Properties

Label 804.2.s.a.161.2
Level 804
Weight 2
Character 804.161
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 161.2
Root \(0.928368 - 0.371662i\)
Character \(\chi\) = 804.161
Dual form 804.2.s.a.5.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.57553 + 0.719520i) q^{3} +(-2.61965 + 2.26994i) q^{7} +(1.96458 + 2.26725i) q^{9} +O(q^{10})\) \(q+(1.57553 + 0.719520i) q^{3} +(-2.61965 + 2.26994i) q^{7} +(1.96458 + 2.26725i) q^{9} +(-3.27007 + 5.08833i) q^{13} +(0.112808 - 0.130187i) q^{19} +(-5.76060 + 1.69146i) q^{21} +(-4.20627 - 2.70320i) q^{25} +(1.46393 + 4.98567i) q^{27} +(4.78165 + 7.44039i) q^{31} +4.66687 q^{37} +(-8.81324 + 5.66393i) q^{39} +(4.37247 - 0.628667i) q^{43} +(0.713735 - 4.96414i) q^{49} +(0.271404 - 0.123946i) q^{57} +(-2.77697 - 9.45750i) q^{61} +(-10.2930 - 1.47991i) q^{63} +(7.66645 + 2.86803i) q^{67} +(12.9519 - 3.80303i) q^{73} +(-4.68209 - 7.28547i) q^{75} +(-6.70847 + 10.4386i) q^{79} +(-1.28083 + 8.90839i) q^{81} +(-2.98376 - 20.7525i) q^{91} +(2.18012 + 15.1630i) q^{93} -18.4910i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{9} + 16q^{19} - 12q^{21} + 10q^{25} - 20q^{37} - 24q^{39} + 10q^{49} - 66q^{57} - 132q^{63} - 16q^{67} + 90q^{73} + 44q^{79} - 18q^{81} + 48q^{91} - 36q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{17}{22}\right)\) \(1\)

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
\(3\) 1.57553 + 0.719520i 0.909632 + 0.415415i
\(4\) 0 0
\(5\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(6\) 0 0
\(7\) −2.61965 + 2.26994i −0.990134 + 0.857956i −0.989860 0.142044i \(-0.954633\pi\)
−0.000273951 1.00000i \(0.500087\pi\)
\(8\) 0 0
\(9\) 1.96458 + 2.26725i 0.654861 + 0.755750i
\(10\) 0 0
\(11\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(12\) 0 0
\(13\) −3.27007 + 5.08833i −0.906954 + 1.41125i 0.00456441 + 0.999990i \(0.498547\pi\)
−0.911519 + 0.411259i \(0.865089\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(18\) 0 0
\(19\) 0.112808 0.130187i 0.0258799 0.0298670i −0.742662 0.669666i \(-0.766437\pi\)
0.768542 + 0.639799i \(0.220983\pi\)
\(20\) 0 0
\(21\) −5.76060 + 1.69146i −1.25707 + 0.369108i
\(22\) 0 0
\(23\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(24\) 0 0
\(25\) −4.20627 2.70320i −0.841254 0.540641i
\(26\) 0 0
\(27\) 1.46393 + 4.98567i 0.281733 + 0.959493i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 4.78165 + 7.44039i 0.858810 + 1.33633i 0.940541 + 0.339680i \(0.110319\pi\)
−0.0817313 + 0.996654i \(0.526045\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.66687 0.767228 0.383614 0.923494i \(-0.374679\pi\)
0.383614 + 0.923494i \(0.374679\pi\)
\(38\) 0 0
\(39\) −8.81324 + 5.66393i −1.41125 + 0.906954i
\(40\) 0 0
\(41\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(42\) 0 0
\(43\) 4.37247 0.628667i 0.666796 0.0958708i 0.199399 0.979918i \(-0.436101\pi\)
0.467397 + 0.884048i \(0.345192\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(48\) 0 0
\(49\) 0.713735 4.96414i 0.101962 0.709163i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.271404 0.123946i 0.0359484 0.0164171i
\(58\) 0 0
\(59\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(60\) 0 0
\(61\) −2.77697 9.45750i −0.355555 1.21091i −0.922124 0.386895i \(-0.873548\pi\)
0.566569 0.824014i \(-0.308270\pi\)
\(62\) 0 0
\(63\) −10.2930 1.47991i −1.29680 0.186452i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.66645 + 2.86803i 0.936605 + 0.350386i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(72\) 0 0
\(73\) 12.9519 3.80303i 1.51591 0.445111i 0.585206 0.810885i \(-0.301014\pi\)
0.930704 + 0.365774i \(0.119196\pi\)
\(74\) 0 0
\(75\) −4.68209 7.28547i −0.540641 0.841254i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.70847 + 10.4386i −0.754762 + 1.17443i 0.225018 + 0.974355i \(0.427756\pi\)
−0.979780 + 0.200078i \(0.935880\pi\)
\(80\) 0 0
\(81\) −1.28083 + 8.90839i −0.142315 + 0.989821i
\(82\) 0 0
\(83\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(90\) 0 0
\(91\) −2.98376 20.7525i −0.312783 2.17545i
\(92\) 0 0
\(93\) 2.18012 + 15.1630i 0.226068 + 1.57233i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.4910i 1.87748i −0.344630 0.938739i \(-0.611996\pi\)
0.344630 0.938739i \(-0.388004\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(102\) 0 0
\(103\) 13.2205 8.49628i 1.30265 0.837164i 0.309154 0.951012i \(-0.399954\pi\)
0.993498 + 0.113848i \(0.0363178\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(108\) 0 0
\(109\) −11.0865 + 17.2509i −1.06189 + 1.65233i −0.372340 + 0.928096i \(0.621444\pi\)
−0.689551 + 0.724238i \(0.742192\pi\)
\(110\) 0 0
\(111\) 7.35278 + 3.35790i 0.697895 + 0.318718i
\(112\) 0 0
\(113\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −17.9608 + 2.58238i −1.66048 + 0.238741i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −9.25379 5.94705i −0.841254 0.540641i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 14.6974 + 16.9617i 1.30418 + 1.50511i 0.721510 + 0.692404i \(0.243448\pi\)
0.582675 + 0.812705i \(0.302006\pi\)
\(128\) 0 0
\(129\) 7.34130 + 2.15560i 0.646365 + 0.189790i
\(130\) 0 0
\(131\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(132\) 0 0
\(133\) 0.597112i 0.0517762i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(138\) 0 0
\(139\) −2.07325 + 7.06083i −0.175850 + 0.598891i 0.823644 + 0.567107i \(0.191937\pi\)
−0.999495 + 0.0317847i \(0.989881\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.69631 7.30760i 0.387345 0.602720i
\(148\) 0 0
\(149\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(150\) 0 0
\(151\) −1.51845 10.5610i −0.123569 0.859444i −0.953460 0.301518i \(-0.902507\pi\)
0.829891 0.557926i \(-0.188403\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.71678 5.94891i 0.216822 0.474775i −0.769699 0.638407i \(-0.779594\pi\)
0.986521 + 0.163632i \(0.0523210\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 22.8634 1.79080 0.895399 0.445265i \(-0.146890\pi\)
0.895399 + 0.445265i \(0.146890\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(168\) 0 0
\(169\) −9.79733 21.4532i −0.753641 1.65024i
\(170\) 0 0
\(171\) 0.516787 0.0395197
\(172\) 0 0
\(173\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(174\) 0 0
\(175\) 17.1551 2.46652i 1.29680 0.186452i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(180\) 0 0
\(181\) 7.10480 15.5574i 0.528096 1.15637i −0.438186 0.898884i \(-0.644379\pi\)
0.966282 0.257485i \(-0.0828937\pi\)
\(182\) 0 0
\(183\) 2.42966 16.8987i 0.179606 1.24918i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −15.1521 9.73769i −1.10216 0.708313i
\(190\) 0 0
\(191\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(192\) 0 0
\(193\) 16.5174 10.6151i 1.18895 0.764089i 0.211936 0.977284i \(-0.432023\pi\)
0.977010 + 0.213195i \(0.0683868\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(198\) 0 0
\(199\) 4.85491 + 5.60287i 0.344156 + 0.397177i 0.901269 0.433260i \(-0.142637\pi\)
−0.557114 + 0.830436i \(0.688091\pi\)
\(200\) 0 0
\(201\) 10.0151 + 10.0348i 0.706411 + 0.707802i
\(202\) 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
\(211\) −3.53943 7.75028i −0.243665 0.533551i 0.747800 0.663924i \(-0.231110\pi\)
−0.991465 + 0.130372i \(0.958383\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −29.4155 8.63717i −1.99685 0.586329i
\(218\) 0 0
\(219\) 23.1425 + 3.32739i 1.56383 + 0.224844i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −0.904960 1.98159i −0.0606006 0.132697i 0.876909 0.480657i \(-0.159602\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 0 0
\(225\) −2.13472 14.8473i −0.142315 0.989821i
\(226\) 0 0
\(227\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(228\) 0 0
\(229\) −15.8836 24.7154i −1.04962 1.63324i −0.726900 0.686743i \(-0.759040\pi\)
−0.322718 0.946495i \(-0.604597\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −18.0802 + 11.6194i −1.17443 + 0.754762i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −25.6763 + 7.53923i −1.65395 + 0.485644i −0.969842 0.243734i \(-0.921628\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(242\) 0 0
\(243\) −8.42776 + 13.1138i −0.540641 + 0.841254i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.293546 + 0.999725i 0.0186779 + 0.0636110i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(258\) 0 0
\(259\) −12.2256 + 10.5935i −0.759659 + 0.658248i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(264\) 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
\(271\) 29.5662 + 13.5024i 1.79602 + 0.820215i 0.964203 + 0.265167i \(0.0854270\pi\)
0.831818 + 0.555048i \(0.187300\pi\)
\(272\) 0 0
\(273\) 10.2308 34.8430i 0.619198 2.10880i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 11.0494 + 12.7517i 0.663893 + 0.766173i 0.983408 0.181408i \(-0.0580655\pi\)
−0.319515 + 0.947581i \(0.603520\pi\)
\(278\) 0 0
\(279\) −7.47528 + 25.4585i −0.447533 + 1.52416i
\(280\) 0 0
\(281\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(282\) 0 0
\(283\) 22.0240 25.4171i 1.30919 1.51089i 0.633125 0.774050i \(-0.281772\pi\)
0.676068 0.736839i \(-0.263682\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.3114 + 4.78945i −0.959493 + 0.281733i
\(290\) 0 0
\(291\) 13.3046 29.1331i 0.779932 1.70781i
\(292\) 0 0
\(293\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −10.0273 + 11.5721i −0.577965 + 0.667007i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −24.0452 + 15.4529i −1.37233 + 0.881943i −0.998954 0.0457370i \(-0.985436\pi\)
−0.373376 + 0.927680i \(0.621800\pi\)
\(308\) 0 0
\(309\) 26.9425 3.87374i 1.53270 0.220370i
\(310\) 0 0
\(311\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(312\) 0 0
\(313\) 1.38404 0.632072i 0.0782308 0.0357268i −0.375915 0.926654i \(-0.622672\pi\)
0.454146 + 0.890927i \(0.349944\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 27.5096 12.5632i 1.52596 0.696881i
\(326\) 0 0
\(327\) −29.8794 + 19.2023i −1.65233 + 1.06189i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 35.9339 + 5.16652i 1.97511 + 0.283977i 0.997182 + 0.0750153i \(0.0239006\pi\)
0.977924 + 0.208962i \(0.0670085\pi\)
\(332\) 0 0
\(333\) 9.16844 + 10.5809i 0.502428 + 0.579832i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −22.7016 + 19.6711i −1.23664 + 1.07155i −0.241771 + 0.970333i \(0.577728\pi\)
−0.994864 + 0.101218i \(0.967726\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −3.71958 5.78778i −0.200839 0.312511i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(348\) 0 0
\(349\) 2.42104 16.8387i 0.129595 0.901356i −0.816472 0.577386i \(-0.804073\pi\)
0.946067 0.323971i \(-0.105018\pi\)
\(350\) 0 0
\(351\) −30.1559 8.85456i −1.60960 0.472622i
\(352\) 0 0
\(353\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(360\) 0 0
\(361\) 2.69976 + 18.7772i 0.142093 + 0.988276i
\(362\) 0 0
\(363\) −10.3006 16.0280i −0.540641 0.841254i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −34.2827 + 15.6564i −1.78954 + 0.817256i −0.819987 + 0.572383i \(0.806019\pi\)
−0.969555 + 0.244874i \(0.921254\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 33.2555i 1.72190i 0.508686 + 0.860952i \(0.330131\pi\)
−0.508686 + 0.860952i \(0.669869\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 28.1982 + 12.8777i 1.44844 + 0.661482i 0.975578 0.219653i \(-0.0704926\pi\)
0.472865 + 0.881135i \(0.343220\pi\)
\(380\) 0 0
\(381\) 10.9519 + 37.2988i 0.561083 + 1.91087i
\(382\) 0 0
\(383\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.0154 + 8.67842i 0.509113 + 0.441149i
\(388\) 0 0
\(389\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 24.3824 + 7.15932i 1.22372 + 0.359316i 0.828876 0.559433i \(-0.188981\pi\)
0.394842 + 0.918749i \(0.370799\pi\)
\(398\) 0 0
\(399\) −0.429634 + 0.940767i −0.0215086 + 0.0470973i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −53.4955 −2.66480
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −18.1387 + 15.7173i −0.896902 + 0.777170i −0.975560 0.219732i \(-0.929482\pi\)
0.0786585 + 0.996902i \(0.474936\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −8.34686 + 9.63279i −0.408748 + 0.471720i
\(418\) 0 0
\(419\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(420\) 0 0
\(421\) −24.3843 + 28.1410i −1.18842 + 1.37151i −0.276567 + 0.960995i \(0.589197\pi\)
−0.911854 + 0.410515i \(0.865349\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 28.7426 + 18.4718i 1.39095 + 0.893912i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 18.3218 + 28.5092i 0.880488 + 1.37007i 0.928545 + 0.371221i \(0.121061\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −34.8444 −1.66303 −0.831516 0.555501i \(-0.812527\pi\)
−0.831516 + 0.555501i \(0.812527\pi\)
\(440\) 0 0
\(441\) 12.6571 8.13424i 0.602720 0.387345i
\(442\) 0 0
\(443\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 5.20651 17.7317i 0.244623 0.833110i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 31.9378 + 20.5252i 1.49399 + 0.960127i 0.995653 + 0.0931420i \(0.0296911\pi\)
0.498334 + 0.866985i \(0.333945\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(462\) 0 0
\(463\) 11.2886 + 38.4455i 0.524626 + 1.78671i 0.612372 + 0.790570i \(0.290216\pi\)
−0.0877454 + 0.996143i \(0.527966\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(468\) 0 0
\(469\) −26.5937 + 9.88912i −1.22798 + 0.456637i
\(470\) 0 0
\(471\) 8.56072 7.41791i 0.394457 0.341799i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −0.826423 + 0.242660i −0.0379189 + 0.0111340i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(480\) 0 0
\(481\) −15.2610 + 23.7465i −0.695841 + 1.08275i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −42.6209 6.12796i −1.93134 0.277684i −0.934421 0.356171i \(-0.884082\pi\)
−0.996915 + 0.0784867i \(0.974991\pi\)
\(488\) 0 0
\(489\) 36.0219 + 16.4507i 1.62897 + 0.743924i
\(490\) 0 0
\(491\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 43.5282i 1.94859i −0.225277 0.974295i \(-0.572329\pi\)
0.225277 0.974295i \(-0.427671\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 40.8495i 1.81419i
\(508\) 0 0
\(509\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(510\) 0 0
\(511\) −25.2969 + 39.3627i −1.11907 + 1.74130i
\(512\) 0 0
\(513\) 0.814213 + 0.371839i 0.0359484 + 0.0164171i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(522\) 0 0
\(523\) −24.2239 15.5678i −1.05924 0.680731i −0.109567 0.993979i \(-0.534946\pi\)
−0.949671 + 0.313249i \(0.898583\pi\)
\(524\) 0 0
\(525\) 28.8030 + 8.45732i 1.25707 + 0.369108i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −15.0618 17.3822i −0.654861 0.755750i
\(530\) 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
\(541\) 13.1056 44.6335i 0.563453 1.91894i 0.275307 0.961356i \(-0.411220\pi\)
0.288145 0.957587i \(-0.406961\pi\)
\(542\) 0 0
\(543\) 22.3876 19.3990i 0.960746 0.832491i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −10.1824 + 34.6779i −0.435366 + 1.48272i 0.391425 + 0.920210i \(0.371982\pi\)
−0.826791 + 0.562510i \(0.809836\pi\)
\(548\) 0 0
\(549\) 15.9869 24.8761i 0.682305 1.06169i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −6.12111 42.5733i −0.260296 1.81040i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(558\) 0 0
\(559\) −11.0994 + 24.3044i −0.469456 + 1.02797i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −16.8662 26.2443i −0.708313 1.10216i
\(568\) 0 0
\(569\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(570\) 0 0
\(571\) −19.7781 43.3079i −0.827686 1.81238i −0.492897 0.870088i \(-0.664062\pi\)
−0.334790 0.942293i \(-0.608665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 25.9876 3.73645i 1.08188 0.155550i 0.421756 0.906709i \(-0.361414\pi\)
0.660121 + 0.751159i \(0.270505\pi\)
\(578\) 0 0
\(579\) 33.6613 4.83977i 1.39892 0.201134i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(588\) 0 0
\(589\) 1.50805 + 0.216825i 0.0621382 + 0.00893413i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3.61768 + 12.3207i 0.148062 + 0.504252i
\(598\) 0 0
\(599\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(600\) 0 0
\(601\) −24.0060 27.7044i −0.979226 1.13009i −0.991493 0.130163i \(-0.958450\pi\)
0.0122663 0.999925i \(-0.496095\pi\)
\(602\) 0 0
\(603\) 8.55882 + 23.0162i 0.348542 + 0.937293i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −2.97960 + 20.7236i −0.120938 + 0.841145i 0.835558 + 0.549402i \(0.185144\pi\)
−0.956497 + 0.291743i \(0.905765\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −19.3503 42.3712i −0.781551 1.71136i −0.699391 0.714739i \(-0.746545\pi\)
−0.0821600 0.996619i \(-0.526182\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(618\) 0 0
\(619\) 19.1595 + 5.62573i 0.770084 + 0.226117i 0.643094 0.765787i \(-0.277650\pi\)
0.126990 + 0.991904i \(0.459469\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 10.3854 + 22.7408i 0.415415 + 0.909632i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −16.1925 25.1960i −0.644612 1.00304i −0.997722 0.0674546i \(-0.978512\pi\)
0.353110 0.935582i \(-0.385124\pi\)
\(632\) 0 0
\(633\) 14.7575i 0.586557i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 22.9252 + 19.8648i 0.908329 + 0.787072i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 35.0716 10.2979i 1.38309 0.406111i 0.496245 0.868183i \(-0.334712\pi\)
0.886843 + 0.462072i \(0.152894\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −40.1303 34.7731i −1.57283 1.36287i
\(652\) 0 0
\(653\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 34.0676 + 21.8939i 1.32910 + 0.854162i
\(658\) 0 0
\(659\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(660\) 0 0
\(661\) 24.7004 21.4030i 0.960733 0.832480i −0.0251892 0.999683i \(-0.508019\pi\)
0.985923 + 0.167203i \(0.0534734\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 3.77318i 0.145880i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.719977 + 0.328802i 0.0277531 + 0.0126744i 0.429243 0.903189i \(-0.358780\pi\)
−0.401490 + 0.915863i \(0.631508\pi\)
\(674\) 0 0
\(675\) 7.31963 24.9284i 0.281733 0.959493i
\(676\) 0 0
\(677\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(678\) 0 0
\(679\) 41.9735 + 48.4400i 1.61079 + 1.85895i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −7.24188 50.3684i −0.276295 1.92167i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −20.4972 + 6.01851i −0.779749 + 0.228955i −0.647300 0.762235i \(-0.724102\pi\)
−0.132448 + 0.991190i \(0.542284\pi\)
\(692\) 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 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(702\) 0 0
\(703\) 0.526459 0.607567i 0.0198558 0.0229148i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 44.8001 28.7913i 1.68250 1.08128i 0.839884 0.542766i \(-0.182623\pi\)
0.842618 0.538512i \(-0.181013\pi\)
\(710\) 0 0
\(711\) −36.8462 + 5.29769i −1.38184 + 0.198679i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(720\) 0 0
\(721\) −15.3470 + 52.2670i −0.571551 + 1.94652i
\(722\) 0 0
\(723\) −45.8783 6.59630i −1.70623 0.245319i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −35.5804 + 16.2490i −1.31961 + 0.602644i −0.945766 0.324850i \(-0.894686\pi\)
−0.373840 + 0.927493i \(0.621959\pi\)
\(728\) 0 0
\(729\) −22.7138 + 14.5973i −0.841254 + 0.540641i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 23.9304 + 3.44067i 0.883888 + 0.127084i 0.569286 0.822140i \(-0.307220\pi\)
0.314602 + 0.949224i \(0.398129\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −17.5576 + 15.2138i −0.645867 + 0.559647i −0.914999 0.403456i \(-0.867809\pi\)
0.269132 + 0.963103i \(0.413263\pi\)
\(740\) 0 0
\(741\) −0.256832 + 1.78631i −0.00943497 + 0.0656217i
\(742\) 0 0
\(743\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −3.49306 + 24.2948i −0.127464 + 0.886528i 0.821290 + 0.570511i \(0.193255\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −48.9909 22.3734i −1.78061 0.813176i −0.975485 0.220065i \(-0.929373\pi\)
−0.805121 0.593111i \(-0.797900\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(762\) 0 0
\(763\) −10.1158 70.3568i −0.366216 2.54709i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 43.3970 19.8187i 1.56494 0.714682i 0.570626 0.821210i \(-0.306701\pi\)
0.994310 + 0.106528i \(0.0339735\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(774\) 0 0
\(775\) 44.2221i 1.58850i
\(776\) 0 0
\(777\) −26.8839 + 7.89384i −0.964456 + 0.283190i
\(778\) 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
\(787\) 27.0832 3.89398i 0.965412 0.138805i 0.358461 0.933545i \(-0.383302\pi\)
0.606951 + 0.794739i \(0.292393\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 57.2038 + 16.7965i 2.03137 + 0.596463i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0