Properties

Label 507.2.f.c.437.2
Level $507$
Weight $2$
Character 507.437
Analytic conductor $4.048$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 437.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 507.437
Dual form 507.2.f.c.239.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205 q^{3} +2.00000i q^{4} +(3.09808 - 3.09808i) q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{3} +2.00000i q^{4} +(3.09808 - 3.09808i) q^{7} +3.00000 q^{9} +3.46410i q^{12} -4.00000 q^{16} +(-2.26795 - 2.26795i) q^{19} +(5.36603 - 5.36603i) q^{21} +5.00000i q^{25} +5.19615 q^{27} +(6.19615 + 6.19615i) q^{28} +(0.830127 + 0.830127i) q^{31} +6.00000i q^{36} +(-8.46410 + 8.46410i) q^{37} -1.73205i q^{43} -6.92820 q^{48} -12.1962i q^{49} +(-3.92820 - 3.92820i) q^{57} +8.66025 q^{61} +(9.29423 - 9.29423i) q^{63} -8.00000i q^{64} +(-11.5622 - 11.5622i) q^{67} +(-7.63397 + 7.63397i) q^{73} +8.66025i q^{75} +(4.53590 - 4.53590i) q^{76} -12.1244 q^{79} +9.00000 q^{81} +(10.7321 + 10.7321i) q^{84} +(1.43782 + 1.43782i) q^{93} +(-7.02628 - 7.02628i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{7} + 12q^{9} + O(q^{10}) \) \( 4q + 2q^{7} + 12q^{9} - 16q^{16} - 16q^{19} + 18q^{21} + 4q^{28} - 14q^{31} - 20q^{37} + 12q^{57} + 6q^{63} - 22q^{67} - 34q^{73} + 32q^{76} + 36q^{81} + 36q^{84} + 30q^{93} + 10q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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 \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 1.73205 1.00000
\(4\) 2.00000i 1.00000i
\(5\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(6\) 0 0
\(7\) 3.09808 3.09808i 1.17096 1.17096i 0.188982 0.981981i \(-0.439481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) 3.46410i 1.00000i
\(13\) 0 0
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −2.26795 2.26795i −0.520303 0.520303i 0.397360 0.917663i \(-0.369927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 5.36603 5.36603i 1.17096 1.17096i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 5.00000i 1.00000i
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 6.19615 + 6.19615i 1.17096 + 1.17096i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0.830127 + 0.830127i 0.149095 + 0.149095i 0.777714 0.628619i \(-0.216379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 6.00000i 1.00000i
\(37\) −8.46410 + 8.46410i −1.39149 + 1.39149i −0.569495 + 0.821995i \(0.692861\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(42\) 0 0
\(43\) 1.73205i 0.264135i −0.991241 0.132068i \(-0.957838\pi\)
0.991241 0.132068i \(-0.0421616\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) −6.92820 −1.00000
\(49\) 12.1962i 1.74231i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.92820 3.92820i −0.520303 0.520303i
\(58\) 0 0
\(59\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) 8.66025 1.10883 0.554416 0.832240i \(-0.312942\pi\)
0.554416 + 0.832240i \(0.312942\pi\)
\(62\) 0 0
\(63\) 9.29423 9.29423i 1.17096 1.17096i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) −11.5622 11.5622i −1.41254 1.41254i −0.740613 0.671932i \(-0.765465\pi\)
−0.671932 0.740613i \(-0.734535\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(72\) 0 0
\(73\) −7.63397 + 7.63397i −0.893489 + 0.893489i −0.994850 0.101361i \(-0.967680\pi\)
0.101361 + 0.994850i \(0.467680\pi\)
\(74\) 0 0
\(75\) 8.66025i 1.00000i
\(76\) 4.53590 4.53590i 0.520303 0.520303i
\(77\) 0 0
\(78\) 0 0
\(79\) −12.1244 −1.36410 −0.682048 0.731307i \(-0.738911\pi\)
−0.682048 + 0.731307i \(0.738911\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 10.7321 + 10.7321i 1.17096 + 1.17096i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.43782 + 1.43782i 0.149095 + 0.149095i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −7.02628 7.02628i −0.713411 0.713411i 0.253837 0.967247i \(-0.418307\pi\)
−0.967247 + 0.253837i \(0.918307\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −10.0000 −1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 15.5885i 1.53598i −0.640464 0.767988i \(-0.721258\pi\)
0.640464 0.767988i \(-0.278742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 10.3923i 1.00000i
\(109\) 13.8301 + 13.8301i 1.32469 + 1.32469i 0.909935 + 0.414751i \(0.136131\pi\)
0.414751 + 0.909935i \(0.363869\pi\)
\(110\) 0 0
\(111\) −14.6603 + 14.6603i −1.39149 + 1.39149i
\(112\) −12.3923 + 12.3923i −1.17096 + 1.17096i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000i 1.00000i
\(122\) 0 0
\(123\) 0 0
\(124\) −1.66025 + 1.66025i −0.149095 + 0.149095i
\(125\) 0 0
\(126\) 0 0
\(127\) 1.00000i 0.0887357i −0.999015 0.0443678i \(-0.985873\pi\)
0.999015 0.0443678i \(-0.0141274\pi\)
\(128\) 0 0
\(129\) 3.00000i 0.264135i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −14.0526 −1.21851
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(138\) 0 0
\(139\) −7.00000 −0.593732 −0.296866 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −12.0000 −1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 21.1244i 1.74231i
\(148\) −16.9282 16.9282i −1.39149 1.39149i
\(149\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(150\) 0 0
\(151\) 10.1244 10.1244i 0.823908 0.823908i −0.162758 0.986666i \(-0.552039\pi\)
0.986666 + 0.162758i \(0.0520389\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −11.0000 −0.877896 −0.438948 0.898513i \(-0.644649\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −9.90192 + 9.90192i −0.775579 + 0.775579i −0.979076 0.203497i \(-0.934769\pi\)
0.203497 + 0.979076i \(0.434769\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −6.80385 6.80385i −0.520303 0.520303i
\(172\) 3.46410 0.264135
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 15.4904 + 15.4904i 1.17096 + 1.17096i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 6.92820i 0.514969i 0.966282 + 0.257485i \(0.0828937\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) 15.0000 1.10883
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 16.0981 16.0981i 1.17096 1.17096i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 13.8564i 1.00000i
\(193\) −3.70577 + 3.70577i −0.266747 + 0.266747i −0.827788 0.561041i \(-0.810401\pi\)
0.561041 + 0.827788i \(0.310401\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 24.3923 1.74231
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) 17.0000i 1.20510i 0.798082 + 0.602549i \(0.205848\pi\)
−0.798082 + 0.602549i \(0.794152\pi\)
\(200\) 0 0
\(201\) −20.0263 20.0263i −1.41254 1.41254i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 25.9808 1.78859 0.894295 0.447478i \(-0.147678\pi\)
0.894295 + 0.447478i \(0.147678\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.14359 0.349170
\(218\) 0 0
\(219\) −13.2224 + 13.2224i −0.893489 + 0.893489i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 19.1962 + 19.1962i 1.28547 + 1.28547i 0.937509 + 0.347960i \(0.113126\pi\)
0.347960 + 0.937509i \(0.386874\pi\)
\(224\) 0 0
\(225\) 15.0000i 1.00000i
\(226\) 0 0
\(227\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(228\) 7.85641 7.85641i 0.520303 0.520303i
\(229\) 0.607695 0.607695i 0.0401576 0.0401576i −0.686743 0.726900i \(-0.740960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 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
\(237\) −21.0000 −1.36410
\(238\) 0 0
\(239\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(240\) 0 0
\(241\) 20.8564 20.8564i 1.34348 1.34348i 0.450910 0.892570i \(-0.351100\pi\)
0.892570 0.450910i \(-0.148900\pi\)
\(242\) 0 0
\(243\) 15.5885 1.00000
\(244\) 17.3205i 1.10883i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 18.5885 + 18.5885i 1.17096 + 1.17096i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 52.4449i 3.25877i
\(260\) 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
\(268\) 23.1244 23.1244i 1.41254 1.41254i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 22.2942 22.2942i 1.35428 1.35428i 0.473466 0.880812i \(-0.343003\pi\)
0.880812 0.473466i \(-0.156997\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 20.7846i 1.24883i 0.781094 + 0.624413i \(0.214662\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 0 0
\(279\) 2.49038 + 2.49038i 0.149095 + 0.149095i
\(280\) 0 0
\(281\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(282\) 0 0
\(283\) 25.0000i 1.48610i 0.669238 + 0.743048i \(0.266621\pi\)
−0.669238 + 0.743048i \(0.733379\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −12.1699 12.1699i −0.713411 0.713411i
\(292\) −15.2679 15.2679i −0.893489 0.893489i
\(293\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −17.3205 −1.00000
\(301\) −5.36603 5.36603i −0.309293 0.309293i
\(302\) 0 0
\(303\) 0 0
\(304\) 9.07180 + 9.07180i 0.520303 + 0.520303i
\(305\) 0 0
\(306\) 0 0
\(307\) 18.3660 18.3660i 1.04820 1.04820i 0.0494267 0.998778i \(-0.484261\pi\)
0.998778 0.0494267i \(-0.0157394\pi\)
\(308\) 0 0
\(309\) 27.0000i 1.53598i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 32.9090 1.86012 0.930062 0.367402i \(-0.119753\pi\)
0.930062 + 0.367402i \(0.119753\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 24.2487i 1.36410i
\(317\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 18.0000i 1.00000i
\(325\) 0 0
\(326\) 0 0
\(327\) 23.9545 + 23.9545i 1.32469 + 1.32469i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5.97372 + 5.97372i 0.328345 + 0.328345i 0.851957 0.523612i \(-0.175416\pi\)
−0.523612 + 0.851957i \(0.675416\pi\)
\(332\) 0 0
\(333\) −25.3923 + 25.3923i −1.39149 + 1.39149i
\(334\) 0 0
\(335\) 0 0
\(336\) −21.4641 + 21.4641i −1.17096 + 1.17096i
\(337\) 29.0000i 1.57973i −0.613280 0.789865i \(-0.710150\pi\)
0.613280 0.789865i \(-0.289850\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −16.0981 16.0981i −0.869214 0.869214i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −26.2224 + 26.2224i −1.40365 + 1.40365i −0.615581 + 0.788074i \(0.711079\pi\)
−0.788074 + 0.615581i \(0.788921\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(360\) 0 0
\(361\) 8.71281i 0.458569i
\(362\) 0 0
\(363\) 19.0526i 1.00000i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 31.0000 1.61819 0.809093 0.587680i \(-0.199959\pi\)
0.809093 + 0.587680i \(0.199959\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −2.87564 + 2.87564i −0.149095 + 0.149095i
\(373\) −36.3731 −1.88333 −0.941663 0.336557i \(-0.890737\pi\)
−0.941663 + 0.336557i \(0.890737\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −24.5622 24.5622i −1.26167 1.26167i −0.950281 0.311393i \(-0.899204\pi\)
−0.311393 0.950281i \(-0.600796\pi\)
\(380\) 0 0
\(381\) 1.73205i 0.0887357i
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.19615i 0.264135i
\(388\) 14.0526 14.0526i 0.713411 0.713411i
\(389\) 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
\(397\) −19.4186 + 19.4186i −0.974591 + 0.974591i −0.999685 0.0250943i \(-0.992011\pi\)
0.0250943 + 0.999685i \(0.492011\pi\)
\(398\) 0 0
\(399\) −24.3397 −1.21851
\(400\) 20.0000i 1.00000i
\(401\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 28.4904 + 28.4904i 1.40876 + 1.40876i 0.766426 + 0.642333i \(0.222033\pi\)
0.642333 + 0.766426i \(0.277967\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 31.1769 1.53598
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −12.1244 −0.593732
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −8.68653 8.68653i −0.423356 0.423356i 0.463002 0.886357i \(-0.346772\pi\)
−0.886357 + 0.463002i \(0.846772\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 26.8301 26.8301i 1.29840 1.29840i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(432\) −20.7846 −1.00000
\(433\) 35.0000i 1.68199i −0.541041 0.840996i \(-0.681970\pi\)
0.541041 0.840996i \(-0.318030\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −27.6603 + 27.6603i −1.32469 + 1.32469i
\(437\) 0 0
\(438\) 0 0
\(439\) 39.8372i 1.90132i 0.310228 + 0.950662i \(0.399595\pi\)
−0.310228 + 0.950662i \(0.600405\pi\)
\(440\) 0 0
\(441\) 36.5885i 1.74231i
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −29.3205 29.3205i −1.39149 1.39149i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −24.7846 24.7846i −1.17096 1.17096i
\(449\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 17.5359 17.5359i 0.823908 0.823908i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 14.4378 + 14.4378i 0.675373 + 0.675373i 0.958950 0.283577i \(-0.0915211\pi\)
−0.283577 + 0.958950i \(0.591521\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(462\) 0 0
\(463\) −20.6340 + 20.6340i −0.958942 + 0.958942i −0.999190 0.0402476i \(-0.987185\pi\)
0.0402476 + 0.999190i \(0.487185\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −71.6410 −3.30807
\(470\) 0 0
\(471\) −19.0526 −0.877896
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 11.3397 11.3397i 0.520303 0.520303i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 23.7321 + 23.7321i 1.07540 + 1.07540i 0.996915 + 0.0784867i \(0.0250088\pi\)
0.0784867 + 0.996915i \(0.474991\pi\)
\(488\) 0 0
\(489\) −17.1506 + 17.1506i −0.775579 + 0.775579i
\(490\) 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
\(496\) −3.32051 3.32051i −0.149095 0.149095i
\(497\) 0 0
\(498\) 0 0
\(499\) 31.5885 + 31.5885i 1.41409 + 1.41409i 0.716258 + 0.697835i \(0.245853\pi\)
0.697835 + 0.716258i \(0.254147\pi\)
\(500\) 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
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(510\) 0 0
\(511\) 47.3013i 2.09248i
\(512\) 0 0
\(513\) −11.7846 11.7846i −0.520303 0.520303i
\(514\) 0 0
\(515\) 0 0
\(516\) 6.00000 0.264135
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 0 0
\(525\) 26.8301 + 26.8301i 1.17096 + 1.17096i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 28.1051i 1.21851i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −30.1506 + 30.1506i −1.29628 + 1.29628i −0.365444 + 0.930834i \(0.619083\pi\)
−0.930834 + 0.365444i \(0.880917\pi\)
\(542\) 0 0
\(543\) 12.0000i 0.514969i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 41.0000 1.75303 0.876517 0.481371i \(-0.159861\pi\)
0.876517 + 0.481371i \(0.159861\pi\)
\(548\) 0 0
\(549\) 25.9808 1.10883
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −37.5622 + 37.5622i −1.59731 + 1.59731i
\(554\) 0 0
\(555\) 0 0
\(556\) 14.0000i 0.593732i
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 27.8827 27.8827i 1.17096 1.17096i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 16.0000i 0.669579i −0.942293 0.334790i \(-0.891335\pi\)
0.942293 0.334790i \(-0.108665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 24.0000i 1.00000i
\(577\) −29.9282 29.9282i −1.24593 1.24593i −0.957503 0.288425i \(-0.906868\pi\)
−0.288425 0.957503i \(-0.593132\pi\)
\(578\) 0 0
\(579\) −6.41858 + 6.41858i −0.266747 + 0.266747i
\(580\) 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 \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 42.2487 1.74231
\(589\) 3.76537i 0.155149i
\(590\) 0 0
\(591\) 0 0
\(592\) 33.8564 33.8564i 1.39149 1.39149i
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 29.4449i 1.20510i
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −41.5692 −1.69564 −0.847822 0.530281i \(-0.822086\pi\)
−0.847822 + 0.530281i \(0.822086\pi\)
\(602\) 0 0
\(603\) −34.6865 34.6865i −1.41254 1.41254i
\(604\) 20.2487 + 20.2487i 0.823908 + 0.823908i
\(605\) 0 0
\(606\) 0 0
\(607\) −20.0000 −0.811775 −0.405887 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −2.04552 2.04552i −0.0826177 0.0826177i 0.664590 0.747208i \(-0.268606\pi\)
−0.747208 + 0.664590i \(0.768606\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(618\) 0 0
\(619\) 14.8827 14.8827i 0.598186 0.598186i −0.341644 0.939829i \(-0.610984\pi\)
0.939829 + 0.341644i \(0.110984\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 22.0000i 0.877896i
\(629\) 0 0
\(630\) 0 0
\(631\) 25.6147 25.6147i 1.01971 1.01971i 0.0199047 0.999802i \(-0.493664\pi\)
0.999802 0.0199047i \(-0.00633628\pi\)
\(632\) 0 0
\(633\) 45.0000 1.78859
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −33.0263 33.0263i −1.30243 1.30243i −0.926750 0.375680i \(-0.877409\pi\)
−0.375680 0.926750i \(-0.622591\pi\)
\(644\) 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
\(651\) 8.90897 0.349170
\(652\) −19.8038 19.8038i −0.775579 0.775579i
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −22.9019 + 22.9019i −0.893489 + 0.893489i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −16.7058 + 16.7058i −0.649779 + 0.649779i −0.952940 0.303160i \(-0.901958\pi\)
0.303160 + 0.952940i \(0.401958\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 33.2487 + 33.2487i 1.28547 + 1.28547i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 50.2295i 1.93620i −0.250557 0.968102i \(-0.580614\pi\)
0.250557 0.968102i \(-0.419386\pi\)
\(674\) 0 0
\(675\) 25.9808i 1.00000i
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −43.5359 −1.67075
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(684\) 13.6077 13.6077i 0.520303 0.520303i
\(685\) 0 0
\(686\) 0 0
\(687\) 1.05256 1.05256i 0.0401576 0.0401576i
\(688\) 6.92820i 0.264135i
\(689\) 0 0
\(690\) 0 0
\(691\) 36.9545 + 36.9545i 1.40581 + 1.40581i 0.779857 + 0.625958i \(0.215292\pi\)
0.625958 + 0.779857i \(0.284708\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −30.9808 + 30.9808i −1.17096 + 1.17096i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 38.3923 1.44799
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 29.0981 29.0981i 1.09280 1.09280i 0.0975728 0.995228i \(-0.468892\pi\)
0.995228 0.0975728i \(-0.0311079\pi\)
\(710\) 0 0
\(711\) −36.3731 −1.36410
\(712\) 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
\(721\) −48.2942 48.2942i −1.79857 1.79857i
\(722\) 0 0
\(723\) 36.1244 36.1244i 1.34348 1.34348i
\(724\) −13.8564 −0.514969
\(725\) 0 0
\(726\) 0 0
\(727\) 49.0000i 1.81731i 0.417548 + 0.908655i \(0.362889\pi\)
−0.417548 + 0.908655i \(0.637111\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 30.0000i 1.10883i
\(733\) 23.3468 + 23.3468i 0.862333 + 0.862333i 0.991609 0.129275i \(-0.0412651\pi\)
−0.129275 + 0.991609i \(0.541265\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −17.9808 + 17.9808i −0.661433 + 0.661433i −0.955718 0.294285i \(-0.904919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 17.3205i 0.632034i 0.948753 + 0.316017i \(0.102346\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 32.1962 + 32.1962i 1.17096 + 1.17096i
\(757\) 48.4974 1.76267 0.881334 0.472493i \(-0.156646\pi\)
0.881334 + 0.472493i \(0.156646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(762\) 0 0
\(763\) 85.6936 3.10232
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 27.7128 1.00000
\(769\) 28.7128 + 28.7128i 1.03541 + 1.03541i 0.999350 + 0.0360609i \(0.0114810\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.41154 7.41154i −0.266747 0.266747i
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0 0
\(775\) −4.15064 + 4.15064i −0.149095 + 0.149095i
\(776\) 0 0
\(777\) 90.8372i 3.25877i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 48.7846i 1.74231i
\(785\) 0 0
\(786\) 0 0
\(787\) 12.6147 12.6147i 0.449667 0.449667i −0.445577 0.895244i \(-0.647001\pi\)
0.895244 + 0.445577i \(0.147001\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −34.0000 −1.20510
\(797\) 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 0
\(803\) 0 0
\(804\) 40.0526 40.0526i 1.41254 1.41254i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0