Properties

Label 69.2.c.a.68.5
Level $69$
Weight $2$
Character 69.68
Analytic conductor $0.551$
Analytic rank $0$
Dimension $6$
CM discriminant -23
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.550967773947\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8869743.1
Defining polynomial: \(x^{6} - 3 x^{3} + 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 68.5
Root \(-1.07255 + 0.921756i\) of defining polynomial
Character \(\chi\) \(=\) 69.68
Dual form 69.2.c.a.68.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.84351i q^{2} +(-1.37328 + 1.05550i) q^{3} -1.39853 q^{4} +(-1.94584 - 2.53166i) q^{6} +1.10881i q^{8} +(0.771819 - 2.89902i) q^{9} +O(q^{10})\) \(q+1.84351i q^{2} +(-1.37328 + 1.05550i) q^{3} -1.39853 q^{4} +(-1.94584 - 2.53166i) q^{6} +1.10881i q^{8} +(0.771819 - 2.89902i) q^{9} +(1.92059 - 1.47616i) q^{12} +7.03677 q^{13} -4.84117 q^{16} +(5.34437 + 1.42286i) q^{18} -4.79583i q^{23} +(-1.17035 - 1.52271i) q^{24} -5.00000 q^{25} +12.9724i q^{26} +(2.00000 + 4.79583i) q^{27} -10.0201i q^{29} -5.83384 q^{31} -6.70714i q^{32} +(-1.07941 + 4.05437i) q^{36} +(-9.66349 + 7.42735i) q^{39} +2.64601i q^{41} +8.84117 q^{46} +13.7071i q^{47} +(6.64830 - 5.10988i) q^{48} +7.00000 q^{49} -9.21756i q^{50} -9.84117 q^{52} +(-8.84117 + 3.68702i) q^{54} +18.4721 q^{58} -9.59166i q^{59} -10.7548i q^{62} +2.68234 q^{64} +(5.06202 + 6.58604i) q^{69} +1.04102i q^{71} +(3.21445 + 0.855799i) q^{72} -9.44264 q^{73} +(6.86642 - 5.27752i) q^{75} +(-13.6924 - 17.8148i) q^{78} +(-7.80859 - 4.47503i) q^{81} -4.87794 q^{82} +(10.5762 + 13.7604i) q^{87} +6.70714i q^{92} +(8.01152 - 6.15765i) q^{93} -25.2692 q^{94} +(7.07941 + 9.21080i) q^{96} +12.9046i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 12q^{4} + 3q^{6} + O(q^{10}) \) \( 6q - 12q^{4} + 3q^{6} - 15q^{12} + 24q^{16} + 21q^{18} - 6q^{24} - 30q^{25} + 12q^{27} - 33q^{36} - 24q^{39} + 69q^{48} + 42q^{49} - 6q^{52} + 30q^{58} - 90q^{64} - 42q^{72} - 51q^{78} + 66q^{82} + 48q^{87} - 6q^{93} - 78q^{94} + 69q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.84351i 1.30356i 0.758408 + 0.651780i \(0.225977\pi\)
−0.758408 + 0.651780i \(0.774023\pi\)
\(3\) −1.37328 + 1.05550i −0.792866 + 0.609396i
\(4\) −1.39853 −0.699267
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.94584 2.53166i −0.794384 1.03355i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.10881i 0.392023i
\(9\) 0.771819 2.89902i 0.257273 0.966339i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.92059 1.47616i 0.554425 0.426131i
\(13\) 7.03677 1.95165 0.975825 0.218554i \(-0.0701339\pi\)
0.975825 + 0.218554i \(0.0701339\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.84117 −1.21029
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 5.34437 + 1.42286i 1.25968 + 0.335370i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.79583i 1.00000i
\(24\) −1.17035 1.52271i −0.238897 0.310822i
\(25\) −5.00000 −1.00000
\(26\) 12.9724i 2.54409i
\(27\) 2.00000 + 4.79583i 0.384900 + 0.922958i
\(28\) 0 0
\(29\) 10.0201i 1.86068i −0.366702 0.930339i \(-0.619513\pi\)
0.366702 0.930339i \(-0.380487\pi\)
\(30\) 0 0
\(31\) −5.83384 −1.04779 −0.523895 0.851783i \(-0.675521\pi\)
−0.523895 + 0.851783i \(0.675521\pi\)
\(32\) 6.70714i 1.18567i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −1.07941 + 4.05437i −0.179902 + 0.675729i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −9.66349 + 7.42735i −1.54740 + 1.18933i
\(40\) 0 0
\(41\) 2.64601i 0.413237i 0.978422 + 0.206618i \(0.0662459\pi\)
−0.978422 + 0.206618i \(0.933754\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 8.84117 1.30356
\(47\) 13.7071i 1.99938i 0.0248485 + 0.999691i \(0.492090\pi\)
−0.0248485 + 0.999691i \(0.507910\pi\)
\(48\) 6.64830 5.10988i 0.959600 0.737548i
\(49\) 7.00000 1.00000
\(50\) 9.21756i 1.30356i
\(51\) 0 0
\(52\) −9.84117 −1.36472
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −8.84117 + 3.68702i −1.20313 + 0.501740i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 18.4721 2.42550
\(59\) 9.59166i 1.24873i −0.781133 0.624364i \(-0.785358\pi\)
0.781133 0.624364i \(-0.214642\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 10.7548i 1.36586i
\(63\) 0 0
\(64\) 2.68234 0.335293
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 5.06202 + 6.58604i 0.609396 + 0.792866i
\(70\) 0 0
\(71\) 1.04102i 0.123546i 0.998090 + 0.0617729i \(0.0196755\pi\)
−0.998090 + 0.0617729i \(0.980325\pi\)
\(72\) 3.21445 + 0.855799i 0.378827 + 0.100857i
\(73\) −9.44264 −1.10518 −0.552588 0.833454i \(-0.686360\pi\)
−0.552588 + 0.833454i \(0.686360\pi\)
\(74\) 0 0
\(75\) 6.86642 5.27752i 0.792866 0.609396i
\(76\) 0 0
\(77\) 0 0
\(78\) −13.6924 17.8148i −1.55036 2.01712i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −7.80859 4.47503i −0.867621 0.497225i
\(82\) −4.87794 −0.538679
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 10.5762 + 13.7604i 1.13389 + 1.47527i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.70714i 0.699267i
\(93\) 8.01152 6.15765i 0.830756 0.638519i
\(94\) −25.2692 −2.60631
\(95\) 0 0
\(96\) 7.07941 + 9.21080i 0.722540 + 0.940074i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 12.9046i 1.30356i
\(99\) 0 0
\(100\) 6.99267 0.699267
\(101\) 19.1833i 1.90881i 0.298511 + 0.954406i \(0.403510\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 7.80244i 0.765092i
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −2.79707 6.70714i −0.269148 0.645394i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 14.0134i 1.30111i
\(117\) 5.43111 20.3997i 0.502107 1.88595i
\(118\) 17.6823 1.62779
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −2.79287 3.63372i −0.251825 0.327641i
\(124\) 8.15883 0.732685
\(125\) 0 0
\(126\) 0 0
\(127\) −22.3133 −1.97998 −0.989990 0.141134i \(-0.954925\pi\)
−0.989990 + 0.141134i \(0.954925\pi\)
\(128\) 8.46935i 0.748591i
\(129\) 0 0
\(130\) 0 0
\(131\) 21.0811i 1.84187i −0.389721 0.920933i \(-0.627428\pi\)
0.389721 0.920933i \(-0.372572\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) −12.1414 + 9.33190i −1.03355 + 0.794384i
\(139\) 19.9074 1.68852 0.844261 0.535932i \(-0.180040\pi\)
0.844261 + 0.535932i \(0.180040\pi\)
\(140\) 0 0
\(141\) −14.4679 18.8237i −1.21842 1.58524i
\(142\) −1.91912 −0.161049
\(143\) 0 0
\(144\) −3.73651 + 14.0346i −0.311375 + 1.16955i
\(145\) 0 0
\(146\) 17.4076i 1.44066i
\(147\) −9.61299 + 7.38853i −0.792866 + 0.609396i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 9.72918 + 12.6583i 0.794384 + 1.03355i
\(151\) 10.6456 0.866324 0.433162 0.901316i \(-0.357398\pi\)
0.433162 + 0.901316i \(0.357398\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 13.5147 10.3874i 1.08204 0.831658i
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 8.24977 14.3952i 0.648163 1.13100i
\(163\) 3.42798 0.268500 0.134250 0.990947i \(-0.457137\pi\)
0.134250 + 0.990947i \(0.457137\pi\)
\(164\) 3.70053i 0.288963i
\(165\) 0 0
\(166\) 0 0
\(167\) 9.59166i 0.742225i −0.928588 0.371113i \(-0.878976\pi\)
0.928588 0.371113i \(-0.121024\pi\)
\(168\) 0 0
\(169\) 36.5162 2.80894
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 19.1833i 1.45848i 0.684257 + 0.729241i \(0.260127\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −25.3674 + 19.4974i −1.92310 + 1.47809i
\(175\) 0 0
\(176\) 0 0
\(177\) 10.1240 + 13.1721i 0.760970 + 0.990074i
\(178\) 0 0
\(179\) 8.41506i 0.628971i −0.949262 0.314486i \(-0.898168\pi\)
0.949262 0.314486i \(-0.101832\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 5.31766 0.392023
\(185\) 0 0
\(186\) 11.3517 + 14.7693i 0.832347 + 1.08294i
\(187\) 0 0
\(188\) 19.1698i 1.39810i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −3.68362 + 2.83122i −0.265842 + 0.204326i
\(193\) −18.7045 −1.34638 −0.673188 0.739471i \(-0.735076\pi\)
−0.673188 + 0.739471i \(0.735076\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −9.78974 −0.699267
\(197\) 24.7681i 1.76466i 0.470634 + 0.882329i \(0.344025\pi\)
−0.470634 + 0.882329i \(0.655975\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 5.54404i 0.392023i
\(201\) 0 0
\(202\) −35.3647 −2.48825
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −13.9032 3.70151i −0.966339 0.257273i
\(208\) −34.0662 −2.36207
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) −1.09880 1.42961i −0.0752884 0.0979553i
\(214\) 0 0
\(215\) 0 0
\(216\) −5.31766 + 2.21762i −0.361821 + 0.150890i
\(217\) 0 0
\(218\) 0 0
\(219\) 12.9674 9.96675i 0.876257 0.673490i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) −3.85909 + 14.4951i −0.257273 + 0.966339i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 11.1103 0.729428
\(233\) 22.6861i 1.48622i −0.669171 0.743108i \(-0.733351\pi\)
0.669171 0.743108i \(-0.266649\pi\)
\(234\) 37.6071 + 10.0123i 2.45845 + 0.654526i
\(235\) 0 0
\(236\) 13.4143i 0.873195i
\(237\) 0 0
\(238\) 0 0
\(239\) 26.3731i 1.70594i 0.521963 + 0.852968i \(0.325200\pi\)
−0.521963 + 0.852968i \(0.674800\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 20.2786i 1.30356i
\(243\) 15.4468 2.09652i 0.990915 0.134492i
\(244\) 0 0
\(245\) 0 0
\(246\) 6.69880 5.14869i 0.427100 0.328269i
\(247\) 0 0
\(248\) 6.46861i 0.410757i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 41.1347i 2.58102i
\(255\) 0 0
\(256\) 20.9780 1.31113
\(257\) 12.1021i 0.754907i 0.926028 + 0.377454i \(0.123200\pi\)
−0.926028 + 0.377454i \(0.876800\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −29.0483 7.73366i −1.79804 0.478702i
\(262\) 38.8633 2.40098
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.3121i 0.933593i 0.884365 + 0.466797i \(0.154592\pi\)
−0.884365 + 0.466797i \(0.845408\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −7.07941 9.21080i −0.426131 0.554425i
\(277\) −2.22505 −0.133690 −0.0668451 0.997763i \(-0.521293\pi\)
−0.0668451 + 0.997763i \(0.521293\pi\)
\(278\) 36.6995i 2.20109i
\(279\) −4.50267 + 16.9124i −0.269568 + 1.01252i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 34.7017 26.6717i 2.06646 1.58828i
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 1.45590i 0.0863916i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −19.4441 5.17669i −1.14575 0.305039i
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 13.2059 0.772814
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) −13.6208 17.7217i −0.794384 1.03355i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 33.7472i 1.95165i
\(300\) −9.60293 + 7.38080i −0.554425 + 0.426131i
\(301\) 0 0
\(302\) 19.6252i 1.12930i
\(303\) −20.2481 26.3442i −1.16322 1.51343i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 11.6250i 0.659196i −0.944121 0.329598i \(-0.893087\pi\)
0.944121 0.329598i \(-0.106913\pi\)
\(312\) −8.23551 10.7150i −0.466244 0.606615i
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 19.1833i 1.07744i 0.842484 + 0.538721i \(0.181092\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 10.9206 + 6.25848i 0.606699 + 0.347693i
\(325\) −35.1839 −1.95165
\(326\) 6.31952i 0.350006i
\(327\) 0 0
\(328\) −2.93392 −0.161998
\(329\) 0 0
\(330\) 0 0
\(331\) 36.3868 2.00000 1.00000 0.000796384i \(-0.000253497\pi\)
1.00000 0.000796384i \(0.000253497\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 17.6823 0.967535
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 67.3180i 3.66162i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −35.3647 −1.90122
\(347\) 9.59166i 0.514907i −0.966291 0.257454i \(-0.917117\pi\)
0.966291 0.257454i \(-0.0828835\pi\)
\(348\) −14.7912 19.2444i −0.792892 1.03161i
\(349\) −25.9220 −1.38758 −0.693788 0.720180i \(-0.744059\pi\)
−0.693788 + 0.720180i \(0.744059\pi\)
\(350\) 0 0
\(351\) 14.0735 + 33.7472i 0.751190 + 1.80129i
\(352\) 0 0
\(353\) 37.4342i 1.99242i 0.0869714 + 0.996211i \(0.472281\pi\)
−0.0869714 + 0.996211i \(0.527719\pi\)
\(354\) −24.2829 + 18.6638i −1.29062 + 0.991970i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 15.5133 0.819901
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 15.1061 11.6106i 0.792866 0.609396i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 23.2174i 1.21029i
\(369\) 7.67082 + 2.04224i 0.399327 + 0.106315i
\(370\) 0 0
\(371\) 0 0
\(372\) −11.2044 + 8.61169i −0.580921 + 0.446495i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −15.1985 −0.783804
\(377\) 70.5088i 3.63139i
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 30.6424 23.5517i 1.56986 1.20659i
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 8.93944 + 11.6308i 0.456189 + 0.593533i
\(385\) 0 0
\(386\) 34.4819i 1.75508i
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 7.76166i 0.392023i
\(393\) 22.2512 + 28.9504i 1.12243 + 1.46035i
\(394\) −45.6604 −2.30034
\(395\) 0 0
\(396\) 0 0
\(397\) 16.2986 0.818003 0.409002 0.912534i \(-0.365877\pi\)
0.409002 + 0.912534i \(0.365877\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 24.2059 1.21029
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −41.0514 −2.04492
\(404\) 26.8285i 1.33477i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 32.7780 1.62077 0.810384 0.585899i \(-0.199258\pi\)
0.810384 + 0.585899i \(0.199258\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 6.82378 25.6307i 0.335370 1.25968i
\(415\) 0 0
\(416\) 47.1966i 2.31400i
\(417\) −27.3385 + 21.0123i −1.33877 + 1.02898i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 7.37405i 0.358963i
\(423\) 39.7370 + 10.5794i 1.93208 + 0.514387i
\(424\) 0 0
\(425\) 0 0
\(426\) 2.63550 2.02565i 0.127691 0.0981429i
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −9.68234 23.2174i −0.465842 1.11705i
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 18.3738 + 23.9056i 0.877935 + 1.14225i
\(439\) −38.7927 −1.85147 −0.925736 0.378170i \(-0.876554\pi\)
−0.925736 + 0.378170i \(0.876554\pi\)
\(440\) 0 0
\(441\) 5.40273 20.2931i 0.257273 0.966339i
\(442\) 0 0
\(443\) 4.25100i 0.201971i 0.994888 + 0.100985i \(0.0321996\pi\)
−0.994888 + 0.100985i \(0.967800\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 14.7481i 0.698342i
\(447\) 0 0
\(448\) 0 0
\(449\) 38.3667i 1.81063i −0.424736 0.905317i \(-0.639633\pi\)
0.424736 0.905317i \(-0.360367\pi\)
\(450\) −26.7218 7.11428i −1.25968 0.335370i
\(451\) 0 0
\(452\) 0 0
\(453\) −14.6194 + 11.2364i −0.686879 + 0.527934i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 19.4761i 0.907094i −0.891232 0.453547i \(-0.850158\pi\)
0.891232 0.453547i \(-0.149842\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 48.5088i 2.25196i
\(465\) 0 0
\(466\) 41.8221 1.93737
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) −7.59560 + 28.5297i −0.351107 + 1.31879i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 10.6353 0.489530
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −48.6192 −2.22379
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 15.3839 0.699267
\(485\) 0 0
\(486\) 3.86496 + 28.4764i 0.175318 + 1.29172i
\(487\) 27.1250 1.22915 0.614575 0.788858i \(-0.289328\pi\)
0.614575 + 0.788858i \(0.289328\pi\)
\(488\) 0 0
\(489\) −4.70759 + 3.61825i −0.212885 + 0.163623i
\(490\) 0 0
\(491\) 35.8292i 1.61695i 0.588531 + 0.808475i \(0.299707\pi\)
−0.588531 + 0.808475i \(0.700293\pi\)
\(492\) 3.90593 + 5.08188i 0.176093 + 0.229109i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 28.2426 1.26813
\(497\) 0 0
\(498\) 0 0
\(499\) −31.5751 −1.41349 −0.706747 0.707466i \(-0.749838\pi\)
−0.706747 + 0.707466i \(0.749838\pi\)
\(500\) 0 0
\(501\) 10.1240 + 13.1721i 0.452309 + 0.588485i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −50.1471 + 38.5430i −2.22711 + 1.71176i
\(508\) 31.2059 1.38454
\(509\) 44.8082i 1.98609i −0.117732 0.993045i \(-0.537562\pi\)
0.117732 0.993045i \(-0.462438\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 21.7345i 0.960539i
\(513\) 0 0
\(514\) −22.3103 −0.984067
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −20.2481 26.3442i −0.888793 1.15638i
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 14.2571 53.5509i 0.624016 2.34386i
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 29.4827i 1.28796i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −27.8064 7.40302i −1.20669 0.321264i
\(532\) 0 0
\(533\) 18.6194i 0.806494i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 8.88214 + 11.5563i 0.383293 + 0.498690i
\(538\) −28.2280 −1.21699
\(539\) 0 0
\(540\) 0 0
\(541\) 39.9956 1.71954 0.859772 0.510677i \(-0.170605\pi\)
0.859772 + 0.510677i \(0.170605\pi\)
\(542\) 29.4962i 1.26697i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −15.0957 −0.645444 −0.322722 0.946494i \(-0.604598\pi\)
−0.322722 + 0.946494i \(0.604598\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) −7.30266 + 5.61282i −0.310822 + 0.238897i
\(553\) 0 0
\(554\) 4.10190i 0.174273i
\(555\) 0 0
\(556\) −27.8412 −1.18073
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) −31.1782 8.30072i −1.31988 0.351397i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 20.2338 + 26.3256i 0.851998 + 1.10851i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −1.15429 −0.0484328
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 23.9792i 1.00000i
\(576\) 2.07028 7.77615i 0.0862617 0.324006i
\(577\) 14.2544 0.593417 0.296708 0.954968i \(-0.404111\pi\)
0.296708 + 0.954968i \(0.404111\pi\)
\(578\) 31.3397i 1.30356i
\(579\) 25.6865 19.7426i 1.06750 0.820476i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 10.4701i 0.433255i
\(585\) 0 0
\(586\) 0 0
\(587\) 23.1632i 0.956046i 0.878347 + 0.478023i \(0.158646\pi\)
−0.878347 + 0.478023i \(0.841354\pi\)
\(588\) 13.4441 10.3331i 0.554425 0.426131i
\(589\) 0 0
\(590\) 0 0
\(591\) −26.1429 34.0137i −1.07538 1.39914i
\(592\) 0 0
\(593\) 38.3667i 1.57553i −0.615976 0.787765i \(-0.711238\pi\)
0.615976 0.787765i \(-0.288762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 62.2133 2.54409
\(599\) 9.59166i 0.391905i −0.980613 0.195952i \(-0.937220\pi\)
0.980613 0.195952i \(-0.0627798\pi\)
\(600\) 5.85177 + 7.61355i 0.238897 + 0.310822i
\(601\) −0.180813 −0.00737552 −0.00368776 0.999993i \(-0.501174\pi\)
−0.00368776 + 0.999993i \(0.501174\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −14.8882 −0.605792
\(605\) 0 0
\(606\) 48.5658 37.3276i 1.97285 1.51633i
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 96.4536i 3.90209i
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 36.8702i 1.48796i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 23.0000 9.59166i 0.922958 0.384900i
\(622\) 21.4309 0.859301
\(623\) 0 0
\(624\) 46.7826 35.9571i 1.87280 1.43943i
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 5.49314 4.22202i 0.218333 0.167810i
\(634\) −35.3647 −1.40451
\(635\) 0 0
\(636\) 0 0
\(637\) 49.2574 1.95165
\(638\) 0 0
\(639\) 3.01792 + 0.803476i 0.119387 + 0.0317850i
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 39.0392i 1.53479i 0.641175 + 0.767395i \(0.278447\pi\)
−0.641175 + 0.767395i \(0.721553\pi\)
\(648\) 4.96195 8.65823i 0.194924 0.340128i
\(649\) 0 0
\(650\) 64.8619i 2.54409i
\(651\) 0 0
\(652\) −4.79415 −0.187753
\(653\) 35.3522i 1.38344i −0.722167 0.691719i \(-0.756854\pi\)
0.722167 0.691719i \(-0.243146\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 12.8098i 0.500137i
\(657\) −7.28800 + 27.3744i −0.284332 + 1.06798i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 67.0795i 2.60712i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −48.0545 −1.86068
\(668\) 13.4143i 0.519014i
\(669\) −10.9863 + 8.44404i −0.424754 + 0.326465i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −51.6633 −1.99147 −0.995737 0.0922433i \(-0.970596\pi\)
−0.995737 + 0.0922433i \(0.970596\pi\)
\(674\) 0 0
\(675\) −10.0000 23.9792i −0.384900 0.922958i
\(676\) −51.0691 −1.96420
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 46.4132i 1.77595i −0.459889 0.887977i \(-0.652111\pi\)
0.459889 0.887977i \(-0.347889\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 26.8285i 1.01987i
\(693\) 0 0
\(694\) 17.6823 0.671212
\(695\) 0 0
\(696\) −15.2576 + 11.7270i −0.578339 + 0.444511i
\(697\) 0 0
\(698\) 47.7876i 1.80879i
\(699\) 23.9453 + 31.1545i 0.905695 + 1.17837i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) −62.2133 + 25.9447i −2.34809 + 0.979221i
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −69.0104 −2.59724
\(707\) 0 0
\(708\) −14.1588 18.4216i −0.532121 0.692326i
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 27.9781i 1.04779i
\(714\) 0 0
\(715\) 0 0
\(716\) 11.7688i 0.439819i
\(717\) −27.8370 36.2178i −1.03959 1.35258i
\(718\) 0 0
\(719\) 47.9583i 1.78854i 0.447524 + 0.894272i \(0.352306\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 35.0267i 1.30356i
\(723\) 0 0
\(724\) 0 0
\(725\) 50.1003i 1.86068i
\(726\) 21.4042 + 27.8483i 0.794384 + 1.03355i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −19.0000 + 19.1833i −0.703704 + 0.710494i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −32.1663 −1.18567
\(737\) 0 0
\(738\) −3.76489 + 14.1412i −0.138587 + 0.520546i
\(739\) 52.8662 1.94471 0.972357 0.233497i \(-0.0750170\pi\)
0.972357 + 0.233497i \(0.0750170\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 6.82766 + 8.88325i 0.250314 + 0.325676i
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 66.3583i 2.41984i
\(753\) 0 0
\(754\) 129.984 4.73373
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.81007i 0.246865i −0.992353 0.123432i \(-0.960610\pi\)
0.992353 0.123432i \(-0.0393902\pi\)
\(762\) 43.4179 + 56.4897i 1.57287 + 2.04641i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 67.4944i 2.43708i
\(768\) −28.8088 + 22.1424i −1.03955 + 0.798995i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −12.7738 16.6196i −0.460038 0.598540i
\(772\) 26.1588 0.941477
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 29.1692 1.04779
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 48.0545 20.0401i 1.71733 0.716175i
\(784\) −33.8882 −1.21029
\(785\) 0 0
\(786\) −53.3703 + 41.0204i −1.90366 + 1.46315i
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 34.6391i 1.23397i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 30.0466i 1.06632i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 33.5357i 1.18567i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 75.6788i 2.66567i
\(807\)