Properties

Label 804.2.ba.a.41.1
Level 804
Weight 2
Character 804.41
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 41.1
Root \(-0.888835 - 0.458227i\)
Character \(\chi\) = 804.41
Dual form 804.2.ba.a.353.1

$q$-expansion

\(f(q)\) \(=\) \(q+(0.936417 + 1.45709i) q^{3} +(-0.502989 - 5.26754i) q^{7} +(-1.24625 + 2.72890i) q^{9} +O(q^{10})\) \(q+(0.936417 + 1.45709i) q^{3} +(-0.502989 - 5.26754i) q^{7} +(-1.24625 + 2.72890i) q^{9} +(4.62817 + 4.85388i) q^{13} +(4.83520 + 0.461706i) q^{19} +(7.20430 - 5.66552i) q^{21} +(4.79746 + 1.40866i) q^{25} +(-5.14326 + 0.739490i) q^{27} +(7.68371 - 8.05844i) q^{31} +(3.82151 - 6.61906i) q^{37} +(-2.73867 + 11.2889i) q^{39} +(4.02392 + 3.48675i) q^{43} +(-20.6205 + 3.97428i) q^{49} +(3.85502 + 7.47769i) q^{57} +(-4.67299 + 11.6726i) q^{61} +(15.0014 + 5.19204i) q^{63} +(3.56930 + 7.36615i) q^{67} +(-15.2494 - 6.10492i) q^{73} +(2.43988 + 8.30945i) q^{75} +(-14.8646 + 3.60611i) q^{79} +(-5.89375 - 6.80175i) q^{81} +(23.2401 - 26.8205i) q^{91} +(18.9371 + 3.64982i) q^{93} +(-11.5396 - 6.66241i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{7} + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{7} + 6q^{9} + 9q^{13} - 8q^{19} + 6q^{21} + 10q^{25} - 3q^{31} + 10q^{37} - 9q^{39} - 5q^{49} + 141q^{57} + 27q^{61} + 147q^{63} + 11q^{67} - 180q^{73} - 166q^{79} - 18q^{81} - 36q^{91} - 3q^{93} + 33q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{53}{66}\right)\) \(1\)

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
\(3\) 0.936417 + 1.45709i 0.540641 + 0.841254i
\(4\) 0 0
\(5\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(6\) 0 0
\(7\) −0.502989 5.26754i −0.190112 1.99094i −0.0953288 0.995446i \(-0.530390\pi\)
−0.0947833 0.995498i \(-0.530216\pi\)
\(8\) 0 0
\(9\) −1.24625 + 2.72890i −0.415415 + 0.909632i
\(10\) 0 0
\(11\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(12\) 0 0
\(13\) 4.62817 + 4.85388i 1.28362 + 1.34622i 0.907726 + 0.419563i \(0.137817\pi\)
0.375896 + 0.926662i \(0.377335\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(18\) 0 0
\(19\) 4.83520 + 0.461706i 1.10927 + 0.105923i 0.633571 0.773685i \(-0.281589\pi\)
0.475700 + 0.879607i \(0.342195\pi\)
\(20\) 0 0
\(21\) 7.20430 5.66552i 1.57211 1.23632i
\(22\) 0 0
\(23\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(24\) 0 0
\(25\) 4.79746 + 1.40866i 0.959493 + 0.281733i
\(26\) 0 0
\(27\) −5.14326 + 0.739490i −0.989821 + 0.142315i
\(28\) 0 0
\(29\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) 7.68371 8.05844i 1.38003 1.44734i 0.680336 0.732900i \(-0.261834\pi\)
0.699699 0.714438i \(-0.253318\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.82151 6.61906i 0.628253 1.08817i −0.359649 0.933088i \(-0.617104\pi\)
0.987902 0.155079i \(-0.0495631\pi\)
\(38\) 0 0
\(39\) −2.73867 + 11.2889i −0.438537 + 1.80768i
\(40\) 0 0
\(41\) 0 0 0.189251 0.981929i \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(42\) 0 0
\(43\) 4.02392 + 3.48675i 0.613643 + 0.531724i 0.905286 0.424803i \(-0.139657\pi\)
−0.291643 + 0.956527i \(0.594202\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(48\) 0 0
\(49\) −20.6205 + 3.97428i −2.94579 + 0.567754i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.85502 + 7.47769i 0.510609 + 0.990444i
\(58\) 0 0
\(59\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(60\) 0 0
\(61\) −4.67299 + 11.6726i −0.598315 + 1.49452i 0.251725 + 0.967799i \(0.419002\pi\)
−0.850039 + 0.526719i \(0.823422\pi\)
\(62\) 0 0
\(63\) 15.0014 + 5.19204i 1.89000 + 0.654136i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.56930 + 7.36615i 0.436059 + 0.899918i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(72\) 0 0
\(73\) −15.2494 6.10492i −1.78480 0.714527i −0.994850 0.101361i \(-0.967680\pi\)
−0.789953 0.613167i \(-0.789895\pi\)
\(74\) 0 0
\(75\) 2.43988 + 8.30945i 0.281733 + 0.959493i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −14.8646 + 3.60611i −1.67240 + 0.405719i −0.956325 0.292306i \(-0.905577\pi\)
−0.716073 + 0.698026i \(0.754062\pi\)
\(80\) 0 0
\(81\) −5.89375 6.80175i −0.654861 0.755750i
\(82\) 0 0
\(83\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(90\) 0 0
\(91\) 23.2401 26.8205i 2.43623 2.81155i
\(92\) 0 0
\(93\) 18.9371 + 3.64982i 1.96368 + 0.378469i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −11.5396 6.66241i −1.17167 0.676466i −0.217599 0.976038i \(-0.569823\pi\)
−0.954074 + 0.299573i \(0.903156\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(102\) 0 0
\(103\) −3.73247 3.55890i −0.367771 0.350669i 0.483547 0.875318i \(-0.339348\pi\)
−0.851319 + 0.524649i \(0.824197\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(108\) 0 0
\(109\) 4.95120 16.8622i 0.474239 1.61511i −0.281032 0.959698i \(-0.590677\pi\)
0.755271 0.655412i \(-0.227505\pi\)
\(110\) 0 0
\(111\) 13.2231 0.629895i 1.25508 0.0597870i
\(112\) 0 0
\(113\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −19.0136 + 6.58066i −1.75780 + 0.608382i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.96107 + 7.59087i −0.723734 + 0.690079i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 14.7475 1.40821i 1.30863 0.124959i 0.582675 0.812705i \(-0.302006\pi\)
0.725951 + 0.687747i \(0.241400\pi\)
\(128\) 0 0
\(129\) −1.31245 + 9.12829i −0.115555 + 0.803701i
\(130\) 0 0
\(131\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(132\) 0 0
\(133\) 25.7019i 2.22863i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(138\) 0 0
\(139\) −16.6350 2.39175i −1.41096 0.202865i −0.605655 0.795727i \(-0.707089\pi\)
−0.805304 + 0.592862i \(0.797998\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −25.1003 26.3244i −2.07024 2.17120i
\(148\) 0 0
\(149\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(150\) 0 0
\(151\) −1.73309 5.00743i −0.141037 0.407499i 0.852256 0.523125i \(-0.175234\pi\)
−0.993293 + 0.115626i \(0.963113\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.17633 + 24.6942i 0.0938812 + 1.97081i 0.210387 + 0.977618i \(0.432527\pi\)
−0.116506 + 0.993190i \(0.537169\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −10.1493 17.5790i −0.794951 1.37690i −0.922870 0.385111i \(-0.874163\pi\)
0.127919 0.991785i \(-0.459170\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(168\) 0 0
\(169\) −1.52167 + 31.9439i −0.117052 + 2.45722i
\(170\) 0 0
\(171\) −7.28579 + 12.6194i −0.557158 + 0.965026i
\(172\) 0 0
\(173\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(174\) 0 0
\(175\) 5.00712 25.9794i 0.378502 1.96386i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(180\) 0 0
\(181\) 5.05145 + 2.60420i 0.375472 + 0.193569i 0.635624 0.771998i \(-0.280743\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) −21.3839 + 4.12141i −1.58074 + 0.304663i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 6.48230 + 26.7204i 0.471518 + 1.94362i
\(190\) 0 0
\(191\) 0 0 −0.458227 0.888835i \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(192\) 0 0
\(193\) 20.5467 6.03304i 1.47898 0.434268i 0.559974 0.828510i \(-0.310811\pi\)
0.919007 + 0.394242i \(0.128993\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.945001 0.327068i \(-0.893939\pi\)
0.945001 + 0.327068i \(0.106061\pi\)
\(198\) 0 0
\(199\) 16.3284 + 22.9300i 1.15749 + 1.62546i 0.633529 + 0.773719i \(0.281606\pi\)
0.523958 + 0.851744i \(0.324455\pi\)
\(200\) 0 0
\(201\) −7.39081 + 12.0986i −0.521308 + 0.853369i
\(202\) 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
\(211\) −2.14450 + 1.10557i −0.147634 + 0.0761104i −0.530460 0.847710i \(-0.677981\pi\)
0.382827 + 0.923820i \(0.374951\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −46.3130 36.4209i −3.14393 2.47241i
\(218\) 0 0
\(219\) −5.38432 27.9365i −0.363839 1.88777i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −9.85346 6.33243i −0.659836 0.424051i 0.167412 0.985887i \(-0.446459\pi\)
−0.827249 + 0.561836i \(0.810095\pi\)
\(224\) 0 0
\(225\) −9.82291 + 11.3362i −0.654861 + 0.755750i
\(226\) 0 0
\(227\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(228\) 0 0
\(229\) 0.135285 + 0.0328197i 0.00893986 + 0.00216879i 0.240226 0.970717i \(-0.422778\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −19.1739 18.2823i −1.24548 1.18756i
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) 1.27966 + 8.90023i 0.0824301 + 0.573314i 0.988619 + 0.150441i \(0.0480694\pi\)
−0.906189 + 0.422873i \(0.861022\pi\)
\(242\) 0 0
\(243\) 4.39178 14.9570i 0.281733 0.959493i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 20.1370 + 25.6063i 1.28129 + 1.62929i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(258\) 0 0
\(259\) −36.7883 16.8007i −2.28592 1.04394i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(264\) 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
\(271\) −17.7628 27.6394i −1.07901 1.67897i −0.593999 0.804466i \(-0.702452\pi\)
−0.485012 0.874508i \(-0.661185\pi\)
\(272\) 0 0
\(273\) 60.8424 + 8.74782i 3.68235 + 0.529442i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.62677 + 10.1312i −0.277996 + 0.608726i −0.996199 0.0871078i \(-0.972238\pi\)
0.718203 + 0.695833i \(0.244965\pi\)
\(278\) 0 0
\(279\) 12.4149 + 31.0108i 0.743258 + 1.85657i
\(280\) 0 0
\(281\) 0 0 −0.690079 0.723734i \(-0.742424\pi\)
0.690079 + 0.723734i \(0.257576\pi\)
\(282\) 0 0
\(283\) 2.38135 + 5.21442i 0.141556 + 0.309965i 0.967110 0.254358i \(-0.0818643\pi\)
−0.825554 + 0.564324i \(0.809137\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.3629 + 10.5087i −0.786053 + 0.618159i
\(290\) 0 0
\(291\) −1.09816 23.0531i −0.0643751 1.35140i
\(292\) 0 0
\(293\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 16.3426 22.9500i 0.941972 1.32282i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.31293 + 9.53404i −0.132006 + 0.544137i 0.866947 + 0.498400i \(0.166079\pi\)
−0.998954 + 0.0457370i \(0.985436\pi\)
\(308\) 0 0
\(309\) 1.69051 8.77118i 0.0961696 0.498975i
\(310\) 0 0
\(311\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(312\) 0 0
\(313\) −17.0433 + 26.5199i −0.963343 + 1.49899i −0.0996196 + 0.995026i \(0.531763\pi\)
−0.863724 + 0.503966i \(0.831874\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 15.3660 + 29.8059i 0.852351 + 1.65333i
\(326\) 0 0
\(327\) 29.2063 8.57574i 1.61511 0.474239i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.691583 0.239359i −0.0380128 0.0131564i 0.307995 0.951388i \(-0.400342\pi\)
−0.346008 + 0.938231i \(0.612463\pi\)
\(332\) 0 0
\(333\) 13.3002 + 18.6775i 0.728845 + 1.02352i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 29.5759 21.0609i 1.61110 1.14726i 0.719448 0.694547i \(-0.244395\pi\)
0.891656 0.452715i \(-0.149544\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.8710 + 71.0802i 1.12693 + 3.83797i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.971812 0.235759i \(-0.0757576\pi\)
−0.971812 + 0.235759i \(0.924242\pi\)
\(348\) 0 0
\(349\) −11.2829 13.0211i −0.603959 0.697006i 0.368620 0.929580i \(-0.379831\pi\)
−0.972579 + 0.232574i \(0.925285\pi\)
\(350\) 0 0
\(351\) −27.3933 21.5423i −1.46214 1.14984i
\(352\) 0 0
\(353\) 0 0 −0.189251 0.981929i \(-0.560606\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(360\) 0 0
\(361\) 4.50934 + 0.869104i 0.237334 + 0.0457423i
\(362\) 0 0
\(363\) −18.5155 4.49181i −0.971812 0.235759i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.46198 + 0.0696427i 0.0763147 + 0.00363532i 0.0857046 0.996321i \(-0.472686\pi\)
−0.00938982 + 0.999956i \(0.502989\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −30.2872 + 17.4863i −1.56821 + 0.905407i −0.571834 + 0.820370i \(0.693768\pi\)
−0.996378 + 0.0850377i \(0.972899\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 28.5873 1.36178i 1.46843 0.0699501i 0.701955 0.712221i \(-0.252311\pi\)
0.766477 + 0.642271i \(0.222008\pi\)
\(380\) 0 0
\(381\) 15.8617 + 20.1698i 0.812618 + 1.03333i
\(382\) 0 0
\(383\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −14.5298 + 6.63552i −0.738590 + 0.337303i
\(388\) 0 0
\(389\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −5.31106 + 36.9392i −0.266554 + 1.85393i 0.213832 + 0.976870i \(0.431405\pi\)
−0.480387 + 0.877057i \(0.659504\pi\)
\(398\) 0 0
\(399\) 37.4500 24.0677i 1.87485 1.20489i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 74.6762 3.71989
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −3.38792 35.4799i −0.167522 1.75437i −0.554414 0.832241i \(-0.687058\pi\)
0.386892 0.922125i \(-0.373549\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −12.0923 26.4784i −0.592161 1.29665i
\(418\) 0 0
\(419\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(420\) 0 0
\(421\) −38.6753 3.69304i −1.88492 0.179988i −0.911854 0.410515i \(-0.865349\pi\)
−0.973066 + 0.230527i \(0.925955\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 63.8362 + 18.7440i 3.08925 + 0.907085i
\(428\) 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
\(433\) 27.1352 28.4586i 1.30404 1.36763i 0.414982 0.909829i \(-0.363788\pi\)
0.889053 0.457804i \(-0.151364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 19.3101 33.4460i 0.921619 1.59629i 0.124708 0.992194i \(-0.460201\pi\)
0.796911 0.604097i \(-0.206466\pi\)
\(440\) 0 0
\(441\) 14.8528 61.2241i 0.707277 2.91543i
\(442\) 0 0
\(443\) 0 0 0.189251 0.981929i \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 5.67340 7.21432i 0.266560 0.338958i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.41533 + 5.83409i 0.0662066 + 0.272907i 0.995287 0.0969697i \(-0.0309150\pi\)
−0.929081 + 0.369877i \(0.879400\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(462\) 0 0
\(463\) 0.878313 2.19392i 0.0408187 0.101960i −0.906558 0.422082i \(-0.861300\pi\)
0.947376 + 0.320122i \(0.103724\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(468\) 0 0
\(469\) 37.0062 22.5065i 1.70879 1.03925i
\(470\) 0 0
\(471\) −34.8802 + 24.8381i −1.60719 + 1.14448i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 22.5463 + 9.02618i 1.03450 + 0.414150i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(480\) 0 0
\(481\) 49.8147 12.0849i 2.27136 0.551025i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 7.65465 + 39.7161i 0.346865 + 1.79971i 0.566429 + 0.824110i \(0.308325\pi\)
−0.219564 + 0.975598i \(0.570463\pi\)
\(488\) 0 0
\(489\) 16.1103 31.2497i 0.728535 1.41316i
\(490\) 0 0
\(491\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −33.7139 19.4647i −1.50924 0.871361i −0.999942 0.0107722i \(-0.996571\pi\)
−0.509300 0.860589i \(-0.670096\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −47.9701 + 27.6956i −2.13043 + 1.23000i
\(508\) 0 0
\(509\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(510\) 0 0
\(511\) −24.4877 + 83.3974i −1.08327 + 3.68928i
\(512\) 0 0
\(513\) −25.2101 + 1.20091i −1.11305 + 0.0530213i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(522\) 0 0
\(523\) −32.0521 + 30.5616i −1.40154 + 1.33636i −0.528312 + 0.849050i \(0.677175\pi\)
−0.873227 + 0.487314i \(0.837977\pi\)
\(524\) 0 0
\(525\) 42.5432 17.0317i 1.85674 0.743325i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −22.8959 + 2.18629i −0.995472 + 0.0950560i
\(530\) 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
\(541\) −45.2705 6.50892i −1.94633 0.279840i −0.947168 0.320739i \(-0.896069\pi\)
−0.999163 + 0.0408986i \(0.986978\pi\)
\(542\) 0 0
\(543\) 0.935697 + 9.79906i 0.0401546 + 0.420518i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −11.1012 27.7295i −0.474653 1.18563i −0.951777 0.306791i \(-0.900745\pi\)
0.477124 0.878836i \(-0.341679\pi\)
\(548\) 0 0
\(549\) −26.0295 27.2990i −1.11091 1.16509i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 26.4721 + 76.4860i 1.12571 + 3.25252i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(558\) 0 0
\(559\) 1.69912 + 35.6689i 0.0718651 + 1.50863i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −32.8640 + 34.4668i −1.38016 + 1.44747i
\(568\) 0 0
\(569\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(570\) 0 0
\(571\) −0.305195 + 6.40683i −0.0127720 + 0.268117i 0.983444 + 0.181210i \(0.0580014\pi\)
−0.996216 + 0.0869073i \(0.972302\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −7.44272 + 38.6165i −0.309844 + 1.60763i 0.409054 + 0.912510i \(0.365859\pi\)
−0.718898 + 0.695115i \(0.755353\pi\)
\(578\) 0 0
\(579\) 28.0310 + 24.2890i 1.16493 + 1.00941i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.618159 0.786053i \(-0.287879\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(588\) 0 0
\(589\) 40.8729 35.4166i 1.68414 1.45931i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.458227 0.888835i \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −18.1210 + 45.2640i −0.741642 + 1.85253i
\(598\) 0 0
\(599\) 0 0 −0.945001 0.327068i \(-0.893939\pi\)
0.945001 + 0.327068i \(0.106061\pi\)
\(600\) 0 0
\(601\) −5.03945 7.07691i −0.205563 0.288673i 0.698905 0.715215i \(-0.253671\pi\)
−0.904468 + 0.426542i \(0.859732\pi\)
\(602\) 0 0
\(603\) −24.5497 + 0.560221i −0.999740 + 0.0228140i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 8.20349 23.7024i 0.332970 0.962052i −0.646928 0.762551i \(-0.723947\pi\)
0.979898 0.199501i \(-0.0639322\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0.569823 0.293764i 0.0230149 0.0118650i −0.446681 0.894693i \(-0.647394\pi\)
0.469696 + 0.882828i \(0.344364\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(618\) 0 0
\(619\) −34.4842 27.1186i −1.38604 1.08999i −0.984738 0.174042i \(-0.944317\pi\)
−0.401297 0.915948i \(-0.631441\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 21.0313 + 13.5160i 0.841254 + 0.540641i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −3.29335 0.798959i −0.131106 0.0318061i 0.169669 0.985501i \(-0.445730\pi\)
−0.300775 + 0.953695i \(0.597245\pi\)
\(632\) 0 0
\(633\) −3.61906 2.08947i −0.143845 0.0830489i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −114.726 81.6959i −4.54560 3.23691i
\(638\) 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
\(643\) 0.374642 + 2.60569i 0.0147745 + 0.102759i 0.995874 0.0907437i \(-0.0289244\pi\)
−0.981100 + 0.193502i \(0.938015\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.998867 0.0475819i \(-0.0151515\pi\)
−0.998867 + 0.0475819i \(0.984848\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 9.70044 101.588i 0.380190 3.98153i
\(652\) 0 0
\(653\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 35.6642 34.0057i 1.39139 1.32669i
\(658\) 0 0
\(659\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(660\) 0 0
\(661\) 22.3588 + 10.2109i 0.869657 + 0.397159i 0.799706 0.600391i \(-0.204989\pi\)
0.0699507 + 0.997550i \(0.477716\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 20.2872i 0.784349i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 17.0015 + 26.4549i 0.655360 + 1.01976i 0.996806 + 0.0798590i \(0.0254470\pi\)
−0.341446 + 0.939901i \(0.610917\pi\)
\(674\) 0 0
\(675\) −25.7163 3.69745i −0.989821 0.142315i
\(676\) 0 0
\(677\) 0 0 −0.0950560 0.995472i \(-0.530303\pi\)
0.0950560 + 0.995472i \(0.469697\pi\)
\(678\) 0 0
\(679\) −29.2902 + 64.1367i −1.12406 + 2.46134i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.690079 0.723734i \(-0.742424\pi\)
0.690079 + 0.723734i \(0.257576\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0.0788615 + 0.227855i 0.00300875 + 0.00869323i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 5.47353 4.30443i 0.208223 0.163748i −0.508601 0.861002i \(-0.669837\pi\)
0.716824 + 0.697254i \(0.245595\pi\)
\(692\) 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 0.690079 0.723734i \(-0.257576\pi\)
−0.690079 + 0.723734i \(0.742424\pi\)
\(702\) 0 0
\(703\) 21.5338 30.2401i 0.812164 1.14053i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 10.5449 43.4665i 0.396020 1.63242i −0.329456 0.944171i \(-0.606865\pi\)
0.725477 0.688247i \(-0.241619\pi\)
\(710\) 0 0
\(711\) 8.68422 45.0580i 0.325684 1.68981i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(720\) 0 0
\(721\) −16.8693 + 21.4510i −0.628245 + 0.798879i
\(722\) 0 0
\(723\) −11.7702 + 10.1989i −0.437738 + 0.379302i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −10.2122 19.8090i −0.378751 0.734675i 0.620002 0.784600i \(-0.287132\pi\)
−0.998753 + 0.0499257i \(0.984102\pi\)
\(728\) 0 0
\(729\) 25.9063 7.60678i 0.959493 0.281733i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 51.1395 + 17.6996i 1.88888 + 0.653748i 0.955702 + 0.294336i \(0.0950985\pi\)
0.933179 + 0.359412i \(0.117023\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −44.0283 + 31.3524i −1.61961 + 1.15332i −0.747101 + 0.664710i \(0.768555\pi\)
−0.872505 + 0.488606i \(0.837506\pi\)
\(740\) 0 0
\(741\) −18.4542 + 53.3198i −0.677930 + 1.95875i
\(742\) 0 0
\(743\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −13.7277 15.8426i −0.500931 0.578105i 0.447822 0.894123i \(-0.352200\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 16.2030 31.4294i 0.588908 1.14232i −0.386577 0.922257i \(-0.626343\pi\)
0.975485 0.220065i \(-0.0706269\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(762\) 0 0
\(763\) −91.3130 17.5991i −3.30575 0.637132i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −14.9937 0.714238i −0.540687 0.0257561i −0.224537 0.974465i \(-0.572087\pi\)
−0.316150 + 0.948709i \(0.602390\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(774\) 0 0
\(775\) 48.2139 27.8363i 1.73190 0.999911i
\(776\) 0 0
\(777\) −9.96909 69.3365i −0.357639 2.48743i
\(778\) 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
\(787\) −11.7683 + 4.07306i −0.419495 + 0.145189i −0.528658 0.848835i \(-0.677304\pi\)
0.109162 + 0.994024i \(0.465183\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −78.2846 + 31.3404i −2.77997 + 1.11293i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0