Properties

Label 2520.2.t.g.1009.5
Level $2520$
Weight $2$
Character 2520.1009
Analytic conductor $20.122$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.t (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.1223013094\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.5
Root \(-1.75233 + 1.75233i\) of defining polynomial
Character \(\chi\) \(=\) 2520.1009
Dual form 2520.2.t.g.1009.6

$q$-expansion

\(f(q)\) \(=\) \(q+(1.75233 - 1.38900i) q^{5} +1.00000i q^{7} +O(q^{10})\) \(q+(1.75233 - 1.38900i) q^{5} +1.00000i q^{7} -5.14134 q^{11} +4.64600i q^{13} -3.86799i q^{17} +0.778008 q^{19} +5.00933i q^{23} +(1.14134 - 4.86799i) q^{25} +9.42401 q^{29} +4.72666 q^{31} +(1.38900 + 1.75233i) q^{35} +6.00000i q^{37} +1.00933 q^{41} +7.00933i q^{43} +11.4240i q^{47} -1.00000 q^{49} +7.55602i q^{53} +(-9.00933 + 7.14134i) q^{55} +12.5140 q^{59} +11.5047 q^{61} +(6.45331 + 8.14134i) q^{65} -11.7360i q^{67} -2.72666 q^{71} -5.00933i q^{73} -5.14134i q^{77} -5.68802 q^{79} -4.67531i q^{83} +(-5.37266 - 6.77801i) q^{85} -2.82936 q^{89} -4.64600 q^{91} +(1.36333 - 1.08066i) q^{95} -1.58532i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + O(q^{10}) \) \( 6q - 14q^{11} - 8q^{19} - 10q^{25} + 6q^{29} + 20q^{31} + 2q^{35} - 36q^{41} - 6q^{49} - 12q^{55} + 12q^{59} + 48q^{61} + 22q^{65} - 8q^{71} - 34q^{79} + 14q^{85} + 10q^{91} + 4q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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 0
\(4\) 0 0
\(5\) 1.75233 1.38900i 0.783667 0.621181i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.14134 −1.55017 −0.775086 0.631856i \(-0.782293\pi\)
−0.775086 + 0.631856i \(0.782293\pi\)
\(12\) 0 0
\(13\) 4.64600i 1.28857i 0.764786 + 0.644284i \(0.222845\pi\)
−0.764786 + 0.644284i \(0.777155\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.86799i 0.938126i −0.883165 0.469063i \(-0.844592\pi\)
0.883165 0.469063i \(-0.155408\pi\)
\(18\) 0 0
\(19\) 0.778008 0.178487 0.0892436 0.996010i \(-0.471555\pi\)
0.0892436 + 0.996010i \(0.471555\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.00933i 1.04452i 0.852787 + 0.522259i \(0.174910\pi\)
−0.852787 + 0.522259i \(0.825090\pi\)
\(24\) 0 0
\(25\) 1.14134 4.86799i 0.228267 0.973599i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.42401 1.74999 0.874997 0.484128i \(-0.160863\pi\)
0.874997 + 0.484128i \(0.160863\pi\)
\(30\) 0 0
\(31\) 4.72666 0.848933 0.424466 0.905444i \(-0.360462\pi\)
0.424466 + 0.905444i \(0.360462\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.38900 + 1.75233i 0.234785 + 0.296198i
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.00933 0.157631 0.0788153 0.996889i \(-0.474886\pi\)
0.0788153 + 0.996889i \(0.474886\pi\)
\(42\) 0 0
\(43\) 7.00933i 1.06891i 0.845196 + 0.534456i \(0.179484\pi\)
−0.845196 + 0.534456i \(0.820516\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.4240i 1.66636i 0.553000 + 0.833181i \(0.313483\pi\)
−0.553000 + 0.833181i \(0.686517\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.55602i 1.03790i 0.854805 + 0.518949i \(0.173677\pi\)
−0.854805 + 0.518949i \(0.826323\pi\)
\(54\) 0 0
\(55\) −9.00933 + 7.14134i −1.21482 + 0.962938i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.5140 1.62918 0.814592 0.580035i \(-0.196961\pi\)
0.814592 + 0.580035i \(0.196961\pi\)
\(60\) 0 0
\(61\) 11.5047 1.47302 0.736511 0.676426i \(-0.236472\pi\)
0.736511 + 0.676426i \(0.236472\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.45331 + 8.14134i 0.800435 + 1.00981i
\(66\) 0 0
\(67\) 11.7360i 1.43378i −0.697187 0.716889i \(-0.745565\pi\)
0.697187 0.716889i \(-0.254435\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.72666 −0.323595 −0.161797 0.986824i \(-0.551729\pi\)
−0.161797 + 0.986824i \(0.551729\pi\)
\(72\) 0 0
\(73\) 5.00933i 0.586298i −0.956067 0.293149i \(-0.905297\pi\)
0.956067 0.293149i \(-0.0947031\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.14134i 0.585910i
\(78\) 0 0
\(79\) −5.68802 −0.639953 −0.319976 0.947426i \(-0.603675\pi\)
−0.319976 + 0.947426i \(0.603675\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.67531i 0.513181i −0.966520 0.256591i \(-0.917401\pi\)
0.966520 0.256591i \(-0.0825992\pi\)
\(84\) 0 0
\(85\) −5.37266 6.77801i −0.582746 0.735178i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.82936 −0.299911 −0.149956 0.988693i \(-0.547913\pi\)
−0.149956 + 0.988693i \(0.547913\pi\)
\(90\) 0 0
\(91\) −4.64600 −0.487033
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.36333 1.08066i 0.139875 0.110873i
\(96\) 0 0
\(97\) 1.58532i 0.160965i −0.996756 0.0804824i \(-0.974354\pi\)
0.996756 0.0804824i \(-0.0256461\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.06068 0.901571 0.450786 0.892632i \(-0.351144\pi\)
0.450786 + 0.892632i \(0.351144\pi\)
\(102\) 0 0
\(103\) 5.14134i 0.506591i −0.967389 0.253295i \(-0.918486\pi\)
0.967389 0.253295i \(-0.0815145\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 3.97070 0.380324 0.190162 0.981753i \(-0.439099\pi\)
0.190162 + 0.981753i \(0.439099\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.54669i 0.427716i 0.976865 + 0.213858i \(0.0686030\pi\)
−0.976865 + 0.213858i \(0.931397\pi\)
\(114\) 0 0
\(115\) 6.95798 + 8.77801i 0.648835 + 0.818553i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.86799 0.354578
\(120\) 0 0
\(121\) 15.4333 1.40303
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.76166 10.1157i −0.425896 0.904772i
\(126\) 0 0
\(127\) 16.1214i 1.43054i 0.698849 + 0.715270i \(0.253696\pi\)
−0.698849 + 0.715270i \(0.746304\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.0513514 −0.00448659 −0.00224330 0.999997i \(-0.500714\pi\)
−0.00224330 + 0.999997i \(0.500714\pi\)
\(132\) 0 0
\(133\) 0.778008i 0.0674618i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.0187i 1.19769i 0.800863 + 0.598847i \(0.204374\pi\)
−0.800863 + 0.598847i \(0.795626\pi\)
\(138\) 0 0
\(139\) −12.6167 −1.07013 −0.535067 0.844810i \(-0.679714\pi\)
−0.535067 + 0.844810i \(0.679714\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 23.8867i 1.99750i
\(144\) 0 0
\(145\) 16.5140 13.0900i 1.37141 1.08706i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −11.4533 −0.938292 −0.469146 0.883121i \(-0.655438\pi\)
−0.469146 + 0.883121i \(0.655438\pi\)
\(150\) 0 0
\(151\) 5.86799 0.477530 0.238765 0.971077i \(-0.423257\pi\)
0.238765 + 0.971077i \(0.423257\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.28267 6.56534i 0.665280 0.527341i
\(156\) 0 0
\(157\) 5.78734i 0.461880i −0.972968 0.230940i \(-0.925820\pi\)
0.972968 0.230940i \(-0.0741801\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.00933 −0.394790
\(162\) 0 0
\(163\) 3.27334i 0.256388i −0.991749 0.128194i \(-0.959082\pi\)
0.991749 0.128194i \(-0.0409180\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 24.8773i 1.92506i 0.271166 + 0.962532i \(0.412591\pi\)
−0.271166 + 0.962532i \(0.587409\pi\)
\(168\) 0 0
\(169\) −8.58532 −0.660409
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.62734i 0.503868i 0.967744 + 0.251934i \(0.0810665\pi\)
−0.967744 + 0.251934i \(0.918933\pi\)
\(174\) 0 0
\(175\) 4.86799 + 1.14134i 0.367986 + 0.0862769i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −10.0187 −0.748830 −0.374415 0.927261i \(-0.622156\pi\)
−0.374415 + 0.927261i \(0.622156\pi\)
\(180\) 0 0
\(181\) 1.78734 0.132852 0.0664258 0.997791i \(-0.478840\pi\)
0.0664258 + 0.997791i \(0.478840\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.33402 + 10.5140i 0.612730 + 0.773004i
\(186\) 0 0
\(187\) 19.8867i 1.45426i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.59465 0.621887 0.310943 0.950428i \(-0.399355\pi\)
0.310943 + 0.950428i \(0.399355\pi\)
\(192\) 0 0
\(193\) 5.17064i 0.372191i 0.982532 + 0.186095i \(0.0595834\pi\)
−0.982532 + 0.186095i \(0.940417\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.9160i 1.70394i −0.523590 0.851971i \(-0.675407\pi\)
0.523590 0.851971i \(-0.324593\pi\)
\(198\) 0 0
\(199\) 15.3107 1.08534 0.542672 0.839945i \(-0.317413\pi\)
0.542672 + 0.839945i \(0.317413\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.42401i 0.661436i
\(204\) 0 0
\(205\) 1.76868 1.40196i 0.123530 0.0979172i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −1.03863 −0.0715025 −0.0357512 0.999361i \(-0.511382\pi\)
−0.0357512 + 0.999361i \(0.511382\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.73599 + 12.2827i 0.663989 + 0.837671i
\(216\) 0 0
\(217\) 4.72666i 0.320866i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 17.9707 1.20884
\(222\) 0 0
\(223\) 9.86799i 0.660810i 0.943839 + 0.330405i \(0.107185\pi\)
−0.943839 + 0.330405i \(0.892815\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.6553i 1.43731i 0.695364 + 0.718657i \(0.255243\pi\)
−0.695364 + 0.718657i \(0.744757\pi\)
\(228\) 0 0
\(229\) −20.2313 −1.33692 −0.668462 0.743747i \(-0.733047\pi\)
−0.668462 + 0.743747i \(0.733047\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.45331i 0.357258i −0.983916 0.178629i \(-0.942834\pi\)
0.983916 0.178629i \(-0.0571663\pi\)
\(234\) 0 0
\(235\) 15.8680 + 20.0187i 1.03511 + 1.30587i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −11.5853 −0.749392 −0.374696 0.927148i \(-0.622253\pi\)
−0.374696 + 0.927148i \(0.622253\pi\)
\(240\) 0 0
\(241\) −5.00933 −0.322679 −0.161340 0.986899i \(-0.551581\pi\)
−0.161340 + 0.986899i \(0.551581\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.75233 + 1.38900i −0.111952 + 0.0887402i
\(246\) 0 0
\(247\) 3.61462i 0.229993i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 23.4206 1.47830 0.739148 0.673543i \(-0.235228\pi\)
0.739148 + 0.673543i \(0.235228\pi\)
\(252\) 0 0
\(253\) 25.7546i 1.61918i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.0093i 0.811500i −0.913984 0.405750i \(-0.867010\pi\)
0.913984 0.405750i \(-0.132990\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.01866i 0.371126i 0.982632 + 0.185563i \(0.0594109\pi\)
−0.982632 + 0.185563i \(0.940589\pi\)
\(264\) 0 0
\(265\) 10.4953 + 13.2406i 0.644723 + 0.813367i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.24065 0.197586 0.0987929 0.995108i \(-0.468502\pi\)
0.0987929 + 0.995108i \(0.468502\pi\)
\(270\) 0 0
\(271\) −29.1307 −1.76956 −0.884782 0.466006i \(-0.845693\pi\)
−0.884782 + 0.466006i \(0.845693\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.86799 + 25.0280i −0.353853 + 1.50924i
\(276\) 0 0
\(277\) 4.44398i 0.267013i −0.991048 0.133507i \(-0.957376\pi\)
0.991048 0.133507i \(-0.0426237\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 24.6974 1.47332 0.736660 0.676263i \(-0.236402\pi\)
0.736660 + 0.676263i \(0.236402\pi\)
\(282\) 0 0
\(283\) 7.73937i 0.460058i −0.973184 0.230029i \(-0.926118\pi\)
0.973184 0.230029i \(-0.0738821\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.00933i 0.0595788i
\(288\) 0 0
\(289\) 2.03863 0.119920
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 25.1086i 1.46686i 0.679764 + 0.733431i \(0.262082\pi\)
−0.679764 + 0.733431i \(0.737918\pi\)
\(294\) 0 0
\(295\) 21.9287 17.3820i 1.27674 1.01202i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −23.2733 −1.34593
\(300\) 0 0
\(301\) −7.00933 −0.404011
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 20.1600 15.9800i 1.15436 0.915014i
\(306\) 0 0
\(307\) 33.9193i 1.93588i −0.251183 0.967940i \(-0.580820\pi\)
0.251183 0.967940i \(-0.419180\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −0.565344 −0.0320577 −0.0160289 0.999872i \(-0.505102\pi\)
−0.0160289 + 0.999872i \(0.505102\pi\)
\(312\) 0 0
\(313\) 27.3400i 1.54535i 0.634804 + 0.772673i \(0.281081\pi\)
−0.634804 + 0.772673i \(0.718919\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.7267i 1.27646i −0.769847 0.638228i \(-0.779668\pi\)
0.769847 0.638228i \(-0.220332\pi\)
\(318\) 0 0
\(319\) −48.4520 −2.71279
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.00933i 0.167444i
\(324\) 0 0
\(325\) 22.6167 + 5.30265i 1.25455 + 0.294138i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −11.4240 −0.629826
\(330\) 0 0
\(331\) 0.462642 0.0254291 0.0127145 0.999919i \(-0.495953\pi\)
0.0127145 + 0.999919i \(0.495953\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −16.3013 20.5653i −0.890637 1.12360i
\(336\) 0 0
\(337\) 10.5653i 0.575531i −0.957701 0.287765i \(-0.907088\pi\)
0.957701 0.287765i \(-0.0929124\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −24.3013 −1.31599
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 31.8760i 1.71119i −0.517643 0.855597i \(-0.673190\pi\)
0.517643 0.855597i \(-0.326810\pi\)
\(348\) 0 0
\(349\) 9.62602 0.515269 0.257635 0.966242i \(-0.417057\pi\)
0.257635 + 0.966242i \(0.417057\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 26.2534i 1.39733i −0.715451 0.698663i \(-0.753779\pi\)
0.715451 0.698663i \(-0.246221\pi\)
\(354\) 0 0
\(355\) −4.77801 + 3.78734i −0.253590 + 0.201011i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4.56534 −0.240950 −0.120475 0.992716i \(-0.538442\pi\)
−0.120475 + 0.992716i \(0.538442\pi\)
\(360\) 0 0
\(361\) −18.3947 −0.968142
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.95798 8.77801i −0.364197 0.459462i
\(366\) 0 0
\(367\) 3.79073i 0.197874i 0.995094 + 0.0989371i \(0.0315443\pi\)
−0.995094 + 0.0989371i \(0.968456\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −7.55602 −0.392289
\(372\) 0 0
\(373\) 20.7453i 1.07415i −0.843534 0.537076i \(-0.819529\pi\)
0.843534 0.537076i \(-0.180471\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 43.7839i 2.25499i
\(378\) 0 0
\(379\) −3.00933 −0.154579 −0.0772894 0.997009i \(-0.524627\pi\)
−0.0772894 + 0.997009i \(0.524627\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 7.00933i 0.358160i 0.983835 + 0.179080i \(0.0573121\pi\)
−0.983835 + 0.179080i \(0.942688\pi\)
\(384\) 0 0
\(385\) −7.14134 9.00933i −0.363956 0.459158i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −19.7653 −1.00214 −0.501070 0.865407i \(-0.667060\pi\)
−0.501070 + 0.865407i \(0.667060\pi\)
\(390\) 0 0
\(391\) 19.3760 0.979889
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.96731 + 7.90069i −0.501510 + 0.397527i
\(396\) 0 0
\(397\) 9.37266i 0.470400i −0.971947 0.235200i \(-0.924425\pi\)
0.971947 0.235200i \(-0.0755745\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 16.1507 0.806526 0.403263 0.915084i \(-0.367876\pi\)
0.403263 + 0.915084i \(0.367876\pi\)
\(402\) 0 0
\(403\) 21.9600i 1.09391i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 30.8480i 1.52908i
\(408\) 0 0
\(409\) −15.2920 −0.756141 −0.378070 0.925777i \(-0.623412\pi\)
−0.378070 + 0.925777i \(0.623412\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 12.5140i 0.615773i
\(414\) 0 0
\(415\) −6.49402 8.19269i −0.318779 0.402163i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −33.1820 −1.62105 −0.810524 0.585705i \(-0.800818\pi\)
−0.810524 + 0.585705i \(0.800818\pi\)
\(420\) 0 0
\(421\) 27.8094 1.35535 0.677673 0.735363i \(-0.262988\pi\)
0.677673 + 0.735363i \(0.262988\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −18.8294 4.41468i −0.913358 0.214143i
\(426\) 0 0
\(427\) 11.5047i 0.556750i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 21.5454 1.03780 0.518902 0.854834i \(-0.326341\pi\)
0.518902 + 0.854834i \(0.326341\pi\)
\(432\) 0 0
\(433\) 36.0187i 1.73095i 0.500955 + 0.865473i \(0.332982\pi\)
−0.500955 + 0.865473i \(0.667018\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.89730i 0.186433i
\(438\) 0 0
\(439\) 21.1893 1.01131 0.505655 0.862736i \(-0.331251\pi\)
0.505655 + 0.862736i \(0.331251\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 11.1120i 0.527949i −0.964530 0.263974i \(-0.914967\pi\)
0.964530 0.263974i \(-0.0850334\pi\)
\(444\) 0 0
\(445\) −4.95798 + 3.92999i −0.235031 + 0.186299i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.961367 0.0453697 0.0226848 0.999743i \(-0.492779\pi\)
0.0226848 + 0.999743i \(0.492779\pi\)
\(450\) 0 0
\(451\) −5.18930 −0.244355
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.14134 + 6.45331i −0.381672 + 0.302536i
\(456\) 0 0
\(457\) 28.2241i 1.32027i 0.751149 + 0.660133i \(0.229500\pi\)
−0.751149 + 0.660133i \(0.770500\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −26.0887 −1.21507 −0.607535 0.794293i \(-0.707842\pi\)
−0.607535 + 0.794293i \(0.707842\pi\)
\(462\) 0 0
\(463\) 2.90663i 0.135082i 0.997716 + 0.0675412i \(0.0215154\pi\)
−0.997716 + 0.0675412i \(0.978485\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.2020i 0.564642i −0.959320 0.282321i \(-0.908896\pi\)
0.959320 0.282321i \(-0.0911043\pi\)
\(468\) 0 0
\(469\) 11.7360 0.541917
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 36.0373i 1.65700i
\(474\) 0 0
\(475\) 0.887968 3.78734i 0.0407428 0.173775i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −17.6519 −0.806538 −0.403269 0.915082i \(-0.632126\pi\)
−0.403269 + 0.915082i \(0.632126\pi\)
\(480\) 0 0
\(481\) −27.8760 −1.27104
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.20202 2.77801i −0.0999884 0.126143i
\(486\) 0 0
\(487\) 1.57467i 0.0713553i −0.999363 0.0356776i \(-0.988641\pi\)
0.999363 0.0356776i \(-0.0113590\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.26270 0.147243 0.0736217 0.997286i \(-0.476544\pi\)
0.0736217 + 0.997286i \(0.476544\pi\)
\(492\) 0 0
\(493\) 36.4520i 1.64172i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.72666i 0.122307i
\(498\) 0 0
\(499\) −42.5293 −1.90387 −0.951936 0.306298i \(-0.900910\pi\)
−0.951936 + 0.306298i \(0.900910\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.6974i 0.833674i −0.908981 0.416837i \(-0.863139\pi\)
0.908981 0.416837i \(-0.136861\pi\)
\(504\) 0 0
\(505\) 15.8773 12.5853i 0.706532 0.560039i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 20.2313 0.896738 0.448369 0.893849i \(-0.352005\pi\)
0.448369 + 0.893849i \(0.352005\pi\)
\(510\) 0 0
\(511\) 5.00933 0.221600
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.14134 9.00933i −0.314685 0.396998i
\(516\) 0 0
\(517\) 58.7347i 2.58315i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −21.8387 −0.956770 −0.478385 0.878150i \(-0.658778\pi\)
−0.478385 + 0.878150i \(0.658778\pi\)
\(522\) 0 0
\(523\) 20.4554i 0.894451i 0.894421 + 0.447226i \(0.147588\pi\)
−0.894421 + 0.447226i \(0.852412\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18.2827i 0.796406i
\(528\) 0 0
\(529\) −2.09337 −0.0910163
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.68934i 0.203118i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.14134 0.221453
\(540\) 0 0
\(541\) 27.3400 1.17544 0.587718 0.809066i \(-0.300026\pi\)
0.587718 + 0.809066i \(0.300026\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.95798 5.51531i 0.298047 0.236250i
\(546\) 0 0
\(547\) 22.6867i 0.970013i −0.874510 0.485007i \(-0.838817\pi\)
0.874510 0.485007i \(-0.161183\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.33195 0.312352
\(552\) 0 0
\(553\) 5.68802i 0.241879i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12.7453i 0.540036i 0.962855 + 0.270018i \(0.0870297\pi\)
−0.962855 + 0.270018i \(0.912970\pi\)
\(558\) 0 0
\(559\) −32.5653 −1.37737
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.20333i 0.219294i −0.993971 0.109647i \(-0.965028\pi\)
0.993971 0.109647i \(-0.0349721\pi\)
\(564\) 0 0
\(565\) 6.31537 + 7.96731i 0.265689 + 0.335187i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7.37605 −0.309220 −0.154610 0.987976i \(-0.549412\pi\)
−0.154610 + 0.987976i \(0.549412\pi\)
\(570\) 0 0
\(571\) 15.8973 0.665281 0.332641 0.943054i \(-0.392060\pi\)
0.332641 + 0.943054i \(0.392060\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 24.3854 + 5.71733i 1.01694 + 0.238429i
\(576\) 0 0
\(577\) 19.9707i 0.831391i −0.909504 0.415695i \(-0.863538\pi\)
0.909504 0.415695i \(-0.136462\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.67531 0.193964
\(582\) 0 0
\(583\) 38.8480i 1.60892i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.8060i 0.569834i 0.958552 + 0.284917i \(0.0919661\pi\)
−0.958552 + 0.284917i \(0.908034\pi\)
\(588\) 0 0
\(589\) 3.67738 0.151524
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 20.0666i 0.824037i 0.911175 + 0.412019i \(0.135176\pi\)
−0.911175 + 0.412019i \(0.864824\pi\)
\(594\) 0 0
\(595\) 6.77801 5.37266i 0.277871 0.220257i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 46.0267 1.88060 0.940299 0.340349i \(-0.110545\pi\)
0.940299 + 0.340349i \(0.110545\pi\)
\(600\) 0 0
\(601\) −1.37605 −0.0561301 −0.0280651 0.999606i \(-0.508935\pi\)
−0.0280651 + 0.999606i \(0.508935\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 27.0443 21.4370i 1.09951 0.871537i
\(606\) 0 0
\(607\) 18.7933i 0.762796i 0.924411 + 0.381398i \(0.124557\pi\)
−0.924411 + 0.381398i \(0.875443\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −53.0759 −2.14722
\(612\) 0 0
\(613\) 24.8480i 1.00360i −0.864983 0.501801i \(-0.832671\pi\)
0.864983 0.501801i \(-0.167329\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 43.1680i 1.73788i −0.494919 0.868939i \(-0.664802\pi\)
0.494919 0.868939i \(-0.335198\pi\)
\(618\) 0 0
\(619\) −31.9486 −1.28412 −0.642062 0.766652i \(-0.721921\pi\)
−0.642062 + 0.766652i \(0.721921\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.82936i 0.113356i
\(624\) 0 0
\(625\) −22.3947 11.1120i −0.895788 0.444481i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 23.2080 0.925362
\(630\) 0 0
\(631\) −8.59465 −0.342148 −0.171074 0.985258i \(-0.554724\pi\)
−0.171074 + 0.985258i \(0.554724\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 22.3926 + 28.2500i 0.888625 + 1.12107i
\(636\) 0 0
\(637\) 4.64600i 0.184081i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −33.2266 −1.31237 −0.656186 0.754599i \(-0.727831\pi\)
−0.656186 + 0.754599i \(0.727831\pi\)
\(642\) 0 0
\(643\) 17.4940i 0.689897i −0.938622 0.344948i \(-0.887896\pi\)
0.938622 0.344948i \(-0.112104\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.91595i 0.389836i −0.980820 0.194918i \(-0.937556\pi\)
0.980820 0.194918i \(-0.0624441\pi\)
\(648\) 0 0
\(649\) −64.3386 −2.52551
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 14.6240i 0.572280i 0.958188 + 0.286140i \(0.0923722\pi\)
−0.958188 + 0.286140i \(0.907628\pi\)
\(654\) 0 0
\(655\) −0.0899847 + 0.0713273i −0.00351599 + 0.00278699i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.2534 0.788959 0.394480 0.918905i \(-0.370925\pi\)
0.394480 + 0.918905i \(0.370925\pi\)
\(660\) 0 0
\(661\) −12.4299 −0.483469 −0.241734 0.970342i \(-0.577716\pi\)
−0.241734 + 0.970342i \(0.577716\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.08066 + 1.36333i 0.0419060 + 0.0528676i
\(666\) 0 0
\(667\) 47.2080i 1.82790i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −59.1493 −2.28344
\(672\) 0 0
\(673\) 47.6774i 1.83783i 0.394458 + 0.918914i \(0.370932\pi\)
−0.394458 + 0.918914i \(0.629068\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00339i 0.230729i −0.993323 0.115364i \(-0.963196\pi\)
0.993323 0.115364i \(-0.0368036\pi\)
\(678\) 0 0
\(679\) 1.58532 0.0608390
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 28.5653i 1.09302i −0.837452 0.546511i \(-0.815956\pi\)
0.837452 0.546511i \(-0.184044\pi\)
\(684\) 0 0
\(685\) 19.4720 + 24.5653i 0.743986 + 0.938594i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −35.1053 −1.33740
\(690\) 0 0
\(691\) 47.8247 1.81934 0.909668 0.415337i \(-0.136336\pi\)
0.909668 + 0.415337i \(0.136336\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −22.1086 + 17.5246i −0.838629 + 0.664747i
\(696\) 0 0
\(697\) 3.90408i 0.147877i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.80005 0.181296 0.0906478 0.995883i \(-0.471106\pi\)
0.0906478 + 0.995883i \(0.471106\pi\)
\(702\) 0 0
\(703\) 4.66805i 0.176059i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.06068i 0.340762i
\(708\) 0 0
\(709\) −11.7653 −0.441855 −0.220927 0.975290i \(-0.570908\pi\)
−0.220927 + 0.975290i \(0.570908\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 23.6774i 0.886725i
\(714\) 0 0
\(715\) −33.1787 41.8573i −1.24081 1.56538i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −40.7640 −1.52024 −0.760120 0.649783i \(-0.774860\pi\)
−0.760120 + 0.649783i \(0.774860\pi\)
\(720\) 0 0
\(721\) 5.14134 0.191473
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 10.7560 45.8760i 0.399466 1.70379i
\(726\) 0 0
\(727\) 22.1214i 0.820436i 0.911988 + 0.410218i \(0.134547\pi\)
−0.911988 + 0.410218i \(0.865453\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 27.1120 1.00277
\(732\) 0 0
\(733\) 13.0500i 0.482014i −0.970523 0.241007i \(-0.922522\pi\)
0.970523 0.241007i \(-0.0774777\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 60.3386i 2.22260i
\(738\) 0 0
\(739\) 34.5734 1.27180 0.635901 0.771771i \(-0.280629\pi\)
0.635901 + 0.771771i \(0.280629\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.55602i 0.130458i −0.997870 0.0652288i \(-0.979222\pi\)
0.997870 0.0652288i \(-0.0207777\pi\)
\(744\) 0 0
\(745\) −20.0700 + 15.9087i −0.735308 + 0.582850i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −23.6226 −0.862002 −0.431001 0.902351i \(-0.641839\pi\)
−0.431001 + 0.902351i \(0.641839\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 10.2827 8.15066i 0.374225 0.296633i
\(756\) 0 0
\(757\) 15.1893i 0.552064i 0.961148 + 0.276032i \(0.0890196\pi\)
−0.961148 + 0.276032i \(0.910980\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −34.4626 −1.24927 −0.624635 0.780917i \(-0.714752\pi\)
−0.624635 + 0.780917i \(0.714752\pi\)
\(762\) 0 0
\(763\) 3.97070i 0.143749i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 58.1400i 2.09931i
\(768\) 0 0
\(769\) −40.3854 −1.45633 −0.728167 0.685400i \(-0.759627\pi\)
−0.728167 + 0.685400i \(0.759627\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 10.7674i 0.387275i −0.981073 0.193638i \(-0.937971\pi\)
0.981073 0.193638i \(-0.0620286\pi\)
\(774\) 0 0
\(775\) 5.39470 23.0093i 0.193783 0.826519i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.785266 0.0281351
\(780\) 0 0
\(781\) 14.0187 0.501627
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −8.03863 10.1413i −0.286911 0.361960i
\(786\) 0 0
\(787\) 9.19269i 0.327684i −0.986487 0.163842i \(-0.947611\pi\)
0.986487 0.163842i \(-0.0523887\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.54669 −0.161662
\(792\) 0 0
\(793\) 53.4507i 1.89809i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.1086i 0.747706i −0.927488 0.373853i \(-0.878036\pi\)
0.927488 0.373853i \(-0.121964\pi\)
\(798\) 0 0
\(799\) 44.1880 1.56326
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 25.7546i 0.908862i
\(804\) 0 0
\(805\) −8.77801 + 6.95798i −0.309384 + 0.245236i
\(806\) 0 0
\(807\) 0 0