Properties

Label 6840.2.a.bn.1.1
Level $6840$
Weight $2$
Character 6840.1
Self dual yes
Analytic conductor $54.618$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.6176749826\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1016.1
Defining polynomial: \(x^{3} - x^{2} - 6 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.321637\) of defining polynomial
Character \(\chi\) \(=\) 6840.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} -4.21819 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} -4.21819 q^{7} -5.57491 q^{11} -2.21819 q^{13} -0.643274 q^{17} -1.00000 q^{19} -1.35673 q^{23} +1.00000 q^{25} -4.86146 q^{29} +2.64327 q^{31} -4.21819 q^{35} -2.21819 q^{37} -3.57491 q^{41} -0.218187 q^{43} +1.35673 q^{47} +10.7931 q^{49} -0.643274 q^{53} -5.57491 q^{55} +4.43637 q^{59} +13.0796 q^{61} -2.21819 q^{65} +1.28655 q^{67} -11.1498 q^{71} +10.0000 q^{73} +23.5160 q^{77} -1.28655 q^{79} -12.3662 q^{83} -0.643274 q^{85} -12.0113 q^{89} +9.35673 q^{91} -1.00000 q^{95} +5.78181 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} + O(q^{10}) \) \( 3 q + 3 q^{5} - 4 q^{11} + 6 q^{13} - 2 q^{17} - 3 q^{19} - 4 q^{23} + 3 q^{25} - 2 q^{29} + 8 q^{31} + 6 q^{37} + 2 q^{41} + 12 q^{43} + 4 q^{47} + 7 q^{49} - 2 q^{53} - 4 q^{55} - 12 q^{59} + 14 q^{61} + 6 q^{65} + 4 q^{67} - 8 q^{71} + 30 q^{73} + 20 q^{77} - 4 q^{79} - 12 q^{83} - 2 q^{85} + 2 q^{89} + 28 q^{91} - 3 q^{95} + 30 q^{97} + O(q^{100}) \)

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
\(7\) −4.21819 −1.59432 −0.797162 0.603765i \(-0.793667\pi\)
−0.797162 + 0.603765i \(0.793667\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.57491 −1.68090 −0.840450 0.541890i \(-0.817709\pi\)
−0.840450 + 0.541890i \(0.817709\pi\)
\(12\) 0 0
\(13\) −2.21819 −0.615214 −0.307607 0.951513i \(-0.599528\pi\)
−0.307607 + 0.951513i \(0.599528\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.643274 −0.156017 −0.0780085 0.996953i \(-0.524856\pi\)
−0.0780085 + 0.996953i \(0.524856\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.35673 −0.282897 −0.141448 0.989946i \(-0.545176\pi\)
−0.141448 + 0.989946i \(0.545176\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.86146 −0.902751 −0.451375 0.892334i \(-0.649066\pi\)
−0.451375 + 0.892334i \(0.649066\pi\)
\(30\) 0 0
\(31\) 2.64327 0.474746 0.237373 0.971419i \(-0.423714\pi\)
0.237373 + 0.971419i \(0.423714\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.21819 −0.713004
\(36\) 0 0
\(37\) −2.21819 −0.364668 −0.182334 0.983237i \(-0.558365\pi\)
−0.182334 + 0.983237i \(0.558365\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.57491 −0.558308 −0.279154 0.960246i \(-0.590054\pi\)
−0.279154 + 0.960246i \(0.590054\pi\)
\(42\) 0 0
\(43\) −0.218187 −0.0332732 −0.0166366 0.999862i \(-0.505296\pi\)
−0.0166366 + 0.999862i \(0.505296\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.35673 0.197899 0.0989494 0.995092i \(-0.468452\pi\)
0.0989494 + 0.995092i \(0.468452\pi\)
\(48\) 0 0
\(49\) 10.7931 1.54187
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.643274 −0.0883605 −0.0441803 0.999024i \(-0.514068\pi\)
−0.0441803 + 0.999024i \(0.514068\pi\)
\(54\) 0 0
\(55\) −5.57491 −0.751721
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.43637 0.577567 0.288783 0.957394i \(-0.406749\pi\)
0.288783 + 0.957394i \(0.406749\pi\)
\(60\) 0 0
\(61\) 13.0796 1.67468 0.837339 0.546685i \(-0.184110\pi\)
0.837339 + 0.546685i \(0.184110\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.21819 −0.275132
\(66\) 0 0
\(67\) 1.28655 0.157177 0.0785885 0.996907i \(-0.474959\pi\)
0.0785885 + 0.996907i \(0.474959\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.1498 −1.32324 −0.661620 0.749839i \(-0.730131\pi\)
−0.661620 + 0.749839i \(0.730131\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 23.5160 2.67990
\(78\) 0 0
\(79\) −1.28655 −0.144748 −0.0723740 0.997378i \(-0.523058\pi\)
−0.0723740 + 0.997378i \(0.523058\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.3662 −1.35737 −0.678683 0.734431i \(-0.737449\pi\)
−0.678683 + 0.734431i \(0.737449\pi\)
\(84\) 0 0
\(85\) −0.643274 −0.0697729
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0113 −1.27319 −0.636597 0.771197i \(-0.719658\pi\)
−0.636597 + 0.771197i \(0.719658\pi\)
\(90\) 0 0
\(91\) 9.35673 0.980851
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 5.78181 0.587054 0.293527 0.955951i \(-0.405171\pi\)
0.293527 + 0.955951i \(0.405171\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.43637 0.640443 0.320222 0.947343i \(-0.396243\pi\)
0.320222 + 0.947343i \(0.396243\pi\)
\(102\) 0 0
\(103\) −1.28655 −0.126767 −0.0633837 0.997989i \(-0.520189\pi\)
−0.0633837 + 0.997989i \(0.520189\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.28655 0.124375 0.0621877 0.998064i \(-0.480192\pi\)
0.0621877 + 0.998064i \(0.480192\pi\)
\(108\) 0 0
\(109\) 18.4364 1.76588 0.882942 0.469482i \(-0.155559\pi\)
0.882942 + 0.469482i \(0.155559\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −13.0796 −1.23043 −0.615215 0.788359i \(-0.710931\pi\)
−0.615215 + 0.788359i \(0.710931\pi\)
\(114\) 0 0
\(115\) −1.35673 −0.126515
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.71345 0.248742
\(120\) 0 0
\(121\) 20.0796 1.82542
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −9.28655 −0.824048 −0.412024 0.911173i \(-0.635178\pi\)
−0.412024 + 0.911173i \(0.635178\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.86146 −0.250007 −0.125004 0.992156i \(-0.539894\pi\)
−0.125004 + 0.992156i \(0.539894\pi\)
\(132\) 0 0
\(133\) 4.21819 0.365763
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.8727 1.27066 0.635332 0.772239i \(-0.280863\pi\)
0.635332 + 0.772239i \(0.280863\pi\)
\(138\) 0 0
\(139\) 0.850175 0.0721109 0.0360555 0.999350i \(-0.488521\pi\)
0.0360555 + 0.999350i \(0.488521\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.3662 1.03411
\(144\) 0 0
\(145\) −4.86146 −0.403722
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 20.1593 1.65151 0.825757 0.564026i \(-0.190748\pi\)
0.825757 + 0.564026i \(0.190748\pi\)
\(150\) 0 0
\(151\) 11.5160 0.937161 0.468580 0.883421i \(-0.344766\pi\)
0.468580 + 0.883421i \(0.344766\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.64327 0.212313
\(156\) 0 0
\(157\) 9.14982 0.730236 0.365118 0.930961i \(-0.381029\pi\)
0.365118 + 0.930961i \(0.381029\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.72292 0.451029
\(162\) 0 0
\(163\) −5.50474 −0.431164 −0.215582 0.976486i \(-0.569165\pi\)
−0.215582 + 0.976486i \(0.569165\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.920352 0.0712190 0.0356095 0.999366i \(-0.488663\pi\)
0.0356095 + 0.999366i \(0.488663\pi\)
\(168\) 0 0
\(169\) −8.07965 −0.621511
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 16.6658 1.26708 0.633540 0.773710i \(-0.281601\pi\)
0.633540 + 0.773710i \(0.281601\pi\)
\(174\) 0 0
\(175\) −4.21819 −0.318865
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 24.0226 1.79553 0.897766 0.440473i \(-0.145189\pi\)
0.897766 + 0.440473i \(0.145189\pi\)
\(180\) 0 0
\(181\) −9.14982 −0.680101 −0.340051 0.940407i \(-0.610444\pi\)
−0.340051 + 0.940407i \(0.610444\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.21819 −0.163084
\(186\) 0 0
\(187\) 3.58620 0.262249
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.2884 1.17858 0.589292 0.807920i \(-0.299407\pi\)
0.589292 + 0.807920i \(0.299407\pi\)
\(192\) 0 0
\(193\) 19.5047 1.40398 0.701991 0.712186i \(-0.252295\pi\)
0.701991 + 0.712186i \(0.252295\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −9.51602 −0.677988 −0.338994 0.940788i \(-0.610087\pi\)
−0.338994 + 0.940788i \(0.610087\pi\)
\(198\) 0 0
\(199\) −2.27708 −0.161418 −0.0807089 0.996738i \(-0.525718\pi\)
−0.0807089 + 0.996738i \(0.525718\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 20.5066 1.43928
\(204\) 0 0
\(205\) −3.57491 −0.249683
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.57491 0.385625
\(210\) 0 0
\(211\) 1.42690 0.0982320 0.0491160 0.998793i \(-0.484360\pi\)
0.0491160 + 0.998793i \(0.484360\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.218187 −0.0148802
\(216\) 0 0
\(217\) −11.1498 −0.756899
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.42690 0.0959839
\(222\) 0 0
\(223\) −6.71345 −0.449566 −0.224783 0.974409i \(-0.572167\pi\)
−0.224783 + 0.974409i \(0.572167\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 13.7931 0.915480 0.457740 0.889086i \(-0.348659\pi\)
0.457740 + 0.889086i \(0.348659\pi\)
\(228\) 0 0
\(229\) −14.5066 −0.958620 −0.479310 0.877646i \(-0.659113\pi\)
−0.479310 + 0.877646i \(0.659113\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.2996 −1.59192 −0.795961 0.605347i \(-0.793034\pi\)
−0.795961 + 0.605347i \(0.793034\pi\)
\(234\) 0 0
\(235\) 1.35673 0.0885030
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.57491 0.360611 0.180306 0.983611i \(-0.442291\pi\)
0.180306 + 0.983611i \(0.442291\pi\)
\(240\) 0 0
\(241\) 24.2996 1.56528 0.782639 0.622476i \(-0.213873\pi\)
0.782639 + 0.622476i \(0.213873\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 10.7931 0.689546
\(246\) 0 0
\(247\) 2.21819 0.141140
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −14.0113 −0.884385 −0.442192 0.896920i \(-0.645799\pi\)
−0.442192 + 0.896920i \(0.645799\pi\)
\(252\) 0 0
\(253\) 7.56363 0.475521
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.9204 0.681193 0.340596 0.940210i \(-0.389371\pi\)
0.340596 + 0.940210i \(0.389371\pi\)
\(258\) 0 0
\(259\) 9.35673 0.581399
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −10.6658 −0.657684 −0.328842 0.944385i \(-0.606658\pi\)
−0.328842 + 0.944385i \(0.606658\pi\)
\(264\) 0 0
\(265\) −0.643274 −0.0395160
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −16.0113 −0.976225 −0.488113 0.872781i \(-0.662314\pi\)
−0.488113 + 0.872781i \(0.662314\pi\)
\(270\) 0 0
\(271\) −30.7360 −1.86708 −0.933540 0.358473i \(-0.883298\pi\)
−0.933540 + 0.358473i \(0.883298\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.57491 −0.336180
\(276\) 0 0
\(277\) 20.2996 1.21969 0.609844 0.792522i \(-0.291232\pi\)
0.609844 + 0.792522i \(0.291232\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 18.5844 1.10865 0.554326 0.832300i \(-0.312976\pi\)
0.554326 + 0.832300i \(0.312976\pi\)
\(282\) 0 0
\(283\) −25.0909 −1.49150 −0.745751 0.666225i \(-0.767909\pi\)
−0.745751 + 0.666225i \(0.767909\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 15.0796 0.890123
\(288\) 0 0
\(289\) −16.5862 −0.975659
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −11.7931 −0.688960 −0.344480 0.938794i \(-0.611945\pi\)
−0.344480 + 0.938794i \(0.611945\pi\)
\(294\) 0 0
\(295\) 4.43637 0.258296
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.00947 0.174042
\(300\) 0 0
\(301\) 0.920352 0.0530482
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 13.0796 0.748938
\(306\) 0 0
\(307\) −8.85018 −0.505106 −0.252553 0.967583i \(-0.581270\pi\)
−0.252553 + 0.967583i \(0.581270\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 21.8709 1.24019 0.620093 0.784528i \(-0.287095\pi\)
0.620093 + 0.784528i \(0.287095\pi\)
\(312\) 0 0
\(313\) −1.14982 −0.0649919 −0.0324960 0.999472i \(-0.510346\pi\)
−0.0324960 + 0.999472i \(0.510346\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −29.1022 −1.63454 −0.817272 0.576252i \(-0.804514\pi\)
−0.817272 + 0.576252i \(0.804514\pi\)
\(318\) 0 0
\(319\) 27.1022 1.51743
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.643274 0.0357927
\(324\) 0 0
\(325\) −2.21819 −0.123043
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.72292 −0.315515
\(330\) 0 0
\(331\) −8.94292 −0.491548 −0.245774 0.969327i \(-0.579042\pi\)
−0.245774 + 0.969327i \(0.579042\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.28655 0.0702917
\(336\) 0 0
\(337\) 20.3775 1.11003 0.555016 0.831840i \(-0.312712\pi\)
0.555016 + 0.831840i \(0.312712\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −14.7360 −0.798000
\(342\) 0 0
\(343\) −16.0000 −0.863919
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.07965 −0.380055 −0.190028 0.981779i \(-0.560858\pi\)
−0.190028 + 0.981779i \(0.560858\pi\)
\(348\) 0 0
\(349\) 36.1593 1.93556 0.967781 0.251792i \(-0.0810199\pi\)
0.967781 + 0.251792i \(0.0810199\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −32.1593 −1.71167 −0.855833 0.517252i \(-0.826955\pi\)
−0.855833 + 0.517252i \(0.826955\pi\)
\(354\) 0 0
\(355\) −11.1498 −0.591771
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.86146 0.151022 0.0755111 0.997145i \(-0.475941\pi\)
0.0755111 + 0.997145i \(0.475941\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) −12.3585 −0.645111 −0.322555 0.946551i \(-0.604542\pi\)
−0.322555 + 0.946551i \(0.604542\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.71345 0.140875
\(372\) 0 0
\(373\) 0.495265 0.0256438 0.0128219 0.999918i \(-0.495919\pi\)
0.0128219 + 0.999918i \(0.495919\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10.7836 0.555385
\(378\) 0 0
\(379\) 2.78363 0.142985 0.0714927 0.997441i \(-0.477224\pi\)
0.0714927 + 0.997441i \(0.477224\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 33.0131 1.68689 0.843445 0.537215i \(-0.180524\pi\)
0.843445 + 0.537215i \(0.180524\pi\)
\(384\) 0 0
\(385\) 23.5160 1.19849
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.4589 −1.54433 −0.772165 0.635422i \(-0.780826\pi\)
−0.772165 + 0.635422i \(0.780826\pi\)
\(390\) 0 0
\(391\) 0.872747 0.0441367
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.28655 −0.0647333
\(396\) 0 0
\(397\) −15.2865 −0.767210 −0.383605 0.923497i \(-0.625318\pi\)
−0.383605 + 0.923497i \(0.625318\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 36.0339 1.79944 0.899722 0.436462i \(-0.143769\pi\)
0.899722 + 0.436462i \(0.143769\pi\)
\(402\) 0 0
\(403\) −5.86328 −0.292071
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.3662 0.612970
\(408\) 0 0
\(409\) −34.0226 −1.68231 −0.841154 0.540796i \(-0.818123\pi\)
−0.841154 + 0.540796i \(0.818123\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −18.7135 −0.920829
\(414\) 0 0
\(415\) −12.3662 −0.607033
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.72474 0.426231 0.213116 0.977027i \(-0.431639\pi\)
0.213116 + 0.977027i \(0.431639\pi\)
\(420\) 0 0
\(421\) 35.0131 1.70643 0.853217 0.521556i \(-0.174648\pi\)
0.853217 + 0.521556i \(0.174648\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.643274 −0.0312034
\(426\) 0 0
\(427\) −55.1724 −2.66998
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 13.8633 0.667771 0.333885 0.942614i \(-0.391640\pi\)
0.333885 + 0.942614i \(0.391640\pi\)
\(432\) 0 0
\(433\) −30.2408 −1.45328 −0.726639 0.687019i \(-0.758919\pi\)
−0.726639 + 0.687019i \(0.758919\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.35673 0.0649010
\(438\) 0 0
\(439\) 31.5862 1.50753 0.753763 0.657146i \(-0.228236\pi\)
0.753763 + 0.657146i \(0.228236\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −18.2295 −0.866108 −0.433054 0.901368i \(-0.642564\pi\)
−0.433054 + 0.901368i \(0.642564\pi\)
\(444\) 0 0
\(445\) −12.0113 −0.569390
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.71164 0.458320 0.229160 0.973389i \(-0.426402\pi\)
0.229160 + 0.973389i \(0.426402\pi\)
\(450\) 0 0
\(451\) 19.9298 0.938459
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.35673 0.438650
\(456\) 0 0
\(457\) −19.8633 −0.929165 −0.464582 0.885530i \(-0.653796\pi\)
−0.464582 + 0.885530i \(0.653796\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 20.2996 0.945449 0.472724 0.881210i \(-0.343271\pi\)
0.472724 + 0.881210i \(0.343271\pi\)
\(462\) 0 0
\(463\) 32.2408 1.49836 0.749178 0.662369i \(-0.230449\pi\)
0.749178 + 0.662369i \(0.230449\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.2295 1.21376 0.606878 0.794795i \(-0.292422\pi\)
0.606878 + 0.794795i \(0.292422\pi\)
\(468\) 0 0
\(469\) −5.42690 −0.250591
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.21637 0.0559288
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −33.1611 −1.51517 −0.757585 0.652737i \(-0.773621\pi\)
−0.757585 + 0.652737i \(0.773621\pi\)
\(480\) 0 0
\(481\) 4.92035 0.224349
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.78181 0.262539
\(486\) 0 0
\(487\) −19.5636 −0.886513 −0.443256 0.896395i \(-0.646177\pi\)
−0.443256 + 0.896395i \(0.646177\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.8615 1.21224 0.606120 0.795373i \(-0.292725\pi\)
0.606120 + 0.795373i \(0.292725\pi\)
\(492\) 0 0
\(493\) 3.12725 0.140844
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 47.0320 2.10968
\(498\) 0 0
\(499\) 40.0226 1.79166 0.895828 0.444401i \(-0.146583\pi\)
0.895828 + 0.444401i \(0.146583\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 17.6527 0.787097 0.393549 0.919304i \(-0.371247\pi\)
0.393549 + 0.919304i \(0.371247\pi\)
\(504\) 0 0
\(505\) 6.43637 0.286415
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16.8615 0.747371 0.373686 0.927555i \(-0.378094\pi\)
0.373686 + 0.927555i \(0.378094\pi\)
\(510\) 0 0
\(511\) −42.1819 −1.86602
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.28655 −0.0566921
\(516\) 0 0
\(517\) −7.56363 −0.332648
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.57491 −0.156620 −0.0783099 0.996929i \(-0.524952\pi\)
−0.0783099 + 0.996929i \(0.524952\pi\)
\(522\) 0 0
\(523\) −28.4364 −1.24344 −0.621718 0.783241i \(-0.713565\pi\)
−0.621718 + 0.783241i \(0.713565\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.70035 −0.0740684
\(528\) 0 0
\(529\) −21.1593 −0.919969
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.92982 0.343479
\(534\) 0 0
\(535\) 1.28655 0.0556224
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −60.1706 −2.59173
\(540\) 0 0
\(541\) 0.573097 0.0246394 0.0123197 0.999924i \(-0.496078\pi\)
0.0123197 + 0.999924i \(0.496078\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 18.4364 0.789727
\(546\) 0 0
\(547\) −8.85018 −0.378406 −0.189203 0.981938i \(-0.560590\pi\)
−0.189203 + 0.981938i \(0.560590\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.86146 0.207105
\(552\) 0 0
\(553\) 5.42690 0.230775
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8.29965 −0.351667 −0.175834 0.984420i \(-0.556262\pi\)
−0.175834 + 0.984420i \(0.556262\pi\)
\(558\) 0 0
\(559\) 0.483979 0.0204701
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −17.3793 −0.732450 −0.366225 0.930526i \(-0.619350\pi\)
−0.366225 + 0.930526i \(0.619350\pi\)
\(564\) 0 0
\(565\) −13.0796 −0.550265
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −19.4346 −0.814739 −0.407370 0.913263i \(-0.633554\pi\)
−0.407370 + 0.913263i \(0.633554\pi\)
\(570\) 0 0
\(571\) 32.1629 1.34598 0.672988 0.739653i \(-0.265010\pi\)
0.672988 + 0.739653i \(0.265010\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.35673 −0.0565794
\(576\) 0 0
\(577\) −31.1688 −1.29757 −0.648786 0.760971i \(-0.724723\pi\)
−0.648786 + 0.760971i \(0.724723\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 52.1629 2.16408
\(582\) 0 0
\(583\) 3.58620 0.148525
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −23.5160 −0.970610 −0.485305 0.874345i \(-0.661291\pi\)
−0.485305 + 0.874345i \(0.661291\pi\)
\(588\) 0 0
\(589\) −2.64327 −0.108914
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 31.0131 1.27356 0.636778 0.771047i \(-0.280267\pi\)
0.636778 + 0.771047i \(0.280267\pi\)
\(594\) 0 0
\(595\) 2.71345 0.111241
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −39.1724 −1.60054 −0.800270 0.599639i \(-0.795311\pi\)
−0.800270 + 0.599639i \(0.795311\pi\)
\(600\) 0 0
\(601\) −3.86328 −0.157586 −0.0787932 0.996891i \(-0.525107\pi\)
−0.0787932 + 0.996891i \(0.525107\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 20.0796 0.816354
\(606\) 0 0
\(607\) 8.02257 0.325626 0.162813 0.986657i \(-0.447943\pi\)
0.162813 + 0.986657i \(0.447943\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.00947 −0.121750
\(612\) 0 0
\(613\) 19.8633 0.802270 0.401135 0.916019i \(-0.368616\pi\)
0.401135 + 0.916019i \(0.368616\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −18.3888 −0.740304 −0.370152 0.928971i \(-0.620694\pi\)
−0.370152 + 0.928971i \(0.620694\pi\)
\(618\) 0 0
\(619\) 12.4364 0.499860 0.249930 0.968264i \(-0.419592\pi\)
0.249930 + 0.968264i \(0.419592\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 50.6658 2.02988
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.42690 0.0568943
\(630\) 0 0
\(631\) 42.6182 1.69661 0.848303 0.529512i \(-0.177625\pi\)
0.848303 + 0.529512i \(0.177625\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −9.28655 −0.368525
\(636\) 0 0
\(637\) −23.9411 −0.948581
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.16111 0.124856 0.0624282 0.998049i \(-0.480116\pi\)
0.0624282 + 0.998049i \(0.480116\pi\)
\(642\) 0 0
\(643\) 36.5368 1.44087 0.720435 0.693523i \(-0.243942\pi\)
0.720435 + 0.693523i \(0.243942\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.6527 0.694001 0.347001 0.937865i \(-0.387200\pi\)
0.347001 + 0.937865i \(0.387200\pi\)
\(648\) 0 0
\(649\) −24.7324 −0.970831
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.51602 0.215859 0.107929 0.994159i \(-0.465578\pi\)
0.107929 + 0.994159i \(0.465578\pi\)
\(654\) 0 0
\(655\) −2.86146 −0.111807
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −18.2996 −0.712853 −0.356427 0.934323i \(-0.616005\pi\)
−0.356427 + 0.934323i \(0.616005\pi\)
\(660\) 0 0
\(661\) 7.72292 0.300387 0.150193 0.988657i \(-0.452010\pi\)
0.150193 + 0.988657i \(0.452010\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.21819 0.163574
\(666\) 0 0
\(667\) 6.59567 0.255385
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −72.9179 −2.81496
\(672\) 0 0
\(673\) 39.9637 1.54049 0.770243 0.637750i \(-0.220135\pi\)
0.770243 + 0.637750i \(0.220135\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 48.2295 1.85361 0.926805 0.375544i \(-0.122544\pi\)
0.926805 + 0.375544i \(0.122544\pi\)
\(678\) 0 0
\(679\) −24.3888 −0.935955
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15.5862 0.596389 0.298195 0.954505i \(-0.403616\pi\)
0.298195 + 0.954505i \(0.403616\pi\)
\(684\) 0 0
\(685\) 14.8727 0.568258
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.42690 0.0543607
\(690\) 0 0
\(691\) 33.8633 1.28822 0.644110 0.764933i \(-0.277228\pi\)
0.644110 + 0.764933i \(0.277228\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.850175 0.0322490
\(696\) 0 0
\(697\) 2.29965 0.0871054
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −31.3317 −1.18338 −0.591691 0.806165i \(-0.701539\pi\)
−0.591691 + 0.806165i \(0.701539\pi\)
\(702\) 0 0
\(703\) 2.21819 0.0836605
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −27.1498 −1.02107
\(708\) 0 0
\(709\) −0.939294 −0.0352759 −0.0176380 0.999844i \(-0.505615\pi\)
−0.0176380 + 0.999844i \(0.505615\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3.58620 −0.134304
\(714\) 0 0
\(715\) 12.3662 0.462470
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 33.1611 1.23670 0.618350 0.785903i \(-0.287801\pi\)
0.618350 + 0.785903i \(0.287801\pi\)
\(720\) 0 0
\(721\) 5.42690 0.202108
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.86146 −0.180550
\(726\) 0 0
\(727\) −32.2408 −1.19574 −0.597872 0.801592i \(-0.703987\pi\)
−0.597872 + 0.801592i \(0.703987\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0.140354 0.00519118
\(732\) 0 0
\(733\) 17.5862 0.649561 0.324781 0.945789i \(-0.394710\pi\)
0.324781 + 0.945789i \(0.394710\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.17240 −0.264199
\(738\) 0 0
\(739\) 14.5731 0.536080 0.268040 0.963408i \(-0.413624\pi\)
0.268040 + 0.963408i \(0.413624\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.5862 1.01204 0.506020 0.862522i \(-0.331116\pi\)
0.506020 + 0.862522i \(0.331116\pi\)
\(744\) 0 0
\(745\) 20.1593 0.738579
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5.42690 −0.198295
\(750\) 0 0
\(751\) −11.5160 −0.420226 −0.210113 0.977677i \(-0.567383\pi\)
−0.210113 + 0.977677i \(0.567383\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 11.5160 0.419111
\(756\) 0 0
\(757\) −13.2902 −0.483040 −0.241520 0.970396i \(-0.577646\pi\)
−0.241520 + 0.970396i \(0.577646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 7.84070 0.284225 0.142113 0.989850i \(-0.454610\pi\)
0.142113 + 0.989850i \(0.454610\pi\)
\(762\) 0 0
\(763\) −77.7681 −2.81539
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −9.84070 −0.355327
\(768\) 0 0
\(769\) −52.2996 −1.88597 −0.942987 0.332830i \(-0.891996\pi\)
−0.942987 + 0.332830i \(0.891996\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25.2426 0.907912 0.453956 0.891024i \(-0.350012\pi\)
0.453956 + 0.891024i \(0.350012\pi\)
\(774\) 0 0
\(775\) 2.64327 0.0949492
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.57491 0.128085
\(780\) 0 0
\(781\) 62.1593 2.22423
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9.14982 0.326571
\(786\) 0 0
\(787\) −17.8633 −0.636757 −0.318379 0.947964i \(-0.603138\pi\)
−0.318379 + 0.947964i \(0.603138\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 55.1724 1.96170
\(792\) 0 0
\(793\) −29.0131 −1.03029
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.50655 −0.230474 −0.115237 0.993338i \(-0.536763\pi\)
−0.115237 + 0.993338i \(0.536763\pi\)
\(798\) 0 0
\(799\) −0.872747 −0.0308756
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −55.7491 −1.96734
\(804\) 0 0
\(805\) 5.72292 0.201707
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −3.84070 −0.135032 −0.0675160 0.997718i \(-0.521507\pi\)
−0.0675160 + 0.997718i \(0.521507\pi\)
\(810\) 0 0
\(811\) −7.44584 −0.261459 −0.130729 0.991418i \(-0.541732\pi\)
−0.130729 + 0.991418i \(0.541732\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.50474 −0.192822
\(816\) 0 0
\(817\) 0.218187 0.00763339
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.2960 1.05734 0.528669 0.848828i \(-0.322691\pi\)
0.528669 + 0.848828i \(0.322691\pi\)
\(822\) 0 0
\(823\) 32.3811 1.12873 0.564367 0.825524i \(-0.309120\pi\)
0.564367 + 0.825524i \(0.309120\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11.9524 0.415625 0.207813 0.978169i \(-0.433366\pi\)
0.207813 + 0.978169i \(0.433366\pi\)
\(828\) 0 0
\(829\) −22.5767 −0.784122 −0.392061 0.919939i \(-0.628238\pi\)
−0.392061 + 0.919939i \(0.628238\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.94292 −0.240558
\(834\) 0 0
\(835\) 0.920352 0.0318501
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 3.14982 0.108744 0.0543720 0.998521i \(-0.482684\pi\)
0.0543720 + 0.998521i \(0.482684\pi\)
\(840\) 0 0
\(841\) −5.36620 −0.185041
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8.07965 −0.277948
\(846\) 0 0
\(847\) −84.6997 −2.91032
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.00947 0.103163
\(852\) 0 0
\(853\) 11.5826 0.396580 0.198290 0.980143i \(-0.436461\pi\)
0.198290 + 0.980143i \(0.436461\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.91088 −0.133593 −0.0667966 0.997767i \(-0.521278\pi\)
−0.0667966 + 0.997767i \(0.521278\pi\)
\(858\) 0 0
\(859\) −5.01310 −0.171045 −0.0855224 0.996336i \(-0.527256\pi\)
−0.0855224 + 0.996336i \(0.527256\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −40.7324 −1.38655 −0.693273 0.720675i \(-0.743832\pi\)
−0.693273 + 0.720675i \(0.743832\pi\)
\(864\) 0 0
\(865\) 16.6658 0.566656
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 7.17240 0.243307
\(870\) 0 0
\(871\) −2.85381 −0.0966975
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.21819 −0.142601
\(876\) 0 0
\(877\) 36.2371 1.22364 0.611820 0.790997i \(-0.290438\pi\)
0.611820 + 0.790997i \(0.290438\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 18.0226 0.607196 0.303598 0.952800i \(-0.401812\pi\)
0.303598 + 0.952800i \(0.401812\pi\)
\(882\) 0 0
\(883\) −52.6771 −1.77273 −0.886363 0.462990i \(-0.846776\pi\)
−0.886363 + 0.462990i \(0.846776\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −32.8727 −1.10376 −0.551879 0.833924i \(-0.686089\pi\)
−0.551879 + 0.833924i \(0.686089\pi\)
\(888\) 0 0
\(889\) 39.1724 1.31380
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.35673 −0.0454011
\(894\) 0 0
\(895\) 24.0226 0.802986
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −12.8502 −0.428577
\(900\) 0 0
\(901\) 0.413802 0.0137857
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9.14982 −0.304150
\(906\) 0 0
\(907\) −26.5957 −0.883095 −0.441547 0.897238i \(-0.645570\pi\)
−0.441547 + 0.897238i \(0.645570\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6.15930 −0.204067 −0.102033 0.994781i \(-0.532535\pi\)
−0.102033 + 0.994781i \(0.532535\pi\)
\(912\) 0 0
\(913\) 68.9405 2.28160
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.0702 0.398592
\(918\) 0 0
\(919\) −33.8858 −1.11779 −0.558895 0.829238i \(-0.688775\pi\)
−0.558895 + 0.829238i \(0.688775\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 24.7324 0.814077
\(924\) 0 0
\(925\) −2.21819 −0.0729335
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 34.8953 1.14488 0.572439 0.819947i \(-0.305997\pi\)
0.572439 + 0.819947i \(0.305997\pi\)
\(930\) 0 0
\(931\) −10.7931 −0.353730
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3.58620 0.117281
\(936\) 0 0
\(937\) −6.29602 −0.205682 −0.102841 0.994698i \(-0.532793\pi\)
−0.102841 + 0.994698i \(0.532793\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −10.8651 −0.354192 −0.177096 0.984194i \(-0.556670\pi\)
−0.177096 + 0.984194i \(0.556670\pi\)
\(942\) 0 0
\(943\) 4.85018 0.157943
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.79310 0.318233 0.159116 0.987260i \(-0.449135\pi\)
0.159116 + 0.987260i \(0.449135\pi\)
\(948\) 0 0
\(949\) −22.1819 −0.720054
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.35310 −0.0438311 −0.0219155 0.999760i \(-0.506976\pi\)
−0.0219155 + 0.999760i \(0.506976\pi\)
\(954\) 0 0
\(955\) 16.2884 0.527079
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −62.7360 −2.02585
\(960\) 0 0
\(961\) −24.0131 −0.774616
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 19.5047 0.627880
\(966\) 0 0
\(967\) 21.9637 0.706304 0.353152 0.935566i \(-0.385110\pi\)
0.353152 + 0.935566i \(0.385110\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 13.7455 0.441114 0.220557 0.975374i \(-0.429213\pi\)
0.220557 + 0.975374i \(0.429213\pi\)
\(972\) 0 0
\(973\) −3.58620 −0.114968
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −48.3698 −1.54749 −0.773744 0.633498i \(-0.781618\pi\)
−0.773744 + 0.633498i \(0.781618\pi\)
\(978\) 0 0
\(979\) 66.9619 2.14011
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −14.0665 −0.448653 −0.224327 0.974514i \(-0.572018\pi\)
−0.224327 + 0.974514i \(0.572018\pi\)
\(984\) 0 0
\(985\) −9.51602 −0.303206
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0.296019 0.00941287
\(990\) 0 0
\(991\) −33.2865 −1.05738 −0.528691 0.848814i \(-0.677317\pi\)
−0.528691 + 0.848814i \(0.677317\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2.27708 −0.0721882
\(996\) 0 0
\(997\) −48.5957 −1.53904 −0.769520 0.638623i \(-0.779505\pi\)
−0.769520 + 0.638623i \(0.779505\pi\)
\(998\) 0 0
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6840.2.a.bn.1.1 3
3.2 odd 2 2280.2.a.t.1.1 3
12.11 even 2 4560.2.a.bq.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.t.1.1 3 3.2 odd 2
4560.2.a.bq.1.3 3 12.11 even 2
6840.2.a.bn.1.1 3 1.1 even 1 trivial