Properties

Label 300.3.g.e.101.2
Level $300$
Weight $3$
Character 300.101
Analytic conductor $8.174$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \(x^{2} + 5\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 101.2
Root \(2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 300.101
Dual form 300.3.g.e.101.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.00000 + 2.23607i) q^{3} +8.00000 q^{7} +(-1.00000 - 8.94427i) q^{9} +O(q^{10})\) \(q+(-2.00000 + 2.23607i) q^{3} +8.00000 q^{7} +(-1.00000 - 8.94427i) q^{9} -8.94427i q^{11} +12.0000 q^{13} +31.3050i q^{17} +6.00000 q^{19} +(-16.0000 + 17.8885i) q^{21} +4.47214i q^{23} +(22.0000 + 15.6525i) q^{27} +26.8328i q^{29} +34.0000 q^{31} +(20.0000 + 17.8885i) q^{33} +44.0000 q^{37} +(-24.0000 + 26.8328i) q^{39} +17.8885i q^{41} -28.0000 q^{43} -4.47214i q^{47} +15.0000 q^{49} +(-70.0000 - 62.6099i) q^{51} +40.2492i q^{53} +(-12.0000 + 13.4164i) q^{57} -98.3870i q^{59} +74.0000 q^{61} +(-8.00000 - 71.5542i) q^{63} -92.0000 q^{67} +(-10.0000 - 8.94427i) q^{69} +53.6656i q^{71} +56.0000 q^{73} -71.5542i q^{77} +78.0000 q^{79} +(-79.0000 + 17.8885i) q^{81} -102.859i q^{83} +(-60.0000 - 53.6656i) q^{87} +17.8885i q^{89} +96.0000 q^{91} +(-68.0000 + 76.0263i) q^{93} -32.0000 q^{97} +(-80.0000 + 8.94427i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} + 16q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} + 16q^{7} - 2q^{9} + 24q^{13} + 12q^{19} - 32q^{21} + 44q^{27} + 68q^{31} + 40q^{33} + 88q^{37} - 48q^{39} - 56q^{43} + 30q^{49} - 140q^{51} - 24q^{57} + 148q^{61} - 16q^{63} - 184q^{67} - 20q^{69} + 112q^{73} + 156q^{79} - 158q^{81} - 120q^{87} + 192q^{91} - 136q^{93} - 64q^{97} - 160q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.00000 + 2.23607i −0.666667 + 0.745356i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 8.00000 1.14286 0.571429 0.820652i \(-0.306389\pi\)
0.571429 + 0.820652i \(0.306389\pi\)
\(8\) 0 0
\(9\) −1.00000 8.94427i −0.111111 0.993808i
\(10\) 0 0
\(11\) 8.94427i 0.813116i −0.913625 0.406558i \(-0.866729\pi\)
0.913625 0.406558i \(-0.133271\pi\)
\(12\) 0 0
\(13\) 12.0000 0.923077 0.461538 0.887120i \(-0.347298\pi\)
0.461538 + 0.887120i \(0.347298\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 31.3050i 1.84147i 0.390191 + 0.920734i \(0.372409\pi\)
−0.390191 + 0.920734i \(0.627591\pi\)
\(18\) 0 0
\(19\) 6.00000 0.315789 0.157895 0.987456i \(-0.449529\pi\)
0.157895 + 0.987456i \(0.449529\pi\)
\(20\) 0 0
\(21\) −16.0000 + 17.8885i −0.761905 + 0.851835i
\(22\) 0 0
\(23\) 4.47214i 0.194441i 0.995263 + 0.0972203i \(0.0309952\pi\)
−0.995263 + 0.0972203i \(0.969005\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 22.0000 + 15.6525i 0.814815 + 0.579721i
\(28\) 0 0
\(29\) 26.8328i 0.925270i 0.886549 + 0.462635i \(0.153096\pi\)
−0.886549 + 0.462635i \(0.846904\pi\)
\(30\) 0 0
\(31\) 34.0000 1.09677 0.548387 0.836225i \(-0.315242\pi\)
0.548387 + 0.836225i \(0.315242\pi\)
\(32\) 0 0
\(33\) 20.0000 + 17.8885i 0.606061 + 0.542077i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 44.0000 1.18919 0.594595 0.804026i \(-0.297313\pi\)
0.594595 + 0.804026i \(0.297313\pi\)
\(38\) 0 0
\(39\) −24.0000 + 26.8328i −0.615385 + 0.688021i
\(40\) 0 0
\(41\) 17.8885i 0.436306i 0.975915 + 0.218153i \(0.0700032\pi\)
−0.975915 + 0.218153i \(0.929997\pi\)
\(42\) 0 0
\(43\) −28.0000 −0.651163 −0.325581 0.945514i \(-0.605560\pi\)
−0.325581 + 0.945514i \(0.605560\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.47214i 0.0951518i −0.998868 0.0475759i \(-0.984850\pi\)
0.998868 0.0475759i \(-0.0151496\pi\)
\(48\) 0 0
\(49\) 15.0000 0.306122
\(50\) 0 0
\(51\) −70.0000 62.6099i −1.37255 1.22765i
\(52\) 0 0
\(53\) 40.2492i 0.759419i 0.925106 + 0.379710i \(0.123976\pi\)
−0.925106 + 0.379710i \(0.876024\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −12.0000 + 13.4164i −0.210526 + 0.235376i
\(58\) 0 0
\(59\) 98.3870i 1.66758i −0.552085 0.833788i \(-0.686167\pi\)
0.552085 0.833788i \(-0.313833\pi\)
\(60\) 0 0
\(61\) 74.0000 1.21311 0.606557 0.795040i \(-0.292550\pi\)
0.606557 + 0.795040i \(0.292550\pi\)
\(62\) 0 0
\(63\) −8.00000 71.5542i −0.126984 1.13578i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −92.0000 −1.37313 −0.686567 0.727066i \(-0.740883\pi\)
−0.686567 + 0.727066i \(0.740883\pi\)
\(68\) 0 0
\(69\) −10.0000 8.94427i −0.144928 0.129627i
\(70\) 0 0
\(71\) 53.6656i 0.755854i 0.925835 + 0.377927i \(0.123363\pi\)
−0.925835 + 0.377927i \(0.876637\pi\)
\(72\) 0 0
\(73\) 56.0000 0.767123 0.383562 0.923515i \(-0.374697\pi\)
0.383562 + 0.923515i \(0.374697\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 71.5542i 0.929275i
\(78\) 0 0
\(79\) 78.0000 0.987342 0.493671 0.869649i \(-0.335655\pi\)
0.493671 + 0.869649i \(0.335655\pi\)
\(80\) 0 0
\(81\) −79.0000 + 17.8885i −0.975309 + 0.220846i
\(82\) 0 0
\(83\) 102.859i 1.23927i −0.784891 0.619633i \(-0.787281\pi\)
0.784891 0.619633i \(-0.212719\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −60.0000 53.6656i −0.689655 0.616846i
\(88\) 0 0
\(89\) 17.8885i 0.200995i 0.994937 + 0.100497i \(0.0320434\pi\)
−0.994937 + 0.100497i \(0.967957\pi\)
\(90\) 0 0
\(91\) 96.0000 1.05495
\(92\) 0 0
\(93\) −68.0000 + 76.0263i −0.731183 + 0.817487i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −32.0000 −0.329897 −0.164948 0.986302i \(-0.552746\pi\)
−0.164948 + 0.986302i \(0.552746\pi\)
\(98\) 0 0
\(99\) −80.0000 + 8.94427i −0.808081 + 0.0903462i
\(100\) 0 0
\(101\) 152.053i 1.50547i −0.658323 0.752736i \(-0.728734\pi\)
0.658323 0.752736i \(-0.271266\pi\)
\(102\) 0 0
\(103\) −104.000 −1.00971 −0.504854 0.863205i \(-0.668454\pi\)
−0.504854 + 0.863205i \(0.668454\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 147.580i 1.37926i −0.724163 0.689628i \(-0.757774\pi\)
0.724163 0.689628i \(-0.242226\pi\)
\(108\) 0 0
\(109\) −74.0000 −0.678899 −0.339450 0.940624i \(-0.610241\pi\)
−0.339450 + 0.940624i \(0.610241\pi\)
\(110\) 0 0
\(111\) −88.0000 + 98.3870i −0.792793 + 0.886369i
\(112\) 0 0
\(113\) 40.2492i 0.356188i 0.984013 + 0.178094i \(0.0569931\pi\)
−0.984013 + 0.178094i \(0.943007\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −12.0000 107.331i −0.102564 0.917361i
\(118\) 0 0
\(119\) 250.440i 2.10453i
\(120\) 0 0
\(121\) 41.0000 0.338843
\(122\) 0 0
\(123\) −40.0000 35.7771i −0.325203 0.290871i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −16.0000 −0.125984 −0.0629921 0.998014i \(-0.520064\pi\)
−0.0629921 + 0.998014i \(0.520064\pi\)
\(128\) 0 0
\(129\) 56.0000 62.6099i 0.434109 0.485348i
\(130\) 0 0
\(131\) 80.4984i 0.614492i 0.951630 + 0.307246i \(0.0994074\pi\)
−0.951630 + 0.307246i \(0.900593\pi\)
\(132\) 0 0
\(133\) 48.0000 0.360902
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 174.413i 1.27309i 0.771240 + 0.636545i \(0.219637\pi\)
−0.771240 + 0.636545i \(0.780363\pi\)
\(138\) 0 0
\(139\) 118.000 0.848921 0.424460 0.905446i \(-0.360464\pi\)
0.424460 + 0.905446i \(0.360464\pi\)
\(140\) 0 0
\(141\) 10.0000 + 8.94427i 0.0709220 + 0.0634346i
\(142\) 0 0
\(143\) 107.331i 0.750568i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −30.0000 + 33.5410i −0.204082 + 0.228170i
\(148\) 0 0
\(149\) 98.3870i 0.660315i −0.943926 0.330158i \(-0.892898\pi\)
0.943926 0.330158i \(-0.107102\pi\)
\(150\) 0 0
\(151\) 34.0000 0.225166 0.112583 0.993642i \(-0.464088\pi\)
0.112583 + 0.993642i \(0.464088\pi\)
\(152\) 0 0
\(153\) 280.000 31.3050i 1.83007 0.204608i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −92.0000 −0.585987 −0.292994 0.956114i \(-0.594651\pi\)
−0.292994 + 0.956114i \(0.594651\pi\)
\(158\) 0 0
\(159\) −90.0000 80.4984i −0.566038 0.506280i
\(160\) 0 0
\(161\) 35.7771i 0.222218i
\(162\) 0 0
\(163\) −68.0000 −0.417178 −0.208589 0.978003i \(-0.566887\pi\)
−0.208589 + 0.978003i \(0.566887\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 67.0820i 0.401689i 0.979623 + 0.200844i \(0.0643686\pi\)
−0.979623 + 0.200844i \(0.935631\pi\)
\(168\) 0 0
\(169\) −25.0000 −0.147929
\(170\) 0 0
\(171\) −6.00000 53.6656i −0.0350877 0.313834i
\(172\) 0 0
\(173\) 76.0263i 0.439458i 0.975561 + 0.219729i \(0.0705174\pi\)
−0.975561 + 0.219729i \(0.929483\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 220.000 + 196.774i 1.24294 + 1.11172i
\(178\) 0 0
\(179\) 259.384i 1.44907i −0.689237 0.724536i \(-0.742054\pi\)
0.689237 0.724536i \(-0.257946\pi\)
\(180\) 0 0
\(181\) −166.000 −0.917127 −0.458564 0.888662i \(-0.651636\pi\)
−0.458564 + 0.888662i \(0.651636\pi\)
\(182\) 0 0
\(183\) −148.000 + 165.469i −0.808743 + 0.904202i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 280.000 1.49733
\(188\) 0 0
\(189\) 176.000 + 125.220i 0.931217 + 0.662539i
\(190\) 0 0
\(191\) 214.663i 1.12389i −0.827175 0.561944i \(-0.810054\pi\)
0.827175 0.561944i \(-0.189946\pi\)
\(192\) 0 0
\(193\) 32.0000 0.165803 0.0829016 0.996558i \(-0.473581\pi\)
0.0829016 + 0.996558i \(0.473581\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.47214i 0.0227012i −0.999936 0.0113506i \(-0.996387\pi\)
0.999936 0.0113506i \(-0.00361309\pi\)
\(198\) 0 0
\(199\) −114.000 −0.572864 −0.286432 0.958101i \(-0.592469\pi\)
−0.286432 + 0.958101i \(0.592469\pi\)
\(200\) 0 0
\(201\) 184.000 205.718i 0.915423 1.02347i
\(202\) 0 0
\(203\) 214.663i 1.05745i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 40.0000 4.47214i 0.193237 0.0216045i
\(208\) 0 0
\(209\) 53.6656i 0.256773i
\(210\) 0 0
\(211\) −6.00000 −0.0284360 −0.0142180 0.999899i \(-0.504526\pi\)
−0.0142180 + 0.999899i \(0.504526\pi\)
\(212\) 0 0
\(213\) −120.000 107.331i −0.563380 0.503903i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 272.000 1.25346
\(218\) 0 0
\(219\) −112.000 + 125.220i −0.511416 + 0.571780i
\(220\) 0 0
\(221\) 375.659i 1.69982i
\(222\) 0 0
\(223\) 272.000 1.21973 0.609865 0.792505i \(-0.291223\pi\)
0.609865 + 0.792505i \(0.291223\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 245.967i 1.08356i 0.840521 + 0.541779i \(0.182249\pi\)
−0.840521 + 0.541779i \(0.817751\pi\)
\(228\) 0 0
\(229\) −154.000 −0.672489 −0.336245 0.941775i \(-0.609157\pi\)
−0.336245 + 0.941775i \(0.609157\pi\)
\(230\) 0 0
\(231\) 160.000 + 143.108i 0.692641 + 0.619517i
\(232\) 0 0
\(233\) 183.358i 0.786942i 0.919337 + 0.393471i \(0.128726\pi\)
−0.919337 + 0.393471i \(0.871274\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −156.000 + 174.413i −0.658228 + 0.735921i
\(238\) 0 0
\(239\) 178.885i 0.748475i 0.927333 + 0.374237i \(0.122095\pi\)
−0.927333 + 0.374237i \(0.877905\pi\)
\(240\) 0 0
\(241\) −206.000 −0.854772 −0.427386 0.904069i \(-0.640565\pi\)
−0.427386 + 0.904069i \(0.640565\pi\)
\(242\) 0 0
\(243\) 118.000 212.426i 0.485597 0.874183i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 72.0000 0.291498
\(248\) 0 0
\(249\) 230.000 + 205.718i 0.923695 + 0.826178i
\(250\) 0 0
\(251\) 26.8328i 0.106904i 0.998570 + 0.0534518i \(0.0170224\pi\)
−0.998570 + 0.0534518i \(0.982978\pi\)
\(252\) 0 0
\(253\) 40.0000 0.158103
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 40.2492i 0.156612i −0.996929 0.0783059i \(-0.975049\pi\)
0.996929 0.0783059i \(-0.0249511\pi\)
\(258\) 0 0
\(259\) 352.000 1.35907
\(260\) 0 0
\(261\) 240.000 26.8328i 0.919540 0.102808i
\(262\) 0 0
\(263\) 210.190i 0.799203i −0.916689 0.399602i \(-0.869149\pi\)
0.916689 0.399602i \(-0.130851\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −40.0000 35.7771i −0.149813 0.133997i
\(268\) 0 0
\(269\) 134.164i 0.498751i 0.968407 + 0.249376i \(0.0802254\pi\)
−0.968407 + 0.249376i \(0.919775\pi\)
\(270\) 0 0
\(271\) −398.000 −1.46863 −0.734317 0.678806i \(-0.762498\pi\)
−0.734317 + 0.678806i \(0.762498\pi\)
\(272\) 0 0
\(273\) −192.000 + 214.663i −0.703297 + 0.786310i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −292.000 −1.05415 −0.527076 0.849818i \(-0.676712\pi\)
−0.527076 + 0.849818i \(0.676712\pi\)
\(278\) 0 0
\(279\) −34.0000 304.105i −0.121864 1.08998i
\(280\) 0 0
\(281\) 53.6656i 0.190981i 0.995430 + 0.0954904i \(0.0304419\pi\)
−0.995430 + 0.0954904i \(0.969558\pi\)
\(282\) 0 0
\(283\) 52.0000 0.183746 0.0918728 0.995771i \(-0.470715\pi\)
0.0918728 + 0.995771i \(0.470715\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 143.108i 0.498635i
\(288\) 0 0
\(289\) −691.000 −2.39100
\(290\) 0 0
\(291\) 64.0000 71.5542i 0.219931 0.245891i
\(292\) 0 0
\(293\) 389.076i 1.32790i −0.747775 0.663952i \(-0.768878\pi\)
0.747775 0.663952i \(-0.231122\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 140.000 196.774i 0.471380 0.662539i
\(298\) 0 0
\(299\) 53.6656i 0.179484i
\(300\) 0 0
\(301\) −224.000 −0.744186
\(302\) 0 0
\(303\) 340.000 + 304.105i 1.12211 + 1.00365i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −492.000 −1.60261 −0.801303 0.598259i \(-0.795859\pi\)
−0.801303 + 0.598259i \(0.795859\pi\)
\(308\) 0 0
\(309\) 208.000 232.551i 0.673139 0.752592i
\(310\) 0 0
\(311\) 482.991i 1.55302i −0.630102 0.776512i \(-0.716987\pi\)
0.630102 0.776512i \(-0.283013\pi\)
\(312\) 0 0
\(313\) −568.000 −1.81470 −0.907348 0.420380i \(-0.861897\pi\)
−0.907348 + 0.420380i \(0.861897\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 612.683i 1.93275i −0.257132 0.966376i \(-0.582777\pi\)
0.257132 0.966376i \(-0.417223\pi\)
\(318\) 0 0
\(319\) 240.000 0.752351
\(320\) 0 0
\(321\) 330.000 + 295.161i 1.02804 + 0.919505i
\(322\) 0 0
\(323\) 187.830i 0.581516i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 148.000 165.469i 0.452599 0.506021i
\(328\) 0 0
\(329\) 35.7771i 0.108745i
\(330\) 0 0
\(331\) 202.000 0.610272 0.305136 0.952309i \(-0.401298\pi\)
0.305136 + 0.952309i \(0.401298\pi\)
\(332\) 0 0
\(333\) −44.0000 393.548i −0.132132 1.18183i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 368.000 1.09199 0.545994 0.837789i \(-0.316152\pi\)
0.545994 + 0.837789i \(0.316152\pi\)
\(338\) 0 0
\(339\) −90.0000 80.4984i −0.265487 0.237459i
\(340\) 0 0
\(341\) 304.105i 0.891804i
\(342\) 0 0
\(343\) −272.000 −0.793003
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 254.912i 0.734616i −0.930099 0.367308i \(-0.880280\pi\)
0.930099 0.367308i \(-0.119720\pi\)
\(348\) 0 0
\(349\) 118.000 0.338109 0.169054 0.985607i \(-0.445929\pi\)
0.169054 + 0.985607i \(0.445929\pi\)
\(350\) 0 0
\(351\) 264.000 + 187.830i 0.752137 + 0.535127i
\(352\) 0 0
\(353\) 31.3050i 0.0886826i −0.999016 0.0443413i \(-0.985881\pi\)
0.999016 0.0443413i \(-0.0141189\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −560.000 500.879i −1.56863 1.40302i
\(358\) 0 0
\(359\) 53.6656i 0.149486i −0.997203 0.0747432i \(-0.976186\pi\)
0.997203 0.0747432i \(-0.0238137\pi\)
\(360\) 0 0
\(361\) −325.000 −0.900277
\(362\) 0 0
\(363\) −82.0000 + 91.6788i −0.225895 + 0.252559i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −352.000 −0.959128 −0.479564 0.877507i \(-0.659205\pi\)
−0.479564 + 0.877507i \(0.659205\pi\)
\(368\) 0 0
\(369\) 160.000 17.8885i 0.433604 0.0484784i
\(370\) 0 0
\(371\) 321.994i 0.867908i
\(372\) 0 0
\(373\) 132.000 0.353887 0.176944 0.984221i \(-0.443379\pi\)
0.176944 + 0.984221i \(0.443379\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 321.994i 0.854095i
\(378\) 0 0
\(379\) −394.000 −1.03958 −0.519789 0.854295i \(-0.673989\pi\)
−0.519789 + 0.854295i \(0.673989\pi\)
\(380\) 0 0
\(381\) 32.0000 35.7771i 0.0839895 0.0939031i
\(382\) 0 0
\(383\) 76.0263i 0.198502i 0.995062 + 0.0992511i \(0.0316447\pi\)
−0.995062 + 0.0992511i \(0.968355\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 28.0000 + 250.440i 0.0723514 + 0.647131i
\(388\) 0 0
\(389\) 277.272i 0.712783i −0.934337 0.356391i \(-0.884007\pi\)
0.934337 0.356391i \(-0.115993\pi\)
\(390\) 0 0
\(391\) −140.000 −0.358056
\(392\) 0 0
\(393\) −180.000 160.997i −0.458015 0.409661i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −652.000 −1.64232 −0.821159 0.570700i \(-0.806672\pi\)
−0.821159 + 0.570700i \(0.806672\pi\)
\(398\) 0 0
\(399\) −96.0000 + 107.331i −0.240602 + 0.269001i
\(400\) 0 0
\(401\) 178.885i 0.446098i −0.974807 0.223049i \(-0.928399\pi\)
0.974807 0.223049i \(-0.0716011\pi\)
\(402\) 0 0
\(403\) 408.000 1.01241
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 393.548i 0.966948i
\(408\) 0 0
\(409\) 206.000 0.503667 0.251834 0.967771i \(-0.418966\pi\)
0.251834 + 0.967771i \(0.418966\pi\)
\(410\) 0 0
\(411\) −390.000 348.827i −0.948905 0.848727i
\(412\) 0 0
\(413\) 787.096i 1.90580i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −236.000 + 263.856i −0.565947 + 0.632748i
\(418\) 0 0
\(419\) 205.718i 0.490974i 0.969400 + 0.245487i \(0.0789479\pi\)
−0.969400 + 0.245487i \(0.921052\pi\)
\(420\) 0 0
\(421\) −38.0000 −0.0902613 −0.0451306 0.998981i \(-0.514370\pi\)
−0.0451306 + 0.998981i \(0.514370\pi\)
\(422\) 0 0
\(423\) −40.0000 + 4.47214i −0.0945626 + 0.0105724i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 592.000 1.38642
\(428\) 0 0
\(429\) 240.000 + 214.663i 0.559441 + 0.500379i
\(430\) 0 0
\(431\) 608.210i 1.41116i 0.708630 + 0.705581i \(0.249314\pi\)
−0.708630 + 0.705581i \(0.750686\pi\)
\(432\) 0 0
\(433\) 272.000 0.628176 0.314088 0.949394i \(-0.398301\pi\)
0.314088 + 0.949394i \(0.398301\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 26.8328i 0.0614023i
\(438\) 0 0
\(439\) 366.000 0.833713 0.416856 0.908972i \(-0.363132\pi\)
0.416856 + 0.908972i \(0.363132\pi\)
\(440\) 0 0
\(441\) −15.0000 134.164i −0.0340136 0.304227i
\(442\) 0 0
\(443\) 576.906i 1.30227i 0.758962 + 0.651135i \(0.225707\pi\)
−0.758962 + 0.651135i \(0.774293\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 220.000 + 196.774i 0.492170 + 0.440210i
\(448\) 0 0
\(449\) 429.325i 0.956181i 0.878311 + 0.478090i \(0.158671\pi\)
−0.878311 + 0.478090i \(0.841329\pi\)
\(450\) 0 0
\(451\) 160.000 0.354767
\(452\) 0 0
\(453\) −68.0000 + 76.0263i −0.150110 + 0.167829i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 104.000 0.227571 0.113786 0.993505i \(-0.463702\pi\)
0.113786 + 0.993505i \(0.463702\pi\)
\(458\) 0 0
\(459\) −490.000 + 688.709i −1.06754 + 1.50046i
\(460\) 0 0
\(461\) 509.823i 1.10591i 0.833212 + 0.552954i \(0.186499\pi\)
−0.833212 + 0.552954i \(0.813501\pi\)
\(462\) 0 0
\(463\) 96.0000 0.207343 0.103672 0.994612i \(-0.466941\pi\)
0.103672 + 0.994612i \(0.466941\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 147.580i 0.316018i −0.987438 0.158009i \(-0.949492\pi\)
0.987438 0.158009i \(-0.0505075\pi\)
\(468\) 0 0
\(469\) −736.000 −1.56930
\(470\) 0 0
\(471\) 184.000 205.718i 0.390658 0.436769i
\(472\) 0 0
\(473\) 250.440i 0.529471i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 360.000 40.2492i 0.754717 0.0843799i
\(478\) 0 0
\(479\) 572.433i 1.19506i −0.801847 0.597530i \(-0.796149\pi\)
0.801847 0.597530i \(-0.203851\pi\)
\(480\) 0 0
\(481\) 528.000 1.09771
\(482\) 0 0
\(483\) −80.0000 71.5542i −0.165631 0.148145i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 648.000 1.33060 0.665298 0.746578i \(-0.268305\pi\)
0.665298 + 0.746578i \(0.268305\pi\)
\(488\) 0 0
\(489\) 136.000 152.053i 0.278119 0.310946i
\(490\) 0 0
\(491\) 134.164i 0.273247i 0.990623 + 0.136623i \(0.0436250\pi\)
−0.990623 + 0.136623i \(0.956375\pi\)
\(492\) 0 0
\(493\) −840.000 −1.70385
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 429.325i 0.863833i
\(498\) 0 0
\(499\) 486.000 0.973948 0.486974 0.873416i \(-0.338101\pi\)
0.486974 + 0.873416i \(0.338101\pi\)
\(500\) 0 0
\(501\) −150.000 134.164i −0.299401 0.267793i
\(502\) 0 0
\(503\) 791.568i 1.57369i 0.617148 + 0.786847i \(0.288288\pi\)
−0.617148 + 0.786847i \(0.711712\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 50.0000 55.9017i 0.0986193 0.110260i
\(508\) 0 0
\(509\) 26.8328i 0.0527167i 0.999653 + 0.0263584i \(0.00839110\pi\)
−0.999653 + 0.0263584i \(0.991609\pi\)
\(510\) 0 0
\(511\) 448.000 0.876712
\(512\) 0 0
\(513\) 132.000 + 93.9149i 0.257310 + 0.183070i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −40.0000 −0.0773694
\(518\) 0 0
\(519\) −170.000 152.053i −0.327553 0.292972i
\(520\) 0 0
\(521\) 983.870i 1.88843i −0.329336 0.944213i \(-0.606825\pi\)
0.329336 0.944213i \(-0.393175\pi\)
\(522\) 0 0
\(523\) 292.000 0.558317 0.279159 0.960245i \(-0.409944\pi\)
0.279159 + 0.960245i \(0.409944\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1064.37i 2.01967i
\(528\) 0 0
\(529\) 509.000 0.962193
\(530\) 0 0
\(531\) −880.000 + 98.3870i −1.65725 + 0.185286i
\(532\) 0 0
\(533\) 214.663i 0.402744i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 580.000 + 518.768i 1.08007 + 0.966048i
\(538\) 0 0
\(539\) 134.164i 0.248913i
\(540\) 0 0
\(541\) −86.0000 −0.158965 −0.0794824 0.996836i \(-0.525327\pi\)
−0.0794824 + 0.996836i \(0.525327\pi\)
\(542\) 0 0
\(543\) 332.000 371.187i 0.611418 0.683586i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 164.000 0.299817 0.149909 0.988700i \(-0.452102\pi\)
0.149909 + 0.988700i \(0.452102\pi\)
\(548\) 0 0
\(549\) −74.0000 661.876i −0.134791 1.20560i
\(550\) 0 0
\(551\) 160.997i 0.292190i
\(552\) 0 0
\(553\) 624.000 1.12839
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 362.243i 0.650347i −0.945654 0.325173i \(-0.894577\pi\)
0.945654 0.325173i \(-0.105423\pi\)
\(558\) 0 0
\(559\) −336.000 −0.601073
\(560\) 0 0
\(561\) −560.000 + 626.099i −0.998217 + 1.11604i
\(562\) 0 0
\(563\) 997.286i 1.77138i −0.464278 0.885689i \(-0.653686\pi\)
0.464278 0.885689i \(-0.346314\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −632.000 + 143.108i −1.11464 + 0.252396i
\(568\) 0 0
\(569\) 626.099i 1.10035i −0.835049 0.550175i \(-0.814561\pi\)
0.835049 0.550175i \(-0.185439\pi\)
\(570\) 0 0
\(571\) 394.000 0.690018 0.345009 0.938599i \(-0.387876\pi\)
0.345009 + 0.938599i \(0.387876\pi\)
\(572\) 0 0
\(573\) 480.000 + 429.325i 0.837696 + 0.749258i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 608.000 1.05373 0.526863 0.849950i \(-0.323368\pi\)
0.526863 + 0.849950i \(0.323368\pi\)
\(578\) 0 0
\(579\) −64.0000 + 71.5542i −0.110535 + 0.123582i
\(580\) 0 0
\(581\) 822.873i 1.41630i
\(582\) 0 0
\(583\) 360.000 0.617496
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 711.070i 1.21136i 0.795707 + 0.605681i \(0.207099\pi\)
−0.795707 + 0.605681i \(0.792901\pi\)
\(588\) 0 0
\(589\) 204.000 0.346350
\(590\) 0 0
\(591\) 10.0000 + 8.94427i 0.0169205 + 0.0151341i
\(592\) 0 0
\(593\) 603.738i 1.01811i −0.860734 0.509054i \(-0.829995\pi\)
0.860734 0.509054i \(-0.170005\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 228.000 254.912i 0.381910 0.426988i
\(598\) 0 0
\(599\) 53.6656i 0.0895920i 0.998996 + 0.0447960i \(0.0142638\pi\)
−0.998996 + 0.0447960i \(0.985736\pi\)
\(600\) 0 0
\(601\) 434.000 0.722130 0.361065 0.932541i \(-0.382413\pi\)
0.361065 + 0.932541i \(0.382413\pi\)
\(602\) 0 0
\(603\) 92.0000 + 822.873i 0.152570 + 1.36463i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −656.000 −1.08072 −0.540362 0.841432i \(-0.681713\pi\)
−0.540362 + 0.841432i \(0.681713\pi\)
\(608\) 0 0
\(609\) −480.000 429.325i −0.788177 0.704967i
\(610\) 0 0
\(611\) 53.6656i 0.0878325i
\(612\) 0 0
\(613\) −844.000 −1.37684 −0.688418 0.725315i \(-0.741694\pi\)
−0.688418 + 0.725315i \(0.741694\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 460.630i 0.746564i 0.927718 + 0.373282i \(0.121768\pi\)
−0.927718 + 0.373282i \(0.878232\pi\)
\(618\) 0 0
\(619\) 1046.00 1.68982 0.844911 0.534907i \(-0.179653\pi\)
0.844911 + 0.534907i \(0.179653\pi\)
\(620\) 0 0
\(621\) −70.0000 + 98.3870i −0.112721 + 0.158433i
\(622\) 0 0
\(623\) 143.108i 0.229708i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 120.000 + 107.331i 0.191388 + 0.171182i
\(628\) 0 0
\(629\) 1377.42i 2.18985i
\(630\) 0 0
\(631\) −46.0000 −0.0729002 −0.0364501 0.999335i \(-0.511605\pi\)
−0.0364501 + 0.999335i \(0.511605\pi\)
\(632\) 0 0
\(633\) 12.0000 13.4164i 0.0189573 0.0211950i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 180.000 0.282575
\(638\) 0 0
\(639\) 480.000 53.6656i 0.751174 0.0839838i
\(640\) 0 0
\(641\) 1144.87i 1.78606i 0.449994 + 0.893032i \(0.351426\pi\)
−0.449994 + 0.893032i \(0.648574\pi\)
\(642\) 0 0
\(643\) −804.000 −1.25039 −0.625194 0.780469i \(-0.714980\pi\)
−0.625194 + 0.780469i \(0.714980\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 576.906i 0.891662i −0.895117 0.445831i \(-0.852908\pi\)
0.895117 0.445831i \(-0.147092\pi\)
\(648\) 0 0
\(649\) −880.000 −1.35593
\(650\) 0 0
\(651\) −544.000 + 608.210i −0.835637 + 0.934271i
\(652\) 0 0
\(653\) 1077.78i 1.65051i 0.564758 + 0.825256i \(0.308969\pi\)
−0.564758 + 0.825256i \(0.691031\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −56.0000 500.879i −0.0852359 0.762373i
\(658\) 0 0
\(659\) 813.929i 1.23510i 0.786533 + 0.617548i \(0.211874\pi\)
−0.786533 + 0.617548i \(0.788126\pi\)
\(660\) 0 0
\(661\) 1082.00 1.63691 0.818457 0.574568i \(-0.194830\pi\)
0.818457 + 0.574568i \(0.194830\pi\)
\(662\) 0 0
\(663\) −840.000 751.319i −1.26697 1.13321i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −120.000 −0.179910
\(668\) 0 0
\(669\) −544.000 + 608.210i −0.813154 + 0.909134i
\(670\) 0 0
\(671\) 661.876i 0.986403i
\(672\) 0 0
\(673\) 1056.00 1.56909 0.784547 0.620070i \(-0.212896\pi\)
0.784547 + 0.620070i \(0.212896\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 612.683i 0.904996i −0.891765 0.452498i \(-0.850533\pi\)
0.891765 0.452498i \(-0.149467\pi\)
\(678\) 0 0
\(679\) −256.000 −0.377025
\(680\) 0 0
\(681\) −550.000 491.935i −0.807636 0.722371i
\(682\) 0 0
\(683\) 317.522i 0.464893i −0.972609 0.232446i \(-0.925327\pi\)
0.972609 0.232446i \(-0.0746730\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 308.000 344.354i 0.448326 0.501244i
\(688\) 0 0
\(689\) 482.991i 0.701002i
\(690\) 0 0
\(691\) 922.000 1.33430 0.667149 0.744924i \(-0.267514\pi\)
0.667149 + 0.744924i \(0.267514\pi\)
\(692\) 0 0
\(693\) −640.000 + 71.5542i −0.923521 + 0.103253i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −560.000 −0.803443
\(698\) 0 0
\(699\) −410.000 366.715i −0.586552 0.524628i
\(700\) 0 0
\(701\) 474.046i 0.676243i 0.941102 + 0.338122i \(0.109791\pi\)
−0.941102 + 0.338122i \(0.890209\pi\)
\(702\) 0 0
\(703\) 264.000 0.375533
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1216.42i 1.72054i
\(708\) 0 0
\(709\) 966.000 1.36248 0.681241 0.732059i \(-0.261441\pi\)
0.681241 + 0.732059i \(0.261441\pi\)
\(710\) 0 0
\(711\) −78.0000 697.653i −0.109705 0.981228i
\(712\) 0 0
\(713\) 152.053i 0.213258i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −400.000 357.771i −0.557880 0.498983i
\(718\) 0 0
\(719\) 1109.09i 1.54254i −0.636505 0.771272i \(-0.719621\pi\)
0.636505 0.771272i \(-0.280379\pi\)
\(720\) 0 0
\(721\) −832.000 −1.15395
\(722\) 0 0
\(723\) 412.000 460.630i 0.569848 0.637109i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 408.000 0.561210 0.280605 0.959823i \(-0.409465\pi\)
0.280605 + 0.959823i \(0.409465\pi\)
\(728\) 0 0
\(729\) 239.000 + 688.709i 0.327846 + 0.944731i
\(730\) 0 0
\(731\) 876.539i 1.19910i
\(732\) 0 0
\(733\) −164.000 −0.223738 −0.111869 0.993723i \(-0.535684\pi\)
−0.111869 + 0.993723i \(0.535684\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 822.873i 1.11652i
\(738\) 0 0
\(739\) −1082.00 −1.46414 −0.732070 0.681229i \(-0.761446\pi\)
−0.732070 + 0.681229i \(0.761446\pi\)
\(740\) 0 0
\(741\) −144.000 + 160.997i −0.194332 + 0.217270i
\(742\) 0 0
\(743\) 1140.39i 1.53485i −0.641138 0.767426i \(-0.721537\pi\)
0.641138 0.767426i \(-0.278463\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −920.000 + 102.859i −1.23159 + 0.137696i
\(748\) 0 0
\(749\) 1180.64i 1.57629i
\(750\) 0 0
\(751\) −958.000 −1.27563 −0.637816 0.770189i \(-0.720162\pi\)
−0.637816 + 0.770189i \(0.720162\pi\)
\(752\) 0 0
\(753\) −60.0000 53.6656i −0.0796813 0.0712691i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −772.000 −1.01982 −0.509908 0.860229i \(-0.670320\pi\)
−0.509908 + 0.860229i \(0.670320\pi\)
\(758\) 0 0
\(759\) −80.0000 + 89.4427i −0.105402 + 0.117843i
\(760\) 0 0
\(761\) 1126.98i 1.48092i 0.672102 + 0.740459i \(0.265392\pi\)
−0.672102 + 0.740459i \(0.734608\pi\)
\(762\) 0 0
\(763\) −592.000 −0.775885
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1180.64i 1.53930i
\(768\) 0 0
\(769\) 1326.00 1.72432 0.862159 0.506638i \(-0.169112\pi\)
0.862159 + 0.506638i \(0.169112\pi\)
\(770\) 0 0
\(771\) 90.0000 + 80.4984i 0.116732 + 0.104408i
\(772\) 0 0
\(773\) 147.580i 0.190919i 0.995433 + 0.0954596i \(0.0304321\pi\)
−0.995433 + 0.0954596i \(0.969568\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −704.000 + 787.096i −0.906049 + 1.01299i
\(778\) 0 0
\(779\) 107.331i 0.137781i
\(780\) 0 0
\(781\) 480.000 0.614597
\(782\) 0 0
\(783\) −420.000 + 590.322i −0.536398 + 0.753923i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1132.00 −1.43837 −0.719187 0.694817i \(-0.755485\pi\)
−0.719187 + 0.694817i \(0.755485\pi\)
\(788\) 0 0
\(789\) 470.000 + 420.381i 0.595691 + 0.532802i
\(790\) 0 0
\(791\) 321.994i 0.407072i
\(792\) 0 0
\(793\) 888.000 1.11980
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 612.683i 0.768736i −0.923180 0.384368i \(-0.874419\pi\)
0.923180 0.384368i \(-0.125581\pi\)
\(798\) 0 0
\(799\) 140.000 0.175219
\(800\) 0 0
\(801\) 160.000