Properties

Label 804.2.s.a.209.2
Level 804
Weight 2
Character 804.209
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 209.2
Root \(0.235759 + 0.971812i\)
Character \(\chi\) = 804.209
Dual form 804.2.s.a.377.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.30900 - 1.13425i) q^{3} +(-4.40541 - 0.633402i) q^{7} +(0.426945 - 2.96946i) q^{9} +O(q^{10})\) \(q+(1.30900 - 1.13425i) q^{3} +(-4.40541 - 0.633402i) q^{7} +(0.426945 - 2.96946i) q^{9} +(0.903862 - 0.412780i) q^{13} +(-1.11815 - 7.77691i) q^{19} +(-6.48510 + 4.16772i) q^{21} +(-2.07708 - 4.54816i) q^{25} +(-2.80925 - 4.37128i) q^{27} +(-5.98538 - 2.73343i) q^{31} -2.14797 q^{37} +(0.714956 - 1.56553i) q^{39} +(0.833910 + 2.84004i) q^{43} +(12.2900 + 3.60866i) q^{49} +(-10.2846 - 8.91168i) q^{57} +(8.17314 + 12.7177i) q^{61} +(-3.76173 + 12.8113i) q^{63} +(4.89885 + 6.55754i) q^{67} +(13.3792 - 8.59828i) q^{73} +(-7.87764 - 3.59760i) q^{75} +(-3.23526 + 1.47749i) q^{79} +(-8.63544 - 2.53559i) q^{81} +(-4.24334 + 1.24596i) q^{91} +(-10.9352 + 3.21088i) q^{93} -1.50648i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{9} + 16q^{19} - 12q^{21} + 10q^{25} - 20q^{37} - 24q^{39} + 10q^{49} - 66q^{57} - 132q^{63} - 16q^{67} + 90q^{73} + 44q^{79} - 18q^{81} + 48q^{91} - 36q^{93} + O(q^{100}) \)

Character values

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

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

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
\(3\) 1.30900 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
\(7\) −4.40541 0.633402i −1.66509 0.239404i −0.755570 0.655068i \(-0.772640\pi\)
−0.909518 + 0.415664i \(0.863549\pi\)
\(8\) 0 0
\(9\) 0.426945 2.96946i 0.142315 0.989821i
\(10\) 0 0
\(11\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(12\) 0 0
\(13\) 0.903862 0.412780i 0.250686 0.114485i −0.286109 0.958197i \(-0.592362\pi\)
0.536795 + 0.843712i \(0.319635\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(18\) 0 0
\(19\) −1.11815 7.77691i −0.256521 1.78414i −0.557158 0.830406i \(-0.688108\pi\)
0.300637 0.953739i \(-0.402801\pi\)
\(20\) 0 0
\(21\) −6.48510 + 4.16772i −1.41517 + 0.909472i
\(22\) 0 0
\(23\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(24\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −5.98538 2.73343i −1.07501 0.490939i −0.202369 0.979309i \(-0.564864\pi\)
−0.872636 + 0.488371i \(0.837591\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.14797 −0.353123 −0.176562 0.984290i \(-0.556497\pi\)
−0.176562 + 0.984290i \(0.556497\pi\)
\(38\) 0 0
\(39\) 0.714956 1.56553i 0.114485 0.250686i
\(40\) 0 0
\(41\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(42\) 0 0
\(43\) 0.833910 + 2.84004i 0.127170 + 0.433102i 0.998322 0.0579125i \(-0.0184445\pi\)
−0.871152 + 0.491014i \(0.836626\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(48\) 0 0
\(49\) 12.2900 + 3.60866i 1.75571 + 0.515524i
\(50\) 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
\(57\) −10.2846 8.91168i −1.36223 1.18038i
\(58\) 0 0
\(59\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(60\) 0 0
\(61\) 8.17314 + 12.7177i 1.04646 + 1.62833i 0.734996 + 0.678072i \(0.237184\pi\)
0.311467 + 0.950257i \(0.399180\pi\)
\(62\) 0 0
\(63\) −3.76173 + 12.8113i −0.473934 + 1.61407i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.89885 + 6.55754i 0.598490 + 0.801131i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(72\) 0 0
\(73\) 13.3792 8.59828i 1.56592 1.00635i 0.585206 0.810885i \(-0.301014\pi\)
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
\(79\) −3.23526 + 1.47749i −0.363995 + 0.166231i −0.589013 0.808124i \(-0.700483\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −8.63544 2.53559i −0.959493 0.281733i
\(82\) 0 0
\(83\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(84\) 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
\(91\) −4.24334 + 1.24596i −0.444823 + 0.130612i
\(92\) 0 0
\(93\) −10.9352 + 3.21088i −1.13393 + 0.332952i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.50648i 0.152960i −0.997071 0.0764798i \(-0.975632\pi\)
0.997071 0.0764798i \(-0.0243681\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(102\) 0 0
\(103\) −0.241998 + 0.529902i −0.0238448 + 0.0522128i −0.921180 0.389138i \(-0.872773\pi\)
0.897335 + 0.441350i \(0.145500\pi\)
\(104\) 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
\(109\) 16.8886 7.71276i 1.61763 0.738749i 0.618733 0.785602i \(-0.287646\pi\)
0.998902 + 0.0468528i \(0.0149192\pi\)
\(110\) 0 0
\(111\) −2.81168 + 2.43633i −0.266873 + 0.231247i
\(112\) 0 0
\(113\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.839836 2.86022i −0.0776429 0.264427i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.56957 10.0060i −0.415415 0.909632i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.52772 17.5807i 0.224299 1.56003i −0.497211 0.867630i \(-0.665643\pi\)
0.721510 0.692404i \(-0.243448\pi\)
\(128\) 0 0
\(129\) 4.31291 + 2.77174i 0.379730 + 0.244038i
\(130\) 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
\(139\) 12.6697 19.7144i 1.07463 1.67215i 0.444118 0.895969i \(-0.353517\pi\)
0.630509 0.776182i \(-0.282846\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 20.1807 9.21621i 1.66447 0.760140i
\(148\) 0 0
\(149\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(150\) 0 0
\(151\) 15.8075 4.64150i 1.28640 0.377720i 0.434141 0.900845i \(-0.357052\pi\)
0.852256 + 0.523125i \(0.175234\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.7772 + 17.0538i 1.17935 + 1.36104i 0.918381 + 0.395698i \(0.129497\pi\)
0.260971 + 0.965347i \(0.415957\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −14.5076 −1.13632 −0.568161 0.822917i \(-0.692345\pi\)
−0.568161 + 0.822917i \(0.692345\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(168\) 0 0
\(169\) −7.86661 + 9.07855i −0.605124 + 0.698350i
\(170\) 0 0
\(171\) −23.5706 −1.80249
\(172\) 0 0
\(173\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(174\) 0 0
\(175\) 6.26955 + 21.3521i 0.473934 + 1.61407i
\(176\) 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
\(181\) 7.72107 + 8.91058i 0.573902 + 0.662318i 0.966282 0.257485i \(-0.0828937\pi\)
−0.392380 + 0.919803i \(0.628348\pi\)
\(182\) 0 0
\(183\) 25.1236 + 7.37696i 1.85719 + 0.545321i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 9.60713 + 21.0367i 0.698816 + 1.53019i
\(190\) 0 0
\(191\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(192\) 0 0
\(193\) −11.5134 + 25.2108i −0.828751 + 1.81471i −0.350068 + 0.936724i \(0.613841\pi\)
−0.478684 + 0.877987i \(0.658886\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(198\) 0 0
\(199\) 3.68451 25.6263i 0.261188 1.81660i −0.262770 0.964858i \(-0.584636\pi\)
0.523958 0.851744i \(-0.324455\pi\)
\(200\) 0 0
\(201\) 13.8505 + 3.02726i 0.976937 + 0.213527i
\(202\) 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
\(211\) −0.229341 + 0.264674i −0.0157885 + 0.0182209i −0.763589 0.645703i \(-0.776565\pi\)
0.747800 + 0.663924i \(0.231110\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 24.6367 + 15.8330i 1.67245 + 1.07482i
\(218\) 0 0
\(219\) 7.76069 26.4305i 0.524419 1.78601i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −17.1509 + 19.7931i −1.14851 + 1.32545i −0.210997 + 0.977487i \(0.567671\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 0 0
\(225\) −14.3924 + 4.22599i −0.959493 + 0.281733i
\(226\) 0 0
\(227\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(228\) 0 0
\(229\) −26.0574 11.9000i −1.72192 0.786377i −0.995024 0.0996338i \(-0.968233\pi\)
−0.726900 0.686743i \(-0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2.55910 + 5.60364i −0.166231 + 0.363995i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 26.0993 16.7730i 1.68120 1.08044i 0.819313 0.573346i \(-0.194355\pi\)
0.861889 0.507097i \(-0.169281\pi\)
\(242\) 0 0
\(243\) −14.1798 + 6.47568i −0.909632 + 0.415415i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.22081 6.56770i −0.268563 0.417893i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(258\) 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
\(271\) −19.0827 + 16.5353i −1.15919 + 1.00445i −0.159340 + 0.987224i \(0.550937\pi\)
−0.999852 + 0.0172215i \(0.994518\pi\)
\(272\) 0 0
\(273\) −4.14129 + 6.44397i −0.250642 + 0.390007i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.38366 + 30.4890i −0.263388 + 1.83191i 0.243508 + 0.969899i \(0.421702\pi\)
−0.506896 + 0.862007i \(0.669207\pi\)
\(278\) 0 0
\(279\) −10.6722 + 16.6063i −0.638931 + 0.994196i
\(280\) 0 0
\(281\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(282\) 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
\(289\) 14.3013 9.19089i 0.841254 0.540641i
\(290\) 0 0
\(291\) −1.70873 1.97197i −0.100167 0.115599i
\(292\) 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
\(301\) −1.87483 13.0397i −0.108063 0.751598i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 10.0262 21.9544i 0.572227 1.25300i −0.373376 0.927680i \(-0.621800\pi\)
0.945603 0.325322i \(-0.105473\pi\)
\(308\) 0 0
\(309\) 0.284268 + 0.968127i 0.0161714 + 0.0550748i
\(310\) 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 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −3.75478 3.25353i −0.208278 0.180474i
\(326\) 0 0
\(327\) 13.3589 29.2519i 0.738749 1.61763i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3.61786 + 12.3213i −0.198856 + 0.677240i 0.798327 + 0.602224i \(0.205719\pi\)
−0.997182 + 0.0750153i \(0.976099\pi\)
\(332\) 0 0
\(333\) −0.917062 + 6.37831i −0.0502547 + 0.349529i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 36.3103 + 5.22063i 1.97795 + 0.284386i 0.994864 + 0.101218i \(0.0322738\pi\)
0.983082 + 0.183168i \(0.0586352\pi\)
\(338\) 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 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(348\) 0 0
\(349\) −34.8112 10.2215i −1.86340 0.547144i −0.999018 0.0443098i \(-0.985891\pi\)
−0.864383 0.502834i \(-0.832291\pi\)
\(350\) 0 0
\(351\) −4.34355 2.79143i −0.231842 0.148996i
\(352\) 0 0
\(353\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(354\) 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
\(361\) −40.9997 + 12.0386i −2.15788 + 0.633610i
\(362\) 0 0
\(363\) −17.3308 7.91472i −0.909632 0.415415i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 25.8607 + 22.4084i 1.34992 + 1.16971i 0.969555 + 0.244874i \(0.0787465\pi\)
0.380363 + 0.924837i \(0.375799\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 36.6272i 1.89649i −0.317546 0.948243i \(-0.602859\pi\)
0.317546 0.948243i \(-0.397141\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.88608 + 1.63430i −0.0968817 + 0.0839484i −0.701955 0.712221i \(-0.747689\pi\)
0.605073 + 0.796170i \(0.293144\pi\)
\(380\) 0 0
\(381\) −16.6322 25.8801i −0.852091 1.32588i
\(382\) 0 0
\(383\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.78943 1.26373i 0.446792 0.0642389i
\(388\) 0 0
\(389\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −22.5179 14.4714i −1.13014 0.726297i −0.164551 0.986369i \(-0.552617\pi\)
−0.965589 + 0.260072i \(0.916254\pi\)
\(398\) 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
\(403\) −6.53826 −0.325694
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −31.8733 4.58269i −1.57604 0.226600i −0.701898 0.712278i \(-0.747664\pi\)
−0.874138 + 0.485678i \(0.838573\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5.77653 40.1766i −0.282878 1.96746i
\(418\) 0 0
\(419\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(420\) 0 0
\(421\) −5.78506 40.2360i −0.281946 1.96098i −0.276567 0.960995i \(-0.589197\pi\)
−0.00537983 0.999986i \(-0.501712\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −27.9506 61.2034i −1.35263 2.96184i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 14.0531 + 6.41784i 0.675349 + 0.308421i 0.723406 0.690423i \(-0.242576\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −18.0823 −0.863019 −0.431509 0.902108i \(-0.642019\pi\)
−0.431509 + 0.902108i \(0.642019\pi\)
\(440\) 0 0
\(441\) 15.9629 34.9540i 0.760140 1.66447i
\(442\) 0 0
\(443\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 15.4273 24.0054i 0.724840 1.12787i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16.5015 36.1333i −0.771908 1.69024i −0.722406 0.691469i \(-0.756964\pi\)
−0.0495022 0.998774i \(-0.515763\pi\)
\(458\) 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
\(463\) 7.77090 + 12.0918i 0.361145 + 0.561952i 0.973516 0.228618i \(-0.0734207\pi\)
−0.612372 + 0.790570i \(0.709784\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(468\) 0 0
\(469\) −17.4279 31.9916i −0.804745 1.47723i
\(470\) 0 0
\(471\) 38.6867 + 5.56231i 1.78259 + 0.256298i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −33.0481 + 21.2387i −1.51635 + 0.974501i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(480\) 0 0
\(481\) −1.94146 + 0.886637i −0.0885231 + 0.0404272i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −4.42884 + 15.0832i −0.200690 + 0.683487i 0.796225 + 0.605000i \(0.206827\pi\)
−0.996915 + 0.0784867i \(0.974991\pi\)
\(488\) 0 0
\(489\) −18.9904 + 16.4553i −0.858775 + 0.744133i
\(490\) 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
\(499\) 42.0596i 1.88285i −0.337229 0.941423i \(-0.609490\pi\)
0.337229 0.941423i \(-0.390510\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 20.8065i 0.924050i
\(508\) 0 0
\(509\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(510\) 0 0
\(511\) −64.3870 + 29.4046i −2.84831 + 1.30078i
\(512\) 0 0
\(513\) −30.8539 + 26.7350i −1.36223 + 1.18038i
\(514\) 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
\(523\) 18.9901 + 41.5824i 0.830377 + 1.81827i 0.445212 + 0.895425i \(0.353128\pi\)
0.385165 + 0.922848i \(0.374145\pi\)
\(524\) 0 0
\(525\) 32.4255 + 20.8386i 1.41517 + 0.909472i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.27324 + 22.7659i −0.142315 + 0.989821i
\(530\) 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
\(541\) 3.41260 5.31011i 0.146719 0.228299i −0.760114 0.649790i \(-0.774857\pi\)
0.906833 + 0.421491i \(0.138493\pi\)
\(542\) 0 0
\(543\) 20.2137 + 2.90629i 0.867453 + 0.124721i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −18.6714 + 29.0533i −0.798333 + 1.24223i 0.168216 + 0.985750i \(0.446199\pi\)
−0.966549 + 0.256481i \(0.917437\pi\)
\(548\) 0 0
\(549\) 41.2541 18.8401i 1.76068 0.804076i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 15.1885 4.45975i 0.645881 0.189648i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(558\) 0 0
\(559\) 1.92605 + 2.22278i 0.0814633 + 0.0940136i
\(560\) 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
\(567\) 36.4366 + 16.6400i 1.53019 + 0.698816i
\(568\) 0 0
\(569\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(570\) 0 0
\(571\) 15.4260 17.8025i 0.645558 0.745013i −0.334790 0.942293i \(-0.608665\pi\)
0.980347 + 0.197280i \(0.0632107\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 10.2906 + 35.0467i 0.428404 + 1.45901i 0.837458 + 0.546501i \(0.184041\pi\)
−0.409054 + 0.912510i \(0.634141\pi\)
\(578\) 0 0
\(579\) 13.5244 + 46.0599i 0.562055 + 1.91418i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(588\) 0 0
\(589\) −14.5651 + 49.6041i −0.600144 + 2.04390i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −24.2437 37.7239i −0.992229 1.54394i
\(598\) 0 0
\(599\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(600\) 0 0
\(601\) 6.31118 43.8952i 0.257438 1.79052i −0.293481 0.955965i \(-0.594814\pi\)
0.550919 0.834559i \(-0.314277\pi\)
\(602\) 0 0
\(603\) 21.5639 11.7473i 0.878150 0.478385i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 45.5003 + 13.3601i 1.84680 + 0.542270i 0.999945 + 0.0104491i \(0.00332612\pi\)
0.846857 + 0.531821i \(0.178492\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −15.2309 + 17.5774i −0.615170 + 0.709944i −0.974782 0.223157i \(-0.928364\pi\)
0.359612 + 0.933102i \(0.382909\pi\)
\(614\) 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
\(619\) 5.31582 + 3.41627i 0.213661 + 0.137312i 0.643094 0.765787i \(-0.277650\pi\)
−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
\(631\) −44.6925 20.4104i −1.77918 0.812524i −0.976312 0.216369i \(-0.930579\pi\)
−0.802869 0.596156i \(-0.796694\pi\)
\(632\) 0 0
\(633\) 0.606587i 0.0241097i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 12.5980 1.81132i 0.499152 0.0717673i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −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 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 50.2080 7.21882i 1.96781 0.282928i
\(652\) 0 0
\(653\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −19.8201 43.4000i −0.773257 1.69320i
\(658\) 0 0
\(659\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(660\) 0 0
\(661\) 43.4964 + 6.25383i 1.69181 + 0.243246i 0.919807 0.392370i \(-0.128345\pi\)
0.772005 + 0.635616i \(0.219254\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 45.3626i 1.75382i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −11.6199 + 10.0687i −0.447913 + 0.388119i −0.849403 0.527744i \(-0.823038\pi\)
0.401490 + 0.915863i \(0.368492\pi\)
\(674\) 0 0
\(675\) −14.0463 + 21.8564i −0.540641 + 0.841254i
\(676\) 0 0
\(677\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(678\) 0 0
\(679\) −0.954206 + 6.63665i −0.0366191 + 0.254691i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −47.6068 + 13.9786i −1.81631 + 0.533317i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 36.9668 23.7571i 1.40628 0.903764i 0.406334 0.913725i \(-0.366807\pi\)
0.999950 + 0.00996078i \(0.00317067\pi\)
\(692\) 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 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(702\) 0 0
\(703\) 2.40175 + 16.7045i 0.0905837 + 0.630023i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 9.24089 20.2347i 0.347049 0.759931i −0.652948 0.757403i \(-0.726468\pi\)
0.999997 0.00252851i \(-0.000804851\pi\)
\(710\) 0 0
\(711\) 3.00609 + 10.2378i 0.112737 + 0.383948i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(720\) 0 0
\(721\) 1.40174 2.18115i 0.0522036 0.0812304i
\(722\) 0 0
\(723\) 15.1391 51.5589i 0.563028 1.91750i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 31.9760 + 27.7073i 1.18592 + 1.02761i 0.998976 + 0.0452379i \(0.0144046\pi\)
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
\(733\) 14.7766 50.3245i 0.545786 1.85878i 0.0344317 0.999407i \(-0.489038\pi\)
0.511354 0.859370i \(-0.329144\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 44.8874 + 6.45384i 1.65121 + 0.237408i 0.904109 0.427302i \(-0.140536\pi\)
0.747101 + 0.664710i \(0.231445\pi\)
\(740\) 0 0
\(741\) −12.9744 3.80964i −0.476628 0.139951i
\(742\) 0 0
\(743\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(744\) 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
\(757\) 41.4569 35.9226i 1.50678 1.30563i 0.701654 0.712517i \(-0.252445\pi\)
0.805121 0.593111i \(-0.202100\pi\)
\(758\) 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
\(763\) −79.2865 + 23.2806i −2.87036 + 0.842815i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 18.7758 + 16.2693i 0.677071 + 0.586686i 0.924021 0.382343i \(-0.124883\pi\)
−0.246949 + 0.969028i \(0.579428\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(774\) 0 0
\(775\) 32.9000i 1.18180i
\(776\) 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
\(787\) −13.9459 47.4952i −0.497116 1.69302i −0.700272 0.713876i \(-0.746938\pi\)
0.203156 0.979146i \(-0.434880\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 12.6370 + 8.12129i 0.448752 + 0.288396i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\)