Properties

Label 804.2.ba.a.233.1
Level 804
Weight 2
Character 804.233
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 233.1
Root \(-0.786053 + 0.618159i\)
Character \(\chi\) = 804.233
Dual form 804.2.ba.a.245.1

$q$-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 \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{66}\right)\) \(1\)

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 \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(6\) 0 0
\(7\) 0.518680 0.543976i 0.196043 0.205604i −0.618374 0.785884i \(-0.712208\pi\)
0.814417 + 0.580280i \(0.197057\pi\)
\(8\) 0 0
\(9\) 2.87848 0.845198i 0.959493 0.281733i
\(10\) 0 0
\(11\) 0 0 −0.0950560 0.995472i \(-0.530303\pi\)
0.0950560 + 0.995472i \(0.469697\pi\)
\(12\) 0 0
\(13\) −1.21014 6.27878i −0.335631 1.74142i −0.621740 0.783223i \(-0.713574\pi\)
0.286109 0.958197i \(-0.407638\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(18\) 0 0
\(19\) 3.30555 3.15184i 0.758346 0.723082i −0.208617 0.977997i \(-0.566896\pi\)
0.966963 + 0.254916i \(0.0820477\pi\)
\(20\) 0 0
\(21\) 0.755148 1.06046i 0.164787 0.231411i
\(22\) 0 0
\(23\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(24\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) −2.02981 + 10.5316i −0.364564 + 1.89154i 0.0817313 + 0.996654i \(0.473955\pi\)
−0.446296 + 0.894886i \(0.647257\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.656310 1.13676i 0.107897 0.186883i −0.807021 0.590522i \(-0.798922\pi\)
0.914918 + 0.403640i \(0.132255\pi\)
\(38\) 0 0
\(39\) −3.62238 10.4662i −0.580045 1.67593i
\(40\) 0 0
\(41\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(42\) 0 0
\(43\) 5.92488 9.21929i 0.903536 1.40593i −0.0103449 0.999946i \(-0.503293\pi\)
0.913881 0.405983i \(-0.133071\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(48\) 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 \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(54\) 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 \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(60\) 0 0
\(61\) −0.304967 + 3.19376i −0.0390471 + 0.408919i 0.954725 + 0.297489i \(0.0961492\pi\)
−0.993772 + 0.111430i \(0.964457\pi\)
\(62\) 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 \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(72\) 0 0
\(73\) −17.0103 1.62429i −1.99090 0.190108i −0.996054 0.0887477i \(-0.971714\pi\)
−0.994850 0.101361i \(-0.967680\pi\)
\(74\) 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 −0.956325 0.292306i \(-0.905577\pi\)
−0.932414 0.361392i \(-0.882301\pi\)
\(80\) 0 0
\(81\) 7.57128 4.86577i 0.841254 0.540641i
\(82\) 0 0
\(83\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(90\) 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 0.797442 0.603396i \(-0.206186\pi\)
−0.123836 + 0.992303i \(0.539520\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.971812 0.235759i \(-0.0757576\pi\)
−0.971812 + 0.235759i \(0.924242\pi\)
\(102\) 0 0
\(103\) 17.8848 + 3.44700i 1.76224 + 0.339643i 0.964645 0.263553i \(-0.0848943\pi\)
0.797593 + 0.603196i \(0.206106\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(108\) 0 0
\(109\) −10.3427 + 8.96204i −0.990655 + 0.858407i −0.989925 0.141592i \(-0.954778\pi\)
−0.000729916 1.00000i \(0.500232\pi\)
\(110\) 0 0
\(111\) 0.844984 2.11067i 0.0802023 0.200336i
\(112\) 0 0
\(113\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(114\) 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 0.577433 0.816438i \(-0.304054\pi\)
−0.788038 + 0.615627i \(0.788903\pi\)
\(128\) 0 0
\(129\) 7.88522 17.2662i 0.694255 1.52021i
\(130\) 0 0
\(131\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(132\) 0 0
\(133\) 3.43294i 0.297674i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(138\) 0 0
\(139\) 20.0359 + 9.15010i 1.69942 + 0.776101i 0.997990 + 0.0633672i \(0.0201839\pi\)
0.701434 + 0.712734i \(0.252543\pi\)
\(140\) 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 \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(150\) 0 0
\(151\) −1.49049 + 0.768401i −0.121294 + 0.0625316i −0.517806 0.855498i \(-0.673251\pi\)
0.396511 + 0.918030i \(0.370221\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −17.9069 7.16885i −1.42913 0.572136i −0.477292 0.878745i \(-0.658382\pi\)
−0.951836 + 0.306608i \(0.900806\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 9.11104 + 15.7808i 0.713632 + 1.23605i 0.963485 + 0.267763i \(0.0862844\pi\)
−0.249853 + 0.968284i \(0.580382\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(168\) 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 \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(174\) 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 \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(180\) 0 0
\(181\) −16.9494 + 13.3291i −1.25984 + 0.990746i −0.260153 + 0.965567i \(0.583773\pi\)
−0.999683 + 0.0251785i \(0.991985\pi\)
\(182\) 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 \(-0.287879\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(192\) 0 0
\(193\) −17.9452 + 20.7099i −1.29173 + 1.49073i −0.521018 + 0.853546i \(0.674447\pi\)
−0.770710 + 0.637187i \(0.780098\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.458227 0.888835i \(-0.348485\pi\)
−0.458227 + 0.888835i \(0.651515\pi\)
\(198\) 0 0
\(199\) −0.501668 + 2.06790i −0.0355623 + 0.146590i −0.986825 0.161793i \(-0.948272\pi\)
0.951263 + 0.308382i \(0.0997876\pi\)
\(200\) 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 0.859935 0.510403i \(-0.170504\pi\)
−0.293279 + 0.956027i \(0.594746\pi\)
\(212\) 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 0.117125 0.993117i \(-0.462632\pi\)
0.167412 0.985887i \(-0.446459\pi\)
\(224\) 0 0
\(225\) 12.6188 + 8.10961i 0.841254 + 0.540641i
\(226\) 0 0
\(227\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(228\) 0 0
\(229\) −15.3520 + 5.31338i −1.01449 + 0.351118i −0.783202 0.621768i \(-0.786415\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(234\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) 3.53199 + 7.73398i 0.227515 + 0.498189i 0.988619 0.150441i \(-0.0480694\pi\)
−0.761103 + 0.648630i \(0.775342\pi\)
\(242\) 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 \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(258\) 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 \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(264\) 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 0.593999 0.804466i \(-0.297548\pi\)
0.796582 + 0.604530i \(0.206639\pi\)
\(272\) 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 −0.885150 + 0.465306i \(0.845944\pi\)
−0.996199 + 0.0871078i \(0.972238\pi\)
\(278\) 0 0
\(279\) 3.05856 + 32.0307i 0.183111 + 1.91763i
\(280\) 0 0
\(281\) 0 0 −0.189251 0.981929i \(-0.560606\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(282\) 0 0
\(283\) −3.68456 1.08188i −0.219024 0.0643113i 0.170379 0.985379i \(-0.445501\pi\)
−0.389404 + 0.921067i \(0.627319\pi\)
\(284\) 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 \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(294\) 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 0.688741 + 0.725007i \(0.258164\pi\)
−0.0932173 + 0.995646i \(0.529715\pi\)
\(308\) 0 0
\(309\) 31.5117 + 1.50109i 1.79264 + 0.0853938i
\(310\) 0 0
\(311\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(312\) 0 0
\(313\) 32.4546 + 4.66627i 1.83444 + 0.263753i 0.970720 0.240212i \(-0.0772171\pi\)
0.863724 + 0.503966i \(0.168126\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(318\) 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 0.0676621 0.997708i \(-0.478446\pi\)
0.773461 0.633844i \(-0.218524\pi\)
\(332\) 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 −0.575002 0.818152i \(-0.694999\pi\)
−0.885981 + 0.463721i \(0.846514\pi\)
\(338\) 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 \(-0.893939\pi\)
0.945001 + 0.327068i \(0.106061\pi\)
\(348\) 0 0
\(349\) 11.5864 7.44612i 0.620205 0.398582i −0.192467 0.981304i \(-0.561649\pi\)
0.812672 + 0.582722i \(0.198012\pi\)
\(350\) 0 0
\(351\) −19.2729 27.0650i −1.02871 1.44463i
\(352\) 0 0
\(353\) 0 0 0.998867 0.0475819i \(-0.0151515\pi\)
−0.998867 + 0.0475819i \(0.984848\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(360\) 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 0.761865 0.647736i \(-0.224284\pi\)
−0.998377 + 0.0569590i \(0.981860\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 30.5668 17.6477i 1.58269 0.913765i 0.588221 0.808700i \(-0.299828\pi\)
0.994465 0.105065i \(-0.0335050\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.449388 1.12252i 0.0230835 0.0576598i −0.916378 0.400313i \(-0.868901\pi\)
0.939462 + 0.342653i \(0.111326\pi\)
\(380\) 0 0
\(381\) −4.62683 3.29475i −0.237040 0.168795i
\(382\) 0 0
\(383\) 0 0 −0.690079 0.723734i \(-0.742424\pi\)
0.690079 + 0.723734i \(0.257576\pi\)
\(384\) 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 \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(390\) 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 0.598239 + 0.801318i \(0.295867\pi\)
−0.997359 + 0.0726328i \(0.976860\pi\)
\(398\) 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 −0.0164592 0.999865i \(-0.505239\pi\)
0.999515 0.0311349i \(-0.00991216\pi\)
\(410\) 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 \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(420\) 0 0
\(421\) 18.4844 17.6248i 0.900874 0.858981i −0.0896992 0.995969i \(-0.528591\pi\)
0.990573 + 0.136988i \(0.0437421\pi\)
\(422\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 0 0
\(433\) 4.86575 25.2459i 0.233833 1.21324i −0.655220 0.755438i \(-0.727424\pi\)
0.889053 0.457804i \(-0.151364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 5.30065 9.18100i 0.252986 0.438185i −0.711360 0.702827i \(-0.751921\pi\)
0.964347 + 0.264642i \(0.0852539\pi\)
\(440\) 0 0
\(441\) 6.31411 + 18.2434i 0.300672 + 0.868734i
\(442\) 0 0
\(443\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(450\) 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 −0.237627 0.971357i \(-0.576370\pi\)
0.787240 0.616647i \(-0.211509\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(462\) 0 0
\(463\) −2.93979 + 30.7869i −0.136624 + 1.43079i 0.623913 + 0.781494i \(0.285542\pi\)
−0.760537 + 0.649295i \(0.775064\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(468\) 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 \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(480\) 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 0.485528 0.874221i \(-0.338628\pi\)
0.566429 + 0.824110i \(0.308325\pi\)
\(488\) 0 0
\(489\) 19.5101 + 24.8091i 0.882276 + 1.12191i
\(490\) 0 0
\(491\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(492\) 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 −0.893720 0.448625i \(-0.851914\pi\)
−0.835381 0.549672i \(-0.814753\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.971812 0.235759i \(-0.0757576\pi\)
−0.971812 + 0.235759i \(0.924242\pi\)
\(504\) 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 \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(510\) 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 \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(522\) 0 0
\(523\) 19.9953 3.85378i 0.874334 0.168514i 0.267707 0.963501i \(-0.413734\pi\)
0.606627 + 0.794987i \(0.292522\pi\)
\(524\) 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 0.101713 0.994814i \(-0.467568\pi\)
−0.685222 + 0.728334i \(0.740295\pi\)
\(542\) 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 −0.601213 0.799089i \(-0.705316\pi\)
0.439120 0.898428i \(-0.355290\pi\)
\(548\) 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 \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(558\) 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 \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(564\) 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 \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(570\) 0 0
\(571\) 37.5196 15.0206i 1.57015 0.628592i 0.586703 0.809802i \(-0.300426\pi\)
0.983444 + 0.181210i \(0.0580014\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −45.4565 2.16536i −1.89238 0.0901450i −0.930940 0.365172i \(-0.881010\pi\)
−0.961437 + 0.275027i \(0.911313\pi\)
\(578\) 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 \(-0.196970\pi\)
−0.814576 + 0.580057i \(0.803030\pi\)
\(588\) 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 \(-0.287879\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(594\) 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 \(-0.348485\pi\)
−0.458227 + 0.888835i \(0.651515\pi\)
\(600\) 0 0
\(601\) 5.17040 21.3127i 0.210905 0.869362i −0.763725 0.645541i \(-0.776632\pi\)
0.974630 0.223821i \(-0.0718530\pi\)
\(602\) 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 −0.956497 0.291743i \(-0.905765\pi\)
−0.792469 0.609912i \(-0.791205\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 38.4524 + 30.2393i 1.55308 + 1.22135i 0.872426 + 0.488746i \(0.162545\pi\)
0.680651 + 0.732608i \(0.261697\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(618\) 0 0
\(619\) −20.5788 28.8989i −0.827132 1.16154i −0.984738 0.174042i \(-0.944317\pi\)
0.157606 0.987502i \(-0.449622\pi\)
\(620\) 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 −0.999617 0.0276903i \(-0.991185\pi\)
−0.768635 + 0.639688i \(0.779064\pi\)
\(632\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(642\) 0 0
\(643\) 20.9808 + 45.9416i 0.827402 + 1.81176i 0.496245 + 0.868183i \(0.334712\pi\)
0.331158 + 0.943575i \(0.392561\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.371662 0.928368i \(-0.378788\pi\)
−0.371662 + 0.928368i \(0.621212\pi\)
\(648\) 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 \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(654\) 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 \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(660\) 0 0
\(661\) 13.5803 + 46.2502i 0.528212 + 1.79893i 0.598163 + 0.801375i \(0.295898\pi\)
−0.0699507 + 0.997550i \(0.522284\pi\)
\(662\) 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 −0.486731 0.873552i \(-0.661811\pi\)
−0.220906 + 0.975295i \(0.570902\pi\)
\(674\) 0 0
\(675\) 23.6329 + 10.7928i 0.909632 + 0.415415i
\(676\) 0 0
\(677\) 0 0 0.690079 0.723734i \(-0.257576\pi\)
−0.690079 + 0.723734i \(0.742424\pi\)
\(678\) 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 \(-0.560606\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(684\) 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 0.245427 + 0.969415i \(0.421072\pi\)
−0.996369 + 0.0851357i \(0.972868\pi\)
\(692\) 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 \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(702\) 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 −0.784488 0.620145i \(-0.787074\pi\)
0.233301 0.972405i \(-0.425047\pi\)
\(710\) 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 \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(720\) 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 0.0973917 0.995246i \(-0.468950\pi\)
0.944231 0.329285i \(-0.106808\pi\)
\(728\) 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 −0.403407 0.915020i \(-0.632174\pi\)
0.979353 0.202158i \(-0.0647955\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −13.1286 3.18497i −0.482945 0.117161i −0.0131073 0.999914i \(-0.504172\pi\)
−0.469837 + 0.882753i \(0.655687\pi\)
\(740\) 0 0
\(741\) −44.9617 23.1794i −1.65171 0.851515i
\(742\) 0 0
\(743\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(744\) 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 −0.681586 + 0.731738i \(0.738709\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 28.3156 + 36.0062i 1.02915 + 1.30867i 0.950150 + 0.311794i \(0.100930\pi\)
0.0789989 + 0.996875i \(0.474828\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(762\) 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 0.996502 0.0835710i \(-0.0266326\pi\)
−0.778873 + 0.627182i \(0.784208\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(774\) 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 0.949544 + 0.313635i \(0.101547\pi\)
−0.295310 + 0.955402i \(0.595423\pi\)
\(788\) 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 \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0