Properties

Label 804.2.ba.a.653.1
Level 804
Weight 2
Character 804.653
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 653.1
Root \(0.0475819 + 0.998867i\)
Character \(\chi\) = 804.653
Dual form 804.2.ba.a.197.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.936417 - 1.45709i) q^{3} +(-0.614593 + 0.437650i) q^{7} +(-1.24625 + 2.72890i) q^{9} +O(q^{10})\) \(q+(-0.936417 - 1.45709i) q^{3} +(-0.614593 + 0.437650i) q^{7} +(-1.24625 + 2.72890i) q^{9} +(-2.32224 + 0.563370i) q^{13} +(-4.99015 + 7.00769i) q^{19} +(1.21321 + 0.485697i) q^{21} +(4.79746 + 1.40866i) q^{25} +(5.14326 - 0.739490i) q^{27} +(-10.3416 - 2.50884i) q^{31} +(5.77342 + 9.99985i) q^{37} +(2.99547 + 2.85617i) q^{39} +(1.51198 + 1.31014i) q^{43} +(-2.10329 + 6.07705i) q^{49} +(14.8837 + 0.708999i) q^{57} +(8.44064 + 10.7332i) q^{61} +(-0.428367 - 2.22258i) q^{63} +(6.98512 - 4.26709i) q^{67} +(-11.3456 + 8.92225i) q^{73} +(-2.43988 - 8.30945i) q^{75} +(-5.50042 - 5.76867i) q^{79} +(-5.89375 - 6.80175i) q^{81} +(1.18067 - 1.36257i) q^{91} +(6.02841 + 17.4180i) q^{93} +(-16.2217 + 9.36562i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{7} + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{7} + 6q^{9} + 9q^{13} - 8q^{19} + 6q^{21} + 10q^{25} - 3q^{31} + 10q^{37} - 9q^{39} - 5q^{49} + 141q^{57} + 27q^{61} + 147q^{63} + 11q^{67} - 180q^{73} - 166q^{79} - 18q^{81} - 36q^{91} - 3q^{93} + 33q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{31}{66}\right)\) \(1\)

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
\(7\) −0.614593 + 0.437650i −0.232294 + 0.165416i −0.690277 0.723545i \(-0.742511\pi\)
0.457983 + 0.888961i \(0.348572\pi\)
\(8\) 0 0
\(9\) −1.24625 + 2.72890i −0.415415 + 0.909632i
\(10\) 0 0
\(11\) 0 0 0.618159 0.786053i \(-0.287879\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(12\) 0 0
\(13\) −2.32224 + 0.563370i −0.644074 + 0.156251i −0.544475 0.838777i \(-0.683271\pi\)
−0.0995988 + 0.995028i \(0.531756\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(18\) 0 0
\(19\) −4.99015 + 7.00769i −1.14482 + 1.60767i −0.440574 + 0.897716i \(0.645225\pi\)
−0.704245 + 0.709957i \(0.748714\pi\)
\(20\) 0 0
\(21\) 1.21321 + 0.485697i 0.264745 + 0.105988i
\(22\) 0 0
\(23\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(24\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) −10.3416 2.50884i −1.85740 0.450601i −0.859260 0.511540i \(-0.829075\pi\)
−0.998141 + 0.0609393i \(0.980590\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.77342 + 9.99985i 0.949144 + 1.64396i 0.747234 + 0.664561i \(0.231381\pi\)
0.201909 + 0.979404i \(0.435285\pi\)
\(38\) 0 0
\(39\) 2.99547 + 2.85617i 0.479659 + 0.457354i
\(40\) 0 0
\(41\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(42\) 0 0
\(43\) 1.51198 + 1.31014i 0.230574 + 0.199794i 0.762484 0.647008i \(-0.223980\pi\)
−0.531909 + 0.846801i \(0.678525\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(48\) 0 0
\(49\) −2.10329 + 6.07705i −0.300470 + 0.868150i
\(50\) 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
\(57\) 14.8837 + 0.708999i 1.97140 + 0.0939092i
\(58\) 0 0
\(59\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(60\) 0 0
\(61\) 8.44064 + 10.7332i 1.08071 + 1.37424i 0.922124 + 0.386895i \(0.126452\pi\)
0.158590 + 0.987345i \(0.449305\pi\)
\(62\) 0 0
\(63\) −0.428367 2.22258i −0.0539692 0.280019i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.98512 4.26709i 0.853369 0.521308i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(72\) 0 0
\(73\) −11.3456 + 8.92225i −1.32790 + 1.04427i −0.333048 + 0.942910i \(0.608077\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) −2.43988 8.30945i −0.281733 0.959493i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.50042 5.76867i −0.618846 0.649027i 0.337479 0.941333i \(-0.390426\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) −5.89375 6.80175i −0.654861 0.755750i
\(82\) 0 0
\(83\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(84\) 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
\(91\) 1.18067 1.36257i 0.123768 0.142836i
\(92\) 0 0
\(93\) 6.02841 + 17.4180i 0.625117 + 1.80616i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −16.2217 + 9.36562i −1.64707 + 0.950934i −0.668837 + 0.743409i \(0.733208\pi\)
−0.978230 + 0.207526i \(0.933459\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(102\) 0 0
\(103\) −2.70165 + 11.1363i −0.266201 + 1.09730i 0.669024 + 0.743241i \(0.266712\pi\)
−0.935225 + 0.354055i \(0.884803\pi\)
\(104\) 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
\(109\) 5.64566 19.2274i 0.540757 1.84165i 0.000729916 1.00000i \(-0.499768\pi\)
0.540027 0.841648i \(-0.318414\pi\)
\(110\) 0 0
\(111\) 9.16439 17.7764i 0.869846 1.68726i
\(112\) 0 0
\(113\) 0 0 0.371662 0.928368i \(-0.378788\pi\)
−0.371662 + 0.928368i \(0.621212\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.35670 7.03925i 0.125427 0.650779i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.59335 10.6899i −0.235759 0.971812i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −12.5560 17.6325i −1.11417 1.56463i −0.784060 0.620685i \(-0.786855\pi\)
−0.330107 0.943944i \(-0.607085\pi\)
\(128\) 0 0
\(129\) 0.493149 3.42993i 0.0434193 0.301988i
\(130\) 0 0
\(131\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(132\) 0 0
\(133\) 6.49081i 0.562825i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(138\) 0 0
\(139\) −7.98138 1.14755i −0.676972 0.0973339i −0.204751 0.978814i \(-0.565638\pi\)
−0.472221 + 0.881480i \(0.656548\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 10.8244 2.62597i 0.892781 0.216586i
\(148\) 0 0
\(149\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(150\) 0 0
\(151\) −11.6313 2.24175i −0.946541 0.182431i −0.307452 0.951564i \(-0.599476\pi\)
−0.639089 + 0.769133i \(0.720688\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.6353 + 5.99843i 0.928601 + 0.478727i 0.855039 0.518564i \(-0.173533\pi\)
0.0735617 + 0.997291i \(0.476563\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −6.22226 + 10.7773i −0.487365 + 0.844142i −0.999894 0.0145283i \(-0.995375\pi\)
0.512529 + 0.858670i \(0.328709\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(168\) 0 0
\(169\) −6.47944 + 3.34038i −0.498419 + 0.256953i
\(170\) 0 0
\(171\) −12.9043 22.3509i −0.986816 1.70922i
\(172\) 0 0
\(173\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(174\) 0 0
\(175\) −3.56499 + 1.23385i −0.269488 + 0.0932706i
\(176\) 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
\(181\) −1.25066 26.2546i −0.0929610 1.95149i −0.260153 0.965567i \(-0.583773\pi\)
0.167192 0.985924i \(-0.446530\pi\)
\(182\) 0 0
\(183\) 7.73524 22.3495i 0.571806 1.65212i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −2.83738 + 2.70543i −0.206389 + 0.196791i
\(190\) 0 0
\(191\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(192\) 0 0
\(193\) −0.617267 + 0.181246i −0.0444319 + 0.0130464i −0.303873 0.952713i \(-0.598280\pi\)
0.259441 + 0.965759i \(0.416462\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.189251 0.981929i \(-0.560606\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(198\) 0 0
\(199\) 5.85768 0.559341i 0.415240 0.0396506i 0.114654 0.993405i \(-0.463424\pi\)
0.300586 + 0.953755i \(0.402818\pi\)
\(200\) 0 0
\(201\) −12.7585 6.18221i −0.899918 0.436059i
\(202\) 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
\(211\) 0.277952 5.83493i 0.0191350 0.401693i −0.968899 0.247457i \(-0.920405\pi\)
0.988034 0.154237i \(-0.0492918\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.45385 2.98407i 0.506000 0.202572i
\(218\) 0 0
\(219\) 23.6247 + 8.17660i 1.59641 + 0.552524i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.9305 + 10.8806i 1.13375 + 0.728618i 0.966340 0.257269i \(-0.0828225\pi\)
0.167412 + 0.985887i \(0.446459\pi\)
\(224\) 0 0
\(225\) −9.82291 + 11.3362i −0.654861 + 0.755750i
\(226\) 0 0
\(227\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(228\) 0 0
\(229\) 11.3723 11.9269i 0.751502 0.788153i −0.231287 0.972886i \(-0.574293\pi\)
0.982789 + 0.184733i \(0.0591420\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.458227 0.888835i \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −3.25481 + 13.4165i −0.211423 + 0.871496i
\(238\) 0 0
\(239\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) 0 0
\(241\) 3.36304 + 23.3905i 0.216633 + 1.50671i 0.750345 + 0.661047i \(0.229888\pi\)
−0.533712 + 0.845666i \(0.679203\pi\)
\(242\) 0 0
\(243\) −4.39178 + 14.9570i −0.281733 + 0.959493i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 7.64042 19.0848i 0.486148 1.21434i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.189251 0.981929i \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(258\) 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 \(-0.454545\pi\)
−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
\(271\) −10.7607 16.7440i −0.653667 1.01713i −0.996961 0.0778984i \(-0.975179\pi\)
0.343294 0.939228i \(-0.388457\pi\)
\(272\) 0 0
\(273\) −3.09100 0.444418i −0.187076 0.0268974i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −12.2396 + 26.8011i −0.735409 + 1.61032i 0.0555527 + 0.998456i \(0.482308\pi\)
−0.790962 + 0.611866i \(0.790419\pi\)
\(278\) 0 0
\(279\) 19.7345 25.0944i 1.18147 1.50237i
\(280\) 0 0
\(281\) 0 0 0.971812 0.235759i \(-0.0757576\pi\)
−0.971812 + 0.235759i \(0.924242\pi\)
\(282\) 0 0
\(283\) −6.16501 13.4995i −0.366472 0.802461i −0.999596 0.0284112i \(-0.990955\pi\)
0.633125 0.774050i \(-0.281772\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 15.7823 + 6.31826i 0.928368 + 0.371662i
\(290\) 0 0
\(291\) 28.8369 + 14.8665i 1.69045 + 0.871487i
\(292\) 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
\(301\) −1.50263 0.143484i −0.0866102 0.00827027i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 25.1102 + 23.9425i 1.43311 + 1.36647i 0.813503 + 0.581560i \(0.197558\pi\)
0.619610 + 0.784910i \(0.287291\pi\)
\(308\) 0 0
\(309\) 18.7566 6.49171i 1.06702 0.369300i
\(310\) 0 0
\(311\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(312\) 0 0
\(313\) −4.59522 + 7.15031i −0.259737 + 0.404159i −0.946491 0.322731i \(-0.895399\pi\)
0.686753 + 0.726891i \(0.259035\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −11.9345 0.568509i −0.662005 0.0315352i
\(326\) 0 0
\(327\) −33.3028 + 9.77858i −1.84165 + 0.540757i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6.20515 + 32.1954i 0.341066 + 1.76962i 0.596323 + 0.802745i \(0.296628\pi\)
−0.255257 + 0.966873i \(0.582160\pi\)
\(332\) 0 0
\(333\) −34.4836 + 3.29279i −1.88969 + 0.180444i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3.18353 33.3394i −0.173418 1.81611i −0.494628 0.869105i \(-0.664696\pi\)
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 −0.690079 0.723734i \(-0.742424\pi\)
0.690079 + 0.723734i \(0.257576\pi\)
\(348\) 0 0
\(349\) 23.7967 + 27.4628i 1.27381 + 1.47005i 0.812672 + 0.582722i \(0.198012\pi\)
0.461136 + 0.887330i \(0.347442\pi\)
\(350\) 0 0
\(351\) −11.5273 + 4.61483i −0.615281 + 0.246321i
\(352\) 0 0
\(353\) 0 0 −0.945001 0.327068i \(-0.893939\pi\)
0.945001 + 0.327068i \(0.106061\pi\)
\(354\) 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
\(361\) −17.9918 51.9838i −0.946935 2.73599i
\(362\) 0 0
\(363\) −13.1478 + 13.7890i −0.690079 + 0.723734i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.29918 + 8.33924i 0.224415 + 0.435305i 0.973982 0.226623i \(-0.0727687\pi\)
−0.749567 + 0.661928i \(0.769738\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 9.10103 + 5.25448i 0.471233 + 0.272067i 0.716756 0.697324i \(-0.245626\pi\)
−0.245523 + 0.969391i \(0.578960\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −17.7302 + 34.3917i −0.910738 + 1.76658i −0.356257 + 0.934388i \(0.615947\pi\)
−0.554481 + 0.832196i \(0.687083\pi\)
\(380\) 0 0
\(381\) −13.9345 + 34.8067i −0.713885 + 1.78320i
\(382\) 0 0
\(383\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.45952 + 2.49328i −0.277523 + 0.126740i
\(388\) 0 0
\(389\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.21269 + 8.43443i −0.0608631 + 0.423312i 0.936496 + 0.350679i \(0.114049\pi\)
−0.997359 + 0.0726328i \(0.976860\pi\)
\(398\) 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
\(403\) 25.4290 1.26671
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −32.2451 + 22.9617i −1.59442 + 1.13538i −0.678073 + 0.734994i \(0.737185\pi\)
−0.916346 + 0.400387i \(0.868876\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.80182 + 12.7042i 0.284116 + 0.622128i
\(418\) 0 0
\(419\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(420\) 0 0
\(421\) 23.1020 32.4423i 1.12592 1.58114i 0.366663 0.930354i \(-0.380500\pi\)
0.759262 0.650785i \(-0.225560\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −9.88492 2.90247i −0.478365 0.140461i
\(428\) 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
\(433\) 39.3086 + 9.53618i 1.88905 + 0.458279i 1.00000 0.000475561i \(0.000151376\pi\)
0.889053 + 0.457804i \(0.151364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −10.7969 18.7007i −0.515306 0.892537i −0.999842 0.0177654i \(-0.994345\pi\)
0.484536 0.874771i \(-0.338989\pi\)
\(440\) 0 0
\(441\) −13.9624 13.3132i −0.664878 0.633960i
\(442\) 0 0
\(443\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 7.62530 + 19.0471i 0.358268 + 0.894910i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 20.3085 19.3641i 0.949991 0.905815i −0.0456614 0.998957i \(-0.514540\pi\)
0.995653 + 0.0931420i \(0.0296911\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(462\) 0 0
\(463\) 21.0311 + 26.7432i 0.977396 + 1.24286i 0.970055 + 0.242887i \(0.0780944\pi\)
0.00734134 + 0.999973i \(0.497663\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(468\) 0 0
\(469\) −2.42552 + 5.67956i −0.112000 + 0.262258i
\(470\) 0 0
\(471\) −2.15525 22.5708i −0.0993087 1.04001i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −33.8115 + 26.5897i −1.55138 + 1.22002i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(480\) 0 0
\(481\) −19.0409 19.9695i −0.868189 0.910530i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 11.0835 + 3.83603i 0.502241 + 0.173827i 0.566429 0.824110i \(-0.308325\pi\)
−0.0641880 + 0.997938i \(0.520446\pi\)
\(488\) 0 0
\(489\) 21.5301 1.02561i 0.973627 0.0463796i
\(490\) 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
\(499\) −18.0987 + 10.4493i −0.810209 + 0.467775i −0.847029 0.531547i \(-0.821611\pi\)
0.0368191 + 0.999322i \(0.488277\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.0950560 0.995472i \(-0.469697\pi\)
−0.0950560 + 0.995472i \(0.530303\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 10.9347 + 6.31316i 0.485628 + 0.280377i
\(508\) 0 0
\(509\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(510\) 0 0
\(511\) 3.06808 10.4489i 0.135724 0.462234i
\(512\) 0 0
\(513\) −20.4835 + 39.7325i −0.904371 + 1.75423i
\(514\) 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
\(523\) 8.87407 + 36.5794i 0.388036 + 1.59951i 0.746117 + 0.665815i \(0.231916\pi\)
−0.358081 + 0.933691i \(0.616569\pi\)
\(524\) 0 0
\(525\) 5.13616 + 4.03912i 0.224160 + 0.176282i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.3413 + 18.7352i 0.580057 + 0.814576i
\(530\) 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
\(541\) 1.88319 + 0.270762i 0.0809647 + 0.0116410i 0.182678 0.983173i \(-0.441523\pi\)
−0.101713 + 0.994814i \(0.532432\pi\)
\(542\) 0 0
\(543\) −37.0843 + 26.4076i −1.59144 + 1.13326i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 19.3982 24.6668i 0.829406 1.05468i −0.168216 0.985750i \(-0.553801\pi\)
0.997622 0.0689253i \(-0.0219570\pi\)
\(548\) 0 0
\(549\) −39.8088 + 9.65750i −1.69900 + 0.412172i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 5.90518 + 1.13813i 0.251114 + 0.0483982i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(558\) 0 0
\(559\) −4.24927 2.19065i −0.179725 0.0926546i
\(560\) 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
\(567\) 6.59904 + 1.60091i 0.277133 + 0.0672318i
\(568\) 0 0
\(569\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(570\) 0 0
\(571\) 40.6586 20.9609i 1.70151 0.877188i 0.718063 0.695978i \(-0.245029\pi\)
0.983444 0.181210i \(-0.0580014\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 28.3125 9.79906i 1.17867 0.407940i 0.333658 0.942694i \(-0.391717\pi\)
0.845008 + 0.534754i \(0.179596\pi\)
\(578\) 0 0
\(579\) 0.842112 + 0.729694i 0.0349970 + 0.0303251i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(588\) 0 0
\(589\) 69.1871 59.9510i 2.85081 2.47024i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6.30024 8.01141i −0.257852 0.327885i
\(598\) 0 0
\(599\) 0 0 −0.189251 0.981929i \(-0.560606\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(600\) 0 0
\(601\) 5.23519 0.499900i 0.213548 0.0203914i 0.0122663 0.999925i \(-0.496095\pi\)
0.201282 + 0.979533i \(0.435489\pi\)
\(602\) 0 0
\(603\) 2.93926 + 24.3795i 0.119696 + 0.992811i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −15.3463 + 2.95776i −0.622888 + 0.120052i −0.490923 0.871203i \(-0.663341\pi\)
−0.131964 + 0.991254i \(0.542128\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 2.33631 49.0452i 0.0943626 1.98092i −0.0821600 0.996619i \(-0.526182\pi\)
0.176523 0.984297i \(-0.443515\pi\)
\(614\) 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
\(619\) −45.2196 + 18.1032i −1.81753 + 0.727629i −0.832792 + 0.553587i \(0.813259\pi\)
−0.984738 + 0.174042i \(0.944317\pi\)
\(620\) 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
\(631\) 30.4929 31.9800i 1.21390 1.27310i 0.262578 0.964911i \(-0.415427\pi\)
0.951324 0.308193i \(-0.0997243\pi\)
\(632\) 0 0
\(633\) −8.76233 + 5.05893i −0.348271 + 0.201074i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.46072 15.2973i 0.0578757 0.606102i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(642\) 0 0
\(643\) 7.08107 + 49.2499i 0.279250 + 1.94223i 0.331158 + 0.943575i \(0.392561\pi\)
−0.0519076 + 0.998652i \(0.516530\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.458227 0.888835i \(-0.348485\pi\)
−0.458227 + 0.888835i \(0.651515\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −11.3280 8.06662i −0.443979 0.316156i
\(652\) 0 0
\(653\) 0 0 0.189251 0.981929i \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −10.2085 42.0802i −0.398273 1.64170i
\(658\) 0 0
\(659\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(660\) 0 0
\(661\) −37.4828 17.1178i −1.45791 0.665806i −0.480470 0.877011i \(-0.659534\pi\)
−0.977442 + 0.211205i \(0.932261\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 34.8582i 1.34769i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 22.5987 + 35.1642i 0.871115 + 1.35548i 0.933930 + 0.357457i \(0.116356\pi\)
−0.0628142 + 0.998025i \(0.520008\pi\)
\(674\) 0 0
\(675\) 25.7163 + 3.69745i 0.989821 + 0.142315i
\(676\) 0 0
\(677\) 0 0 0.814576 0.580057i \(-0.196970\pi\)
−0.814576 + 0.580057i \(0.803030\pi\)
\(678\) 0 0
\(679\) 5.87090 12.8555i 0.225304 0.493348i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.971812 0.235759i \(-0.0757576\pi\)
−0.971812 + 0.235759i \(0.924242\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −28.0279 5.40193i −1.06933 0.206096i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 20.7167 + 8.29372i 0.788101 + 0.315508i 0.730572 0.682836i \(-0.239254\pi\)
0.0575291 + 0.998344i \(0.481678\pi\)
\(692\) 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 −0.971812 0.235759i \(-0.924242\pi\)
0.971812 + 0.235759i \(0.0757576\pi\)
\(702\) 0 0
\(703\) −98.8860 9.44247i −3.72956 0.356129i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −5.58152 5.32196i −0.209618 0.199871i 0.577995 0.816040i \(-0.303835\pi\)
−0.787614 + 0.616169i \(0.788684\pi\)
\(710\) 0 0
\(711\) 22.5970 7.82089i 0.847453 0.293306i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(720\) 0 0
\(721\) −3.21340 8.02669i −0.119673 0.298929i
\(722\) 0 0
\(723\) 30.9329 26.8035i 1.15041 0.996834i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 51.7184 + 2.46365i 1.91813 + 0.0913717i 0.972362 0.233478i \(-0.0750106\pi\)
0.945766 + 0.324850i \(0.105314\pi\)
\(728\) 0 0
\(729\) 25.9063 7.60678i 0.959493 0.281733i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −6.97341 36.1815i −0.257569 1.33639i −0.848296 0.529522i \(-0.822371\pi\)
0.590727 0.806871i \(-0.298841\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −0.178224 1.86645i −0.00655608 0.0686584i 0.991602 0.129329i \(-0.0412822\pi\)
−0.998158 + 0.0606703i \(0.980676\pi\)
\(740\) 0 0
\(741\) −34.9630 + 6.73857i −1.28440 + 0.247547i
\(742\) 0 0
\(743\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(744\) 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 −0.948753 0.316017i \(-0.897654\pi\)
−0.177779 0.984070i \(-0.556891\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −22.0244 + 1.04915i −0.800492 + 0.0381321i −0.443841 0.896106i \(-0.646384\pi\)
−0.356651 + 0.934238i \(0.616081\pi\)
\(758\) 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
\(763\) 4.94506 + 14.2878i 0.179023 + 0.517254i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −19.0135 36.8810i −0.685643 1.32996i −0.932592 0.360933i \(-0.882458\pi\)
0.246949 0.969028i \(-0.420572\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(774\) 0 0
\(775\) −46.0792 26.6038i −1.65521 0.955638i
\(776\) 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
\(787\) 3.61485 18.7556i 0.128856 0.668566i −0.858848 0.512230i \(-0.828820\pi\)
0.987704 0.156336i \(-0.0499683\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −25.6479 20.1698i −0.910785 0.716249i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0