Properties

Label 804.2.ba.a.785.1
Level 804
Weight 2
Character 804.785
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 785.1
Root \(0.235759 - 0.971812i\)
Character \(\chi\) = 804.785
Dual form 804.2.ba.a.677.1

$q$-expansion

\(f(q)\) \(=\) \(q+(0.487975 + 1.66189i) q^{3} +(-5.07182 + 0.241600i) q^{7} +(-2.52376 + 1.62192i) q^{9} +O(q^{10})\) \(q+(0.487975 + 1.66189i) q^{3} +(-5.07182 + 0.241600i) q^{7} +(-2.52376 + 1.62192i) q^{9} +(-0.619962 - 1.54859i) q^{13} +(0.411674 - 8.64209i) q^{19} +(-2.87643 - 8.31091i) q^{21} +(0.711574 - 4.94911i) q^{25} +(-3.92699 - 3.40276i) q^{27} +(-3.51128 + 8.77075i) q^{31} +(-2.71727 - 4.70645i) q^{37} +(2.27106 - 1.78598i) q^{39} +(-11.9291 + 5.44783i) q^{43} +(18.6966 - 1.78531i) q^{49} +(14.5631 - 3.53297i) q^{57} +(-2.88395 + 14.9634i) q^{61} +(12.4082 - 8.83583i) q^{63} +(-5.21774 - 6.30676i) q^{67} +(-16.0693 - 3.09710i) q^{73} +(8.57211 - 1.23248i) q^{75} +(-9.02089 + 11.4710i) q^{79} +(3.73874 - 8.18669i) q^{81} +(3.51847 + 7.70438i) q^{91} +(-16.2895 - 1.55545i) q^{93} +(17.0516 - 9.84473i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{7} + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{7} + 6q^{9} + 9q^{13} - 8q^{19} + 6q^{21} + 10q^{25} - 3q^{31} + 10q^{37} - 9q^{39} - 5q^{49} + 141q^{57} + 27q^{61} + 147q^{63} + 11q^{67} - 180q^{73} - 166q^{79} - 18q^{81} - 36q^{91} - 3q^{93} + 33q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{43}{66}\right)\) \(1\)

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
\(3\) 0.487975 + 1.66189i 0.281733 + 0.959493i
\(4\) 0 0
\(5\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(6\) 0 0
\(7\) −5.07182 + 0.241600i −1.91697 + 0.0913164i −0.971876 0.235493i \(-0.924330\pi\)
−0.945090 + 0.326809i \(0.894027\pi\)
\(8\) 0 0
\(9\) −2.52376 + 1.62192i −0.841254 + 0.540641i
\(10\) 0 0
\(11\) 0 0 −0.189251 0.981929i \(-0.560606\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(12\) 0 0
\(13\) −0.619962 1.54859i −0.171946 0.429501i 0.817215 0.576333i \(-0.195517\pi\)
−0.989162 + 0.146831i \(0.953093\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(18\) 0 0
\(19\) 0.411674 8.64209i 0.0944444 1.98263i −0.0753671 0.997156i \(-0.524013\pi\)
0.169812 0.985477i \(-0.445684\pi\)
\(20\) 0 0
\(21\) −2.87643 8.31091i −0.627689 1.81359i
\(22\) 0 0
\(23\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(24\) 0 0
\(25\) 0.711574 4.94911i 0.142315 0.989821i
\(26\) 0 0
\(27\) −3.92699 3.40276i −0.755750 0.654861i
\(28\) 0 0
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) −3.51128 + 8.77075i −0.630645 + 1.57527i 0.176099 + 0.984372i \(0.443652\pi\)
−0.806744 + 0.590901i \(0.798772\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.71727 4.70645i −0.446716 0.773735i 0.551454 0.834205i \(-0.314073\pi\)
−0.998170 + 0.0604704i \(0.980740\pi\)
\(38\) 0 0
\(39\) 2.27106 1.78598i 0.363661 0.285986i
\(40\) 0 0
\(41\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(42\) 0 0
\(43\) −11.9291 + 5.44783i −1.81917 + 0.830786i −0.905286 + 0.424803i \(0.860343\pi\)
−0.913881 + 0.405983i \(0.866929\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(48\) 0 0
\(49\) 18.6966 1.78531i 2.67095 0.255045i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 14.5631 3.53297i 1.92893 0.467953i
\(58\) 0 0
\(59\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(60\) 0 0
\(61\) −2.88395 + 14.9634i −0.369252 + 1.91586i 0.0311325 + 0.999515i \(0.490089\pi\)
−0.400385 + 0.916347i \(0.631124\pi\)
\(62\) 0 0
\(63\) 12.4082 8.83583i 1.56329 1.11321i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.21774 6.30676i −0.637449 0.770493i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(72\) 0 0
\(73\) −16.0693 3.09710i −1.88076 0.362488i −0.885915 0.463848i \(-0.846468\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 8.57211 1.23248i 0.989821 0.142315i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −9.02089 + 11.4710i −1.01493 + 1.29059i −0.0586047 + 0.998281i \(0.518665\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) 3.73874 8.18669i 0.415415 0.909632i
\(82\) 0 0
\(83\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(90\) 0 0
\(91\) 3.51847 + 7.70438i 0.368836 + 0.807638i
\(92\) 0 0
\(93\) −16.2895 1.55545i −1.68914 0.161293i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 17.0516 9.84473i 1.73132 0.999581i 0.851187 0.524862i \(-0.175883\pi\)
0.880138 0.474718i \(-0.157450\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.458227 0.888835i \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(102\) 0 0
\(103\) −17.3585 6.94931i −1.71039 0.684736i −0.710566 0.703631i \(-0.751561\pi\)
−0.999821 + 0.0188949i \(0.993985\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(108\) 0 0
\(109\) 8.59955 + 1.23643i 0.823688 + 0.118428i 0.541255 0.840859i \(-0.317949\pi\)
0.282433 + 0.959287i \(0.408859\pi\)
\(110\) 0 0
\(111\) 6.49564 6.81243i 0.616539 0.646607i
\(112\) 0 0
\(113\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.07633 + 2.90274i 0.376857 + 0.268358i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.2120 + 4.08829i −0.928368 + 0.371662i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0.442367 + 9.28643i 0.0392538 + 0.824038i 0.929559 + 0.368674i \(0.120188\pi\)
−0.890305 + 0.455364i \(0.849509\pi\)
\(128\) 0 0
\(129\) −14.8748 17.1664i −1.30965 1.51142i
\(130\) 0 0
\(131\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(132\) 0 0
\(133\) 43.9306i 3.80926i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(138\) 0 0
\(139\) −1.12922 + 0.978478i −0.0957795 + 0.0829934i −0.701434 0.712734i \(-0.747457\pi\)
0.605655 + 0.795727i \(0.292911\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 12.0905 + 30.2006i 0.997207 + 2.49090i
\(148\) 0 0
\(149\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(150\) 0 0
\(151\) −4.94024 + 6.93759i −0.402031 + 0.564573i −0.965115 0.261826i \(-0.915675\pi\)
0.563084 + 0.826399i \(0.309615\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.391208 + 0.373016i 0.0312218 + 0.0297699i 0.705529 0.708681i \(-0.250709\pi\)
−0.674308 + 0.738451i \(0.735558\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 7.55598 13.0873i 0.591830 1.02508i −0.402156 0.915571i \(-0.631739\pi\)
0.993986 0.109509i \(-0.0349277\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(168\) 0 0
\(169\) 7.39477 7.05089i 0.568828 0.542376i
\(170\) 0 0
\(171\) 12.9778 + 22.4783i 0.992440 + 1.71896i
\(172\) 0 0
\(173\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(174\) 0 0
\(175\) −2.41327 + 25.2729i −0.182426 + 1.91045i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(180\) 0 0
\(181\) 5.56346 22.9329i 0.413529 1.70459i −0.260153 0.965567i \(-0.583773\pi\)
0.673682 0.739022i \(-0.264712\pi\)
\(182\) 0 0
\(183\) −26.2748 + 2.50894i −1.94229 + 0.185466i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 20.7391 + 16.3094i 1.50855 + 1.18633i
\(190\) 0 0
\(191\) 0 0 0.971812 0.235759i \(-0.0757576\pi\)
−0.971812 + 0.235759i \(0.924242\pi\)
\(192\) 0 0
\(193\) 2.06021 + 14.3291i 0.148297 + 1.03143i 0.919007 + 0.394242i \(0.128993\pi\)
−0.770710 + 0.637187i \(0.780098\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.814576 0.580057i \(-0.196970\pi\)
−0.814576 + 0.580057i \(0.803030\pi\)
\(198\) 0 0
\(199\) 24.4820 + 12.6214i 1.73548 + 0.894704i 0.966977 + 0.254864i \(0.0820306\pi\)
0.768507 + 0.639841i \(0.221000\pi\)
\(200\) 0 0
\(201\) 7.93501 11.7489i 0.559692 0.828700i
\(202\) 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
\(211\) 4.16868 + 17.1835i 0.286984 + 1.18296i 0.914574 + 0.404419i \(0.132526\pi\)
−0.627590 + 0.778544i \(0.715958\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 15.6896 45.3320i 1.06508 3.07734i
\(218\) 0 0
\(219\) −2.69436 28.2167i −0.182068 1.90670i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 12.5627 + 3.68874i 0.841259 + 0.247016i 0.673847 0.738871i \(-0.264641\pi\)
0.167412 + 0.985887i \(0.446459\pi\)
\(224\) 0 0
\(225\) 6.23123 + 13.6445i 0.415415 + 0.909632i
\(226\) 0 0
\(227\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(228\) 0 0
\(229\) −16.9823 21.5947i −1.12222 1.42702i −0.890934 0.454133i \(-0.849949\pi\)
−0.231287 0.972886i \(-0.574293\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.690079 0.723734i \(-0.742424\pi\)
0.690079 + 0.723734i \(0.257576\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −23.4655 9.39417i −1.52425 0.610217i
\(238\) 0 0
\(239\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) 0 0
\(241\) −15.2563 + 17.6067i −0.982743 + 1.13415i 0.00821459 + 0.999966i \(0.497385\pi\)
−0.990957 + 0.134179i \(0.957160\pi\)
\(242\) 0 0
\(243\) 15.4298 + 2.21847i 0.989821 + 0.142315i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −13.6383 + 4.72025i −0.867783 + 0.300343i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(258\) 0 0
\(259\) 14.9186 + 23.2137i 0.926994 + 1.44243i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(264\) 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
\(271\) 0.722570 + 2.46085i 0.0438930 + 0.149486i 0.978524 0.206133i \(-0.0660881\pi\)
−0.934631 + 0.355619i \(0.884270\pi\)
\(272\) 0 0
\(273\) −11.0869 + 9.60686i −0.671010 + 0.581434i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.55562 0.999733i 0.0934679 0.0600682i −0.493072 0.869989i \(-0.664126\pi\)
0.586540 + 0.809920i \(0.300490\pi\)
\(278\) 0 0
\(279\) −5.36385 27.8303i −0.321125 1.66616i
\(280\) 0 0
\(281\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(282\) 0 0
\(283\) −11.0217 7.08324i −0.655174 0.421055i 0.170379 0.985379i \(-0.445501\pi\)
−0.825554 + 0.564324i \(0.809137\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −5.56016 16.0650i −0.327068 0.945001i
\(290\) 0 0
\(291\) 24.6816 + 23.5339i 1.44686 + 1.37958i
\(292\) 0 0
\(293\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 59.1859 30.5124i 3.41142 1.75871i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −14.8494 + 11.6777i −0.847501 + 0.666482i −0.944395 0.328813i \(-0.893351\pi\)
0.0968945 + 0.995295i \(0.469109\pi\)
\(308\) 0 0
\(309\) 3.07846 32.2391i 0.175127 1.83402i
\(310\) 0 0
\(311\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(312\) 0 0
\(313\) −3.21722 + 10.9568i −0.181848 + 0.619317i 0.817227 + 0.576316i \(0.195510\pi\)
−0.999075 + 0.0430013i \(0.986308\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −8.10529 + 1.96632i −0.449600 + 0.109072i
\(326\) 0 0
\(327\) 2.14156 + 14.8949i 0.118428 + 0.823688i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 22.0054 15.6700i 1.20952 0.861298i 0.216166 0.976357i \(-0.430645\pi\)
0.993359 + 0.115058i \(0.0367055\pi\)
\(332\) 0 0
\(333\) 14.4912 + 7.47075i 0.794114 + 0.409394i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4.63979 8.99993i 0.252745 0.490258i −0.727993 0.685585i \(-0.759547\pi\)
0.980738 + 0.195327i \(0.0625769\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −59.2134 + 8.51361i −3.19723 + 0.459692i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.618159 0.786053i \(-0.287879\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(348\) 0 0
\(349\) −14.6508 + 32.0808i −0.784241 + 1.71725i −0.0917948 + 0.995778i \(0.529260\pi\)
−0.692446 + 0.721470i \(0.743467\pi\)
\(350\) 0 0
\(351\) −2.83489 + 8.19087i −0.151315 + 0.437197i
\(352\) 0 0
\(353\) 0 0 −0.0950560 0.995472i \(-0.530303\pi\)
0.0950560 + 0.995472i \(0.469697\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(360\) 0 0
\(361\) −55.6024 5.30938i −2.92644 0.279441i
\(362\) 0 0
\(363\) −11.7775 14.9763i −0.618159 0.786053i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −15.1338 15.8719i −0.789978 0.828506i 0.198463 0.980108i \(-0.436405\pi\)
−0.988441 + 0.151603i \(0.951557\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −9.74675 5.62729i −0.504667 0.291370i 0.225971 0.974134i \(-0.427444\pi\)
−0.730639 + 0.682764i \(0.760778\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 20.8427 21.8592i 1.07062 1.12283i 0.0785782 0.996908i \(-0.474962\pi\)
0.992040 0.125924i \(-0.0401895\pi\)
\(380\) 0 0
\(381\) −15.2172 + 5.26672i −0.779599 + 0.269822i
\(382\) 0 0
\(383\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 21.2702 33.0970i 1.08122 1.68242i
\(388\) 0 0
\(389\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 20.8706 + 24.0860i 1.04747 + 1.20884i 0.977422 + 0.211298i \(0.0677690\pi\)
0.0700455 + 0.997544i \(0.477686\pi\)
\(398\) 0 0
\(399\) −73.0078 + 21.4370i −3.65496 + 1.07319i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 15.7592 0.785020
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 19.6222 0.934721i 0.970256 0.0462190i 0.443535 0.896257i \(-0.353724\pi\)
0.526721 + 0.850038i \(0.323421\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2.17716 1.39917i −0.106616 0.0685178i
\(418\) 0 0
\(419\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(420\) 0 0
\(421\) 0.196061 4.11582i 0.00955541 0.200593i −0.989041 0.147638i \(-0.952833\pi\)
0.998597 0.0529550i \(-0.0168640\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 11.0117 76.5882i 0.532895 3.70636i
\(428\) 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
\(433\) −1.46297 + 3.65431i −0.0703056 + 0.175615i −0.959359 0.282189i \(-0.908940\pi\)
0.889053 + 0.457804i \(0.151364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −20.8188 36.0592i −0.993626 1.72101i −0.594437 0.804142i \(-0.702625\pi\)
−0.399190 0.916868i \(-0.630708\pi\)
\(440\) 0 0
\(441\) −44.2902 + 34.8302i −2.10906 + 1.65858i
\(442\) 0 0
\(443\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −13.9402 4.82476i −0.654969 0.226687i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −32.2646 25.3731i −1.50927 1.18691i −0.927652 0.373447i \(-0.878176\pi\)
−0.581622 0.813459i \(-0.697582\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(462\) 0 0
\(463\) 8.11299 42.0942i 0.377043 1.95628i 0.102362 0.994747i \(-0.467360\pi\)
0.274681 0.961535i \(-0.411428\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(468\) 0 0
\(469\) 27.9871 + 30.7261i 1.29233 + 1.41880i
\(470\) 0 0
\(471\) −0.429012 + 0.832167i −0.0197678 + 0.0383443i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −42.4777 8.18691i −1.94901 0.375641i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(480\) 0 0
\(481\) −5.60375 + 7.12575i −0.255509 + 0.324906i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −3.21593 33.6788i −0.145728 1.52613i −0.712157 0.702020i \(-0.752282\pi\)
0.566429 0.824110i \(-0.308325\pi\)
\(488\) 0 0
\(489\) 25.4369 + 6.17092i 1.15029 + 0.279058i
\(490\) 0 0
\(491\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −31.2732 + 18.0556i −1.39998 + 0.808280i −0.994390 0.105774i \(-0.966268\pi\)
−0.405592 + 0.914054i \(0.632935\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.458227 0.888835i \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 15.3263 + 8.84863i 0.680664 + 0.392981i
\(508\) 0 0
\(509\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(510\) 0 0
\(511\) 82.2486 + 11.8256i 3.63846 + 0.523132i
\(512\) 0 0
\(513\) −31.0236 + 32.5366i −1.36972 + 1.43653i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(522\) 0 0
\(523\) 16.3549 6.54751i 0.715150 0.286303i 0.0145869 0.999894i \(-0.495357\pi\)
0.700563 + 0.713591i \(0.252932\pi\)
\(524\) 0 0
\(525\) −43.1784 + 8.32195i −1.88446 + 0.363200i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.09438 + 22.9739i 0.0475819 + 0.998867i
\(530\) 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
\(541\) 31.3555 27.1697i 1.34808 1.16812i 0.377865 0.925861i \(-0.376658\pi\)
0.970213 0.242255i \(-0.0778870\pi\)
\(542\) 0 0
\(543\) 40.8268 1.94482i 1.75205 0.0834602i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.50444 18.1828i −0.149839 0.777439i −0.976630 0.214929i \(-0.931048\pi\)
0.826791 0.562510i \(-0.190164\pi\)
\(548\) 0 0
\(549\) −16.9910 42.4415i −0.725159 1.81136i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 42.9809 60.3582i 1.82773 2.56669i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(558\) 0 0
\(559\) 15.8320 + 15.0958i 0.669623 + 0.638484i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −16.9843 + 42.4247i −0.713272 + 1.78167i
\(568\) 0 0
\(569\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(570\) 0 0
\(571\) 28.9434 27.5975i 1.21124 1.15492i 0.227799 0.973708i \(-0.426847\pi\)
0.983444 0.181210i \(-0.0580014\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −4.56288 + 47.7847i −0.189955 + 1.98930i −0.0545549 + 0.998511i \(0.517374\pi\)
−0.135400 + 0.990791i \(0.543232\pi\)
\(578\) 0 0
\(579\) −22.8080 + 10.4161i −0.947869 + 0.432877i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.945001 0.327068i \(-0.893939\pi\)
0.945001 + 0.327068i \(0.106061\pi\)
\(588\) 0 0
\(589\) 74.3522 + 33.9555i 3.06363 + 1.39911i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.971812 0.235759i \(-0.0757576\pi\)
−0.971812 + 0.235759i \(0.924242\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −9.02870 + 46.8453i −0.369520 + 1.91725i
\(598\) 0 0
\(599\) 0 0 0.814576 0.580057i \(-0.196970\pi\)
−0.814576 + 0.580057i \(0.803030\pi\)
\(600\) 0 0
\(601\) 29.6846 + 15.3035i 1.21086 + 0.624242i 0.940918 0.338635i \(-0.109965\pi\)
0.269942 + 0.962877i \(0.412995\pi\)
\(602\) 0 0
\(603\) 23.3974 + 7.45397i 0.952816 + 0.303549i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −22.2812 31.2896i −0.904368 1.27001i −0.962384 0.271691i \(-0.912417\pi\)
0.0580166 0.998316i \(-0.481522\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 3.43376 + 14.1541i 0.138688 + 0.571680i 0.998168 + 0.0605099i \(0.0192727\pi\)
−0.859479 + 0.511170i \(0.829212\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(618\) 0 0
\(619\) −7.91841 + 22.8787i −0.318268 + 0.919574i 0.666471 + 0.745531i \(0.267804\pi\)
−0.984738 + 0.174042i \(0.944317\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −23.9873 7.04331i −0.959493 0.281733i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −22.0452 28.0328i −0.877606 1.11597i −0.992458 0.122585i \(-0.960882\pi\)
0.114852 0.993383i \(-0.463361\pi\)
\(632\) 0 0
\(633\) −26.5229 + 15.3130i −1.05419 + 0.608638i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −14.3559 27.8466i −0.568803 1.10332i
\(638\) 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
\(643\) 7.68998 8.87471i 0.303263 0.349984i −0.583579 0.812056i \(-0.698348\pi\)
0.886843 + 0.462072i \(0.152894\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.690079 0.723734i \(-0.257576\pi\)
−0.690079 + 0.723734i \(0.742424\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 82.9929 + 3.95344i 3.25275 + 0.154947i
\(652\) 0 0
\(653\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 45.5782 18.2468i 1.77818 0.711874i
\(658\) 0 0
\(659\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(660\) 0 0
\(661\) 5.87143 + 9.13612i 0.228372 + 0.355354i 0.936462 0.350768i \(-0.114079\pi\)
−0.708090 + 0.706122i \(0.750443\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 22.6778i 0.876775i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.16734 + 3.97560i 0.0449977 + 0.153248i 0.978927 0.204209i \(-0.0654621\pi\)
−0.933930 + 0.357457i \(0.883644\pi\)
\(674\) 0 0
\(675\) −19.6349 + 17.0138i −0.755750 + 0.654861i
\(676\) 0 0
\(677\) 0 0 0.998867 0.0475819i \(-0.0151515\pi\)
−0.998867 + 0.0475819i \(0.984848\pi\)
\(678\) 0 0
\(679\) −84.1040 + 54.0503i −3.22761 + 2.07426i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 27.6011 38.7604i 1.05305 1.47880i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 16.4496 + 47.5280i 0.625772 + 1.80805i 0.588142 + 0.808758i \(0.299860\pi\)
0.0376301 + 0.999292i \(0.488019\pi\)
\(692\) 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 0.371662 0.928368i \(-0.378788\pi\)
−0.371662 + 0.928368i \(0.621212\pi\)
\(702\) 0 0
\(703\) −41.7922 + 21.5454i −1.57622 + 0.812599i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 5.85259 4.60253i 0.219799 0.172852i −0.502188 0.864758i \(-0.667472\pi\)
0.721987 + 0.691907i \(0.243229\pi\)
\(710\) 0 0
\(711\) 4.16150 43.5812i 0.156069 1.63442i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(720\) 0 0
\(721\) 89.7183 + 31.0518i 3.34128 + 1.15643i
\(722\) 0 0
\(723\) −36.7050 16.7626i −1.36508 0.623409i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −34.2255 + 8.30301i −1.26935 + 0.307942i −0.813213 0.581967i \(-0.802283\pi\)
−0.456140 + 0.889908i \(0.650768\pi\)
\(728\) 0 0
\(729\) 3.84250 + 26.7252i 0.142315 + 0.989821i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 7.77669 5.53775i 0.287238 0.204542i −0.427351 0.904086i \(-0.640553\pi\)
0.714589 + 0.699544i \(0.246614\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 23.0054 44.6243i 0.846268 1.64153i 0.0819999 0.996632i \(-0.473869\pi\)
0.764268 0.644899i \(-0.223100\pi\)
\(740\) 0 0
\(741\) −14.4997 20.3620i −0.532659 0.748015i
\(742\) 0 0
\(743\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.42864 + 5.31797i −0.0886222 + 0.194056i −0.948753 0.316017i \(-0.897654\pi\)
0.860131 + 0.510073i \(0.170382\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −51.0605 12.3871i −1.85582 0.450218i −0.857854 0.513893i \(-0.828203\pi\)
−0.997971 + 0.0636752i \(0.979718\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(762\) 0 0
\(763\) −43.9141 4.19328i −1.58980 0.151807i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −29.2118 30.6365i −1.05341 1.10478i −0.994310 0.106528i \(-0.966026\pi\)
−0.0590966 0.998252i \(-0.518822\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(774\) 0 0
\(775\) 40.9089 + 23.6187i 1.46949 + 0.848410i
\(776\) 0 0
\(777\) −31.2988 + 36.1207i −1.12284 + 1.29582i
\(778\) 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
\(787\) 44.7504 + 31.8666i 1.59518 + 1.13592i 0.915431 + 0.402475i \(0.131850\pi\)
0.679747 + 0.733447i \(0.262090\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 24.9601 4.81066i 0.886357 0.170831i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0