Properties

Label 8280.2.p.a.1241.3
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 1241.3
Character \(\chi\) \(=\) 8280.1241
Dual form 8280.2.p.a.1241.46

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} -4.29385i q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -4.29385i q^{7} +0.813503 q^{11} +1.00050 q^{13} +7.09518 q^{17} +3.82420i q^{19} +(-1.12608 + 4.66175i) q^{23} +1.00000 q^{25} +0.328907i q^{29} -4.61002 q^{31} +4.29385i q^{35} +10.4272i q^{37} -3.42126i q^{41} +7.16889i q^{43} -0.556882i q^{47} -11.4371 q^{49} +3.55889 q^{53} -0.813503 q^{55} +12.4450i q^{59} -9.65842i q^{61} -1.00050 q^{65} +1.86503i q^{67} -0.163405i q^{71} -8.45188 q^{73} -3.49306i q^{77} +15.2342i q^{79} +5.16406 q^{83} -7.09518 q^{85} +3.52407 q^{89} -4.29599i q^{91} -3.82420i q^{95} +7.94278i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 48 q^{5}+O(q^{10}) \) 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
\(7\) 4.29385i 1.62292i −0.584406 0.811461i \(-0.698672\pi\)
0.584406 0.811461i \(-0.301328\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.813503 0.245280 0.122640 0.992451i \(-0.460864\pi\)
0.122640 + 0.992451i \(0.460864\pi\)
\(12\) 0 0
\(13\) 1.00050 0.277488 0.138744 0.990328i \(-0.455693\pi\)
0.138744 + 0.990328i \(0.455693\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.09518 1.72083 0.860417 0.509590i \(-0.170203\pi\)
0.860417 + 0.509590i \(0.170203\pi\)
\(18\) 0 0
\(19\) 3.82420i 0.877331i 0.898650 + 0.438666i \(0.144549\pi\)
−0.898650 + 0.438666i \(0.855451\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.12608 + 4.66175i −0.234805 + 0.972043i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.328907i 0.0610765i 0.999534 + 0.0305383i \(0.00972214\pi\)
−0.999534 + 0.0305383i \(0.990278\pi\)
\(30\) 0 0
\(31\) −4.61002 −0.827983 −0.413992 0.910281i \(-0.635866\pi\)
−0.413992 + 0.910281i \(0.635866\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.29385i 0.725793i
\(36\) 0 0
\(37\) 10.4272i 1.71422i 0.515133 + 0.857110i \(0.327743\pi\)
−0.515133 + 0.857110i \(0.672257\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.42126i 0.534311i −0.963653 0.267155i \(-0.913916\pi\)
0.963653 0.267155i \(-0.0860837\pi\)
\(42\) 0 0
\(43\) 7.16889i 1.09325i 0.837379 + 0.546623i \(0.184087\pi\)
−0.837379 + 0.546623i \(0.815913\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.556882i 0.0812295i −0.999175 0.0406148i \(-0.987068\pi\)
0.999175 0.0406148i \(-0.0129316\pi\)
\(48\) 0 0
\(49\) −11.4371 −1.63388
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.55889 0.488852 0.244426 0.969668i \(-0.421401\pi\)
0.244426 + 0.969668i \(0.421401\pi\)
\(54\) 0 0
\(55\) −0.813503 −0.109693
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.4450i 1.62021i 0.586287 + 0.810104i \(0.300589\pi\)
−0.586287 + 0.810104i \(0.699411\pi\)
\(60\) 0 0
\(61\) 9.65842i 1.23663i −0.785929 0.618317i \(-0.787815\pi\)
0.785929 0.618317i \(-0.212185\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.00050 −0.124097
\(66\) 0 0
\(67\) 1.86503i 0.227849i 0.993489 + 0.113925i \(0.0363423\pi\)
−0.993489 + 0.113925i \(0.963658\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.163405i 0.0193927i −0.999953 0.00969633i \(-0.996914\pi\)
0.999953 0.00969633i \(-0.00308649\pi\)
\(72\) 0 0
\(73\) −8.45188 −0.989217 −0.494609 0.869116i \(-0.664689\pi\)
−0.494609 + 0.869116i \(0.664689\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.49306i 0.398071i
\(78\) 0 0
\(79\) 15.2342i 1.71398i 0.515336 + 0.856988i \(0.327667\pi\)
−0.515336 + 0.856988i \(0.672333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.16406 0.566829 0.283414 0.958998i \(-0.408533\pi\)
0.283414 + 0.958998i \(0.408533\pi\)
\(84\) 0 0
\(85\) −7.09518 −0.769581
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.52407 0.373550 0.186775 0.982403i \(-0.440196\pi\)
0.186775 + 0.982403i \(0.440196\pi\)
\(90\) 0 0
\(91\) 4.29599i 0.450342i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.82420i 0.392355i
\(96\) 0 0
\(97\) 7.94278i 0.806467i 0.915097 + 0.403233i \(0.132114\pi\)
−0.915097 + 0.403233i \(0.867886\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.35867i 0.732215i −0.930573 0.366107i \(-0.880690\pi\)
0.930573 0.366107i \(-0.119310\pi\)
\(102\) 0 0
\(103\) 5.03295i 0.495912i 0.968771 + 0.247956i \(0.0797588\pi\)
−0.968771 + 0.247956i \(0.920241\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.80673 −0.561357 −0.280679 0.959802i \(-0.590560\pi\)
−0.280679 + 0.959802i \(0.590560\pi\)
\(108\) 0 0
\(109\) 2.93260i 0.280892i 0.990088 + 0.140446i \(0.0448537\pi\)
−0.990088 + 0.140446i \(0.955146\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −10.1479 −0.954636 −0.477318 0.878731i \(-0.658391\pi\)
−0.477318 + 0.878731i \(0.658391\pi\)
\(114\) 0 0
\(115\) 1.12608 4.66175i 0.105008 0.434711i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 30.4657i 2.79278i
\(120\) 0 0
\(121\) −10.3382 −0.939838
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.47190 −0.396817 −0.198409 0.980119i \(-0.563577\pi\)
−0.198409 + 0.980119i \(0.563577\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.97112i 0.871181i 0.900145 + 0.435590i \(0.143460\pi\)
−0.900145 + 0.435590i \(0.856540\pi\)
\(132\) 0 0
\(133\) 16.4205 1.42384
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.36208 −0.543549 −0.271775 0.962361i \(-0.587611\pi\)
−0.271775 + 0.962361i \(0.587611\pi\)
\(138\) 0 0
\(139\) 13.2054 1.12007 0.560034 0.828470i \(-0.310788\pi\)
0.560034 + 0.828470i \(0.310788\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.813908 0.0680624
\(144\) 0 0
\(145\) 0.328907i 0.0273142i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 13.6514 1.11836 0.559182 0.829045i \(-0.311115\pi\)
0.559182 + 0.829045i \(0.311115\pi\)
\(150\) 0 0
\(151\) 2.28358 0.185835 0.0929177 0.995674i \(-0.470381\pi\)
0.0929177 + 0.995674i \(0.470381\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.61002 0.370285
\(156\) 0 0
\(157\) 23.1675i 1.84897i 0.381219 + 0.924485i \(0.375504\pi\)
−0.381219 + 0.924485i \(0.624496\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 20.0169 + 4.83523i 1.57755 + 0.381070i
\(162\) 0 0
\(163\) 13.2846 1.04053 0.520265 0.854005i \(-0.325833\pi\)
0.520265 + 0.854005i \(0.325833\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.37549i 0.338585i −0.985566 0.169293i \(-0.945852\pi\)
0.985566 0.169293i \(-0.0541483\pi\)
\(168\) 0 0
\(169\) −11.9990 −0.923000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 25.6230i 1.94808i −0.226380 0.974039i \(-0.572689\pi\)
0.226380 0.974039i \(-0.427311\pi\)
\(174\) 0 0
\(175\) 4.29385i 0.324585i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.7392i 1.02692i 0.858115 + 0.513458i \(0.171636\pi\)
−0.858115 + 0.513458i \(0.828364\pi\)
\(180\) 0 0
\(181\) 22.6528i 1.68377i −0.539658 0.841884i \(-0.681446\pi\)
0.539658 0.841884i \(-0.318554\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.4272i 0.766623i
\(186\) 0 0
\(187\) 5.77195 0.422087
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.9372 1.51496 0.757482 0.652857i \(-0.226430\pi\)
0.757482 + 0.652857i \(0.226430\pi\)
\(192\) 0 0
\(193\) −11.0000 −0.791800 −0.395900 0.918294i \(-0.629567\pi\)
−0.395900 + 0.918294i \(0.629567\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.7476i 0.836982i 0.908221 + 0.418491i \(0.137441\pi\)
−0.908221 + 0.418491i \(0.862559\pi\)
\(198\) 0 0
\(199\) 25.4484i 1.80399i 0.431747 + 0.901995i \(0.357897\pi\)
−0.431747 + 0.901995i \(0.642103\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.41228 0.0991225
\(204\) 0 0
\(205\) 3.42126i 0.238951i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.11100i 0.215192i
\(210\) 0 0
\(211\) 4.31144 0.296812 0.148406 0.988927i \(-0.452586\pi\)
0.148406 + 0.988927i \(0.452586\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 7.16889i 0.488914i
\(216\) 0 0
\(217\) 19.7947i 1.34375i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.09872 0.477511
\(222\) 0 0
\(223\) 19.6974 1.31904 0.659519 0.751688i \(-0.270760\pi\)
0.659519 + 0.751688i \(0.270760\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −13.2783 −0.881312 −0.440656 0.897676i \(-0.645254\pi\)
−0.440656 + 0.897676i \(0.645254\pi\)
\(228\) 0 0
\(229\) 13.1695i 0.870263i −0.900367 0.435132i \(-0.856702\pi\)
0.900367 0.435132i \(-0.143298\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.20321i 0.471898i 0.971765 + 0.235949i \(0.0758198\pi\)
−0.971765 + 0.235949i \(0.924180\pi\)
\(234\) 0 0
\(235\) 0.556882i 0.0363270i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 14.0367i 0.907962i 0.891011 + 0.453981i \(0.149997\pi\)
−0.891011 + 0.453981i \(0.850003\pi\)
\(240\) 0 0
\(241\) 1.84347i 0.118748i −0.998236 0.0593740i \(-0.981090\pi\)
0.998236 0.0593740i \(-0.0189105\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 11.4371 0.730692
\(246\) 0 0
\(247\) 3.82610i 0.243449i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.77543 0.553900 0.276950 0.960884i \(-0.410676\pi\)
0.276950 + 0.960884i \(0.410676\pi\)
\(252\) 0 0
\(253\) −0.916072 + 3.79235i −0.0575929 + 0.238423i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 15.7886i 0.984867i −0.870350 0.492433i \(-0.836108\pi\)
0.870350 0.492433i \(-0.163892\pi\)
\(258\) 0 0
\(259\) 44.7728 2.78205
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.6791 −0.781825 −0.390912 0.920428i \(-0.627840\pi\)
−0.390912 + 0.920428i \(0.627840\pi\)
\(264\) 0 0
\(265\) −3.55889 −0.218621
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.58747i 0.218732i 0.994002 + 0.109366i \(0.0348821\pi\)
−0.994002 + 0.109366i \(0.965118\pi\)
\(270\) 0 0
\(271\) 17.6817 1.07408 0.537042 0.843555i \(-0.319542\pi\)
0.537042 + 0.843555i \(0.319542\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.813503 0.0490561
\(276\) 0 0
\(277\) 15.3191 0.920435 0.460217 0.887806i \(-0.347771\pi\)
0.460217 + 0.887806i \(0.347771\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.4562 1.33963 0.669813 0.742530i \(-0.266374\pi\)
0.669813 + 0.742530i \(0.266374\pi\)
\(282\) 0 0
\(283\) 24.0624i 1.43036i 0.698940 + 0.715181i \(0.253656\pi\)
−0.698940 + 0.715181i \(0.746344\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −14.6904 −0.867145
\(288\) 0 0
\(289\) 33.3416 1.96127
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 10.0094 0.584756 0.292378 0.956303i \(-0.405553\pi\)
0.292378 + 0.956303i \(0.405553\pi\)
\(294\) 0 0
\(295\) 12.4450i 0.724579i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.12664 + 4.66408i −0.0651555 + 0.269730i
\(300\) 0 0
\(301\) 30.7821 1.77425
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.65842i 0.553039i
\(306\) 0 0
\(307\) 3.95099 0.225495 0.112747 0.993624i \(-0.464035\pi\)
0.112747 + 0.993624i \(0.464035\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 34.3304i 1.94670i −0.229325 0.973350i \(-0.573652\pi\)
0.229325 0.973350i \(-0.426348\pi\)
\(312\) 0 0
\(313\) 21.1115i 1.19329i 0.802504 + 0.596647i \(0.203501\pi\)
−0.802504 + 0.596647i \(0.796499\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 28.9788i 1.62761i −0.581138 0.813805i \(-0.697392\pi\)
0.581138 0.813805i \(-0.302608\pi\)
\(318\) 0 0
\(319\) 0.267567i 0.0149809i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 27.1334i 1.50974i
\(324\) 0 0
\(325\) 1.00050 0.0554977
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.39117 −0.131829
\(330\) 0 0
\(331\) −21.7836 −1.19734 −0.598668 0.800997i \(-0.704303\pi\)
−0.598668 + 0.800997i \(0.704303\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.86503i 0.101897i
\(336\) 0 0
\(337\) 2.82800i 0.154051i 0.997029 + 0.0770254i \(0.0245423\pi\)
−0.997029 + 0.0770254i \(0.975458\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3.75026 −0.203088
\(342\) 0 0
\(343\) 19.0524i 1.02874i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.04369i 0.431808i −0.976415 0.215904i \(-0.930730\pi\)
0.976415 0.215904i \(-0.0692698\pi\)
\(348\) 0 0
\(349\) −6.49701 −0.347777 −0.173889 0.984765i \(-0.555633\pi\)
−0.173889 + 0.984765i \(0.555633\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 32.8289i 1.74731i 0.486548 + 0.873654i \(0.338256\pi\)
−0.486548 + 0.873654i \(0.661744\pi\)
\(354\) 0 0
\(355\) 0.163405i 0.00867266i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −27.4672 −1.44966 −0.724832 0.688925i \(-0.758083\pi\)
−0.724832 + 0.688925i \(0.758083\pi\)
\(360\) 0 0
\(361\) 4.37550 0.230290
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 8.45188 0.442391
\(366\) 0 0
\(367\) 6.84036i 0.357064i 0.983934 + 0.178532i \(0.0571348\pi\)
−0.983934 + 0.178532i \(0.942865\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 15.2814i 0.793368i
\(372\) 0 0
\(373\) 21.2371i 1.09962i −0.835290 0.549809i \(-0.814701\pi\)
0.835290 0.549809i \(-0.185299\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.329071i 0.0169480i
\(378\) 0 0
\(379\) 17.9444i 0.921742i 0.887467 + 0.460871i \(0.152463\pi\)
−0.887467 + 0.460871i \(0.847537\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.3781 0.632489 0.316244 0.948678i \(-0.397578\pi\)
0.316244 + 0.948678i \(0.397578\pi\)
\(384\) 0 0
\(385\) 3.49306i 0.178023i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13.9600 0.707801 0.353900 0.935283i \(-0.384855\pi\)
0.353900 + 0.935283i \(0.384855\pi\)
\(390\) 0 0
\(391\) −7.98977 + 33.0760i −0.404060 + 1.67272i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 15.2342i 0.766514i
\(396\) 0 0
\(397\) −13.2916 −0.667087 −0.333543 0.942735i \(-0.608244\pi\)
−0.333543 + 0.942735i \(0.608244\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.0805 1.05271 0.526355 0.850265i \(-0.323558\pi\)
0.526355 + 0.850265i \(0.323558\pi\)
\(402\) 0 0
\(403\) −4.61231 −0.229756
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 8.48255i 0.420465i
\(408\) 0 0
\(409\) 23.4381 1.15894 0.579469 0.814994i \(-0.303260\pi\)
0.579469 + 0.814994i \(0.303260\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 53.4372 2.62947
\(414\) 0 0
\(415\) −5.16406 −0.253494
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.681486 −0.0332928 −0.0166464 0.999861i \(-0.505299\pi\)
−0.0166464 + 0.999861i \(0.505299\pi\)
\(420\) 0 0
\(421\) 12.4676i 0.607633i 0.952731 + 0.303816i \(0.0982610\pi\)
−0.952731 + 0.303816i \(0.901739\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.09518 0.344167
\(426\) 0 0
\(427\) −41.4718 −2.00696
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.4770 0.745501 0.372751 0.927932i \(-0.378415\pi\)
0.372751 + 0.927932i \(0.378415\pi\)
\(432\) 0 0
\(433\) 21.3471i 1.02588i −0.858426 0.512938i \(-0.828557\pi\)
0.858426 0.512938i \(-0.171443\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −17.8275 4.30637i −0.852803 0.206001i
\(438\) 0 0
\(439\) −8.69629 −0.415051 −0.207526 0.978230i \(-0.566541\pi\)
−0.207526 + 0.978230i \(0.566541\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 14.5706i 0.692272i −0.938184 0.346136i \(-0.887494\pi\)
0.938184 0.346136i \(-0.112506\pi\)
\(444\) 0 0
\(445\) −3.52407 −0.167057
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 32.6200i 1.53943i 0.638386 + 0.769717i \(0.279602\pi\)
−0.638386 + 0.769717i \(0.720398\pi\)
\(450\) 0 0
\(451\) 2.78320i 0.131056i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.29599i 0.201399i
\(456\) 0 0
\(457\) 11.9080i 0.557035i 0.960431 + 0.278517i \(0.0898430\pi\)
−0.960431 + 0.278517i \(0.910157\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 23.5935i 1.09886i 0.835540 + 0.549429i \(0.185155\pi\)
−0.835540 + 0.549429i \(0.814845\pi\)
\(462\) 0 0
\(463\) −37.7038 −1.75224 −0.876122 0.482090i \(-0.839878\pi\)
−0.876122 + 0.482090i \(0.839878\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.80548 0.314920 0.157460 0.987525i \(-0.449669\pi\)
0.157460 + 0.987525i \(0.449669\pi\)
\(468\) 0 0
\(469\) 8.00815 0.369782
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.83191i 0.268152i
\(474\) 0 0
\(475\) 3.82420i 0.175466i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0.183570 0.00838754 0.00419377 0.999991i \(-0.498665\pi\)
0.00419377 + 0.999991i \(0.498665\pi\)
\(480\) 0 0
\(481\) 10.4324i 0.475676i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.94278i 0.360663i
\(486\) 0 0
\(487\) −16.8484 −0.763476 −0.381738 0.924271i \(-0.624674\pi\)
−0.381738 + 0.924271i \(0.624674\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 30.5373i 1.37813i 0.724700 + 0.689065i \(0.241979\pi\)
−0.724700 + 0.689065i \(0.758021\pi\)
\(492\) 0 0
\(493\) 2.33366i 0.105103i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.701639 −0.0314728
\(498\) 0 0
\(499\) −30.5806 −1.36898 −0.684488 0.729024i \(-0.739974\pi\)
−0.684488 + 0.729024i \(0.739974\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0.671716 0.0299503 0.0149752 0.999888i \(-0.495233\pi\)
0.0149752 + 0.999888i \(0.495233\pi\)
\(504\) 0 0
\(505\) 7.35867i 0.327456i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 34.8846i 1.54623i −0.634265 0.773116i \(-0.718697\pi\)
0.634265 0.773116i \(-0.281303\pi\)
\(510\) 0 0
\(511\) 36.2911i 1.60542i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.03295i 0.221778i
\(516\) 0 0
\(517\) 0.453025i 0.0199240i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.5506 −0.637473 −0.318736 0.947843i \(-0.603258\pi\)
−0.318736 + 0.947843i \(0.603258\pi\)
\(522\) 0 0
\(523\) 22.7614i 0.995289i −0.867381 0.497644i \(-0.834198\pi\)
0.867381 0.497644i \(-0.165802\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −32.7089 −1.42482
\(528\) 0 0
\(529\) −20.4639 10.4990i −0.889734 0.456480i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.42296i 0.148265i
\(534\) 0 0
\(535\) 5.80673 0.251047
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −9.30415 −0.400758
\(540\) 0 0
\(541\) −14.1629 −0.608909 −0.304455 0.952527i \(-0.598474\pi\)
−0.304455 + 0.952527i \(0.598474\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.93260i 0.125619i
\(546\) 0 0
\(547\) 22.9864 0.982827 0.491413 0.870926i \(-0.336480\pi\)
0.491413 + 0.870926i \(0.336480\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.25781 −0.0535843
\(552\) 0 0
\(553\) 65.4132 2.78165
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.43773 −0.188032 −0.0940162 0.995571i \(-0.529971\pi\)
−0.0940162 + 0.995571i \(0.529971\pi\)
\(558\) 0 0
\(559\) 7.17246i 0.303363i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20.5077 0.864295 0.432148 0.901803i \(-0.357756\pi\)
0.432148 + 0.901803i \(0.357756\pi\)
\(564\) 0 0
\(565\) 10.1479 0.426926
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11.6778 0.489559 0.244779 0.969579i \(-0.421284\pi\)
0.244779 + 0.969579i \(0.421284\pi\)
\(570\) 0 0
\(571\) 29.3337i 1.22758i 0.789471 + 0.613788i \(0.210355\pi\)
−0.789471 + 0.613788i \(0.789645\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.12608 + 4.66175i −0.0469609 + 0.194409i
\(576\) 0 0
\(577\) 17.6579 0.735109 0.367555 0.930002i \(-0.380195\pi\)
0.367555 + 0.930002i \(0.380195\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 22.1737i 0.919919i
\(582\) 0 0
\(583\) 2.89517 0.119906
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.2214i 0.504430i −0.967671 0.252215i \(-0.918841\pi\)
0.967671 0.252215i \(-0.0811590\pi\)
\(588\) 0 0
\(589\) 17.6296i 0.726416i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 10.6127i 0.435812i −0.975970 0.217906i \(-0.930077\pi\)
0.975970 0.217906i \(-0.0699226\pi\)
\(594\) 0 0
\(595\) 30.4657i 1.24897i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.07408i 0.0847445i 0.999102 + 0.0423722i \(0.0134915\pi\)
−0.999102 + 0.0423722i \(0.986508\pi\)
\(600\) 0 0
\(601\) −40.5425 −1.65376 −0.826882 0.562375i \(-0.809888\pi\)
−0.826882 + 0.562375i \(0.809888\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.3382 0.420308
\(606\) 0 0
\(607\) −35.2667 −1.43143 −0.715716 0.698391i \(-0.753900\pi\)
−0.715716 + 0.698391i \(0.753900\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.557159i 0.0225402i
\(612\) 0 0
\(613\) 12.7368i 0.514433i −0.966354 0.257216i \(-0.917195\pi\)
0.966354 0.257216i \(-0.0828053\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 38.9494 1.56804 0.784022 0.620733i \(-0.213165\pi\)
0.784022 + 0.620733i \(0.213165\pi\)
\(618\) 0 0
\(619\) 23.4023i 0.940618i 0.882502 + 0.470309i \(0.155857\pi\)
−0.882502 + 0.470309i \(0.844143\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 15.1318i 0.606243i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 73.9829i 2.94989i
\(630\) 0 0
\(631\) 13.2157i 0.526109i −0.964781 0.263054i \(-0.915270\pi\)
0.964781 0.263054i \(-0.0847298\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.47190 0.177462
\(636\) 0 0
\(637\) −11.4428 −0.453382
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.4461 0.807572 0.403786 0.914854i \(-0.367694\pi\)
0.403786 + 0.914854i \(0.367694\pi\)
\(642\) 0 0
\(643\) 26.0848i 1.02869i 0.857585 + 0.514343i \(0.171964\pi\)
−0.857585 + 0.514343i \(0.828036\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 37.0363i 1.45605i −0.685551 0.728025i \(-0.740439\pi\)
0.685551 0.728025i \(-0.259561\pi\)
\(648\) 0 0
\(649\) 10.1241i 0.397405i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.48658i 0.253839i 0.991913 + 0.126920i \(0.0405091\pi\)
−0.991913 + 0.126920i \(0.959491\pi\)
\(654\) 0 0
\(655\) 9.97112i 0.389604i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 28.7980 1.12181 0.560905 0.827880i \(-0.310453\pi\)
0.560905 + 0.827880i \(0.310453\pi\)
\(660\) 0 0
\(661\) 23.6161i 0.918561i 0.888291 + 0.459280i \(0.151893\pi\)
−0.888291 + 0.459280i \(0.848107\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −16.4205 −0.636761
\(666\) 0 0
\(667\) −1.53328 0.370377i −0.0593690 0.0143410i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 7.85715i 0.303322i
\(672\) 0 0
\(673\) −31.8499 −1.22772 −0.613861 0.789414i \(-0.710385\pi\)
−0.613861 + 0.789414i \(0.710385\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 35.4831 1.36373 0.681864 0.731479i \(-0.261170\pi\)
0.681864 + 0.731479i \(0.261170\pi\)
\(678\) 0 0
\(679\) 34.1051 1.30883
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 47.5707i 1.82024i 0.414344 + 0.910120i \(0.364011\pi\)
−0.414344 + 0.910120i \(0.635989\pi\)
\(684\) 0 0
\(685\) 6.36208 0.243083
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.56067 0.135651
\(690\) 0 0
\(691\) 34.0730 1.29620 0.648098 0.761557i \(-0.275565\pi\)
0.648098 + 0.761557i \(0.275565\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −13.2054 −0.500910
\(696\) 0 0
\(697\) 24.2745i 0.919460i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10.7857 −0.407372 −0.203686 0.979036i \(-0.565292\pi\)
−0.203686 + 0.979036i \(0.565292\pi\)
\(702\) 0 0
\(703\) −39.8757 −1.50394
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −31.5970 −1.18833
\(708\) 0 0
\(709\) 8.57056i 0.321874i −0.986965 0.160937i \(-0.948548\pi\)
0.986965 0.160937i \(-0.0514516\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.19126 21.4908i 0.194414 0.804835i
\(714\) 0 0
\(715\) −0.813908 −0.0304384
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0.199893i 0.00745476i −0.999993 0.00372738i \(-0.998814\pi\)
0.999993 0.00372738i \(-0.00118646\pi\)
\(720\) 0 0
\(721\) 21.6107 0.804826
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.328907i 0.0122153i
\(726\) 0 0
\(727\) 16.7842i 0.622492i −0.950329 0.311246i \(-0.899254\pi\)
0.950329 0.311246i \(-0.100746\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 50.8646i 1.88129i
\(732\) 0 0
\(733\) 20.7005i 0.764592i 0.924040 + 0.382296i \(0.124866\pi\)
−0.924040 + 0.382296i \(0.875134\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.51721i 0.0558870i
\(738\) 0 0
\(739\) 29.0959 1.07031 0.535155 0.844754i \(-0.320253\pi\)
0.535155 + 0.844754i \(0.320253\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −36.6338 −1.34396 −0.671982 0.740567i \(-0.734557\pi\)
−0.671982 + 0.740567i \(0.734557\pi\)
\(744\) 0 0
\(745\) −13.6514 −0.500148
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 24.9332i 0.911040i
\(750\) 0 0
\(751\) 6.74192i 0.246016i 0.992406 + 0.123008i \(0.0392541\pi\)
−0.992406 + 0.123008i \(0.960746\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.28358 −0.0831082
\(756\) 0 0
\(757\) 17.6311i 0.640812i −0.947280 0.320406i \(-0.896181\pi\)
0.947280 0.320406i \(-0.103819\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 28.4539i 1.03145i −0.856753 0.515726i \(-0.827522\pi\)
0.856753 0.515726i \(-0.172478\pi\)
\(762\) 0 0
\(763\) 12.5922 0.455867
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.4512i 0.449588i
\(768\) 0 0
\(769\) 51.5377i 1.85850i −0.369454 0.929249i \(-0.620455\pi\)
0.369454 0.929249i \(-0.379545\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 38.8535 1.39746 0.698732 0.715384i \(-0.253748\pi\)
0.698732 + 0.715384i \(0.253748\pi\)
\(774\) 0 0
\(775\) −4.61002 −0.165597
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 13.0836 0.468768
\(780\) 0 0
\(781\) 0.132931i 0.00475664i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 23.1675i 0.826884i
\(786\) 0 0
\(787\) 20.4094i 0.727517i 0.931493 + 0.363759i \(0.118507\pi\)
−0.931493 + 0.363759i \(0.881493\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 43.5737i 1.54930i
\(792\) 0 0
\(793\) 9.66323i 0.343151i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 20.8255 0.737675 0.368838 0.929494i \(-0.379756\pi\)
0.368838 + 0.929494i \(0.379756\pi\)
\(798\) 0 0
\(799\) 3.95118i 0.139783i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.87562 −0.242636
\(804\) 0 0
\(805\) −20.0169 4.83523i −0.705502 0.170420i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.85576i 0.241036i 0.992711 + 0.120518i \(0.0384555\pi\)
−0.992711 + 0.120518i