Properties

Label 2352.3.m.f.1471.4
Level $2352$
Weight $3$
Character 2352.1471
Analytic conductor $64.087$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
Defining polynomial: \(x^{4} - x^{3} - x^{2} - 2 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.4
Root \(1.39564 - 0.228425i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1471
Dual form 2352.3.m.f.1471.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205i q^{3} +3.58258 q^{5} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{3} +3.58258 q^{5} -3.00000 q^{9} +0.913701i q^{11} +1.16515 q^{13} +6.20520i q^{15} +26.7477 q^{17} -17.5112i q^{19} +27.1805i q^{23} -12.1652 q^{25} -5.19615i q^{27} -2.00000 q^{29} -45.6054i q^{31} -1.58258 q^{33} +47.4955 q^{37} +2.01810i q^{39} +42.5735 q^{41} +14.6192i q^{43} -10.7477 q^{45} +8.37420i q^{47} +46.3284i q^{51} -41.8258 q^{53} +3.27340i q^{55} +30.3303 q^{57} +27.0296i q^{59} +11.0091 q^{61} +4.17424 q^{65} -71.8722i q^{67} -47.0780 q^{69} -55.6561i q^{71} +95.4955 q^{73} -21.0707i q^{75} +63.7998i q^{79} +9.00000 q^{81} +32.5118i q^{83} +95.8258 q^{85} -3.46410i q^{87} +120.904 q^{89} +78.9909 q^{93} -62.7352i q^{95} -107.495 q^{97} -2.74110i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} - 12q^{9} + O(q^{10}) \) \( 4q - 4q^{5} - 12q^{9} - 32q^{13} + 52q^{17} - 12q^{25} - 8q^{29} + 12q^{33} + 80q^{37} - 68q^{41} + 12q^{45} + 16q^{53} + 48q^{57} + 264q^{61} + 200q^{65} - 60q^{69} + 272q^{73} + 36q^{81} + 200q^{85} + 172q^{89} + 96q^{93} - 320q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 1.73205i 0.577350i
\(4\) 0 0
\(5\) 3.58258 0.716515 0.358258 0.933623i \(-0.383371\pi\)
0.358258 + 0.933623i \(0.383371\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) 0.913701i 0.0830637i 0.999137 + 0.0415318i \(0.0132238\pi\)
−0.999137 + 0.0415318i \(0.986776\pi\)
\(12\) 0 0
\(13\) 1.16515 0.0896270 0.0448135 0.998995i \(-0.485731\pi\)
0.0448135 + 0.998995i \(0.485731\pi\)
\(14\) 0 0
\(15\) 6.20520i 0.413680i
\(16\) 0 0
\(17\) 26.7477 1.57340 0.786698 0.617338i \(-0.211789\pi\)
0.786698 + 0.617338i \(0.211789\pi\)
\(18\) 0 0
\(19\) − 17.5112i − 0.921643i −0.887493 0.460821i \(-0.847555\pi\)
0.887493 0.460821i \(-0.152445\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 27.1805i 1.18176i 0.806759 + 0.590881i \(0.201220\pi\)
−0.806759 + 0.590881i \(0.798780\pi\)
\(24\) 0 0
\(25\) −12.1652 −0.486606
\(26\) 0 0
\(27\) − 5.19615i − 0.192450i
\(28\) 0 0
\(29\) −2.00000 −0.0689655 −0.0344828 0.999405i \(-0.510978\pi\)
−0.0344828 + 0.999405i \(0.510978\pi\)
\(30\) 0 0
\(31\) − 45.6054i − 1.47114i −0.677447 0.735571i \(-0.736914\pi\)
0.677447 0.735571i \(-0.263086\pi\)
\(32\) 0 0
\(33\) −1.58258 −0.0479568
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 47.4955 1.28366 0.641830 0.766847i \(-0.278175\pi\)
0.641830 + 0.766847i \(0.278175\pi\)
\(38\) 0 0
\(39\) 2.01810i 0.0517462i
\(40\) 0 0
\(41\) 42.5735 1.03838 0.519189 0.854660i \(-0.326234\pi\)
0.519189 + 0.854660i \(0.326234\pi\)
\(42\) 0 0
\(43\) 14.6192i 0.339982i 0.985446 + 0.169991i \(0.0543738\pi\)
−0.985446 + 0.169991i \(0.945626\pi\)
\(44\) 0 0
\(45\) −10.7477 −0.238838
\(46\) 0 0
\(47\) 8.37420i 0.178175i 0.996024 + 0.0890873i \(0.0283950\pi\)
−0.996024 + 0.0890873i \(0.971605\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 46.3284i 0.908400i
\(52\) 0 0
\(53\) −41.8258 −0.789165 −0.394583 0.918860i \(-0.629111\pi\)
−0.394583 + 0.918860i \(0.629111\pi\)
\(54\) 0 0
\(55\) 3.27340i 0.0595164i
\(56\) 0 0
\(57\) 30.3303 0.532111
\(58\) 0 0
\(59\) 27.0296i 0.458129i 0.973411 + 0.229065i \(0.0735667\pi\)
−0.973411 + 0.229065i \(0.926433\pi\)
\(60\) 0 0
\(61\) 11.0091 0.180477 0.0902385 0.995920i \(-0.471237\pi\)
0.0902385 + 0.995920i \(0.471237\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.17424 0.0642191
\(66\) 0 0
\(67\) − 71.8722i − 1.07272i −0.843989 0.536360i \(-0.819799\pi\)
0.843989 0.536360i \(-0.180201\pi\)
\(68\) 0 0
\(69\) −47.0780 −0.682290
\(70\) 0 0
\(71\) − 55.6561i − 0.783889i −0.919989 0.391945i \(-0.871803\pi\)
0.919989 0.391945i \(-0.128197\pi\)
\(72\) 0 0
\(73\) 95.4955 1.30816 0.654078 0.756427i \(-0.273057\pi\)
0.654078 + 0.756427i \(0.273057\pi\)
\(74\) 0 0
\(75\) − 21.0707i − 0.280942i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 63.7998i 0.807593i 0.914849 + 0.403796i \(0.132310\pi\)
−0.914849 + 0.403796i \(0.867690\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 32.5118i 0.391709i 0.980633 + 0.195854i \(0.0627480\pi\)
−0.980633 + 0.195854i \(0.937252\pi\)
\(84\) 0 0
\(85\) 95.8258 1.12736
\(86\) 0 0
\(87\) − 3.46410i − 0.0398173i
\(88\) 0 0
\(89\) 120.904 1.35847 0.679235 0.733921i \(-0.262312\pi\)
0.679235 + 0.733921i \(0.262312\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 78.9909 0.849365
\(94\) 0 0
\(95\) − 62.7352i − 0.660371i
\(96\) 0 0
\(97\) −107.495 −1.10820 −0.554100 0.832450i \(-0.686938\pi\)
−0.554100 + 0.832450i \(0.686938\pi\)
\(98\) 0 0
\(99\) − 2.74110i − 0.0276879i
\(100\) 0 0
\(101\) 104.417 1.03384 0.516918 0.856035i \(-0.327079\pi\)
0.516918 + 0.856035i \(0.327079\pi\)
\(102\) 0 0
\(103\) 119.305i 1.15830i 0.815220 + 0.579151i \(0.196616\pi\)
−0.815220 + 0.579151i \(0.803384\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 67.0019i − 0.626186i −0.949722 0.313093i \(-0.898635\pi\)
0.949722 0.313093i \(-0.101365\pi\)
\(108\) 0 0
\(109\) 161.303 1.47984 0.739922 0.672693i \(-0.234862\pi\)
0.739922 + 0.672693i \(0.234862\pi\)
\(110\) 0 0
\(111\) 82.2645i 0.741122i
\(112\) 0 0
\(113\) 32.6788 0.289193 0.144596 0.989491i \(-0.453812\pi\)
0.144596 + 0.989491i \(0.453812\pi\)
\(114\) 0 0
\(115\) 97.3762i 0.846750i
\(116\) 0 0
\(117\) −3.49545 −0.0298757
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 120.165 0.993100
\(122\) 0 0
\(123\) 73.7394i 0.599508i
\(124\) 0 0
\(125\) −133.147 −1.06518
\(126\) 0 0
\(127\) 56.4902i 0.444805i 0.974955 + 0.222402i \(0.0713899\pi\)
−0.974955 + 0.222402i \(0.928610\pi\)
\(128\) 0 0
\(129\) −25.3212 −0.196288
\(130\) 0 0
\(131\) 140.471i 1.07230i 0.844123 + 0.536149i \(0.180122\pi\)
−0.844123 + 0.536149i \(0.819878\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) − 18.6156i − 0.137893i
\(136\) 0 0
\(137\) −228.156 −1.66537 −0.832686 0.553745i \(-0.813198\pi\)
−0.832686 + 0.553745i \(0.813198\pi\)
\(138\) 0 0
\(139\) − 176.256i − 1.26803i −0.773320 0.634015i \(-0.781406\pi\)
0.773320 0.634015i \(-0.218594\pi\)
\(140\) 0 0
\(141\) −14.5045 −0.102869
\(142\) 0 0
\(143\) 1.06460i 0.00744475i
\(144\) 0 0
\(145\) −7.16515 −0.0494148
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 274.835 1.84453 0.922265 0.386559i \(-0.126337\pi\)
0.922265 + 0.386559i \(0.126337\pi\)
\(150\) 0 0
\(151\) − 175.875i − 1.16473i −0.812926 0.582367i \(-0.802127\pi\)
0.812926 0.582367i \(-0.197873\pi\)
\(152\) 0 0
\(153\) −80.2432 −0.524465
\(154\) 0 0
\(155\) − 163.385i − 1.05410i
\(156\) 0 0
\(157\) −55.9818 −0.356572 −0.178286 0.983979i \(-0.557055\pi\)
−0.178286 + 0.983979i \(0.557055\pi\)
\(158\) 0 0
\(159\) − 72.4443i − 0.455625i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 237.386i 1.45636i 0.685387 + 0.728179i \(0.259633\pi\)
−0.685387 + 0.728179i \(0.740367\pi\)
\(164\) 0 0
\(165\) −5.66970 −0.0343618
\(166\) 0 0
\(167\) − 237.386i − 1.42147i −0.703457 0.710737i \(-0.748361\pi\)
0.703457 0.710737i \(-0.251639\pi\)
\(168\) 0 0
\(169\) −167.642 −0.991967
\(170\) 0 0
\(171\) 52.5336i 0.307214i
\(172\) 0 0
\(173\) 173.078 1.00045 0.500226 0.865895i \(-0.333250\pi\)
0.500226 + 0.865895i \(0.333250\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −46.8167 −0.264501
\(178\) 0 0
\(179\) 136.284i 0.761363i 0.924706 + 0.380681i \(0.124311\pi\)
−0.924706 + 0.380681i \(0.875689\pi\)
\(180\) 0 0
\(181\) −80.5045 −0.444776 −0.222388 0.974958i \(-0.571385\pi\)
−0.222388 + 0.974958i \(0.571385\pi\)
\(182\) 0 0
\(183\) 19.0683i 0.104198i
\(184\) 0 0
\(185\) 170.156 0.919762
\(186\) 0 0
\(187\) 24.4394i 0.130692i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 131.183i 0.686823i 0.939185 + 0.343411i \(0.111583\pi\)
−0.939185 + 0.343411i \(0.888417\pi\)
\(192\) 0 0
\(193\) −171.495 −0.888577 −0.444289 0.895884i \(-0.646544\pi\)
−0.444289 + 0.895884i \(0.646544\pi\)
\(194\) 0 0
\(195\) 7.23000i 0.0370769i
\(196\) 0 0
\(197\) 57.1652 0.290178 0.145089 0.989419i \(-0.453653\pi\)
0.145089 + 0.989419i \(0.453653\pi\)
\(198\) 0 0
\(199\) − 6.70601i − 0.0336985i −0.999858 0.0168493i \(-0.994636\pi\)
0.999858 0.0168493i \(-0.00536354\pi\)
\(200\) 0 0
\(201\) 124.486 0.619335
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 152.523 0.744013
\(206\) 0 0
\(207\) − 81.5415i − 0.393920i
\(208\) 0 0
\(209\) 16.0000 0.0765550
\(210\) 0 0
\(211\) 178.767i 0.847236i 0.905841 + 0.423618i \(0.139240\pi\)
−0.905841 + 0.423618i \(0.860760\pi\)
\(212\) 0 0
\(213\) 96.3992 0.452579
\(214\) 0 0
\(215\) 52.3744i 0.243602i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 165.403i 0.755265i
\(220\) 0 0
\(221\) 31.1652 0.141019
\(222\) 0 0
\(223\) 114.506i 0.513480i 0.966480 + 0.256740i \(0.0826484\pi\)
−0.966480 + 0.256740i \(0.917352\pi\)
\(224\) 0 0
\(225\) 36.4955 0.162202
\(226\) 0 0
\(227\) 307.813i 1.35600i 0.735061 + 0.678001i \(0.237154\pi\)
−0.735061 + 0.678001i \(0.762846\pi\)
\(228\) 0 0
\(229\) 356.156 1.55527 0.777633 0.628718i \(-0.216420\pi\)
0.777633 + 0.628718i \(0.216420\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 213.514 0.916368 0.458184 0.888857i \(-0.348500\pi\)
0.458184 + 0.888857i \(0.348500\pi\)
\(234\) 0 0
\(235\) 30.0012i 0.127665i
\(236\) 0 0
\(237\) −110.505 −0.466264
\(238\) 0 0
\(239\) 402.988i 1.68614i 0.537801 + 0.843072i \(0.319255\pi\)
−0.537801 + 0.843072i \(0.680745\pi\)
\(240\) 0 0
\(241\) 344.468 1.42933 0.714664 0.699468i \(-0.246579\pi\)
0.714664 + 0.699468i \(0.246579\pi\)
\(242\) 0 0
\(243\) 15.5885i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 20.4032i − 0.0826041i
\(248\) 0 0
\(249\) −56.3121 −0.226153
\(250\) 0 0
\(251\) − 491.601i − 1.95857i −0.202491 0.979284i \(-0.564904\pi\)
0.202491 0.979284i \(-0.435096\pi\)
\(252\) 0 0
\(253\) −24.8348 −0.0981615
\(254\) 0 0
\(255\) 165.975i 0.650883i
\(256\) 0 0
\(257\) −335.372 −1.30495 −0.652475 0.757811i \(-0.726269\pi\)
−0.652475 + 0.757811i \(0.726269\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.0229885
\(262\) 0 0
\(263\) 484.601i 1.84259i 0.388865 + 0.921295i \(0.372867\pi\)
−0.388865 + 0.921295i \(0.627133\pi\)
\(264\) 0 0
\(265\) −149.844 −0.565449
\(266\) 0 0
\(267\) 209.412i 0.784313i
\(268\) 0 0
\(269\) 27.9311 0.103833 0.0519165 0.998651i \(-0.483467\pi\)
0.0519165 + 0.998651i \(0.483467\pi\)
\(270\) 0 0
\(271\) − 16.5262i − 0.0609823i −0.999535 0.0304912i \(-0.990293\pi\)
0.999535 0.0304912i \(-0.00970714\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 11.1153i − 0.0404193i
\(276\) 0 0
\(277\) −359.459 −1.29769 −0.648843 0.760922i \(-0.724747\pi\)
−0.648843 + 0.760922i \(0.724747\pi\)
\(278\) 0 0
\(279\) 136.816i 0.490381i
\(280\) 0 0
\(281\) 259.147 0.922231 0.461116 0.887340i \(-0.347449\pi\)
0.461116 + 0.887340i \(0.347449\pi\)
\(282\) 0 0
\(283\) 177.241i 0.626294i 0.949705 + 0.313147i \(0.101383\pi\)
−0.949705 + 0.313147i \(0.898617\pi\)
\(284\) 0 0
\(285\) 108.661 0.381265
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 426.441 1.47557
\(290\) 0 0
\(291\) − 186.188i − 0.639820i
\(292\) 0 0
\(293\) 296.069 1.01047 0.505237 0.862981i \(-0.331405\pi\)
0.505237 + 0.862981i \(0.331405\pi\)
\(294\) 0 0
\(295\) 96.8356i 0.328256i
\(296\) 0 0
\(297\) 4.74773 0.0159856
\(298\) 0 0
\(299\) 31.6694i 0.105918i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 180.856i 0.596885i
\(304\) 0 0
\(305\) 39.4409 0.129314
\(306\) 0 0
\(307\) 170.854i 0.556527i 0.960505 + 0.278263i \(0.0897588\pi\)
−0.960505 + 0.278263i \(0.910241\pi\)
\(308\) 0 0
\(309\) −206.642 −0.668746
\(310\) 0 0
\(311\) − 419.967i − 1.35038i −0.737645 0.675188i \(-0.764062\pi\)
0.737645 0.675188i \(-0.235938\pi\)
\(312\) 0 0
\(313\) 238.624 0.762378 0.381189 0.924497i \(-0.375515\pi\)
0.381189 + 0.924497i \(0.375515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 585.441 1.84682 0.923408 0.383819i \(-0.125391\pi\)
0.923408 + 0.383819i \(0.125391\pi\)
\(318\) 0 0
\(319\) − 1.82740i − 0.00572853i
\(320\) 0 0
\(321\) 116.051 0.361529
\(322\) 0 0
\(323\) − 468.385i − 1.45011i
\(324\) 0 0
\(325\) −14.1742 −0.0436131
\(326\) 0 0
\(327\) 279.385i 0.854389i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 71.4112i 0.215744i 0.994165 + 0.107872i \(0.0344037\pi\)
−0.994165 + 0.107872i \(0.965596\pi\)
\(332\) 0 0
\(333\) −142.486 −0.427887
\(334\) 0 0
\(335\) − 257.488i − 0.768620i
\(336\) 0 0
\(337\) 416.955 1.23725 0.618627 0.785685i \(-0.287689\pi\)
0.618627 + 0.785685i \(0.287689\pi\)
\(338\) 0 0
\(339\) 56.6013i 0.166966i
\(340\) 0 0
\(341\) 41.6697 0.122199
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −168.661 −0.488871
\(346\) 0 0
\(347\) − 546.812i − 1.57583i −0.615785 0.787914i \(-0.711161\pi\)
0.615785 0.787914i \(-0.288839\pi\)
\(348\) 0 0
\(349\) 97.6151 0.279700 0.139850 0.990173i \(-0.455338\pi\)
0.139850 + 0.990173i \(0.455338\pi\)
\(350\) 0 0
\(351\) − 6.05430i − 0.0172487i
\(352\) 0 0
\(353\) −315.858 −0.894783 −0.447391 0.894338i \(-0.647647\pi\)
−0.447391 + 0.894338i \(0.647647\pi\)
\(354\) 0 0
\(355\) − 199.392i − 0.561668i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 31.2797i − 0.0871301i −0.999051 0.0435650i \(-0.986128\pi\)
0.999051 0.0435650i \(-0.0138716\pi\)
\(360\) 0 0
\(361\) 54.3576 0.150575
\(362\) 0 0
\(363\) 208.132i 0.573367i
\(364\) 0 0
\(365\) 342.120 0.937314
\(366\) 0 0
\(367\) 501.437i 1.36631i 0.730271 + 0.683157i \(0.239394\pi\)
−0.730271 + 0.683157i \(0.760606\pi\)
\(368\) 0 0
\(369\) −127.720 −0.346126
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −257.267 −0.689723 −0.344861 0.938654i \(-0.612074\pi\)
−0.344861 + 0.938654i \(0.612074\pi\)
\(374\) 0 0
\(375\) − 230.617i − 0.614980i
\(376\) 0 0
\(377\) −2.33030 −0.00618117
\(378\) 0 0
\(379\) − 169.169i − 0.446356i −0.974778 0.223178i \(-0.928357\pi\)
0.974778 0.223178i \(-0.0716431\pi\)
\(380\) 0 0
\(381\) −97.8439 −0.256808
\(382\) 0 0
\(383\) 436.858i 1.14062i 0.821429 + 0.570311i \(0.193177\pi\)
−0.821429 + 0.570311i \(0.806823\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 43.8576i − 0.113327i
\(388\) 0 0
\(389\) −215.982 −0.555223 −0.277612 0.960693i \(-0.589543\pi\)
−0.277612 + 0.960693i \(0.589543\pi\)
\(390\) 0 0
\(391\) 727.017i 1.85938i
\(392\) 0 0
\(393\) −243.303 −0.619092
\(394\) 0 0
\(395\) 228.568i 0.578652i
\(396\) 0 0
\(397\) −256.606 −0.646363 −0.323181 0.946337i \(-0.604752\pi\)
−0.323181 + 0.946337i \(0.604752\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 557.441 1.39013 0.695063 0.718948i \(-0.255376\pi\)
0.695063 + 0.718948i \(0.255376\pi\)
\(402\) 0 0
\(403\) − 53.1372i − 0.131854i
\(404\) 0 0
\(405\) 32.2432 0.0796128
\(406\) 0 0
\(407\) 43.3966i 0.106626i
\(408\) 0 0
\(409\) −16.1561 −0.0395014 −0.0197507 0.999805i \(-0.506287\pi\)
−0.0197507 + 0.999805i \(0.506287\pi\)
\(410\) 0 0
\(411\) − 395.178i − 0.961503i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 116.476i 0.280665i
\(416\) 0 0
\(417\) 305.285 0.732098
\(418\) 0 0
\(419\) − 582.652i − 1.39058i −0.718730 0.695289i \(-0.755276\pi\)
0.718730 0.695289i \(-0.244724\pi\)
\(420\) 0 0
\(421\) −662.762 −1.57426 −0.787128 0.616789i \(-0.788433\pi\)
−0.787128 + 0.616789i \(0.788433\pi\)
\(422\) 0 0
\(423\) − 25.1226i − 0.0593915i
\(424\) 0 0
\(425\) −325.390 −0.765624
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.84394 −0.00429823
\(430\) 0 0
\(431\) 286.336i 0.664354i 0.943217 + 0.332177i \(0.107783\pi\)
−0.943217 + 0.332177i \(0.892217\pi\)
\(432\) 0 0
\(433\) −554.900 −1.28152 −0.640762 0.767739i \(-0.721382\pi\)
−0.640762 + 0.767739i \(0.721382\pi\)
\(434\) 0 0
\(435\) − 12.4104i − 0.0285297i
\(436\) 0 0
\(437\) 475.964 1.08916
\(438\) 0 0
\(439\) 364.843i 0.831078i 0.909575 + 0.415539i \(0.136407\pi\)
−0.909575 + 0.415539i \(0.863593\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 445.241i − 1.00506i −0.864560 0.502529i \(-0.832403\pi\)
0.864560 0.502529i \(-0.167597\pi\)
\(444\) 0 0
\(445\) 433.147 0.973364
\(446\) 0 0
\(447\) 476.028i 1.06494i
\(448\) 0 0
\(449\) 256.955 0.572282 0.286141 0.958188i \(-0.407627\pi\)
0.286141 + 0.958188i \(0.407627\pi\)
\(450\) 0 0
\(451\) 38.8994i 0.0862515i
\(452\) 0 0
\(453\) 304.624 0.672460
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −643.982 −1.40915 −0.704575 0.709629i \(-0.748862\pi\)
−0.704575 + 0.709629i \(0.748862\pi\)
\(458\) 0 0
\(459\) − 138.985i − 0.302800i
\(460\) 0 0
\(461\) −30.8856 −0.0669970 −0.0334985 0.999439i \(-0.510665\pi\)
−0.0334985 + 0.999439i \(0.510665\pi\)
\(462\) 0 0
\(463\) 435.794i 0.941239i 0.882336 + 0.470619i \(0.155970\pi\)
−0.882336 + 0.470619i \(0.844030\pi\)
\(464\) 0 0
\(465\) 282.991 0.608583
\(466\) 0 0
\(467\) 22.0084i 0.0471272i 0.999722 + 0.0235636i \(0.00750123\pi\)
−0.999722 + 0.0235636i \(0.992499\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) − 96.9634i − 0.205867i
\(472\) 0 0
\(473\) −13.3576 −0.0282401
\(474\) 0 0
\(475\) 213.027i 0.448477i
\(476\) 0 0
\(477\) 125.477 0.263055
\(478\) 0 0
\(479\) − 662.677i − 1.38346i −0.722157 0.691729i \(-0.756849\pi\)
0.722157 0.691729i \(-0.243151\pi\)
\(480\) 0 0
\(481\) 55.3394 0.115051
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −385.111 −0.794042
\(486\) 0 0
\(487\) − 24.5024i − 0.0503129i −0.999684 0.0251565i \(-0.991992\pi\)
0.999684 0.0251565i \(-0.00800840\pi\)
\(488\) 0 0
\(489\) −411.165 −0.840829
\(490\) 0 0
\(491\) − 684.136i − 1.39335i −0.717386 0.696676i \(-0.754661\pi\)
0.717386 0.696676i \(-0.245339\pi\)
\(492\) 0 0
\(493\) −53.4955 −0.108510
\(494\) 0 0
\(495\) − 9.82020i − 0.0198388i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 818.768i 1.64082i 0.571777 + 0.820409i \(0.306254\pi\)
−0.571777 + 0.820409i \(0.693746\pi\)
\(500\) 0 0
\(501\) 411.165 0.820689
\(502\) 0 0
\(503\) − 693.298i − 1.37833i −0.724606 0.689163i \(-0.757978\pi\)
0.724606 0.689163i \(-0.242022\pi\)
\(504\) 0 0
\(505\) 374.083 0.740759
\(506\) 0 0
\(507\) − 290.365i − 0.572712i
\(508\) 0 0
\(509\) 87.5462 0.171996 0.0859982 0.996295i \(-0.472592\pi\)
0.0859982 + 0.996295i \(0.472592\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −90.9909 −0.177370
\(514\) 0 0
\(515\) 427.419i 0.829941i
\(516\) 0 0
\(517\) −7.65151 −0.0147998
\(518\) 0 0
\(519\) 299.780i 0.577611i
\(520\) 0 0
\(521\) −190.573 −0.365784 −0.182892 0.983133i \(-0.558546\pi\)
−0.182892 + 0.983133i \(0.558546\pi\)
\(522\) 0 0
\(523\) − 767.379i − 1.46726i −0.679547 0.733632i \(-0.737824\pi\)
0.679547 0.733632i \(-0.262176\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1219.84i − 2.31469i
\(528\) 0 0
\(529\) −209.780 −0.396560
\(530\) 0 0
\(531\) − 81.0888i − 0.152710i
\(532\) 0 0
\(533\) 49.6046 0.0930667
\(534\) 0 0
\(535\) − 240.040i − 0.448672i
\(536\) 0 0
\(537\) −236.051 −0.439573
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −491.495 −0.908494 −0.454247 0.890876i \(-0.650092\pi\)
−0.454247 + 0.890876i \(0.650092\pi\)
\(542\) 0 0
\(543\) − 139.438i − 0.256792i
\(544\) 0 0
\(545\) 577.880 1.06033
\(546\) 0 0
\(547\) − 1043.66i − 1.90798i −0.299839 0.953990i \(-0.596933\pi\)
0.299839 0.953990i \(-0.403067\pi\)
\(548\) 0 0
\(549\) −33.0273 −0.0601590
\(550\) 0 0
\(551\) 35.0224i 0.0635616i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 294.719i 0.531025i
\(556\) 0 0
\(557\) 149.826 0.268987 0.134493 0.990914i \(-0.457059\pi\)
0.134493 + 0.990914i \(0.457059\pi\)
\(558\) 0 0
\(559\) 17.0336i 0.0304715i
\(560\) 0 0
\(561\) −42.3303 −0.0754551
\(562\) 0 0
\(563\) − 863.309i − 1.53341i −0.642000 0.766704i \(-0.721895\pi\)
0.642000 0.766704i \(-0.278105\pi\)
\(564\) 0 0
\(565\) 117.074 0.207211
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −765.056 −1.34456 −0.672281 0.740296i \(-0.734685\pi\)
−0.672281 + 0.740296i \(0.734685\pi\)
\(570\) 0 0
\(571\) − 124.850i − 0.218652i −0.994006 0.109326i \(-0.965131\pi\)
0.994006 0.109326i \(-0.0348692\pi\)
\(572\) 0 0
\(573\) −227.216 −0.396537
\(574\) 0 0
\(575\) − 330.655i − 0.575052i
\(576\) 0 0
\(577\) −930.900 −1.61334 −0.806672 0.590999i \(-0.798734\pi\)
−0.806672 + 0.590999i \(0.798734\pi\)
\(578\) 0 0
\(579\) − 297.039i − 0.513020i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 38.2162i − 0.0655510i
\(584\) 0 0
\(585\) −12.5227 −0.0214064
\(586\) 0 0
\(587\) − 333.681i − 0.568452i −0.958757 0.284226i \(-0.908263\pi\)
0.958757 0.284226i \(-0.0917366\pi\)
\(588\) 0 0
\(589\) −798.606 −1.35587
\(590\) 0 0
\(591\) 99.0129i 0.167535i
\(592\) 0 0
\(593\) −244.977 −0.413114 −0.206557 0.978435i \(-0.566226\pi\)
−0.206557 + 0.978435i \(0.566226\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11.6151 0.0194559
\(598\) 0 0
\(599\) 597.598i 0.997660i 0.866700 + 0.498830i \(0.166237\pi\)
−0.866700 + 0.498830i \(0.833763\pi\)
\(600\) 0 0
\(601\) 236.955 0.394267 0.197134 0.980377i \(-0.436837\pi\)
0.197134 + 0.980377i \(0.436837\pi\)
\(602\) 0 0
\(603\) 215.617i 0.357573i
\(604\) 0 0
\(605\) 430.501 0.711571
\(606\) 0 0
\(607\) − 579.236i − 0.954261i −0.878833 0.477130i \(-0.841677\pi\)
0.878833 0.477130i \(-0.158323\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.75721i 0.0159693i
\(612\) 0 0
\(613\) 10.0000 0.0163132 0.00815661 0.999967i \(-0.497404\pi\)
0.00815661 + 0.999967i \(0.497404\pi\)
\(614\) 0 0
\(615\) 264.177i 0.429556i
\(616\) 0 0
\(617\) −1160.32 −1.88059 −0.940294 0.340364i \(-0.889450\pi\)
−0.940294 + 0.340364i \(0.889450\pi\)
\(618\) 0 0
\(619\) − 516.279i − 0.834053i −0.908894 0.417027i \(-0.863072\pi\)
0.908894 0.417027i \(-0.136928\pi\)
\(620\) 0 0
\(621\) 141.234 0.227430
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −172.880 −0.276608
\(626\) 0 0
\(627\) 27.7128i 0.0441991i
\(628\) 0 0
\(629\) 1270.40 2.01971
\(630\) 0 0
\(631\) 1212.67i 1.92183i 0.276845 + 0.960915i \(0.410711\pi\)
−0.276845 + 0.960915i \(0.589289\pi\)
\(632\) 0 0
\(633\) −309.633 −0.489152
\(634\) 0 0
\(635\) 202.381i 0.318709i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 166.968i 0.261296i
\(640\) 0 0
\(641\) −40.1197 −0.0625892 −0.0312946 0.999510i \(-0.509963\pi\)
−0.0312946 + 0.999510i \(0.509963\pi\)
\(642\) 0 0
\(643\) 198.852i 0.309256i 0.987973 + 0.154628i \(0.0494179\pi\)
−0.987973 + 0.154628i \(0.950582\pi\)
\(644\) 0 0
\(645\) −90.7152 −0.140644
\(646\) 0 0
\(647\) − 211.484i − 0.326869i −0.986554 0.163435i \(-0.947743\pi\)
0.986554 0.163435i \(-0.0522573\pi\)
\(648\) 0 0
\(649\) −24.6970 −0.0380539
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −994.000 −1.52221 −0.761103 0.648632i \(-0.775342\pi\)
−0.761103 + 0.648632i \(0.775342\pi\)
\(654\) 0 0
\(655\) 503.248i 0.768318i
\(656\) 0 0
\(657\) −286.486 −0.436052
\(658\) 0 0
\(659\) − 900.356i − 1.36625i −0.730303 0.683123i \(-0.760621\pi\)
0.730303 0.683123i \(-0.239379\pi\)
\(660\) 0 0
\(661\) −683.945 −1.03471 −0.517357 0.855770i \(-0.673084\pi\)
−0.517357 + 0.855770i \(0.673084\pi\)
\(662\) 0 0
\(663\) 53.9796i 0.0814172i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 54.3610i − 0.0815008i
\(668\) 0 0
\(669\) −198.330 −0.296458
\(670\) 0 0
\(671\) 10.0590i 0.0149911i
\(672\) 0 0
\(673\) 122.211 0.181591 0.0907954 0.995870i \(-0.471059\pi\)
0.0907954 + 0.995870i \(0.471059\pi\)
\(674\) 0 0
\(675\) 63.2120i 0.0936474i
\(676\) 0 0
\(677\) −191.931 −0.283502 −0.141751 0.989902i \(-0.545273\pi\)
−0.141751 + 0.989902i \(0.545273\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −533.147 −0.782888
\(682\) 0 0
\(683\) − 303.340i − 0.444129i −0.975032 0.222065i \(-0.928720\pi\)
0.975032 0.222065i \(-0.0712796\pi\)
\(684\) 0 0
\(685\) −817.386 −1.19326
\(686\) 0 0
\(687\) 616.880i 0.897934i
\(688\) 0 0
\(689\) −48.7333 −0.0707305
\(690\) 0 0
\(691\) − 329.725i − 0.477170i −0.971122 0.238585i \(-0.923316\pi\)
0.971122 0.238585i \(-0.0766836\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 631.451i − 0.908563i
\(696\) 0 0
\(697\) 1138.74 1.63378
\(698\) 0 0
\(699\) 369.816i 0.529065i
\(700\) 0 0
\(701\) 712.918 1.01700 0.508501 0.861061i \(-0.330200\pi\)
0.508501 + 0.861061i \(0.330200\pi\)
\(702\) 0 0
\(703\) − 831.703i − 1.18308i
\(704\) 0 0
\(705\) −51.9636 −0.0737073
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −975.248 −1.37553 −0.687763 0.725935i \(-0.741407\pi\)
−0.687763 + 0.725935i \(0.741407\pi\)
\(710\) 0 0
\(711\) − 191.399i − 0.269198i
\(712\) 0 0
\(713\) 1239.58 1.73854
\(714\) 0 0
\(715\) 3.81401i 0.00533428i
\(716\) 0 0
\(717\) −697.996 −0.973495
\(718\) 0 0
\(719\) 30.2864i 0.0421230i 0.999778 + 0.0210615i \(0.00670457\pi\)
−0.999778 + 0.0210615i \(0.993295\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 596.636i 0.825223i
\(724\) 0 0
\(725\) 24.3303 0.0335590
\(726\) 0 0
\(727\) 1292.40i 1.77771i 0.458186 + 0.888856i \(0.348499\pi\)
−0.458186 + 0.888856i \(0.651501\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) 391.031i 0.534926i
\(732\) 0 0
\(733\) −1053.37 −1.43706 −0.718532 0.695494i \(-0.755186\pi\)
−0.718532 + 0.695494i \(0.755186\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 65.6697 0.0891041
\(738\) 0 0
\(739\) 167.341i 0.226443i 0.993570 + 0.113222i \(0.0361170\pi\)
−0.993570 + 0.113222i \(0.963883\pi\)
\(740\) 0 0
\(741\) 35.3394 0.0476915
\(742\) 0 0
\(743\) 618.001i 0.831765i 0.909418 + 0.415883i \(0.136527\pi\)
−0.909418 + 0.415883i \(0.863473\pi\)
\(744\) 0 0
\(745\) 984.617 1.32163
\(746\) 0 0
\(747\) − 97.5355i − 0.130570i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 1215.11i − 1.61798i −0.587820 0.808992i \(-0.700014\pi\)
0.587820 0.808992i \(-0.299986\pi\)
\(752\) 0 0
\(753\) 851.477 1.13078
\(754\) 0 0
\(755\) − 630.085i − 0.834550i
\(756\) 0 0
\(757\) 1011.80 1.33659 0.668296 0.743896i \(-0.267024\pi\)
0.668296 + 0.743896i \(0.267024\pi\)
\(758\) 0 0
\(759\) − 43.0152i − 0.0566735i
\(760\) 0 0
\(761\) −209.042 −0.274693 −0.137347 0.990523i \(-0.543857\pi\)
−0.137347 + 0.990523i \(0.543857\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −287.477 −0.375787
\(766\) 0 0
\(767\) 31.4936i 0.0410607i
\(768\) 0 0
\(769\) −7.00909 −0.00911455 −0.00455728 0.999990i \(-0.501451\pi\)
−0.00455728 + 0.999990i \(0.501451\pi\)
\(770\) 0 0
\(771\) − 580.881i − 0.753413i
\(772\) 0 0
\(773\) −4.59167 −0.00594006 −0.00297003 0.999996i \(-0.500945\pi\)
−0.00297003 + 0.999996i \(0.500945\pi\)
\(774\) 0 0
\(775\) 554.797i 0.715867i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 745.513i − 0.957013i
\(780\) 0 0
\(781\) 50.8530 0.0651127
\(782\) 0 0
\(783\) 10.3923i 0.0132724i
\(784\) 0 0
\(785\) −200.559 −0.255489
\(786\) 0 0
\(787\) 732.642i 0.930930i 0.885066 + 0.465465i \(0.154113\pi\)
−0.885066 + 0.465465i \(0.845887\pi\)
\(788\) 0 0
\(789\) −839.354 −1.06382
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 12.8273 0.0161756
\(794\) 0 0
\(795\) − 259.537i − 0.326462i
\(796\) 0 0
\(797\) −78.8856 −0.0989782 −0.0494891 0.998775i \(-0.515759\pi\)
−0.0494891 + 0.998775i \(0.515759\pi\)
\(798\) 0 0
\(799\) 223.991i 0.280339i
\(800\) 0 0
\(801\) −362.711 −0.452823
\(802\) 0 0
\(803\) 87.2542i 0.108660i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 48.3780i 0.0599480i
\(808\) 0 0
\(809\) −1022.83 −1.26431 −0.632155 0.774842i \(-0.717830\pi\)
−0.632155 + 0.774842i \(0.717830\pi\)
\(810\) 0 0
\(811\) − 73.3182i − 0.0904047i −0.998978 0.0452024i \(-0.985607\pi\)
0.998978 0.0452024i \(-0.0143933\pi\)
\(812\) 0 0
\(813\) 28.6242 0.0352082
\(814\) 0 0
\(815\) 850.454i 1.04350i
\(816\) 0 0
\(817\) 256.000 0.313341
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 336.395 0.409739 0.204869 0.978789i \(-0.434323\pi\)
0.204869 + 0.978789i \(0.434323\pi\)
\(822\) 0 0
\(823\) − 333.937i − 0.405755i −0.979204 0.202878i \(-0.934971\pi\)
0.979204 0.202878i \(-0.0650294\pi\)
\(824\) 0 0
\(825\) 19.2523 0.0233361
\(826\) 0 0
\(827\) − 1652.01i − 1.99759i −0.0490993 0.998794i \(-0.515635\pi\)
0.0490993 0.998794i \(-0.484365\pi\)
\(828\) 0 0
\(829\) −1011.07 −1.21963 −0.609816 0.792543i \(-0.708756\pi\)
−0.609816 + 0.792543i \(0.708756\pi\)
\(830\) 0 0
\(831\) − 622.601i − 0.749219i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 850.454i − 1.01851i
\(836\) 0 0
\(837\) −236.973 −0.283122
\(838\) 0 0
\(839\) − 337.718i − 0.402524i −0.979537 0.201262i \(-0.935496\pi\)
0.979537 0.201262i \(-0.0645042\pi\)
\(840\) 0 0
\(841\) −837.000 −0.995244
\(842\) 0 0
\(843\) 448.856i 0.532450i
\(844\) 0 0
\(845\) −600.592 −0.710759
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −306.991 −0.361591
\(850\) 0 0
\(851\) 1290.95i 1.51698i
\(852\) 0 0
\(853\) 1285.51 1.50704 0.753521 0.657424i \(-0.228354\pi\)
0.753521 + 0.657424i \(0.228354\pi\)
\(854\) 0 0
\(855\) 188.206i 0.220124i
\(856\) 0 0
\(857\) −1169.04 −1.36411 −0.682055 0.731301i \(-0.738913\pi\)
−0.682055 + 0.731301i \(0.738913\pi\)
\(858\) 0 0
\(859\) − 1283.62i − 1.49432i −0.664642 0.747162i \(-0.731416\pi\)
0.664642 0.747162i \(-0.268584\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 1019.52i − 1.18137i −0.806904 0.590683i \(-0.798858\pi\)
0.806904 0.590683i \(-0.201142\pi\)
\(864\) 0 0
\(865\) 620.065 0.716838
\(866\) 0 0
\(867\) 738.617i 0.851923i
\(868\) 0 0
\(869\) −58.2939 −0.0670816
\(870\) 0 0
\(871\) − 83.7420i − 0.0961447i
\(872\) 0 0
\(873\) 322.486 0.369400
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 301.194 0.343437 0.171718 0.985146i \(-0.445068\pi\)
0.171718 + 0.985146i \(0.445068\pi\)
\(878\) 0 0
\(879\) 512.806i 0.583398i
\(880\) 0 0
\(881\) 1530.67 1.73743 0.868715 0.495313i \(-0.164947\pi\)
0.868715 + 0.495313i \(0.164947\pi\)
\(882\) 0 0
\(883\) − 319.301i − 0.361609i −0.983519 0.180805i \(-0.942130\pi\)
0.983519 0.180805i \(-0.0578702\pi\)
\(884\) 0 0
\(885\) −167.724 −0.189519
\(886\) 0 0
\(887\) 1315.17i 1.48271i 0.671110 + 0.741357i \(0.265818\pi\)
−0.671110 + 0.741357i \(0.734182\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 8.22330i 0.00922930i
\(892\) 0 0
\(893\) 146.642 0.164213
\(894\) 0 0
\(895\) 488.248i 0.545528i
\(896\) 0 0
\(897\) −54.8530 −0.0611517
\(898\) 0 0
\(899\) 91.2108i 0.101458i
\(900\) 0 0
\(901\) −1118.74 −1.24167
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −288.414 −0.318689
\(906\) 0 0
\(907\) − 1585.93i − 1.74854i −0.485441 0.874270i \(-0.661341\pi\)
0.485441 0.874270i \(-0.338659\pi\)
\(908\) 0 0
\(909\) −313.252 −0.344612
\(910\) 0 0
\(911\) 350.279i 0.384499i 0.981346 + 0.192250i \(0.0615783\pi\)
−0.981346 + 0.192250i \(0.938422\pi\)
\(912\) 0 0
\(913\) −29.7061 −0.0325368
\(914\) 0 0
\(915\) 68.3136i 0.0746597i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 431.552i 0.469588i 0.972045 + 0.234794i \(0.0754416\pi\)
−0.972045 + 0.234794i \(0.924558\pi\)
\(920\) 0 0
\(921\) −295.927 −0.321311
\(922\) 0 0
\(923\) − 64.8478i − 0.0702577i
\(924\) 0 0
\(925\) −577.789 −0.624637
\(926\) 0 0
\(927\) − 357.915i − 0.386101i
\(928\) 0 0
\(929\) −620.802 −0.668248 −0.334124 0.942529i \(-0.608440\pi\)
−0.334124 + 0.942529i \(0.608440\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 727.405 0.779640
\(934\) 0 0
\(935\) 87.5560i 0.0936428i
\(936\) 0 0
\(937\) 298.936 0.319036 0.159518 0.987195i \(-0.449006\pi\)
0.159518 + 0.987195i \(0.449006\pi\)
\(938\) 0 0
\(939\) 413.309i 0.440159i
\(940\) 0 0
\(941\) −786.951 −0.836292 −0.418146 0.908380i \(-0.637320\pi\)
−0.418146 + 0.908380i \(0.637320\pi\)
\(942\) 0 0
\(943\) 1157.17i 1.22711i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 629.775i − 0.665021i −0.943100 0.332511i \(-0.892104\pi\)
0.943100 0.332511i \(-0.107896\pi\)
\(948\) 0 0
\(949\) 111.267 0.117246
\(950\) 0 0
\(951\) 1014.01i 1.06626i
\(952\) 0 0
\(953\) 465.855 0.488830 0.244415 0.969671i \(-0.421404\pi\)
0.244415 + 0.969671i \(0.421404\pi\)
\(954\) 0 0
\(955\) 469.974i 0.492119i
\(956\) 0 0
\(957\) 3.16515 0.00330737
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −1118.85 −1.16426
\(962\) 0 0
\(963\) 201.006i 0.208729i
\(964\) 0 0
\(965\) −614.395 −0.636679
\(966\) 0 0
\(967\) 362.538i 0.374910i 0.982273 + 0.187455i \(0.0600239\pi\)
−0.982273 + 0.187455i \(0.939976\pi\)
\(968\) 0 0
\(969\) 811.267 0.837220
\(970\) 0 0
\(971\) − 1530.12i − 1.57582i −0.615792 0.787908i \(-0.711164\pi\)
0.615792 0.787908i \(-0.288836\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) − 24.5505i − 0.0251800i
\(976\) 0 0
\(977\) −841.441 −0.861250 −0.430625 0.902531i \(-0.641707\pi\)
−0.430625 + 0.902531i \(0.641707\pi\)
\(978\) 0 0
\(979\) 110.470i 0.112839i
\(980\) 0 0
\(981\) −483.909 −0.493281
\(982\) 0 0
\(983\) − 1611.63i − 1.63951i −0.572717 0.819753i \(-0.694111\pi\)
0.572717 0.819753i \(-0.305889\pi\)
\(984\) 0 0
\(985\) 204.798 0.207917
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −397.358 −0.401777
\(990\) 0 0
\(991\) 368.180i 0.371523i 0.982595 + 0.185762i \(0.0594753\pi\)
−0.982595 + 0.185762i \(0.940525\pi\)
\(992\) 0 0
\(993\) −123.688 −0.124560
\(994\) 0 0
\(995\) − 24.0248i − 0.0241455i
\(996\) 0 0
\(997\) −924.642 −0.927425 −0.463712 0.885986i \(-0.653483\pi\)
−0.463712 + 0.885986i \(0.653483\pi\)
\(998\) 0 0
\(999\) − 246.794i − 0.247041i
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 2352.3.m.f.1471.4 4
4.3 odd 2 inner 2352.3.m.f.1471.2 4
7.6 odd 2 336.3.m.c.127.1 4
21.20 even 2 1008.3.m.b.127.4 4
28.27 even 2 336.3.m.c.127.3 yes 4
56.13 odd 2 1344.3.m.a.127.4 4
56.27 even 2 1344.3.m.a.127.2 4
84.83 odd 2 1008.3.m.b.127.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.3.m.c.127.1 4 7.6 odd 2
336.3.m.c.127.3 yes 4 28.27 even 2
1008.3.m.b.127.3 4 84.83 odd 2
1008.3.m.b.127.4 4 21.20 even 2
1344.3.m.a.127.2 4 56.27 even 2
1344.3.m.a.127.4 4 56.13 odd 2
2352.3.m.f.1471.2 4 4.3 odd 2 inner
2352.3.m.f.1471.4 4 1.1 even 1 trivial