Properties

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

$q$-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 \(0.358875\pi\)
−0.428976 + 0.903316i \(0.641125\pi\)
\(6\) 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 \(-0.887055\pi\)
0.937706 0.347429i \(-0.112945\pi\)
\(14\) 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 \(0.179949\pi\)
−0.844414 + 0.535690i \(0.820051\pi\)
\(24\) 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 \(0.126512\pi\)
−0.922051 + 0.387068i \(0.873488\pi\)
\(30\) 0 0
\(31\) 16.7824i 0.541367i 0.962668 + 0.270684i \(0.0872497\pi\)
−0.962668 + 0.270684i \(0.912750\pi\)
\(32\) 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 \(0.0704759\pi\)
−0.975589 + 0.219602i \(0.929524\pi\)
\(38\) 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 \(0.0210081\pi\)
−0.997823 + 0.0659509i \(0.978992\pi\)
\(48\) 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 \(0.0240918\pi\)
−0.997137 + 0.0756145i \(0.975908\pi\)
\(54\) 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 \(-0.960153\pi\)
0.992175 0.124857i \(-0.0398472\pi\)
\(62\) 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 \(0.0394062\pi\)
−0.992347 + 0.123482i \(0.960594\pi\)
\(72\) 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 \(0.171416\pi\)
−0.858470 + 0.512864i \(0.828584\pi\)
\(80\) 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 \(0.0191304\pi\)
−0.998195 + 0.0600636i \(0.980870\pi\)
\(102\) 0 0
\(103\) 106.582i 1.03478i 0.855750 + 0.517389i \(0.173096\pi\)
−0.855750 + 0.517389i \(0.826904\pi\)
\(104\) 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 \(-0.795080\pi\)
0.799835 0.600220i \(-0.204920\pi\)
\(110\) 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 \(0.148775\pi\)
−0.892748 + 0.450557i \(0.851225\pi\)
\(128\) 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 \(0.321133\pi\)
−0.532819 + 0.846229i \(0.678867\pi\)
\(150\) 0 0
\(151\) 234.486i 1.55289i 0.630186 + 0.776444i \(0.282979\pi\)
−0.630186 + 0.776444i \(0.717021\pi\)
\(152\) 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 \(-0.989763\pi\)
0.999483 0.0321554i \(-0.0102372\pi\)
\(158\) 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 \(-0.651736\pi\)
0.458843 0.888517i \(-0.348264\pi\)
\(168\) 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 \(-0.963091\pi\)
0.993285 0.115694i \(-0.0369091\pi\)
\(174\) 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 \(-0.964339\pi\)
0.993731 0.111797i \(-0.0356608\pi\)
\(182\) 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 \(-0.865246\pi\)
0.911721 0.410810i \(-0.134754\pi\)
\(192\) 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 \(0.120210\pi\)
−0.929533 + 0.368738i \(0.879790\pi\)
\(198\) 0 0
\(199\) − 390.508i − 1.96235i −0.193122 0.981175i \(-0.561861\pi\)
0.193122 0.981175i \(-0.438139\pi\)
\(200\) 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 \(-0.992446\pi\)
0.999718 0.0237287i \(-0.00755379\pi\)
\(224\) 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 \(0.169102\pi\)
−0.862174 + 0.506612i \(0.830898\pi\)
\(230\) 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 \(0.240644\pi\)
−0.727583 + 0.686020i \(0.759356\pi\)
\(240\) 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 \(0.254672\pi\)
−0.696653 + 0.717408i \(0.745328\pi\)
\(264\) 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 \(0.0166704\pi\)
−0.998629 + 0.0523478i \(0.983330\pi\)
\(270\) 0 0
\(271\) − 399.715i − 1.47496i −0.675367 0.737482i \(-0.736015\pi\)
0.675367 0.737482i \(-0.263985\pi\)
\(272\) 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 \(-0.940502\pi\)
0.982582 0.185831i \(-0.0594979\pi\)
\(278\) 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 \(-0.877971\pi\)
0.927411 0.374043i \(-0.122029\pi\)
\(294\) 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 \(-0.962934\pi\)
0.993228 0.116182i \(-0.0370656\pi\)
\(312\) 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 \(-0.944925\pi\)
0.985069 0.172161i \(-0.0550749\pi\)
\(318\) 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 \(0.122678\pi\)
−0.926646 + 0.375934i \(0.877322\pi\)
\(350\) 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 \(-0.830195\pi\)
0.861053 0.508515i \(-0.169805\pi\)
\(360\) 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 \(0.250648\pi\)
−0.705666 + 0.708544i \(0.749352\pi\)
\(368\) 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 \(0.249524\pi\)
−0.708164 + 0.706048i \(0.750476\pi\)
\(374\) 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 \(-0.864450\pi\)
0.910690 0.413089i \(-0.135550\pi\)
\(384\) 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 \(-0.962210\pi\)
0.992961 0.118441i \(-0.0377898\pi\)
\(390\) 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 \(-0.749533\pi\)
0.706068 0.708144i \(-0.250467\pi\)
\(398\) 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 \(-0.765103\pi\)
0.739848 0.672774i \(-0.234897\pi\)
\(422\) 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 \(0.109034\pi\)
−0.941905 + 0.335880i \(0.890966\pi\)
\(432\) 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 \(-0.986078\pi\)
0.999044 0.0437232i \(-0.0139220\pi\)
\(440\) 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 \(0.310341\pi\)
−0.561196 + 0.827683i \(0.689659\pi\)
\(462\) 0 0
\(463\) 123.988i 0.267792i 0.990995 + 0.133896i \(0.0427488\pi\)
−0.990995 + 0.133896i \(0.957251\pi\)
\(464\) 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 \(-0.960364\pi\)
0.992258 0.124197i \(-0.0396355\pi\)
\(480\) 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 \(-0.906305\pi\)
0.956990 0.290120i \(-0.0936952\pi\)
\(488\) 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 \(0.257301\pi\)
−0.690703 + 0.723138i \(0.742699\pi\)
\(504\) 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 \(0.214074\pi\)
−0.782246 + 0.622969i \(0.785926\pi\)
\(510\) 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 \(-0.877590\pi\)
0.926963 0.375152i \(-0.122410\pi\)
\(542\) 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 \(0.0104153\pi\)
−0.999465 + 0.0327147i \(0.989585\pi\)
\(558\) 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 \(-0.958754\pi\)
0.991616 0.129217i \(-0.0412464\pi\)
\(600\) 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 \(-0.774229\pi\)
0.758831 0.651288i \(-0.225771\pi\)
\(608\) 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 \(0.206743\pi\)
−0.796386 + 0.604789i \(0.793257\pi\)
\(614\) 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 \(-0.931523\pi\)
0.976950 0.213470i \(-0.0684766\pi\)
\(632\) 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 \(0.317117\pi\)
−0.543452 + 0.839440i \(0.682883\pi\)
\(648\) 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 \(0.368026\pi\)
−0.402832 + 0.915274i \(0.631974\pi\)
\(654\) 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 \(-0.729368\pi\)
0.659820 0.751423i \(-0.270632\pi\)
\(662\) 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 \(0.265953\pi\)
−0.670795 + 0.741643i \(0.734047\pi\)
\(678\) 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 \(0.233997\pi\)
−0.741748 + 0.670679i \(0.766003\pi\)
\(702\) 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 \(-0.732097\pi\)
0.666239 0.745738i \(-0.267903\pi\)
\(710\) 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 \(0.255688\pi\)
−0.694360 + 0.719628i \(0.744312\pi\)
\(720\) 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 \(0.110610\pi\)
−0.940230 + 0.340540i \(0.889390\pi\)
\(728\) 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 \(-0.873476\pi\)
0.922036 0.387103i \(-0.126524\pi\)
\(734\) 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 \(0.163771\pi\)
−0.870537 + 0.492102i \(0.836229\pi\)
\(744\) 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 \(0.146293\pi\)
−0.896233 + 0.443583i \(0.853707\pi\)
\(752\) 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 \(0.0503791\pi\)
−0.987501 + 0.157611i \(0.949621\pi\)
\(758\) 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 \(0.133965\pi\)
−0.912737 + 0.408548i \(0.866035\pi\)
\(774\) 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 \(-0.956216\pi\)
0.990555 0.137118i \(-0.0437839\pi\)
\(798\) 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 \(0.250412\pi\)
−0.706190 + 0.708022i \(0.749588\pi\)
\(822\) 0 0
\(823\) − 1041.65i − 1.26567i −0.774286 0.632835i \(-0.781891\pi\)
0.774286 0.632835i \(-0.218109\pi\)
\(824\) 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 \(0.222221\pi\)
−0.766046 + 0.642785i \(0.777779\pi\)
\(830\) 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 \(0.0583434\pi\)
−0.983249 + 0.182267i \(0.941657\pi\)
\(840\) 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 \(0.0306502\pi\)
−0.995368 + 0.0961417i \(0.969350\pi\)
\(854\) 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 \(0.0516890\pi\)
−0.986844 + 0.161673i \(0.948311\pi\)
\(864\) 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 \(0.125576\pi\)
−0.923185 + 0.384355i \(0.874424\pi\)
\(878\) 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 \(0.00127047\pi\)
−0.999992 + 0.00399128i \(0.998730\pi\)
\(888\) 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 \(0.0354376\pi\)
−0.993809 + 0.111101i \(0.964562\pi\)
\(912\) 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 \(0.334017\pi\)
−0.498138 + 0.867098i \(0.665983\pi\)
\(920\) 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 \(-0.830917\pi\)
0.862205 0.506559i \(-0.169083\pi\)
\(942\) 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 \(-0.739864\pi\)
0.684235 0.729262i \(-0.260136\pi\)
\(968\) 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 \(-0.972530\pi\)
0.996278 0.0861939i \(-0.0274704\pi\)
\(984\) 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 \(0.324070\pi\)
−0.524988 + 0.851109i \(0.675930\pi\)
\(992\) 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 \(0.336321\pi\)
−0.491849 + 0.870680i \(0.663679\pi\)
\(998\) 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.15.4 4
3.2 odd 2 2016.3.g.a.1135.1 4
4.3 odd 2 56.3.g.a.43.2 yes 4
7.6 odd 2 1568.3.g.h.687.1 4
8.3 odd 2 inner 224.3.g.a.15.3 4
8.5 even 2 56.3.g.a.43.1 4
12.11 even 2 504.3.g.a.379.3 4
16.3 odd 4 1792.3.d.g.1023.6 8
16.5 even 4 1792.3.d.g.1023.5 8
16.11 odd 4 1792.3.d.g.1023.3 8
16.13 even 4 1792.3.d.g.1023.4 8
24.5 odd 2 504.3.g.a.379.4 4
24.11 even 2 2016.3.g.a.1135.4 4
28.3 even 6 392.3.k.j.275.2 8
28.11 odd 6 392.3.k.i.275.2 8
28.19 even 6 392.3.k.j.67.4 8
28.23 odd 6 392.3.k.i.67.4 8
28.27 even 2 392.3.g.h.99.2 4
56.5 odd 6 392.3.k.j.67.2 8
56.13 odd 2 392.3.g.h.99.1 4
56.27 even 2 1568.3.g.h.687.2 4
56.37 even 6 392.3.k.i.67.2 8
56.45 odd 6 392.3.k.j.275.4 8
56.53 even 6 392.3.k.i.275.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.g.a.43.1 4 8.5 even 2
56.3.g.a.43.2 yes 4 4.3 odd 2
224.3.g.a.15.3 4 8.3 odd 2 inner
224.3.g.a.15.4 4 1.1 even 1 trivial
392.3.g.h.99.1 4 56.13 odd 2
392.3.g.h.99.2 4 28.27 even 2
392.3.k.i.67.2 8 56.37 even 6
392.3.k.i.67.4 8 28.23 odd 6
392.3.k.i.275.2 8 28.11 odd 6
392.3.k.i.275.4 8 56.53 even 6
392.3.k.j.67.2 8 56.5 odd 6
392.3.k.j.67.4 8 28.19 even 6
392.3.k.j.275.2 8 28.3 even 6
392.3.k.j.275.4 8 56.45 odd 6
504.3.g.a.379.3 4 12.11 even 2
504.3.g.a.379.4 4 24.5 odd 2
1568.3.g.h.687.1 4 7.6 odd 2
1568.3.g.h.687.2 4 56.27 even 2
1792.3.d.g.1023.3 8 16.11 odd 4
1792.3.d.g.1023.4 8 16.13 even 4
1792.3.d.g.1023.5 8 16.5 even 4
1792.3.d.g.1023.6 8 16.3 odd 4
2016.3.g.a.1135.1 4 3.2 odd 2
2016.3.g.a.1135.4 4 24.11 even 2