Properties

Label 39.4.f.a.5.2
Level $39$
Weight $4$
Character 39.5
Analytic conductor $2.301$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 39.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.30107449022\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 5.2
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 39.5
Dual form 39.4.f.a.8.2

$q$-expansion

\(f(q)\) \(=\) \(q+5.19615 q^{3} -8.00000i q^{4} +(5.58846 + 5.58846i) q^{7} +27.0000 q^{9} +O(q^{10})\) \(q+5.19615 q^{3} -8.00000i q^{4} +(5.58846 + 5.58846i) q^{7} +27.0000 q^{9} -41.5692i q^{12} +(-31.1769 + 35.0000i) q^{13} -64.0000 q^{16} +(-105.942 + 105.942i) q^{19} +(29.0385 + 29.0385i) q^{21} -125.000i q^{25} +140.296 q^{27} +(44.7077 - 44.7077i) q^{28} +(76.0577 - 76.0577i) q^{31} -216.000i q^{36} +(273.238 + 273.238i) q^{37} +(-162.000 + 181.865i) q^{39} -218.238i q^{43} -332.554 q^{48} -280.538i q^{49} +(280.000 + 249.415i) q^{52} +(-550.492 + 550.492i) q^{57} -935.307 q^{61} +(150.888 + 150.888i) q^{63} +512.000i q^{64} +(767.358 - 767.358i) q^{67} +(-782.061 - 782.061i) q^{73} -649.519i q^{75} +(847.538 + 847.538i) q^{76} +1091.19 q^{79} +729.000 q^{81} +(232.308 - 232.308i) q^{84} +(-369.827 + 21.3651i) q^{91} +(395.207 - 395.207i) q^{93} +(-1350.89 + 1350.89i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{7} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 40 q^{7} + 108 q^{9} - 256 q^{16} - 112 q^{19} + 324 q^{21} - 320 q^{28} + 616 q^{31} + 220 q^{37} - 648 q^{39} + 1120 q^{52} - 1620 q^{57} - 1080 q^{63} + 1760 q^{67} - 2380 q^{73} + 896 q^{76} + 2916 q^{81} + 2592 q^{84} - 544 q^{91} - 1620 q^{93} - 2660 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\) \(14\) \(28\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 5.19615 1.00000
\(4\) 8.00000i 1.00000i
\(5\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0 0
\(7\) 5.58846 + 5.58846i 0.301748 + 0.301748i 0.841698 0.539949i \(-0.181557\pi\)
−0.539949 + 0.841698i \(0.681557\pi\)
\(8\) 0 0
\(9\) 27.0000 1.00000
\(10\) 0 0
\(11\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(12\) 41.5692i 1.00000i
\(13\) −31.1769 + 35.0000i −0.665148 + 0.746712i
\(14\) 0 0
\(15\) 0 0
\(16\) −64.0000 −1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −105.942 + 105.942i −1.27920 + 1.27920i −0.338086 + 0.941115i \(0.609780\pi\)
−0.941115 + 0.338086i \(0.890220\pi\)
\(20\) 0 0
\(21\) 29.0385 + 29.0385i 0.301748 + 0.301748i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 125.000i 1.00000i
\(26\) 0 0
\(27\) 140.296 1.00000
\(28\) 44.7077 44.7077i 0.301748 0.301748i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 76.0577 76.0577i 0.440657 0.440657i −0.451576 0.892233i \(-0.649138\pi\)
0.892233 + 0.451576i \(0.149138\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 216.000i 1.00000i
\(37\) 273.238 + 273.238i 1.21406 + 1.21406i 0.969680 + 0.244377i \(0.0785834\pi\)
0.244377 + 0.969680i \(0.421417\pi\)
\(38\) 0 0
\(39\) −162.000 + 181.865i −0.665148 + 0.746712i
\(40\) 0 0
\(41\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(42\) 0 0
\(43\) 218.238i 0.773978i −0.922084 0.386989i \(-0.873515\pi\)
0.922084 0.386989i \(-0.126485\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) −332.554 −1.00000
\(49\) 280.538i 0.817896i
\(50\) 0 0
\(51\) 0 0
\(52\) 280.000 + 249.415i 0.746712 + 0.665148i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −550.492 + 550.492i −1.27920 + 1.27920i
\(58\) 0 0
\(59\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(60\) 0 0
\(61\) −935.307 −1.96318 −0.981589 0.191006i \(-0.938825\pi\)
−0.981589 + 0.191006i \(0.938825\pi\)
\(62\) 0 0
\(63\) 150.888 + 150.888i 0.301748 + 0.301748i
\(64\) 512.000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 767.358 767.358i 1.39922 1.39922i 0.596912 0.802307i \(-0.296394\pi\)
0.802307 0.596912i \(-0.203606\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(72\) 0 0
\(73\) −782.061 782.061i −1.25388 1.25388i −0.953966 0.299916i \(-0.903041\pi\)
−0.299916 0.953966i \(-0.596959\pi\)
\(74\) 0 0
\(75\) 649.519i 1.00000i
\(76\) 847.538 + 847.538i 1.27920 + 1.27920i
\(77\) 0 0
\(78\) 0 0
\(79\) 1091.19 1.55403 0.777017 0.629480i \(-0.216732\pi\)
0.777017 + 0.629480i \(0.216732\pi\)
\(80\) 0 0
\(81\) 729.000 1.00000
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 232.308 232.308i 0.301748 0.301748i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(90\) 0 0
\(91\) −369.827 + 21.3651i −0.426026 + 0.0246118i
\(92\) 0 0
\(93\) 395.207 395.207i 0.440657 0.440657i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1350.89 + 1350.89i −1.41404 + 1.41404i −0.696088 + 0.717957i \(0.745078\pi\)
−0.717957 + 0.696088i \(0.754922\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1000.00 −1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 1028.84i 0.984218i −0.870534 0.492109i \(-0.836226\pi\)
0.870534 0.492109i \(-0.163774\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1122.37i 1.00000i
\(109\) −768.192 + 768.192i −0.675041 + 0.675041i −0.958874 0.283833i \(-0.908394\pi\)
0.283833 + 0.958874i \(0.408394\pi\)
\(110\) 0 0
\(111\) 1419.79 + 1419.79i 1.21406 + 1.21406i
\(112\) −357.661 357.661i −0.301748 0.301748i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −841.777 + 945.000i −0.665148 + 0.746712i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1331.00i 1.00000i
\(122\) 0 0
\(123\) 0 0
\(124\) −608.462 608.462i −0.440657 0.440657i
\(125\) 0 0
\(126\) 0 0
\(127\) 380.000i 0.265508i −0.991149 0.132754i \(-0.957618\pi\)
0.991149 0.132754i \(-0.0423821\pi\)
\(128\) 0 0
\(129\) 1134.00i 0.773978i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −1184.11 −0.771994
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(138\) 0 0
\(139\) 2576.00 1.57190 0.785948 0.618293i \(-0.212175\pi\)
0.785948 + 0.618293i \(0.212175\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −1728.00 −1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 1457.72i 0.817896i
\(148\) 2185.91 2185.91i 1.21406 1.21406i
\(149\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(150\) 0 0
\(151\) 2510.79 + 2510.79i 1.35315 + 1.35315i 0.882119 + 0.471027i \(0.156117\pi\)
0.471027 + 0.882119i \(0.343883\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1454.92 + 1296.00i 0.746712 + 0.665148i
\(157\) 3850.00 1.95709 0.978546 0.206028i \(-0.0660539\pi\)
0.978546 + 0.206028i \(0.0660539\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2900.31 2900.31i −1.39368 1.39368i −0.816897 0.576783i \(-0.804308\pi\)
−0.576783 0.816897i \(-0.695692\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(168\) 0 0
\(169\) −253.000 2182.38i −0.115157 0.993347i
\(170\) 0 0
\(171\) −2860.44 + 2860.44i −1.27920 + 1.27920i
\(172\) −1745.91 −0.773978
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 698.557 698.557i 0.301748 0.301748i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 3429.46i 1.40834i 0.710031 + 0.704171i \(0.248681\pi\)
−0.710031 + 0.704171i \(0.751319\pi\)
\(182\) 0 0
\(183\) −4860.00 −1.96318
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 784.039 + 784.039i 0.301748 + 0.301748i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 2660.43i 1.00000i
\(193\) −2043.86 2043.86i −0.762281 0.762281i 0.214453 0.976734i \(-0.431203\pi\)
−0.976734 + 0.214453i \(0.931203\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2244.31 −0.817896
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0 0
\(199\) 5236.00i 1.86518i 0.360942 + 0.932588i \(0.382455\pi\)
−0.360942 + 0.932588i \(0.617545\pi\)
\(200\) 0 0
\(201\) 3987.31 3987.31i 1.39922 1.39922i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1995.32 2240.00i 0.665148 0.746712i
\(209\) 0 0
\(210\) 0 0
\(211\) 1091.19 0.356023 0.178011 0.984028i \(-0.443034\pi\)
0.178011 + 0.984028i \(0.443034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 850.091 0.265935
\(218\) 0 0
\(219\) −4063.71 4063.71i −1.25388 1.25388i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1305.04 + 1305.04i −0.391893 + 0.391893i −0.875362 0.483469i \(-0.839377\pi\)
0.483469 + 0.875362i \(0.339377\pi\)
\(224\) 0 0
\(225\) 3375.00i 1.00000i
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 4403.94 + 4403.94i 1.27920 + 1.27920i
\(229\) 417.038 + 417.038i 0.120343 + 0.120343i 0.764714 0.644370i \(-0.222880\pi\)
−0.644370 + 0.764714i \(0.722880\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 5670.00 1.55403
\(238\) 0 0
\(239\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(240\) 0 0
\(241\) −3065.46 3065.46i −0.819352 0.819352i 0.166662 0.986014i \(-0.446701\pi\)
−0.986014 + 0.166662i \(0.946701\pi\)
\(242\) 0 0
\(243\) 3788.00 1.00000
\(244\) 7482.46i 1.96318i
\(245\) 0 0
\(246\) 0 0
\(247\) −405.026 7010.93i −0.104337 1.80605i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 1207.11 1207.11i 0.301748 0.301748i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4096.00 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 3053.96i 0.732679i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −6138.86 6138.86i −1.39922 1.39922i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −4036.71 4036.71i −0.904844 0.904844i 0.0910064 0.995850i \(-0.470992\pi\)
−0.995850 + 0.0910064i \(0.970992\pi\)
\(272\) 0 0
\(273\) −1921.68 + 111.017i −0.426026 + 0.0246118i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8293.06i 1.79885i 0.437074 + 0.899425i \(0.356015\pi\)
−0.437074 + 0.899425i \(0.643985\pi\)
\(278\) 0 0
\(279\) 2053.56 2053.56i 0.440657 0.440657i
\(280\) 0 0
\(281\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(282\) 0 0
\(283\) 5600.00i 1.17627i 0.808761 + 0.588137i \(0.200138\pi\)
−0.808761 + 0.588137i \(0.799862\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4913.00 −1.00000
\(290\) 0 0
\(291\) −7019.44 + 7019.44i −1.41404 + 1.41404i
\(292\) −6256.49 + 6256.49i −1.25388 + 1.25388i
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −5196.15 −1.00000
\(301\) 1219.62 1219.62i 0.233547 0.233547i
\(302\) 0 0
\(303\) 0 0
\(304\) 6780.31 6780.31i 1.27920 1.27920i
\(305\) 0 0
\(306\) 0 0
\(307\) 4524.99 + 4524.99i 0.841221 + 0.841221i 0.989018 0.147797i \(-0.0472182\pi\)
−0.147797 + 0.989018i \(0.547218\pi\)
\(308\) 0 0
\(309\) 5346.00i 0.984218i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 4738.89 0.855776 0.427888 0.903832i \(-0.359258\pi\)
0.427888 + 0.903832i \(0.359258\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 8729.54i 1.55403i
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 5832.00i 1.00000i
\(325\) 4375.00 + 3897.11i 0.746712 + 0.665148i
\(326\) 0 0
\(327\) −3991.64 + 3991.64i −0.675041 + 0.675041i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5505.56 5505.56i 0.914238 0.914238i −0.0823644 0.996602i \(-0.526247\pi\)
0.996602 + 0.0823644i \(0.0262471\pi\)
\(332\) 0 0
\(333\) 7377.44 + 7377.44i 1.21406 + 1.21406i
\(334\) 0 0
\(335\) 0 0
\(336\) −1858.46 1858.46i −0.301748 0.301748i
\(337\) 4930.00i 0.796897i −0.917191 0.398448i \(-0.869549\pi\)
0.917191 0.398448i \(-0.130451\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 3484.62 3484.62i 0.548547 0.548547i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 8607.04 + 8607.04i 1.32013 + 1.32013i 0.913670 + 0.406456i \(0.133236\pi\)
0.406456 + 0.913670i \(0.366764\pi\)
\(350\) 0 0
\(351\) −4374.00 + 4910.36i −0.665148 + 0.746712i
\(352\) 0 0
\(353\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(360\) 0 0
\(361\) 15588.5i 2.27271i
\(362\) 0 0
\(363\) 6916.08i 1.00000i
\(364\) 170.921 + 2958.61i 0.0246118 + 0.426026i
\(365\) 0 0
\(366\) 0 0
\(367\) −4340.00 −0.617292 −0.308646 0.951177i \(-0.599876\pi\)
−0.308646 + 0.951177i \(0.599876\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −3161.66 3161.66i −0.440657 0.440657i
\(373\) −7420.11 −1.03002 −0.515011 0.857183i \(-0.672212\pi\)
−0.515011 + 0.857183i \(0.672212\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1709.56 1709.56i 0.231699 0.231699i −0.581702 0.813402i \(-0.697613\pi\)
0.813402 + 0.581702i \(0.197613\pi\)
\(380\) 0 0
\(381\) 1974.54i 0.265508i
\(382\) 0 0
\(383\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5892.44i 0.773978i
\(388\) 10807.1 + 10807.1i 1.41404 + 1.41404i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −8482.76 8482.76i −1.07239 1.07239i −0.997167 0.0752196i \(-0.976034\pi\)
−0.0752196 0.997167i \(-0.523966\pi\)
\(398\) 0 0
\(399\) −6152.81 −0.771994
\(400\) 8000.00i 1.00000i
\(401\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(402\) 0 0
\(403\) 290.775 + 5033.26i 0.0359418 + 0.622146i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −11293.7 + 11293.7i −1.36537 + 1.36537i −0.498458 + 0.866914i \(0.666100\pi\)
−0.866914 + 0.498458i \(0.833900\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8230.71 −0.984218
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 13385.3 1.57190
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −9660.19 + 9660.19i −1.11831 + 1.11831i −0.126322 + 0.991989i \(0.540317\pi\)
−0.991989 + 0.126322i \(0.959683\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −5226.93 5226.93i −0.592386 0.592386i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(432\) −8978.95 −1.00000
\(433\) 2590.00i 0.287454i −0.989617 0.143727i \(-0.954091\pi\)
0.989617 0.143727i \(-0.0459087\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6145.54 + 6145.54i 0.675041 + 0.675041i
\(437\) 0 0
\(438\) 0 0
\(439\) 10756.0i 1.16938i −0.811257 0.584690i \(-0.801216\pi\)
0.811257 0.584690i \(-0.198784\pi\)
\(440\) 0 0
\(441\) 7574.53i 0.817896i
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 11358.3 11358.3i 1.21406 1.21406i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −2861.29 + 2861.29i −0.301748 + 0.301748i
\(449\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 13046.4 + 13046.4i 1.35315 + 1.35315i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13775.1 13775.1i 1.41000 1.41000i 0.650491 0.759514i \(-0.274563\pi\)
0.759514 0.650491i \(-0.225437\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) 0 0
\(463\) −11090.3 11090.3i −1.11320 1.11320i −0.992716 0.120482i \(-0.961556\pi\)
−0.120482 0.992716i \(-0.538444\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 7560.00 + 6734.21i 0.746712 + 0.665148i
\(469\) 8576.69 0.844424
\(470\) 0 0
\(471\) 20005.2 1.95709
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 13242.8 + 13242.8i 1.27920 + 1.27920i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(480\) 0 0
\(481\) −18082.1 + 1044.61i −1.71408 + 0.0990235i
\(482\) 0 0
\(483\) 0 0
\(484\) 10648.0 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 7940.26 7940.26i 0.738824 0.738824i −0.233526 0.972351i \(-0.575026\pi\)
0.972351 + 0.233526i \(0.0750265\pi\)
\(488\) 0 0
\(489\) −15070.5 15070.5i −1.39368 1.39368i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −4867.69 + 4867.69i −0.440657 + 0.440657i
\(497\) 0 0
\(498\) 0 0
\(499\) −15751.9 + 15751.9i −1.41313 + 1.41313i −0.678938 + 0.734195i \(0.737560\pi\)
−0.734195 + 0.678938i \(0.762440\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1314.63 11340.0i −0.115157 0.993347i
\(508\) −3040.00 −0.265508
\(509\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(510\) 0 0
\(511\) 8741.03i 0.756713i
\(512\) 0 0
\(513\) −14863.3 + 14863.3i −1.27920 + 1.27920i
\(514\) 0 0
\(515\) 0 0
\(516\) −9072.00 −0.773978
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 12040.0 1.00664 0.503320 0.864100i \(-0.332112\pi\)
0.503320 + 0.864100i \(0.332112\pi\)
\(524\) 0 0
\(525\) 3629.81 3629.81i 0.301748 0.301748i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 9472.86i 0.771994i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 5883.04 + 5883.04i 0.467526 + 0.467526i 0.901112 0.433586i \(-0.142752\pi\)
−0.433586 + 0.901112i \(0.642752\pi\)
\(542\) 0 0
\(543\) 17820.0i 1.40834i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1640.00 0.128193 0.0640963 0.997944i \(-0.479584\pi\)
0.0640963 + 0.997944i \(0.479584\pi\)
\(548\) 0 0
\(549\) −25253.3 −1.96318
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 6098.08 + 6098.08i 0.468927 + 0.468927i
\(554\) 0 0
\(555\) 0 0
\(556\) 20608.0i 1.57190i
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) 7638.34 + 6804.00i 0.577938 + 0.514810i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4073.99 + 4073.99i 0.301748 + 0.301748i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 23312.0i 1.70854i −0.519829 0.854270i \(-0.674004\pi\)
0.519829 0.854270i \(-0.325996\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 13824.0i 1.00000i
\(577\) 1807.50 1807.50i 0.130411 0.130411i −0.638888 0.769300i \(-0.720605\pi\)
0.769300 + 0.638888i \(0.220605\pi\)
\(578\) 0 0
\(579\) −10620.2 10620.2i −0.762281 0.762281i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(588\) −11661.8 −0.817896
\(589\) 16115.5i 1.12738i
\(590\) 0 0
\(591\) 0 0
\(592\) −17487.3 17487.3i −1.21406 1.21406i
\(593\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 27207.1i 1.86518i
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 3117.69 0.211603 0.105801 0.994387i \(-0.466259\pi\)
0.105801 + 0.994387i \(0.466259\pi\)
\(602\) 0 0
\(603\) 20718.7 20718.7i 1.39922 1.39922i
\(604\) 20086.3 20086.3i 1.35315 1.35315i
\(605\) 0 0
\(606\) 0 0
\(607\) 28420.0 1.90038 0.950191 0.311667i \(-0.100887\pi\)
0.950191 + 0.311667i \(0.100887\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −3744.59 + 3744.59i −0.246725 + 0.246725i −0.819625 0.572900i \(-0.805818\pi\)
0.572900 + 0.819625i \(0.305818\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(618\) 0 0
\(619\) 21044.3 + 21044.3i 1.36646 + 1.36646i 0.865424 + 0.501040i \(0.167049\pi\)
0.501040 + 0.865424i \(0.332951\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 10368.0 11639.4i 0.665148 0.746712i
\(625\) −15625.0 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 30800.0i 1.95709i
\(629\) 0 0
\(630\) 0 0
\(631\) 14876.3 + 14876.3i 0.938535 + 0.938535i 0.998217 0.0596825i \(-0.0190088\pi\)
−0.0596825 + 0.998217i \(0.519009\pi\)
\(632\) 0 0
\(633\) 5670.00 0.356023
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 9818.84 + 8746.32i 0.610733 + 0.544022i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 8338.15 8338.15i 0.511391 0.511391i −0.403561 0.914953i \(-0.632228\pi\)
0.914953 + 0.403561i \(0.132228\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 4417.20 0.265935
\(652\) −23202.5 + 23202.5i −1.39368 + 1.39368i
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −21115.7 21115.7i −1.25388 1.25388i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −23803.0 23803.0i −1.40065 1.40065i −0.798032 0.602615i \(-0.794125\pi\)
−0.602615 0.798032i \(-0.705875\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −6781.19 + 6781.19i −0.391893 + 0.391893i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 25315.7i 1.45000i 0.688751 + 0.724998i \(0.258159\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(674\) 0 0
\(675\) 17537.0i 1.00000i
\(676\) −17459.1 + 2024.00i −0.993347 + 0.115157i
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −15098.8 −0.853371
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 22883.5 + 22883.5i 1.27920 + 1.27920i
\(685\) 0 0
\(686\) 0 0
\(687\) 2166.99 + 2166.99i 0.120343 + 0.120343i
\(688\) 13967.3i 0.773978i
\(689\) 0 0
\(690\) 0 0
\(691\) −24325.9 + 24325.9i −1.33922 + 1.33922i −0.442408 + 0.896814i \(0.645876\pi\)
−0.896814 + 0.442408i \(0.854124\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −5588.46 5588.46i −0.301748 0.301748i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −57895.0 −3.10605
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 12617.0 + 12617.0i 0.668326 + 0.668326i 0.957328 0.289003i \(-0.0933236\pi\)
−0.289003 + 0.957328i \(0.593324\pi\)
\(710\) 0 0
\(711\) 29462.2 1.55403
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 5749.62 5749.62i 0.296986 0.296986i
\(722\) 0 0
\(723\) −15928.6 15928.6i −0.819352 0.819352i
\(724\) 27435.7 1.40834
\(725\) 0 0
\(726\) 0 0
\(727\) 10780.0i 0.549942i −0.961452 0.274971i \(-0.911332\pi\)
0.961452 0.274971i \(-0.0886683\pi\)
\(728\) 0 0
\(729\) 19683.0 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 38880.0i 1.96318i
\(733\) 25888.2 25888.2i 1.30451 1.30451i 0.379184 0.925321i \(-0.376205\pi\)
0.925321 0.379184i \(-0.123795\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −28236.7 28236.7i −1.40555 1.40555i −0.780910 0.624644i \(-0.785244\pi\)
−0.624644 0.780910i \(-0.714756\pi\)
\(740\) 0 0
\(741\) −2104.58 36429.9i −0.104337 1.80605i
\(742\) 0 0
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 33827.0i 1.64363i 0.569757 + 0.821813i \(0.307037\pi\)
−0.569757 + 0.821813i \(0.692963\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 6272.31 6272.31i 0.301748 0.301748i
\(757\) 3928.29 0.188608 0.0943039 0.995543i \(-0.469937\pi\)
0.0943039 + 0.995543i \(0.469937\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(762\) 0 0
\(763\) −8586.02 −0.407385
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 21283.4 1.00000
\(769\) 18897.3 18897.3i 0.886156 0.886156i −0.107995 0.994151i \(-0.534443\pi\)
0.994151 + 0.107995i \(0.0344431\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −16350.9 + 16350.9i −0.762281 + 0.762281i
\(773\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 0 0
\(775\) −9507.21 9507.21i −0.440657 0.440657i
\(776\) 0 0
\(777\) 15868.9i 0.732679i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 17954.5i 0.817896i
\(785\) 0 0
\(786\) 0 0
\(787\) 25768.6 + 25768.6i 1.16715 + 1.16715i 0.982874 + 0.184281i \(0.0589958\pi\)
0.184281 + 0.982874i \(0.441004\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 29160.0 32735.8i 1.30580 1.46593i
\(794\) 0 0
\(795\) 0 0
\(796\) 41888.0 1.86518
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0