Properties

Label 39.4.f.a.5.1
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 5.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 39.5
Dual form 39.4.f.a.8.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.19615 q^{3} -8.00000i q^{4} +(-25.5885 - 25.5885i) q^{7} +27.0000 q^{9} +O(q^{10})\) \(q-5.19615 q^{3} -8.00000i q^{4} +(-25.5885 - 25.5885i) q^{7} +27.0000 q^{9} +41.5692i q^{12} +(31.1769 + 35.0000i) q^{13} -64.0000 q^{16} +(49.9423 - 49.9423i) q^{19} +(132.962 + 132.962i) q^{21} -125.000i q^{25} -140.296 q^{27} +(-204.708 + 204.708i) q^{28} +(231.942 - 231.942i) q^{31} -216.000i q^{36} +(-163.238 - 163.238i) q^{37} +(-162.000 - 181.865i) q^{39} +218.238i q^{43} +332.554 q^{48} +966.538i q^{49} +(280.000 - 249.415i) q^{52} +(-259.508 + 259.508i) q^{57} +935.307 q^{61} +(-690.888 - 690.888i) q^{63} +512.000i q^{64} +(112.642 - 112.642i) q^{67} +(-407.939 - 407.939i) q^{73} +649.519i q^{75} +(-399.538 - 399.538i) q^{76} -1091.19 q^{79} +729.000 q^{81} +(1063.69 - 1063.69i) q^{84} +(97.8269 - 1693.37i) q^{91} +(-1205.21 + 1205.21i) q^{93} +(20.8921 - 20.8921i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{7} + 108 q^{9}+O(q^{10}) \) 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
\(7\) −25.5885 25.5885i −1.38165 1.38165i −0.841698 0.539949i \(-0.818443\pi\)
−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
\(19\) 49.9423 49.9423i 0.603029 0.603029i −0.338086 0.941115i \(-0.609780\pi\)
0.941115 + 0.338086i \(0.109780\pi\)
\(20\) 0 0
\(21\) 132.962 + 132.962i 1.38165 + 1.38165i
\(22\) 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
\(28\) −204.708 + 204.708i −1.38165 + 1.38165i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 231.942 231.942i 1.34381 1.34381i 0.451576 0.892233i \(-0.350862\pi\)
0.892233 0.451576i \(-0.149138\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 216.000i 1.00000i
\(37\) −163.238 163.238i −0.725303 0.725303i 0.244377 0.969680i \(-0.421417\pi\)
−0.969680 + 0.244377i \(0.921417\pi\)
\(38\) 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 \(0.126485\pi\)
−0.922084 + 0.386989i \(0.873515\pi\)
\(44\) 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
\(49\) 966.538i 2.81790i
\(50\) 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
\(57\) −259.508 + 259.508i −0.603029 + 0.603029i
\(58\) 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 \(-0.0611749\pi\)
0.981589 + 0.191006i \(0.0611749\pi\)
\(62\) 0 0
\(63\) −690.888 690.888i −1.38165 1.38165i
\(64\) 512.000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 112.642 112.642i 0.205395 0.205395i −0.596912 0.802307i \(-0.703606\pi\)
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
\(73\) −407.939 407.939i −0.654049 0.654049i 0.299916 0.953966i \(-0.403041\pi\)
−0.953966 + 0.299916i \(0.903041\pi\)
\(74\) 0 0
\(75\) 649.519i 1.00000i
\(76\) −399.538 399.538i −0.603029 0.603029i
\(77\) 0 0
\(78\) 0 0
\(79\) −1091.19 −1.55403 −0.777017 0.629480i \(-0.783268\pi\)
−0.777017 + 0.629480i \(0.783268\pi\)
\(80\) 0 0
\(81\) 729.000 1.00000
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 1063.69 1063.69i 1.38165 1.38165i
\(85\) 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
\(91\) 97.8269 1693.37i 0.112693 1.95069i
\(92\) 0 0
\(93\) −1205.21 + 1205.21i −1.34381 + 1.34381i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 20.8921 20.8921i 0.0218688 0.0218688i −0.696088 0.717957i \(-0.745078\pi\)
0.717957 + 0.696088i \(0.245078\pi\)
\(98\) 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 \(0.163774\pi\)
−0.870534 + 0.492109i \(0.836226\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1122.37i 1.00000i
\(109\) 1414.19 1414.19i 1.24271 1.24271i 0.283833 0.958874i \(-0.408394\pi\)
0.958874 0.283833i \(-0.0916061\pi\)
\(110\) 0 0
\(111\) 848.212 + 848.212i 0.725303 + 0.725303i
\(112\) 1637.66 + 1637.66i 1.38165 + 1.38165i
\(113\) 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
\(124\) −1855.54 1855.54i −1.34381 1.34381i
\(125\) 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
\(133\) −2555.89 −1.66635
\(134\) 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
\(147\) 5022.28i 2.81790i
\(148\) −1305.91 + 1305.91i −0.725303 + 0.725303i
\(149\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(150\) 0 0
\(151\) −762.788 762.788i −0.411091 0.411091i 0.471027 0.882119i \(-0.343883\pi\)
−0.882119 + 0.471027i \(0.843883\pi\)
\(152\) 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
\(163\) −499.689 499.689i −0.240114 0.240114i 0.576783 0.816897i \(-0.304308\pi\)
−0.816897 + 0.576783i \(0.804308\pi\)
\(164\) 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
\(171\) 1348.44 1348.44i 0.603029 0.603029i
\(172\) 1745.91 0.773978
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −3198.56 + 3198.56i −1.38165 + 1.38165i
\(176\) 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 \(-0.751319\pi\)
0.710031 0.704171i \(-0.248681\pi\)
\(182\) 0 0
\(183\) −4860.00 −1.96318
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3589.96 + 3589.96i 1.38165 + 1.38165i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 2660.43i 1.00000i
\(193\) 3193.86 + 3193.86i 1.19119 + 1.19119i 0.976734 + 0.214453i \(0.0687968\pi\)
0.214453 + 0.976734i \(0.431203\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 7732.31 2.81790
\(197\) 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
\(201\) −585.307 + 585.307i −0.205395 + 0.205395i
\(202\) 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 \(-0.556966\pi\)
−0.178011 + 0.984028i \(0.556966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −11870.1 −3.71334
\(218\) 0 0
\(219\) 2119.71 + 2119.71i 0.654049 + 0.654049i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4525.04 4525.04i 1.35883 1.35883i 0.483469 0.875362i \(-0.339377\pi\)
0.875362 0.483469i \(-0.160623\pi\)
\(224\) 0 0
\(225\) 3375.00i 1.00000i
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 2076.06 + 2076.06i 0.603029 + 0.603029i
\(229\) −4883.04 4883.04i −1.40908 1.40908i −0.764714 0.644370i \(-0.777120\pi\)
−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
\(241\) −4312.54 4312.54i −1.15268 1.15268i −0.986014 0.166662i \(-0.946701\pi\)
−0.166662 0.986014i \(-0.553299\pi\)
\(242\) 0 0
\(243\) −3788.00 −1.00000
\(244\) 7482.46i 1.96318i
\(245\) 0 0
\(246\) 0 0
\(247\) 3305.03 + 190.934i 0.851392 + 0.0491855i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −5527.11 + 5527.11i −1.38165 + 1.38165i
\(253\) 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
\(259\) 8354.04i 2.00423i
\(260\) 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
\(268\) −901.139 901.139i −0.205395 0.205395i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 4848.71 + 4848.71i 1.08686 + 1.08686i 0.995850 + 0.0910064i \(0.0290084\pi\)
0.0910064 + 0.995850i \(0.470992\pi\)
\(272\) 0 0
\(273\) −508.323 + 8798.98i −0.112693 + 1.95069i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8293.06i 1.79885i −0.437074 0.899425i \(-0.643985\pi\)
0.437074 0.899425i \(-0.356015\pi\)
\(278\) 0 0
\(279\) 6262.44 6262.44i 1.34381 1.34381i
\(280\) 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
\(291\) −108.559 + 108.559i −0.0218688 + 0.0218688i
\(292\) −3263.51 + 3263.51i −0.654049 + 0.654049i
\(293\) 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
\(301\) 5584.38 5584.38i 1.06936 1.06936i
\(302\) 0 0
\(303\) 0 0
\(304\) −3196.31 + 3196.31i −0.603029 + 0.603029i
\(305\) 0 0
\(306\) 0 0
\(307\) 6115.01 + 6115.01i 1.13681 + 1.13681i 0.989018 + 0.147797i \(0.0472182\pi\)
0.147797 + 0.989018i \(0.452782\pi\)
\(308\) 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 \(-0.640742\pi\)
−0.427888 + 0.903832i \(0.640742\pi\)
\(314\) 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
\(327\) −7348.36 + 7348.36i −1.24271 + 1.24271i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −6497.56 + 6497.56i −1.07897 + 1.07897i −0.0823644 + 0.996602i \(0.526247\pi\)
−0.996602 + 0.0823644i \(0.973753\pi\)
\(332\) 0 0
\(333\) −4407.44 4407.44i −0.725303 0.725303i
\(334\) 0 0
\(335\) 0 0
\(336\) −8509.54 8509.54i −1.38165 1.38165i
\(337\) 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
\(343\) 15955.4 15955.4i 2.51169 2.51169i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 3306.96 + 3306.96i 0.507214 + 0.507214i 0.913670 0.406456i \(-0.133236\pi\)
−0.406456 + 0.913670i \(0.633236\pi\)
\(350\) 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
\(361\) 1870.54i 0.272713i
\(362\) 0 0
\(363\) 6916.08i 1.00000i
\(364\) −13546.9 782.615i −1.95069 0.112693i
\(365\) 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
\(372\) 9641.66 + 9641.66i 1.34381 + 1.34381i
\(373\) 7420.11 1.03002 0.515011 0.857183i \(-0.327788\pi\)
0.515011 + 0.857183i \(0.327788\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −10293.6 + 10293.6i −1.39510 + 1.39510i −0.581702 + 0.813402i \(0.697613\pi\)
−0.813402 + 0.581702i \(0.802387\pi\)
\(380\) 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
\(388\) −167.137 167.137i −0.0218688 0.0218688i
\(389\) 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
\(397\) 7292.76 + 7292.76i 0.921947 + 0.921947i 0.997167 0.0752196i \(-0.0239658\pi\)
−0.0752196 + 0.997167i \(0.523966\pi\)
\(398\) 0 0
\(399\) 13280.8 1.66635
\(400\) 8000.00i 1.00000i
\(401\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(402\) 0 0
\(403\) 15349.2 + 886.735i 1.89727 + 0.109607i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 3047.69 3047.69i 0.368456 0.368456i −0.498458 0.866914i \(-0.666100\pi\)
0.866914 + 0.498458i \(0.166100\pi\)
\(410\) 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
\(421\) −7477.81 + 7477.81i −0.865668 + 0.865668i −0.991989 0.126322i \(-0.959683\pi\)
0.126322 + 0.991989i \(0.459683\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −23933.1 23933.1i −2.71242 2.71242i
\(428\) 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
\(436\) −11313.5 11313.5i −1.24271 1.24271i
\(437\) 0 0
\(438\) 0 0
\(439\) 10756.0i 1.16938i 0.811257 + 0.584690i \(0.198784\pi\)
−0.811257 + 0.584690i \(0.801216\pi\)
\(440\) 0 0
\(441\) 26096.5i 2.81790i
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 6785.69 6785.69i 0.725303 0.725303i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 13101.3 13101.3i 1.38165 1.38165i
\(449\) 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
\(453\) 3963.56 + 3963.56i 0.411091 + 0.411091i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1065.11 + 1065.11i −0.109023 + 0.109023i −0.759514 0.650491i \(-0.774563\pi\)
0.650491 + 0.759514i \(0.274563\pi\)
\(458\) 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
\(463\) −8689.69 8689.69i −0.872233 0.872233i 0.120482 0.992716i \(-0.461556\pi\)
−0.992716 + 0.120482i \(0.961556\pi\)
\(464\) 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
\(469\) −5764.69 −0.567566
\(470\) 0 0
\(471\) −20005.2 −1.95709
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −6242.79 6242.79i −0.603029 0.603029i
\(476\) 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
\(481\) 624.074 10802.6i 0.0591587 1.02403i
\(482\) 0 0
\(483\) 0 0
\(484\) 10648.0 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 12959.7 12959.7i 1.20588 1.20588i 0.233526 0.972351i \(-0.424974\pi\)
0.972351 0.233526i \(-0.0750265\pi\)
\(488\) 0 0
\(489\) 2596.46 + 2596.46i 0.240114 + 0.240114i
\(490\) 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
\(496\) −14844.3 + 14844.3i −1.34381 + 1.34381i
\(497\) 0 0
\(498\) 0 0
\(499\) 615.940 615.940i 0.0552570 0.0552570i −0.678938 0.734195i \(-0.737560\pi\)
0.734195 + 0.678938i \(0.237560\pi\)
\(500\) 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
\(511\) 20877.0i 1.80733i
\(512\) 0 0
\(513\) −7006.71 + 7006.71i −0.603029 + 0.603029i
\(514\) 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
\(525\) 16620.2 16620.2i 1.38165 1.38165i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 20447.1i 1.66635i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 16795.0 + 16795.0i 1.33470 + 1.33470i 0.901112 + 0.433586i \(0.142752\pi\)
0.433586 + 0.901112i \(0.357248\pi\)
\(542\) 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
\(553\) 27921.9 + 27921.9i 2.14713 + 2.14713i
\(554\) 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
\(567\) −18654.0 18654.0i −1.38165 1.38165i
\(568\) 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
\(577\) −19517.5 + 19517.5i −1.40819 + 1.40819i −0.638888 + 0.769300i \(0.720605\pi\)
−0.769300 + 0.638888i \(0.779395\pi\)
\(578\) 0 0
\(579\) −16595.8 16595.8i −1.19119 1.19119i
\(580\) 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 \(0.750000\pi\)
\(588\) −40178.2 −2.81790
\(589\) 23167.5i 1.62071i
\(590\) 0 0
\(591\) 0 0
\(592\) 10447.3 + 10447.3i 0.725303 + 0.725303i
\(593\) 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 \(-0.533741\pi\)
−0.105801 + 0.994387i \(0.533741\pi\)
\(602\) 0 0
\(603\) 3041.34 3041.34i 0.205395 0.205395i
\(604\) −6102.30 + 6102.30i −0.411091 + 0.411091i
\(605\) 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
\(613\) 21134.6 21134.6i 1.39253 1.39253i 0.572900 0.819625i \(-0.305818\pi\)
0.819625 0.572900i \(-0.194182\pi\)
\(614\) 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
\(619\) 5611.71 + 5611.71i 0.364384 + 0.364384i 0.865424 0.501040i \(-0.167049\pi\)
−0.501040 + 0.865424i \(0.667049\pi\)
\(620\) 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
\(631\) −16768.3 16768.3i −1.05790 1.05790i −0.998217 0.0596825i \(-0.980991\pi\)
−0.0596825 0.998217i \(-0.519009\pi\)
\(632\) 0 0
\(633\) 5670.00 0.356023
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −33828.8 + 30133.7i −2.10416 + 1.87432i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −21498.2 + 21498.2i −1.31851 + 1.31851i −0.403561 + 0.914953i \(0.632228\pi\)
−0.914953 + 0.403561i \(0.867772\pi\)
\(644\) 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
\(651\) 61678.8 3.71334
\(652\) −3997.51 + 3997.51i −0.240114 + 0.240114i
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −11014.3 11014.3i −0.654049 0.654049i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 3320.96 + 3320.96i 0.195416 + 0.195416i 0.798032 0.602615i \(-0.205875\pi\)
−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
\(669\) −23512.8 + 23512.8i −1.35883 + 1.35883i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 25315.7i 1.45000i −0.688751 0.724998i \(-0.741841\pi\)
0.688751 0.724998i \(-0.258159\pi\)
\(674\) 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
\(679\) −1069.19 −0.0604299
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) −10787.5 10787.5i −0.603029 0.603029i
\(685\) 0 0
\(686\) 0 0
\(687\) 25373.0 + 25373.0i 1.40908 + 1.40908i
\(688\) 13967.3i 0.773978i
\(689\) 0 0
\(690\) 0 0
\(691\) 8253.94 8253.94i 0.454406 0.454406i −0.442408 0.896814i \(-0.645876\pi\)
0.896814 + 0.442408i \(0.145876\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 25588.5 + 25588.5i 1.38165 + 1.38165i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −16305.0 −0.874758
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 23529.0 + 23529.0i 1.24633 + 1.24633i 0.957328 + 0.289003i \(0.0933236\pi\)
0.289003 + 0.957328i \(0.406676\pi\)
\(710\) 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
\(721\) 26326.4 26326.4i 1.35984 1.35984i
\(722\) 0 0
\(723\) 22408.6 + 22408.6i 1.15268 + 1.15268i
\(724\) −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
\(733\) −10838.2 + 10838.2i −0.546137 + 0.546137i −0.925321 0.379184i \(-0.876205\pi\)
0.379184 + 0.925321i \(0.376205\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −3139.29 3139.29i −0.156266 0.156266i 0.624644 0.780910i \(-0.285244\pi\)
−0.780910 + 0.624644i \(0.785244\pi\)
\(740\) 0 0
\(741\) −17173.4 992.120i −0.851392 0.0491855i
\(742\) 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 \(-0.692963\pi\)
0.569757 0.821813i \(-0.307037\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 28719.7 28719.7i 1.38165 1.38165i
\(757\) −3928.29 −0.188608 −0.0943039 0.995543i \(-0.530063\pi\)
−0.0943039 + 0.995543i \(0.530063\pi\)
\(758\) 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
\(763\) −72374.0 −3.43396
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −21283.4 −1.00000
\(769\) −23503.3 + 23503.3i −1.10215 + 1.10215i −0.107995 + 0.994151i \(0.534443\pi\)
−0.994151 + 0.107995i \(0.965557\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 25550.9 25550.9i 1.19119 1.19119i
\(773\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 0 0
\(775\) −28992.8 28992.8i −1.34381 1.34381i
\(776\) 0 0
\(777\) 43408.9i 2.00423i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 61858.5i 2.81790i
\(785\) 0 0
\(786\) 0 0
\(787\) 17631.4 + 17631.4i 0.798592 + 0.798592i 0.982874 0.184281i \(-0.0589958\pi\)
−0.184281 + 0.982874i \(0.558996\pi\)
\(788\) 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
\(802\)