Properties

Label 8280.2.p.a.1241.2
Level $8280$
Weight $2$
Character 8280.1241
Analytic conductor $66.116$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(66.1161328736\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1241.2
Character \(\chi\) \(=\) 8280.1241
Dual form 8280.2.p.a.1241.47

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} -4.64793i q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -4.64793i q^{7} +3.84841 q^{11} -6.83108 q^{13} -4.29829 q^{17} +6.20683i q^{19} +(2.23033 - 4.24566i) q^{23} +1.00000 q^{25} +1.90324i q^{29} +5.07055 q^{31} +4.64793i q^{35} -2.26790i q^{37} -10.8603i q^{41} +3.10929i q^{43} +3.10323i q^{47} -14.6033 q^{49} -8.56168 q^{53} -3.84841 q^{55} +8.43405i q^{59} +8.26761i q^{61} +6.83108 q^{65} -12.6821i q^{67} -10.5482i q^{71} -5.77129 q^{73} -17.8872i q^{77} -2.14676i q^{79} -14.9978 q^{83} +4.29829 q^{85} -2.95587 q^{89} +31.7504i q^{91} -6.20683i q^{95} +3.87584i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 48 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times\).

\(n\) \(1657\) \(2071\) \(3961\) \(4141\) \(4601\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(-1\)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.64793i 1.75675i −0.477969 0.878377i \(-0.658627\pi\)
0.477969 0.878377i \(-0.341373\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.84841 1.16034 0.580170 0.814495i \(-0.302986\pi\)
0.580170 + 0.814495i \(0.302986\pi\)
\(12\) 0 0
\(13\) −6.83108 −1.89460 −0.947300 0.320349i \(-0.896200\pi\)
−0.947300 + 0.320349i \(0.896200\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.29829 −1.04249 −0.521244 0.853408i \(-0.674532\pi\)
−0.521244 + 0.853408i \(0.674532\pi\)
\(18\) 0 0
\(19\) 6.20683i 1.42394i 0.702208 + 0.711972i \(0.252198\pi\)
−0.702208 + 0.711972i \(0.747802\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.23033 4.24566i 0.465056 0.885281i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.90324i 0.353423i 0.984263 + 0.176712i \(0.0565460\pi\)
−0.984263 + 0.176712i \(0.943454\pi\)
\(30\) 0 0
\(31\) 5.07055 0.910698 0.455349 0.890313i \(-0.349515\pi\)
0.455349 + 0.890313i \(0.349515\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.64793i 0.785644i
\(36\) 0 0
\(37\) 2.26790i 0.372841i −0.982470 0.186420i \(-0.940311\pi\)
0.982470 0.186420i \(-0.0596887\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.8603i 1.69609i −0.529927 0.848043i \(-0.677781\pi\)
0.529927 0.848043i \(-0.322219\pi\)
\(42\) 0 0
\(43\) 3.10929i 0.474162i 0.971490 + 0.237081i \(0.0761906\pi\)
−0.971490 + 0.237081i \(0.923809\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.10323i 0.452652i 0.974052 + 0.226326i \(0.0726715\pi\)
−0.974052 + 0.226326i \(0.927328\pi\)
\(48\) 0 0
\(49\) −14.6033 −2.08618
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.56168 −1.17604 −0.588018 0.808848i \(-0.700092\pi\)
−0.588018 + 0.808848i \(0.700092\pi\)
\(54\) 0 0
\(55\) −3.84841 −0.518920
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.43405i 1.09802i 0.835816 + 0.549010i \(0.184995\pi\)
−0.835816 + 0.549010i \(0.815005\pi\)
\(60\) 0 0
\(61\) 8.26761i 1.05856i 0.848447 + 0.529280i \(0.177538\pi\)
−0.848447 + 0.529280i \(0.822462\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.83108 0.847291
\(66\) 0 0
\(67\) 12.6821i 1.54936i −0.632353 0.774680i \(-0.717911\pi\)
0.632353 0.774680i \(-0.282089\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.5482i 1.25184i −0.779886 0.625922i \(-0.784723\pi\)
0.779886 0.625922i \(-0.215277\pi\)
\(72\) 0 0
\(73\) −5.77129 −0.675479 −0.337739 0.941240i \(-0.609662\pi\)
−0.337739 + 0.941240i \(0.609662\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 17.8872i 2.03843i
\(78\) 0 0
\(79\) 2.14676i 0.241530i −0.992681 0.120765i \(-0.961465\pi\)
0.992681 0.120765i \(-0.0385347\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −14.9978 −1.64622 −0.823109 0.567883i \(-0.807762\pi\)
−0.823109 + 0.567883i \(0.807762\pi\)
\(84\) 0 0
\(85\) 4.29829 0.466215
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.95587 −0.313322 −0.156661 0.987652i \(-0.550073\pi\)
−0.156661 + 0.987652i \(0.550073\pi\)
\(90\) 0 0
\(91\) 31.7504i 3.32834i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.20683i 0.636807i
\(96\) 0 0
\(97\) 3.87584i 0.393532i 0.980450 + 0.196766i \(0.0630439\pi\)
−0.980450 + 0.196766i \(0.936956\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.965924i 0.0961130i −0.998845 0.0480565i \(-0.984697\pi\)
0.998845 0.0480565i \(-0.0153028\pi\)
\(102\) 0 0
\(103\) 4.95069i 0.487806i 0.969800 + 0.243903i \(0.0784278\pi\)
−0.969800 + 0.243903i \(0.921572\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.8916 1.34295 0.671475 0.741027i \(-0.265661\pi\)
0.671475 + 0.741027i \(0.265661\pi\)
\(108\) 0 0
\(109\) 5.90475i 0.565573i 0.959183 + 0.282786i \(0.0912588\pi\)
−0.959183 + 0.282786i \(0.908741\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.33678 −0.313898 −0.156949 0.987607i \(-0.550166\pi\)
−0.156949 + 0.987607i \(0.550166\pi\)
\(114\) 0 0
\(115\) −2.23033 + 4.24566i −0.207979 + 0.395910i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 19.9782i 1.83139i
\(120\) 0 0
\(121\) 3.81029 0.346390
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 5.39573 0.478794 0.239397 0.970922i \(-0.423050\pi\)
0.239397 + 0.970922i \(0.423050\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 20.2823i 1.77208i 0.463612 + 0.886038i \(0.346553\pi\)
−0.463612 + 0.886038i \(0.653447\pi\)
\(132\) 0 0
\(133\) 28.8489 2.50152
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −21.4696 −1.83427 −0.917135 0.398577i \(-0.869504\pi\)
−0.917135 + 0.398577i \(0.869504\pi\)
\(138\) 0 0
\(139\) −7.68441 −0.651783 −0.325892 0.945407i \(-0.605664\pi\)
−0.325892 + 0.945407i \(0.605664\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −26.2888 −2.19838
\(144\) 0 0
\(145\) 1.90324i 0.158056i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −0.834989 −0.0684049 −0.0342025 0.999415i \(-0.510889\pi\)
−0.0342025 + 0.999415i \(0.510889\pi\)
\(150\) 0 0
\(151\) 8.85860 0.720902 0.360451 0.932778i \(-0.382623\pi\)
0.360451 + 0.932778i \(0.382623\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.07055 −0.407276
\(156\) 0 0
\(157\) 22.5413i 1.79899i 0.436932 + 0.899494i \(0.356065\pi\)
−0.436932 + 0.899494i \(0.643935\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −19.7335 10.3664i −1.55522 0.816989i
\(162\) 0 0
\(163\) −6.61191 −0.517884 −0.258942 0.965893i \(-0.583374\pi\)
−0.258942 + 0.965893i \(0.583374\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.89912i 0.224340i 0.993689 + 0.112170i \(0.0357802\pi\)
−0.993689 + 0.112170i \(0.964220\pi\)
\(168\) 0 0
\(169\) 33.6636 2.58951
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.97514i 0.378253i −0.981953 0.189127i \(-0.939434\pi\)
0.981953 0.189127i \(-0.0605656\pi\)
\(174\) 0 0
\(175\) 4.64793i 0.351351i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.9352i 0.817337i 0.912683 + 0.408668i \(0.134007\pi\)
−0.912683 + 0.408668i \(0.865993\pi\)
\(180\) 0 0
\(181\) 19.3798i 1.44049i 0.693722 + 0.720243i \(0.255970\pi\)
−0.693722 + 0.720243i \(0.744030\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.26790i 0.166740i
\(186\) 0 0
\(187\) −16.5416 −1.20964
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.36656 0.533026 0.266513 0.963831i \(-0.414129\pi\)
0.266513 + 0.963831i \(0.414129\pi\)
\(192\) 0 0
\(193\) 10.5605 0.760164 0.380082 0.924953i \(-0.375896\pi\)
0.380082 + 0.924953i \(0.375896\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.07854i 0.646820i −0.946259 0.323410i \(-0.895171\pi\)
0.946259 0.323410i \(-0.104829\pi\)
\(198\) 0 0
\(199\) 2.82488i 0.200250i 0.994975 + 0.100125i \(0.0319243\pi\)
−0.994975 + 0.100125i \(0.968076\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 8.84614 0.620877
\(204\) 0 0
\(205\) 10.8603i 0.758513i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 23.8864i 1.65226i
\(210\) 0 0
\(211\) 21.7417 1.49676 0.748381 0.663269i \(-0.230831\pi\)
0.748381 + 0.663269i \(0.230831\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.10929i 0.212052i
\(216\) 0 0
\(217\) 23.5676i 1.59987i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 29.3619 1.97510
\(222\) 0 0
\(223\) 24.0275 1.60900 0.804500 0.593952i \(-0.202433\pi\)
0.804500 + 0.593952i \(0.202433\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.82958 −0.586040 −0.293020 0.956106i \(-0.594660\pi\)
−0.293020 + 0.956106i \(0.594660\pi\)
\(228\) 0 0
\(229\) 19.9336i 1.31725i 0.752472 + 0.658625i \(0.228861\pi\)
−0.752472 + 0.658625i \(0.771139\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.6603i 1.09145i 0.837963 + 0.545727i \(0.183746\pi\)
−0.837963 + 0.545727i \(0.816254\pi\)
\(234\) 0 0
\(235\) 3.10323i 0.202432i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.31719i 0.214571i 0.994228 + 0.107286i \(0.0342159\pi\)
−0.994228 + 0.107286i \(0.965784\pi\)
\(240\) 0 0
\(241\) 12.6323i 0.813715i −0.913492 0.406857i \(-0.866625\pi\)
0.913492 0.406857i \(-0.133375\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 14.6033 0.932969
\(246\) 0 0
\(247\) 42.3993i 2.69780i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −0.681418 −0.0430107 −0.0215054 0.999769i \(-0.506846\pi\)
−0.0215054 + 0.999769i \(0.506846\pi\)
\(252\) 0 0
\(253\) 8.58323 16.3391i 0.539623 1.02723i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 20.8424i 1.30012i 0.759885 + 0.650058i \(0.225255\pi\)
−0.759885 + 0.650058i \(0.774745\pi\)
\(258\) 0 0
\(259\) −10.5411 −0.654990
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −5.61790 −0.346415 −0.173207 0.984885i \(-0.555413\pi\)
−0.173207 + 0.984885i \(0.555413\pi\)
\(264\) 0 0
\(265\) 8.56168 0.525940
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 22.4317i 1.36768i −0.729631 0.683841i \(-0.760308\pi\)
0.729631 0.683841i \(-0.239692\pi\)
\(270\) 0 0
\(271\) 28.2038 1.71326 0.856630 0.515931i \(-0.172554\pi\)
0.856630 + 0.515931i \(0.172554\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.84841 0.232068
\(276\) 0 0
\(277\) 25.1174 1.50916 0.754580 0.656208i \(-0.227841\pi\)
0.754580 + 0.656208i \(0.227841\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.4945 1.34191 0.670954 0.741499i \(-0.265885\pi\)
0.670954 + 0.741499i \(0.265885\pi\)
\(282\) 0 0
\(283\) 3.78317i 0.224886i −0.993658 0.112443i \(-0.964132\pi\)
0.993658 0.112443i \(-0.0358676\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −50.4777 −2.97961
\(288\) 0 0
\(289\) 1.47528 0.0867813
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.45783 0.0851673 0.0425836 0.999093i \(-0.486441\pi\)
0.0425836 + 0.999093i \(0.486441\pi\)
\(294\) 0 0
\(295\) 8.43405i 0.491049i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −15.2356 + 29.0024i −0.881095 + 1.67725i
\(300\) 0 0
\(301\) 14.4517 0.832985
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.26761i 0.473402i
\(306\) 0 0
\(307\) −34.1122 −1.94688 −0.973442 0.228934i \(-0.926476\pi\)
−0.973442 + 0.228934i \(0.926476\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 11.6163i 0.658698i −0.944208 0.329349i \(-0.893171\pi\)
0.944208 0.329349i \(-0.106829\pi\)
\(312\) 0 0
\(313\) 1.12369i 0.0635145i 0.999496 + 0.0317573i \(0.0101103\pi\)
−0.999496 + 0.0317573i \(0.989890\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.4146i 1.03427i 0.855905 + 0.517134i \(0.173001\pi\)
−0.855905 + 0.517134i \(0.826999\pi\)
\(318\) 0 0
\(319\) 7.32446i 0.410091i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 26.6787i 1.48444i
\(324\) 0 0
\(325\) −6.83108 −0.378920
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 14.4236 0.795198
\(330\) 0 0
\(331\) −17.2424 −0.947726 −0.473863 0.880599i \(-0.657141\pi\)
−0.473863 + 0.880599i \(0.657141\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.6821i 0.692895i
\(336\) 0 0
\(337\) 29.3360i 1.59804i 0.601307 + 0.799018i \(0.294647\pi\)
−0.601307 + 0.799018i \(0.705353\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 19.5136 1.05672
\(342\) 0 0
\(343\) 35.3395i 1.90815i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.31183i 0.285154i 0.989784 + 0.142577i \(0.0455388\pi\)
−0.989784 + 0.142577i \(0.954461\pi\)
\(348\) 0 0
\(349\) −25.2846 −1.35345 −0.676727 0.736234i \(-0.736602\pi\)
−0.676727 + 0.736234i \(0.736602\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 20.0786i 1.06867i 0.845271 + 0.534337i \(0.179439\pi\)
−0.845271 + 0.534337i \(0.820561\pi\)
\(354\) 0 0
\(355\) 10.5482i 0.559842i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −27.1966 −1.43538 −0.717690 0.696363i \(-0.754800\pi\)
−0.717690 + 0.696363i \(0.754800\pi\)
\(360\) 0 0
\(361\) −19.5247 −1.02762
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.77129 0.302083
\(366\) 0 0
\(367\) 21.7769i 1.13674i −0.822772 0.568372i \(-0.807574\pi\)
0.822772 0.568372i \(-0.192426\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 39.7941i 2.06601i
\(372\) 0 0
\(373\) 8.34201i 0.431933i −0.976401 0.215967i \(-0.930710\pi\)
0.976401 0.215967i \(-0.0692902\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 13.0012i 0.669595i
\(378\) 0 0
\(379\) 31.2838i 1.60694i 0.595344 + 0.803471i \(0.297016\pi\)
−0.595344 + 0.803471i \(0.702984\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 27.0099 1.38014 0.690070 0.723743i \(-0.257580\pi\)
0.690070 + 0.723743i \(0.257580\pi\)
\(384\) 0 0
\(385\) 17.8872i 0.911614i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 18.2140 0.923485 0.461743 0.887014i \(-0.347224\pi\)
0.461743 + 0.887014i \(0.347224\pi\)
\(390\) 0 0
\(391\) −9.58660 + 18.2491i −0.484815 + 0.922895i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.14676i 0.108015i
\(396\) 0 0
\(397\) 17.5817 0.882399 0.441199 0.897409i \(-0.354553\pi\)
0.441199 + 0.897409i \(0.354553\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −30.1783 −1.50703 −0.753515 0.657431i \(-0.771643\pi\)
−0.753515 + 0.657431i \(0.771643\pi\)
\(402\) 0 0
\(403\) −34.6373 −1.72541
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 8.72783i 0.432622i
\(408\) 0 0
\(409\) −3.55891 −0.175977 −0.0879885 0.996121i \(-0.528044\pi\)
−0.0879885 + 0.996121i \(0.528044\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 39.2009 1.92895
\(414\) 0 0
\(415\) 14.9978 0.736211
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −27.4135 −1.33924 −0.669619 0.742705i \(-0.733542\pi\)
−0.669619 + 0.742705i \(0.733542\pi\)
\(420\) 0 0
\(421\) 29.0950i 1.41800i 0.705206 + 0.709002i \(0.250855\pi\)
−0.705206 + 0.709002i \(0.749145\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.29829 −0.208498
\(426\) 0 0
\(427\) 38.4273 1.85963
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9.81593 −0.472817 −0.236408 0.971654i \(-0.575970\pi\)
−0.236408 + 0.971654i \(0.575970\pi\)
\(432\) 0 0
\(433\) 28.6704i 1.37781i 0.724851 + 0.688905i \(0.241908\pi\)
−0.724851 + 0.688905i \(0.758092\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 26.3521 + 13.8433i 1.26059 + 0.662214i
\(438\) 0 0
\(439\) 32.7417 1.56267 0.781337 0.624109i \(-0.214538\pi\)
0.781337 + 0.624109i \(0.214538\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 35.3787i 1.68089i 0.541894 + 0.840447i \(0.317707\pi\)
−0.541894 + 0.840447i \(0.682293\pi\)
\(444\) 0 0
\(445\) 2.95587 0.140122
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12.2277i 0.577061i 0.957471 + 0.288530i \(0.0931666\pi\)
−0.957471 + 0.288530i \(0.906833\pi\)
\(450\) 0 0
\(451\) 41.7947i 1.96804i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 31.7504i 1.48848i
\(456\) 0 0
\(457\) 28.2564i 1.32178i 0.750483 + 0.660890i \(0.229821\pi\)
−0.750483 + 0.660890i \(0.770179\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 41.9856i 1.95547i −0.209854 0.977733i \(-0.567299\pi\)
0.209854 0.977733i \(-0.432701\pi\)
\(462\) 0 0
\(463\) −24.9376 −1.15895 −0.579474 0.814991i \(-0.696742\pi\)
−0.579474 + 0.814991i \(0.696742\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.52227 0.301815 0.150907 0.988548i \(-0.451781\pi\)
0.150907 + 0.988548i \(0.451781\pi\)
\(468\) 0 0
\(469\) −58.9454 −2.72184
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11.9658i 0.550189i
\(474\) 0 0
\(475\) 6.20683i 0.284789i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.49609 −0.159741 −0.0798703 0.996805i \(-0.525451\pi\)
−0.0798703 + 0.996805i \(0.525451\pi\)
\(480\) 0 0
\(481\) 15.4922i 0.706384i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.87584i 0.175993i
\(486\) 0 0
\(487\) 12.9238 0.585633 0.292816 0.956169i \(-0.405408\pi\)
0.292816 + 0.956169i \(0.405408\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 22.5515i 1.01773i −0.860845 0.508867i \(-0.830064\pi\)
0.860845 0.508867i \(-0.169936\pi\)
\(492\) 0 0
\(493\) 8.18068i 0.368439i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −49.0274 −2.19918
\(498\) 0 0
\(499\) 35.2380 1.57747 0.788735 0.614733i \(-0.210736\pi\)
0.788735 + 0.614733i \(0.210736\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.67358 0.119209 0.0596046 0.998222i \(-0.481016\pi\)
0.0596046 + 0.998222i \(0.481016\pi\)
\(504\) 0 0
\(505\) 0.965924i 0.0429830i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.925217i 0.0410095i −0.999790 0.0205048i \(-0.993473\pi\)
0.999790 0.0205048i \(-0.00652733\pi\)
\(510\) 0 0
\(511\) 26.8246i 1.18665i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.95069i 0.218153i
\(516\) 0 0
\(517\) 11.9425i 0.525231i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 22.0897 0.967767 0.483884 0.875132i \(-0.339226\pi\)
0.483884 + 0.875132i \(0.339226\pi\)
\(522\) 0 0
\(523\) 30.9483i 1.35327i 0.736316 + 0.676637i \(0.236564\pi\)
−0.736316 + 0.676637i \(0.763436\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −21.7947 −0.949392
\(528\) 0 0
\(529\) −13.0513 18.9384i −0.567446 0.823411i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 74.1872i 3.21340i
\(534\) 0 0
\(535\) −13.8916 −0.600586
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −56.1994 −2.42068
\(540\) 0 0
\(541\) −9.40237 −0.404240 −0.202120 0.979361i \(-0.564783\pi\)
−0.202120 + 0.979361i \(0.564783\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.90475i 0.252932i
\(546\) 0 0
\(547\) −15.9774 −0.683143 −0.341571 0.939856i \(-0.610959\pi\)
−0.341571 + 0.939856i \(0.610959\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −11.8131 −0.503255
\(552\) 0 0
\(553\) −9.97801 −0.424308
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −26.3096 −1.11477 −0.557387 0.830253i \(-0.688196\pi\)
−0.557387 + 0.830253i \(0.688196\pi\)
\(558\) 0 0
\(559\) 21.2398i 0.898346i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −16.5075 −0.695710 −0.347855 0.937548i \(-0.613090\pi\)
−0.347855 + 0.937548i \(0.613090\pi\)
\(564\) 0 0
\(565\) 3.33678 0.140379
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7.10502 −0.297858 −0.148929 0.988848i \(-0.547583\pi\)
−0.148929 + 0.988848i \(0.547583\pi\)
\(570\) 0 0
\(571\) 40.2844i 1.68585i −0.538031 0.842925i \(-0.680832\pi\)
0.538031 0.842925i \(-0.319168\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.23033 4.24566i 0.0930112 0.177056i
\(576\) 0 0
\(577\) −32.6969 −1.36119 −0.680594 0.732661i \(-0.738278\pi\)
−0.680594 + 0.732661i \(0.738278\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 69.7086i 2.89200i
\(582\) 0 0
\(583\) −32.9489 −1.36460
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 32.3737i 1.33620i −0.744070 0.668102i \(-0.767107\pi\)
0.744070 0.668102i \(-0.232893\pi\)
\(588\) 0 0
\(589\) 31.4720i 1.29678i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12.5763i 0.516446i −0.966085 0.258223i \(-0.916863\pi\)
0.966085 0.258223i \(-0.0831368\pi\)
\(594\) 0 0
\(595\) 19.9782i 0.819024i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 31.4007i 1.28300i −0.767124 0.641499i \(-0.778313\pi\)
0.767124 0.641499i \(-0.221687\pi\)
\(600\) 0 0
\(601\) −36.0180 −1.46920 −0.734602 0.678499i \(-0.762631\pi\)
−0.734602 + 0.678499i \(0.762631\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.81029 −0.154910
\(606\) 0 0
\(607\) 27.4609 1.11460 0.557302 0.830310i \(-0.311837\pi\)
0.557302 + 0.830310i \(0.311837\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 21.1984i 0.857595i
\(612\) 0 0
\(613\) 34.4453i 1.39123i 0.718413 + 0.695617i \(0.244869\pi\)
−0.718413 + 0.695617i \(0.755131\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −14.6321 −0.589068 −0.294534 0.955641i \(-0.595164\pi\)
−0.294534 + 0.955641i \(0.595164\pi\)
\(618\) 0 0
\(619\) 4.45456i 0.179044i −0.995985 0.0895220i \(-0.971466\pi\)
0.995985 0.0895220i \(-0.0285339\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 13.7387i 0.550429i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9.74810i 0.388682i
\(630\) 0 0
\(631\) 18.8420i 0.750088i 0.927007 + 0.375044i \(0.122372\pi\)
−0.927007 + 0.375044i \(0.877628\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.39573 −0.214123
\(636\) 0 0
\(637\) 99.7561 3.95248
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −17.7464 −0.700942 −0.350471 0.936574i \(-0.613979\pi\)
−0.350471 + 0.936574i \(0.613979\pi\)
\(642\) 0 0
\(643\) 44.9439i 1.77241i 0.463289 + 0.886207i \(0.346669\pi\)
−0.463289 + 0.886207i \(0.653331\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.2710i 0.993505i 0.867892 + 0.496752i \(0.165474\pi\)
−0.867892 + 0.496752i \(0.834526\pi\)
\(648\) 0 0
\(649\) 32.4577i 1.27408i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 43.1865i 1.69002i −0.534751 0.845010i \(-0.679595\pi\)
0.534751 0.845010i \(-0.320405\pi\)
\(654\) 0 0
\(655\) 20.2823i 0.792497i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.79943 0.186959 0.0934796 0.995621i \(-0.470201\pi\)
0.0934796 + 0.995621i \(0.470201\pi\)
\(660\) 0 0
\(661\) 4.06371i 0.158060i 0.996872 + 0.0790300i \(0.0251823\pi\)
−0.996872 + 0.0790300i \(0.974818\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −28.8489 −1.11871
\(666\) 0 0
\(667\) 8.08052 + 4.24486i 0.312879 + 0.164362i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 31.8172i 1.22829i
\(672\) 0 0
\(673\) 45.4339 1.75135 0.875674 0.482903i \(-0.160418\pi\)
0.875674 + 0.482903i \(0.160418\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −38.6560 −1.48567 −0.742835 0.669474i \(-0.766519\pi\)
−0.742835 + 0.669474i \(0.766519\pi\)
\(678\) 0 0
\(679\) 18.0146 0.691339
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 35.3394i 1.35223i −0.736798 0.676113i \(-0.763663\pi\)
0.736798 0.676113i \(-0.236337\pi\)
\(684\) 0 0
\(685\) 21.4696 0.820310
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 58.4854 2.22812
\(690\) 0 0
\(691\) −19.2810 −0.733483 −0.366742 0.930323i \(-0.619527\pi\)
−0.366742 + 0.930323i \(0.619527\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.68441 0.291486
\(696\) 0 0
\(697\) 46.6805i 1.76815i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −8.59946 −0.324797 −0.162399 0.986725i \(-0.551923\pi\)
−0.162399 + 0.986725i \(0.551923\pi\)
\(702\) 0 0
\(703\) 14.0765 0.530905
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.48955 −0.168847
\(708\) 0 0
\(709\) 1.77246i 0.0665660i 0.999446 + 0.0332830i \(0.0105963\pi\)
−0.999446 + 0.0332830i \(0.989404\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 11.3090 21.5278i 0.423525 0.806224i
\(714\) 0 0
\(715\) 26.2888 0.983145
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 18.4697i 0.688804i 0.938822 + 0.344402i \(0.111918\pi\)
−0.938822 + 0.344402i \(0.888082\pi\)
\(720\) 0 0
\(721\) 23.0105 0.856954
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.90324i 0.0706846i
\(726\) 0 0
\(727\) 14.1803i 0.525917i 0.964807 + 0.262959i \(0.0846983\pi\)
−0.964807 + 0.262959i \(0.915302\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 13.3646i 0.494308i
\(732\) 0 0
\(733\) 21.4452i 0.792096i −0.918230 0.396048i \(-0.870381\pi\)
0.918230 0.396048i \(-0.129619\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 48.8058i 1.79779i
\(738\) 0 0
\(739\) −31.2095 −1.14806 −0.574031 0.818834i \(-0.694621\pi\)
−0.574031 + 0.818834i \(0.694621\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −7.65948 −0.280999 −0.140499 0.990081i \(-0.544871\pi\)
−0.140499 + 0.990081i \(0.544871\pi\)
\(744\) 0 0
\(745\) 0.834989 0.0305916
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 64.5672i 2.35923i
\(750\) 0 0
\(751\) 0.296355i 0.0108141i −0.999985 0.00540706i \(-0.998279\pi\)
0.999985 0.00540706i \(-0.00172113\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8.85860 −0.322397
\(756\) 0 0
\(757\) 5.41555i 0.196831i −0.995145 0.0984157i \(-0.968623\pi\)
0.995145 0.0984157i \(-0.0313775\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 28.0323i 1.01617i −0.861307 0.508085i \(-0.830354\pi\)
0.861307 0.508085i \(-0.169646\pi\)
\(762\) 0 0
\(763\) 27.4449 0.993572
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 57.6136i 2.08031i
\(768\) 0 0
\(769\) 10.2138i 0.368321i −0.982896 0.184160i \(-0.941043\pi\)
0.982896 0.184160i \(-0.0589566\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −10.4077 −0.374338 −0.187169 0.982328i \(-0.559931\pi\)
−0.187169 + 0.982328i \(0.559931\pi\)
\(774\) 0 0
\(775\) 5.07055 0.182140
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 67.4077 2.41513
\(780\) 0 0
\(781\) 40.5939i 1.45256i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 22.5413i 0.804532i
\(786\) 0 0
\(787\) 1.93489i 0.0689712i 0.999405 + 0.0344856i \(0.0109793\pi\)
−0.999405 + 0.0344856i \(0.989021\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 15.5091i 0.551441i
\(792\) 0 0
\(793\) 56.4767i 2.00555i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −24.0936 −0.853439 −0.426720 0.904384i \(-0.640331\pi\)
−0.426720 + 0.904384i \(0.640331\pi\)
\(798\) 0 0
\(799\) 13.3386i 0.471885i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −22.2103 −0.783786
\(804\) 0 0
\(805\) 19.7335 + 10.3664i 0.695516 + 0.365368i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.81387i 0.0989303i 0.998776 + 0.0494652i \(0.0157517\pi\)