Properties

Label 3150.2.b.c.251.8
Level 3150
Weight 2
Character 3150.251
Analytic conductor 25.153
Analytic rank 0
Dimension 8
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.8
Root \(-0.437016 - 0.437016i\)
Character \(\chi\) = 3150.251
Dual form 3150.2.b.c.251.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +(2.12132 + 1.58114i) q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +(2.12132 + 1.58114i) q^{7} -1.00000i q^{8} -1.41421i q^{11} -3.16228i q^{13} +(-1.58114 + 2.12132i) q^{14} +1.00000 q^{16} -4.47214 q^{17} +1.41421 q^{22} -6.00000i q^{23} +3.16228 q^{26} +(-2.12132 - 1.58114i) q^{28} -2.82843i q^{29} +1.00000i q^{32} -4.47214i q^{34} -4.24264 q^{37} -9.48683 q^{41} -8.48528 q^{43} +1.41421i q^{44} +6.00000 q^{46} -4.47214 q^{47} +(2.00000 + 6.70820i) q^{49} +3.16228i q^{52} -6.00000i q^{53} +(1.58114 - 2.12132i) q^{56} +2.82843 q^{58} +9.48683 q^{59} -13.4164i q^{61} -1.00000 q^{64} +4.47214 q^{68} -5.65685i q^{71} +6.32456i q^{73} -4.24264i q^{74} +(2.23607 - 3.00000i) q^{77} +4.00000 q^{79} -9.48683i q^{82} -8.94427 q^{83} -8.48528i q^{86} -1.41421 q^{88} -9.48683 q^{89} +(5.00000 - 6.70820i) q^{91} +6.00000i q^{92} -4.47214i q^{94} +12.6491i q^{97} +(-6.70820 + 2.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + O(q^{10}) \) \( 8q - 8q^{4} + 8q^{16} + 48q^{46} + 16q^{49} - 8q^{64} + 32q^{79} + 40q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.12132 + 1.58114i 0.801784 + 0.597614i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421i 0.426401i −0.977008 0.213201i \(-0.931611\pi\)
0.977008 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) 3.16228i 0.877058i −0.898717 0.438529i \(-0.855500\pi\)
0.898717 0.438529i \(-0.144500\pi\)
\(14\) −1.58114 + 2.12132i −0.422577 + 0.566947i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.41421 0.301511
\(23\) 6.00000i 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.16228 0.620174
\(27\) 0 0
\(28\) −2.12132 1.58114i −0.400892 0.298807i
\(29\) 2.82843i 0.525226i −0.964901 0.262613i \(-0.915416\pi\)
0.964901 0.262613i \(-0.0845842\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 4.47214i 0.766965i
\(35\) 0 0
\(36\) 0 0
\(37\) −4.24264 −0.697486 −0.348743 0.937218i \(-0.613391\pi\)
−0.348743 + 0.937218i \(0.613391\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.48683 −1.48159 −0.740797 0.671729i \(-0.765552\pi\)
−0.740797 + 0.671729i \(0.765552\pi\)
\(42\) 0 0
\(43\) −8.48528 −1.29399 −0.646997 0.762493i \(-0.723975\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 1.41421i 0.213201i
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −4.47214 −0.652328 −0.326164 0.945313i \(-0.605756\pi\)
−0.326164 + 0.945313i \(0.605756\pi\)
\(48\) 0 0
\(49\) 2.00000 + 6.70820i 0.285714 + 0.958315i
\(50\) 0 0
\(51\) 0 0
\(52\) 3.16228i 0.438529i
\(53\) 6.00000i 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.58114 2.12132i 0.211289 0.283473i
\(57\) 0 0
\(58\) 2.82843 0.371391
\(59\) 9.48683 1.23508 0.617540 0.786539i \(-0.288129\pi\)
0.617540 + 0.786539i \(0.288129\pi\)
\(60\) 0 0
\(61\) 13.4164i 1.71780i −0.512148 0.858898i \(-0.671150\pi\)
0.512148 0.858898i \(-0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 4.47214 0.542326
\(69\) 0 0
\(70\) 0 0
\(71\) 5.65685i 0.671345i −0.941979 0.335673i \(-0.891036\pi\)
0.941979 0.335673i \(-0.108964\pi\)
\(72\) 0 0
\(73\) 6.32456i 0.740233i 0.928985 + 0.370117i \(0.120682\pi\)
−0.928985 + 0.370117i \(0.879318\pi\)
\(74\) 4.24264i 0.493197i
\(75\) 0 0
\(76\) 0 0
\(77\) 2.23607 3.00000i 0.254824 0.341882i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 9.48683i 1.04765i
\(83\) −8.94427 −0.981761 −0.490881 0.871227i \(-0.663325\pi\)
−0.490881 + 0.871227i \(0.663325\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.48528i 0.914991i
\(87\) 0 0
\(88\) −1.41421 −0.150756
\(89\) −9.48683 −1.00560 −0.502801 0.864402i \(-0.667697\pi\)
−0.502801 + 0.864402i \(0.667697\pi\)
\(90\) 0 0
\(91\) 5.00000 6.70820i 0.524142 0.703211i
\(92\) 6.00000i 0.625543i
\(93\) 0 0
\(94\) 4.47214i 0.461266i
\(95\) 0 0
\(96\) 0 0
\(97\) 12.6491i 1.28432i 0.766570 + 0.642161i \(0.221962\pi\)
−0.766570 + 0.642161i \(0.778038\pi\)
\(98\) −6.70820 + 2.00000i −0.677631 + 0.202031i
\(99\) 0 0
\(100\) 0 0
\(101\) −18.9737 −1.88795 −0.943975 0.330017i \(-0.892946\pi\)
−0.943975 + 0.330017i \(0.892946\pi\)
\(102\) 0 0
\(103\) 15.8114i 1.55794i 0.627060 + 0.778971i \(0.284258\pi\)
−0.627060 + 0.778971i \(0.715742\pi\)
\(104\) −3.16228 −0.310087
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000i 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.12132 + 1.58114i 0.200446 + 0.149404i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.82843i 0.262613i
\(117\) 0 0
\(118\) 9.48683i 0.873334i
\(119\) −9.48683 7.07107i −0.869657 0.648204i
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 13.4164 1.21466
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −21.2132 −1.88237 −0.941184 0.337895i \(-0.890285\pi\)
−0.941184 + 0.337895i \(0.890285\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 9.48683 0.828868 0.414434 0.910079i \(-0.363979\pi\)
0.414434 + 0.910079i \(0.363979\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 4.47214i 0.383482i
\(137\) 18.0000i 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) 13.4164i 1.13796i −0.822350 0.568982i \(-0.807337\pi\)
0.822350 0.568982i \(-0.192663\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 5.65685 0.474713
\(143\) −4.47214 −0.373979
\(144\) 0 0
\(145\) 0 0
\(146\) −6.32456 −0.523424
\(147\) 0 0
\(148\) 4.24264 0.348743
\(149\) 11.3137i 0.926855i −0.886135 0.463428i \(-0.846619\pi\)
0.886135 0.463428i \(-0.153381\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 3.00000 + 2.23607i 0.241747 + 0.180187i
\(155\) 0 0
\(156\) 0 0
\(157\) 15.8114i 1.26189i −0.775829 0.630943i \(-0.782668\pi\)
0.775829 0.630943i \(-0.217332\pi\)
\(158\) 4.00000i 0.318223i
\(159\) 0 0
\(160\) 0 0
\(161\) 9.48683 12.7279i 0.747667 1.00310i
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 9.48683 0.740797
\(165\) 0 0
\(166\) 8.94427i 0.694210i
\(167\) 8.94427 0.692129 0.346064 0.938211i \(-0.387518\pi\)
0.346064 + 0.938211i \(0.387518\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 8.48528 0.646997
\(173\) −8.94427 −0.680020 −0.340010 0.940422i \(-0.610431\pi\)
−0.340010 + 0.940422i \(0.610431\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.41421i 0.106600i
\(177\) 0 0
\(178\) 9.48683i 0.711068i
\(179\) 18.3848i 1.37414i 0.726590 + 0.687071i \(0.241104\pi\)
−0.726590 + 0.687071i \(0.758896\pi\)
\(180\) 0 0
\(181\) 13.4164i 0.997234i −0.866822 0.498617i \(-0.833841\pi\)
0.866822 0.498617i \(-0.166159\pi\)
\(182\) 6.70820 + 5.00000i 0.497245 + 0.370625i
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 0 0
\(187\) 6.32456i 0.462497i
\(188\) 4.47214 0.326164
\(189\) 0 0
\(190\) 0 0
\(191\) 22.6274i 1.63726i −0.574320 0.818631i \(-0.694733\pi\)
0.574320 0.818631i \(-0.305267\pi\)
\(192\) 0 0
\(193\) 8.48528 0.610784 0.305392 0.952227i \(-0.401213\pi\)
0.305392 + 0.952227i \(0.401213\pi\)
\(194\) −12.6491 −0.908153
\(195\) 0 0
\(196\) −2.00000 6.70820i −0.142857 0.479157i
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 0 0
\(199\) 26.8328i 1.90213i −0.308994 0.951064i \(-0.599992\pi\)
0.308994 0.951064i \(-0.400008\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 18.9737i 1.33498i
\(203\) 4.47214 6.00000i 0.313882 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) −15.8114 −1.10163
\(207\) 0 0
\(208\) 3.16228i 0.219265i
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 10.0000i 0.677285i
\(219\) 0 0
\(220\) 0 0
\(221\) 14.1421i 0.951303i
\(222\) 0 0
\(223\) 22.1359i 1.48233i −0.671322 0.741166i \(-0.734273\pi\)
0.671322 0.741166i \(-0.265727\pi\)
\(224\) −1.58114 + 2.12132i −0.105644 + 0.141737i
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −17.8885 −1.18730 −0.593652 0.804722i \(-0.702314\pi\)
−0.593652 + 0.804722i \(0.702314\pi\)
\(228\) 0 0
\(229\) 13.4164i 0.886581i 0.896378 + 0.443291i \(0.146189\pi\)
−0.896378 + 0.443291i \(0.853811\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.82843 −0.185695
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −9.48683 −0.617540
\(237\) 0 0
\(238\) 7.07107 9.48683i 0.458349 0.614940i
\(239\) 11.3137i 0.731823i −0.930650 0.365911i \(-0.880757\pi\)
0.930650 0.365911i \(-0.119243\pi\)
\(240\) 0 0
\(241\) 13.4164i 0.864227i 0.901819 + 0.432113i \(0.142232\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 9.00000i 0.578542i
\(243\) 0 0
\(244\) 13.4164i 0.858898i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 9.48683 0.598804 0.299402 0.954127i \(-0.403213\pi\)
0.299402 + 0.954127i \(0.403213\pi\)
\(252\) 0 0
\(253\) −8.48528 −0.533465
\(254\) 21.2132i 1.33103i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −4.47214 −0.278964 −0.139482 0.990225i \(-0.544544\pi\)
−0.139482 + 0.990225i \(0.544544\pi\)
\(258\) 0 0
\(259\) −9.00000 6.70820i −0.559233 0.416828i
\(260\) 0 0
\(261\) 0 0
\(262\) 9.48683i 0.586098i
\(263\) 6.00000i 0.369976i −0.982741 0.184988i \(-0.940775\pi\)
0.982741 0.184988i \(-0.0592246\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.9737 1.15684 0.578422 0.815737i \(-0.303669\pi\)
0.578422 + 0.815737i \(0.303669\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −4.47214 −0.271163
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) 21.2132 1.27458 0.637289 0.770625i \(-0.280056\pi\)
0.637289 + 0.770625i \(0.280056\pi\)
\(278\) 13.4164 0.804663
\(279\) 0 0
\(280\) 0 0
\(281\) 1.41421i 0.0843649i −0.999110 0.0421825i \(-0.986569\pi\)
0.999110 0.0421825i \(-0.0134311\pi\)
\(282\) 0 0
\(283\) 6.32456i 0.375956i 0.982173 + 0.187978i \(0.0601933\pi\)
−0.982173 + 0.187978i \(0.939807\pi\)
\(284\) 5.65685i 0.335673i
\(285\) 0 0
\(286\) 4.47214i 0.264443i
\(287\) −20.1246 15.0000i −1.18792 0.885422i
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 0 0
\(292\) 6.32456i 0.370117i
\(293\) 4.47214 0.261265 0.130632 0.991431i \(-0.458299\pi\)
0.130632 + 0.991431i \(0.458299\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4.24264i 0.246598i
\(297\) 0 0
\(298\) 11.3137 0.655386
\(299\) −18.9737 −1.09728
\(300\) 0 0
\(301\) −18.0000 13.4164i −1.03750 0.773309i
\(302\) 20.0000i 1.15087i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.6491i 0.721923i 0.932581 + 0.360961i \(0.117551\pi\)
−0.932581 + 0.360961i \(0.882449\pi\)
\(308\) −2.23607 + 3.00000i −0.127412 + 0.170941i
\(309\) 0 0
\(310\) 0 0
\(311\) −18.9737 −1.07590 −0.537949 0.842977i \(-0.680801\pi\)
−0.537949 + 0.842977i \(0.680801\pi\)
\(312\) 0 0
\(313\) 31.6228i 1.78743i −0.448640 0.893713i \(-0.648091\pi\)
0.448640 0.893713i \(-0.351909\pi\)
\(314\) 15.8114 0.892288
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 0 0
\(322\) 12.7279 + 9.48683i 0.709299 + 0.528681i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 9.48683i 0.523823i
\(329\) −9.48683 7.07107i −0.523026 0.389841i
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 8.94427 0.490881
\(333\) 0 0
\(334\) 8.94427i 0.489409i
\(335\) 0 0
\(336\) 0 0
\(337\) −25.4558 −1.38667 −0.693334 0.720616i \(-0.743859\pi\)
−0.693334 + 0.720616i \(0.743859\pi\)
\(338\) 3.00000i 0.163178i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −6.36396 + 17.3925i −0.343622 + 0.939108i
\(344\) 8.48528i 0.457496i
\(345\) 0 0
\(346\) 8.94427i 0.480847i
\(347\) 12.0000i 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) 13.4164i 0.718164i 0.933306 + 0.359082i \(0.116910\pi\)
−0.933306 + 0.359082i \(0.883090\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.41421 0.0753778
\(353\) 31.3050 1.66619 0.833097 0.553127i \(-0.186565\pi\)
0.833097 + 0.553127i \(0.186565\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 9.48683 0.502801
\(357\) 0 0
\(358\) −18.3848 −0.971666
\(359\) 31.1127i 1.64207i 0.570881 + 0.821033i \(0.306602\pi\)
−0.570881 + 0.821033i \(0.693398\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 13.4164 0.705151
\(363\) 0 0
\(364\) −5.00000 + 6.70820i −0.262071 + 0.351605i
\(365\) 0 0
\(366\) 0 0
\(367\) 22.1359i 1.15549i 0.816218 + 0.577743i \(0.196067\pi\)
−0.816218 + 0.577743i \(0.803933\pi\)
\(368\) 6.00000i 0.312772i
\(369\) 0 0
\(370\) 0 0
\(371\) 9.48683 12.7279i 0.492532 0.660801i
\(372\) 0 0
\(373\) 21.2132 1.09838 0.549189 0.835698i \(-0.314937\pi\)
0.549189 + 0.835698i \(0.314937\pi\)
\(374\) −6.32456 −0.327035
\(375\) 0 0
\(376\) 4.47214i 0.230633i
\(377\) −8.94427 −0.460653
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 22.6274 1.15772
\(383\) 4.47214 0.228515 0.114258 0.993451i \(-0.463551\pi\)
0.114258 + 0.993451i \(0.463551\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.48528i 0.431889i
\(387\) 0 0
\(388\) 12.6491i 0.642161i
\(389\) 11.3137i 0.573628i −0.957986 0.286814i \(-0.907404\pi\)
0.957986 0.286814i \(-0.0925961\pi\)
\(390\) 0 0
\(391\) 26.8328i 1.35699i
\(392\) 6.70820 2.00000i 0.338815 0.101015i
\(393\) 0 0
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) 0 0
\(397\) 22.1359i 1.11097i 0.831526 + 0.555486i \(0.187468\pi\)
−0.831526 + 0.555486i \(0.812532\pi\)
\(398\) 26.8328 1.34501
\(399\) 0 0
\(400\) 0 0
\(401\) 1.41421i 0.0706225i −0.999376 0.0353112i \(-0.988758\pi\)
0.999376 0.0353112i \(-0.0112422\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 18.9737 0.943975
\(405\) 0 0
\(406\) 6.00000 + 4.47214i 0.297775 + 0.221948i
\(407\) 6.00000i 0.297409i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 15.8114i 0.778971i
\(413\) 20.1246 + 15.0000i 0.990267 + 0.738102i
\(414\) 0 0
\(415\) 0 0
\(416\) 3.16228 0.155043
\(417\) 0 0
\(418\) 0 0
\(419\) 9.48683 0.463462 0.231731 0.972780i \(-0.425561\pi\)
0.231731 + 0.972780i \(0.425561\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 20.0000i 0.973585i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 21.2132 28.4605i 1.02658 1.37730i
\(428\) 12.0000i 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 36.7696i 1.77113i 0.464518 + 0.885564i \(0.346227\pi\)
−0.464518 + 0.885564i \(0.653773\pi\)
\(432\) 0 0
\(433\) 6.32456i 0.303939i 0.988385 + 0.151969i \(0.0485615\pi\)
−0.988385 + 0.151969i \(0.951438\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −14.1421 −0.672673
\(443\) 24.0000i 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 22.1359 1.04817
\(447\) 0 0
\(448\) −2.12132 1.58114i −0.100223 0.0747018i
\(449\) 18.3848i 0.867631i 0.901002 + 0.433816i \(0.142833\pi\)
−0.901002 + 0.433816i \(0.857167\pi\)
\(450\) 0 0
\(451\) 13.4164i 0.631754i
\(452\) 6.00000i 0.282216i
\(453\) 0 0
\(454\) 17.8885i 0.839551i
\(455\) 0 0
\(456\) 0 0
\(457\) −16.9706 −0.793849 −0.396925 0.917851i \(-0.629923\pi\)
−0.396925 + 0.917851i \(0.629923\pi\)
\(458\) −13.4164 −0.626908
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −21.2132 −0.985861 −0.492931 0.870069i \(-0.664074\pi\)
−0.492931 + 0.870069i \(0.664074\pi\)
\(464\) 2.82843i 0.131306i
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 8.94427 0.413892 0.206946 0.978352i \(-0.433648\pi\)
0.206946 + 0.978352i \(0.433648\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 9.48683i 0.436667i
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) 0 0
\(476\) 9.48683 + 7.07107i 0.434828 + 0.324102i
\(477\) 0 0
\(478\) 11.3137 0.517477
\(479\) −18.9737 −0.866929 −0.433464 0.901171i \(-0.642709\pi\)
−0.433464 + 0.901171i \(0.642709\pi\)
\(480\) 0 0
\(481\) 13.4164i 0.611736i
\(482\) −13.4164 −0.611101
\(483\) 0 0
\(484\) −9.00000 −0.409091
\(485\) 0 0
\(486\) 0 0
\(487\) 4.24264 0.192252 0.0961262 0.995369i \(-0.469355\pi\)
0.0961262 + 0.995369i \(0.469355\pi\)
\(488\) −13.4164 −0.607332
\(489\) 0 0
\(490\) 0 0
\(491\) 15.5563i 0.702048i 0.936366 + 0.351024i \(0.114166\pi\)
−0.936366 + 0.351024i \(0.885834\pi\)
\(492\) 0 0
\(493\) 12.6491i 0.569687i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.94427 12.0000i 0.401205 0.538274i
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 9.48683i 0.423418i
\(503\) −22.3607 −0.997013 −0.498507 0.866886i \(-0.666118\pi\)
−0.498507 + 0.866886i \(0.666118\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8.48528i 0.377217i
\(507\) 0 0
\(508\) 21.2132 0.941184
\(509\) 18.9737 0.840993 0.420496 0.907294i \(-0.361856\pi\)
0.420496 + 0.907294i \(0.361856\pi\)
\(510\) 0 0
\(511\) −10.0000 + 13.4164i −0.442374 + 0.593507i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 4.47214i 0.197257i
\(515\) 0 0
\(516\) 0 0
\(517\) 6.32456i 0.278154i
\(518\) 6.70820 9.00000i 0.294742 0.395437i
\(519\) 0 0
\(520\) 0 0
\(521\) 28.4605 1.24688 0.623439 0.781872i \(-0.285735\pi\)
0.623439 + 0.781872i \(0.285735\pi\)
\(522\) 0 0
\(523\) 6.32456i 0.276553i 0.990394 + 0.138277i \(0.0441563\pi\)
−0.990394 + 0.138277i \(0.955844\pi\)
\(524\) −9.48683 −0.414434
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) 0 0
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 30.0000i 1.29944i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 18.9737i 0.818013i
\(539\) 9.48683 2.82843i 0.408627 0.121829i
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 4.47214i 0.191741i
\(545\) 0 0
\(546\) 0 0
\(547\) −42.4264 −1.81402 −0.907011 0.421107i \(-0.861642\pi\)
−0.907011 + 0.421107i \(0.861642\pi\)
\(548\) 18.0000i 0.768922i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.48528 + 6.32456i 0.360831 + 0.268947i
\(554\) 21.2132i 0.901263i
\(555\) 0 0
\(556\) 13.4164i 0.568982i
\(557\) 18.0000i 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) 26.8328i 1.13491i
\(560\) 0 0
\(561\) 0 0
\(562\) 1.41421 0.0596550
\(563\) −35.7771 −1.50782 −0.753912 0.656975i \(-0.771836\pi\)
−0.753912 + 0.656975i \(0.771836\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −6.32456 −0.265841
\(567\) 0 0
\(568\) −5.65685 −0.237356
\(569\) 9.89949i 0.415008i 0.978234 + 0.207504i \(0.0665341\pi\)
−0.978234 + 0.207504i \(0.933466\pi\)
\(570\) 0 0
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 4.47214 0.186989
\(573\) 0 0
\(574\) 15.0000 20.1246i 0.626088 0.839985i
\(575\) 0 0
\(576\) 0 0
\(577\) 25.2982i 1.05318i −0.850120 0.526589i \(-0.823471\pi\)
0.850120 0.526589i \(-0.176529\pi\)
\(578\) 3.00000i 0.124784i
\(579\) 0 0
\(580\) 0 0
\(581\) −18.9737 14.1421i −0.787160 0.586715i
\(582\) 0 0
\(583\) −8.48528 −0.351424
\(584\) 6.32456 0.261712
\(585\) 0 0
\(586\) 4.47214i 0.184742i
\(587\) 8.94427 0.369170 0.184585 0.982817i \(-0.440906\pi\)
0.184585 + 0.982817i \(0.440906\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −4.24264 −0.174371
\(593\) −22.3607 −0.918243 −0.459122 0.888373i \(-0.651836\pi\)
−0.459122 + 0.888373i \(0.651836\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 11.3137i 0.463428i
\(597\) 0 0
\(598\) 18.9737i 0.775891i
\(599\) 2.82843i 0.115566i −0.998329 0.0577832i \(-0.981597\pi\)
0.998329 0.0577832i \(-0.0184032\pi\)
\(600\) 0 0
\(601\) 13.4164i 0.547267i 0.961834 + 0.273633i \(0.0882255\pi\)
−0.961834 + 0.273633i \(0.911775\pi\)
\(602\) 13.4164 18.0000i 0.546812 0.733625i
\(603\) 0 0
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) 0 0
\(607\) 15.8114i 0.641764i −0.947119 0.320882i \(-0.896021\pi\)
0.947119 0.320882i \(-0.103979\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 14.1421i 0.572130i
\(612\) 0 0
\(613\) 21.2132 0.856793 0.428397 0.903591i \(-0.359079\pi\)
0.428397 + 0.903591i \(0.359079\pi\)
\(614\) −12.6491 −0.510477
\(615\) 0 0
\(616\) −3.00000 2.23607i −0.120873 0.0900937i
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 0 0
\(619\) 40.2492i 1.61775i 0.587979 + 0.808876i \(0.299924\pi\)
−0.587979 + 0.808876i \(0.700076\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 18.9737i 0.760775i
\(623\) −20.1246 15.0000i −0.806276 0.600962i
\(624\) 0 0
\(625\) 0 0
\(626\) 31.6228 1.26390
\(627\) 0 0
\(628\) 15.8114i 0.630943i
\(629\) 18.9737 0.756530
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 4.00000i 0.159111i
\(633\) 0 0
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) 0 0
\(637\) 21.2132 6.32456i 0.840498 0.250588i
\(638\) 4.00000i 0.158362i
\(639\) 0 0
\(640\) 0 0
\(641\) 15.5563i 0.614439i 0.951639 + 0.307219i \(0.0993986\pi\)
−0.951639 + 0.307219i \(0.900601\pi\)
\(642\) 0 0
\(643\) 6.32456i 0.249416i 0.992193 + 0.124708i \(0.0397994\pi\)
−0.992193 + 0.124708i \(0.960201\pi\)
\(644\) −9.48683 + 12.7279i −0.373834 + 0.501550i
\(645\) 0 0
\(646\) 0 0
\(647\) 22.3607 0.879089 0.439545 0.898221i \(-0.355140\pi\)
0.439545 + 0.898221i \(0.355140\pi\)
\(648\) 0 0
\(649\) 13.4164i 0.526640i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.0000i 0.939193i 0.882881 + 0.469596i \(0.155601\pi\)
−0.882881 + 0.469596i \(0.844399\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −9.48683 −0.370399
\(657\) 0 0
\(658\) 7.07107 9.48683i 0.275659 0.369835i
\(659\) 32.5269i 1.26707i −0.773715 0.633534i \(-0.781604\pi\)
0.773715 0.633534i \(-0.218396\pi\)
\(660\) 0 0
\(661\) 40.2492i 1.56551i −0.622328 0.782757i \(-0.713813\pi\)
0.622328 0.782757i \(-0.286187\pi\)
\(662\) 10.0000i 0.388661i
\(663\) 0 0
\(664\) 8.94427i 0.347105i
\(665\) 0 0
\(666\) 0 0
\(667\) −16.9706 −0.657103
\(668\) −8.94427 −0.346064
\(669\) 0 0
\(670\) 0 0
\(671\) −18.9737 −0.732470
\(672\) 0 0
\(673\) 8.48528 0.327084 0.163542 0.986536i \(-0.447708\pi\)
0.163542 + 0.986536i \(0.447708\pi\)
\(674\) 25.4558i 0.980522i
\(675\) 0 0
\(676\) −3.00000 −0.115385
\(677\) 22.3607 0.859391 0.429695 0.902974i \(-0.358621\pi\)
0.429695 + 0.902974i \(0.358621\pi\)
\(678\) 0 0
\(679\) −20.0000 + 26.8328i −0.767530 + 1.02975i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 36.0000i 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −17.3925 6.36396i −0.664050 0.242977i
\(687\) 0 0
\(688\) −8.48528 −0.323498
\(689\) −18.9737 −0.722839
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 8.94427 0.340010
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 42.4264 1.60701
\(698\) −13.4164 −0.507819
\(699\) 0 0
\(700\) 0 0
\(701\) 36.7696i 1.38877i 0.719605 + 0.694383i \(0.244323\pi\)
−0.719605 + 0.694383i \(0.755677\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.41421i 0.0533002i
\(705\) 0 0
\(706\) 31.3050i 1.17818i
\(707\) −40.2492 30.0000i −1.51373 1.12827i
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 9.48683i 0.355534i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 18.3848i 0.687071i
\(717\) 0 0
\(718\) −31.1127 −1.16112
\(719\) 37.9473 1.41520 0.707598 0.706615i \(-0.249779\pi\)
0.707598 + 0.706615i \(0.249779\pi\)
\(720\) 0 0
\(721\) −25.0000 + 33.5410i −0.931049 + 1.24913i
\(722\) 19.0000i 0.707107i
\(723\) 0 0
\(724\) 13.4164i 0.498617i
\(725\) 0 0
\(726\) 0 0
\(727\) 3.16228i 0.117282i 0.998279 + 0.0586412i \(0.0186768\pi\)
−0.998279 + 0.0586412i \(0.981323\pi\)
\(728\) −6.70820 5.00000i −0.248623 0.185312i
\(729\) 0 0
\(730\) 0 0
\(731\) 37.9473 1.40353
\(732\) 0 0
\(733\) 41.1096i 1.51842i −0.650847 0.759209i \(-0.725586\pi\)
0.650847 0.759209i \(-0.274414\pi\)
\(734\) −22.1359 −0.817053
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 0 0
\(738\) 0 0
\(739\) 34.0000 1.25071 0.625355 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 12.7279 + 9.48683i 0.467257 + 0.348273i
\(743\) 24.0000i 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 21.2132i 0.776671i
\(747\) 0 0
\(748\) 6.32456i 0.231249i
\(749\) 18.9737 25.4558i 0.693283 0.930136i
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) −4.47214 −0.163082
\(753\) 0 0
\(754\) 8.94427i 0.325731i
\(755\) 0 0
\(756\) 0 0
\(757\) −46.6690 −1.69622 −0.848108 0.529824i \(-0.822258\pi\)
−0.848108 + 0.529824i \(0.822258\pi\)
\(758\) 20.0000i 0.726433i
\(759\) 0 0
\(760\) 0 0
\(761\) 9.48683 0.343897 0.171949 0.985106i \(-0.444994\pi\)
0.171949 + 0.985106i \(0.444994\pi\)
\(762\) 0 0
\(763\) 21.2132 + 15.8114i 0.767970 + 0.572411i
\(764\) 22.6274i 0.818631i
\(765\) 0 0
\(766\) 4.47214i 0.161585i
\(767\) 30.0000i 1.08324i
\(768\) 0 0
\(769\) 40.2492i 1.45142i −0.687999 0.725712i \(-0.741510\pi\)
0.687999 0.725712i \(-0.258490\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8.48528 −0.305392
\(773\) 31.3050 1.12596 0.562980 0.826470i \(-0.309655\pi\)
0.562980 + 0.826470i \(0.309655\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12.6491 0.454077
\(777\) 0 0
\(778\) 11.3137 0.405616
\(779\) 0 0
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) −26.8328 −0.959540
\(783\) 0 0
\(784\) 2.00000 + 6.70820i 0.0714286 + 0.239579i
\(785\) 0 0
\(786\) 0 0
\(787\) 31.6228i 1.12723i 0.826038 + 0.563615i \(0.190590\pi\)
−0.826038 + 0.563615i \(0.809410\pi\)
\(788\) 12.0000i 0.427482i
\(789\) 0 0
\(790\) 0 0
\(791\) −9.48683 + 12.7279i −0.337313 + 0.452553i
\(792\) 0 0
\(793\) −42.4264 −1.50661
\(794\) −22.1359 −0.785575
\(795\) 0 0
\(796\) 26.8328i 0.951064i
\(797\) 8.94427 0.316822 0.158411 0.987373i \(-0.449363\pi\)
0.158411 + 0.987373i \(0.449363\pi\)
\(798\) 0 0
\(799\) 20.0000 0.707549
\(800\) 0 0
\(801\) 0 0
\(802\) 1.41421 0.0499376
\(803\) 8.94427 0.315637
\(804\) 0 0
\(805\) 0 0
\(806\)