Properties

Label 804.2.ba.a.245.1
Level 804
Weight 2
Character 804.245
Analytic conductor 6.420
Analytic rank 0
Dimension 20
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.ba (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{66}]$

Embedding invariants

Embedding label 245.1
Root \(-0.786053 - 0.618159i\)
Character \(\chi\) = 804.245
Dual form 804.2.ba.a.233.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.71442 + 0.246497i) q^{3} +(0.518680 + 0.543976i) q^{7} +(2.87848 + 0.845198i) q^{9} +O(q^{10})\) \(q+(1.71442 + 0.246497i) q^{3} +(0.518680 + 0.543976i) q^{7} +(2.87848 + 0.845198i) q^{9} +(-1.21014 + 6.27878i) q^{13} +(3.30555 + 3.15184i) q^{19} +(0.755148 + 1.06046i) q^{21} +(3.27430 - 3.77875i) q^{25} +(4.72659 + 2.15856i) q^{27} +(-2.02981 - 10.5316i) q^{31} +(0.656310 + 1.13676i) q^{37} +(-3.62238 + 10.4662i) q^{39} +(5.92488 + 9.21929i) q^{43} +(0.306193 - 6.42777i) q^{49} +(4.89019 + 6.21839i) q^{57} +(-0.304967 - 3.19376i) q^{61} +(1.03324 + 2.00421i) q^{63} +(-0.979754 - 8.12650i) q^{67} +(-17.0103 + 1.62429i) q^{73} +(6.54498 - 5.67126i) q^{75} +(-16.7875 + 5.81020i) q^{79} +(7.57128 + 4.86577i) q^{81} +(-4.04318 + 2.59839i) q^{91} +(-0.883932 - 18.5560i) q^{93} +(6.63425 - 3.83028i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{7} + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{7} + 6q^{9} + 9q^{13} - 8q^{19} + 6q^{21} + 10q^{25} - 3q^{31} + 10q^{37} - 9q^{39} - 5q^{49} + 141q^{57} + 27q^{61} + 147q^{63} + 11q^{67} - 180q^{73} - 166q^{79} - 18q^{81} - 36q^{91} - 3q^{93} + 33q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{61}{66}\right)\) \(1\)

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
\(3\) 1.71442 + 0.246497i 0.989821 + 0.142315i
\(4\) 0 0
\(5\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(6\) 0 0
\(7\) 0.518680 + 0.543976i 0.196043 + 0.205604i 0.814417 0.580280i \(-0.197057\pi\)
−0.618374 + 0.785884i \(0.712208\pi\)
\(8\) 0 0
\(9\) 2.87848 + 0.845198i 0.959493 + 0.281733i
\(10\) 0 0
\(11\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(12\) 0 0
\(13\) −1.21014 + 6.27878i −0.335631 + 1.74142i 0.286109 + 0.958197i \(0.407638\pi\)
−0.621740 + 0.783223i \(0.713574\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(18\) 0 0
\(19\) 3.30555 + 3.15184i 0.758346 + 0.723082i 0.966963 0.254916i \(-0.0820477\pi\)
−0.208617 + 0.977997i \(0.566896\pi\)
\(20\) 0 0
\(21\) 0.755148 + 1.06046i 0.164787 + 0.231411i
\(22\) 0 0
\(23\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(24\) 0 0
\(25\) 3.27430 3.77875i 0.654861 0.755750i
\(26\) 0 0
\(27\) 4.72659 + 2.15856i 0.909632 + 0.415415i
\(28\) 0 0
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) −2.02981 10.5316i −0.364564 1.89154i −0.446296 0.894886i \(-0.647257\pi\)
0.0817313 0.996654i \(-0.473955\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.656310 + 1.13676i 0.107897 + 0.186883i 0.914918 0.403640i \(-0.132255\pi\)
−0.807021 + 0.590522i \(0.798922\pi\)
\(38\) 0 0
\(39\) −3.62238 + 10.4662i −0.580045 + 1.67593i
\(40\) 0 0
\(41\) 0 0 0.998867 0.0475819i \(-0.0151515\pi\)
−0.998867 + 0.0475819i \(0.984848\pi\)
\(42\) 0 0
\(43\) 5.92488 + 9.21929i 0.903536 + 1.40593i 0.913881 + 0.405983i \(0.133071\pi\)
−0.0103449 + 0.999946i \(0.503293\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(48\) 0 0
\(49\) 0.306193 6.42777i 0.0437418 0.918253i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.89019 + 6.21839i 0.647722 + 0.823646i
\(58\) 0 0
\(59\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(60\) 0 0
\(61\) −0.304967 3.19376i −0.0390471 0.408919i −0.993772 0.111430i \(-0.964457\pi\)
0.954725 0.297489i \(-0.0961492\pi\)
\(62\) 0 0
\(63\) 1.03324 + 2.00421i 0.130176 + 0.252507i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.979754 8.12650i −0.119696 0.992811i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(72\) 0 0
\(73\) −17.0103 + 1.62429i −1.99090 + 0.190108i −0.994850 + 0.101361i \(0.967680\pi\)
−0.996054 + 0.0887477i \(0.971714\pi\)
\(74\) 0 0
\(75\) 6.54498 5.67126i 0.755750 0.654861i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −16.7875 + 5.81020i −1.88874 + 0.653699i −0.932414 + 0.361392i \(0.882301\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) 7.57128 + 4.86577i 0.841254 + 0.540641i
\(82\) 0 0
\(83\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(90\) 0 0
\(91\) −4.04318 + 2.59839i −0.423841 + 0.272386i
\(92\) 0 0
\(93\) −0.883932 18.5560i −0.0916595 1.92417i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.63425 3.83028i 0.673606 0.388906i −0.123836 0.992303i \(-0.539520\pi\)
0.797442 + 0.603396i \(0.206186\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.971812 0.235759i \(-0.924242\pi\)
0.971812 + 0.235759i \(0.0757576\pi\)
\(102\) 0 0
\(103\) 17.8848 3.44700i 1.76224 0.339643i 0.797593 0.603196i \(-0.206106\pi\)
0.964645 + 0.263553i \(0.0848943\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(108\) 0 0
\(109\) −10.3427 8.96204i −0.990655 0.858407i −0.000729916 1.00000i \(-0.500232\pi\)
−0.989925 + 0.141592i \(0.954778\pi\)
\(110\) 0 0
\(111\) 0.844984 + 2.11067i 0.0802023 + 0.200336i
\(112\) 0 0
\(113\) 0 0 0.814576 0.580057i \(-0.196970\pi\)
−0.814576 + 0.580057i \(0.803030\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −8.79016 + 17.0505i −0.812651 + 1.57632i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.8012 2.08176i −0.981929 0.189251i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2.37340 + 2.26303i −0.210605 + 0.200811i −0.788038 0.615627i \(-0.788903\pi\)
0.577433 + 0.816438i \(0.304054\pi\)
\(128\) 0 0
\(129\) 7.88522 + 17.2662i 0.694255 + 1.52021i
\(130\) 0 0
\(131\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(132\) 0 0
\(133\) 3.43294i 0.297674i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(138\) 0 0
\(139\) 20.0359 9.15010i 1.69942 0.776101i 0.701434 0.712734i \(-0.252543\pi\)
0.997990 0.0633672i \(-0.0201839\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.10937 10.9444i 0.173978 0.902682i
\(148\) 0 0
\(149\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(150\) 0 0
\(151\) −1.49049 0.768401i −0.121294 0.0625316i 0.396511 0.918030i \(-0.370221\pi\)
−0.517806 + 0.855498i \(0.673251\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −17.9069 + 7.16885i −1.42913 + 0.572136i −0.951836 0.306608i \(-0.900806\pi\)
−0.477292 + 0.878745i \(0.658382\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 9.11104 15.7808i 0.713632 1.23605i −0.249853 0.968284i \(-0.580382\pi\)
0.963485 0.267763i \(-0.0862844\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(168\) 0 0
\(169\) −25.8899 10.3647i −1.99153 0.797288i
\(170\) 0 0
\(171\) 6.85104 + 11.8664i 0.523912 + 0.907443i
\(172\) 0 0
\(173\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(174\) 0 0
\(175\) 3.75387 0.178819i 0.283766 0.0135174i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(180\) 0 0
\(181\) −16.9494 13.3291i −1.25984 0.990746i −0.999683 0.0251785i \(-0.991985\pi\)
−0.260153 0.965567i \(-0.583773\pi\)
\(182\) 0 0
\(183\) 0.264409 5.55063i 0.0195457 0.410314i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.27738 + 3.69075i 0.0929159 + 0.268463i
\(190\) 0 0
\(191\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(192\) 0 0
\(193\) −17.9452 20.7099i −1.29173 1.49073i −0.770710 0.637187i \(-0.780098\pi\)
−0.521018 0.853546i \(-0.674447\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.458227 0.888835i \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(198\) 0 0
\(199\) −0.501668 2.06790i −0.0355623 0.146590i 0.951263 0.308382i \(-0.0997876\pi\)
−0.986825 + 0.161793i \(0.948272\pi\)
\(200\) 0 0
\(201\) 0.323444 14.1738i 0.0228140 0.999740i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 8.23116 6.47306i 0.566656 0.445624i −0.293279 0.956027i \(-0.594746\pi\)
0.859935 + 0.510403i \(0.170504\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.67614 6.56673i 0.317437 0.445778i
\(218\) 0 0
\(219\) −29.5632 1.40827i −1.99769 0.0951619i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.24906 + 29.5528i 0.284538 + 1.97900i 0.167412 + 0.985887i \(0.446459\pi\)
0.117125 + 0.993117i \(0.462632\pi\)
\(224\) 0 0
\(225\) 12.6188 8.10961i 0.841254 0.540641i
\(226\) 0 0
\(227\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(228\) 0 0
\(229\) −15.3520 5.31338i −1.01449 0.351118i −0.231287 0.972886i \(-0.574293\pi\)
−0.783202 + 0.621768i \(0.786415\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.371662 0.928368i \(-0.378788\pi\)
−0.371662 + 0.928368i \(0.621212\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −30.2130 + 5.82308i −1.96254 + 0.378250i
\(238\) 0 0
\(239\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) 0 0
\(241\) 3.53199 7.73398i 0.227515 0.498189i −0.761103 0.648630i \(-0.775342\pi\)
0.988619 + 0.150441i \(0.0480694\pi\)
\(242\) 0 0
\(243\) 11.7810 + 10.2083i 0.755750 + 0.654861i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −23.7899 + 16.9407i −1.51371 + 1.07791i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.458227 0.888835i \(-0.348485\pi\)
−0.458227 + 0.888835i \(0.651515\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(258\) 0 0
\(259\) −0.277957 + 0.946634i −0.0172714 + 0.0588210i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 22.8919 + 3.29135i 1.39058 + 0.199935i 0.796582 0.604530i \(-0.206639\pi\)
0.593999 + 0.804466i \(0.297548\pi\)
\(272\) 0 0
\(273\) −7.57221 + 3.45811i −0.458291 + 0.209294i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −31.3119 9.19400i −1.88135 0.552414i −0.996199 0.0871078i \(-0.972238\pi\)
−0.885150 0.465306i \(-0.845944\pi\)
\(278\) 0 0
\(279\) 3.05856 32.0307i 0.183111 1.91763i
\(280\) 0 0
\(281\) 0 0 0.189251 0.981929i \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(282\) 0 0
\(283\) −3.68456 + 1.08188i −0.219024 + 0.0643113i −0.389404 0.921067i \(-0.627319\pi\)
0.170379 + 0.985379i \(0.445501\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 9.86097 + 13.8478i 0.580057 + 0.814576i
\(290\) 0 0
\(291\) 12.3180 4.93140i 0.722097 0.289084i
\(292\) 0 0
\(293\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −1.94196 + 8.00486i −0.111933 + 0.461392i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 10.4344 30.1483i 0.595524 1.72065i −0.0932173 0.995646i \(-0.529715\pi\)
0.688741 0.725007i \(-0.258164\pi\)
\(308\) 0 0
\(309\) 31.5117 1.50109i 1.79264 0.0853938i
\(310\) 0 0
\(311\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(312\) 0 0
\(313\) 32.4546 4.66627i 1.83444 0.263753i 0.863724 0.503966i \(-0.168126\pi\)
0.970720 + 0.240212i \(0.0772171\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 19.7636 + 25.1314i 1.09629 + 1.39404i
\(326\) 0 0
\(327\) −15.5227 17.9142i −0.858407 0.990655i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 15.3029 + 29.6835i 0.841123 + 1.63155i 0.773461 + 0.633844i \(0.218524\pi\)
0.0676621 + 0.997708i \(0.478446\pi\)
\(332\) 0 0
\(333\) 0.928386 + 3.82686i 0.0508752 + 0.209711i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −26.8201 + 6.50648i −1.46098 + 0.354431i −0.885981 0.463721i \(-0.846514\pi\)
−0.575002 + 0.818152i \(0.694999\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 7.63165 6.61286i 0.412070 0.357061i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(348\) 0 0
\(349\) 11.5864 + 7.44612i 0.620205 + 0.398582i 0.812672 0.582722i \(-0.198012\pi\)
−0.192467 + 0.981304i \(0.561649\pi\)
\(350\) 0 0
\(351\) −19.2729 + 27.0650i −1.02871 + 1.44463i
\(352\) 0 0
\(353\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(360\) 0 0
\(361\) 0.0885397 + 1.85868i 0.00465998 + 0.0978251i
\(362\) 0 0
\(363\) −18.0047 6.23148i −0.945001 0.327068i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4.53090 + 11.3176i −0.236511 + 0.590777i −0.998377 0.0569590i \(-0.981860\pi\)
0.761865 + 0.647736i \(0.224284\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 30.5668 + 17.6477i 1.58269 + 0.913765i 0.994465 + 0.105065i \(0.0335050\pi\)
0.588221 + 0.808700i \(0.299828\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.449388 + 1.12252i 0.0230835 + 0.0576598i 0.939462 0.342653i \(-0.111326\pi\)
−0.916378 + 0.400313i \(0.868901\pi\)
\(380\) 0 0
\(381\) −4.62683 + 3.29475i −0.237040 + 0.168795i
\(382\) 0 0
\(383\) 0 0 0.690079 0.723734i \(-0.257576\pi\)
−0.690079 + 0.723734i \(0.742424\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.26252 + 31.5452i 0.470840 + 1.60353i
\(388\) 0 0
\(389\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7.95241 17.4133i −0.399120 0.873951i −0.997359 0.0726328i \(-0.976860\pi\)
0.598239 0.801318i \(-0.295867\pi\)
\(398\) 0 0
\(399\) −0.846208 + 5.88550i −0.0423634 + 0.294644i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 68.5823 3.41633
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 19.8811 + 20.8507i 0.983056 + 1.03100i 0.999515 + 0.0311349i \(0.00991216\pi\)
−0.0164592 + 0.999865i \(0.505239\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 36.6055 10.7483i 1.79258 0.526348i
\(418\) 0 0
\(419\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(420\) 0 0
\(421\) 18.4844 + 17.6248i 0.900874 + 0.858981i 0.990573 0.136988i \(-0.0437421\pi\)
−0.0896992 + 0.995969i \(0.528591\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.57915 1.82244i 0.0764204 0.0881939i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 0 0
\(433\) 4.86575 + 25.2459i 0.233833 + 1.21324i 0.889053 + 0.457804i \(0.151364\pi\)
−0.655220 + 0.755438i \(0.727424\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 5.30065 + 9.18100i 0.252986 + 0.438185i 0.964347 0.264642i \(-0.0852539\pi\)
−0.711360 + 0.702827i \(0.751921\pi\)
\(440\) 0 0
\(441\) 6.31411 18.2434i 0.300672 0.868734i
\(442\) 0 0
\(443\) 0 0 0.998867 0.0475819i \(-0.0151515\pi\)
−0.998867 + 0.0475819i \(0.984848\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −2.36592 1.68476i −0.111161 0.0791570i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.7494 + 33.9476i 0.549613 + 1.58800i 0.787240 + 0.616647i \(0.211509\pi\)
−0.237627 + 0.971357i \(0.576370\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(462\) 0 0
\(463\) −2.93979 30.7869i −0.136624 1.43079i −0.760537 0.649295i \(-0.775064\pi\)
0.623913 0.781494i \(-0.285542\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(468\) 0 0
\(469\) 3.91245 4.74802i 0.180660 0.219243i
\(470\) 0 0
\(471\) −32.4671 + 7.87643i −1.49600 + 0.362927i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 22.7334 2.17078i 1.04308 0.0996020i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(480\) 0 0
\(481\) −7.93171 + 2.74519i −0.361655 + 0.125170i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 23.2147 + 1.10585i 1.05196 + 0.0501109i 0.566429 0.824110i \(-0.308325\pi\)
0.485528 + 0.874221i \(0.338628\pi\)
\(488\) 0 0
\(489\) 19.5101 24.8091i 0.882276 1.12191i
\(490\) 0 0
\(491\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −38.6252 + 22.3003i −1.72910 + 0.998297i −0.835381 + 0.549672i \(0.814753\pi\)
−0.893720 + 0.448625i \(0.851914\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.971812 0.235759i \(-0.924242\pi\)
0.971812 + 0.235759i \(0.0757576\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −41.8313 24.1513i −1.85779 1.07260i
\(508\) 0 0
\(509\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(510\) 0 0
\(511\) −9.70647 8.41071i −0.429389 0.372068i
\(512\) 0 0
\(513\) 8.82055 + 22.0327i 0.389437 + 0.972767i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(522\) 0 0
\(523\) 19.9953 + 3.85378i 0.874334 + 0.168514i 0.606627 0.794987i \(-0.292522\pi\)
0.267707 + 0.963501i \(0.413734\pi\)
\(524\) 0 0
\(525\) 6.47978 + 0.618744i 0.282801 + 0.0270042i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 16.6459 15.8718i 0.723734 0.690079i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −13.5721 + 6.19815i −0.583509 + 0.266479i −0.685222 0.728334i \(-0.740295\pi\)
0.101713 + 0.994814i \(0.467568\pi\)
\(542\) 0 0
\(543\) −25.7728 27.0297i −1.10601 1.15995i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.79104 + 39.7016i −0.162093 + 1.69752i 0.439120 + 0.898428i \(0.355290\pi\)
−0.601213 + 0.799089i \(0.705316\pi\)
\(548\) 0 0
\(549\) 1.82152 9.45094i 0.0777405 0.403356i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −11.8679 6.11835i −0.504676 0.260179i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(558\) 0 0
\(559\) −65.0558 + 26.0444i −2.75157 + 1.10156i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.28021 + 6.64237i 0.0537639 + 0.278954i
\(568\) 0 0
\(569\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(570\) 0 0
\(571\) 37.5196 + 15.0206i 1.57015 + 0.628592i 0.983444 0.181210i \(-0.0580014\pi\)
0.586703 + 0.809802i \(0.300426\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −45.4565 + 2.16536i −1.89238 + 0.0901450i −0.961437 0.275027i \(-0.911313\pi\)
−0.930940 + 0.365172i \(0.881010\pi\)
\(578\) 0 0
\(579\) −25.6608 39.9290i −1.06643 1.65939i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(588\) 0 0
\(589\) 26.4844 41.2106i 1.09127 1.69805i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.350339 3.66892i −0.0143384 0.150159i
\(598\) 0 0
\(599\) 0 0 −0.458227 0.888835i \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(600\) 0 0
\(601\) 5.17040 + 21.3127i 0.210905 + 0.869362i 0.974630 + 0.223821i \(0.0718530\pi\)
−0.763725 + 0.645541i \(0.776632\pi\)
\(602\) 0 0
\(603\) 4.04830 24.2201i 0.164860 0.986317i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −43.0899 + 22.2144i −1.74897 + 0.901655i −0.792469 + 0.609912i \(0.791205\pi\)
−0.956497 + 0.291743i \(0.905765\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 38.4524 30.2393i 1.55308 1.22135i 0.680651 0.732608i \(-0.261697\pi\)
0.872426 0.488746i \(-0.162545\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(618\) 0 0
\(619\) −20.5788 + 28.8989i −0.827132 + 1.16154i 0.157606 + 0.987502i \(0.449622\pi\)
−0.984738 + 0.174042i \(0.944317\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −3.55787 24.7455i −0.142315 0.989821i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −44.4180 15.3732i −1.76825 0.611998i −0.768635 0.639688i \(-0.779064\pi\)
−0.999617 + 0.0276903i \(0.991185\pi\)
\(632\) 0 0
\(633\) 15.7073 9.06859i 0.624308 0.360444i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 39.9880 + 9.70099i 1.58438 + 0.384367i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(642\) 0 0
\(643\) 20.9808 45.9416i 0.827402 1.81176i 0.331158 0.943575i \(-0.392561\pi\)
0.496245 0.868183i \(-0.334712\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 9.63556 10.1055i 0.377647 0.396065i
\(652\) 0 0
\(653\) 0 0 0.458227 0.888835i \(-0.348485\pi\)
−0.458227 + 0.888835i \(0.651515\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −50.3366 9.70159i −1.96382 0.378495i
\(658\) 0 0
\(659\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(660\) 0 0
\(661\) 13.5803 46.2502i 0.528212 1.79893i −0.0699507 0.997550i \(-0.522284\pi\)
0.598163 0.801375i \(-0.295898\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 51.7134i 1.99935i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −18.3577 2.63944i −0.707637 0.101743i −0.220906 0.975295i \(-0.570902\pi\)
−0.486731 + 0.873552i \(0.661811\pi\)
\(674\) 0 0
\(675\) 23.6329 10.7928i 0.909632 0.415415i
\(676\) 0 0
\(677\) 0 0 −0.690079 0.723734i \(-0.742424\pi\)
0.690079 + 0.723734i \(0.257576\pi\)
\(678\) 0 0
\(679\) 5.52464 + 1.62218i 0.212016 + 0.0622535i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.189251 0.981929i \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −25.0100 12.8936i −0.954193 0.491921i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −19.7399 27.7208i −0.750942 1.05455i −0.996369 0.0851357i \(-0.972868\pi\)
0.245427 0.969415i \(-0.421072\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 −0.189251 0.981929i \(-0.560606\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(702\) 0 0
\(703\) −1.41342 + 5.82622i −0.0533083 + 0.219740i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −14.6765 + 42.4049i −0.551187 + 1.59255i 0.233301 + 0.972405i \(0.425047\pi\)
−0.784488 + 0.620145i \(0.787074\pi\)
\(710\) 0 0
\(711\) −53.2332 + 2.53581i −1.99640 + 0.0951002i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(720\) 0 0
\(721\) 11.1516 + 7.94099i 0.415306 + 0.295738i
\(722\) 0 0
\(723\) 7.96172 12.3887i 0.296099 0.460740i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28.0852 + 35.7132i 1.04162 + 1.32453i 0.944231 + 0.329285i \(0.106808\pi\)
0.0973917 + 0.995246i \(0.468950\pi\)
\(728\) 0 0
\(729\) 17.6812 + 20.4052i 0.654861 + 0.755750i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 15.5931 + 30.2465i 0.575946 + 1.11718i 0.979353 + 0.202158i \(0.0647955\pi\)
−0.403407 + 0.915020i \(0.632174\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −13.1286 + 3.18497i −0.482945 + 0.117161i −0.469837 0.882753i \(-0.655687\pi\)
−0.0131073 + 0.999914i \(0.504172\pi\)
\(740\) 0 0
\(741\) −44.9617 + 23.1794i −1.65171 + 0.851515i
\(742\) 0 0
\(743\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −44.6784 28.7131i −1.63034 1.04776i −0.948753 0.316017i \(-0.897654\pi\)
−0.681586 0.731738i \(-0.738709\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 28.3156 36.0062i 1.02915 1.30867i 0.0789989 0.996875i \(-0.474828\pi\)
0.950150 0.311794i \(-0.100930\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(762\) 0 0
\(763\) −0.489441 10.2746i −0.0177190 0.371967i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.03503 15.0748i 0.217629 0.543611i −0.778873 0.627182i \(-0.784208\pi\)
0.996502 + 0.0835710i \(0.0266326\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(774\) 0 0
\(775\) −46.4427 26.8137i −1.66827 0.963176i
\(776\) 0 0
\(777\) −0.709877 + 1.55441i −0.0254667 + 0.0557643i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18.3536 35.6009i 0.654234 1.26904i −0.295310 0.955402i \(-0.595423\pi\)
0.949544 0.313635i \(-0.101547\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 20.4220 + 1.95006i 0.725206 + 0.0692488i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0