Properties

Label 1148.2.d.a.1065.13
Level $1148$
Weight $2$
Character 1148.1065
Analytic conductor $9.167$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 39 x^{18} + 531 x^{16} + 3488 x^{14} + 12661 x^{12} + 27027 x^{10} + 34540 x^{8} + 25909 x^{6} + 10677 x^{4} + 2092 x^{2} + 144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1065.13
Root \(-0.379831i\) of defining polynomial
Character \(\chi\) \(=\) 1148.1065
Dual form 1148.2.d.a.1065.8

$q$-expansion

\(f(q)\) \(=\) \(q+1.03684i q^{3} +2.44552 q^{5} +1.00000i q^{7} +1.92496 q^{9} +O(q^{10})\) \(q+1.03684i q^{3} +2.44552 q^{5} +1.00000i q^{7} +1.92496 q^{9} +5.06947i q^{11} -2.54571i q^{13} +2.53562i q^{15} +5.30064i q^{17} -1.14984i q^{19} -1.03684 q^{21} -5.57153 q^{23} +0.980584 q^{25} +5.10640i q^{27} -8.69224i q^{29} +8.34429 q^{31} -5.25624 q^{33} +2.44552i q^{35} +11.0486 q^{37} +2.63950 q^{39} +(-6.22148 - 1.51433i) q^{41} -5.96616 q^{43} +4.70754 q^{45} +4.18168i q^{47} -1.00000 q^{49} -5.49592 q^{51} +6.93136i q^{53} +12.3975i q^{55} +1.19220 q^{57} +7.10317 q^{59} +1.10590 q^{61} +1.92496i q^{63} -6.22560i q^{65} +2.43504i q^{67} -5.77679i q^{69} -6.81857i q^{71} -5.66952 q^{73} +1.01671i q^{75} -5.06947 q^{77} +4.13643i q^{79} +0.480357 q^{81} -12.5457 q^{83} +12.9628i q^{85} +9.01247 q^{87} +2.55515i q^{89} +2.54571 q^{91} +8.65171i q^{93} -2.81195i q^{95} -6.85832i q^{97} +9.75854i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{5} - 20q^{9} + O(q^{10}) \) \( 20q + 4q^{5} - 20q^{9} + 4q^{21} + 8q^{31} + 20q^{37} + 4q^{39} - 16q^{41} + 20q^{43} - 4q^{45} - 20q^{49} + 52q^{51} - 36q^{57} + 20q^{59} - 4q^{61} - 12q^{73} + 8q^{77} + 20q^{81} - 48q^{83} + 44q^{87} - 4q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 1.03684i 0.598620i 0.954156 + 0.299310i \(0.0967565\pi\)
−0.954156 + 0.299310i \(0.903243\pi\)
\(4\) 0 0
\(5\) 2.44552 1.09367 0.546836 0.837240i \(-0.315832\pi\)
0.546836 + 0.837240i \(0.315832\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 1.92496 0.641654
\(10\) 0 0
\(11\) 5.06947i 1.52850i 0.644918 + 0.764252i \(0.276892\pi\)
−0.644918 + 0.764252i \(0.723108\pi\)
\(12\) 0 0
\(13\) 2.54571i 0.706053i −0.935613 0.353027i \(-0.885152\pi\)
0.935613 0.353027i \(-0.114848\pi\)
\(14\) 0 0
\(15\) 2.53562i 0.654694i
\(16\) 0 0
\(17\) 5.30064i 1.28559i 0.766037 + 0.642797i \(0.222226\pi\)
−0.766037 + 0.642797i \(0.777774\pi\)
\(18\) 0 0
\(19\) 1.14984i 0.263791i −0.991264 0.131895i \(-0.957894\pi\)
0.991264 0.131895i \(-0.0421063\pi\)
\(20\) 0 0
\(21\) −1.03684 −0.226257
\(22\) 0 0
\(23\) −5.57153 −1.16174 −0.580872 0.813995i \(-0.697288\pi\)
−0.580872 + 0.813995i \(0.697288\pi\)
\(24\) 0 0
\(25\) 0.980584 0.196117
\(26\) 0 0
\(27\) 5.10640i 0.982727i
\(28\) 0 0
\(29\) 8.69224i 1.61411i −0.590477 0.807054i \(-0.701061\pi\)
0.590477 0.807054i \(-0.298939\pi\)
\(30\) 0 0
\(31\) 8.34429 1.49868 0.749340 0.662186i \(-0.230371\pi\)
0.749340 + 0.662186i \(0.230371\pi\)
\(32\) 0 0
\(33\) −5.25624 −0.914994
\(34\) 0 0
\(35\) 2.44552i 0.413369i
\(36\) 0 0
\(37\) 11.0486 1.81637 0.908187 0.418564i \(-0.137466\pi\)
0.908187 + 0.418564i \(0.137466\pi\)
\(38\) 0 0
\(39\) 2.63950 0.422658
\(40\) 0 0
\(41\) −6.22148 1.51433i −0.971632 0.236498i
\(42\) 0 0
\(43\) −5.96616 −0.909832 −0.454916 0.890534i \(-0.650331\pi\)
−0.454916 + 0.890534i \(0.650331\pi\)
\(44\) 0 0
\(45\) 4.70754 0.701758
\(46\) 0 0
\(47\) 4.18168i 0.609961i 0.952359 + 0.304980i \(0.0986499\pi\)
−0.952359 + 0.304980i \(0.901350\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −5.49592 −0.769582
\(52\) 0 0
\(53\) 6.93136i 0.952096i 0.879419 + 0.476048i \(0.157931\pi\)
−0.879419 + 0.476048i \(0.842069\pi\)
\(54\) 0 0
\(55\) 12.3975i 1.67168i
\(56\) 0 0
\(57\) 1.19220 0.157910
\(58\) 0 0
\(59\) 7.10317 0.924754 0.462377 0.886684i \(-0.346997\pi\)
0.462377 + 0.886684i \(0.346997\pi\)
\(60\) 0 0
\(61\) 1.10590 0.141597 0.0707983 0.997491i \(-0.477445\pi\)
0.0707983 + 0.997491i \(0.477445\pi\)
\(62\) 0 0
\(63\) 1.92496i 0.242522i
\(64\) 0 0
\(65\) 6.22560i 0.772190i
\(66\) 0 0
\(67\) 2.43504i 0.297487i 0.988876 + 0.148744i \(0.0475229\pi\)
−0.988876 + 0.148744i \(0.952477\pi\)
\(68\) 0 0
\(69\) 5.77679i 0.695444i
\(70\) 0 0
\(71\) 6.81857i 0.809215i −0.914490 0.404608i \(-0.867408\pi\)
0.914490 0.404608i \(-0.132592\pi\)
\(72\) 0 0
\(73\) −5.66952 −0.663567 −0.331784 0.943356i \(-0.607650\pi\)
−0.331784 + 0.943356i \(0.607650\pi\)
\(74\) 0 0
\(75\) 1.01671i 0.117400i
\(76\) 0 0
\(77\) −5.06947 −0.577720
\(78\) 0 0
\(79\) 4.13643i 0.465385i 0.972550 + 0.232693i \(0.0747536\pi\)
−0.972550 + 0.232693i \(0.925246\pi\)
\(80\) 0 0
\(81\) 0.480357 0.0533730
\(82\) 0 0
\(83\) −12.5457 −1.37707 −0.688536 0.725203i \(-0.741746\pi\)
−0.688536 + 0.725203i \(0.741746\pi\)
\(84\) 0 0
\(85\) 12.9628i 1.40602i
\(86\) 0 0
\(87\) 9.01247 0.966238
\(88\) 0 0
\(89\) 2.55515i 0.270846i 0.990788 + 0.135423i \(0.0432393\pi\)
−0.990788 + 0.135423i \(0.956761\pi\)
\(90\) 0 0
\(91\) 2.54571 0.266863
\(92\) 0 0
\(93\) 8.65171i 0.897140i
\(94\) 0 0
\(95\) 2.81195i 0.288500i
\(96\) 0 0
\(97\) 6.85832i 0.696356i −0.937428 0.348178i \(-0.886800\pi\)
0.937428 0.348178i \(-0.113200\pi\)
\(98\) 0 0
\(99\) 9.75854i 0.980770i
\(100\) 0 0
\(101\) 0.0728344i 0.00724729i −0.999993 0.00362365i \(-0.998847\pi\)
0.999993 0.00362365i \(-0.00115345\pi\)
\(102\) 0 0
\(103\) 1.65441 0.163014 0.0815069 0.996673i \(-0.474027\pi\)
0.0815069 + 0.996673i \(0.474027\pi\)
\(104\) 0 0
\(105\) −2.53562 −0.247451
\(106\) 0 0
\(107\) 13.3535 1.29093 0.645465 0.763790i \(-0.276664\pi\)
0.645465 + 0.763790i \(0.276664\pi\)
\(108\) 0 0
\(109\) 2.30709i 0.220979i 0.993877 + 0.110489i \(0.0352418\pi\)
−0.993877 + 0.110489i \(0.964758\pi\)
\(110\) 0 0
\(111\) 11.4556i 1.08732i
\(112\) 0 0
\(113\) 12.5497 1.18057 0.590287 0.807194i \(-0.299015\pi\)
0.590287 + 0.807194i \(0.299015\pi\)
\(114\) 0 0
\(115\) −13.6253 −1.27057
\(116\) 0 0
\(117\) 4.90040i 0.453042i
\(118\) 0 0
\(119\) −5.30064 −0.485909
\(120\) 0 0
\(121\) −14.6996 −1.33632
\(122\) 0 0
\(123\) 1.57012 6.45069i 0.141573 0.581639i
\(124\) 0 0
\(125\) −9.82957 −0.879184
\(126\) 0 0
\(127\) 5.73087 0.508532 0.254266 0.967134i \(-0.418166\pi\)
0.254266 + 0.967134i \(0.418166\pi\)
\(128\) 0 0
\(129\) 6.18596i 0.544644i
\(130\) 0 0
\(131\) −4.51320 −0.394320 −0.197160 0.980371i \(-0.563172\pi\)
−0.197160 + 0.980371i \(0.563172\pi\)
\(132\) 0 0
\(133\) 1.14984 0.0997035
\(134\) 0 0
\(135\) 12.4878i 1.07478i
\(136\) 0 0
\(137\) 3.22565i 0.275585i 0.990461 + 0.137793i \(0.0440008\pi\)
−0.990461 + 0.137793i \(0.955999\pi\)
\(138\) 0 0
\(139\) 4.48715 0.380595 0.190298 0.981726i \(-0.439055\pi\)
0.190298 + 0.981726i \(0.439055\pi\)
\(140\) 0 0
\(141\) −4.33574 −0.365135
\(142\) 0 0
\(143\) 12.9054 1.07921
\(144\) 0 0
\(145\) 21.2571i 1.76530i
\(146\) 0 0
\(147\) 1.03684i 0.0855172i
\(148\) 0 0
\(149\) 1.78471i 0.146209i 0.997324 + 0.0731044i \(0.0232906\pi\)
−0.997324 + 0.0731044i \(0.976709\pi\)
\(150\) 0 0
\(151\) 8.90375i 0.724577i −0.932066 0.362288i \(-0.881996\pi\)
0.932066 0.362288i \(-0.118004\pi\)
\(152\) 0 0
\(153\) 10.2035i 0.824906i
\(154\) 0 0
\(155\) 20.4062 1.63906
\(156\) 0 0
\(157\) 0.205961i 0.0164375i 0.999966 + 0.00821873i \(0.00261613\pi\)
−0.999966 + 0.00821873i \(0.997384\pi\)
\(158\) 0 0
\(159\) −7.18672 −0.569944
\(160\) 0 0
\(161\) 5.57153i 0.439098i
\(162\) 0 0
\(163\) −1.00934 −0.0790572 −0.0395286 0.999218i \(-0.512586\pi\)
−0.0395286 + 0.999218i \(0.512586\pi\)
\(164\) 0 0
\(165\) −12.8543 −1.00070
\(166\) 0 0
\(167\) 22.7851i 1.76316i −0.472034 0.881580i \(-0.656480\pi\)
0.472034 0.881580i \(-0.343520\pi\)
\(168\) 0 0
\(169\) 6.51935 0.501489
\(170\) 0 0
\(171\) 2.21339i 0.169262i
\(172\) 0 0
\(173\) −2.64615 −0.201183 −0.100591 0.994928i \(-0.532073\pi\)
−0.100591 + 0.994928i \(0.532073\pi\)
\(174\) 0 0
\(175\) 0.980584i 0.0741252i
\(176\) 0 0
\(177\) 7.36485i 0.553576i
\(178\) 0 0
\(179\) 12.1448i 0.907747i 0.891066 + 0.453873i \(0.149958\pi\)
−0.891066 + 0.453873i \(0.850042\pi\)
\(180\) 0 0
\(181\) 20.4684i 1.52141i −0.649100 0.760703i \(-0.724854\pi\)
0.649100 0.760703i \(-0.275146\pi\)
\(182\) 0 0
\(183\) 1.14665i 0.0847626i
\(184\) 0 0
\(185\) 27.0195 1.98652
\(186\) 0 0
\(187\) −26.8714 −1.96503
\(188\) 0 0
\(189\) −5.10640 −0.371436
\(190\) 0 0
\(191\) 13.5091i 0.977487i 0.872428 + 0.488744i \(0.162545\pi\)
−0.872428 + 0.488744i \(0.837455\pi\)
\(192\) 0 0
\(193\) 14.1767i 1.02046i −0.860037 0.510232i \(-0.829560\pi\)
0.860037 0.510232i \(-0.170440\pi\)
\(194\) 0 0
\(195\) 6.45495 0.462249
\(196\) 0 0
\(197\) 22.0306 1.56961 0.784807 0.619740i \(-0.212762\pi\)
0.784807 + 0.619740i \(0.212762\pi\)
\(198\) 0 0
\(199\) 14.9190i 1.05758i −0.848752 0.528791i \(-0.822646\pi\)
0.848752 0.528791i \(-0.177354\pi\)
\(200\) 0 0
\(201\) −2.52475 −0.178082
\(202\) 0 0
\(203\) 8.69224 0.610076
\(204\) 0 0
\(205\) −15.2148 3.70332i −1.06265 0.258651i
\(206\) 0 0
\(207\) −10.7250 −0.745437
\(208\) 0 0
\(209\) 5.82907 0.403205
\(210\) 0 0
\(211\) 24.6004i 1.69356i −0.531940 0.846782i \(-0.678537\pi\)
0.531940 0.846782i \(-0.321463\pi\)
\(212\) 0 0
\(213\) 7.06977 0.484413
\(214\) 0 0
\(215\) −14.5904 −0.995057
\(216\) 0 0
\(217\) 8.34429i 0.566448i
\(218\) 0 0
\(219\) 5.87839i 0.397225i
\(220\) 0 0
\(221\) 13.4939 0.907698
\(222\) 0 0
\(223\) 2.63720 0.176600 0.0883001 0.996094i \(-0.471857\pi\)
0.0883001 + 0.996094i \(0.471857\pi\)
\(224\) 0 0
\(225\) 1.88759 0.125839
\(226\) 0 0
\(227\) 13.8673i 0.920403i −0.887815 0.460201i \(-0.847777\pi\)
0.887815 0.460201i \(-0.152223\pi\)
\(228\) 0 0
\(229\) 18.3429i 1.21214i −0.795413 0.606068i \(-0.792746\pi\)
0.795413 0.606068i \(-0.207254\pi\)
\(230\) 0 0
\(231\) 5.25624i 0.345835i
\(232\) 0 0
\(233\) 4.30305i 0.281902i 0.990017 + 0.140951i \(0.0450160\pi\)
−0.990017 + 0.140951i \(0.954984\pi\)
\(234\) 0 0
\(235\) 10.2264i 0.667097i
\(236\) 0 0
\(237\) −4.28882 −0.278589
\(238\) 0 0
\(239\) 18.6883i 1.20885i −0.796664 0.604423i \(-0.793404\pi\)
0.796664 0.604423i \(-0.206596\pi\)
\(240\) 0 0
\(241\) 12.9190 0.832188 0.416094 0.909322i \(-0.363399\pi\)
0.416094 + 0.909322i \(0.363399\pi\)
\(242\) 0 0
\(243\) 15.8173i 1.01468i
\(244\) 0 0
\(245\) −2.44552 −0.156239
\(246\) 0 0
\(247\) −2.92715 −0.186250
\(248\) 0 0
\(249\) 13.0079i 0.824343i
\(250\) 0 0
\(251\) −24.3743 −1.53849 −0.769245 0.638954i \(-0.779367\pi\)
−0.769245 + 0.638954i \(0.779367\pi\)
\(252\) 0 0
\(253\) 28.2447i 1.77573i
\(254\) 0 0
\(255\) −13.4404 −0.841670
\(256\) 0 0
\(257\) 14.1856i 0.884872i −0.896800 0.442436i \(-0.854114\pi\)
0.896800 0.442436i \(-0.145886\pi\)
\(258\) 0 0
\(259\) 11.0486i 0.686525i
\(260\) 0 0
\(261\) 16.7322i 1.03570i
\(262\) 0 0
\(263\) 7.24943i 0.447019i −0.974702 0.223510i \(-0.928249\pi\)
0.974702 0.223510i \(-0.0717514\pi\)
\(264\) 0 0
\(265\) 16.9508i 1.04128i
\(266\) 0 0
\(267\) −2.64929 −0.162134
\(268\) 0 0
\(269\) −29.5342 −1.80073 −0.900366 0.435133i \(-0.856701\pi\)
−0.900366 + 0.435133i \(0.856701\pi\)
\(270\) 0 0
\(271\) −9.81750 −0.596371 −0.298185 0.954508i \(-0.596381\pi\)
−0.298185 + 0.954508i \(0.596381\pi\)
\(272\) 0 0
\(273\) 2.63950i 0.159750i
\(274\) 0 0
\(275\) 4.97105i 0.299765i
\(276\) 0 0
\(277\) −9.02683 −0.542370 −0.271185 0.962527i \(-0.587415\pi\)
−0.271185 + 0.962527i \(0.587415\pi\)
\(278\) 0 0
\(279\) 16.0624 0.961633
\(280\) 0 0
\(281\) 20.2704i 1.20923i −0.796518 0.604615i \(-0.793327\pi\)
0.796518 0.604615i \(-0.206673\pi\)
\(282\) 0 0
\(283\) 21.2024 1.26035 0.630176 0.776452i \(-0.282983\pi\)
0.630176 + 0.776452i \(0.282983\pi\)
\(284\) 0 0
\(285\) 2.91555 0.172702
\(286\) 0 0
\(287\) 1.51433 6.22148i 0.0893878 0.367242i
\(288\) 0 0
\(289\) −11.0968 −0.652750
\(290\) 0 0
\(291\) 7.11098 0.416853
\(292\) 0 0
\(293\) 19.6438i 1.14761i 0.818994 + 0.573803i \(0.194532\pi\)
−0.818994 + 0.573803i \(0.805468\pi\)
\(294\) 0 0
\(295\) 17.3710 1.01138
\(296\) 0 0
\(297\) −25.8868 −1.50210
\(298\) 0 0
\(299\) 14.1835i 0.820253i
\(300\) 0 0
\(301\) 5.96616i 0.343884i
\(302\) 0 0
\(303\) 0.0755177 0.00433838
\(304\) 0 0
\(305\) 2.70452 0.154860
\(306\) 0 0
\(307\) −24.9629 −1.42471 −0.712353 0.701822i \(-0.752370\pi\)
−0.712353 + 0.701822i \(0.752370\pi\)
\(308\) 0 0
\(309\) 1.71536i 0.0975833i
\(310\) 0 0
\(311\) 19.5023i 1.10587i 0.833223 + 0.552937i \(0.186493\pi\)
−0.833223 + 0.552937i \(0.813507\pi\)
\(312\) 0 0
\(313\) 23.5846i 1.33308i −0.745470 0.666539i \(-0.767775\pi\)
0.745470 0.666539i \(-0.232225\pi\)
\(314\) 0 0
\(315\) 4.70754i 0.265240i
\(316\) 0 0
\(317\) 23.2861i 1.30788i −0.756547 0.653940i \(-0.773115\pi\)
0.756547 0.653940i \(-0.226885\pi\)
\(318\) 0 0
\(319\) 44.0651 2.46717
\(320\) 0 0
\(321\) 13.8454i 0.772777i
\(322\) 0 0
\(323\) 6.09487 0.339127
\(324\) 0 0
\(325\) 2.49629i 0.138469i
\(326\) 0 0
\(327\) −2.39208 −0.132282
\(328\) 0 0
\(329\) −4.18168 −0.230543
\(330\) 0 0
\(331\) 6.30854i 0.346749i −0.984856 0.173374i \(-0.944533\pi\)
0.984856 0.173374i \(-0.0554671\pi\)
\(332\) 0 0
\(333\) 21.2681 1.16548
\(334\) 0 0
\(335\) 5.95494i 0.325353i
\(336\) 0 0
\(337\) 21.0885 1.14876 0.574381 0.818588i \(-0.305243\pi\)
0.574381 + 0.818588i \(0.305243\pi\)
\(338\) 0 0
\(339\) 13.0120i 0.706715i
\(340\) 0 0
\(341\) 42.3012i 2.29074i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 14.1273i 0.760587i
\(346\) 0 0
\(347\) 24.8209i 1.33245i 0.745749 + 0.666227i \(0.232092\pi\)
−0.745749 + 0.666227i \(0.767908\pi\)
\(348\) 0 0
\(349\) −8.51924 −0.456025 −0.228012 0.973658i \(-0.573223\pi\)
−0.228012 + 0.973658i \(0.573223\pi\)
\(350\) 0 0
\(351\) 12.9994 0.693858
\(352\) 0 0
\(353\) −26.3620 −1.40311 −0.701554 0.712617i \(-0.747510\pi\)
−0.701554 + 0.712617i \(0.747510\pi\)
\(354\) 0 0
\(355\) 16.6750i 0.885015i
\(356\) 0 0
\(357\) 5.49592i 0.290875i
\(358\) 0 0
\(359\) 10.2119 0.538963 0.269481 0.963006i \(-0.413148\pi\)
0.269481 + 0.963006i \(0.413148\pi\)
\(360\) 0 0
\(361\) 17.6779 0.930415
\(362\) 0 0
\(363\) 15.2411i 0.799951i
\(364\) 0 0
\(365\) −13.8649 −0.725724
\(366\) 0 0
\(367\) 15.9741 0.833842 0.416921 0.908943i \(-0.363109\pi\)
0.416921 + 0.908943i \(0.363109\pi\)
\(368\) 0 0
\(369\) −11.9761 2.91502i −0.623451 0.151750i
\(370\) 0 0
\(371\) −6.93136 −0.359858
\(372\) 0 0
\(373\) −29.2093 −1.51240 −0.756200 0.654340i \(-0.772946\pi\)
−0.756200 + 0.654340i \(0.772946\pi\)
\(374\) 0 0
\(375\) 10.1917i 0.526297i
\(376\) 0 0
\(377\) −22.1279 −1.13965
\(378\) 0 0
\(379\) 20.6013 1.05822 0.529108 0.848554i \(-0.322526\pi\)
0.529108 + 0.848554i \(0.322526\pi\)
\(380\) 0 0
\(381\) 5.94200i 0.304418i
\(382\) 0 0
\(383\) 9.48638i 0.484731i 0.970185 + 0.242366i \(0.0779233\pi\)
−0.970185 + 0.242366i \(0.922077\pi\)
\(384\) 0 0
\(385\) −12.3975 −0.631836
\(386\) 0 0
\(387\) −11.4846 −0.583797
\(388\) 0 0
\(389\) 9.88002 0.500937 0.250468 0.968125i \(-0.419415\pi\)
0.250468 + 0.968125i \(0.419415\pi\)
\(390\) 0 0
\(391\) 29.5326i 1.49353i
\(392\) 0 0
\(393\) 4.67947i 0.236048i
\(394\) 0 0
\(395\) 10.1157i 0.508978i
\(396\) 0 0
\(397\) 35.1575i 1.76450i 0.470778 + 0.882251i \(0.343973\pi\)
−0.470778 + 0.882251i \(0.656027\pi\)
\(398\) 0 0
\(399\) 1.19220i 0.0596845i
\(400\) 0 0
\(401\) −29.9291 −1.49459 −0.747295 0.664493i \(-0.768648\pi\)
−0.747295 + 0.664493i \(0.768648\pi\)
\(402\) 0 0
\(403\) 21.2422i 1.05815i
\(404\) 0 0
\(405\) 1.17472 0.0583725
\(406\) 0 0
\(407\) 56.0105i 2.77634i
\(408\) 0 0
\(409\) −24.6460 −1.21866 −0.609332 0.792915i \(-0.708562\pi\)
−0.609332 + 0.792915i \(0.708562\pi\)
\(410\) 0 0
\(411\) −3.34448 −0.164971
\(412\) 0 0
\(413\) 7.10317i 0.349524i
\(414\) 0 0
\(415\) −30.6808 −1.50606
\(416\) 0 0
\(417\) 4.65246i 0.227832i
\(418\) 0 0
\(419\) 1.57574 0.0769801 0.0384900 0.999259i \(-0.487745\pi\)
0.0384900 + 0.999259i \(0.487745\pi\)
\(420\) 0 0
\(421\) 3.89300i 0.189733i −0.995490 0.0948665i \(-0.969758\pi\)
0.995490 0.0948665i \(-0.0302424\pi\)
\(422\) 0 0
\(423\) 8.04957i 0.391383i
\(424\) 0 0
\(425\) 5.19772i 0.252127i
\(426\) 0 0
\(427\) 1.10590i 0.0535185i
\(428\) 0 0
\(429\) 13.3809i 0.646034i
\(430\) 0 0
\(431\) −36.5130 −1.75877 −0.879385 0.476112i \(-0.842046\pi\)
−0.879385 + 0.476112i \(0.842046\pi\)
\(432\) 0 0
\(433\) 4.87501 0.234278 0.117139 0.993116i \(-0.462628\pi\)
0.117139 + 0.993116i \(0.462628\pi\)
\(434\) 0 0
\(435\) 22.0402 1.05675
\(436\) 0 0
\(437\) 6.40635i 0.306457i
\(438\) 0 0
\(439\) 26.1080i 1.24607i 0.782195 + 0.623033i \(0.214100\pi\)
−0.782195 + 0.623033i \(0.785900\pi\)
\(440\) 0 0
\(441\) −1.92496 −0.0916648
\(442\) 0 0
\(443\) 14.6787 0.697405 0.348702 0.937233i \(-0.386622\pi\)
0.348702 + 0.937233i \(0.386622\pi\)
\(444\) 0 0
\(445\) 6.24868i 0.296216i
\(446\) 0 0
\(447\) −1.85046 −0.0875236
\(448\) 0 0
\(449\) 35.7567 1.68746 0.843732 0.536764i \(-0.180354\pi\)
0.843732 + 0.536764i \(0.180354\pi\)
\(450\) 0 0
\(451\) 7.67684 31.5396i 0.361488 1.48514i
\(452\) 0 0
\(453\) 9.23177 0.433746
\(454\) 0 0
\(455\) 6.22560 0.291861
\(456\) 0 0
\(457\) 19.2988i 0.902761i −0.892332 0.451380i \(-0.850932\pi\)
0.892332 0.451380i \(-0.149068\pi\)
\(458\) 0 0
\(459\) −27.0672 −1.26339
\(460\) 0 0
\(461\) 13.1664 0.613219 0.306609 0.951835i \(-0.400805\pi\)
0.306609 + 0.951835i \(0.400805\pi\)
\(462\) 0 0
\(463\) 5.48571i 0.254942i −0.991842 0.127471i \(-0.959314\pi\)
0.991842 0.127471i \(-0.0406860\pi\)
\(464\) 0 0
\(465\) 21.1579i 0.981176i
\(466\) 0 0
\(467\) 0.0926038 0.00428519 0.00214260 0.999998i \(-0.499318\pi\)
0.00214260 + 0.999998i \(0.499318\pi\)
\(468\) 0 0
\(469\) −2.43504 −0.112440
\(470\) 0 0
\(471\) −0.213549 −0.00983980
\(472\) 0 0
\(473\) 30.2453i 1.39068i
\(474\) 0 0
\(475\) 1.12751i 0.0517338i
\(476\) 0 0
\(477\) 13.3426i 0.610916i
\(478\) 0 0
\(479\) 18.5595i 0.848007i −0.905661 0.424003i \(-0.860624\pi\)
0.905661 0.424003i \(-0.139376\pi\)
\(480\) 0 0
\(481\) 28.1265i 1.28246i
\(482\) 0 0
\(483\) 5.77679 0.262853
\(484\) 0 0
\(485\) 16.7722i 0.761585i
\(486\) 0 0
\(487\) −11.0124 −0.499021 −0.249510 0.968372i \(-0.580270\pi\)
−0.249510 + 0.968372i \(0.580270\pi\)
\(488\) 0 0
\(489\) 1.04652i 0.0473253i
\(490\) 0 0
\(491\) 27.9332 1.26061 0.630304 0.776349i \(-0.282930\pi\)
0.630304 + 0.776349i \(0.282930\pi\)
\(492\) 0 0
\(493\) 46.0744 2.07509
\(494\) 0 0
\(495\) 23.8647i 1.07264i
\(496\) 0 0
\(497\) 6.81857 0.305855
\(498\) 0 0
\(499\) 35.9775i 1.61057i −0.592885 0.805287i \(-0.702011\pi\)
0.592885 0.805287i \(-0.297989\pi\)
\(500\) 0 0
\(501\) 23.6245 1.05546
\(502\) 0 0
\(503\) 16.3821i 0.730442i −0.930921 0.365221i \(-0.880993\pi\)
0.930921 0.365221i \(-0.119007\pi\)
\(504\) 0 0
\(505\) 0.178118i 0.00792616i
\(506\) 0 0
\(507\) 6.75953i 0.300201i
\(508\) 0 0
\(509\) 29.7528i 1.31877i 0.751805 + 0.659385i \(0.229183\pi\)
−0.751805 + 0.659385i \(0.770817\pi\)
\(510\) 0 0
\(511\) 5.66952i 0.250805i
\(512\) 0 0
\(513\) 5.87153 0.259234
\(514\) 0 0
\(515\) 4.04590 0.178283
\(516\) 0 0
\(517\) −21.1989 −0.932327
\(518\) 0 0
\(519\) 2.74363i 0.120432i
\(520\) 0 0
\(521\) 42.2378i 1.85047i −0.379394 0.925235i \(-0.623867\pi\)
0.379394 0.925235i \(-0.376133\pi\)
\(522\) 0 0
\(523\) −43.9610 −1.92228 −0.961140 0.276062i \(-0.910971\pi\)
−0.961140 + 0.276062i \(0.910971\pi\)
\(524\) 0 0
\(525\) −1.01671 −0.0443729
\(526\) 0 0
\(527\) 44.2301i 1.92669i
\(528\) 0 0
\(529\) 8.04193 0.349649
\(530\) 0 0
\(531\) 13.6733 0.593371
\(532\) 0 0
\(533\) −3.85504 + 15.8381i −0.166980 + 0.686024i
\(534\) 0 0
\(535\) 32.6562 1.41185
\(536\) 0 0
\(537\) −12.5923 −0.543396
\(538\) 0 0
\(539\) 5.06947i 0.218358i
\(540\) 0 0
\(541\) 34.0851 1.46543 0.732717 0.680534i \(-0.238252\pi\)
0.732717 + 0.680534i \(0.238252\pi\)
\(542\) 0 0
\(543\) 21.2225 0.910745
\(544\) 0 0
\(545\) 5.64203i 0.241678i
\(546\) 0 0
\(547\) 37.2593i 1.59309i 0.604578 + 0.796546i \(0.293342\pi\)
−0.604578 + 0.796546i \(0.706658\pi\)
\(548\) 0 0
\(549\) 2.12882 0.0908559
\(550\) 0 0
\(551\) −9.99465 −0.425787
\(552\) 0 0
\(553\) −4.13643 −0.175899
\(554\) 0 0
\(555\) 28.0150i 1.18917i
\(556\) 0 0
\(557\) 41.4925i 1.75810i 0.476734 + 0.879048i \(0.341821\pi\)
−0.476734 + 0.879048i \(0.658179\pi\)
\(558\) 0 0
\(559\) 15.1881i 0.642390i
\(560\) 0 0
\(561\) 27.8614i 1.17631i
\(562\) 0 0
\(563\) 18.5817i 0.783127i 0.920151 + 0.391563i \(0.128066\pi\)
−0.920151 + 0.391563i \(0.871934\pi\)
\(564\) 0 0
\(565\) 30.6905 1.29116
\(566\) 0 0
\(567\) 0.480357i 0.0201731i
\(568\) 0 0
\(569\) 44.1911 1.85259 0.926293 0.376804i \(-0.122977\pi\)
0.926293 + 0.376804i \(0.122977\pi\)
\(570\) 0 0
\(571\) 35.8113i 1.49866i −0.662199 0.749328i \(-0.730377\pi\)
0.662199 0.749328i \(-0.269623\pi\)
\(572\) 0 0
\(573\) −14.0068 −0.585144
\(574\) 0 0
\(575\) −5.46335 −0.227838
\(576\) 0 0
\(577\) 10.7811i 0.448821i −0.974495 0.224411i \(-0.927954\pi\)
0.974495 0.224411i \(-0.0720457\pi\)
\(578\) 0 0
\(579\) 14.6990 0.610870
\(580\) 0 0
\(581\) 12.5457i 0.520484i
\(582\) 0 0
\(583\) −35.1384 −1.45528
\(584\) 0 0
\(585\) 11.9840i 0.495479i
\(586\) 0 0
\(587\) 20.6643i 0.852909i 0.904509 + 0.426454i \(0.140238\pi\)
−0.904509 + 0.426454i \(0.859762\pi\)
\(588\) 0 0
\(589\) 9.59457i 0.395338i
\(590\) 0 0
\(591\) 22.8422i 0.939603i
\(592\) 0 0
\(593\) 5.15963i 0.211881i 0.994372 + 0.105940i \(0.0337852\pi\)
−0.994372 + 0.105940i \(0.966215\pi\)
\(594\) 0 0
\(595\) −12.9628 −0.531424
\(596\) 0 0
\(597\) 15.4686 0.633090
\(598\) 0 0
\(599\) 24.0629 0.983183 0.491591 0.870826i \(-0.336415\pi\)
0.491591 + 0.870826i \(0.336415\pi\)
\(600\) 0 0
\(601\) 30.4468i 1.24195i 0.783830 + 0.620976i \(0.213263\pi\)
−0.783830 + 0.620976i \(0.786737\pi\)
\(602\) 0 0
\(603\) 4.68735i 0.190884i
\(604\) 0 0
\(605\) −35.9481 −1.46150
\(606\) 0 0
\(607\) −1.12398 −0.0456210 −0.0228105 0.999740i \(-0.507261\pi\)
−0.0228105 + 0.999740i \(0.507261\pi\)
\(608\) 0 0
\(609\) 9.01247i 0.365204i
\(610\) 0 0
\(611\) 10.6454 0.430665
\(612\) 0 0
\(613\) −18.0651 −0.729643 −0.364821 0.931077i \(-0.618870\pi\)
−0.364821 + 0.931077i \(0.618870\pi\)
\(614\) 0 0
\(615\) 3.83975 15.7753i 0.154834 0.636122i
\(616\) 0 0
\(617\) 0.700580 0.0282043 0.0141021 0.999901i \(-0.495511\pi\)
0.0141021 + 0.999901i \(0.495511\pi\)
\(618\) 0 0
\(619\) −3.83466 −0.154128 −0.0770640 0.997026i \(-0.524555\pi\)
−0.0770640 + 0.997026i \(0.524555\pi\)
\(620\) 0 0
\(621\) 28.4505i 1.14168i
\(622\) 0 0
\(623\) −2.55515 −0.102370
\(624\) 0 0
\(625\) −28.9414 −1.15766
\(626\) 0 0
\(627\) 6.04381i 0.241367i
\(628\) 0 0
\(629\) 58.5645i 2.33512i
\(630\) 0 0
\(631\) −43.8159 −1.74429 −0.872143 0.489252i \(-0.837270\pi\)
−0.872143 + 0.489252i \(0.837270\pi\)
\(632\) 0 0
\(633\) 25.5067 1.01380
\(634\) 0 0
\(635\) 14.0150 0.556167
\(636\) 0 0
\(637\) 2.54571i 0.100865i
\(638\) 0 0
\(639\) 13.1255i 0.519236i
\(640\) 0 0
\(641\) 26.6857i 1.05402i 0.849858 + 0.527012i \(0.176688\pi\)
−0.849858 + 0.527012i \(0.823312\pi\)
\(642\) 0 0
\(643\) 10.3703i 0.408964i 0.978870 + 0.204482i \(0.0655510\pi\)
−0.978870 + 0.204482i \(0.934449\pi\)
\(644\) 0 0
\(645\) 15.1279i 0.595661i
\(646\) 0 0
\(647\) −14.2927 −0.561903 −0.280951 0.959722i \(-0.590650\pi\)
−0.280951 + 0.959722i \(0.590650\pi\)
\(648\) 0 0
\(649\) 36.0093i 1.41349i
\(650\) 0 0
\(651\) −8.65171 −0.339087
\(652\) 0 0
\(653\) 39.3548i 1.54007i 0.638000 + 0.770037i \(0.279762\pi\)
−0.638000 + 0.770037i \(0.720238\pi\)
\(654\) 0 0
\(655\) −11.0371 −0.431257
\(656\) 0 0
\(657\) −10.9136 −0.425780
\(658\) 0 0
\(659\) 15.6534i 0.609768i −0.952389 0.304884i \(-0.901382\pi\)
0.952389 0.304884i \(-0.0986178\pi\)
\(660\) 0 0
\(661\) −7.42227 −0.288693 −0.144347 0.989527i \(-0.546108\pi\)
−0.144347 + 0.989527i \(0.546108\pi\)
\(662\) 0 0
\(663\) 13.9910i 0.543366i
\(664\) 0 0
\(665\) 2.81195 0.109043
\(666\) 0 0
\(667\) 48.4291i 1.87518i
\(668\) 0 0
\(669\) 2.73436i 0.105716i
\(670\) 0 0
\(671\) 5.60635i 0.216431i
\(672\) 0 0
\(673\) 38.0277i 1.46586i 0.680304 + 0.732930i \(0.261848\pi\)
−0.680304 + 0.732930i \(0.738152\pi\)
\(674\) 0 0
\(675\) 5.00726i 0.192729i
\(676\) 0 0
\(677\) −16.2475 −0.624444 −0.312222 0.950009i \(-0.601073\pi\)
−0.312222 + 0.950009i \(0.601073\pi\)
\(678\) 0 0
\(679\) 6.85832 0.263198
\(680\) 0 0
\(681\) 14.3781 0.550972
\(682\) 0 0
\(683\) 21.4141i 0.819387i 0.912223 + 0.409694i \(0.134364\pi\)
−0.912223 + 0.409694i \(0.865636\pi\)
\(684\) 0 0
\(685\) 7.88839i 0.301400i
\(686\) 0 0
\(687\) 19.0187 0.725609
\(688\) 0 0
\(689\) 17.6452 0.672230
\(690\) 0 0
\(691\) 7.95535i 0.302636i 0.988485 + 0.151318i \(0.0483517\pi\)
−0.988485 + 0.151318i \(0.951648\pi\)
\(692\) 0 0
\(693\) −9.75854 −0.370696
\(694\) 0 0
\(695\) 10.9734 0.416246
\(696\) 0 0
\(697\) 8.02689 32.9778i 0.304040 1.24912i
\(698\) 0 0
\(699\) −4.46158 −0.168752
\(700\) 0 0
\(701\) 31.1962 1.17826 0.589132 0.808037i \(-0.299470\pi\)
0.589132 + 0.808037i \(0.299470\pi\)
\(702\) 0 0
\(703\) 12.7041i 0.479143i
\(704\) 0 0
\(705\) −10.6031 −0.399338
\(706\) 0 0
\(707\) 0.0728344 0.00273922
\(708\) 0 0
\(709\) 49.0697i 1.84285i 0.388554 + 0.921426i \(0.372975\pi\)
−0.388554 + 0.921426i \(0.627025\pi\)
\(710\) 0 0
\(711\) 7.96247i 0.298616i
\(712\) 0 0
\(713\) −46.4905 −1.74108
\(714\) 0 0
\(715\) 31.5605 1.18030
\(716\) 0 0
\(717\) 19.3768 0.723640
\(718\) 0 0
\(719\) 33.7020i 1.25687i −0.777861 0.628436i \(-0.783695\pi\)
0.777861 0.628436i \(-0.216305\pi\)
\(720\) 0 0
\(721\) 1.65441i 0.0616134i
\(722\) 0 0
\(723\) 13.3950i 0.498165i
\(724\) 0 0
\(725\) 8.52347i 0.316554i
\(726\) 0 0
\(727\) 39.1952i 1.45367i −0.686812 0.726835i \(-0.740990\pi\)
0.686812 0.726835i \(-0.259010\pi\)
\(728\) 0 0
\(729\) −14.9589 −0.554034
\(730\) 0 0
\(731\) 31.6245i 1.16967i
\(732\) 0 0
\(733\) 37.2355 1.37533 0.687663 0.726030i \(-0.258637\pi\)
0.687663 + 0.726030i \(0.258637\pi\)
\(734\) 0 0
\(735\) 2.53562i 0.0935277i
\(736\) 0 0
\(737\) −12.3444 −0.454710
\(738\) 0 0
\(739\) −27.9374 −1.02769 −0.513846 0.857882i \(-0.671780\pi\)
−0.513846 + 0.857882i \(0.671780\pi\)
\(740\) 0 0
\(741\) 3.03499i 0.111493i
\(742\) 0 0
\(743\) −11.8458 −0.434582 −0.217291 0.976107i \(-0.569722\pi\)
−0.217291 + 0.976107i \(0.569722\pi\)
\(744\) 0 0
\(745\) 4.36454i 0.159904i
\(746\) 0 0
\(747\) −24.1500 −0.883603
\(748\) 0 0
\(749\) 13.3535i 0.487925i
\(750\) 0 0
\(751\) 44.7271i 1.63211i −0.577971 0.816057i \(-0.696155\pi\)
0.577971 0.816057i \(-0.303845\pi\)
\(752\) 0 0
\(753\) 25.2722i 0.920972i
\(754\) 0 0
\(755\) 21.7743i 0.792449i
\(756\) 0 0
\(757\) 1.18991i 0.0432481i −0.999766 0.0216241i \(-0.993116\pi\)
0.999766 0.0216241i \(-0.00688369\pi\)
\(758\) 0 0
\(759\) 29.2853 1.06299
\(760\) 0 0
\(761\) 14.8342 0.537741 0.268870 0.963176i \(-0.413350\pi\)
0.268870 + 0.963176i \(0.413350\pi\)
\(762\) 0 0
\(763\) −2.30709 −0.0835221
\(764\) 0 0
\(765\) 24.9529i 0.902176i
\(766\) 0 0
\(767\) 18.0826i 0.652925i
\(768\) 0 0
\(769\) 9.17776 0.330959 0.165479 0.986213i \(-0.447083\pi\)
0.165479 + 0.986213i \(0.447083\pi\)
\(770\) 0 0
\(771\) 14.7082 0.529702
\(772\) 0 0
\(773\) 13.7536i 0.494681i 0.968929 + 0.247341i \(0.0795567\pi\)
−0.968929 + 0.247341i \(0.920443\pi\)
\(774\) 0 0
\(775\) 8.18229 0.293916
\(776\) 0 0
\(777\) −11.4556 −0.410968
\(778\) 0 0
\(779\) −1.74123 + 7.15369i −0.0623859 + 0.256307i
\(780\) 0 0
\(781\) 34.5666 1.23689
\(782\) 0 0
\(783\) 44.3861 1.58623
\(784\) 0 0
\(785\) 0.503682i 0.0179772i
\(786\) 0 0
\(787\) −2.33979 −0.0834045 −0.0417022 0.999130i \(-0.513278\pi\)
−0.0417022 + 0.999130i \(0.513278\pi\)
\(788\) 0 0
\(789\) 7.51651 0.267595
\(790\) 0 0
\(791\) 12.5497i 0.446215i
\(792\) 0 0
\(793\) 2.81531i 0.0999747i
\(794\) 0 0
\(795\) −17.5753 −0.623331
\(796\) 0 0
\(797\) 37.4073 1.32504 0.662518 0.749046i \(-0.269488\pi\)
0.662518 + 0.749046i \(0.269488\pi\)
\(798\) 0 0
\(799\) −22.1656 −0.784161
\(800\) 0 0
\(801\) 4.91857i 0.173789i
\(802\) 0 0
\(803\) 28.7415i 1.01426i
\(804\) 0 0
\(805\) 13.6253i 0.480229i
\(806\) 0