Properties

Label 1520.2.d.h.609.5
Level $1520$
Weight $2$
Character 1520.609
Analytic conductor $12.137$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.16516096.1
Defining polynomial: \(x^{6} + 9 x^{4} + 13 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.5
Root \(2.68667i\) of defining polynomial
Character \(\chi\) \(=\) 1520.609
Dual form 1520.2.d.h.609.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.31446i q^{3} +(1.94827 - 1.09737i) q^{5} -1.45033i q^{7} -2.35673 q^{9} +O(q^{10})\) \(q+2.31446i q^{3} +(1.94827 - 1.09737i) q^{5} -1.45033i q^{7} -2.35673 q^{9} +3.89655 q^{11} -3.05888i q^{13} +(2.53982 + 4.50920i) q^{15} +3.92301i q^{17} -1.00000 q^{19} +3.35673 q^{21} -5.37334i q^{23} +(2.59155 - 4.27596i) q^{25} +1.48883i q^{27} -6.00000 q^{29} +8.43637 q^{31} +9.01841i q^{33} +(-1.59155 - 2.82564i) q^{35} -5.95953i q^{37} +7.07965 q^{39} +10.4364 q^{41} -1.45033i q^{43} +(-4.59155 + 2.58620i) q^{45} -4.90686i q^{47} +4.89655 q^{49} -9.07965 q^{51} +4.23127i q^{53} +(7.59155 - 4.27596i) q^{55} -2.31446i q^{57} +3.35673 q^{59} +10.3329 q^{61} +3.41802i q^{63} +(-3.35673 - 5.95953i) q^{65} +9.84404i q^{67} +12.4364 q^{69} -8.64327 q^{71} +2.43418i q^{73} +(9.89655 + 5.99804i) q^{75} -5.65127i q^{77} -12.4364 q^{79} -10.5160 q^{81} +12.6635i q^{83} +(4.30500 + 7.64310i) q^{85} -13.8868i q^{87} -12.3662 q^{89} -4.43637 q^{91} +19.5256i q^{93} +(-1.94827 + 1.09737i) q^{95} +3.05888i q^{97} -9.18310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{5} - 14q^{9} + O(q^{10}) \) \( 6q - q^{5} - 14q^{9} - 2q^{11} - 10q^{15} - 6q^{19} + 20q^{21} + 3q^{25} - 36q^{29} + 3q^{35} - 8q^{39} + 12q^{41} - 15q^{45} + 4q^{49} - 4q^{51} + 33q^{55} + 20q^{59} - 14q^{61} - 20q^{65} + 24q^{69} - 52q^{71} + 34q^{75} - 24q^{79} + 38q^{81} + 13q^{85} - 24q^{89} + 24q^{91} + q^{95} - 30q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 2.31446i 1.33625i 0.744047 + 0.668127i \(0.232904\pi\)
−0.744047 + 0.668127i \(0.767096\pi\)
\(4\) 0 0
\(5\) 1.94827 1.09737i 0.871295 0.490760i
\(6\) 0 0
\(7\) 1.45033i 0.548172i −0.961705 0.274086i \(-0.911625\pi\)
0.961705 0.274086i \(-0.0883754\pi\)
\(8\) 0 0
\(9\) −2.35673 −0.785575
\(10\) 0 0
\(11\) 3.89655 1.17485 0.587427 0.809277i \(-0.300141\pi\)
0.587427 + 0.809277i \(0.300141\pi\)
\(12\) 0 0
\(13\) 3.05888i 0.848380i −0.905573 0.424190i \(-0.860559\pi\)
0.905573 0.424190i \(-0.139441\pi\)
\(14\) 0 0
\(15\) 2.53982 + 4.50920i 0.655780 + 1.16427i
\(16\) 0 0
\(17\) 3.92301i 0.951469i 0.879589 + 0.475735i \(0.157818\pi\)
−0.879589 + 0.475735i \(0.842182\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 3.35673 0.732498
\(22\) 0 0
\(23\) 5.37334i 1.12042i −0.828351 0.560209i \(-0.810721\pi\)
0.828351 0.560209i \(-0.189279\pi\)
\(24\) 0 0
\(25\) 2.59155 4.27596i 0.518310 0.855193i
\(26\) 0 0
\(27\) 1.48883i 0.286526i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.43637 1.51522 0.757609 0.652709i \(-0.226368\pi\)
0.757609 + 0.652709i \(0.226368\pi\)
\(32\) 0 0
\(33\) 9.01841i 1.56990i
\(34\) 0 0
\(35\) −1.59155 2.82564i −0.269021 0.477620i
\(36\) 0 0
\(37\) 5.95953i 0.979741i −0.871795 0.489871i \(-0.837044\pi\)
0.871795 0.489871i \(-0.162956\pi\)
\(38\) 0 0
\(39\) 7.07965 1.13365
\(40\) 0 0
\(41\) 10.4364 1.62989 0.814944 0.579540i \(-0.196768\pi\)
0.814944 + 0.579540i \(0.196768\pi\)
\(42\) 0 0
\(43\) 1.45033i 0.221173i −0.993867 0.110586i \(-0.964727\pi\)
0.993867 0.110586i \(-0.0352729\pi\)
\(44\) 0 0
\(45\) −4.59155 + 2.58620i −0.684468 + 0.385529i
\(46\) 0 0
\(47\) 4.90686i 0.715739i −0.933772 0.357869i \(-0.883503\pi\)
0.933772 0.357869i \(-0.116497\pi\)
\(48\) 0 0
\(49\) 4.89655 0.699507
\(50\) 0 0
\(51\) −9.07965 −1.27140
\(52\) 0 0
\(53\) 4.23127i 0.581209i 0.956843 + 0.290605i \(0.0938564\pi\)
−0.956843 + 0.290605i \(0.906144\pi\)
\(54\) 0 0
\(55\) 7.59155 4.27596i 1.02364 0.576571i
\(56\) 0 0
\(57\) 2.31446i 0.306558i
\(58\) 0 0
\(59\) 3.35673 0.437008 0.218504 0.975836i \(-0.429882\pi\)
0.218504 + 0.975836i \(0.429882\pi\)
\(60\) 0 0
\(61\) 10.3329 1.32300 0.661498 0.749947i \(-0.269921\pi\)
0.661498 + 0.749947i \(0.269921\pi\)
\(62\) 0 0
\(63\) 3.41802i 0.430631i
\(64\) 0 0
\(65\) −3.35673 5.95953i −0.416351 0.739189i
\(66\) 0 0
\(67\) 9.84404i 1.20264i 0.799008 + 0.601320i \(0.205358\pi\)
−0.799008 + 0.601320i \(0.794642\pi\)
\(68\) 0 0
\(69\) 12.4364 1.49716
\(70\) 0 0
\(71\) −8.64327 −1.02577 −0.512884 0.858458i \(-0.671423\pi\)
−0.512884 + 0.858458i \(0.671423\pi\)
\(72\) 0 0
\(73\) 2.43418i 0.284899i 0.989802 + 0.142449i \(0.0454978\pi\)
−0.989802 + 0.142449i \(0.954502\pi\)
\(74\) 0 0
\(75\) 9.89655 + 5.99804i 1.14276 + 0.692594i
\(76\) 0 0
\(77\) 5.65127i 0.644022i
\(78\) 0 0
\(79\) −12.4364 −1.39920 −0.699601 0.714534i \(-0.746639\pi\)
−0.699601 + 0.714534i \(0.746639\pi\)
\(80\) 0 0
\(81\) −10.5160 −1.16845
\(82\) 0 0
\(83\) 12.6635i 1.39000i 0.719011 + 0.694999i \(0.244595\pi\)
−0.719011 + 0.694999i \(0.755405\pi\)
\(84\) 0 0
\(85\) 4.30500 + 7.64310i 0.466943 + 0.829011i
\(86\) 0 0
\(87\) 13.8868i 1.48882i
\(88\) 0 0
\(89\) −12.3662 −1.31081 −0.655407 0.755276i \(-0.727503\pi\)
−0.655407 + 0.755276i \(0.727503\pi\)
\(90\) 0 0
\(91\) −4.43637 −0.465058
\(92\) 0 0
\(93\) 19.5256i 2.02472i
\(94\) 0 0
\(95\) −1.94827 + 1.09737i −0.199889 + 0.112588i
\(96\) 0 0
\(97\) 3.05888i 0.310582i 0.987869 + 0.155291i \(0.0496315\pi\)
−0.987869 + 0.155291i \(0.950369\pi\)
\(98\) 0 0
\(99\) −9.18310 −0.922936
\(100\) 0 0
\(101\) 3.35673 0.334007 0.167003 0.985956i \(-0.446591\pi\)
0.167003 + 0.985956i \(0.446591\pi\)
\(102\) 0 0
\(103\) 13.0611i 1.28695i 0.765466 + 0.643476i \(0.222508\pi\)
−0.765466 + 0.643476i \(0.777492\pi\)
\(104\) 0 0
\(105\) 6.53982 3.68358i 0.638221 0.359480i
\(106\) 0 0
\(107\) 5.77099i 0.557903i 0.960305 + 0.278951i \(0.0899868\pi\)
−0.960305 + 0.278951i \(0.910013\pi\)
\(108\) 0 0
\(109\) −6.64327 −0.636310 −0.318155 0.948039i \(-0.603063\pi\)
−0.318155 + 0.948039i \(0.603063\pi\)
\(110\) 0 0
\(111\) 13.7931 1.30918
\(112\) 0 0
\(113\) 9.41606i 0.885789i 0.896574 + 0.442894i \(0.146048\pi\)
−0.896574 + 0.442894i \(0.853952\pi\)
\(114\) 0 0
\(115\) −5.89655 10.4687i −0.549856 0.976215i
\(116\) 0 0
\(117\) 7.20893i 0.666466i
\(118\) 0 0
\(119\) 5.68965 0.521569
\(120\) 0 0
\(121\) 4.18310 0.380282
\(122\) 0 0
\(123\) 24.1546i 2.17794i
\(124\) 0 0
\(125\) 0.356726 11.1746i 0.0319065 0.999491i
\(126\) 0 0
\(127\) 11.0934i 0.984383i −0.870487 0.492192i \(-0.836196\pi\)
0.870487 0.492192i \(-0.163804\pi\)
\(128\) 0 0
\(129\) 3.35673 0.295543
\(130\) 0 0
\(131\) −4.61000 −0.402778 −0.201389 0.979511i \(-0.564545\pi\)
−0.201389 + 0.979511i \(0.564545\pi\)
\(132\) 0 0
\(133\) 1.45033i 0.125759i
\(134\) 0 0
\(135\) 1.63380 + 2.90066i 0.140615 + 0.249649i
\(136\) 0 0
\(137\) 13.1808i 1.12612i −0.826417 0.563058i \(-0.809625\pi\)
0.826417 0.563058i \(-0.190375\pi\)
\(138\) 0 0
\(139\) 1.18310 0.100349 0.0501745 0.998740i \(-0.484022\pi\)
0.0501745 + 0.998740i \(0.484022\pi\)
\(140\) 0 0
\(141\) 11.3567 0.956409
\(142\) 0 0
\(143\) 11.9191i 0.996722i
\(144\) 0 0
\(145\) −11.6896 + 6.58423i −0.970772 + 0.546791i
\(146\) 0 0
\(147\) 11.3329i 0.934719i
\(148\) 0 0
\(149\) −5.46018 −0.447315 −0.223658 0.974668i \(-0.571800\pi\)
−0.223658 + 0.974668i \(0.571800\pi\)
\(150\) 0 0
\(151\) 5.07965 0.413376 0.206688 0.978407i \(-0.433732\pi\)
0.206688 + 0.978407i \(0.433732\pi\)
\(152\) 0 0
\(153\) 9.24546i 0.747451i
\(154\) 0 0
\(155\) 16.4364 9.25784i 1.32020 0.743608i
\(156\) 0 0
\(157\) 6.11775i 0.488250i 0.969744 + 0.244125i \(0.0785007\pi\)
−0.969744 + 0.244125i \(0.921499\pi\)
\(158\) 0 0
\(159\) −9.79310 −0.776643
\(160\) 0 0
\(161\) −7.79310 −0.614182
\(162\) 0 0
\(163\) 16.4365i 1.28740i −0.765277 0.643701i \(-0.777398\pi\)
0.765277 0.643701i \(-0.222602\pi\)
\(164\) 0 0
\(165\) 9.89655 + 17.5703i 0.770445 + 1.36785i
\(166\) 0 0
\(167\) 3.80329i 0.294308i 0.989114 + 0.147154i \(0.0470112\pi\)
−0.989114 + 0.147154i \(0.952989\pi\)
\(168\) 0 0
\(169\) 3.64327 0.280252
\(170\) 0 0
\(171\) 2.35673 0.180223
\(172\) 0 0
\(173\) 11.3838i 0.865491i 0.901516 + 0.432746i \(0.142455\pi\)
−0.901516 + 0.432746i \(0.857545\pi\)
\(174\) 0 0
\(175\) −6.20155 3.75860i −0.468793 0.284123i
\(176\) 0 0
\(177\) 7.76901i 0.583954i
\(178\) 0 0
\(179\) 10.0702 0.752680 0.376340 0.926482i \(-0.377182\pi\)
0.376340 + 0.926482i \(0.377182\pi\)
\(180\) 0 0
\(181\) 0.573097 0.0425980 0.0212990 0.999773i \(-0.493220\pi\)
0.0212990 + 0.999773i \(0.493220\pi\)
\(182\) 0 0
\(183\) 23.9151i 1.76786i
\(184\) 0 0
\(185\) −6.53982 11.6108i −0.480817 0.853643i
\(186\) 0 0
\(187\) 15.2862i 1.11784i
\(188\) 0 0
\(189\) 2.15930 0.157066
\(190\) 0 0
\(191\) 3.18310 0.230321 0.115160 0.993347i \(-0.463262\pi\)
0.115160 + 0.993347i \(0.463262\pi\)
\(192\) 0 0
\(193\) 3.05888i 0.220183i −0.993921 0.110091i \(-0.964886\pi\)
0.993921 0.110091i \(-0.0351143\pi\)
\(194\) 0 0
\(195\) 13.7931 7.76901i 0.987744 0.556350i
\(196\) 0 0
\(197\) 21.4933i 1.53134i 0.643235 + 0.765669i \(0.277592\pi\)
−0.643235 + 0.765669i \(0.722408\pi\)
\(198\) 0 0
\(199\) −4.81690 −0.341461 −0.170731 0.985318i \(-0.554613\pi\)
−0.170731 + 0.985318i \(0.554613\pi\)
\(200\) 0 0
\(201\) −22.7836 −1.60703
\(202\) 0 0
\(203\) 8.70197i 0.610758i
\(204\) 0 0
\(205\) 20.3329 11.4526i 1.42011 0.799883i
\(206\) 0 0
\(207\) 12.6635i 0.880173i
\(208\) 0 0
\(209\) −3.89655 −0.269530
\(210\) 0 0
\(211\) −10.5066 −0.723301 −0.361650 0.932314i \(-0.617787\pi\)
−0.361650 + 0.932314i \(0.617787\pi\)
\(212\) 0 0
\(213\) 20.0045i 1.37069i
\(214\) 0 0
\(215\) −1.59155 2.82564i −0.108543 0.192707i
\(216\) 0 0
\(217\) 12.2355i 0.830600i
\(218\) 0 0
\(219\) −5.63380 −0.380697
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 16.8947i 1.13136i −0.824626 0.565678i \(-0.808615\pi\)
0.824626 0.565678i \(-0.191385\pi\)
\(224\) 0 0
\(225\) −6.10757 + 10.0773i −0.407171 + 0.671818i
\(226\) 0 0
\(227\) 17.1342i 1.13724i −0.822602 0.568618i \(-0.807478\pi\)
0.822602 0.568618i \(-0.192522\pi\)
\(228\) 0 0
\(229\) −25.0464 −1.65511 −0.827555 0.561384i \(-0.810269\pi\)
−0.827555 + 0.561384i \(0.810269\pi\)
\(230\) 0 0
\(231\) 13.0796 0.860578
\(232\) 0 0
\(233\) 19.2986i 1.26429i −0.774849 0.632147i \(-0.782174\pi\)
0.774849 0.632147i \(-0.217826\pi\)
\(234\) 0 0
\(235\) −5.38465 9.55991i −0.351256 0.623620i
\(236\) 0 0
\(237\) 28.7835i 1.86969i
\(238\) 0 0
\(239\) 18.7693 1.21408 0.607042 0.794669i \(-0.292356\pi\)
0.607042 + 0.794669i \(0.292356\pi\)
\(240\) 0 0
\(241\) −14.4364 −0.929929 −0.464964 0.885329i \(-0.653933\pi\)
−0.464964 + 0.885329i \(0.653933\pi\)
\(242\) 0 0
\(243\) 19.8724i 1.27482i
\(244\) 0 0
\(245\) 9.53982 5.37334i 0.609477 0.343290i
\(246\) 0 0
\(247\) 3.05888i 0.194632i
\(248\) 0 0
\(249\) −29.3091 −1.85739
\(250\) 0 0
\(251\) 10.9762 0.692811 0.346406 0.938085i \(-0.387402\pi\)
0.346406 + 0.938085i \(0.387402\pi\)
\(252\) 0 0
\(253\) 20.9375i 1.31633i
\(254\) 0 0
\(255\) −17.6896 + 9.96375i −1.10777 + 0.623954i
\(256\) 0 0
\(257\) 17.6392i 1.10030i −0.835066 0.550150i \(-0.814570\pi\)
0.835066 0.550150i \(-0.185430\pi\)
\(258\) 0 0
\(259\) −8.64327 −0.537067
\(260\) 0 0
\(261\) 14.1404 0.875266
\(262\) 0 0
\(263\) 1.68976i 0.104195i −0.998642 0.0520975i \(-0.983409\pi\)
0.998642 0.0520975i \(-0.0165907\pi\)
\(264\) 0 0
\(265\) 4.64327 + 8.24367i 0.285234 + 0.506405i
\(266\) 0 0
\(267\) 28.6211i 1.75158i
\(268\) 0 0
\(269\) 27.1022 1.65245 0.826226 0.563339i \(-0.190484\pi\)
0.826226 + 0.563339i \(0.190484\pi\)
\(270\) 0 0
\(271\) −23.9524 −1.45500 −0.727502 0.686105i \(-0.759319\pi\)
−0.727502 + 0.686105i \(0.759319\pi\)
\(272\) 0 0
\(273\) 10.2678i 0.621436i
\(274\) 0 0
\(275\) 10.0981 16.6615i 0.608938 1.00473i
\(276\) 0 0
\(277\) 8.23549i 0.494822i 0.968911 + 0.247411i \(0.0795799\pi\)
−0.968911 + 0.247411i \(0.920420\pi\)
\(278\) 0 0
\(279\) −19.8822 −1.19032
\(280\) 0 0
\(281\) −10.4364 −0.622582 −0.311291 0.950315i \(-0.600761\pi\)
−0.311291 + 0.950315i \(0.600761\pi\)
\(282\) 0 0
\(283\) 10.4687i 0.622302i −0.950361 0.311151i \(-0.899286\pi\)
0.950361 0.311151i \(-0.100714\pi\)
\(284\) 0 0
\(285\) −2.53982 4.50920i −0.150446 0.267102i
\(286\) 0 0
\(287\) 15.1362i 0.893459i
\(288\) 0 0
\(289\) 1.61000 0.0947059
\(290\) 0 0
\(291\) −7.07965 −0.415016
\(292\) 0 0
\(293\) 16.4668i 0.961999i 0.876721 + 0.481000i \(0.159726\pi\)
−0.876721 + 0.481000i \(0.840274\pi\)
\(294\) 0 0
\(295\) 6.53982 3.68358i 0.380763 0.214466i
\(296\) 0 0
\(297\) 5.80131i 0.336626i
\(298\) 0 0
\(299\) −16.4364 −0.950540
\(300\) 0 0
\(301\) −2.10345 −0.121241
\(302\) 0 0
\(303\) 7.76901i 0.446318i
\(304\) 0 0
\(305\) 20.1314 11.3391i 1.15272 0.649273i
\(306\) 0 0
\(307\) 7.87634i 0.449526i −0.974413 0.224763i \(-0.927839\pi\)
0.974413 0.224763i \(-0.0721609\pi\)
\(308\) 0 0
\(309\) −30.2295 −1.71969
\(310\) 0 0
\(311\) −3.89655 −0.220953 −0.110477 0.993879i \(-0.535238\pi\)
−0.110477 + 0.993879i \(0.535238\pi\)
\(312\) 0 0
\(313\) 7.76901i 0.439130i −0.975598 0.219565i \(-0.929536\pi\)
0.975598 0.219565i \(-0.0704638\pi\)
\(314\) 0 0
\(315\) 3.75084 + 6.65925i 0.211336 + 0.375206i
\(316\) 0 0
\(317\) 19.0510i 1.07001i 0.844849 + 0.535005i \(0.179690\pi\)
−0.844849 + 0.535005i \(0.820310\pi\)
\(318\) 0 0
\(319\) −23.3793 −1.30899
\(320\) 0 0
\(321\) −13.3567 −0.745500
\(322\) 0 0
\(323\) 3.92301i 0.218282i
\(324\) 0 0
\(325\) −13.0796 7.92723i −0.725528 0.439724i
\(326\) 0 0
\(327\) 15.3756i 0.850272i
\(328\) 0 0
\(329\) −7.11655 −0.392348
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 14.0450i 0.769660i
\(334\) 0 0
\(335\) 10.8026 + 19.1789i 0.590207 + 1.04785i
\(336\) 0 0
\(337\) 6.89249i 0.375458i −0.982221 0.187729i \(-0.939887\pi\)
0.982221 0.187729i \(-0.0601127\pi\)
\(338\) 0 0
\(339\) −21.7931 −1.18364
\(340\) 0 0
\(341\) 32.8727 1.78016
\(342\) 0 0
\(343\) 17.2539i 0.931623i
\(344\) 0 0
\(345\) 24.2295 13.6473i 1.30447 0.734747i
\(346\) 0 0
\(347\) 30.5503i 1.64002i 0.572346 + 0.820012i \(0.306033\pi\)
−0.572346 + 0.820012i \(0.693967\pi\)
\(348\) 0 0
\(349\) −16.7693 −0.897640 −0.448820 0.893622i \(-0.648156\pi\)
−0.448820 + 0.893622i \(0.648156\pi\)
\(350\) 0 0
\(351\) 4.55416 0.243083
\(352\) 0 0
\(353\) 29.0999i 1.54883i −0.632676 0.774417i \(-0.718044\pi\)
0.632676 0.774417i \(-0.281956\pi\)
\(354\) 0 0
\(355\) −16.8395 + 9.48489i −0.893746 + 0.503406i
\(356\) 0 0
\(357\) 13.1685i 0.696949i
\(358\) 0 0
\(359\) −11.6896 −0.616956 −0.308478 0.951231i \(-0.599820\pi\)
−0.308478 + 0.951231i \(0.599820\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 9.68161i 0.508153i
\(364\) 0 0
\(365\) 2.67120 + 4.74244i 0.139817 + 0.248231i
\(366\) 0 0
\(367\) 4.20095i 0.219288i 0.993971 + 0.109644i \(0.0349710\pi\)
−0.993971 + 0.109644i \(0.965029\pi\)
\(368\) 0 0
\(369\) −24.5957 −1.28040
\(370\) 0 0
\(371\) 6.13672 0.318603
\(372\) 0 0
\(373\) 16.9456i 0.877412i 0.898631 + 0.438706i \(0.144563\pi\)
−0.898631 + 0.438706i \(0.855437\pi\)
\(374\) 0 0
\(375\) 25.8633 + 0.825627i 1.33557 + 0.0426352i
\(376\) 0 0
\(377\) 18.3533i 0.945241i
\(378\) 0 0
\(379\) 10.3662 0.532476 0.266238 0.963907i \(-0.414219\pi\)
0.266238 + 0.963907i \(0.414219\pi\)
\(380\) 0 0
\(381\) 25.6753 1.31539
\(382\) 0 0
\(383\) 20.5907i 1.05214i 0.850442 + 0.526068i \(0.176334\pi\)
−0.850442 + 0.526068i \(0.823666\pi\)
\(384\) 0 0
\(385\) −6.20155 11.0102i −0.316060 0.561133i
\(386\) 0 0
\(387\) 3.41802i 0.173748i
\(388\) 0 0
\(389\) −8.10345 −0.410861 −0.205431 0.978672i \(-0.565859\pi\)
−0.205431 + 0.978672i \(0.565859\pi\)
\(390\) 0 0
\(391\) 21.0796 1.06604
\(392\) 0 0
\(393\) 10.6697i 0.538213i
\(394\) 0 0
\(395\) −24.2295 + 13.6473i −1.21912 + 0.686672i
\(396\) 0 0
\(397\) 3.46891i 0.174100i 0.996204 + 0.0870499i \(0.0277440\pi\)
−0.996204 + 0.0870499i \(0.972256\pi\)
\(398\) 0 0
\(399\) −3.35673 −0.168046
\(400\) 0 0
\(401\) −15.9524 −0.796625 −0.398312 0.917250i \(-0.630404\pi\)
−0.398312 + 0.917250i \(0.630404\pi\)
\(402\) 0 0
\(403\) 25.8058i 1.28548i
\(404\) 0 0
\(405\) −20.4881 + 11.5400i −1.01806 + 0.573427i
\(406\) 0 0
\(407\) 23.2216i 1.15105i
\(408\) 0 0
\(409\) 3.92982 0.194317 0.0971586 0.995269i \(-0.469025\pi\)
0.0971586 + 0.995269i \(0.469025\pi\)
\(410\) 0 0
\(411\) 30.5066 1.50478
\(412\) 0 0
\(413\) 4.86835i 0.239556i
\(414\) 0 0
\(415\) 13.8965 + 24.6719i 0.682155 + 1.21110i
\(416\) 0 0
\(417\) 2.73823i 0.134092i
\(418\) 0 0
\(419\) −34.2996 −1.67565 −0.837824 0.545941i \(-0.816172\pi\)
−0.837824 + 0.545941i \(0.816172\pi\)
\(420\) 0 0
\(421\) −26.0891 −1.27151 −0.635753 0.771893i \(-0.719310\pi\)
−0.635753 + 0.771893i \(0.719310\pi\)
\(422\) 0 0
\(423\) 11.5641i 0.562267i
\(424\) 0 0
\(425\) 16.7746 + 10.1667i 0.813690 + 0.493156i
\(426\) 0 0
\(427\) 14.9861i 0.725229i
\(428\) 0 0
\(429\) 27.5862 1.33187
\(430\) 0 0
\(431\) −10.2996 −0.496117 −0.248058 0.968745i \(-0.579792\pi\)
−0.248058 + 0.968745i \(0.579792\pi\)
\(432\) 0 0
\(433\) 36.2319i 1.74119i 0.491999 + 0.870596i \(0.336266\pi\)
−0.491999 + 0.870596i \(0.663734\pi\)
\(434\) 0 0
\(435\) −15.2389 27.0552i −0.730651 1.29720i
\(436\) 0 0
\(437\) 5.37334i 0.257042i
\(438\) 0 0
\(439\) −24.0891 −1.14971 −0.574855 0.818255i \(-0.694942\pi\)
−0.574855 + 0.818255i \(0.694942\pi\)
\(440\) 0 0
\(441\) −11.5398 −0.549515
\(442\) 0 0
\(443\) 5.52337i 0.262423i −0.991354 0.131212i \(-0.958113\pi\)
0.991354 0.131212i \(-0.0418868\pi\)
\(444\) 0 0
\(445\) −24.0927 + 13.5703i −1.14211 + 0.643295i
\(446\) 0 0
\(447\) 12.6374i 0.597727i
\(448\) 0 0
\(449\) −7.92982 −0.374231 −0.187116 0.982338i \(-0.559914\pi\)
−0.187116 + 0.982338i \(0.559914\pi\)
\(450\) 0 0
\(451\) 40.6658 1.91488
\(452\) 0 0
\(453\) 11.7566i 0.552375i
\(454\) 0 0
\(455\) −8.64327 + 4.86835i −0.405203 + 0.228232i
\(456\) 0 0
\(457\) 22.6534i 1.05968i −0.848098 0.529840i \(-0.822252\pi\)
0.848098 0.529840i \(-0.177748\pi\)
\(458\) 0 0
\(459\) −5.84070 −0.272621
\(460\) 0 0
\(461\) 13.3900 0.623634 0.311817 0.950142i \(-0.399062\pi\)
0.311817 + 0.950142i \(0.399062\pi\)
\(462\) 0 0
\(463\) 16.9029i 0.785546i −0.919635 0.392773i \(-0.871516\pi\)
0.919635 0.392773i \(-0.128484\pi\)
\(464\) 0 0
\(465\) 21.4269 + 38.0413i 0.993649 + 1.76412i
\(466\) 0 0
\(467\) 22.2501i 1.02961i 0.857306 + 0.514807i \(0.172136\pi\)
−0.857306 + 0.514807i \(0.827864\pi\)
\(468\) 0 0
\(469\) 14.2771 0.659254
\(470\) 0 0
\(471\) −14.1593 −0.652426
\(472\) 0 0
\(473\) 5.65127i 0.259846i
\(474\) 0 0
\(475\) −2.59155 + 4.27596i −0.118908 + 0.196195i
\(476\) 0 0
\(477\) 9.97194i 0.456584i
\(478\) 0 0
\(479\) 0.366196 0.0167319 0.00836597 0.999965i \(-0.497337\pi\)
0.00836597 + 0.999965i \(0.497337\pi\)
\(480\) 0 0
\(481\) −18.2295 −0.831192
\(482\) 0 0
\(483\) 18.0368i 0.820704i
\(484\) 0 0
\(485\) 3.35673 + 5.95953i 0.152421 + 0.270608i
\(486\) 0 0
\(487\) 26.8461i 1.21651i 0.793740 + 0.608257i \(0.208131\pi\)
−0.793740 + 0.608257i \(0.791869\pi\)
\(488\) 0 0
\(489\) 38.0415 1.72030
\(490\) 0 0
\(491\) 23.7266 1.07076 0.535382 0.844610i \(-0.320168\pi\)
0.535382 + 0.844610i \(0.320168\pi\)
\(492\) 0 0
\(493\) 23.5381i 1.06010i
\(494\) 0 0
\(495\) −17.8912 + 10.0773i −0.804150 + 0.452940i
\(496\) 0 0
\(497\) 12.5356i 0.562298i
\(498\) 0 0
\(499\) 6.81690 0.305166 0.152583 0.988291i \(-0.451241\pi\)
0.152583 + 0.988291i \(0.451241\pi\)
\(500\) 0 0
\(501\) −8.80257 −0.393270
\(502\) 0 0
\(503\) 23.4102i 1.04381i −0.853004 0.521904i \(-0.825222\pi\)
0.853004 0.521904i \(-0.174778\pi\)
\(504\) 0 0
\(505\) 6.53982 3.68358i 0.291018 0.163917i
\(506\) 0 0
\(507\) 8.43221i 0.374488i
\(508\) 0 0
\(509\) −16.9204 −0.749981 −0.374991 0.927029i \(-0.622354\pi\)
−0.374991 + 0.927029i \(0.622354\pi\)
\(510\) 0 0
\(511\) 3.53035 0.156174
\(512\) 0 0
\(513\) 1.48883i 0.0657336i
\(514\) 0 0
\(515\) 14.3329 + 25.4467i 0.631584 + 1.12131i
\(516\) 0 0
\(517\) 19.1198i 0.840888i
\(518\) 0 0
\(519\) −26.3473 −1.15652
\(520\) 0 0
\(521\) −3.49345 −0.153051 −0.0765254 0.997068i \(-0.524383\pi\)
−0.0765254 + 0.997068i \(0.524383\pi\)
\(522\) 0 0
\(523\) 14.9271i 0.652714i −0.945247 0.326357i \(-0.894179\pi\)
0.945247 0.326357i \(-0.105821\pi\)
\(524\) 0 0
\(525\) 8.69912 14.3532i 0.379661 0.626427i
\(526\) 0 0
\(527\) 33.0960i 1.44168i
\(528\) 0 0
\(529\) −5.87275 −0.255337
\(530\) 0 0
\(531\) −7.91088 −0.343303
\(532\) 0 0
\(533\) 31.9236i 1.38276i
\(534\) 0 0
\(535\) 6.33292 + 11.2435i 0.273796 + 0.486098i
\(536\) 0 0
\(537\) 23.3070i 1.00577i
\(538\) 0 0
\(539\) 19.0796 0.821819
\(540\) 0 0
\(541\) −12.6991 −0.545978 −0.272989 0.962017i \(-0.588012\pi\)
−0.272989 + 0.962017i \(0.588012\pi\)
\(542\) 0 0
\(543\) 1.32641i 0.0569217i
\(544\) 0 0
\(545\) −12.9429 + 7.29014i −0.554414 + 0.312275i
\(546\) 0 0
\(547\) 31.8162i 1.36036i 0.733043 + 0.680182i \(0.238099\pi\)
−0.733043 + 0.680182i \(0.761901\pi\)
\(548\) 0 0
\(549\) −24.3519 −1.03931
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 18.0368i 0.767003i
\(554\) 0 0
\(555\) 26.8727 15.1362i 1.14068 0.642494i
\(556\) 0 0
\(557\) 9.64731i 0.408770i −0.978891 0.204385i \(-0.934481\pi\)
0.978891 0.204385i \(-0.0655194\pi\)
\(558\) 0 0
\(559\) −4.43637 −0.187639
\(560\) 0 0
\(561\) −35.3793 −1.49372
\(562\) 0 0
\(563\) 4.28216i 0.180471i −0.995920 0.0902357i \(-0.971238\pi\)
0.995920 0.0902357i \(-0.0287620\pi\)
\(564\) 0 0
\(565\) 10.3329 + 18.3451i 0.434709 + 0.771783i
\(566\) 0 0
\(567\) 15.2517i 0.640510i
\(568\) 0 0
\(569\) −42.2295 −1.77035 −0.885176 0.465257i \(-0.845962\pi\)
−0.885176 + 0.465257i \(0.845962\pi\)
\(570\) 0 0
\(571\) 19.2200 0.804332 0.402166 0.915567i \(-0.368257\pi\)
0.402166 + 0.915567i \(0.368257\pi\)
\(572\) 0 0
\(573\) 7.36715i 0.307767i
\(574\) 0 0
\(575\) −22.9762 13.9253i −0.958174 0.580724i
\(576\) 0 0
\(577\) 40.7919i 1.69819i −0.528239 0.849096i \(-0.677148\pi\)
0.528239 0.849096i \(-0.322852\pi\)
\(578\) 0 0
\(579\) 7.07965 0.294220
\(580\) 0 0
\(581\) 18.3662 0.761958
\(582\) 0 0
\(583\) 16.4873i 0.682836i
\(584\) 0 0
\(585\) 7.91088 + 14.0450i 0.327075 + 0.580689i
\(586\) 0 0
\(587\) 31.8851i 1.31604i 0.753001 + 0.658019i \(0.228605\pi\)
−0.753001 + 0.658019i \(0.771395\pi\)
\(588\) 0 0
\(589\) −8.43637 −0.347615
\(590\) 0 0
\(591\) −49.7455 −2.04626
\(592\) 0 0
\(593\) 38.8973i 1.59732i 0.601783 + 0.798660i \(0.294457\pi\)
−0.601783 + 0.798660i \(0.705543\pi\)
\(594\) 0 0
\(595\) 11.0850 6.24366i 0.454441 0.255965i
\(596\) 0 0
\(597\) 11.1485i 0.456279i
\(598\) 0 0
\(599\) 28.1629 1.15071 0.575353 0.817905i \(-0.304865\pi\)
0.575353 + 0.817905i \(0.304865\pi\)
\(600\) 0 0
\(601\) −5.56363 −0.226945 −0.113473 0.993541i \(-0.536197\pi\)
−0.113473 + 0.993541i \(0.536197\pi\)
\(602\) 0 0
\(603\) 23.1997i 0.944764i
\(604\) 0 0
\(605\) 8.14982 4.59042i 0.331337 0.186627i
\(606\) 0 0
\(607\) 33.9986i 1.37996i −0.723828 0.689980i \(-0.757619\pi\)
0.723828 0.689980i \(-0.242381\pi\)
\(608\) 0 0
\(609\) −20.1404 −0.816128
\(610\) 0 0
\(611\) −15.0095 −0.607218
\(612\) 0 0
\(613\) 17.5703i 0.709659i −0.934931 0.354830i \(-0.884539\pi\)
0.934931 0.354830i \(-0.115461\pi\)
\(614\) 0 0
\(615\) 26.5066 + 47.0597i 1.06885 + 1.89763i
\(616\) 0 0
\(617\) 13.0791i 0.526544i −0.964722 0.263272i \(-0.915198\pi\)
0.964722 0.263272i \(-0.0848016\pi\)
\(618\) 0 0
\(619\) −18.9393 −0.761234 −0.380617 0.924733i \(-0.624288\pi\)
−0.380617 + 0.924733i \(0.624288\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) 17.9350i 0.718552i
\(624\) 0 0
\(625\) −11.5677 22.1627i −0.462710 0.886510i
\(626\) 0 0
\(627\) 9.01841i 0.360161i
\(628\) 0 0
\(629\) 23.3793 0.932194
\(630\) 0 0
\(631\) −31.6896 −1.26154 −0.630772 0.775968i \(-0.717262\pi\)
−0.630772 + 0.775968i \(0.717262\pi\)
\(632\) 0 0
\(633\) 24.3170i 0.966514i
\(634\) 0 0
\(635\) −12.1736 21.6131i −0.483096 0.857688i
\(636\) 0 0
\(637\) 14.9779i 0.593448i
\(638\) 0 0
\(639\) 20.3698 0.805818
\(640\) 0 0
\(641\) 47.9750 1.89490 0.947449 0.319908i \(-0.103652\pi\)
0.947449 + 0.319908i \(0.103652\pi\)
\(642\) 0 0
\(643\) 0.200927i 0.00792378i −0.999992 0.00396189i \(-0.998739\pi\)
0.999992 0.00396189i \(-0.00126111\pi\)
\(644\) 0 0
\(645\) 6.53982 3.68358i 0.257505 0.145041i
\(646\) 0 0
\(647\) 1.58798i 0.0624299i −0.999513 0.0312150i \(-0.990062\pi\)
0.999513 0.0312150i \(-0.00993765\pi\)
\(648\) 0 0
\(649\) 13.0796 0.513421
\(650\) 0 0
\(651\) 28.3186 1.10989
\(652\) 0 0
\(653\) 33.5624i 1.31340i 0.754152 + 0.656700i \(0.228048\pi\)
−0.754152 + 0.656700i \(0.771952\pi\)
\(654\) 0 0
\(655\) −8.98155 + 5.05889i −0.350938 + 0.197667i
\(656\) 0 0
\(657\) 5.73669i 0.223809i
\(658\) 0 0
\(659\) 15.3567 0.598213 0.299107 0.954220i \(-0.403311\pi\)
0.299107 + 0.954220i \(0.403311\pi\)
\(660\) 0 0
\(661\) 12.8026 0.497962 0.248981 0.968508i \(-0.419904\pi\)
0.248981 + 0.968508i \(0.419904\pi\)
\(662\) 0 0
\(663\) 27.7735i 1.07863i
\(664\) 0 0
\(665\) 1.59155 + 2.82564i 0.0617176 + 0.109573i
\(666\) 0 0
\(667\) 32.2400i 1.24834i
\(668\) 0 0
\(669\) 39.1022 1.51178
\(670\) 0 0
\(671\) 40.2627 1.55433
\(672\) 0 0
\(673\) 7.82545i 0.301649i −0.988561 0.150824i \(-0.951807\pi\)
0.988561 0.150824i \(-0.0481928\pi\)
\(674\) 0 0
\(675\) 6.36620 + 3.85838i 0.245035 + 0.148509i
\(676\) 0 0
\(677\) 21.6516i 0.832137i 0.909333 + 0.416069i \(0.136592\pi\)
−0.909333 + 0.416069i \(0.863408\pi\)
\(678\) 0 0
\(679\) 4.43637 0.170252
\(680\) 0 0
\(681\) 39.6564 1.51964
\(682\) 0 0
\(683\) 6.38751i 0.244411i 0.992505 + 0.122206i \(0.0389967\pi\)
−0.992505 + 0.122206i \(0.961003\pi\)
\(684\) 0 0
\(685\) −14.4643 25.6799i −0.552652 0.981179i
\(686\) 0 0
\(687\) 57.9688i 2.21165i
\(688\) 0 0
\(689\) 12.9429 0.493086
\(690\) 0 0
\(691\) −44.4958 −1.69270 −0.846351 0.532626i \(-0.821205\pi\)
−0.846351 + 0.532626i \(0.821205\pi\)
\(692\) 0 0
\(693\) 13.3185i 0.505928i
\(694\) 0 0
\(695\) 2.30500 1.29830i 0.0874336 0.0492473i
\(696\) 0 0
\(697\) 40.9420i 1.55079i
\(698\) 0 0
\(699\) 44.6658 1.68942
\(700\) 0 0
\(701\) 17.5160 0.661571 0.330785 0.943706i \(-0.392686\pi\)
0.330785 + 0.943706i \(0.392686\pi\)
\(702\) 0 0
\(703\) 5.95953i 0.224768i
\(704\) 0 0
\(705\) 22.1260 12.4626i 0.833314 0.469367i
\(706\) 0 0
\(707\) 4.86835i 0.183093i
\(708\) 0 0
\(709\) −11.4269 −0.429146 −0.214573 0.976708i \(-0.568836\pi\)
−0.214573 + 0.976708i \(0.568836\pi\)
\(710\) 0 0
\(711\) 29.3091 1.09918
\(712\) 0 0
\(713\) 45.3315i 1.69768i
\(714\) 0 0
\(715\) −13.0796 23.2216i −0.489151 0.868439i
\(716\) 0 0
\(717\) 43.4408i 1.62233i
\(718\) 0 0
\(719\) 15.8965 0.592841 0.296421 0.955057i \(-0.404207\pi\)
0.296421 + 0.955057i \(0.404207\pi\)
\(720\) 0 0
\(721\) 18.9429 0.705471
\(722\) 0 0
\(723\) 33.4124i 1.24262i
\(724\) 0 0
\(725\) −15.5493 + 25.6558i −0.577486 + 0.952832i
\(726\) 0 0
\(727\) 41.9905i 1.55734i −0.627434 0.778670i \(-0.715895\pi\)
0.627434 0.778670i \(-0.284105\pi\)
\(728\) 0 0
\(729\) 14.4458 0.535031
\(730\) 0 0
\(731\) 5.68965 0.210439
\(732\) 0 0
\(733\) 0.632884i 0.0233761i 0.999932 + 0.0116881i \(0.00372051\pi\)
−0.999932 + 0.0116881i \(0.996279\pi\)
\(734\) 0 0
\(735\) 12.4364 + 22.0795i 0.458723 + 0.814416i
\(736\) 0 0
\(737\) 38.3578i 1.41293i
\(738\) 0 0
\(739\) −29.5493 −1.08699 −0.543494 0.839413i \(-0.682899\pi\)
−0.543494 + 0.839413i \(0.682899\pi\)
\(740\) 0 0
\(741\) −7.07965 −0.260077
\(742\) 0 0
\(743\) 31.3374i 1.14966i −0.818274 0.574829i \(-0.805069\pi\)
0.818274 0.574829i \(-0.194931\pi\)
\(744\) 0 0
\(745\) −10.6379 + 5.99184i −0.389743 + 0.219524i
\(746\) 0 0
\(747\) 29.8443i 1.09195i
\(748\) 0 0
\(749\) 8.36983 0.305827
\(750\) 0 0
\(751\) −25.0131 −0.912741 −0.456371 0.889790i \(-0.650851\pi\)
−0.456371 + 0.889790i \(0.650851\pi\)
\(752\) 0 0
\(753\) 25.4040i 0.925772i
\(754\) 0 0
\(755\) 9.89655 5.57426i 0.360172 0.202868i
\(756\) 0 0
\(757\) 32.0900i 1.16633i −0.812354 0.583165i \(-0.801814\pi\)
0.812354 0.583165i \(-0.198186\pi\)
\(758\) 0 0
\(759\) 48.4589 1.75895
\(760\) 0 0
\(761\) −40.4922 −1.46784 −0.733921 0.679235i \(-0.762312\pi\)
−0.733921 + 0.679235i \(0.762312\pi\)
\(762\) 0 0
\(763\) 9.63492i 0.348808i
\(764\) 0 0
\(765\) −10.1457 18.0127i −0.366819 0.651250i
\(766\) 0 0
\(767\) 10.2678i 0.370749i
\(768\) 0 0
\(769\) −3.09398 −0.111572 −0.0557859 0.998443i \(-0.517766\pi\)
−0.0557859 + 0.998443i \(0.517766\pi\)
\(770\) 0 0
\(771\) 40.8251 1.47028
\(772\) 0 0
\(773\) 1.96350i 0.0706220i 0.999376 + 0.0353110i \(0.0112422\pi\)
−0.999376 + 0.0353110i \(0.988758\pi\)
\(774\) 0 0
\(775\) 21.8633 36.0736i 0.785352 1.29580i
\(776\) 0 0
\(777\) 20.0045i 0.717658i
\(778\) 0 0
\(779\) −10.4364 −0.373922
\(780\) 0 0
\(781\) −33.6789 −1.20513
\(782\) 0 0
\(783\) 8.93300i 0.319239i
\(784\) 0 0
\(785\) 6.71345 + 11.9191i 0.239613 + 0.425410i
\(786\) 0 0
\(787\) 0.107331i 0.00382595i 0.999998 + 0.00191297i \(0.000608919\pi\)
−0.999998 + 0.00191297i \(0.999391\pi\)
\(788\) 0 0
\(789\) 3.91088 0.139231
\(790\) 0 0
\(791\) 13.6564 0.485565
\(792\) 0 0
\(793\) 31.6071i 1.12240i
\(794\) 0 0
\(795\) −19.0796 + 10.7467i −0.676685 + 0.381145i
\(796\) 0 0
\(797\) 8.32068i 0.294734i −0.989082 0.147367i \(-0.952920\pi\)
0.989082 0.147367i \(-0.0470798\pi\)
\(798\) 0 0
\(799\) 19.2496 0.681003
\(800\) 0 0
\(801\) 29.1437 1.02974