Properties

Label 3150.2.b.c.251.4
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 251.4
Root \(1.14412 - 1.14412i\)
Character \(\chi\) = 3150.251
Dual form 3150.2.b.c.251.7

$q$-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 \(0.0683888\pi\)
−0.977008 + 0.213201i \(0.931611\pi\)
\(12\) 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 \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 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 \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 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 \(0.0845842\pi\)
−0.964901 + 0.262613i \(0.915416\pi\)
\(30\) 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 \(-0.234448\pi\)
0.740797 + 0.671729i \(0.234448\pi\)
\(42\) 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 \(-0.394244\pi\)
0.326164 + 0.945313i \(0.394244\pi\)
\(48\) 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 \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 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 \(-0.711871\pi\)
−0.617540 + 0.786539i \(0.711871\pi\)
\(60\) 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 \(0.108964\pi\)
−0.941979 + 0.335673i \(0.891036\pi\)
\(72\) 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 \(-0.336675\pi\)
0.490881 + 0.871227i \(0.336675\pi\)
\(84\) 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 \(-0.332303\pi\)
0.502801 + 0.864402i \(0.332303\pi\)
\(90\) 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 \(-0.107054\pi\)
0.943975 + 0.330017i \(0.107054\pi\)
\(102\) 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 \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 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 \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 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 \(-0.636021\pi\)
−0.414434 + 0.910079i \(0.636021\pi\)
\(132\) 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 \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 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 \(0.153381\pi\)
−0.886135 + 0.463428i \(0.846619\pi\)
\(150\) 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 \(-0.612482\pi\)
−0.346064 + 0.938211i \(0.612482\pi\)
\(168\) 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 \(-0.389569\pi\)
0.340010 + 0.940422i \(0.389569\pi\)
\(174\) 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 \(-0.758896\pi\)
0.726590 0.687071i \(-0.241104\pi\)
\(180\) 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 \(0.305267\pi\)
−0.574320 + 0.818631i \(0.694733\pi\)
\(192\) 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 \(-0.859401\pi\)
0.904024 0.427482i \(-0.140599\pi\)
\(198\) 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 \(-0.297686\pi\)
0.593652 + 0.804722i \(0.297686\pi\)
\(228\) 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 \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 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 \(0.119243\pi\)
−0.930650 + 0.365911i \(0.880757\pi\)
\(240\) 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 \(-0.596787\pi\)
−0.299402 + 0.954127i \(0.596787\pi\)
\(252\) 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 \(-0.455456\pi\)
0.139482 + 0.990225i \(0.455456\pi\)
\(258\) 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 \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.9737 −1.15684 −0.578422 0.815737i \(-0.696331\pi\)
−0.578422 + 0.815737i \(0.696331\pi\)
\(270\) 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 \(0.0134311\pi\)
−0.999110 + 0.0421825i \(0.986569\pi\)
\(282\) 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 \(-0.541701\pi\)
−0.130632 + 0.991431i \(0.541701\pi\)
\(294\) 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 \(-0.319199\pi\)
0.537949 + 0.842977i \(0.319199\pi\)
\(312\) 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 \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) 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 \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 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 \(-0.813435\pi\)
−0.833097 + 0.553127i \(0.813435\pi\)
\(354\) 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 \(-0.693398\pi\)
0.570881 0.821033i \(-0.306602\pi\)
\(360\) 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 \(-0.536449\pi\)
−0.114258 + 0.993451i \(0.536449\pi\)
\(384\) 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 \(0.0925961\pi\)
−0.957986 + 0.286814i \(0.907404\pi\)
\(390\) 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 \(0.0112422\pi\)
−0.999376 + 0.0353112i \(0.988758\pi\)
\(402\) 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 \(-0.574439\pi\)
−0.231731 + 0.972780i \(0.574439\pi\)
\(420\) 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 \(-0.653773\pi\)
0.464518 0.885564i \(-0.346227\pi\)
\(432\) 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 \(0.193110\pi\)
−0.821549 + 0.570137i \(0.806890\pi\)
\(444\) 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 \(-0.857167\pi\)
0.901002 0.433816i \(-0.142833\pi\)
\(450\) 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 \(-0.566352\pi\)
−0.206946 + 0.978352i \(0.566352\pi\)
\(468\) 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 \(-0.357291\pi\)
0.433464 + 0.901171i \(0.357291\pi\)
\(480\) 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 \(-0.885834\pi\)
0.936366 0.351024i \(-0.114166\pi\)
\(492\) 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 \(-0.333882\pi\)
0.498507 + 0.866886i \(0.333882\pi\)
\(504\) 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 \(-0.638144\pi\)
−0.420496 + 0.907294i \(0.638144\pi\)
\(510\) 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 \(-0.714265\pi\)
−0.623439 + 0.781872i \(0.714265\pi\)
\(522\) 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 \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 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 \(-0.228164\pi\)
0.753912 + 0.656975i \(0.228164\pi\)
\(564\) 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 \(-0.933466\pi\)
0.978234 0.207504i \(-0.0665341\pi\)
\(570\) 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 \(-0.559094\pi\)
−0.184585 + 0.982817i \(0.559094\pi\)
\(588\) 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 \(-0.348164\pi\)
0.459122 + 0.888373i \(0.348164\pi\)
\(594\) 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 \(0.0184032\pi\)
−0.998329 + 0.0577832i \(0.981597\pi\)
\(600\) 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 \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 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 \(-0.900601\pi\)
0.951639 0.307219i \(-0.0993986\pi\)
\(642\) 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 \(-0.644860\pi\)
−0.439545 + 0.898221i \(0.644860\pi\)
\(648\) 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 \(-0.844399\pi\)
0.882881 0.469596i \(-0.155601\pi\)
\(654\) 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 \(0.218396\pi\)
−0.773715 + 0.633534i \(0.781604\pi\)
\(660\) 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 \(-0.641379\pi\)
−0.429695 + 0.902974i \(0.641379\pi\)
\(678\) 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 \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) 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 \(-0.755677\pi\)
0.719605 0.694383i \(-0.244323\pi\)
\(702\) 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 \(-0.750221\pi\)
−0.707598 + 0.706615i \(0.750221\pi\)
\(720\) 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 \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 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 \(-0.555006\pi\)
−0.171949 + 0.985106i \(0.555006\pi\)
\(762\) 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 \(-0.690345\pi\)
−0.562980 + 0.826470i \(0.690345\pi\)
\(774\) 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 \(-0.550637\pi\)
−0.158411 + 0.987373i \(0.550637\pi\)
\(798\) 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
\(806\) 0