Properties

Label 1280.3.e.f.639.5
Level $1280$
Weight $3$
Character 1280.639
Analytic conductor $34.877$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1280.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.8774738381\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
Defining polynomial: \(x^{6} + 9 x^{4} + 14 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 639.5
Root \(1.37720i\) of defining polynomial
Character \(\chi\) \(=\) 1280.639
Dual form 1280.3.e.f.639.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.75441i q^{3} +(-2.54778 + 4.30219i) q^{5} +3.84997 q^{7} +1.41325 q^{9} +O(q^{10})\) \(q+2.75441i q^{3} +(-2.54778 + 4.30219i) q^{5} +3.84997 q^{7} +1.41325 q^{9} -6.19112 q^{11} +16.1132 q^{13} +(-11.8500 - 7.01762i) q^{15} +5.20875i q^{17} +36.2264 q^{19} +10.6044i q^{21} +22.0411 q^{23} +(-12.0176 - 21.9221i) q^{25} +28.6823i q^{27} +20.0352i q^{29} -26.4175i q^{31} -17.0529i q^{33} +(-9.80888 + 16.5633i) q^{35} +69.3219 q^{37} +44.3822i q^{39} -11.6220 q^{41} +25.8542i q^{43} +(-3.60065 + 6.08006i) q^{45} +66.1853 q^{47} -34.1777 q^{49} -14.3470 q^{51} -39.5751 q^{53} +(15.7736 - 26.6354i) q^{55} +99.7821i q^{57} -27.7736 q^{59} +54.1954i q^{61} +5.44096 q^{63} +(-41.0529 + 69.3219i) q^{65} -107.507i q^{67} +60.7101i q^{69} -70.7997i q^{71} +37.4351i q^{73} +(60.3822 - 33.1014i) q^{75} -23.8356 q^{77} -97.6530i q^{79} -66.2835 q^{81} +126.163i q^{83} +(-22.4090 - 13.2707i) q^{85} -55.1852 q^{87} -133.635 q^{89} +62.0352 q^{91} +72.7645 q^{93} +(-92.2969 + 155.853i) q^{95} -6.40900i q^{97} -8.74960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{5} - 12 q^{7} - 18 q^{9} + O(q^{10}) \) \( 6 q - 8 q^{5} - 12 q^{7} - 18 q^{9} - 8 q^{11} - 36 q^{15} + 24 q^{19} + 68 q^{23} + 10 q^{25} - 88 q^{35} + 208 q^{37} + 68 q^{41} + 232 q^{45} + 268 q^{47} - 62 q^{49} - 192 q^{51} - 64 q^{53} + 288 q^{55} - 360 q^{59} - 172 q^{63} + 304 q^{75} - 400 q^{77} + 238 q^{81} - 304 q^{85} - 584 q^{87} - 76 q^{89} + 208 q^{91} + 320 q^{93} - 32 q^{95} + 856 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 2.75441i 0.918135i 0.888401 + 0.459068i \(0.151816\pi\)
−0.888401 + 0.459068i \(0.848184\pi\)
\(4\) 0 0
\(5\) −2.54778 + 4.30219i −0.509556 + 0.860437i
\(6\) 0 0
\(7\) 3.84997 0.549995 0.274998 0.961445i \(-0.411323\pi\)
0.274998 + 0.961445i \(0.411323\pi\)
\(8\) 0 0
\(9\) 1.41325 0.157028
\(10\) 0 0
\(11\) −6.19112 −0.562829 −0.281415 0.959586i \(-0.590804\pi\)
−0.281415 + 0.959586i \(0.590804\pi\)
\(12\) 0 0
\(13\) 16.1132 1.23948 0.619738 0.784809i \(-0.287239\pi\)
0.619738 + 0.784809i \(0.287239\pi\)
\(14\) 0 0
\(15\) −11.8500 7.01762i −0.789998 0.467842i
\(16\) 0 0
\(17\) 5.20875i 0.306397i 0.988195 + 0.153198i \(0.0489574\pi\)
−0.988195 + 0.153198i \(0.951043\pi\)
\(18\) 0 0
\(19\) 36.2264 1.90665 0.953326 0.301944i \(-0.0976357\pi\)
0.953326 + 0.301944i \(0.0976357\pi\)
\(20\) 0 0
\(21\) 10.6044i 0.504970i
\(22\) 0 0
\(23\) 22.0411 0.958308 0.479154 0.877731i \(-0.340943\pi\)
0.479154 + 0.877731i \(0.340943\pi\)
\(24\) 0 0
\(25\) −12.0176 21.9221i −0.480705 0.876882i
\(26\) 0 0
\(27\) 28.6823i 1.06231i
\(28\) 0 0
\(29\) 20.0352i 0.690871i 0.938443 + 0.345435i \(0.112269\pi\)
−0.938443 + 0.345435i \(0.887731\pi\)
\(30\) 0 0
\(31\) 26.4175i 0.852177i −0.904681 0.426089i \(-0.859891\pi\)
0.904681 0.426089i \(-0.140109\pi\)
\(32\) 0 0
\(33\) 17.0529i 0.516754i
\(34\) 0 0
\(35\) −9.80888 + 16.5633i −0.280254 + 0.473237i
\(36\) 0 0
\(37\) 69.3219 1.87357 0.936783 0.349911i \(-0.113788\pi\)
0.936783 + 0.349911i \(0.113788\pi\)
\(38\) 0 0
\(39\) 44.3822i 1.13801i
\(40\) 0 0
\(41\) −11.6220 −0.283463 −0.141732 0.989905i \(-0.545267\pi\)
−0.141732 + 0.989905i \(0.545267\pi\)
\(42\) 0 0
\(43\) 25.8542i 0.601261i 0.953741 + 0.300630i \(0.0971971\pi\)
−0.953741 + 0.300630i \(0.902803\pi\)
\(44\) 0 0
\(45\) −3.60065 + 6.08006i −0.0800144 + 0.135112i
\(46\) 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
\(51\) −14.3470 −0.281314
\(52\) 0 0
\(53\) −39.5751 −0.746699 −0.373350 0.927691i \(-0.621791\pi\)
−0.373350 + 0.927691i \(0.621791\pi\)
\(54\) 0 0
\(55\) 15.7736 26.6354i 0.286793 0.484280i
\(56\) 0 0
\(57\) 99.7821i 1.75056i
\(58\) 0 0
\(59\) −27.7736 −0.470739 −0.235370 0.971906i \(-0.575630\pi\)
−0.235370 + 0.971906i \(0.575630\pi\)
\(60\) 0 0
\(61\) 54.1954i 0.888449i 0.895916 + 0.444224i \(0.146521\pi\)
−0.895916 + 0.444224i \(0.853479\pi\)
\(62\) 0 0
\(63\) 5.44096 0.0863645
\(64\) 0 0
\(65\) −41.0529 + 69.3219i −0.631583 + 1.06649i
\(66\) 0 0
\(67\) 107.507i 1.60459i −0.596931 0.802293i \(-0.703613\pi\)
0.596931 0.802293i \(-0.296387\pi\)
\(68\) 0 0
\(69\) 60.7101i 0.879857i
\(70\) 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
\(75\) 60.3822 33.1014i 0.805097 0.441352i
\(76\) 0 0
\(77\) −23.8356 −0.309554
\(78\) 0 0
\(79\) 97.6530i 1.23611i −0.786133 0.618057i \(-0.787920\pi\)
0.786133 0.618057i \(-0.212080\pi\)
\(80\) 0 0
\(81\) −66.2835 −0.818315
\(82\) 0 0
\(83\) 126.163i 1.52003i 0.649904 + 0.760017i \(0.274809\pi\)
−0.649904 + 0.760017i \(0.725191\pi\)
\(84\) 0 0
\(85\) −22.4090 13.2707i −0.263635 0.156126i
\(86\) 0 0
\(87\) −55.1852 −0.634313
\(88\) 0 0
\(89\) −133.635 −1.50151 −0.750757 0.660579i \(-0.770311\pi\)
−0.750757 + 0.660579i \(0.770311\pi\)
\(90\) 0 0
\(91\) 62.0352 0.681706
\(92\) 0 0
\(93\) 72.7645 0.782414
\(94\) 0 0
\(95\) −92.2969 + 155.853i −0.971546 + 1.64055i
\(96\) 0 0
\(97\) 6.40900i 0.0660722i −0.999454 0.0330361i \(-0.989482\pi\)
0.999454 0.0330361i \(-0.0105176\pi\)
\(98\) 0 0
\(99\) −8.74960 −0.0883798
\(100\) 0 0
\(101\) 121.564i 1.20361i 0.798644 + 0.601803i \(0.205551\pi\)
−0.798644 + 0.601803i \(0.794449\pi\)
\(102\) 0 0
\(103\) −9.95891 −0.0966884 −0.0483442 0.998831i \(-0.515394\pi\)
−0.0483442 + 0.998831i \(0.515394\pi\)
\(104\) 0 0
\(105\) −45.6220 27.0176i −0.434495 0.257311i
\(106\) 0 0
\(107\) 134.842i 1.26020i −0.776512 0.630102i \(-0.783013\pi\)
0.776512 0.630102i \(-0.216987\pi\)
\(108\) 0 0
\(109\) 28.2306i 0.258996i 0.991580 + 0.129498i \(0.0413367\pi\)
−0.991580 + 0.129498i \(0.958663\pi\)
\(110\) 0 0
\(111\) 190.941i 1.72019i
\(112\) 0 0
\(113\) 190.052i 1.68188i 0.541130 + 0.840939i \(0.317997\pi\)
−0.541130 + 0.840939i \(0.682003\pi\)
\(114\) 0 0
\(115\) −56.1559 + 94.8249i −0.488312 + 0.824564i
\(116\) 0 0
\(117\) 22.7719 0.194632
\(118\) 0 0
\(119\) 20.0535i 0.168517i
\(120\) 0 0
\(121\) −82.6700 −0.683223
\(122\) 0 0
\(123\) 32.0117i 0.260258i
\(124\) 0 0
\(125\) 124.931 + 4.15055i 0.999449 + 0.0332044i
\(126\) 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
\(131\) −111.985 −0.854848 −0.427424 0.904051i \(-0.640579\pi\)
−0.427424 + 0.904051i \(0.640579\pi\)
\(132\) 0 0
\(133\) 139.470 1.04865
\(134\) 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
\(139\) −222.332 −1.59951 −0.799756 0.600325i \(-0.795038\pi\)
−0.799756 + 0.600325i \(0.795038\pi\)
\(140\) 0 0
\(141\) 182.301i 1.29292i
\(142\) 0 0
\(143\) −99.7587 −0.697614
\(144\) 0 0
\(145\) −86.1954 51.0454i −0.594451 0.352037i
\(146\) 0 0
\(147\) 94.1394i 0.640404i
\(148\) 0 0
\(149\) 20.0981i 0.134887i 0.997723 + 0.0674434i \(0.0214842\pi\)
−0.997723 + 0.0674434i \(0.978516\pi\)
\(150\) 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
\(155\) 113.653 + 67.3060i 0.733245 + 0.434232i
\(156\) 0 0
\(157\) 16.0342 0.102129 0.0510643 0.998695i \(-0.483739\pi\)
0.0510643 + 0.998695i \(0.483739\pi\)
\(158\) 0 0
\(159\) 109.006i 0.685571i
\(160\) 0 0
\(161\) 84.8575 0.527065
\(162\) 0 0
\(163\) 179.157i 1.09912i −0.835454 0.549561i \(-0.814795\pi\)
0.835454 0.549561i \(-0.185205\pi\)
\(164\) 0 0
\(165\) 73.3646 + 43.4470i 0.444634 + 0.263315i
\(166\) 0 0
\(167\) 137.800 0.825149 0.412574 0.910924i \(-0.364630\pi\)
0.412574 + 0.910924i \(0.364630\pi\)
\(168\) 0 0
\(169\) 90.6347 0.536300
\(170\) 0 0
\(171\) 51.1969 0.299397
\(172\) 0 0
\(173\) −62.8895 −0.363523 −0.181762 0.983343i \(-0.558180\pi\)
−0.181762 + 0.983343i \(0.558180\pi\)
\(174\) 0 0
\(175\) −46.2675 84.3992i −0.264385 0.482281i
\(176\) 0 0
\(177\) 76.4998i 0.432203i
\(178\) 0 0
\(179\) 238.020 1.32972 0.664861 0.746967i \(-0.268491\pi\)
0.664861 + 0.746967i \(0.268491\pi\)
\(180\) 0 0
\(181\) 186.718i 1.03159i 0.856712 + 0.515795i \(0.172503\pi\)
−0.856712 + 0.515795i \(0.827497\pi\)
\(182\) 0 0
\(183\) −149.276 −0.815716
\(184\) 0 0
\(185\) −176.617 + 298.236i −0.954687 + 1.61209i
\(186\) 0 0
\(187\) 32.2480i 0.172449i
\(188\) 0 0
\(189\) 110.426i 0.584264i
\(190\) 0 0
\(191\) 123.447i 0.646319i −0.946344 0.323160i \(-0.895255\pi\)
0.946344 0.323160i \(-0.104745\pi\)
\(192\) 0 0
\(193\) 162.355i 0.841220i −0.907242 0.420610i \(-0.861816\pi\)
0.907242 0.420610i \(-0.138184\pi\)
\(194\) 0 0
\(195\) −190.941 113.076i −0.979183 0.579878i
\(196\) 0 0
\(197\) −113.540 −0.576344 −0.288172 0.957579i \(-0.593048\pi\)
−0.288172 + 0.957579i \(0.593048\pi\)
\(198\) 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
\(203\) 77.1351i 0.379976i
\(204\) 0 0
\(205\) 29.6103 50.0000i 0.144441 0.243902i
\(206\) 0 0
\(207\) 31.1496 0.150481
\(208\) 0 0
\(209\) −224.282 −1.07312
\(210\) 0 0
\(211\) 130.731 0.619580 0.309790 0.950805i \(-0.399741\pi\)
0.309790 + 0.950805i \(0.399741\pi\)
\(212\) 0 0
\(213\) 195.011 0.915546
\(214\) 0 0
\(215\) −111.230 65.8709i −0.517347 0.306376i
\(216\) 0 0
\(217\) 101.707i 0.468694i
\(218\) 0 0
\(219\) −103.112 −0.470829
\(220\) 0 0
\(221\) 83.9295i 0.379772i
\(222\) 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
\(227\) 14.9957i 0.0660602i 0.999454 + 0.0330301i \(0.0105157\pi\)
−0.999454 + 0.0330301i \(0.989484\pi\)
\(228\) 0 0
\(229\) 144.106i 0.629283i −0.949211 0.314641i \(-0.898116\pi\)
0.949211 0.314641i \(-0.101884\pi\)
\(230\) 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
\(235\) −168.626 + 284.741i −0.717556 + 1.21167i
\(236\) 0 0
\(237\) 268.976 1.13492
\(238\) 0 0
\(239\) 1.65300i 0.00691630i −0.999994 0.00345815i \(-0.998899\pi\)
0.999994 0.00345815i \(-0.00110077\pi\)
\(240\) 0 0
\(241\) 206.928 0.858622 0.429311 0.903157i \(-0.358756\pi\)
0.429311 + 0.903157i \(0.358756\pi\)
\(242\) 0 0
\(243\) 75.5692i 0.310984i
\(244\) 0 0
\(245\) 87.0774 147.039i 0.355418 0.600159i
\(246\) 0 0
\(247\) 583.722 2.36325
\(248\) 0 0
\(249\) −347.503 −1.39560
\(250\) 0 0
\(251\) −74.1206 −0.295301 −0.147651 0.989040i \(-0.547171\pi\)
−0.147651 + 0.989040i \(0.547171\pi\)
\(252\) 0 0
\(253\) −136.459 −0.539364
\(254\) 0 0
\(255\) 36.5530 61.7235i 0.143345 0.242053i
\(256\) 0 0
\(257\) 274.682i 1.06880i 0.845231 + 0.534402i \(0.179463\pi\)
−0.845231 + 0.534402i \(0.820537\pi\)
\(258\) 0 0
\(259\) 266.887 1.03045
\(260\) 0 0
\(261\) 28.3148i 0.108486i
\(262\) 0 0
\(263\) 75.5382 0.287218 0.143609 0.989635i \(-0.454129\pi\)
0.143609 + 0.989635i \(0.454129\pi\)
\(264\) 0 0
\(265\) 100.829 170.259i 0.380485 0.642488i
\(266\) 0 0
\(267\) 368.084i 1.37859i
\(268\) 0 0
\(269\) 314.087i 1.16761i −0.811895 0.583804i \(-0.801564\pi\)
0.811895 0.583804i \(-0.198436\pi\)
\(270\) 0 0
\(271\) 128.158i 0.472908i −0.971643 0.236454i \(-0.924015\pi\)
0.971643 0.236454i \(-0.0759852\pi\)
\(272\) 0 0
\(273\) 170.870i 0.625898i
\(274\) 0 0
\(275\) 74.4026 + 135.722i 0.270555 + 0.493535i
\(276\) 0 0
\(277\) −242.118 −0.874073 −0.437037 0.899444i \(-0.643972\pi\)
−0.437037 + 0.899444i \(0.643972\pi\)
\(278\) 0 0
\(279\) 37.3345i 0.133815i
\(280\) 0 0
\(281\) −28.8562 −0.102691 −0.0513456 0.998681i \(-0.516351\pi\)
−0.0513456 + 0.998681i \(0.516351\pi\)
\(282\) 0 0
\(283\) 269.993i 0.954039i 0.878893 + 0.477020i \(0.158283\pi\)
−0.878893 + 0.477020i \(0.841717\pi\)
\(284\) 0 0
\(285\) −429.281 254.223i −1.50625 0.892011i
\(286\) 0 0
\(287\) −44.7443 −0.155904
\(288\) 0 0
\(289\) 261.869 0.906121
\(290\) 0 0
\(291\) 17.6530 0.0606632
\(292\) 0 0
\(293\) −353.448 −1.20631 −0.603154 0.797625i \(-0.706089\pi\)
−0.603154 + 0.797625i \(0.706089\pi\)
\(294\) 0 0
\(295\) 70.7611 119.487i 0.239868 0.405042i
\(296\) 0 0
\(297\) 177.576i 0.597898i
\(298\) 0 0
\(299\) 355.152 1.18780
\(300\) 0 0
\(301\) 99.5379i 0.330691i
\(302\) 0 0
\(303\) −334.837 −1.10507
\(304\) 0 0
\(305\) −233.159 138.078i −0.764454 0.452715i
\(306\) 0 0
\(307\) 260.946i 0.849985i 0.905197 + 0.424993i \(0.139723\pi\)
−0.905197 + 0.424993i \(0.860277\pi\)
\(308\) 0 0
\(309\) 27.4309i 0.0887730i
\(310\) 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
\(315\) −13.8624 + 23.4080i −0.0440076 + 0.0743113i
\(316\) 0 0
\(317\) −9.22805 −0.0291106 −0.0145553 0.999894i \(-0.504633\pi\)
−0.0145553 + 0.999894i \(0.504633\pi\)
\(318\) 0 0
\(319\) 124.041i 0.388842i
\(320\) 0 0
\(321\) 371.409 1.15704
\(322\) 0 0
\(323\) 188.694i 0.584192i
\(324\) 0 0
\(325\) −193.642 353.234i −0.595822 1.08687i
\(326\) 0 0
\(327\) −77.7586 −0.237794
\(328\) 0 0
\(329\) 254.811 0.774502
\(330\) 0 0
\(331\) 53.3799 0.161268 0.0806342 0.996744i \(-0.474305\pi\)
0.0806342 + 0.996744i \(0.474305\pi\)
\(332\) 0 0
\(333\) 97.9692 0.294202
\(334\) 0 0
\(335\) 462.516 + 273.905i 1.38065 + 0.817626i
\(336\) 0 0
\(337\) 350.458i 1.03994i 0.854186 + 0.519968i \(0.174056\pi\)
−0.854186 + 0.519968i \(0.825944\pi\)
\(338\) 0 0
\(339\) −523.481 −1.54419
\(340\) 0 0
\(341\) 163.554i 0.479630i
\(342\) 0 0
\(343\) −320.232 −0.933620
\(344\) 0 0
\(345\) −261.186 154.676i −0.757062 0.448336i
\(346\) 0 0
\(347\) 70.8302i 0.204122i 0.994778 + 0.102061i \(0.0325436\pi\)
−0.994778 + 0.102061i \(0.967456\pi\)
\(348\) 0 0
\(349\) 373.045i 1.06890i −0.845201 0.534449i \(-0.820519\pi\)
0.845201 0.534449i \(-0.179481\pi\)
\(350\) 0 0
\(351\) 462.163i 1.31670i
\(352\) 0 0
\(353\) 543.568i 1.53985i −0.638132 0.769927i \(-0.720293\pi\)
0.638132 0.769927i \(-0.279707\pi\)
\(354\) 0 0
\(355\) 304.594 + 180.382i 0.858010 + 0.508119i
\(356\) 0 0
\(357\) −55.2355 −0.154721
\(358\) 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
\(363\) 227.707i 0.627291i
\(364\) 0 0
\(365\) −161.053 95.3765i −0.441241 0.261305i
\(366\) 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
\(371\) −152.363 −0.410681
\(372\) 0 0
\(373\) −160.000 −0.428954 −0.214477 0.976729i \(-0.568805\pi\)
−0.214477 + 0.976729i \(0.568805\pi\)
\(374\) 0 0
\(375\) −11.4323 + 344.111i −0.0304862 + 0.917629i
\(376\) 0 0
\(377\) 322.832i 0.856317i
\(378\) 0 0
\(379\) 192.796 0.508698 0.254349 0.967113i \(-0.418139\pi\)
0.254349 + 0.967113i \(0.418139\pi\)
\(380\) 0 0
\(381\) 165.442i 0.434232i
\(382\) 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
\(387\) 36.5384i 0.0944146i
\(388\) 0 0
\(389\) 522.159i 1.34231i −0.741317 0.671155i \(-0.765798\pi\)
0.741317 0.671155i \(-0.234202\pi\)
\(390\) 0 0
\(391\) 114.806i 0.293623i
\(392\) 0 0
\(393\) 308.452i 0.784866i
\(394\) 0 0
\(395\) 420.121 + 248.798i 1.06360 + 0.629870i
\(396\) 0 0
\(397\) −357.537 −0.900598 −0.450299 0.892878i \(-0.648683\pi\)
−0.450299 + 0.892878i \(0.648683\pi\)
\(398\) 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
\(403\) 425.670i 1.05625i
\(404\) 0 0
\(405\) 168.876 285.164i 0.416977 0.704108i
\(406\) 0 0
\(407\) −429.181 −1.05450
\(408\) 0 0
\(409\) 63.2015 0.154527 0.0772634 0.997011i \(-0.475382\pi\)
0.0772634 + 0.997011i \(0.475382\pi\)
\(410\) 0 0
\(411\) 117.459 0.285787
\(412\) 0 0
\(413\) −106.928 −0.258905
\(414\) 0 0
\(415\) −542.776 321.435i −1.30789 0.774542i
\(416\) 0 0
\(417\) 612.393i 1.46857i
\(418\) 0 0
\(419\) −673.390 −1.60714 −0.803568 0.595213i \(-0.797068\pi\)
−0.803568 + 0.595213i \(0.797068\pi\)
\(420\) 0 0
\(421\) 84.6877i 0.201158i 0.994929 + 0.100579i \(0.0320696\pi\)
−0.994929 + 0.100579i \(0.967930\pi\)
\(422\) 0 0
\(423\) 93.5363 0.221126
\(424\) 0 0
\(425\) 114.186 62.5968i 0.268674 0.147286i
\(426\) 0 0
\(427\) 208.650i 0.488643i
\(428\) 0 0
\(429\) 274.776i 0.640504i
\(430\) 0 0
\(431\) 672.158i 1.55953i −0.626072 0.779766i \(-0.715338\pi\)
0.626072 0.779766i \(-0.284662\pi\)
\(432\) 0 0
\(433\) 562.185i 1.29835i −0.760640 0.649174i \(-0.775115\pi\)
0.760640 0.649174i \(-0.224885\pi\)
\(434\) 0 0
\(435\) 140.600 237.417i 0.323218 0.545786i
\(436\) 0 0
\(437\) 798.469 1.82716
\(438\) 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
\(443\) 461.625i 1.04204i −0.853544 0.521021i \(-0.825551\pi\)
0.853544 0.521021i \(-0.174449\pi\)
\(444\) 0 0
\(445\) 340.472 574.922i 0.765106 1.29196i
\(446\) 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
\(451\) 71.9532 0.159542
\(452\) 0 0
\(453\) −237.023 −0.523229
\(454\) 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
\(459\) −149.399 −0.325488
\(460\) 0 0
\(461\) 406.436i 0.881639i 0.897596 + 0.440820i \(0.145312\pi\)
−0.897596 + 0.440820i \(0.854688\pi\)
\(462\) 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
\(467\) 594.738i 1.27353i −0.771058 0.636765i \(-0.780272\pi\)
0.771058 0.636765i \(-0.219728\pi\)
\(468\) 0 0
\(469\) 413.899i 0.882515i
\(470\) 0 0
\(471\) 44.1647i 0.0937679i
\(472\) 0 0
\(473\) 160.067i 0.338407i
\(474\) 0 0
\(475\) −435.355 794.157i −0.916537 1.67191i
\(476\) 0 0
\(477\) −55.9294 −0.117252
\(478\) 0 0
\(479\) 534.894i 1.11669i 0.829609 + 0.558344i \(0.188563\pi\)
−0.829609 + 0.558344i \(0.811437\pi\)
\(480\) 0 0
\(481\) 1117.00 2.32224
\(482\) 0 0
\(483\) 233.732i 0.483917i
\(484\) 0 0
\(485\) 27.5727 + 16.3287i 0.0568510 + 0.0336675i
\(486\) 0 0
\(487\) −264.298 −0.542706 −0.271353 0.962480i \(-0.587471\pi\)
−0.271353 + 0.962480i \(0.587471\pi\)
\(488\) 0 0
\(489\) 493.471 1.00914
\(490\) 0 0
\(491\) −539.150 −1.09807 −0.549033 0.835801i \(-0.685004\pi\)
−0.549033 + 0.835801i \(0.685004\pi\)
\(492\) 0 0
\(493\) −104.359 −0.211681
\(494\) 0 0
\(495\) 22.2921 37.6424i 0.0450345 0.0760453i
\(496\) 0 0
\(497\) 272.577i 0.548444i
\(498\) 0 0
\(499\) 138.218 0.276991 0.138495 0.990363i \(-0.455773\pi\)
0.138495 + 0.990363i \(0.455773\pi\)
\(500\) 0 0
\(501\) 379.557i 0.757598i
\(502\) 0 0
\(503\) 389.170 0.773697 0.386848 0.922143i \(-0.373564\pi\)
0.386848 + 0.922143i \(0.373564\pi\)
\(504\) 0 0
\(505\) −522.992 309.719i −1.03563 0.613305i
\(506\) 0 0
\(507\) 249.645i 0.492396i
\(508\) 0 0
\(509\) 468.599i 0.920627i 0.887756 + 0.460314i \(0.152263\pi\)
−0.887756 + 0.460314i \(0.847737\pi\)
\(510\) 0 0
\(511\) 144.124i 0.282043i
\(512\) 0 0
\(513\) 1039.06i 2.02545i
\(514\) 0 0
\(515\) 25.3731 42.8451i 0.0492682 0.0831943i
\(516\) 0 0
\(517\) −409.761 −0.792575
\(518\) 0 0
\(519\) 173.223i 0.333764i
\(520\) 0 0
\(521\) 931.151 1.78724 0.893619 0.448826i \(-0.148158\pi\)
0.893619 + 0.448826i \(0.148158\pi\)
\(522\) 0 0
\(523\) 227.656i 0.435289i 0.976028 + 0.217645i \(0.0698374\pi\)
−0.976028 + 0.217645i \(0.930163\pi\)
\(524\) 0 0
\(525\) 232.470 127.439i 0.442799 0.242742i
\(526\) 0 0
\(527\) 137.602 0.261104
\(528\) 0 0
\(529\) −43.1902 −0.0816451
\(530\) 0 0
\(531\) −39.2511 −0.0739191
\(532\) 0 0
\(533\) −187.267 −0.351346
\(534\) 0 0
\(535\) 580.115 + 343.548i 1.08433 + 0.642145i
\(536\) 0 0
\(537\) 655.605i 1.22087i
\(538\) 0 0
\(539\) 211.599 0.392576
\(540\) 0 0
\(541\) 388.174i 0.717511i −0.933431 0.358756i \(-0.883201\pi\)
0.933431 0.358756i \(-0.116799\pi\)
\(542\) 0 0
\(543\) −514.296 −0.947139
\(544\) 0 0
\(545\) −121.453 71.9254i −0.222850 0.131973i
\(546\) 0 0
\(547\) 473.059i 0.864824i 0.901676 + 0.432412i \(0.142337\pi\)
−0.901676 + 0.432412i \(0.857663\pi\)
\(548\) 0 0
\(549\) 76.5916i 0.139511i
\(550\) 0 0
\(551\) 725.804i 1.31725i
\(552\) 0 0
\(553\) 375.961i 0.679857i
\(554\) 0 0
\(555\) −821.463 486.475i −1.48011 0.876532i
\(556\) 0 0
\(557\) −419.101 −0.752426 −0.376213 0.926533i \(-0.622774\pi\)
−0.376213 + 0.926533i \(0.622774\pi\)
\(558\) 0 0
\(559\) 416.594i 0.745248i
\(560\) 0 0
\(561\) 88.8241 0.158332
\(562\) 0 0
\(563\) 145.910i 0.259165i −0.991569 0.129582i \(-0.958636\pi\)
0.991569 0.129582i \(-0.0413637\pi\)
\(564\) 0 0
\(565\) −817.640 484.211i −1.44715 0.857011i
\(566\) 0 0
\(567\) −255.189 −0.450069
\(568\) 0 0
\(569\) −950.513 −1.67050 −0.835249 0.549872i \(-0.814677\pi\)
−0.835249 + 0.549872i \(0.814677\pi\)
\(570\) 0 0
\(571\) −404.107 −0.707717 −0.353859 0.935299i \(-0.615131\pi\)
−0.353859 + 0.935299i \(0.615131\pi\)
\(572\) 0 0
\(573\) 340.023 0.593408
\(574\) 0 0
\(575\) −264.882 483.186i −0.460664 0.840324i
\(576\) 0 0
\(577\) 847.944i 1.46957i −0.678298 0.734787i \(-0.737282\pi\)
0.678298 0.734787i \(-0.262718\pi\)
\(578\) 0 0
\(579\) 447.193 0.772354
\(580\) 0 0
\(581\) 485.723i 0.836011i
\(582\) 0 0
\(583\) 245.014 0.420264
\(584\) 0 0
\(585\) −58.0179 + 97.9692i −0.0991760 + 0.167469i
\(586\) 0 0
\(587\) 658.243i 1.12137i 0.828030 + 0.560684i \(0.189462\pi\)
−0.828030 + 0.560684i \(0.810538\pi\)
\(588\) 0 0
\(589\) 957.010i 1.62480i
\(590\) 0 0
\(591\) 312.735i 0.529162i
\(592\) 0 0
\(593\) 282.430i 0.476274i 0.971232 + 0.238137i \(0.0765367\pi\)
−0.971232 + 0.238137i \(0.923463\pi\)
\(594\) 0 0
\(595\) −86.2739 51.0920i −0.144998 0.0858688i
\(596\) 0 0
\(597\) −897.739 −1.50375
\(598\) 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 \(-0.423121\pi\)
0.239182 + 0.970975i \(0.423121\pi\)
\(602\) 0 0
\(603\) 151.934i 0.251964i
\(604\) 0 0
\(605\) 210.625 355.662i 0.348141 0.587871i
\(606\) 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
\(611\) 1066.46 1.74543
\(612\) 0 0
\(613\) 975.340 1.59109 0.795547 0.605892i \(-0.207184\pi\)
0.795547 + 0.605892i \(0.207184\pi\)
\(614\) 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
\(619\) 845.837 1.36646 0.683228 0.730205i \(-0.260575\pi\)
0.683228 + 0.730205i \(0.260575\pi\)
\(620\) 0 0
\(621\) 632.190i 1.01802i
\(622\) 0 0
\(623\) −514.489 −0.825826
\(624\) 0 0
\(625\) −336.153 + 526.902i −0.537846 + 0.843043i
\(626\) 0 0
\(627\) 617.764i 0.985269i
\(628\) 0 0
\(629\) 361.080i 0.574055i
\(630\) 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
\(635\) −153.032 + 258.409i −0.240995 + 0.406944i
\(636\) 0 0
\(637\) −550.712 −0.864541
\(638\) 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
\(643\) 118.227i 0.183867i 0.995765 + 0.0919335i \(0.0293047\pi\)
−0.995765 + 0.0919335i \(0.970695\pi\)
\(644\) 0 0
\(645\) 181.435 306.372i 0.281295 0.474995i
\(646\) 0 0
\(647\) −783.464 −1.21092 −0.605459 0.795876i \(-0.707011\pi\)
−0.605459 + 0.795876i \(0.707011\pi\)
\(648\) 0 0
\(649\) 171.950 0.264946
\(650\) 0 0
\(651\) 280.141 0.430324
\(652\) 0 0
\(653\) 28.4352 0.0435455 0.0217727 0.999763i \(-0.493069\pi\)
0.0217727 + 0.999763i \(0.493069\pi\)
\(654\) 0 0
\(655\) 285.314 481.781i 0.435593 0.735543i
\(656\) 0 0
\(657\) 52.9051i 0.0805253i
\(658\) 0 0
\(659\) −594.296 −0.901814 −0.450907 0.892571i \(-0.648899\pi\)
−0.450907 + 0.892571i \(0.648899\pi\)
\(660\) 0 0
\(661\) 495.511i 0.749638i −0.927098 0.374819i \(-0.877705\pi\)
0.927098 0.374819i \(-0.122295\pi\)
\(662\) 0 0
\(663\) −231.176 −0.348682
\(664\) 0 0
\(665\) −355.340 + 600.028i −0.534346 + 0.902297i
\(666\) 0 0
\(667\) 441.599i 0.662067i
\(668\) 0 0
\(669\) 257.231i 0.384501i
\(670\) 0 0
\(671\) 335.530i 0.500045i
\(672\) 0 0
\(673\) 168.874i 0.250927i 0.992098 + 0.125464i \(0.0400418\pi\)
−0.992098 + 0.125464i \(0.959958\pi\)
\(674\) 0 0
\(675\) 628.775 344.693i 0.931519 0.510657i
\(676\) 0 0
\(677\) 895.899 1.32334 0.661668 0.749797i \(-0.269849\pi\)
0.661668 + 0.749797i \(0.269849\pi\)
\(678\) 0 0
\(679\) 24.6745i 0.0363394i
\(680\) 0 0
\(681\) −41.3042 −0.0606522
\(682\) 0 0
\(683\) 359.410i 0.526223i 0.964765 + 0.263112i \(0.0847487\pi\)
−0.964765 + 0.263112i \(0.915251\pi\)
\(684\) 0 0
\(685\) 183.462 + 108.647i 0.267828 + 0.158609i
\(686\) 0 0
\(687\) 396.926 0.577767
\(688\) 0 0
\(689\) −637.680 −0.925516
\(690\) 0 0
\(691\) −515.701 −0.746311 −0.373155 0.927769i \(-0.621724\pi\)
−0.373155 + 0.927769i \(0.621724\pi\)
\(692\) 0 0
\(693\) −33.6857 −0.0486085
\(694\) 0 0
\(695\) 566.454 956.514i 0.815041 1.37628i
\(696\) 0 0
\(697\) 60.5360i 0.0868523i
\(698\) 0 0
\(699\) 348.510 0.498584
\(700\) 0 0
\(701\) 1370.37i 1.95488i 0.211221 + 0.977438i \(0.432256\pi\)
−0.211221 + 0.977438i \(0.567744\pi\)
\(702\) 0 0
\(703\) 2511.28 3.57224
\(704\) 0 0
\(705\) −784.293 464.463i −1.11247 0.658813i
\(706\) 0 0
\(707\) 468.018i 0.661978i
\(708\) 0 0
\(709\) 662.128i 0.933890i −0.884286 0.466945i \(-0.845355\pi\)
0.884286 0.466945i \(-0.154645\pi\)
\(710\) 0 0
\(711\) 138.008i 0.194104i
\(712\) 0 0
\(713\) 582.270i 0.816649i
\(714\) 0 0
\(715\) 254.163 429.181i 0.355473 0.600253i
\(716\) 0 0
\(717\) 4.55302 0.00635010
\(718\) 0 0
\(719\) 370.003i 0.514608i −0.966331 0.257304i \(-0.917166\pi\)
0.966331 0.257304i \(-0.0828341\pi\)
\(720\) 0 0
\(721\) −38.3415 −0.0531782
\(722\) 0 0
\(723\) 569.964i 0.788331i
\(724\) 0 0
\(725\) 439.214 240.776i 0.605812 0.332105i
\(726\) 0 0
\(727\) −607.695 −0.835894 −0.417947 0.908471i \(-0.637250\pi\)
−0.417947 + 0.908471i \(0.637250\pi\)
\(728\) 0 0
\(729\) −804.700 −1.10384
\(730\) 0 0
\(731\) −134.668 −0.184224
\(732\) 0 0
\(733\) −1066.76 −1.45533 −0.727666 0.685931i \(-0.759395\pi\)
−0.727666 + 0.685931i \(0.759395\pi\)
\(734\) 0 0
\(735\) 405.005 + 239.847i 0.551027 + 0.326322i
\(736\) 0 0
\(737\) 665.591i 0.903108i
\(738\) 0 0
\(739\) −558.366 −0.755570 −0.377785 0.925893i \(-0.623314\pi\)
−0.377785 + 0.925893i \(0.623314\pi\)
\(740\) 0 0
\(741\) 1607.81i 2.16978i
\(742\) 0 0
\(743\) 1112.00 1.49664 0.748319 0.663339i \(-0.230861\pi\)
0.748319 + 0.663339i \(0.230861\pi\)
\(744\) 0 0
\(745\) −86.4659 51.2056i −0.116062 0.0687324i
\(746\) 0 0
\(747\) 178.299i 0.238687i
\(748\) 0 0
\(749\) 519.137i 0.693107i
\(750\) 0 0
\(751\) 1207.34i 1.60764i 0.594873 + 0.803820i \(0.297202\pi\)
−0.594873 + 0.803820i \(0.702798\pi\)
\(752\) 0 0
\(753\) 204.158i 0.271127i
\(754\) 0 0
\(755\) −370.213 219.242i −0.490348 0.290387i
\(756\) 0 0
\(757\) 87.7776 0.115955 0.0579773 0.998318i \(-0.481535\pi\)
0.0579773 + 0.998318i \(0.481535\pi\)
\(758\) 0 0
\(759\) 375.864i 0.495209i
\(760\) 0 0
\(761\) −67.0202 −0.0880686 −0.0440343 0.999030i \(-0.514021\pi\)
−0.0440343 + 0.999030i \(0.514021\pi\)
\(762\) 0 0
\(763\) 108.687i 0.142447i
\(764\) 0 0
\(765\) −31.6695 18.7549i −0.0413980 0.0245162i
\(766\) 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
\(771\) −756.587 −0.981306
\(772\) 0 0
\(773\) −244.756 −0.316631 −0.158316 0.987389i \(-0.550606\pi\)
−0.158316 + 0.987389i \(0.550606\pi\)
\(774\) 0 0
\(775\) −579.126 + 317.475i −0.747259 + 0.409646i
\(776\) 0 0
\(777\) 735.116i 0.946095i
\(778\) 0 0
\(779\) −421.023 −0.540466
\(780\) 0 0
\(781\) 438.330i 0.561242i
\(782\) 0 0
\(783\) −574.657 −0.733917
\(784\) 0 0
\(785\) −40.8516 + 68.9821i −0.0520403 + 0.0878753i
\(786\) 0 0
\(787\) 1120.38i 1.42361i 0.702376 + 0.711806i \(0.252123\pi\)
−0.702376 + 0.711806i \(0.747877\pi\)
\(788\) 0 0
\(789\) 208.063i 0.263705i
\(790\) 0 0
\(791\) 731.695i 0.925025i
\(792\) 0 0
\(793\) 873.260i 1.10121i
\(794\) 0 0
\(795\) 468.963 + 277.723i 0.589891 + 0.349337i
\(796\) 0 0
\(797\) 1344.51 1.68696 0.843480 0.537161i \(-0.180503\pi\)
0.843480 + 0.537161i \(0.180503\pi\)
\(798\) 0 0
\(799\) 344.742i 0.431467i
\(800\) 0