Properties

Label 6012.2.h.a.3005.3
Level 6012
Weight 2
Character 6012.3005
Analytic conductor 48.006
Analytic rank 0
Dimension 56
CM no
Inner twists 4

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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 3005.3
Character \(\chi\) \(=\) 6012.3005
Dual form 6012.2.h.a.3005.4

$q$-expansion

\(f(q)\) \(=\) \(q-3.84494 q^{5} -2.14597 q^{7} +O(q^{10})\) \(q-3.84494 q^{5} -2.14597 q^{7} +0.389573i q^{11} +5.51059i q^{13} -6.90150 q^{17} -6.97103 q^{19} -4.97768 q^{23} +9.78356 q^{25} +7.02613i q^{29} +0.566791 q^{31} +8.25113 q^{35} -7.77849i q^{37} -8.87875 q^{41} +3.56851i q^{43} +12.8085i q^{47} -2.39480 q^{49} -1.41777 q^{53} -1.49789i q^{55} -7.04798 q^{59} +2.28271 q^{61} -21.1879i q^{65} +12.4511i q^{67} -2.35431 q^{71} +13.7908i q^{73} -0.836013i q^{77} -6.60434i q^{79} +10.0867 q^{83} +26.5359 q^{85} -13.1160i q^{89} -11.8256i q^{91} +26.8032 q^{95} -13.0804 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56q + O(q^{10}) \) \( 56q + 8q^{19} + 64q^{25} - 8q^{31} + 56q^{49} - 8q^{61} + 32q^{85} - 48q^{97} + O(q^{100}) \)

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
\(5\) −3.84494 −1.71951 −0.859755 0.510707i \(-0.829384\pi\)
−0.859755 + 0.510707i \(0.829384\pi\)
\(6\) 0 0
\(7\) −2.14597 −0.811101 −0.405551 0.914073i \(-0.632920\pi\)
−0.405551 + 0.914073i \(0.632920\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.389573i 0.117461i 0.998274 + 0.0587304i \(0.0187052\pi\)
−0.998274 + 0.0587304i \(0.981295\pi\)
\(12\) 0 0
\(13\) 5.51059i 1.52836i 0.645002 + 0.764181i \(0.276857\pi\)
−0.645002 + 0.764181i \(0.723143\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.90150 −1.67386 −0.836930 0.547310i \(-0.815652\pi\)
−0.836930 + 0.547310i \(0.815652\pi\)
\(18\) 0 0
\(19\) −6.97103 −1.59926 −0.799632 0.600490i \(-0.794972\pi\)
−0.799632 + 0.600490i \(0.794972\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.97768 −1.03792 −0.518959 0.854799i \(-0.673680\pi\)
−0.518959 + 0.854799i \(0.673680\pi\)
\(24\) 0 0
\(25\) 9.78356 1.95671
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.02613i 1.30472i 0.757910 + 0.652359i \(0.226221\pi\)
−0.757910 + 0.652359i \(0.773779\pi\)
\(30\) 0 0
\(31\) 0.566791 0.101799 0.0508993 0.998704i \(-0.483791\pi\)
0.0508993 + 0.998704i \(0.483791\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8.25113 1.39470
\(36\) 0 0
\(37\) 7.77849i 1.27878i −0.768884 0.639388i \(-0.779188\pi\)
0.768884 0.639388i \(-0.220812\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.87875 −1.38663 −0.693314 0.720636i \(-0.743850\pi\)
−0.693314 + 0.720636i \(0.743850\pi\)
\(42\) 0 0
\(43\) 3.56851i 0.544193i 0.962270 + 0.272096i \(0.0877170\pi\)
−0.962270 + 0.272096i \(0.912283\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.8085i 1.86831i 0.356865 + 0.934156i \(0.383845\pi\)
−0.356865 + 0.934156i \(0.616155\pi\)
\(48\) 0 0
\(49\) −2.39480 −0.342115
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.41777 −0.194746 −0.0973728 0.995248i \(-0.531044\pi\)
−0.0973728 + 0.995248i \(0.531044\pi\)
\(54\) 0 0
\(55\) 1.49789i 0.201975i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.04798 −0.917569 −0.458785 0.888548i \(-0.651715\pi\)
−0.458785 + 0.888548i \(0.651715\pi\)
\(60\) 0 0
\(61\) 2.28271 0.292271 0.146136 0.989265i \(-0.453316\pi\)
0.146136 + 0.989265i \(0.453316\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 21.1879i 2.62803i
\(66\) 0 0
\(67\) 12.4511i 1.52114i 0.649257 + 0.760569i \(0.275080\pi\)
−0.649257 + 0.760569i \(0.724920\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.35431 −0.279405 −0.139703 0.990193i \(-0.544615\pi\)
−0.139703 + 0.990193i \(0.544615\pi\)
\(72\) 0 0
\(73\) 13.7908i 1.61409i 0.590489 + 0.807045i \(0.298935\pi\)
−0.590489 + 0.807045i \(0.701065\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.836013i 0.0952725i
\(78\) 0 0
\(79\) 6.60434i 0.743046i −0.928424 0.371523i \(-0.878836\pi\)
0.928424 0.371523i \(-0.121164\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.0867 1.10716 0.553578 0.832797i \(-0.313262\pi\)
0.553578 + 0.832797i \(0.313262\pi\)
\(84\) 0 0
\(85\) 26.5359 2.87822
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.1160i 1.39030i −0.718867 0.695148i \(-0.755339\pi\)
0.718867 0.695148i \(-0.244661\pi\)
\(90\) 0 0
\(91\) 11.8256i 1.23966i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 26.8032 2.74995
\(96\) 0 0
\(97\) −13.0804 −1.32812 −0.664059 0.747680i \(-0.731168\pi\)
−0.664059 + 0.747680i \(0.731168\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.2424 −1.01916 −0.509581 0.860423i \(-0.670199\pi\)
−0.509581 + 0.860423i \(0.670199\pi\)
\(102\) 0 0
\(103\) 4.81035i 0.473978i 0.971512 + 0.236989i \(0.0761605\pi\)
−0.971512 + 0.236989i \(0.923840\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 17.4053i 1.68264i −0.540540 0.841318i \(-0.681780\pi\)
0.540540 0.841318i \(-0.318220\pi\)
\(108\) 0 0
\(109\) 0.344407i 0.0329882i 0.999864 + 0.0164941i \(0.00525048\pi\)
−0.999864 + 0.0164941i \(0.994750\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.14643 −0.295992 −0.147996 0.988988i \(-0.547282\pi\)
−0.147996 + 0.988988i \(0.547282\pi\)
\(114\) 0 0
\(115\) 19.1389 1.78471
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 14.8104 1.35767
\(120\) 0 0
\(121\) 10.8482 0.986203
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −18.3925 −1.64508
\(126\) 0 0
\(127\) 6.31278 0.560168 0.280084 0.959975i \(-0.409638\pi\)
0.280084 + 0.959975i \(0.409638\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −17.3867 −1.51909 −0.759544 0.650456i \(-0.774578\pi\)
−0.759544 + 0.650456i \(0.774578\pi\)
\(132\) 0 0
\(133\) 14.9596 1.29717
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.61256i 0.735821i 0.929861 + 0.367911i \(0.119927\pi\)
−0.929861 + 0.367911i \(0.880073\pi\)
\(138\) 0 0
\(139\) 13.3981i 1.13641i 0.822887 + 0.568205i \(0.192362\pi\)
−0.822887 + 0.568205i \(0.807638\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.14678 −0.179523
\(144\) 0 0
\(145\) 27.0150i 2.24348i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −22.5320 −1.84590 −0.922948 0.384925i \(-0.874227\pi\)
−0.922948 + 0.384925i \(0.874227\pi\)
\(150\) 0 0
\(151\) 5.63828i 0.458837i −0.973328 0.229418i \(-0.926318\pi\)
0.973328 0.229418i \(-0.0736824\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.17928 −0.175044
\(156\) 0 0
\(157\) 21.8304 1.74225 0.871126 0.491059i \(-0.163390\pi\)
0.871126 + 0.491059i \(0.163390\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 10.6820 0.841857
\(162\) 0 0
\(163\) 14.5361i 1.13856i 0.822145 + 0.569278i \(0.192777\pi\)
−0.822145 + 0.569278i \(0.807223\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.86706 12.3307i 0.299242 0.954177i
\(168\) 0 0
\(169\) −17.3666 −1.33589
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.41761i 0.335865i −0.985798 0.167932i \(-0.946291\pi\)
0.985798 0.167932i \(-0.0537091\pi\)
\(174\) 0 0
\(175\) −20.9953 −1.58709
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.15992i 0.385670i 0.981231 + 0.192835i \(0.0617683\pi\)
−0.981231 + 0.192835i \(0.938232\pi\)
\(180\) 0 0
\(181\) −7.15190 −0.531597 −0.265798 0.964029i \(-0.585636\pi\)
−0.265798 + 0.964029i \(0.585636\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 29.9078i 2.19887i
\(186\) 0 0
\(187\) 2.68864i 0.196613i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.4303i 0.754709i −0.926069 0.377354i \(-0.876834\pi\)
0.926069 0.377354i \(-0.123166\pi\)
\(192\) 0 0
\(193\) 5.17362i 0.372405i −0.982511 0.186203i \(-0.940382\pi\)
0.982511 0.186203i \(-0.0596181\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.6148 1.18375 0.591877 0.806028i \(-0.298387\pi\)
0.591877 + 0.806028i \(0.298387\pi\)
\(198\) 0 0
\(199\) −0.121544 −0.00861603 −0.00430802 0.999991i \(-0.501371\pi\)
−0.00430802 + 0.999991i \(0.501371\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 15.0779i 1.05826i
\(204\) 0 0
\(205\) 34.1383 2.38432
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.71573i 0.187851i
\(210\) 0 0
\(211\) 17.7050 1.21886 0.609431 0.792839i \(-0.291398\pi\)
0.609431 + 0.792839i \(0.291398\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 13.7207i 0.935745i
\(216\) 0 0
\(217\) −1.21632 −0.0825690
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 38.0314i 2.55827i
\(222\) 0 0
\(223\) −5.69361 −0.381272 −0.190636 0.981661i \(-0.561055\pi\)
−0.190636 + 0.981661i \(0.561055\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.7274 −0.712004 −0.356002 0.934485i \(-0.615860\pi\)
−0.356002 + 0.934485i \(0.615860\pi\)
\(228\) 0 0
\(229\) −10.8467 −0.716773 −0.358386 0.933573i \(-0.616673\pi\)
−0.358386 + 0.933573i \(0.616673\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.490036i 0.0321033i −0.999871 0.0160517i \(-0.994890\pi\)
0.999871 0.0160517i \(-0.00510962\pi\)
\(234\) 0 0
\(235\) 49.2479i 3.21258i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.7501i 1.01879i 0.860532 + 0.509396i \(0.170131\pi\)
−0.860532 + 0.509396i \(0.829869\pi\)
\(240\) 0 0
\(241\) 5.55929i 0.358105i 0.983839 + 0.179053i \(0.0573032\pi\)
−0.983839 + 0.179053i \(0.942697\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 9.20788 0.588270
\(246\) 0 0
\(247\) 38.4145i 2.44426i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 15.0853i 0.952176i 0.879398 + 0.476088i \(0.157946\pi\)
−0.879398 + 0.476088i \(0.842054\pi\)
\(252\) 0 0
\(253\) 1.93917i 0.121915i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.73533 0.544895 0.272447 0.962171i \(-0.412167\pi\)
0.272447 + 0.962171i \(0.412167\pi\)
\(258\) 0 0
\(259\) 16.6924i 1.03722i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 28.7921i 1.77540i −0.460423 0.887700i \(-0.652302\pi\)
0.460423 0.887700i \(-0.347698\pi\)
\(264\) 0 0
\(265\) 5.45124 0.334867
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 21.4007 1.30483 0.652413 0.757864i \(-0.273757\pi\)
0.652413 + 0.757864i \(0.273757\pi\)
\(270\) 0 0
\(271\) 18.7899i 1.14141i −0.821156 0.570704i \(-0.806670\pi\)
0.821156 0.570704i \(-0.193330\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.81141i 0.229837i
\(276\) 0 0
\(277\) 16.7280i 1.00509i −0.864551 0.502545i \(-0.832397\pi\)
0.864551 0.502545i \(-0.167603\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 28.6978i 1.71197i 0.517005 + 0.855983i \(0.327047\pi\)
−0.517005 + 0.855983i \(0.672953\pi\)
\(282\) 0 0
\(283\) −4.08638 −0.242910 −0.121455 0.992597i \(-0.538756\pi\)
−0.121455 + 0.992597i \(0.538756\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 19.0535 1.12470
\(288\) 0 0
\(289\) 30.6308 1.80181
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.1321i 0.650341i −0.945655 0.325171i \(-0.894578\pi\)
0.945655 0.325171i \(-0.105422\pi\)
\(294\) 0 0
\(295\) 27.0991 1.57777
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 27.4300i 1.58632i
\(300\) 0 0
\(301\) 7.65793i 0.441396i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8.77689 −0.502563
\(306\) 0 0
\(307\) 32.8026i 1.87214i −0.351811 0.936071i \(-0.614434\pi\)
0.351811 0.936071i \(-0.385566\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.24173i 0.183822i 0.995767 + 0.0919109i \(0.0292975\pi\)
−0.995767 + 0.0919109i \(0.970702\pi\)
\(312\) 0 0
\(313\) 21.5298i 1.21693i −0.793579 0.608467i \(-0.791785\pi\)
0.793579 0.608467i \(-0.208215\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.4153i 0.697312i −0.937251 0.348656i \(-0.886638\pi\)
0.937251 0.348656i \(-0.113362\pi\)
\(318\) 0 0
\(319\) −2.73719 −0.153253
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 48.1106 2.67695
\(324\) 0 0
\(325\) 53.9132i 2.99057i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 27.4867i 1.51539i
\(330\) 0 0
\(331\) 9.46960i 0.520496i −0.965542 0.260248i \(-0.916196\pi\)
0.965542 0.260248i \(-0.0838043\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 47.8735i 2.61561i
\(336\) 0 0
\(337\) −31.2526 −1.70244 −0.851219 0.524811i \(-0.824136\pi\)
−0.851219 + 0.524811i \(0.824136\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.220806i 0.0119573i
\(342\) 0 0
\(343\) 20.1610 1.08859
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −13.8933 −0.745833 −0.372916 0.927865i \(-0.621642\pi\)
−0.372916 + 0.927865i \(0.621642\pi\)
\(348\) 0 0
\(349\) 20.4127i 1.09267i 0.837568 + 0.546333i \(0.183977\pi\)
−0.837568 + 0.546333i \(0.816023\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.97509i 0.105123i −0.998618 0.0525617i \(-0.983261\pi\)
0.998618 0.0525617i \(-0.0167386\pi\)
\(354\) 0 0
\(355\) 9.05219 0.480440
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.84437i 0.361232i 0.983554 + 0.180616i \(0.0578092\pi\)
−0.983554 + 0.180616i \(0.942191\pi\)
\(360\) 0 0
\(361\) 29.5953 1.55765
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 53.0248i 2.77544i
\(366\) 0 0
\(367\) −28.5647 −1.49107 −0.745533 0.666469i \(-0.767805\pi\)
−0.745533 + 0.666469i \(0.767805\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.04249 0.157958
\(372\) 0 0
\(373\) 16.3680i 0.847501i −0.905779 0.423750i \(-0.860713\pi\)
0.905779 0.423750i \(-0.139287\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −38.7181 −1.99408
\(378\) 0 0
\(379\) 12.4090i 0.637408i −0.947854 0.318704i \(-0.896752\pi\)
0.947854 0.318704i \(-0.103248\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 26.7233i 1.36550i 0.730653 + 0.682749i \(0.239216\pi\)
−0.730653 + 0.682749i \(0.760784\pi\)
\(384\) 0 0
\(385\) 3.21442i 0.163822i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0.794694 0.0402926 0.0201463 0.999797i \(-0.493587\pi\)
0.0201463 + 0.999797i \(0.493587\pi\)
\(390\) 0 0
\(391\) 34.3535 1.73733
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 25.3933i 1.27767i
\(396\) 0 0
\(397\) 28.4180 1.42626 0.713130 0.701032i \(-0.247277\pi\)
0.713130 + 0.701032i \(0.247277\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.657033 0.0328107 0.0164053 0.999865i \(-0.494778\pi\)
0.0164053 + 0.999865i \(0.494778\pi\)
\(402\) 0 0
\(403\) 3.12335i 0.155585i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.03029 0.150206
\(408\) 0 0
\(409\) −11.1429 −0.550981 −0.275491 0.961304i \(-0.588840\pi\)
−0.275491 + 0.961304i \(0.588840\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 15.1248 0.744241
\(414\) 0 0
\(415\) −38.7827 −1.90377
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.61669i 0.225540i 0.993621 + 0.112770i \(0.0359724\pi\)
−0.993621 + 0.112770i \(0.964028\pi\)
\(420\) 0 0
\(421\) 11.3568 0.553497 0.276749 0.960942i \(-0.410743\pi\)
0.276749 + 0.960942i \(0.410743\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −67.5213 −3.27526
\(426\) 0 0
\(427\) −4.89864 −0.237062
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.6836i 0.851789i 0.904773 + 0.425895i \(0.140041\pi\)
−0.904773 + 0.425895i \(0.859959\pi\)
\(432\) 0 0
\(433\) −14.9960 −0.720660 −0.360330 0.932825i \(-0.617336\pi\)
−0.360330 + 0.932825i \(0.617336\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 34.6996 1.65991
\(438\) 0 0
\(439\) 24.5369i 1.17108i 0.810642 + 0.585542i \(0.199118\pi\)
−0.810642 + 0.585542i \(0.800882\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.39883 0.113972 0.0569858 0.998375i \(-0.481851\pi\)
0.0569858 + 0.998375i \(0.481851\pi\)
\(444\) 0 0
\(445\) 50.4303i 2.39063i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 11.2497i 0.530906i 0.964124 + 0.265453i \(0.0855215\pi\)
−0.964124 + 0.265453i \(0.914478\pi\)
\(450\) 0 0
\(451\) 3.45892i 0.162874i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 45.4686i 2.13160i
\(456\) 0 0
\(457\) 30.2665i 1.41581i 0.706309 + 0.707903i \(0.250359\pi\)
−0.706309 + 0.707903i \(0.749641\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.77808i 0.175963i 0.996122 + 0.0879814i \(0.0280416\pi\)
−0.996122 + 0.0879814i \(0.971958\pi\)
\(462\) 0 0
\(463\) 40.9705i 1.90406i 0.306005 + 0.952030i \(0.401008\pi\)
−0.306005 + 0.952030i \(0.598992\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 36.7738i 1.70169i 0.525417 + 0.850845i \(0.323909\pi\)
−0.525417 + 0.850845i \(0.676091\pi\)
\(468\) 0 0
\(469\) 26.7196i 1.23380i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.39020 −0.0639213
\(474\) 0 0
\(475\) −68.2016 −3.12930
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −34.4612 −1.57457 −0.787285 0.616589i \(-0.788514\pi\)
−0.787285 + 0.616589i \(0.788514\pi\)
\(480\) 0 0
\(481\) 42.8641 1.95443
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 50.2935 2.28371
\(486\) 0 0
\(487\) 17.7395i 0.803855i 0.915672 + 0.401927i \(0.131660\pi\)
−0.915672 + 0.401927i \(0.868340\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.84701i 0.354130i 0.984199 + 0.177065i \(0.0566604\pi\)
−0.984199 + 0.177065i \(0.943340\pi\)
\(492\) 0 0
\(493\) 48.4908i 2.18392i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.05229 0.226626
\(498\) 0 0
\(499\) 31.5691i 1.41323i 0.707600 + 0.706613i \(0.249778\pi\)
−0.707600 + 0.706613i \(0.750222\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.4593i 0.555532i 0.960649 + 0.277766i \(0.0895939\pi\)
−0.960649 + 0.277766i \(0.910406\pi\)
\(504\) 0 0
\(505\) 39.3816 1.75246
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 39.7294i 1.76097i 0.474071 + 0.880486i \(0.342784\pi\)
−0.474071 + 0.880486i \(0.657216\pi\)
\(510\) 0 0
\(511\) 29.5947i 1.30919i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 18.4955i 0.815009i
\(516\) 0 0
\(517\) −4.98985 −0.219453
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.8198 0.561648 0.280824 0.959759i \(-0.409392\pi\)
0.280824 + 0.959759i \(0.409392\pi\)
\(522\) 0 0
\(523\) 43.6321 1.90790 0.953950 0.299965i \(-0.0969749\pi\)
0.953950 + 0.299965i \(0.0969749\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.91171 −0.170397
\(528\) 0 0
\(529\) 1.77731 0.0772744
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 48.9272i 2.11927i
\(534\) 0 0
\(535\) 66.9224i 2.89331i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.932952i 0.0401851i
\(540\) 0 0
\(541\) 4.44318i 0.191027i −0.995428 0.0955135i \(-0.969551\pi\)
0.995428 0.0955135i \(-0.0304493\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.32423i 0.0567236i
\(546\) 0 0
\(547\) 0.608706i 0.0260264i −0.999915 0.0130132i \(-0.995858\pi\)
0.999915 0.0130132i \(-0.00414234\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 48.9794i 2.08659i
\(552\) 0 0
\(553\) 14.1727i 0.602685i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24.0939i 1.02089i −0.859910 0.510446i \(-0.829481\pi\)
0.859910 0.510446i \(-0.170519\pi\)
\(558\) 0 0
\(559\) −19.6646 −0.831724
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 37.0205i 1.56023i 0.625638 + 0.780114i \(0.284839\pi\)
−0.625638 + 0.780114i \(0.715161\pi\)
\(564\) 0 0
\(565\) 12.0979 0.508960
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −12.6770 −0.531447 −0.265724 0.964049i \(-0.585611\pi\)
−0.265724 + 0.964049i \(0.585611\pi\)
\(570\) 0 0
\(571\) 36.6157i 1.53232i 0.642649 + 0.766160i \(0.277835\pi\)
−0.642649 + 0.766160i \(0.722165\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −48.6995 −2.03091
\(576\) 0 0
\(577\) −35.0676 −1.45988 −0.729942 0.683509i \(-0.760453\pi\)
−0.729942 + 0.683509i \(0.760453\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −21.6457 −0.898016
\(582\) 0 0
\(583\) 0.552325i 0.0228750i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 35.5617 1.46779 0.733894 0.679264i \(-0.237701\pi\)
0.733894 + 0.679264i \(0.237701\pi\)
\(588\) 0 0
\(589\) −3.95112 −0.162803
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8.86416 0.364007 0.182004 0.983298i \(-0.441742\pi\)
0.182004 + 0.983298i \(0.441742\pi\)
\(594\) 0 0
\(595\) −56.9452 −2.33453
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 22.9526i 0.937816i 0.883247 + 0.468908i \(0.155352\pi\)
−0.883247 + 0.468908i \(0.844648\pi\)
\(600\) 0 0
\(601\) −40.5175 −1.65275 −0.826373 0.563124i \(-0.809599\pi\)
−0.826373 + 0.563124i \(0.809599\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −41.7108 −1.69579
\(606\) 0 0
\(607\) 39.3296i 1.59634i −0.602434 0.798169i \(-0.705802\pi\)
0.602434 0.798169i \(-0.294198\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −70.5824 −2.85546
\(612\) 0 0
\(613\) −1.06987 −0.0432116 −0.0216058 0.999767i \(-0.506878\pi\)
−0.0216058 + 0.999767i \(0.506878\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 44.8837i 1.80695i −0.428640 0.903476i \(-0.641007\pi\)
0.428640 0.903476i \(-0.358993\pi\)
\(618\) 0 0
\(619\) 27.7345i 1.11475i −0.830262 0.557373i \(-0.811809\pi\)
0.830262 0.557373i \(-0.188191\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 28.1466i 1.12767i
\(624\) 0 0
\(625\) 21.8003 0.872012
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 53.6833i 2.14049i
\(630\) 0 0
\(631\) −24.3534 −0.969495 −0.484747 0.874654i \(-0.661088\pi\)
−0.484747 + 0.874654i \(0.661088\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −24.2722 −0.963215
\(636\) 0 0
\(637\) 13.1968i 0.522876i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 25.8492 1.02098 0.510492 0.859883i \(-0.329463\pi\)
0.510492 + 0.859883i \(0.329463\pi\)
\(642\) 0 0
\(643\) 7.67452i 0.302654i 0.988484 + 0.151327i \(0.0483546\pi\)
−0.988484 + 0.151327i \(0.951645\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −27.2381 −1.07084 −0.535420 0.844586i \(-0.679847\pi\)
−0.535420 + 0.844586i \(0.679847\pi\)
\(648\) 0 0
\(649\) 2.74571i 0.107778i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11.5414i 0.451650i −0.974168 0.225825i \(-0.927492\pi\)
0.974168 0.225825i \(-0.0725077\pi\)
\(654\) 0 0
\(655\) 66.8510 2.61208
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.76163 0.380259 0.190130 0.981759i \(-0.439109\pi\)
0.190130 + 0.981759i \(0.439109\pi\)
\(660\) 0 0
\(661\) 1.42557i 0.0554482i −0.999616 0.0277241i \(-0.991174\pi\)
0.999616 0.0277241i \(-0.00882598\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −57.5189 −2.23049
\(666\) 0 0
\(667\) 34.9738i 1.35419i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.889284i 0.0343304i
\(672\) 0 0
\(673\) 15.6409i 0.602911i −0.953480 0.301456i \(-0.902527\pi\)
0.953480 0.301456i \(-0.0974725\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 32.2251i 1.23851i −0.785190 0.619255i \(-0.787435\pi\)
0.785190 0.619255i \(-0.212565\pi\)
\(678\) 0 0
\(679\) 28.0703 1.07724
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −19.8122 −0.758093 −0.379047 0.925378i \(-0.623748\pi\)
−0.379047 + 0.925378i \(0.623748\pi\)
\(684\) 0 0
\(685\) 33.1148i 1.26525i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7.81275i 0.297642i
\(690\) 0 0
\(691\) 1.57096i 0.0597621i 0.999553 + 0.0298811i \(0.00951285\pi\)
−0.999553 + 0.0298811i \(0.990487\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 51.5148i 1.95407i
\(696\) 0 0
\(697\) 61.2767 2.32102
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 32.3493i 1.22182i −0.791701 0.610909i \(-0.790804\pi\)
0.791701 0.610909i \(-0.209196\pi\)
\(702\) 0 0
\(703\) 54.2241i 2.04510i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 21.9800 0.826643
\(708\) 0 0
\(709\) 0.370820i 0.0139264i −0.999976 0.00696322i \(-0.997784\pi\)
0.999976 0.00696322i \(-0.00221648\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.82130 −0.105659
\(714\) 0 0
\(715\) 8.25423 0.308691
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −7.74411 −0.288806 −0.144403 0.989519i \(-0.546126\pi\)
−0.144403 + 0.989519i \(0.546126\pi\)
\(720\) 0 0
\(721\) 10.3229i 0.384444i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 68.7406i 2.55296i
\(726\) 0 0
\(727\) 10.4401i 0.387203i −0.981080 0.193602i \(-0.937983\pi\)
0.981080 0.193602i \(-0.0620169\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 24.6281i 0.910903i
\(732\) 0 0
\(733\) −12.6827 −0.468445 −0.234222 0.972183i \(-0.575254\pi\)
−0.234222 + 0.972183i \(0.575254\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.85060 −0.178674
\(738\) 0 0
\(739\) 28.8453i 1.06109i 0.847656 + 0.530547i \(0.178013\pi\)
−0.847656 + 0.530547i \(0.821987\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 19.3349i 0.709329i 0.934994 + 0.354665i \(0.115405\pi\)
−0.934994 + 0.354665i \(0.884595\pi\)
\(744\) 0 0
\(745\) 86.6343 3.17403
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 37.3513i 1.36479i
\(750\) 0 0
\(751\) 14.2753i 0.520913i 0.965486 + 0.260457i \(0.0838731\pi\)
−0.965486 + 0.260457i \(0.916127\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21.6789i 0.788974i
\(756\) 0 0
\(757\) 9.35269 0.339929 0.169965 0.985450i \(-0.445635\pi\)
0.169965 + 0.985450i \(0.445635\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.70767i 0.243153i 0.992582 + 0.121577i \(0.0387950\pi\)
−0.992582 + 0.121577i \(0.961205\pi\)
\(762\) 0 0
\(763\) 0.739088i 0.0267568i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 38.8385i 1.40238i
\(768\) 0 0
\(769\) 8.17188i 0.294686i 0.989085 + 0.147343i \(0.0470721\pi\)
−0.989085 + 0.147343i \(0.952928\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.64691 0.239073 0.119536 0.992830i \(-0.461859\pi\)
0.119536 + 0.992830i \(0.461859\pi\)
\(774\) 0 0
\(775\) 5.54523 0.199191
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 61.8941 2.21759
\(780\) 0 0
\(781\) 0.917177i 0.0328192i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −83.9364 −2.99582
\(786\) 0 0
\(787\) 14.9964i 0.534563i 0.963618 + 0.267282i \(0.0861254\pi\)
−0.963618 + 0.267282i \(0.913875\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.75216 0.240079
\(792\) 0 0
\(793\) 12.5791i 0.446697i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −13.1470 −0.465690 −0.232845 0.972514i \(-0.574803\pi\)
−0.232845 + 0.972514i \(0.574803\pi\)
\(798\) 0 0