Properties

Label 224.3.g.b.15.3
Level 224
Weight 3
Character 224.15
Analytic conductor 6.104
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.292213762624.3
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 15.3
Root \(1.85837 + 0.739226i\) of \(x^{8} - x^{7} - 2 x^{6} - 2 x^{5} + 24 x^{4} - 8 x^{3} - 32 x^{2} - 64 x + 256\)
Character \(\chi\) \(=\) 224.15
Dual form 224.3.g.b.15.4

$q$-expansion

\(f(q)\) \(=\) \(q-0.0974366 q^{3} -3.46547i q^{5} -2.64575i q^{7} -8.99051 q^{9} +O(q^{10})\) \(q-0.0974366 q^{3} -3.46547i q^{5} -2.64575i q^{7} -8.99051 q^{9} +2.92866 q^{11} -19.1586i q^{13} +0.337664i q^{15} -14.3897 q^{17} -8.09744 q^{19} +0.257793i q^{21} -16.7598i q^{23} +12.9905 q^{25} +1.75293 q^{27} -27.1649i q^{29} -44.8923i q^{31} -0.285359 q^{33} -9.16878 q^{35} +39.5687i q^{37} +1.86675i q^{39} +45.8766 q^{41} -61.0334 q^{43} +31.1563i q^{45} +46.2793i q^{47} -7.00000 q^{49} +1.40209 q^{51} +9.69424i q^{53} -10.1492i q^{55} +0.788986 q^{57} +114.554 q^{59} -7.48032i q^{61} +23.7866i q^{63} -66.3935 q^{65} +12.0590 q^{67} +1.63302i q^{69} +129.187i q^{71} -18.2854 q^{73} -1.26575 q^{75} -7.74851i q^{77} +42.6168i q^{79} +80.7438 q^{81} +109.670 q^{83} +49.8673i q^{85} +2.64685i q^{87} -80.9162 q^{89} -50.6889 q^{91} +4.37415i q^{93} +28.0614i q^{95} +162.086 q^{97} -26.3301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} + 48q^{9} + O(q^{10}) \) \( 8q + 8q^{3} + 48q^{9} + 32q^{11} - 80q^{17} - 56q^{19} - 16q^{25} + 32q^{27} + 32q^{33} - 56q^{35} + 128q^{41} - 56q^{49} + 368q^{51} + 56q^{57} - 104q^{59} - 72q^{65} - 304q^{67} - 112q^{73} - 72q^{75} + 48q^{81} - 72q^{83} - 512q^{89} + 56q^{91} + 64q^{97} - 256q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) −0.0974366 −0.0324789 −0.0162394 0.999868i \(-0.505169\pi\)
−0.0162394 + 0.999868i \(0.505169\pi\)
\(4\) 0 0
\(5\) − 3.46547i − 0.693094i −0.938033 0.346547i \(-0.887354\pi\)
0.938033 0.346547i \(-0.112646\pi\)
\(6\) 0 0
\(7\) − 2.64575i − 0.377964i
\(8\) 0 0
\(9\) −8.99051 −0.998945
\(10\) 0 0
\(11\) 2.92866 0.266242 0.133121 0.991100i \(-0.457500\pi\)
0.133121 + 0.991100i \(0.457500\pi\)
\(12\) 0 0
\(13\) − 19.1586i − 1.47374i −0.676036 0.736869i \(-0.736304\pi\)
0.676036 0.736869i \(-0.263696\pi\)
\(14\) 0 0
\(15\) 0.337664i 0.0225109i
\(16\) 0 0
\(17\) −14.3897 −0.846456 −0.423228 0.906023i \(-0.639103\pi\)
−0.423228 + 0.906023i \(0.639103\pi\)
\(18\) 0 0
\(19\) −8.09744 −0.426181 −0.213090 0.977032i \(-0.568353\pi\)
−0.213090 + 0.977032i \(0.568353\pi\)
\(20\) 0 0
\(21\) 0.257793i 0.0122759i
\(22\) 0 0
\(23\) − 16.7598i − 0.728687i −0.931265 0.364344i \(-0.881293\pi\)
0.931265 0.364344i \(-0.118707\pi\)
\(24\) 0 0
\(25\) 12.9905 0.519620
\(26\) 0 0
\(27\) 1.75293 0.0649234
\(28\) 0 0
\(29\) − 27.1649i − 0.936720i −0.883538 0.468360i \(-0.844845\pi\)
0.883538 0.468360i \(-0.155155\pi\)
\(30\) 0 0
\(31\) − 44.8923i − 1.44814i −0.689728 0.724069i \(-0.742270\pi\)
0.689728 0.724069i \(-0.257730\pi\)
\(32\) 0 0
\(33\) −0.285359 −0.00864723
\(34\) 0 0
\(35\) −9.16878 −0.261965
\(36\) 0 0
\(37\) 39.5687i 1.06943i 0.845034 + 0.534713i \(0.179580\pi\)
−0.845034 + 0.534713i \(0.820420\pi\)
\(38\) 0 0
\(39\) 1.86675i 0.0478653i
\(40\) 0 0
\(41\) 45.8766 1.11894 0.559471 0.828850i \(-0.311004\pi\)
0.559471 + 0.828850i \(0.311004\pi\)
\(42\) 0 0
\(43\) −61.0334 −1.41938 −0.709690 0.704514i \(-0.751165\pi\)
−0.709690 + 0.704514i \(0.751165\pi\)
\(44\) 0 0
\(45\) 31.1563i 0.692363i
\(46\) 0 0
\(47\) 46.2793i 0.984666i 0.870407 + 0.492333i \(0.163856\pi\)
−0.870407 + 0.492333i \(0.836144\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 1.40209 0.0274919
\(52\) 0 0
\(53\) 9.69424i 0.182910i 0.995809 + 0.0914551i \(0.0291518\pi\)
−0.995809 + 0.0914551i \(0.970848\pi\)
\(54\) 0 0
\(55\) − 10.1492i − 0.184531i
\(56\) 0 0
\(57\) 0.788986 0.0138419
\(58\) 0 0
\(59\) 114.554 1.94159 0.970796 0.239907i \(-0.0771170\pi\)
0.970796 + 0.239907i \(0.0771170\pi\)
\(60\) 0 0
\(61\) − 7.48032i − 0.122628i −0.998119 0.0613141i \(-0.980471\pi\)
0.998119 0.0613141i \(-0.0195291\pi\)
\(62\) 0 0
\(63\) 23.7866i 0.377566i
\(64\) 0 0
\(65\) −66.3935 −1.02144
\(66\) 0 0
\(67\) 12.0590 0.179985 0.0899925 0.995942i \(-0.471316\pi\)
0.0899925 + 0.995942i \(0.471316\pi\)
\(68\) 0 0
\(69\) 1.63302i 0.0236669i
\(70\) 0 0
\(71\) 129.187i 1.81953i 0.415124 + 0.909765i \(0.363738\pi\)
−0.415124 + 0.909765i \(0.636262\pi\)
\(72\) 0 0
\(73\) −18.2854 −0.250484 −0.125242 0.992126i \(-0.539971\pi\)
−0.125242 + 0.992126i \(0.539971\pi\)
\(74\) 0 0
\(75\) −1.26575 −0.0168767
\(76\) 0 0
\(77\) − 7.74851i − 0.100630i
\(78\) 0 0
\(79\) 42.6168i 0.539454i 0.962937 + 0.269727i \(0.0869334\pi\)
−0.962937 + 0.269727i \(0.913067\pi\)
\(80\) 0 0
\(81\) 80.7438 0.996836
\(82\) 0 0
\(83\) 109.670 1.32133 0.660663 0.750683i \(-0.270275\pi\)
0.660663 + 0.750683i \(0.270275\pi\)
\(84\) 0 0
\(85\) 49.8673i 0.586674i
\(86\) 0 0
\(87\) 2.64685i 0.0304236i
\(88\) 0 0
\(89\) −80.9162 −0.909170 −0.454585 0.890703i \(-0.650212\pi\)
−0.454585 + 0.890703i \(0.650212\pi\)
\(90\) 0 0
\(91\) −50.6889 −0.557020
\(92\) 0 0
\(93\) 4.37415i 0.0470339i
\(94\) 0 0
\(95\) 28.0614i 0.295384i
\(96\) 0 0
\(97\) 162.086 1.67099 0.835495 0.549498i \(-0.185181\pi\)
0.835495 + 0.549498i \(0.185181\pi\)
\(98\) 0 0
\(99\) −26.3301 −0.265961
\(100\) 0 0
\(101\) 106.827i 1.05769i 0.848717 + 0.528847i \(0.177375\pi\)
−0.848717 + 0.528847i \(0.822625\pi\)
\(102\) 0 0
\(103\) − 126.626i − 1.22938i −0.788768 0.614691i \(-0.789281\pi\)
0.788768 0.614691i \(-0.210719\pi\)
\(104\) 0 0
\(105\) 0.893374 0.00850832
\(106\) 0 0
\(107\) −87.0191 −0.813263 −0.406632 0.913592i \(-0.633297\pi\)
−0.406632 + 0.913592i \(0.633297\pi\)
\(108\) 0 0
\(109\) − 189.921i − 1.74240i −0.490930 0.871199i \(-0.663343\pi\)
0.490930 0.871199i \(-0.336657\pi\)
\(110\) 0 0
\(111\) − 3.85544i − 0.0347337i
\(112\) 0 0
\(113\) −40.1848 −0.355617 −0.177809 0.984065i \(-0.556901\pi\)
−0.177809 + 0.984065i \(0.556901\pi\)
\(114\) 0 0
\(115\) −58.0806 −0.505049
\(116\) 0 0
\(117\) 172.245i 1.47218i
\(118\) 0 0
\(119\) 38.0717i 0.319930i
\(120\) 0 0
\(121\) −112.423 −0.929115
\(122\) 0 0
\(123\) −4.47006 −0.0363420
\(124\) 0 0
\(125\) − 131.655i − 1.05324i
\(126\) 0 0
\(127\) − 153.657i − 1.20989i −0.796266 0.604947i \(-0.793194\pi\)
0.796266 0.604947i \(-0.206806\pi\)
\(128\) 0 0
\(129\) 5.94688 0.0460999
\(130\) 0 0
\(131\) −61.8649 −0.472251 −0.236126 0.971723i \(-0.575878\pi\)
−0.236126 + 0.971723i \(0.575878\pi\)
\(132\) 0 0
\(133\) 21.4238i 0.161081i
\(134\) 0 0
\(135\) − 6.07474i − 0.0449981i
\(136\) 0 0
\(137\) 105.943 0.773307 0.386653 0.922225i \(-0.373631\pi\)
0.386653 + 0.922225i \(0.373631\pi\)
\(138\) 0 0
\(139\) 185.384 1.33370 0.666848 0.745194i \(-0.267643\pi\)
0.666848 + 0.745194i \(0.267643\pi\)
\(140\) 0 0
\(141\) − 4.50930i − 0.0319808i
\(142\) 0 0
\(143\) − 56.1090i − 0.392371i
\(144\) 0 0
\(145\) −94.1392 −0.649236
\(146\) 0 0
\(147\) 0.682056 0.00463984
\(148\) 0 0
\(149\) 47.4096i 0.318185i 0.987264 + 0.159093i \(0.0508568\pi\)
−0.987264 + 0.159093i \(0.949143\pi\)
\(150\) 0 0
\(151\) − 114.576i − 0.758780i −0.925237 0.379390i \(-0.876134\pi\)
0.925237 0.379390i \(-0.123866\pi\)
\(152\) 0 0
\(153\) 129.371 0.845563
\(154\) 0 0
\(155\) −155.573 −1.00370
\(156\) 0 0
\(157\) − 294.095i − 1.87322i −0.350378 0.936608i \(-0.613947\pi\)
0.350378 0.936608i \(-0.386053\pi\)
\(158\) 0 0
\(159\) − 0.944573i − 0.00594071i
\(160\) 0 0
\(161\) −44.3423 −0.275418
\(162\) 0 0
\(163\) −171.021 −1.04921 −0.524603 0.851347i \(-0.675786\pi\)
−0.524603 + 0.851347i \(0.675786\pi\)
\(164\) 0 0
\(165\) 0.988902i 0.00599335i
\(166\) 0 0
\(167\) − 120.657i − 0.722499i −0.932469 0.361249i \(-0.882350\pi\)
0.932469 0.361249i \(-0.117650\pi\)
\(168\) 0 0
\(169\) −198.052 −1.17190
\(170\) 0 0
\(171\) 72.8001 0.425731
\(172\) 0 0
\(173\) − 108.339i − 0.626236i −0.949714 0.313118i \(-0.898626\pi\)
0.949714 0.313118i \(-0.101374\pi\)
\(174\) 0 0
\(175\) − 34.3696i − 0.196398i
\(176\) 0 0
\(177\) −11.1617 −0.0630607
\(178\) 0 0
\(179\) 161.438 0.901886 0.450943 0.892553i \(-0.351088\pi\)
0.450943 + 0.892553i \(0.351088\pi\)
\(180\) 0 0
\(181\) − 7.14696i − 0.0394860i −0.999805 0.0197430i \(-0.993715\pi\)
0.999805 0.0197430i \(-0.00628479\pi\)
\(182\) 0 0
\(183\) 0.728856i 0.00398282i
\(184\) 0 0
\(185\) 137.124 0.741212
\(186\) 0 0
\(187\) −42.1427 −0.225362
\(188\) 0 0
\(189\) − 4.63782i − 0.0245388i
\(190\) 0 0
\(191\) 73.2983i 0.383761i 0.981418 + 0.191880i \(0.0614586\pi\)
−0.981418 + 0.191880i \(0.938541\pi\)
\(192\) 0 0
\(193\) −85.0705 −0.440780 −0.220390 0.975412i \(-0.570733\pi\)
−0.220390 + 0.975412i \(0.570733\pi\)
\(194\) 0 0
\(195\) 6.46916 0.0331752
\(196\) 0 0
\(197\) 140.460i 0.712996i 0.934296 + 0.356498i \(0.116029\pi\)
−0.934296 + 0.356498i \(0.883971\pi\)
\(198\) 0 0
\(199\) 143.082i 0.719006i 0.933144 + 0.359503i \(0.117054\pi\)
−0.933144 + 0.359503i \(0.882946\pi\)
\(200\) 0 0
\(201\) −1.17499 −0.00584571
\(202\) 0 0
\(203\) −71.8715 −0.354047
\(204\) 0 0
\(205\) − 158.984i − 0.775532i
\(206\) 0 0
\(207\) 150.679i 0.727919i
\(208\) 0 0
\(209\) −23.7146 −0.113467
\(210\) 0 0
\(211\) 111.955 0.530591 0.265295 0.964167i \(-0.414531\pi\)
0.265295 + 0.964167i \(0.414531\pi\)
\(212\) 0 0
\(213\) − 12.5875i − 0.0590962i
\(214\) 0 0
\(215\) 211.509i 0.983765i
\(216\) 0 0
\(217\) −118.774 −0.547345
\(218\) 0 0
\(219\) 1.78166 0.00813545
\(220\) 0 0
\(221\) 275.687i 1.24745i
\(222\) 0 0
\(223\) − 311.438i − 1.39658i −0.715814 0.698291i \(-0.753944\pi\)
0.715814 0.698291i \(-0.246056\pi\)
\(224\) 0 0
\(225\) −116.791 −0.519072
\(226\) 0 0
\(227\) −74.3581 −0.327569 −0.163784 0.986496i \(-0.552370\pi\)
−0.163784 + 0.986496i \(0.552370\pi\)
\(228\) 0 0
\(229\) − 78.2710i − 0.341795i −0.985289 0.170897i \(-0.945333\pi\)
0.985289 0.170897i \(-0.0546667\pi\)
\(230\) 0 0
\(231\) 0.754988i 0.00326835i
\(232\) 0 0
\(233\) 93.0573 0.399388 0.199694 0.979858i \(-0.436005\pi\)
0.199694 + 0.979858i \(0.436005\pi\)
\(234\) 0 0
\(235\) 160.380 0.682466
\(236\) 0 0
\(237\) − 4.15244i − 0.0175208i
\(238\) 0 0
\(239\) 291.605i 1.22011i 0.792361 + 0.610053i \(0.208852\pi\)
−0.792361 + 0.610053i \(0.791148\pi\)
\(240\) 0 0
\(241\) 223.748 0.928413 0.464207 0.885727i \(-0.346340\pi\)
0.464207 + 0.885727i \(0.346340\pi\)
\(242\) 0 0
\(243\) −23.6438 −0.0972996
\(244\) 0 0
\(245\) 24.2583i 0.0990135i
\(246\) 0 0
\(247\) 155.135i 0.628079i
\(248\) 0 0
\(249\) −10.6859 −0.0429151
\(250\) 0 0
\(251\) 310.605 1.23747 0.618734 0.785600i \(-0.287646\pi\)
0.618734 + 0.785600i \(0.287646\pi\)
\(252\) 0 0
\(253\) − 49.0838i − 0.194007i
\(254\) 0 0
\(255\) − 4.85889i − 0.0190545i
\(256\) 0 0
\(257\) 175.472 0.682769 0.341385 0.939924i \(-0.389104\pi\)
0.341385 + 0.939924i \(0.389104\pi\)
\(258\) 0 0
\(259\) 104.689 0.404205
\(260\) 0 0
\(261\) 244.226i 0.935732i
\(262\) 0 0
\(263\) − 312.127i − 1.18680i −0.804910 0.593398i \(-0.797786\pi\)
0.804910 0.593398i \(-0.202214\pi\)
\(264\) 0 0
\(265\) 33.5951 0.126774
\(266\) 0 0
\(267\) 7.88419 0.0295288
\(268\) 0 0
\(269\) 142.817i 0.530918i 0.964122 + 0.265459i \(0.0855234\pi\)
−0.964122 + 0.265459i \(0.914477\pi\)
\(270\) 0 0
\(271\) 266.117i 0.981982i 0.871165 + 0.490991i \(0.163365\pi\)
−0.871165 + 0.490991i \(0.836635\pi\)
\(272\) 0 0
\(273\) 4.93895 0.0180914
\(274\) 0 0
\(275\) 38.0448 0.138345
\(276\) 0 0
\(277\) 366.740i 1.32397i 0.749516 + 0.661986i \(0.230286\pi\)
−0.749516 + 0.661986i \(0.769714\pi\)
\(278\) 0 0
\(279\) 403.604i 1.44661i
\(280\) 0 0
\(281\) 147.977 0.526607 0.263303 0.964713i \(-0.415188\pi\)
0.263303 + 0.964713i \(0.415188\pi\)
\(282\) 0 0
\(283\) −327.739 −1.15809 −0.579043 0.815297i \(-0.696574\pi\)
−0.579043 + 0.815297i \(0.696574\pi\)
\(284\) 0 0
\(285\) − 2.73421i − 0.00959372i
\(286\) 0 0
\(287\) − 121.378i − 0.422920i
\(288\) 0 0
\(289\) −81.9352 −0.283513
\(290\) 0 0
\(291\) −15.7931 −0.0542718
\(292\) 0 0
\(293\) 259.881i 0.886966i 0.896283 + 0.443483i \(0.146257\pi\)
−0.896283 + 0.443483i \(0.853743\pi\)
\(294\) 0 0
\(295\) − 396.983i − 1.34571i
\(296\) 0 0
\(297\) 5.13375 0.0172853
\(298\) 0 0
\(299\) −321.094 −1.07389
\(300\) 0 0
\(301\) 161.479i 0.536475i
\(302\) 0 0
\(303\) − 10.4089i − 0.0343527i
\(304\) 0 0
\(305\) −25.9228 −0.0849929
\(306\) 0 0
\(307\) −290.462 −0.946131 −0.473065 0.881027i \(-0.656853\pi\)
−0.473065 + 0.881027i \(0.656853\pi\)
\(308\) 0 0
\(309\) 12.3380i 0.0399289i
\(310\) 0 0
\(311\) 74.9081i 0.240862i 0.992722 + 0.120431i \(0.0384276\pi\)
−0.992722 + 0.120431i \(0.961572\pi\)
\(312\) 0 0
\(313\) 284.507 0.908969 0.454485 0.890755i \(-0.349823\pi\)
0.454485 + 0.890755i \(0.349823\pi\)
\(314\) 0 0
\(315\) 82.4319 0.261689
\(316\) 0 0
\(317\) 12.2631i 0.0386850i 0.999813 + 0.0193425i \(0.00615729\pi\)
−0.999813 + 0.0193425i \(0.993843\pi\)
\(318\) 0 0
\(319\) − 79.5567i − 0.249394i
\(320\) 0 0
\(321\) 8.47885 0.0264139
\(322\) 0 0
\(323\) 116.520 0.360743
\(324\) 0 0
\(325\) − 248.880i − 0.765784i
\(326\) 0 0
\(327\) 18.5053i 0.0565911i
\(328\) 0 0
\(329\) 122.443 0.372169
\(330\) 0 0
\(331\) −194.466 −0.587510 −0.293755 0.955881i \(-0.594905\pi\)
−0.293755 + 0.955881i \(0.594905\pi\)
\(332\) 0 0
\(333\) − 355.743i − 1.06830i
\(334\) 0 0
\(335\) − 41.7901i − 0.124747i
\(336\) 0 0
\(337\) 0.596077 0.00176877 0.000884387 1.00000i \(-0.499718\pi\)
0.000884387 1.00000i \(0.499718\pi\)
\(338\) 0 0
\(339\) 3.91547 0.0115500
\(340\) 0 0
\(341\) − 131.474i − 0.385555i
\(342\) 0 0
\(343\) 18.5203i 0.0539949i
\(344\) 0 0
\(345\) 5.65918 0.0164034
\(346\) 0 0
\(347\) 204.867 0.590394 0.295197 0.955436i \(-0.404615\pi\)
0.295197 + 0.955436i \(0.404615\pi\)
\(348\) 0 0
\(349\) − 128.396i − 0.367898i −0.982936 0.183949i \(-0.941112\pi\)
0.982936 0.183949i \(-0.0588882\pi\)
\(350\) 0 0
\(351\) − 33.5837i − 0.0956801i
\(352\) 0 0
\(353\) −190.841 −0.540627 −0.270314 0.962772i \(-0.587127\pi\)
−0.270314 + 0.962772i \(0.587127\pi\)
\(354\) 0 0
\(355\) 447.692 1.26111
\(356\) 0 0
\(357\) − 3.70957i − 0.0103910i
\(358\) 0 0
\(359\) − 215.704i − 0.600847i −0.953806 0.300424i \(-0.902872\pi\)
0.953806 0.300424i \(-0.0971281\pi\)
\(360\) 0 0
\(361\) −295.432 −0.818370
\(362\) 0 0
\(363\) 10.9541 0.0301766
\(364\) 0 0
\(365\) 63.3674i 0.173609i
\(366\) 0 0
\(367\) 454.789i 1.23921i 0.784915 + 0.619604i \(0.212707\pi\)
−0.784915 + 0.619604i \(0.787293\pi\)
\(368\) 0 0
\(369\) −412.454 −1.11776
\(370\) 0 0
\(371\) 25.6485 0.0691335
\(372\) 0 0
\(373\) − 360.748i − 0.967153i −0.875302 0.483576i \(-0.839338\pi\)
0.875302 0.483576i \(-0.160662\pi\)
\(374\) 0 0
\(375\) 12.8280i 0.0342080i
\(376\) 0 0
\(377\) −520.441 −1.38048
\(378\) 0 0
\(379\) 268.427 0.708250 0.354125 0.935198i \(-0.384779\pi\)
0.354125 + 0.935198i \(0.384779\pi\)
\(380\) 0 0
\(381\) 14.9718i 0.0392960i
\(382\) 0 0
\(383\) − 581.532i − 1.51836i −0.650881 0.759180i \(-0.725600\pi\)
0.650881 0.759180i \(-0.274400\pi\)
\(384\) 0 0
\(385\) −26.8522 −0.0697461
\(386\) 0 0
\(387\) 548.721 1.41788
\(388\) 0 0
\(389\) − 512.278i − 1.31691i −0.752620 0.658455i \(-0.771210\pi\)
0.752620 0.658455i \(-0.228790\pi\)
\(390\) 0 0
\(391\) 241.169i 0.616801i
\(392\) 0 0
\(393\) 6.02790 0.0153382
\(394\) 0 0
\(395\) 147.687 0.373892
\(396\) 0 0
\(397\) − 81.3250i − 0.204849i −0.994741 0.102424i \(-0.967340\pi\)
0.994741 0.102424i \(-0.0326600\pi\)
\(398\) 0 0
\(399\) − 2.08746i − 0.00523173i
\(400\) 0 0
\(401\) 527.441 1.31531 0.657657 0.753318i \(-0.271548\pi\)
0.657657 + 0.753318i \(0.271548\pi\)
\(402\) 0 0
\(403\) −860.073 −2.13418
\(404\) 0 0
\(405\) − 279.815i − 0.690902i
\(406\) 0 0
\(407\) 115.883i 0.284726i
\(408\) 0 0
\(409\) −58.6727 −0.143454 −0.0717270 0.997424i \(-0.522851\pi\)
−0.0717270 + 0.997424i \(0.522851\pi\)
\(410\) 0 0
\(411\) −10.3227 −0.0251161
\(412\) 0 0
\(413\) − 303.081i − 0.733853i
\(414\) 0 0
\(415\) − 380.058i − 0.915803i
\(416\) 0 0
\(417\) −18.0632 −0.0433169
\(418\) 0 0
\(419\) −760.704 −1.81552 −0.907761 0.419487i \(-0.862210\pi\)
−0.907761 + 0.419487i \(0.862210\pi\)
\(420\) 0 0
\(421\) − 46.2918i − 0.109957i −0.998488 0.0549784i \(-0.982491\pi\)
0.998488 0.0549784i \(-0.0175090\pi\)
\(422\) 0 0
\(423\) − 416.074i − 0.983627i
\(424\) 0 0
\(425\) −186.930 −0.439835
\(426\) 0 0
\(427\) −19.7911 −0.0463491
\(428\) 0 0
\(429\) 5.46707i 0.0127437i
\(430\) 0 0
\(431\) 336.176i 0.779991i 0.920817 + 0.389996i \(0.127523\pi\)
−0.920817 + 0.389996i \(0.872477\pi\)
\(432\) 0 0
\(433\) −372.694 −0.860725 −0.430363 0.902656i \(-0.641614\pi\)
−0.430363 + 0.902656i \(0.641614\pi\)
\(434\) 0 0
\(435\) 9.17260 0.0210864
\(436\) 0 0
\(437\) 135.711i 0.310553i
\(438\) 0 0
\(439\) − 397.478i − 0.905418i −0.891658 0.452709i \(-0.850458\pi\)
0.891658 0.452709i \(-0.149542\pi\)
\(440\) 0 0
\(441\) 62.9335 0.142706
\(442\) 0 0
\(443\) −273.530 −0.617450 −0.308725 0.951151i \(-0.599902\pi\)
−0.308725 + 0.951151i \(0.599902\pi\)
\(444\) 0 0
\(445\) 280.413i 0.630141i
\(446\) 0 0
\(447\) − 4.61943i − 0.0103343i
\(448\) 0 0
\(449\) 428.702 0.954792 0.477396 0.878688i \(-0.341581\pi\)
0.477396 + 0.878688i \(0.341581\pi\)
\(450\) 0 0
\(451\) 134.357 0.297909
\(452\) 0 0
\(453\) 11.1639i 0.0246443i
\(454\) 0 0
\(455\) 175.661i 0.386068i
\(456\) 0 0
\(457\) 10.0500 0.0219913 0.0109956 0.999940i \(-0.496500\pi\)
0.0109956 + 0.999940i \(0.496500\pi\)
\(458\) 0 0
\(459\) −25.2243 −0.0549548
\(460\) 0 0
\(461\) 825.802i 1.79133i 0.444732 + 0.895664i \(0.353299\pi\)
−0.444732 + 0.895664i \(0.646701\pi\)
\(462\) 0 0
\(463\) − 114.707i − 0.247748i −0.992298 0.123874i \(-0.960468\pi\)
0.992298 0.123874i \(-0.0395318\pi\)
\(464\) 0 0
\(465\) 15.1585 0.0325989
\(466\) 0 0
\(467\) 201.727 0.431964 0.215982 0.976397i \(-0.430705\pi\)
0.215982 + 0.976397i \(0.430705\pi\)
\(468\) 0 0
\(469\) − 31.9051i − 0.0680280i
\(470\) 0 0
\(471\) 28.6556i 0.0608399i
\(472\) 0 0
\(473\) −178.746 −0.377899
\(474\) 0 0
\(475\) −105.190 −0.221452
\(476\) 0 0
\(477\) − 87.1561i − 0.182717i
\(478\) 0 0
\(479\) − 597.538i − 1.24747i −0.781636 0.623735i \(-0.785615\pi\)
0.781636 0.623735i \(-0.214385\pi\)
\(480\) 0 0
\(481\) 758.081 1.57605
\(482\) 0 0
\(483\) 4.32056 0.00894526
\(484\) 0 0
\(485\) − 561.705i − 1.15815i
\(486\) 0 0
\(487\) 345.125i 0.708675i 0.935118 + 0.354337i \(0.115294\pi\)
−0.935118 + 0.354337i \(0.884706\pi\)
\(488\) 0 0
\(489\) 16.6637 0.0340770
\(490\) 0 0
\(491\) −373.498 −0.760689 −0.380344 0.924845i \(-0.624195\pi\)
−0.380344 + 0.924845i \(0.624195\pi\)
\(492\) 0 0
\(493\) 390.896i 0.792892i
\(494\) 0 0
\(495\) 91.2464i 0.184336i
\(496\) 0 0
\(497\) 341.796 0.687718
\(498\) 0 0
\(499\) 850.317 1.70404 0.852021 0.523508i \(-0.175377\pi\)
0.852021 + 0.523508i \(0.175377\pi\)
\(500\) 0 0
\(501\) 11.7564i 0.0234659i
\(502\) 0 0
\(503\) − 459.256i − 0.913033i −0.889715 0.456517i \(-0.849097\pi\)
0.889715 0.456517i \(-0.150903\pi\)
\(504\) 0 0
\(505\) 370.206 0.733082
\(506\) 0 0
\(507\) 19.2975 0.0380620
\(508\) 0 0
\(509\) 673.910i 1.32399i 0.749509 + 0.661994i \(0.230290\pi\)
−0.749509 + 0.661994i \(0.769710\pi\)
\(510\) 0 0
\(511\) 48.3785i 0.0946742i
\(512\) 0 0
\(513\) −14.1943 −0.0276691
\(514\) 0 0
\(515\) −438.820 −0.852078
\(516\) 0 0
\(517\) 135.536i 0.262159i
\(518\) 0 0
\(519\) 10.5562i 0.0203394i
\(520\) 0 0
\(521\) −486.473 −0.933729 −0.466864 0.884329i \(-0.654616\pi\)
−0.466864 + 0.884329i \(0.654616\pi\)
\(522\) 0 0
\(523\) 680.087 1.30036 0.650179 0.759781i \(-0.274694\pi\)
0.650179 + 0.759781i \(0.274694\pi\)
\(524\) 0 0
\(525\) 3.34886i 0.00637878i
\(526\) 0 0
\(527\) 645.988i 1.22578i
\(528\) 0 0
\(529\) 248.109 0.469015
\(530\) 0 0
\(531\) −1029.90 −1.93954
\(532\) 0 0
\(533\) − 878.931i − 1.64903i
\(534\) 0 0
\(535\) 301.562i 0.563668i
\(536\) 0 0
\(537\) −15.7299 −0.0292922
\(538\) 0 0
\(539\) −20.5006 −0.0380346
\(540\) 0 0
\(541\) 794.999i 1.46950i 0.678339 + 0.734749i \(0.262700\pi\)
−0.678339 + 0.734749i \(0.737300\pi\)
\(542\) 0 0
\(543\) 0.696375i 0.00128246i
\(544\) 0 0
\(545\) −658.167 −1.20765
\(546\) 0 0
\(547\) 736.752 1.34690 0.673448 0.739235i \(-0.264813\pi\)
0.673448 + 0.739235i \(0.264813\pi\)
\(548\) 0 0
\(549\) 67.2518i 0.122499i
\(550\) 0 0
\(551\) 219.966i 0.399212i
\(552\) 0 0
\(553\) 112.754 0.203894
\(554\) 0 0
\(555\) −13.3609 −0.0240737
\(556\) 0 0
\(557\) 415.758i 0.746423i 0.927746 + 0.373212i \(0.121743\pi\)
−0.927746 + 0.373212i \(0.878257\pi\)
\(558\) 0 0
\(559\) 1169.31i 2.09179i
\(560\) 0 0
\(561\) 4.10624 0.00731950
\(562\) 0 0
\(563\) 96.4996 0.171402 0.0857012 0.996321i \(-0.472687\pi\)
0.0857012 + 0.996321i \(0.472687\pi\)
\(564\) 0 0
\(565\) 139.259i 0.246476i
\(566\) 0 0
\(567\) − 213.628i − 0.376769i
\(568\) 0 0
\(569\) 347.953 0.611517 0.305759 0.952109i \(-0.401090\pi\)
0.305759 + 0.952109i \(0.401090\pi\)
\(570\) 0 0
\(571\) 13.7251 0.0240370 0.0120185 0.999928i \(-0.496174\pi\)
0.0120185 + 0.999928i \(0.496174\pi\)
\(572\) 0 0
\(573\) − 7.14193i − 0.0124641i
\(574\) 0 0
\(575\) − 217.718i − 0.378641i
\(576\) 0 0
\(577\) −827.320 −1.43383 −0.716915 0.697161i \(-0.754446\pi\)
−0.716915 + 0.697161i \(0.754446\pi\)
\(578\) 0 0
\(579\) 8.28898 0.0143160
\(580\) 0 0
\(581\) − 290.160i − 0.499414i
\(582\) 0 0
\(583\) 28.3911i 0.0486983i
\(584\) 0 0
\(585\) 596.912 1.02036
\(586\) 0 0
\(587\) 675.987 1.15160 0.575798 0.817592i \(-0.304692\pi\)
0.575798 + 0.817592i \(0.304692\pi\)
\(588\) 0 0
\(589\) 363.512i 0.617169i
\(590\) 0 0
\(591\) − 13.6860i − 0.0231573i
\(592\) 0 0
\(593\) −414.116 −0.698341 −0.349171 0.937059i \(-0.613537\pi\)
−0.349171 + 0.937059i \(0.613537\pi\)
\(594\) 0 0
\(595\) 131.936 0.221742
\(596\) 0 0
\(597\) − 13.9414i − 0.0233525i
\(598\) 0 0
\(599\) 723.303i 1.20752i 0.797167 + 0.603759i \(0.206331\pi\)
−0.797167 + 0.603759i \(0.793669\pi\)
\(600\) 0 0
\(601\) 68.7503 0.114393 0.0571966 0.998363i \(-0.481784\pi\)
0.0571966 + 0.998363i \(0.481784\pi\)
\(602\) 0 0
\(603\) −108.416 −0.179795
\(604\) 0 0
\(605\) 389.599i 0.643965i
\(606\) 0 0
\(607\) 141.263i 0.232724i 0.993207 + 0.116362i \(0.0371232\pi\)
−0.993207 + 0.116362i \(0.962877\pi\)
\(608\) 0 0
\(609\) 7.00292 0.0114990
\(610\) 0 0
\(611\) 886.646 1.45114
\(612\) 0 0
\(613\) − 96.7370i − 0.157809i −0.996882 0.0789046i \(-0.974858\pi\)
0.996882 0.0789046i \(-0.0251422\pi\)
\(614\) 0 0
\(615\) 15.4909i 0.0251884i
\(616\) 0 0
\(617\) 580.418 0.940709 0.470355 0.882478i \(-0.344126\pi\)
0.470355 + 0.882478i \(0.344126\pi\)
\(618\) 0 0
\(619\) −157.945 −0.255161 −0.127581 0.991828i \(-0.540721\pi\)
−0.127581 + 0.991828i \(0.540721\pi\)
\(620\) 0 0
\(621\) − 29.3788i − 0.0473089i
\(622\) 0 0
\(623\) 214.084i 0.343634i
\(624\) 0 0
\(625\) −131.484 −0.210375
\(626\) 0 0
\(627\) 2.31067 0.00368528
\(628\) 0 0
\(629\) − 569.384i − 0.905221i
\(630\) 0 0
\(631\) 771.793i 1.22313i 0.791195 + 0.611564i \(0.209459\pi\)
−0.791195 + 0.611564i \(0.790541\pi\)
\(632\) 0 0
\(633\) −10.9085 −0.0172330
\(634\) 0 0
\(635\) −532.493 −0.838571
\(636\) 0 0
\(637\) 134.110i 0.210534i
\(638\) 0 0
\(639\) − 1161.45i − 1.81761i
\(640\) 0 0
\(641\) −586.903 −0.915605 −0.457802 0.889054i \(-0.651363\pi\)
−0.457802 + 0.889054i \(0.651363\pi\)
\(642\) 0 0
\(643\) −865.328 −1.34577 −0.672883 0.739749i \(-0.734944\pi\)
−0.672883 + 0.739749i \(0.734944\pi\)
\(644\) 0 0
\(645\) − 20.6087i − 0.0319515i
\(646\) 0 0
\(647\) − 132.883i − 0.205383i −0.994713 0.102691i \(-0.967255\pi\)
0.994713 0.102691i \(-0.0327454\pi\)
\(648\) 0 0
\(649\) 335.489 0.516933
\(650\) 0 0
\(651\) 11.5729 0.0177771
\(652\) 0 0
\(653\) − 364.309i − 0.557900i −0.960306 0.278950i \(-0.910014\pi\)
0.960306 0.278950i \(-0.0899865\pi\)
\(654\) 0 0
\(655\) 214.391i 0.327315i
\(656\) 0 0
\(657\) 164.395 0.250220
\(658\) 0 0
\(659\) 18.8972 0.0286756 0.0143378 0.999897i \(-0.495436\pi\)
0.0143378 + 0.999897i \(0.495436\pi\)
\(660\) 0 0
\(661\) 339.106i 0.513019i 0.966542 + 0.256510i \(0.0825725\pi\)
−0.966542 + 0.256510i \(0.917427\pi\)
\(662\) 0 0
\(663\) − 26.8620i − 0.0405159i
\(664\) 0 0
\(665\) 74.2436 0.111644
\(666\) 0 0
\(667\) −455.278 −0.682576
\(668\) 0 0
\(669\) 30.3454i 0.0453594i
\(670\) 0 0
\(671\) − 21.9073i − 0.0326487i
\(672\) 0 0
\(673\) 674.869 1.00278 0.501389 0.865222i \(-0.332823\pi\)
0.501389 + 0.865222i \(0.332823\pi\)
\(674\) 0 0
\(675\) 22.7715 0.0337355
\(676\) 0 0
\(677\) − 988.747i − 1.46048i −0.683189 0.730242i \(-0.739407\pi\)
0.683189 0.730242i \(-0.260593\pi\)
\(678\) 0 0
\(679\) − 428.839i − 0.631575i
\(680\) 0 0
\(681\) 7.24520 0.0106391
\(682\) 0 0
\(683\) 518.125 0.758602 0.379301 0.925273i \(-0.376165\pi\)
0.379301 + 0.925273i \(0.376165\pi\)
\(684\) 0 0
\(685\) − 367.143i − 0.535975i
\(686\) 0 0
\(687\) 7.62646i 0.0111011i
\(688\) 0 0
\(689\) 185.728 0.269562
\(690\) 0 0
\(691\) 617.021 0.892940 0.446470 0.894799i \(-0.352681\pi\)
0.446470 + 0.894799i \(0.352681\pi\)
\(692\) 0 0
\(693\) 69.6630i 0.100524i
\(694\) 0 0
\(695\) − 642.442i − 0.924377i
\(696\) 0 0
\(697\) −660.153 −0.947135
\(698\) 0 0
\(699\) −9.06718 −0.0129717
\(700\) 0 0
\(701\) − 97.6954i − 0.139366i −0.997569 0.0696829i \(-0.977801\pi\)
0.997569 0.0696829i \(-0.0221987\pi\)
\(702\) 0 0
\(703\) − 320.405i − 0.455768i
\(704\) 0 0
\(705\) −15.6268 −0.0221657
\(706\) 0 0
\(707\) 282.638 0.399771
\(708\) 0 0
\(709\) 1249.74i 1.76269i 0.472476 + 0.881343i \(0.343360\pi\)
−0.472476 + 0.881343i \(0.656640\pi\)
\(710\) 0 0
\(711\) − 383.147i − 0.538885i
\(712\) 0 0
\(713\) −752.386 −1.05524
\(714\) 0 0
\(715\) −194.444 −0.271950
\(716\) 0 0
\(717\) − 28.4130i − 0.0396277i
\(718\) 0 0
\(719\) 424.744i 0.590743i 0.955382 + 0.295372i \(0.0954435\pi\)
−0.955382 + 0.295372i \(0.904557\pi\)
\(720\) 0 0
\(721\) −335.022 −0.464663
\(722\) 0 0
\(723\) −21.8012 −0.0301538
\(724\) 0 0
\(725\) − 352.886i − 0.486739i
\(726\) 0 0
\(727\) 79.1445i 0.108865i 0.998517 + 0.0544323i \(0.0173349\pi\)
−0.998517 + 0.0544323i \(0.982665\pi\)
\(728\) 0 0
\(729\) −724.390 −0.993676
\(730\) 0 0
\(731\) 878.255 1.20144
\(732\) 0 0
\(733\) − 663.766i − 0.905548i −0.891625 0.452774i \(-0.850434\pi\)
0.891625 0.452774i \(-0.149566\pi\)
\(734\) 0 0
\(735\) − 2.36365i − 0.00321584i
\(736\) 0 0
\(737\) 35.3167 0.0479196
\(738\) 0 0
\(739\) −832.112 −1.12600 −0.562998 0.826458i \(-0.690352\pi\)
−0.562998 + 0.826458i \(0.690352\pi\)
\(740\) 0 0
\(741\) − 15.1159i − 0.0203993i
\(742\) 0 0
\(743\) − 283.217i − 0.381180i −0.981670 0.190590i \(-0.938960\pi\)
0.981670 0.190590i \(-0.0610401\pi\)
\(744\) 0 0
\(745\) 164.297 0.220532
\(746\) 0 0
\(747\) −985.989 −1.31993
\(748\) 0 0
\(749\) 230.231i 0.307385i
\(750\) 0 0
\(751\) 374.981i 0.499309i 0.968335 + 0.249654i \(0.0803170\pi\)
−0.968335 + 0.249654i \(0.919683\pi\)
\(752\) 0 0
\(753\) −30.2642 −0.0401916
\(754\) 0 0
\(755\) −397.059 −0.525906
\(756\) 0 0
\(757\) − 63.0951i − 0.0833488i −0.999131 0.0416744i \(-0.986731\pi\)
0.999131 0.0416744i \(-0.0132692\pi\)
\(758\) 0 0
\(759\) 4.78255i 0.00630113i
\(760\) 0 0
\(761\) 467.505 0.614330 0.307165 0.951656i \(-0.400620\pi\)
0.307165 + 0.951656i \(0.400620\pi\)
\(762\) 0 0
\(763\) −502.485 −0.658564
\(764\) 0 0
\(765\) − 448.332i − 0.586055i
\(766\) 0 0
\(767\) − 2194.69i − 2.86140i
\(768\) 0 0
\(769\) 900.573 1.17110 0.585548 0.810638i \(-0.300879\pi\)
0.585548 + 0.810638i \(0.300879\pi\)
\(770\) 0 0
\(771\) −17.0974 −0.0221756
\(772\) 0 0
\(773\) − 1222.76i − 1.58184i −0.611920 0.790919i \(-0.709603\pi\)
0.611920 0.790919i \(-0.290397\pi\)
\(774\) 0 0
\(775\) − 583.173i − 0.752482i
\(776\) 0 0
\(777\) −10.2005 −0.0131281
\(778\) 0 0
\(779\) −371.483 −0.476872
\(780\) 0 0
\(781\) 378.344i 0.484435i
\(782\) 0 0
\(783\) − 47.6182i − 0.0608151i
\(784\) 0 0
\(785\) −1019.18 −1.29832
\(786\) 0 0
\(787\) −862.942 −1.09650 −0.548248 0.836316i \(-0.684705\pi\)
−0.548248 + 0.836316i \(0.684705\pi\)
\(788\) 0 0
\(789\) 30.4126i 0.0385457i
\(790\) 0 0
\(791\) 106.319i 0.134411i
\(792\) 0 0
\(793\) −143.312 −0.180722
\(794\) 0 0
\(795\) −3.27339 −0.00411747
\(796\) 0 0
\(797\) 1078.34i 1.35300i 0.736442 + 0.676500i \(0.236504\pi\)
−0.736442 + 0.676500i \(0.763496\pi\)
\(798\) 0 0
\(799\) − 665.947i − 0.833476i
\(800\) 0 0
\(801\) 727.477 0.908211
\(802\) 0 0
\(803\) −53.5516 −0.0666894
\(804\) 0 0
\(805\) 153.667i 0.190891i
\(806\) 0 0
\(807\) − 13.9156i − 0.0172436i
\(808\) 0 0
\(809\) 333.388 0.412099 0.206050 0.978542i \(-0.433939\pi\)
0.206050 + 0.978542i \(0.433939\pi\)
\(810\) 0 0
\(811\) 1246.04 1.53642 0.768211 0.640197i \(-0.221147\pi\)
0.768211 + 0.640197i \(0.221147\pi\)
\(812\) 0 0
\(813\) − 25.9295i − 0.0318936i
\(814\) 0 0
\(815\) 592.667i 0.727199i
\(816\) 0 0
\(817\) 494.214 0.604913
\(818\) 0 0
\(819\) 455.719 0.556433
\(820\) 0 0
\(821\) − 1458.68i − 1.77671i −0.459162 0.888353i \(-0.651850\pi\)
0.459162 0.888353i \(-0.348150\pi\)
\(822\) 0 0
\(823\) 464.047i 0.563848i 0.959437 + 0.281924i \(0.0909725\pi\)
−0.959437 + 0.281924i \(0.909027\pi\)
\(824\) 0 0
\(825\) −3.70695 −0.00449328
\(826\) 0 0
\(827\) −1077.41 −1.30279 −0.651394 0.758739i \(-0.725816\pi\)
−0.651394 + 0.758739i \(0.725816\pi\)
\(828\) 0 0
\(829\) 35.9354i 0.0433479i 0.999765 + 0.0216740i \(0.00689958\pi\)
−0.999765 + 0.0216740i \(0.993100\pi\)
\(830\) 0 0
\(831\) − 35.7339i − 0.0430011i
\(832\) 0 0
\(833\) 100.728 0.120922
\(834\) 0 0
\(835\) −418.134 −0.500760
\(836\) 0 0
\(837\) − 78.6932i − 0.0940181i
\(838\) 0 0
\(839\) − 734.676i − 0.875656i −0.899059 0.437828i \(-0.855748\pi\)
0.899059 0.437828i \(-0.144252\pi\)
\(840\) 0 0
\(841\) 103.069 0.122555
\(842\) 0 0
\(843\) −14.4183 −0.0171036
\(844\) 0 0
\(845\) 686.342i 0.812239i
\(846\) 0 0
\(847\) 297.443i 0.351173i
\(848\) 0 0
\(849\) 31.9337 0.0376133
\(850\) 0 0
\(851\) 663.164 0.779276
\(852\) 0 0
\(853\) 402.566i 0.471942i 0.971760 + 0.235971i \(0.0758270\pi\)
−0.971760 + 0.235971i \(0.924173\pi\)
\(854\) 0 0
\(855\) − 252.287i − 0.295072i
\(856\) 0 0
\(857\) 552.003 0.644110 0.322055 0.946721i \(-0.395626\pi\)
0.322055 + 0.946721i \(0.395626\pi\)
\(858\) 0 0
\(859\) 330.117 0.384303 0.192152 0.981365i \(-0.438453\pi\)
0.192152 + 0.981365i \(0.438453\pi\)
\(860\) 0 0
\(861\) 11.8267i 0.0137360i
\(862\) 0 0
\(863\) 1116.67i 1.29394i 0.762516 + 0.646969i \(0.223964\pi\)
−0.762516 + 0.646969i \(0.776036\pi\)
\(864\) 0 0
\(865\) −375.445 −0.434041
\(866\) 0 0
\(867\) 7.98348 0.00920817
\(868\) 0 0
\(869\) 124.810i 0.143625i
\(870\) 0 0
\(871\) − 231.033i − 0.265251i
\(872\) 0 0
\(873\) −1457.24 −1.66923
\(874\) 0 0
\(875\) −348.326 −0.398087
\(876\) 0 0
\(877\) 0.747441i 0 0.000852270i 1.00000 0.000426135i \(0.000135643\pi\)
−1.00000 0.000426135i \(0.999864\pi\)
\(878\) 0 0
\(879\) − 25.3219i − 0.0288076i
\(880\) 0 0
\(881\) −247.826 −0.281301 −0.140650 0.990059i \(-0.544919\pi\)
−0.140650 + 0.990059i \(0.544919\pi\)
\(882\) 0 0
\(883\) 1613.74 1.82757 0.913784 0.406200i \(-0.133146\pi\)
0.913784 + 0.406200i \(0.133146\pi\)
\(884\) 0 0
\(885\) 38.6807i 0.0437070i
\(886\) 0 0
\(887\) 1110.59i 1.25207i 0.779795 + 0.626034i \(0.215323\pi\)
−0.779795 + 0.626034i \(0.784677\pi\)
\(888\) 0 0
\(889\) −406.537 −0.457297
\(890\) 0 0
\(891\) 236.471 0.265400
\(892\) 0 0
\(893\) − 374.744i − 0.419646i
\(894\) 0 0
\(895\) − 559.458i − 0.625092i
\(896\) 0 0
\(897\) 31.2863 0.0348788
\(898\) 0 0
\(899\) −1219.49 −1.35650
\(900\) 0 0
\(901\) − 139.498i − 0.154825i
\(902\) 0 0
\(903\) − 15.7340i − 0.0174241i
\(904\) 0 0
\(905\) −24.7676 −0.0273675
\(906\) 0 0
\(907\) 518.009 0.571123 0.285562 0.958360i \(-0.407820\pi\)
0.285562 + 0.958360i \(0.407820\pi\)
\(908\) 0 0
\(909\) − 960.430i − 1.05658i
\(910\) 0 0
\(911\) 1065.61i 1.16972i 0.811135 + 0.584860i \(0.198850\pi\)
−0.811135 + 0.584860i \(0.801150\pi\)
\(912\) 0 0
\(913\) 321.186 0.351792
\(914\) 0 0
\(915\) 2.52583 0.00276047
\(916\) 0 0
\(917\) 163.679i 0.178494i
\(918\) 0 0
\(919\) 1183.66i 1.28799i 0.765030 + 0.643994i \(0.222724\pi\)
−0.765030 + 0.643994i \(0.777276\pi\)
\(920\) 0 0
\(921\) 28.3016 0.0307292
\(922\) 0 0
\(923\) 2475.03 2.68151
\(924\) 0 0
\(925\) 514.018i 0.555695i
\(926\) 0 0
\(927\) 1138.43i 1.22809i
\(928\) 0 0
\(929\) −379.019 −0.407986 −0.203993 0.978972i \(-0.565392\pi\)
−0.203993 + 0.978972i \(0.565392\pi\)
\(930\) 0 0
\(931\) 56.6821 0.0608830
\(932\) 0 0
\(933\) − 7.29878i − 0.00782292i
\(934\) 0 0
\(935\) 146.044i 0.156197i
\(936\) 0 0
\(937\) 1316.09 1.40458 0.702289 0.711892i \(-0.252161\pi\)
0.702289 + 0.711892i \(0.252161\pi\)
\(938\) 0 0
\(939\) −27.7214 −0.0295223
\(940\) 0 0
\(941\) − 497.031i − 0.528194i −0.964496 0.264097i \(-0.914926\pi\)
0.964496 0.264097i \(-0.0850739\pi\)
\(942\) 0 0
\(943\) − 768.883i − 0.815359i
\(944\) 0 0
\(945\) −16.0723 −0.0170077
\(946\) 0 0
\(947\) −808.487 −0.853735 −0.426867 0.904314i \(-0.640383\pi\)
−0.426867 + 0.904314i \(0.640383\pi\)
\(948\) 0 0
\(949\) 350.322i 0.369148i
\(950\) 0 0
\(951\) − 1.19488i − 0.00125644i
\(952\) 0 0
\(953\) −1345.58 −1.41194 −0.705969 0.708243i \(-0.749488\pi\)
−0.705969 + 0.708243i \(0.749488\pi\)
\(954\) 0 0
\(955\) 254.013 0.265982
\(956\) 0 0
\(957\) 7.75173i 0.00810004i
\(958\) 0 0
\(959\) − 280.299i − 0.292283i
\(960\) 0 0
\(961\) −1054.32 −1.09710
\(962\) 0 0
\(963\) 782.346 0.812405
\(964\) 0 0
\(965\) 294.809i 0.305502i
\(966\) 0 0
\(967\) 140.279i 0.145066i 0.997366 + 0.0725330i \(0.0231083\pi\)
−0.997366 + 0.0725330i \(0.976892\pi\)
\(968\) 0 0
\(969\) −11.3533 −0.0117165
\(970\) 0 0
\(971\) −1438.67 −1.48164 −0.740820 0.671704i \(-0.765563\pi\)
−0.740820 + 0.671704i \(0.765563\pi\)
\(972\) 0 0
\(973\) − 490.479i − 0.504090i
\(974\) 0 0
\(975\) 24.2500i 0.0248718i
\(976\) 0 0
\(977\) 902.190 0.923428 0.461714 0.887029i \(-0.347235\pi\)
0.461714 + 0.887029i \(0.347235\pi\)
\(978\) 0 0
\(979\) −236.976 −0.242059
\(980\) 0 0
\(981\) 1707.49i 1.74056i
\(982\) 0 0
\(983\) 503.193i 0.511896i 0.966691 + 0.255948i \(0.0823875\pi\)
−0.966691 + 0.255948i \(0.917612\pi\)
\(984\) 0 0
\(985\) 486.761 0.494173
\(986\) 0 0
\(987\) −11.9305 −0.0120876
\(988\) 0 0
\(989\) 1022.91i 1.03428i
\(990\) 0 0
\(991\) 906.322i 0.914553i 0.889325 + 0.457277i \(0.151175\pi\)
−0.889325 + 0.457277i \(0.848825\pi\)
\(992\) 0 0
\(993\) 18.9481 0.0190816
\(994\) 0 0
\(995\) 495.847 0.498339
\(996\) 0 0
\(997\) 608.625i 0.610457i 0.952279 + 0.305228i \(0.0987328\pi\)
−0.952279 + 0.305228i \(0.901267\pi\)
\(998\) 0 0
\(999\) 69.3613i 0.0694308i
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 224.3.g.b.15.3 8
3.2 odd 2 2016.3.g.b.1135.5 8
4.3 odd 2 56.3.g.b.43.8 yes 8
7.6 odd 2 1568.3.g.m.687.6 8
8.3 odd 2 inner 224.3.g.b.15.4 8
8.5 even 2 56.3.g.b.43.7 8
12.11 even 2 504.3.g.b.379.1 8
16.3 odd 4 1792.3.d.j.1023.9 16
16.5 even 4 1792.3.d.j.1023.10 16
16.11 odd 4 1792.3.d.j.1023.8 16
16.13 even 4 1792.3.d.j.1023.7 16
24.5 odd 2 504.3.g.b.379.2 8
24.11 even 2 2016.3.g.b.1135.4 8
28.3 even 6 392.3.k.n.275.2 16
28.11 odd 6 392.3.k.o.275.2 16
28.19 even 6 392.3.k.n.67.4 16
28.23 odd 6 392.3.k.o.67.4 16
28.27 even 2 392.3.g.m.99.8 8
56.5 odd 6 392.3.k.n.67.2 16
56.13 odd 2 392.3.g.m.99.7 8
56.27 even 2 1568.3.g.m.687.5 8
56.37 even 6 392.3.k.o.67.2 16
56.45 odd 6 392.3.k.n.275.4 16
56.53 even 6 392.3.k.o.275.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.g.b.43.7 8 8.5 even 2
56.3.g.b.43.8 yes 8 4.3 odd 2
224.3.g.b.15.3 8 1.1 even 1 trivial
224.3.g.b.15.4 8 8.3 odd 2 inner
392.3.g.m.99.7 8 56.13 odd 2
392.3.g.m.99.8 8 28.27 even 2
392.3.k.n.67.2 16 56.5 odd 6
392.3.k.n.67.4 16 28.19 even 6
392.3.k.n.275.2 16 28.3 even 6
392.3.k.n.275.4 16 56.45 odd 6
392.3.k.o.67.2 16 56.37 even 6
392.3.k.o.67.4 16 28.23 odd 6
392.3.k.o.275.2 16 28.11 odd 6
392.3.k.o.275.4 16 56.53 even 6
504.3.g.b.379.1 8 12.11 even 2
504.3.g.b.379.2 8 24.5 odd 2
1568.3.g.m.687.5 8 56.27 even 2
1568.3.g.m.687.6 8 7.6 odd 2
1792.3.d.j.1023.7 16 16.13 even 4
1792.3.d.j.1023.8 16 16.11 odd 4
1792.3.d.j.1023.9 16 16.3 odd 4
1792.3.d.j.1023.10 16 16.5 even 4
2016.3.g.b.1135.4 8 24.11 even 2
2016.3.g.b.1135.5 8 3.2 odd 2