Properties

Label 4560.2.d.e.2431.4
Level $4560$
Weight $2$
Character 4560.2431
Analytic conductor $36.412$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9821011968.3
Defining polynomial: \(x^{6} + 20 x^{4} + 118 x^{2} + 192\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.4
Root \(1.63858i\) of defining polynomial
Character \(\chi\) \(=\) 4560.2431
Dual form 4560.2.d.e.2431.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.63858i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.63858i q^{7} +1.00000 q^{9} +1.02200i q^{11} +1.63858i q^{13} +1.00000 q^{15} +3.31507 q^{17} +(4.31507 - 0.616577i) q^{19} -1.63858i q^{21} -8.09257i q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.02200i q^{29} -6.30476 q^{31} -1.02200i q^{33} -1.63858i q^{35} -9.11457i q^{37} -1.63858i q^{39} +2.25515i q^{41} -9.11457i q^{43} -1.00000 q^{45} +6.85941i q^{47} +4.31507 q^{49} -3.31507 q^{51} +8.09257i q^{53} -1.02200i q^{55} +(-4.31507 + 0.616577i) q^{57} -6.63014 q^{59} -4.30476 q^{61} +1.63858i q^{63} -1.63858i q^{65} +0.989692 q^{67} +8.09257i q^{69} -6.63014 q^{71} +16.2500 q^{73} -1.00000 q^{75} -1.67462 q^{77} -1.61983 q^{79} +1.00000 q^{81} +10.1366i q^{83} -3.31507 q^{85} -1.02200i q^{87} -5.22084i q^{89} -2.68493 q^{91} +6.30476 q^{93} +(-4.31507 + 0.616577i) q^{95} +19.0569i q^{97} +1.02200i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} - 6q^{5} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} - 6q^{5} + 6q^{9} + 6q^{15} - 4q^{17} + 2q^{19} + 6q^{25} - 6q^{27} - 12q^{31} - 6q^{45} + 2q^{49} + 4q^{51} - 2q^{57} + 8q^{59} + 4q^{67} + 8q^{71} - 6q^{75} - 32q^{77} + 40q^{79} + 6q^{81} + 4q^{85} - 40q^{91} + 12q^{93} - 2q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(1711\) \(1921\) \(2737\) \(3041\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.63858i 0.619323i 0.950847 + 0.309662i \(0.100216\pi\)
−0.950847 + 0.309662i \(0.899784\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.02200i 0.308144i 0.988060 + 0.154072i \(0.0492388\pi\)
−0.988060 + 0.154072i \(0.950761\pi\)
\(12\) 0 0
\(13\) 1.63858i 0.454459i 0.973841 + 0.227230i \(0.0729668\pi\)
−0.973841 + 0.227230i \(0.927033\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 3.31507 0.804023 0.402011 0.915635i \(-0.368311\pi\)
0.402011 + 0.915635i \(0.368311\pi\)
\(18\) 0 0
\(19\) 4.31507 0.616577i 0.989945 0.141452i
\(20\) 0 0
\(21\) 1.63858i 0.357566i
\(22\) 0 0
\(23\) 8.09257i 1.68742i −0.536802 0.843708i \(-0.680368\pi\)
0.536802 0.843708i \(-0.319632\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.02200i 0.189780i 0.995488 + 0.0948902i \(0.0302500\pi\)
−0.995488 + 0.0948902i \(0.969750\pi\)
\(30\) 0 0
\(31\) −6.30476 −1.13237 −0.566184 0.824279i \(-0.691581\pi\)
−0.566184 + 0.824279i \(0.691581\pi\)
\(32\) 0 0
\(33\) 1.02200i 0.177907i
\(34\) 0 0
\(35\) 1.63858i 0.276970i
\(36\) 0 0
\(37\) 9.11457i 1.49843i −0.662330 0.749213i \(-0.730432\pi\)
0.662330 0.749213i \(-0.269568\pi\)
\(38\) 0 0
\(39\) 1.63858i 0.262382i
\(40\) 0 0
\(41\) 2.25515i 0.352196i 0.984373 + 0.176098i \(0.0563475\pi\)
−0.984373 + 0.176098i \(0.943652\pi\)
\(42\) 0 0
\(43\) 9.11457i 1.38996i −0.719030 0.694979i \(-0.755414\pi\)
0.719030 0.694979i \(-0.244586\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 6.85941i 1.00055i 0.865867 + 0.500274i \(0.166767\pi\)
−0.865867 + 0.500274i \(0.833233\pi\)
\(48\) 0 0
\(49\) 4.31507 0.616439
\(50\) 0 0
\(51\) −3.31507 −0.464203
\(52\) 0 0
\(53\) 8.09257i 1.11160i 0.831316 + 0.555800i \(0.187588\pi\)
−0.831316 + 0.555800i \(0.812412\pi\)
\(54\) 0 0
\(55\) 1.02200i 0.137806i
\(56\) 0 0
\(57\) −4.31507 + 0.616577i −0.571545 + 0.0816676i
\(58\) 0 0
\(59\) −6.63014 −0.863171 −0.431585 0.902072i \(-0.642046\pi\)
−0.431585 + 0.902072i \(0.642046\pi\)
\(60\) 0 0
\(61\) −4.30476 −0.551168 −0.275584 0.961277i \(-0.588871\pi\)
−0.275584 + 0.961277i \(0.588871\pi\)
\(62\) 0 0
\(63\) 1.63858i 0.206441i
\(64\) 0 0
\(65\) 1.63858i 0.203240i
\(66\) 0 0
\(67\) 0.989692 0.120910 0.0604550 0.998171i \(-0.480745\pi\)
0.0604550 + 0.998171i \(0.480745\pi\)
\(68\) 0 0
\(69\) 8.09257i 0.974231i
\(70\) 0 0
\(71\) −6.63014 −0.786853 −0.393426 0.919356i \(-0.628710\pi\)
−0.393426 + 0.919356i \(0.628710\pi\)
\(72\) 0 0
\(73\) 16.2500 1.90192 0.950958 0.309321i \(-0.100102\pi\)
0.950958 + 0.309321i \(0.100102\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −1.67462 −0.190841
\(78\) 0 0
\(79\) −1.61983 −0.182245 −0.0911227 0.995840i \(-0.529046\pi\)
−0.0911227 + 0.995840i \(0.529046\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.1366i 1.11263i 0.830971 + 0.556316i \(0.187786\pi\)
−0.830971 + 0.556316i \(0.812214\pi\)
\(84\) 0 0
\(85\) −3.31507 −0.359570
\(86\) 0 0
\(87\) 1.02200i 0.109570i
\(88\) 0 0
\(89\) 5.22084i 0.553408i −0.960955 0.276704i \(-0.910758\pi\)
0.960955 0.276704i \(-0.0892421\pi\)
\(90\) 0 0
\(91\) −2.68493 −0.281457
\(92\) 0 0
\(93\) 6.30476 0.653773
\(94\) 0 0
\(95\) −4.31507 + 0.616577i −0.442717 + 0.0632595i
\(96\) 0 0
\(97\) 19.0569i 1.93493i 0.253002 + 0.967466i \(0.418582\pi\)
−0.253002 + 0.967466i \(0.581418\pi\)
\(98\) 0 0
\(99\) 1.02200i 0.102715i
\(100\) 0 0
\(101\) 10.6095 1.05569 0.527844 0.849342i \(-0.323001\pi\)
0.527844 + 0.849342i \(0.323001\pi\)
\(102\) 0 0
\(103\) 10.6301 1.04742 0.523709 0.851897i \(-0.324548\pi\)
0.523709 + 0.851897i \(0.324548\pi\)
\(104\) 0 0
\(105\) 1.63858i 0.159909i
\(106\) 0 0
\(107\) −7.64045 −0.738630 −0.369315 0.929304i \(-0.620408\pi\)
−0.369315 + 0.929304i \(0.620408\pi\)
\(108\) 0 0
\(109\) 19.4623i 1.86415i −0.362267 0.932074i \(-0.617997\pi\)
0.362267 0.932074i \(-0.382003\pi\)
\(110\) 0 0
\(111\) 9.11457i 0.865116i
\(112\) 0 0
\(113\) 8.90341i 0.837562i −0.908087 0.418781i \(-0.862457\pi\)
0.908087 0.418781i \(-0.137543\pi\)
\(114\) 0 0
\(115\) 8.09257i 0.754636i
\(116\) 0 0
\(117\) 1.63858i 0.151486i
\(118\) 0 0
\(119\) 5.43199i 0.497950i
\(120\) 0 0
\(121\) 9.95552 0.905047
\(122\) 0 0
\(123\) 2.25515i 0.203340i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 14.2500 1.26448 0.632240 0.774772i \(-0.282136\pi\)
0.632240 + 0.774772i \(0.282136\pi\)
\(128\) 0 0
\(129\) 9.11457i 0.802493i
\(130\) 0 0
\(131\) 14.2414i 1.24428i −0.782906 0.622141i \(-0.786263\pi\)
0.782906 0.622141i \(-0.213737\pi\)
\(132\) 0 0
\(133\) 1.01031 + 7.07057i 0.0876048 + 0.613096i
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 18.2500 1.55920 0.779600 0.626277i \(-0.215422\pi\)
0.779600 + 0.626277i \(0.215422\pi\)
\(138\) 0 0
\(139\) 4.51031i 0.382559i 0.981536 + 0.191280i \(0.0612637\pi\)
−0.981536 + 0.191280i \(0.938736\pi\)
\(140\) 0 0
\(141\) 6.85941i 0.577667i
\(142\) 0 0
\(143\) −1.67462 −0.140039
\(144\) 0 0
\(145\) 1.02200i 0.0848723i
\(146\) 0 0
\(147\) −4.31507 −0.355901
\(148\) 0 0
\(149\) −0.630141 −0.0516231 −0.0258116 0.999667i \(-0.508217\pi\)
−0.0258116 + 0.999667i \(0.508217\pi\)
\(150\) 0 0
\(151\) 11.2945 0.919130 0.459565 0.888144i \(-0.348005\pi\)
0.459565 + 0.888144i \(0.348005\pi\)
\(152\) 0 0
\(153\) 3.31507 0.268008
\(154\) 0 0
\(155\) 6.30476 0.506411
\(156\) 0 0
\(157\) 5.01031 0.399866 0.199933 0.979810i \(-0.435928\pi\)
0.199933 + 0.979810i \(0.435928\pi\)
\(158\) 0 0
\(159\) 8.09257i 0.641782i
\(160\) 0 0
\(161\) 13.2603 1.04506
\(162\) 0 0
\(163\) 22.3340i 1.74934i 0.484723 + 0.874668i \(0.338920\pi\)
−0.484723 + 0.874668i \(0.661080\pi\)
\(164\) 0 0
\(165\) 1.02200i 0.0795625i
\(166\) 0 0
\(167\) −9.31507 −0.720822 −0.360411 0.932794i \(-0.617364\pi\)
−0.360411 + 0.932794i \(0.617364\pi\)
\(168\) 0 0
\(169\) 10.3151 0.793467
\(170\) 0 0
\(171\) 4.31507 0.616577i 0.329982 0.0471508i
\(172\) 0 0
\(173\) 7.67025i 0.583159i −0.956547 0.291579i \(-0.905819\pi\)
0.956547 0.291579i \(-0.0941807\pi\)
\(174\) 0 0
\(175\) 1.63858i 0.123865i
\(176\) 0 0
\(177\) 6.63014 0.498352
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 8.59830i 0.639106i 0.947568 + 0.319553i \(0.103533\pi\)
−0.947568 + 0.319553i \(0.896467\pi\)
\(182\) 0 0
\(183\) 4.30476 0.318217
\(184\) 0 0
\(185\) 9.11457i 0.670116i
\(186\) 0 0
\(187\) 3.38800i 0.247755i
\(188\) 0 0
\(189\) 1.63858i 0.119189i
\(190\) 0 0
\(191\) 20.4843i 1.48219i −0.671400 0.741095i \(-0.734307\pi\)
0.671400 0.741095i \(-0.265693\pi\)
\(192\) 0 0
\(193\) 12.7032i 0.914395i 0.889365 + 0.457198i \(0.151147\pi\)
−0.889365 + 0.457198i \(0.848853\pi\)
\(194\) 0 0
\(195\) 1.63858i 0.117341i
\(196\) 0 0
\(197\) 15.3151 1.09115 0.545577 0.838061i \(-0.316311\pi\)
0.545577 + 0.838061i \(0.316311\pi\)
\(198\) 0 0
\(199\) 18.2291i 1.29223i −0.763241 0.646114i \(-0.776393\pi\)
0.763241 0.646114i \(-0.223607\pi\)
\(200\) 0 0
\(201\) −0.989692 −0.0698075
\(202\) 0 0
\(203\) −1.67462 −0.117535
\(204\) 0 0
\(205\) 2.25515i 0.157507i
\(206\) 0 0
\(207\) 8.09257i 0.562472i
\(208\) 0 0
\(209\) 0.630141 + 4.40999i 0.0435878 + 0.305046i
\(210\) 0 0
\(211\) −8.98969 −0.618876 −0.309438 0.950920i \(-0.600141\pi\)
−0.309438 + 0.950920i \(0.600141\pi\)
\(212\) 0 0
\(213\) 6.63014 0.454290
\(214\) 0 0
\(215\) 9.11457i 0.621608i
\(216\) 0 0
\(217\) 10.3308i 0.701302i
\(218\) 0 0
\(219\) −16.2500 −1.09807
\(220\) 0 0
\(221\) 5.43199i 0.365395i
\(222\) 0 0
\(223\) 23.8904 1.59982 0.799911 0.600119i \(-0.204880\pi\)
0.799911 + 0.600119i \(0.204880\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −27.1849 −1.80432 −0.902162 0.431398i \(-0.858021\pi\)
−0.902162 + 0.431398i \(0.858021\pi\)
\(228\) 0 0
\(229\) −12.6849 −0.838244 −0.419122 0.907930i \(-0.637662\pi\)
−0.419122 + 0.907930i \(0.637662\pi\)
\(230\) 0 0
\(231\) 1.67462 0.110182
\(232\) 0 0
\(233\) 21.2809 1.39416 0.697079 0.716994i \(-0.254483\pi\)
0.697079 + 0.716994i \(0.254483\pi\)
\(234\) 0 0
\(235\) 6.85941i 0.447459i
\(236\) 0 0
\(237\) 1.61983 0.105219
\(238\) 0 0
\(239\) 5.64315i 0.365025i −0.983204 0.182512i \(-0.941577\pi\)
0.983204 0.182512i \(-0.0584230\pi\)
\(240\) 0 0
\(241\) 17.8068i 1.14704i −0.819192 0.573519i \(-0.805578\pi\)
0.819192 0.573519i \(-0.194422\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −4.31507 −0.275680
\(246\) 0 0
\(247\) 1.01031 + 7.07057i 0.0642844 + 0.449890i
\(248\) 0 0
\(249\) 10.1366i 0.642379i
\(250\) 0 0
\(251\) 15.0523i 0.950092i 0.879961 + 0.475046i \(0.157569\pi\)
−0.879961 + 0.475046i \(0.842431\pi\)
\(252\) 0 0
\(253\) 8.27059 0.519968
\(254\) 0 0
\(255\) 3.31507 0.207598
\(256\) 0 0
\(257\) 19.7674i 1.23306i −0.787333 0.616528i \(-0.788539\pi\)
0.787333 0.616528i \(-0.211461\pi\)
\(258\) 0 0
\(259\) 14.9349 0.928010
\(260\) 0 0
\(261\) 1.02200i 0.0632601i
\(262\) 0 0
\(263\) 19.7674i 1.21891i −0.792821 0.609455i \(-0.791388\pi\)
0.792821 0.609455i \(-0.208612\pi\)
\(264\) 0 0
\(265\) 8.09257i 0.497122i
\(266\) 0 0
\(267\) 5.22084i 0.319510i
\(268\) 0 0
\(269\) 26.7271i 1.62958i 0.579755 + 0.814791i \(0.303148\pi\)
−0.579755 + 0.814791i \(0.696852\pi\)
\(270\) 0 0
\(271\) 5.74346i 0.348890i 0.984667 + 0.174445i \(0.0558132\pi\)
−0.984667 + 0.174445i \(0.944187\pi\)
\(272\) 0 0
\(273\) 2.68493 0.162499
\(274\) 0 0
\(275\) 1.02200i 0.0616288i
\(276\) 0 0
\(277\) 18.6095 1.11814 0.559069 0.829121i \(-0.311159\pi\)
0.559069 + 0.829121i \(0.311159\pi\)
\(278\) 0 0
\(279\) −6.30476 −0.377456
\(280\) 0 0
\(281\) 13.4306i 0.801203i 0.916252 + 0.400601i \(0.131199\pi\)
−0.916252 + 0.400601i \(0.868801\pi\)
\(282\) 0 0
\(283\) 3.68257i 0.218906i −0.993992 0.109453i \(-0.965090\pi\)
0.993992 0.109453i \(-0.0349099\pi\)
\(284\) 0 0
\(285\) 4.31507 0.616577i 0.255603 0.0365229i
\(286\) 0 0
\(287\) −3.69524 −0.218123
\(288\) 0 0
\(289\) −6.01031 −0.353548
\(290\) 0 0
\(291\) 19.0569i 1.11713i
\(292\) 0 0
\(293\) 6.04857i 0.353361i 0.984268 + 0.176681i \(0.0565360\pi\)
−0.984268 + 0.176681i \(0.943464\pi\)
\(294\) 0 0
\(295\) 6.63014 0.386022
\(296\) 0 0
\(297\) 1.02200i 0.0593024i
\(298\) 0 0
\(299\) 13.2603 0.766862
\(300\) 0 0
\(301\) 14.9349 0.860833
\(302\) 0 0
\(303\) −10.6095 −0.609501
\(304\) 0 0
\(305\) 4.30476 0.246490
\(306\) 0 0
\(307\) 22.6301 1.29157 0.645785 0.763519i \(-0.276530\pi\)
0.645785 + 0.763519i \(0.276530\pi\)
\(308\) 0 0
\(309\) −10.6301 −0.604728
\(310\) 0 0
\(311\) 30.0043i 1.70139i 0.525663 + 0.850693i \(0.323817\pi\)
−0.525663 + 0.850693i \(0.676183\pi\)
\(312\) 0 0
\(313\) 22.8801 1.29326 0.646630 0.762804i \(-0.276178\pi\)
0.646630 + 0.762804i \(0.276178\pi\)
\(314\) 0 0
\(315\) 1.63858i 0.0923233i
\(316\) 0 0
\(317\) 19.9680i 1.12152i 0.827980 + 0.560758i \(0.189490\pi\)
−0.827980 + 0.560758i \(0.810510\pi\)
\(318\) 0 0
\(319\) −1.04448 −0.0584797
\(320\) 0 0
\(321\) 7.64045 0.426448
\(322\) 0 0
\(323\) 14.3048 2.04400i 0.795938 0.113731i
\(324\) 0 0
\(325\) 1.63858i 0.0908918i
\(326\) 0 0
\(327\) 19.4623i 1.07627i
\(328\) 0 0
\(329\) −11.2397 −0.619663
\(330\) 0 0
\(331\) 19.9452 1.09629 0.548144 0.836384i \(-0.315335\pi\)
0.548144 + 0.836384i \(0.315335\pi\)
\(332\) 0 0
\(333\) 9.11457i 0.499475i
\(334\) 0 0
\(335\) −0.989692 −0.0540726
\(336\) 0 0
\(337\) 33.6974i 1.83561i 0.397028 + 0.917807i \(0.370042\pi\)
−0.397028 + 0.917807i \(0.629958\pi\)
\(338\) 0 0
\(339\) 8.90341i 0.483567i
\(340\) 0 0
\(341\) 6.44346i 0.348933i
\(342\) 0 0
\(343\) 18.5406i 1.00110i
\(344\) 0 0
\(345\) 8.09257i 0.435689i
\(346\) 0 0
\(347\) 20.3903i 1.09461i −0.836933 0.547305i \(-0.815654\pi\)
0.836933 0.547305i \(-0.184346\pi\)
\(348\) 0 0
\(349\) −12.2706 −0.656830 −0.328415 0.944534i \(-0.606514\pi\)
−0.328415 + 0.944534i \(0.606514\pi\)
\(350\) 0 0
\(351\) 1.63858i 0.0874607i
\(352\) 0 0
\(353\) 19.2603 1.02512 0.512561 0.858651i \(-0.328697\pi\)
0.512561 + 0.858651i \(0.328697\pi\)
\(354\) 0 0
\(355\) 6.63014 0.351891
\(356\) 0 0
\(357\) 5.43199i 0.287492i
\(358\) 0 0
\(359\) 9.84199i 0.519440i 0.965684 + 0.259720i \(0.0836303\pi\)
−0.965684 + 0.259720i \(0.916370\pi\)
\(360\) 0 0
\(361\) 18.2397 5.32115i 0.959982 0.280060i
\(362\) 0 0
\(363\) −9.95552 −0.522529
\(364\) 0 0
\(365\) −16.2500 −0.850562
\(366\) 0 0
\(367\) 7.27119i 0.379553i 0.981827 + 0.189777i \(0.0607763\pi\)
−0.981827 + 0.189777i \(0.939224\pi\)
\(368\) 0 0
\(369\) 2.25515i 0.117399i
\(370\) 0 0
\(371\) −13.2603 −0.688440
\(372\) 0 0
\(373\) 15.3574i 0.795176i −0.917564 0.397588i \(-0.869847\pi\)
0.917564 0.397588i \(-0.130153\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −1.67462 −0.0862474
\(378\) 0 0
\(379\) 19.9452 1.02452 0.512258 0.858831i \(-0.328809\pi\)
0.512258 + 0.858831i \(0.328809\pi\)
\(380\) 0 0
\(381\) −14.2500 −0.730048
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 1.67462 0.0853466
\(386\) 0 0
\(387\) 9.11457i 0.463319i
\(388\) 0 0
\(389\) −25.8904 −1.31270 −0.656348 0.754458i \(-0.727900\pi\)
−0.656348 + 0.754458i \(0.727900\pi\)
\(390\) 0 0
\(391\) 26.8274i 1.35672i
\(392\) 0 0
\(393\) 14.2414i 0.718386i
\(394\) 0 0
\(395\) 1.61983 0.0815026
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) −1.01031 7.07057i −0.0505787 0.353971i
\(400\) 0 0
\(401\) 27.0386i 1.35024i 0.737707 + 0.675121i \(0.235909\pi\)
−0.737707 + 0.675121i \(0.764091\pi\)
\(402\) 0 0
\(403\) 10.3308i 0.514615i
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 9.31507 0.461731
\(408\) 0 0
\(409\) 23.5503i 1.16449i 0.813015 + 0.582243i \(0.197825\pi\)
−0.813015 + 0.582243i \(0.802175\pi\)
\(410\) 0 0
\(411\) −18.2500 −0.900205
\(412\) 0 0
\(413\) 10.8640i 0.534582i
\(414\) 0 0
\(415\) 10.1366i 0.497584i
\(416\) 0 0
\(417\) 4.51031i 0.220871i
\(418\) 0 0
\(419\) 12.6968i 0.620281i −0.950691 0.310140i \(-0.899624\pi\)
0.950691 0.310140i \(-0.100376\pi\)
\(420\) 0 0
\(421\) 20.2731i 0.988052i 0.869447 + 0.494026i \(0.164475\pi\)
−0.869447 + 0.494026i \(0.835525\pi\)
\(422\) 0 0
\(423\) 6.85941i 0.333516i
\(424\) 0 0
\(425\) 3.31507 0.160805
\(426\) 0 0
\(427\) 7.05368i 0.341351i
\(428\) 0 0
\(429\) 1.67462 0.0808515
\(430\) 0 0
\(431\) −6.63014 −0.319363 −0.159681 0.987169i \(-0.551047\pi\)
−0.159681 + 0.987169i \(0.551047\pi\)
\(432\) 0 0
\(433\) 3.68257i 0.176973i −0.996077 0.0884866i \(-0.971797\pi\)
0.996077 0.0884866i \(-0.0282030\pi\)
\(434\) 0 0
\(435\) 1.02200i 0.0490011i
\(436\) 0 0
\(437\) −4.98969 34.9200i −0.238689 1.67045i
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 4.31507 0.205480
\(442\) 0 0
\(443\) 5.23773i 0.248852i 0.992229 + 0.124426i \(0.0397089\pi\)
−0.992229 + 0.124426i \(0.960291\pi\)
\(444\) 0 0
\(445\) 5.22084i 0.247491i
\(446\) 0 0
\(447\) 0.630141 0.0298046
\(448\) 0 0
\(449\) 41.6020i 1.96332i −0.190637 0.981661i \(-0.561055\pi\)
0.190637 0.981661i \(-0.438945\pi\)
\(450\) 0 0
\(451\) −2.30476 −0.108527
\(452\) 0 0
\(453\) −11.2945 −0.530660
\(454\) 0 0
\(455\) 2.68493 0.125871
\(456\) 0 0
\(457\) 2.98969 0.139852 0.0699259 0.997552i \(-0.477724\pi\)
0.0699259 + 0.997552i \(0.477724\pi\)
\(458\) 0 0
\(459\) −3.31507 −0.154734
\(460\) 0 0
\(461\) 27.2190 1.26772 0.633859 0.773449i \(-0.281470\pi\)
0.633859 + 0.773449i \(0.281470\pi\)
\(462\) 0 0
\(463\) 9.92541i 0.461273i −0.973040 0.230636i \(-0.925919\pi\)
0.973040 0.230636i \(-0.0740808\pi\)
\(464\) 0 0
\(465\) −6.30476 −0.292376
\(466\) 0 0
\(467\) 26.7440i 1.23757i −0.785562 0.618783i \(-0.787626\pi\)
0.785562 0.618783i \(-0.212374\pi\)
\(468\) 0 0
\(469\) 1.62168i 0.0748824i
\(470\) 0 0
\(471\) −5.01031 −0.230863
\(472\) 0 0
\(473\) 9.31507 0.428307
\(474\) 0 0
\(475\) 4.31507 0.616577i 0.197989 0.0282905i
\(476\) 0 0
\(477\) 8.09257i 0.370533i
\(478\) 0 0
\(479\) 0.100311i 0.00458330i 0.999997 + 0.00229165i \(0.000729456\pi\)
−0.999997 + 0.00229165i \(0.999271\pi\)
\(480\) 0 0
\(481\) 14.9349 0.680973
\(482\) 0 0
\(483\) −13.2603 −0.603364
\(484\) 0 0
\(485\) 19.0569i 0.865328i
\(486\) 0 0
\(487\) 9.64045 0.436850 0.218425 0.975854i \(-0.429908\pi\)
0.218425 + 0.975854i \(0.429908\pi\)
\(488\) 0 0
\(489\) 22.3340i 1.00998i
\(490\) 0 0
\(491\) 3.06599i 0.138366i 0.997604 + 0.0691832i \(0.0220393\pi\)
−0.997604 + 0.0691832i \(0.977961\pi\)
\(492\) 0 0
\(493\) 3.38800i 0.152588i
\(494\) 0 0
\(495\) 1.02200i 0.0459354i
\(496\) 0 0
\(497\) 10.8640i 0.487316i
\(498\) 0 0
\(499\) 16.9960i 0.760844i 0.924813 + 0.380422i \(0.124221\pi\)
−0.924813 + 0.380422i \(0.875779\pi\)
\(500\) 0 0
\(501\) 9.31507 0.416167
\(502\) 0 0
\(503\) 19.7674i 0.881385i 0.897658 + 0.440692i \(0.145267\pi\)
−0.897658 + 0.440692i \(0.854733\pi\)
\(504\) 0 0
\(505\) −10.6095 −0.472118
\(506\) 0 0
\(507\) −10.3151 −0.458108
\(508\) 0 0
\(509\) 24.1837i 1.07193i −0.844242 0.535963i \(-0.819949\pi\)
0.844242 0.535963i \(-0.180051\pi\)
\(510\) 0 0
\(511\) 26.6268i 1.17790i
\(512\) 0 0
\(513\) −4.31507 + 0.616577i −0.190515 + 0.0272225i
\(514\) 0 0
\(515\) −10.6301 −0.468420
\(516\) 0 0
\(517\) −7.01031 −0.308313
\(518\) 0 0
\(519\) 7.67025i 0.336687i
\(520\) 0 0
\(521\) 3.79977i 0.166471i 0.996530 + 0.0832355i \(0.0265254\pi\)
−0.996530 + 0.0832355i \(0.973475\pi\)
\(522\) 0 0
\(523\) 34.1404 1.49286 0.746428 0.665467i \(-0.231767\pi\)
0.746428 + 0.665467i \(0.231767\pi\)
\(524\) 0 0
\(525\) 1.63858i 0.0715133i
\(526\) 0 0
\(527\) −20.9007 −0.910450
\(528\) 0 0
\(529\) −42.4896 −1.84738
\(530\) 0 0
\(531\) −6.63014 −0.287724
\(532\) 0 0
\(533\) −3.69524 −0.160059
\(534\) 0 0
\(535\) 7.64045 0.330325
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.40999i 0.189952i
\(540\) 0 0
\(541\) −29.2190 −1.25623 −0.628113 0.778123i \(-0.716172\pi\)
−0.628113 + 0.778123i \(0.716172\pi\)
\(542\) 0 0
\(543\) 8.59830i 0.368988i
\(544\) 0 0
\(545\) 19.4623i 0.833673i
\(546\) 0 0
\(547\) −12.6095 −0.539144 −0.269572 0.962980i \(-0.586882\pi\)
−0.269572 + 0.962980i \(0.586882\pi\)
\(548\) 0 0
\(549\) −4.30476 −0.183723
\(550\) 0 0
\(551\) 0.630141 + 4.40999i 0.0268449 + 0.187872i
\(552\) 0 0
\(553\) 2.65422i 0.112869i
\(554\) 0 0
\(555\) 9.11457i 0.386892i
\(556\) 0 0
\(557\) −41.4896 −1.75797 −0.878986 0.476847i \(-0.841780\pi\)
−0.878986 + 0.476847i \(0.841780\pi\)
\(558\) 0 0
\(559\) 14.9349 0.631679
\(560\) 0 0
\(561\) 3.38800i 0.143041i
\(562\) 0 0
\(563\) −13.9246 −0.586852 −0.293426 0.955982i \(-0.594795\pi\)
−0.293426 + 0.955982i \(0.594795\pi\)
\(564\) 0 0
\(565\) 8.90341i 0.374569i
\(566\) 0 0
\(567\) 1.63858i 0.0688137i
\(568\) 0 0
\(569\) 27.6488i 1.15910i 0.814937 + 0.579549i \(0.196771\pi\)
−0.814937 + 0.579549i \(0.803229\pi\)
\(570\) 0 0
\(571\) 16.3858i 0.685722i 0.939386 + 0.342861i \(0.111396\pi\)
−0.939386 + 0.342861i \(0.888604\pi\)
\(572\) 0 0
\(573\) 20.4843i 0.855743i
\(574\) 0 0
\(575\) 8.09257i 0.337483i
\(576\) 0 0
\(577\) −3.64045 −0.151554 −0.0757769 0.997125i \(-0.524144\pi\)
−0.0757769 + 0.997125i \(0.524144\pi\)
\(578\) 0 0
\(579\) 12.7032i 0.527926i
\(580\) 0 0
\(581\) −16.6095 −0.689079
\(582\) 0 0
\(583\) −8.27059 −0.342533
\(584\) 0 0
\(585\) 1.63858i 0.0677468i
\(586\) 0 0
\(587\) 40.2411i 1.66093i 0.557071 + 0.830465i \(0.311925\pi\)
−0.557071 + 0.830465i \(0.688075\pi\)
\(588\) 0 0
\(589\) −27.2055 + 3.88737i −1.12098 + 0.160176i
\(590\) 0 0
\(591\) −15.3151 −0.629978
\(592\) 0 0
\(593\) 8.27059 0.339632 0.169816 0.985476i \(-0.445683\pi\)
0.169816 + 0.985476i \(0.445683\pi\)
\(594\) 0 0
\(595\) 5.43199i 0.222690i
\(596\) 0 0
\(597\) 18.2291i 0.746069i
\(598\) 0 0
\(599\) −7.39048 −0.301967 −0.150983 0.988536i \(-0.548244\pi\)
−0.150983 + 0.988536i \(0.548244\pi\)
\(600\) 0 0
\(601\) 4.89884i 0.199828i −0.994996 0.0999138i \(-0.968143\pi\)
0.994996 0.0999138i \(-0.0318567\pi\)
\(602\) 0 0
\(603\) 0.989692 0.0403034
\(604\) 0 0
\(605\) −9.95552 −0.404749
\(606\) 0 0
\(607\) 27.5103 1.11661 0.558303 0.829637i \(-0.311453\pi\)
0.558303 + 0.829637i \(0.311453\pi\)
\(608\) 0 0
\(609\) 1.67462 0.0678591
\(610\) 0 0
\(611\) −11.2397 −0.454708
\(612\) 0 0
\(613\) 19.5992 0.791605 0.395802 0.918336i \(-0.370467\pi\)
0.395802 + 0.918336i \(0.370467\pi\)
\(614\) 0 0
\(615\) 2.25515i 0.0909365i
\(616\) 0 0
\(617\) −9.94521 −0.400379 −0.200190 0.979757i \(-0.564156\pi\)
−0.200190 + 0.979757i \(0.564156\pi\)
\(618\) 0 0
\(619\) 32.5709i 1.30913i 0.756004 + 0.654567i \(0.227149\pi\)
−0.756004 + 0.654567i \(0.772851\pi\)
\(620\) 0 0
\(621\) 8.09257i 0.324744i
\(622\) 0 0
\(623\) 8.55474 0.342738
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −0.630141 4.40999i −0.0251654 0.176118i
\(628\) 0 0
\(629\) 30.2154i 1.20477i
\(630\) 0 0
\(631\) 27.0491i 1.07681i 0.842687 + 0.538404i \(0.180973\pi\)
−0.842687 + 0.538404i \(0.819027\pi\)
\(632\) 0 0
\(633\) 8.98969 0.357308
\(634\) 0 0
\(635\) −14.2500 −0.565493
\(636\) 0 0
\(637\) 7.07057i 0.280146i
\(638\) 0 0
\(639\) −6.63014 −0.262284
\(640\) 0 0
\(641\) 24.0729i 0.950822i −0.879764 0.475411i \(-0.842299\pi\)
0.879764 0.475411i \(-0.157701\pi\)
\(642\) 0 0
\(643\) 2.56026i 0.100967i 0.998725 + 0.0504835i \(0.0160762\pi\)
−0.998725 + 0.0504835i \(0.983924\pi\)
\(644\) 0 0
\(645\) 9.11457i 0.358886i
\(646\) 0 0
\(647\) 19.1572i 0.753146i −0.926387 0.376573i \(-0.877102\pi\)
0.926387 0.376573i \(-0.122898\pi\)
\(648\) 0 0
\(649\) 6.77599i 0.265981i
\(650\) 0 0
\(651\) 10.3308i 0.404897i
\(652\) 0 0
\(653\) 39.8150 1.55808 0.779041 0.626973i \(-0.215706\pi\)
0.779041 + 0.626973i \(0.215706\pi\)
\(654\) 0 0
\(655\) 14.2414i 0.556459i
\(656\) 0 0
\(657\) 16.2500 0.633972
\(658\) 0 0
\(659\) −0.760335 −0.0296184 −0.0148092 0.999890i \(-0.504714\pi\)
−0.0148092 + 0.999890i \(0.504714\pi\)
\(660\) 0 0
\(661\) 35.6474i 1.38652i −0.720686 0.693262i \(-0.756173\pi\)
0.720686 0.693262i \(-0.243827\pi\)
\(662\) 0 0
\(663\) 5.43199i 0.210961i
\(664\) 0 0
\(665\) −1.01031 7.07057i −0.0391781 0.274185i
\(666\) 0 0
\(667\) 8.27059 0.320239
\(668\) 0 0
\(669\) −23.8904 −0.923657
\(670\) 0 0
\(671\) 4.39946i 0.169839i
\(672\) 0 0
\(673\) 13.8255i 0.532934i −0.963844 0.266467i \(-0.914144\pi\)
0.963844 0.266467i \(-0.0858563\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 44.5508i 1.71223i 0.516788 + 0.856114i \(0.327128\pi\)
−0.516788 + 0.856114i \(0.672872\pi\)
\(678\) 0 0
\(679\) −31.2261 −1.19835
\(680\) 0 0
\(681\) 27.1849 1.04173
\(682\) 0 0
\(683\) 26.8389 1.02696 0.513481 0.858101i \(-0.328356\pi\)
0.513481 + 0.858101i \(0.328356\pi\)
\(684\) 0 0
\(685\) −18.2500 −0.697296
\(686\) 0 0
\(687\) 12.6849 0.483960
\(688\) 0 0
\(689\) −13.2603 −0.505176
\(690\) 0 0
\(691\) 48.1331i 1.83107i −0.402239 0.915535i \(-0.631768\pi\)
0.402239 0.915535i \(-0.368232\pi\)
\(692\) 0 0
\(693\) −1.67462 −0.0636136
\(694\) 0 0
\(695\) 4.51031i 0.171086i
\(696\) 0 0
\(697\) 7.47599i 0.283173i
\(698\) 0 0
\(699\) −21.2809 −0.804917
\(700\) 0 0
\(701\) −25.8904 −0.977868 −0.488934 0.872321i \(-0.662614\pi\)
−0.488934 + 0.872321i \(0.662614\pi\)
\(702\) 0 0
\(703\) −5.61983 39.3300i −0.211956 1.48336i
\(704\) 0 0
\(705\) 6.85941i 0.258340i
\(706\) 0 0
\(707\) 17.3845i 0.653812i
\(708\) 0 0
\(709\) −26.2158 −0.984555 −0.492278 0.870438i \(-0.663835\pi\)
−0.492278 + 0.870438i \(0.663835\pi\)
\(710\) 0 0
\(711\) −1.61983 −0.0607485
\(712\) 0 0
\(713\) 51.0217i 1.91078i
\(714\) 0 0
\(715\) 1.67462 0.0626273
\(716\) 0 0
\(717\) 5.64315i 0.210747i
\(718\) 0 0
\(719\) 5.84377i 0.217936i 0.994045 + 0.108968i \(0.0347546\pi\)
−0.994045 + 0.108968i \(0.965245\pi\)
\(720\) 0 0
\(721\) 17.4183i 0.648691i
\(722\) 0 0
\(723\) 17.8068i 0.662243i
\(724\) 0 0
\(725\) 1.02200i 0.0379561i
\(726\) 0 0
\(727\) 17.8237i 0.661045i −0.943798 0.330522i \(-0.892775\pi\)
0.943798 0.330522i \(-0.107225\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 30.2154i 1.11756i
\(732\) 0 0
\(733\) 3.75003 0.138510 0.0692552 0.997599i \(-0.477938\pi\)
0.0692552 + 0.997599i \(0.477938\pi\)
\(734\) 0 0
\(735\) 4.31507 0.159164
\(736\) 0 0
\(737\) 1.01146i 0.0372577i
\(738\) 0 0
\(739\) 32.7588i 1.20505i −0.798099 0.602526i \(-0.794161\pi\)
0.798099 0.602526i \(-0.205839\pi\)
\(740\) 0 0
\(741\) −1.01031 7.07057i −0.0371146 0.259744i
\(742\) 0 0
\(743\) −47.2397 −1.73306 −0.866528 0.499129i \(-0.833653\pi\)
−0.866528 + 0.499129i \(0.833653\pi\)
\(744\) 0 0
\(745\) 0.630141 0.0230866
\(746\) 0 0
\(747\) 10.1366i 0.370877i
\(748\) 0 0
\(749\) 12.5195i 0.457451i
\(750\) 0 0
\(751\) −1.96583 −0.0717341 −0.0358670 0.999357i \(-0.511419\pi\)
−0.0358670 + 0.999357i \(0.511419\pi\)
\(752\) 0 0
\(753\) 15.0523i 0.548536i
\(754\) 0 0
\(755\) −11.2945 −0.411047
\(756\) 0 0
\(757\) −26.6095 −0.967140 −0.483570 0.875306i \(-0.660660\pi\)
−0.483570 + 0.875306i \(0.660660\pi\)
\(758\) 0 0
\(759\) −8.27059 −0.300203
\(760\) 0 0
\(761\) 38.3904 1.39165 0.695825 0.718211i \(-0.255039\pi\)
0.695825 + 0.718211i \(0.255039\pi\)
\(762\) 0 0
\(763\) 31.8904 1.15451
\(764\) 0 0
\(765\) −3.31507 −0.119857
\(766\) 0 0
\(767\) 10.8640i 0.392276i
\(768\) 0 0
\(769\) −11.2809 −0.406800 −0.203400 0.979096i \(-0.565199\pi\)
−0.203400 + 0.979096i \(0.565199\pi\)
\(770\) 0 0
\(771\) 19.7674i 0.711905i
\(772\) 0 0
\(773\) 11.5703i 0.416156i 0.978112 + 0.208078i \(0.0667208\pi\)
−0.978112 + 0.208078i \(0.933279\pi\)
\(774\) 0 0
\(775\) −6.30476 −0.226474
\(776\) 0 0
\(777\) −14.9349 −0.535787
\(778\) 0 0
\(779\) 1.39048 + 9.73114i 0.0498190 + 0.348654i
\(780\) 0 0
\(781\) 6.77599i 0.242464i
\(782\) 0 0
\(783\) 1.02200i 0.0365232i
\(784\) 0 0
\(785\) −5.01031 −0.178826
\(786\) 0 0
\(787\) 19.6198 0.699372 0.349686 0.936867i \(-0.386288\pi\)
0.349686 + 0.936867i \(0.386288\pi\)
\(788\) 0 0
\(789\) 19.7674i 0.703738i
\(790\) 0 0
\(791\) 14.5889 0.518722
\(792\) 0 0
\(793\) 7.05368i 0.250483i
\(794\) 0 0
\(795\) 8.09257i 0.287014i
\(796\) 0 0
\(797\) 41.8966i 1.48405i −0.670370 0.742027i \(-0.733865\pi\)
0.670370 0.742027i \(-0.266135\pi\)
\(798\) 0 0
\(799\) 22.7394i 0.804463i
\(800\) 0 0
\(801\) 5.22084i 0.184469i
\(802\) 0 0
\(803\) 16.6074i 0.586064i
\(804\)