Properties

Label 224.3.g.b.15.7
Level 224
Weight 3
Character 224.15
Analytic conductor 6.104
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.292213762624.3
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 15.7
Root \(1.37098 - 1.45617i\) of \(x^{8} - x^{7} - 2 x^{6} - 2 x^{5} + 24 x^{4} - 8 x^{3} - 32 x^{2} - 64 x + 256\)
Character \(\chi\) \(=\) 224.15
Dual form 224.3.g.b.15.8

$q$-expansion

\(f(q)\) \(=\) \(q+5.22363 q^{3} -6.26788i q^{5} -2.64575i q^{7} +18.2863 q^{9} +O(q^{10})\) \(q+5.22363 q^{3} -6.26788i q^{5} -2.64575i q^{7} +18.2863 q^{9} -9.80688 q^{11} -2.41653i q^{13} -32.7411i q^{15} +6.89452 q^{17} -2.77637 q^{19} -13.8204i q^{21} +42.8332i q^{23} -14.2863 q^{25} +48.5083 q^{27} +37.3505i q^{29} -7.16835i q^{31} -51.2275 q^{33} -16.5833 q^{35} +0.202653i q^{37} -12.6231i q^{39} +63.5494 q^{41} +35.3384 q^{43} -114.616i q^{45} -37.9129i q^{47} -7.00000 q^{49} +36.0144 q^{51} +54.6651i q^{53} +61.4684i q^{55} -14.5027 q^{57} -104.795 q^{59} -43.7668i q^{61} -48.3810i q^{63} -15.1465 q^{65} -31.1021 q^{67} +223.745i q^{69} +23.1294i q^{71} -69.2275 q^{73} -74.6264 q^{75} +25.9466i q^{77} +19.9328i q^{79} +88.8125 q^{81} +5.11617 q^{83} -43.2140i q^{85} +195.105i q^{87} -17.9889 q^{89} -6.39353 q^{91} -37.4448i q^{93} +17.4019i q^{95} +12.4864 q^{97} -179.332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} + 48q^{9} + O(q^{10}) \) \( 8q + 8q^{3} + 48q^{9} + 32q^{11} - 80q^{17} - 56q^{19} - 16q^{25} + 32q^{27} + 32q^{33} - 56q^{35} + 128q^{41} - 56q^{49} + 368q^{51} + 56q^{57} - 104q^{59} - 72q^{65} - 304q^{67} - 112q^{73} - 72q^{75} + 48q^{81} - 72q^{83} - 512q^{89} + 56q^{91} + 64q^{97} - 256q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 5.22363 1.74121 0.870605 0.491982i \(-0.163728\pi\)
0.870605 + 0.491982i \(0.163728\pi\)
\(4\) 0 0
\(5\) − 6.26788i − 1.25358i −0.779190 0.626788i \(-0.784369\pi\)
0.779190 0.626788i \(-0.215631\pi\)
\(6\) 0 0
\(7\) − 2.64575i − 0.377964i
\(8\) 0 0
\(9\) 18.2863 2.03181
\(10\) 0 0
\(11\) −9.80688 −0.891535 −0.445767 0.895149i \(-0.647069\pi\)
−0.445767 + 0.895149i \(0.647069\pi\)
\(12\) 0 0
\(13\) − 2.41653i − 0.185887i −0.995671 0.0929434i \(-0.970372\pi\)
0.995671 0.0929434i \(-0.0296276\pi\)
\(14\) 0 0
\(15\) − 32.7411i − 2.18274i
\(16\) 0 0
\(17\) 6.89452 0.405560 0.202780 0.979224i \(-0.435002\pi\)
0.202780 + 0.979224i \(0.435002\pi\)
\(18\) 0 0
\(19\) −2.77637 −0.146125 −0.0730624 0.997327i \(-0.523277\pi\)
−0.0730624 + 0.997327i \(0.523277\pi\)
\(20\) 0 0
\(21\) − 13.8204i − 0.658116i
\(22\) 0 0
\(23\) 42.8332i 1.86231i 0.364617 + 0.931157i \(0.381200\pi\)
−0.364617 + 0.931157i \(0.618800\pi\)
\(24\) 0 0
\(25\) −14.2863 −0.571453
\(26\) 0 0
\(27\) 48.5083 1.79660
\(28\) 0 0
\(29\) 37.3505i 1.28795i 0.765048 + 0.643974i \(0.222715\pi\)
−0.765048 + 0.643974i \(0.777285\pi\)
\(30\) 0 0
\(31\) − 7.16835i − 0.231237i −0.993294 0.115619i \(-0.963115\pi\)
0.993294 0.115619i \(-0.0368850\pi\)
\(32\) 0 0
\(33\) −51.2275 −1.55235
\(34\) 0 0
\(35\) −16.5833 −0.473807
\(36\) 0 0
\(37\) 0.202653i 0.00547709i 0.999996 + 0.00273855i \(0.000871708\pi\)
−0.999996 + 0.00273855i \(0.999128\pi\)
\(38\) 0 0
\(39\) − 12.6231i − 0.323668i
\(40\) 0 0
\(41\) 63.5494 1.54999 0.774993 0.631970i \(-0.217753\pi\)
0.774993 + 0.631970i \(0.217753\pi\)
\(42\) 0 0
\(43\) 35.3384 0.821823 0.410911 0.911675i \(-0.365211\pi\)
0.410911 + 0.911675i \(0.365211\pi\)
\(44\) 0 0
\(45\) − 114.616i − 2.54703i
\(46\) 0 0
\(47\) − 37.9129i − 0.806657i −0.915055 0.403329i \(-0.867853\pi\)
0.915055 0.403329i \(-0.132147\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 36.0144 0.706165
\(52\) 0 0
\(53\) 54.6651i 1.03142i 0.856764 + 0.515709i \(0.172471\pi\)
−0.856764 + 0.515709i \(0.827529\pi\)
\(54\) 0 0
\(55\) 61.4684i 1.11761i
\(56\) 0 0
\(57\) −14.5027 −0.254434
\(58\) 0 0
\(59\) −104.795 −1.77619 −0.888093 0.459665i \(-0.847970\pi\)
−0.888093 + 0.459665i \(0.847970\pi\)
\(60\) 0 0
\(61\) − 43.7668i − 0.717489i −0.933436 0.358745i \(-0.883205\pi\)
0.933436 0.358745i \(-0.116795\pi\)
\(62\) 0 0
\(63\) − 48.3810i − 0.767953i
\(64\) 0 0
\(65\) −15.1465 −0.233023
\(66\) 0 0
\(67\) −31.1021 −0.464210 −0.232105 0.972691i \(-0.574561\pi\)
−0.232105 + 0.972691i \(0.574561\pi\)
\(68\) 0 0
\(69\) 223.745i 3.24268i
\(70\) 0 0
\(71\) 23.1294i 0.325766i 0.986645 + 0.162883i \(0.0520794\pi\)
−0.986645 + 0.162883i \(0.947921\pi\)
\(72\) 0 0
\(73\) −69.2275 −0.948322 −0.474161 0.880438i \(-0.657249\pi\)
−0.474161 + 0.880438i \(0.657249\pi\)
\(74\) 0 0
\(75\) −74.6264 −0.995019
\(76\) 0 0
\(77\) 25.9466i 0.336968i
\(78\) 0 0
\(79\) 19.9328i 0.252315i 0.992010 + 0.126157i \(0.0402644\pi\)
−0.992010 + 0.126157i \(0.959736\pi\)
\(80\) 0 0
\(81\) 88.8125 1.09645
\(82\) 0 0
\(83\) 5.11617 0.0616406 0.0308203 0.999525i \(-0.490188\pi\)
0.0308203 + 0.999525i \(0.490188\pi\)
\(84\) 0 0
\(85\) − 43.2140i − 0.508400i
\(86\) 0 0
\(87\) 195.105i 2.24259i
\(88\) 0 0
\(89\) −17.9889 −0.202122 −0.101061 0.994880i \(-0.532224\pi\)
−0.101061 + 0.994880i \(0.532224\pi\)
\(90\) 0 0
\(91\) −6.39353 −0.0702586
\(92\) 0 0
\(93\) − 37.4448i − 0.402632i
\(94\) 0 0
\(95\) 17.4019i 0.183178i
\(96\) 0 0
\(97\) 12.4864 0.128726 0.0643629 0.997927i \(-0.479498\pi\)
0.0643629 + 0.997927i \(0.479498\pi\)
\(98\) 0 0
\(99\) −179.332 −1.81143
\(100\) 0 0
\(101\) 68.0753i 0.674013i 0.941502 + 0.337006i \(0.109414\pi\)
−0.941502 + 0.337006i \(0.890586\pi\)
\(102\) 0 0
\(103\) − 58.2931i − 0.565952i −0.959127 0.282976i \(-0.908678\pi\)
0.959127 0.282976i \(-0.0913217\pi\)
\(104\) 0 0
\(105\) −86.6248 −0.824998
\(106\) 0 0
\(107\) −135.868 −1.26979 −0.634897 0.772597i \(-0.718957\pi\)
−0.634897 + 0.772597i \(0.718957\pi\)
\(108\) 0 0
\(109\) − 44.4981i − 0.408239i −0.978946 0.204120i \(-0.934567\pi\)
0.978946 0.204120i \(-0.0654332\pi\)
\(110\) 0 0
\(111\) 1.05858i 0.00953677i
\(112\) 0 0
\(113\) −133.391 −1.18045 −0.590224 0.807240i \(-0.700961\pi\)
−0.590224 + 0.807240i \(0.700961\pi\)
\(114\) 0 0
\(115\) 268.474 2.33455
\(116\) 0 0
\(117\) − 44.1894i − 0.377687i
\(118\) 0 0
\(119\) − 18.2412i − 0.153287i
\(120\) 0 0
\(121\) −24.8251 −0.205166
\(122\) 0 0
\(123\) 331.959 2.69885
\(124\) 0 0
\(125\) − 67.1521i − 0.537217i
\(126\) 0 0
\(127\) − 130.977i − 1.03131i −0.856795 0.515657i \(-0.827548\pi\)
0.856795 0.515657i \(-0.172452\pi\)
\(128\) 0 0
\(129\) 184.595 1.43097
\(130\) 0 0
\(131\) 53.3311 0.407108 0.203554 0.979064i \(-0.434751\pi\)
0.203554 + 0.979064i \(0.434751\pi\)
\(132\) 0 0
\(133\) 7.34558i 0.0552299i
\(134\) 0 0
\(135\) − 304.044i − 2.25218i
\(136\) 0 0
\(137\) −57.7179 −0.421299 −0.210649 0.977562i \(-0.567558\pi\)
−0.210649 + 0.977562i \(0.567558\pi\)
\(138\) 0 0
\(139\) 172.422 1.24045 0.620224 0.784425i \(-0.287042\pi\)
0.620224 + 0.784425i \(0.287042\pi\)
\(140\) 0 0
\(141\) − 198.043i − 1.40456i
\(142\) 0 0
\(143\) 23.6986i 0.165725i
\(144\) 0 0
\(145\) 234.108 1.61454
\(146\) 0 0
\(147\) −36.5654 −0.248744
\(148\) 0 0
\(149\) − 219.529i − 1.47335i −0.676249 0.736673i \(-0.736396\pi\)
0.676249 0.736673i \(-0.263604\pi\)
\(150\) 0 0
\(151\) − 185.668i − 1.22959i −0.788687 0.614795i \(-0.789239\pi\)
0.788687 0.614795i \(-0.210761\pi\)
\(152\) 0 0
\(153\) 126.075 0.824022
\(154\) 0 0
\(155\) −44.9304 −0.289873
\(156\) 0 0
\(157\) 188.182i 1.19861i 0.800521 + 0.599305i \(0.204556\pi\)
−0.800521 + 0.599305i \(0.795444\pi\)
\(158\) 0 0
\(159\) 285.550i 1.79591i
\(160\) 0 0
\(161\) 113.326 0.703889
\(162\) 0 0
\(163\) −54.5154 −0.334450 −0.167225 0.985919i \(-0.553481\pi\)
−0.167225 + 0.985919i \(0.553481\pi\)
\(164\) 0 0
\(165\) 321.088i 1.94599i
\(166\) 0 0
\(167\) 266.435i 1.59542i 0.603042 + 0.797709i \(0.293955\pi\)
−0.603042 + 0.797709i \(0.706045\pi\)
\(168\) 0 0
\(169\) 163.160 0.965446
\(170\) 0 0
\(171\) −50.7696 −0.296898
\(172\) 0 0
\(173\) − 114.835i − 0.663786i −0.943317 0.331893i \(-0.892313\pi\)
0.943317 0.331893i \(-0.107687\pi\)
\(174\) 0 0
\(175\) 37.7980i 0.215989i
\(176\) 0 0
\(177\) −547.410 −3.09271
\(178\) 0 0
\(179\) −112.849 −0.630439 −0.315220 0.949019i \(-0.602078\pi\)
−0.315220 + 0.949019i \(0.602078\pi\)
\(180\) 0 0
\(181\) − 60.9470i − 0.336724i −0.985725 0.168362i \(-0.946152\pi\)
0.985725 0.168362i \(-0.0538477\pi\)
\(182\) 0 0
\(183\) − 228.622i − 1.24930i
\(184\) 0 0
\(185\) 1.27020 0.00686595
\(186\) 0 0
\(187\) −67.6138 −0.361571
\(188\) 0 0
\(189\) − 128.341i − 0.679052i
\(190\) 0 0
\(191\) − 178.459i − 0.934342i −0.884167 0.467171i \(-0.845273\pi\)
0.884167 0.467171i \(-0.154727\pi\)
\(192\) 0 0
\(193\) 221.588 1.14812 0.574062 0.818812i \(-0.305367\pi\)
0.574062 + 0.818812i \(0.305367\pi\)
\(194\) 0 0
\(195\) −79.1198 −0.405742
\(196\) 0 0
\(197\) 242.298i 1.22994i 0.788550 + 0.614970i \(0.210832\pi\)
−0.788550 + 0.614970i \(0.789168\pi\)
\(198\) 0 0
\(199\) − 297.047i − 1.49270i −0.665555 0.746349i \(-0.731805\pi\)
0.665555 0.746349i \(-0.268195\pi\)
\(200\) 0 0
\(201\) −162.466 −0.808287
\(202\) 0 0
\(203\) 98.8201 0.486798
\(204\) 0 0
\(205\) − 398.320i − 1.94302i
\(206\) 0 0
\(207\) 783.262i 3.78388i
\(208\) 0 0
\(209\) 27.2275 0.130275
\(210\) 0 0
\(211\) 141.020 0.668341 0.334171 0.942513i \(-0.391544\pi\)
0.334171 + 0.942513i \(0.391544\pi\)
\(212\) 0 0
\(213\) 120.820i 0.567228i
\(214\) 0 0
\(215\) − 221.497i − 1.03022i
\(216\) 0 0
\(217\) −18.9657 −0.0873994
\(218\) 0 0
\(219\) −361.619 −1.65123
\(220\) 0 0
\(221\) − 16.6608i − 0.0753883i
\(222\) 0 0
\(223\) − 40.8267i − 0.183079i −0.995801 0.0915396i \(-0.970821\pi\)
0.995801 0.0915396i \(-0.0291788\pi\)
\(224\) 0 0
\(225\) −261.244 −1.16108
\(226\) 0 0
\(227\) −8.18598 −0.0360616 −0.0180308 0.999837i \(-0.505740\pi\)
−0.0180308 + 0.999837i \(0.505740\pi\)
\(228\) 0 0
\(229\) 332.252i 1.45088i 0.688285 + 0.725440i \(0.258364\pi\)
−0.688285 + 0.725440i \(0.741636\pi\)
\(230\) 0 0
\(231\) 135.535i 0.586733i
\(232\) 0 0
\(233\) 329.260 1.41314 0.706568 0.707646i \(-0.250243\pi\)
0.706568 + 0.707646i \(0.250243\pi\)
\(234\) 0 0
\(235\) −237.633 −1.01121
\(236\) 0 0
\(237\) 104.122i 0.439333i
\(238\) 0 0
\(239\) 137.719i 0.576230i 0.957596 + 0.288115i \(0.0930286\pi\)
−0.957596 + 0.288115i \(0.906971\pi\)
\(240\) 0 0
\(241\) 201.854 0.837567 0.418783 0.908086i \(-0.362457\pi\)
0.418783 + 0.908086i \(0.362457\pi\)
\(242\) 0 0
\(243\) 27.3492 0.112548
\(244\) 0 0
\(245\) 43.8752i 0.179082i
\(246\) 0 0
\(247\) 6.70917i 0.0271627i
\(248\) 0 0
\(249\) 26.7250 0.107329
\(250\) 0 0
\(251\) 269.203 1.07252 0.536261 0.844052i \(-0.319836\pi\)
0.536261 + 0.844052i \(0.319836\pi\)
\(252\) 0 0
\(253\) − 420.061i − 1.66032i
\(254\) 0 0
\(255\) − 225.734i − 0.885232i
\(256\) 0 0
\(257\) −242.359 −0.943032 −0.471516 0.881858i \(-0.656293\pi\)
−0.471516 + 0.881858i \(0.656293\pi\)
\(258\) 0 0
\(259\) 0.536168 0.00207015
\(260\) 0 0
\(261\) 683.002i 2.61687i
\(262\) 0 0
\(263\) − 33.8470i − 0.128696i −0.997928 0.0643479i \(-0.979503\pi\)
0.997928 0.0643479i \(-0.0204967\pi\)
\(264\) 0 0
\(265\) 342.634 1.29296
\(266\) 0 0
\(267\) −93.9673 −0.351937
\(268\) 0 0
\(269\) 165.598i 0.615606i 0.951450 + 0.307803i \(0.0995938\pi\)
−0.951450 + 0.307803i \(0.900406\pi\)
\(270\) 0 0
\(271\) − 148.308i − 0.547263i −0.961835 0.273632i \(-0.911775\pi\)
0.961835 0.273632i \(-0.0882249\pi\)
\(272\) 0 0
\(273\) −33.3975 −0.122335
\(274\) 0 0
\(275\) 140.104 0.509470
\(276\) 0 0
\(277\) − 478.358i − 1.72693i −0.504413 0.863463i \(-0.668291\pi\)
0.504413 0.863463i \(-0.331709\pi\)
\(278\) 0 0
\(279\) − 131.083i − 0.469830i
\(280\) 0 0
\(281\) −226.066 −0.804506 −0.402253 0.915528i \(-0.631773\pi\)
−0.402253 + 0.915528i \(0.631773\pi\)
\(282\) 0 0
\(283\) 254.628 0.899745 0.449873 0.893093i \(-0.351469\pi\)
0.449873 + 0.893093i \(0.351469\pi\)
\(284\) 0 0
\(285\) 90.9014i 0.318952i
\(286\) 0 0
\(287\) − 168.136i − 0.585840i
\(288\) 0 0
\(289\) −241.466 −0.835521
\(290\) 0 0
\(291\) 65.2244 0.224139
\(292\) 0 0
\(293\) − 149.558i − 0.510437i −0.966883 0.255218i \(-0.917853\pi\)
0.966883 0.255218i \(-0.0821474\pi\)
\(294\) 0 0
\(295\) 656.842i 2.22658i
\(296\) 0 0
\(297\) −475.715 −1.60173
\(298\) 0 0
\(299\) 103.508 0.346180
\(300\) 0 0
\(301\) − 93.4966i − 0.310620i
\(302\) 0 0
\(303\) 355.600i 1.17360i
\(304\) 0 0
\(305\) −274.325 −0.899427
\(306\) 0 0
\(307\) −271.779 −0.885272 −0.442636 0.896701i \(-0.645957\pi\)
−0.442636 + 0.896701i \(0.645957\pi\)
\(308\) 0 0
\(309\) − 304.502i − 0.985442i
\(310\) 0 0
\(311\) 534.180i 1.71762i 0.512293 + 0.858811i \(0.328796\pi\)
−0.512293 + 0.858811i \(0.671204\pi\)
\(312\) 0 0
\(313\) −556.232 −1.77710 −0.888550 0.458780i \(-0.848287\pi\)
−0.888550 + 0.458780i \(0.848287\pi\)
\(314\) 0 0
\(315\) −303.247 −0.962688
\(316\) 0 0
\(317\) 387.459i 1.22227i 0.791527 + 0.611134i \(0.209286\pi\)
−0.791527 + 0.611134i \(0.790714\pi\)
\(318\) 0 0
\(319\) − 366.292i − 1.14825i
\(320\) 0 0
\(321\) −709.724 −2.21098
\(322\) 0 0
\(323\) −19.1417 −0.0592624
\(324\) 0 0
\(325\) 34.5233i 0.106225i
\(326\) 0 0
\(327\) − 232.442i − 0.710830i
\(328\) 0 0
\(329\) −100.308 −0.304888
\(330\) 0 0
\(331\) 383.205 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(332\) 0 0
\(333\) 3.70577i 0.0111284i
\(334\) 0 0
\(335\) 194.944i 0.581923i
\(336\) 0 0
\(337\) 563.726 1.67278 0.836388 0.548138i \(-0.184663\pi\)
0.836388 + 0.548138i \(0.184663\pi\)
\(338\) 0 0
\(339\) −696.783 −2.05541
\(340\) 0 0
\(341\) 70.2992i 0.206156i
\(342\) 0 0
\(343\) 18.5203i 0.0539949i
\(344\) 0 0
\(345\) 1402.41 4.06495
\(346\) 0 0
\(347\) −51.5890 −0.148671 −0.0743357 0.997233i \(-0.523684\pi\)
−0.0743357 + 0.997233i \(0.523684\pi\)
\(348\) 0 0
\(349\) 586.383i 1.68018i 0.542447 + 0.840090i \(0.317498\pi\)
−0.542447 + 0.840090i \(0.682502\pi\)
\(350\) 0 0
\(351\) − 117.222i − 0.333965i
\(352\) 0 0
\(353\) 303.844 0.860746 0.430373 0.902651i \(-0.358382\pi\)
0.430373 + 0.902651i \(0.358382\pi\)
\(354\) 0 0
\(355\) 144.972 0.408373
\(356\) 0 0
\(357\) − 95.2852i − 0.266905i
\(358\) 0 0
\(359\) 116.130i 0.323481i 0.986833 + 0.161741i \(0.0517108\pi\)
−0.986833 + 0.161741i \(0.948289\pi\)
\(360\) 0 0
\(361\) −353.292 −0.978648
\(362\) 0 0
\(363\) −129.677 −0.357237
\(364\) 0 0
\(365\) 433.910i 1.18879i
\(366\) 0 0
\(367\) − 476.288i − 1.29779i −0.760879 0.648894i \(-0.775232\pi\)
0.760879 0.648894i \(-0.224768\pi\)
\(368\) 0 0
\(369\) 1162.08 3.14928
\(370\) 0 0
\(371\) 144.630 0.389839
\(372\) 0 0
\(373\) − 49.2857i − 0.132133i −0.997815 0.0660666i \(-0.978955\pi\)
0.997815 0.0660666i \(-0.0210450\pi\)
\(374\) 0 0
\(375\) − 350.778i − 0.935407i
\(376\) 0 0
\(377\) 90.2585 0.239412
\(378\) 0 0
\(379\) −167.511 −0.441983 −0.220991 0.975276i \(-0.570929\pi\)
−0.220991 + 0.975276i \(0.570929\pi\)
\(380\) 0 0
\(381\) − 684.174i − 1.79573i
\(382\) 0 0
\(383\) − 513.207i − 1.33997i −0.742376 0.669983i \(-0.766301\pi\)
0.742376 0.669983i \(-0.233699\pi\)
\(384\) 0 0
\(385\) 162.630 0.422416
\(386\) 0 0
\(387\) 646.209 1.66979
\(388\) 0 0
\(389\) − 709.398i − 1.82365i −0.410584 0.911823i \(-0.634675\pi\)
0.410584 0.911823i \(-0.365325\pi\)
\(390\) 0 0
\(391\) 295.315i 0.755281i
\(392\) 0 0
\(393\) 278.582 0.708860
\(394\) 0 0
\(395\) 124.937 0.316295
\(396\) 0 0
\(397\) − 309.404i − 0.779355i −0.920951 0.389677i \(-0.872587\pi\)
0.920951 0.389677i \(-0.127413\pi\)
\(398\) 0 0
\(399\) 38.3706i 0.0961669i
\(400\) 0 0
\(401\) −528.073 −1.31689 −0.658445 0.752629i \(-0.728785\pi\)
−0.658445 + 0.752629i \(0.728785\pi\)
\(402\) 0 0
\(403\) −17.3225 −0.0429839
\(404\) 0 0
\(405\) − 556.666i − 1.37448i
\(406\) 0 0
\(407\) − 1.98739i − 0.00488302i
\(408\) 0 0
\(409\) −612.830 −1.49836 −0.749181 0.662366i \(-0.769552\pi\)
−0.749181 + 0.662366i \(0.769552\pi\)
\(410\) 0 0
\(411\) −301.497 −0.733569
\(412\) 0 0
\(413\) 277.261i 0.671335i
\(414\) 0 0
\(415\) − 32.0676i − 0.0772712i
\(416\) 0 0
\(417\) 900.670 2.15988
\(418\) 0 0
\(419\) 237.642 0.567165 0.283583 0.958948i \(-0.408477\pi\)
0.283583 + 0.958948i \(0.408477\pi\)
\(420\) 0 0
\(421\) 394.516i 0.937092i 0.883439 + 0.468546i \(0.155222\pi\)
−0.883439 + 0.468546i \(0.844778\pi\)
\(422\) 0 0
\(423\) − 693.287i − 1.63898i
\(424\) 0 0
\(425\) −98.4973 −0.231758
\(426\) 0 0
\(427\) −115.796 −0.271185
\(428\) 0 0
\(429\) 123.793i 0.288561i
\(430\) 0 0
\(431\) − 363.359i − 0.843059i −0.906815 0.421530i \(-0.861493\pi\)
0.906815 0.421530i \(-0.138507\pi\)
\(432\) 0 0
\(433\) 119.733 0.276520 0.138260 0.990396i \(-0.455849\pi\)
0.138260 + 0.990396i \(0.455849\pi\)
\(434\) 0 0
\(435\) 1222.89 2.81125
\(436\) 0 0
\(437\) − 118.921i − 0.272130i
\(438\) 0 0
\(439\) 871.477i 1.98514i 0.121672 + 0.992570i \(0.461174\pi\)
−0.121672 + 0.992570i \(0.538826\pi\)
\(440\) 0 0
\(441\) −128.004 −0.290259
\(442\) 0 0
\(443\) 343.956 0.776424 0.388212 0.921570i \(-0.373093\pi\)
0.388212 + 0.921570i \(0.373093\pi\)
\(444\) 0 0
\(445\) 112.752i 0.253376i
\(446\) 0 0
\(447\) − 1146.74i − 2.56541i
\(448\) 0 0
\(449\) −242.849 −0.540866 −0.270433 0.962739i \(-0.587167\pi\)
−0.270433 + 0.962739i \(0.587167\pi\)
\(450\) 0 0
\(451\) −623.222 −1.38187
\(452\) 0 0
\(453\) − 969.861i − 2.14097i
\(454\) 0 0
\(455\) 40.0739i 0.0880745i
\(456\) 0 0
\(457\) −42.2571 −0.0924662 −0.0462331 0.998931i \(-0.514722\pi\)
−0.0462331 + 0.998931i \(0.514722\pi\)
\(458\) 0 0
\(459\) 334.441 0.728631
\(460\) 0 0
\(461\) − 816.370i − 1.77087i −0.464766 0.885434i \(-0.653861\pi\)
0.464766 0.885434i \(-0.346139\pi\)
\(462\) 0 0
\(463\) 115.161i 0.248727i 0.992237 + 0.124363i \(0.0396889\pi\)
−0.992237 + 0.124363i \(0.960311\pi\)
\(464\) 0 0
\(465\) −234.700 −0.504730
\(466\) 0 0
\(467\) −603.424 −1.29213 −0.646064 0.763283i \(-0.723586\pi\)
−0.646064 + 0.763283i \(0.723586\pi\)
\(468\) 0 0
\(469\) 82.2884i 0.175455i
\(470\) 0 0
\(471\) 982.992i 2.08703i
\(472\) 0 0
\(473\) −346.559 −0.732684
\(474\) 0 0
\(475\) 39.6641 0.0835034
\(476\) 0 0
\(477\) 999.624i 2.09565i
\(478\) 0 0
\(479\) 158.595i 0.331097i 0.986202 + 0.165548i \(0.0529394\pi\)
−0.986202 + 0.165548i \(0.947061\pi\)
\(480\) 0 0
\(481\) 0.489715 0.00101812
\(482\) 0 0
\(483\) 591.974 1.22562
\(484\) 0 0
\(485\) − 78.2633i − 0.161368i
\(486\) 0 0
\(487\) − 106.987i − 0.219687i −0.993949 0.109843i \(-0.964965\pi\)
0.993949 0.109843i \(-0.0350349\pi\)
\(488\) 0 0
\(489\) −284.768 −0.582348
\(490\) 0 0
\(491\) −616.591 −1.25579 −0.627893 0.778299i \(-0.716083\pi\)
−0.627893 + 0.778299i \(0.716083\pi\)
\(492\) 0 0
\(493\) 257.514i 0.522340i
\(494\) 0 0
\(495\) 1124.03i 2.27077i
\(496\) 0 0
\(497\) 61.1947 0.123128
\(498\) 0 0
\(499\) −554.090 −1.11040 −0.555200 0.831717i \(-0.687358\pi\)
−0.555200 + 0.831717i \(0.687358\pi\)
\(500\) 0 0
\(501\) 1391.76i 2.77796i
\(502\) 0 0
\(503\) 148.158i 0.294548i 0.989096 + 0.147274i \(0.0470499\pi\)
−0.989096 + 0.147274i \(0.952950\pi\)
\(504\) 0 0
\(505\) 426.688 0.844926
\(506\) 0 0
\(507\) 852.290 1.68104
\(508\) 0 0
\(509\) 182.889i 0.359310i 0.983730 + 0.179655i \(0.0574981\pi\)
−0.983730 + 0.179655i \(0.942502\pi\)
\(510\) 0 0
\(511\) 183.159i 0.358432i
\(512\) 0 0
\(513\) −134.677 −0.262528
\(514\) 0 0
\(515\) −365.374 −0.709464
\(516\) 0 0
\(517\) 371.807i 0.719163i
\(518\) 0 0
\(519\) − 599.856i − 1.15579i
\(520\) 0 0
\(521\) 98.2461 0.188572 0.0942861 0.995545i \(-0.469943\pi\)
0.0942861 + 0.995545i \(0.469943\pi\)
\(522\) 0 0
\(523\) 574.764 1.09898 0.549488 0.835502i \(-0.314823\pi\)
0.549488 + 0.835502i \(0.314823\pi\)
\(524\) 0 0
\(525\) 197.443i 0.376082i
\(526\) 0 0
\(527\) − 49.4223i − 0.0937805i
\(528\) 0 0
\(529\) −1305.69 −2.46822
\(530\) 0 0
\(531\) −1916.31 −3.60888
\(532\) 0 0
\(533\) − 153.569i − 0.288122i
\(534\) 0 0
\(535\) 851.604i 1.59178i
\(536\) 0 0
\(537\) −589.480 −1.09773
\(538\) 0 0
\(539\) 68.6482 0.127362
\(540\) 0 0
\(541\) 370.654i 0.685128i 0.939495 + 0.342564i \(0.111295\pi\)
−0.939495 + 0.342564i \(0.888705\pi\)
\(542\) 0 0
\(543\) − 318.365i − 0.586307i
\(544\) 0 0
\(545\) −278.909 −0.511759
\(546\) 0 0
\(547\) 56.5966 0.103467 0.0517336 0.998661i \(-0.483525\pi\)
0.0517336 + 0.998661i \(0.483525\pi\)
\(548\) 0 0
\(549\) − 800.334i − 1.45780i
\(550\) 0 0
\(551\) − 103.699i − 0.188201i
\(552\) 0 0
\(553\) 52.7374 0.0953659
\(554\) 0 0
\(555\) 6.63506 0.0119551
\(556\) 0 0
\(557\) 151.525i 0.272037i 0.990706 + 0.136019i \(0.0434307\pi\)
−0.990706 + 0.136019i \(0.956569\pi\)
\(558\) 0 0
\(559\) − 85.3962i − 0.152766i
\(560\) 0 0
\(561\) −353.189 −0.629571
\(562\) 0 0
\(563\) 318.048 0.564917 0.282458 0.959280i \(-0.408850\pi\)
0.282458 + 0.959280i \(0.408850\pi\)
\(564\) 0 0
\(565\) 836.076i 1.47978i
\(566\) 0 0
\(567\) − 234.976i − 0.414419i
\(568\) 0 0
\(569\) 356.654 0.626808 0.313404 0.949620i \(-0.398531\pi\)
0.313404 + 0.949620i \(0.398531\pi\)
\(570\) 0 0
\(571\) −831.014 −1.45537 −0.727683 0.685914i \(-0.759403\pi\)
−0.727683 + 0.685914i \(0.759403\pi\)
\(572\) 0 0
\(573\) − 932.205i − 1.62689i
\(574\) 0 0
\(575\) − 611.929i − 1.06422i
\(576\) 0 0
\(577\) 771.483 1.33706 0.668530 0.743685i \(-0.266924\pi\)
0.668530 + 0.743685i \(0.266924\pi\)
\(578\) 0 0
\(579\) 1157.49 1.99912
\(580\) 0 0
\(581\) − 13.5361i − 0.0232980i
\(582\) 0 0
\(583\) − 536.094i − 0.919544i
\(584\) 0 0
\(585\) −276.974 −0.473460
\(586\) 0 0
\(587\) −144.376 −0.245956 −0.122978 0.992409i \(-0.539244\pi\)
−0.122978 + 0.992409i \(0.539244\pi\)
\(588\) 0 0
\(589\) 19.9020i 0.0337894i
\(590\) 0 0
\(591\) 1265.68i 2.14158i
\(592\) 0 0
\(593\) 838.926 1.41471 0.707357 0.706856i \(-0.249887\pi\)
0.707357 + 0.706856i \(0.249887\pi\)
\(594\) 0 0
\(595\) −114.334 −0.192157
\(596\) 0 0
\(597\) − 1551.66i − 2.59910i
\(598\) 0 0
\(599\) 711.341i 1.18755i 0.804632 + 0.593774i \(0.202363\pi\)
−0.804632 + 0.593774i \(0.797637\pi\)
\(600\) 0 0
\(601\) −356.394 −0.593002 −0.296501 0.955033i \(-0.595820\pi\)
−0.296501 + 0.955033i \(0.595820\pi\)
\(602\) 0 0
\(603\) −568.742 −0.943188
\(604\) 0 0
\(605\) 155.601i 0.257191i
\(606\) 0 0
\(607\) 60.3719i 0.0994595i 0.998763 + 0.0497298i \(0.0158360\pi\)
−0.998763 + 0.0497298i \(0.984164\pi\)
\(608\) 0 0
\(609\) 516.199 0.847618
\(610\) 0 0
\(611\) −91.6176 −0.149947
\(612\) 0 0
\(613\) − 482.989i − 0.787911i −0.919130 0.393955i \(-0.871106\pi\)
0.919130 0.393955i \(-0.128894\pi\)
\(614\) 0 0
\(615\) − 2080.68i − 3.38321i
\(616\) 0 0
\(617\) 712.490 1.15476 0.577382 0.816474i \(-0.304074\pi\)
0.577382 + 0.816474i \(0.304074\pi\)
\(618\) 0 0
\(619\) 93.0817 0.150374 0.0751872 0.997169i \(-0.476045\pi\)
0.0751872 + 0.997169i \(0.476045\pi\)
\(620\) 0 0
\(621\) 2077.77i 3.34584i
\(622\) 0 0
\(623\) 47.5941i 0.0763951i
\(624\) 0 0
\(625\) −778.059 −1.24489
\(626\) 0 0
\(627\) 142.227 0.226837
\(628\) 0 0
\(629\) 1.39719i 0.00222129i
\(630\) 0 0
\(631\) 610.573i 0.967628i 0.875171 + 0.483814i \(0.160749\pi\)
−0.875171 + 0.483814i \(0.839251\pi\)
\(632\) 0 0
\(633\) 736.636 1.16372
\(634\) 0 0
\(635\) −820.947 −1.29283
\(636\) 0 0
\(637\) 16.9157i 0.0265553i
\(638\) 0 0
\(639\) 422.952i 0.661897i
\(640\) 0 0
\(641\) 590.153 0.920676 0.460338 0.887744i \(-0.347728\pi\)
0.460338 + 0.887744i \(0.347728\pi\)
\(642\) 0 0
\(643\) 257.971 0.401199 0.200600 0.979673i \(-0.435711\pi\)
0.200600 + 0.979673i \(0.435711\pi\)
\(644\) 0 0
\(645\) − 1157.02i − 1.79383i
\(646\) 0 0
\(647\) 379.964i 0.587271i 0.955918 + 0.293635i \(0.0948651\pi\)
−0.955918 + 0.293635i \(0.905135\pi\)
\(648\) 0 0
\(649\) 1027.71 1.58353
\(650\) 0 0
\(651\) −99.0696 −0.152181
\(652\) 0 0
\(653\) 952.773i 1.45907i 0.683943 + 0.729535i \(0.260264\pi\)
−0.683943 + 0.729535i \(0.739736\pi\)
\(654\) 0 0
\(655\) − 334.273i − 0.510340i
\(656\) 0 0
\(657\) −1265.92 −1.92681
\(658\) 0 0
\(659\) 963.119 1.46149 0.730743 0.682653i \(-0.239174\pi\)
0.730743 + 0.682653i \(0.239174\pi\)
\(660\) 0 0
\(661\) − 71.6817i − 0.108444i −0.998529 0.0542222i \(-0.982732\pi\)
0.998529 0.0542222i \(-0.0172679\pi\)
\(662\) 0 0
\(663\) − 87.0299i − 0.131267i
\(664\) 0 0
\(665\) 46.0412 0.0692349
\(666\) 0 0
\(667\) −1599.84 −2.39856
\(668\) 0 0
\(669\) − 213.263i − 0.318779i
\(670\) 0 0
\(671\) 429.216i 0.639667i
\(672\) 0 0
\(673\) −712.783 −1.05911 −0.529556 0.848275i \(-0.677642\pi\)
−0.529556 + 0.848275i \(0.677642\pi\)
\(674\) 0 0
\(675\) −693.005 −1.02667
\(676\) 0 0
\(677\) − 767.527i − 1.13372i −0.823815 0.566859i \(-0.808159\pi\)
0.823815 0.566859i \(-0.191841\pi\)
\(678\) 0 0
\(679\) − 33.0359i − 0.0486538i
\(680\) 0 0
\(681\) −42.7605 −0.0627908
\(682\) 0 0
\(683\) 484.354 0.709156 0.354578 0.935026i \(-0.384625\pi\)
0.354578 + 0.935026i \(0.384625\pi\)
\(684\) 0 0
\(685\) 361.769i 0.528130i
\(686\) 0 0
\(687\) 1735.56i 2.52629i
\(688\) 0 0
\(689\) 132.100 0.191727
\(690\) 0 0
\(691\) 574.851 0.831912 0.415956 0.909385i \(-0.363447\pi\)
0.415956 + 0.909385i \(0.363447\pi\)
\(692\) 0 0
\(693\) 474.467i 0.684657i
\(694\) 0 0
\(695\) − 1080.72i − 1.55500i
\(696\) 0 0
\(697\) 438.143 0.628612
\(698\) 0 0
\(699\) 1719.94 2.46057
\(700\) 0 0
\(701\) 143.138i 0.204191i 0.994775 + 0.102096i \(0.0325548\pi\)
−0.994775 + 0.102096i \(0.967445\pi\)
\(702\) 0 0
\(703\) − 0.562638i 0 0.000800339i
\(704\) 0 0
\(705\) −1241.31 −1.76072
\(706\) 0 0
\(707\) 180.110 0.254753
\(708\) 0 0
\(709\) − 255.311i − 0.360101i −0.983657 0.180050i \(-0.942374\pi\)
0.983657 0.180050i \(-0.0576261\pi\)
\(710\) 0 0
\(711\) 364.498i 0.512656i
\(712\) 0 0
\(713\) 307.044 0.430636
\(714\) 0 0
\(715\) 148.540 0.207748
\(716\) 0 0
\(717\) 719.393i 1.00334i
\(718\) 0 0
\(719\) − 415.630i − 0.578067i −0.957319 0.289034i \(-0.906666\pi\)
0.957319 0.289034i \(-0.0933340\pi\)
\(720\) 0 0
\(721\) −154.229 −0.213910
\(722\) 0 0
\(723\) 1054.41 1.45838
\(724\) 0 0
\(725\) − 533.601i − 0.736001i
\(726\) 0 0
\(727\) − 896.838i − 1.23361i −0.787114 0.616807i \(-0.788426\pi\)
0.787114 0.616807i \(-0.211574\pi\)
\(728\) 0 0
\(729\) −656.451 −0.900481
\(730\) 0 0
\(731\) 243.641 0.333299
\(732\) 0 0
\(733\) − 509.059i − 0.694487i −0.937775 0.347244i \(-0.887118\pi\)
0.937775 0.347244i \(-0.112882\pi\)
\(734\) 0 0
\(735\) 229.188i 0.311820i
\(736\) 0 0
\(737\) 305.014 0.413859
\(738\) 0 0
\(739\) −741.427 −1.00328 −0.501642 0.865075i \(-0.667271\pi\)
−0.501642 + 0.865075i \(0.667271\pi\)
\(740\) 0 0
\(741\) 35.0463i 0.0472959i
\(742\) 0 0
\(743\) − 1344.98i − 1.81021i −0.425191 0.905104i \(-0.639793\pi\)
0.425191 0.905104i \(-0.360207\pi\)
\(744\) 0 0
\(745\) −1375.98 −1.84695
\(746\) 0 0
\(747\) 93.5560 0.125242
\(748\) 0 0
\(749\) 359.473i 0.479937i
\(750\) 0 0
\(751\) − 27.6931i − 0.0368749i −0.999830 0.0184375i \(-0.994131\pi\)
0.999830 0.0184375i \(-0.00586916\pi\)
\(752\) 0 0
\(753\) 1406.22 1.86749
\(754\) 0 0
\(755\) −1163.74 −1.54138
\(756\) 0 0
\(757\) 1341.69i 1.77238i 0.463318 + 0.886192i \(0.346659\pi\)
−0.463318 + 0.886192i \(0.653341\pi\)
\(758\) 0 0
\(759\) − 2194.24i − 2.89096i
\(760\) 0 0
\(761\) −112.001 −0.147176 −0.0735881 0.997289i \(-0.523445\pi\)
−0.0735881 + 0.997289i \(0.523445\pi\)
\(762\) 0 0
\(763\) −117.731 −0.154300
\(764\) 0 0
\(765\) − 790.226i − 1.03297i
\(766\) 0 0
\(767\) 253.240i 0.330169i
\(768\) 0 0
\(769\) 140.749 0.183028 0.0915142 0.995804i \(-0.470829\pi\)
0.0915142 + 0.995804i \(0.470829\pi\)
\(770\) 0 0
\(771\) −1265.99 −1.64202
\(772\) 0 0
\(773\) − 1325.14i − 1.71428i −0.515087 0.857138i \(-0.672240\pi\)
0.515087 0.857138i \(-0.327760\pi\)
\(774\) 0 0
\(775\) 102.409i 0.132141i
\(776\) 0 0
\(777\) 2.80074 0.00360456
\(778\) 0 0
\(779\) −176.437 −0.226491
\(780\) 0 0
\(781\) − 226.827i − 0.290432i
\(782\) 0 0
\(783\) 1811.81i 2.31393i
\(784\) 0 0
\(785\) 1179.50 1.50255
\(786\) 0 0
\(787\) 327.801 0.416519 0.208260 0.978074i \(-0.433220\pi\)
0.208260 + 0.978074i \(0.433220\pi\)
\(788\) 0 0
\(789\) − 176.804i − 0.224086i
\(790\) 0 0
\(791\) 352.918i 0.446167i
\(792\) 0 0
\(793\) −105.764 −0.133372
\(794\) 0 0
\(795\) 1789.80 2.25132
\(796\) 0 0
\(797\) 393.650i 0.493915i 0.969026 + 0.246958i \(0.0794308\pi\)
−0.969026 + 0.246958i \(0.920569\pi\)
\(798\) 0 0
\(799\) − 261.391i − 0.327148i
\(800\) 0 0
\(801\) −328.950 −0.410675
\(802\) 0 0
\(803\) 678.906 0.845462
\(804\) 0 0
\(805\) − 710.314i − 0.882378i
\(806\) 0 0
\(807\) 865.023i 1.07190i
\(808\) 0 0
\(809\) 416.641 0.515008 0.257504 0.966277i \(-0.417100\pi\)
0.257504 + 0.966277i \(0.417100\pi\)
\(810\) 0 0
\(811\) 748.707 0.923190 0.461595 0.887091i \(-0.347277\pi\)
0.461595 + 0.887091i \(0.347277\pi\)
\(812\) 0 0
\(813\) − 774.708i − 0.952901i
\(814\) 0 0
\(815\) 341.696i 0.419259i
\(816\) 0 0
\(817\) −98.1124 −0.120089
\(818\) 0 0
\(819\) −116.914 −0.142752
\(820\) 0 0
\(821\) 554.169i 0.674993i 0.941327 + 0.337496i \(0.109580\pi\)
−0.941327 + 0.337496i \(0.890420\pi\)
\(822\) 0 0
\(823\) − 121.452i − 0.147572i −0.997274 0.0737861i \(-0.976492\pi\)
0.997274 0.0737861i \(-0.0235082\pi\)
\(824\) 0 0
\(825\) 731.853 0.887094
\(826\) 0 0
\(827\) 1516.61 1.83386 0.916932 0.399043i \(-0.130657\pi\)
0.916932 + 0.399043i \(0.130657\pi\)
\(828\) 0 0
\(829\) − 325.042i − 0.392089i −0.980595 0.196044i \(-0.937190\pi\)
0.980595 0.196044i \(-0.0628097\pi\)
\(830\) 0 0
\(831\) − 2498.77i − 3.00694i
\(832\) 0 0
\(833\) −48.2617 −0.0579372
\(834\) 0 0
\(835\) 1669.98 1.99998
\(836\) 0 0
\(837\) − 347.724i − 0.415441i
\(838\) 0 0
\(839\) 1165.70i 1.38939i 0.719303 + 0.694696i \(0.244461\pi\)
−0.719303 + 0.694696i \(0.755539\pi\)
\(840\) 0 0
\(841\) −554.058 −0.658808
\(842\) 0 0
\(843\) −1180.89 −1.40081
\(844\) 0 0
\(845\) − 1022.67i − 1.21026i
\(846\) 0 0
\(847\) 65.6810i 0.0775454i
\(848\) 0 0
\(849\) 1330.08 1.56665
\(850\) 0 0
\(851\) −8.68026 −0.0102001
\(852\) 0 0
\(853\) − 151.949i − 0.178134i −0.996026 0.0890672i \(-0.971611\pi\)
0.996026 0.0890672i \(-0.0283886\pi\)
\(854\) 0 0
\(855\) 318.218i 0.372184i
\(856\) 0 0
\(857\) 412.018 0.480768 0.240384 0.970678i \(-0.422727\pi\)
0.240384 + 0.970678i \(0.422727\pi\)
\(858\) 0 0
\(859\) −159.993 −0.186255 −0.0931274 0.995654i \(-0.529686\pi\)
−0.0931274 + 0.995654i \(0.529686\pi\)
\(860\) 0 0
\(861\) − 878.280i − 1.02007i
\(862\) 0 0
\(863\) − 992.910i − 1.15053i −0.817966 0.575266i \(-0.804898\pi\)
0.817966 0.575266i \(-0.195102\pi\)
\(864\) 0 0
\(865\) −719.772 −0.832107
\(866\) 0 0
\(867\) −1261.33 −1.45482
\(868\) 0 0
\(869\) − 195.479i − 0.224947i
\(870\) 0 0
\(871\) 75.1590i 0.0862905i
\(872\) 0 0
\(873\) 228.330 0.261547
\(874\) 0 0
\(875\) −177.668 −0.203049
\(876\) 0 0
\(877\) − 825.096i − 0.940817i −0.882449 0.470408i \(-0.844107\pi\)
0.882449 0.470408i \(-0.155893\pi\)
\(878\) 0 0
\(879\) − 781.236i − 0.888778i
\(880\) 0 0
\(881\) 1352.83 1.53556 0.767780 0.640714i \(-0.221361\pi\)
0.767780 + 0.640714i \(0.221361\pi\)
\(882\) 0 0
\(883\) 1013.40 1.14768 0.573838 0.818969i \(-0.305454\pi\)
0.573838 + 0.818969i \(0.305454\pi\)
\(884\) 0 0
\(885\) 3431.10i 3.87695i
\(886\) 0 0
\(887\) 233.760i 0.263540i 0.991280 + 0.131770i \(0.0420661\pi\)
−0.991280 + 0.131770i \(0.957934\pi\)
\(888\) 0 0
\(889\) −346.532 −0.389800
\(890\) 0 0
\(891\) −870.974 −0.977524
\(892\) 0 0
\(893\) 105.260i 0.117873i
\(894\) 0 0
\(895\) 707.322i 0.790304i
\(896\) 0 0
\(897\) 540.686 0.602772
\(898\) 0 0
\(899\) 267.741 0.297821
\(900\) 0 0
\(901\) 376.890i 0.418302i
\(902\) 0 0
\(903\) − 488.392i − 0.540854i
\(904\) 0 0
\(905\) −382.008 −0.422109
\(906\) 0 0
\(907\) 203.590 0.224465 0.112233 0.993682i \(-0.464200\pi\)
0.112233 + 0.993682i \(0.464200\pi\)
\(908\) 0 0
\(909\) 1244.85i 1.36947i
\(910\) 0 0
\(911\) − 712.022i − 0.781583i −0.920479 0.390792i \(-0.872201\pi\)
0.920479 0.390792i \(-0.127799\pi\)
\(912\) 0 0
\(913\) −50.1737 −0.0549548
\(914\) 0 0
\(915\) −1432.97 −1.56609
\(916\) 0 0
\(917\) − 141.101i − 0.153872i
\(918\) 0 0
\(919\) 786.783i 0.856129i 0.903748 + 0.428065i \(0.140804\pi\)
−0.903748 + 0.428065i \(0.859196\pi\)
\(920\) 0 0
\(921\) −1419.67 −1.54145
\(922\) 0 0
\(923\) 55.8929 0.0605557
\(924\) 0 0
\(925\) − 2.89516i − 0.00312990i
\(926\) 0 0
\(927\) − 1065.97i − 1.14991i
\(928\) 0 0
\(929\) −1657.99 −1.78471 −0.892353 0.451338i \(-0.850947\pi\)
−0.892353 + 0.451338i \(0.850947\pi\)
\(930\) 0 0
\(931\) 19.4346 0.0208750
\(932\) 0 0
\(933\) 2790.36i 2.99074i
\(934\) 0 0
\(935\) 423.795i 0.453257i
\(936\) 0 0
\(937\) 333.736 0.356175 0.178088 0.984015i \(-0.443009\pi\)
0.178088 + 0.984015i \(0.443009\pi\)
\(938\) 0 0
\(939\) −2905.55 −3.09430
\(940\) 0 0
\(941\) 543.324i 0.577390i 0.957421 + 0.288695i \(0.0932214\pi\)
−0.957421 + 0.288695i \(0.906779\pi\)
\(942\) 0 0
\(943\) 2722.03i 2.88656i
\(944\) 0 0
\(945\) −804.425 −0.851243
\(946\) 0 0
\(947\) 359.728 0.379861 0.189930 0.981798i \(-0.439174\pi\)
0.189930 + 0.981798i \(0.439174\pi\)
\(948\) 0 0
\(949\) 167.290i 0.176281i
\(950\) 0 0
\(951\) 2023.94i 2.12823i
\(952\) 0 0
\(953\) −904.225 −0.948820 −0.474410 0.880304i \(-0.657339\pi\)
−0.474410 + 0.880304i \(0.657339\pi\)
\(954\) 0 0
\(955\) −1118.56 −1.17127
\(956\) 0 0
\(957\) − 1913.37i − 1.99934i
\(958\) 0 0
\(959\) 152.707i 0.159236i
\(960\) 0 0
\(961\) 909.615 0.946529
\(962\) 0 0
\(963\) −2484.52 −2.57998
\(964\) 0 0
\(965\) − 1388.89i − 1.43926i
\(966\) 0 0
\(967\) 920.961i 0.952390i 0.879340 + 0.476195i \(0.157984\pi\)
−0.879340 + 0.476195i \(0.842016\pi\)
\(968\) 0 0
\(969\) −99.9894 −0.103188
\(970\) 0 0
\(971\) −1327.31 −1.36695 −0.683476 0.729973i \(-0.739532\pi\)
−0.683476 + 0.729973i \(0.739532\pi\)
\(972\) 0 0
\(973\) − 456.186i − 0.468845i
\(974\) 0 0
\(975\) 180.337i 0.184961i
\(976\) 0 0
\(977\) −688.490 −0.704698 −0.352349 0.935869i \(-0.614617\pi\)
−0.352349 + 0.935869i \(0.614617\pi\)
\(978\) 0 0
\(979\) 176.415 0.180199
\(980\) 0 0
\(981\) − 813.706i − 0.829466i
\(982\) 0 0
\(983\) − 1687.08i − 1.71625i −0.513438 0.858127i \(-0.671628\pi\)
0.513438 0.858127i \(-0.328372\pi\)
\(984\) 0 0
\(985\) 1518.70 1.54182
\(986\) 0 0
\(987\) −523.972 −0.530874
\(988\) 0 0
\(989\) 1513.66i 1.53049i
\(990\) 0 0
\(991\) − 1139.22i − 1.14957i −0.818304 0.574785i \(-0.805086\pi\)
0.818304 0.574785i \(-0.194914\pi\)
\(992\) 0 0
\(993\) 2001.72 2.01583
\(994\) 0 0
\(995\) −1861.85 −1.87121
\(996\) 0 0
\(997\) − 206.085i − 0.206705i −0.994645 0.103353i \(-0.967043\pi\)
0.994645 0.103353i \(-0.0329570\pi\)
\(998\) 0 0
\(999\) 9.83033i 0.00984017i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 224.3.g.b.15.7 8
3.2 odd 2 2016.3.g.b.1135.8 8
4.3 odd 2 56.3.g.b.43.5 8
7.6 odd 2 1568.3.g.m.687.2 8
8.3 odd 2 inner 224.3.g.b.15.8 8
8.5 even 2 56.3.g.b.43.6 yes 8
12.11 even 2 504.3.g.b.379.4 8
16.3 odd 4 1792.3.d.j.1023.1 16
16.5 even 4 1792.3.d.j.1023.2 16
16.11 odd 4 1792.3.d.j.1023.16 16
16.13 even 4 1792.3.d.j.1023.15 16
24.5 odd 2 504.3.g.b.379.3 8
24.11 even 2 2016.3.g.b.1135.1 8
28.3 even 6 392.3.k.n.275.6 16
28.11 odd 6 392.3.k.o.275.6 16
28.19 even 6 392.3.k.n.67.1 16
28.23 odd 6 392.3.k.o.67.1 16
28.27 even 2 392.3.g.m.99.5 8
56.5 odd 6 392.3.k.n.67.6 16
56.13 odd 2 392.3.g.m.99.6 8
56.27 even 2 1568.3.g.m.687.1 8
56.37 even 6 392.3.k.o.67.6 16
56.45 odd 6 392.3.k.n.275.1 16
56.53 even 6 392.3.k.o.275.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.g.b.43.5 8 4.3 odd 2
56.3.g.b.43.6 yes 8 8.5 even 2
224.3.g.b.15.7 8 1.1 even 1 trivial
224.3.g.b.15.8 8 8.3 odd 2 inner
392.3.g.m.99.5 8 28.27 even 2
392.3.g.m.99.6 8 56.13 odd 2
392.3.k.n.67.1 16 28.19 even 6
392.3.k.n.67.6 16 56.5 odd 6
392.3.k.n.275.1 16 56.45 odd 6
392.3.k.n.275.6 16 28.3 even 6
392.3.k.o.67.1 16 28.23 odd 6
392.3.k.o.67.6 16 56.37 even 6
392.3.k.o.275.1 16 56.53 even 6
392.3.k.o.275.6 16 28.11 odd 6
504.3.g.b.379.3 8 24.5 odd 2
504.3.g.b.379.4 8 12.11 even 2
1568.3.g.m.687.1 8 56.27 even 2
1568.3.g.m.687.2 8 7.6 odd 2
1792.3.d.j.1023.1 16 16.3 odd 4
1792.3.d.j.1023.2 16 16.5 even 4
1792.3.d.j.1023.15 16 16.13 even 4
1792.3.d.j.1023.16 16 16.11 odd 4
2016.3.g.b.1135.1 8 24.11 even 2
2016.3.g.b.1135.8 8 3.2 odd 2