Properties

Label 160.3.h.b.159.6
Level $160$
Weight $3$
Character 160.159
Analytic conductor $4.360$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 160.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35968422976\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
Defining polynomial: \(x^{6} + 9 x^{4} + 14 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 159.6
Root \(-0.273891i\) of defining polynomial
Character \(\chi\) \(=\) 160.159
Dual form 160.3.h.b.159.5

$q$-expansion

\(f(q)\) \(=\) \(q+5.30219 q^{3} +(-1.54778 + 4.75441i) q^{5} -0.206625 q^{7} +19.1132 q^{9} +O(q^{10})\) \(q+5.30219 q^{3} +(-1.54778 + 4.75441i) q^{5} -0.206625 q^{7} +19.1132 q^{9} +15.0176i q^{11} -11.6999i q^{13} +(-8.20662 + 25.2087i) q^{15} -18.1911i q^{17} -19.3999i q^{19} -1.09556 q^{21} -27.2242 q^{23} +(-20.2087 - 14.7176i) q^{25} +53.6220 q^{27} +44.4175 q^{29} +20.3822i q^{31} +79.6262i q^{33} +(0.319810 - 0.982377i) q^{35} -18.1089i q^{37} -62.0352i q^{39} -32.3043 q^{41} -4.06244 q^{43} +(-29.5830 + 90.8718i) q^{45} +5.37588 q^{47} -48.9573 q^{49} -96.4527i q^{51} -79.1703i q^{53} +(-71.3999 - 23.2440i) q^{55} -102.862i q^{57} +83.3999i q^{59} -36.7486 q^{61} -3.94925 q^{63} +(55.6262 + 18.1089i) q^{65} -4.51518 q^{67} -144.348 q^{69} +41.6530i q^{71} +41.5910i q^{73} +(-107.151 - 78.0352i) q^{75} -3.10301i q^{77} -15.5473i q^{79} +112.295 q^{81} -50.9862 q^{83} +(86.4880 + 28.1559i) q^{85} +235.510 q^{87} +10.8885 q^{89} +2.41749i q^{91} +108.070i q^{93} +(92.2349 + 30.0268i) q^{95} -12.1559i q^{97} +287.035i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{3} - 2q^{5} + 12q^{7} + 18q^{9} + O(q^{10}) \) \( 6q + 4q^{3} - 2q^{5} + 12q^{7} + 18q^{9} - 36q^{15} + 8q^{21} - 68q^{23} - 10q^{25} + 184q^{27} + 44q^{29} - 108q^{35} - 68q^{41} - 76q^{43} - 6q^{45} + 268q^{47} - 62q^{49} - 288q^{55} - 100q^{61} - 172q^{63} + 308q^{67} - 184q^{69} - 284q^{75} + 238q^{81} - 204q^{83} - 32q^{85} + 584q^{87} + 76q^{89} - 32q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 5.30219 1.76740 0.883698 0.468058i \(-0.155046\pi\)
0.883698 + 0.468058i \(0.155046\pi\)
\(4\) 0 0
\(5\) −1.54778 + 4.75441i −0.309556 + 0.950881i
\(6\) 0 0
\(7\) −0.206625 −0.0295178 −0.0147589 0.999891i \(-0.504698\pi\)
−0.0147589 + 0.999891i \(0.504698\pi\)
\(8\) 0 0
\(9\) 19.1132 2.12369
\(10\) 0 0
\(11\) 15.0176i 1.36524i 0.730774 + 0.682619i \(0.239159\pi\)
−0.730774 + 0.682619i \(0.760841\pi\)
\(12\) 0 0
\(13\) 11.6999i 0.899995i −0.893030 0.449998i \(-0.851425\pi\)
0.893030 0.449998i \(-0.148575\pi\)
\(14\) 0 0
\(15\) −8.20662 + 25.2087i −0.547108 + 1.68058i
\(16\) 0 0
\(17\) 18.1911i 1.07007i −0.844831 0.535033i \(-0.820299\pi\)
0.844831 0.535033i \(-0.179701\pi\)
\(18\) 0 0
\(19\) 19.3999i 1.02105i −0.859864 0.510523i \(-0.829452\pi\)
0.859864 0.510523i \(-0.170548\pi\)
\(20\) 0 0
\(21\) −1.09556 −0.0521696
\(22\) 0 0
\(23\) −27.2242 −1.18366 −0.591831 0.806062i \(-0.701595\pi\)
−0.591831 + 0.806062i \(0.701595\pi\)
\(24\) 0 0
\(25\) −20.2087 14.7176i −0.808350 0.588702i
\(26\) 0 0
\(27\) 53.6220 1.98600
\(28\) 0 0
\(29\) 44.4175 1.53164 0.765819 0.643056i \(-0.222334\pi\)
0.765819 + 0.643056i \(0.222334\pi\)
\(30\) 0 0
\(31\) 20.3822i 0.657492i 0.944418 + 0.328746i \(0.106626\pi\)
−0.944418 + 0.328746i \(0.893374\pi\)
\(32\) 0 0
\(33\) 79.6262i 2.41292i
\(34\) 0 0
\(35\) 0.319810 0.982377i 0.00913742 0.0280679i
\(36\) 0 0
\(37\) 18.1089i 0.489431i −0.969595 0.244715i \(-0.921305\pi\)
0.969595 0.244715i \(-0.0786945\pi\)
\(38\) 0 0
\(39\) 62.0352i 1.59065i
\(40\) 0 0
\(41\) −32.3043 −0.787910 −0.393955 0.919130i \(-0.628893\pi\)
−0.393955 + 0.919130i \(0.628893\pi\)
\(42\) 0 0
\(43\) −4.06244 −0.0944753 −0.0472377 0.998884i \(-0.515042\pi\)
−0.0472377 + 0.998884i \(0.515042\pi\)
\(44\) 0 0
\(45\) −29.5830 + 90.8718i −0.657401 + 2.01937i
\(46\) 0 0
\(47\) 5.37588 0.114380 0.0571902 0.998363i \(-0.481786\pi\)
0.0571902 + 0.998363i \(0.481786\pi\)
\(48\) 0 0
\(49\) −48.9573 −0.999129
\(50\) 0 0
\(51\) 96.4527i 1.89123i
\(52\) 0 0
\(53\) 79.1703i 1.49378i −0.664948 0.746890i \(-0.731546\pi\)
0.664948 0.746890i \(-0.268454\pi\)
\(54\) 0 0
\(55\) −71.3999 23.2440i −1.29818 0.422618i
\(56\) 0 0
\(57\) 102.862i 1.80459i
\(58\) 0 0
\(59\) 83.3999i 1.41356i 0.707435 + 0.706779i \(0.249852\pi\)
−0.707435 + 0.706779i \(0.750148\pi\)
\(60\) 0 0
\(61\) −36.7486 −0.602435 −0.301218 0.953555i \(-0.597393\pi\)
−0.301218 + 0.953555i \(0.597393\pi\)
\(62\) 0 0
\(63\) −3.94925 −0.0626866
\(64\) 0 0
\(65\) 55.6262 + 18.1089i 0.855788 + 0.278599i
\(66\) 0 0
\(67\) −4.51518 −0.0673907 −0.0336954 0.999432i \(-0.510728\pi\)
−0.0336954 + 0.999432i \(0.510728\pi\)
\(68\) 0 0
\(69\) −144.348 −2.09200
\(70\) 0 0
\(71\) 41.6530i 0.586662i 0.956011 + 0.293331i \(0.0947638\pi\)
−0.956011 + 0.293331i \(0.905236\pi\)
\(72\) 0 0
\(73\) 41.5910i 0.569740i 0.958566 + 0.284870i \(0.0919504\pi\)
−0.958566 + 0.284870i \(0.908050\pi\)
\(74\) 0 0
\(75\) −107.151 78.0352i −1.42867 1.04047i
\(76\) 0 0
\(77\) 3.10301i 0.0402988i
\(78\) 0 0
\(79\) 15.5473i 0.196801i −0.995147 0.0984004i \(-0.968627\pi\)
0.995147 0.0984004i \(-0.0313726\pi\)
\(80\) 0 0
\(81\) 112.295 1.38636
\(82\) 0 0
\(83\) −50.9862 −0.614291 −0.307146 0.951663i \(-0.599374\pi\)
−0.307146 + 0.951663i \(0.599374\pi\)
\(84\) 0 0
\(85\) 86.4880 + 28.1559i 1.01751 + 0.331246i
\(86\) 0 0
\(87\) 235.510 2.70701
\(88\) 0 0
\(89\) 10.8885 0.122343 0.0611713 0.998127i \(-0.480516\pi\)
0.0611713 + 0.998127i \(0.480516\pi\)
\(90\) 0 0
\(91\) 2.41749i 0.0265659i
\(92\) 0 0
\(93\) 108.070i 1.16205i
\(94\) 0 0
\(95\) 92.2349 + 30.0268i 0.970893 + 0.316071i
\(96\) 0 0
\(97\) 12.1559i 0.125318i −0.998035 0.0626592i \(-0.980042\pi\)
0.998035 0.0626592i \(-0.0199581\pi\)
\(98\) 0 0
\(99\) 287.035i 2.89934i
\(100\) 0 0
\(101\) 127.723 1.26459 0.632294 0.774728i \(-0.282113\pi\)
0.632294 + 0.774728i \(0.282113\pi\)
\(102\) 0 0
\(103\) 4.77575 0.0463665 0.0231833 0.999731i \(-0.492620\pi\)
0.0231833 + 0.999731i \(0.492620\pi\)
\(104\) 0 0
\(105\) 1.69569 5.20875i 0.0161494 0.0496071i
\(106\) 0 0
\(107\) 107.213 1.00199 0.500997 0.865449i \(-0.332967\pi\)
0.500997 + 0.865449i \(0.332967\pi\)
\(108\) 0 0
\(109\) 53.6689 0.492376 0.246188 0.969222i \(-0.420822\pi\)
0.246188 + 0.969222i \(0.420822\pi\)
\(110\) 0 0
\(111\) 96.0170i 0.865018i
\(112\) 0 0
\(113\) 20.5063i 0.181471i 0.995875 + 0.0907356i \(0.0289218\pi\)
−0.995875 + 0.0907356i \(0.971078\pi\)
\(114\) 0 0
\(115\) 42.1372 129.435i 0.366410 1.12552i
\(116\) 0 0
\(117\) 223.623i 1.91131i
\(118\) 0 0
\(119\) 3.75873i 0.0315860i
\(120\) 0 0
\(121\) −104.529 −0.863876
\(122\) 0 0
\(123\) −171.283 −1.39255
\(124\) 0 0
\(125\) 101.252 73.3010i 0.810016 0.586408i
\(126\) 0 0
\(127\) −138.477 −1.09037 −0.545184 0.838316i \(-0.683540\pi\)
−0.545184 + 0.838316i \(0.683540\pi\)
\(128\) 0 0
\(129\) −21.5398 −0.166975
\(130\) 0 0
\(131\) 219.105i 1.67256i 0.548304 + 0.836279i \(0.315274\pi\)
−0.548304 + 0.836279i \(0.684726\pi\)
\(132\) 0 0
\(133\) 4.00849i 0.0301390i
\(134\) 0 0
\(135\) −82.9951 + 254.941i −0.614779 + 1.88845i
\(136\) 0 0
\(137\) 59.7821i 0.436366i −0.975908 0.218183i \(-0.929987\pi\)
0.975908 0.218183i \(-0.0700129\pi\)
\(138\) 0 0
\(139\) 26.6524i 0.191744i −0.995394 0.0958718i \(-0.969436\pi\)
0.995394 0.0958718i \(-0.0305639\pi\)
\(140\) 0 0
\(141\) 28.5039 0.202155
\(142\) 0 0
\(143\) 175.705 1.22871
\(144\) 0 0
\(145\) −68.7486 + 211.179i −0.474128 + 1.45641i
\(146\) 0 0
\(147\) −259.581 −1.76586
\(148\) 0 0
\(149\) 143.463 0.962838 0.481419 0.876490i \(-0.340121\pi\)
0.481419 + 0.876490i \(0.340121\pi\)
\(150\) 0 0
\(151\) 83.4937i 0.552939i 0.961023 + 0.276469i \(0.0891644\pi\)
−0.961023 + 0.276469i \(0.910836\pi\)
\(152\) 0 0
\(153\) 347.690i 2.27249i
\(154\) 0 0
\(155\) −96.9055 31.5473i −0.625197 0.203531i
\(156\) 0 0
\(157\) 169.673i 1.08072i 0.841434 + 0.540360i \(0.181712\pi\)
−0.841434 + 0.540360i \(0.818288\pi\)
\(158\) 0 0
\(159\) 419.776i 2.64010i
\(160\) 0 0
\(161\) 5.62520 0.0349391
\(162\) 0 0
\(163\) 275.478 1.69005 0.845026 0.534726i \(-0.179585\pi\)
0.845026 + 0.534726i \(0.179585\pi\)
\(164\) 0 0
\(165\) −378.575 123.244i −2.29440 0.746933i
\(166\) 0 0
\(167\) 132.481 0.793299 0.396650 0.917970i \(-0.370173\pi\)
0.396650 + 0.917970i \(0.370173\pi\)
\(168\) 0 0
\(169\) 32.1115 0.190009
\(170\) 0 0
\(171\) 370.793i 2.16838i
\(172\) 0 0
\(173\) 272.614i 1.57580i 0.615801 + 0.787901i \(0.288832\pi\)
−0.615801 + 0.787901i \(0.711168\pi\)
\(174\) 0 0
\(175\) 4.17562 + 3.04101i 0.0238607 + 0.0173772i
\(176\) 0 0
\(177\) 442.202i 2.49831i
\(178\) 0 0
\(179\) 157.523i 0.880014i −0.897994 0.440007i \(-0.854976\pi\)
0.897994 0.440007i \(-0.145024\pi\)
\(180\) 0 0
\(181\) −335.063 −1.85118 −0.925590 0.378529i \(-0.876430\pi\)
−0.925590 + 0.378529i \(0.876430\pi\)
\(182\) 0 0
\(183\) −194.848 −1.06474
\(184\) 0 0
\(185\) 86.0972 + 28.0287i 0.465391 + 0.151506i
\(186\) 0 0
\(187\) 273.187 1.46090
\(188\) 0 0
\(189\) −11.0796 −0.0586223
\(190\) 0 0
\(191\) 298.575i 1.56322i 0.623766 + 0.781611i \(0.285602\pi\)
−0.623766 + 0.781611i \(0.714398\pi\)
\(192\) 0 0
\(193\) 191.915i 0.994376i −0.867643 0.497188i \(-0.834366\pi\)
0.867643 0.497188i \(-0.165634\pi\)
\(194\) 0 0
\(195\) 294.941 + 96.0170i 1.51252 + 0.492395i
\(196\) 0 0
\(197\) 59.2472i 0.300747i 0.988629 + 0.150374i \(0.0480477\pi\)
−0.988629 + 0.150374i \(0.951952\pi\)
\(198\) 0 0
\(199\) 309.100i 1.55327i −0.629953 0.776633i \(-0.716926\pi\)
0.629953 0.776633i \(-0.283074\pi\)
\(200\) 0 0
\(201\) −23.9403 −0.119106
\(202\) 0 0
\(203\) −9.17775 −0.0452106
\(204\) 0 0
\(205\) 50.0000 153.588i 0.243902 0.749209i
\(206\) 0 0
\(207\) −520.342 −2.51373
\(208\) 0 0
\(209\) 291.340 1.39397
\(210\) 0 0
\(211\) 205.693i 0.974850i −0.873165 0.487425i \(-0.837936\pi\)
0.873165 0.487425i \(-0.162064\pi\)
\(212\) 0 0
\(213\) 220.852i 1.03686i
\(214\) 0 0
\(215\) 6.28777 19.3145i 0.0292454 0.0898348i
\(216\) 0 0
\(217\) 4.21147i 0.0194077i
\(218\) 0 0
\(219\) 220.523i 1.00696i
\(220\) 0 0
\(221\) −212.835 −0.963054
\(222\) 0 0
\(223\) −228.723 −1.02567 −0.512833 0.858488i \(-0.671404\pi\)
−0.512833 + 0.858488i \(0.671404\pi\)
\(224\) 0 0
\(225\) −386.254 281.299i −1.71668 1.25022i
\(226\) 0 0
\(227\) −282.403 −1.24407 −0.622033 0.782991i \(-0.713693\pi\)
−0.622033 + 0.782991i \(0.713693\pi\)
\(228\) 0 0
\(229\) 49.2525 0.215076 0.107538 0.994201i \(-0.465703\pi\)
0.107538 + 0.994201i \(0.465703\pi\)
\(230\) 0 0
\(231\) 16.4527i 0.0712240i
\(232\) 0 0
\(233\) 124.273i 0.533363i 0.963785 + 0.266681i \(0.0859271\pi\)
−0.963785 + 0.266681i \(0.914073\pi\)
\(234\) 0 0
\(235\) −8.32069 + 25.5591i −0.0354072 + 0.108762i
\(236\) 0 0
\(237\) 82.4345i 0.347825i
\(238\) 0 0
\(239\) 80.4527i 0.336622i 0.985734 + 0.168311i \(0.0538313\pi\)
−0.985734 + 0.168311i \(0.946169\pi\)
\(240\) 0 0
\(241\) −1.20979 −0.00501988 −0.00250994 0.999997i \(-0.500799\pi\)
−0.00250994 + 0.999997i \(0.500799\pi\)
\(242\) 0 0
\(243\) 112.812 0.464247
\(244\) 0 0
\(245\) 75.7752 232.763i 0.309287 0.950053i
\(246\) 0 0
\(247\) −226.977 −0.918936
\(248\) 0 0
\(249\) −270.338 −1.08570
\(250\) 0 0
\(251\) 211.853i 0.844034i 0.906588 + 0.422017i \(0.138678\pi\)
−0.906588 + 0.422017i \(0.861322\pi\)
\(252\) 0 0
\(253\) 408.843i 1.61598i
\(254\) 0 0
\(255\) 458.575 + 149.288i 1.79834 + 0.585442i
\(256\) 0 0
\(257\) 182.646i 0.710685i −0.934736 0.355342i \(-0.884364\pi\)
0.934736 0.355342i \(-0.115636\pi\)
\(258\) 0 0
\(259\) 3.74175i 0.0144469i
\(260\) 0 0
\(261\) 848.960 3.25272
\(262\) 0 0
\(263\) 74.6636 0.283892 0.141946 0.989874i \(-0.454664\pi\)
0.141946 + 0.989874i \(0.454664\pi\)
\(264\) 0 0
\(265\) 376.408 + 122.538i 1.42041 + 0.462409i
\(266\) 0 0
\(267\) 57.7329 0.216228
\(268\) 0 0
\(269\) −184.089 −0.684344 −0.342172 0.939637i \(-0.611163\pi\)
−0.342172 + 0.939637i \(0.611163\pi\)
\(270\) 0 0
\(271\) 234.746i 0.866222i 0.901341 + 0.433111i \(0.142584\pi\)
−0.901341 + 0.433111i \(0.857416\pi\)
\(272\) 0 0
\(273\) 12.8180i 0.0469524i
\(274\) 0 0
\(275\) 221.023 303.487i 0.803719 1.10359i
\(276\) 0 0
\(277\) 452.208i 1.63252i 0.577684 + 0.816260i \(0.303957\pi\)
−0.577684 + 0.816260i \(0.696043\pi\)
\(278\) 0 0
\(279\) 389.570i 1.39631i
\(280\) 0 0
\(281\) −196.110 −0.697900 −0.348950 0.937141i \(-0.613462\pi\)
−0.348950 + 0.937141i \(0.613462\pi\)
\(282\) 0 0
\(283\) 418.449 1.47862 0.739309 0.673366i \(-0.235152\pi\)
0.739309 + 0.673366i \(0.235152\pi\)
\(284\) 0 0
\(285\) 489.046 + 159.207i 1.71595 + 0.558623i
\(286\) 0 0
\(287\) 6.67486 0.0232574
\(288\) 0 0
\(289\) −41.9170 −0.145042
\(290\) 0 0
\(291\) 64.4527i 0.221487i
\(292\) 0 0
\(293\) 286.666i 0.978383i −0.872176 0.489191i \(-0.837292\pi\)
0.872176 0.489191i \(-0.162708\pi\)
\(294\) 0 0
\(295\) −396.517 129.085i −1.34412 0.437575i
\(296\) 0 0
\(297\) 805.275i 2.71136i
\(298\) 0 0
\(299\) 318.522i 1.06529i
\(300\) 0 0
\(301\) 0.839400 0.00278870
\(302\) 0 0
\(303\) 677.214 2.23503
\(304\) 0 0
\(305\) 56.8787 174.718i 0.186488 0.572844i
\(306\) 0 0
\(307\) −261.715 −0.852493 −0.426247 0.904607i \(-0.640164\pi\)
−0.426247 + 0.904607i \(0.640164\pi\)
\(308\) 0 0
\(309\) 25.3219 0.0819480
\(310\) 0 0
\(311\) 578.904i 1.86143i −0.365747 0.930714i \(-0.619187\pi\)
0.365747 0.930714i \(-0.380813\pi\)
\(312\) 0 0
\(313\) 99.3124i 0.317292i −0.987336 0.158646i \(-0.949287\pi\)
0.987336 0.158646i \(-0.0507129\pi\)
\(314\) 0 0
\(315\) 6.11258 18.7764i 0.0194050 0.0596075i
\(316\) 0 0
\(317\) 191.623i 0.604489i 0.953230 + 0.302245i \(0.0977359\pi\)
−0.953230 + 0.302245i \(0.902264\pi\)
\(318\) 0 0
\(319\) 667.045i 2.09105i
\(320\) 0 0
\(321\) 568.466 1.77092
\(322\) 0 0
\(323\) −352.905 −1.09259
\(324\) 0 0
\(325\) −172.194 + 236.441i −0.529829 + 0.727511i
\(326\) 0 0
\(327\) 284.563 0.870222
\(328\) 0 0
\(329\) −1.11079 −0.00337626
\(330\) 0 0
\(331\) 530.187i 1.60177i 0.598816 + 0.800886i \(0.295638\pi\)
−0.598816 + 0.800886i \(0.704362\pi\)
\(332\) 0 0
\(333\) 346.120i 1.03940i
\(334\) 0 0
\(335\) 6.98851 21.4670i 0.0208612 0.0640806i
\(336\) 0 0
\(337\) 487.427i 1.44637i −0.690653 0.723186i \(-0.742677\pi\)
0.690653 0.723186i \(-0.257323\pi\)
\(338\) 0 0
\(339\) 108.728i 0.320731i
\(340\) 0 0
\(341\) −306.093 −0.897633
\(342\) 0 0
\(343\) 20.2404 0.0590099
\(344\) 0 0
\(345\) 223.419 686.289i 0.647592 1.98924i
\(346\) 0 0
\(347\) −310.497 −0.894804 −0.447402 0.894333i \(-0.647651\pi\)
−0.447402 + 0.894333i \(0.647651\pi\)
\(348\) 0 0
\(349\) −253.004 −0.724941 −0.362471 0.931995i \(-0.618067\pi\)
−0.362471 + 0.931995i \(0.618067\pi\)
\(350\) 0 0
\(351\) 627.374i 1.78739i
\(352\) 0 0
\(353\) 322.639i 0.913992i 0.889469 + 0.456996i \(0.151075\pi\)
−0.889469 + 0.456996i \(0.848925\pi\)
\(354\) 0 0
\(355\) −198.035 64.4697i −0.557846 0.181605i
\(356\) 0 0
\(357\) 19.9295i 0.0558250i
\(358\) 0 0
\(359\) 254.975i 0.710236i −0.934822 0.355118i \(-0.884441\pi\)
0.934822 0.355118i \(-0.115559\pi\)
\(360\) 0 0
\(361\) −15.3550 −0.0425347
\(362\) 0 0
\(363\) −554.232 −1.52681
\(364\) 0 0
\(365\) −197.740 64.3738i −0.541755 0.176366i
\(366\) 0 0
\(367\) −207.935 −0.566581 −0.283291 0.959034i \(-0.591426\pi\)
−0.283291 + 0.959034i \(0.591426\pi\)
\(368\) 0 0
\(369\) −617.438 −1.67327
\(370\) 0 0
\(371\) 16.3585i 0.0440931i
\(372\) 0 0
\(373\) 203.826i 0.546450i −0.961950 0.273225i \(-0.911910\pi\)
0.961950 0.273225i \(-0.0880903\pi\)
\(374\) 0 0
\(375\) 536.857 388.656i 1.43162 1.03642i
\(376\) 0 0
\(377\) 519.682i 1.37847i
\(378\) 0 0
\(379\) 454.099i 1.19815i −0.800692 0.599076i \(-0.795535\pi\)
0.800692 0.599076i \(-0.204465\pi\)
\(380\) 0 0
\(381\) −734.229 −1.92711
\(382\) 0 0
\(383\) −541.569 −1.41402 −0.707009 0.707205i \(-0.749956\pi\)
−0.707009 + 0.707205i \(0.749956\pi\)
\(384\) 0 0
\(385\) 14.7530 + 4.80278i 0.0383194 + 0.0124748i
\(386\) 0 0
\(387\) −77.6462 −0.200636
\(388\) 0 0
\(389\) −423.431 −1.08851 −0.544256 0.838919i \(-0.683188\pi\)
−0.544256 + 0.838919i \(0.683188\pi\)
\(390\) 0 0
\(391\) 495.240i 1.26660i
\(392\) 0 0
\(393\) 1161.74i 2.95607i
\(394\) 0 0
\(395\) 73.9180 + 24.0638i 0.187134 + 0.0609209i
\(396\) 0 0
\(397\) 11.7772i 0.0296654i −0.999890 0.0148327i \(-0.995278\pi\)
0.999890 0.0148327i \(-0.00472157\pi\)
\(398\) 0 0
\(399\) 21.2538i 0.0532676i
\(400\) 0 0
\(401\) 127.442 0.317809 0.158905 0.987294i \(-0.449204\pi\)
0.158905 + 0.987294i \(0.449204\pi\)
\(402\) 0 0
\(403\) 238.471 0.591739
\(404\) 0 0
\(405\) −173.808 + 533.897i −0.429156 + 1.31826i
\(406\) 0 0
\(407\) 271.953 0.668190
\(408\) 0 0
\(409\) 608.012 1.48658 0.743290 0.668969i \(-0.233264\pi\)
0.743290 + 0.668969i \(0.233264\pi\)
\(410\) 0 0
\(411\) 316.976i 0.771231i
\(412\) 0 0
\(413\) 17.2325i 0.0417251i
\(414\) 0 0
\(415\) 78.9155 242.409i 0.190158 0.584118i
\(416\) 0 0
\(417\) 141.316i 0.338887i
\(418\) 0 0
\(419\) 565.630i 1.34995i −0.737840 0.674976i \(-0.764154\pi\)
0.737840 0.674976i \(-0.235846\pi\)
\(420\) 0 0
\(421\) 711.356 1.68968 0.844841 0.535018i \(-0.179695\pi\)
0.844841 + 0.535018i \(0.179695\pi\)
\(422\) 0 0
\(423\) 102.750 0.242908
\(424\) 0 0
\(425\) −267.729 + 367.620i −0.629950 + 0.864988i
\(426\) 0 0
\(427\) 7.59316 0.0177826
\(428\) 0 0
\(429\) 931.622 2.17161
\(430\) 0 0
\(431\) 309.254i 0.717526i −0.933429 0.358763i \(-0.883199\pi\)
0.933429 0.358763i \(-0.116801\pi\)
\(432\) 0 0
\(433\) 187.374i 0.432735i −0.976312 0.216368i \(-0.930579\pi\)
0.976312 0.216368i \(-0.0694210\pi\)
\(434\) 0 0
\(435\) −364.518 + 1119.71i −0.837972 + 2.57404i
\(436\) 0 0
\(437\) 528.147i 1.20857i
\(438\) 0 0
\(439\) 289.657i 0.659811i −0.944014 0.329906i \(-0.892983\pi\)
0.944014 0.329906i \(-0.107017\pi\)
\(440\) 0 0
\(441\) −935.730 −2.12184
\(442\) 0 0
\(443\) −295.516 −0.667079 −0.333539 0.942736i \(-0.608243\pi\)
−0.333539 + 0.942736i \(0.608243\pi\)
\(444\) 0 0
\(445\) −16.8530 + 51.7683i −0.0378719 + 0.116333i
\(446\) 0 0
\(447\) 760.667 1.70172
\(448\) 0 0
\(449\) −604.409 −1.34612 −0.673061 0.739587i \(-0.735021\pi\)
−0.673061 + 0.739587i \(0.735021\pi\)
\(450\) 0 0
\(451\) 485.134i 1.07569i
\(452\) 0 0
\(453\) 442.699i 0.977262i
\(454\) 0 0
\(455\) −11.4937 3.74175i −0.0252610 0.00822363i
\(456\) 0 0
\(457\) 392.507i 0.858877i 0.903096 + 0.429438i \(0.141288\pi\)
−0.903096 + 0.429438i \(0.858712\pi\)
\(458\) 0 0
\(459\) 975.444i 2.12515i
\(460\) 0 0
\(461\) −400.277 −0.868279 −0.434139 0.900846i \(-0.642947\pi\)
−0.434139 + 0.900846i \(0.642947\pi\)
\(462\) 0 0
\(463\) 732.679 1.58246 0.791230 0.611518i \(-0.209441\pi\)
0.791230 + 0.611518i \(0.209441\pi\)
\(464\) 0 0
\(465\) −513.811 167.269i −1.10497 0.359719i
\(466\) 0 0
\(467\) 592.126 1.26794 0.633968 0.773360i \(-0.281425\pi\)
0.633968 + 0.773360i \(0.281425\pi\)
\(468\) 0 0
\(469\) 0.932947 0.00198923
\(470\) 0 0
\(471\) 899.639i 1.91006i
\(472\) 0 0
\(473\) 61.0082i 0.128981i
\(474\) 0 0
\(475\) −285.519 + 392.047i −0.601092 + 0.825362i
\(476\) 0 0
\(477\) 1513.20i 3.17232i
\(478\) 0 0
\(479\) 309.151i 0.645409i −0.946500 0.322705i \(-0.895408\pi\)
0.946500 0.322705i \(-0.104592\pi\)
\(480\) 0 0
\(481\) −211.873 −0.440485
\(482\) 0 0
\(483\) 29.8259 0.0617513
\(484\) 0 0
\(485\) 57.7940 + 18.8146i 0.119163 + 0.0387931i
\(486\) 0 0
\(487\) −570.769 −1.17201 −0.586005 0.810307i \(-0.699300\pi\)
−0.586005 + 0.810307i \(0.699300\pi\)
\(488\) 0 0
\(489\) 1460.64 2.98699
\(490\) 0 0
\(491\) 301.659i 0.614378i 0.951649 + 0.307189i \(0.0993883\pi\)
−0.951649 + 0.307189i \(0.900612\pi\)
\(492\) 0 0
\(493\) 808.004i 1.63895i
\(494\) 0 0
\(495\) −1364.68 444.267i −2.75693 0.897509i
\(496\) 0 0
\(497\) 8.60653i 0.0173170i
\(498\) 0 0
\(499\) 517.758i 1.03759i 0.854898 + 0.518795i \(0.173619\pi\)
−0.854898 + 0.518795i \(0.826381\pi\)
\(500\) 0 0
\(501\) 702.439 1.40207
\(502\) 0 0
\(503\) −406.671 −0.808491 −0.404246 0.914650i \(-0.632466\pi\)
−0.404246 + 0.914650i \(0.632466\pi\)
\(504\) 0 0
\(505\) −197.688 + 607.249i −0.391461 + 1.20247i
\(506\) 0 0
\(507\) 170.261 0.335821
\(508\) 0 0
\(509\) 627.097 1.23202 0.616009 0.787739i \(-0.288748\pi\)
0.616009 + 0.787739i \(0.288748\pi\)
\(510\) 0 0
\(511\) 8.59372i 0.0168175i
\(512\) 0 0
\(513\) 1040.26i 2.02780i
\(514\) 0 0
\(515\) −7.39182 + 22.7059i −0.0143530 + 0.0440891i
\(516\) 0 0
\(517\) 80.7330i 0.156157i
\(518\) 0 0
\(519\) 1445.45i 2.78507i
\(520\) 0 0
\(521\) −111.743 −0.214478 −0.107239 0.994233i \(-0.534201\pi\)
−0.107239 + 0.994233i \(0.534201\pi\)
\(522\) 0 0
\(523\) 769.813 1.47192 0.735959 0.677027i \(-0.236732\pi\)
0.735959 + 0.677027i \(0.236732\pi\)
\(524\) 0 0
\(525\) 22.1399 + 16.1240i 0.0421713 + 0.0307124i
\(526\) 0 0
\(527\) 370.776 0.703560
\(528\) 0 0
\(529\) 212.160 0.401058
\(530\) 0 0
\(531\) 1594.04i 3.00195i
\(532\) 0 0
\(533\) 377.958i 0.709115i
\(534\) 0 0
\(535\) −165.943 + 509.736i −0.310174 + 0.952778i
\(536\) 0 0
\(537\) 835.214i 1.55533i
\(538\) 0 0
\(539\) 735.222i 1.36405i
\(540\) 0 0
\(541\) −225.558 −0.416927 −0.208463 0.978030i \(-0.566846\pi\)
−0.208463 + 0.978030i \(0.566846\pi\)
\(542\) 0 0
\(543\) −1776.57 −3.27177
\(544\) 0 0
\(545\) −83.0678 + 255.164i −0.152418 + 0.468191i
\(546\) 0 0
\(547\) 882.346 1.61306 0.806532 0.591190i \(-0.201342\pi\)
0.806532 + 0.591190i \(0.201342\pi\)
\(548\) 0 0
\(549\) −702.382 −1.27938
\(550\) 0 0
\(551\) 861.694i 1.56387i
\(552\) 0 0
\(553\) 3.21245i 0.00580913i
\(554\) 0 0
\(555\) 456.504 + 148.613i 0.822529 + 0.267772i
\(556\) 0 0
\(557\) 303.119i 0.544199i 0.962269 + 0.272100i \(0.0877180\pi\)
−0.962269 + 0.272100i \(0.912282\pi\)
\(558\) 0 0
\(559\) 47.5303i 0.0850273i
\(560\) 0 0
\(561\) 1448.49 2.58198
\(562\) 0 0
\(563\) −344.003 −0.611017 −0.305508 0.952189i \(-0.598826\pi\)
−0.305508 + 0.952189i \(0.598826\pi\)
\(564\) 0 0
\(565\) −97.4950 31.7392i −0.172558 0.0561756i
\(566\) 0 0
\(567\) −23.2029 −0.0409223
\(568\) 0 0
\(569\) −228.925 −0.402329 −0.201165 0.979557i \(-0.564473\pi\)
−0.201165 + 0.979557i \(0.564473\pi\)
\(570\) 0 0
\(571\) 371.169i 0.650033i −0.945708 0.325017i \(-0.894630\pi\)
0.945708 0.325017i \(-0.105370\pi\)
\(572\) 0 0
\(573\) 1583.10i 2.76283i
\(574\) 0 0
\(575\) 550.168 + 400.674i 0.956814 + 0.696825i
\(576\) 0 0
\(577\) 580.289i 1.00570i −0.864374 0.502850i \(-0.832285\pi\)
0.864374 0.502850i \(-0.167715\pi\)
\(578\) 0 0
\(579\) 1017.57i 1.75746i
\(580\) 0 0
\(581\) 10.5350 0.0181325
\(582\) 0 0
\(583\) 1188.95 2.03936
\(584\) 0 0
\(585\) 1063.19 + 346.120i 1.81743 + 0.591657i
\(586\) 0 0
\(587\) 65.0801 0.110869 0.0554345 0.998462i \(-0.482346\pi\)
0.0554345 + 0.998462i \(0.482346\pi\)
\(588\) 0 0
\(589\) 395.413 0.671329
\(590\) 0 0
\(591\) 314.140i 0.531539i
\(592\) 0 0
\(593\) 1002.90i 1.69124i 0.533787 + 0.845619i \(0.320768\pi\)
−0.533787 + 0.845619i \(0.679232\pi\)
\(594\) 0 0
\(595\) −17.8705 5.81770i −0.0300345 0.00977764i
\(596\) 0 0
\(597\) 1638.91i 2.74524i
\(598\) 0 0
\(599\) 888.567i 1.48342i 0.670722 + 0.741709i \(0.265984\pi\)
−0.670722 + 0.741709i \(0.734016\pi\)
\(600\) 0 0
\(601\) 132.065 0.219742 0.109871 0.993946i \(-0.464956\pi\)
0.109871 + 0.993946i \(0.464956\pi\)
\(602\) 0 0
\(603\) −86.2995 −0.143117
\(604\) 0 0
\(605\) 161.788 496.973i 0.267418 0.821443i
\(606\) 0 0
\(607\) 700.090 1.15336 0.576680 0.816970i \(-0.304348\pi\)
0.576680 + 0.816970i \(0.304348\pi\)
\(608\) 0 0
\(609\) −48.6621 −0.0799050
\(610\) 0 0
\(611\) 62.8975i 0.102942i
\(612\) 0 0
\(613\) 727.420i 1.18666i 0.804961 + 0.593328i \(0.202186\pi\)
−0.804961 + 0.593328i \(0.797814\pi\)
\(614\) 0 0
\(615\) 265.109 814.351i 0.431072 1.32415i
\(616\) 0 0
\(617\) 99.7137i 0.161611i 0.996730 + 0.0808053i \(0.0257492\pi\)
−0.996730 + 0.0808053i \(0.974251\pi\)
\(618\) 0 0
\(619\) 721.349i 1.16535i 0.812707 + 0.582673i \(0.197993\pi\)
−0.812707 + 0.582673i \(0.802007\pi\)
\(620\) 0 0
\(621\) −1459.82 −2.35075
\(622\) 0 0
\(623\) −2.24983 −0.00361129
\(624\) 0 0
\(625\) 191.787 + 594.847i 0.306859 + 0.951755i
\(626\) 0 0
\(627\) 1544.74 2.46370
\(628\) 0 0
\(629\) −329.422 −0.523723
\(630\) 0 0
\(631\) 796.856i 1.26285i −0.775438 0.631423i \(-0.782471\pi\)
0.775438 0.631423i \(-0.217529\pi\)
\(632\) 0 0
\(633\) 1090.62i 1.72295i
\(634\) 0 0
\(635\) 214.332 658.375i 0.337530 1.03681i
\(636\) 0 0
\(637\) 572.797i 0.899211i
\(638\) 0 0
\(639\) 796.121i 1.24589i
\(640\) 0 0
\(641\) 205.013 0.319833 0.159917 0.987131i \(-0.448877\pi\)
0.159917 + 0.987131i \(0.448877\pi\)
\(642\) 0 0
\(643\) −495.044 −0.769897 −0.384949 0.922938i \(-0.625781\pi\)
−0.384949 + 0.922938i \(0.625781\pi\)
\(644\) 0 0
\(645\) 33.3389 102.409i 0.0516882 0.158774i
\(646\) 0 0
\(647\) −1121.84 −1.73391 −0.866953 0.498389i \(-0.833925\pi\)
−0.866953 + 0.498389i \(0.833925\pi\)
\(648\) 0 0
\(649\) −1252.47 −1.92984
\(650\) 0 0
\(651\) 22.3300i 0.0343011i
\(652\) 0 0
\(653\) 622.987i 0.954038i 0.878893 + 0.477019i \(0.158283\pi\)
−0.878893 + 0.477019i \(0.841717\pi\)
\(654\) 0 0
\(655\) −1041.71 339.127i −1.59040 0.517751i
\(656\) 0 0
\(657\) 794.936i 1.20995i
\(658\) 0 0
\(659\) 264.030i 0.400653i −0.979729 0.200326i \(-0.935800\pi\)
0.979729 0.200326i \(-0.0642002\pi\)
\(660\) 0 0
\(661\) 1285.15 1.94425 0.972124 0.234469i \(-0.0753352\pi\)
0.972124 + 0.234469i \(0.0753352\pi\)
\(662\) 0 0
\(663\) −1128.49 −1.70210
\(664\) 0 0
\(665\) −19.0580 6.20427i −0.0286586 0.00932972i
\(666\) 0 0
\(667\) −1209.23 −1.81294
\(668\) 0 0
\(669\) −1212.73 −1.81276
\(670\) 0 0
\(671\) 551.876i 0.822468i
\(672\) 0 0
\(673\) 1244.16i 1.84868i −0.381565 0.924342i \(-0.624615\pi\)
0.381565 0.924342i \(-0.375385\pi\)
\(674\) 0 0
\(675\) −1083.63 789.185i −1.60538 1.16916i
\(676\) 0 0
\(677\) 963.335i 1.42295i −0.702713 0.711473i \(-0.748028\pi\)
0.702713 0.711473i \(-0.251972\pi\)
\(678\) 0 0
\(679\) 2.51170i 0.00369912i
\(680\) 0 0
\(681\) −1497.35 −2.19876
\(682\) 0 0
\(683\) 770.819 1.12858 0.564289 0.825577i \(-0.309150\pi\)
0.564289 + 0.825577i \(0.309150\pi\)
\(684\) 0 0
\(685\) 284.228 + 92.5296i 0.414932 + 0.135080i
\(686\) 0 0
\(687\) 261.146 0.380125
\(688\) 0 0
\(689\) −926.287 −1.34439
\(690\) 0 0
\(691\) 408.765i 0.591555i −0.955257 0.295778i \(-0.904421\pi\)
0.955257 0.295778i \(-0.0955788\pi\)
\(692\) 0 0
\(693\) 59.3084i 0.0855821i
\(694\) 0 0
\(695\) 126.716 + 41.2520i 0.182325 + 0.0593554i
\(696\) 0 0
\(697\) 587.652i 0.843116i
\(698\) 0 0
\(699\) 658.921i 0.942663i
\(700\) 0 0
\(701\) 1335.62 1.90530 0.952650 0.304068i \(-0.0983450\pi\)
0.952650 + 0.304068i \(0.0983450\pi\)
\(702\) 0 0
\(703\) −351.311 −0.499731
\(704\) 0 0
\(705\) −44.1178 + 135.519i −0.0625785 + 0.192226i
\(706\) 0 0
\(707\) −26.3908 −0.0373279
\(708\) 0 0
\(709\) 362.956 0.511927 0.255964 0.966686i \(-0.417607\pi\)
0.255964 + 0.966686i \(0.417607\pi\)
\(710\) 0 0
\(711\) 297.158i 0.417943i
\(712\) 0 0
\(713\) 554.891i 0.778249i
\(714\) 0 0
\(715\) −271.953 + 835.374i −0.380354 + 1.16836i
\(716\) 0 0
\(717\) 426.575i 0.594945i
\(718\) 0 0
\(719\) 648.098i 0.901388i 0.892679 + 0.450694i \(0.148823\pi\)
−0.892679 + 0.450694i \(0.851177\pi\)
\(720\) 0 0
\(721\) −0.986788 −0.00136864
\(722\) 0 0
\(723\) −6.41453 −0.00887211
\(724\) 0 0
\(725\) −897.622 653.717i −1.23810 0.901679i
\(726\) 0 0
\(727\) 431.123 0.593017 0.296508 0.955030i \(-0.404178\pi\)
0.296508 + 0.955030i \(0.404178\pi\)
\(728\) 0 0
\(729\) −412.506 −0.565852
\(730\) 0 0
\(731\) 73.9003i 0.101095i
\(732\) 0 0
\(733\) 1464.87i 1.99846i −0.0392126 0.999231i \(-0.512485\pi\)
0.0392126 0.999231i \(-0.487515\pi\)
\(734\) 0 0
\(735\) 401.774 1234.15i 0.546632 1.67912i
\(736\) 0 0
\(737\) 67.8073i 0.0920044i
\(738\) 0 0
\(739\) 99.1951i 0.134229i −0.997745 0.0671144i \(-0.978621\pi\)
0.997745 0.0671144i \(-0.0213793\pi\)
\(740\) 0 0
\(741\) −1203.48 −1.62412
\(742\) 0 0
\(743\) 602.719 0.811196 0.405598 0.914052i \(-0.367063\pi\)
0.405598 + 0.914052i \(0.367063\pi\)
\(744\) 0 0
\(745\) −222.049 + 682.081i −0.298053 + 0.915545i
\(746\) 0 0
\(747\) −974.508 −1.30456
\(748\) 0 0
\(749\) −22.1529 −0.0295767
\(750\) 0 0
\(751\) 541.472i 0.721001i 0.932759 + 0.360501i \(0.117394\pi\)
−0.932759 + 0.360501i \(0.882606\pi\)
\(752\) 0 0
\(753\) 1123.28i 1.49174i
\(754\) 0 0
\(755\) −396.963 129.230i −0.525779 0.171166i
\(756\) 0 0
\(757\) 1092.79i 1.44358i 0.692113 + 0.721789i \(0.256680\pi\)
−0.692113 + 0.721789i \(0.743320\pi\)
\(758\) 0 0
\(759\) 2167.76i 2.85608i
\(760\) 0 0
\(761\) −18.7706 −0.0246657 −0.0123328 0.999924i \(-0.503926\pi\)
−0.0123328 + 0.999924i \(0.503926\pi\)
\(762\) 0 0
\(763\) −11.0893 −0.0145338
\(764\) 0 0
\(765\) 1653.06 + 538.149i 2.16086 + 0.703462i
\(766\) 0 0
\(767\) 975.773 1.27219
\(768\) 0 0
\(769\) 39.0830 0.0508231 0.0254116 0.999677i \(-0.491910\pi\)
0.0254116 + 0.999677i \(0.491910\pi\)
\(770\) 0 0
\(771\) 968.423i 1.25606i
\(772\) 0 0
\(773\) 31.2171i 0.0403843i −0.999796 0.0201922i \(-0.993572\pi\)
0.999796 0.0201922i \(-0.00642780\pi\)
\(774\) 0 0
\(775\) 299.977 411.900i 0.387067 0.531484i
\(776\) 0 0
\(777\) 19.8395i 0.0255334i
\(778\) 0 0
\(779\) 626.699i 0.804492i
\(780\) 0 0
\(781\) −625.529 −0.800933
\(782\) 0 0
\(783\) 2381.75 3.04183
\(784\) 0 0
\(785\) −806.695 262.617i −1.02764 0.334544i
\(786\) 0 0
\(787\) −988.563 −1.25612 −0.628058 0.778167i \(-0.716150\pi\)
−0.628058 + 0.778167i \(0.716150\pi\)
\(788\) 0 0
\(789\) 395.880 0.501750
\(790\) 0 0
\(791\) 4.23710i 0.00535663i
\(792\) 0 0
\(793\) 429.956i 0.542189i
\(794\) 0 0
\(795\) 1995.78 + 649.721i 2.51042 + 0.817259i
\(796\) 0 0
\(797\) 118.920i 0.149210i −0.997213 0.0746048i \(-0.976230\pi\)
0.997213 0.0746048i \(-0.0237695\pi\)
\(798\) 0 0
\(799\) 97.7933i 0.122395i
\(800\) 0 0
\(801\)