Properties

Label 224.3.g.a.15.3
Level 224
Weight 3
Character 224.15
Analytic conductor 6.104
Analytic rank 0
Dimension 4
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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-7})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 15.3
Root \(-0.707107 + 1.87083i\) of \(x^{4} + 6 x^{2} + 16\)
Character \(\chi\) \(=\) 224.15
Dual form 224.3.g.a.15.4

$q$-expansion

\(f(q)\) \(=\) \(q-0.585786 q^{3} -9.03316i q^{5} +2.64575i q^{7} -8.65685 q^{9} +O(q^{10})\) \(q-0.585786 q^{3} -9.03316i q^{5} +2.64575i q^{7} -8.65685 q^{9} -12.4853 q^{11} +9.03316i q^{13} +5.29150i q^{15} +12.3431 q^{17} -28.8701 q^{19} -1.54985i q^{21} -24.6418i q^{23} -56.5980 q^{25} +10.3431 q^{27} -22.4499i q^{29} -16.7824i q^{31} +7.31371 q^{33} +23.8995 q^{35} -16.2506i q^{37} -5.29150i q^{39} +6.97056 q^{41} +22.8284 q^{43} +78.1987i q^{45} -6.19938i q^{47} -7.00000 q^{49} -7.23045 q^{51} -8.01514i q^{53} +112.782i q^{55} +16.9117 q^{57} -30.4437 q^{59} +15.2325i q^{61} -22.9039i q^{63} +81.5980 q^{65} +78.6274 q^{67} +14.4348i q^{69} -17.5345i q^{71} +46.6863 q^{73} +33.1543 q^{75} -33.0329i q^{77} -81.0325i q^{79} +71.8528 q^{81} -40.3848 q^{83} -111.498i q^{85} +13.1509i q^{87} +111.941 q^{89} -23.8995 q^{91} +9.83089i q^{93} +260.788i q^{95} -164.108 q^{97} +108.083 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{3} - 12q^{9} + O(q^{10}) \) \( 4q - 8q^{3} - 12q^{9} - 16q^{11} + 72q^{17} - 8q^{19} - 68q^{25} + 64q^{27} - 16q^{33} + 56q^{35} - 40q^{41} + 80q^{43} - 28q^{49} - 176q^{51} - 136q^{57} - 184q^{59} + 168q^{65} + 224q^{67} + 232q^{73} - 88q^{75} - 52q^{81} - 88q^{83} + 312q^{89} - 56q^{91} - 136q^{97} + 240q^{99} + O(q^{100}) \)

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
\(3\) −0.585786 −0.195262 −0.0976311 0.995223i \(-0.531127\pi\)
−0.0976311 + 0.995223i \(0.531127\pi\)
\(4\) 0 0
\(5\) − 9.03316i − 1.80663i −0.428976 0.903316i \(-0.641125\pi\)
0.428976 0.903316i \(-0.358875\pi\)
\(6\) 0 0
\(7\) 2.64575i 0.377964i
\(8\) 0 0
\(9\) −8.65685 −0.961873
\(10\) 0 0
\(11\) −12.4853 −1.13503 −0.567513 0.823365i \(-0.692094\pi\)
−0.567513 + 0.823365i \(0.692094\pi\)
\(12\) 0 0
\(13\) 9.03316i 0.694858i 0.937706 + 0.347429i \(0.112945\pi\)
−0.937706 + 0.347429i \(0.887055\pi\)
\(14\) 0 0
\(15\) 5.29150i 0.352767i
\(16\) 0 0
\(17\) 12.3431 0.726067 0.363034 0.931776i \(-0.381741\pi\)
0.363034 + 0.931776i \(0.381741\pi\)
\(18\) 0 0
\(19\) −28.8701 −1.51948 −0.759738 0.650229i \(-0.774673\pi\)
−0.759738 + 0.650229i \(0.774673\pi\)
\(20\) 0 0
\(21\) − 1.54985i − 0.0738022i
\(22\) 0 0
\(23\) − 24.6418i − 1.07138i −0.844414 0.535690i \(-0.820051\pi\)
0.844414 0.535690i \(-0.179949\pi\)
\(24\) 0 0
\(25\) −56.5980 −2.26392
\(26\) 0 0
\(27\) 10.3431 0.383079
\(28\) 0 0
\(29\) − 22.4499i − 0.774136i −0.922051 0.387068i \(-0.873488\pi\)
0.922051 0.387068i \(-0.126512\pi\)
\(30\) 0 0
\(31\) − 16.7824i − 0.541367i −0.962668 0.270684i \(-0.912750\pi\)
0.962668 0.270684i \(-0.0872497\pi\)
\(32\) 0 0
\(33\) 7.31371 0.221628
\(34\) 0 0
\(35\) 23.8995 0.682843
\(36\) 0 0
\(37\) − 16.2506i − 0.439204i −0.975589 0.219602i \(-0.929524\pi\)
0.975589 0.219602i \(-0.0704759\pi\)
\(38\) 0 0
\(39\) − 5.29150i − 0.135680i
\(40\) 0 0
\(41\) 6.97056 0.170014 0.0850069 0.996380i \(-0.472909\pi\)
0.0850069 + 0.996380i \(0.472909\pi\)
\(42\) 0 0
\(43\) 22.8284 0.530894 0.265447 0.964126i \(-0.414481\pi\)
0.265447 + 0.964126i \(0.414481\pi\)
\(44\) 0 0
\(45\) 78.1987i 1.73775i
\(46\) 0 0
\(47\) − 6.19938i − 0.131902i −0.997823 0.0659509i \(-0.978992\pi\)
0.997823 0.0659509i \(-0.0210081\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) −7.23045 −0.141773
\(52\) 0 0
\(53\) − 8.01514i − 0.151229i −0.997137 0.0756145i \(-0.975908\pi\)
0.997137 0.0756145i \(-0.0240918\pi\)
\(54\) 0 0
\(55\) 112.782i 2.05057i
\(56\) 0 0
\(57\) 16.9117 0.296696
\(58\) 0 0
\(59\) −30.4437 −0.515994 −0.257997 0.966146i \(-0.583062\pi\)
−0.257997 + 0.966146i \(0.583062\pi\)
\(60\) 0 0
\(61\) 15.2325i 0.249714i 0.992175 + 0.124857i \(0.0398472\pi\)
−0.992175 + 0.124857i \(0.960153\pi\)
\(62\) 0 0
\(63\) − 22.9039i − 0.363554i
\(64\) 0 0
\(65\) 81.5980 1.25535
\(66\) 0 0
\(67\) 78.6274 1.17354 0.586772 0.809752i \(-0.300399\pi\)
0.586772 + 0.809752i \(0.300399\pi\)
\(68\) 0 0
\(69\) 14.4348i 0.209200i
\(70\) 0 0
\(71\) − 17.5345i − 0.246965i −0.992347 0.123482i \(-0.960594\pi\)
0.992347 0.123482i \(-0.0394062\pi\)
\(72\) 0 0
\(73\) 46.6863 0.639538 0.319769 0.947495i \(-0.396395\pi\)
0.319769 + 0.947495i \(0.396395\pi\)
\(74\) 0 0
\(75\) 33.1543 0.442058
\(76\) 0 0
\(77\) − 33.0329i − 0.428999i
\(78\) 0 0
\(79\) − 81.0325i − 1.02573i −0.858470 0.512864i \(-0.828584\pi\)
0.858470 0.512864i \(-0.171416\pi\)
\(80\) 0 0
\(81\) 71.8528 0.887072
\(82\) 0 0
\(83\) −40.3848 −0.486564 −0.243282 0.969956i \(-0.578224\pi\)
−0.243282 + 0.969956i \(0.578224\pi\)
\(84\) 0 0
\(85\) − 111.498i − 1.31174i
\(86\) 0 0
\(87\) 13.1509i 0.151159i
\(88\) 0 0
\(89\) 111.941 1.25777 0.628883 0.777500i \(-0.283513\pi\)
0.628883 + 0.777500i \(0.283513\pi\)
\(90\) 0 0
\(91\) −23.8995 −0.262632
\(92\) 0 0
\(93\) 9.83089i 0.105709i
\(94\) 0 0
\(95\) 260.788i 2.74514i
\(96\) 0 0
\(97\) −164.108 −1.69183 −0.845916 0.533317i \(-0.820945\pi\)
−0.845916 + 0.533317i \(0.820945\pi\)
\(98\) 0 0
\(99\) 108.083 1.09175
\(100\) 0 0
\(101\) − 12.1329i − 0.120127i −0.998195 0.0600636i \(-0.980870\pi\)
0.998195 0.0600636i \(-0.0191304\pi\)
\(102\) 0 0
\(103\) − 106.582i − 1.03478i −0.855750 0.517389i \(-0.826904\pi\)
0.855750 0.517389i \(-0.173096\pi\)
\(104\) 0 0
\(105\) −14.0000 −0.133333
\(106\) 0 0
\(107\) 63.5980 0.594374 0.297187 0.954819i \(-0.403952\pi\)
0.297187 + 0.954819i \(0.403952\pi\)
\(108\) 0 0
\(109\) 130.848i 1.20044i 0.799835 + 0.600220i \(0.204920\pi\)
−0.799835 + 0.600220i \(0.795080\pi\)
\(110\) 0 0
\(111\) 9.51936i 0.0857600i
\(112\) 0 0
\(113\) −138.225 −1.22323 −0.611617 0.791154i \(-0.709481\pi\)
−0.611617 + 0.791154i \(0.709481\pi\)
\(114\) 0 0
\(115\) −222.593 −1.93559
\(116\) 0 0
\(117\) − 78.1987i − 0.668365i
\(118\) 0 0
\(119\) 32.6569i 0.274428i
\(120\) 0 0
\(121\) 34.8823 0.288283
\(122\) 0 0
\(123\) −4.08326 −0.0331972
\(124\) 0 0
\(125\) 285.430i 2.28344i
\(126\) 0 0
\(127\) − 114.442i − 0.901114i −0.892748 0.450557i \(-0.851225\pi\)
0.892748 0.450557i \(-0.148775\pi\)
\(128\) 0 0
\(129\) −13.3726 −0.103663
\(130\) 0 0
\(131\) 168.350 1.28512 0.642558 0.766237i \(-0.277873\pi\)
0.642558 + 0.766237i \(0.277873\pi\)
\(132\) 0 0
\(133\) − 76.3830i − 0.574308i
\(134\) 0 0
\(135\) − 93.4313i − 0.692084i
\(136\) 0 0
\(137\) 34.6863 0.253185 0.126592 0.991955i \(-0.459596\pi\)
0.126592 + 0.991955i \(0.459596\pi\)
\(138\) 0 0
\(139\) −107.664 −0.774561 −0.387281 0.921962i \(-0.626586\pi\)
−0.387281 + 0.921962i \(0.626586\pi\)
\(140\) 0 0
\(141\) 3.63151i 0.0257554i
\(142\) 0 0
\(143\) − 112.782i − 0.788682i
\(144\) 0 0
\(145\) −202.794 −1.39858
\(146\) 0 0
\(147\) 4.10051 0.0278946
\(148\) 0 0
\(149\) − 252.176i − 1.69246i −0.532819 0.846229i \(-0.678867\pi\)
0.532819 0.846229i \(-0.321133\pi\)
\(150\) 0 0
\(151\) − 234.486i − 1.55289i −0.630186 0.776444i \(-0.717021\pi\)
0.630186 0.776444i \(-0.282979\pi\)
\(152\) 0 0
\(153\) −106.853 −0.698384
\(154\) 0 0
\(155\) −151.598 −0.978051
\(156\) 0 0
\(157\) 10.0968i 0.0643109i 0.999483 + 0.0321554i \(0.0102372\pi\)
−0.999483 + 0.0321554i \(0.989763\pi\)
\(158\) 0 0
\(159\) 4.69516i 0.0295293i
\(160\) 0 0
\(161\) 65.1960 0.404944
\(162\) 0 0
\(163\) −104.534 −0.641313 −0.320657 0.947196i \(-0.603904\pi\)
−0.320657 + 0.947196i \(0.603904\pi\)
\(164\) 0 0
\(165\) − 66.0659i − 0.400399i
\(166\) 0 0
\(167\) 296.765i 1.77703i 0.458843 + 0.888517i \(0.348264\pi\)
−0.458843 + 0.888517i \(0.651736\pi\)
\(168\) 0 0
\(169\) 87.4020 0.517172
\(170\) 0 0
\(171\) 249.924 1.46154
\(172\) 0 0
\(173\) 40.0301i 0.231388i 0.993285 + 0.115694i \(0.0369091\pi\)
−0.993285 + 0.115694i \(0.963091\pi\)
\(174\) 0 0
\(175\) − 149.744i − 0.855681i
\(176\) 0 0
\(177\) 17.8335 0.100754
\(178\) 0 0
\(179\) −294.794 −1.64689 −0.823447 0.567394i \(-0.807952\pi\)
−0.823447 + 0.567394i \(0.807952\pi\)
\(180\) 0 0
\(181\) 40.4706i 0.223595i 0.993731 + 0.111797i \(0.0356608\pi\)
−0.993731 + 0.111797i \(0.964339\pi\)
\(182\) 0 0
\(183\) − 8.92302i − 0.0487596i
\(184\) 0 0
\(185\) −146.794 −0.793481
\(186\) 0 0
\(187\) −154.108 −0.824105
\(188\) 0 0
\(189\) 27.3654i 0.144790i
\(190\) 0 0
\(191\) 156.929i 0.821619i 0.911721 + 0.410810i \(0.134754\pi\)
−0.911721 + 0.410810i \(0.865246\pi\)
\(192\) 0 0
\(193\) −261.304 −1.35390 −0.676952 0.736027i \(-0.736700\pi\)
−0.676952 + 0.736027i \(0.736700\pi\)
\(194\) 0 0
\(195\) −47.7990 −0.245123
\(196\) 0 0
\(197\) − 145.283i − 0.737475i −0.929533 0.368738i \(-0.879790\pi\)
0.929533 0.368738i \(-0.120210\pi\)
\(198\) 0 0
\(199\) 390.508i 1.96235i 0.193122 + 0.981175i \(0.438139\pi\)
−0.193122 + 0.981175i \(0.561861\pi\)
\(200\) 0 0
\(201\) −46.0589 −0.229149
\(202\) 0 0
\(203\) 59.3970 0.292596
\(204\) 0 0
\(205\) − 62.9662i − 0.307152i
\(206\) 0 0
\(207\) 213.320i 1.03053i
\(208\) 0 0
\(209\) 360.451 1.72464
\(210\) 0 0
\(211\) 164.049 0.777482 0.388741 0.921347i \(-0.372910\pi\)
0.388741 + 0.921347i \(0.372910\pi\)
\(212\) 0 0
\(213\) 10.2715i 0.0482229i
\(214\) 0 0
\(215\) − 206.213i − 0.959129i
\(216\) 0 0
\(217\) 44.4020 0.204618
\(218\) 0 0
\(219\) −27.3482 −0.124878
\(220\) 0 0
\(221\) 111.498i 0.504514i
\(222\) 0 0
\(223\) 10.5830i 0.0474574i 0.999718 + 0.0237287i \(0.00755379\pi\)
−0.999718 + 0.0237287i \(0.992446\pi\)
\(224\) 0 0
\(225\) 489.960 2.17760
\(226\) 0 0
\(227\) 213.806 0.941877 0.470939 0.882166i \(-0.343915\pi\)
0.470939 + 0.882166i \(0.343915\pi\)
\(228\) 0 0
\(229\) − 232.028i − 1.01322i −0.862174 0.506612i \(-0.830898\pi\)
0.862174 0.506612i \(-0.169102\pi\)
\(230\) 0 0
\(231\) 19.3503i 0.0837673i
\(232\) 0 0
\(233\) −192.863 −0.827738 −0.413869 0.910336i \(-0.635823\pi\)
−0.413869 + 0.910336i \(0.635823\pi\)
\(234\) 0 0
\(235\) −56.0000 −0.238298
\(236\) 0 0
\(237\) 47.4678i 0.200286i
\(238\) 0 0
\(239\) − 327.917i − 1.37204i −0.727583 0.686020i \(-0.759356\pi\)
0.727583 0.686020i \(-0.240644\pi\)
\(240\) 0 0
\(241\) 71.8721 0.298225 0.149112 0.988820i \(-0.452358\pi\)
0.149112 + 0.988820i \(0.452358\pi\)
\(242\) 0 0
\(243\) −135.179 −0.556291
\(244\) 0 0
\(245\) 63.2321i 0.258090i
\(246\) 0 0
\(247\) − 260.788i − 1.05582i
\(248\) 0 0
\(249\) 23.6569 0.0950074
\(250\) 0 0
\(251\) 256.919 1.02358 0.511790 0.859110i \(-0.328982\pi\)
0.511790 + 0.859110i \(0.328982\pi\)
\(252\) 0 0
\(253\) 307.659i 1.21604i
\(254\) 0 0
\(255\) 65.3138i 0.256133i
\(256\) 0 0
\(257\) 319.352 1.24262 0.621308 0.783566i \(-0.286602\pi\)
0.621308 + 0.783566i \(0.286602\pi\)
\(258\) 0 0
\(259\) 42.9949 0.166004
\(260\) 0 0
\(261\) 194.346i 0.744620i
\(262\) 0 0
\(263\) − 377.357i − 1.43482i −0.696653 0.717408i \(-0.745328\pi\)
0.696653 0.717408i \(-0.254672\pi\)
\(264\) 0 0
\(265\) −72.4020 −0.273215
\(266\) 0 0
\(267\) −65.5736 −0.245594
\(268\) 0 0
\(269\) − 28.1631i − 0.104696i −0.998629 0.0523478i \(-0.983330\pi\)
0.998629 0.0523478i \(-0.0166704\pi\)
\(270\) 0 0
\(271\) 399.715i 1.47496i 0.675367 + 0.737482i \(0.263985\pi\)
−0.675367 + 0.737482i \(0.736015\pi\)
\(272\) 0 0
\(273\) 14.0000 0.0512821
\(274\) 0 0
\(275\) 706.642 2.56961
\(276\) 0 0
\(277\) 102.951i 0.371663i 0.982582 + 0.185831i \(0.0594979\pi\)
−0.982582 + 0.185831i \(0.940502\pi\)
\(278\) 0 0
\(279\) 145.283i 0.520726i
\(280\) 0 0
\(281\) −150.235 −0.534646 −0.267323 0.963607i \(-0.586139\pi\)
−0.267323 + 0.963607i \(0.586139\pi\)
\(282\) 0 0
\(283\) 178.561 0.630959 0.315480 0.948932i \(-0.397835\pi\)
0.315480 + 0.948932i \(0.397835\pi\)
\(284\) 0 0
\(285\) − 152.766i − 0.536021i
\(286\) 0 0
\(287\) 18.4424i 0.0642591i
\(288\) 0 0
\(289\) −136.647 −0.472826
\(290\) 0 0
\(291\) 96.1320 0.330351
\(292\) 0 0
\(293\) 219.189i 0.748085i 0.927411 + 0.374043i \(0.122029\pi\)
−0.927411 + 0.374043i \(0.877971\pi\)
\(294\) 0 0
\(295\) 275.002i 0.932211i
\(296\) 0 0
\(297\) −129.137 −0.434805
\(298\) 0 0
\(299\) 222.593 0.744458
\(300\) 0 0
\(301\) 60.3983i 0.200659i
\(302\) 0 0
\(303\) 7.10726i 0.0234563i
\(304\) 0 0
\(305\) 137.598 0.451141
\(306\) 0 0
\(307\) −316.669 −1.03150 −0.515748 0.856741i \(-0.672486\pi\)
−0.515748 + 0.856741i \(0.672486\pi\)
\(308\) 0 0
\(309\) 62.4344i 0.202053i
\(310\) 0 0
\(311\) 72.2653i 0.232364i 0.993228 + 0.116182i \(0.0370656\pi\)
−0.993228 + 0.116182i \(0.962934\pi\)
\(312\) 0 0
\(313\) 81.9512 0.261825 0.130913 0.991394i \(-0.458209\pi\)
0.130913 + 0.991394i \(0.458209\pi\)
\(314\) 0 0
\(315\) −206.894 −0.656808
\(316\) 0 0
\(317\) 109.150i 0.344322i 0.985069 + 0.172161i \(0.0550749\pi\)
−0.985069 + 0.172161i \(0.944925\pi\)
\(318\) 0 0
\(319\) 280.294i 0.878664i
\(320\) 0 0
\(321\) −37.2548 −0.116059
\(322\) 0 0
\(323\) −356.347 −1.10324
\(324\) 0 0
\(325\) − 511.259i − 1.57310i
\(326\) 0 0
\(327\) − 76.6489i − 0.234400i
\(328\) 0 0
\(329\) 16.4020 0.0498542
\(330\) 0 0
\(331\) −321.740 −0.972025 −0.486012 0.873952i \(-0.661549\pi\)
−0.486012 + 0.873952i \(0.661549\pi\)
\(332\) 0 0
\(333\) 140.679i 0.422459i
\(334\) 0 0
\(335\) − 710.254i − 2.12016i
\(336\) 0 0
\(337\) −164.049 −0.486792 −0.243396 0.969927i \(-0.578261\pi\)
−0.243396 + 0.969927i \(0.578261\pi\)
\(338\) 0 0
\(339\) 80.9706 0.238851
\(340\) 0 0
\(341\) 209.533i 0.614466i
\(342\) 0 0
\(343\) − 18.5203i − 0.0539949i
\(344\) 0 0
\(345\) 130.392 0.377948
\(346\) 0 0
\(347\) 330.309 0.951898 0.475949 0.879473i \(-0.342105\pi\)
0.475949 + 0.879473i \(0.342105\pi\)
\(348\) 0 0
\(349\) − 262.402i − 0.751869i −0.926646 0.375934i \(-0.877322\pi\)
0.926646 0.375934i \(-0.122678\pi\)
\(350\) 0 0
\(351\) 93.4313i 0.266186i
\(352\) 0 0
\(353\) −578.098 −1.63767 −0.818835 0.574029i \(-0.805380\pi\)
−0.818835 + 0.574029i \(0.805380\pi\)
\(354\) 0 0
\(355\) −158.392 −0.446174
\(356\) 0 0
\(357\) − 19.1300i − 0.0535853i
\(358\) 0 0
\(359\) 365.114i 1.01703i 0.861053 + 0.508515i \(0.169805\pi\)
−0.861053 + 0.508515i \(0.830195\pi\)
\(360\) 0 0
\(361\) 472.480 1.30881
\(362\) 0 0
\(363\) −20.4335 −0.0562908
\(364\) 0 0
\(365\) − 421.725i − 1.15541i
\(366\) 0 0
\(367\) − 520.071i − 1.41709i −0.705666 0.708544i \(-0.749352\pi\)
0.705666 0.708544i \(-0.250648\pi\)
\(368\) 0 0
\(369\) −60.3431 −0.163532
\(370\) 0 0
\(371\) 21.2061 0.0571592
\(372\) 0 0
\(373\) − 526.711i − 1.41210i −0.708164 0.706048i \(-0.750476\pi\)
0.708164 0.706048i \(-0.249524\pi\)
\(374\) 0 0
\(375\) − 167.201i − 0.445869i
\(376\) 0 0
\(377\) 202.794 0.537915
\(378\) 0 0
\(379\) −121.976 −0.321835 −0.160918 0.986968i \(-0.551445\pi\)
−0.160918 + 0.986968i \(0.551445\pi\)
\(380\) 0 0
\(381\) 67.0383i 0.175954i
\(382\) 0 0
\(383\) 316.427i 0.826179i 0.910690 + 0.413089i \(0.135550\pi\)
−0.910690 + 0.413089i \(0.864450\pi\)
\(384\) 0 0
\(385\) −298.392 −0.775044
\(386\) 0 0
\(387\) −197.622 −0.510652
\(388\) 0 0
\(389\) 92.1474i 0.236883i 0.992961 + 0.118441i \(0.0377898\pi\)
−0.992961 + 0.118441i \(0.962210\pi\)
\(390\) 0 0
\(391\) − 304.157i − 0.777895i
\(392\) 0 0
\(393\) −98.6173 −0.250935
\(394\) 0 0
\(395\) −731.980 −1.85311
\(396\) 0 0
\(397\) 562.267i 1.41629i 0.706068 + 0.708144i \(0.250467\pi\)
−0.706068 + 0.708144i \(0.749533\pi\)
\(398\) 0 0
\(399\) 44.7441i 0.112141i
\(400\) 0 0
\(401\) 81.2061 0.202509 0.101254 0.994861i \(-0.467714\pi\)
0.101254 + 0.994861i \(0.467714\pi\)
\(402\) 0 0
\(403\) 151.598 0.376174
\(404\) 0 0
\(405\) − 649.058i − 1.60261i
\(406\) 0 0
\(407\) 202.893i 0.498508i
\(408\) 0 0
\(409\) 450.735 1.10204 0.551021 0.834491i \(-0.314238\pi\)
0.551021 + 0.834491i \(0.314238\pi\)
\(410\) 0 0
\(411\) −20.3188 −0.0494374
\(412\) 0 0
\(413\) − 80.5463i − 0.195027i
\(414\) 0 0
\(415\) 364.802i 0.879041i
\(416\) 0 0
\(417\) 63.0681 0.151242
\(418\) 0 0
\(419\) −624.988 −1.49162 −0.745809 0.666160i \(-0.767937\pi\)
−0.745809 + 0.666160i \(0.767937\pi\)
\(420\) 0 0
\(421\) 566.476i 1.34555i 0.739848 + 0.672774i \(0.234897\pi\)
−0.739848 + 0.672774i \(0.765103\pi\)
\(422\) 0 0
\(423\) 53.6671i 0.126873i
\(424\) 0 0
\(425\) −698.597 −1.64376
\(426\) 0 0
\(427\) −40.3015 −0.0943829
\(428\) 0 0
\(429\) 66.0659i 0.154000i
\(430\) 0 0
\(431\) − 289.528i − 0.671760i −0.941905 0.335880i \(-0.890966\pi\)
0.941905 0.335880i \(-0.109034\pi\)
\(432\) 0 0
\(433\) −597.696 −1.38036 −0.690180 0.723638i \(-0.742468\pi\)
−0.690180 + 0.723638i \(0.742468\pi\)
\(434\) 0 0
\(435\) 118.794 0.273090
\(436\) 0 0
\(437\) 711.409i 1.62794i
\(438\) 0 0
\(439\) 38.3890i 0.0874464i 0.999044 + 0.0437232i \(0.0139220\pi\)
−0.999044 + 0.0437232i \(0.986078\pi\)
\(440\) 0 0
\(441\) 60.5980 0.137410
\(442\) 0 0
\(443\) −599.058 −1.35228 −0.676138 0.736775i \(-0.736348\pi\)
−0.676138 + 0.736775i \(0.736348\pi\)
\(444\) 0 0
\(445\) − 1011.18i − 2.27232i
\(446\) 0 0
\(447\) 147.721i 0.330473i
\(448\) 0 0
\(449\) −460.039 −1.02459 −0.512293 0.858811i \(-0.671204\pi\)
−0.512293 + 0.858811i \(0.671204\pi\)
\(450\) 0 0
\(451\) −87.0294 −0.192970
\(452\) 0 0
\(453\) 137.359i 0.303220i
\(454\) 0 0
\(455\) 215.888i 0.474479i
\(456\) 0 0
\(457\) 266.323 0.582764 0.291382 0.956607i \(-0.405885\pi\)
0.291382 + 0.956607i \(0.405885\pi\)
\(458\) 0 0
\(459\) 127.667 0.278142
\(460\) 0 0
\(461\) − 763.123i − 1.65537i −0.561196 0.827683i \(-0.689659\pi\)
0.561196 0.827683i \(-0.310341\pi\)
\(462\) 0 0
\(463\) − 123.988i − 0.267792i −0.990995 0.133896i \(-0.957251\pi\)
0.990995 0.133896i \(-0.0427488\pi\)
\(464\) 0 0
\(465\) 88.8040 0.190976
\(466\) 0 0
\(467\) 768.718 1.64608 0.823038 0.567986i \(-0.192277\pi\)
0.823038 + 0.567986i \(0.192277\pi\)
\(468\) 0 0
\(469\) 208.029i 0.443558i
\(470\) 0 0
\(471\) − 5.91457i − 0.0125575i
\(472\) 0 0
\(473\) −285.019 −0.602578
\(474\) 0 0
\(475\) 1633.99 3.43997
\(476\) 0 0
\(477\) 69.3859i 0.145463i
\(478\) 0 0
\(479\) 118.981i 0.248394i 0.992258 + 0.124197i \(0.0396355\pi\)
−0.992258 + 0.124197i \(0.960364\pi\)
\(480\) 0 0
\(481\) 146.794 0.305185
\(482\) 0 0
\(483\) −38.1909 −0.0790702
\(484\) 0 0
\(485\) 1482.41i 3.05652i
\(486\) 0 0
\(487\) 282.577i 0.580240i 0.956990 + 0.290120i \(0.0936952\pi\)
−0.956990 + 0.290120i \(0.906305\pi\)
\(488\) 0 0
\(489\) 61.2346 0.125224
\(490\) 0 0
\(491\) 388.049 0.790323 0.395162 0.918612i \(-0.370689\pi\)
0.395162 + 0.918612i \(0.370689\pi\)
\(492\) 0 0
\(493\) − 277.103i − 0.562075i
\(494\) 0 0
\(495\) − 976.333i − 1.97239i
\(496\) 0 0
\(497\) 46.3919 0.0933439
\(498\) 0 0
\(499\) 27.7157 0.0555425 0.0277713 0.999614i \(-0.491159\pi\)
0.0277713 + 0.999614i \(0.491159\pi\)
\(500\) 0 0
\(501\) − 173.841i − 0.346988i
\(502\) 0 0
\(503\) − 727.477i − 1.44628i −0.690703 0.723138i \(-0.742699\pi\)
0.690703 0.723138i \(-0.257301\pi\)
\(504\) 0 0
\(505\) −109.598 −0.217026
\(506\) 0 0
\(507\) −51.1989 −0.100984
\(508\) 0 0
\(509\) − 634.183i − 1.24594i −0.782246 0.622969i \(-0.785926\pi\)
0.782246 0.622969i \(-0.214074\pi\)
\(510\) 0 0
\(511\) 123.520i 0.241723i
\(512\) 0 0
\(513\) −298.607 −0.582080
\(514\) 0 0
\(515\) −962.774 −1.86946
\(516\) 0 0
\(517\) 77.4010i 0.149712i
\(518\) 0 0
\(519\) − 23.4491i − 0.0451813i
\(520\) 0 0
\(521\) 833.127 1.59909 0.799546 0.600605i \(-0.205073\pi\)
0.799546 + 0.600605i \(0.205073\pi\)
\(522\) 0 0
\(523\) −876.434 −1.67578 −0.837891 0.545838i \(-0.816211\pi\)
−0.837891 + 0.545838i \(0.816211\pi\)
\(524\) 0 0
\(525\) 87.7181i 0.167082i
\(526\) 0 0
\(527\) − 207.147i − 0.393069i
\(528\) 0 0
\(529\) −78.2162 −0.147857
\(530\) 0 0
\(531\) 263.546 0.496321
\(532\) 0 0
\(533\) 62.9662i 0.118135i
\(534\) 0 0
\(535\) − 574.491i − 1.07381i
\(536\) 0 0
\(537\) 172.686 0.321576
\(538\) 0 0
\(539\) 87.3970 0.162147
\(540\) 0 0
\(541\) 405.915i 0.750305i 0.926963 + 0.375152i \(0.122410\pi\)
−0.926963 + 0.375152i \(0.877590\pi\)
\(542\) 0 0
\(543\) − 23.7072i − 0.0436596i
\(544\) 0 0
\(545\) 1181.97 2.16875
\(546\) 0 0
\(547\) 606.024 1.10791 0.553953 0.832548i \(-0.313119\pi\)
0.553953 + 0.832548i \(0.313119\pi\)
\(548\) 0 0
\(549\) − 131.866i − 0.240193i
\(550\) 0 0
\(551\) 648.131i 1.17628i
\(552\) 0 0
\(553\) 214.392 0.387689
\(554\) 0 0
\(555\) 85.9899 0.154937
\(556\) 0 0
\(557\) − 36.4442i − 0.0654294i −0.999465 0.0327147i \(-0.989585\pi\)
0.999465 0.0327147i \(-0.0104153\pi\)
\(558\) 0 0
\(559\) 206.213i 0.368896i
\(560\) 0 0
\(561\) 90.2742 0.160917
\(562\) 0 0
\(563\) 186.389 0.331064 0.165532 0.986204i \(-0.447066\pi\)
0.165532 + 0.986204i \(0.447066\pi\)
\(564\) 0 0
\(565\) 1248.61i 2.20993i
\(566\) 0 0
\(567\) 190.105i 0.335282i
\(568\) 0 0
\(569\) −670.891 −1.17907 −0.589536 0.807742i \(-0.700689\pi\)
−0.589536 + 0.807742i \(0.700689\pi\)
\(570\) 0 0
\(571\) 677.082 1.18578 0.592892 0.805282i \(-0.297986\pi\)
0.592892 + 0.805282i \(0.297986\pi\)
\(572\) 0 0
\(573\) − 91.9271i − 0.160431i
\(574\) 0 0
\(575\) 1394.67i 2.42552i
\(576\) 0 0
\(577\) 927.901 1.60815 0.804073 0.594530i \(-0.202662\pi\)
0.804073 + 0.594530i \(0.202662\pi\)
\(578\) 0 0
\(579\) 153.068 0.264366
\(580\) 0 0
\(581\) − 106.848i − 0.183904i
\(582\) 0 0
\(583\) 100.071i 0.171649i
\(584\) 0 0
\(585\) −706.382 −1.20749
\(586\) 0 0
\(587\) 321.120 0.547053 0.273526 0.961865i \(-0.411810\pi\)
0.273526 + 0.961865i \(0.411810\pi\)
\(588\) 0 0
\(589\) 484.508i 0.822595i
\(590\) 0 0
\(591\) 85.1046i 0.144001i
\(592\) 0 0
\(593\) −219.255 −0.369738 −0.184869 0.982763i \(-0.559186\pi\)
−0.184869 + 0.982763i \(0.559186\pi\)
\(594\) 0 0
\(595\) 294.995 0.495790
\(596\) 0 0
\(597\) − 228.754i − 0.383173i
\(598\) 0 0
\(599\) 154.802i 0.258434i 0.991616 + 0.129217i \(0.0412464\pi\)
−0.991616 + 0.129217i \(0.958754\pi\)
\(600\) 0 0
\(601\) 205.862 0.342533 0.171266 0.985225i \(-0.445214\pi\)
0.171266 + 0.985225i \(0.445214\pi\)
\(602\) 0 0
\(603\) −680.666 −1.12880
\(604\) 0 0
\(605\) − 315.097i − 0.520821i
\(606\) 0 0
\(607\) 790.663i 1.30258i 0.758831 + 0.651288i \(0.225771\pi\)
−0.758831 + 0.651288i \(0.774229\pi\)
\(608\) 0 0
\(609\) −34.7939 −0.0571329
\(610\) 0 0
\(611\) 56.0000 0.0916530
\(612\) 0 0
\(613\) − 741.471i − 1.20958i −0.796386 0.604789i \(-0.793257\pi\)
0.796386 0.604789i \(-0.206743\pi\)
\(614\) 0 0
\(615\) 36.8848i 0.0599752i
\(616\) 0 0
\(617\) 171.578 0.278084 0.139042 0.990286i \(-0.455598\pi\)
0.139042 + 0.990286i \(0.455598\pi\)
\(618\) 0 0
\(619\) 540.198 0.872695 0.436347 0.899778i \(-0.356272\pi\)
0.436347 + 0.899778i \(0.356272\pi\)
\(620\) 0 0
\(621\) − 254.873i − 0.410424i
\(622\) 0 0
\(623\) 296.168i 0.475391i
\(624\) 0 0
\(625\) 1163.38 1.86141
\(626\) 0 0
\(627\) −211.147 −0.336758
\(628\) 0 0
\(629\) − 200.583i − 0.318892i
\(630\) 0 0
\(631\) 269.399i 0.426940i 0.976950 + 0.213470i \(0.0684766\pi\)
−0.976950 + 0.213470i \(0.931523\pi\)
\(632\) 0 0
\(633\) −96.0975 −0.151813
\(634\) 0 0
\(635\) −1033.77 −1.62798
\(636\) 0 0
\(637\) − 63.2321i − 0.0992655i
\(638\) 0 0
\(639\) 151.794i 0.237549i
\(640\) 0 0
\(641\) 36.1867 0.0564535 0.0282268 0.999602i \(-0.491014\pi\)
0.0282268 + 0.999602i \(0.491014\pi\)
\(642\) 0 0
\(643\) −266.297 −0.414148 −0.207074 0.978325i \(-0.566394\pi\)
−0.207074 + 0.978325i \(0.566394\pi\)
\(644\) 0 0
\(645\) 120.797i 0.187282i
\(646\) 0 0
\(647\) − 1086.24i − 1.67888i −0.543452 0.839440i \(-0.682883\pi\)
0.543452 0.839440i \(-0.317117\pi\)
\(648\) 0 0
\(649\) 380.098 0.585666
\(650\) 0 0
\(651\) −26.0101 −0.0399541
\(652\) 0 0
\(653\) − 1195.35i − 1.83055i −0.402832 0.915274i \(-0.631974\pi\)
0.402832 0.915274i \(-0.368026\pi\)
\(654\) 0 0
\(655\) − 1520.74i − 2.32173i
\(656\) 0 0
\(657\) −404.156 −0.615154
\(658\) 0 0
\(659\) 685.220 1.03979 0.519894 0.854231i \(-0.325971\pi\)
0.519894 + 0.854231i \(0.325971\pi\)
\(660\) 0 0
\(661\) 993.382i 1.50285i 0.659820 + 0.751423i \(0.270632\pi\)
−0.659820 + 0.751423i \(0.729368\pi\)
\(662\) 0 0
\(663\) − 65.3138i − 0.0985125i
\(664\) 0 0
\(665\) −689.980 −1.03756
\(666\) 0 0
\(667\) −553.206 −0.829394
\(668\) 0 0
\(669\) − 6.19938i − 0.00926664i
\(670\) 0 0
\(671\) − 190.183i − 0.283432i
\(672\) 0 0
\(673\) 106.569 0.158349 0.0791743 0.996861i \(-0.474772\pi\)
0.0791743 + 0.996861i \(0.474772\pi\)
\(674\) 0 0
\(675\) −585.401 −0.867261
\(676\) 0 0
\(677\) − 1004.18i − 1.48329i −0.670795 0.741643i \(-0.734047\pi\)
0.670795 0.741643i \(-0.265953\pi\)
\(678\) 0 0
\(679\) − 434.188i − 0.639452i
\(680\) 0 0
\(681\) −125.245 −0.183913
\(682\) 0 0
\(683\) 678.225 0.993009 0.496505 0.868034i \(-0.334617\pi\)
0.496505 + 0.868034i \(0.334617\pi\)
\(684\) 0 0
\(685\) − 313.327i − 0.457411i
\(686\) 0 0
\(687\) 135.919i 0.197844i
\(688\) 0 0
\(689\) 72.4020 0.105083
\(690\) 0 0
\(691\) 365.175 0.528473 0.264236 0.964458i \(-0.414880\pi\)
0.264236 + 0.964458i \(0.414880\pi\)
\(692\) 0 0
\(693\) 285.961i 0.412643i
\(694\) 0 0
\(695\) 972.546i 1.39935i
\(696\) 0 0
\(697\) 86.0387 0.123441
\(698\) 0 0
\(699\) 112.976 0.161626
\(700\) 0 0
\(701\) − 940.292i − 1.34136i −0.741748 0.670679i \(-0.766003\pi\)
0.741748 0.670679i \(-0.233997\pi\)
\(702\) 0 0
\(703\) 469.155i 0.667361i
\(704\) 0 0
\(705\) 32.8040 0.0465306
\(706\) 0 0
\(707\) 32.1005 0.0454038
\(708\) 0 0
\(709\) 1057.46i 1.49148i 0.666239 + 0.745738i \(0.267903\pi\)
−0.666239 + 0.745738i \(0.732097\pi\)
\(710\) 0 0
\(711\) 701.487i 0.986620i
\(712\) 0 0
\(713\) −413.547 −0.580010
\(714\) 0 0
\(715\) −1018.77 −1.42486
\(716\) 0 0
\(717\) 192.090i 0.267907i
\(718\) 0 0
\(719\) − 1034.82i − 1.43926i −0.694360 0.719628i \(-0.744312\pi\)
0.694360 0.719628i \(-0.255688\pi\)
\(720\) 0 0
\(721\) 281.990 0.391109
\(722\) 0 0
\(723\) −42.1017 −0.0582320
\(724\) 0 0
\(725\) 1270.62i 1.75258i
\(726\) 0 0
\(727\) − 495.145i − 0.681080i −0.940230 0.340540i \(-0.889390\pi\)
0.940230 0.340540i \(-0.110610\pi\)
\(728\) 0 0
\(729\) −567.489 −0.778449
\(730\) 0 0
\(731\) 281.775 0.385465
\(732\) 0 0
\(733\) 567.494i 0.774207i 0.922036 + 0.387103i \(0.126524\pi\)
−0.922036 + 0.387103i \(0.873476\pi\)
\(734\) 0 0
\(735\) − 37.0405i − 0.0503953i
\(736\) 0 0
\(737\) −981.685 −1.33200
\(738\) 0 0
\(739\) 544.701 0.737078 0.368539 0.929612i \(-0.379858\pi\)
0.368539 + 0.929612i \(0.379858\pi\)
\(740\) 0 0
\(741\) 152.766i 0.206162i
\(742\) 0 0
\(743\) − 731.264i − 0.984205i −0.870537 0.492102i \(-0.836229\pi\)
0.870537 0.492102i \(-0.163771\pi\)
\(744\) 0 0
\(745\) −2277.95 −3.05765
\(746\) 0 0
\(747\) 349.605 0.468012
\(748\) 0 0
\(749\) 168.264i 0.224652i
\(750\) 0 0
\(751\) − 666.262i − 0.887166i −0.896233 0.443583i \(-0.853707\pi\)
0.896233 0.443583i \(-0.146293\pi\)
\(752\) 0 0
\(753\) −150.500 −0.199867
\(754\) 0 0
\(755\) −2118.15 −2.80550
\(756\) 0 0
\(757\) − 238.623i − 0.315222i −0.987501 0.157611i \(-0.949621\pi\)
0.987501 0.157611i \(-0.0503791\pi\)
\(758\) 0 0
\(759\) − 180.223i − 0.237447i
\(760\) 0 0
\(761\) 614.930 0.808055 0.404028 0.914747i \(-0.367610\pi\)
0.404028 + 0.914747i \(0.367610\pi\)
\(762\) 0 0
\(763\) −346.191 −0.453723
\(764\) 0 0
\(765\) 965.219i 1.26172i
\(766\) 0 0
\(767\) − 275.002i − 0.358543i
\(768\) 0 0
\(769\) 178.950 0.232705 0.116353 0.993208i \(-0.462880\pi\)
0.116353 + 0.993208i \(0.462880\pi\)
\(770\) 0 0
\(771\) −187.072 −0.242636
\(772\) 0 0
\(773\) − 631.615i − 0.817095i −0.912737 0.408548i \(-0.866035\pi\)
0.912737 0.408548i \(-0.133965\pi\)
\(774\) 0 0
\(775\) 949.849i 1.22561i
\(776\) 0 0
\(777\) −25.1859 −0.0324142
\(778\) 0 0
\(779\) −201.241 −0.258332
\(780\) 0 0
\(781\) 218.923i 0.280311i
\(782\) 0 0
\(783\) − 232.203i − 0.296556i
\(784\) 0 0
\(785\) 91.2061 0.116186
\(786\) 0 0
\(787\) −456.655 −0.580247 −0.290124 0.956989i \(-0.593696\pi\)
−0.290124 + 0.956989i \(0.593696\pi\)
\(788\) 0 0
\(789\) 221.050i 0.280165i
\(790\) 0 0
\(791\) − 365.710i − 0.462339i
\(792\) 0 0
\(793\) −137.598 −0.173516
\(794\) 0 0
\(795\) 42.4121 0.0533486
\(796\) 0 0
\(797\) 218.566i 0.274236i 0.990555 + 0.137118i \(0.0437839\pi\)
−0.990555 + 0.137118i \(0.956216\pi\)
\(798\) 0 0
\(799\) − 76.5199i − 0.0957695i
\(800\) 0 0
\(801\) −969.058 −1.20981
\(802\) 0 0
\(803\) −582.891 −0.725892
\(804\) 0 0
\(805\) − 588.926i − 0.731585i
\(806\) 0 0
\(807\) 16.4976i 0.0204431i
\(808\) 0 0
\(809\) 1347.46 1.66559 0.832794 0.553584i \(-0.186740\pi\)
0.832794 + 0.553584i \(0.186740\pi\)
\(810\) 0 0
\(811\) 672.620 0.829371 0.414686 0.909965i \(-0.363892\pi\)
0.414686 + 0.909965i \(0.363892\pi\)
\(812\) 0 0
\(813\) − 234.148i − 0.288005i
\(814\) 0 0
\(815\) 944.273i 1.15862i
\(816\) 0 0
\(817\) −659.058 −0.806681
\(818\) 0 0
\(819\) 206.894 0.252618
\(820\) 0 0
\(821\) − 1162.57i − 1.41604i −0.706190 0.708022i \(-0.749588\pi\)
0.706190 0.708022i \(-0.250412\pi\)
\(822\) 0 0
\(823\) 1041.65i 1.26567i 0.774286 + 0.632835i \(0.218109\pi\)
−0.774286 + 0.632835i \(0.781891\pi\)
\(824\) 0 0
\(825\) −413.941 −0.501747
\(826\) 0 0
\(827\) −278.432 −0.336678 −0.168339 0.985729i \(-0.553840\pi\)
−0.168339 + 0.985729i \(0.553840\pi\)
\(828\) 0 0
\(829\) − 1065.74i − 1.28557i −0.766046 0.642785i \(-0.777779\pi\)
0.766046 0.642785i \(-0.222221\pi\)
\(830\) 0 0
\(831\) − 60.3071i − 0.0725717i
\(832\) 0 0
\(833\) −86.4020 −0.103724
\(834\) 0 0
\(835\) 2680.72 3.21045
\(836\) 0 0
\(837\) − 173.583i − 0.207387i
\(838\) 0 0
\(839\) − 305.844i − 0.364533i −0.983249 0.182267i \(-0.941657\pi\)
0.983249 0.182267i \(-0.0583434\pi\)
\(840\) 0 0
\(841\) 337.000 0.400713
\(842\) 0 0
\(843\) 88.0059 0.104396
\(844\) 0 0
\(845\) − 789.516i − 0.934339i
\(846\) 0 0
\(847\) 92.2898i 0.108961i
\(848\) 0 0
\(849\) −104.599 −0.123202
\(850\) 0 0
\(851\) −400.442 −0.470555
\(852\) 0 0
\(853\) − 164.018i − 0.192283i −0.995368 0.0961417i \(-0.969350\pi\)
0.995368 0.0961417i \(-0.0306502\pi\)
\(854\) 0 0
\(855\) − 2257.60i − 2.64047i
\(856\) 0 0
\(857\) 851.068 0.993078 0.496539 0.868014i \(-0.334604\pi\)
0.496539 + 0.868014i \(0.334604\pi\)
\(858\) 0 0
\(859\) 1179.69 1.37333 0.686666 0.726973i \(-0.259073\pi\)
0.686666 + 0.726973i \(0.259073\pi\)
\(860\) 0 0
\(861\) − 10.8033i − 0.0125474i
\(862\) 0 0
\(863\) − 279.048i − 0.323346i −0.986844 0.161673i \(-0.948311\pi\)
0.986844 0.161673i \(-0.0516890\pi\)
\(864\) 0 0
\(865\) 361.598 0.418032
\(866\) 0 0
\(867\) 80.0458 0.0923250
\(868\) 0 0
\(869\) 1011.71i 1.16423i
\(870\) 0 0
\(871\) 710.254i 0.815447i
\(872\) 0 0
\(873\) 1420.66 1.62733
\(874\) 0 0
\(875\) −755.176 −0.863058
\(876\) 0 0
\(877\) − 674.159i − 0.768711i −0.923185 0.384355i \(-0.874424\pi\)
0.923185 0.384355i \(-0.125576\pi\)
\(878\) 0 0
\(879\) − 128.398i − 0.146073i
\(880\) 0 0
\(881\) −1001.29 −1.13654 −0.568271 0.822841i \(-0.692387\pi\)
−0.568271 + 0.822841i \(0.692387\pi\)
\(882\) 0 0
\(883\) −882.010 −0.998879 −0.499439 0.866349i \(-0.666461\pi\)
−0.499439 + 0.866349i \(0.666461\pi\)
\(884\) 0 0
\(885\) − 161.093i − 0.182026i
\(886\) 0 0
\(887\) − 7.08053i − 0.00798256i −0.999992 0.00399128i \(-0.998730\pi\)
0.999992 0.00399128i \(-0.00127047\pi\)
\(888\) 0 0
\(889\) 302.784 0.340589
\(890\) 0 0
\(891\) −897.103 −1.00685
\(892\) 0 0
\(893\) 178.976i 0.200422i
\(894\) 0 0
\(895\) 2662.92i 2.97533i
\(896\) 0 0
\(897\) −130.392 −0.145364
\(898\) 0 0
\(899\) −376.764 −0.419092
\(900\) 0 0
\(901\) − 98.9320i − 0.109802i
\(902\) 0 0
\(903\) − 35.3805i − 0.0391811i
\(904\) 0 0
\(905\) 365.578 0.403953
\(906\) 0 0
\(907\) −450.372 −0.496551 −0.248275 0.968689i \(-0.579864\pi\)
−0.248275 + 0.968689i \(0.579864\pi\)
\(908\) 0 0
\(909\) 105.032i 0.115547i
\(910\) 0 0
\(911\) − 202.426i − 0.222201i −0.993809 0.111101i \(-0.964562\pi\)
0.993809 0.111101i \(-0.0354376\pi\)
\(912\) 0 0
\(913\) 504.215 0.552262
\(914\) 0 0
\(915\) −80.6030 −0.0880907
\(916\) 0 0
\(917\) 445.413i 0.485728i
\(918\) 0 0
\(919\) − 1593.73i − 1.73420i −0.498138 0.867098i \(-0.665983\pi\)
0.498138 0.867098i \(-0.334017\pi\)
\(920\) 0 0
\(921\) 185.500 0.201412
\(922\) 0 0
\(923\) 158.392 0.171606
\(924\) 0 0
\(925\) 919.749i 0.994323i
\(926\) 0 0
\(927\) 922.666i 0.995325i
\(928\) 0 0
\(929\) −1039.40 −1.11884 −0.559419 0.828885i \(-0.688976\pi\)
−0.559419 + 0.828885i \(0.688976\pi\)
\(930\) 0 0
\(931\) 202.090 0.217068
\(932\) 0 0
\(933\) − 42.3320i − 0.0453719i
\(934\) 0 0
\(935\) 1392.08i 1.48885i
\(936\) 0 0
\(937\) 881.765 0.941051 0.470525 0.882386i \(-0.344064\pi\)
0.470525 + 0.882386i \(0.344064\pi\)
\(938\) 0 0
\(939\) −48.0059 −0.0511245
\(940\) 0 0
\(941\) 953.344i 1.01312i 0.862205 + 0.506559i \(0.169083\pi\)
−0.862205 + 0.506559i \(0.830917\pi\)
\(942\) 0 0
\(943\) − 171.767i − 0.182149i
\(944\) 0 0
\(945\) 247.196 0.261583
\(946\) 0 0
\(947\) −16.8957 −0.0178413 −0.00892063 0.999960i \(-0.502840\pi\)
−0.00892063 + 0.999960i \(0.502840\pi\)
\(948\) 0 0
\(949\) 421.725i 0.444389i
\(950\) 0 0
\(951\) − 63.9386i − 0.0672330i
\(952\) 0 0
\(953\) 1526.31 1.60159 0.800794 0.598940i \(-0.204411\pi\)
0.800794 + 0.598940i \(0.204411\pi\)
\(954\) 0 0
\(955\) 1417.57 1.48436
\(956\) 0 0
\(957\) − 164.192i − 0.171570i
\(958\) 0 0
\(959\) 91.7713i 0.0956948i
\(960\) 0 0
\(961\) 679.352 0.706921
\(962\) 0 0
\(963\) −550.558 −0.571712
\(964\) 0 0
\(965\) 2360.40i 2.44601i
\(966\) 0 0
\(967\) 1410.39i 1.45852i 0.684235 + 0.729262i \(0.260136\pi\)
−0.684235 + 0.729262i \(0.739864\pi\)
\(968\) 0 0
\(969\) 208.743 0.215421
\(970\) 0 0
\(971\) 596.497 0.614312 0.307156 0.951659i \(-0.400623\pi\)
0.307156 + 0.951659i \(0.400623\pi\)
\(972\) 0 0
\(973\) − 284.852i − 0.292757i
\(974\) 0 0
\(975\) 299.488i 0.307168i
\(976\) 0 0
\(977\) 146.686 0.150140 0.0750698 0.997178i \(-0.476082\pi\)
0.0750698 + 0.997178i \(0.476082\pi\)
\(978\) 0 0
\(979\) −1397.62 −1.42760
\(980\) 0 0
\(981\) − 1132.73i − 1.15467i
\(982\) 0 0
\(983\) 169.457i 0.172388i 0.996278 + 0.0861939i \(0.0274704\pi\)
−0.996278 + 0.0861939i \(0.972530\pi\)
\(984\) 0 0
\(985\) −1312.36 −1.33235
\(986\) 0 0
\(987\) −9.60808 −0.00973463
\(988\) 0 0
\(989\) − 562.533i − 0.568789i
\(990\) 0 0
\(991\) − 1686.90i − 1.70222i −0.524988 0.851109i \(-0.675930\pi\)
0.524988 0.851109i \(-0.324070\pi\)
\(992\) 0 0
\(993\) 188.471 0.189800
\(994\) 0 0
\(995\) 3527.52 3.54524
\(996\) 0 0
\(997\) − 1736.14i − 1.74136i −0.491849 0.870680i \(-0.663679\pi\)
0.491849 0.870680i \(-0.336321\pi\)
\(998\) 0 0
\(999\) − 168.082i − 0.168250i
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.a.15.3 4
3.2 odd 2 2016.3.g.a.1135.4 4
4.3 odd 2 56.3.g.a.43.1 4
7.6 odd 2 1568.3.g.h.687.2 4
8.3 odd 2 inner 224.3.g.a.15.4 4
8.5 even 2 56.3.g.a.43.2 yes 4
12.11 even 2 504.3.g.a.379.4 4
16.3 odd 4 1792.3.d.g.1023.5 8
16.5 even 4 1792.3.d.g.1023.6 8
16.11 odd 4 1792.3.d.g.1023.4 8
16.13 even 4 1792.3.d.g.1023.3 8
24.5 odd 2 504.3.g.a.379.3 4
24.11 even 2 2016.3.g.a.1135.1 4
28.3 even 6 392.3.k.j.275.4 8
28.11 odd 6 392.3.k.i.275.4 8
28.19 even 6 392.3.k.j.67.2 8
28.23 odd 6 392.3.k.i.67.2 8
28.27 even 2 392.3.g.h.99.1 4
56.5 odd 6 392.3.k.j.67.4 8
56.13 odd 2 392.3.g.h.99.2 4
56.27 even 2 1568.3.g.h.687.1 4
56.37 even 6 392.3.k.i.67.4 8
56.45 odd 6 392.3.k.j.275.2 8
56.53 even 6 392.3.k.i.275.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.g.a.43.1 4 4.3 odd 2
56.3.g.a.43.2 yes 4 8.5 even 2
224.3.g.a.15.3 4 1.1 even 1 trivial
224.3.g.a.15.4 4 8.3 odd 2 inner
392.3.g.h.99.1 4 28.27 even 2
392.3.g.h.99.2 4 56.13 odd 2
392.3.k.i.67.2 8 28.23 odd 6
392.3.k.i.67.4 8 56.37 even 6
392.3.k.i.275.2 8 56.53 even 6
392.3.k.i.275.4 8 28.11 odd 6
392.3.k.j.67.2 8 28.19 even 6
392.3.k.j.67.4 8 56.5 odd 6
392.3.k.j.275.2 8 56.45 odd 6
392.3.k.j.275.4 8 28.3 even 6
504.3.g.a.379.3 4 24.5 odd 2
504.3.g.a.379.4 4 12.11 even 2
1568.3.g.h.687.1 4 56.27 even 2
1568.3.g.h.687.2 4 7.6 odd 2
1792.3.d.g.1023.3 8 16.13 even 4
1792.3.d.g.1023.4 8 16.11 odd 4
1792.3.d.g.1023.5 8 16.3 odd 4
1792.3.d.g.1023.6 8 16.5 even 4
2016.3.g.a.1135.1 4 24.11 even 2
2016.3.g.a.1135.4 4 3.2 odd 2