 0
\(167\) 15.8496i 1.22648i 0.789898 + 0.613238i \(0.210133\pi\)
−0.789898 + 0.613238i \(0.789867\pi\)
\(168\) 0 0
\(169\) 4.22425 0.324943
\(170\) 0 0
\(171\) −2.96239 −0.226540
\(172\) 0 0
\(173\) 23.9756i 1.82283i 0.411490 + 0.911414i \(0.365008\pi\)
−0.411490 + 0.911414i \(0.634992\pi\)
\(174\) 0 0
\(175\) −4.96239 + 0.612127i −0.375121 + 0.0462724i
\(176\) 0 0
\(177\) 3.22425i 0.242350i
\(178\) 0 0
\(179\) −12.2374 −0.914668 −0.457334 0.889295i \(-0.651196\pi\)
−0.457334 + 0.889295i \(0.651196\pi\)
\(180\) 0 0
\(181\) 8.70052 0.646705 0.323352 0.946279i \(-0.395190\pi\)
0.323352 + 0.946279i \(0.395190\pi\)
\(182\) 0 0
\(183\) 14.6253i 1.08113i
\(184\) 0 0
\(185\) −9.92478 8.77575i −0.729684 0.645206i
\(186\) 0 0
\(187\) 1.29948i 0.0950271i
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −13.5369 −0.979496 −0.489748 0.871864i \(-0.662911\pi\)
−0.489748 + 0.871864i \(0.662911\pi\)
\(192\) 0 0
\(193\) 1.92478i 0.138548i −0.997598 0.0692742i \(-0.977932\pi\)
0.997598 0.0692742i \(-0.0220684\pi\)
\(194\) 0 0
\(195\) 4.38787 4.96239i 0.314222 0.355364i
\(196\) 0 0
\(197\) 20.1114i 1.43288i −0.697649 0.716440i \(-0.745771\pi\)
0.697649 0.716440i \(-0.254229\pi\)
\(198\) 0 0
\(199\) 8.26187 0.585668 0.292834 0.956163i \(-0.405402\pi\)
0.292834 + 0.956163i \(0.405402\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.22425i 0.0859258i
\(204\) 0 0
\(205\) 1.53690 1.73813i 0.107342 0.121397i
\(206\) 0 0
\(207\) 0.962389i 0.0668906i
\(208\) 0 0
\(209\) 1.14903 0.0794801
\(210\) 0 0
\(211\) −18.1768 −1.25134 −0.625671 0.780087i \(-0.715175\pi\)
−0.625671 + 0.780087i \(0.715175\pi\)
\(212\) 0 0
\(213\) 5.53690i 0.379382i
\(214\) 0 0
\(215\) 17.9248 + 15.8496i 1.22246 + 1.08093i
\(216\) 0 0
\(217\) 2.96239i 0.201100i
\(218\) 0 0
\(219\) −6.18664 −0.418055
\(220\) 0 0
\(221\) −9.92478 −0.667613
\(222\) 0 0
\(223\) 21.4010i 1.43312i 0.697525 + 0.716560i \(0.254284\pi\)
−0.697525 + 0.716560i \(0.745716\pi\)
\(224\) 0 0
\(225\) 0.612127 + 4.96239i 0.0408085 + 0.330826i
\(226\) 0 0
\(227\) 18.5501i 1.23121i −0.788054 0.615606i \(-0.788911\pi\)
0.788054 0.615606i \(-0.211089\pi\)
\(228\) 0 0
\(229\) 28.5501 1.88664 0.943321 0.331881i \(-0.107683\pi\)
0.943321 + 0.331881i \(0.107683\pi\)
\(230\) 0 0
\(231\) 0.387873 0.0255202
\(232\) 0 0
\(233\) 18.9624i 1.24227i 0.783705 + 0.621134i \(0.213328\pi\)
−0.783705 + 0.621134i \(0.786672\pi\)
\(234\) 0 0
\(235\) −5.40105 4.77575i −0.352325 0.311535i
\(236\) 0 0
\(237\) 13.9248i 0.904511i
\(238\) 0 0
\(239\) 18.1622 1.17482 0.587408 0.809291i \(-0.300149\pi\)
0.587408 + 0.809291i \(0.300149\pi\)
\(240\) 0 0
\(241\) 14.3733 0.925865 0.462932 0.886394i \(-0.346797\pi\)
0.462932 + 0.886394i \(0.346797\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) −1.48119 + 1.67513i −0.0946300 + 0.107020i
\(246\) 0 0
\(247\) 8.77575i 0.558387i
\(248\) 0 0
\(249\) −3.22425 −0.204329
\(250\) 0 0
\(251\) −8.62530 −0.544424 −0.272212 0.962237i \(-0.587755\pi\)
−0.272212 + 0.962237i \(0.587755\pi\)
\(252\) 0 0
\(253\) 0.373285i 0.0234682i
\(254\) 0 0
\(255\) 4.96239 5.61213i 0.310757 0.351445i
\(256\) 0 0
\(257\) 27.8251i 1.73568i 0.496841 + 0.867842i \(0.334493\pi\)
−0.496841 + 0.867842i \(0.665507\pi\)
\(258\) 0 0
\(259\) −5.92478 −0.368148
\(260\) 0 0
\(261\) 1.22425 0.0757794
\(262\) 0 0
\(263\) 0.589104i 0.0363257i −0.999835 0.0181629i \(-0.994218\pi\)
0.999835 0.0181629i \(-0.00578173\pi\)
\(264\) 0 0
\(265\) −9.48612 8.38787i −0.582728 0.515263i
\(266\) 0 0
\(267\) 3.73813i 0.228770i
\(268\) 0 0
\(269\) −15.7381 −0.959571 −0.479786 0.877386i \(-0.659286\pi\)
−0.479786 + 0.877386i \(0.659286\pi\)
\(270\) 0 0
\(271\) −7.11283 −0.432074 −0.216037 0.976385i \(-0.569313\pi\)
−0.216037 + 0.976385i \(0.569313\pi\)
\(272\) 0 0
\(273\) 2.96239i 0.179292i
\(274\) 0 0
\(275\) −0.237428 1.92478i −0.0143174 0.116068i
\(276\) 0 0
\(277\) 5.92478i 0.355985i −0.984032 0.177993i \(-0.943040\pi\)
0.984032 0.177993i \(-0.0569603\pi\)
\(278\) 0 0
\(279\) −2.96239 −0.177354
\(280\) 0 0
\(281\) 29.3258 1.74943 0.874716 0.484636i \(-0.161048\pi\)
0.874716 + 0.484636i \(0.161048\pi\)
\(282\) 0 0
\(283\) 16.1016i 0.957139i −0.878050 0.478570i \(-0.841155\pi\)
0.878050 0.478570i \(-0.158845\pi\)
\(284\) 0 0
\(285\) 4.96239 + 4.38787i 0.293947 + 0.259915i
\(286\) 0 0
\(287\) 1.03761i 0.0612483i
\(288\) 0 0
\(289\) 5.77575 0.339750
\(290\) 0 0
\(291\) −7.73813 −0.453617
\(292\) 0 0
\(293\) 9.27504i 0.541854i 0.962600 + 0.270927i \(0.0873301\pi\)
−0.962600 + 0.270927i \(0.912670\pi\)
\(294\) 0 0
\(295\) −4.77575 + 5.40105i −0.278055 + 0.314461i
\(296\) 0 0
\(297\) 0.387873i 0.0225067i
\(298\) 0 0
\(299\) −2.85097 −0.164876
\(300\) 0 0
\(301\) 10.7005 0.616768
\(302\) 0 0
\(303\) 7.58769i 0.435901i
\(304\) 0 0
\(305\) 21.6629 24.4993i 1.24041 1.40283i
\(306\) 0 0
\(307\) 10.7005i 0.610711i −0.952238 0.305356i \(-0.901225\pi\)
0.952238 0.305356i \(-0.0987753\pi\)
\(308\) 0 0
\(309\) −14.7005 −0.836284
\(310\) 0 0
\(311\) 12.1016 0.686217 0.343109 0.939296i \(-0.388520\pi\)
0.343109 + 0.939296i \(0.388520\pi\)
\(312\) 0 0
\(313\) 28.3634i 1.60320i 0.597863 + 0.801598i \(0.296017\pi\)
−0.597863 + 0.801598i \(0.703983\pi\)
\(314\) 0 0
\(315\) 1.67513 + 1.48119i 0.0943829 + 0.0834558i
\(316\) 0 0
\(317\) 17.5125i 0.983598i 0.870709 + 0.491799i \(0.163661\pi\)
−0.870709 + 0.491799i \(0.836339\pi\)
\(318\) 0 0
\(319\) −0.474855 −0.0265868
\(320\) 0 0
\(321\) 16.4387 0.917516
\(322\) 0 0
\(323\) 9.92478i 0.552229i
\(324\) 0 0
\(325\) −14.7005 + 1.81336i −0.815438 + 0.100587i
\(326\) 0 0
\(327\) 2.77575i 0.153499i
\(328\) 0 0
\(329\) −3.22425 −0.177759
\(330\) 0 0
\(331\) −19.2243 −1.05666 −0.528330 0.849039i \(-0.677182\pi\)
−0.528330 + 0.849039i \(0.677182\pi\)
\(332\) 0 0
\(333\) 5.92478i 0.324676i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.0000i 0.871576i −0.900049 0.435788i \(-0.856470\pi\)
0.900049 0.435788i \(-0.143530\pi\)
\(338\) 0 0
\(339\) −4.26187 −0.231473
\(340\) 0 0
\(341\) 1.14903 0.0622235
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 1.42548 1.61213i 0.0767455 0.0867940i
\(346\) 0 0
\(347\) 1.11283i 0.0597401i 0.999554 + 0.0298700i \(0.00950934\pi\)
−0.999554 + 0.0298700i \(0.990491\pi\)
\(348\) 0 0
\(349\) 32.0263 1.71433 0.857166 0.515041i \(-0.172223\pi\)
0.857166 + 0.515041i \(0.172223\pi\)
\(350\) 0 0
\(351\) −2.96239 −0.158121
\(352\) 0 0
\(353\) 6.20123i 0.330058i 0.986289 + 0.165029i \(0.0527718\pi\)
−0.986289 + 0.165029i \(0.947228\pi\)
\(354\) 0 0
\(355\) 8.20123 9.27504i 0.435276 0.492268i
\(356\) 0 0
\(357\) 3.35026i 0.177315i
\(358\) 0 0
\(359\) 1.68735 0.0890549 0.0445275 0.999008i \(-0.485822\pi\)
0.0445275 + 0.999008i \(0.485822\pi\)
\(360\) 0 0
\(361\) −10.2243 −0.538119
\(362\) 0 0
\(363\) 10.8496i 0.569454i
\(364\) 0 0
\(365\) 10.3634 + 9.16362i 0.542447 + 0.479646i
\(366\) 0 0
\(367\) 3.84955i 0.200945i −0.994940 0.100473i \(-0.967965\pi\)
0.994940 0.100473i \(-0.0320355\pi\)
\(368\) 0 0
\(369\) −1.03761 −0.0540159
\(370\) 0 0
\(371\) −5.66291 −0.294004
\(372\) 0 0
\(373\) 24.8773i 1.28810i −0.764984 0.644049i \(-0.777253\pi\)
0.764984 0.644049i \(-0.222747\pi\)
\(374\) 0 0
\(375\) 6.32487 9.21933i 0.326615 0.476084i
\(376\) 0 0
\(377\) 3.62672i 0.186785i
\(378\) 0 0
\(379\) −2.70052 −0.138717 −0.0693583 0.997592i \(-0.522095\pi\)
−0.0693583 + 0.997592i \(0.522095\pi\)
\(380\) 0 0
\(381\) −14.5501 −0.745423
\(382\) 0 0
\(383\) 0.523730i 0.0267614i −0.999910 0.0133807i \(-0.995741\pi\)
0.999910 0.0133807i \(-0.00425933\pi\)
\(384\) 0 0
\(385\) −0.649738 0.574515i −0.0331137 0.0292800i
\(386\) 0 0
\(387\) 10.7005i 0.543938i
\(388\) 0 0
\(389\) −6.77575 −0.343544 −0.171772 0.985137i \(-0.554949\pi\)
−0.171772 + 0.985137i \(0.554949\pi\)
\(390\) 0 0
\(391\) −3.22425 −0.163058
\(392\) 0 0
\(393\) 4.00000i 0.201773i
\(394\) 0 0
\(395\) 20.6253 23.3258i 1.03777 1.17365i
\(396\) 0 0
\(397\) 0.635150i 0.0318773i −0.999873 0.0159386i \(-0.994926\pi\)
0.999873 0.0159386i \(-0.00507364\pi\)
\(398\) 0 0
\(399\) 2.96239 0.148305
\(400\) 0 0
\(401\) 10.3733 0.518017 0.259009 0.965875i \(-0.416604\pi\)
0.259009 + 0.965875i \(0.416604\pi\)
\(402\) 0 0
\(403\) 8.77575i 0.437151i
\(404\) 0 0
\(405\) 1.48119 1.67513i 0.0736011 0.0832379i
\(406\) 0 0
\(407\) 2.29806i 0.113911i
\(408\) 0 0
\(409\) −15.7743 −0.779991 −0.389995 0.920817i \(-0.627523\pi\)
−0.389995 + 0.920817i \(0.627523\pi\)
\(410\) 0 0
\(411\) 10.4387 0.514901
\(412\) 0 0
\(413\) 3.22425i 0.158655i
\(414\) 0 0
\(415\) 5.40105 + 4.77575i 0.265127 + 0.234432i
\(416\) 0 0
\(417\) 5.66291i 0.277314i
\(418\) 0 0
\(419\) −25.1490 −1.22861 −0.614305 0.789068i \(-0.710564\pi\)
−0.614305 + 0.789068i \(0.710564\pi\)
\(420\) 0 0
\(421\) 20.1768 0.983357 0.491678 0.870777i \(-0.336384\pi\)
0.491678 + 0.870777i \(0.336384\pi\)
\(422\) 0 0
\(423\) 3.22425i 0.156769i
\(424\) 0 0
\(425\) −16.6253 + 2.05079i −0.806446 + 0.0994777i
\(426\) 0 0
\(427\) 14.6253i 0.707768i
\(428\) 0 0
\(429\) 1.14903 0.0554757
\(430\) 0 0
\(431\) −38.3127 −1.84546 −0.922728 0.385452i \(-0.874045\pi\)
−0.922728 + 0.385452i \(0.874045\pi\)
\(432\) 0 0
\(433\) 12.8872i 0.619318i −0.950848 0.309659i \(-0.899785\pi\)
0.950848 0.309659i \(-0.100215\pi\)
\(434\) 0 0
\(435\) −2.05079 1.81336i −0.0983276 0.0869438i
\(436\) 0 0
\(437\) 2.85097i 0.136380i
\(438\) 0 0
\(439\) 29.6629 1.41573 0.707867 0.706346i \(-0.249658\pi\)
0.707867 + 0.706346i \(0.249658\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 24.8119i 1.17885i −0.807823 0.589425i \(-0.799354\pi\)
0.807823 0.589425i \(-0.200646\pi\)
\(444\) 0 0
\(445\) −5.53690 + 6.26187i −0.262474 + 0.296841i
\(446\) 0 0
\(447\) 20.5501i 0.971985i
\(448\) 0 0
\(449\) 36.6516 1.72970 0.864849 0.502032i \(-0.167414\pi\)
0.864849 + 0.502032i \(0.167414\pi\)
\(450\) 0 0
\(451\) 0.402462 0.0189512
\(452\) 0 0
\(453\) 22.7005i 1.06656i
\(454\) 0 0
\(455\) −4.38787 + 4.96239i −0.205707 + 0.232640i
\(456\) 0 0
\(457\) 15.4763i 0.723949i −0.932188 0.361975i \(-0.882103\pi\)
0.932188 0.361975i \(-0.117897\pi\)
\(458\) 0 0
\(459\) −3.35026 −0.156377
\(460\) 0 0
\(461\) 41.2605 1.92169 0.960845 0.277085i \(-0.0893684\pi\)
0.960845 + 0.277085i \(0.0893684\pi\)
\(462\) 0 0
\(463\) 15.5975i 0.724879i 0.932007 + 0.362440i \(0.118056\pi\)
−0.932007 + 0.362440i \(0.881944\pi\)
\(464\) 0 0
\(465\) 4.96239 + 4.38787i 0.230125 + 0.203483i
\(466\) 0 0
\(467\) 18.5501i 0.858395i 0.903211 + 0.429198i \(0.141204\pi\)
−0.903211 + 0.429198i \(0.858796\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −2.18664 −0.100755
\(472\) 0 0
\(473\) 4.15045i 0.190838i
\(474\) 0 0
\(475\) −1.81336 14.7005i −0.0832026 0.674506i
\(476\) 0 0
\(477\) 5.66291i 0.259287i
\(478\) 0 0
\(479\) 41.5026 1.89630 0.948151 0.317819i \(-0.102950\pi\)
0.948151 + 0.317819i \(0.102950\pi\)
\(480\) 0 0
\(481\) −17.5515 −0.800279
\(482\) 0 0
\(483\) 0.962389i 0.0437902i
\(484\) 0 0
\(485\) 12.9624 + 11.4617i 0.588592 + 0.520448i
\(486\) 0 0
\(487\) 13.6728i 0.619572i 0.950806 + 0.309786i \(0.100257\pi\)
−0.950806 + 0.309786i \(0.899743\pi\)
\(488\) 0 0
\(489\) −7.22425 −0.326692
\(490\) 0 0
\(491\) 23.3404 1.05334 0.526669 0.850070i \(-0.323441\pi\)
0.526669 + 0.850070i \(0.323441\pi\)
\(492\) 0 0
\(493\) 4.10157i 0.184725i
\(494\) 0 0
\(495\) −0.574515 + 0.649738i −0.0258226 + 0.0292036i
\(496\) 0 0
\(497\) 5.53690i 0.248364i
\(498\) 0 0
\(499\) 6.85097 0.306691 0.153346 0.988173i \(-0.450995\pi\)
0.153346 + 0.988173i \(0.450995\pi\)
\(500\) 0 0
\(501\) −15.8496 −0.708106
\(502\) 0 0
\(503\) 3.32582i 0.148291i 0.997247 + 0.0741456i \(0.0236230\pi\)
−0.997247 + 0.0741456i \(0.976377\pi\)
\(504\) 0 0
\(505\) −11.2388 + 12.7104i −0.500122 + 0.565604i
\(506\) 0 0
\(507\) 4.22425i 0.187606i
\(508\) 0 0
\(509\) 6.18664 0.274218 0.137109 0.990556i \(-0.456219\pi\)
0.137109 + 0.990556i \(0.456219\pi\)
\(510\) 0 0
\(511\) 6.18664 0.273681
\(512\) 0 0
\(513\) 2.96239i 0.130793i
\(514\) 0 0
\(515\) 24.6253 + 21.7743i 1.08512 + 0.959492i
\(516\) 0 0
\(517\) 1.25060i 0.0550014i
\(518\) 0 0
\(519\) −23.9756 −1.05241
\(520\) 0 0
\(521\) 34.4387 1.50879 0.754393 0.656424i \(-0.227932\pi\)
0.754393 + 0.656424i \(0.227932\pi\)
\(522\) 0 0
\(523\) 6.59895i 0.288552i 0.989537 + 0.144276i \(0.0460853\pi\)
−0.989537 + 0.144276i \(0.953915\pi\)
\(524\) 0 0
\(525\) −0.612127 4.96239i −0.0267154 0.216576i
\(526\) 0 0
\(527\) 9.92478i 0.432330i
\(528\) 0 0
\(529\) 22.0738 0.959731
\(530\) 0 0
\(531\) 3.22425 0.139921
\(532\) 0 0
\(533\) 3.07381i 0.133141i
\(534\) 0 0
\(535\) −27.5369 24.3488i −1.19052 1.05269i
\(536\) 0 0
\(537\) 12.2374i 0.528084i
\(538\) 0 0
\(539\) −0.387873 −0.0167069
\(540\) 0 0
\(541\) −9.07381 −0.390113 −0.195057 0.980792i \(-0.562489\pi\)
−0.195057 + 0.980792i \(0.562489\pi\)
\(542\) 0 0
\(543\) 8.70052i 0.373375i
\(544\) 0 0
\(545\) 4.11142 4.64974i 0.176114 0.199173i
\(546\) 0 0
\(547\) 19.8496i 0.848706i 0.905497 + 0.424353i \(0.139498\pi\)
−0.905497 + 0.424353i \(0.860502\pi\)
\(548\) 0 0
\(549\) −14.6253 −0.624193
\(550\) 0 0
\(551\) −3.62672 −0.154503
\(552\) 0 0
\(553\) 13.9248i 0.592142i
\(554\) 0 0
\(555\) 8.77575 9.92478i 0.372510 0.421283i
\(556\) 0 0
\(557\) 0.785595i 0.0332867i −0.999861 0.0166434i \(-0.994702\pi\)
0.999861 0.0166434i \(-0.00529799\pi\)
\(558\) 0 0
\(559\) 31.6991 1.34073
\(560\) 0 0
\(561\) 1.29948 0.0548639
\(562\) 0 0
\(563\) 32.2228i 1.35803i −0.734124 0.679015i \(-0.762407\pi\)
0.734124 0.679015i \(-0.237593\pi\)
\(564\) 0 0
\(565\) 7.13918 + 6.31265i 0.300348 + 0.265575i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) −16.0752 −0.673908 −0.336954 0.941521i \(-0.609397\pi\)
−0.336954 + 0.941521i \(0.609397\pi\)
\(570\) 0 0
\(571\) −39.8496 −1.66765 −0.833826 0.552027i \(-0.813854\pi\)
−0.833826 + 0.552027i \(0.813854\pi\)
\(572\) 0 0
\(573\) 13.5369i 0.565512i
\(574\) 0 0
\(575\) −4.77575 + 0.589104i −0.199162 + 0.0245673i
\(576\) 0 0
\(577\) 30.1378i 1.25465i 0.778757 + 0.627326i \(0.215851\pi\)
−0.778757 + 0.627326i \(0.784149\pi\)
\(578\) 0 0
\(579\) 1.92478 0.0799910
\(580\) 0 0
\(581\) 3.22425 0.133765
\(582\) 0 0
\(583\) 2.19649i 0.0909694i
\(584\) 0 0
\(585\) 4.96239 + 4.38787i 0.205170 + 0.181416i
\(586\) 0 0
\(587\) 33.7743i 1.39402i 0.717063 + 0.697008i \(0.245486\pi\)
−0.717063 + 0.697008i \(0.754514\pi\)
\(588\) 0 0
\(589\) 8.77575 0.361598
\(590\) 0 0
\(591\) 20.1114 0.827273
\(592\) 0 0
\(593\) 7.19982i 0.295661i 0.989013 + 0.147831i \(0.0472290\pi\)
−0.989013 + 0.147831i \(0.952771\pi\)
\(594\) 0 0
\(595\) −4.96239 + 5.61213i −0.203438 + 0.230075i
\(596\) 0 0
\(597\) 8.26187i 0.338136i
\(598\) 0 0
\(599\) 29.5369 1.20685 0.603423 0.797422i \(-0.293803\pi\)
0.603423 + 0.797422i \(0.293803\pi\)
\(600\) 0 0
\(601\) 8.59895 0.350759 0.175379 0.984501i \(-0.443885\pi\)
0.175379 + 0.984501i \(0.443885\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −16.0703 + 18.1744i −0.653351 + 0.738895i
\(606\) 0 0
\(607\) 37.4010i 1.51806i −0.651055 0.759031i \(-0.725673\pi\)
0.651055 0.759031i \(-0.274327\pi\)
\(608\) 0 0
\(609\) −1.22425 −0.0496093
\(610\) 0 0
\(611\) −9.55149 −0.386412
\(612\) 0 0
\(613\) 14.5990i 0.589646i −0.955552 0.294823i \(-0.904739\pi\)
0.955552 0.294823i \(-0.0952607\pi\)
\(614\) 0 0
\(615\) 1.73813 + 1.53690i 0.0700884 + 0.0619740i
\(616\) 0 0
\(617\) 22.8119i 0.918374i −0.888340 0.459187i \(-0.848141\pi\)
0.888340 0.459187i \(-0.151859\pi\)
\(618\) 0 0
\(619\) −0.261865 −0.0105252 −0.00526262 0.999986i \(-0.501675\pi\)
−0.00526262 + 0.999986i \(0.501675\pi\)
\(620\) 0 0
\(621\) −0.962389 −0.0386193
\(622\) 0 0
\(623\) 3.73813i 0.149765i
\(624\) 0 0
\(625\) −24.2506 + 6.07522i −0.970024 + 0.243009i
\(626\) 0 0
\(627\) 1.14903i 0.0458879i
\(628\) 0 0
\(629\) −19.8496 −0.791454
\(630\) 0 0
\(631\) −5.92478 −0.235862 −0.117931 0.993022i \(-0.537626\pi\)
−0.117931 + 0.993022i \(0.537626\pi\)
\(632\) 0 0
\(633\) 18.1768i 0.722463i
\(634\) 0 0
\(635\) 24.3733 + 21.5515i 0.967224 + 0.855245i
\(636\) 0 0
\(637\) 2.96239i 0.117374i
\(638\) 0 0
\(639\) −5.53690 −0.219037
\(640\) 0 0
\(641\) 6.87732 0.271638 0.135819 0.990734i \(-0.456633\pi\)
0.135819 + 0.990734i \(0.456633\pi\)
\(642\) 0 0
\(643\) 10.7005i 0.421987i −0.977487 0.210994i \(-0.932330\pi\)
0.977487 0.210994i \(-0.0676700\pi\)
\(644\) 0 0
\(645\) −15.8496 + 17.9248i −0.624076 + 0.705787i
\(646\) 0 0
\(647\) 45.2506i 1.77898i 0.456950 + 0.889492i \(0.348942\pi\)
−0.456950 + 0.889492i \(0.651058\pi\)
\(648\) 0 0
\(649\) −1.25060 −0.0490904
\(650\) 0 0
\(651\) 2.96239 0.116105
\(652\) 0 0
\(653\) 36.0625i 1.41124i 0.708592 + 0.705618i \(0.249331\pi\)
−0.708592 + 0.705618i \(0.750669\pi\)
\(654\) 0 0
\(655\) −5.92478 + 6.70052i −0.231500 + 0.261811i
\(656\) 0 0
\(657\) 6.18664i 0.241364i
\(658\) 0 0
\(659\) −43.4617 −1.69303 −0.846513 0.532367i \(-0.821302\pi\)
−0.846513 + 0.532367i \(0.821302\pi\)
\(660\) 0 0
\(661\) −22.4749 −0.874171 −0.437085 0.899420i \(-0.643989\pi\)
−0.437085 + 0.899420i \(0.643989\pi\)
\(662\) 0 0
\(663\) 9.92478i 0.385446i
\(664\) 0 0
\(665\) −4.96239 4.38787i −0.192433 0.170154i
\(666\) 0 0
\(667\) 1.17821i 0.0456204i
\(668\) 0 0
\(669\) −21.4010 −0.827412
\(670\) 0 0
\(671\) 5.67276 0.218995
\(672\) 0 0
\(673\) 47.5487i 1.83287i 0.400188 + 0.916433i \(0.368945\pi\)
−0.400188 + 0.916433i \(0.631055\pi\)
\(674\) 0 0
\(675\) −4.96239 + 0.612127i −0.191002 + 0.0235608i
\(676\) 0 0
\(677\) 5.67750i 0.218204i −0.994031 0.109102i \(-0.965202\pi\)
0.994031 0.109102i \(-0.0347975\pi\)
\(678\) 0 0
\(679\) 7.73813 0.296962
\(680\) 0 0
\(681\) 18.5501 0.710841
\(682\) 0 0
\(683\) 12.2882i 0.470195i 0.971972 + 0.235098i \(0.0755410\pi\)
−0.971972 + 0.235098i \(0.924459\pi\)
\(684\) 0 0
\(685\) −17.4861 15.4617i −0.668110 0.590760i
\(686\) 0 0
\(687\) 28.5501i 1.08925i
\(688\) 0 0
\(689\) −16.7757 −0.639105
\(690\) 0 0
\(691\) −1.81336 −0.0689834 −0.0344917 0.999405i \(-0.510981\pi\)
−0.0344917 + 0.999405i \(0.510981\pi\)
\(692\) 0 0
\(693\) 0.387873i 0.0147341i
\(694\) 0 0
\(695\) −8.38787 + 9.48612i −0.318170 + 0.359829i
\(696\) 0 0
\(697\) 3.47627i 0.131673i
\(698\) 0 0
\(699\) −18.9624 −0.717223
\(700\) 0 0
\(701\) 29.5778 1.11714 0.558570 0.829458i \(-0.311350\pi\)
0.558570 + 0.829458i \(0.311350\pi\)
\(702\) 0 0
\(703\) 17.5515i 0.661967i
\(704\) 0 0
\(705\) 4.77575 5.40105i 0.179865 0.203415i
\(706\) 0 0
\(707\) 7.58769i 0.285364i
\(708\) 0 0
\(709\) −20.8021 −0.781239 −0.390620 0.920552i \(-0.627739\pi\)
−0.390620 + 0.920552i \(0.627739\pi\)
\(710\) 0 0
\(711\) −13.9248 −0.522220
\(712\) 0 0
\(713\) 2.85097i 0.106770i
\(714\) 0 0
\(715\) −1.92478 1.70194i −0.0719826 0.0636489i
\(716\) 0 0
\(717\) 18.1622i 0.678280i
\(718\) 0 0
\(719\) −19.3258 −0.720732 −0.360366 0.932811i \(-0.617348\pi\)
−0.360366 + 0.932811i \(0.617348\pi\)
\(720\) 0 0
\(721\) 14.7005 0.547476
\(722\) 0 0
\(723\) 14.3733i 0.534548i
\(724\) 0 0
\(725\) 0.749399 + 6.07522i 0.0278320 + 0.225628i
\(726\) 0 0
\(727\) 21.6531i 0.803068i 0.915844 + 0.401534i \(0.131523\pi\)
−0.915844 + 0.401534i \(0.868477\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 35.8496 1.32594
\(732\) 0 0
\(733\) 48.0625i 1.77523i −0.460586 0.887615i \(-0.652361\pi\)
0.460586 0.887615i \(-0.347639\pi\)
\(734\) 0 0
\(735\) −1.67513 1.48119i −0.0617881 0.0546347i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 30.2981 1.11453 0.557266 0.830334i \(-0.311850\pi\)
0.557266 + 0.830334i \(0.311850\pi\)
\(740\) 0 0
\(741\) 8.77575 0.322385
\(742\) 0 0
\(743\) 40.9624i 1.50276i −0.659867 0.751382i \(-0.729388\pi\)
0.659867 0.751382i \(-0.270612\pi\)
\(744\) 0 0
\(745\) −30.4387 + 34.4241i −1.11519 + 1.26120i
\(746\) 0 0
\(747\) 3.22425i 0.117969i
\(748\) 0 0
\(749\) −16.4387 −0.600656
\(750\) 0 0
\(751\) 9.52232 0.347474 0.173737 0.984792i \(-0.444416\pi\)
0.173737 + 0.984792i \(0.444416\pi\)
\(752\) 0 0
\(753\) 8.62530i 0.314323i
\(754\) 0 0
\(755\) −33.6239 + 38.0263i −1.22370 + 1.38392i
\(756\) 0 0
\(757\) 31.3258i 1.13856i −0.822145 0.569278i \(-0.807223\pi\)
0.822145 0.569278i \(-0.192777\pi\)
\(758\) 0 0
\(759\) 0.373285 0.0135494
\(760\) 0 0
\(761\) −54.2393 −1.96617 −0.983087 0.183138i \(-0.941375\pi\)
−0.983087 + 0.183138i \(0.941375\pi\)
\(762\) 0 0
\(763\) 2.77575i 0.100489i
\(764\) 0 0
\(765\) 5.61213 + 4.96239i 0.202907 + 0.179416i
\(766\) 0 0
\(767\) 9.55149i 0.344884i
\(768\) 0 0
\(769\) −35.1002 −1.26574 −0.632872 0.774256i \(-0.718124\pi\)
−0.632872 + 0.774256i \(0.718124\pi\)
\(770\) 0 0
\(771\) −27.8251 −1.00210
\(772\) 0 0
\(773\) 19.8740i 0.714818i 0.933948 + 0.357409i \(0.116340\pi\)
−0.933948 + 0.357409i \(0.883660\pi\)
\(774\) 0 0
\(775\) −1.81336 14.7005i −0.0651377 0.528058i
\(776\) 0 0
\(777\) 5.92478i 0.212550i
\(778\) 0 0
\(779\) 3.07381 0.110131
\(780\) 0 0
\(781\) 2.14762 0.0768478
\(782\) 0 0
\(783\) 1.22425i 0.0437513i
\(784\) 0 0
\(785\) 3.66291 + 3.23884i 0.130735 + 0.115599i
\(786\) 0 0
\(787\) 13.2506i 0.472333i −0.971713 0.236166i \(-0.924109\pi\)
0.971713 0.236166i \(-0.0758911\pi\)
\(788\) 0 0
\(789\) 0.589104 0.0209727
\(790\) 0 0
\(791\) 4.26187 0.151534
\(792\) 0 0
\(793\) 43.3258i 1.53855i
\(794\) 0 0
\(795\) 8.38787 9.48612i 0.297487 0.336438i
\(796\) 0 0
\(797\) 15.7235i 0.556957i 0.960443 + 0.278478i \(0.0898300\pi\)
−0.960443 + 0.278478i \(0.910170\pi\)
\(798\) 0 0
\(799\) −10.8021 −0.382151
\(800\) 0 0
\(801\) 3.73813 0.132081