Properties

Label 112.2.f.a.111.1
Level 112
Weight 2
Character 112.111
Analytic conductor 0.894
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 112.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 111.1
Root \(0.500000 + 0.866025i\)
Character \(\chi\) = 112.111
Dual form 112.2.f.a.111.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{3} -3.46410i q^{5} +(2.00000 - 1.73205i) q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} -3.46410i q^{5} +(2.00000 - 1.73205i) q^{7} +1.00000 q^{9} -3.46410i q^{11} +3.46410i q^{13} +6.92820i q^{15} -2.00000 q^{19} +(-4.00000 + 3.46410i) q^{21} +3.46410i q^{23} -7.00000 q^{25} +4.00000 q^{27} +6.00000 q^{29} +8.00000 q^{31} +6.92820i q^{33} +(-6.00000 - 6.92820i) q^{35} -2.00000 q^{37} -6.92820i q^{39} -6.92820i q^{41} +10.3923i q^{43} -3.46410i q^{45} +(1.00000 - 6.92820i) q^{49} +6.00000 q^{53} -12.0000 q^{55} +4.00000 q^{57} -6.00000 q^{59} +3.46410i q^{61} +(2.00000 - 1.73205i) q^{63} +12.0000 q^{65} -3.46410i q^{67} -6.92820i q^{69} -3.46410i q^{71} +6.92820i q^{73} +14.0000 q^{75} +(-6.00000 - 6.92820i) q^{77} -3.46410i q^{79} -11.0000 q^{81} +6.00000 q^{83} -12.0000 q^{87} -6.92820i q^{89} +(6.00000 + 6.92820i) q^{91} -16.0000 q^{93} +6.92820i q^{95} +13.8564i q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} + 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} + 4q^{7} + 2q^{9} - 4q^{19} - 8q^{21} - 14q^{25} + 8q^{27} + 12q^{29} + 16q^{31} - 12q^{35} - 4q^{37} + 2q^{49} + 12q^{53} - 24q^{55} + 8q^{57} - 12q^{59} + 4q^{63} + 24q^{65} + 28q^{75} - 12q^{77} - 22q^{81} + 12q^{83} - 24q^{87} + 12q^{91} - 32q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 3.46410i 1.54919i −0.632456 0.774597i \(-0.717953\pi\)
0.632456 0.774597i \(-0.282047\pi\)
\(6\) 0 0
\(7\) 2.00000 1.73205i 0.755929 0.654654i
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.46410i 1.04447i −0.852803 0.522233i \(-0.825099\pi\)
0.852803 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 6.92820i 1.78885i
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −4.00000 + 3.46410i −0.872872 + 0.755929i
\(22\) 0 0
\(23\) 3.46410i 0.722315i 0.932505 + 0.361158i \(0.117618\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 6.92820i 1.20605i
\(34\) 0 0
\(35\) −6.00000 6.92820i −1.01419 1.17108i
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 6.92820i 1.10940i
\(40\) 0 0
\(41\) 6.92820i 1.08200i −0.841021 0.541002i \(-0.818045\pi\)
0.841021 0.541002i \(-0.181955\pi\)
\(42\) 0 0
\(43\) 10.3923i 1.58481i 0.609994 + 0.792406i \(0.291172\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 3.46410i 0.516398i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 6.92820i 0.142857 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 3.46410i 0.443533i 0.975100 + 0.221766i \(0.0711822\pi\)
−0.975100 + 0.221766i \(0.928818\pi\)
\(62\) 0 0
\(63\) 2.00000 1.73205i 0.251976 0.218218i
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 3.46410i 0.423207i −0.977356 0.211604i \(-0.932131\pi\)
0.977356 0.211604i \(-0.0678686\pi\)
\(68\) 0 0
\(69\) 6.92820i 0.834058i
\(70\) 0 0
\(71\) 3.46410i 0.411113i −0.978645 0.205557i \(-0.934100\pi\)
0.978645 0.205557i \(-0.0659005\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) 0 0
\(75\) 14.0000 1.61658
\(76\) 0 0
\(77\) −6.00000 6.92820i −0.683763 0.789542i
\(78\) 0 0
\(79\) 3.46410i 0.389742i −0.980829 0.194871i \(-0.937571\pi\)
0.980829 0.194871i \(-0.0624288\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) 6.92820i 0.734388i −0.930144 0.367194i \(-0.880318\pi\)
0.930144 0.367194i \(-0.119682\pi\)
\(90\) 0 0
\(91\) 6.00000 + 6.92820i 0.628971 + 0.726273i
\(92\) 0 0
\(93\) −16.0000 −1.65912
\(94\) 0 0
\(95\) 6.92820i 0.710819i
\(96\) 0 0
\(97\) 13.8564i 1.40690i 0.710742 + 0.703452i \(0.248359\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 3.46410i 0.348155i
\(100\) 0 0
\(101\) 10.3923i 1.03407i 0.855963 + 0.517036i \(0.172965\pi\)
−0.855963 + 0.517036i \(0.827035\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 12.0000 + 13.8564i 1.17108 + 1.35225i
\(106\) 0 0
\(107\) 10.3923i 1.00466i 0.864675 + 0.502331i \(0.167524\pi\)
−0.864675 + 0.502331i \(0.832476\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 12.0000 1.11901
\(116\) 0 0
\(117\) 3.46410i 0.320256i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 13.8564i 1.24939i
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) 10.3923i 0.922168i −0.887357 0.461084i \(-0.847461\pi\)
0.887357 0.461084i \(-0.152539\pi\)
\(128\) 0 0
\(129\) 20.7846i 1.82998i
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) −4.00000 + 3.46410i −0.346844 + 0.300376i
\(134\) 0 0
\(135\) 13.8564i 1.19257i
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 20.7846i 1.72607i
\(146\) 0 0
\(147\) −2.00000 + 13.8564i −0.164957 + 1.14286i
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 3.46410i 0.281905i 0.990016 + 0.140952i \(0.0450164\pi\)
−0.990016 + 0.140952i \(0.954984\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 27.7128i 2.22595i
\(156\) 0 0
\(157\) 10.3923i 0.829396i −0.909959 0.414698i \(-0.863887\pi\)
0.909959 0.414698i \(-0.136113\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 6.00000 + 6.92820i 0.472866 + 0.546019i
\(162\) 0 0
\(163\) 17.3205i 1.35665i −0.734763 0.678323i \(-0.762707\pi\)
0.734763 0.678323i \(-0.237293\pi\)
\(164\) 0 0
\(165\) 24.0000 1.86840
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 3.46410i 0.263371i 0.991292 + 0.131685i \(0.0420389\pi\)
−0.991292 + 0.131685i \(0.957961\pi\)
\(174\) 0 0
\(175\) −14.0000 + 12.1244i −1.05830 + 0.916515i
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 10.3923i 0.776757i 0.921500 + 0.388379i \(0.126965\pi\)
−0.921500 + 0.388379i \(0.873035\pi\)
\(180\) 0 0
\(181\) 17.3205i 1.28742i −0.765268 0.643712i \(-0.777394\pi\)
0.765268 0.643712i \(-0.222606\pi\)
\(182\) 0 0
\(183\) 6.92820i 0.512148i
\(184\) 0 0
\(185\) 6.92820i 0.509372i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 8.00000 6.92820i 0.581914 0.503953i
\(190\) 0 0
\(191\) 24.2487i 1.75458i 0.479965 + 0.877288i \(0.340649\pi\)
−0.479965 + 0.877288i \(0.659351\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) −24.0000 −1.71868
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 6.92820i 0.488678i
\(202\) 0 0
\(203\) 12.0000 10.3923i 0.842235 0.729397i
\(204\) 0 0
\(205\) −24.0000 −1.67623
\(206\) 0 0
\(207\) 3.46410i 0.240772i
\(208\) 0 0
\(209\) 6.92820i 0.479234i
\(210\) 0 0
\(211\) 3.46410i 0.238479i −0.992866 0.119239i \(-0.961954\pi\)
0.992866 0.119239i \(-0.0380456\pi\)
\(212\) 0 0
\(213\) 6.92820i 0.474713i
\(214\) 0 0
\(215\) 36.0000 2.45518
\(216\) 0 0
\(217\) 16.0000 13.8564i 1.08615 0.940634i
\(218\) 0 0
\(219\) 13.8564i 0.936329i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) −7.00000 −0.466667
\(226\) 0 0
\(227\) 6.00000 0.398234 0.199117 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) 0 0
\(229\) 10.3923i 0.686743i 0.939200 + 0.343371i \(0.111569\pi\)
−0.939200 + 0.343371i \(0.888431\pi\)
\(230\) 0 0
\(231\) 12.0000 + 13.8564i 0.789542 + 0.911685i
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 6.92820i 0.450035i
\(238\) 0 0
\(239\) 10.3923i 0.672222i −0.941822 0.336111i \(-0.890888\pi\)
0.941822 0.336111i \(-0.109112\pi\)
\(240\) 0 0
\(241\) 27.7128i 1.78514i 0.450910 + 0.892570i \(0.351100\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) −24.0000 3.46410i −1.53330 0.221313i
\(246\) 0 0
\(247\) 6.92820i 0.440831i
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −30.0000 −1.89358 −0.946792 0.321847i \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.7128i 1.72868i 0.502910 + 0.864339i \(0.332263\pi\)
−0.502910 + 0.864339i \(0.667737\pi\)
\(258\) 0 0
\(259\) −4.00000 + 3.46410i −0.248548 + 0.215249i
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 24.2487i 1.49524i 0.664127 + 0.747620i \(0.268803\pi\)
−0.664127 + 0.747620i \(0.731197\pi\)
\(264\) 0 0
\(265\) 20.7846i 1.27679i
\(266\) 0 0
\(267\) 13.8564i 0.847998i
\(268\) 0 0
\(269\) 3.46410i 0.211210i 0.994408 + 0.105605i \(0.0336779\pi\)
−0.994408 + 0.105605i \(0.966322\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) −12.0000 13.8564i −0.726273 0.838628i
\(274\) 0 0
\(275\) 24.2487i 1.46225i
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 0 0
\(285\) 13.8564i 0.820783i
\(286\) 0 0
\(287\) −12.0000 13.8564i −0.708338 0.817918i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 27.7128i 1.62455i
\(292\) 0 0
\(293\) 10.3923i 0.607125i 0.952812 + 0.303562i \(0.0981761\pi\)
−0.952812 + 0.303562i \(0.901824\pi\)
\(294\) 0 0
\(295\) 20.7846i 1.21013i
\(296\) 0 0
\(297\) 13.8564i 0.804030i
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 18.0000 + 20.7846i 1.03750 + 1.19800i
\(302\) 0 0
\(303\) 20.7846i 1.19404i
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 6.92820i 0.391605i 0.980643 + 0.195803i \(0.0627312\pi\)
−0.980643 + 0.195803i \(0.937269\pi\)
\(314\) 0 0
\(315\) −6.00000 6.92820i −0.338062 0.390360i
\(316\) 0 0
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 20.7846i 1.16371i
\(320\) 0 0
\(321\) 20.7846i 1.16008i
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 24.2487i 1.34508i
\(326\) 0 0
\(327\) −28.0000 −1.54840
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3.46410i 0.190404i −0.995458 0.0952021i \(-0.969650\pi\)
0.995458 0.0952021i \(-0.0303497\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 36.0000 1.95525
\(340\) 0 0
\(341\) 27.7128i 1.50073i
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) −24.0000 −1.29212
\(346\) 0 0
\(347\) 17.3205i 0.929814i −0.885360 0.464907i \(-0.846088\pi\)
0.885360 0.464907i \(-0.153912\pi\)
\(348\) 0 0
\(349\) 3.46410i 0.185429i 0.995693 + 0.0927146i \(0.0295544\pi\)
−0.995693 + 0.0927146i \(0.970446\pi\)
\(350\) 0 0
\(351\) 13.8564i 0.739600i
\(352\) 0 0
\(353\) 27.7128i 1.47500i −0.675345 0.737502i \(-0.736005\pi\)
0.675345 0.737502i \(-0.263995\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 31.1769i 1.64545i 0.568436 + 0.822727i \(0.307549\pi\)
−0.568436 + 0.822727i \(0.692451\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 24.0000 1.25622
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 6.92820i 0.360668i
\(370\) 0 0
\(371\) 12.0000 10.3923i 0.623009 0.539542i
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 13.8564i 0.715542i
\(376\) 0 0
\(377\) 20.7846i 1.07046i
\(378\) 0 0
\(379\) 24.2487i 1.24557i 0.782392 + 0.622786i \(0.213999\pi\)
−0.782392 + 0.622786i \(0.786001\pi\)
\(380\) 0 0
\(381\) 20.7846i 1.06483i
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) −24.0000 + 20.7846i −1.22315 + 1.05928i
\(386\) 0 0
\(387\) 10.3923i 0.528271i
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 36.0000 1.81596
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 10.3923i 0.521575i −0.965396 0.260787i \(-0.916018\pi\)
0.965396 0.260787i \(-0.0839822\pi\)
\(398\) 0 0
\(399\) 8.00000 6.92820i 0.400501 0.346844i
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 27.7128i 1.38047i
\(404\) 0 0
\(405\) 38.1051i 1.89346i
\(406\) 0 0
\(407\) 6.92820i 0.343418i
\(408\) 0 0
\(409\) 6.92820i 0.342578i 0.985221 + 0.171289i \(0.0547931\pi\)
−0.985221 + 0.171289i \(0.945207\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) −12.0000 + 10.3923i −0.590481 + 0.511372i
\(414\) 0 0
\(415\) 20.7846i 1.02028i
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6.00000 + 6.92820i 0.290360 + 0.335279i
\(428\) 0 0
\(429\) −24.0000 −1.15873
\(430\) 0 0
\(431\) 38.1051i 1.83546i −0.397206 0.917729i \(-0.630020\pi\)
0.397206 0.917729i \(-0.369980\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 41.5692i 1.99309i
\(436\) 0 0
\(437\) 6.92820i 0.331421i
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 1.00000 6.92820i 0.0476190 0.329914i
\(442\) 0 0
\(443\) 3.46410i 0.164584i −0.996608 0.0822922i \(-0.973776\pi\)
0.996608 0.0822922i \(-0.0262241\pi\)
\(444\) 0 0
\(445\) −24.0000 −1.13771
\(446\) 0 0
\(447\) −12.0000 −0.567581
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 0 0
\(453\) 6.92820i 0.325515i
\(454\) 0 0
\(455\) 24.0000 20.7846i 1.12514 0.974398i
\(456\) 0 0
\(457\) −34.0000 −1.59045 −0.795226 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 38.1051i 1.77473i −0.461065 0.887366i \(-0.652533\pi\)
0.461065 0.887366i \(-0.347467\pi\)
\(462\) 0 0
\(463\) 24.2487i 1.12693i 0.826139 + 0.563467i \(0.190533\pi\)
−0.826139 + 0.563467i \(0.809467\pi\)
\(464\) 0 0
\(465\) 55.4256i 2.57030i
\(466\) 0 0
\(467\) 30.0000 1.38823 0.694117 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) 0 0
\(469\) −6.00000 6.92820i −0.277054 0.319915i
\(470\) 0 0
\(471\) 20.7846i 0.957704i
\(472\) 0 0
\(473\) 36.0000 1.65528
\(474\) 0 0
\(475\) 14.0000 0.642364
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 6.92820i 0.315899i
\(482\) 0 0
\(483\) −12.0000 13.8564i −0.546019 0.630488i
\(484\) 0 0
\(485\) 48.0000 2.17957
\(486\) 0 0
\(487\) 24.2487i 1.09881i −0.835555 0.549407i \(-0.814854\pi\)
0.835555 0.549407i \(-0.185146\pi\)
\(488\) 0 0
\(489\) 34.6410i 1.56652i
\(490\) 0 0
\(491\) 17.3205i 0.781664i −0.920462 0.390832i \(-0.872187\pi\)
0.920462 0.390832i \(-0.127813\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 0 0
\(497\) −6.00000 6.92820i −0.269137 0.310772i
\(498\) 0 0
\(499\) 10.3923i 0.465223i 0.972570 + 0.232612i \(0.0747271\pi\)
−0.972570 + 0.232612i \(0.925273\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) 0 0
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) 0 0
\(507\) −2.00000 −0.0888231
\(508\) 0 0
\(509\) 31.1769i 1.38189i 0.722906 + 0.690946i \(0.242806\pi\)
−0.722906 + 0.690946i \(0.757194\pi\)
\(510\) 0 0
\(511\) 12.0000 + 13.8564i 0.530849 + 0.612971i
\(512\) 0 0
\(513\) −8.00000 −0.353209
\(514\) 0 0
\(515\) 13.8564i 0.610586i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 6.92820i 0.304114i
\(520\) 0 0
\(521\) 34.6410i 1.51765i −0.651294 0.758825i \(-0.725774\pi\)
0.651294 0.758825i \(-0.274226\pi\)
\(522\) 0 0
\(523\) 2.00000 0.0874539 0.0437269 0.999044i \(-0.486077\pi\)
0.0437269 + 0.999044i \(0.486077\pi\)
\(524\) 0 0
\(525\) 28.0000 24.2487i 1.22202 1.05830i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) 36.0000 1.55642
\(536\) 0 0
\(537\) 20.7846i 0.896922i
\(538\) 0 0
\(539\) −24.0000 3.46410i −1.03375 0.149209i
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 34.6410i 1.48659i
\(544\) 0 0
\(545\) 48.4974i 2.07740i
\(546\) 0 0
\(547\) 17.3205i 0.740571i −0.928918 0.370286i \(-0.879260\pi\)
0.928918 0.370286i \(-0.120740\pi\)
\(548\) 0 0
\(549\) 3.46410i 0.147844i
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −6.00000 6.92820i −0.255146 0.294617i
\(554\) 0 0
\(555\) 13.8564i 0.588172i
\(556\) 0 0
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) −36.0000 −1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −18.0000 −0.758610 −0.379305 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) 0 0
\(565\) 62.3538i 2.62325i
\(566\) 0 0
\(567\) −22.0000 + 19.0526i −0.923913 + 0.800132i
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 45.0333i 1.88459i −0.334790 0.942293i \(-0.608665\pi\)
0.334790 0.942293i \(-0.391335\pi\)
\(572\) 0 0
\(573\) 48.4974i 2.02601i
\(574\) 0 0
\(575\) 24.2487i 1.01124i
\(576\) 0 0
\(577\) 27.7128i 1.15370i −0.816850 0.576850i \(-0.804282\pi\)
0.816850 0.576850i \(-0.195718\pi\)
\(578\) 0 0
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 12.0000 10.3923i 0.497844 0.431145i
\(582\) 0 0
\(583\) 20.7846i 0.860811i
\(584\) 0 0
\(585\) 12.0000 0.496139
\(586\) 0 0
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 36.0000 1.48084
\(592\) 0 0
\(593\) 13.8564i 0.569014i −0.958674 0.284507i \(-0.908170\pi\)
0.958674 0.284507i \(-0.0918300\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 40.0000 1.63709
\(598\) 0 0
\(599\) 24.2487i 0.990775i 0.868672 + 0.495388i \(0.164974\pi\)
−0.868672 + 0.495388i \(0.835026\pi\)
\(600\) 0 0
\(601\) 20.7846i 0.847822i 0.905704 + 0.423911i \(0.139343\pi\)
−0.905704 + 0.423911i \(0.860657\pi\)
\(602\) 0 0
\(603\) 3.46410i 0.141069i
\(604\) 0 0
\(605\) 3.46410i 0.140836i
\(606\) 0 0
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 0 0
\(609\) −24.0000 + 20.7846i −0.972529 + 0.842235i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 0 0
\(615\) 48.0000 1.93555
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 13.8564i 0.556038i
\(622\) 0 0
\(623\) −12.0000 13.8564i −0.480770 0.555145i
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 13.8564i 0.553372i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 3.46410i 0.137904i −0.997620 0.0689519i \(-0.978035\pi\)
0.997620 0.0689519i \(-0.0219655\pi\)
\(632\) 0 0
\(633\) 6.92820i 0.275371i
\(634\) 0 0
\(635\) −36.0000 −1.42862
\(636\) 0 0
\(637\) 24.0000 + 3.46410i 0.950915 + 0.137253i
\(638\) 0 0
\(639\) 3.46410i 0.137038i
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) 0 0
\(645\) −72.0000 −2.83500
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 20.7846i 0.815867i
\(650\) 0 0
\(651\) −32.0000 + 27.7128i −1.25418 + 1.08615i
\(652\) 0 0
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) 62.3538i 2.43637i
\(656\) 0 0
\(657\) 6.92820i 0.270295i
\(658\) 0 0
\(659\) 3.46410i 0.134942i −0.997721 0.0674711i \(-0.978507\pi\)
0.997721 0.0674711i \(-0.0214931\pi\)
\(660\) 0 0
\(661\) 3.46410i 0.134738i −0.997728 0.0673690i \(-0.978540\pi\)
0.997728 0.0673690i \(-0.0214605\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 12.0000 + 13.8564i 0.465340 + 0.537328i
\(666\) 0 0
\(667\) 20.7846i 0.804783i
\(668\) 0 0
\(669\) −32.0000 −1.23719
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) −28.0000 −1.07772
\(676\) 0 0
\(677\) 3.46410i 0.133136i −0.997782 0.0665681i \(-0.978795\pi\)
0.997782 0.0665681i \(-0.0212050\pi\)
\(678\) 0 0
\(679\) 24.0000 + 27.7128i 0.921035 + 1.06352i
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 38.1051i 1.45805i 0.684486 + 0.729026i \(0.260027\pi\)
−0.684486 + 0.729026i \(0.739973\pi\)
\(684\) 0 0
\(685\) 20.7846i 0.794139i
\(686\) 0 0
\(687\) 20.7846i 0.792982i
\(688\) 0 0
\(689\) 20.7846i 0.791831i
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 0 0
\(693\) −6.00000 6.92820i −0.227921 0.263181i
\(694\) 0 0
\(695\) 6.92820i 0.262802i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.0000 + 20.7846i 0.676960 + 0.781686i
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 3.46410i 0.129914i
\(712\) 0 0
\(713\) 27.7128i 1.03785i
\(714\) 0 0
\(715\) 41.5692i 1.55460i
\(716\) 0 0
\(717\) 20.7846i 0.776215i
\(718\) 0 0
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) −8.00000 + 6.92820i −0.297936 + 0.258020i
\(722\) 0 0
\(723\) 55.4256i 2.06130i
\(724\) 0 0
\(725\) −42.0000 −1.55984
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 51.9615i 1.91924i −0.281295 0.959621i \(-0.590764\pi\)
0.281295 0.959621i \(-0.409236\pi\)
\(734\) 0 0
\(735\) 48.0000 + 6.92820i 1.77051 + 0.255551i
\(736\) 0 0
\(737\) −12.0000 −0.442026
\(738\) 0 0
\(739\) 10.3923i 0.382287i 0.981562 + 0.191144i \(0.0612196\pi\)
−0.981562 + 0.191144i \(0.938780\pi\)
\(740\) 0 0
\(741\) 13.8564i 0.509028i
\(742\) 0 0
\(743\) 24.2487i 0.889599i −0.895630 0.444799i \(-0.853275\pi\)
0.895630 0.444799i \(-0.146725\pi\)
\(744\) 0 0
\(745\) 20.7846i 0.761489i
\(746\) 0 0
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) 18.0000 + 20.7846i 0.657706 + 0.759453i
\(750\) 0 0
\(751\) 45.0333i 1.64329i 0.570000 + 0.821645i \(0.306943\pi\)
−0.570000 + 0.821645i \(0.693057\pi\)
\(752\) 0 0
\(753\) 60.0000 2.18652
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) −24.0000 −0.871145
\(760\) 0 0
\(761\) 20.7846i 0.753442i −0.926327 0.376721i \(-0.877052\pi\)
0.926327 0.376721i \(-0.122948\pi\)
\(762\) 0 0
\(763\) 28.0000 24.2487i 1.01367 0.877862i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.7846i 0.750489i
\(768\) 0 0
\(769\) 13.8564i 0.499675i 0.968288 + 0.249837i \(0.0803772\pi\)
−0.968288 + 0.249837i \(0.919623\pi\)
\(770\) 0 0
\(771\) 55.4256i 1.99611i
\(772\) 0 0
\(773\) 3.46410i 0.124595i −0.998058 0.0622975i \(-0.980157\pi\)
0.998058 0.0622975i \(-0.0198428\pi\)
\(774\) 0 0
\(775\) −56.0000 −2.01158
\(776\) 0 0
\(777\) 8.00000 6.92820i 0.286998 0.248548i
\(778\) 0 0
\(779\) 13.8564i 0.496457i
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 0 0
\(785\) −36.0000 −1.28490
\(786\) 0 0
\(787\) −34.0000 −1.21197 −0.605985 0.795476i \(-0.707221\pi\)
−0.605985 + 0.795476i \(0.707221\pi\)
\(788\) 0 0
\(789\) 48.4974i 1.72655i
\(790\) 0 0
\(791\) −36.0000 + 31.1769i −1.28001 + 1.10852i
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 0 0
\(795\) 41.5692i 1.47431i
\(796\) 0 0
\(797\) 17.3205i 0.613524i 0.951786 + 0.306762i \(0.0992455\pi\)
−0.951786 + 0.306762i \(0.900754\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.92820i 0.244796i
\(802\) 0 0
\(803\) 24.0000 0.846942
\(804\) 0 0
\(805\) 24.0000 20.7846i 0.845889 0.732561i
\(806\) 0 0
\(807\) 6.92820i 0.243884i
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −22.0000 −0.772524 −0.386262 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0