Properties

Label 840.2.t.e.169.2
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 169.2
Root \(1.45161 + 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 840.169
Dual form 840.2.t.e.169.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +(-0.311108 - 2.21432i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(-0.311108 - 2.21432i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -3.80642 q^{11} -0.622216i q^{13} +(-2.21432 + 0.311108i) q^{15} -4.42864i q^{17} -0.622216 q^{19} +1.00000 q^{21} -2.62222i q^{23} +(-4.80642 + 1.37778i) q^{25} +1.00000i q^{27} -9.61285 q^{29} -0.622216 q^{31} +3.80642i q^{33} +(2.21432 - 0.311108i) q^{35} -1.24443i q^{37} -0.622216 q^{39} +4.62222 q^{41} +4.85728i q^{43} +(0.311108 + 2.21432i) q^{45} +11.6128i q^{47} -1.00000 q^{49} -4.42864 q^{51} -13.4795i q^{53} +(1.18421 + 8.42864i) q^{55} +0.622216i q^{57} -11.6128 q^{59} -8.10171 q^{61} -1.00000i q^{63} +(-1.37778 + 0.193576i) q^{65} -2.62222 q^{69} +2.56199 q^{71} -10.9906i q^{73} +(1.37778 + 4.80642i) q^{75} -3.80642i q^{77} +6.75557 q^{79} +1.00000 q^{81} -11.6128i q^{83} +(-9.80642 + 1.37778i) q^{85} +9.61285i q^{87} +8.23506 q^{89} +0.622216 q^{91} +0.622216i q^{93} +(0.193576 + 1.37778i) q^{95} +4.23506i q^{97} +3.80642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 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
\(5\) −0.311108 2.21432i −0.139132 0.990274i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.80642 −1.14768 −0.573840 0.818967i \(-0.694547\pi\)
−0.573840 + 0.818967i \(0.694547\pi\)
\(12\) 0 0
\(13\) 0.622216i 0.172572i −0.996270 0.0862858i \(-0.972500\pi\)
0.996270 0.0862858i \(-0.0274998\pi\)
\(14\) 0 0
\(15\) −2.21432 + 0.311108i −0.571735 + 0.0803277i
\(16\) 0 0
\(17\) 4.42864i 1.07410i −0.843550 0.537051i \(-0.819538\pi\)
0.843550 0.537051i \(-0.180462\pi\)
\(18\) 0 0
\(19\) −0.622216 −0.142746 −0.0713730 0.997450i \(-0.522738\pi\)
−0.0713730 + 0.997450i \(0.522738\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 2.62222i 0.546770i −0.961905 0.273385i \(-0.911857\pi\)
0.961905 0.273385i \(-0.0881433\pi\)
\(24\) 0 0
\(25\) −4.80642 + 1.37778i −0.961285 + 0.275557i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −9.61285 −1.78506 −0.892531 0.450987i \(-0.851072\pi\)
−0.892531 + 0.450987i \(0.851072\pi\)
\(30\) 0 0
\(31\) −0.622216 −0.111753 −0.0558766 0.998438i \(-0.517795\pi\)
−0.0558766 + 0.998438i \(0.517795\pi\)
\(32\) 0 0
\(33\) 3.80642i 0.662613i
\(34\) 0 0
\(35\) 2.21432 0.311108i 0.374288 0.0525868i
\(36\) 0 0
\(37\) 1.24443i 0.204583i −0.994754 0.102292i \(-0.967383\pi\)
0.994754 0.102292i \(-0.0326175\pi\)
\(38\) 0 0
\(39\) −0.622216 −0.0996342
\(40\) 0 0
\(41\) 4.62222 0.721869 0.360934 0.932591i \(-0.382458\pi\)
0.360934 + 0.932591i \(0.382458\pi\)
\(42\) 0 0
\(43\) 4.85728i 0.740728i 0.928887 + 0.370364i \(0.120767\pi\)
−0.928887 + 0.370364i \(0.879233\pi\)
\(44\) 0 0
\(45\) 0.311108 + 2.21432i 0.0463772 + 0.330091i
\(46\) 0 0
\(47\) 11.6128i 1.69391i 0.531666 + 0.846954i \(0.321566\pi\)
−0.531666 + 0.846954i \(0.678434\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −4.42864 −0.620134
\(52\) 0 0
\(53\) 13.4795i 1.85155i −0.378074 0.925775i \(-0.623413\pi\)
0.378074 0.925775i \(-0.376587\pi\)
\(54\) 0 0
\(55\) 1.18421 + 8.42864i 0.159679 + 1.13652i
\(56\) 0 0
\(57\) 0.622216i 0.0824145i
\(58\) 0 0
\(59\) −11.6128 −1.51186 −0.755932 0.654650i \(-0.772816\pi\)
−0.755932 + 0.654650i \(0.772816\pi\)
\(60\) 0 0
\(61\) −8.10171 −1.03732 −0.518659 0.854981i \(-0.673569\pi\)
−0.518659 + 0.854981i \(0.673569\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) −1.37778 + 0.193576i −0.170893 + 0.0240102i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −2.62222 −0.315678
\(70\) 0 0
\(71\) 2.56199 0.304053 0.152026 0.988376i \(-0.451420\pi\)
0.152026 + 0.988376i \(0.451420\pi\)
\(72\) 0 0
\(73\) 10.9906i 1.28636i −0.765717 0.643178i \(-0.777615\pi\)
0.765717 0.643178i \(-0.222385\pi\)
\(74\) 0 0
\(75\) 1.37778 + 4.80642i 0.159093 + 0.554998i
\(76\) 0 0
\(77\) 3.80642i 0.433782i
\(78\) 0 0
\(79\) 6.75557 0.760061 0.380030 0.924974i \(-0.375914\pi\)
0.380030 + 0.924974i \(0.375914\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.6128i 1.27468i −0.770585 0.637338i \(-0.780036\pi\)
0.770585 0.637338i \(-0.219964\pi\)
\(84\) 0 0
\(85\) −9.80642 + 1.37778i −1.06366 + 0.149442i
\(86\) 0 0
\(87\) 9.61285i 1.03061i
\(88\) 0 0
\(89\) 8.23506 0.872915 0.436457 0.899725i \(-0.356233\pi\)
0.436457 + 0.899725i \(0.356233\pi\)
\(90\) 0 0
\(91\) 0.622216 0.0652259
\(92\) 0 0
\(93\) 0.622216i 0.0645208i
\(94\) 0 0
\(95\) 0.193576 + 1.37778i 0.0198605 + 0.141358i
\(96\) 0 0
\(97\) 4.23506i 0.430006i 0.976613 + 0.215003i \(0.0689761\pi\)
−0.976613 + 0.215003i \(0.931024\pi\)
\(98\) 0 0
\(99\) 3.80642 0.382560
\(100\) 0 0
\(101\) 18.7239 1.86310 0.931550 0.363613i \(-0.118457\pi\)
0.931550 + 0.363613i \(0.118457\pi\)
\(102\) 0 0
\(103\) 0.857279i 0.0844702i 0.999108 + 0.0422351i \(0.0134479\pi\)
−0.999108 + 0.0422351i \(0.986552\pi\)
\(104\) 0 0
\(105\) −0.311108 2.21432i −0.0303610 0.216095i
\(106\) 0 0
\(107\) 11.0923i 1.07234i −0.844111 0.536169i \(-0.819871\pi\)
0.844111 0.536169i \(-0.180129\pi\)
\(108\) 0 0
\(109\) −5.61285 −0.537613 −0.268807 0.963194i \(-0.586629\pi\)
−0.268807 + 0.963194i \(0.586629\pi\)
\(110\) 0 0
\(111\) −1.24443 −0.118116
\(112\) 0 0
\(113\) 16.2351i 1.52727i −0.645650 0.763633i \(-0.723414\pi\)
0.645650 0.763633i \(-0.276586\pi\)
\(114\) 0 0
\(115\) −5.80642 + 0.815792i −0.541452 + 0.0760730i
\(116\) 0 0
\(117\) 0.622216i 0.0575239i
\(118\) 0 0
\(119\) 4.42864 0.405973
\(120\) 0 0
\(121\) 3.48886 0.317169
\(122\) 0 0
\(123\) 4.62222i 0.416771i
\(124\) 0 0
\(125\) 4.54617 + 10.2143i 0.406622 + 0.913597i
\(126\) 0 0
\(127\) 15.3461i 1.36175i 0.732400 + 0.680875i \(0.238400\pi\)
−0.732400 + 0.680875i \(0.761600\pi\)
\(128\) 0 0
\(129\) 4.85728 0.427660
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 0.622216i 0.0539529i
\(134\) 0 0
\(135\) 2.21432 0.311108i 0.190578 0.0267759i
\(136\) 0 0
\(137\) 17.0923i 1.46030i −0.683288 0.730149i \(-0.739451\pi\)
0.683288 0.730149i \(-0.260549\pi\)
\(138\) 0 0
\(139\) 13.4795 1.14332 0.571658 0.820492i \(-0.306300\pi\)
0.571658 + 0.820492i \(0.306300\pi\)
\(140\) 0 0
\(141\) 11.6128 0.977978
\(142\) 0 0
\(143\) 2.36842i 0.198057i
\(144\) 0 0
\(145\) 2.99063 + 21.2859i 0.248358 + 1.76770i
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) 9.34614 0.765666 0.382833 0.923818i \(-0.374949\pi\)
0.382833 + 0.923818i \(0.374949\pi\)
\(150\) 0 0
\(151\) −7.14272 −0.581266 −0.290633 0.956835i \(-0.593866\pi\)
−0.290633 + 0.956835i \(0.593866\pi\)
\(152\) 0 0
\(153\) 4.42864i 0.358034i
\(154\) 0 0
\(155\) 0.193576 + 1.37778i 0.0155484 + 0.110666i
\(156\) 0 0
\(157\) 6.99063i 0.557913i −0.960304 0.278957i \(-0.910011\pi\)
0.960304 0.278957i \(-0.0899886\pi\)
\(158\) 0 0
\(159\) −13.4795 −1.06899
\(160\) 0 0
\(161\) 2.62222 0.206660
\(162\) 0 0
\(163\) 15.6128i 1.22289i −0.791286 0.611446i \(-0.790588\pi\)
0.791286 0.611446i \(-0.209412\pi\)
\(164\) 0 0
\(165\) 8.42864 1.18421i 0.656169 0.0921905i
\(166\) 0 0
\(167\) 1.51114i 0.116935i −0.998289 0.0584677i \(-0.981379\pi\)
0.998289 0.0584677i \(-0.0186215\pi\)
\(168\) 0 0
\(169\) 12.6128 0.970219
\(170\) 0 0
\(171\) 0.622216 0.0475820
\(172\) 0 0
\(173\) 6.53035i 0.496493i 0.968697 + 0.248247i \(0.0798543\pi\)
−0.968697 + 0.248247i \(0.920146\pi\)
\(174\) 0 0
\(175\) −1.37778 4.80642i −0.104151 0.363331i
\(176\) 0 0
\(177\) 11.6128i 0.872875i
\(178\) 0 0
\(179\) 6.29529 0.470532 0.235266 0.971931i \(-0.424404\pi\)
0.235266 + 0.971931i \(0.424404\pi\)
\(180\) 0 0
\(181\) −6.85728 −0.509698 −0.254849 0.966981i \(-0.582026\pi\)
−0.254849 + 0.966981i \(0.582026\pi\)
\(182\) 0 0
\(183\) 8.10171i 0.598896i
\(184\) 0 0
\(185\) −2.75557 + 0.387152i −0.202593 + 0.0284640i
\(186\) 0 0
\(187\) 16.8573i 1.23273i
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −10.5620 −0.764239 −0.382119 0.924113i \(-0.624806\pi\)
−0.382119 + 0.924113i \(0.624806\pi\)
\(192\) 0 0
\(193\) 5.24443i 0.377502i −0.982025 0.188751i \(-0.939556\pi\)
0.982025 0.188751i \(-0.0604440\pi\)
\(194\) 0 0
\(195\) 0.193576 + 1.37778i 0.0138623 + 0.0986652i
\(196\) 0 0
\(197\) 17.7462i 1.26436i 0.774820 + 0.632182i \(0.217841\pi\)
−0.774820 + 0.632182i \(0.782159\pi\)
\(198\) 0 0
\(199\) 20.2351 1.43443 0.717213 0.696854i \(-0.245418\pi\)
0.717213 + 0.696854i \(0.245418\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.61285i 0.674690i
\(204\) 0 0
\(205\) −1.43801 10.2351i −0.100435 0.714848i
\(206\) 0 0
\(207\) 2.62222i 0.182257i
\(208\) 0 0
\(209\) 2.36842 0.163827
\(210\) 0 0
\(211\) 21.3274 1.46824 0.734120 0.679020i \(-0.237595\pi\)
0.734120 + 0.679020i \(0.237595\pi\)
\(212\) 0 0
\(213\) 2.56199i 0.175545i
\(214\) 0 0
\(215\) 10.7556 1.51114i 0.733524 0.103059i
\(216\) 0 0
\(217\) 0.622216i 0.0422387i
\(218\) 0 0
\(219\) −10.9906 −0.742678
\(220\) 0 0
\(221\) −2.75557 −0.185360
\(222\) 0 0
\(223\) 9.71456i 0.650535i 0.945622 + 0.325267i \(0.105454\pi\)
−0.945622 + 0.325267i \(0.894546\pi\)
\(224\) 0 0
\(225\) 4.80642 1.37778i 0.320428 0.0918523i
\(226\) 0 0
\(227\) 11.3461i 0.753070i −0.926402 0.376535i \(-0.877115\pi\)
0.926402 0.376535i \(-0.122885\pi\)
\(228\) 0 0
\(229\) −1.34614 −0.0889555 −0.0444778 0.999010i \(-0.514162\pi\)
−0.0444778 + 0.999010i \(0.514162\pi\)
\(230\) 0 0
\(231\) −3.80642 −0.250444
\(232\) 0 0
\(233\) 15.3778i 1.00743i −0.863869 0.503716i \(-0.831966\pi\)
0.863869 0.503716i \(-0.168034\pi\)
\(234\) 0 0
\(235\) 25.7146 3.61285i 1.67743 0.235676i
\(236\) 0 0
\(237\) 6.75557i 0.438821i
\(238\) 0 0
\(239\) −7.53972 −0.487704 −0.243852 0.969812i \(-0.578411\pi\)
−0.243852 + 0.969812i \(0.578411\pi\)
\(240\) 0 0
\(241\) 23.9813 1.54477 0.772385 0.635155i \(-0.219064\pi\)
0.772385 + 0.635155i \(0.219064\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0.311108 + 2.21432i 0.0198759 + 0.141468i
\(246\) 0 0
\(247\) 0.387152i 0.0246339i
\(248\) 0 0
\(249\) −11.6128 −0.735934
\(250\) 0 0
\(251\) 14.1017 0.890092 0.445046 0.895508i \(-0.353187\pi\)
0.445046 + 0.895508i \(0.353187\pi\)
\(252\) 0 0
\(253\) 9.98126i 0.627517i
\(254\) 0 0
\(255\) 1.37778 + 9.80642i 0.0862802 + 0.614102i
\(256\) 0 0
\(257\) 17.0192i 1.06163i 0.847488 + 0.530815i \(0.178114\pi\)
−0.847488 + 0.530815i \(0.821886\pi\)
\(258\) 0 0
\(259\) 1.24443 0.0773252
\(260\) 0 0
\(261\) 9.61285 0.595020
\(262\) 0 0
\(263\) 12.6035i 0.777164i −0.921414 0.388582i \(-0.872965\pi\)
0.921414 0.388582i \(-0.127035\pi\)
\(264\) 0 0
\(265\) −29.8479 + 4.19358i −1.83354 + 0.257609i
\(266\) 0 0
\(267\) 8.23506i 0.503978i
\(268\) 0 0
\(269\) −3.76494 −0.229552 −0.114776 0.993391i \(-0.536615\pi\)
−0.114776 + 0.993391i \(0.536615\pi\)
\(270\) 0 0
\(271\) −17.8666 −1.08532 −0.542661 0.839952i \(-0.682583\pi\)
−0.542661 + 0.839952i \(0.682583\pi\)
\(272\) 0 0
\(273\) 0.622216i 0.0376582i
\(274\) 0 0
\(275\) 18.2953 5.24443i 1.10325 0.316251i
\(276\) 0 0
\(277\) 1.24443i 0.0747706i −0.999301 0.0373853i \(-0.988097\pi\)
0.999301 0.0373853i \(-0.0119029\pi\)
\(278\) 0 0
\(279\) 0.622216 0.0372511
\(280\) 0 0
\(281\) −8.95899 −0.534448 −0.267224 0.963634i \(-0.586106\pi\)
−0.267224 + 0.963634i \(0.586106\pi\)
\(282\) 0 0
\(283\) 30.5718i 1.81731i −0.417551 0.908654i \(-0.637111\pi\)
0.417551 0.908654i \(-0.362889\pi\)
\(284\) 0 0
\(285\) 1.37778 0.193576i 0.0816129 0.0114665i
\(286\) 0 0
\(287\) 4.62222i 0.272841i
\(288\) 0 0
\(289\) −2.61285 −0.153697
\(290\) 0 0
\(291\) 4.23506 0.248264
\(292\) 0 0
\(293\) 5.67307i 0.331424i 0.986174 + 0.165712i \(0.0529923\pi\)
−0.986174 + 0.165712i \(0.947008\pi\)
\(294\) 0 0
\(295\) 3.61285 + 25.7146i 0.210348 + 1.49716i
\(296\) 0 0
\(297\) 3.80642i 0.220871i
\(298\) 0 0
\(299\) −1.63158 −0.0943569
\(300\) 0 0
\(301\) −4.85728 −0.279969
\(302\) 0 0
\(303\) 18.7239i 1.07566i
\(304\) 0 0
\(305\) 2.52051 + 17.9398i 0.144324 + 1.02723i
\(306\) 0 0
\(307\) 4.85728i 0.277220i −0.990347 0.138610i \(-0.955737\pi\)
0.990347 0.138610i \(-0.0442634\pi\)
\(308\) 0 0
\(309\) 0.857279 0.0487689
\(310\) 0 0
\(311\) −34.5718 −1.96039 −0.980195 0.198037i \(-0.936543\pi\)
−0.980195 + 0.198037i \(0.936543\pi\)
\(312\) 0 0
\(313\) 6.33677i 0.358176i 0.983833 + 0.179088i \(0.0573146\pi\)
−0.983833 + 0.179088i \(0.942685\pi\)
\(314\) 0 0
\(315\) −2.21432 + 0.311108i −0.124763 + 0.0175289i
\(316\) 0 0
\(317\) 15.9684i 0.896872i 0.893815 + 0.448436i \(0.148019\pi\)
−0.893815 + 0.448436i \(0.851981\pi\)
\(318\) 0 0
\(319\) 36.5906 2.04868
\(320\) 0 0
\(321\) −11.0923 −0.619114
\(322\) 0 0
\(323\) 2.75557i 0.153324i
\(324\) 0 0
\(325\) 0.857279 + 2.99063i 0.0475533 + 0.165890i
\(326\) 0 0
\(327\) 5.61285i 0.310391i
\(328\) 0 0
\(329\) −11.6128 −0.640237
\(330\) 0 0
\(331\) −27.6128 −1.51774 −0.758870 0.651243i \(-0.774248\pi\)
−0.758870 + 0.651243i \(0.774248\pi\)
\(332\) 0 0
\(333\) 1.24443i 0.0681944i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.0000i 0.871576i 0.900049 + 0.435788i \(0.143530\pi\)
−0.900049 + 0.435788i \(0.856470\pi\)
\(338\) 0 0
\(339\) −16.2351 −0.881768
\(340\) 0 0
\(341\) 2.36842 0.128257
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0.815792 + 5.80642i 0.0439208 + 0.312607i
\(346\) 0 0
\(347\) 11.8666i 0.637035i −0.947917 0.318517i \(-0.896815\pi\)
0.947917 0.318517i \(-0.103185\pi\)
\(348\) 0 0
\(349\) −21.8163 −1.16780 −0.583899 0.811826i \(-0.698474\pi\)
−0.583899 + 0.811826i \(0.698474\pi\)
\(350\) 0 0
\(351\) 0.622216 0.0332114
\(352\) 0 0
\(353\) 2.79706i 0.148872i 0.997226 + 0.0744361i \(0.0237157\pi\)
−0.997226 + 0.0744361i \(0.976284\pi\)
\(354\) 0 0
\(355\) −0.797056 5.67307i −0.0423033 0.301095i
\(356\) 0 0
\(357\) 4.42864i 0.234388i
\(358\) 0 0
\(359\) 13.0509 0.688798 0.344399 0.938823i \(-0.388083\pi\)
0.344399 + 0.938823i \(0.388083\pi\)
\(360\) 0 0
\(361\) −18.6128 −0.979624
\(362\) 0 0
\(363\) 3.48886i 0.183118i
\(364\) 0 0
\(365\) −24.3368 + 3.41927i −1.27384 + 0.178973i
\(366\) 0 0
\(367\) 10.4889i 0.547514i −0.961799 0.273757i \(-0.911734\pi\)
0.961799 0.273757i \(-0.0882664\pi\)
\(368\) 0 0
\(369\) −4.62222 −0.240623
\(370\) 0 0
\(371\) 13.4795 0.699820
\(372\) 0 0
\(373\) 30.1847i 1.56290i −0.623966 0.781452i \(-0.714479\pi\)
0.623966 0.781452i \(-0.285521\pi\)
\(374\) 0 0
\(375\) 10.2143 4.54617i 0.527465 0.234763i
\(376\) 0 0
\(377\) 5.98126i 0.308051i
\(378\) 0 0
\(379\) 12.8573 0.660434 0.330217 0.943905i \(-0.392878\pi\)
0.330217 + 0.943905i \(0.392878\pi\)
\(380\) 0 0
\(381\) 15.3461 0.786207
\(382\) 0 0
\(383\) 24.4701i 1.25037i 0.780479 + 0.625183i \(0.214975\pi\)
−0.780479 + 0.625183i \(0.785025\pi\)
\(384\) 0 0
\(385\) −8.42864 + 1.18421i −0.429563 + 0.0603528i
\(386\) 0 0
\(387\) 4.85728i 0.246909i
\(388\) 0 0
\(389\) 1.61285 0.0817746 0.0408873 0.999164i \(-0.486982\pi\)
0.0408873 + 0.999164i \(0.486982\pi\)
\(390\) 0 0
\(391\) −11.6128 −0.587287
\(392\) 0 0
\(393\) 4.00000i 0.201773i
\(394\) 0 0
\(395\) −2.10171 14.9590i −0.105749 0.752668i
\(396\) 0 0
\(397\) 22.2163i 1.11501i 0.830175 + 0.557503i \(0.188240\pi\)
−0.830175 + 0.557503i \(0.811760\pi\)
\(398\) 0 0
\(399\) −0.622216 −0.0311497
\(400\) 0 0
\(401\) 19.9813 0.997817 0.498908 0.866655i \(-0.333734\pi\)
0.498908 + 0.866655i \(0.333734\pi\)
\(402\) 0 0
\(403\) 0.387152i 0.0192854i
\(404\) 0 0
\(405\) −0.311108 2.21432i −0.0154591 0.110030i
\(406\) 0 0
\(407\) 4.73683i 0.234796i
\(408\) 0 0
\(409\) 5.73329 0.283493 0.141747 0.989903i \(-0.454728\pi\)
0.141747 + 0.989903i \(0.454728\pi\)
\(410\) 0 0
\(411\) −17.0923 −0.843103
\(412\) 0 0
\(413\) 11.6128i 0.571431i
\(414\) 0 0
\(415\) −25.7146 + 3.61285i −1.26228 + 0.177348i
\(416\) 0 0
\(417\) 13.4795i 0.660094i
\(418\) 0 0
\(419\) −26.3684 −1.28818 −0.644091 0.764949i \(-0.722764\pi\)
−0.644091 + 0.764949i \(0.722764\pi\)
\(420\) 0 0
\(421\) −19.3274 −0.941960 −0.470980 0.882144i \(-0.656100\pi\)
−0.470980 + 0.882144i \(0.656100\pi\)
\(422\) 0 0
\(423\) 11.6128i 0.564636i
\(424\) 0 0
\(425\) 6.10171 + 21.2859i 0.295976 + 1.03252i
\(426\) 0 0
\(427\) 8.10171i 0.392069i
\(428\) 0 0
\(429\) 2.36842 0.114348
\(430\) 0 0
\(431\) −26.9491 −1.29809 −0.649047 0.760748i \(-0.724832\pi\)
−0.649047 + 0.760748i \(0.724832\pi\)
\(432\) 0 0
\(433\) 2.13335i 0.102522i 0.998685 + 0.0512612i \(0.0163241\pi\)
−0.998685 + 0.0512612i \(0.983676\pi\)
\(434\) 0 0
\(435\) 21.2859 2.99063i 1.02058 0.143390i
\(436\) 0 0
\(437\) 1.63158i 0.0780492i
\(438\) 0 0
\(439\) 10.5205 0.502116 0.251058 0.967972i \(-0.419221\pi\)
0.251058 + 0.967972i \(0.419221\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 6.88892i 0.327303i 0.986518 + 0.163651i \(0.0523272\pi\)
−0.986518 + 0.163651i \(0.947673\pi\)
\(444\) 0 0
\(445\) −2.56199 18.2351i −0.121450 0.864425i
\(446\) 0 0
\(447\) 9.34614i 0.442057i
\(448\) 0 0
\(449\) −39.9180 −1.88385 −0.941923 0.335829i \(-0.890984\pi\)
−0.941923 + 0.335829i \(0.890984\pi\)
\(450\) 0 0
\(451\) −17.5941 −0.828474
\(452\) 0 0
\(453\) 7.14272i 0.335594i
\(454\) 0 0
\(455\) −0.193576 1.37778i −0.00907499 0.0645915i
\(456\) 0 0
\(457\) 8.47013i 0.396216i −0.980180 0.198108i \(-0.936520\pi\)
0.980180 0.198108i \(-0.0634796\pi\)
\(458\) 0 0
\(459\) 4.42864 0.206711
\(460\) 0 0
\(461\) 40.1146 1.86832 0.934162 0.356849i \(-0.116149\pi\)
0.934162 + 0.356849i \(0.116149\pi\)
\(462\) 0 0
\(463\) 33.5941i 1.56125i −0.624999 0.780625i \(-0.714901\pi\)
0.624999 0.780625i \(-0.285099\pi\)
\(464\) 0 0
\(465\) 1.37778 0.193576i 0.0638932 0.00897688i
\(466\) 0 0
\(467\) 11.3461i 0.525037i 0.964927 + 0.262518i \(0.0845530\pi\)
−0.964927 + 0.262518i \(0.915447\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −6.99063 −0.322111
\(472\) 0 0
\(473\) 18.4889i 0.850119i
\(474\) 0 0
\(475\) 2.99063 0.857279i 0.137220 0.0393347i
\(476\) 0 0
\(477\) 13.4795i 0.617184i
\(478\) 0 0
\(479\) −36.2864 −1.65797 −0.828984 0.559273i \(-0.811081\pi\)
−0.828984 + 0.559273i \(0.811081\pi\)
\(480\) 0 0
\(481\) −0.774305 −0.0353053
\(482\) 0 0
\(483\) 2.62222i 0.119315i
\(484\) 0 0
\(485\) 9.37778 1.31756i 0.425823 0.0598274i
\(486\) 0 0
\(487\) 38.8385i 1.75994i −0.475027 0.879971i \(-0.657562\pi\)
0.475027 0.879971i \(-0.342438\pi\)
\(488\) 0 0
\(489\) −15.6128 −0.706037
\(490\) 0 0
\(491\) −28.7467 −1.29732 −0.648660 0.761079i \(-0.724670\pi\)
−0.648660 + 0.761079i \(0.724670\pi\)
\(492\) 0 0
\(493\) 42.5718i 1.91734i
\(494\) 0 0
\(495\) −1.18421 8.42864i −0.0532262 0.378839i
\(496\) 0 0
\(497\) 2.56199i 0.114921i
\(498\) 0 0
\(499\) 5.63158 0.252104 0.126052 0.992024i \(-0.459769\pi\)
0.126052 + 0.992024i \(0.459769\pi\)
\(500\) 0 0
\(501\) −1.51114 −0.0675126
\(502\) 0 0
\(503\) 34.9590i 1.55874i 0.626561 + 0.779372i \(0.284462\pi\)
−0.626561 + 0.779372i \(0.715538\pi\)
\(504\) 0 0
\(505\) −5.82516 41.4608i −0.259216 1.84498i
\(506\) 0 0
\(507\) 12.6128i 0.560156i
\(508\) 0 0
\(509\) 10.9906 0.487151 0.243576 0.969882i \(-0.421680\pi\)
0.243576 + 0.969882i \(0.421680\pi\)
\(510\) 0 0
\(511\) 10.9906 0.486197
\(512\) 0 0
\(513\) 0.622216i 0.0274715i
\(514\) 0 0
\(515\) 1.89829 0.266706i 0.0836486 0.0117525i
\(516\) 0 0
\(517\) 44.2034i 1.94406i
\(518\) 0 0
\(519\) 6.53035 0.286651
\(520\) 0 0
\(521\) 6.90766 0.302630 0.151315 0.988486i \(-0.451649\pi\)
0.151315 + 0.988486i \(0.451649\pi\)
\(522\) 0 0
\(523\) 37.7146i 1.64914i −0.565758 0.824571i \(-0.691416\pi\)
0.565758 0.824571i \(-0.308584\pi\)
\(524\) 0 0
\(525\) −4.80642 + 1.37778i −0.209770 + 0.0601314i
\(526\) 0 0
\(527\) 2.75557i 0.120034i
\(528\) 0 0
\(529\) 16.1240 0.701043
\(530\) 0 0
\(531\) 11.6128 0.503955
\(532\) 0 0
\(533\) 2.87601i 0.124574i
\(534\) 0 0
\(535\) −24.5620 + 3.45091i −1.06191 + 0.149196i
\(536\) 0 0
\(537\) 6.29529i 0.271662i
\(538\) 0 0
\(539\) 3.80642 0.163954
\(540\) 0 0
\(541\) −3.12399 −0.134311 −0.0671553 0.997743i \(-0.521392\pi\)
−0.0671553 + 0.997743i \(0.521392\pi\)
\(542\) 0 0
\(543\) 6.85728i 0.294274i
\(544\) 0 0
\(545\) 1.74620 + 12.4286i 0.0747990 + 0.532384i
\(546\) 0 0
\(547\) 5.51114i 0.235639i −0.993035 0.117820i \(-0.962410\pi\)
0.993035 0.117820i \(-0.0375905\pi\)
\(548\) 0 0
\(549\) 8.10171 0.345773
\(550\) 0 0
\(551\) 5.98126 0.254810
\(552\) 0 0
\(553\) 6.75557i 0.287276i
\(554\) 0 0
\(555\) 0.387152 + 2.75557i 0.0164337 + 0.116967i
\(556\) 0 0
\(557\) 36.7052i 1.55525i 0.628729 + 0.777624i \(0.283575\pi\)
−0.628729 + 0.777624i \(0.716425\pi\)
\(558\) 0 0
\(559\) 3.02227 0.127829
\(560\) 0 0
\(561\) 16.8573 0.711715
\(562\) 0 0
\(563\) 27.4924i 1.15867i 0.815091 + 0.579333i \(0.196687\pi\)
−0.815091 + 0.579333i \(0.803313\pi\)
\(564\) 0 0
\(565\) −35.9496 + 5.05086i −1.51241 + 0.212491i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) −23.2444 −0.974457 −0.487229 0.873274i \(-0.661992\pi\)
−0.487229 + 0.873274i \(0.661992\pi\)
\(570\) 0 0
\(571\) −25.5111 −1.06761 −0.533804 0.845608i \(-0.679238\pi\)
−0.533804 + 0.845608i \(0.679238\pi\)
\(572\) 0 0
\(573\) 10.5620i 0.441234i
\(574\) 0 0
\(575\) 3.61285 + 12.6035i 0.150666 + 0.525601i
\(576\) 0 0
\(577\) 26.0701i 1.08531i 0.839955 + 0.542656i \(0.182581\pi\)
−0.839955 + 0.542656i \(0.817419\pi\)
\(578\) 0 0
\(579\) −5.24443 −0.217951
\(580\) 0 0
\(581\) 11.6128 0.481782
\(582\) 0 0
\(583\) 51.3087i 2.12499i
\(584\) 0 0
\(585\) 1.37778 0.193576i 0.0569644 0.00800339i
\(586\) 0 0
\(587\) 12.2667i 0.506301i −0.967427 0.253151i \(-0.918533\pi\)
0.967427 0.253151i \(-0.0814668\pi\)
\(588\) 0 0
\(589\) 0.387152 0.0159523
\(590\) 0 0
\(591\) 17.7462 0.729981
\(592\) 0 0
\(593\) 14.9175i 0.612588i 0.951937 + 0.306294i \(0.0990891\pi\)
−0.951937 + 0.306294i \(0.900911\pi\)
\(594\) 0 0
\(595\) −1.37778 9.80642i −0.0564837 0.402024i
\(596\) 0 0
\(597\) 20.2351i 0.828166i
\(598\) 0 0
\(599\) 26.5620 1.08529 0.542647 0.839961i \(-0.317422\pi\)
0.542647 + 0.839961i \(0.317422\pi\)
\(600\) 0 0
\(601\) 39.7146 1.61999 0.809995 0.586436i \(-0.199470\pi\)
0.809995 + 0.586436i \(0.199470\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.08541 7.72546i −0.0441283 0.314084i
\(606\) 0 0
\(607\) 6.28544i 0.255118i 0.991831 + 0.127559i \(0.0407143\pi\)
−0.991831 + 0.127559i \(0.959286\pi\)
\(608\) 0 0
\(609\) −9.61285 −0.389532
\(610\) 0 0
\(611\) 7.22570 0.292320
\(612\) 0 0
\(613\) 45.7146i 1.84639i 0.384328 + 0.923197i \(0.374433\pi\)
−0.384328 + 0.923197i \(0.625567\pi\)
\(614\) 0 0
\(615\) −10.2351 + 1.43801i −0.412718 + 0.0579861i
\(616\) 0 0
\(617\) 4.88892i 0.196821i 0.995146 + 0.0984103i \(0.0313758\pi\)
−0.995146 + 0.0984103i \(0.968624\pi\)
\(618\) 0 0
\(619\) −12.2351 −0.491769 −0.245884 0.969299i \(-0.579078\pi\)
−0.245884 + 0.969299i \(0.579078\pi\)
\(620\) 0 0
\(621\) 2.62222 0.105226
\(622\) 0 0
\(623\) 8.23506i 0.329931i
\(624\) 0 0
\(625\) 21.2034 13.2444i 0.848137 0.529777i
\(626\) 0 0
\(627\) 2.36842i 0.0945854i
\(628\) 0 0
\(629\) −5.51114 −0.219743
\(630\) 0 0
\(631\) 1.24443 0.0495400 0.0247700 0.999693i \(-0.492115\pi\)
0.0247700 + 0.999693i \(0.492115\pi\)
\(632\) 0 0
\(633\) 21.3274i 0.847688i
\(634\) 0 0
\(635\) 33.9813 4.77430i 1.34851 0.189462i
\(636\) 0 0
\(637\) 0.622216i 0.0246531i
\(638\) 0 0
\(639\) −2.56199 −0.101351
\(640\) 0 0
\(641\) −48.1847 −1.90318 −0.951590 0.307369i \(-0.900551\pi\)
−0.951590 + 0.307369i \(0.900551\pi\)
\(642\) 0 0
\(643\) 4.85728i 0.191552i −0.995403 0.0957762i \(-0.969467\pi\)
0.995403 0.0957762i \(-0.0305333\pi\)
\(644\) 0 0
\(645\) −1.51114 10.7556i −0.0595010 0.423500i
\(646\) 0 0
\(647\) 0.203420i 0.00799728i 0.999992 + 0.00399864i \(0.00127281\pi\)
−0.999992 + 0.00399864i \(0.998727\pi\)
\(648\) 0 0
\(649\) 44.2034 1.73514
\(650\) 0 0
\(651\) −0.622216 −0.0243866
\(652\) 0 0
\(653\) 27.3145i 1.06890i 0.845200 + 0.534449i \(0.179481\pi\)
−0.845200 + 0.534449i \(0.820519\pi\)
\(654\) 0 0
\(655\) 1.24443 + 8.85728i 0.0486240 + 0.346083i
\(656\) 0 0
\(657\) 10.9906i 0.428785i
\(658\) 0 0
\(659\) −33.3176 −1.29787 −0.648934 0.760845i \(-0.724785\pi\)
−0.648934 + 0.760845i \(0.724785\pi\)
\(660\) 0 0
\(661\) 14.5906 0.567508 0.283754 0.958897i \(-0.408420\pi\)
0.283754 + 0.958897i \(0.408420\pi\)
\(662\) 0 0
\(663\) 2.75557i 0.107017i
\(664\) 0 0
\(665\) −1.37778 + 0.193576i −0.0534282 + 0.00750656i
\(666\) 0 0
\(667\) 25.2070i 0.976017i
\(668\) 0 0
\(669\) 9.71456 0.375587
\(670\) 0 0
\(671\) 30.8385 1.19051
\(672\) 0 0
\(673\) 4.53341i 0.174750i −0.996175 0.0873751i \(-0.972152\pi\)
0.996175 0.0873751i \(-0.0278479\pi\)
\(674\) 0 0
\(675\) −1.37778 4.80642i −0.0530309 0.184999i
\(676\) 0 0
\(677\) 27.2672i 1.04796i −0.851730 0.523981i \(-0.824446\pi\)
0.851730 0.523981i \(-0.175554\pi\)
\(678\) 0 0
\(679\) −4.23506 −0.162527
\(680\) 0 0
\(681\) −11.3461 −0.434785
\(682\) 0 0
\(683\) 29.5812i 1.13189i 0.824442 + 0.565947i \(0.191489\pi\)
−0.824442 + 0.565947i \(0.808511\pi\)
\(684\) 0 0
\(685\) −37.8479 + 5.31756i −1.44609 + 0.203174i
\(686\) 0 0
\(687\) 1.34614i 0.0513585i
\(688\) 0 0
\(689\) −8.38715 −0.319525
\(690\) 0 0
\(691\) 2.99063 0.113769 0.0568845 0.998381i \(-0.481883\pi\)
0.0568845 + 0.998381i \(0.481883\pi\)
\(692\) 0 0
\(693\) 3.80642i 0.144594i
\(694\) 0 0
\(695\) −4.19358 29.8479i −0.159071 1.13220i
\(696\) 0 0
\(697\) 20.4701i 0.775361i
\(698\) 0 0
\(699\) −15.3778 −0.581641
\(700\) 0 0
\(701\) −41.0420 −1.55013 −0.775067 0.631879i \(-0.782284\pi\)
−0.775067 + 0.631879i \(0.782284\pi\)
\(702\) 0 0
\(703\) 0.774305i 0.0292035i
\(704\) 0 0
\(705\) −3.61285 25.7146i −0.136068 0.968466i
\(706\) 0 0
\(707\) 18.7239i 0.704186i
\(708\) 0 0
\(709\) 41.4291 1.55590 0.777952 0.628324i \(-0.216259\pi\)
0.777952 + 0.628324i \(0.216259\pi\)
\(710\) 0 0
\(711\) −6.75557 −0.253354
\(712\) 0 0
\(713\) 1.63158i 0.0611033i
\(714\) 0 0
\(715\) 5.24443 0.736833i 0.196131 0.0275560i
\(716\) 0 0
\(717\) 7.53972i 0.281576i
\(718\) 0 0
\(719\) 18.9590 0.707051 0.353525 0.935425i \(-0.384983\pi\)
0.353525 + 0.935425i \(0.384983\pi\)
\(720\) 0 0
\(721\) −0.857279 −0.0319267
\(722\) 0 0
\(723\) 23.9813i 0.891873i
\(724\) 0 0
\(725\) 46.2034 13.2444i 1.71595 0.491886i
\(726\) 0 0
\(727\) 41.7975i 1.55018i 0.631848 + 0.775092i \(0.282297\pi\)
−0.631848 + 0.775092i \(0.717703\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 21.5111 0.795618
\(732\) 0 0
\(733\) 15.3145i 0.565654i −0.959171 0.282827i \(-0.908728\pi\)
0.959171 0.282827i \(-0.0912722\pi\)
\(734\) 0 0
\(735\) 2.21432 0.311108i 0.0816764 0.0114754i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 32.7368 1.20424 0.602122 0.798404i \(-0.294322\pi\)
0.602122 + 0.798404i \(0.294322\pi\)
\(740\) 0 0
\(741\) 0.387152 0.0142224
\(742\) 0 0
\(743\) 37.3778i 1.37126i 0.727951 + 0.685629i \(0.240473\pi\)
−0.727951 + 0.685629i \(0.759527\pi\)
\(744\) 0 0
\(745\) −2.90766 20.6953i −0.106528 0.758219i
\(746\) 0 0
\(747\) 11.6128i 0.424892i
\(748\) 0 0
\(749\) 11.0923 0.405305
\(750\) 0 0
\(751\) 20.3497 0.742570 0.371285 0.928519i \(-0.378917\pi\)
0.371285 + 0.928519i \(0.378917\pi\)
\(752\) 0 0
\(753\) 14.1017i 0.513895i
\(754\) 0 0
\(755\) 2.22216 + 15.8163i 0.0808725 + 0.575613i
\(756\) 0 0
\(757\) 6.95899i 0.252929i −0.991971 0.126464i \(-0.959637\pi\)
0.991971 0.126464i \(-0.0403629\pi\)
\(758\) 0 0
\(759\) 9.98126 0.362297
\(760\) 0 0
\(761\) 48.6419 1.76327 0.881634 0.471934i \(-0.156444\pi\)
0.881634 + 0.471934i \(0.156444\pi\)
\(762\) 0 0
\(763\) 5.61285i 0.203199i
\(764\) 0 0
\(765\) 9.80642 1.37778i 0.354552 0.0498139i
\(766\) 0 0
\(767\) 7.22570i 0.260905i
\(768\) 0 0
\(769\) 24.6923 0.890426 0.445213 0.895425i \(-0.353128\pi\)
0.445213 + 0.895425i \(0.353128\pi\)
\(770\) 0 0
\(771\) 17.0192 0.612932
\(772\) 0 0
\(773\) 36.0415i 1.29632i −0.761503 0.648161i \(-0.775538\pi\)
0.761503 0.648161i \(-0.224462\pi\)
\(774\) 0 0
\(775\) 2.99063 0.857279i 0.107427 0.0307944i
\(776\) 0 0
\(777\) 1.24443i 0.0446437i
\(778\) 0 0
\(779\) −2.87601 −0.103044
\(780\) 0 0
\(781\) −9.75203 −0.348955
\(782\) 0 0
\(783\) 9.61285i 0.343535i
\(784\) 0 0
\(785\) −15.4795 + 2.17484i −0.552487 + 0.0776234i
\(786\) 0 0
\(787\) 32.2034i 1.14793i −0.818881 0.573964i \(-0.805405\pi\)
0.818881 0.573964i \(-0.194595\pi\)
\(788\) 0 0
\(789\) −12.6035 −0.448696
\(790\) 0 0
\(791\) 16.2351 0.577252
\(792\) 0 0
\(793\) 5.04101i 0.179012i
\(794\) 0 0
\(795\) 4.19358 + 29.8479i 0.148731 + 1.05860i
\(796\) 0 0
\(797\) 17.5526i 0.621746i −0.950451 0.310873i \(-0.899379\pi\)
0.950451 0.310873i \(-0.100621\pi\)
\(798\) 0 0
\(799\) 51.4291 1.81943
\(800\) 0 0
\(801\) −8.23506 −0.290972
\(802\)