Properties

Label 804.2.ba.a.701.1
Level 804
Weight 2
Character 804.701
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 701.1
Root \(-0.327068 + 0.945001i\)
Character \(\chi\) = 804.701
Dual form 804.2.ba.a.281.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.30900 - 1.13425i) q^{3} +(1.65416 + 4.13190i) q^{7} +(0.426945 - 2.96946i) q^{9} +O(q^{10})\) \(q+(1.30900 - 1.13425i) q^{3} +(1.65416 + 4.13190i) q^{7} +(0.426945 - 2.96946i) q^{9} +(-0.540736 + 5.66284i) q^{13} +(7.29407 + 2.92011i) q^{19} +(6.85191 + 3.53240i) q^{21} +(-2.07708 - 4.54816i) q^{25} +(-2.80925 - 4.37128i) q^{27} +(0.426797 + 4.46962i) q^{31} +(1.07398 - 1.86019i) q^{37} +(5.71527 + 8.02597i) q^{39} +(-3.53426 - 12.0366i) q^{43} +(-9.27019 + 8.83911i) q^{49} +(12.8601 - 4.45091i) q^{57} +(10.9516 - 0.521689i) q^{61} +(12.9758 - 3.14788i) q^{63} +(3.22957 - 7.52130i) q^{67} +(-0.120810 - 2.53611i) q^{73} +(-7.87764 - 3.59760i) q^{75} +(-13.7353 - 9.78084i) q^{79} +(-8.63544 - 2.53559i) q^{81} +(-24.2927 + 7.13299i) q^{91} +(5.62836 + 5.36663i) q^{93} +(14.0777 + 8.12776i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{7} + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{7} + 6q^{9} + 9q^{13} - 8q^{19} + 6q^{21} + 10q^{25} - 3q^{31} + 10q^{37} - 9q^{39} - 5q^{49} + 141q^{57} + 27q^{61} + 147q^{63} + 11q^{67} - 180q^{73} - 166q^{79} - 18q^{81} - 36q^{91} - 3q^{93} + 33q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{47}{66}\right)\) \(1\)

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
\(3\) 1.30900 1.13425i 0.755750 0.654861i
\(4\) 0 0
\(5\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(6\) 0 0
\(7\) 1.65416 + 4.13190i 0.625215 + 1.56171i 0.814735 + 0.579834i \(0.196882\pi\)
−0.189520 + 0.981877i \(0.560693\pi\)
\(8\) 0 0
\(9\) 0.426945 2.96946i 0.142315 0.989821i
\(10\) 0 0
\(11\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(12\) 0 0
\(13\) −0.540736 + 5.66284i −0.149973 + 1.57059i 0.536795 + 0.843712i \(0.319635\pi\)
−0.686768 + 0.726876i \(0.740971\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(18\) 0 0
\(19\) 7.29407 + 2.92011i 1.67338 + 0.669918i 0.997732 0.0673101i \(-0.0214417\pi\)
0.675643 + 0.737229i \(0.263866\pi\)
\(20\) 0 0
\(21\) 6.85191 + 3.53240i 1.49521 + 0.770834i
\(22\) 0 0
\(23\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(24\) 0 0
\(25\) −2.07708 4.54816i −0.415415 0.909632i
\(26\) 0 0
\(27\) −2.80925 4.37128i −0.540641 0.841254i
\(28\) 0 0
\(29\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) 0.426797 + 4.46962i 0.0766551 + 0.802768i 0.949291 + 0.314398i \(0.101803\pi\)
−0.872636 + 0.488371i \(0.837591\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.07398 1.86019i 0.176562 0.305814i −0.764139 0.645052i \(-0.776836\pi\)
0.940701 + 0.339238i \(0.110169\pi\)
\(38\) 0 0
\(39\) 5.71527 + 8.02597i 0.915175 + 1.28518i
\(40\) 0 0
\(41\) 0 0 0.690079 0.723734i \(-0.257576\pi\)
−0.690079 + 0.723734i \(0.742424\pi\)
\(42\) 0 0
\(43\) −3.53426 12.0366i −0.538970 1.83556i −0.549315 0.835616i \(-0.685111\pi\)
0.0103449 0.999946i \(-0.496707\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(48\) 0 0
\(49\) −9.27019 + 8.83911i −1.32431 + 1.26273i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 12.8601 4.45091i 1.70336 0.589537i
\(58\) 0 0
\(59\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(60\) 0 0
\(61\) 10.9516 0.521689i 1.40221 0.0667954i 0.667213 0.744867i \(-0.267487\pi\)
0.734996 + 0.678072i \(0.237184\pi\)
\(62\) 0 0
\(63\) 12.9758 3.14788i 1.63479 0.396596i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.22957 7.52130i 0.394555 0.918873i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(72\) 0 0
\(73\) −0.120810 2.53611i −0.0141397 0.296829i −0.994850 0.101361i \(-0.967680\pi\)
0.980710 0.195468i \(-0.0626226\pi\)
\(74\) 0 0
\(75\) −7.87764 3.59760i −0.909632 0.415415i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13.7353 9.78084i −1.54534 1.10043i −0.956325 0.292306i \(-0.905577\pi\)
−0.589013 0.808124i \(-0.700483\pi\)
\(80\) 0 0
\(81\) −8.63544 2.53559i −0.959493 0.281733i
\(82\) 0 0
\(83\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(90\) 0 0
\(91\) −24.2927 + 7.13299i −2.54657 + 0.747741i
\(92\) 0 0
\(93\) 5.62836 + 5.36663i 0.583633 + 0.556493i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0777 + 8.12776i 1.42937 + 0.825249i 0.997071 0.0764798i \(-0.0243681\pi\)
0.432302 + 0.901729i \(0.357701\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(102\) 0 0
\(103\) 0.579907 0.0553745i 0.0571400 0.00545621i −0.0664469 0.997790i \(-0.521166\pi\)
0.123587 + 0.992334i \(0.460560\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(108\) 0 0
\(109\) −15.9710 + 7.29371i −1.52974 + 0.698611i −0.989717 0.143037i \(-0.954313\pi\)
−0.540027 + 0.841648i \(0.681586\pi\)
\(110\) 0 0
\(111\) −0.704087 3.65315i −0.0668290 0.346742i
\(112\) 0 0
\(113\) 0 0 0.458227 0.888835i \(-0.348485\pi\)
−0.458227 + 0.888835i \(0.651515\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 16.5847 + 4.02341i 1.53326 + 0.371965i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.9502 + 1.04562i 0.995472 + 0.0950560i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2.91119 + 1.16547i −0.258326 + 0.103418i −0.497211 0.867630i \(-0.665643\pi\)
0.238885 + 0.971048i \(0.423218\pi\)
\(128\) 0 0
\(129\) −18.2789 11.7471i −1.60936 1.03428i
\(130\) 0 0
\(131\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(132\) 0 0
\(133\) 34.9687i 3.03217i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(138\) 0 0
\(139\) −7.55788 + 11.7603i −0.641051 + 0.997495i 0.356939 + 0.934128i \(0.383820\pi\)
−0.997990 + 0.0633672i \(0.979816\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2.10887 + 22.0851i −0.173937 + 1.82155i
\(148\) 0 0
\(149\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(150\) 0 0
\(151\) 5.66550 23.3535i 0.461052 1.90048i 0.0269113 0.999638i \(-0.491433\pi\)
0.434141 0.900845i \(-0.357052\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −22.1577 + 4.27054i −1.76837 + 0.340826i −0.966500 0.256666i \(-0.917376\pi\)
−0.801875 + 0.597492i \(0.796164\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −12.7256 22.0414i −0.996748 1.72642i −0.568161 0.822917i \(-0.692345\pi\)
−0.428587 0.903501i \(-0.640988\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(168\) 0 0
\(169\) −19.0103 3.66393i −1.46233 0.281841i
\(170\) 0 0
\(171\) 11.7853 20.4128i 0.901246 1.56100i
\(172\) 0 0
\(173\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(174\) 0 0
\(175\) 15.3567 16.1057i 1.16086 1.21747i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(180\) 0 0
\(181\) −8.77893 + 25.3651i −0.652533 + 1.88537i −0.260153 + 0.965567i \(0.583773\pi\)
−0.392380 + 0.919803i \(0.628348\pi\)
\(182\) 0 0
\(183\) 13.7439 13.1048i 1.01598 0.968732i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 13.4147 18.8384i 0.975778 1.37029i
\(190\) 0 0
\(191\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(192\) 0 0
\(193\) 6.46337 14.1528i 0.465243 1.01874i −0.521018 0.853546i \(-0.674447\pi\)
0.986261 0.165195i \(-0.0528253\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.971812 0.235759i \(-0.0757576\pi\)
−0.971812 + 0.235759i \(0.924242\pi\)
\(198\) 0 0
\(199\) −17.8081 14.0044i −1.26238 0.992748i −0.999611 0.0278890i \(-0.991121\pi\)
−0.262770 0.964858i \(-0.584636\pi\)
\(200\) 0 0
\(201\) −4.30355 13.5085i −0.303549 0.952816i
\(202\) 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
\(211\) 8.28554 + 23.9395i 0.570400 + 1.64806i 0.747800 + 0.663924i \(0.231110\pi\)
−0.177400 + 0.984139i \(0.556769\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −17.7620 + 9.15697i −1.20577 + 0.621615i
\(218\) 0 0
\(219\) −3.03472 3.18273i −0.205068 0.215069i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.7168 19.2922i 1.11944 1.29190i 0.167412 0.985887i \(-0.446459\pi\)
0.952027 0.306015i \(-0.0989957\pi\)
\(224\) 0 0
\(225\) −14.3924 + 4.22599i −0.959493 + 0.281733i
\(226\) 0 0
\(227\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(228\) 0 0
\(229\) −18.5574 + 13.2147i −1.22631 + 0.873252i −0.995024 0.0996338i \(-0.968233\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.189251 0.981929i \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −29.0733 + 2.77617i −1.88852 + 0.180331i
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) −13.9403 + 8.95889i −0.897974 + 0.577093i −0.906189 0.422873i \(-0.861022\pi\)
0.00821459 + 0.999966i \(0.497385\pi\)
\(242\) 0 0
\(243\) −14.1798 + 6.47568i −0.909632 + 0.415415i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −20.4803 + 39.7262i −1.30313 + 2.52772i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.971812 0.235759i \(-0.924242\pi\)
0.971812 + 0.235759i \(0.0757576\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(258\) 0 0
\(259\) 9.46267 + 1.36053i 0.587981 + 0.0845390i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(264\) 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
\(271\) 23.3703 20.2504i 1.41964 1.23013i 0.485012 0.874508i \(-0.338815\pi\)
0.934631 0.355619i \(-0.115730\pi\)
\(272\) 0 0
\(273\) −23.7085 + 36.8911i −1.43490 + 2.23275i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.74693 26.0605i 0.225131 1.56582i −0.493072 0.869989i \(-0.664126\pi\)
0.718203 0.695833i \(-0.244965\pi\)
\(278\) 0 0
\(279\) 13.4546 + 0.640922i 0.805506 + 0.0383710i
\(280\) 0 0
\(281\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(282\) 0 0
\(283\) −4.63073 32.2074i −0.275268 1.91453i −0.389404 0.921067i \(-0.627319\pi\)
0.114135 0.993465i \(-0.463590\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.1102 7.78985i −0.888835 0.458227i
\(290\) 0 0
\(291\) 27.6466 5.32844i 1.62067 0.312359i
\(292\) 0 0
\(293\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 43.8877 34.5137i 2.52965 1.98933i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5.76274 + 8.09264i 0.328897 + 0.461871i 0.945603 0.325322i \(-0.105473\pi\)
−0.616706 + 0.787193i \(0.711533\pi\)
\(308\) 0 0
\(309\) 0.696288 0.730246i 0.0396105 0.0415423i
\(310\) 0 0
\(311\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(312\) 0 0
\(313\) 19.4379 + 16.8430i 1.09869 + 0.952024i 0.999075 0.0430013i \(-0.0136920\pi\)
0.0996196 + 0.995026i \(0.468237\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 26.8786 9.30279i 1.49096 0.516026i
\(326\) 0 0
\(327\) −12.6331 + 27.6626i −0.698611 + 1.52974i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −34.8929 + 8.46493i −1.91789 + 0.465274i −0.920705 + 0.390259i \(0.872385\pi\)
−0.997182 + 0.0750153i \(0.976099\pi\)
\(332\) 0 0
\(333\) −5.06524 3.98335i −0.277574 0.218286i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 12.1518 15.4523i 0.661952 0.841740i −0.332913 0.942958i \(-0.608031\pi\)
0.994864 + 0.101218i \(0.0322738\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −23.5171 10.7399i −1.26980 0.579900i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(348\) 0 0
\(349\) 24.8239 + 7.28896i 1.32879 + 0.390169i 0.867659 0.497161i \(-0.165624\pi\)
0.461136 + 0.887330i \(0.347442\pi\)
\(350\) 0 0
\(351\) 26.2729 13.5446i 1.40234 0.722959i
\(352\) 0 0
\(353\) 0 0 −0.690079 0.723734i \(-0.742424\pi\)
0.690079 + 0.723734i \(0.257576\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(360\) 0 0
\(361\) 30.9256 + 29.4875i 1.62766 + 1.55197i
\(362\) 0 0
\(363\) 15.5198 11.0516i 0.814576 0.580057i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 6.47592 33.6003i 0.338040 1.75392i −0.272711 0.962096i \(-0.587920\pi\)
0.610751 0.791823i \(-0.290868\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 25.0593 14.4680i 1.29752 0.749124i 0.317546 0.948243i \(-0.397141\pi\)
0.979975 + 0.199119i \(0.0638080\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 6.60447 + 34.2673i 0.339249 + 1.76019i 0.605073 + 0.796170i \(0.293144\pi\)
−0.265824 + 0.964022i \(0.585644\pi\)
\(380\) 0 0
\(381\) −2.48881 + 4.82762i −0.127506 + 0.247326i
\(382\) 0 0
\(383\) 0 0 0.371662 0.928368i \(-0.378788\pi\)
−0.371662 + 0.928368i \(0.621212\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −37.2511 + 5.35591i −1.89358 + 0.272256i
\(388\) 0 0
\(389\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 32.7668 + 21.0579i 1.64452 + 1.05687i 0.936496 + 0.350679i \(0.114049\pi\)
0.708023 + 0.706189i \(0.249587\pi\)
\(398\) 0 0
\(399\) 39.6633 + 45.7739i 1.98565 + 2.29156i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −25.5415 −1.27231
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 11.9679 + 29.8945i 0.591777 + 1.47819i 0.857678 + 0.514186i \(0.171906\pi\)
−0.265902 + 0.964000i \(0.585670\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 3.44589 + 23.9667i 0.168746 + 1.17366i
\(418\) 0 0
\(419\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(420\) 0 0
\(421\) −14.3493 5.74460i −0.699342 0.279974i −0.00537983 0.999986i \(-0.501712\pi\)
−0.693963 + 0.720011i \(0.744137\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 20.2713 + 44.3879i 0.980996 + 2.14808i
\(428\) 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
\(433\) −1.46854 15.3793i −0.0705736 0.739080i −0.959627 0.281276i \(-0.909242\pi\)
0.889053 0.457804i \(-0.151364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 11.8484 20.5221i 0.565494 0.979465i −0.431509 0.902108i \(-0.642019\pi\)
0.997004 0.0773561i \(-0.0246478\pi\)
\(440\) 0 0
\(441\) 22.2896 + 31.3013i 1.06141 + 1.49054i
\(442\) 0 0
\(443\) 0 0 0.690079 0.723734i \(-0.257576\pi\)
−0.690079 + 0.723734i \(0.742424\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −19.0727 36.9958i −0.896112 1.73821i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.57665 5.02271i 0.167309 0.234952i −0.722406 0.691469i \(-0.756964\pi\)
0.889715 + 0.456517i \(0.150903\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(462\) 0 0
\(463\) −14.3572 + 0.683919i −0.667237 + 0.0317844i −0.378468 0.925614i \(-0.623549\pi\)
−0.288769 + 0.957399i \(0.593246\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(468\) 0 0
\(469\) 36.4195 + 0.902802i 1.68169 + 0.0416875i
\(470\) 0 0
\(471\) −24.1604 + 30.7225i −1.11325 + 1.41562i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1.86923 39.2399i −0.0857660 1.80045i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(480\) 0 0
\(481\) 9.95323 + 7.08766i 0.453828 + 0.323170i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 30.0712 + 31.5377i 1.36265 + 1.42911i 0.796225 + 0.605000i \(0.206827\pi\)
0.566429 + 0.824110i \(0.308325\pi\)
\(488\) 0 0
\(489\) −41.6583 14.4181i −1.88386 0.652009i
\(490\) 0 0
\(491\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 36.4247 + 21.0298i 1.63059 + 0.941423i 0.983910 + 0.178663i \(0.0571771\pi\)
0.646682 + 0.762760i \(0.276156\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −29.0402 + 16.7664i −1.28972 + 0.744621i
\(508\) 0 0
\(509\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(510\) 0 0
\(511\) 10.2791 4.69430i 0.454720 0.207664i
\(512\) 0 0
\(513\) −7.72628 40.0878i −0.341124 1.76992i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(522\) 0 0
\(523\) −45.5065 4.34534i −1.98986 0.190009i −0.998067 0.0621473i \(-0.980205\pi\)
−0.991792 0.127861i \(-0.959189\pi\)
\(524\) 0 0
\(525\) 1.83401 38.5006i 0.0800428 1.68031i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 21.3525 8.54824i 0.928368 0.371662i
\(530\) 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
\(541\) −23.2854 + 36.2327i −1.00112 + 1.55777i −0.182678 + 0.983173i \(0.558477\pi\)
−0.818438 + 0.574595i \(0.805160\pi\)
\(542\) 0 0
\(543\) 17.2788 + 43.1603i 0.741504 + 1.85219i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −44.5388 2.12164i −1.90434 0.0907150i −0.937793 0.347196i \(-0.887134\pi\)
−0.966549 + 0.256481i \(0.917437\pi\)
\(548\) 0 0
\(549\) 3.12659 32.7431i 0.133440 1.39744i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 17.6931 72.9318i 0.752385 3.10138i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(558\) 0 0
\(559\) 70.0723 13.5053i 2.96374 0.571215i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −3.80760 39.8750i −0.159904 1.67459i
\(568\) 0 0
\(569\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(570\) 0 0
\(571\) 46.9260 + 9.04424i 1.96379 + 0.378490i 0.983444 + 0.181210i \(0.0580014\pi\)
0.980347 + 0.197280i \(0.0632107\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 6.04672 6.34161i 0.251728 0.264005i −0.585730 0.810506i \(-0.699192\pi\)
0.837458 + 0.546501i \(0.184041\pi\)
\(578\) 0 0
\(579\) −7.59232 25.8571i −0.315526 1.07458i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.458227 0.888835i \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(588\) 0 0
\(589\) −9.93869 + 33.8481i −0.409517 + 1.39469i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −39.1953 + 1.86710i −1.60416 + 0.0764153i
\(598\) 0 0
\(599\) 0 0 0.971812 0.235759i \(-0.0757576\pi\)
−0.971812 + 0.235759i \(0.924242\pi\)
\(600\) 0 0
\(601\) −31.6662 24.9025i −1.29169 1.01580i −0.998209 0.0598308i \(-0.980944\pi\)
−0.293481 0.955965i \(-0.594814\pi\)
\(602\) 0 0
\(603\) −20.9554 12.8013i −0.853369 0.521308i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 2.85662 + 11.7751i 0.115947 + 0.477938i 0.999945 + 0.0104491i \(0.00332612\pi\)
−0.883999 + 0.467489i \(0.845159\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −8.57890 24.7871i −0.346498 1.00114i −0.974782 0.223157i \(-0.928364\pi\)
0.628284 0.777984i \(-0.283758\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(618\) 0 0
\(619\) −35.1842 + 18.1387i −1.41417 + 0.729056i −0.984738 0.174042i \(-0.944317\pi\)
−0.429433 + 0.903099i \(0.641287\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −16.3715 + 18.8937i −0.654861 + 0.755750i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 40.0222 28.4997i 1.59326 1.13455i 0.675537 0.737326i \(-0.263912\pi\)
0.917720 0.397227i \(-0.130027\pi\)
\(632\) 0 0
\(633\) 37.9992 + 21.9388i 1.51033 + 0.871990i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −45.0417 57.2752i −1.78462 2.26933i
\(638\) 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
\(643\) −24.8979 + 16.0009i −0.981877 + 0.631014i −0.929969 0.367638i \(-0.880167\pi\)
−0.0519076 + 0.998652i \(0.516530\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.189251 0.981929i \(-0.560606\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −12.8641 + 32.1331i −0.504185 + 1.25939i
\(652\) 0 0
\(653\) 0 0 −0.971812 0.235759i \(-0.924242\pi\)
0.971812 + 0.235759i \(0.0757576\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −7.58245 0.724037i −0.295820 0.0282474i
\(658\) 0 0
\(659\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(660\) 0 0
\(661\) −44.6368 6.41780i −1.73617 0.249623i −0.799706 0.600391i \(-0.795011\pi\)
−0.936462 + 0.350768i \(0.885921\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 44.2145i 1.70943i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 38.2430 33.1377i 1.47416 1.27736i 0.592416 0.805632i \(-0.298174\pi\)
0.881742 0.471732i \(-0.156371\pi\)
\(674\) 0 0
\(675\) −14.0463 + 21.8564i −0.540641 + 0.841254i
\(676\) 0 0
\(677\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(678\) 0 0
\(679\) −10.2963 + 71.6122i −0.395135 + 2.74822i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −9.30284 + 38.3468i −0.354925 + 1.46302i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −39.0577 20.1356i −1.48582 0.765996i −0.491349 0.870963i \(-0.663496\pi\)
−0.994476 + 0.104967i \(0.966526\pi\)
\(692\) 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 −0.0950560 0.995472i \(-0.530303\pi\)
0.0950560 + 0.995472i \(0.469697\pi\)
\(702\) 0 0
\(703\) 13.2657 10.4322i 0.500324 0.393459i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 12.9033 + 18.1202i 0.484595 + 0.680519i 0.982404 0.186768i \(-0.0598012\pi\)
−0.497809 + 0.867287i \(0.665862\pi\)
\(710\) 0 0
\(711\) −34.9080 + 36.6105i −1.30915 + 1.37300i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(720\) 0 0
\(721\) 1.18806 + 2.30452i 0.0442458 + 0.0858248i
\(722\) 0 0
\(723\) −8.08618 + 27.5390i −0.300728 + 1.02419i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −7.37071 + 2.55103i −0.273364 + 0.0946123i −0.460311 0.887758i \(-0.652262\pi\)
0.186947 + 0.982370i \(0.440141\pi\)
\(728\) 0 0
\(729\) −11.2162 + 24.5601i −0.415415 + 0.909632i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −50.9706 + 12.3653i −1.88264 + 0.456724i −0.999913 0.0131610i \(-0.995811\pi\)
−0.882728 + 0.469885i \(0.844295\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 30.0719 38.2396i 1.10621 1.40667i 0.202105 0.979364i \(-0.435222\pi\)
0.904109 0.427302i \(-0.140536\pi\)
\(740\) 0 0
\(741\) 18.2509 + 75.2312i 0.670463 + 2.76369i
\(742\) 0 0
\(743\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −43.1905 12.6819i −1.57604 0.462768i −0.627290 0.778785i \(-0.715836\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22.3615 + 7.73939i 0.812743 + 0.281293i 0.701654 0.712517i \(-0.252445\pi\)
0.111088 + 0.993811i \(0.464566\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(762\) 0 0
\(763\) −56.5555 53.9255i −2.04745 1.95224i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 5.77611 29.9693i 0.208292 1.08072i −0.715729 0.698378i \(-0.753905\pi\)
0.924021 0.382343i \(-0.124883\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(774\) 0 0
\(775\) 19.4421 11.2249i 0.698380 0.403210i
\(776\) 0 0
\(777\) 13.9298 8.95212i 0.499728 0.321156i
\(778\) 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
\(787\) 48.1050 + 11.6702i 1.71476 + 0.415996i 0.968371 0.249515i \(-0.0802713\pi\)
0.746388 + 0.665511i \(0.231786\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.96768 + 62.2992i −0.105385 + 2.21231i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0