Properties

Label 300.2.e.b.251.1
Level $300$
Weight $2$
Character 300.251
Analytic conductor $2.396$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.39551206064\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 251.1
Root \(0.707107 + 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 300.251
Dual form 300.2.e.b.251.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{2} +(0.707107 - 1.58114i) q^{3} +2.00000 q^{4} +(-1.00000 + 2.23607i) q^{6} -3.16228i q^{7} -2.82843 q^{8} +(-2.00000 - 2.23607i) q^{9} +O(q^{10})\) \(q-1.41421 q^{2} +(0.707107 - 1.58114i) q^{3} +2.00000 q^{4} +(-1.00000 + 2.23607i) q^{6} -3.16228i q^{7} -2.82843 q^{8} +(-2.00000 - 2.23607i) q^{9} +(1.41421 - 3.16228i) q^{12} +4.47214i q^{14} +4.00000 q^{16} +(2.82843 + 3.16228i) q^{18} +(-5.00000 - 2.23607i) q^{21} -1.41421 q^{23} +(-2.00000 + 4.47214i) q^{24} +(-4.94975 + 1.58114i) q^{27} -6.32456i q^{28} -8.94427i q^{29} -5.65685 q^{32} +(-4.00000 - 4.47214i) q^{36} -4.47214i q^{41} +(7.07107 + 3.16228i) q^{42} +3.16228i q^{43} +2.00000 q^{46} +9.89949 q^{47} +(2.82843 - 6.32456i) q^{48} -3.00000 q^{49} +(7.00000 - 2.23607i) q^{54} +8.94427i q^{56} +12.6491i q^{58} +8.00000 q^{61} +(-7.07107 + 6.32456i) q^{63} +8.00000 q^{64} +15.8114i q^{67} +(-1.00000 + 2.23607i) q^{69} +(5.65685 + 6.32456i) q^{72} +(-1.00000 + 8.94427i) q^{81} +6.32456i q^{82} +15.5563 q^{83} +(-10.0000 - 4.47214i) q^{84} -4.47214i q^{86} +(-14.1421 - 6.32456i) q^{87} +17.8885i q^{89} -2.82843 q^{92} -14.0000 q^{94} +(-4.00000 + 8.94427i) q^{96} +4.24264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} - 4q^{6} - 8q^{9} + O(q^{10}) \) \( 4q + 8q^{4} - 4q^{6} - 8q^{9} + 16q^{16} - 20q^{21} - 8q^{24} - 16q^{36} + 8q^{46} - 12q^{49} + 28q^{54} + 32q^{61} + 32q^{64} - 4q^{69} - 4q^{81} - 40q^{84} - 56q^{94} - 16q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −1.00000
\(3\) 0.707107 1.58114i 0.408248 0.912871i
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) −1.00000 + 2.23607i −0.408248 + 0.912871i
\(7\) 3.16228i 1.19523i −0.801784 0.597614i \(-0.796115\pi\)
0.801784 0.597614i \(-0.203885\pi\)
\(8\) −2.82843 −1.00000
\(9\) −2.00000 2.23607i −0.666667 0.745356i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.41421 3.16228i 0.408248 0.912871i
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 4.47214i 1.19523i
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 2.82843 + 3.16228i 0.666667 + 0.745356i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −5.00000 2.23607i −1.09109 0.487950i
\(22\) 0 0
\(23\) −1.41421 −0.294884 −0.147442 0.989071i \(-0.547104\pi\)
−0.147442 + 0.989071i \(0.547104\pi\)
\(24\) −2.00000 + 4.47214i −0.408248 + 0.912871i
\(25\) 0 0
\(26\) 0 0
\(27\) −4.94975 + 1.58114i −0.952579 + 0.304290i
\(28\) 6.32456i 1.19523i
\(29\) 8.94427i 1.66091i −0.557086 0.830455i \(-0.688081\pi\)
0.557086 0.830455i \(-0.311919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −5.65685 −1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −4.00000 4.47214i −0.666667 0.745356i
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.47214i 0.698430i −0.937043 0.349215i \(-0.886448\pi\)
0.937043 0.349215i \(-0.113552\pi\)
\(42\) 7.07107 + 3.16228i 1.09109 + 0.487950i
\(43\) 3.16228i 0.482243i 0.970495 + 0.241121i \(0.0775152\pi\)
−0.970495 + 0.241121i \(0.922485\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 9.89949 1.44399 0.721995 0.691898i \(-0.243225\pi\)
0.721995 + 0.691898i \(0.243225\pi\)
\(48\) 2.82843 6.32456i 0.408248 0.912871i
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 7.00000 2.23607i 0.952579 0.304290i
\(55\) 0 0
\(56\) 8.94427i 1.19523i
\(57\) 0 0
\(58\) 12.6491i 1.66091i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) −7.07107 + 6.32456i −0.890871 + 0.796819i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 15.8114i 1.93167i 0.259161 + 0.965834i \(0.416554\pi\)
−0.259161 + 0.965834i \(0.583446\pi\)
\(68\) 0 0
\(69\) −1.00000 + 2.23607i −0.120386 + 0.269191i
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 5.65685 + 6.32456i 0.666667 + 0.745356i
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −1.00000 + 8.94427i −0.111111 + 0.993808i
\(82\) 6.32456i 0.698430i
\(83\) 15.5563 1.70753 0.853766 0.520658i \(-0.174313\pi\)
0.853766 + 0.520658i \(0.174313\pi\)
\(84\) −10.0000 4.47214i −1.09109 0.487950i
\(85\) 0 0
\(86\) 4.47214i 0.482243i
\(87\) −14.1421 6.32456i −1.51620 0.678064i
\(88\) 0 0
\(89\) 17.8885i 1.89618i 0.317999 + 0.948091i \(0.396989\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.82843 −0.294884
\(93\) 0 0
\(94\) −14.0000 −1.44399
\(95\) 0 0
\(96\) −4.00000 + 8.94427i −0.408248 + 0.912871i
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 4.24264 0.428571
\(99\) 0 0
\(100\) 0 0
\(101\) 8.94427i 0.889988i 0.895533 + 0.444994i \(0.146794\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) 15.8114i 1.55794i −0.627060 0.778971i \(-0.715742\pi\)
0.627060 0.778971i \(-0.284258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.3848 1.77732 0.888662 0.458563i \(-0.151636\pi\)
0.888662 + 0.458563i \(0.151636\pi\)
\(108\) −9.89949 + 3.16228i −0.952579 + 0.304290i
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 12.6491i 1.19523i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 17.8885i 1.66091i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −11.3137 −1.02430
\(123\) −7.07107 3.16228i −0.637577 0.285133i
\(124\) 0 0
\(125\) 0 0
\(126\) 10.0000 8.94427i 0.890871 0.796819i
\(127\) 22.1359i 1.96425i −0.188237 0.982124i \(-0.560277\pi\)
0.188237 0.982124i \(-0.439723\pi\)
\(128\) −11.3137 −1.00000
\(129\) 5.00000 + 2.23607i 0.440225 + 0.196875i
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 22.3607i 1.93167i
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 1.41421 3.16228i 0.120386 0.269191i
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 7.00000 15.6525i 0.589506 1.31818i
\(142\) 0 0
\(143\) 0 0
\(144\) −8.00000 8.94427i −0.666667 0.745356i
\(145\) 0 0
\(146\) 0 0
\(147\) −2.12132 + 4.74342i −0.174964 + 0.391230i
\(148\) 0 0
\(149\) 4.47214i 0.366372i 0.983078 + 0.183186i \(0.0586410\pi\)
−0.983078 + 0.183186i \(0.941359\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.47214i 0.352454i
\(162\) 1.41421 12.6491i 0.111111 0.993808i
\(163\) 22.1359i 1.73382i 0.498464 + 0.866910i \(0.333898\pi\)
−0.498464 + 0.866910i \(0.666102\pi\)
\(164\) 8.94427i 0.698430i
\(165\) 0 0
\(166\) −22.0000 −1.70753
\(167\) −24.0416 −1.86040 −0.930199 0.367057i \(-0.880366\pi\)
−0.930199 + 0.367057i \(0.880366\pi\)
\(168\) 14.1421 + 6.32456i 1.09109 + 0.487950i
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 6.32456i 0.482243i
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 20.0000 + 8.94427i 1.51620 + 0.678064i
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 25.2982i 1.89618i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 5.65685 12.6491i 0.418167 0.935049i
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 19.7990 1.44399
\(189\) 5.00000 + 15.6525i 0.363696 + 1.13855i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 5.65685 12.6491i 0.408248 0.912871i
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 25.0000 + 11.1803i 1.76336 + 0.788600i
\(202\) 12.6491i 0.889988i
\(203\) −28.2843 −1.98517
\(204\) 0 0
\(205\) 0 0
\(206\) 22.3607i 1.55794i
\(207\) 2.82843 + 3.16228i 0.196589 + 0.219793i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −26.0000 −1.77732
\(215\) 0 0
\(216\) 14.0000 4.47214i 0.952579 0.304290i
\(217\) 0 0
\(218\) −22.6274 −1.53252
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 3.16228i 0.211762i 0.994379 + 0.105881i \(0.0337662\pi\)
−0.994379 + 0.105881i \(0.966234\pi\)
\(224\) 17.8885i 1.19523i
\(225\) 0 0
\(226\) 0 0
\(227\) 9.89949 0.657053 0.328526 0.944495i \(-0.393448\pi\)
0.328526 + 0.944495i \(0.393448\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 25.2982i 1.66091i
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 15.5563 1.00000
\(243\) 13.4350 + 7.90569i 0.861858 + 0.507151i
\(244\) 16.0000 1.02430
\(245\) 0 0
\(246\) 10.0000 + 4.47214i 0.637577 + 0.285133i
\(247\) 0 0
\(248\) 0 0
\(249\) 11.0000 24.5967i 0.697097 1.55876i
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −14.1421 + 12.6491i −0.890871 + 0.796819i
\(253\) 0 0
\(254\) 31.3050i 1.96425i
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) −7.07107 3.16228i −0.440225 0.196875i
\(259\) 0 0
\(260\) 0 0
\(261\) −20.0000 + 17.8885i −1.23797 + 1.10727i
\(262\) 0 0
\(263\) 15.5563 0.959246 0.479623 0.877475i \(-0.340774\pi\)
0.479623 + 0.877475i \(0.340774\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 28.2843 + 12.6491i 1.73097 + 0.774113i
\(268\) 31.6228i 1.93167i
\(269\) 22.3607i 1.36335i −0.731653 0.681677i \(-0.761251\pi\)
0.731653 0.681677i \(-0.238749\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −2.00000 + 4.47214i −0.120386 + 0.269191i
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 31.3050i 1.86750i −0.357930 0.933748i \(-0.616517\pi\)
0.357930 0.933748i \(-0.383483\pi\)
\(282\) −9.89949 + 22.1359i −0.589506 + 1.31818i
\(283\) 15.8114i 0.939889i −0.882696 0.469945i \(-0.844274\pi\)
0.882696 0.469945i \(-0.155726\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −14.1421 −0.834784
\(288\) 11.3137 + 12.6491i 0.666667 + 0.745356i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 3.00000 6.70820i 0.174964 0.391230i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 6.32456i 0.366372i
\(299\) 0 0
\(300\) 0 0
\(301\) 10.0000 0.576390
\(302\) 0 0
\(303\) 14.1421 + 6.32456i 0.812444 + 0.363336i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 34.7851i 1.98529i 0.121070 + 0.992644i \(0.461367\pi\)
−0.121070 + 0.992644i \(0.538633\pi\)
\(308\) 0 0
\(309\) −25.0000 11.1803i −1.42220 0.636027i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 13.0000 29.0689i 0.725589 1.62247i
\(322\) 6.32456i 0.352454i
\(323\) 0 0
\(324\) −2.00000 + 17.8885i −0.111111 + 0.993808i
\(325\) 0 0
\(326\) 31.3050i 1.73382i
\(327\) 11.3137 25.2982i 0.625650 1.39899i
\(328\) 12.6491i 0.698430i
\(329\) 31.3050i 1.72590i
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 31.1127 1.70753
\(333\) 0 0
\(334\) 34.0000 1.86040
\(335\) 0 0
\(336\) −20.0000 8.94427i −1.09109 0.487950i
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 18.3848 1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 12.6491i 0.682988i
\(344\) 8.94427i 0.482243i
\(345\) 0 0
\(346\) 0 0
\(347\) −24.0416 −1.29062 −0.645311 0.763920i \(-0.723272\pi\)
−0.645311 + 0.763920i \(0.723272\pi\)
\(348\) −28.2843 12.6491i −1.51620 0.678064i
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 35.7771i 1.89618i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) −2.82843 −0.148659
\(363\) −7.77817 + 17.3925i −0.408248 + 0.912871i
\(364\) 0 0
\(365\) 0 0
\(366\) −8.00000 + 17.8885i −0.418167 + 0.935049i
\(367\) 3.16228i 0.165070i −0.996588 0.0825348i \(-0.973698\pi\)
0.996588 0.0825348i \(-0.0263016\pi\)
\(368\) −5.65685 −0.294884
\(369\) −10.0000 + 8.94427i −0.520579 + 0.465620i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −28.0000 −1.44399
\(377\) 0 0
\(378\) −7.07107 22.1359i −0.363696 1.13855i
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −35.0000 15.6525i −1.79310 0.801901i
\(382\) 0 0
\(383\) −26.8701 −1.37300 −0.686498 0.727132i \(-0.740853\pi\)
−0.686498 + 0.727132i \(0.740853\pi\)
\(384\) −8.00000 + 17.8885i −0.408248 + 0.912871i
\(385\) 0 0
\(386\) 0 0
\(387\) 7.07107 6.32456i 0.359443 0.321495i
\(388\) 0 0
\(389\) 31.3050i 1.58722i 0.608424 + 0.793612i \(0.291802\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 8.48528 0.428571
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 35.7771i 1.78662i 0.449439 + 0.893311i \(0.351624\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −35.3553 15.8114i −1.76336 0.788600i
\(403\) 0 0
\(404\) 17.8885i 0.889988i
\(405\) 0 0
\(406\) 40.0000 1.98517
\(407\) 0 0
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 31.6228i 1.55794i
\(413\) 0 0
\(414\) −4.00000 4.47214i −0.196589 0.219793i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 0 0
\(423\) −19.7990 22.1359i −0.962660 1.07629i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 25.2982i 1.22427i
\(428\) 36.7696 1.77732
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −19.7990 + 6.32456i −0.952579 + 0.304290i
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 32.0000 1.53252
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 6.00000 + 6.70820i 0.285714 + 0.319438i
\(442\) 0 0
\(443\) 41.0122 1.94855 0.974274 0.225367i \(-0.0723580\pi\)
0.974274 + 0.225367i \(0.0723580\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 4.47214i 0.211762i
\(447\) 7.07107 + 3.16228i 0.334450 + 0.149571i
\(448\) 25.2982i 1.19523i
\(449\) 22.3607i 1.05527i −0.849473 0.527633i \(-0.823080\pi\)
0.849473 0.527633i \(-0.176920\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) −14.0000 −0.657053
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 19.7990 0.925146
\(459\) 0 0
\(460\) 0 0
\(461\) 8.94427i 0.416576i 0.978068 + 0.208288i \(0.0667892\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 0 0
\(463\) 41.1096i 1.91053i 0.295758 + 0.955263i \(0.404428\pi\)
−0.295758 + 0.955263i \(0.595572\pi\)
\(464\) 35.7771i 1.66091i
\(465\) 0 0
\(466\) 0 0
\(467\) −32.5269 −1.50517 −0.752583 0.658497i \(-0.771192\pi\)
−0.752583 + 0.658497i \(0.771192\pi\)
\(468\) 0 0
\(469\) 50.0000 2.30879
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 39.5980 1.80364
\(483\) 7.07107 + 3.16228i 0.321745 + 0.143889i
\(484\) −22.0000 −1.00000
\(485\) 0 0
\(486\) −19.0000 11.1803i −0.861858 0.507151i
\(487\) 22.1359i 1.00308i −0.865136 0.501538i \(-0.832768\pi\)
0.865136 0.501538i \(-0.167232\pi\)
\(488\) −22.6274 −1.02430
\(489\) 35.0000 + 15.6525i 1.58275 + 0.707829i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −14.1421 6.32456i −0.637577 0.285133i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −15.5563 + 34.7851i −0.697097 + 1.55876i
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) −17.0000 + 38.0132i −0.759504 + 1.69830i
\(502\) 0 0
\(503\) −43.8406 −1.95476 −0.977378 0.211498i \(-0.932166\pi\)
−0.977378 + 0.211498i \(0.932166\pi\)
\(504\) 20.0000 17.8885i 0.890871 0.796819i
\(505\) 0 0
\(506\) 0 0
\(507\) −9.19239 + 20.5548i −0.408248 + 0.912871i
\(508\) 44.2719i 1.96425i
\(509\) 44.7214i 1.98224i 0.132973 + 0.991120i \(0.457548\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −22.6274 −1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 10.0000 + 4.47214i 0.440225 + 0.196875i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.8885i 0.783711i −0.920027 0.391856i \(-0.871833\pi\)
0.920027 0.391856i \(-0.128167\pi\)
\(522\) 28.2843 25.2982i 1.23797 1.10727i
\(523\) 34.7851i 1.52104i −0.649312 0.760522i \(-0.724943\pi\)
0.649312 0.760522i \(-0.275057\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −22.0000 −0.959246
\(527\) 0 0
\(528\) 0 0
\(529\) −21.0000 −0.913043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) −40.0000 17.8885i −1.73097 0.774113i
\(535\) 0 0
\(536\) 44.7214i 1.93167i
\(537\) 0 0
\(538\) 31.6228i 1.36335i
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) 1.41421 3.16228i 0.0606897 0.135706i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.16228i 0.135209i −0.997712 0.0676046i \(-0.978464\pi\)
0.997712 0.0676046i \(-0.0215356\pi\)
\(548\) 0 0
\(549\) −16.0000 17.8885i −0.682863 0.763464i
\(550\) 0 0
\(551\) 0 0
\(552\) 2.82843 6.32456i 0.120386 0.269191i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 44.2719i 1.86750i
\(563\) −1.41421 −0.0596020 −0.0298010 0.999556i \(-0.509487\pi\)
−0.0298010 + 0.999556i \(0.509487\pi\)
\(564\) 14.0000 31.3050i 0.589506 1.31818i
\(565\) 0 0
\(566\) 22.3607i 0.939889i
\(567\) 28.2843 + 3.16228i 1.18783 + 0.132803i
\(568\) 0 0
\(569\) 31.3050i 1.31237i 0.754599 + 0.656186i \(0.227831\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 20.0000 0.834784
\(575\) 0 0
\(576\) −16.0000 17.8885i −0.666667 0.745356i
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −24.0416 −1.00000
\(579\) 0 0
\(580\) 0 0
\(581\) 49.1935i 2.04089i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.89949 0.408596 0.204298 0.978909i \(-0.434509\pi\)
0.204298 + 0.978909i \(0.434509\pi\)
\(588\) −4.24264 + 9.48683i −0.174964 + 0.391230i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8.94427i 0.366372i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) −14.1421 −0.576390
\(603\) 35.3553 31.6228i 1.43978 1.28778i
\(604\) 0 0
\(605\) 0 0
\(606\) −20.0000 8.94427i −0.812444 0.363336i
\(607\) 15.8114i 0.641764i 0.947119 + 0.320882i \(0.103979\pi\)
−0.947119 + 0.320882i \(0.896021\pi\)
\(608\) 0 0
\(609\) −20.0000 + 44.7214i −0.810441 + 1.81220i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 49.1935i 1.98529i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 35.3553 + 15.8114i 1.42220 + 0.636027i
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 7.00000 2.23607i 0.280900 0.0897303i
\(622\) 0 0
\(623\) 56.5685 2.26637
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 49.1935i 1.94303i 0.236986 + 0.971513i \(0.423841\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) −18.3848 + 41.1096i −0.725589 + 1.62247i
\(643\) 41.1096i 1.62120i 0.585597 + 0.810602i \(0.300860\pi\)
−0.585597 + 0.810602i \(0.699140\pi\)
\(644\) 8.94427i 0.352454i
\(645\) 0 0
\(646\) 0 0
\(647\) 18.3848 0.722780 0.361390 0.932415i \(-0.382302\pi\)
0.361390 + 0.932415i \(0.382302\pi\)
\(648\) 2.82843 25.2982i 0.111111 0.993808i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 44.2719i 1.73382i
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) −16.0000 + 35.7771i −0.625650 + 1.39899i
\(655\) 0 0
\(656\) 17.8885i 0.698430i
\(657\) 0 0
\(658\) 44.2719i 1.72590i
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 32.0000 1.24466 0.622328 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −44.0000 −1.70753
\(665\) 0 0
\(666\) 0 0
\(667\) 12.6491i 0.489776i
\(668\) −48.0833 −1.86040
\(669\) 5.00000 + 2.23607i 0.193311 + 0.0864514i
\(670\) 0 0
\(671\) 0 0
\(672\) 28.2843 + 12.6491i 1.09109 + 0.487950i
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −26.0000 −1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 7.00000 15.6525i 0.268241 0.599804i
\(682\) 0 0
\(683\) −43.8406 −1.67751 −0.838757 0.544505i \(-0.816717\pi\)
−0.838757 + 0.544505i \(0.816717\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 17.8885i 0.682988i
\(687\) −9.89949 + 22.1359i −0.377689 + 0.844539i
\(688\) 12.6491i 0.482243i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 34.0000 1.29062
\(695\) 0 0
\(696\) 40.0000 + 17.8885i 1.51620 + 0.678064i
\(697\) 0 0
\(698\) 36.7696 1.39175
\(699\) 0 0
\(700\) 0 0
\(701\) 22.3607i 0.844551i 0.906467 + 0.422276i \(0.138769\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 28.2843 1.06374
\(708\) 0 0
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 50.5964i 1.89618i
\(713\) 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
\(721\) −50.0000 −1.86210
\(722\) −26.8701 −1.00000
\(723\) −19.7990 + 44.2719i −0.736332 + 1.64649i
\(724\) 4.00000 0.148659
\(725\) 0 0
\(726\) 11.0000 24.5967i 0.408248 0.912871i
\(727\) 53.7587i 1.99380i 0.0786754 + 0.996900i \(0.474931\pi\)
−0.0786754 + 0.996900i \(0.525069\pi\)
\(728\) 0 0
\(729\) 22.0000 15.6525i 0.814815 0.579721i
\(730\) 0 0
\(731\) 0 0
\(732\) 11.3137 25.2982i 0.418167 0.935049i
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 4.47214i 0.165070i
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 0 0
\(738\) 14.1421 12.6491i 0.520579 0.465620i
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −26.8701 −0.985767 −0.492883 0.870095i \(-0.664057\pi\)
−0.492883 + 0.870095i \(0.664057\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −31.1127 34.7851i −1.13835 1.27272i
\(748\) 0 0
\(749\) 58.1378i 2.12431i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 39.5980 1.44399
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 10.0000 + 31.3050i 0.363696 + 1.13855i
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 35.7771i 1.29692i 0.761249 + 0.648459i \(0.224586\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 49.4975 + 22.1359i 1.79310 + 0.801901i
\(763\) 50.5964i 1.83171i
\(764\) 0 0
\(765\) 0 0
\(766\) 38.0000 1.37300
\(767\) 0 0
\(768\) 11.3137 25.2982i 0.408248 0.912871i
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −10.0000 + 8.94427i −0.359443 + 0.321495i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 44.2719i 1.58722i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 14.1421 + 44.2719i 0.505399 + 1.58215i
\(784\) −12.0000 −0.428571
\(785\) 0 0
\(786\) 0 0
\(787\) 41.1096i 1.46540i −0.680552 0.732700i \(-0.738260\pi\)
0.680552 0.732700i \(-0.261740\pi\)
\(788\) 0 0
\(789\) 11.0000 24.5967i 0.391610 0.875667i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 40.0000 35.7771i 1.41333 1.26412i
\(802\) 50.5964i 1.78662i
\(803\) 0 0
\(804\) 50.0000 + 22.3607i 1.76336 + 0.788600i
\(805\) 0 0
\(806\) 0 0
\(807\) −35.3553 15.8114i