Properties

Label 6012.2.h.a.3005.9
Level $6012$
Weight $2$
Character 6012.3005
Analytic conductor $48.006$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6012.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0060616952\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3005.9
Character \(\chi\) \(=\) 6012.3005
Dual form 6012.2.h.a.3005.10

$q$-expansion

\(f(q)\) \(=\) \(q-3.04368 q^{5} +0.233711 q^{7} +O(q^{10})\) \(q-3.04368 q^{5} +0.233711 q^{7} -2.11221i q^{11} +1.71375i q^{13} +5.18919 q^{17} -1.94672 q^{19} -2.99706 q^{23} +4.26397 q^{25} -2.73646i q^{29} -8.50175 q^{31} -0.711342 q^{35} +8.21183i q^{37} +3.33880 q^{41} -0.222843i q^{43} -8.47244i q^{47} -6.94538 q^{49} +9.15818 q^{53} +6.42890i q^{55} -10.8700 q^{59} +10.7341 q^{61} -5.21611i q^{65} -0.936932i q^{67} +9.56802 q^{71} +9.59038i q^{73} -0.493649i q^{77} -5.20452i q^{79} -3.37637 q^{83} -15.7942 q^{85} -16.1716i q^{89} +0.400524i q^{91} +5.92520 q^{95} -11.2646 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56q + O(q^{10}) \) \( 56q + 8q^{19} + 64q^{25} - 8q^{31} + 56q^{49} - 8q^{61} + 32q^{85} - 48q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6012\mathbb{Z}\right)^\times\).

\(n\) \(3007\) \(3341\) \(4681\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −3.04368 −1.36117 −0.680587 0.732668i \(-0.738275\pi\)
−0.680587 + 0.732668i \(0.738275\pi\)
\(6\) 0 0
\(7\) 0.233711 0.0883346 0.0441673 0.999024i \(-0.485937\pi\)
0.0441673 + 0.999024i \(0.485937\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.11221i 0.636857i −0.947947 0.318428i \(-0.896845\pi\)
0.947947 0.318428i \(-0.103155\pi\)
\(12\) 0 0
\(13\) 1.71375i 0.475310i 0.971350 + 0.237655i \(0.0763787\pi\)
−0.971350 + 0.237655i \(0.923621\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.18919 1.25856 0.629281 0.777178i \(-0.283349\pi\)
0.629281 + 0.777178i \(0.283349\pi\)
\(18\) 0 0
\(19\) −1.94672 −0.446609 −0.223305 0.974749i \(-0.571684\pi\)
−0.223305 + 0.974749i \(0.571684\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.99706 −0.624931 −0.312465 0.949929i \(-0.601155\pi\)
−0.312465 + 0.949929i \(0.601155\pi\)
\(24\) 0 0
\(25\) 4.26397 0.852793
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.73646i 0.508149i −0.967185 0.254074i \(-0.918229\pi\)
0.967185 0.254074i \(-0.0817708\pi\)
\(30\) 0 0
\(31\) −8.50175 −1.52696 −0.763480 0.645832i \(-0.776511\pi\)
−0.763480 + 0.645832i \(0.776511\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.711342 −0.120239
\(36\) 0 0
\(37\) 8.21183i 1.35002i 0.737810 + 0.675008i \(0.235860\pi\)
−0.737810 + 0.675008i \(0.764140\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.33880 0.521433 0.260717 0.965415i \(-0.416041\pi\)
0.260717 + 0.965415i \(0.416041\pi\)
\(42\) 0 0
\(43\) 0.222843i 0.0339833i −0.999856 0.0169916i \(-0.994591\pi\)
0.999856 0.0169916i \(-0.00540887\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.47244i 1.23583i −0.786244 0.617916i \(-0.787977\pi\)
0.786244 0.617916i \(-0.212023\pi\)
\(48\) 0 0
\(49\) −6.94538 −0.992197
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.15818 1.25797 0.628987 0.777416i \(-0.283470\pi\)
0.628987 + 0.777416i \(0.283470\pi\)
\(54\) 0 0
\(55\) 6.42890i 0.866872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.8700 −1.41516 −0.707579 0.706634i \(-0.750213\pi\)
−0.707579 + 0.706634i \(0.750213\pi\)
\(60\) 0 0
\(61\) 10.7341 1.37436 0.687179 0.726489i \(-0.258849\pi\)
0.687179 + 0.726489i \(0.258849\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.21611i 0.646979i
\(66\) 0 0
\(67\) 0.936932i 0.114464i −0.998361 0.0572322i \(-0.981772\pi\)
0.998361 0.0572322i \(-0.0182275\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.56802 1.13551 0.567757 0.823196i \(-0.307811\pi\)
0.567757 + 0.823196i \(0.307811\pi\)
\(72\) 0 0
\(73\) 9.59038i 1.12247i 0.827657 + 0.561234i \(0.189673\pi\)
−0.827657 + 0.561234i \(0.810327\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.493649i 0.0562565i
\(78\) 0 0
\(79\) 5.20452i 0.585554i −0.956181 0.292777i \(-0.905421\pi\)
0.956181 0.292777i \(-0.0945794\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.37637 −0.370605 −0.185303 0.982682i \(-0.559327\pi\)
−0.185303 + 0.982682i \(0.559327\pi\)
\(84\) 0 0
\(85\) −15.7942 −1.71312
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.1716i 1.71419i −0.515161 0.857093i \(-0.672268\pi\)
0.515161 0.857093i \(-0.327732\pi\)
\(90\) 0 0
\(91\) 0.400524i 0.0419863i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.92520 0.607912
\(96\) 0 0
\(97\) −11.2646 −1.14374 −0.571872 0.820343i \(-0.693782\pi\)
−0.571872 + 0.820343i \(0.693782\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.22423 0.121816 0.0609080 0.998143i \(-0.480600\pi\)
0.0609080 + 0.998143i \(0.480600\pi\)
\(102\) 0 0
\(103\) 0.764815i 0.0753594i 0.999290 + 0.0376797i \(0.0119967\pi\)
−0.999290 + 0.0376797i \(0.988003\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.90034i 0.957102i 0.878060 + 0.478551i \(0.158838\pi\)
−0.878060 + 0.478551i \(0.841162\pi\)
\(108\) 0 0
\(109\) 0.0922689i 0.00883776i −0.999990 0.00441888i \(-0.998593\pi\)
0.999990 0.00441888i \(-0.00140658\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.87054 −0.458182 −0.229091 0.973405i \(-0.573575\pi\)
−0.229091 + 0.973405i \(0.573575\pi\)
\(114\) 0 0
\(115\) 9.12209 0.850639
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.21277 0.111175
\(120\) 0 0
\(121\) 6.53855 0.594414
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.24025 0.200374
\(126\) 0 0
\(127\) 1.79810 0.159555 0.0797777 0.996813i \(-0.474579\pi\)
0.0797777 + 0.996813i \(0.474579\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 11.0140 0.962299 0.481150 0.876639i \(-0.340219\pi\)
0.481150 + 0.876639i \(0.340219\pi\)
\(132\) 0 0
\(133\) −0.454972 −0.0394510
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.63132i 0.566552i −0.959038 0.283276i \(-0.908579\pi\)
0.959038 0.283276i \(-0.0914212\pi\)
\(138\) 0 0
\(139\) 21.6598i 1.83716i 0.395236 + 0.918580i \(0.370663\pi\)
−0.395236 + 0.918580i \(0.629337\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.61981 0.302704
\(144\) 0 0
\(145\) 8.32891i 0.691678i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 17.6000 1.44184 0.720922 0.693016i \(-0.243718\pi\)
0.720922 + 0.693016i \(0.243718\pi\)
\(150\) 0 0
\(151\) 23.5281i 1.91469i 0.288953 + 0.957343i \(0.406693\pi\)
−0.288953 + 0.957343i \(0.593307\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 25.8766 2.07846
\(156\) 0 0
\(157\) −18.4566 −1.47300 −0.736499 0.676439i \(-0.763522\pi\)
−0.736499 + 0.676439i \(0.763522\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.700448 −0.0552030
\(162\) 0 0
\(163\) 16.5832i 1.29889i −0.760407 0.649447i \(-0.775000\pi\)
0.760407 0.649447i \(-0.225000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.8060 1.73367i −0.990960 0.134155i
\(168\) 0 0
\(169\) 10.0631 0.774081
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 25.6712i 1.95175i 0.218332 + 0.975874i \(0.429938\pi\)
−0.218332 + 0.975874i \(0.570062\pi\)
\(174\) 0 0
\(175\) 0.996537 0.0753311
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 17.9325i 1.34034i 0.742208 + 0.670169i \(0.233778\pi\)
−0.742208 + 0.670169i \(0.766222\pi\)
\(180\) 0 0
\(181\) 21.2042 1.57610 0.788049 0.615612i \(-0.211091\pi\)
0.788049 + 0.615612i \(0.211091\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 24.9942i 1.83761i
\(186\) 0 0
\(187\) 10.9607i 0.801524i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 26.4287i 1.91231i 0.292856 + 0.956157i \(0.405394\pi\)
−0.292856 + 0.956157i \(0.594606\pi\)
\(192\) 0 0
\(193\) 13.7744i 0.991506i 0.868464 + 0.495753i \(0.165108\pi\)
−0.868464 + 0.495753i \(0.834892\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −9.33812 −0.665314 −0.332657 0.943048i \(-0.607945\pi\)
−0.332657 + 0.943048i \(0.607945\pi\)
\(198\) 0 0
\(199\) 16.7492 1.18732 0.593659 0.804717i \(-0.297683\pi\)
0.593659 + 0.804717i \(0.297683\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.639543i 0.0448871i
\(204\) 0 0
\(205\) −10.1622 −0.709761
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.11190i 0.284426i
\(210\) 0 0
\(211\) 13.9376 0.959504 0.479752 0.877404i \(-0.340727\pi\)
0.479752 + 0.877404i \(0.340727\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.678263i 0.0462571i
\(216\) 0 0
\(217\) −1.98696 −0.134883
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.89298i 0.598207i
\(222\) 0 0
\(223\) 12.9899 0.869865 0.434933 0.900463i \(-0.356772\pi\)
0.434933 + 0.900463i \(0.356772\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.33538 −0.619611 −0.309805 0.950800i \(-0.600264\pi\)
−0.309805 + 0.950800i \(0.600264\pi\)
\(228\) 0 0
\(229\) 26.0496 1.72141 0.860703 0.509107i \(-0.170024\pi\)
0.860703 + 0.509107i \(0.170024\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.10871i 0.0726338i 0.999340 + 0.0363169i \(0.0115626\pi\)
−0.999340 + 0.0363169i \(0.988437\pi\)
\(234\) 0 0
\(235\) 25.7874i 1.68218i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.4479i 0.805184i 0.915379 + 0.402592i \(0.131891\pi\)
−0.915379 + 0.402592i \(0.868109\pi\)
\(240\) 0 0
\(241\) 11.4845i 0.739784i −0.929075 0.369892i \(-0.879395\pi\)
0.929075 0.369892i \(-0.120605\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 21.1395 1.35055
\(246\) 0 0
\(247\) 3.33620i 0.212278i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.3240i 1.03036i −0.857082 0.515181i \(-0.827725\pi\)
0.857082 0.515181i \(-0.172275\pi\)
\(252\) 0 0
\(253\) 6.33044i 0.397991i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.6077 1.16072 0.580359 0.814361i \(-0.302912\pi\)
0.580359 + 0.814361i \(0.302912\pi\)
\(258\) 0 0
\(259\) 1.91920i 0.119253i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.3510i 1.00825i 0.863631 + 0.504124i \(0.168185\pi\)
−0.863631 + 0.504124i \(0.831815\pi\)
\(264\) 0 0
\(265\) −27.8745 −1.71232
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 22.7864 1.38931 0.694657 0.719341i \(-0.255556\pi\)
0.694657 + 0.719341i \(0.255556\pi\)
\(270\) 0 0
\(271\) 13.2759i 0.806453i 0.915100 + 0.403226i \(0.132111\pi\)
−0.915100 + 0.403226i \(0.867889\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9.00641i 0.543107i
\(276\) 0 0
\(277\) 24.9639i 1.49994i 0.661473 + 0.749969i \(0.269932\pi\)
−0.661473 + 0.749969i \(0.730068\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 13.7440i 0.819898i −0.912108 0.409949i \(-0.865546\pi\)
0.912108 0.409949i \(-0.134454\pi\)
\(282\) 0 0
\(283\) 4.64056 0.275853 0.137926 0.990442i \(-0.455956\pi\)
0.137926 + 0.990442i \(0.455956\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.780316 0.0460606
\(288\) 0 0
\(289\) 9.92764 0.583979
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.10870i 0.473715i 0.971544 + 0.236858i \(0.0761175\pi\)
−0.971544 + 0.236858i \(0.923883\pi\)
\(294\) 0 0
\(295\) 33.0849 1.92627
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.13622i 0.297035i
\(300\) 0 0
\(301\) 0.0520810i 0.00300190i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −32.6710 −1.87074
\(306\) 0 0
\(307\) 19.9446i 1.13830i −0.822234 0.569150i \(-0.807272\pi\)
0.822234 0.569150i \(-0.192728\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.07490i 0.174362i 0.996193 + 0.0871808i \(0.0277858\pi\)
−0.996193 + 0.0871808i \(0.972214\pi\)
\(312\) 0 0
\(313\) 8.69801i 0.491641i 0.969315 + 0.245820i \(0.0790573\pi\)
−0.969315 + 0.245820i \(0.920943\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.795282i 0.0446675i 0.999751 + 0.0223337i \(0.00710964\pi\)
−0.999751 + 0.0223337i \(0.992890\pi\)
\(318\) 0 0
\(319\) −5.78000 −0.323618
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −10.1019 −0.562085
\(324\) 0 0
\(325\) 7.30738i 0.405341i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.98011i 0.109167i
\(330\) 0 0
\(331\) 14.1043i 0.775240i 0.921819 + 0.387620i \(0.126703\pi\)
−0.921819 + 0.387620i \(0.873297\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.85172i 0.155806i
\(336\) 0 0
\(337\) −12.0058 −0.653998 −0.326999 0.945025i \(-0.606037\pi\)
−0.326999 + 0.945025i \(0.606037\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 17.9575i 0.972454i
\(342\) 0 0
\(343\) −3.25919 −0.175980
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 26.5883 1.42734 0.713668 0.700484i \(-0.247032\pi\)
0.713668 + 0.700484i \(0.247032\pi\)
\(348\) 0 0
\(349\) 29.7666i 1.59337i 0.604393 + 0.796686i \(0.293416\pi\)
−0.604393 + 0.796686i \(0.706584\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 31.1203i 1.65637i −0.560456 0.828184i \(-0.689374\pi\)
0.560456 0.828184i \(-0.310626\pi\)
\(354\) 0 0
\(355\) −29.1219 −1.54563
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 21.2836i 1.12330i 0.827374 + 0.561652i \(0.189834\pi\)
−0.827374 + 0.561652i \(0.810166\pi\)
\(360\) 0 0
\(361\) −15.2103 −0.800540
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 29.1900i 1.52787i
\(366\) 0 0
\(367\) 7.18368 0.374985 0.187492 0.982266i \(-0.439964\pi\)
0.187492 + 0.982266i \(0.439964\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.14037 0.111123
\(372\) 0 0
\(373\) 12.3158i 0.637689i 0.947807 + 0.318844i \(0.103295\pi\)
−0.947807 + 0.318844i \(0.896705\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.68962 0.241528
\(378\) 0 0
\(379\) 12.8067i 0.657834i 0.944359 + 0.328917i \(0.106684\pi\)
−0.944359 + 0.328917i \(0.893316\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 26.0935i 1.33332i 0.745363 + 0.666659i \(0.232276\pi\)
−0.745363 + 0.666659i \(0.767724\pi\)
\(384\) 0 0
\(385\) 1.50251i 0.0765748i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 15.1524 0.768257 0.384129 0.923280i \(-0.374502\pi\)
0.384129 + 0.923280i \(0.374502\pi\)
\(390\) 0 0
\(391\) −15.5523 −0.786514
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 15.8409i 0.797041i
\(396\) 0 0
\(397\) 21.1906 1.06353 0.531764 0.846893i \(-0.321530\pi\)
0.531764 + 0.846893i \(0.321530\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0819 −0.902965 −0.451482 0.892280i \(-0.649105\pi\)
−0.451482 + 0.892280i \(0.649105\pi\)
\(402\) 0 0
\(403\) 14.5699i 0.725778i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 17.3451 0.859767
\(408\) 0 0
\(409\) −2.21828 −0.109687 −0.0548434 0.998495i \(-0.517466\pi\)
−0.0548434 + 0.998495i \(0.517466\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.54045 −0.125007
\(414\) 0 0
\(415\) 10.2766 0.504458
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.24647i 0.0608939i −0.999536 0.0304469i \(-0.990307\pi\)
0.999536 0.0304469i \(-0.00969306\pi\)
\(420\) 0 0
\(421\) 4.24823 0.207046 0.103523 0.994627i \(-0.466988\pi\)
0.103523 + 0.994627i \(0.466988\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 22.1265 1.07329
\(426\) 0 0
\(427\) 2.50867 0.121403
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 22.3570i 1.07690i −0.842658 0.538448i \(-0.819011\pi\)
0.842658 0.538448i \(-0.180989\pi\)
\(432\) 0 0
\(433\) 11.6183 0.558341 0.279171 0.960241i \(-0.409940\pi\)
0.279171 + 0.960241i \(0.409940\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.83445 0.279100
\(438\) 0 0
\(439\) 39.7712i 1.89817i −0.315014 0.949087i \(-0.602009\pi\)
0.315014 0.949087i \(-0.397991\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 16.7204 0.794408 0.397204 0.917730i \(-0.369981\pi\)
0.397204 + 0.917730i \(0.369981\pi\)
\(444\) 0 0
\(445\) 49.2211i 2.33330i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14.8333i 0.700028i −0.936744 0.350014i \(-0.886177\pi\)
0.936744 0.350014i \(-0.113823\pi\)
\(450\) 0 0
\(451\) 7.05226i 0.332078i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.21906i 0.0571506i
\(456\) 0 0
\(457\) 29.3212i 1.37159i 0.727795 + 0.685794i \(0.240545\pi\)
−0.727795 + 0.685794i \(0.759455\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.232415i 0.0108247i 0.999985 + 0.00541233i \(0.00172281\pi\)
−0.999985 + 0.00541233i \(0.998277\pi\)
\(462\) 0 0
\(463\) 27.5746i 1.28150i 0.767749 + 0.640750i \(0.221377\pi\)
−0.767749 + 0.640750i \(0.778623\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.6372i 0.538507i 0.963069 + 0.269254i \(0.0867770\pi\)
−0.963069 + 0.269254i \(0.913223\pi\)
\(468\) 0 0
\(469\) 0.218972i 0.0101112i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.470693 −0.0216425
\(474\) 0 0
\(475\) −8.30076 −0.380865
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −10.9459 −0.500129 −0.250065 0.968229i \(-0.580452\pi\)
−0.250065 + 0.968229i \(0.580452\pi\)
\(480\) 0 0
\(481\) −14.0730 −0.641676
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 34.2857 1.55683
\(486\) 0 0
\(487\) 16.4976i 0.747579i −0.927514 0.373789i \(-0.878058\pi\)
0.927514 0.373789i \(-0.121942\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 24.7588i 1.11735i −0.829387 0.558674i \(-0.811310\pi\)
0.829387 0.558674i \(-0.188690\pi\)
\(492\) 0 0
\(493\) 14.2000i 0.639537i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.23615 0.100305
\(498\) 0 0
\(499\) 26.5947i 1.19054i −0.803525 0.595272i \(-0.797045\pi\)
0.803525 0.595272i \(-0.202955\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 39.4288i 1.75804i −0.476782 0.879022i \(-0.658197\pi\)
0.476782 0.879022i \(-0.341803\pi\)
\(504\) 0 0
\(505\) −3.72617 −0.165813
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 33.1756i 1.47048i 0.677806 + 0.735241i \(0.262931\pi\)
−0.677806 + 0.735241i \(0.737069\pi\)
\(510\) 0 0
\(511\) 2.24138i 0.0991528i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.32785i 0.102577i
\(516\) 0 0
\(517\) −17.8956 −0.787048
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5.16160 −0.226134 −0.113067 0.993587i \(-0.536067\pi\)
−0.113067 + 0.993587i \(0.536067\pi\)
\(522\) 0 0
\(523\) 6.41931 0.280697 0.140348 0.990102i \(-0.455178\pi\)
0.140348 + 0.990102i \(0.455178\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −44.1171 −1.92177
\(528\) 0 0
\(529\) −14.0176 −0.609462
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5.72188i 0.247842i
\(534\) 0 0
\(535\) 30.1334i 1.30278i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 14.6701i 0.631887i
\(540\) 0 0
\(541\) 21.4097i 0.920476i 0.887796 + 0.460238i \(0.152236\pi\)
−0.887796 + 0.460238i \(0.847764\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.280837i 0.0120297i
\(546\) 0 0
\(547\) 4.02545i 0.172116i −0.996290 0.0860579i \(-0.972573\pi\)
0.996290 0.0860579i \(-0.0274270\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5.32714i 0.226944i
\(552\) 0 0
\(553\) 1.21636i 0.0517247i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 36.5466i 1.54853i −0.632861 0.774265i \(-0.718120\pi\)
0.632861 0.774265i \(-0.281880\pi\)
\(558\) 0 0
\(559\) 0.381898 0.0161526
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 14.6535i 0.617571i 0.951132 + 0.308786i \(0.0999225\pi\)
−0.951132 + 0.308786i \(0.900077\pi\)
\(564\) 0 0
\(565\) 14.8243 0.623665
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 33.3881 1.39970 0.699850 0.714290i \(-0.253250\pi\)
0.699850 + 0.714290i \(0.253250\pi\)
\(570\) 0 0
\(571\) 25.9499i 1.08597i −0.839743 0.542984i \(-0.817294\pi\)
0.839743 0.542984i \(-0.182706\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12.7794 −0.532936
\(576\) 0 0
\(577\) 22.5037 0.936840 0.468420 0.883506i \(-0.344824\pi\)
0.468420 + 0.883506i \(0.344824\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −0.789097 −0.0327373
\(582\) 0 0
\(583\) 19.3440i 0.801149i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.12151 0.128838 0.0644192 0.997923i \(-0.479481\pi\)
0.0644192 + 0.997923i \(0.479481\pi\)
\(588\) 0 0
\(589\) 16.5506 0.681954
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −5.01684 −0.206017 −0.103009 0.994680i \(-0.532847\pi\)
−0.103009 + 0.994680i \(0.532847\pi\)
\(594\) 0 0
\(595\) −3.69128 −0.151328
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 41.9042i 1.71216i −0.516844 0.856080i \(-0.672893\pi\)
0.516844 0.856080i \(-0.327107\pi\)
\(600\) 0 0
\(601\) −16.0514 −0.654751 −0.327375 0.944894i \(-0.606164\pi\)
−0.327375 + 0.944894i \(0.606164\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −19.9012 −0.809100
\(606\) 0 0
\(607\) 6.91136i 0.280523i 0.990114 + 0.140262i \(0.0447944\pi\)
−0.990114 + 0.140262i \(0.955206\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 14.5197 0.587403
\(612\) 0 0
\(613\) 22.9693 0.927723 0.463861 0.885908i \(-0.346464\pi\)
0.463861 + 0.885908i \(0.346464\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.885982i 0.0356683i −0.999841 0.0178342i \(-0.994323\pi\)
0.999841 0.0178342i \(-0.00567709\pi\)
\(618\) 0 0
\(619\) 16.5451i 0.665005i −0.943102 0.332503i \(-0.892107\pi\)
0.943102 0.332503i \(-0.107893\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.77949i 0.151422i
\(624\) 0 0
\(625\) −28.1384 −1.12554
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 42.6127i 1.69908i
\(630\) 0 0
\(631\) −32.2675 −1.28455 −0.642274 0.766475i \(-0.722009\pi\)
−0.642274 + 0.766475i \(0.722009\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.47283 −0.217183
\(636\) 0 0
\(637\) 11.9027i 0.471601i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7.26760 −0.287053 −0.143526 0.989646i \(-0.545844\pi\)
−0.143526 + 0.989646i \(0.545844\pi\)
\(642\) 0 0
\(643\) 11.1194i 0.438507i 0.975668 + 0.219254i \(0.0703622\pi\)
−0.975668 + 0.219254i \(0.929638\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −38.0483 −1.49583 −0.747917 0.663792i \(-0.768946\pi\)
−0.747917 + 0.663792i \(0.768946\pi\)
\(648\) 0 0
\(649\) 22.9598i 0.901253i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 49.0410i 1.91912i 0.281498 + 0.959562i \(0.409169\pi\)
−0.281498 + 0.959562i \(0.590831\pi\)
\(654\) 0 0
\(655\) −33.5231 −1.30986
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −0.928331 −0.0361626 −0.0180813 0.999837i \(-0.505756\pi\)
−0.0180813 + 0.999837i \(0.505756\pi\)
\(660\) 0 0
\(661\) 6.60700i 0.256983i −0.991711 0.128491i \(-0.958987\pi\)
0.991711 0.128491i \(-0.0410134\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.38479 0.0536997
\(666\) 0 0
\(667\) 8.20135i 0.317558i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 22.6727i 0.875268i
\(672\) 0 0
\(673\) 18.6494i 0.718881i 0.933168 + 0.359440i \(0.117032\pi\)
−0.933168 + 0.359440i \(0.882968\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 40.2712i 1.54775i −0.633340 0.773874i \(-0.718316\pi\)
0.633340 0.773874i \(-0.281684\pi\)
\(678\) 0 0
\(679\) −2.63266 −0.101032
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −15.3335 −0.586720 −0.293360 0.956002i \(-0.594773\pi\)
−0.293360 + 0.956002i \(0.594773\pi\)
\(684\) 0 0
\(685\) 20.1836i 0.771175i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 15.6949i 0.597927i
\(690\) 0 0
\(691\) 20.9822i 0.798201i 0.916907 + 0.399101i \(0.130678\pi\)
−0.916907 + 0.399101i \(0.869322\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 65.9254i 2.50069i
\(696\) 0 0
\(697\) 17.3257 0.656256
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 8.00186i 0.302226i 0.988516 + 0.151113i \(0.0482858\pi\)
−0.988516 + 0.151113i \(0.951714\pi\)
\(702\) 0 0
\(703\) 15.9862i 0.602930i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.286118 0.0107606
\(708\) 0 0
\(709\) 19.1316i 0.718502i 0.933241 + 0.359251i \(0.116968\pi\)
−0.933241 + 0.359251i \(0.883032\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 25.4803 0.954243
\(714\) 0 0
\(715\) −11.0175 −0.412033
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0.562268 0.0209691 0.0104845 0.999945i \(-0.496663\pi\)
0.0104845 + 0.999945i \(0.496663\pi\)
\(720\) 0 0
\(721\) 0.178746i 0.00665684i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 11.6682i 0.433346i
\(726\) 0 0
\(727\) 23.7745i 0.881747i 0.897569 + 0.440874i \(0.145331\pi\)
−0.897569 + 0.440874i \(0.854669\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.15638i 0.0427701i
\(732\) 0 0
\(733\) 14.6661 0.541705 0.270853 0.962621i \(-0.412694\pi\)
0.270853 + 0.962621i \(0.412694\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.97900 −0.0728974
\(738\) 0 0
\(739\) 11.2062i 0.412225i 0.978528 + 0.206113i \(0.0660813\pi\)
−0.978528 + 0.206113i \(0.933919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18.3715i 0.673985i −0.941507 0.336992i \(-0.890590\pi\)
0.941507 0.336992i \(-0.109410\pi\)
\(744\) 0 0
\(745\) −53.5686 −1.96260
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.31382i 0.0845453i
\(750\) 0 0
\(751\) 28.1684i 1.02788i 0.857826 + 0.513940i \(0.171815\pi\)
−0.857826 + 0.513940i \(0.828185\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 71.6118i 2.60622i
\(756\) 0 0
\(757\) 16.5297 0.600783 0.300392 0.953816i \(-0.402883\pi\)
0.300392 + 0.953816i \(0.402883\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.7874i 0.789792i −0.918726 0.394896i \(-0.870781\pi\)
0.918726 0.394896i \(-0.129219\pi\)
\(762\) 0 0
\(763\) 0.0215643i 0.000780680i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.6285i 0.672638i
\(768\) 0 0
\(769\) 47.7462i 1.72177i −0.508799 0.860885i \(-0.669910\pi\)
0.508799 0.860885i \(-0.330090\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −2.34511 −0.0843478 −0.0421739 0.999110i \(-0.513428\pi\)
−0.0421739 + 0.999110i \(0.513428\pi\)
\(774\) 0 0
\(775\) −36.2512 −1.30218
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.49972 −0.232877
\(780\) 0 0
\(781\) 20.2097i 0.723160i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 56.1759 2.00500
\(786\) 0 0
\(787\) 28.3179i 1.00942i 0.863288 + 0.504711i \(0.168401\pi\)
−0.863288 + 0.504711i \(0.831599\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.13830 −0.0404733
\(792\) 0 0
\(793\) 18.3955i 0.653245i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.3671 0.402642 0.201321 0.979525i \(-0.435477\pi\)
0.201321 + 0.979525i \(0.435477\pi\)
\(798\) 0 0