Properties

Label 224.3.g.b.15.4
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 15.4
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.15.3

$q$-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 \(0.112646\pi\)
−0.938033 + 0.346547i \(0.887354\pi\)
\(6\) 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 \(0.263696\pi\)
−0.676036 + 0.736869i \(0.736304\pi\)
\(14\) 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 \(0.118707\pi\)
−0.931265 + 0.364344i \(0.881293\pi\)
\(24\) 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 \(0.155155\pi\)
−0.883538 + 0.468360i \(0.844845\pi\)
\(30\) 0 0
\(31\) 44.8923i 1.44814i 0.689728 + 0.724069i \(0.257730\pi\)
−0.689728 + 0.724069i \(0.742270\pi\)
\(32\) 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 \(-0.820420\pi\)
0.845034 0.534713i \(-0.179580\pi\)
\(38\) 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 \(-0.836144\pi\)
0.870407 0.492333i \(-0.163856\pi\)
\(48\) 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 \(-0.970848\pi\)
0.995809 0.0914551i \(-0.0291518\pi\)
\(54\) 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 \(0.0195291\pi\)
−0.998119 + 0.0613141i \(0.980471\pi\)
\(62\) 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 \(-0.636262\pi\)
0.415124 0.909765i \(-0.363738\pi\)
\(72\) 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 \(-0.913067\pi\)
0.962937 0.269727i \(-0.0869334\pi\)
\(80\) 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 \(-0.822625\pi\)
0.848717 0.528847i \(-0.177375\pi\)
\(102\) 0 0
\(103\) 126.626i 1.22938i 0.788768 + 0.614691i \(0.210719\pi\)
−0.788768 + 0.614691i \(0.789281\pi\)
\(104\) 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 \(0.336657\pi\)
−0.490930 + 0.871199i \(0.663343\pi\)
\(110\) 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 \(0.206806\pi\)
−0.796266 + 0.604947i \(0.793194\pi\)
\(128\) 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 \(-0.949143\pi\)
0.987264 0.159093i \(-0.0508568\pi\)
\(150\) 0 0
\(151\) 114.576i 0.758780i 0.925237 + 0.379390i \(0.123866\pi\)
−0.925237 + 0.379390i \(0.876134\pi\)
\(152\) 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 \(0.386053\pi\)
−0.350378 + 0.936608i \(0.613947\pi\)
\(158\) 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 \(0.117650\pi\)
−0.932469 + 0.361249i \(0.882350\pi\)
\(168\) 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 \(0.101374\pi\)
−0.949714 + 0.313118i \(0.898626\pi\)
\(174\) 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 \(0.00628479\pi\)
−0.999805 + 0.0197430i \(0.993715\pi\)
\(182\) 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 \(-0.938541\pi\)
0.981418 0.191880i \(-0.0614586\pi\)
\(192\) 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 \(-0.883971\pi\)
0.934296 0.356498i \(-0.116029\pi\)
\(198\) 0 0
\(199\) − 143.082i − 0.719006i −0.933144 0.359503i \(-0.882946\pi\)
0.933144 0.359503i \(-0.117054\pi\)
\(200\) 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 \(0.246056\pi\)
−0.715814 + 0.698291i \(0.753944\pi\)
\(224\) 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 \(0.0546667\pi\)
−0.985289 + 0.170897i \(0.945333\pi\)
\(230\) 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 \(-0.791148\pi\)
0.792361 0.610053i \(-0.208852\pi\)
\(240\) 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 \(0.202214\pi\)
−0.804910 + 0.593398i \(0.797786\pi\)
\(264\) 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 \(-0.914477\pi\)
0.964122 0.265459i \(-0.0855234\pi\)
\(270\) 0 0
\(271\) − 266.117i − 0.981982i −0.871165 0.490991i \(-0.836635\pi\)
0.871165 0.490991i \(-0.163365\pi\)
\(272\) 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 \(-0.769714\pi\)
0.749516 0.661986i \(-0.230286\pi\)
\(278\) 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 \(-0.853743\pi\)
0.896283 0.443483i \(-0.146257\pi\)
\(294\) 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 \(-0.961572\pi\)
0.992722 0.120431i \(-0.0384276\pi\)
\(312\) 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 \(-0.993843\pi\)
0.999813 0.0193425i \(-0.00615729\pi\)
\(318\) 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 \(0.0588882\pi\)
−0.982936 + 0.183949i \(0.941112\pi\)
\(350\) 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 \(0.0971281\pi\)
−0.953806 + 0.300424i \(0.902872\pi\)
\(360\) 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 \(-0.787293\pi\)
0.784915 0.619604i \(-0.212707\pi\)
\(368\) 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 \(0.160662\pi\)
−0.875302 + 0.483576i \(0.839338\pi\)
\(374\) 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 \(0.274400\pi\)
−0.650881 + 0.759180i \(0.725600\pi\)
\(384\) 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 \(0.228790\pi\)
−0.752620 + 0.658455i \(0.771210\pi\)
\(390\) 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 \(0.0326600\pi\)
−0.994741 + 0.102424i \(0.967340\pi\)
\(398\) 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 \(0.0175090\pi\)
−0.998488 + 0.0549784i \(0.982491\pi\)
\(422\) 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 \(-0.872477\pi\)
0.920817 0.389996i \(-0.127523\pi\)
\(432\) 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 \(0.149542\pi\)
−0.891658 + 0.452709i \(0.850458\pi\)
\(440\) 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 \(-0.646701\pi\)
0.444732 0.895664i \(-0.353299\pi\)
\(462\) 0 0
\(463\) 114.707i 0.247748i 0.992298 + 0.123874i \(0.0395318\pi\)
−0.992298 + 0.123874i \(0.960468\pi\)
\(464\) 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 \(0.214385\pi\)
−0.781636 + 0.623735i \(0.785615\pi\)
\(480\) 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 \(-0.884706\pi\)
0.935118 0.354337i \(-0.115294\pi\)
\(488\) 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 \(0.150903\pi\)
−0.889715 + 0.456517i \(0.849097\pi\)
\(504\) 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 \(-0.769710\pi\)
0.749509 0.661994i \(-0.230290\pi\)
\(510\) 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 \(-0.737300\pi\)
0.678339 0.734749i \(-0.262700\pi\)
\(542\) 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 \(-0.878257\pi\)
0.927746 0.373212i \(-0.121743\pi\)
\(558\) 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 \(-0.793669\pi\)
0.797167 0.603759i \(-0.206331\pi\)
\(600\) 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 \(-0.962877\pi\)
0.993207 0.116362i \(-0.0371232\pi\)
\(608\) 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 \(0.0251422\pi\)
−0.996882 + 0.0789046i \(0.974858\pi\)
\(614\) 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 \(-0.790541\pi\)
0.791195 0.611564i \(-0.209459\pi\)
\(632\) 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 \(0.0327454\pi\)
−0.994713 + 0.102691i \(0.967255\pi\)
\(648\) 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 \(0.0899865\pi\)
−0.960306 + 0.278950i \(0.910014\pi\)
\(654\) 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 \(-0.917427\pi\)
0.966542 0.256510i \(-0.0825725\pi\)
\(662\) 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 \(0.260593\pi\)
−0.683189 + 0.730242i \(0.739407\pi\)
\(678\) 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 \(0.0221987\pi\)
−0.997569 + 0.0696829i \(0.977801\pi\)
\(702\) 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 \(-0.656640\pi\)
0.472476 0.881343i \(-0.343360\pi\)
\(710\) 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 \(-0.904557\pi\)
0.955382 0.295372i \(-0.0954435\pi\)
\(720\) 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 \(-0.982665\pi\)
0.998517 0.0544323i \(-0.0173349\pi\)
\(728\) 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 \(0.149566\pi\)
−0.891625 + 0.452774i \(0.850434\pi\)
\(734\) 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 \(0.0610401\pi\)
−0.981670 + 0.190590i \(0.938960\pi\)
\(744\) 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 \(-0.919683\pi\)
0.968335 0.249654i \(-0.0803170\pi\)
\(752\) 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 \(0.0132692\pi\)
−0.999131 + 0.0416744i \(0.986731\pi\)
\(758\) 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 \(0.290397\pi\)
−0.611920 + 0.790919i \(0.709603\pi\)
\(774\) 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 \(-0.763496\pi\)
0.736442 0.676500i \(-0.236504\pi\)
\(798\) 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 \(0.348150\pi\)
−0.459162 + 0.888353i \(0.651850\pi\)
\(822\) 0 0
\(823\) − 464.047i − 0.563848i −0.959437 0.281924i \(-0.909027\pi\)
0.959437 0.281924i \(-0.0909725\pi\)
\(824\) 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 \(-0.993100\pi\)
0.999765 0.0216740i \(-0.00689958\pi\)
\(830\) 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 \(0.144252\pi\)
−0.899059 + 0.437828i \(0.855748\pi\)
\(840\) 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 \(-0.924173\pi\)
0.971760 0.235971i \(-0.0758270\pi\)
\(854\) 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 \(-0.776036\pi\)
0.762516 0.646969i \(-0.223964\pi\)
\(864\) 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 \(-0.999864\pi\)
1.00000 0.000426135i \(-0.000135643\pi\)
\(878\) 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 \(-0.784677\pi\)
0.779795 0.626034i \(-0.215323\pi\)
\(888\) 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 \(-0.801150\pi\)
0.811135 0.584860i \(-0.198850\pi\)
\(912\) 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 \(-0.777276\pi\)
0.765030 0.643994i \(-0.222724\pi\)
\(920\) 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 \(0.0850739\pi\)
−0.964496 + 0.264097i \(0.914926\pi\)
\(942\) 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 \(-0.976892\pi\)
0.997366 0.0725330i \(-0.0231083\pi\)
\(968\) 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 \(-0.917612\pi\)
0.966691 0.255948i \(-0.0823875\pi\)
\(984\) 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 \(-0.848825\pi\)
0.889325 0.457277i \(-0.151175\pi\)
\(992\) 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 \(-0.901267\pi\)
0.952279 0.305228i \(-0.0987328\pi\)
\(998\) 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.15.4 8
3.2 odd 2 2016.3.g.b.1135.4 8
4.3 odd 2 56.3.g.b.43.7 8
7.6 odd 2 1568.3.g.m.687.5 8
8.3 odd 2 inner 224.3.g.b.15.3 8
8.5 even 2 56.3.g.b.43.8 yes 8
12.11 even 2 504.3.g.b.379.2 8
16.3 odd 4 1792.3.d.j.1023.10 16
16.5 even 4 1792.3.d.j.1023.9 16
16.11 odd 4 1792.3.d.j.1023.7 16
16.13 even 4 1792.3.d.j.1023.8 16
24.5 odd 2 504.3.g.b.379.1 8
24.11 even 2 2016.3.g.b.1135.5 8
28.3 even 6 392.3.k.n.275.4 16
28.11 odd 6 392.3.k.o.275.4 16
28.19 even 6 392.3.k.n.67.2 16
28.23 odd 6 392.3.k.o.67.2 16
28.27 even 2 392.3.g.m.99.7 8
56.5 odd 6 392.3.k.n.67.4 16
56.13 odd 2 392.3.g.m.99.8 8
56.27 even 2 1568.3.g.m.687.6 8
56.37 even 6 392.3.k.o.67.4 16
56.45 odd 6 392.3.k.n.275.2 16
56.53 even 6 392.3.k.o.275.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.g.b.43.7 8 4.3 odd 2
56.3.g.b.43.8 yes 8 8.5 even 2
224.3.g.b.15.3 8 8.3 odd 2 inner
224.3.g.b.15.4 8 1.1 even 1 trivial
392.3.g.m.99.7 8 28.27 even 2
392.3.g.m.99.8 8 56.13 odd 2
392.3.k.n.67.2 16 28.19 even 6
392.3.k.n.67.4 16 56.5 odd 6
392.3.k.n.275.2 16 56.45 odd 6
392.3.k.n.275.4 16 28.3 even 6
392.3.k.o.67.2 16 28.23 odd 6
392.3.k.o.67.4 16 56.37 even 6
392.3.k.o.275.2 16 56.53 even 6
392.3.k.o.275.4 16 28.11 odd 6
504.3.g.b.379.1 8 24.5 odd 2
504.3.g.b.379.2 8 12.11 even 2
1568.3.g.m.687.5 8 7.6 odd 2
1568.3.g.m.687.6 8 56.27 even 2
1792.3.d.j.1023.7 16 16.11 odd 4
1792.3.d.j.1023.8 16 16.13 even 4
1792.3.d.j.1023.9 16 16.5 even 4
1792.3.d.j.1023.10 16 16.3 odd 4
2016.3.g.b.1135.4 8 3.2 odd 2
2016.3.g.b.1135.5 8 24.11 even 2