Properties

Label 507.2.f.c.239.1
Level $507$
Weight $2$
Character 507.239
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 239.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 507.239
Dual form 507.2.f.c.437.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.00000i q^{4} +(-2.09808 - 2.09808i) q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.00000i q^{4} +(-2.09808 - 2.09808i) q^{7} +3.00000 q^{9} +3.46410i q^{12} -4.00000 q^{16} +(-5.73205 + 5.73205i) q^{19} +(3.63397 + 3.63397i) q^{21} -5.00000i q^{25} -5.19615 q^{27} +(-4.19615 + 4.19615i) q^{28} +(-7.83013 + 7.83013i) q^{31} -6.00000i q^{36} +(-1.53590 - 1.53590i) q^{37} -1.73205i q^{43} +6.92820 q^{48} +1.80385i q^{49} +(9.92820 - 9.92820i) q^{57} -8.66025 q^{61} +(-6.29423 - 6.29423i) q^{63} +8.00000i q^{64} +(0.562178 - 0.562178i) q^{67} +(-9.36603 - 9.36603i) q^{73} +8.66025i q^{75} +(11.4641 + 11.4641i) q^{76} +12.1244 q^{79} +9.00000 q^{81} +(7.26795 - 7.26795i) q^{84} +(13.5622 - 13.5622i) q^{93} +(12.0263 - 12.0263i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{7} + 12q^{9} + O(q^{10}) \) \( 4q + 2q^{7} + 12q^{9} - 16q^{16} - 16q^{19} + 18q^{21} + 4q^{28} - 14q^{31} - 20q^{37} + 12q^{57} + 6q^{63} - 22q^{67} - 34q^{73} + 32q^{76} + 36q^{81} + 36q^{84} + 30q^{93} + 10q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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 \(0.750000\pi\)
\(3\) −1.73205 −1.00000
\(4\) 2.00000i 1.00000i
\(5\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0 0
\(7\) −2.09808 2.09808i −0.792998 0.792998i 0.188982 0.981981i \(-0.439481\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(12\) 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
\(19\) −5.73205 + 5.73205i −1.31502 + 1.31502i −0.397360 + 0.917663i \(0.630073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 3.63397 + 3.63397i 0.792998 + 0.792998i
\(22\) 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
\(28\) −4.19615 + 4.19615i −0.792998 + 0.792998i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −7.83013 + 7.83013i −1.40633 + 1.40633i −0.628619 + 0.777714i \(0.716379\pi\)
−0.777714 + 0.628619i \(0.783621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 6.00000i 1.00000i
\(37\) −1.53590 1.53590i −0.252500 0.252500i 0.569495 0.821995i \(-0.307139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(42\) 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 \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) 6.92820 1.00000
\(49\) 1.80385i 0.257693i
\(50\) 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
\(57\) 9.92820 9.92820i 1.31502 1.31502i
\(58\) 0 0
\(59\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(60\) 0 0
\(61\) −8.66025 −1.10883 −0.554416 0.832240i \(-0.687058\pi\)
−0.554416 + 0.832240i \(0.687058\pi\)
\(62\) 0 0
\(63\) −6.29423 6.29423i −0.792998 0.792998i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.562178 0.562178i 0.0686810 0.0686810i −0.671932 0.740613i \(-0.734535\pi\)
0.740613 + 0.671932i \(0.234535\pi\)
\(68\) 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
\(73\) −9.36603 9.36603i −1.09621 1.09621i −0.994850 0.101361i \(-0.967680\pi\)
−0.101361 0.994850i \(-0.532320\pi\)
\(74\) 0 0
\(75\) 8.66025i 1.00000i
\(76\) 11.4641 + 11.4641i 1.31502 + 1.31502i
\(77\) 0 0
\(78\) 0 0
\(79\) 12.1244 1.36410 0.682048 0.731307i \(-0.261089\pi\)
0.682048 + 0.731307i \(0.261089\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 7.26795 7.26795i 0.792998 0.792998i
\(85\) 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
\(91\) 0 0
\(92\) 0 0
\(93\) 13.5622 13.5622i 1.40633 1.40633i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0263 12.0263i 1.22108 1.22108i 0.253837 0.967247i \(-0.418307\pi\)
0.967247 0.253837i \(-0.0816925\pi\)
\(98\) 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
\(109\) 5.16987 5.16987i 0.495184 0.495184i −0.414751 0.909935i \(-0.636131\pi\)
0.909935 + 0.414751i \(0.136131\pi\)
\(110\) 0 0
\(111\) 2.66025 + 2.66025i 0.252500 + 0.252500i
\(112\) 8.39230 + 8.39230i 0.792998 + 0.792998i
\(113\) 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
\(124\) 15.6603 + 15.6603i 1.40633 + 1.40633i
\(125\) 0 0
\(126\) 0 0
\(127\) 1.00000i 0.0887357i 0.999015 + 0.0443678i \(0.0141274\pi\)
−0.999015 + 0.0443678i \(0.985873\pi\)
\(128\) 0 0
\(129\) 3.00000i 0.264135i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 24.0526 2.08562
\(134\) 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
\(139\) −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
\(147\) 3.12436i 0.257693i
\(148\) −3.07180 + 3.07180i −0.252500 + 0.252500i
\(149\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(150\) 0 0
\(151\) −14.1244 14.1244i −1.14942 1.14942i −0.986666 0.162758i \(-0.947961\pi\)
−0.162758 0.986666i \(-0.552039\pi\)
\(152\) 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
\(163\) −15.0981 15.0981i −1.18257 1.18257i −0.979076 0.203497i \(-0.934769\pi\)
−0.203497 0.979076i \(-0.565231\pi\)
\(164\) 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
\(169\) 0 0
\(170\) 0 0
\(171\) −17.1962 + 17.1962i −1.31502 + 1.31502i
\(172\) −3.46410 −0.264135
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −10.4904 + 10.4904i −0.792998 + 0.792998i
\(176\) 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
\(189\) 10.9019 + 10.9019i 0.792998 + 0.792998i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 13.8564i 1.00000i
\(193\) −19.2942 19.2942i −1.38883 1.38883i −0.827788 0.561041i \(-0.810401\pi\)
−0.561041 0.827788i \(-0.689599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 3.60770 0.257693
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0 0
\(199\) 17.0000i 1.20510i −0.798082 0.602549i \(-0.794152\pi\)
0.798082 0.602549i \(-0.205848\pi\)
\(200\) 0 0
\(201\) −0.973721 + 0.973721i −0.0686810 + 0.0686810i
\(202\) 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 \(-0.852322\pi\)
−0.894295 + 0.447478i \(0.852322\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 32.8564 2.23044
\(218\) 0 0
\(219\) 16.2224 + 16.2224i 1.09621 + 1.09621i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 8.80385 8.80385i 0.589549 0.589549i −0.347960 0.937509i \(-0.613126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 0 0
\(225\) 15.0000i 1.00000i
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) −19.8564 19.8564i −1.31502 1.31502i
\(229\) 21.3923 + 21.3923i 1.41364 + 1.41364i 0.726900 + 0.686743i \(0.240960\pi\)
0.686743 + 0.726900i \(0.259040\pi\)
\(230\) 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 \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(240\) 0 0
\(241\) −6.85641 6.85641i −0.441660 0.441660i 0.450910 0.892570i \(-0.351100\pi\)
−0.892570 + 0.450910i \(0.851100\pi\)
\(242\) 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 \(0.5\pi\)
\(252\) −12.5885 + 12.5885i −0.792998 + 0.792998i
\(253\) 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
\(259\) 6.44486i 0.400464i
\(260\) 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
\(268\) −1.12436 1.12436i −0.0686810 0.0686810i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 6.70577 + 6.70577i 0.407347 + 0.407347i 0.880812 0.473466i \(-0.156997\pi\)
−0.473466 + 0.880812i \(0.656997\pi\)
\(272\) 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
\(279\) −23.4904 + 23.4904i −1.40633 + 1.40633i
\(280\) 0 0
\(281\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(282\) 0 0
\(283\) 25.0000i 1.48610i −0.669238 0.743048i \(-0.733379\pi\)
0.669238 0.743048i \(-0.266621\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −20.8301 + 20.8301i −1.22108 + 1.22108i
\(292\) −18.7321 + 18.7321i −1.09621 + 1.09621i
\(293\) 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
\(300\) 17.3205 1.00000
\(301\) −3.63397 + 3.63397i −0.209459 + 0.209459i
\(302\) 0 0
\(303\) 0 0
\(304\) 22.9282 22.9282i 1.31502 1.31502i
\(305\) 0 0
\(306\) 0 0
\(307\) 16.6340 + 16.6340i 0.949351 + 0.949351i 0.998778 0.0494267i \(-0.0157394\pi\)
−0.0494267 + 0.998778i \(0.515739\pi\)
\(308\) 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 \(-0.880247\pi\)
−0.930062 + 0.367402i \(0.880247\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 24.2487i 1.36410i
\(317\) 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
\(324\) 18.0000i 1.00000i
\(325\) 0 0
\(326\) 0 0
\(327\) −8.95448 + 8.95448i −0.495184 + 0.495184i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 25.0263 25.0263i 1.37557 1.37557i 0.523612 0.851957i \(-0.324584\pi\)
0.851957 0.523612i \(-0.175416\pi\)
\(332\) 0 0
\(333\) −4.60770 4.60770i −0.252500 0.252500i
\(334\) 0 0
\(335\) 0 0
\(336\) −14.5359 14.5359i −0.792998 0.792998i
\(337\) 29.0000i 1.57973i 0.613280 + 0.789865i \(0.289850\pi\)
−0.613280 + 0.789865i \(0.710150\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −10.9019 + 10.9019i −0.588649 + 0.588649i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 3.22243 + 3.22243i 0.172493 + 0.172493i 0.788074 0.615581i \(-0.211079\pi\)
−0.615581 + 0.788074i \(0.711079\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 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
\(361\) 46.7128i 2.45857i
\(362\) 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
\(372\) −27.1244 27.1244i −1.40633 1.40633i
\(373\) 36.3731 1.88333 0.941663 0.336557i \(-0.109263\pi\)
0.941663 + 0.336557i \(0.109263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −12.4378 + 12.4378i −0.638888 + 0.638888i −0.950281 0.311393i \(-0.899204\pi\)
0.311393 + 0.950281i \(0.399204\pi\)
\(380\) 0 0
\(381\) 1.73205i 0.0887357i
\(382\) 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
\(387\) 5.19615i 0.264135i
\(388\) −24.0526 24.0526i −1.22108 1.22108i
\(389\) 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
\(397\) 20.4186 + 20.4186i 1.02478 + 1.02478i 0.999685 + 0.0250943i \(0.00798860\pi\)
0.0250943 + 0.999685i \(0.492011\pi\)
\(398\) 0 0
\(399\) −41.6603 −2.08562
\(400\) 20.0000i 1.00000i
\(401\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.50962 2.50962i 0.124093 0.124093i −0.642333 0.766426i \(-0.722033\pi\)
0.766426 + 0.642333i \(0.222033\pi\)
\(410\) 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
\(421\) 27.6865 27.6865i 1.34936 1.34936i 0.463002 0.886357i \(-0.346772\pi\)
0.886357 0.463002i \(-0.153228\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 18.1699 + 18.1699i 0.879302 + 0.879302i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(432\) 20.7846 1.00000
\(433\) 35.0000i 1.68199i 0.541041 + 0.840996i \(0.318030\pi\)
−0.541041 + 0.840996i \(0.681970\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.3397 10.3397i −0.495184 0.495184i
\(437\) 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
\(441\) 5.41154i 0.257693i
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 5.32051 5.32051i 0.252500 0.252500i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 16.7846 16.7846i 0.792998 0.792998i
\(449\) 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
\(453\) 24.4641 + 24.4641i 1.14942 + 1.14942i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.5622 26.5622i 1.24253 1.24253i 0.283577 0.958950i \(-0.408479\pi\)
0.958950 0.283577i \(-0.0915211\pi\)
\(458\) 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
\(463\) −22.3660 22.3660i −1.03944 1.03944i −0.999190 0.0402476i \(-0.987185\pi\)
−0.0402476 0.999190i \(-0.512815\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −2.35898 −0.108928
\(470\) 0 0
\(471\) 19.0526 0.877896
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 28.6603 + 28.6603i 1.31502 + 1.31502i
\(476\) 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
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 20.2679 20.2679i 0.918428 0.918428i −0.0784867 0.996915i \(-0.525009\pi\)
0.996915 + 0.0784867i \(0.0250088\pi\)
\(488\) 0 0
\(489\) 26.1506 + 26.1506i 1.18257 + 1.18257i
\(490\) 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
\(496\) 31.3205 31.3205i 1.40633 1.40633i
\(497\) 0 0
\(498\) 0 0
\(499\) 0.411543 0.411543i 0.0184232 0.0184232i −0.697835 0.716258i \(-0.745853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 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 \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(510\) 0 0
\(511\) 39.3013i 1.73859i
\(512\) 0 0
\(513\) 29.7846 29.7846i 1.31502 1.31502i
\(514\) 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
\(525\) 18.1699 18.1699i 0.792998 0.792998i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 48.1051i 2.08562i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 13.1506 + 13.1506i 0.565390 + 0.565390i 0.930834 0.365444i \(-0.119083\pi\)
−0.365444 + 0.930834i \(0.619083\pi\)
\(542\) 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
\(553\) −25.4378 25.4378i −1.08173 1.08173i
\(554\) 0 0
\(555\) 0 0
\(556\) 14.0000i 0.593732i
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 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
\(567\) −18.8827 18.8827i −0.792998 0.792998i
\(568\) 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 \(0.108665\pi\)
−0.942293 + 0.334790i \(0.891335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 24.0000i 1.00000i
\(577\) −16.0718 + 16.0718i −0.669078 + 0.669078i −0.957503 0.288425i \(-0.906868\pi\)
0.288425 + 0.957503i \(0.406868\pi\)
\(578\) 0 0
\(579\) 33.4186 + 33.4186i 1.38883 + 1.38883i
\(580\) 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 \(0.750000\pi\)
\(588\) −6.24871 −0.257693
\(589\) 89.7654i 3.69872i
\(590\) 0 0
\(591\) 0 0
\(592\) 6.14359 + 6.14359i 0.252500 + 0.252500i
\(593\) 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
\(597\) 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 \(-0.177914\pi\)
0.847822 + 0.530281i \(0.177914\pi\)
\(602\) 0 0
\(603\) 1.68653 1.68653i 0.0686810 0.0686810i
\(604\) −28.2487 + 28.2487i −1.14942 + 1.14942i
\(605\) 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
\(613\) −34.9545 + 34.9545i −1.41180 + 1.41180i −0.664590 + 0.747208i \(0.731394\pi\)
−0.747208 + 0.664590i \(0.768606\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(618\) 0 0
\(619\) −31.8827 31.8827i −1.28147 1.28147i −0.939829 0.341644i \(-0.889016\pi\)
−0.341644 0.939829i \(-0.610984\pi\)
\(620\) 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
\(631\) −24.6147 24.6147i −0.979897 0.979897i 0.0199047 0.999802i \(-0.493664\pi\)
−0.999802 + 0.0199047i \(0.993664\pi\)
\(632\) 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
\(643\) −13.9737 + 13.9737i −0.551070 + 0.551070i −0.926750 0.375680i \(-0.877409\pi\)
0.375680 + 0.926750i \(0.377409\pi\)
\(644\) 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
\(651\) −56.9090 −2.23044
\(652\) −30.1962 + 30.1962i −1.18257 + 1.18257i
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −28.0981 28.0981i −1.09621 1.09621i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −32.2942 32.2942i −1.25610 1.25610i −0.952940 0.303160i \(-0.901958\pi\)
−0.303160 0.952940i \(-0.598042\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −15.2487 + 15.2487i −0.589549 + 0.589549i
\(670\) 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
\(679\) −50.4641 −1.93663
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 34.3923 + 34.3923i 1.31502 + 1.31502i
\(685\) 0 0
\(686\) 0 0
\(687\) −37.0526 37.0526i −1.41364 1.41364i
\(688\) 6.92820i 0.264135i
\(689\) 0 0
\(690\) 0 0
\(691\) 4.04552 4.04552i 0.153899 0.153899i −0.625958 0.779857i \(-0.715292\pi\)
0.779857 + 0.625958i \(0.215292\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 20.9808 + 20.9808i 0.792998 + 0.792998i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 17.6077 0.664087
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 23.9019 + 23.9019i 0.897656 + 0.897656i 0.995228 0.0975728i \(-0.0311079\pi\)
−0.0975728 + 0.995228i \(0.531108\pi\)
\(710\) 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
\(721\) −32.7058 + 32.7058i −1.21803 + 1.21803i
\(722\) 0 0
\(723\) 11.8756 + 11.8756i 0.441660 + 0.441660i
\(724\) 13.8564 0.514969
\(725\) 0 0
\(726\) 0 0
\(727\) 49.0000i 1.81731i −0.417548 0.908655i \(-0.637111\pi\)
0.417548 0.908655i \(-0.362889\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 30.0000i 1.10883i
\(733\) −30.3468 + 30.3468i −1.12088 + 1.12088i −0.129275 + 0.991609i \(0.541265\pi\)
−0.991609 + 0.129275i \(0.958735\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 33.9808 + 33.9808i 1.25000 + 1.25000i 0.955718 + 0.294285i \(0.0950814\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−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
\(751\) 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
\(756\) 21.8038 21.8038i 0.792998 0.792998i
\(757\) −48.4974 −1.76267 −0.881334 0.472493i \(-0.843354\pi\)
−0.881334 + 0.472493i \(0.843354\pi\)
\(758\) 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
\(763\) −21.6936 −0.785360
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −27.7128 −1.00000
\(769\) −26.7128 + 26.7128i −0.963289 + 0.963289i −0.999350 0.0360609i \(-0.988519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −38.5885 + 38.5885i −1.38883 + 1.38883i
\(773\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 0 0
\(775\) 39.1506 + 39.1506i 1.40633 + 1.40633i
\(776\) 0 0
\(777\) 11.1628i 0.400464i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 7.21539i 0.257693i
\(785\) 0 0
\(786\) 0 0
\(787\) −37.6147 37.6147i −1.34082 1.34082i −0.895244 0.445577i \(-0.852999\pi\)
−0.445577 0.895244i \(-0.647001\pi\)
\(788\) 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
\(804\) 1.94744 + 1.94744i 0.0686810 + 0.0686810i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\)