Properties

Label 224.3.d.b.127.2
Level 224
Weight 3
Character 224.127
Analytic conductor 6.104
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1997017344.2
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.2
Root \(-1.92812i\) of \(x^{8} + 14 x^{6} + 53 x^{4} + 56 x^{2} + 4\)
Character \(\chi\) \(=\) 224.127
Dual form 224.3.d.b.127.7

$q$-expansion

\(f(q)\) \(=\) \(q-3.85623i q^{3} +0.490168 q^{5} -2.64575i q^{7} -5.87054 q^{9} +O(q^{10})\) \(q-3.85623i q^{3} +0.490168 q^{5} -2.64575i q^{7} -5.87054 q^{9} -15.5633i q^{11} +3.50983 q^{13} -1.89020i q^{15} -24.1463 q^{17} -3.56870i q^{19} -10.2026 q^{21} +19.5741i q^{23} -24.7597 q^{25} -12.0679i q^{27} -10.9803 q^{29} -21.1767i q^{31} -60.0159 q^{33} -1.29686i q^{35} +58.4212 q^{37} -13.5347i q^{39} +54.1285 q^{41} +35.6420i q^{43} -2.87755 q^{45} -64.2248i q^{47} -7.00000 q^{49} +93.1140i q^{51} +87.4015 q^{53} -7.62865i q^{55} -13.7617 q^{57} -66.6954i q^{59} +16.8615 q^{61} +15.5320i q^{63} +1.72041 q^{65} -21.2420i q^{67} +75.4822 q^{69} +64.2140i q^{71} +99.4587 q^{73} +95.4793i q^{75} -41.1767 q^{77} +139.441i q^{79} -99.3716 q^{81} -6.03134i q^{83} -11.8358 q^{85} +42.3427i q^{87} -23.9821 q^{89} -9.28614i q^{91} -81.6624 q^{93} -1.74926i q^{95} +171.509 q^{97} +91.3652i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 40q^{9} + O(q^{10}) \) \( 8q - 40q^{9} + 32q^{13} - 16q^{17} + 104q^{25} - 80q^{29} - 176q^{37} + 144q^{41} + 256q^{45} - 56q^{49} + 48q^{53} - 400q^{57} - 192q^{61} - 304q^{65} + 576q^{69} + 272q^{73} - 112q^{77} + 504q^{81} - 160q^{85} - 80q^{89} - 608q^{93} + 528q^{97} + O(q^{100}) \)

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
\(3\) − 3.85623i − 1.28541i −0.766113 0.642706i \(-0.777812\pi\)
0.766113 0.642706i \(-0.222188\pi\)
\(4\) 0 0
\(5\) 0.490168 0.0980336 0.0490168 0.998798i \(-0.484391\pi\)
0.0490168 + 0.998798i \(0.484391\pi\)
\(6\) 0 0
\(7\) − 2.64575i − 0.377964i
\(8\) 0 0
\(9\) −5.87054 −0.652282
\(10\) 0 0
\(11\) − 15.5633i − 1.41485i −0.706789 0.707425i \(-0.749857\pi\)
0.706789 0.707425i \(-0.250143\pi\)
\(12\) 0 0
\(13\) 3.50983 0.269987 0.134994 0.990846i \(-0.456899\pi\)
0.134994 + 0.990846i \(0.456899\pi\)
\(14\) 0 0
\(15\) − 1.89020i − 0.126013i
\(16\) 0 0
\(17\) −24.1463 −1.42037 −0.710187 0.704013i \(-0.751389\pi\)
−0.710187 + 0.704013i \(0.751389\pi\)
\(18\) 0 0
\(19\) − 3.56870i − 0.187826i −0.995580 0.0939132i \(-0.970062\pi\)
0.995580 0.0939132i \(-0.0299376\pi\)
\(20\) 0 0
\(21\) −10.2026 −0.485840
\(22\) 0 0
\(23\) 19.5741i 0.851046i 0.904948 + 0.425523i \(0.139910\pi\)
−0.904948 + 0.425523i \(0.860090\pi\)
\(24\) 0 0
\(25\) −24.7597 −0.990389
\(26\) 0 0
\(27\) − 12.0679i − 0.446961i
\(28\) 0 0
\(29\) −10.9803 −0.378632 −0.189316 0.981916i \(-0.560627\pi\)
−0.189316 + 0.981916i \(0.560627\pi\)
\(30\) 0 0
\(31\) − 21.1767i − 0.683120i −0.939860 0.341560i \(-0.889045\pi\)
0.939860 0.341560i \(-0.110955\pi\)
\(32\) 0 0
\(33\) −60.0159 −1.81866
\(34\) 0 0
\(35\) − 1.29686i − 0.0370532i
\(36\) 0 0
\(37\) 58.4212 1.57895 0.789475 0.613783i \(-0.210353\pi\)
0.789475 + 0.613783i \(0.210353\pi\)
\(38\) 0 0
\(39\) − 13.5347i − 0.347044i
\(40\) 0 0
\(41\) 54.1285 1.32021 0.660103 0.751175i \(-0.270513\pi\)
0.660103 + 0.751175i \(0.270513\pi\)
\(42\) 0 0
\(43\) 35.6420i 0.828884i 0.910076 + 0.414442i \(0.136023\pi\)
−0.910076 + 0.414442i \(0.863977\pi\)
\(44\) 0 0
\(45\) −2.87755 −0.0639455
\(46\) 0 0
\(47\) − 64.2248i − 1.36648i −0.730192 0.683242i \(-0.760569\pi\)
0.730192 0.683242i \(-0.239431\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 93.1140i 1.82576i
\(52\) 0 0
\(53\) 87.4015 1.64908 0.824542 0.565800i \(-0.191433\pi\)
0.824542 + 0.565800i \(0.191433\pi\)
\(54\) 0 0
\(55\) − 7.62865i − 0.138703i
\(56\) 0 0
\(57\) −13.7617 −0.241434
\(58\) 0 0
\(59\) − 66.6954i − 1.13043i −0.824944 0.565215i \(-0.808793\pi\)
0.824944 0.565215i \(-0.191207\pi\)
\(60\) 0 0
\(61\) 16.8615 0.276418 0.138209 0.990403i \(-0.455865\pi\)
0.138209 + 0.990403i \(0.455865\pi\)
\(62\) 0 0
\(63\) 15.5320i 0.246539i
\(64\) 0 0
\(65\) 1.72041 0.0264678
\(66\) 0 0
\(67\) − 21.2420i − 0.317045i −0.987355 0.158523i \(-0.949327\pi\)
0.987355 0.158523i \(-0.0506731\pi\)
\(68\) 0 0
\(69\) 75.4822 1.09394
\(70\) 0 0
\(71\) 64.2140i 0.904423i 0.891911 + 0.452212i \(0.149365\pi\)
−0.891911 + 0.452212i \(0.850635\pi\)
\(72\) 0 0
\(73\) 99.4587 1.36245 0.681224 0.732075i \(-0.261448\pi\)
0.681224 + 0.732075i \(0.261448\pi\)
\(74\) 0 0
\(75\) 95.4793i 1.27306i
\(76\) 0 0
\(77\) −41.1767 −0.534763
\(78\) 0 0
\(79\) 139.441i 1.76507i 0.470243 + 0.882537i \(0.344166\pi\)
−0.470243 + 0.882537i \(0.655834\pi\)
\(80\) 0 0
\(81\) −99.3716 −1.22681
\(82\) 0 0
\(83\) − 6.03134i − 0.0726668i −0.999340 0.0363334i \(-0.988432\pi\)
0.999340 0.0363334i \(-0.0115678\pi\)
\(84\) 0 0
\(85\) −11.8358 −0.139244
\(86\) 0 0
\(87\) 42.3427i 0.486698i
\(88\) 0 0
\(89\) −23.9821 −0.269462 −0.134731 0.990882i \(-0.543017\pi\)
−0.134731 + 0.990882i \(0.543017\pi\)
\(90\) 0 0
\(91\) − 9.28614i − 0.102046i
\(92\) 0 0
\(93\) −81.6624 −0.878091
\(94\) 0 0
\(95\) − 1.74926i − 0.0184133i
\(96\) 0 0
\(97\) 171.509 1.76813 0.884064 0.467365i \(-0.154797\pi\)
0.884064 + 0.467365i \(0.154797\pi\)
\(98\) 0 0
\(99\) 91.3652i 0.922881i
\(100\) 0 0
\(101\) −146.427 −1.44977 −0.724887 0.688867i \(-0.758108\pi\)
−0.724887 + 0.688867i \(0.758108\pi\)
\(102\) 0 0
\(103\) 118.849i 1.15388i 0.816787 + 0.576939i \(0.195753\pi\)
−0.816787 + 0.576939i \(0.804247\pi\)
\(104\) 0 0
\(105\) −5.00100 −0.0476286
\(106\) 0 0
\(107\) − 142.434i − 1.33116i −0.746326 0.665581i \(-0.768184\pi\)
0.746326 0.665581i \(-0.231816\pi\)
\(108\) 0 0
\(109\) 170.835 1.56730 0.783649 0.621204i \(-0.213356\pi\)
0.783649 + 0.621204i \(0.213356\pi\)
\(110\) 0 0
\(111\) − 225.286i − 2.02960i
\(112\) 0 0
\(113\) 24.9436 0.220740 0.110370 0.993891i \(-0.464796\pi\)
0.110370 + 0.993891i \(0.464796\pi\)
\(114\) 0 0
\(115\) 9.59458i 0.0834311i
\(116\) 0 0
\(117\) −20.6046 −0.176108
\(118\) 0 0
\(119\) 63.8852i 0.536851i
\(120\) 0 0
\(121\) −121.218 −1.00180
\(122\) 0 0
\(123\) − 208.732i − 1.69701i
\(124\) 0 0
\(125\) −24.3906 −0.195125
\(126\) 0 0
\(127\) − 68.8755i − 0.542326i −0.962533 0.271163i \(-0.912592\pi\)
0.962533 0.271163i \(-0.0874083\pi\)
\(128\) 0 0
\(129\) 137.444 1.06546
\(130\) 0 0
\(131\) 101.326i 0.773481i 0.922189 + 0.386741i \(0.126399\pi\)
−0.922189 + 0.386741i \(0.873601\pi\)
\(132\) 0 0
\(133\) −9.44190 −0.0709917
\(134\) 0 0
\(135\) − 5.91532i − 0.0438172i
\(136\) 0 0
\(137\) 63.8820 0.466292 0.233146 0.972442i \(-0.425098\pi\)
0.233146 + 0.972442i \(0.425098\pi\)
\(138\) 0 0
\(139\) − 199.256i − 1.43349i −0.697333 0.716747i \(-0.745630\pi\)
0.697333 0.716747i \(-0.254370\pi\)
\(140\) 0 0
\(141\) −247.666 −1.75649
\(142\) 0 0
\(143\) − 54.6247i − 0.381991i
\(144\) 0 0
\(145\) −5.38221 −0.0371187
\(146\) 0 0
\(147\) 26.9936i 0.183630i
\(148\) 0 0
\(149\) −271.585 −1.82272 −0.911359 0.411612i \(-0.864966\pi\)
−0.911359 + 0.411612i \(0.864966\pi\)
\(150\) 0 0
\(151\) 131.329i 0.869729i 0.900496 + 0.434864i \(0.143204\pi\)
−0.900496 + 0.434864i \(0.856796\pi\)
\(152\) 0 0
\(153\) 141.752 0.926484
\(154\) 0 0
\(155\) − 10.3802i − 0.0669687i
\(156\) 0 0
\(157\) 133.685 0.851494 0.425747 0.904842i \(-0.360011\pi\)
0.425747 + 0.904842i \(0.360011\pi\)
\(158\) 0 0
\(159\) − 337.041i − 2.11975i
\(160\) 0 0
\(161\) 51.7881 0.321665
\(162\) 0 0
\(163\) 29.8167i 0.182925i 0.995809 + 0.0914623i \(0.0291541\pi\)
−0.995809 + 0.0914623i \(0.970846\pi\)
\(164\) 0 0
\(165\) −29.4179 −0.178290
\(166\) 0 0
\(167\) − 259.708i − 1.55514i −0.628796 0.777570i \(-0.716452\pi\)
0.628796 0.777570i \(-0.283548\pi\)
\(168\) 0 0
\(169\) −156.681 −0.927107
\(170\) 0 0
\(171\) 20.9502i 0.122516i
\(172\) 0 0
\(173\) −84.9760 −0.491190 −0.245595 0.969372i \(-0.578983\pi\)
−0.245595 + 0.969372i \(0.578983\pi\)
\(174\) 0 0
\(175\) 65.5081i 0.374332i
\(176\) 0 0
\(177\) −257.193 −1.45307
\(178\) 0 0
\(179\) 19.3648i 0.108183i 0.998536 + 0.0540916i \(0.0172263\pi\)
−0.998536 + 0.0540916i \(0.982774\pi\)
\(180\) 0 0
\(181\) −183.350 −1.01298 −0.506491 0.862245i \(-0.669058\pi\)
−0.506491 + 0.862245i \(0.669058\pi\)
\(182\) 0 0
\(183\) − 65.0219i − 0.355311i
\(184\) 0 0
\(185\) 28.6362 0.154790
\(186\) 0 0
\(187\) 375.798i 2.00961i
\(188\) 0 0
\(189\) −31.9288 −0.168935
\(190\) 0 0
\(191\) 93.1822i 0.487865i 0.969792 + 0.243932i \(0.0784375\pi\)
−0.969792 + 0.243932i \(0.921562\pi\)
\(192\) 0 0
\(193\) 276.855 1.43448 0.717242 0.696824i \(-0.245404\pi\)
0.717242 + 0.696824i \(0.245404\pi\)
\(194\) 0 0
\(195\) − 6.63429i − 0.0340220i
\(196\) 0 0
\(197\) −177.712 −0.902089 −0.451045 0.892501i \(-0.648948\pi\)
−0.451045 + 0.892501i \(0.648948\pi\)
\(198\) 0 0
\(199\) 227.421i 1.14282i 0.820666 + 0.571408i \(0.193603\pi\)
−0.820666 + 0.571408i \(0.806397\pi\)
\(200\) 0 0
\(201\) −81.9143 −0.407534
\(202\) 0 0
\(203\) 29.0512i 0.143110i
\(204\) 0 0
\(205\) 26.5320 0.129425
\(206\) 0 0
\(207\) − 114.910i − 0.555122i
\(208\) 0 0
\(209\) −55.5409 −0.265746
\(210\) 0 0
\(211\) 325.518i 1.54274i 0.636387 + 0.771370i \(0.280428\pi\)
−0.636387 + 0.771370i \(0.719572\pi\)
\(212\) 0 0
\(213\) 247.624 1.16256
\(214\) 0 0
\(215\) 17.4706i 0.0812584i
\(216\) 0 0
\(217\) −56.0284 −0.258195
\(218\) 0 0
\(219\) − 383.536i − 1.75131i
\(220\) 0 0
\(221\) −84.7496 −0.383482
\(222\) 0 0
\(223\) 106.001i 0.475342i 0.971346 + 0.237671i \(0.0763840\pi\)
−0.971346 + 0.237671i \(0.923616\pi\)
\(224\) 0 0
\(225\) 145.353 0.646013
\(226\) 0 0
\(227\) − 282.765i − 1.24566i −0.782357 0.622830i \(-0.785983\pi\)
0.782357 0.622830i \(-0.214017\pi\)
\(228\) 0 0
\(229\) 144.716 0.631949 0.315975 0.948768i \(-0.397669\pi\)
0.315975 + 0.948768i \(0.397669\pi\)
\(230\) 0 0
\(231\) 158.787i 0.687390i
\(232\) 0 0
\(233\) 36.9174 0.158444 0.0792219 0.996857i \(-0.474756\pi\)
0.0792219 + 0.996857i \(0.474756\pi\)
\(234\) 0 0
\(235\) − 31.4809i − 0.133961i
\(236\) 0 0
\(237\) 537.716 2.26885
\(238\) 0 0
\(239\) − 65.5138i − 0.274116i −0.990563 0.137058i \(-0.956235\pi\)
0.990563 0.137058i \(-0.0437647\pi\)
\(240\) 0 0
\(241\) −155.844 −0.646656 −0.323328 0.946287i \(-0.604802\pi\)
−0.323328 + 0.946287i \(0.604802\pi\)
\(242\) 0 0
\(243\) 274.589i 1.12999i
\(244\) 0 0
\(245\) −3.43118 −0.0140048
\(246\) 0 0
\(247\) − 12.5255i − 0.0507107i
\(248\) 0 0
\(249\) −23.2583 −0.0934067
\(250\) 0 0
\(251\) − 230.946i − 0.920105i −0.887892 0.460052i \(-0.847831\pi\)
0.887892 0.460052i \(-0.152169\pi\)
\(252\) 0 0
\(253\) 304.638 1.20410
\(254\) 0 0
\(255\) 45.6415i 0.178986i
\(256\) 0 0
\(257\) 337.195 1.31204 0.656022 0.754742i \(-0.272238\pi\)
0.656022 + 0.754742i \(0.272238\pi\)
\(258\) 0 0
\(259\) − 154.568i − 0.596787i
\(260\) 0 0
\(261\) 64.4605 0.246975
\(262\) 0 0
\(263\) 395.803i 1.50495i 0.658619 + 0.752476i \(0.271141\pi\)
−0.658619 + 0.752476i \(0.728859\pi\)
\(264\) 0 0
\(265\) 42.8414 0.161666
\(266\) 0 0
\(267\) 92.4806i 0.346369i
\(268\) 0 0
\(269\) 31.6881 0.117800 0.0588998 0.998264i \(-0.481241\pi\)
0.0588998 + 0.998264i \(0.481241\pi\)
\(270\) 0 0
\(271\) − 69.0163i − 0.254673i −0.991860 0.127336i \(-0.959357\pi\)
0.991860 0.127336i \(-0.0406428\pi\)
\(272\) 0 0
\(273\) −35.8095 −0.131170
\(274\) 0 0
\(275\) 385.344i 1.40125i
\(276\) 0 0
\(277\) 54.5783 0.197034 0.0985168 0.995135i \(-0.468590\pi\)
0.0985168 + 0.995135i \(0.468590\pi\)
\(278\) 0 0
\(279\) 124.319i 0.445587i
\(280\) 0 0
\(281\) −140.453 −0.499831 −0.249916 0.968268i \(-0.580403\pi\)
−0.249916 + 0.968268i \(0.580403\pi\)
\(282\) 0 0
\(283\) 35.7205i 0.126221i 0.998007 + 0.0631105i \(0.0201021\pi\)
−0.998007 + 0.0631105i \(0.979898\pi\)
\(284\) 0 0
\(285\) −6.74557 −0.0236687
\(286\) 0 0
\(287\) − 143.210i − 0.498991i
\(288\) 0 0
\(289\) 294.046 1.01746
\(290\) 0 0
\(291\) − 661.377i − 2.27277i
\(292\) 0 0
\(293\) 73.9179 0.252279 0.126140 0.992012i \(-0.459741\pi\)
0.126140 + 0.992012i \(0.459741\pi\)
\(294\) 0 0
\(295\) − 32.6919i − 0.110820i
\(296\) 0 0
\(297\) −187.817 −0.632382
\(298\) 0 0
\(299\) 68.7017i 0.229771i
\(300\) 0 0
\(301\) 94.2999 0.313289
\(302\) 0 0
\(303\) 564.658i 1.86356i
\(304\) 0 0
\(305\) 8.26498 0.0270983
\(306\) 0 0
\(307\) 507.360i 1.65264i 0.563202 + 0.826319i \(0.309569\pi\)
−0.563202 + 0.826319i \(0.690431\pi\)
\(308\) 0 0
\(309\) 458.311 1.48321
\(310\) 0 0
\(311\) 242.948i 0.781185i 0.920564 + 0.390592i \(0.127730\pi\)
−0.920564 + 0.390592i \(0.872270\pi\)
\(312\) 0 0
\(313\) −208.239 −0.665301 −0.332651 0.943050i \(-0.607943\pi\)
−0.332651 + 0.943050i \(0.607943\pi\)
\(314\) 0 0
\(315\) 7.61328i 0.0241691i
\(316\) 0 0
\(317\) −10.6307 −0.0335355 −0.0167677 0.999859i \(-0.505338\pi\)
−0.0167677 + 0.999859i \(0.505338\pi\)
\(318\) 0 0
\(319\) 170.891i 0.535708i
\(320\) 0 0
\(321\) −549.260 −1.71109
\(322\) 0 0
\(323\) 86.1711i 0.266784i
\(324\) 0 0
\(325\) −86.9025 −0.267392
\(326\) 0 0
\(327\) − 658.781i − 2.01462i
\(328\) 0 0
\(329\) −169.923 −0.516482
\(330\) 0 0
\(331\) − 122.813i − 0.371035i −0.982641 0.185517i \(-0.940604\pi\)
0.982641 0.185517i \(-0.0593961\pi\)
\(332\) 0 0
\(333\) −342.964 −1.02992
\(334\) 0 0
\(335\) − 10.4122i − 0.0310811i
\(336\) 0 0
\(337\) −518.410 −1.53831 −0.769154 0.639063i \(-0.779322\pi\)
−0.769154 + 0.639063i \(0.779322\pi\)
\(338\) 0 0
\(339\) − 96.1885i − 0.283742i
\(340\) 0 0
\(341\) −329.581 −0.966512
\(342\) 0 0
\(343\) 18.5203i 0.0539949i
\(344\) 0 0
\(345\) 36.9989 0.107243
\(346\) 0 0
\(347\) 231.720i 0.667780i 0.942612 + 0.333890i \(0.108361\pi\)
−0.942612 + 0.333890i \(0.891639\pi\)
\(348\) 0 0
\(349\) 170.720 0.489169 0.244584 0.969628i \(-0.421348\pi\)
0.244584 + 0.969628i \(0.421348\pi\)
\(350\) 0 0
\(351\) − 42.3564i − 0.120674i
\(352\) 0 0
\(353\) −200.237 −0.567244 −0.283622 0.958936i \(-0.591536\pi\)
−0.283622 + 0.958936i \(0.591536\pi\)
\(354\) 0 0
\(355\) 31.4757i 0.0886638i
\(356\) 0 0
\(357\) 246.356 0.690074
\(358\) 0 0
\(359\) 86.9601i 0.242229i 0.992639 + 0.121114i \(0.0386468\pi\)
−0.992639 + 0.121114i \(0.961353\pi\)
\(360\) 0 0
\(361\) 348.264 0.964721
\(362\) 0 0
\(363\) 467.443i 1.28772i
\(364\) 0 0
\(365\) 48.7515 0.133566
\(366\) 0 0
\(367\) − 672.949i − 1.83365i −0.399290 0.916825i \(-0.630743\pi\)
0.399290 0.916825i \(-0.369257\pi\)
\(368\) 0 0
\(369\) −317.763 −0.861147
\(370\) 0 0
\(371\) − 231.243i − 0.623295i
\(372\) 0 0
\(373\) 692.328 1.85611 0.928053 0.372448i \(-0.121481\pi\)
0.928053 + 0.372448i \(0.121481\pi\)
\(374\) 0 0
\(375\) 94.0560i 0.250816i
\(376\) 0 0
\(377\) −38.5391 −0.102226
\(378\) 0 0
\(379\) − 193.386i − 0.510253i −0.966908 0.255126i \(-0.917883\pi\)
0.966908 0.255126i \(-0.0821171\pi\)
\(380\) 0 0
\(381\) −265.600 −0.697112
\(382\) 0 0
\(383\) − 518.606i − 1.35406i −0.735954 0.677031i \(-0.763266\pi\)
0.735954 0.677031i \(-0.236734\pi\)
\(384\) 0 0
\(385\) −20.1835 −0.0524247
\(386\) 0 0
\(387\) − 209.238i − 0.540666i
\(388\) 0 0
\(389\) −531.999 −1.36761 −0.683803 0.729666i \(-0.739675\pi\)
−0.683803 + 0.729666i \(0.739675\pi\)
\(390\) 0 0
\(391\) − 472.642i − 1.20880i
\(392\) 0 0
\(393\) 390.737 0.994242
\(394\) 0 0
\(395\) 68.3494i 0.173036i
\(396\) 0 0
\(397\) −490.068 −1.23443 −0.617214 0.786795i \(-0.711739\pi\)
−0.617214 + 0.786795i \(0.711739\pi\)
\(398\) 0 0
\(399\) 36.4102i 0.0912535i
\(400\) 0 0
\(401\) 712.938 1.77790 0.888951 0.458003i \(-0.151435\pi\)
0.888951 + 0.458003i \(0.151435\pi\)
\(402\) 0 0
\(403\) − 74.3268i − 0.184434i
\(404\) 0 0
\(405\) −48.7088 −0.120269
\(406\) 0 0
\(407\) − 909.228i − 2.23398i
\(408\) 0 0
\(409\) −307.818 −0.752612 −0.376306 0.926495i \(-0.622806\pi\)
−0.376306 + 0.926495i \(0.622806\pi\)
\(410\) 0 0
\(411\) − 246.344i − 0.599377i
\(412\) 0 0
\(413\) −176.459 −0.427262
\(414\) 0 0
\(415\) − 2.95637i − 0.00712378i
\(416\) 0 0
\(417\) −768.377 −1.84263
\(418\) 0 0
\(419\) 176.023i 0.420103i 0.977690 + 0.210051i \(0.0673631\pi\)
−0.977690 + 0.210051i \(0.932637\pi\)
\(420\) 0 0
\(421\) 263.167 0.625101 0.312550 0.949901i \(-0.398817\pi\)
0.312550 + 0.949901i \(0.398817\pi\)
\(422\) 0 0
\(423\) 377.034i 0.891333i
\(424\) 0 0
\(425\) 597.857 1.40672
\(426\) 0 0
\(427\) − 44.6114i − 0.104476i
\(428\) 0 0
\(429\) −210.646 −0.491016
\(430\) 0 0
\(431\) − 256.629i − 0.595426i −0.954655 0.297713i \(-0.903776\pi\)
0.954655 0.297713i \(-0.0962238\pi\)
\(432\) 0 0
\(433\) −636.797 −1.47066 −0.735331 0.677708i \(-0.762973\pi\)
−0.735331 + 0.677708i \(0.762973\pi\)
\(434\) 0 0
\(435\) 20.7551i 0.0477128i
\(436\) 0 0
\(437\) 69.8540 0.159849
\(438\) 0 0
\(439\) − 64.4215i − 0.146746i −0.997305 0.0733730i \(-0.976624\pi\)
0.997305 0.0733730i \(-0.0233763\pi\)
\(440\) 0 0
\(441\) 41.0938 0.0931831
\(442\) 0 0
\(443\) − 449.411i − 1.01447i −0.861807 0.507236i \(-0.830667\pi\)
0.861807 0.507236i \(-0.169333\pi\)
\(444\) 0 0
\(445\) −11.7553 −0.0264163
\(446\) 0 0
\(447\) 1047.30i 2.34294i
\(448\) 0 0
\(449\) 362.900 0.808241 0.404120 0.914706i \(-0.367578\pi\)
0.404120 + 0.914706i \(0.367578\pi\)
\(450\) 0 0
\(451\) − 842.420i − 1.86789i
\(452\) 0 0
\(453\) 506.435 1.11796
\(454\) 0 0
\(455\) − 4.55177i − 0.0100039i
\(456\) 0 0
\(457\) −61.1286 −0.133761 −0.0668803 0.997761i \(-0.521305\pi\)
−0.0668803 + 0.997761i \(0.521305\pi\)
\(458\) 0 0
\(459\) 291.397i 0.634851i
\(460\) 0 0
\(461\) −564.752 −1.22506 −0.612530 0.790448i \(-0.709848\pi\)
−0.612530 + 0.790448i \(0.709848\pi\)
\(462\) 0 0
\(463\) 581.095i 1.25506i 0.778590 + 0.627532i \(0.215935\pi\)
−0.778590 + 0.627532i \(0.784065\pi\)
\(464\) 0 0
\(465\) −40.0283 −0.0860824
\(466\) 0 0
\(467\) 125.780i 0.269336i 0.990891 + 0.134668i \(0.0429968\pi\)
−0.990891 + 0.134668i \(0.957003\pi\)
\(468\) 0 0
\(469\) −56.2012 −0.119832
\(470\) 0 0
\(471\) − 515.519i − 1.09452i
\(472\) 0 0
\(473\) 554.709 1.17275
\(474\) 0 0
\(475\) 88.3601i 0.186021i
\(476\) 0 0
\(477\) −513.094 −1.07567
\(478\) 0 0
\(479\) 393.505i 0.821513i 0.911745 + 0.410756i \(0.134735\pi\)
−0.911745 + 0.410756i \(0.865265\pi\)
\(480\) 0 0
\(481\) 205.048 0.426296
\(482\) 0 0
\(483\) − 199.707i − 0.413472i
\(484\) 0 0
\(485\) 84.0680 0.173336
\(486\) 0 0
\(487\) 149.606i 0.307198i 0.988133 + 0.153599i \(0.0490865\pi\)
−0.988133 + 0.153599i \(0.950914\pi\)
\(488\) 0 0
\(489\) 114.980 0.235133
\(490\) 0 0
\(491\) − 79.1826i − 0.161268i −0.996744 0.0806340i \(-0.974305\pi\)
0.996744 0.0806340i \(-0.0256945\pi\)
\(492\) 0 0
\(493\) 265.135 0.537799
\(494\) 0 0
\(495\) 44.7843i 0.0904733i
\(496\) 0 0
\(497\) 169.894 0.341840
\(498\) 0 0
\(499\) − 244.391i − 0.489762i −0.969553 0.244881i \(-0.921251\pi\)
0.969553 0.244881i \(-0.0787489\pi\)
\(500\) 0 0
\(501\) −1001.50 −1.99899
\(502\) 0 0
\(503\) 278.539i 0.553755i 0.960905 + 0.276878i \(0.0892997\pi\)
−0.960905 + 0.276878i \(0.910700\pi\)
\(504\) 0 0
\(505\) −71.7739 −0.142127
\(506\) 0 0
\(507\) 604.199i 1.19171i
\(508\) 0 0
\(509\) 444.338 0.872963 0.436481 0.899713i \(-0.356225\pi\)
0.436481 + 0.899713i \(0.356225\pi\)
\(510\) 0 0
\(511\) − 263.143i − 0.514957i
\(512\) 0 0
\(513\) −43.0669 −0.0839510
\(514\) 0 0
\(515\) 58.2562i 0.113119i
\(516\) 0 0
\(517\) −999.552 −1.93337
\(518\) 0 0
\(519\) 327.687i 0.631382i
\(520\) 0 0
\(521\) 127.704 0.245113 0.122557 0.992462i \(-0.460891\pi\)
0.122557 + 0.992462i \(0.460891\pi\)
\(522\) 0 0
\(523\) 937.135i 1.79184i 0.444211 + 0.895922i \(0.353484\pi\)
−0.444211 + 0.895922i \(0.646516\pi\)
\(524\) 0 0
\(525\) 252.615 0.481171
\(526\) 0 0
\(527\) 511.341i 0.970286i
\(528\) 0 0
\(529\) 145.856 0.275721
\(530\) 0 0
\(531\) 391.538i 0.737359i
\(532\) 0 0
\(533\) 189.982 0.356439
\(534\) 0 0
\(535\) − 69.8167i − 0.130499i
\(536\) 0 0
\(537\) 74.6751 0.139060
\(538\) 0 0
\(539\) 108.943i 0.202121i
\(540\) 0 0
\(541\) −540.845 −0.999713 −0.499857 0.866108i \(-0.666614\pi\)
−0.499857 + 0.866108i \(0.666614\pi\)
\(542\) 0 0
\(543\) 707.040i 1.30210i
\(544\) 0 0
\(545\) 83.7380 0.153648
\(546\) 0 0
\(547\) − 667.995i − 1.22120i −0.791940 0.610599i \(-0.790929\pi\)
0.791940 0.610599i \(-0.209071\pi\)
\(548\) 0 0
\(549\) −98.9862 −0.180303
\(550\) 0 0
\(551\) 39.1855i 0.0711171i
\(552\) 0 0
\(553\) 368.926 0.667135
\(554\) 0 0
\(555\) − 110.428i − 0.198969i
\(556\) 0 0
\(557\) 746.470 1.34016 0.670081 0.742288i \(-0.266259\pi\)
0.670081 + 0.742288i \(0.266259\pi\)
\(558\) 0 0
\(559\) 125.097i 0.223788i
\(560\) 0 0
\(561\) 1449.16 2.58318
\(562\) 0 0
\(563\) 518.437i 0.920846i 0.887700 + 0.460423i \(0.152302\pi\)
−0.887700 + 0.460423i \(0.847698\pi\)
\(564\) 0 0
\(565\) 12.2266 0.0216399
\(566\) 0 0
\(567\) 262.913i 0.463691i
\(568\) 0 0
\(569\) −215.471 −0.378683 −0.189341 0.981911i \(-0.560635\pi\)
−0.189341 + 0.981911i \(0.560635\pi\)
\(570\) 0 0
\(571\) 4.73397i 0.00829067i 0.999991 + 0.00414534i \(0.00131951\pi\)
−0.999991 + 0.00414534i \(0.998680\pi\)
\(572\) 0 0
\(573\) 359.332 0.627107
\(574\) 0 0
\(575\) − 484.649i − 0.842867i
\(576\) 0 0
\(577\) −72.0226 −0.124823 −0.0624113 0.998051i \(-0.519879\pi\)
−0.0624113 + 0.998051i \(0.519879\pi\)
\(578\) 0 0
\(579\) − 1067.62i − 1.84390i
\(580\) 0 0
\(581\) −15.9574 −0.0274655
\(582\) 0 0
\(583\) − 1360.26i − 2.33321i
\(584\) 0 0
\(585\) −10.0997 −0.0172645
\(586\) 0 0
\(587\) 272.118i 0.463575i 0.972766 + 0.231787i \(0.0744573\pi\)
−0.972766 + 0.231787i \(0.925543\pi\)
\(588\) 0 0
\(589\) −75.5734 −0.128308
\(590\) 0 0
\(591\) 685.297i 1.15956i
\(592\) 0 0
\(593\) 842.019 1.41993 0.709966 0.704236i \(-0.248710\pi\)
0.709966 + 0.704236i \(0.248710\pi\)
\(594\) 0 0
\(595\) 31.3145i 0.0526294i
\(596\) 0 0
\(597\) 876.987 1.46899
\(598\) 0 0
\(599\) 478.633i 0.799054i 0.916721 + 0.399527i \(0.130826\pi\)
−0.916721 + 0.399527i \(0.869174\pi\)
\(600\) 0 0
\(601\) 206.471 0.343546 0.171773 0.985137i \(-0.445050\pi\)
0.171773 + 0.985137i \(0.445050\pi\)
\(602\) 0 0
\(603\) 124.702i 0.206803i
\(604\) 0 0
\(605\) −59.4170 −0.0982099
\(606\) 0 0
\(607\) 292.279i 0.481514i 0.970585 + 0.240757i \(0.0773956\pi\)
−0.970585 + 0.240757i \(0.922604\pi\)
\(608\) 0 0
\(609\) 112.028 0.183955
\(610\) 0 0
\(611\) − 225.418i − 0.368933i
\(612\) 0 0
\(613\) 827.863 1.35051 0.675255 0.737584i \(-0.264033\pi\)
0.675255 + 0.737584i \(0.264033\pi\)
\(614\) 0 0
\(615\) − 102.314i − 0.166364i
\(616\) 0 0
\(617\) −667.348 −1.08160 −0.540801 0.841151i \(-0.681879\pi\)
−0.540801 + 0.841151i \(0.681879\pi\)
\(618\) 0 0
\(619\) 750.661i 1.21270i 0.795198 + 0.606350i \(0.207367\pi\)
−0.795198 + 0.606350i \(0.792633\pi\)
\(620\) 0 0
\(621\) 236.219 0.380384
\(622\) 0 0
\(623\) 63.4507i 0.101847i
\(624\) 0 0
\(625\) 607.038 0.971261
\(626\) 0 0
\(627\) 214.179i 0.341593i
\(628\) 0 0
\(629\) −1410.66 −2.24270
\(630\) 0 0
\(631\) − 1076.70i − 1.70633i −0.521637 0.853167i \(-0.674679\pi\)
0.521637 0.853167i \(-0.325321\pi\)
\(632\) 0 0
\(633\) 1255.27 1.98305
\(634\) 0 0
\(635\) − 33.7605i − 0.0531662i
\(636\) 0 0
\(637\) −24.5688 −0.0385696
\(638\) 0 0
\(639\) − 376.971i − 0.589939i
\(640\) 0 0
\(641\) −298.632 −0.465885 −0.232942 0.972491i \(-0.574835\pi\)
−0.232942 + 0.972491i \(0.574835\pi\)
\(642\) 0 0
\(643\) − 504.242i − 0.784202i −0.919922 0.392101i \(-0.871748\pi\)
0.919922 0.392101i \(-0.128252\pi\)
\(644\) 0 0
\(645\) 67.3706 0.104451
\(646\) 0 0
\(647\) 76.2691i 0.117881i 0.998261 + 0.0589406i \(0.0187723\pi\)
−0.998261 + 0.0589406i \(0.981228\pi\)
\(648\) 0 0
\(649\) −1038.00 −1.59939
\(650\) 0 0
\(651\) 216.058i 0.331887i
\(652\) 0 0
\(653\) −199.148 −0.304974 −0.152487 0.988305i \(-0.548728\pi\)
−0.152487 + 0.988305i \(0.548728\pi\)
\(654\) 0 0
\(655\) 49.6668i 0.0758272i
\(656\) 0 0
\(657\) −583.876 −0.888700
\(658\) 0 0
\(659\) 488.851i 0.741807i 0.928671 + 0.370904i \(0.120952\pi\)
−0.928671 + 0.370904i \(0.879048\pi\)
\(660\) 0 0
\(661\) −477.111 −0.721802 −0.360901 0.932604i \(-0.617531\pi\)
−0.360901 + 0.932604i \(0.617531\pi\)
\(662\) 0 0
\(663\) 326.814i 0.492933i
\(664\) 0 0
\(665\) −4.62811 −0.00695957
\(666\) 0 0
\(667\) − 214.930i − 0.322234i
\(668\) 0 0
\(669\) 408.765 0.611010
\(670\) 0 0
\(671\) − 262.422i − 0.391090i
\(672\) 0 0
\(673\) −437.908 −0.650680 −0.325340 0.945597i \(-0.605479\pi\)
−0.325340 + 0.945597i \(0.605479\pi\)
\(674\) 0 0
\(675\) 298.799i 0.442665i
\(676\) 0 0
\(677\) 509.653 0.752810 0.376405 0.926455i \(-0.377160\pi\)
0.376405 + 0.926455i \(0.377160\pi\)
\(678\) 0 0
\(679\) − 453.769i − 0.668290i
\(680\) 0 0
\(681\) −1090.41 −1.60119
\(682\) 0 0
\(683\) 794.509i 1.16326i 0.813452 + 0.581632i \(0.197586\pi\)
−0.813452 + 0.581632i \(0.802414\pi\)
\(684\) 0 0
\(685\) 31.3129 0.0457123
\(686\) 0 0
\(687\) − 558.060i − 0.812315i
\(688\) 0 0
\(689\) 306.765 0.445232
\(690\) 0 0
\(691\) 19.0808i 0.0276133i 0.999905 + 0.0138066i \(0.00439493\pi\)
−0.999905 + 0.0138066i \(0.995605\pi\)
\(692\) 0 0
\(693\) 241.730 0.348816
\(694\) 0 0
\(695\) − 97.6688i − 0.140531i
\(696\) 0 0
\(697\) −1307.00 −1.87519
\(698\) 0 0
\(699\) − 142.362i − 0.203665i
\(700\) 0 0
\(701\) 333.075 0.475143 0.237572 0.971370i \(-0.423649\pi\)
0.237572 + 0.971370i \(0.423649\pi\)
\(702\) 0 0
\(703\) − 208.488i − 0.296568i
\(704\) 0 0
\(705\) −121.398 −0.172195
\(706\) 0 0
\(707\) 387.410i 0.547963i
\(708\) 0 0
\(709\) −692.721 −0.977039 −0.488519 0.872553i \(-0.662463\pi\)
−0.488519 + 0.872553i \(0.662463\pi\)
\(710\) 0 0
\(711\) − 818.593i − 1.15133i
\(712\) 0 0
\(713\) 414.515 0.581367
\(714\) 0 0
\(715\) − 26.7753i − 0.0374479i
\(716\) 0 0
\(717\) −252.636 −0.352352
\(718\) 0 0
\(719\) 187.332i 0.260545i 0.991478 + 0.130272i \(0.0415852\pi\)
−0.991478 + 0.130272i \(0.958415\pi\)
\(720\) 0 0
\(721\) 314.446 0.436125
\(722\) 0 0
\(723\) 600.971i 0.831219i
\(724\) 0 0
\(725\) 271.870 0.374993
\(726\) 0 0
\(727\) 669.583i 0.921022i 0.887654 + 0.460511i \(0.152334\pi\)
−0.887654 + 0.460511i \(0.847666\pi\)
\(728\) 0 0
\(729\) 164.534 0.225698
\(730\) 0 0
\(731\) − 860.624i − 1.17732i
\(732\) 0 0
\(733\) −585.685 −0.799024 −0.399512 0.916728i \(-0.630820\pi\)
−0.399512 + 0.916728i \(0.630820\pi\)
\(734\) 0 0
\(735\) 13.2314i 0.0180019i
\(736\) 0 0
\(737\) −330.597 −0.448571
\(738\) 0 0
\(739\) − 318.797i − 0.431389i −0.976461 0.215695i \(-0.930798\pi\)
0.976461 0.215695i \(-0.0692016\pi\)
\(740\) 0 0
\(741\) −48.3014 −0.0651841
\(742\) 0 0
\(743\) 1440.64i 1.93895i 0.245185 + 0.969476i \(0.421151\pi\)
−0.245185 + 0.969476i \(0.578849\pi\)
\(744\) 0 0
\(745\) −133.122 −0.178688
\(746\) 0 0
\(747\) 35.4072i 0.0473992i
\(748\) 0 0
\(749\) −376.846 −0.503132
\(750\) 0 0
\(751\) − 238.758i − 0.317920i −0.987285 0.158960i \(-0.949186\pi\)
0.987285 0.158960i \(-0.0508141\pi\)
\(752\) 0 0
\(753\) −890.583 −1.18271
\(754\) 0 0
\(755\) 64.3733i 0.0852626i
\(756\) 0 0
\(757\) −975.208 −1.28825 −0.644127 0.764919i \(-0.722779\pi\)
−0.644127 + 0.764919i \(0.722779\pi\)
\(758\) 0 0
\(759\) − 1174.75i − 1.54777i
\(760\) 0 0
\(761\) −135.752 −0.178386 −0.0891929 0.996014i \(-0.528429\pi\)
−0.0891929 + 0.996014i \(0.528429\pi\)
\(762\) 0 0
\(763\) − 451.988i − 0.592383i
\(764\) 0 0
\(765\) 69.4823 0.0908265
\(766\) 0 0
\(767\) − 234.090i − 0.305202i
\(768\) 0 0
\(769\) −1375.28 −1.78841 −0.894203 0.447661i \(-0.852257\pi\)
−0.894203 + 0.447661i \(0.852257\pi\)
\(770\) 0 0
\(771\) − 1300.30i − 1.68652i
\(772\) 0 0
\(773\) −844.987 −1.09313 −0.546564 0.837418i \(-0.684064\pi\)
−0.546564 + 0.837418i \(0.684064\pi\)
\(774\) 0 0
\(775\) 524.330i 0.676555i
\(776\) 0 0
\(777\) −596.050 −0.767117
\(778\) 0 0
\(779\) − 193.168i − 0.247970i
\(780\) 0 0
\(781\) 999.385 1.27962
\(782\) 0 0
\(783\) 132.510i 0.169234i
\(784\) 0 0
\(785\) 65.5279 0.0834750
\(786\) 0 0
\(787\) − 706.124i − 0.897235i −0.893724 0.448617i \(-0.851917\pi\)
0.893724 0.448617i \(-0.148083\pi\)
\(788\) 0 0
\(789\) 1526.31 1.93448
\(790\) 0 0
\(791\) − 65.9947i − 0.0834319i
\(792\) 0 0
\(793\) 59.1811 0.0746294
\(794\) 0 0
\(795\) − 165.206i − 0.207807i
\(796\) 0 0
\(797\) 1103.57 1.38465 0.692325 0.721586i \(-0.256586\pi\)
0.692325 + 0.721586i \(0.256586\pi\)
\(798\) 0 0
\(799\) 1550.79i 1.94092i
\(800\) 0 0
\(801\) 140.788 0.175765
\(802\) 0 0
\(803\) − 1547.91i − 1.92766i
\(804\) 0 0
\(805\) 25.3849 0.0315340
\(806\) 0 0
\(807\) − 122.197i − 0.151421i
\(808\) 0 0
\(809\) −890.340 −1.10054 −0.550272 0.834986i \(-0.685476\pi\)
−0.550272 + 0.834986i \(0.685476\pi\)
\(810\) 0 0
\(811\) − 326.603i − 0.402717i −0.979518 0.201358i \(-0.935464\pi\)
0.979518 0.201358i \(-0.0645356\pi\)
\(812\) 0 0
\(813\) −266.143 −0.327359
\(814\) 0 0
\(815\) 14.6152i 0.0179328i
\(816\) 0 0
\(817\) 127.196 0.155686
\(818\) 0 0
\(819\) 54.5147i 0.0665625i
\(820\) 0 0
\(821\) 177.210 0.215847 0.107924 0.994159i \(-0.465580\pi\)
0.107924 + 0.994159i \(0.465580\pi\)
\(822\) 0 0
\(823\) − 1474.36i − 1.79144i −0.444614 0.895722i \(-0.646659\pi\)
0.444614 0.895722i \(-0.353341\pi\)
\(824\) 0 0
\(825\) 1485.98 1.80118
\(826\) 0 0
\(827\) − 339.749i − 0.410821i −0.978676 0.205411i \(-0.934147\pi\)
0.978676 0.205411i \(-0.0658530\pi\)
\(828\) 0 0
\(829\) 1063.05 1.28233 0.641166 0.767402i \(-0.278451\pi\)
0.641166 + 0.767402i \(0.278451\pi\)
\(830\) 0 0
\(831\) − 210.467i − 0.253269i
\(832\) 0 0
\(833\) 169.024 0.202910
\(834\) 0 0
\(835\) − 127.301i − 0.152456i
\(836\) 0 0
\(837\) −255.559 −0.305328
\(838\) 0 0
\(839\) − 670.497i − 0.799162i −0.916698 0.399581i \(-0.869156\pi\)
0.916698 0.399581i \(-0.130844\pi\)
\(840\) 0 0
\(841\) −720.432 −0.856638
\(842\) 0 0
\(843\) 541.618i 0.642489i
\(844\) 0 0
\(845\) −76.8000 −0.0908876
\(846\) 0 0
\(847\) 320.712i 0.378644i
\(848\) 0 0
\(849\) 137.747 0.162246
\(850\) 0 0
\(851\) 1143.54i 1.34376i
\(852\) 0 0
\(853\) −165.395 −0.193898 −0.0969489 0.995289i \(-0.530908\pi\)
−0.0969489 + 0.995289i \(0.530908\pi\)
\(854\) 0 0
\(855\) 10.2691i 0.0120107i
\(856\) 0 0
\(857\) −1088.22 −1.26981 −0.634904 0.772591i \(-0.718960\pi\)
−0.634904 + 0.772591i \(0.718960\pi\)
\(858\) 0 0
\(859\) − 335.213i − 0.390236i −0.980780 0.195118i \(-0.937491\pi\)
0.980780 0.195118i \(-0.0625091\pi\)
\(860\) 0 0
\(861\) −552.253 −0.641409
\(862\) 0 0
\(863\) 527.699i 0.611471i 0.952117 + 0.305735i \(0.0989023\pi\)
−0.952117 + 0.305735i \(0.901098\pi\)
\(864\) 0 0
\(865\) −41.6525 −0.0481532
\(866\) 0 0
\(867\) − 1133.91i − 1.30785i
\(868\) 0 0
\(869\) 2170.16 2.49731
\(870\) 0 0
\(871\) − 74.5560i − 0.0855982i
\(872\) 0 0
\(873\) −1006.85 −1.15332
\(874\) 0 0
\(875\) 64.5315i 0.0737503i
\(876\) 0 0
\(877\) 1257.89 1.43431 0.717154 0.696915i \(-0.245444\pi\)
0.717154 + 0.696915i \(0.245444\pi\)
\(878\) 0 0
\(879\) − 285.045i − 0.324283i
\(880\) 0 0
\(881\) 1111.12 1.26120 0.630599 0.776109i \(-0.282809\pi\)
0.630599 + 0.776109i \(0.282809\pi\)
\(882\) 0 0
\(883\) 358.772i 0.406310i 0.979147 + 0.203155i \(0.0651195\pi\)
−0.979147 + 0.203155i \(0.934880\pi\)
\(884\) 0 0
\(885\) −126.068 −0.142449
\(886\) 0 0
\(887\) 1570.56i 1.77064i 0.464978 + 0.885322i \(0.346062\pi\)
−0.464978 + 0.885322i \(0.653938\pi\)
\(888\) 0 0
\(889\) −182.227 −0.204980
\(890\) 0 0
\(891\) 1546.55i 1.73575i
\(892\) 0 0
\(893\) −229.199 −0.256662
\(894\) 0 0
\(895\) 9.49200i 0.0106056i
\(896\) 0 0
\(897\) 264.930 0.295351
\(898\) 0 0
\(899\) 232.528i 0.258651i
\(900\) 0 0
\(901\) −2110.43 −2.34232
\(902\) 0 0
\(903\) − 363.642i − 0.402705i
\(904\) 0 0
\(905\) −89.8722 −0.0993063
\(906\) 0 0
\(907\) − 901.249i − 0.993659i −0.867848 0.496829i \(-0.834497\pi\)
0.867848 0.496829i \(-0.165503\pi\)
\(908\) 0 0
\(909\) 859.607 0.945662
\(910\) 0 0
\(911\) − 901.747i − 0.989843i −0.868938 0.494922i \(-0.835197\pi\)
0.868938 0.494922i \(-0.164803\pi\)
\(912\) 0 0
\(913\) −93.8678 −0.102813
\(914\) 0 0
\(915\) − 31.8717i − 0.0348324i
\(916\) 0 0
\(917\) 268.084 0.292348
\(918\) 0 0
\(919\) 1165.98i 1.26875i 0.773025 + 0.634375i \(0.218743\pi\)
−0.773025 + 0.634375i \(0.781257\pi\)
\(920\) 0 0
\(921\) 1956.50 2.12432
\(922\) 0 0
\(923\) 225.380i 0.244183i
\(924\) 0 0
\(925\) −1446.49 −1.56378
\(926\) 0 0
\(927\) − 697.710i − 0.752654i
\(928\) 0 0
\(929\) −1400.58 −1.50762 −0.753811 0.657091i \(-0.771787\pi\)
−0.753811 + 0.657091i \(0.771787\pi\)
\(930\) 0 0
\(931\) 24.9809i 0.0268323i
\(932\) 0 0
\(933\) 936.866 1.00414
\(934\) 0 0
\(935\) 184.204i 0.197010i
\(936\) 0 0
\(937\) 1252.37 1.33657 0.668285 0.743905i \(-0.267028\pi\)
0.668285 + 0.743905i \(0.267028\pi\)
\(938\) 0 0
\(939\) 803.020i 0.855186i
\(940\) 0 0
\(941\) −335.582 −0.356622 −0.178311 0.983974i \(-0.557063\pi\)
−0.178311 + 0.983974i \(0.557063\pi\)
\(942\) 0 0
\(943\) 1059.51i 1.12356i
\(944\) 0 0
\(945\) −15.6505 −0.0165613
\(946\) 0 0
\(947\) 1588.48i 1.67738i 0.544610 + 0.838689i \(0.316678\pi\)
−0.544610 + 0.838689i \(0.683322\pi\)
\(948\) 0 0
\(949\) 349.083 0.367843
\(950\) 0 0
\(951\) 40.9946i 0.0431069i
\(952\) 0 0
\(953\) −99.1693 −0.104060 −0.0520301 0.998646i \(-0.516569\pi\)
−0.0520301 + 0.998646i \(0.516569\pi\)
\(954\) 0 0
\(955\) 45.6749i 0.0478271i
\(956\) 0 0
\(957\) 658.995 0.688605
\(958\) 0 0
\(959\) − 169.016i − 0.176242i
\(960\) 0 0
\(961\) 512.546 0.533347
\(962\) 0 0
\(963\) 836.166i 0.868292i
\(964\) 0 0
\(965\) 135.706 0.140628
\(966\) 0 0
\(967\) 1322.78i 1.36792i 0.729518 + 0.683962i \(0.239744\pi\)
−0.729518 + 0.683962i \(0.760256\pi\)
\(968\) 0 0
\(969\) 332.296 0.342927
\(970\) 0 0
\(971\) − 1571.76i − 1.61870i −0.587327 0.809350i \(-0.699820\pi\)
0.587327 0.809350i \(-0.300180\pi\)
\(972\) 0 0
\(973\) −527.181 −0.541810
\(974\) 0 0
\(975\) 335.116i 0.343709i
\(976\) 0 0
\(977\) −15.9043 −0.0162787 −0.00813936 0.999967i \(-0.502591\pi\)
−0.00813936 + 0.999967i \(0.502591\pi\)
\(978\) 0 0
\(979\) 373.242i 0.381248i
\(980\) 0 0
\(981\) −1002.90 −1.02232
\(982\) 0 0
\(983\) − 202.919i − 0.206428i −0.994659 0.103214i \(-0.967087\pi\)
0.994659 0.103214i \(-0.0329127\pi\)
\(984\) 0 0
\(985\) −87.1085 −0.0884350
\(986\) 0 0
\(987\) 655.262i 0.663892i
\(988\) 0 0
\(989\) −697.659 −0.705418
\(990\) 0 0
\(991\) − 760.920i − 0.767830i −0.923368 0.383915i \(-0.874576\pi\)
0.923368 0.383915i \(-0.125424\pi\)
\(992\) 0 0
\(993\) −473.594 −0.476932
\(994\) 0 0
\(995\) 111.474i 0.112034i
\(996\) 0 0
\(997\) 860.114 0.862702 0.431351 0.902184i \(-0.358037\pi\)
0.431351 + 0.902184i \(0.358037\pi\)
\(998\) 0 0
\(999\) − 705.023i − 0.705728i
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.d.b.127.2 8
3.2 odd 2 2016.3.m.c.127.3 8
4.3 odd 2 inner 224.3.d.b.127.7 yes 8
7.6 odd 2 1568.3.d.n.1471.7 8
8.3 odd 2 448.3.d.e.127.2 8
8.5 even 2 448.3.d.e.127.7 8
12.11 even 2 2016.3.m.c.127.4 8
16.3 odd 4 1792.3.g.f.127.1 8
16.5 even 4 1792.3.g.f.127.2 8
16.11 odd 4 1792.3.g.d.127.8 8
16.13 even 4 1792.3.g.d.127.7 8
28.27 even 2 1568.3.d.n.1471.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.3.d.b.127.2 8 1.1 even 1 trivial
224.3.d.b.127.7 yes 8 4.3 odd 2 inner
448.3.d.e.127.2 8 8.3 odd 2
448.3.d.e.127.7 8 8.5 even 2
1568.3.d.n.1471.2 8 28.27 even 2
1568.3.d.n.1471.7 8 7.6 odd 2
1792.3.g.d.127.7 8 16.13 even 4
1792.3.g.d.127.8 8 16.11 odd 4
1792.3.g.f.127.1 8 16.3 odd 4
1792.3.g.f.127.2 8 16.5 even 4
2016.3.m.c.127.3 8 3.2 odd 2
2016.3.m.c.127.4 8 12.11 even 2