Properties

Label 1560.2.l.d.1249.6
Level $1560$
Weight $2$
Character 1560.1249
Analytic conductor $12.457$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.4566627153\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.57815240704.2
Defining polynomial: \(x^{8} - 2 x^{7} + 2 x^{6} + 89 x^{4} - 170 x^{3} + 162 x^{2} - 72 x + 16\)
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 1249.6
Root \(0.594137 + 0.594137i\) of defining polynomial
Character \(\chi\) \(=\) 1560.1249
Dual form 1560.2.l.d.1249.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +(-0.594137 - 2.15569i) q^{5} -4.92778i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(-0.594137 - 2.15569i) q^{5} -4.92778i q^{7} -1.00000 q^{9} -1.38360 q^{11} -1.00000i q^{13} +(2.15569 - 0.594137i) q^{15} +0.195329i q^{17} +4.92778 q^{21} +2.19533i q^{23} +(-4.29400 + 2.56155i) q^{25} -1.00000i q^{27} -7.49966 q^{29} -1.38360i q^{33} +(-10.6228 + 2.92778i) q^{35} +6.05088i q^{37} +1.00000 q^{39} -2.11605 q^{41} -3.49966i q^{43} +(0.594137 + 2.15569i) q^{45} +2.31138i q^{47} -17.2830 q^{49} -0.195329 q^{51} +6.05088i q^{53} +(0.822051 + 2.98262i) q^{55} -9.43449 q^{59} -3.69498 q^{61} +4.92778i q^{63} +(-2.15569 + 0.594137i) q^{65} -12.6228i q^{67} -2.19533 q^{69} +13.9716 q^{71} -4.73245i q^{73} +(-2.56155 - 4.29400i) q^{75} +6.81809i q^{77} +8.07153 q^{79} +1.00000 q^{81} +13.3997i q^{83} +(0.421068 - 0.116052i) q^{85} -7.49966i q^{87} -3.02770 q^{89} -4.92778 q^{91} +6.01612i q^{97} +1.38360 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 8 q^{9} + O(q^{10}) \) \( 8 q - 2 q^{5} - 8 q^{9} + 2 q^{11} - 2 q^{15} + 14 q^{21} - 16 q^{29} - 8 q^{35} + 8 q^{39} + 14 q^{41} + 2 q^{45} - 18 q^{49} + 6 q^{51} + 10 q^{55} - 4 q^{59} + 22 q^{61} + 2 q^{65} - 10 q^{69} + 30 q^{71} - 4 q^{75} + 2 q^{79} + 8 q^{81} + 24 q^{85} - 18 q^{89} - 14 q^{91} - 2 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −0.594137 2.15569i −0.265706 0.964054i
\(6\) 0 0
\(7\) 4.92778i 1.86252i −0.364349 0.931262i \(-0.618709\pi\)
0.364349 0.931262i \(-0.381291\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.38360 −0.417172 −0.208586 0.978004i \(-0.566886\pi\)
−0.208586 + 0.978004i \(0.566886\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 2.15569 0.594137i 0.556597 0.153406i
\(16\) 0 0
\(17\) 0.195329i 0.0473741i 0.999719 + 0.0236871i \(0.00754053\pi\)
−0.999719 + 0.0236871i \(0.992459\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 4.92778 1.07533
\(22\) 0 0
\(23\) 2.19533i 0.457758i 0.973455 + 0.228879i \(0.0735060\pi\)
−0.973455 + 0.228879i \(0.926494\pi\)
\(24\) 0 0
\(25\) −4.29400 + 2.56155i −0.858800 + 0.512311i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −7.49966 −1.39265 −0.696326 0.717726i \(-0.745183\pi\)
−0.696326 + 0.717726i \(0.745183\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 1.38360i 0.240854i
\(34\) 0 0
\(35\) −10.6228 + 2.92778i −1.79557 + 0.494885i
\(36\) 0 0
\(37\) 6.05088i 0.994759i 0.867533 + 0.497379i \(0.165704\pi\)
−0.867533 + 0.497379i \(0.834296\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −2.11605 −0.330472 −0.165236 0.986254i \(-0.552839\pi\)
−0.165236 + 0.986254i \(0.552839\pi\)
\(42\) 0 0
\(43\) 3.49966i 0.533692i −0.963739 0.266846i \(-0.914018\pi\)
0.963739 0.266846i \(-0.0859816\pi\)
\(44\) 0 0
\(45\) 0.594137 + 2.15569i 0.0885688 + 0.321351i
\(46\) 0 0
\(47\) 2.31138i 0.337150i 0.985689 + 0.168575i \(0.0539165\pi\)
−0.985689 + 0.168575i \(0.946084\pi\)
\(48\) 0 0
\(49\) −17.2830 −2.46900
\(50\) 0 0
\(51\) −0.195329 −0.0273515
\(52\) 0 0
\(53\) 6.05088i 0.831153i 0.909558 + 0.415576i \(0.136420\pi\)
−0.909558 + 0.415576i \(0.863580\pi\)
\(54\) 0 0
\(55\) 0.822051 + 2.98262i 0.110845 + 0.402176i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.43449 −1.22827 −0.614133 0.789203i \(-0.710494\pi\)
−0.614133 + 0.789203i \(0.710494\pi\)
\(60\) 0 0
\(61\) −3.69498 −0.473094 −0.236547 0.971620i \(-0.576016\pi\)
−0.236547 + 0.971620i \(0.576016\pi\)
\(62\) 0 0
\(63\) 4.92778i 0.620842i
\(64\) 0 0
\(65\) −2.15569 + 0.594137i −0.267380 + 0.0736937i
\(66\) 0 0
\(67\) 12.6228i 1.54212i −0.636765 0.771058i \(-0.719728\pi\)
0.636765 0.771058i \(-0.280272\pi\)
\(68\) 0 0
\(69\) −2.19533 −0.264287
\(70\) 0 0
\(71\) 13.9716 1.65812 0.829062 0.559156i \(-0.188875\pi\)
0.829062 + 0.559156i \(0.188875\pi\)
\(72\) 0 0
\(73\) 4.73245i 0.553891i −0.960886 0.276946i \(-0.910678\pi\)
0.960886 0.276946i \(-0.0893222\pi\)
\(74\) 0 0
\(75\) −2.56155 4.29400i −0.295783 0.495829i
\(76\) 0 0
\(77\) 6.81809i 0.776993i
\(78\) 0 0
\(79\) 8.07153 0.908119 0.454059 0.890971i \(-0.349975\pi\)
0.454059 + 0.890971i \(0.349975\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.3997i 1.47081i 0.677627 + 0.735406i \(0.263008\pi\)
−0.677627 + 0.735406i \(0.736992\pi\)
\(84\) 0 0
\(85\) 0.421068 0.116052i 0.0456712 0.0125876i
\(86\) 0 0
\(87\) 7.49966i 0.804047i
\(88\) 0 0
\(89\) −3.02770 −0.320936 −0.160468 0.987041i \(-0.551300\pi\)
−0.160468 + 0.987041i \(0.551300\pi\)
\(90\) 0 0
\(91\) −4.92778 −0.516571
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.01612i 0.610845i 0.952217 + 0.305422i \(0.0987977\pi\)
−0.952217 + 0.305422i \(0.901202\pi\)
\(98\) 0 0
\(99\) 1.38360 0.139057
\(100\) 0 0
\(101\) −3.10900 −0.309357 −0.154678 0.987965i \(-0.549434\pi\)
−0.154678 + 0.987965i \(0.549434\pi\)
\(102\) 0 0
\(103\) 16.8690i 1.66215i −0.556161 0.831075i \(-0.687726\pi\)
0.556161 0.831075i \(-0.312274\pi\)
\(104\) 0 0
\(105\) −2.92778 10.6228i −0.285722 1.03668i
\(106\) 0 0
\(107\) 17.1946i 1.66227i −0.556072 0.831134i \(-0.687692\pi\)
0.556072 0.831134i \(-0.312308\pi\)
\(108\) 0 0
\(109\) −8.76721 −0.839746 −0.419873 0.907583i \(-0.637925\pi\)
−0.419873 + 0.907583i \(0.637925\pi\)
\(110\) 0 0
\(111\) −6.05088 −0.574324
\(112\) 0 0
\(113\) 7.46490i 0.702238i −0.936331 0.351119i \(-0.885801\pi\)
0.936331 0.351119i \(-0.114199\pi\)
\(114\) 0 0
\(115\) 4.73245 1.30433i 0.441303 0.121629i
\(116\) 0 0
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) 0.962536 0.0882355
\(120\) 0 0
\(121\) −9.08564 −0.825967
\(122\) 0 0
\(123\) 2.11605i 0.190798i
\(124\) 0 0
\(125\) 8.07314 + 7.73462i 0.722084 + 0.691806i
\(126\) 0 0
\(127\) 11.1231i 0.987016i −0.869741 0.493508i \(-0.835714\pi\)
0.869741 0.493508i \(-0.164286\pi\)
\(128\) 0 0
\(129\) 3.49966 0.308127
\(130\) 0 0
\(131\) −1.82080 −0.159084 −0.0795418 0.996832i \(-0.525346\pi\)
−0.0795418 + 0.996832i \(0.525346\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.15569 + 0.594137i −0.185532 + 0.0511352i
\(136\) 0 0
\(137\) 8.70204i 0.743465i −0.928340 0.371733i \(-0.878764\pi\)
0.928340 0.371733i \(-0.121236\pi\)
\(138\) 0 0
\(139\) −6.81809 −0.578303 −0.289151 0.957283i \(-0.593373\pi\)
−0.289151 + 0.957283i \(0.593373\pi\)
\(140\) 0 0
\(141\) −2.31138 −0.194653
\(142\) 0 0
\(143\) 1.38360i 0.115703i
\(144\) 0 0
\(145\) 4.45583 + 16.1669i 0.370036 + 1.34259i
\(146\) 0 0
\(147\) 17.2830i 1.42548i
\(148\) 0 0
\(149\) 7.48537 0.613225 0.306613 0.951834i \(-0.400804\pi\)
0.306613 + 0.951834i \(0.400804\pi\)
\(150\) 0 0
\(151\) 12.8549 1.04611 0.523057 0.852298i \(-0.324791\pi\)
0.523057 + 0.852298i \(0.324791\pi\)
\(152\) 0 0
\(153\) 0.195329i 0.0157914i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 17.7318i 1.41515i −0.706639 0.707574i \(-0.749790\pi\)
0.706639 0.707574i \(-0.250210\pi\)
\(158\) 0 0
\(159\) −6.05088 −0.479866
\(160\) 0 0
\(161\) 10.8181 0.852585
\(162\) 0 0
\(163\) 18.1385i 1.42072i −0.703838 0.710360i \(-0.748532\pi\)
0.703838 0.710360i \(-0.251468\pi\)
\(164\) 0 0
\(165\) −2.98262 + 0.822051i −0.232197 + 0.0639966i
\(166\) 0 0
\(167\) 4.06517i 0.314572i −0.987553 0.157286i \(-0.949726\pi\)
0.987553 0.157286i \(-0.0502745\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 23.2455i 1.76732i 0.468125 + 0.883662i \(0.344930\pi\)
−0.468125 + 0.883662i \(0.655070\pi\)
\(174\) 0 0
\(175\) 12.6228 + 21.1599i 0.954191 + 1.59954i
\(176\) 0 0
\(177\) 9.43449i 0.709139i
\(178\) 0 0
\(179\) −14.1224 −1.05556 −0.527779 0.849381i \(-0.676975\pi\)
−0.527779 + 0.849381i \(0.676975\pi\)
\(180\) 0 0
\(181\) 3.67433 0.273111 0.136556 0.990632i \(-0.456397\pi\)
0.136556 + 0.990632i \(0.456397\pi\)
\(182\) 0 0
\(183\) 3.69498i 0.273141i
\(184\) 0 0
\(185\) 13.0438 3.59506i 0.959001 0.264314i
\(186\) 0 0
\(187\) 0.270257i 0.0197632i
\(188\) 0 0
\(189\) −4.92778 −0.358443
\(190\) 0 0
\(191\) −16.8549 −1.21958 −0.609788 0.792565i \(-0.708745\pi\)
−0.609788 + 0.792565i \(0.708745\pi\)
\(192\) 0 0
\(193\) 6.53712i 0.470552i −0.971929 0.235276i \(-0.924401\pi\)
0.971929 0.235276i \(-0.0755994\pi\)
\(194\) 0 0
\(195\) −0.594137 2.15569i −0.0425471 0.154372i
\(196\) 0 0
\(197\) 25.7890i 1.83739i 0.394967 + 0.918695i \(0.370756\pi\)
−0.394967 + 0.918695i \(0.629244\pi\)
\(198\) 0 0
\(199\) 16.9646 1.20259 0.601293 0.799029i \(-0.294653\pi\)
0.601293 + 0.799029i \(0.294653\pi\)
\(200\) 0 0
\(201\) 12.6228 0.890341
\(202\) 0 0
\(203\) 36.9566i 2.59385i
\(204\) 0 0
\(205\) 1.25723 + 4.56155i 0.0878085 + 0.318593i
\(206\) 0 0
\(207\) 2.19533i 0.152586i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −26.4577 −1.82142 −0.910710 0.413046i \(-0.864465\pi\)
−0.910710 + 0.413046i \(0.864465\pi\)
\(212\) 0 0
\(213\) 13.9716i 0.957319i
\(214\) 0 0
\(215\) −7.54417 + 2.07928i −0.514508 + 0.141805i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 4.73245 0.319789
\(220\) 0 0
\(221\) 0.195329 0.0131392
\(222\) 0 0
\(223\) 2.02065i 0.135313i 0.997709 + 0.0676564i \(0.0215522\pi\)
−0.997709 + 0.0676564i \(0.978448\pi\)
\(224\) 0 0
\(225\) 4.29400 2.56155i 0.286267 0.170770i
\(226\) 0 0
\(227\) 17.7904i 1.18079i −0.807115 0.590395i \(-0.798972\pi\)
0.807115 0.590395i \(-0.201028\pi\)
\(228\) 0 0
\(229\) 18.3339 1.21154 0.605768 0.795641i \(-0.292866\pi\)
0.605768 + 0.795641i \(0.292866\pi\)
\(230\) 0 0
\(231\) −6.81809 −0.448597
\(232\) 0 0
\(233\) 23.0361i 1.50914i 0.656217 + 0.754572i \(0.272156\pi\)
−0.656217 + 0.754572i \(0.727844\pi\)
\(234\) 0 0
\(235\) 4.98262 1.37328i 0.325030 0.0895828i
\(236\) 0 0
\(237\) 8.07153i 0.524302i
\(238\) 0 0
\(239\) 0.986402 0.0638051 0.0319025 0.999491i \(-0.489843\pi\)
0.0319025 + 0.999491i \(0.489843\pi\)
\(240\) 0 0
\(241\) 1.52031 0.0979316 0.0489658 0.998800i \(-0.484407\pi\)
0.0489658 + 0.998800i \(0.484407\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 10.2685 + 37.2568i 0.656028 + 2.38025i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −13.3997 −0.849173
\(250\) 0 0
\(251\) 11.6763 0.737005 0.368502 0.929627i \(-0.379871\pi\)
0.368502 + 0.929627i \(0.379871\pi\)
\(252\) 0 0
\(253\) 3.03746i 0.190964i
\(254\) 0 0
\(255\) 0.116052 + 0.421068i 0.00726746 + 0.0263683i
\(256\) 0 0
\(257\) 3.00069i 0.187178i −0.995611 0.0935889i \(-0.970166\pi\)
0.995611 0.0935889i \(-0.0298339\pi\)
\(258\) 0 0
\(259\) 29.8174 1.85276
\(260\) 0 0
\(261\) 7.49966 0.464217
\(262\) 0 0
\(263\) 25.8683i 1.59511i −0.603248 0.797553i \(-0.706127\pi\)
0.603248 0.797553i \(-0.293873\pi\)
\(264\) 0 0
\(265\) 13.0438 3.59506i 0.801276 0.220843i
\(266\) 0 0
\(267\) 3.02770i 0.185293i
\(268\) 0 0
\(269\) −11.1090 −0.677328 −0.338664 0.940907i \(-0.609975\pi\)
−0.338664 + 0.940907i \(0.609975\pi\)
\(270\) 0 0
\(271\) −13.9859 −0.849582 −0.424791 0.905291i \(-0.639653\pi\)
−0.424791 + 0.905291i \(0.639653\pi\)
\(272\) 0 0
\(273\) 4.92778i 0.298243i
\(274\) 0 0
\(275\) 5.94120 3.54417i 0.358268 0.213722i
\(276\) 0 0
\(277\) 10.0529i 0.604020i 0.953305 + 0.302010i \(0.0976576\pi\)
−0.953305 + 0.302010i \(0.902342\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −19.6318 −1.17114 −0.585568 0.810623i \(-0.699129\pi\)
−0.585568 + 0.810623i \(0.699129\pi\)
\(282\) 0 0
\(283\) 25.5667i 1.51978i 0.650051 + 0.759890i \(0.274747\pi\)
−0.650051 + 0.759890i \(0.725253\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.4274i 0.615512i
\(288\) 0 0
\(289\) 16.9618 0.997756
\(290\) 0 0
\(291\) −6.01612 −0.352671
\(292\) 0 0
\(293\) 6.71614i 0.392361i −0.980568 0.196181i \(-0.937146\pi\)
0.980568 0.196181i \(-0.0628539\pi\)
\(294\) 0 0
\(295\) 5.60538 + 20.3378i 0.326358 + 1.18411i
\(296\) 0 0
\(297\) 1.38360i 0.0802848i
\(298\) 0 0
\(299\) 2.19533 0.126959
\(300\) 0 0
\(301\) −17.2455 −0.994015
\(302\) 0 0
\(303\) 3.10900i 0.178607i
\(304\) 0 0
\(305\) 2.19533 + 7.96524i 0.125704 + 0.456088i
\(306\) 0 0
\(307\) 14.2689i 0.814368i 0.913346 + 0.407184i \(0.133489\pi\)
−0.913346 + 0.407184i \(0.866511\pi\)
\(308\) 0 0
\(309\) 16.8690 0.959642
\(310\) 0 0
\(311\) −7.15786 −0.405885 −0.202943 0.979191i \(-0.565050\pi\)
−0.202943 + 0.979191i \(0.565050\pi\)
\(312\) 0 0
\(313\) 21.6568i 1.22412i −0.790813 0.612058i \(-0.790342\pi\)
0.790813 0.612058i \(-0.209658\pi\)
\(314\) 0 0
\(315\) 10.6228 2.92778i 0.598525 0.164962i
\(316\) 0 0
\(317\) 29.0232i 1.63010i −0.579388 0.815052i \(-0.696708\pi\)
0.579388 0.815052i \(-0.303292\pi\)
\(318\) 0 0
\(319\) 10.3765 0.580975
\(320\) 0 0
\(321\) 17.1946 0.959711
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 2.56155 + 4.29400i 0.142089 + 0.238188i
\(326\) 0 0
\(327\) 8.76721i 0.484828i
\(328\) 0 0
\(329\) 11.3900 0.627949
\(330\) 0 0
\(331\) 21.8683 1.20199 0.600995 0.799253i \(-0.294771\pi\)
0.600995 + 0.799253i \(0.294771\pi\)
\(332\) 0 0
\(333\) 6.05088i 0.331586i
\(334\) 0 0
\(335\) −27.2108 + 7.49966i −1.48668 + 0.409750i
\(336\) 0 0
\(337\) 22.2991i 1.21471i −0.794431 0.607355i \(-0.792231\pi\)
0.794431 0.607355i \(-0.207769\pi\)
\(338\) 0 0
\(339\) 7.46490 0.405438
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 50.6723i 2.73605i
\(344\) 0 0
\(345\) 1.30433 + 4.73245i 0.0702226 + 0.254786i
\(346\) 0 0
\(347\) 22.9199i 1.23040i 0.788370 + 0.615201i \(0.210925\pi\)
−0.788370 + 0.615201i \(0.789075\pi\)
\(348\) 0 0
\(349\) −6.75310 −0.361485 −0.180743 0.983530i \(-0.557850\pi\)
−0.180743 + 0.983530i \(0.557850\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 3.07859i 0.163857i −0.996638 0.0819283i \(-0.973892\pi\)
0.996638 0.0819283i \(-0.0261079\pi\)
\(354\) 0 0
\(355\) −8.30105 30.1185i −0.440574 1.59852i
\(356\) 0 0
\(357\) 0.962536i 0.0509428i
\(358\) 0 0
\(359\) 13.0091 0.686592 0.343296 0.939227i \(-0.388457\pi\)
0.343296 + 0.939227i \(0.388457\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 9.08564i 0.476872i
\(364\) 0 0
\(365\) −10.2017 + 2.81173i −0.533981 + 0.147172i
\(366\) 0 0
\(367\) 27.4041i 1.43048i −0.698878 0.715241i \(-0.746317\pi\)
0.698878 0.715241i \(-0.253683\pi\)
\(368\) 0 0
\(369\) 2.11605 0.110157
\(370\) 0 0
\(371\) 29.8174 1.54804
\(372\) 0 0
\(373\) 19.9573i 1.03335i −0.856181 0.516675i \(-0.827169\pi\)
0.856181 0.516675i \(-0.172831\pi\)
\(374\) 0 0
\(375\) −7.73462 + 8.07314i −0.399414 + 0.416895i
\(376\) 0 0
\(377\) 7.49966i 0.386252i
\(378\) 0 0
\(379\) 29.2042 1.50012 0.750060 0.661370i \(-0.230025\pi\)
0.750060 + 0.661370i \(0.230025\pi\)
\(380\) 0 0
\(381\) 11.1231 0.569854
\(382\) 0 0
\(383\) 15.5764i 0.795918i −0.917403 0.397959i \(-0.869719\pi\)
0.917403 0.397959i \(-0.130281\pi\)
\(384\) 0 0
\(385\) 14.6977 4.05088i 0.749064 0.206452i
\(386\) 0 0
\(387\) 3.49966i 0.177897i
\(388\) 0 0
\(389\) −20.0529 −1.01672 −0.508361 0.861144i \(-0.669749\pi\)
−0.508361 + 0.861144i \(0.669749\pi\)
\(390\) 0 0
\(391\) −0.428810 −0.0216859
\(392\) 0 0
\(393\) 1.82080i 0.0918470i
\(394\) 0 0
\(395\) −4.79560 17.3997i −0.241293 0.875475i
\(396\) 0 0
\(397\) 30.9057i 1.55112i −0.631277 0.775558i \(-0.717469\pi\)
0.631277 0.775558i \(-0.282531\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.9902 −0.598764 −0.299382 0.954133i \(-0.596780\pi\)
−0.299382 + 0.954133i \(0.596780\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.594137 2.15569i −0.0295229 0.107117i
\(406\) 0 0
\(407\) 8.37202i 0.414986i
\(408\) 0 0
\(409\) −39.2169 −1.93915 −0.969577 0.244788i \(-0.921282\pi\)
−0.969577 + 0.244788i \(0.921282\pi\)
\(410\) 0 0
\(411\) 8.70204 0.429240
\(412\) 0 0
\(413\) 46.4910i 2.28767i
\(414\) 0 0
\(415\) 28.8857 7.96128i 1.41794 0.390804i
\(416\) 0 0
\(417\) 6.81809i 0.333883i
\(418\) 0 0
\(419\) 17.8621 0.872621 0.436310 0.899796i \(-0.356285\pi\)
0.436310 + 0.899796i \(0.356285\pi\)
\(420\) 0 0
\(421\) 28.8690 1.40699 0.703494 0.710701i \(-0.251622\pi\)
0.703494 + 0.710701i \(0.251622\pi\)
\(422\) 0 0
\(423\) 2.31138i 0.112383i
\(424\) 0 0
\(425\) −0.500344 0.838741i −0.0242703 0.0406849i
\(426\) 0 0
\(427\) 18.2081i 0.881150i
\(428\) 0 0
\(429\) −1.38360 −0.0668010
\(430\) 0 0
\(431\) 17.0373 0.820657 0.410329 0.911938i \(-0.365414\pi\)
0.410329 + 0.911938i \(0.365414\pi\)
\(432\) 0 0
\(433\) 29.1554i 1.40112i 0.713595 + 0.700558i \(0.247066\pi\)
−0.713595 + 0.700558i \(0.752934\pi\)
\(434\) 0 0
\(435\) −16.1669 + 4.45583i −0.775145 + 0.213641i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 23.2087 1.10769 0.553847 0.832619i \(-0.313159\pi\)
0.553847 + 0.832619i \(0.313159\pi\)
\(440\) 0 0
\(441\) 17.2830 0.822999
\(442\) 0 0
\(443\) 14.9484i 0.710221i 0.934824 + 0.355111i \(0.115557\pi\)
−0.934824 + 0.355111i \(0.884443\pi\)
\(444\) 0 0
\(445\) 1.79887 + 6.52679i 0.0852748 + 0.309400i
\(446\) 0 0
\(447\) 7.48537i 0.354046i
\(448\) 0 0
\(449\) 42.2023 1.99165 0.995826 0.0912763i \(-0.0290947\pi\)
0.995826 + 0.0912763i \(0.0290947\pi\)
\(450\) 0 0
\(451\) 2.92778 0.137864
\(452\) 0 0
\(453\) 12.8549i 0.603974i
\(454\) 0 0
\(455\) 2.92778 + 10.6228i 0.137256 + 0.498003i
\(456\) 0 0
\(457\) 15.1599i 0.709149i −0.935028 0.354575i \(-0.884626\pi\)
0.935028 0.354575i \(-0.115374\pi\)
\(458\) 0 0
\(459\) 0.195329 0.00911716
\(460\) 0 0
\(461\) −0.848501 −0.0395186 −0.0197593 0.999805i \(-0.506290\pi\)
−0.0197593 + 0.999805i \(0.506290\pi\)
\(462\) 0 0
\(463\) 13.8109i 0.641845i −0.947105 0.320922i \(-0.896007\pi\)
0.947105 0.320922i \(-0.103993\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.5165i 0.486644i 0.969946 + 0.243322i \(0.0782372\pi\)
−0.969946 + 0.243322i \(0.921763\pi\)
\(468\) 0 0
\(469\) −62.2022 −2.87223
\(470\) 0 0
\(471\) 17.7318 0.817036
\(472\) 0 0
\(473\) 4.84214i 0.222642i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.05088i 0.277051i
\(478\) 0 0
\(479\) 6.41905 0.293294 0.146647 0.989189i \(-0.453152\pi\)
0.146647 + 0.989189i \(0.453152\pi\)
\(480\) 0 0
\(481\) 6.05088 0.275897
\(482\) 0 0
\(483\) 10.8181i 0.492240i
\(484\) 0 0
\(485\) 12.9689 3.57440i 0.588887 0.162305i
\(486\) 0 0
\(487\) 23.5492i 1.06711i 0.845764 + 0.533557i \(0.179145\pi\)
−0.845764 + 0.533557i \(0.820855\pi\)
\(488\) 0 0
\(489\) 18.1385 0.820253
\(490\) 0 0
\(491\) 3.99346 0.180222 0.0901111 0.995932i \(-0.471278\pi\)
0.0901111 + 0.995932i \(0.471278\pi\)
\(492\) 0 0
\(493\) 1.46490i 0.0659756i
\(494\) 0 0
\(495\) −0.822051 2.98262i −0.0369484 0.134059i
\(496\) 0 0
\(497\) 68.8490i 3.08830i
\(498\) 0 0
\(499\) 5.30231 0.237364 0.118682 0.992932i \(-0.462133\pi\)
0.118682 + 0.992932i \(0.462133\pi\)
\(500\) 0 0
\(501\) 4.06517 0.181618
\(502\) 0 0
\(503\) 34.8408i 1.55347i −0.629826 0.776736i \(-0.716874\pi\)
0.629826 0.776736i \(-0.283126\pi\)
\(504\) 0 0
\(505\) 1.84717 + 6.70204i 0.0821981 + 0.298237i
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) −42.2526 −1.87281 −0.936406 0.350918i \(-0.885870\pi\)
−0.936406 + 0.350918i \(0.885870\pi\)
\(510\) 0 0
\(511\) −23.3205 −1.03164
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −36.3643 + 10.0225i −1.60240 + 0.441644i
\(516\) 0 0
\(517\) 3.19803i 0.140649i
\(518\) 0 0
\(519\) −23.2455 −1.02037
\(520\) 0 0
\(521\) −2.65098 −0.116141 −0.0580707 0.998312i \(-0.518495\pi\)
−0.0580707 + 0.998312i \(0.518495\pi\)
\(522\) 0 0
\(523\) 31.6014i 1.38183i 0.722934 + 0.690917i \(0.242793\pi\)
−0.722934 + 0.690917i \(0.757207\pi\)
\(524\) 0 0
\(525\) −21.1599 + 12.6228i −0.923493 + 0.550902i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 18.1805 0.790458
\(530\) 0 0
\(531\) 9.43449 0.409422
\(532\) 0 0
\(533\) 2.11605i 0.0916564i
\(534\) 0 0
\(535\) −37.0663 + 10.2160i −1.60252 + 0.441675i
\(536\) 0 0
\(537\) 14.1224i 0.609427i
\(538\) 0 0
\(539\) 23.9128 1.03000
\(540\) 0 0
\(541\) 1.54852 0.0665761 0.0332881 0.999446i \(-0.489402\pi\)
0.0332881 + 0.999446i \(0.489402\pi\)
\(542\) 0 0
\(543\) 3.67433i 0.157681i
\(544\) 0 0
\(545\) 5.20893 + 18.8994i 0.223126 + 0.809561i
\(546\) 0 0
\(547\) 33.2376i 1.42114i −0.703628 0.710569i \(-0.748438\pi\)
0.703628 0.710569i \(-0.251562\pi\)
\(548\) 0 0
\(549\) 3.69498 0.157698
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 39.7747i 1.69139i
\(554\) 0 0
\(555\) 3.59506 + 13.0438i 0.152602 + 0.553680i
\(556\) 0 0
\(557\) 1.64594i 0.0697407i 0.999392 + 0.0348703i \(0.0111018\pi\)
−0.999392 + 0.0348703i \(0.988898\pi\)
\(558\) 0 0
\(559\) −3.49966 −0.148020
\(560\) 0 0
\(561\) 0.270257 0.0114103
\(562\) 0 0
\(563\) 18.7486i 0.790158i 0.918647 + 0.395079i \(0.129283\pi\)
−0.918647 + 0.395079i \(0.870717\pi\)
\(564\) 0 0
\(565\) −16.0920 + 4.43518i −0.676996 + 0.186589i
\(566\) 0 0
\(567\) 4.92778i 0.206947i
\(568\) 0 0
\(569\) 41.6543 1.74624 0.873120 0.487505i \(-0.162093\pi\)
0.873120 + 0.487505i \(0.162093\pi\)
\(570\) 0 0
\(571\) −27.3184 −1.14324 −0.571620 0.820518i \(-0.693685\pi\)
−0.571620 + 0.820518i \(0.693685\pi\)
\(572\) 0 0
\(573\) 16.8549i 0.704122i
\(574\) 0 0
\(575\) −5.62345 9.42674i −0.234514 0.393122i
\(576\) 0 0
\(577\) 24.0715i 1.00211i −0.865415 0.501056i \(-0.832945\pi\)
0.865415 0.501056i \(-0.167055\pi\)
\(578\) 0 0
\(579\) 6.53712 0.271673
\(580\) 0 0
\(581\) 66.0309 2.73942
\(582\) 0 0
\(583\) 8.37202i 0.346734i
\(584\) 0 0
\(585\) 2.15569 0.594137i 0.0891268 0.0245646i
\(586\) 0 0
\(587\) 23.6872i 0.977677i 0.872374 + 0.488839i \(0.162579\pi\)
−0.872374 + 0.488839i \(0.837421\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −25.7890 −1.06082
\(592\) 0 0
\(593\) 38.8632i 1.59592i −0.602709 0.797961i \(-0.705912\pi\)
0.602709 0.797961i \(-0.294088\pi\)
\(594\) 0 0
\(595\) −0.571878 2.07493i −0.0234447 0.0850638i
\(596\) 0 0
\(597\) 16.9646i 0.694313i
\(598\) 0 0
\(599\) −18.4783 −0.755003 −0.377502 0.926009i \(-0.623217\pi\)
−0.377502 + 0.926009i \(0.623217\pi\)
\(600\) 0 0
\(601\) −2.08702 −0.0851313 −0.0425656 0.999094i \(-0.513553\pi\)
−0.0425656 + 0.999094i \(0.513553\pi\)
\(602\) 0 0
\(603\) 12.6228i 0.514039i
\(604\) 0 0
\(605\) 5.39812 + 19.5858i 0.219465 + 0.796277i
\(606\) 0 0
\(607\) 3.59120i 0.145762i −0.997341 0.0728812i \(-0.976781\pi\)
0.997341 0.0728812i \(-0.0232194\pi\)
\(608\) 0 0
\(609\) −36.9566 −1.49756
\(610\) 0 0
\(611\) 2.31138 0.0935084
\(612\) 0 0
\(613\) 24.7895i 1.00124i 0.865667 + 0.500620i \(0.166894\pi\)
−0.865667 + 0.500620i \(0.833106\pi\)
\(614\) 0 0
\(615\) −4.56155 + 1.25723i −0.183940 + 0.0506962i
\(616\) 0 0
\(617\) 14.7574i 0.594112i −0.954860 0.297056i \(-0.903995\pi\)
0.954860 0.297056i \(-0.0960049\pi\)
\(618\) 0 0
\(619\) −25.4081 −1.02124 −0.510619 0.859807i \(-0.670584\pi\)
−0.510619 + 0.859807i \(0.670584\pi\)
\(620\) 0 0
\(621\) 2.19533 0.0880955
\(622\) 0 0
\(623\) 14.9199i 0.597751i
\(624\) 0 0
\(625\) 11.8769 21.9986i 0.475076 0.879945i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.18191 −0.0471258
\(630\) 0 0
\(631\) 9.46490 0.376792 0.188396 0.982093i \(-0.439671\pi\)
0.188396 + 0.982093i \(0.439671\pi\)
\(632\) 0 0
\(633\) 26.4577i 1.05160i
\(634\) 0 0
\(635\) −23.9780 + 6.60865i −0.951537 + 0.262256i
\(636\) 0 0
\(637\) 17.2830i 0.684777i
\(638\) 0 0
\(639\) −13.9716 −0.552708
\(640\) 0 0
\(641\) 7.89686 0.311907 0.155954 0.987764i \(-0.450155\pi\)
0.155954 + 0.987764i \(0.450155\pi\)
\(642\) 0 0
\(643\) 16.1204i 0.635727i −0.948137 0.317863i \(-0.897035\pi\)
0.948137 0.317863i \(-0.102965\pi\)
\(644\) 0 0
\(645\) −2.07928 7.54417i −0.0818714 0.297052i
\(646\) 0 0
\(647\) 2.99616i 0.117791i 0.998264 + 0.0588956i \(0.0187579\pi\)
−0.998264 + 0.0588956i \(0.981242\pi\)
\(648\) 0 0
\(649\) 13.0536 0.512398
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21.5330i 0.842653i −0.906909 0.421326i \(-0.861565\pi\)
0.906909 0.421326i \(-0.138435\pi\)
\(654\) 0 0
\(655\) 1.08180 + 3.92507i 0.0422695 + 0.153365i
\(656\) 0 0
\(657\) 4.73245i 0.184630i
\(658\) 0 0
\(659\) −17.4429 −0.679478 −0.339739 0.940520i \(-0.610339\pi\)
−0.339739 + 0.940520i \(0.610339\pi\)
\(660\) 0 0
\(661\) −22.4642 −0.873756 −0.436878 0.899521i \(-0.643916\pi\)
−0.436878 + 0.899521i \(0.643916\pi\)
\(662\) 0 0
\(663\) 0.195329i 0.00758593i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.4642i 0.637497i
\(668\) 0 0
\(669\) −2.02065 −0.0781229
\(670\) 0 0
\(671\) 5.11239 0.197362
\(672\) 0 0
\(673\) 46.8122i 1.80448i 0.431237 + 0.902239i \(0.358077\pi\)
−0.431237 + 0.902239i \(0.641923\pi\)
\(674\) 0 0
\(675\) 2.56155 + 4.29400i 0.0985942 + 0.165276i
\(676\) 0 0
\(677\) 31.2682i 1.20173i −0.799349 0.600867i \(-0.794822\pi\)
0.799349 0.600867i \(-0.205178\pi\)
\(678\) 0 0
\(679\) 29.6461 1.13771
\(680\) 0 0
\(681\) 17.7904 0.681729
\(682\) 0 0
\(683\) 44.1797i 1.69049i −0.534381 0.845244i \(-0.679455\pi\)
0.534381 0.845244i \(-0.320545\pi\)
\(684\) 0 0
\(685\) −18.7589 + 5.17021i −0.716741 + 0.197543i
\(686\) 0 0
\(687\) 18.3339i 0.699481i
\(688\) 0 0
\(689\) 6.05088 0.230520
\(690\) 0 0
\(691\) −25.8360 −0.982849 −0.491425 0.870920i \(-0.663524\pi\)
−0.491425 + 0.870920i \(0.663524\pi\)
\(692\) 0 0
\(693\) 6.81809i 0.258998i
\(694\) 0 0
\(695\) 4.05088 + 14.6977i 0.153659 + 0.557515i
\(696\) 0 0
\(697\) 0.413325i 0.0156558i
\(698\) 0 0
\(699\) −23.0361 −0.871305
\(700\) 0 0
\(701\) 15.5887 0.588777 0.294388 0.955686i \(-0.404884\pi\)
0.294388 + 0.955686i \(0.404884\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 1.37328 + 4.98262i 0.0517206 + 0.187656i
\(706\) 0 0
\(707\) 15.3205i 0.576185i
\(708\) 0 0
\(709\) 4.05541 0.152304 0.0761521 0.997096i \(-0.475737\pi\)
0.0761521 + 0.997096i \(0.475737\pi\)
\(710\) 0 0
\(711\) −8.07153 −0.302706
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 2.98262 0.822051i 0.111544 0.0307430i
\(716\) 0 0
\(717\) 0.986402i 0.0368379i
\(718\) 0 0
\(719\) −13.4081 −0.500038 −0.250019 0.968241i \(-0.580437\pi\)
−0.250019 + 0.968241i \(0.580437\pi\)
\(720\) 0 0
\(721\) −83.1265 −3.09579
\(722\) 0 0
\(723\) 1.52031i 0.0565408i
\(724\) 0 0
\(725\) 32.2035 19.2108i 1.19601 0.713470i
\(726\) 0 0
\(727\) 12.9787i 0.481352i −0.970606 0.240676i \(-0.922631\pi\)
0.970606 0.240676i \(-0.0773691\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0.683583 0.0252832
\(732\) 0 0
\(733\) 49.2410i 1.81876i 0.415969 + 0.909379i \(0.363443\pi\)
−0.415969 + 0.909379i \(0.636557\pi\)
\(734\) 0 0
\(735\) −37.2568 + 10.2685i −1.37424 + 0.378758i
\(736\) 0 0
\(737\) 17.4649i 0.643328i
\(738\) 0 0
\(739\) 11.3359 0.416999 0.208500 0.978022i \(-0.433142\pi\)
0.208500 + 0.978022i \(0.433142\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.07790i 0.296349i 0.988961 + 0.148175i \(0.0473398\pi\)
−0.988961 + 0.148175i \(0.952660\pi\)
\(744\) 0 0
\(745\) −4.44734 16.1361i −0.162938 0.591182i
\(746\) 0 0
\(747\) 13.3997i 0.490270i
\(748\) 0 0
\(749\) −84.7314 −3.09602
\(750\) 0 0
\(751\) 18.2403 0.665598 0.332799 0.942998i \(-0.392007\pi\)
0.332799 + 0.942998i \(0.392007\pi\)
\(752\) 0 0
\(753\) 11.6763i 0.425510i
\(754\) 0 0
\(755\) −7.63756 27.7111i −0.277959 1.00851i
\(756\) 0 0
\(757\) 5.71111i 0.207574i 0.994600 + 0.103787i \(0.0330960\pi\)
−0.994600 + 0.103787i \(0.966904\pi\)
\(758\) 0 0
\(759\) 3.03746 0.110253
\(760\) 0 0
\(761\) 12.6543 0.458718 0.229359 0.973342i \(-0.426337\pi\)
0.229359 + 0.973342i \(0.426337\pi\)
\(762\) 0 0
\(763\) 43.2028i 1.56405i
\(764\) 0 0
\(765\) −0.421068 + 0.116052i −0.0152237 + 0.00419587i
\(766\) 0 0
\(767\) 9.43449i 0.340660i
\(768\) 0 0
\(769\) 21.4222 0.772505 0.386252 0.922393i \(-0.373769\pi\)
0.386252 + 0.922393i \(0.373769\pi\)
\(770\) 0 0
\(771\) 3.00069 0.108067
\(772\) 0 0
\(773\) 36.0547i 1.29680i −0.761300 0.648399i \(-0.775439\pi\)
0.761300 0.648399i \(-0.224561\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 29.8174i 1.06969i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −19.3312 −0.691723
\(782\) 0 0
\(783\) 7.49966i 0.268016i
\(784\) 0 0
\(785\) −38.2242 + 10.5351i −1.36428 + 0.376014i
\(786\) 0 0
\(787\) 40.8394i 1.45577i −0.685701 0.727883i \(-0.740504\pi\)
0.685701 0.727883i \(-0.259496\pi\)
\(788\) 0 0
\(789\) 25.8683 0.920935
\(790\) 0 0
\(791\) −36.7853 −1.30794
\(792\) 0 0
\(793\) 3.69498i 0.131213i
\(794\) 0 0
\(795\) 3.59506 + 13.0438i 0.127504 + 0.462617i
\(796\) 0 0
\(797\) 25.4691i 0.902161i −0.892483 0.451080i \(-0.851039\pi\)
0.892483 0.451080i \(-0.148961\pi\)
\(798\) 0 0
\(799\) −0.451479 −0.0159722
\(800\) 0 0
\(801\) 3.02770 0.106979
\(802\) 0 0
\(803\) 6.54783i 0.231068i
\(804\) 0