Properties

Label 804.2.ba.a.605.1
Level 804
Weight 2
Character 804.605
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 605.1
Root \(0.723734 - 0.690079i\)
Character \(\chi\) = 804.605
Dual form 804.2.ba.a.101.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.487975 + 1.66189i) q^{3} +(1.00786 + 1.95498i) q^{7} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})\) \(q+(-0.487975 + 1.66189i) q^{3} +(1.00786 + 1.95498i) q^{7} +(-2.52376 - 1.62192i) q^{9} +(4.37316 + 5.56092i) q^{13} +(-5.95519 - 3.07011i) q^{19} +(-3.74078 + 0.720976i) q^{21} +(0.711574 + 4.94911i) q^{25} +(3.92699 - 3.40276i) q^{27} +(-4.43776 + 5.64307i) q^{31} +(4.14042 + 7.17141i) q^{37} +(-11.3756 + 4.55412i) q^{39} +(-11.4806 - 5.24303i) q^{43} +(1.25423 - 1.76132i) q^{49} +(8.00818 - 8.39874i) q^{57} +(-13.8682 - 4.79981i) q^{61} +(0.627224 - 6.56859i) q^{63} +(6.78321 + 4.58128i) q^{67} +(-2.35389 + 6.80112i) q^{73} +(-8.57211 - 1.23248i) q^{75} +(-4.14114 + 10.3441i) q^{79} +(3.73874 + 8.18669i) q^{81} +(-6.46396 + 14.1541i) q^{91} +(-7.21265 - 10.1287i) q^{93} +(14.6117 - 8.43607i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{7} + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{7} + 6q^{9} + 9q^{13} - 8q^{19} + 6q^{21} + 10q^{25} - 3q^{31} + 10q^{37} - 9q^{39} - 5q^{49} + 141q^{57} + 27q^{61} + 147q^{63} + 11q^{67} - 180q^{73} - 166q^{79} - 18q^{81} - 36q^{91} - 3q^{93} + 33q^{97} + O(q^{100}) \)

Character values

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

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

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 \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(6\) 0 0
\(7\) 1.00786 + 1.95498i 0.380937 + 0.738914i 0.998880 0.0473083i \(-0.0150643\pi\)
−0.617944 + 0.786222i \(0.712034\pi\)
\(8\) 0 0
\(9\) −2.52376 1.62192i −0.841254 0.540641i
\(10\) 0 0
\(11\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(12\) 0 0
\(13\) 4.37316 + 5.56092i 1.21290 + 1.54232i 0.758731 + 0.651404i \(0.225820\pi\)
0.454165 + 0.890918i \(0.349938\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(18\) 0 0
\(19\) −5.95519 3.07011i −1.36621 0.704333i −0.389934 0.920843i \(-0.627502\pi\)
−0.976280 + 0.216510i \(0.930533\pi\)
\(20\) 0 0
\(21\) −3.74078 + 0.720976i −0.816305 + 0.157330i
\(22\) 0 0
\(23\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(24\) 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
\(31\) −4.43776 + 5.64307i −0.797045 + 1.01353i 0.202369 + 0.979309i \(0.435136\pi\)
−0.999414 + 0.0342160i \(0.989107\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.14042 + 7.17141i 0.680680 + 1.17897i 0.974774 + 0.223196i \(0.0716490\pi\)
−0.294093 + 0.955777i \(0.595018\pi\)
\(38\) 0 0
\(39\) −11.3756 + 4.55412i −1.82156 + 0.729242i
\(40\) 0 0
\(41\) 0 0 0.814576 0.580057i \(-0.196970\pi\)
−0.814576 + 0.580057i \(0.803030\pi\)
\(42\) 0 0
\(43\) −11.4806 5.24303i −1.75078 0.799554i −0.988296 0.152547i \(-0.951253\pi\)
−0.762484 0.647008i \(-0.776020\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(48\) 0 0
\(49\) 1.25423 1.76132i 0.179176 0.251617i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00818 8.39874i 1.06071 1.11244i
\(58\) 0 0
\(59\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(60\) 0 0
\(61\) −13.8682 4.79981i −1.77564 0.614553i −0.775771 0.631015i \(-0.782639\pi\)
−0.999864 + 0.0164616i \(0.994760\pi\)
\(62\) 0 0
\(63\) 0.627224 6.56859i 0.0790228 0.827564i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.78321 + 4.58128i 0.828700 + 0.559692i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(72\) 0 0
\(73\) −2.35389 + 6.80112i −0.275502 + 0.796011i 0.719348 + 0.694650i \(0.244441\pi\)
−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
\(79\) −4.14114 + 10.3441i −0.465914 + 1.16380i 0.490410 + 0.871492i \(0.336847\pi\)
−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 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(90\) 0 0
\(91\) −6.46396 + 14.1541i −0.677607 + 1.48375i
\(92\) 0 0
\(93\) −7.21265 10.1287i −0.747917 1.05030i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.6117 8.43607i 1.48359 0.856553i 0.483767 0.875197i \(-0.339268\pi\)
0.999826 + 0.0186441i \(0.00593496\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.998867 0.0475819i \(-0.0151515\pi\)
−0.998867 + 0.0475819i \(0.984848\pi\)
\(102\) 0 0
\(103\) 11.7533 + 9.24286i 1.15808 + 0.910726i 0.997090 0.0762298i \(-0.0242882\pi\)
0.160992 + 0.986956i \(0.448531\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(108\) 0 0
\(109\) 19.8266 2.85064i 1.89905 0.273042i 0.909329 0.416079i \(-0.136596\pi\)
0.989717 + 0.143037i \(0.0456869\pi\)
\(110\) 0 0
\(111\) −13.9385 + 3.38145i −1.32299 + 0.320953i
\(112\) 0 0
\(113\) 0 0 −0.189251 0.981929i \(-0.560606\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.01742 21.1274i −0.186510 1.95322i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 8.64658 6.79975i 0.786053 0.618159i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 13.0703 6.73819i 1.15980 0.597918i 0.232631 0.972565i \(-0.425267\pi\)
0.927167 + 0.374648i \(0.122236\pi\)
\(128\) 0 0
\(129\) 14.3156 16.5211i 1.26042 1.45460i
\(130\) 0 0
\(131\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(132\) 0 0
\(133\) 14.7366i 1.27782i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(138\) 0 0
\(139\) −8.66506 7.50832i −0.734961 0.636847i 0.204751 0.978814i \(-0.434362\pi\)
−0.939712 + 0.341967i \(0.888907\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.31509 + 2.94387i 0.190945 + 0.242806i
\(148\) 0 0
\(149\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(150\) 0 0
\(151\) 21.5089 2.05386i 1.75037 0.167140i 0.829891 0.557926i \(-0.188403\pi\)
0.920482 + 0.390785i \(0.127796\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.42589 + 22.3658i 0.433033 + 1.78499i 0.601060 + 0.799204i \(0.294745\pi\)
−0.168027 + 0.985782i \(0.553740\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.82939 4.90065i 0.221615 0.383849i −0.733683 0.679492i \(-0.762200\pi\)
0.955299 + 0.295643i \(0.0955338\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(168\) 0 0
\(169\) −8.73448 + 36.0040i −0.671883 + 2.76954i
\(170\) 0 0
\(171\) 10.0500 + 17.4071i 0.768542 + 1.33115i
\(172\) 0 0
\(173\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(174\) 0 0
\(175\) −8.95825 + 6.37914i −0.677180 + 0.482218i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(180\) 0 0
\(181\) 9.30913 8.87624i 0.691942 0.659766i −0.260153 0.965567i \(-0.583773\pi\)
0.952095 + 0.305802i \(0.0989245\pi\)
\(182\) 0 0
\(183\) 14.7441 20.7052i 1.08991 1.53057i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 10.6102 + 4.24768i 0.771779 + 0.308974i
\(190\) 0 0
\(191\) 0 0 0.690079 0.723734i \(-0.257576\pi\)
−0.690079 + 0.723734i \(0.742424\pi\)
\(192\) 0 0
\(193\) 3.92593 27.3055i 0.282595 1.96549i 0.0231538 0.999732i \(-0.492629\pi\)
0.259441 0.965759i \(-0.416462\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(198\) 0 0
\(199\) 0.638541 13.4046i 0.0452650 0.950228i −0.856004 0.516969i \(-0.827060\pi\)
0.901269 0.433260i \(-0.142637\pi\)
\(200\) 0 0
\(201\) −10.9236 + 9.03739i −0.770493 + 0.637449i
\(202\) 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
\(211\) 14.6918 + 14.0086i 1.01142 + 0.964390i 0.999368 0.0355373i \(-0.0113142\pi\)
0.0120548 + 0.999927i \(0.496163\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −15.5048 2.98830i −1.05253 0.202859i
\(218\) 0 0
\(219\) −10.1541 7.23068i −0.686149 0.488604i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −3.35641 + 0.985532i −0.224762 + 0.0659961i −0.392174 0.919891i \(-0.628277\pi\)
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 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(228\) 0 0
\(229\) −11.1266 27.7928i −0.735265 1.83660i −0.503978 0.863716i \(-0.668131\pi\)
−0.231287 0.972886i \(-0.574293\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.971812 0.235759i \(-0.924242\pi\)
0.971812 + 0.235759i \(0.0757576\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −15.1699 11.9298i −0.985393 0.774921i
\(238\) 0 0
\(239\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) 0 0
\(241\) 18.4249 + 21.2635i 1.18685 + 1.36970i 0.913014 + 0.407929i \(0.133749\pi\)
0.273841 + 0.961775i \(0.411706\pi\)
\(242\) 0 0
\(243\) −15.4298 + 2.21847i −0.989821 + 0.142315i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8.97032 46.5424i −0.570768 2.96142i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.0950560 0.995472i \(-0.530303\pi\)
0.0950560 + 0.995472i \(0.469697\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(258\) 0 0
\(259\) −9.84701 + 15.3222i −0.611864 + 0.952078i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(264\) 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
\(271\) 8.11176 27.6261i 0.492755 1.67817i −0.218988 0.975728i \(-0.570276\pi\)
0.711742 0.702440i \(-0.247906\pi\)
\(272\) 0 0
\(273\) −20.3683 17.6492i −1.23275 1.06818i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 20.1115 + 12.9249i 1.20838 + 0.776580i 0.980387 0.197082i \(-0.0631466\pi\)
0.227995 + 0.973662i \(0.426783\pi\)
\(278\) 0 0
\(279\) 20.3525 7.04406i 1.21847 0.421717i
\(280\) 0 0
\(281\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(282\) 0 0
\(283\) −24.2361 + 15.5756i −1.44069 + 0.925874i −0.441091 + 0.897462i \(0.645408\pi\)
−0.999596 + 0.0284112i \(0.990955\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 16.6928 3.21727i 0.981929 0.189251i
\(290\) 0 0
\(291\) 6.88968 + 28.3996i 0.403880 + 1.66482i
\(292\) 0 0
\(293\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −1.32088 27.7287i −0.0761343 1.59826i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −32.1929 + 12.8881i −1.83735 + 0.735562i −0.865100 + 0.501599i \(0.832745\pi\)
−0.972245 + 0.233964i \(0.924830\pi\)
\(308\) 0 0
\(309\) −21.0959 + 15.0223i −1.20010 + 0.854591i
\(310\) 0 0
\(311\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(312\) 0 0
\(313\) 9.91913 + 33.7814i 0.560662 + 1.90944i 0.375915 + 0.926654i \(0.377328\pi\)
0.184747 + 0.982786i \(0.440853\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −24.4098 + 25.6002i −1.35401 + 1.42005i
\(326\) 0 0
\(327\) −4.93745 + 34.3407i −0.273042 + 1.89905i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.907508 + 9.50386i −0.0498812 + 0.522379i 0.935655 + 0.352915i \(0.114810\pi\)
−0.985536 + 0.169464i \(0.945796\pi\)
\(332\) 0 0
\(333\) 1.18205 24.8144i 0.0647761 1.35982i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.227062 + 0.0108163i 0.0123689 + 0.000589202i 0.0537654 0.998554i \(-0.482878\pi\)
−0.0413966 + 0.999143i \(0.513181\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 19.9471 + 2.86797i 1.07704 + 0.154856i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.371662 0.928368i \(-0.378788\pi\)
−0.371662 + 0.928368i \(0.621212\pi\)
\(348\) 0 0
\(349\) −1.42476 3.11980i −0.0762659 0.166999i 0.867659 0.497161i \(-0.165624\pi\)
−0.943924 + 0.330162i \(0.892897\pi\)
\(350\) 0 0
\(351\) 36.0958 + 6.95690i 1.92665 + 0.371332i
\(352\) 0 0
\(353\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(360\) 0 0
\(361\) 15.0176 + 21.0893i 0.790401 + 1.10996i
\(362\) 0 0
\(363\) 7.08112 + 17.6878i 0.371662 + 0.928368i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −27.1879 6.59572i −1.41920 0.344294i −0.548516 0.836140i \(-0.684807\pi\)
−0.870682 + 0.491846i \(0.836322\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −33.1572 19.1433i −1.71681 0.991203i −0.924582 0.380982i \(-0.875586\pi\)
−0.792231 0.610221i \(-0.791081\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −9.52769 + 2.31139i −0.489405 + 0.118728i −0.472865 0.881135i \(-0.656780\pi\)
−0.0165397 + 0.999863i \(0.505265\pi\)
\(380\) 0 0
\(381\) 4.82017 + 25.0094i 0.246945 + 1.28127i
\(382\) 0 0
\(383\) 0 0 0.458227 0.888835i \(-0.348485\pi\)
−0.458227 + 0.888835i \(0.651515\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 20.4706 + 31.8528i 1.04058 + 1.61917i
\(388\) 0 0
\(389\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.82791 + 2.10952i −0.0917400 + 0.105874i −0.799763 0.600315i \(-0.795042\pi\)
0.708023 + 0.706189i \(0.249587\pi\)
\(398\) 0 0
\(399\) 24.4905 + 7.19107i 1.22606 + 0.360004i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −50.7877 −2.52991
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −17.3175 33.5913i −0.856297 1.66098i −0.744869 0.667211i \(-0.767488\pi\)
−0.111428 0.993773i \(-0.535542\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 16.7063 10.7365i 0.818113 0.525769i
\(418\) 0 0
\(419\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(420\) 0 0
\(421\) −6.71011 3.45930i −0.327031 0.168596i 0.286890 0.957963i \(-0.407378\pi\)
−0.613921 + 0.789367i \(0.710409\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4.59365 31.9496i −0.222303 1.54615i
\(428\) 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
\(433\) 4.88071 6.20633i 0.234552 0.298257i −0.654501 0.756061i \(-0.727121\pi\)
0.889053 + 0.457804i \(0.151364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 15.4184 + 26.7054i 0.735879 + 1.27458i 0.954336 + 0.298734i \(0.0965643\pi\)
−0.218457 + 0.975847i \(0.570102\pi\)
\(440\) 0 0
\(441\) −6.02210 + 2.41088i −0.286766 + 0.114804i
\(442\) 0 0
\(443\) 0 0 0.814576 0.580057i \(-0.196970\pi\)
−0.814576 + 0.580057i \(0.803030\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −7.08255 + 36.7477i −0.332767 + 1.72656i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 25.9353 + 10.3829i 1.21320 + 0.485693i 0.887953 0.459935i \(-0.152127\pi\)
0.325251 + 0.945628i \(0.394551\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(462\) 0 0
\(463\) 20.0749 + 6.94799i 0.932959 + 0.322900i 0.750922 0.660391i \(-0.229610\pi\)
0.182037 + 0.983292i \(0.441731\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(468\) 0 0
\(469\) −2.11978 + 17.8784i −0.0978822 + 0.825546i
\(470\) 0 0
\(471\) −39.8172 1.89673i −1.83468 0.0873966i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 10.9568 31.6575i 0.502731 1.45255i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(480\) 0 0
\(481\) −21.7730 + 54.3862i −0.992762 + 2.47980i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −0.597939 0.425791i −0.0270952 0.0192944i 0.566429 0.824110i \(-0.308325\pi\)
−0.593524 + 0.804816i \(0.702264\pi\)
\(488\) 0 0
\(489\) 6.76368 + 7.09354i 0.305864 + 0.320781i
\(490\) 0 0
\(491\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 23.5882 13.6187i 1.05595 0.609656i 0.131644 0.991297i \(-0.457974\pi\)
0.924310 + 0.381641i \(0.124641\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.998867 0.0475819i \(-0.0151515\pi\)
−0.998867 + 0.0475819i \(0.984848\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −55.5725 32.0848i −2.46806 1.42494i
\(508\) 0 0
\(509\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(510\) 0 0
\(511\) −15.6685 + 2.25279i −0.693132 + 0.0996574i
\(512\) 0 0
\(513\) −33.8328 + 8.20776i −1.49376 + 0.362381i
\(514\) 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 \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(522\) 0 0
\(523\) 24.4281 19.2105i 1.06817 0.840014i 0.0805251 0.996753i \(-0.474340\pi\)
0.987640 + 0.156738i \(0.0500979\pi\)
\(524\) 0 0
\(525\) −6.23003 18.0005i −0.271901 0.785606i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −20.4432 + 10.5392i −0.888835 + 0.458227i
\(530\) 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
\(541\) 8.51685 + 7.37989i 0.366168 + 0.317286i 0.818438 0.574595i \(-0.194840\pi\)
−0.452270 + 0.891881i \(0.649386\pi\)
\(542\) 0 0
\(543\) 10.2087 + 19.8021i 0.438098 + 0.849791i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −19.2182 + 6.65147i −0.821710 + 0.284396i −0.705394 0.708816i \(-0.749230\pi\)
−0.116316 + 0.993212i \(0.537109\pi\)
\(548\) 0 0
\(549\) 27.2150 + 34.6067i 1.16151 + 1.47698i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −24.3962 + 2.32955i −1.03743 + 0.0990626i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(558\) 0 0
\(559\) −21.0505 86.7715i −0.890343 3.67004i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −12.2367 + 15.5602i −0.513893 + 0.653468i
\(568\) 0 0
\(569\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(570\) 0 0
\(571\) 4.59655 18.9472i 0.192359 0.792917i −0.791085 0.611706i \(-0.790483\pi\)
0.983444 0.181210i \(-0.0580014\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −3.99179 + 2.84254i −0.166180 + 0.118336i −0.660121 0.751159i \(-0.729495\pi\)
0.493941 + 0.869495i \(0.335556\pi\)
\(578\) 0 0
\(579\) 43.4629 + 19.8489i 1.80626 + 0.824891i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.189251 0.981929i \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(588\) 0 0
\(589\) 43.7526 19.9811i 1.80279 0.823308i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.690079 0.723734i \(-0.257576\pi\)
−0.690079 + 0.723734i \(0.742424\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 21.9654 + 7.60231i 0.898985 + 0.311142i
\(598\) 0 0
\(599\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(600\) 0 0
\(601\) −1.46453 + 30.7443i −0.0597395 + 1.25409i 0.747660 + 0.664082i \(0.231177\pi\)
−0.807400 + 0.590005i \(0.799126\pi\)
\(602\) 0 0
\(603\) −9.68871 22.5639i −0.394555 0.918873i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 43.8313 + 4.18538i 1.77906 + 0.169879i 0.932201 0.361942i \(-0.117886\pi\)
0.846857 + 0.531821i \(0.178492\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −32.7891 31.2643i −1.32434 1.26275i −0.940687 0.339275i \(-0.889818\pi\)
−0.383651 0.923478i \(-0.625333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(618\) 0 0
\(619\) −45.6357 8.79555i −1.83425 0.353523i −0.849514 0.527566i \(-0.823105\pi\)
−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
\(631\) 4.04006 + 10.0916i 0.160832 + 0.401739i 0.986793 0.161989i \(-0.0517908\pi\)
−0.825960 + 0.563728i \(0.809367\pi\)
\(632\) 0 0
\(633\) −30.4499 + 17.5803i −1.21028 + 0.698753i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 15.2795 0.727852i 0.605396 0.0288385i
\(638\) 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
\(643\) −29.4531 33.9907i −1.16152 1.34046i −0.929969 0.367638i \(-0.880167\pi\)
−0.231548 0.972824i \(-0.574379\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.971812 0.235759i \(-0.0757576\pi\)
−0.971812 + 0.235759i \(0.924242\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 12.5322 24.3090i 0.491174 0.952745i
\(652\) 0 0
\(653\) 0 0 −0.0950560 0.995472i \(-0.530303\pi\)
0.0950560 + 0.995472i \(0.469697\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 16.9715 13.3466i 0.662123 0.520699i
\(658\) 0 0
\(659\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(660\) 0 0
\(661\) −16.6907 + 25.9712i −0.649192 + 1.01016i 0.348160 + 0.937435i \(0.386807\pi\)
−0.997352 + 0.0727275i \(0.976830\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 6.05891i 0.234251i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 14.5887 49.6845i 0.562352 1.91520i 0.220906 0.975295i \(-0.429098\pi\)
0.341446 0.939901i \(-0.389083\pi\)
\(674\) 0 0
\(675\) 19.6349 + 17.0138i 0.755750 + 0.654861i
\(676\) 0 0
\(677\) 0 0 −0.458227 0.888835i \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(678\) 0 0
\(679\) 31.2190 + 20.0632i 1.19807 + 0.769955i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 51.6181 4.92893i 1.96935 0.188051i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 51.6211 9.94916i 1.96376 0.378484i 0.979995 0.199023i \(-0.0637767\pi\)
0.983765 0.179461i \(-0.0574354\pi\)
\(692\) 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 0.618159 0.786053i \(-0.287879\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(702\) 0 0
\(703\) −2.63992 55.4187i −0.0995664 2.09016i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −32.2813 + 12.9235i −1.21235 + 0.485352i −0.887674 0.460472i \(-0.847680\pi\)
−0.324677 + 0.945825i \(0.605256\pi\)
\(710\) 0 0
\(711\) 27.2285 19.3893i 1.02115 0.727157i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(720\) 0 0
\(721\) −6.22396 + 32.2930i −0.231792 + 1.20265i
\(722\) 0 0
\(723\) −44.3286 + 20.2442i −1.64860 + 0.752889i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 37.2130 39.0279i 1.38015 1.44746i 0.688379 0.725351i \(-0.258323\pi\)
0.691775 0.722113i \(-0.256829\pi\)
\(728\) 0 0
\(729\) 3.84250 26.7252i 0.142315 0.989821i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 4.23163 44.3156i 0.156299 1.63683i −0.488559 0.872531i \(-0.662477\pi\)
0.644858 0.764303i \(-0.276917\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −17.0880 0.814004i −0.628594 0.0299436i −0.269132 0.963103i \(-0.586737\pi\)
−0.359462 + 0.933160i \(0.617040\pi\)
\(740\) 0 0
\(741\) 81.7257 + 7.80386i 3.00227 + 0.286682i
\(742\) 0 0
\(743\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −15.5186 33.9810i −0.566282 1.23998i −0.948753 0.316017i \(-0.897654\pi\)
0.382472 0.923967i \(-0.375073\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −6.00925 6.30233i −0.218410 0.229062i 0.605409 0.795914i \(-0.293009\pi\)
−0.823819 + 0.566852i \(0.808161\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(762\) 0 0
\(763\) 25.5555 + 35.8877i 0.925171 + 1.29922i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 49.2826 + 11.9558i 1.77717 + 0.431138i 0.984046 0.177916i \(-0.0569356\pi\)
0.793129 + 0.609054i \(0.208451\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(774\) 0 0
\(775\) −31.0860 17.9475i −1.11664 0.644693i
\(776\) 0 0
\(777\) −20.6588 23.8415i −0.741131 0.855310i
\(778\) 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
\(787\) 5.31582 + 55.6698i 0.189488 + 1.98441i 0.175309 + 0.984514i \(0.443908\pi\)
0.0141799 + 0.999899i \(0.495486\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −33.9562 98.1101i −1.20582 3.48399i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0