Properties

Label 804.2.s.a.53.1
Level 804
Weight 2
Character 804.53
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.s (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{33})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label 53.1
Root \(0.981929 + 0.189251i\)
Character \(\chi\) = 804.53
Dual form 804.2.s.a.713.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.936417 + 1.45709i) q^{3} +(0.686312 - 0.313428i) q^{7} +(-1.24625 - 2.72890i) q^{9} +O(q^{10})\) \(q+(-0.936417 + 1.45709i) q^{3} +(0.686312 - 0.313428i) q^{7} +(-1.24625 - 2.72890i) q^{9} +(1.32339 + 4.50707i) q^{13} +(-3.57376 + 7.82544i) q^{19} +(-0.185980 + 1.29352i) q^{21} +(4.79746 - 1.40866i) q^{25} +(5.14326 + 0.739490i) q^{27} +(-0.698783 + 2.37984i) q^{31} -11.5468 q^{37} +(-7.80647 - 2.29219i) q^{39} +(-9.23919 + 8.00581i) q^{43} +(-4.21124 + 4.86003i) q^{49} +(-8.05587 - 12.5352i) q^{57} +(13.2610 + 1.90664i) q^{61} +(-1.71063 - 1.48227i) q^{63} +(-7.18797 + 3.91575i) q^{67} +(2.15444 - 14.9844i) q^{73} +(-2.43988 + 8.30945i) q^{75} +(4.99958 + 17.0270i) q^{79} +(-5.89375 + 6.80175i) q^{81} +(2.32090 + 2.67847i) q^{91} +(-2.81329 - 3.24671i) q^{93} -4.08778i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{9} + 16q^{19} - 12q^{21} + 10q^{25} - 20q^{37} - 24q^{39} + 10q^{49} - 66q^{57} - 132q^{63} - 16q^{67} + 90q^{73} + 44q^{79} - 18q^{81} + 48q^{91} - 36q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{19}{22}\right)\) \(1\)

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 \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(6\) 0 0
\(7\) 0.686312 0.313428i 0.259402 0.118465i −0.281470 0.959570i \(-0.590822\pi\)
0.540871 + 0.841105i \(0.318095\pi\)
\(8\) 0 0
\(9\) −1.24625 2.72890i −0.415415 0.909632i
\(10\) 0 0
\(11\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(12\) 0 0
\(13\) 1.32339 + 4.50707i 0.367044 + 1.25004i 0.911519 + 0.411259i \(0.134911\pi\)
−0.544475 + 0.838777i \(0.683271\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(18\) 0 0
\(19\) −3.57376 + 7.82544i −0.819876 + 1.79528i −0.262718 + 0.964873i \(0.584619\pi\)
−0.557158 + 0.830406i \(0.688108\pi\)
\(20\) 0 0
\(21\) −0.185980 + 1.29352i −0.0405842 + 0.282269i
\(22\) 0 0
\(23\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(24\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −0.698783 + 2.37984i −0.125505 + 0.427431i −0.998141 0.0609393i \(-0.980590\pi\)
0.872636 + 0.488371i \(0.162409\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −11.5468 −1.89829 −0.949144 0.314844i \(-0.898048\pi\)
−0.949144 + 0.314844i \(0.898048\pi\)
\(38\) 0 0
\(39\) −7.80647 2.29219i −1.25004 0.367044i
\(40\) 0 0
\(41\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(42\) 0 0
\(43\) −9.23919 + 8.00581i −1.40896 + 1.22087i −0.467397 + 0.884048i \(0.654808\pi\)
−0.941567 + 0.336826i \(0.890646\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(48\) 0 0
\(49\) −4.21124 + 4.86003i −0.601605 + 0.694290i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −8.05587 12.5352i −1.06703 1.66032i
\(58\) 0 0
\(59\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(60\) 0 0
\(61\) 13.2610 + 1.90664i 1.69789 + 0.244120i 0.922124 0.386895i \(-0.126452\pi\)
0.775771 + 0.631015i \(0.217361\pi\)
\(62\) 0 0
\(63\) −1.71063 1.48227i −0.215519 0.186748i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.18797 + 3.91575i −0.878150 + 0.478385i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(72\) 0 0
\(73\) 2.15444 14.9844i 0.252158 1.75379i −0.333048 0.942910i \(-0.608077\pi\)
0.585206 0.810885i \(-0.301014\pi\)
\(74\) 0 0
\(75\) −2.43988 + 8.30945i −0.281733 + 0.959493i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.99958 + 17.0270i 0.562497 + 1.91569i 0.337479 + 0.941333i \(0.390426\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −5.89375 + 6.80175i −0.654861 + 0.755750i
\(82\) 0 0
\(83\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(90\) 0 0
\(91\) 2.32090 + 2.67847i 0.243297 + 0.280780i
\(92\) 0 0
\(93\) −2.81329 3.24671i −0.291725 0.336668i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.08778i 0.415051i −0.978230 0.207526i \(-0.933459\pi\)
0.978230 0.207526i \(-0.0665410\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(102\) 0 0
\(103\) 10.9952 + 3.22847i 1.08339 + 0.318111i 0.774233 0.632901i \(-0.218136\pi\)
0.309154 + 0.951012i \(0.399954\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(108\) 0 0
\(109\) −1.39108 4.73759i −0.133242 0.453779i 0.865660 0.500632i \(-0.166899\pi\)
−0.998902 + 0.0468528i \(0.985081\pi\)
\(110\) 0 0
\(111\) 10.8127 16.8248i 1.02629 1.59694i
\(112\) 0 0
\(113\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 10.6500 9.22832i 0.984597 0.853158i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.5544 3.09906i 0.959493 0.281733i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.75548 + 14.7924i 0.599452 + 1.31262i 0.929559 + 0.368674i \(0.120188\pi\)
−0.330107 + 0.943944i \(0.607085\pi\)
\(128\) 0 0
\(129\) −3.01347 20.9591i −0.265321 1.84535i
\(130\) 0 0
\(131\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(132\) 0 0
\(133\) 6.49081i 0.562825i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(138\) 0 0
\(139\) 22.9848 3.30472i 1.94955 0.280303i 0.950053 0.312087i \(-0.101028\pi\)
0.999495 + 0.0317847i \(0.0101191\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.13804 10.6872i −0.258821 0.881464i
\(148\) 0 0
\(149\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(150\) 0 0
\(151\) −16.0907 18.5696i −1.30944 1.51118i −0.670352 0.742043i \(-0.733857\pi\)
−0.639089 0.769133i \(-0.720688\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −11.0125 7.07728i −0.878890 0.564828i 0.0215699 0.999767i \(-0.493134\pi\)
−0.900460 + 0.434939i \(0.856770\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 13.0871 1.02506 0.512529 0.858670i \(-0.328709\pi\)
0.512529 + 0.858670i \(0.328709\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(168\) 0 0
\(169\) −7.62599 + 4.90093i −0.586615 + 0.376995i
\(170\) 0 0
\(171\) 25.8086 1.97363
\(172\) 0 0
\(173\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(174\) 0 0
\(175\) 2.85104 2.47044i 0.215519 0.186748i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(180\) 0 0
\(181\) 15.2493 + 9.80015i 1.13347 + 0.728440i 0.966282 0.257485i \(-0.0828937\pi\)
0.167192 + 0.985924i \(0.446530\pi\)
\(182\) 0 0
\(183\) −15.1960 + 17.5371i −1.12332 + 1.29638i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3.76166 1.10452i 0.273621 0.0803422i
\(190\) 0 0
\(191\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(192\) 0 0
\(193\) 23.3902 + 6.86797i 1.68366 + 0.494367i 0.977010 0.213195i \(-0.0683868\pi\)
0.706651 + 0.707562i \(0.250205\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(198\) 0 0
\(199\) −8.70465 19.0605i −0.617056 1.35116i −0.917642 0.397409i \(-0.869909\pi\)
0.300586 0.953755i \(-0.402818\pi\)
\(200\) 0 0
\(201\) 1.02532 14.1403i 0.0723206 0.997381i
\(202\) 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
\(211\) 18.2761 11.7453i 1.25818 0.808583i 0.270146 0.962819i \(-0.412928\pi\)
0.988034 + 0.154237i \(0.0492918\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.266325 + 1.85233i 0.0180793 + 0.125744i
\(218\) 0 0
\(219\) 19.8163 + 17.1709i 1.33906 + 1.16030i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −24.5424 + 15.7724i −1.64348 + 1.05620i −0.705971 + 0.708241i \(0.749489\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 0 0
\(225\) −9.82291 11.3362i −0.654861 0.755750i
\(226\) 0 0
\(227\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(228\) 0 0
\(229\) 3.87229 13.1878i 0.255888 0.871476i −0.726900 0.686743i \(-0.759040\pi\)
0.982789 0.184733i \(-0.0591420\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −29.4916 8.65953i −1.91569 0.562497i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 0.800564 5.56804i 0.0515688 0.358669i −0.947656 0.319294i \(-0.896554\pi\)
0.999225 0.0393750i \(-0.0125367\pi\)
\(242\) 0 0
\(243\) −4.39178 14.9570i −0.281733 0.959493i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −39.9993 5.75103i −2.54509 0.365929i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(258\) 0 0
\(259\) −7.92473 + 3.61910i −0.492419 + 0.224880i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(264\) 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
\(271\) 17.6600 27.4795i 1.07277 1.66926i 0.431019 0.902343i \(-0.358154\pi\)
0.641748 0.766915i \(-0.278209\pi\)
\(272\) 0 0
\(273\) −6.07611 + 0.873613i −0.367743 + 0.0528735i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.547691 + 1.19928i 0.0329076 + 0.0720575i 0.925372 0.379060i \(-0.123753\pi\)
−0.892464 + 0.451118i \(0.851025\pi\)
\(278\) 0 0
\(279\) 7.36518 1.05895i 0.440942 0.0633979i
\(280\) 0 0
\(281\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(282\) 0 0
\(283\) −6.16501 + 13.4995i −0.366472 + 0.802461i 0.633125 + 0.774050i \(0.281772\pi\)
−0.999596 + 0.0284112i \(0.990955\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.41935 + 16.8270i −0.142315 + 0.989821i
\(290\) 0 0
\(291\) 5.95628 + 3.82787i 0.349163 + 0.224394i
\(292\) 0 0
\(293\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −3.83172 + 8.39030i −0.220857 + 0.483609i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.5542 + 3.68624i 0.716504 + 0.210385i 0.619610 0.784910i \(-0.287291\pi\)
0.0968945 + 0.995295i \(0.469109\pi\)
\(308\) 0 0
\(309\) −15.0003 + 12.9978i −0.853335 + 0.739419i
\(310\) 0 0
\(311\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(312\) 0 0
\(313\) −4.59522 7.15031i −0.259737 0.404159i 0.686753 0.726891i \(-0.259035\pi\)
−0.946491 + 0.322731i \(0.895399\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 12.6979 + 19.7583i 0.704352 + 1.09599i
\(326\) 0 0
\(327\) 8.20575 + 2.40943i 0.453779 + 0.133242i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.06287 1.78749i −0.113386 0.0982491i 0.596323 0.802745i \(-0.296628\pi\)
−0.709708 + 0.704496i \(0.751173\pi\)
\(332\) 0 0
\(333\) 14.3902 + 31.5101i 0.788577 + 1.72674i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3.38039 + 1.54377i −0.184142 + 0.0840947i −0.505352 0.862913i \(-0.668637\pi\)
0.321211 + 0.947008i \(0.395910\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −2.85491 + 9.72294i −0.154151 + 0.524989i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(348\) 0 0
\(349\) −16.8265 + 19.4188i −0.900702 + 1.03947i 0.0983163 + 0.995155i \(0.468654\pi\)
−0.999018 + 0.0443098i \(0.985891\pi\)
\(350\) 0 0
\(351\) 3.47364 + 24.1597i 0.185409 + 1.28955i
\(352\) 0 0
\(353\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(360\) 0 0
\(361\) −36.0234 41.5732i −1.89597 2.18806i
\(362\) 0 0
\(363\) −5.36773 + 18.2808i −0.281733 + 0.959493i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 5.07241 + 7.89282i 0.264777 + 0.412002i 0.948030 0.318180i \(-0.103072\pi\)
−0.683253 + 0.730182i \(0.739435\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 37.4441i 1.93878i −0.245523 0.969391i \(-0.578960\pi\)
0.245523 0.969391i \(-0.421040\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 12.4923 19.4384i 0.641687 0.998485i −0.356257 0.934388i \(-0.615947\pi\)
0.997944 0.0640964i \(-0.0204165\pi\)
\(380\) 0 0
\(381\) −27.8799 4.00853i −1.42833 0.205363i
\(382\) 0 0
\(383\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 33.3613 + 15.2356i 1.69585 + 0.774469i
\(388\) 0 0
\(389\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.40415 + 37.5867i 0.271227 + 1.88642i 0.435778 + 0.900054i \(0.356473\pi\)
−0.164551 + 0.986369i \(0.552617\pi\)
\(398\) 0 0
\(399\) −9.45772 6.07811i −0.473478 0.304286i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −11.6509 −0.580370
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 36.0080 16.4443i 1.78048 0.813117i 0.804918 0.593386i \(-0.202209\pi\)
0.975560 0.219732i \(-0.0705182\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −16.7081 + 36.5857i −0.818200 + 1.79161i
\(418\) 0 0
\(419\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(420\) 0 0
\(421\) −4.71469 + 10.3237i −0.229780 + 0.503148i −0.989041 0.147638i \(-0.952833\pi\)
0.759262 + 0.650785i \(0.225560\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 9.69877 2.84781i 0.469356 0.137815i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −11.3958 + 38.8104i −0.547645 + 1.86511i −0.0480569 + 0.998845i \(0.515303\pi\)
−0.499588 + 0.866263i \(0.666515\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −41.8980 −1.99968 −0.999842 0.0177654i \(-0.994345\pi\)
−0.999842 + 0.0177654i \(0.994345\pi\)
\(440\) 0 0
\(441\) 18.5107 + 5.43525i 0.881464 + 0.258821i
\(442\) 0 0
\(443\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 42.1253 6.05670i 1.97922 0.284569i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 40.2669 11.8234i 1.88361 0.553077i 0.887953 0.459935i \(-0.152127\pi\)
0.995653 0.0931420i \(-0.0296911\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(462\) 0 0
\(463\) −33.6758 4.84185i −1.56505 0.225020i −0.695374 0.718648i \(-0.744761\pi\)
−0.869673 + 0.493629i \(0.835670\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(468\) 0 0
\(469\) −3.70588 + 4.94034i −0.171122 + 0.228124i
\(470\) 0 0
\(471\) 20.6245 9.41890i 0.950328 0.434000i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −6.12158 + 42.5765i −0.280877 + 1.95354i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(480\) 0 0
\(481\) −15.2810 52.0424i −0.696754 2.37293i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −23.4165 20.2905i −1.06110 0.919451i −0.0641880 0.997938i \(-0.520446\pi\)
−0.996915 + 0.0784867i \(0.974991\pi\)
\(488\) 0 0
\(489\) −12.2550 + 19.0691i −0.554188 + 0.862334i
\(490\) 0 0
\(491\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 20.8986i 0.935549i 0.883848 + 0.467775i \(0.154944\pi\)
−0.883848 + 0.467775i \(0.845056\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 15.7011i 0.697311i
\(508\) 0 0
\(509\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(510\) 0 0
\(511\) −3.21793 10.9593i −0.142353 0.484809i
\(512\) 0 0
\(513\) −24.1676 + 37.6055i −1.06703 + 1.66032i
\(514\) 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 \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(522\) 0 0
\(523\) −36.1157 + 10.6045i −1.57923 + 0.463704i −0.949671 0.313249i \(-0.898583\pi\)
−0.629559 + 0.776952i \(0.716765\pi\)
\(524\) 0 0
\(525\) 0.929901 + 6.46760i 0.0405842 + 0.282269i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 9.55455 + 20.9215i 0.415415 + 0.909632i
\(530\) 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
\(541\) 38.9015 5.59319i 1.67250 0.240470i 0.760114 0.649790i \(-0.225143\pi\)
0.912391 + 0.409320i \(0.134234\pi\)
\(542\) 0 0
\(543\) −28.5595 + 13.0427i −1.22560 + 0.559715i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −14.2044 + 2.04228i −0.607336 + 0.0873217i −0.439120 0.898428i \(-0.644710\pi\)
−0.168216 + 0.985750i \(0.553801\pi\)
\(548\) 0 0
\(549\) −11.3234 38.5640i −0.483271 1.64587i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.76802 + 10.1188i 0.372854 + 0.430296i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(558\) 0 0
\(559\) −48.3098 31.0468i −2.04329 1.31314i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.91309 + 6.51539i −0.0803422 + 0.273621i
\(568\) 0 0
\(569\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(570\) 0 0
\(571\) 9.15855 5.88585i 0.383274 0.246315i −0.334790 0.942293i \(-0.608665\pi\)
0.718063 + 0.695978i \(0.245029\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 35.9009 31.1083i 1.49458 1.29506i 0.649568 0.760303i \(-0.274950\pi\)
0.845008 0.534754i \(-0.179596\pi\)
\(578\) 0 0
\(579\) −31.9102 + 27.6504i −1.32614 + 1.14911i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(588\) 0 0
\(589\) −16.1260 13.9732i −0.664460 0.575758i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 35.9241 + 5.16511i 1.47028 + 0.211394i
\(598\) 0 0
\(599\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(600\) 0 0
\(601\) 18.6298 + 40.7936i 0.759926 + 1.66401i 0.747660 + 0.664082i \(0.231177\pi\)
0.0122663 + 0.999925i \(0.496095\pi\)
\(602\) 0 0
\(603\) 19.6436 + 14.7352i 0.799951 + 0.600065i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 21.3848 24.6793i 0.867981 1.00170i −0.131964 0.991254i \(-0.542128\pi\)
0.999945 0.0104491i \(-0.00332612\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −25.3245 + 16.2751i −1.02285 + 0.657344i −0.940687 0.339275i \(-0.889818\pi\)
−0.0821600 + 0.996619i \(0.526182\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(618\) 0 0
\(619\) −4.71961 32.8256i −0.189697 1.31937i −0.832792 0.553587i \(-0.813259\pi\)
0.643094 0.765787i \(-0.277650\pi\)
\(620\) 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
\(631\) 12.4491 42.3976i 0.495589 1.68782i −0.208759 0.977967i \(-0.566942\pi\)
0.704348 0.709855i \(-0.251239\pi\)
\(632\) 0 0
\(633\) 37.6286i 1.49560i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −27.4776 12.5486i −1.08870 0.497194i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 7.08107 49.2499i 0.279250 1.94223i −0.0519076 0.998652i \(-0.516530\pi\)
0.331158 0.943575i \(-0.392561\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −2.94841 1.34649i −0.115557 0.0527732i
\(652\) 0 0
\(653\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −43.5759 + 12.7950i −1.70006 + 0.499182i
\(658\) 0 0
\(659\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(660\) 0 0
\(661\) −5.48821 + 2.50638i −0.213467 + 0.0974870i −0.519279 0.854605i \(-0.673799\pi\)
0.305812 + 0.952092i \(0.401072\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 50.5302i 1.95361i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −25.6908 + 39.9756i −0.990306 + 1.54095i −0.157398 + 0.987535i \(0.550311\pi\)
−0.832908 + 0.553411i \(0.813326\pi\)
\(674\) 0 0
\(675\) 25.7163 3.69745i 0.989821 0.142315i
\(676\) 0 0
\(677\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(678\) 0 0
\(679\) −1.28123 2.80549i −0.0491689 0.107665i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 15.5898 + 17.9916i 0.594788 + 0.686422i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −3.17578 + 22.0881i −0.120813 + 0.840269i 0.835827 + 0.548994i \(0.184989\pi\)
−0.956639 + 0.291276i \(0.905920\pi\)
\(692\) 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 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(702\) 0 0
\(703\) 41.2656 90.3590i 1.55636 3.40795i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 7.39971 + 2.17275i 0.277902 + 0.0815994i 0.417714 0.908579i \(-0.362832\pi\)
−0.139812 + 0.990178i \(0.544650\pi\)
\(710\) 0 0
\(711\) 40.2342 34.8632i 1.50890 1.30747i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(720\) 0 0
\(721\) 8.55801 1.23046i 0.318717 0.0458246i
\(722\) 0 0
\(723\) 7.36350 + 6.38051i 0.273851 + 0.237294i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 6.93991 + 10.7987i 0.257387 + 0.400502i 0.945766 0.324850i \(-0.105314\pi\)
−0.688379 + 0.725351i \(0.741677\pi\)
\(728\) 0 0
\(729\) 25.9063 + 7.60678i 0.959493 + 0.281733i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −27.8474 24.1299i −1.02857 0.891258i −0.0344317 0.999407i \(-0.510962\pi\)
−0.994135 + 0.108149i \(0.965508\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 41.9518 19.1588i 1.54322 0.704766i 0.551621 0.834095i \(-0.314010\pi\)
0.991602 + 0.129329i \(0.0412822\pi\)
\(740\) 0 0
\(741\) 45.8358 52.8973i 1.68382 1.94323i
\(742\) 0 0
\(743\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −30.8719 + 35.6281i −1.12653 + 1.30009i −0.177779 + 0.984070i \(0.556891\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 17.6450 27.4562i 0.641320 0.997913i −0.356651 0.934238i \(-0.616081\pi\)
0.997971 0.0636752i \(-0.0202822\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(762\) 0 0
\(763\) −2.43961 2.81546i −0.0883199 0.101927i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 28.4469 + 44.2643i 1.02582 + 1.59621i 0.778873 + 0.627182i \(0.215792\pi\)
0.246949 + 0.969028i \(0.420572\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(774\) 0 0
\(775\) 12.4015i 0.445476i
\(776\) 0 0
\(777\) 2.14748 14.9361i 0.0770405 0.535828i
\(778\) 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
\(787\) 14.4354 12.5084i 0.514567 0.445875i −0.358461 0.933545i \(-0.616698\pi\)
0.873028 + 0.487670i \(0.162153\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 8.95615 + 62.2914i 0.318042 + 2.21203i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0