Properties

Label 1216.2.h.d.1215.8
Level $1216$
Weight $2$
Character 1216.1215
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14453810176.1
Defining polynomial: \(x^{8} + 3 x^{6} + 6 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1215.8
Root \(0.331077 - 1.37491i\) of defining polynomial
Character \(\chi\) \(=\) 1216.1215
Dual form 1216.2.h.d.1215.7

$q$-expansion

\(f(q)\) \(=\) \(q+2.35829 q^{3} +2.56155 q^{5} +4.15286i q^{7} +2.56155 q^{9} +O(q^{10})\) \(q+2.35829 q^{3} +2.56155 q^{5} +4.15286i q^{7} +2.56155 q^{9} +2.33205i q^{11} -4.29400i q^{13} +6.04090 q^{15} -1.00000 q^{17} +(3.68260 - 2.33205i) q^{19} +9.79366i q^{21} -1.82081i q^{23} +1.56155 q^{25} -1.03399 q^{27} -1.20565i q^{29} +1.32431 q^{31} +5.49966i q^{33} +10.6378i q^{35} +5.49966i q^{37} -10.1265i q^{39} +5.49966i q^{41} -1.30957i q^{43} +6.56155 q^{45} +6.99614i q^{47} -10.2462 q^{49} -2.35829 q^{51} -9.79366i q^{53} +5.97366i q^{55} +(8.68466 - 5.49966i) q^{57} +6.33122 q^{59} -11.6847 q^{61} +10.6378i q^{63} -10.9993i q^{65} -0.290319 q^{67} -4.29400i q^{69} -2.06798 q^{71} -0.123106 q^{73} +3.68260 q^{75} -9.68466 q^{77} +13.4061 q^{79} -10.1231 q^{81} -11.9473i q^{83} -2.56155 q^{85} -2.84329i q^{87} -14.0877i q^{89} +17.8324 q^{91} +3.12311 q^{93} +(9.43318 - 5.97366i) q^{95} -2.41131i q^{97} +5.97366i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{5} + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{5} + 4q^{9} - 8q^{17} - 4q^{25} + 36q^{45} - 16q^{49} + 20q^{57} - 44q^{61} + 32q^{73} - 28q^{77} - 48q^{81} - 4q^{85} - 8q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 2.35829 1.36156 0.680781 0.732487i \(-0.261641\pi\)
0.680781 + 0.732487i \(0.261641\pi\)
\(4\) 0 0
\(5\) 2.56155 1.14556 0.572781 0.819709i \(-0.305865\pi\)
0.572781 + 0.819709i \(0.305865\pi\)
\(6\) 0 0
\(7\) 4.15286i 1.56963i 0.619729 + 0.784816i \(0.287243\pi\)
−0.619729 + 0.784816i \(0.712757\pi\)
\(8\) 0 0
\(9\) 2.56155 0.853851
\(10\) 0 0
\(11\) 2.33205i 0.703139i 0.936162 + 0.351569i \(0.114352\pi\)
−0.936162 + 0.351569i \(0.885648\pi\)
\(12\) 0 0
\(13\) 4.29400i 1.19094i −0.803377 0.595471i \(-0.796966\pi\)
0.803377 0.595471i \(-0.203034\pi\)
\(14\) 0 0
\(15\) 6.04090 1.55975
\(16\) 0 0
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) 3.68260 2.33205i 0.844847 0.535008i
\(20\) 0 0
\(21\) 9.79366i 2.13715i
\(22\) 0 0
\(23\) 1.82081i 0.379665i −0.981817 0.189832i \(-0.939206\pi\)
0.981817 0.189832i \(-0.0607944\pi\)
\(24\) 0 0
\(25\) 1.56155 0.312311
\(26\) 0 0
\(27\) −1.03399 −0.198991
\(28\) 0 0
\(29\) 1.20565i 0.223884i −0.993715 0.111942i \(-0.964293\pi\)
0.993715 0.111942i \(-0.0357071\pi\)
\(30\) 0 0
\(31\) 1.32431 0.237853 0.118926 0.992903i \(-0.462055\pi\)
0.118926 + 0.992903i \(0.462055\pi\)
\(32\) 0 0
\(33\) 5.49966i 0.957367i
\(34\) 0 0
\(35\) 10.6378i 1.79811i
\(36\) 0 0
\(37\) 5.49966i 0.904138i 0.891983 + 0.452069i \(0.149314\pi\)
−0.891983 + 0.452069i \(0.850686\pi\)
\(38\) 0 0
\(39\) 10.1265i 1.62154i
\(40\) 0 0
\(41\) 5.49966i 0.858902i 0.903090 + 0.429451i \(0.141293\pi\)
−0.903090 + 0.429451i \(0.858707\pi\)
\(42\) 0 0
\(43\) 1.30957i 0.199707i −0.995002 0.0998536i \(-0.968163\pi\)
0.995002 0.0998536i \(-0.0318375\pi\)
\(44\) 0 0
\(45\) 6.56155 0.978139
\(46\) 0 0
\(47\) 6.99614i 1.02049i 0.860028 + 0.510246i \(0.170446\pi\)
−0.860028 + 0.510246i \(0.829554\pi\)
\(48\) 0 0
\(49\) −10.2462 −1.46374
\(50\) 0 0
\(51\) −2.35829 −0.330227
\(52\) 0 0
\(53\) 9.79366i 1.34526i −0.739978 0.672631i \(-0.765164\pi\)
0.739978 0.672631i \(-0.234836\pi\)
\(54\) 0 0
\(55\) 5.97366i 0.805489i
\(56\) 0 0
\(57\) 8.68466 5.49966i 1.15031 0.728447i
\(58\) 0 0
\(59\) 6.33122 0.824254 0.412127 0.911126i \(-0.364786\pi\)
0.412127 + 0.911126i \(0.364786\pi\)
\(60\) 0 0
\(61\) −11.6847 −1.49607 −0.748034 0.663661i \(-0.769002\pi\)
−0.748034 + 0.663661i \(0.769002\pi\)
\(62\) 0 0
\(63\) 10.6378i 1.34023i
\(64\) 0 0
\(65\) 10.9993i 1.36430i
\(66\) 0 0
\(67\) −0.290319 −0.0354681 −0.0177341 0.999843i \(-0.505645\pi\)
−0.0177341 + 0.999843i \(0.505645\pi\)
\(68\) 0 0
\(69\) 4.29400i 0.516937i
\(70\) 0 0
\(71\) −2.06798 −0.245423 −0.122712 0.992442i \(-0.539159\pi\)
−0.122712 + 0.992442i \(0.539159\pi\)
\(72\) 0 0
\(73\) −0.123106 −0.0144084 −0.00720421 0.999974i \(-0.502293\pi\)
−0.00720421 + 0.999974i \(0.502293\pi\)
\(74\) 0 0
\(75\) 3.68260 0.425230
\(76\) 0 0
\(77\) −9.68466 −1.10367
\(78\) 0 0
\(79\) 13.4061 1.50830 0.754152 0.656700i \(-0.228048\pi\)
0.754152 + 0.656700i \(0.228048\pi\)
\(80\) 0 0
\(81\) −10.1231 −1.12479
\(82\) 0 0
\(83\) 11.9473i 1.31139i −0.755026 0.655695i \(-0.772376\pi\)
0.755026 0.655695i \(-0.227624\pi\)
\(84\) 0 0
\(85\) −2.56155 −0.277839
\(86\) 0 0
\(87\) 2.84329i 0.304832i
\(88\) 0 0
\(89\) 14.0877i 1.49329i −0.665223 0.746644i \(-0.731664\pi\)
0.665223 0.746644i \(-0.268336\pi\)
\(90\) 0 0
\(91\) 17.8324 1.86934
\(92\) 0 0
\(93\) 3.12311 0.323851
\(94\) 0 0
\(95\) 9.43318 5.97366i 0.967824 0.612885i
\(96\) 0 0
\(97\) 2.41131i 0.244831i −0.992479 0.122416i \(-0.960936\pi\)
0.992479 0.122416i \(-0.0390641\pi\)
\(98\) 0 0
\(99\) 5.97366i 0.600376i
\(100\) 0 0
\(101\) 8.24621 0.820529 0.410264 0.911967i \(-0.365436\pi\)
0.410264 + 0.911967i \(0.365436\pi\)
\(102\) 0 0
\(103\) −12.0818 −1.19045 −0.595227 0.803557i \(-0.702938\pi\)
−0.595227 + 0.803557i \(0.702938\pi\)
\(104\) 0 0
\(105\) 25.0870i 2.44824i
\(106\) 0 0
\(107\) −5.75058 −0.555929 −0.277965 0.960591i \(-0.589660\pi\)
−0.277965 + 0.960591i \(0.589660\pi\)
\(108\) 0 0
\(109\) 12.2050i 1.16902i −0.811385 0.584512i \(-0.801286\pi\)
0.811385 0.584512i \(-0.198714\pi\)
\(110\) 0 0
\(111\) 12.9698i 1.23104i
\(112\) 0 0
\(113\) 5.49966i 0.517364i −0.965963 0.258682i \(-0.916712\pi\)
0.965963 0.258682i \(-0.0832882\pi\)
\(114\) 0 0
\(115\) 4.66410i 0.434929i
\(116\) 0 0
\(117\) 10.9993i 1.01689i
\(118\) 0 0
\(119\) 4.15286i 0.380692i
\(120\) 0 0
\(121\) 5.56155 0.505596
\(122\) 0 0
\(123\) 12.9698i 1.16945i
\(124\) 0 0
\(125\) −8.80776 −0.787790
\(126\) 0 0
\(127\) −6.04090 −0.536043 −0.268021 0.963413i \(-0.586370\pi\)
−0.268021 + 0.963413i \(0.586370\pi\)
\(128\) 0 0
\(129\) 3.08835i 0.271914i
\(130\) 0 0
\(131\) 5.97366i 0.521921i 0.965349 + 0.260961i \(0.0840393\pi\)
−0.965349 + 0.260961i \(0.915961\pi\)
\(132\) 0 0
\(133\) 9.68466 + 15.2933i 0.839766 + 1.32610i
\(134\) 0 0
\(135\) −2.64861 −0.227956
\(136\) 0 0
\(137\) −12.1231 −1.03575 −0.517873 0.855457i \(-0.673276\pi\)
−0.517873 + 0.855457i \(0.673276\pi\)
\(138\) 0 0
\(139\) 18.9435i 1.60676i 0.595464 + 0.803382i \(0.296968\pi\)
−0.595464 + 0.803382i \(0.703032\pi\)
\(140\) 0 0
\(141\) 16.4990i 1.38946i
\(142\) 0 0
\(143\) 10.0138 0.837397
\(144\) 0 0
\(145\) 3.08835i 0.256473i
\(146\) 0 0
\(147\) −24.1636 −1.99298
\(148\) 0 0
\(149\) −2.80776 −0.230021 −0.115010 0.993364i \(-0.536690\pi\)
−0.115010 + 0.993364i \(0.536690\pi\)
\(150\) 0 0
\(151\) 8.85254 0.720409 0.360205 0.932873i \(-0.382707\pi\)
0.360205 + 0.932873i \(0.382707\pi\)
\(152\) 0 0
\(153\) −2.56155 −0.207089
\(154\) 0 0
\(155\) 3.39228 0.272475
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 0 0
\(159\) 23.0963i 1.83166i
\(160\) 0 0
\(161\) 7.56155 0.595934
\(162\) 0 0
\(163\) 17.6339i 1.38119i −0.723240 0.690597i \(-0.757348\pi\)
0.723240 0.690597i \(-0.242652\pi\)
\(164\) 0 0
\(165\) 14.0877i 1.09672i
\(166\) 0 0
\(167\) 10.7575 0.832439 0.416220 0.909264i \(-0.363355\pi\)
0.416220 + 0.909264i \(0.363355\pi\)
\(168\) 0 0
\(169\) −5.43845 −0.418342
\(170\) 0 0
\(171\) 9.43318 5.97366i 0.721373 0.456817i
\(172\) 0 0
\(173\) 2.41131i 0.183328i 0.995790 + 0.0916642i \(0.0292186\pi\)
−0.995790 + 0.0916642i \(0.970781\pi\)
\(174\) 0 0
\(175\) 6.48490i 0.490213i
\(176\) 0 0
\(177\) 14.9309 1.12227
\(178\) 0 0
\(179\) −8.10887 −0.606085 −0.303043 0.952977i \(-0.598002\pi\)
−0.303043 + 0.952977i \(0.598002\pi\)
\(180\) 0 0
\(181\) 19.5873i 1.45591i −0.685623 0.727957i \(-0.740470\pi\)
0.685623 0.727957i \(-0.259530\pi\)
\(182\) 0 0
\(183\) −27.5559 −2.03699
\(184\) 0 0
\(185\) 14.0877i 1.03575i
\(186\) 0 0
\(187\) 2.33205i 0.170536i
\(188\) 0 0
\(189\) 4.29400i 0.312343i
\(190\) 0 0
\(191\) 7.79447i 0.563988i 0.959416 + 0.281994i \(0.0909959\pi\)
−0.959416 + 0.281994i \(0.909004\pi\)
\(192\) 0 0
\(193\) 5.49966i 0.395874i 0.980215 + 0.197937i \(0.0634241\pi\)
−0.980215 + 0.197937i \(0.936576\pi\)
\(194\) 0 0
\(195\) 25.9396i 1.85757i
\(196\) 0 0
\(197\) 22.4924 1.60252 0.801259 0.598317i \(-0.204164\pi\)
0.801259 + 0.598317i \(0.204164\pi\)
\(198\) 0 0
\(199\) 5.17534i 0.366870i −0.983032 0.183435i \(-0.941278\pi\)
0.983032 0.183435i \(-0.0587217\pi\)
\(200\) 0 0
\(201\) −0.684658 −0.0482921
\(202\) 0 0
\(203\) 5.00691 0.351416
\(204\) 0 0
\(205\) 14.0877i 0.983925i
\(206\) 0 0
\(207\) 4.66410i 0.324177i
\(208\) 0 0
\(209\) 5.43845 + 8.58800i 0.376185 + 0.594045i
\(210\) 0 0
\(211\) −25.1976 −1.73467 −0.867336 0.497723i \(-0.834170\pi\)
−0.867336 + 0.497723i \(0.834170\pi\)
\(212\) 0 0
\(213\) −4.87689 −0.334159
\(214\) 0 0
\(215\) 3.35453i 0.228777i
\(216\) 0 0
\(217\) 5.49966i 0.373341i
\(218\) 0 0
\(219\) −0.290319 −0.0196180
\(220\) 0 0
\(221\) 4.29400i 0.288846i
\(222\) 0 0
\(223\) −22.2586 −1.49055 −0.745274 0.666758i \(-0.767681\pi\)
−0.745274 + 0.666758i \(0.767681\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) 24.6169 1.63388 0.816942 0.576720i \(-0.195668\pi\)
0.816942 + 0.576720i \(0.195668\pi\)
\(228\) 0 0
\(229\) 6.56155 0.433600 0.216800 0.976216i \(-0.430438\pi\)
0.216800 + 0.976216i \(0.430438\pi\)
\(230\) 0 0
\(231\) −22.8393 −1.50271
\(232\) 0 0
\(233\) 10.8078 0.708040 0.354020 0.935238i \(-0.384814\pi\)
0.354020 + 0.935238i \(0.384814\pi\)
\(234\) 0 0
\(235\) 17.9210i 1.16904i
\(236\) 0 0
\(237\) 31.6155 2.05365
\(238\) 0 0
\(239\) 9.83943i 0.636460i 0.948014 + 0.318230i \(0.103088\pi\)
−0.948014 + 0.318230i \(0.896912\pi\)
\(240\) 0 0
\(241\) 22.6757i 1.46067i 0.683090 + 0.730334i \(0.260635\pi\)
−0.683090 + 0.730334i \(0.739365\pi\)
\(242\) 0 0
\(243\) −20.7713 −1.33248
\(244\) 0 0
\(245\) −26.2462 −1.67681
\(246\) 0 0
\(247\) −10.0138 15.8131i −0.637164 1.00616i
\(248\) 0 0
\(249\) 28.1753i 1.78554i
\(250\) 0 0
\(251\) 21.5626i 1.36102i −0.732739 0.680510i \(-0.761758\pi\)
0.732739 0.680510i \(-0.238242\pi\)
\(252\) 0 0
\(253\) 4.24621 0.266957
\(254\) 0 0
\(255\) −6.04090 −0.378296
\(256\) 0 0
\(257\) 17.1760i 1.07141i −0.844405 0.535705i \(-0.820046\pi\)
0.844405 0.535705i \(-0.179954\pi\)
\(258\) 0 0
\(259\) −22.8393 −1.41916
\(260\) 0 0
\(261\) 3.08835i 0.191164i
\(262\) 0 0
\(263\) 14.2794i 0.880504i 0.897874 + 0.440252i \(0.145111\pi\)
−0.897874 + 0.440252i \(0.854889\pi\)
\(264\) 0 0
\(265\) 25.0870i 1.54108i
\(266\) 0 0
\(267\) 33.2228i 2.03321i
\(268\) 0 0
\(269\) 5.49966i 0.335320i −0.985845 0.167660i \(-0.946379\pi\)
0.985845 0.167660i \(-0.0536211\pi\)
\(270\) 0 0
\(271\) 22.0738i 1.34089i −0.741959 0.670445i \(-0.766103\pi\)
0.741959 0.670445i \(-0.233897\pi\)
\(272\) 0 0
\(273\) 42.0540 2.54522
\(274\) 0 0
\(275\) 3.64162i 0.219598i
\(276\) 0 0
\(277\) −13.0540 −0.784337 −0.392169 0.919893i \(-0.628275\pi\)
−0.392169 + 0.919893i \(0.628275\pi\)
\(278\) 0 0
\(279\) 3.39228 0.203091
\(280\) 0 0
\(281\) 28.1753i 1.68080i 0.541968 + 0.840399i \(0.317679\pi\)
−0.541968 + 0.840399i \(0.682321\pi\)
\(282\) 0 0
\(283\) 5.97366i 0.355097i 0.984112 + 0.177549i \(0.0568167\pi\)
−0.984112 + 0.177549i \(0.943183\pi\)
\(284\) 0 0
\(285\) 22.2462 14.0877i 1.31775 0.834481i
\(286\) 0 0
\(287\) −22.8393 −1.34816
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 5.68658i 0.333353i
\(292\) 0 0
\(293\) 4.29400i 0.250858i 0.992103 + 0.125429i \(0.0400308\pi\)
−0.992103 + 0.125429i \(0.959969\pi\)
\(294\) 0 0
\(295\) 16.2177 0.944233
\(296\) 0 0
\(297\) 2.41131i 0.139918i
\(298\) 0 0
\(299\) −7.81855 −0.452159
\(300\) 0 0
\(301\) 5.43845 0.313467
\(302\) 0 0
\(303\) 19.4470 1.11720
\(304\) 0 0
\(305\) −29.9309 −1.71384
\(306\) 0 0
\(307\) −13.4061 −0.765126 −0.382563 0.923929i \(-0.624959\pi\)
−0.382563 + 0.923929i \(0.624959\pi\)
\(308\) 0 0
\(309\) −28.4924 −1.62088
\(310\) 0 0
\(311\) 4.15286i 0.235487i 0.993044 + 0.117743i \(0.0375660\pi\)
−0.993044 + 0.117743i \(0.962434\pi\)
\(312\) 0 0
\(313\) −0.438447 −0.0247825 −0.0123913 0.999923i \(-0.503944\pi\)
−0.0123913 + 0.999923i \(0.503944\pi\)
\(314\) 0 0
\(315\) 27.2492i 1.53532i
\(316\) 0 0
\(317\) 4.29400i 0.241175i −0.992703 0.120588i \(-0.961522\pi\)
0.992703 0.120588i \(-0.0384779\pi\)
\(318\) 0 0
\(319\) 2.81164 0.157422
\(320\) 0 0
\(321\) −13.5616 −0.756932
\(322\) 0 0
\(323\) −3.68260 + 2.33205i −0.204905 + 0.129759i
\(324\) 0 0
\(325\) 6.70531i 0.371944i
\(326\) 0 0
\(327\) 28.7829i 1.59170i
\(328\) 0 0
\(329\) −29.0540 −1.60180
\(330\) 0 0
\(331\) −23.8733 −1.31219 −0.656097 0.754677i \(-0.727794\pi\)
−0.656097 + 0.754677i \(0.727794\pi\)
\(332\) 0 0
\(333\) 14.0877i 0.771999i
\(334\) 0 0
\(335\) −0.743668 −0.0406309
\(336\) 0 0
\(337\) 6.17669i 0.336466i −0.985747 0.168233i \(-0.946194\pi\)
0.985747 0.168233i \(-0.0538061\pi\)
\(338\) 0 0
\(339\) 12.9698i 0.704423i
\(340\) 0 0
\(341\) 3.08835i 0.167243i
\(342\) 0 0
\(343\) 13.4810i 0.727908i
\(344\) 0 0
\(345\) 10.9993i 0.592183i
\(346\) 0 0
\(347\) 29.8683i 1.60342i 0.597716 + 0.801708i \(0.296075\pi\)
−0.597716 + 0.801708i \(0.703925\pi\)
\(348\) 0 0
\(349\) −9.05398 −0.484648 −0.242324 0.970195i \(-0.577910\pi\)
−0.242324 + 0.970195i \(0.577910\pi\)
\(350\) 0 0
\(351\) 4.43994i 0.236987i
\(352\) 0 0
\(353\) 18.6847 0.994484 0.497242 0.867612i \(-0.334346\pi\)
0.497242 + 0.867612i \(0.334346\pi\)
\(354\) 0 0
\(355\) −5.29723 −0.281148
\(356\) 0 0
\(357\) 9.79366i 0.518335i
\(358\) 0 0
\(359\) 6.19782i 0.327108i 0.986534 + 0.163554i \(0.0522958\pi\)
−0.986534 + 0.163554i \(0.947704\pi\)
\(360\) 0 0
\(361\) 8.12311 17.1760i 0.427532 0.904000i
\(362\) 0 0
\(363\) 13.1158 0.688400
\(364\) 0 0
\(365\) −0.315342 −0.0165057
\(366\) 0 0
\(367\) 1.02248i 0.0533730i 0.999644 + 0.0266865i \(0.00849559\pi\)
−0.999644 + 0.0266865i \(0.991504\pi\)
\(368\) 0 0
\(369\) 14.0877i 0.733374i
\(370\) 0 0
\(371\) 40.6716 2.11157
\(372\) 0 0
\(373\) 6.70531i 0.347188i −0.984817 0.173594i \(-0.944462\pi\)
0.984817 0.173594i \(-0.0555380\pi\)
\(374\) 0 0
\(375\) −20.7713 −1.07263
\(376\) 0 0
\(377\) −5.17708 −0.266633
\(378\) 0 0
\(379\) 15.7644 0.809762 0.404881 0.914369i \(-0.367313\pi\)
0.404881 + 0.914369i \(0.367313\pi\)
\(380\) 0 0
\(381\) −14.2462 −0.729856
\(382\) 0 0
\(383\) 22.2586 1.13736 0.568682 0.822558i \(-0.307454\pi\)
0.568682 + 0.822558i \(0.307454\pi\)
\(384\) 0 0
\(385\) −24.8078 −1.26432
\(386\) 0 0
\(387\) 3.35453i 0.170520i
\(388\) 0 0
\(389\) −14.8078 −0.750783 −0.375392 0.926866i \(-0.622492\pi\)
−0.375392 + 0.926866i \(0.622492\pi\)
\(390\) 0 0
\(391\) 1.82081i 0.0920822i
\(392\) 0 0
\(393\) 14.0877i 0.710628i
\(394\) 0 0
\(395\) 34.3404 1.72785
\(396\) 0 0
\(397\) 3.93087 0.197285 0.0986423 0.995123i \(-0.468550\pi\)
0.0986423 + 0.995123i \(0.468550\pi\)
\(398\) 0 0
\(399\) 22.8393 + 36.0661i 1.14339 + 1.80557i
\(400\) 0 0
\(401\) 7.91096i 0.395055i 0.980297 + 0.197527i \(0.0632911\pi\)
−0.980297 + 0.197527i \(0.936709\pi\)
\(402\) 0 0
\(403\) 5.68658i 0.283269i
\(404\) 0 0
\(405\) −25.9309 −1.28852
\(406\) 0 0
\(407\) −12.8255 −0.635734
\(408\) 0 0
\(409\) 27.4983i 1.35970i −0.733350 0.679851i \(-0.762044\pi\)
0.733350 0.679851i \(-0.237956\pi\)
\(410\) 0 0
\(411\) −28.5899 −1.41023
\(412\) 0 0
\(413\) 26.2926i 1.29378i
\(414\) 0 0
\(415\) 30.6037i 1.50228i
\(416\) 0 0
\(417\) 44.6743i 2.18771i
\(418\) 0 0
\(419\) 39.9319i 1.95080i 0.220440 + 0.975401i \(0.429251\pi\)
−0.220440 + 0.975401i \(0.570749\pi\)
\(420\) 0 0
\(421\) 23.2043i 1.13091i 0.824780 + 0.565454i \(0.191299\pi\)
−0.824780 + 0.565454i \(0.808701\pi\)
\(422\) 0 0
\(423\) 17.9210i 0.871348i
\(424\) 0 0
\(425\) −1.56155 −0.0757464
\(426\) 0 0
\(427\) 48.5247i 2.34827i
\(428\) 0 0
\(429\) 23.6155 1.14017
\(430\) 0 0
\(431\) 5.29723 0.255158 0.127579 0.991828i \(-0.459279\pi\)
0.127579 + 0.991828i \(0.459279\pi\)
\(432\) 0 0
\(433\) 18.9103i 0.908770i −0.890805 0.454385i \(-0.849859\pi\)
0.890805 0.454385i \(-0.150141\pi\)
\(434\) 0 0
\(435\) 7.28323i 0.349204i
\(436\) 0 0
\(437\) −4.24621 6.70531i −0.203124 0.320758i
\(438\) 0 0
\(439\) −19.6100 −0.935935 −0.467968 0.883746i \(-0.655014\pi\)
−0.467968 + 0.883746i \(0.655014\pi\)
\(440\) 0 0
\(441\) −26.2462 −1.24982
\(442\) 0 0
\(443\) 12.2344i 0.581275i −0.956833 0.290637i \(-0.906133\pi\)
0.956833 0.290637i \(-0.0938673\pi\)
\(444\) 0 0
\(445\) 36.0863i 1.71065i
\(446\) 0 0
\(447\) −6.62153 −0.313188
\(448\) 0 0
\(449\) 14.7647i 0.696789i 0.937348 + 0.348395i \(0.113273\pi\)
−0.937348 + 0.348395i \(0.886727\pi\)
\(450\) 0 0
\(451\) −12.8255 −0.603927
\(452\) 0 0
\(453\) 20.8769 0.980882
\(454\) 0 0
\(455\) 45.6786 2.14144
\(456\) 0 0
\(457\) 3.87689 0.181353 0.0906767 0.995880i \(-0.471097\pi\)
0.0906767 + 0.995880i \(0.471097\pi\)
\(458\) 0 0
\(459\) 1.03399 0.0482624
\(460\) 0 0
\(461\) −31.6847 −1.47570 −0.737851 0.674964i \(-0.764159\pi\)
−0.737851 + 0.674964i \(0.764159\pi\)
\(462\) 0 0
\(463\) 16.3243i 0.758656i 0.925262 + 0.379328i \(0.123845\pi\)
−0.925262 + 0.379328i \(0.876155\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 0 0
\(467\) 2.33205i 0.107914i 0.998543 + 0.0539572i \(0.0171834\pi\)
−0.998543 + 0.0539572i \(0.982817\pi\)
\(468\) 0 0
\(469\) 1.20565i 0.0556719i
\(470\) 0 0
\(471\) −14.1498 −0.651987
\(472\) 0 0
\(473\) 3.05398 0.140422
\(474\) 0 0
\(475\) 5.75058 3.64162i 0.263855 0.167089i
\(476\) 0 0
\(477\) 25.0870i 1.14865i
\(478\) 0 0
\(479\) 41.5286i 1.89749i 0.316045 + 0.948744i \(0.397645\pi\)
−0.316045 + 0.948744i \(0.602355\pi\)
\(480\) 0 0
\(481\) 23.6155 1.07678
\(482\) 0 0
\(483\) 17.8324 0.811401
\(484\) 0 0
\(485\) 6.17669i 0.280469i
\(486\) 0 0
\(487\) −18.8664 −0.854916 −0.427458 0.904035i \(-0.640591\pi\)
−0.427458 + 0.904035i \(0.640591\pi\)
\(488\) 0 0
\(489\) 41.5859i 1.88058i
\(490\) 0 0
\(491\) 21.2755i 0.960151i 0.877227 + 0.480075i \(0.159391\pi\)
−0.877227 + 0.480075i \(0.840609\pi\)
\(492\) 0 0
\(493\) 1.20565i 0.0542999i
\(494\) 0 0
\(495\) 15.3019i 0.687767i
\(496\) 0 0
\(497\) 8.58800i 0.385225i
\(498\) 0 0
\(499\) 1.30957i 0.0586243i −0.999570 0.0293122i \(-0.990668\pi\)
0.999570 0.0293122i \(-0.00933169\pi\)
\(500\) 0 0
\(501\) 25.3693 1.13342
\(502\) 0 0
\(503\) 16.3873i 0.730672i −0.930876 0.365336i \(-0.880954\pi\)
0.930876 0.365336i \(-0.119046\pi\)
\(504\) 0 0
\(505\) 21.1231 0.939966
\(506\) 0 0
\(507\) −12.8255 −0.569599
\(508\) 0 0
\(509\) 8.58800i 0.380657i 0.981721 + 0.190328i \(0.0609552\pi\)
−0.981721 + 0.190328i \(0.939045\pi\)
\(510\) 0 0
\(511\) 0.511240i 0.0226159i
\(512\) 0 0
\(513\) −3.80776 + 2.41131i −0.168117 + 0.106462i
\(514\) 0 0
\(515\) −30.9481 −1.36374
\(516\) 0 0
\(517\) −16.3153 −0.717548
\(518\) 0 0
\(519\) 5.68658i 0.249613i
\(520\) 0 0
\(521\) 22.6757i 0.993439i −0.867911 0.496719i \(-0.834538\pi\)
0.867911 0.496719i \(-0.165462\pi\)
\(522\) 0 0
\(523\) 2.19526 0.0959922 0.0479961 0.998848i \(-0.484716\pi\)
0.0479961 + 0.998848i \(0.484716\pi\)
\(524\) 0 0
\(525\) 15.2933i 0.667455i
\(526\) 0 0
\(527\) −1.32431 −0.0576877
\(528\) 0 0
\(529\) 19.6847 0.855855
\(530\) 0 0
\(531\) 16.2177 0.703790
\(532\) 0 0
\(533\) 23.6155 1.02290
\(534\) 0 0
\(535\) −14.7304 −0.636851
\(536\) 0 0
\(537\) −19.1231 −0.825223
\(538\) 0 0
\(539\) 23.8947i 1.02922i
\(540\) 0 0
\(541\) 32.8078 1.41052 0.705258 0.708951i \(-0.250831\pi\)
0.705258 + 0.708951i \(0.250831\pi\)
\(542\) 0 0
\(543\) 46.1927i 1.98232i
\(544\) 0 0
\(545\) 31.2637i 1.33919i
\(546\) 0 0
\(547\) 0.580639 0.0248263 0.0124132 0.999923i \(-0.496049\pi\)
0.0124132 + 0.999923i \(0.496049\pi\)
\(548\) 0 0
\(549\) −29.9309 −1.27742
\(550\) 0 0
\(551\) −2.81164 4.43994i −0.119780 0.189148i
\(552\) 0 0
\(553\) 55.6736i 2.36748i
\(554\) 0 0
\(555\) 33.2228i 1.41023i
\(556\) 0 0
\(557\) −9.93087 −0.420784 −0.210392 0.977617i \(-0.567474\pi\)
−0.210392 + 0.977617i \(0.567474\pi\)
\(558\) 0 0
\(559\) −5.62329 −0.237840
\(560\) 0 0
\(561\) 5.49966i 0.232196i
\(562\) 0 0
\(563\) 21.3519 0.899877 0.449938 0.893060i \(-0.351446\pi\)
0.449938 + 0.893060i \(0.351446\pi\)
\(564\) 0 0
\(565\) 14.0877i 0.592672i
\(566\) 0 0
\(567\) 42.0398i 1.76551i
\(568\) 0 0
\(569\) 41.5859i 1.74337i 0.490064 + 0.871687i \(0.336973\pi\)
−0.490064 + 0.871687i \(0.663027\pi\)
\(570\) 0 0
\(571\) 15.5889i 0.652377i 0.945305 + 0.326188i \(0.105764\pi\)
−0.945305 + 0.326188i \(0.894236\pi\)
\(572\) 0 0
\(573\) 18.3817i 0.767905i
\(574\) 0 0
\(575\) 2.84329i 0.118573i
\(576\) 0 0
\(577\) 27.0000 1.12402 0.562012 0.827129i \(-0.310027\pi\)
0.562012 + 0.827129i \(0.310027\pi\)
\(578\) 0 0
\(579\) 12.9698i 0.539007i
\(580\) 0 0
\(581\) 49.6155 2.05840
\(582\) 0 0
\(583\) 22.8393 0.945906
\(584\) 0 0
\(585\) 28.1753i 1.16491i
\(586\) 0 0
\(587\) 5.97366i 0.246559i 0.992372 + 0.123280i \(0.0393412\pi\)
−0.992372 + 0.123280i \(0.960659\pi\)
\(588\) 0 0
\(589\) 4.87689 3.08835i 0.200949 0.127253i
\(590\) 0 0
\(591\) 53.0438 2.18193
\(592\) 0 0
\(593\) 12.7386 0.523113 0.261556 0.965188i \(-0.415764\pi\)
0.261556 + 0.965188i \(0.415764\pi\)
\(594\) 0 0
\(595\) 10.6378i 0.436106i
\(596\) 0 0
\(597\) 12.2050i 0.499516i
\(598\) 0 0
\(599\) 24.9073 1.01768 0.508841 0.860860i \(-0.330074\pi\)
0.508841 + 0.860860i \(0.330074\pi\)
\(600\) 0 0
\(601\) 35.4092i 1.44437i 0.691698 + 0.722187i \(0.256863\pi\)
−0.691698 + 0.722187i \(0.743137\pi\)
\(602\) 0 0
\(603\) −0.743668 −0.0302845
\(604\) 0 0
\(605\) 14.2462 0.579191
\(606\) 0 0
\(607\) −31.5288 −1.27971 −0.639857 0.768494i \(-0.721006\pi\)
−0.639857 + 0.768494i \(0.721006\pi\)
\(608\) 0 0
\(609\) 11.8078 0.478475
\(610\) 0 0
\(611\) 30.0414 1.21535
\(612\) 0 0
\(613\) 29.3002 1.18342 0.591712 0.806150i \(-0.298452\pi\)
0.591712 + 0.806150i \(0.298452\pi\)
\(614\) 0 0
\(615\) 33.2228i 1.33967i
\(616\) 0 0
\(617\) 41.5464 1.67259 0.836297 0.548276i \(-0.184716\pi\)
0.836297 + 0.548276i \(0.184716\pi\)
\(618\) 0 0
\(619\) 6.70906i 0.269660i 0.990869 + 0.134830i \(0.0430488\pi\)
−0.990869 + 0.134830i \(0.956951\pi\)
\(620\) 0 0
\(621\) 1.88269i 0.0755499i
\(622\) 0 0
\(623\) 58.5040 2.34391
\(624\) 0 0
\(625\) −30.3693 −1.21477
\(626\) 0 0
\(627\) 12.8255 + 20.2530i 0.512200 + 0.808828i
\(628\) 0 0
\(629\) 5.49966i 0.219286i
\(630\) 0 0
\(631\) 9.61528i 0.382778i −0.981514 0.191389i \(-0.938701\pi\)
0.981514 0.191389i \(-0.0612992\pi\)
\(632\) 0 0
\(633\) −59.4233 −2.36186
\(634\) 0 0
\(635\) −15.4741 −0.614070
\(636\) 0 0
\(637\) 43.9972i 1.74323i
\(638\) 0 0
\(639\) −5.29723 −0.209555
\(640\) 0 0
\(641\) 26.8212i 1.05938i −0.848193 0.529688i \(-0.822309\pi\)
0.848193 0.529688i \(-0.177691\pi\)
\(642\) 0 0
\(643\) 14.2794i 0.563124i −0.959543 0.281562i \(-0.909148\pi\)
0.959543 0.281562i \(-0.0908524\pi\)
\(644\) 0 0
\(645\) 7.91096i 0.311494i
\(646\) 0 0
\(647\) 32.1374i 1.26345i 0.775191 + 0.631726i \(0.217653\pi\)
−0.775191 + 0.631726i \(0.782347\pi\)
\(648\) 0 0
\(649\) 14.7647i 0.579565i
\(650\) 0 0
\(651\) 12.9698i 0.508327i
\(652\) 0 0
\(653\) 25.1922 0.985848 0.492924 0.870072i \(-0.335928\pi\)
0.492924 + 0.870072i \(0.335928\pi\)
\(654\) 0 0
\(655\) 15.3019i 0.597893i
\(656\) 0 0
\(657\) −0.315342 −0.0123026
\(658\) 0 0
\(659\) 41.9960 1.63593 0.817965 0.575268i \(-0.195102\pi\)
0.817965 + 0.575268i \(0.195102\pi\)
\(660\) 0 0
\(661\) 37.2919i 1.45049i 0.688492 + 0.725244i \(0.258273\pi\)
−0.688492 + 0.725244i \(0.741727\pi\)
\(662\) 0 0
\(663\) 10.1265i 0.393281i
\(664\) 0 0
\(665\) 24.8078 + 39.1746i 0.962004 + 1.51913i
\(666\) 0 0
\(667\) −2.19526 −0.0850010
\(668\) 0 0
\(669\) −52.4924 −2.02947
\(670\) 0 0
\(671\) 27.2492i 1.05194i
\(672\) 0 0
\(673\) 24.4099i 0.940934i 0.882418 + 0.470467i \(0.155914\pi\)
−0.882418 + 0.470467i \(0.844086\pi\)
\(674\) 0 0
\(675\) −1.61463 −0.0621470
\(676\) 0 0
\(677\) 40.3803i 1.55194i 0.630770 + 0.775970i \(0.282739\pi\)
−0.630770 + 0.775970i \(0.717261\pi\)
\(678\) 0 0
\(679\) 10.0138 0.384295
\(680\) 0 0
\(681\) 58.0540 2.22463
\(682\) 0 0
\(683\) −31.5288 −1.20642 −0.603208 0.797584i \(-0.706111\pi\)
−0.603208 + 0.797584i \(0.706111\pi\)
\(684\) 0 0
\(685\) −31.0540 −1.18651
\(686\) 0 0
\(687\) 15.4741 0.590373
\(688\) 0 0
\(689\) −42.0540 −1.60213
\(690\) 0 0
\(691\) 8.01862i 0.305043i 0.988300 + 0.152521i \(0.0487393\pi\)
−0.988300 + 0.152521i \(0.951261\pi\)
\(692\) 0 0
\(693\) −24.8078 −0.942369
\(694\) 0 0
\(695\) 48.5247i 1.84065i
\(696\) 0 0
\(697\) 5.49966i 0.208314i
\(698\) 0 0
\(699\) 25.4879 0.964041
\(700\) 0 0
\(701\) −4.63068 −0.174898 −0.0874492 0.996169i \(-0.527872\pi\)
−0.0874492 + 0.996169i \(0.527872\pi\)
\(702\) 0 0
\(703\) 12.8255 + 20.2530i 0.483721 + 0.763858i
\(704\) 0 0
\(705\) 42.2630i 1.59172i
\(706\) 0 0
\(707\) 34.2453i 1.28793i
\(708\) 0 0
\(709\) −12.6307 −0.474355 −0.237178 0.971466i \(-0.576222\pi\)
−0.237178 + 0.971466i \(0.576222\pi\)
\(710\) 0 0
\(711\) 34.3404 1.28787
\(712\) 0 0
\(713\) 2.41131i 0.0903042i
\(714\) 0 0
\(715\) 25.6509 0.959290
\(716\) 0 0
\(717\) 23.2043i 0.866580i
\(718\) 0 0
\(719\) 26.4509i 0.986450i 0.869902 + 0.493225i \(0.164182\pi\)
−0.869902 + 0.493225i \(0.835818\pi\)
\(720\) 0 0
\(721\) 50.1739i 1.86858i
\(722\) 0 0
\(723\) 53.4759i 1.98879i
\(724\) 0 0
\(725\) 1.88269i 0.0699215i
\(726\) 0 0
\(727\) 44.6589i 1.65631i 0.560500 + 0.828154i \(0.310609\pi\)
−0.560500 + 0.828154i \(0.689391\pi\)
\(728\) 0 0
\(729\) −18.6155 −0.689464
\(730\) 0 0
\(731\) 1.30957i 0.0484361i
\(732\) 0 0
\(733\) 5.12311 0.189226 0.0946131 0.995514i \(-0.469839\pi\)
0.0946131 + 0.995514i \(0.469839\pi\)
\(734\) 0 0
\(735\) −61.8963 −2.28308
\(736\) 0 0
\(737\) 0.677039i 0.0249390i
\(738\) 0 0
\(739\) 25.2042i 0.927152i −0.886057 0.463576i \(-0.846566\pi\)
0.886057 0.463576i \(-0.153434\pi\)
\(740\) 0 0
\(741\) −23.6155 37.2919i −0.867538 1.36995i
\(742\) 0 0
\(743\) 11.9188 0.437257 0.218628 0.975808i \(-0.429842\pi\)
0.218628 + 0.975808i \(0.429842\pi\)
\(744\) 0 0
\(745\) −7.19224 −0.263503
\(746\) 0 0
\(747\) 30.6037i 1.11973i
\(748\) 0 0
\(749\) 23.8813i 0.872604i
\(750\) 0 0
\(751\) −35.6647 −1.30142 −0.650712 0.759324i \(-0.725530\pi\)
−0.650712 + 0.759324i \(0.725530\pi\)
\(752\) 0 0
\(753\) 50.8510i 1.85311i
\(754\) 0 0
\(755\) 22.6762 0.825273
\(756\) 0 0
\(757\) 44.8078 1.62857 0.814283 0.580468i \(-0.197130\pi\)
0.814283 + 0.580468i \(0.197130\pi\)
\(758\) 0 0
\(759\) 10.0138 0.363479
\(760\) 0 0
\(761\) 30.6155 1.10981 0.554906 0.831913i \(-0.312754\pi\)
0.554906 + 0.831913i \(0.312754\pi\)
\(762\) 0 0
\(763\) 50.6855 1.83494
\(764\) 0 0
\(765\) −6.56155 −0.237233
\(766\) 0 0
\(767\) 27.1862i 0.981638i
\(768\) 0 0
\(769\) −28.2311 −1.01804 −0.509019 0.860755i \(-0.669992\pi\)
−0.509019 + 0.860755i \(0.669992\pi\)
\(770\) 0 0
\(771\) 40.5061i 1.45879i
\(772\) 0 0
\(773\) 41.0573i 1.47673i −0.674402 0.738365i \(-0.735598\pi\)
0.674402 0.738365i \(-0.264402\pi\)
\(774\) 0 0
\(775\) 2.06798 0.0742839
\(776\) 0 0
\(777\) −53.8617 −1.93228
\(778\) 0 0
\(779\) 12.8255 + 20.2530i 0.459520 + 0.725640i
\(780\) 0 0
\(781\) 4.82262i 0.172567i
\(782\) 0 0
\(783\) 1.24663i 0.0445510i
\(784\) 0 0
\(785\) −15.3693 −0.548554
\(786\) 0 0
\(787\) −26.3588 −0.939591 −0.469796 0.882775i \(-0.655672\pi\)
−0.469796 + 0.882775i \(0.655672\pi\)
\(788\) 0 0
\(789\) 33.6750i 1.19886i
\(790\) 0 0
\(791\) 22.8393 0.812071
\(792\) 0 0
\(793\) 50.1739i 1.78173i
\(794\) 0 0
\(795\) 59.1625i 2.09828i
\(796\) 0 0
\(797\) 23.2043i 0.821938i −0.911649 0.410969i \(-0.865191\pi\)
0.911649 0.410969i \(-0.134809\pi\)
\(798\) 0 0
\(799\) 6.99614i 0.247506i
\(800\) 0 0
\(801\) 36.0863i 1.27505i
\(802\) 0 0
\(803\) 0.287088i 0.0101311i
\(804\) 0 0
\(805\) 19.3693 0.682679
\(806\) 0