Properties

Label 160.3.h.b.159.1
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 159.1
Root \(-2.65109i\) of defining polynomial
Character \(\chi\) \(=\) 160.159
Dual form 160.3.h.b.159.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.75441 q^{3} +(4.30219 - 2.54778i) q^{5} -3.84997 q^{7} -1.41325 q^{9} +O(q^{10})\) \(q-2.75441 q^{3} +(4.30219 - 2.54778i) q^{5} -3.84997 q^{7} -1.41325 q^{9} -6.19112i q^{11} -16.1132i q^{13} +(-11.8500 + 7.01762i) q^{15} -5.20875i q^{17} -36.2264i q^{19} +10.6044 q^{21} -22.0411 q^{23} +(12.0176 - 21.9221i) q^{25} +28.6823 q^{27} -20.0352 q^{29} +26.4175i q^{31} +17.0529i q^{33} +(-16.5633 + 9.80888i) q^{35} +69.3219i q^{37} +44.3822i q^{39} +11.6220 q^{41} +25.8542 q^{43} +(-6.08006 + 3.60065i) q^{45} +66.1853 q^{47} -34.1777 q^{49} +14.3470i q^{51} -39.5751i q^{53} +(-15.7736 - 26.6354i) q^{55} +99.7821i q^{57} -27.7736i q^{59} -54.1954 q^{61} +5.44096 q^{63} +(-41.0529 - 69.3219i) q^{65} +107.507 q^{67} +60.7101 q^{69} -70.7997i q^{71} +37.4351i q^{73} +(-33.1014 + 60.3822i) q^{75} +23.8356i q^{77} +97.6530i q^{79} -66.2835 q^{81} -126.163 q^{83} +(-13.2707 - 22.4090i) q^{85} +55.1852 q^{87} +133.635 q^{89} +62.0352i q^{91} -72.7645i q^{93} +(-92.2969 - 155.853i) q^{95} +6.40900i q^{97} +8.74960i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{3} - 2q^{5} + 12q^{7} + 18q^{9} + O(q^{10}) \) \( 6q + 4q^{3} - 2q^{5} + 12q^{7} + 18q^{9} - 36q^{15} + 8q^{21} - 68q^{23} - 10q^{25} + 184q^{27} + 44q^{29} - 108q^{35} - 68q^{41} - 76q^{43} - 6q^{45} + 268q^{47} - 62q^{49} - 288q^{55} - 100q^{61} - 172q^{63} + 308q^{67} - 184q^{69} - 284q^{75} + 238q^{81} - 204q^{83} - 32q^{85} + 584q^{87} + 76q^{89} - 32q^{95} + O(q^{100}) \)

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
\(3\) −2.75441 −0.918135 −0.459068 0.888401i \(-0.651816\pi\)
−0.459068 + 0.888401i \(0.651816\pi\)
\(4\) 0 0
\(5\) 4.30219 2.54778i 0.860437 0.509556i
\(6\) 0 0
\(7\) −3.84997 −0.549995 −0.274998 0.961445i \(-0.588677\pi\)
−0.274998 + 0.961445i \(0.588677\pi\)
\(8\) 0 0
\(9\) −1.41325 −0.157028
\(10\) 0 0
\(11\) 6.19112i 0.562829i −0.959586 0.281415i \(-0.909196\pi\)
0.959586 0.281415i \(-0.0908037\pi\)
\(12\) 0 0
\(13\) 16.1132i 1.23948i −0.784809 0.619738i \(-0.787239\pi\)
0.784809 0.619738i \(-0.212761\pi\)
\(14\) 0 0
\(15\) −11.8500 + 7.01762i −0.789998 + 0.467842i
\(16\) 0 0
\(17\) 5.20875i 0.306397i −0.988195 0.153198i \(-0.951043\pi\)
0.988195 0.153198i \(-0.0489574\pi\)
\(18\) 0 0
\(19\) 36.2264i 1.90665i −0.301944 0.953326i \(-0.597636\pi\)
0.301944 0.953326i \(-0.402364\pi\)
\(20\) 0 0
\(21\) 10.6044 0.504970
\(22\) 0 0
\(23\) −22.0411 −0.958308 −0.479154 0.877731i \(-0.659057\pi\)
−0.479154 + 0.877731i \(0.659057\pi\)
\(24\) 0 0
\(25\) 12.0176 21.9221i 0.480705 0.876882i
\(26\) 0 0
\(27\) 28.6823 1.06231
\(28\) 0 0
\(29\) −20.0352 −0.690871 −0.345435 0.938443i \(-0.612269\pi\)
−0.345435 + 0.938443i \(0.612269\pi\)
\(30\) 0 0
\(31\) 26.4175i 0.852177i 0.904681 + 0.426089i \(0.140109\pi\)
−0.904681 + 0.426089i \(0.859891\pi\)
\(32\) 0 0
\(33\) 17.0529i 0.516754i
\(34\) 0 0
\(35\) −16.5633 + 9.80888i −0.473237 + 0.280254i
\(36\) 0 0
\(37\) 69.3219i 1.87357i 0.349911 + 0.936783i \(0.386212\pi\)
−0.349911 + 0.936783i \(0.613788\pi\)
\(38\) 0 0
\(39\) 44.3822i 1.13801i
\(40\) 0 0
\(41\) 11.6220 0.283463 0.141732 0.989905i \(-0.454733\pi\)
0.141732 + 0.989905i \(0.454733\pi\)
\(42\) 0 0
\(43\) 25.8542 0.601261 0.300630 0.953741i \(-0.402803\pi\)
0.300630 + 0.953741i \(0.402803\pi\)
\(44\) 0 0
\(45\) −6.08006 + 3.60065i −0.135112 + 0.0800144i
\(46\) 0 0
\(47\) 66.1853 1.40820 0.704099 0.710102i \(-0.251351\pi\)
0.704099 + 0.710102i \(0.251351\pi\)
\(48\) 0 0
\(49\) −34.1777 −0.697505
\(50\) 0 0
\(51\) 14.3470i 0.281314i
\(52\) 0 0
\(53\) 39.5751i 0.746699i −0.927691 0.373350i \(-0.878209\pi\)
0.927691 0.373350i \(-0.121791\pi\)
\(54\) 0 0
\(55\) −15.7736 26.6354i −0.286793 0.484280i
\(56\) 0 0
\(57\) 99.7821i 1.75056i
\(58\) 0 0
\(59\) 27.7736i 0.470739i −0.971906 0.235370i \(-0.924370\pi\)
0.971906 0.235370i \(-0.0756301\pi\)
\(60\) 0 0
\(61\) −54.1954 −0.888449 −0.444224 0.895916i \(-0.646521\pi\)
−0.444224 + 0.895916i \(0.646521\pi\)
\(62\) 0 0
\(63\) 5.44096 0.0863645
\(64\) 0 0
\(65\) −41.0529 69.3219i −0.631583 1.06649i
\(66\) 0 0
\(67\) 107.507 1.60459 0.802293 0.596931i \(-0.203613\pi\)
0.802293 + 0.596931i \(0.203613\pi\)
\(68\) 0 0
\(69\) 60.7101 0.879857
\(70\) 0 0
\(71\) 70.7997i 0.997179i −0.866838 0.498590i \(-0.833851\pi\)
0.866838 0.498590i \(-0.166149\pi\)
\(72\) 0 0
\(73\) 37.4351i 0.512810i 0.966569 + 0.256405i \(0.0825380\pi\)
−0.966569 + 0.256405i \(0.917462\pi\)
\(74\) 0 0
\(75\) −33.1014 + 60.3822i −0.441352 + 0.805097i
\(76\) 0 0
\(77\) 23.8356i 0.309554i
\(78\) 0 0
\(79\) 97.6530i 1.23611i 0.786133 + 0.618057i \(0.212080\pi\)
−0.786133 + 0.618057i \(0.787920\pi\)
\(80\) 0 0
\(81\) −66.2835 −0.818315
\(82\) 0 0
\(83\) −126.163 −1.52003 −0.760017 0.649904i \(-0.774809\pi\)
−0.760017 + 0.649904i \(0.774809\pi\)
\(84\) 0 0
\(85\) −13.2707 22.4090i −0.156126 0.263635i
\(86\) 0 0
\(87\) 55.1852 0.634313
\(88\) 0 0
\(89\) 133.635 1.50151 0.750757 0.660579i \(-0.229689\pi\)
0.750757 + 0.660579i \(0.229689\pi\)
\(90\) 0 0
\(91\) 62.0352i 0.681706i
\(92\) 0 0
\(93\) 72.7645i 0.782414i
\(94\) 0 0
\(95\) −92.2969 155.853i −0.971546 1.64055i
\(96\) 0 0
\(97\) 6.40900i 0.0660722i 0.999454 + 0.0330361i \(0.0105176\pi\)
−0.999454 + 0.0330361i \(0.989482\pi\)
\(98\) 0 0
\(99\) 8.74960i 0.0883798i
\(100\) 0 0
\(101\) 121.564 1.20361 0.601803 0.798644i \(-0.294449\pi\)
0.601803 + 0.798644i \(0.294449\pi\)
\(102\) 0 0
\(103\) 9.95891 0.0966884 0.0483442 0.998831i \(-0.484606\pi\)
0.0483442 + 0.998831i \(0.484606\pi\)
\(104\) 0 0
\(105\) 45.6220 27.0176i 0.434495 0.257311i
\(106\) 0 0
\(107\) −134.842 −1.26020 −0.630102 0.776512i \(-0.716987\pi\)
−0.630102 + 0.776512i \(0.716987\pi\)
\(108\) 0 0
\(109\) −28.2306 −0.258996 −0.129498 0.991580i \(-0.541337\pi\)
−0.129498 + 0.991580i \(0.541337\pi\)
\(110\) 0 0
\(111\) 190.941i 1.72019i
\(112\) 0 0
\(113\) 190.052i 1.68188i −0.541130 0.840939i \(-0.682003\pi\)
0.541130 0.840939i \(-0.317997\pi\)
\(114\) 0 0
\(115\) −94.8249 + 56.1559i −0.824564 + 0.488312i
\(116\) 0 0
\(117\) 22.7719i 0.194632i
\(118\) 0 0
\(119\) 20.0535i 0.168517i
\(120\) 0 0
\(121\) 82.6700 0.683223
\(122\) 0 0
\(123\) −32.0117 −0.260258
\(124\) 0 0
\(125\) −4.15055 124.931i −0.0332044 0.999449i
\(126\) 0 0
\(127\) 60.0646 0.472950 0.236475 0.971638i \(-0.424008\pi\)
0.236475 + 0.971638i \(0.424008\pi\)
\(128\) 0 0
\(129\) −71.2130 −0.552039
\(130\) 0 0
\(131\) 111.985i 0.854848i 0.904051 + 0.427424i \(0.140579\pi\)
−0.904051 + 0.427424i \(0.859421\pi\)
\(132\) 0 0
\(133\) 139.470i 1.04865i
\(134\) 0 0
\(135\) 123.397 73.0763i 0.914049 0.541306i
\(136\) 0 0
\(137\) 42.6439i 0.311269i −0.987815 0.155635i \(-0.950258\pi\)
0.987815 0.155635i \(-0.0497422\pi\)
\(138\) 0 0
\(139\) 222.332i 1.59951i −0.600325 0.799756i \(-0.704962\pi\)
0.600325 0.799756i \(-0.295038\pi\)
\(140\) 0 0
\(141\) −182.301 −1.29292
\(142\) 0 0
\(143\) −99.7587 −0.697614
\(144\) 0 0
\(145\) −86.1954 + 51.0454i −0.594451 + 0.352037i
\(146\) 0 0
\(147\) 94.1394 0.640404
\(148\) 0 0
\(149\) 20.0981 0.134887 0.0674434 0.997723i \(-0.478516\pi\)
0.0674434 + 0.997723i \(0.478516\pi\)
\(150\) 0 0
\(151\) 86.0522i 0.569882i 0.958545 + 0.284941i \(0.0919741\pi\)
−0.958545 + 0.284941i \(0.908026\pi\)
\(152\) 0 0
\(153\) 7.36126i 0.0481128i
\(154\) 0 0
\(155\) 67.3060 + 113.653i 0.434232 + 0.733245i
\(156\) 0 0
\(157\) 16.0342i 0.102129i −0.998695 0.0510643i \(-0.983739\pi\)
0.998695 0.0510643i \(-0.0162614\pi\)
\(158\) 0 0
\(159\) 109.006i 0.685571i
\(160\) 0 0
\(161\) 84.8575 0.527065
\(162\) 0 0
\(163\) 179.157 1.09912 0.549561 0.835454i \(-0.314795\pi\)
0.549561 + 0.835454i \(0.314795\pi\)
\(164\) 0 0
\(165\) 43.4470 + 73.3646i 0.263315 + 0.444634i
\(166\) 0 0
\(167\) −137.800 −0.825149 −0.412574 0.910924i \(-0.635370\pi\)
−0.412574 + 0.910924i \(0.635370\pi\)
\(168\) 0 0
\(169\) −90.6347 −0.536300
\(170\) 0 0
\(171\) 51.1969i 0.299397i
\(172\) 0 0
\(173\) 62.8895i 0.363523i 0.983343 + 0.181762i \(0.0581799\pi\)
−0.983343 + 0.181762i \(0.941820\pi\)
\(174\) 0 0
\(175\) −46.2675 + 84.3992i −0.264385 + 0.482281i
\(176\) 0 0
\(177\) 76.4998i 0.432203i
\(178\) 0 0
\(179\) 238.020i 1.32972i −0.746967 0.664861i \(-0.768491\pi\)
0.746967 0.664861i \(-0.231509\pi\)
\(180\) 0 0
\(181\) 186.718 1.03159 0.515795 0.856712i \(-0.327497\pi\)
0.515795 + 0.856712i \(0.327497\pi\)
\(182\) 0 0
\(183\) 149.276 0.815716
\(184\) 0 0
\(185\) 176.617 + 298.236i 0.954687 + 1.61209i
\(186\) 0 0
\(187\) −32.2480 −0.172449
\(188\) 0 0
\(189\) −110.426 −0.584264
\(190\) 0 0
\(191\) 123.447i 0.646319i 0.946344 + 0.323160i \(0.104745\pi\)
−0.946344 + 0.323160i \(0.895255\pi\)
\(192\) 0 0
\(193\) 162.355i 0.841220i 0.907242 + 0.420610i \(0.138184\pi\)
−0.907242 + 0.420610i \(0.861816\pi\)
\(194\) 0 0
\(195\) 113.076 + 190.941i 0.579878 + 0.979183i
\(196\) 0 0
\(197\) 113.540i 0.576344i −0.957579 0.288172i \(-0.906952\pi\)
0.957579 0.288172i \(-0.0930475\pi\)
\(198\) 0 0
\(199\) 325.928i 1.63783i 0.573915 + 0.818915i \(0.305424\pi\)
−0.573915 + 0.818915i \(0.694576\pi\)
\(200\) 0 0
\(201\) −296.118 −1.47323
\(202\) 0 0
\(203\) 77.1351 0.379976
\(204\) 0 0
\(205\) 50.0000 29.6103i 0.243902 0.144441i
\(206\) 0 0
\(207\) 31.1496 0.150481
\(208\) 0 0
\(209\) −224.282 −1.07312
\(210\) 0 0
\(211\) 130.731i 0.619580i −0.950805 0.309790i \(-0.899741\pi\)
0.950805 0.309790i \(-0.100259\pi\)
\(212\) 0 0
\(213\) 195.011i 0.915546i
\(214\) 0 0
\(215\) 111.230 65.8709i 0.517347 0.306376i
\(216\) 0 0
\(217\) 101.707i 0.468694i
\(218\) 0 0
\(219\) 103.112i 0.470829i
\(220\) 0 0
\(221\) −83.9295 −0.379772
\(222\) 0 0
\(223\) 93.3889 0.418784 0.209392 0.977832i \(-0.432851\pi\)
0.209392 + 0.977832i \(0.432851\pi\)
\(224\) 0 0
\(225\) −16.9839 + 30.9813i −0.0754840 + 0.137695i
\(226\) 0 0
\(227\) −14.9957 −0.0660602 −0.0330301 0.999454i \(-0.510516\pi\)
−0.0330301 + 0.999454i \(0.510516\pi\)
\(228\) 0 0
\(229\) −144.106 −0.629283 −0.314641 0.949211i \(-0.601884\pi\)
−0.314641 + 0.949211i \(0.601884\pi\)
\(230\) 0 0
\(231\) 65.6530i 0.284212i
\(232\) 0 0
\(233\) 126.528i 0.543040i −0.962433 0.271520i \(-0.912474\pi\)
0.962433 0.271520i \(-0.0875263\pi\)
\(234\) 0 0
\(235\) 284.741 168.626i 1.21167 0.717556i
\(236\) 0 0
\(237\) 268.976i 1.13492i
\(238\) 0 0
\(239\) 1.65300i 0.00691630i 0.999994 + 0.00345815i \(0.00110077\pi\)
−0.999994 + 0.00345815i \(0.998899\pi\)
\(240\) 0 0
\(241\) 206.928 0.858622 0.429311 0.903157i \(-0.358756\pi\)
0.429311 + 0.903157i \(0.358756\pi\)
\(242\) 0 0
\(243\) −75.5692 −0.310984
\(244\) 0 0
\(245\) −147.039 + 87.0774i −0.600159 + 0.355418i
\(246\) 0 0
\(247\) −583.722 −2.36325
\(248\) 0 0
\(249\) 347.503 1.39560
\(250\) 0 0
\(251\) 74.1206i 0.295301i −0.989040 0.147651i \(-0.952829\pi\)
0.989040 0.147651i \(-0.0471711\pi\)
\(252\) 0 0
\(253\) 136.459i 0.539364i
\(254\) 0 0
\(255\) 36.5530 + 61.7235i 0.143345 + 0.242053i
\(256\) 0 0
\(257\) 274.682i 1.06880i −0.845231 0.534402i \(-0.820537\pi\)
0.845231 0.534402i \(-0.179463\pi\)
\(258\) 0 0
\(259\) 266.887i 1.03045i
\(260\) 0 0
\(261\) 28.3148 0.108486
\(262\) 0 0
\(263\) −75.5382 −0.287218 −0.143609 0.989635i \(-0.545871\pi\)
−0.143609 + 0.989635i \(0.545871\pi\)
\(264\) 0 0
\(265\) −100.829 170.259i −0.380485 0.642488i
\(266\) 0 0
\(267\) −368.084 −1.37859
\(268\) 0 0
\(269\) 314.087 1.16761 0.583804 0.811895i \(-0.301564\pi\)
0.583804 + 0.811895i \(0.301564\pi\)
\(270\) 0 0
\(271\) 128.158i 0.472908i 0.971643 + 0.236454i \(0.0759852\pi\)
−0.971643 + 0.236454i \(0.924015\pi\)
\(272\) 0 0
\(273\) 170.870i 0.625898i
\(274\) 0 0
\(275\) −135.722 74.4026i −0.493535 0.270555i
\(276\) 0 0
\(277\) 242.118i 0.874073i −0.899444 0.437037i \(-0.856028\pi\)
0.899444 0.437037i \(-0.143972\pi\)
\(278\) 0 0
\(279\) 37.3345i 0.133815i
\(280\) 0 0
\(281\) 28.8562 0.102691 0.0513456 0.998681i \(-0.483649\pi\)
0.0513456 + 0.998681i \(0.483649\pi\)
\(282\) 0 0
\(283\) 269.993 0.954039 0.477020 0.878893i \(-0.341717\pi\)
0.477020 + 0.878893i \(0.341717\pi\)
\(284\) 0 0
\(285\) 254.223 + 429.281i 0.892011 + 1.50625i
\(286\) 0 0
\(287\) −44.7443 −0.155904
\(288\) 0 0
\(289\) 261.869 0.906121
\(290\) 0 0
\(291\) 17.6530i 0.0606632i
\(292\) 0 0
\(293\) 353.448i 1.20631i −0.797625 0.603154i \(-0.793911\pi\)
0.797625 0.603154i \(-0.206089\pi\)
\(294\) 0 0
\(295\) −70.7611 119.487i −0.239868 0.405042i
\(296\) 0 0
\(297\) 177.576i 0.597898i
\(298\) 0 0
\(299\) 355.152i 1.18780i
\(300\) 0 0
\(301\) −99.5379 −0.330691
\(302\) 0 0
\(303\) −334.837 −1.10507
\(304\) 0 0
\(305\) −233.159 + 138.078i −0.764454 + 0.452715i
\(306\) 0 0
\(307\) −260.946 −0.849985 −0.424993 0.905197i \(-0.639723\pi\)
−0.424993 + 0.905197i \(0.639723\pi\)
\(308\) 0 0
\(309\) −27.4309 −0.0887730
\(310\) 0 0
\(311\) 141.570i 0.455208i −0.973754 0.227604i \(-0.926911\pi\)
0.973754 0.227604i \(-0.0730892\pi\)
\(312\) 0 0
\(313\) 365.950i 1.16917i 0.811333 + 0.584584i \(0.198742\pi\)
−0.811333 + 0.584584i \(0.801258\pi\)
\(314\) 0 0
\(315\) 23.4080 13.8624i 0.0743113 0.0440076i
\(316\) 0 0
\(317\) 9.22805i 0.0291106i 0.999894 + 0.0145553i \(0.00463326\pi\)
−0.999894 + 0.0145553i \(0.995367\pi\)
\(318\) 0 0
\(319\) 124.041i 0.388842i
\(320\) 0 0
\(321\) 371.409 1.15704
\(322\) 0 0
\(323\) −188.694 −0.584192
\(324\) 0 0
\(325\) −353.234 193.642i −1.08687 0.595822i
\(326\) 0 0
\(327\) 77.7586 0.237794
\(328\) 0 0
\(329\) −254.811 −0.774502
\(330\) 0 0
\(331\) 53.3799i 0.161268i 0.996744 + 0.0806342i \(0.0256946\pi\)
−0.996744 + 0.0806342i \(0.974305\pi\)
\(332\) 0 0
\(333\) 97.9692i 0.294202i
\(334\) 0 0
\(335\) 462.516 273.905i 1.38065 0.817626i
\(336\) 0 0
\(337\) 350.458i 1.03994i −0.854186 0.519968i \(-0.825944\pi\)
0.854186 0.519968i \(-0.174056\pi\)
\(338\) 0 0
\(339\) 523.481i 1.54419i
\(340\) 0 0
\(341\) 163.554 0.479630
\(342\) 0 0
\(343\) 320.232 0.933620
\(344\) 0 0
\(345\) 261.186 154.676i 0.757062 0.448336i
\(346\) 0 0
\(347\) 70.8302 0.204122 0.102061 0.994778i \(-0.467456\pi\)
0.102061 + 0.994778i \(0.467456\pi\)
\(348\) 0 0
\(349\) 373.045 1.06890 0.534449 0.845201i \(-0.320519\pi\)
0.534449 + 0.845201i \(0.320519\pi\)
\(350\) 0 0
\(351\) 462.163i 1.31670i
\(352\) 0 0
\(353\) 543.568i 1.53985i 0.638132 + 0.769927i \(0.279707\pi\)
−0.638132 + 0.769927i \(0.720293\pi\)
\(354\) 0 0
\(355\) −180.382 304.594i −0.508119 0.858010i
\(356\) 0 0
\(357\) 55.2355i 0.154721i
\(358\) 0 0
\(359\) 500.805i 1.39500i −0.716585 0.697500i \(-0.754296\pi\)
0.716585 0.697500i \(-0.245704\pi\)
\(360\) 0 0
\(361\) −951.350 −2.63532
\(362\) 0 0
\(363\) −227.707 −0.627291
\(364\) 0 0
\(365\) 95.3765 + 161.053i 0.261305 + 0.441241i
\(366\) 0 0
\(367\) 142.499 0.388281 0.194140 0.980974i \(-0.437808\pi\)
0.194140 + 0.980974i \(0.437808\pi\)
\(368\) 0 0
\(369\) −16.4248 −0.0445116
\(370\) 0 0
\(371\) 152.363i 0.410681i
\(372\) 0 0
\(373\) 160.000i 0.428954i −0.976729 0.214477i \(-0.931195\pi\)
0.976729 0.214477i \(-0.0688047\pi\)
\(374\) 0 0
\(375\) 11.4323 + 344.111i 0.0304862 + 0.917629i
\(376\) 0 0
\(377\) 322.832i 0.856317i
\(378\) 0 0
\(379\) 192.796i 0.508698i 0.967113 + 0.254349i \(0.0818611\pi\)
−0.967113 + 0.254349i \(0.918139\pi\)
\(380\) 0 0
\(381\) −165.442 −0.434232
\(382\) 0 0
\(383\) 605.286 1.58038 0.790191 0.612861i \(-0.209981\pi\)
0.790191 + 0.612861i \(0.209981\pi\)
\(384\) 0 0
\(385\) 60.7280 + 102.545i 0.157735 + 0.266352i
\(386\) 0 0
\(387\) −36.5384 −0.0944146
\(388\) 0 0
\(389\) −522.159 −1.34231 −0.671155 0.741317i \(-0.734202\pi\)
−0.671155 + 0.741317i \(0.734202\pi\)
\(390\) 0 0
\(391\) 114.806i 0.293623i
\(392\) 0 0
\(393\) 308.452i 0.784866i
\(394\) 0 0
\(395\) 248.798 + 420.121i 0.629870 + 1.06360i
\(396\) 0 0
\(397\) 357.537i 0.900598i 0.892878 + 0.450299i \(0.148683\pi\)
−0.892878 + 0.450299i \(0.851317\pi\)
\(398\) 0 0
\(399\) 384.158i 0.962802i
\(400\) 0 0
\(401\) 262.506 0.654629 0.327315 0.944915i \(-0.393856\pi\)
0.327315 + 0.944915i \(0.393856\pi\)
\(402\) 0 0
\(403\) 425.670 1.05625
\(404\) 0 0
\(405\) −285.164 + 168.876i −0.704108 + 0.416977i
\(406\) 0 0
\(407\) 429.181 1.05450
\(408\) 0 0
\(409\) −63.2015 −0.154527 −0.0772634 0.997011i \(-0.524618\pi\)
−0.0772634 + 0.997011i \(0.524618\pi\)
\(410\) 0 0
\(411\) 117.459i 0.285787i
\(412\) 0 0
\(413\) 106.928i 0.258905i
\(414\) 0 0
\(415\) −542.776 + 321.435i −1.30789 + 0.774542i
\(416\) 0 0
\(417\) 612.393i 1.46857i
\(418\) 0 0
\(419\) 673.390i 1.60714i 0.595213 + 0.803568i \(0.297068\pi\)
−0.595213 + 0.803568i \(0.702932\pi\)
\(420\) 0 0
\(421\) 84.6877 0.201158 0.100579 0.994929i \(-0.467930\pi\)
0.100579 + 0.994929i \(0.467930\pi\)
\(422\) 0 0
\(423\) −93.5363 −0.221126
\(424\) 0 0
\(425\) −114.186 62.5968i −0.268674 0.147286i
\(426\) 0 0
\(427\) 208.650 0.488643
\(428\) 0 0
\(429\) 274.776 0.640504
\(430\) 0 0
\(431\) 672.158i 1.55953i 0.626072 + 0.779766i \(0.284662\pi\)
−0.626072 + 0.779766i \(0.715338\pi\)
\(432\) 0 0
\(433\) 562.185i 1.29835i 0.760640 + 0.649174i \(0.224885\pi\)
−0.760640 + 0.649174i \(0.775115\pi\)
\(434\) 0 0
\(435\) 237.417 140.600i 0.545786 0.323218i
\(436\) 0 0
\(437\) 798.469i 1.82716i
\(438\) 0 0
\(439\) 384.842i 0.876633i −0.898821 0.438316i \(-0.855575\pi\)
0.898821 0.438316i \(-0.144425\pi\)
\(440\) 0 0
\(441\) 48.3017 0.109528
\(442\) 0 0
\(443\) −461.625 −1.04204 −0.521021 0.853544i \(-0.674449\pi\)
−0.521021 + 0.853544i \(0.674449\pi\)
\(444\) 0 0
\(445\) 574.922 340.472i 1.29196 0.765106i
\(446\) 0 0
\(447\) −55.3584 −0.123844
\(448\) 0 0
\(449\) 48.7390 0.108550 0.0542750 0.998526i \(-0.482715\pi\)
0.0542750 + 0.998526i \(0.482715\pi\)
\(450\) 0 0
\(451\) 71.9532i 0.159542i
\(452\) 0 0
\(453\) 237.023i 0.523229i
\(454\) 0 0
\(455\) 158.052 + 266.887i 0.347368 + 0.586565i
\(456\) 0 0
\(457\) 762.588i 1.66868i −0.551248 0.834341i \(-0.685848\pi\)
0.551248 0.834341i \(-0.314152\pi\)
\(458\) 0 0
\(459\) 149.399i 0.325488i
\(460\) 0 0
\(461\) −406.436 −0.881639 −0.440820 0.897596i \(-0.645312\pi\)
−0.440820 + 0.897596i \(0.645312\pi\)
\(462\) 0 0
\(463\) −260.743 −0.563159 −0.281580 0.959538i \(-0.590858\pi\)
−0.281580 + 0.959538i \(0.590858\pi\)
\(464\) 0 0
\(465\) −185.388 313.046i −0.398684 0.673218i
\(466\) 0 0
\(467\) 594.738 1.27353 0.636765 0.771058i \(-0.280272\pi\)
0.636765 + 0.771058i \(0.280272\pi\)
\(468\) 0 0
\(469\) −413.899 −0.882515
\(470\) 0 0
\(471\) 44.1647i 0.0937679i
\(472\) 0 0
\(473\) 160.067i 0.338407i
\(474\) 0 0
\(475\) −794.157 435.355i −1.67191 0.916537i
\(476\) 0 0
\(477\) 55.9294i 0.117252i
\(478\) 0 0
\(479\) 534.894i 1.11669i −0.829609 0.558344i \(-0.811437\pi\)
0.829609 0.558344i \(-0.188563\pi\)
\(480\) 0 0
\(481\) 1117.00 2.32224
\(482\) 0 0
\(483\) −233.732 −0.483917
\(484\) 0 0
\(485\) 16.3287 + 27.5727i 0.0336675 + 0.0568510i
\(486\) 0 0
\(487\) 264.298 0.542706 0.271353 0.962480i \(-0.412529\pi\)
0.271353 + 0.962480i \(0.412529\pi\)
\(488\) 0 0
\(489\) −493.471 −1.00914
\(490\) 0 0
\(491\) 539.150i 1.09807i −0.835801 0.549033i \(-0.814996\pi\)
0.835801 0.549033i \(-0.185004\pi\)
\(492\) 0 0
\(493\) 104.359i 0.211681i
\(494\) 0 0
\(495\) 22.2921 + 37.6424i 0.0450345 + 0.0760453i
\(496\) 0 0
\(497\) 272.577i 0.548444i
\(498\) 0 0
\(499\) 138.218i 0.276991i −0.990363 0.138495i \(-0.955773\pi\)
0.990363 0.138495i \(-0.0442266\pi\)
\(500\) 0 0
\(501\) 379.557 0.757598
\(502\) 0 0
\(503\) −389.170 −0.773697 −0.386848 0.922143i \(-0.626436\pi\)
−0.386848 + 0.922143i \(0.626436\pi\)
\(504\) 0 0
\(505\) 522.992 309.719i 1.03563 0.613305i
\(506\) 0 0
\(507\) 249.645 0.492396
\(508\) 0 0
\(509\) −468.599 −0.920627 −0.460314 0.887756i \(-0.652263\pi\)
−0.460314 + 0.887756i \(0.652263\pi\)
\(510\) 0 0
\(511\) 144.124i 0.282043i
\(512\) 0 0
\(513\) 1039.06i 2.02545i
\(514\) 0 0
\(515\) 42.8451 25.3731i 0.0831943 0.0492682i
\(516\) 0 0
\(517\) 409.761i 0.792575i
\(518\) 0 0
\(519\) 173.223i 0.333764i
\(520\) 0 0
\(521\) −931.151 −1.78724 −0.893619 0.448826i \(-0.851842\pi\)
−0.893619 + 0.448826i \(0.851842\pi\)
\(522\) 0 0
\(523\) 227.656 0.435289 0.217645 0.976028i \(-0.430163\pi\)
0.217645 + 0.976028i \(0.430163\pi\)
\(524\) 0 0
\(525\) 127.439 232.470i 0.242742 0.442799i
\(526\) 0 0
\(527\) 137.602 0.261104
\(528\) 0 0
\(529\) −43.1902 −0.0816451
\(530\) 0 0
\(531\) 39.2511i 0.0739191i
\(532\) 0 0
\(533\) 187.267i 0.351346i
\(534\) 0 0
\(535\) −580.115 + 343.548i −1.08433 + 0.642145i
\(536\) 0 0
\(537\) 655.605i 1.22087i
\(538\) 0 0
\(539\) 211.599i 0.392576i
\(540\) 0 0
\(541\) 388.174 0.717511 0.358756 0.933431i \(-0.383201\pi\)
0.358756 + 0.933431i \(0.383201\pi\)
\(542\) 0 0
\(543\) −514.296 −0.947139
\(544\) 0 0
\(545\) −121.453 + 71.9254i −0.222850 + 0.131973i
\(546\) 0 0
\(547\) −473.059 −0.864824 −0.432412 0.901676i \(-0.642337\pi\)
−0.432412 + 0.901676i \(0.642337\pi\)
\(548\) 0 0
\(549\) 76.5916 0.139511
\(550\) 0 0
\(551\) 725.804i 1.31725i
\(552\) 0 0
\(553\) 375.961i 0.679857i
\(554\) 0 0
\(555\) −486.475 821.463i −0.876532 1.48011i
\(556\) 0 0
\(557\) 419.101i 0.752426i 0.926533 + 0.376213i \(0.122774\pi\)
−0.926533 + 0.376213i \(0.877226\pi\)
\(558\) 0 0
\(559\) 416.594i 0.745248i
\(560\) 0 0
\(561\) 88.8241 0.158332
\(562\) 0 0
\(563\) 145.910 0.259165 0.129582 0.991569i \(-0.458636\pi\)
0.129582 + 0.991569i \(0.458636\pi\)
\(564\) 0 0
\(565\) −484.211 817.640i −0.857011 1.44715i
\(566\) 0 0
\(567\) 255.189 0.450069
\(568\) 0 0
\(569\) 950.513 1.67050 0.835249 0.549872i \(-0.185323\pi\)
0.835249 + 0.549872i \(0.185323\pi\)
\(570\) 0 0
\(571\) 404.107i 0.707717i −0.935299 0.353859i \(-0.884869\pi\)
0.935299 0.353859i \(-0.115131\pi\)
\(572\) 0 0
\(573\) 340.023i 0.593408i
\(574\) 0 0
\(575\) −264.882 + 483.186i −0.460664 + 0.840324i
\(576\) 0 0
\(577\) 847.944i 1.46957i 0.678298 + 0.734787i \(0.262718\pi\)
−0.678298 + 0.734787i \(0.737282\pi\)
\(578\) 0 0
\(579\) 447.193i 0.772354i
\(580\) 0 0
\(581\) 485.723 0.836011
\(582\) 0 0
\(583\) −245.014 −0.420264
\(584\) 0 0
\(585\) 58.0179 + 97.9692i 0.0991760 + 0.167469i
\(586\) 0 0
\(587\) 658.243 1.12137 0.560684 0.828030i \(-0.310538\pi\)
0.560684 + 0.828030i \(0.310538\pi\)
\(588\) 0 0
\(589\) 957.010 1.62480
\(590\) 0 0
\(591\) 312.735i 0.529162i
\(592\) 0 0
\(593\) 282.430i 0.476274i −0.971232 0.238137i \(-0.923463\pi\)
0.971232 0.238137i \(-0.0765367\pi\)
\(594\) 0 0
\(595\) 51.0920 + 86.2739i 0.0858688 + 0.144998i
\(596\) 0 0
\(597\) 897.739i 1.50375i
\(598\) 0 0
\(599\) 498.597i 0.832382i 0.909277 + 0.416191i \(0.136635\pi\)
−0.909277 + 0.416191i \(0.863365\pi\)
\(600\) 0 0
\(601\) −287.496 −0.478363 −0.239182 0.970975i \(-0.576879\pi\)
−0.239182 + 0.970975i \(0.576879\pi\)
\(602\) 0 0
\(603\) −151.934 −0.251964
\(604\) 0 0
\(605\) 355.662 210.625i 0.587871 0.348141i
\(606\) 0 0
\(607\) −844.260 −1.39087 −0.695437 0.718587i \(-0.744789\pi\)
−0.695437 + 0.718587i \(0.744789\pi\)
\(608\) 0 0
\(609\) −212.461 −0.348869
\(610\) 0 0
\(611\) 1066.46i 1.74543i
\(612\) 0 0
\(613\) 975.340i 1.59109i 0.605892 + 0.795547i \(0.292816\pi\)
−0.605892 + 0.795547i \(0.707184\pi\)
\(614\) 0 0
\(615\) −137.720 + 81.5588i −0.223935 + 0.132616i
\(616\) 0 0
\(617\) 319.229i 0.517389i 0.965959 + 0.258695i \(0.0832923\pi\)
−0.965959 + 0.258695i \(0.916708\pi\)
\(618\) 0 0
\(619\) 845.837i 1.36646i 0.730205 + 0.683228i \(0.239425\pi\)
−0.730205 + 0.683228i \(0.760575\pi\)
\(620\) 0 0
\(621\) −632.190 −1.01802
\(622\) 0 0
\(623\) −514.489 −0.825826
\(624\) 0 0
\(625\) −336.153 526.902i −0.537846 0.843043i
\(626\) 0 0
\(627\) 617.764 0.985269
\(628\) 0 0
\(629\) 361.080 0.574055
\(630\) 0 0
\(631\) 322.653i 0.511335i −0.966765 0.255668i \(-0.917705\pi\)
0.966765 0.255668i \(-0.0822953\pi\)
\(632\) 0 0
\(633\) 360.087i 0.568858i
\(634\) 0 0
\(635\) 258.409 153.032i 0.406944 0.240995i
\(636\) 0 0
\(637\) 550.712i 0.864541i
\(638\) 0 0
\(639\) 100.058i 0.156585i
\(640\) 0 0
\(641\) 167.659 0.261558 0.130779 0.991412i \(-0.458252\pi\)
0.130779 + 0.991412i \(0.458252\pi\)
\(642\) 0 0
\(643\) −118.227 −0.183867 −0.0919335 0.995765i \(-0.529305\pi\)
−0.0919335 + 0.995765i \(0.529305\pi\)
\(644\) 0 0
\(645\) −306.372 + 181.435i −0.474995 + 0.281295i
\(646\) 0 0
\(647\) 783.464 1.21092 0.605459 0.795876i \(-0.292989\pi\)
0.605459 + 0.795876i \(0.292989\pi\)
\(648\) 0 0
\(649\) −171.950 −0.264946
\(650\) 0 0
\(651\) 280.141i 0.430324i
\(652\) 0 0
\(653\) 28.4352i 0.0435455i −0.999763 0.0217727i \(-0.993069\pi\)
0.999763 0.0217727i \(-0.00693102\pi\)
\(654\) 0 0
\(655\) 285.314 + 481.781i 0.435593 + 0.735543i
\(656\) 0 0
\(657\) 52.9051i 0.0805253i
\(658\) 0 0
\(659\) 594.296i 0.901814i 0.892571 + 0.450907i \(0.148899\pi\)
−0.892571 + 0.450907i \(0.851101\pi\)
\(660\) 0 0
\(661\) −495.511 −0.749638 −0.374819 0.927098i \(-0.622295\pi\)
−0.374819 + 0.927098i \(0.622295\pi\)
\(662\) 0 0
\(663\) 231.176 0.348682
\(664\) 0 0
\(665\) 355.340 + 600.028i 0.534346 + 0.902297i
\(666\) 0 0
\(667\) 441.599 0.662067
\(668\) 0 0
\(669\) −257.231 −0.384501
\(670\) 0 0
\(671\) 335.530i 0.500045i
\(672\) 0 0
\(673\) 168.874i 0.250927i −0.992098 0.125464i \(-0.959958\pi\)
0.992098 0.125464i \(-0.0400418\pi\)
\(674\) 0 0
\(675\) 344.693 628.775i 0.510657 0.931519i
\(676\) 0 0
\(677\) 895.899i 1.32334i 0.749797 + 0.661668i \(0.230151\pi\)
−0.749797 + 0.661668i \(0.769849\pi\)
\(678\) 0 0
\(679\) 24.6745i 0.0363394i
\(680\) 0 0
\(681\) 41.3042 0.0606522
\(682\) 0 0
\(683\) 359.410 0.526223 0.263112 0.964765i \(-0.415251\pi\)
0.263112 + 0.964765i \(0.415251\pi\)
\(684\) 0 0
\(685\) −108.647 183.462i −0.158609 0.267828i
\(686\) 0 0
\(687\) 396.926 0.577767
\(688\) 0 0
\(689\) −637.680 −0.925516
\(690\) 0 0
\(691\) 515.701i 0.746311i 0.927769 + 0.373155i \(0.121724\pi\)
−0.927769 + 0.373155i \(0.878276\pi\)
\(692\) 0 0
\(693\) 33.6857i 0.0486085i
\(694\) 0 0
\(695\) −566.454 956.514i −0.815041 1.37628i
\(696\) 0 0
\(697\) 60.5360i 0.0868523i
\(698\) 0 0
\(699\) 348.510i 0.498584i
\(700\) 0 0
\(701\) −1370.37 −1.95488 −0.977438 0.211221i \(-0.932256\pi\)
−0.977438 + 0.211221i \(0.932256\pi\)
\(702\) 0 0
\(703\) 2511.28 3.57224
\(704\) 0 0
\(705\) −784.293 + 464.463i −1.11247 + 0.658813i
\(706\) 0 0
\(707\) −468.018 −0.661978
\(708\) 0 0
\(709\) −662.128 −0.933890 −0.466945 0.884286i \(-0.654645\pi\)
−0.466945 + 0.884286i \(0.654645\pi\)
\(710\) 0 0
\(711\) 138.008i 0.194104i
\(712\) 0 0
\(713\) 582.270i 0.816649i
\(714\) 0 0
\(715\) −429.181 + 254.163i −0.600253 + 0.355473i
\(716\) 0 0
\(717\) 4.55302i 0.00635010i
\(718\) 0 0
\(719\) 370.003i 0.514608i 0.966331 + 0.257304i \(0.0828341\pi\)
−0.966331 + 0.257304i \(0.917166\pi\)
\(720\) 0 0
\(721\) −38.3415 −0.0531782
\(722\) 0 0
\(723\) −569.964 −0.788331
\(724\) 0 0
\(725\) −240.776 + 439.214i −0.332105 + 0.605812i
\(726\) 0 0
\(727\) 607.695 0.835894 0.417947 0.908471i \(-0.362750\pi\)
0.417947 + 0.908471i \(0.362750\pi\)
\(728\) 0 0
\(729\) 804.700 1.10384
\(730\) 0 0
\(731\) 134.668i 0.184224i
\(732\) 0 0
\(733\) 1066.76i 1.45533i 0.685931 + 0.727666i \(0.259395\pi\)
−0.685931 + 0.727666i \(0.740605\pi\)
\(734\) 0 0
\(735\) 405.005 239.847i 0.551027 0.326322i
\(736\) 0 0
\(737\) 665.591i 0.903108i
\(738\) 0 0
\(739\) 558.366i 0.755570i 0.925893 + 0.377785i \(0.123314\pi\)
−0.925893 + 0.377785i \(0.876686\pi\)
\(740\) 0 0
\(741\) 1607.81 2.16978
\(742\) 0 0
\(743\) −1112.00 −1.49664 −0.748319 0.663339i \(-0.769139\pi\)
−0.748319 + 0.663339i \(0.769139\pi\)
\(744\) 0 0
\(745\) 86.4659 51.2056i 0.116062 0.0687324i
\(746\) 0 0
\(747\) 178.299 0.238687
\(748\) 0 0
\(749\) 519.137 0.693107
\(750\) 0 0
\(751\) 1207.34i 1.60764i −0.594873 0.803820i \(-0.702798\pi\)
0.594873 0.803820i \(-0.297202\pi\)
\(752\) 0 0
\(753\) 204.158i 0.271127i
\(754\) 0 0
\(755\) 219.242 + 370.213i 0.290387 + 0.490348i
\(756\) 0 0
\(757\) 87.7776i 0.115955i 0.998318 + 0.0579773i \(0.0184651\pi\)
−0.998318 + 0.0579773i \(0.981535\pi\)
\(758\) 0 0
\(759\) 375.864i 0.495209i
\(760\) 0 0
\(761\) 67.0202 0.0880686 0.0440343 0.999030i \(-0.485979\pi\)
0.0440343 + 0.999030i \(0.485979\pi\)
\(762\) 0 0
\(763\) 108.687 0.142447
\(764\) 0 0
\(765\) 18.7549 + 31.6695i 0.0245162 + 0.0413980i
\(766\) 0 0
\(767\) −447.522 −0.583470
\(768\) 0 0
\(769\) 342.869 0.445863 0.222932 0.974834i \(-0.428437\pi\)
0.222932 + 0.974834i \(0.428437\pi\)
\(770\) 0 0
\(771\) 756.587i 0.981306i
\(772\) 0 0
\(773\) 244.756i 0.316631i −0.987389 0.158316i \(-0.949394\pi\)
0.987389 0.158316i \(-0.0506063\pi\)
\(774\) 0 0
\(775\) 579.126 + 317.475i 0.747259 + 0.409646i
\(776\) 0 0
\(777\) 735.116i 0.946095i
\(778\) 0 0
\(779\) 421.023i 0.540466i
\(780\) 0 0
\(781\) −438.330 −0.561242
\(782\) 0 0
\(783\) −574.657 −0.733917
\(784\) 0 0
\(785\) −40.8516 68.9821i −0.0520403 0.0878753i
\(786\) 0 0
\(787\) −1120.38 −1.42361 −0.711806 0.702376i \(-0.752123\pi\)
−0.711806 + 0.702376i \(0.752123\pi\)
\(788\) 0 0
\(789\) 208.063 0.263705
\(790\) 0 0
\(791\) 731.695i 0.925025i
\(792\) 0 0
\(793\) 873.260i 1.10121i
\(794\) 0 0
\(795\) 277.723 + 468.963i 0.349337 + 0.589891i
\(796\) 0 0
\(797\) 1344.51i 1.68696i −0.537161 0.843480i \(-0.680503\pi\)
0.537161 0.843480i \(-0.319497\pi\)
\(798\) 0 0
\(799\) 344.742i 0.431467i
\(800\) 0 0
\(801\) −188.859