Properties

Label 804.2.s.a.125.2
Level 804
Weight 2
Character 804.125
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 125.2
Root \(0.580057 - 0.814576i\)
Character \(\chi\) = 804.125
Dual form 804.2.s.a.521.2

$q$-expansion

\(f(q)\) \(=\) \(q+(0.487975 - 1.66189i) q^{3} +(2.74514 - 4.27152i) q^{7} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})\) \(q+(0.487975 - 1.66189i) q^{3} +(2.74514 - 4.27152i) q^{7} +(-2.52376 - 1.62192i) q^{9} +(-6.83933 + 0.983347i) q^{13} +(7.27844 - 4.67757i) q^{19} +(-5.75924 - 6.64652i) q^{21} +(0.711574 + 4.94911i) q^{25} +(-3.92699 + 3.40276i) q^{27} +(-9.72847 - 1.39874i) q^{31} +5.43454 q^{37} +(-1.70321 + 11.8461i) q^{39} +(5.85766 + 2.67510i) q^{43} +(-7.80220 - 17.0844i) q^{49} +(-4.22191 - 14.3785i) q^{57} +(8.00476 - 6.93617i) q^{61} +(-13.8562 + 6.32789i) q^{63} +(-2.85294 - 7.67208i) q^{67} +(-2.56926 - 2.96508i) q^{73} +(8.57211 + 1.23248i) q^{75} +(1.47911 - 0.212664i) q^{79} +(3.73874 + 8.18669i) q^{81} +(-14.5745 + 31.9138i) q^{91} +(-7.07181 + 15.4851i) q^{93} +9.35087i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{9} + 16q^{19} - 12q^{21} + 10q^{25} - 20q^{37} - 24q^{39} + 10q^{49} - 66q^{57} - 132q^{63} - 16q^{67} + 90q^{73} + 44q^{79} - 18q^{81} + 48q^{91} - 36q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{15}{22}\right)\) \(1\)

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
\(3\) 0.487975 1.66189i 0.281733 0.959493i
\(4\) 0 0
\(5\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(6\) 0 0
\(7\) 2.74514 4.27152i 1.03757 1.61448i 0.281995 0.959416i \(-0.409004\pi\)
0.755570 0.655068i \(-0.227360\pi\)
\(8\) 0 0
\(9\) −2.52376 1.62192i −0.841254 0.540641i
\(10\) 0 0
\(11\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(12\) 0 0
\(13\) −6.83933 + 0.983347i −1.89689 + 0.272731i −0.989162 0.146831i \(-0.953093\pi\)
−0.907726 + 0.419563i \(0.862183\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(18\) 0 0
\(19\) 7.27844 4.67757i 1.66979 1.07311i 0.768542 0.639799i \(-0.220983\pi\)
0.901246 0.433308i \(-0.142654\pi\)
\(20\) 0 0
\(21\) −5.75924 6.64652i −1.25677 1.45039i
\(22\) 0 0
\(23\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(24\) 0 0
\(25\) 0.711574 + 4.94911i 0.142315 + 0.989821i
\(26\) 0 0
\(27\) −3.92699 + 3.40276i −0.755750 + 0.654861i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −9.72847 1.39874i −1.74728 0.251222i −0.806744 0.590901i \(-0.798772\pi\)
−0.940541 + 0.339680i \(0.889681\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.43454 0.893432 0.446716 0.894676i \(-0.352593\pi\)
0.446716 + 0.894676i \(0.352593\pi\)
\(38\) 0 0
\(39\) −1.70321 + 11.8461i −0.272731 + 1.89689i
\(40\) 0 0
\(41\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(42\) 0 0
\(43\) 5.85766 + 2.67510i 0.893285 + 0.407949i 0.808532 0.588453i \(-0.200263\pi\)
0.0847529 + 0.996402i \(0.472990\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(48\) 0 0
\(49\) −7.80220 17.0844i −1.11460 2.44063i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.22191 14.3785i −0.559205 1.90448i
\(58\) 0 0
\(59\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(60\) 0 0
\(61\) 8.00476 6.93617i 1.02490 0.888085i 0.0311325 0.999515i \(-0.490089\pi\)
0.993772 + 0.111430i \(0.0355432\pi\)
\(62\) 0 0
\(63\) −13.8562 + 6.32789i −1.74571 + 0.797240i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.85294 7.67208i −0.348542 0.937293i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(72\) 0 0
\(73\) −2.56926 2.96508i −0.300709 0.347037i 0.585206 0.810885i \(-0.301014\pi\)
−0.885915 + 0.463848i \(0.846468\pi\)
\(74\) 0 0
\(75\) 8.57211 + 1.23248i 0.989821 + 0.142315i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.47911 0.212664i 0.166413 0.0239266i −0.0586047 0.998281i \(-0.518665\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 3.73874 + 8.18669i 0.415415 + 0.909632i
\(82\) 0 0
\(83\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(90\) 0 0
\(91\) −14.5745 + 31.9138i −1.52783 + 3.34547i
\(92\) 0 0
\(93\) −7.07181 + 15.4851i −0.733312 + 1.60573i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.35087i 0.949437i 0.880138 + 0.474718i \(0.157450\pi\)
−0.880138 + 0.474718i \(0.842550\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(102\) 0 0
\(103\) 2.66099 18.5076i 0.262195 1.82361i −0.254079 0.967183i \(-0.581772\pi\)
0.516274 0.856423i \(-0.327319\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(108\) 0 0
\(109\) 11.9764 1.72194i 1.14713 0.164932i 0.457578 0.889170i \(-0.348717\pi\)
0.689551 + 0.724238i \(0.257808\pi\)
\(110\) 0 0
\(111\) 2.65192 9.03160i 0.251709 0.857242i
\(112\) 0 0
\(113\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 18.8557 + 8.61113i 1.74321 + 0.796099i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.56546 + 10.8880i 0.142315 + 0.989821i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.0480 + 7.10014i 0.980354 + 0.630036i 0.929559 0.368674i \(-0.120188\pi\)
0.0507955 + 0.998709i \(0.483824\pi\)
\(128\) 0 0
\(129\) 7.30412 8.42940i 0.643092 0.742168i
\(130\) 0 0
\(131\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(132\) 0 0
\(133\) 43.9306i 3.80926i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(138\) 0 0
\(139\) 15.9664 + 13.8350i 1.35426 + 1.17347i 0.967963 + 0.251093i \(0.0807900\pi\)
0.386293 + 0.922376i \(0.373755\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −32.1997 + 4.62962i −2.65579 + 0.381845i
\(148\) 0 0
\(149\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(150\) 0 0
\(151\) −6.52472 + 14.2871i −0.530974 + 1.16267i 0.434141 + 0.900845i \(0.357052\pi\)
−0.965115 + 0.261826i \(0.915675\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.518645 0.152288i −0.0413924 0.0121539i 0.260971 0.965347i \(-0.415957\pi\)
−0.302363 + 0.953193i \(0.597776\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 25.3807 1.98797 0.993986 0.109509i \(-0.0349277\pi\)
0.993986 + 0.109509i \(0.0349277\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(168\) 0 0
\(169\) 33.3360 9.78834i 2.56431 0.752949i
\(170\) 0 0
\(171\) −25.9557 −1.98488
\(172\) 0 0
\(173\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(174\) 0 0
\(175\) 23.0936 + 10.5465i 1.74571 + 0.797240i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(180\) 0 0
\(181\) 22.0635 + 6.47842i 1.63996 + 0.481537i 0.966282 0.257485i \(-0.0828937\pi\)
0.673682 + 0.739022i \(0.264712\pi\)
\(182\) 0 0
\(183\) −7.62102 16.6877i −0.563362 1.23359i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3.75481 + 26.1153i 0.273122 + 1.89961i
\(190\) 0 0
\(191\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(192\) 0 0
\(193\) 1.89281 13.1648i 0.136248 0.947623i −0.800927 0.598762i \(-0.795660\pi\)
0.937175 0.348861i \(-0.113431\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(198\) 0 0
\(199\) 7.13430 + 4.58494i 0.505737 + 0.325018i 0.768507 0.639841i \(-0.221000\pi\)
−0.262770 + 0.964858i \(0.584636\pi\)
\(200\) 0 0
\(201\) −14.1423 + 0.997492i −0.997522 + 0.0703577i
\(202\) 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
\(211\) 27.6370 8.11495i 1.90261 0.558656i 0.914574 0.404419i \(-0.132526\pi\)
0.988034 0.154237i \(-0.0492918\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −32.6808 + 37.7156i −2.21852 + 2.56030i
\(218\) 0 0
\(219\) −6.18138 + 2.82294i −0.417699 + 0.190757i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −28.5868 + 8.39384i −1.91431 + 0.562093i −0.937509 + 0.347960i \(0.886874\pi\)
−0.976804 + 0.214133i \(0.931307\pi\)
\(224\) 0 0
\(225\) 6.23123 13.6445i 0.415415 0.909632i
\(226\) 0 0
\(227\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(228\) 0 0
\(229\) −24.4823 3.52002i −1.61783 0.232609i −0.726900 0.686743i \(-0.759040\pi\)
−0.890934 + 0.454133i \(0.849949\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.368345 2.56189i 0.0239266 0.166413i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 19.2681 + 22.2365i 1.24116 + 1.43238i 0.861889 + 0.507097i \(0.169281\pi\)
0.379276 + 0.925284i \(0.376173\pi\)
\(242\) 0 0
\(243\) 15.4298 2.21847i 0.989821 0.142315i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −45.1799 + 39.1486i −2.87473 + 2.49097i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(258\) 0 0
\(259\) 14.9186 23.2137i 0.926994 1.44243i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(264\) 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
\(271\) −8.36996 + 28.5055i −0.508439 + 1.73158i 0.159340 + 0.987224i \(0.449063\pi\)
−0.667779 + 0.744360i \(0.732755\pi\)
\(272\) 0 0
\(273\) 45.9252 + 39.7944i 2.77952 + 2.40847i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −24.9912 16.0609i −1.50158 0.965005i −0.994682 0.102998i \(-0.967156\pi\)
−0.506896 0.862007i \(-0.669207\pi\)
\(278\) 0 0
\(279\) 22.2837 + 19.3089i 1.33409 + 1.15599i
\(280\) 0 0
\(281\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(282\) 0 0
\(283\) −11.0217 + 7.08324i −0.655174 + 0.421055i −0.825554 0.564324i \(-0.809137\pi\)
0.170379 + 0.985379i \(0.445501\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −11.1326 12.8477i −0.654861 0.755750i
\(290\) 0 0
\(291\) 15.5401 + 4.56299i 0.910978 + 0.267487i
\(292\) 0 0
\(293\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 27.5069 17.6776i 1.58547 1.01892i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.98190 34.6498i 0.284332 1.97757i 0.0968945 0.995295i \(-0.469109\pi\)
0.187437 0.982277i \(-0.439982\pi\)
\(308\) 0 0
\(309\) −29.4591 13.4535i −1.67587 0.765344i
\(310\) 0 0
\(311\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(312\) 0 0
\(313\) −3.21722 10.9568i −0.181848 0.619317i −0.999075 0.0430013i \(-0.986308\pi\)
0.817227 0.576316i \(-0.195510\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −9.73338 33.1488i −0.539911 1.83877i
\(326\) 0 0
\(327\) 2.98249 20.7437i 0.164932 1.14713i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −6.91635 + 3.15859i −0.380157 + 0.173612i −0.596323 0.802745i \(-0.703372\pi\)
0.216166 + 0.976357i \(0.430645\pi\)
\(332\) 0 0
\(333\) −13.7155 8.81440i −0.751603 0.483026i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −19.2608 + 29.9704i −1.04920 + 1.63259i −0.321211 + 0.947008i \(0.604090\pi\)
−0.727993 + 0.685585i \(0.759547\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −59.2134 8.51361i −3.19723 0.459692i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(348\) 0 0
\(349\) 2.88747 + 6.32267i 0.154562 + 0.338445i 0.971034 0.238941i \(-0.0768002\pi\)
−0.816472 + 0.577386i \(0.804073\pi\)
\(350\) 0 0
\(351\) 23.5119 27.1342i 1.25497 1.44831i
\(352\) 0 0
\(353\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(360\) 0 0
\(361\) 23.2031 50.8077i 1.22122 2.67409i
\(362\) 0 0
\(363\) 18.8586 + 2.71146i 0.989821 + 0.142315i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −6.17855 21.0422i −0.322518 1.09839i −0.948030 0.318180i \(-0.896928\pi\)
0.625513 0.780214i \(-0.284890\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 26.3727i 1.36553i −0.730639 0.682764i \(-0.760778\pi\)
0.730639 0.682764i \(-0.239222\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −10.2498 + 34.9075i −0.526495 + 1.79308i 0.0785782 + 0.996908i \(0.474962\pi\)
−0.605073 + 0.796170i \(0.706856\pi\)
\(380\) 0 0
\(381\) 17.1908 14.8959i 0.880712 0.763142i
\(382\) 0 0
\(383\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −10.4445 16.2520i −0.530925 0.826135i
\(388\) 0 0
\(389\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.13170 3.61417i 0.157175 0.181390i −0.671700 0.740823i \(-0.734436\pi\)
0.828876 + 0.559433i \(0.188981\pi\)
\(398\) 0 0
\(399\) −73.0078 21.4370i −3.65496 1.07319i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 67.9116 3.38292
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −10.6206 + 16.5260i −0.525155 + 0.817157i −0.997949 0.0640160i \(-0.979609\pi\)
0.472794 + 0.881173i \(0.343245\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 30.7835 19.7833i 1.50747 0.968794i
\(418\) 0 0
\(419\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(420\) 0 0
\(421\) −31.4792 + 20.2304i −1.53420 + 0.985971i −0.545159 + 0.838333i \(0.683531\pi\)
−0.989041 + 0.147638i \(0.952833\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −7.65379 53.2333i −0.370393 2.57614i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 3.89621 + 0.560190i 0.187240 + 0.0269210i 0.235297 0.971924i \(-0.424394\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −16.7279 −0.798379 −0.399190 0.916868i \(-0.630708\pi\)
−0.399190 + 0.916868i \(0.630708\pi\)
\(440\) 0 0
\(441\) −8.01874 + 55.7716i −0.381845 + 2.65579i
\(442\) 0 0
\(443\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 20.5598 + 17.8151i 0.965982 + 0.837028i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.39561 + 30.5722i 0.205618 + 1.43011i 0.787240 + 0.616647i \(0.211509\pi\)
−0.581622 + 0.813459i \(0.697582\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(462\) 0 0
\(463\) 32.3981 28.0731i 1.50567 1.30467i 0.695374 0.718648i \(-0.255239\pi\)
0.810296 0.586021i \(-0.199307\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(468\) 0 0
\(469\) −40.6032 8.87453i −1.87488 0.409788i
\(470\) 0 0
\(471\) −0.506172 + 0.787619i −0.0233232 + 0.0362916i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 28.3289 + 32.6933i 1.29982 + 1.50007i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(480\) 0 0
\(481\) −37.1686 + 5.34403i −1.69474 + 0.243667i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −37.7159 + 17.2243i −1.70907 + 0.780507i −0.712157 + 0.702020i \(0.752282\pi\)
−0.996915 + 0.0784867i \(0.974991\pi\)
\(488\) 0 0
\(489\) 12.3852 42.1800i 0.560076 1.90744i
\(490\) 0 0
\(491\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 36.1112i 1.61656i 0.588798 + 0.808280i \(0.299601\pi\)
−0.588798 + 0.808280i \(0.700399\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 60.1773i 2.67257i
\(508\) 0 0
\(509\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(510\) 0 0
\(511\) −19.7184 + 2.83508i −0.872290 + 0.125416i
\(512\) 0 0
\(513\) −12.6657 + 43.1355i −0.559205 + 1.90448i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(522\) 0 0
\(523\) −2.50714 17.4375i −0.109629 0.762489i −0.968269 0.249910i \(-0.919599\pi\)
0.858640 0.512579i \(-0.171310\pi\)
\(524\) 0 0
\(525\) 28.7962 33.2326i 1.25677 1.45039i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 19.3488 + 12.4347i 0.841254 + 0.540641i
\(530\) 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
\(541\) −29.4478 25.5166i −1.26606 1.09705i −0.990751 0.135690i \(-0.956675\pi\)
−0.275307 0.961356i \(-0.588780\pi\)
\(542\) 0 0
\(543\) 21.5328 33.5058i 0.924063 1.43787i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 35.1110 + 30.4239i 1.50124 + 1.30083i 0.826791 + 0.562510i \(0.190164\pi\)
0.674449 + 0.738321i \(0.264381\pi\)
\(548\) 0 0
\(549\) −31.4520 + 4.52212i −1.34234 + 0.192999i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 3.15197 6.90185i 0.134035 0.293496i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(558\) 0 0
\(559\) −42.6930 12.5358i −1.80572 0.530208i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 45.2330 + 6.50352i 1.89961 + 0.273122i
\(568\) 0 0
\(569\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(570\) 0 0
\(571\) −2.55660 + 0.750686i −0.106990 + 0.0314152i −0.334790 0.942293i \(-0.608665\pi\)
0.227799 + 0.973708i \(0.426847\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −23.3689 10.6722i −0.972859 0.444290i −0.135400 0.990791i \(-0.543232\pi\)
−0.837458 + 0.546501i \(0.815959\pi\)
\(578\) 0 0
\(579\) −20.9548 9.56974i −0.870852 0.397705i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(588\) 0 0
\(589\) −77.3508 + 35.3249i −3.18718 + 1.45554i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11.1010 9.61910i 0.454335 0.393683i
\(598\) 0 0
\(599\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(600\) 0 0
\(601\) −12.1053 7.77957i −0.493784 0.317335i 0.269942 0.962877i \(-0.412995\pi\)
−0.763725 + 0.645541i \(0.776632\pi\)
\(602\) 0 0
\(603\) −5.24337 + 23.9897i −0.213527 + 0.976937i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −3.12466 6.84205i −0.126826 0.277710i 0.835558 0.549402i \(-0.185144\pi\)
−0.962384 + 0.271691i \(0.912417\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 46.3137 13.5989i 1.87059 0.549256i 0.872426 0.488746i \(-0.162545\pi\)
0.998168 0.0605099i \(-0.0192727\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(618\) 0 0
\(619\) 32.5816 37.6012i 1.30956 1.51132i 0.643094 0.765787i \(-0.277650\pi\)
0.666471 0.745531i \(-0.267804\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −23.9873 + 7.04331i −0.959493 + 0.281733i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 35.2997 + 5.07533i 1.40526 + 0.202046i 0.802869 0.596156i \(-0.203306\pi\)
0.602390 + 0.798202i \(0.294215\pi\)
\(632\) 0 0
\(633\) 49.8895i 1.98293i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 70.1617 + 109.174i 2.77991 + 4.32562i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 7.68998 + 8.87471i 0.303263 + 0.349984i 0.886843 0.462072i \(-0.152894\pi\)
−0.583579 + 0.812056i \(0.698348\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 46.7318 + 72.7162i 1.83157 + 2.84997i
\(652\) 0 0
\(653\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.67506 + 11.6503i 0.0653503 + 0.454521i
\(658\) 0 0
\(659\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(660\) 0 0
\(661\) −26.4678 + 41.1848i −1.02948 + 1.60190i −0.257474 + 0.966285i \(0.582890\pi\)
−0.772005 + 0.635616i \(0.780746\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 51.6041i 1.99513i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −13.2024 + 44.9633i −0.508915 + 1.73321i 0.157398 + 0.987535i \(0.449689\pi\)
−0.666314 + 0.745672i \(0.732129\pi\)
\(674\) 0 0
\(675\) −19.6349 17.0138i −0.755750 0.654861i
\(676\) 0 0
\(677\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(678\) 0 0
\(679\) 39.9424 + 25.6694i 1.53285 + 0.985103i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −17.7966 + 38.9692i −0.678984 + 1.48677i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 32.9356 + 38.0098i 1.25293 + 1.44596i 0.846597 + 0.532234i \(0.178648\pi\)
0.406334 + 0.913725i \(0.366807\pi\)
\(692\) 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 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(702\) 0 0
\(703\) 39.5549 25.4204i 1.49184 0.958749i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.05961 7.36976i 0.0397945 0.276777i −0.960202 0.279306i \(-0.909896\pi\)
0.999997 + 0.00252851i \(0.000804851\pi\)
\(710\) 0 0
\(711\) −4.07785 1.86229i −0.152931 0.0698413i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(720\) 0 0
\(721\) −71.7508 62.1724i −2.67214 2.31542i
\(722\) 0 0
\(723\) 46.3570 21.1705i 1.72404 0.787341i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −14.9248 50.8293i −0.553531 1.88515i −0.456140 0.889908i \(-0.650768\pi\)
−0.0973917 0.995246i \(-0.531050\pi\)
\(728\) 0 0
\(729\) 3.84250 26.7252i 0.142315 0.989821i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −8.68418 + 3.96593i −0.320757 + 0.146485i −0.569286 0.822140i \(-0.692780\pi\)
0.248529 + 0.968625i \(0.420053\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −3.80151 + 5.91526i −0.139841 + 0.217596i −0.904109 0.427302i \(-0.859464\pi\)
0.764268 + 0.644899i \(0.223100\pi\)
\(740\) 0 0
\(741\) 43.0141 + 94.1877i 1.58016 + 3.46007i
\(742\) 0 0
\(743\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.42864 5.31797i −0.0886222 0.194056i 0.860131 0.510073i \(-0.170382\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −3.41166 + 11.6190i −0.123999 + 0.422301i −0.997971 0.0636752i \(-0.979718\pi\)
0.873972 + 0.485977i \(0.161536\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(762\) 0 0
\(763\) 25.5215 55.8843i 0.923940 2.02315i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −2.78000 9.46783i −0.100250 0.341419i 0.894060 0.447947i \(-0.147845\pi\)
−0.994310 + 0.106528i \(0.966026\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(774\) 0 0
\(775\) 49.1426i 1.76525i
\(776\) 0 0
\(777\) −31.2988 36.1207i −1.12284 1.29582i
\(778\) 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
\(787\) −49.9725 22.8217i −1.78133 0.813504i −0.975057 0.221955i \(-0.928756\pi\)
−0.806269 0.591549i \(-0.798517\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −47.9265 + 55.3102i −1.70192 + 1.96412i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\)