Properties

Label 507.2.k.b.89.1
Level $507$
Weight $2$
Character 507.89
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.k (of order \(12\), degree \(4\), not minimal)

Newform invariants

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

Embedding invariants

Embedding label 89.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 507.89
Dual form 507.2.k.b.188.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.866025 - 1.50000i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(1.13397 + 4.23205i) q^{7} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\) \(q+(-0.866025 - 1.50000i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(1.13397 + 4.23205i) q^{7} +(-1.50000 + 2.59808i) q^{9} +3.46410i q^{12} +(2.00000 + 3.46410i) q^{16} +(3.09808 - 0.830127i) q^{19} +(5.36603 - 5.36603i) q^{21} +5.00000i q^{25} +5.19615 q^{27} +(2.26795 - 8.46410i) q^{28} +(0.830127 + 0.830127i) q^{31} +(5.19615 - 3.00000i) q^{36} +(11.5622 + 3.09808i) q^{37} +(1.50000 + 0.866025i) q^{43} +(3.46410 - 6.00000i) q^{48} +(-10.5622 + 6.09808i) q^{49} +(-3.92820 - 3.92820i) q^{57} +(-4.33013 + 7.50000i) q^{61} +(-12.6962 - 3.40192i) q^{63} -8.00000i q^{64} +(-4.23205 + 15.7942i) q^{67} +(-7.63397 + 7.63397i) q^{73} +(7.50000 - 4.33013i) q^{75} +(-6.19615 - 1.66025i) q^{76} -12.1244 q^{79} +(-4.50000 - 7.79423i) q^{81} +(-14.6603 + 3.92820i) q^{84} +(0.526279 - 1.96410i) q^{93} +(9.59808 - 2.57180i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{7} - 6q^{9} + O(q^{10}) \) \( 4q + 8q^{7} - 6q^{9} + 8q^{16} + 2q^{19} + 18q^{21} + 16q^{28} - 14q^{31} + 22q^{37} + 6q^{43} - 18q^{49} + 12q^{57} - 30q^{63} - 10q^{67} - 34q^{73} + 30q^{75} - 4q^{76} - 18q^{81} - 24q^{84} - 36q^{93} + 28q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(e\left(\frac{7}{12}\right)\)

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 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(3\) −0.866025 1.50000i −0.500000 0.866025i
\(4\) −1.73205 1.00000i −0.866025 0.500000i
\(5\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(6\) 0 0
\(7\) 1.13397 + 4.23205i 0.428602 + 1.59956i 0.755929 + 0.654654i \(0.227186\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) 0 0
\(9\) −1.50000 + 2.59808i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(12\) 3.46410i 1.00000i
\(13\) 0 0
\(14\) 0 0
\(15\) 0 0
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 3.09808 0.830127i 0.710747 0.190444i 0.114708 0.993399i \(-0.463407\pi\)
0.596040 + 0.802955i \(0.296740\pi\)
\(20\) 0 0
\(21\) 5.36603 5.36603i 1.17096 1.17096i
\(22\) 0 0
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 5.00000i 1.00000i
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 2.26795 8.46410i 0.428602 1.59956i
\(29\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 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
\(36\) 5.19615 3.00000i 0.866025 0.500000i
\(37\) 11.5622 + 3.09808i 1.90081 + 0.509321i 0.996616 + 0.0821995i \(0.0261945\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(42\) 0 0
\(43\) 1.50000 + 0.866025i 0.228748 + 0.132068i 0.609994 0.792406i \(-0.291172\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 3.46410 6.00000i 0.500000 0.866025i
\(49\) −10.5622 + 6.09808i −1.50888 + 0.871154i
\(50\) 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 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(60\) 0 0
\(61\) −4.33013 + 7.50000i −0.554416 + 0.960277i 0.443533 + 0.896258i \(0.353725\pi\)
−0.997949 + 0.0640184i \(0.979608\pi\)
\(62\) 0 0
\(63\) −12.6962 3.40192i −1.59956 0.428602i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) −4.23205 + 15.7942i −0.517027 + 1.92957i −0.211604 + 0.977356i \(0.567869\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(72\) 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
\(75\) 7.50000 4.33013i 0.866025 0.500000i
\(76\) −6.19615 1.66025i −0.710747 0.190444i
\(77\) 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
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) −14.6603 + 3.92820i −1.59956 + 0.428602i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0.526279 1.96410i 0.0545726 0.203668i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.59808 2.57180i 0.974537 0.261126i 0.263795 0.964579i \(-0.415026\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.00000 8.66025i 0.500000 0.866025i
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) −9.00000 5.19615i −0.866025 0.500000i
\(109\) 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
\(111\) −5.36603 20.0263i −0.509321 1.90081i
\(112\) −12.3923 + 12.3923i −1.17096 + 1.17096i
\(113\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 9.52628 + 5.50000i 0.866025 + 0.500000i
\(122\) 0 0
\(123\) 0 0
\(124\) −0.607695 2.26795i −0.0545726 0.203668i
\(125\) 0 0
\(126\) 0 0
\(127\) −0.866025 + 0.500000i −0.0768473 + 0.0443678i −0.537931 0.842989i \(-0.680794\pi\)
0.461084 + 0.887357i \(0.347461\pi\)
\(128\) 0 0
\(129\) 3.00000i 0.264135i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 7.02628 + 12.1699i 0.609256 + 1.05526i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(138\) 0 0
\(139\) 3.50000 6.06218i 0.296866 0.514187i −0.678551 0.734553i \(-0.737392\pi\)
0.975417 + 0.220366i \(0.0707252\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −12.0000 −1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 18.2942 + 10.5622i 1.50888 + 0.871154i
\(148\) −16.9282 16.9282i −1.39149 1.39149i
\(149\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(150\) 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
\(163\) −3.62436 13.5263i −0.283881 1.05946i −0.949653 0.313304i \(-0.898564\pi\)
0.665771 0.746156i \(-0.268103\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −2.49038 + 9.29423i −0.190444 + 0.710747i
\(172\) −1.73205 3.00000i −0.132068 0.228748i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) −21.1603 + 5.66987i −1.59956 + 0.428602i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 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
\(189\) 5.89230 + 21.9904i 0.428602 + 1.59956i
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) −12.0000 + 6.92820i −0.866025 + 0.500000i
\(193\) 5.06218 + 1.35641i 0.364384 + 0.0976363i 0.436365 0.899770i \(-0.356266\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 24.3923 1.74231
\(197\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(198\) 0 0
\(199\) −14.7224 8.50000i −1.04365 0.602549i −0.122782 0.992434i \(-0.539182\pi\)
−0.920864 + 0.389885i \(0.872515\pi\)
\(200\) 0 0
\(201\) 27.3564 7.33013i 1.92957 0.517027i
\(202\) 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
\(211\) −12.9904 22.5000i −0.894295 1.54896i −0.834675 0.550743i \(-0.814345\pi\)
−0.0596196 0.998221i \(-0.518989\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.57180 + 4.45448i −0.174585 + 0.302390i
\(218\) 0 0
\(219\) 18.0622 + 4.83975i 1.22053 + 0.327040i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 7.02628 26.2224i 0.470514 1.75598i −0.167412 0.985887i \(-0.553541\pi\)
0.637927 0.770097i \(-0.279792\pi\)
\(224\) 0 0
\(225\) −12.9904 7.50000i −0.866025 0.500000i
\(226\) 0 0
\(227\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(228\) 2.87564 + 10.7321i 0.190444 + 0.710747i
\(229\) 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
\(237\) 10.5000 + 18.1865i 0.682048 + 1.18134i
\(238\) 0 0
\(239\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(240\) 0 0
\(241\) 7.63397 + 28.4904i 0.491748 + 1.83523i 0.547533 + 0.836784i \(0.315567\pi\)
−0.0557856 + 0.998443i \(0.517766\pi\)
\(242\) 0 0
\(243\) −7.79423 + 13.5000i −0.500000 + 0.866025i
\(244\) 15.0000 8.66025i 0.960277 0.554416i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(252\) 18.5885 + 18.5885i 1.17096 + 1.17096i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 + 13.8564i −0.500000 + 0.866025i
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) 52.4449i 3.25877i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 23.1244 23.1244i 1.41254 1.41254i
\(269\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(270\) 0 0
\(271\) −30.4545 8.16025i −1.84998 0.495700i −0.850439 0.526073i \(-0.823664\pi\)
−0.999539 + 0.0303728i \(0.990331\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18.0000 10.3923i −1.08152 0.624413i −0.150210 0.988654i \(-0.547995\pi\)
−0.931305 + 0.364241i \(0.881328\pi\)
\(278\) 0 0
\(279\) −3.40192 + 0.911543i −0.203668 + 0.0545726i
\(280\) 0 0
\(281\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(282\) 0 0
\(283\) 21.6506 12.5000i 1.28700 0.743048i 0.308879 0.951101i \(-0.400046\pi\)
0.978117 + 0.208053i \(0.0667128\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 + 14.7224i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) −12.1699 12.1699i −0.713411 0.713411i
\(292\) 20.8564 5.58846i 1.22053 0.327040i
\(293\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −17.3205 −1.00000
\(301\) −1.96410 + 7.33013i −0.113209 + 0.422501i
\(302\) 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
\(309\) −23.3827 + 13.5000i −1.33019 + 0.767988i
\(310\) 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
\(316\) 21.0000 + 12.1244i 1.18134 + 0.682048i
\(317\) 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
\(327\) 8.76795 32.7224i 0.484869 1.80955i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −8.16025 + 2.18653i −0.448528 + 0.120183i −0.476011 0.879440i \(-0.657918\pi\)
0.0274825 + 0.999622i \(0.491251\pi\)
\(332\) 0 0
\(333\) −25.3923 + 25.3923i −1.39149 + 1.39149i
\(334\) 0 0
\(335\) 0 0
\(336\) 29.3205 + 7.85641i 1.59956 + 0.428602i
\(337\) 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 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) 35.8205 + 9.59808i 1.91743 + 0.513773i 0.990282 + 0.139072i \(0.0444119\pi\)
0.927146 + 0.374701i \(0.122255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(354\) 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
\(361\) −7.54552 + 4.35641i −0.397132 + 0.229285i
\(362\) 0 0
\(363\) 19.0526i 1.00000i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −15.5000 26.8468i −0.809093 1.40139i −0.913493 0.406855i \(-0.866625\pi\)
0.104399 0.994535i \(-0.466708\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −2.87564 + 2.87564i −0.149095 + 0.149095i
\(373\) 18.1865 31.5000i 0.941663 1.63101i 0.179364 0.983783i \(-0.442596\pi\)
0.762299 0.647225i \(-0.224071\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −8.99038 + 33.5526i −0.461805 + 1.72348i 0.205466 + 0.978664i \(0.434129\pi\)
−0.667271 + 0.744815i \(0.732538\pi\)
\(380\) 0 0
\(381\) 1.50000 + 0.866025i 0.0768473 + 0.0443678i
\(382\) 0 0
\(383\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.50000 + 2.59808i −0.228748 + 0.132068i
\(388\) −19.1962 5.14359i −0.974537 0.261126i
\(389\) 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
\(397\) −7.10770 26.5263i −0.356725 1.33132i −0.878300 0.478110i \(-0.841322\pi\)
0.521575 0.853206i \(-0.325345\pi\)
\(398\) 0 0
\(399\) 12.1699 21.0788i 0.609256 1.05526i
\(400\) −17.3205 + 10.0000i −0.866025 + 0.500000i
\(401\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −38.9186 + 10.4282i −1.92440 + 0.515641i −0.939490 + 0.342578i \(0.888700\pi\)
−0.984911 + 0.173064i \(0.944633\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −15.5885 + 27.0000i −0.767988 + 1.33019i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −12.1244 −0.593732
\(418\) 0 0
\(419\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(420\) 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
\(427\) −36.6506 9.82051i −1.77365 0.475248i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(432\) 10.3923 + 18.0000i 0.500000 + 0.866025i
\(433\) 30.3109 + 17.5000i 1.45665 + 0.840996i 0.998845 0.0480569i \(-0.0153029\pi\)
0.457804 + 0.889053i \(0.348636\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.1244 37.7846i −0.484869 1.80955i
\(437\) 0 0
\(438\) 0 0
\(439\) 34.5000 19.9186i 1.64660 0.950662i 0.668184 0.743996i \(-0.267072\pi\)
0.978412 0.206666i \(-0.0662612\pi\)
\(440\) 0 0
\(441\) 36.5885i 1.74231i
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −10.7321 + 40.0526i −0.509321 + 1.90081i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 33.8564 9.07180i 1.59956 0.428602i
\(449\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −23.9545 6.41858i −1.12548 0.301571i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.28461 19.7224i 0.247204 0.922576i −0.725059 0.688686i \(-0.758188\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(462\) 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
\(471\) 9.52628 + 16.5000i 0.438948 + 0.760280i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.15064 + 15.4904i 0.190444 + 0.710747i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −11.0000 19.0526i −0.500000 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) −32.4186 + 8.68653i −1.46903 + 0.393624i −0.902597 0.430486i \(-0.858342\pi\)
−0.566429 + 0.824110i \(0.691675\pi\)
\(488\) 0 0
\(489\) −17.1506 + 17.1506i −0.775579 + 0.775579i
\(490\) 0 0
\(491\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.21539 + 4.53590i −0.0545726 + 0.203668i
\(497\) 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 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(510\) 0 0
\(511\) −40.9641 23.6506i −1.81215 1.04624i
\(512\) 0 0
\(513\) 16.0981 4.31347i 0.710747 0.190444i
\(514\) 0 0
\(515\) 0 0
\(516\) −3.00000 + 5.19615i −0.132068 + 0.228748i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 4.00000 + 6.92820i 0.174908 + 0.302949i 0.940129 0.340818i \(-0.110704\pi\)
−0.765222 + 0.643767i \(0.777371\pi\)
\(524\) 0 0
\(525\) 26.8301 + 26.8301i 1.17096 + 1.17096i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 19.9186i 0.500000 0.866025i
\(530\) 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
\(543\) 10.3923 6.00000i 0.445976 0.257485i
\(544\) 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
\(549\) −12.9904 22.5000i −0.554416 0.960277i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −13.7487 51.3109i −0.584655 2.18196i
\(554\) 0 0
\(555\) 0 0
\(556\) −12.1244 + 7.00000i −0.514187 + 0.296866i
\(557\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 27.8827 27.8827i 1.17096 1.17096i
\(568\) 0 0
\(569\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(570\) 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
\(576\) 20.7846 + 12.0000i 0.866025 + 0.500000i
\(577\) −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
\(579\) −2.34936 8.76795i −0.0976363 0.364384i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(588\) −21.1244 36.5885i −0.871154 1.50888i
\(589\) 3.26091 + 1.88269i 0.134363 + 0.0775747i
\(590\) 0 0
\(591\) 0 0
\(592\) 12.3923 + 46.2487i 0.509321 + 1.90081i
\(593\) 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
\(601\) 20.7846 + 36.0000i 0.847822 + 1.46847i 0.883148 + 0.469095i \(0.155420\pi\)
−0.0353259 + 0.999376i \(0.511247\pi\)
\(602\) 0 0
\(603\) −34.6865 34.6865i −1.41254 1.41254i
\(604\) −27.6603 + 7.41154i −1.12548 + 0.301571i
\(605\) 0 0
\(606\) 0 0
\(607\) 10.0000 17.3205i 0.405887 0.703018i −0.588537 0.808470i \(-0.700296\pi\)
0.994424 + 0.105453i \(0.0336291\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −0.748711 + 2.79423i −0.0302402 + 0.112858i −0.979396 0.201948i \(-0.935273\pi\)
0.949156 + 0.314806i \(0.101939\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(618\) 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
\(628\) 19.0526 + 11.0000i 0.760280 + 0.438948i
\(629\) 0 0
\(630\) 0 0
\(631\) 9.37564 + 34.9904i 0.373239 + 1.39295i 0.855901 + 0.517139i \(0.173003\pi\)
−0.482663 + 0.875806i \(0.660330\pi\)
\(632\) 0 0
\(633\) −22.5000 + 38.9711i −0.894295 + 1.54896i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(642\) 0 0
\(643\) 45.1147 12.0885i 1.77915 0.476722i 0.788723 0.614749i \(-0.210743\pi\)
0.990429 + 0.138027i \(0.0440759\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 8.90897 0.349170
\(652\) −7.24871 + 27.0526i −0.283881 + 1.05946i
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −8.38269 31.2846i −0.327040 1.22053i
\(658\) 0 0
\(659\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(660\) 0 0
\(661\) 22.8205 + 6.11474i 0.887615 + 0.237836i 0.673690 0.739014i \(-0.264708\pi\)
0.213925 + 0.976850i \(0.431375\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −45.4186 + 12.1699i −1.75598 + 0.470514i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −43.5000 + 25.1147i −1.67680 + 0.968102i −0.713123 + 0.701039i \(0.752720\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) 0 0
\(675\) 25.9808i 1.00000i
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 21.7679 + 37.7032i 0.835377 + 1.44692i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(684\) 13.6077 13.6077i 0.520303 0.520303i
\(685\) 0 0
\(686\) 0 0
\(687\) −1.43782 0.385263i −0.0548563 0.0146987i
\(688\) 6.92820i 0.264135i
\(689\) 0 0
\(690\) 0 0
\(691\) 13.5263 50.4808i 0.514564 1.92038i 0.152167 0.988355i \(-0.451375\pi\)
0.362397 0.932024i \(-0.381959\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 42.3205 + 11.3397i 1.59956 + 0.428602i
\(701\) 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
\(709\) 10.6506 + 39.7487i 0.399993 + 1.49279i 0.813107 + 0.582115i \(0.197775\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 18.1865 31.5000i 0.682048 1.18134i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 65.9711 17.6769i 2.45689 0.658323i
\(722\) 0 0
\(723\) 36.1244 36.1244i 1.34348 1.34348i
\(724\) 6.92820 12.0000i 0.257485 0.445976i
\(725\) 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
\(732\) −25.9808 15.0000i −0.960277 0.554416i
\(733\) 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
\(739\) 24.5622 + 6.58142i 0.903534 + 0.242101i 0.680534 0.732717i \(-0.261748\pi\)
0.223001 + 0.974818i \(0.428415\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 15.0000 8.66025i 0.547358 0.316017i −0.200698 0.979653i \(-0.564321\pi\)
0.748056 + 0.663636i \(0.230988\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 11.7846 43.9808i 0.428602 1.59956i
\(757\) −24.2487 42.0000i −0.881334 1.52652i −0.849858 0.527011i \(-0.823312\pi\)
−0.0314762 0.999505i \(-0.510021\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(762\) 0 0
\(763\) −42.8468 + 74.2128i −1.55116 + 2.68668i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 27.7128 1.00000
\(769\) 10.5096 39.2224i 0.378987 1.41440i −0.468445 0.883493i \(-0.655186\pi\)
0.847432 0.530904i \(-0.178148\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.41154 7.41154i −0.266747 0.266747i
\(773\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(774\) 0 0
\(775\) −4.15064 + 4.15064i −0.149095 + 0.149095i
\(776\) 0 0
\(777\) 78.6673 45.4186i 2.82217 1.62938i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −42.2487 24.3923i −1.50888 0.871154i
\(785\) 0 0
\(786\) 0 0
\(787\) 4.61731 + 17.2321i 0.164589 + 0.614256i 0.998092 + 0.0617409i \(0.0196653\pi\)
−0.833503 + 0.552515i \(0.813668\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 17.0000 + 29.4449i 0.602549 + 1.04365i
\(797\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −54.7128 14.6603i −1.92957 0.517027i
\(805\) 0 0
\(806\)