Properties

Label 3150.2.b.b.251.1
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.1
Root \(1.14412 - 1.14412i\)
Character \(\chi\) = 3150.251
Dual form 3150.2.b.b.251.6

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +(-1.41421 - 2.23607i) q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +(-1.41421 - 2.23607i) q^{7} +1.00000i q^{8} -5.65685i q^{11} -4.47214i q^{13} +(-2.23607 + 1.41421i) q^{14} +1.00000 q^{16} +3.16228 q^{17} -3.16228i q^{19} -5.65685 q^{22} -4.00000i q^{23} -4.47214 q^{26} +(1.41421 + 2.23607i) q^{28} +2.82843i q^{29} +6.32456i q^{31} -1.00000i q^{32} -3.16228i q^{34} +9.89949 q^{37} -3.16228 q^{38} +4.47214 q^{41} -1.41421 q^{43} +5.65685i q^{44} -4.00000 q^{46} -9.48683 q^{47} +(-3.00000 + 6.32456i) q^{49} +4.47214i q^{52} -4.00000i q^{53} +(2.23607 - 1.41421i) q^{56} +2.82843 q^{58} -4.47214 q^{59} +9.48683i q^{61} +6.32456 q^{62} -1.00000 q^{64} +7.07107 q^{67} -3.16228 q^{68} -1.41421i q^{71} -13.4164i q^{73} -9.89949i q^{74} +3.16228i q^{76} +(-12.6491 + 8.00000i) q^{77} -6.00000 q^{79} -4.47214i q^{82} -12.6491 q^{83} +1.41421i q^{86} +5.65685 q^{88} +4.47214 q^{89} +(-10.0000 + 6.32456i) q^{91} +4.00000i q^{92} +9.48683i q^{94} +(6.32456 + 3.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + O(q^{10}) \) \( 8q - 8q^{4} + 8q^{16} - 32q^{46} - 24q^{49} - 8q^{64} - 48q^{79} - 80q^{91} + O(q^{100}) \)

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
\(7\) −1.41421 2.23607i −0.534522 0.845154i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685i 1.70561i −0.522233 0.852803i \(-0.674901\pi\)
0.522233 0.852803i \(-0.325099\pi\)
\(12\) 0 0
\(13\) 4.47214i 1.24035i −0.784465 0.620174i \(-0.787062\pi\)
0.784465 0.620174i \(-0.212938\pi\)
\(14\) −2.23607 + 1.41421i −0.597614 + 0.377964i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.16228 0.766965 0.383482 0.923548i \(-0.374725\pi\)
0.383482 + 0.923548i \(0.374725\pi\)
\(18\) 0 0
\(19\) 3.16228i 0.725476i −0.931891 0.362738i \(-0.881842\pi\)
0.931891 0.362738i \(-0.118158\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.65685 −1.20605
\(23\) 4.00000i 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.47214 −0.877058
\(27\) 0 0
\(28\) 1.41421 + 2.23607i 0.267261 + 0.422577i
\(29\) 2.82843i 0.525226i 0.964901 + 0.262613i \(0.0845842\pi\)
−0.964901 + 0.262613i \(0.915416\pi\)
\(30\) 0 0
\(31\) 6.32456i 1.13592i 0.823055 + 0.567962i \(0.192268\pi\)
−0.823055 + 0.567962i \(0.807732\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 3.16228i 0.542326i
\(35\) 0 0
\(36\) 0 0
\(37\) 9.89949 1.62747 0.813733 0.581238i \(-0.197432\pi\)
0.813733 + 0.581238i \(0.197432\pi\)
\(38\) −3.16228 −0.512989
\(39\) 0 0
\(40\) 0 0
\(41\) 4.47214 0.698430 0.349215 0.937043i \(-0.386448\pi\)
0.349215 + 0.937043i \(0.386448\pi\)
\(42\) 0 0
\(43\) −1.41421 −0.215666 −0.107833 0.994169i \(-0.534391\pi\)
−0.107833 + 0.994169i \(0.534391\pi\)
\(44\) 5.65685i 0.852803i
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −9.48683 −1.38380 −0.691898 0.721995i \(-0.743225\pi\)
−0.691898 + 0.721995i \(0.743225\pi\)
\(48\) 0 0
\(49\) −3.00000 + 6.32456i −0.428571 + 0.903508i
\(50\) 0 0
\(51\) 0 0
\(52\) 4.47214i 0.620174i
\(53\) 4.00000i 0.549442i −0.961524 0.274721i \(-0.911414\pi\)
0.961524 0.274721i \(-0.0885855\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.23607 1.41421i 0.298807 0.188982i
\(57\) 0 0
\(58\) 2.82843 0.371391
\(59\) −4.47214 −0.582223 −0.291111 0.956689i \(-0.594025\pi\)
−0.291111 + 0.956689i \(0.594025\pi\)
\(60\) 0 0
\(61\) 9.48683i 1.21466i 0.794448 + 0.607332i \(0.207760\pi\)
−0.794448 + 0.607332i \(0.792240\pi\)
\(62\) 6.32456 0.803219
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 7.07107 0.863868 0.431934 0.901905i \(-0.357831\pi\)
0.431934 + 0.901905i \(0.357831\pi\)
\(68\) −3.16228 −0.383482
\(69\) 0 0
\(70\) 0 0
\(71\) 1.41421i 0.167836i −0.996473 0.0839181i \(-0.973257\pi\)
0.996473 0.0839181i \(-0.0267434\pi\)
\(72\) 0 0
\(73\) 13.4164i 1.57027i −0.619324 0.785136i \(-0.712593\pi\)
0.619324 0.785136i \(-0.287407\pi\)
\(74\) 9.89949i 1.15079i
\(75\) 0 0
\(76\) 3.16228i 0.362738i
\(77\) −12.6491 + 8.00000i −1.44150 + 0.911685i
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.47214i 0.493865i
\(83\) −12.6491 −1.38842 −0.694210 0.719772i \(-0.744246\pi\)
−0.694210 + 0.719772i \(0.744246\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.41421i 0.152499i
\(87\) 0 0
\(88\) 5.65685 0.603023
\(89\) 4.47214 0.474045 0.237023 0.971504i \(-0.423828\pi\)
0.237023 + 0.971504i \(0.423828\pi\)
\(90\) 0 0
\(91\) −10.0000 + 6.32456i −1.04828 + 0.662994i
\(92\) 4.00000i 0.417029i
\(93\) 0 0
\(94\) 9.48683i 0.978492i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 6.32456 + 3.00000i 0.638877 + 0.303046i
\(99\) 0 0
\(100\) 0 0
\(101\) −17.8885 −1.77998 −0.889988 0.455983i \(-0.849288\pi\)
−0.889988 + 0.455983i \(0.849288\pi\)
\(102\) 0 0
\(103\) 8.94427i 0.881305i 0.897678 + 0.440653i \(0.145253\pi\)
−0.897678 + 0.440653i \(0.854747\pi\)
\(104\) 4.47214 0.438529
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) 18.0000i 1.74013i −0.492941 0.870063i \(-0.664078\pi\)
0.492941 0.870063i \(-0.335922\pi\)
\(108\) 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
\(112\) −1.41421 2.23607i −0.133631 0.211289i
\(113\) 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
\(118\) 4.47214i 0.411693i
\(119\) −4.47214 7.07107i −0.409960 0.648204i
\(120\) 0 0
\(121\) −21.0000 −1.90909
\(122\) 9.48683 0.858898
\(123\) 0 0
\(124\) 6.32456i 0.567962i
\(125\) 0 0
\(126\) 0 0
\(127\) −14.1421 −1.25491 −0.627456 0.778652i \(-0.715904\pi\)
−0.627456 + 0.778652i \(0.715904\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −17.8885 −1.56293 −0.781465 0.623949i \(-0.785527\pi\)
−0.781465 + 0.623949i \(0.785527\pi\)
\(132\) 0 0
\(133\) −7.07107 + 4.47214i −0.613139 + 0.387783i
\(134\) 7.07107i 0.610847i
\(135\) 0 0
\(136\) 3.16228i 0.271163i
\(137\) 2.00000i 0.170872i −0.996344 0.0854358i \(-0.972772\pi\)
0.996344 0.0854358i \(-0.0272282\pi\)
\(138\) 0 0
\(139\) 15.8114i 1.34110i 0.741862 + 0.670552i \(0.233943\pi\)
−0.741862 + 0.670552i \(0.766057\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.41421 −0.118678
\(143\) −25.2982 −2.11554
\(144\) 0 0
\(145\) 0 0
\(146\) −13.4164 −1.11035
\(147\) 0 0
\(148\) −9.89949 −0.813733
\(149\) 16.9706i 1.39028i −0.718873 0.695141i \(-0.755342\pi\)
0.718873 0.695141i \(-0.244658\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 3.16228 0.256495
\(153\) 0 0
\(154\) 8.00000 + 12.6491i 0.644658 + 1.01929i
\(155\) 0 0
\(156\) 0 0
\(157\) 4.47214i 0.356915i 0.983948 + 0.178458i \(0.0571108\pi\)
−0.983948 + 0.178458i \(0.942889\pi\)
\(158\) 6.00000i 0.477334i
\(159\) 0 0
\(160\) 0 0
\(161\) −8.94427 + 5.65685i −0.704907 + 0.445823i
\(162\) 0 0
\(163\) 21.2132 1.66155 0.830773 0.556611i \(-0.187899\pi\)
0.830773 + 0.556611i \(0.187899\pi\)
\(164\) −4.47214 −0.349215
\(165\) 0 0
\(166\) 12.6491i 0.981761i
\(167\) −9.48683 −0.734113 −0.367057 0.930199i \(-0.619634\pi\)
−0.367057 + 0.930199i \(0.619634\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) 1.41421 0.107833
\(173\) 25.2982 1.92339 0.961694 0.274125i \(-0.0883882\pi\)
0.961694 + 0.274125i \(0.0883882\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.65685i 0.426401i
\(177\) 0 0
\(178\) 4.47214i 0.335201i
\(179\) 2.82843i 0.211407i 0.994398 + 0.105703i \(0.0337094\pi\)
−0.994398 + 0.105703i \(0.966291\pi\)
\(180\) 0 0
\(181\) 3.16228i 0.235050i −0.993070 0.117525i \(-0.962504\pi\)
0.993070 0.117525i \(-0.0374961\pi\)
\(182\) 6.32456 + 10.0000i 0.468807 + 0.741249i
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) 17.8885i 1.30814i
\(188\) 9.48683 0.691898
\(189\) 0 0
\(190\) 0 0
\(191\) 1.41421i 0.102329i 0.998690 + 0.0511645i \(0.0162933\pi\)
−0.998690 + 0.0511645i \(0.983707\pi\)
\(192\) 0 0
\(193\) 8.48528 0.610784 0.305392 0.952227i \(-0.401213\pi\)
0.305392 + 0.952227i \(0.401213\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 3.00000 6.32456i 0.214286 0.451754i
\(197\) 12.0000i 0.854965i −0.904024 0.427482i \(-0.859401\pi\)
0.904024 0.427482i \(-0.140599\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 17.8885i 1.25863i
\(203\) 6.32456 4.00000i 0.443897 0.280745i
\(204\) 0 0
\(205\) 0 0
\(206\) 8.94427 0.623177
\(207\) 0 0
\(208\) 4.47214i 0.310087i
\(209\) −17.8885 −1.23738
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 4.00000i 0.274721i
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) 0 0
\(217\) 14.1421 8.94427i 0.960031 0.607177i
\(218\) 10.0000i 0.677285i
\(219\) 0 0
\(220\) 0 0
\(221\) 14.1421i 0.951303i
\(222\) 0 0
\(223\) 8.94427i 0.598953i −0.954104 0.299476i \(-0.903188\pi\)
0.954104 0.299476i \(-0.0968120\pi\)
\(224\) −2.23607 + 1.41421i −0.149404 + 0.0944911i
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 18.9737 1.25933 0.629663 0.776868i \(-0.283193\pi\)
0.629663 + 0.776868i \(0.283193\pi\)
\(228\) 0 0
\(229\) 15.8114i 1.04485i 0.852686 + 0.522423i \(0.174972\pi\)
−0.852686 + 0.522423i \(0.825028\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.82843 −0.185695
\(233\) 14.0000i 0.917170i 0.888650 + 0.458585i \(0.151644\pi\)
−0.888650 + 0.458585i \(0.848356\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4.47214 0.291111
\(237\) 0 0
\(238\) −7.07107 + 4.47214i −0.458349 + 0.289886i
\(239\) 4.24264i 0.274434i 0.990541 + 0.137217i \(0.0438157\pi\)
−0.990541 + 0.137217i \(0.956184\pi\)
\(240\) 0 0
\(241\) 25.2982i 1.62960i 0.579741 + 0.814801i \(0.303154\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 21.0000i 1.34993i
\(243\) 0 0
\(244\) 9.48683i 0.607332i
\(245\) 0 0
\(246\) 0 0
\(247\) −14.1421 −0.899843
\(248\) −6.32456 −0.401610
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −22.6274 −1.42257
\(254\) 14.1421i 0.887357i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −15.8114 −0.986287 −0.493144 0.869948i \(-0.664152\pi\)
−0.493144 + 0.869948i \(0.664152\pi\)
\(258\) 0 0
\(259\) −14.0000 22.1359i −0.869918 1.37546i
\(260\) 0 0
\(261\) 0 0
\(262\) 17.8885i 1.10516i
\(263\) 16.0000i 0.986602i 0.869859 + 0.493301i \(0.164210\pi\)
−0.869859 + 0.493301i \(0.835790\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.47214 + 7.07107i 0.274204 + 0.433555i
\(267\) 0 0
\(268\) −7.07107 −0.431934
\(269\) 17.8885 1.09068 0.545342 0.838214i \(-0.316400\pi\)
0.545342 + 0.838214i \(0.316400\pi\)
\(270\) 0 0
\(271\) 6.32456i 0.384189i 0.981376 + 0.192095i \(0.0615281\pi\)
−0.981376 + 0.192095i \(0.938472\pi\)
\(272\) 3.16228 0.191741
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) 0 0
\(277\) 21.2132 1.27458 0.637289 0.770625i \(-0.280056\pi\)
0.637289 + 0.770625i \(0.280056\pi\)
\(278\) 15.8114 0.948304
\(279\) 0 0
\(280\) 0 0
\(281\) 12.7279i 0.759284i −0.925133 0.379642i \(-0.876047\pi\)
0.925133 0.379642i \(-0.123953\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 1.41421i 0.0839181i
\(285\) 0 0
\(286\) 25.2982i 1.49592i
\(287\) −6.32456 10.0000i −0.373327 0.590281i
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 13.4164i 0.785136i
\(293\) 6.32456 0.369484 0.184742 0.982787i \(-0.440855\pi\)
0.184742 + 0.982787i \(0.440855\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 9.89949i 0.575396i
\(297\) 0 0
\(298\) −16.9706 −0.983078
\(299\) −17.8885 −1.03452
\(300\) 0 0
\(301\) 2.00000 + 3.16228i 0.115278 + 0.182271i
\(302\) 10.0000i 0.575435i
\(303\) 0 0
\(304\) 3.16228i 0.181369i
\(305\) 0 0
\(306\) 0 0
\(307\) 26.8328i 1.53143i −0.643180 0.765715i \(-0.722385\pi\)
0.643180 0.765715i \(-0.277615\pi\)
\(308\) 12.6491 8.00000i 0.720750 0.455842i
\(309\) 0 0
\(310\) 0 0
\(311\) 8.94427 0.507183 0.253592 0.967311i \(-0.418388\pi\)
0.253592 + 0.967311i \(0.418388\pi\)
\(312\) 0 0
\(313\) 13.4164i 0.758340i 0.925327 + 0.379170i \(0.123790\pi\)
−0.925327 + 0.379170i \(0.876210\pi\)
\(314\) 4.47214 0.252377
\(315\) 0 0
\(316\) 6.00000 0.337526
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 16.0000 0.895828
\(320\) 0 0
\(321\) 0 0
\(322\) 5.65685 + 8.94427i 0.315244 + 0.498445i
\(323\) 10.0000i 0.556415i
\(324\) 0 0
\(325\) 0 0
\(326\) 21.2132i 1.17489i
\(327\) 0 0
\(328\) 4.47214i 0.246932i
\(329\) 13.4164 + 21.2132i 0.739671 + 1.16952i
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 12.6491 0.694210
\(333\) 0 0
\(334\) 9.48683i 0.519096i
\(335\) 0 0
\(336\) 0 0
\(337\) −25.4558 −1.38667 −0.693334 0.720616i \(-0.743859\pi\)
−0.693334 + 0.720616i \(0.743859\pi\)
\(338\) 7.00000i 0.380750i
\(339\) 0 0
\(340\) 0 0
\(341\) 35.7771 1.93744
\(342\) 0 0
\(343\) 18.3848 2.23607i 0.992685 0.120736i
\(344\) 1.41421i 0.0762493i
\(345\) 0 0
\(346\) 25.2982i 1.36004i
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 0 0
\(349\) 15.8114i 0.846364i 0.906045 + 0.423182i \(0.139087\pi\)
−0.906045 + 0.423182i \(0.860913\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −5.65685 −0.301511
\(353\) −3.16228 −0.168311 −0.0841555 0.996453i \(-0.526819\pi\)
−0.0841555 + 0.996453i \(0.526819\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −4.47214 −0.237023
\(357\) 0 0
\(358\) 2.82843 0.149487
\(359\) 18.3848i 0.970311i 0.874428 + 0.485156i \(0.161237\pi\)
−0.874428 + 0.485156i \(0.838763\pi\)
\(360\) 0 0
\(361\) 9.00000 0.473684
\(362\) −3.16228 −0.166206
\(363\) 0 0
\(364\) 10.0000 6.32456i 0.524142 0.331497i
\(365\) 0 0
\(366\) 0 0
\(367\) 13.4164i 0.700331i 0.936688 + 0.350165i \(0.113875\pi\)
−0.936688 + 0.350165i \(0.886125\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 0 0
\(370\) 0 0
\(371\) −8.94427 + 5.65685i −0.464363 + 0.293689i
\(372\) 0 0
\(373\) −21.2132 −1.09838 −0.549189 0.835698i \(-0.685063\pi\)
−0.549189 + 0.835698i \(0.685063\pi\)
\(374\) −17.8885 −0.924995
\(375\) 0 0
\(376\) 9.48683i 0.489246i
\(377\) 12.6491 0.651462
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.41421 0.0723575
\(383\) −9.48683 −0.484755 −0.242377 0.970182i \(-0.577927\pi\)
−0.242377 + 0.970182i \(0.577927\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.48528i 0.431889i
\(387\) 0 0
\(388\) 0 0
\(389\) 16.9706i 0.860442i −0.902724 0.430221i \(-0.858436\pi\)
0.902724 0.430221i \(-0.141564\pi\)
\(390\) 0 0
\(391\) 12.6491i 0.639693i
\(392\) −6.32456 3.00000i −0.319438 0.151523i
\(393\) 0 0
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) 0 0
\(397\) 4.47214i 0.224450i −0.993683 0.112225i \(-0.964202\pi\)
0.993683 0.112225i \(-0.0357978\pi\)
\(398\) 0 0
\(399\) 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
\(403\) 28.2843 1.40894
\(404\) 17.8885 0.889988
\(405\) 0 0
\(406\) −4.00000 6.32456i −0.198517 0.313882i
\(407\) 56.0000i 2.77582i
\(408\) 0 0
\(409\) 37.9473i 1.87637i −0.346128 0.938187i \(-0.612504\pi\)
0.346128 0.938187i \(-0.387496\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.94427i 0.440653i
\(413\) 6.32456 + 10.0000i 0.311211 + 0.492068i
\(414\) 0 0
\(415\) 0 0
\(416\) −4.47214 −0.219265
\(417\) 0 0
\(418\) 17.8885i 0.874957i
\(419\) −13.4164 −0.655434 −0.327717 0.944776i \(-0.606279\pi\)
−0.327717 + 0.944776i \(0.606279\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 20.0000i 0.973585i
\(423\) 0 0
\(424\) 4.00000 0.194257
\(425\) 0 0
\(426\) 0 0
\(427\) 21.2132 13.4164i 1.02658 0.649265i
\(428\) 18.0000i 0.870063i
\(429\) 0 0
\(430\) 0 0
\(431\) 15.5563i 0.749323i −0.927162 0.374661i \(-0.877759\pi\)
0.927162 0.374661i \(-0.122241\pi\)
\(432\) 0 0
\(433\) 26.8328i 1.28950i 0.764392 + 0.644751i \(0.223039\pi\)
−0.764392 + 0.644751i \(0.776961\pi\)
\(434\) −8.94427 14.1421i −0.429339 0.678844i
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) −12.6491 −0.605089
\(438\) 0 0
\(439\) 6.32456i 0.301855i −0.988545 0.150927i \(-0.951774\pi\)
0.988545 0.150927i \(-0.0482259\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −14.1421 −0.672673
\(443\) 4.00000i 0.190046i 0.995475 + 0.0950229i \(0.0302924\pi\)
−0.995475 + 0.0950229i \(0.969708\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −8.94427 −0.423524
\(447\) 0 0
\(448\) 1.41421 + 2.23607i 0.0668153 + 0.105644i
\(449\) 9.89949i 0.467186i 0.972334 + 0.233593i \(0.0750483\pi\)
−0.972334 + 0.233593i \(0.924952\pi\)
\(450\) 0 0
\(451\) 25.2982i 1.19125i
\(452\) 6.00000i 0.282216i
\(453\) 0 0
\(454\) 18.9737i 0.890478i
\(455\) 0 0
\(456\) 0 0
\(457\) −16.9706 −0.793849 −0.396925 0.917851i \(-0.629923\pi\)
−0.396925 + 0.917851i \(0.629923\pi\)
\(458\) 15.8114 0.738818
\(459\) 0 0
\(460\) 0 0
\(461\) −8.94427 −0.416576 −0.208288 0.978068i \(-0.566789\pi\)
−0.208288 + 0.978068i \(0.566789\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 2.82843i 0.131306i
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) −12.6491 −0.585331 −0.292666 0.956215i \(-0.594542\pi\)
−0.292666 + 0.956215i \(0.594542\pi\)
\(468\) 0 0
\(469\) −10.0000 15.8114i −0.461757 0.730102i
\(470\) 0 0
\(471\) 0 0
\(472\) 4.47214i 0.205847i
\(473\) 8.00000i 0.367840i
\(474\) 0 0
\(475\) 0 0
\(476\) 4.47214 + 7.07107i 0.204980 + 0.324102i
\(477\) 0 0
\(478\) 4.24264 0.194054
\(479\) 17.8885 0.817348 0.408674 0.912680i \(-0.365991\pi\)
0.408674 + 0.912680i \(0.365991\pi\)
\(480\) 0 0
\(481\) 44.2719i 2.01862i
\(482\) 25.2982 1.15230
\(483\) 0 0
\(484\) 21.0000 0.954545
\(485\) 0 0
\(486\) 0 0
\(487\) 25.4558 1.15351 0.576757 0.816916i \(-0.304318\pi\)
0.576757 + 0.816916i \(0.304318\pi\)
\(488\) −9.48683 −0.429449
\(489\) 0 0
\(490\) 0 0
\(491\) 19.7990i 0.893516i 0.894655 + 0.446758i \(0.147421\pi\)
−0.894655 + 0.446758i \(0.852579\pi\)
\(492\) 0 0
\(493\) 8.94427i 0.402830i
\(494\) 14.1421i 0.636285i
\(495\) 0 0
\(496\) 6.32456i 0.283981i
\(497\) −3.16228 + 2.00000i −0.141848 + 0.0897123i
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −9.48683 −0.422997 −0.211498 0.977378i \(-0.567834\pi\)
−0.211498 + 0.977378i \(0.567834\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 22.6274i 1.00591i
\(507\) 0 0
\(508\) 14.1421 0.627456
\(509\) −13.4164 −0.594672 −0.297336 0.954773i \(-0.596098\pi\)
−0.297336 + 0.954773i \(0.596098\pi\)
\(510\) 0 0
\(511\) −30.0000 + 18.9737i −1.32712 + 0.839346i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 15.8114i 0.697410i
\(515\) 0 0
\(516\) 0 0
\(517\) 53.6656i 2.36021i
\(518\) −22.1359 + 14.0000i −0.972598 + 0.615125i
\(519\) 0 0
\(520\) 0 0
\(521\) 13.4164 0.587784 0.293892 0.955839i \(-0.405049\pi\)
0.293892 + 0.955839i \(0.405049\pi\)
\(522\) 0 0
\(523\) 17.8885i 0.782211i −0.920346 0.391106i \(-0.872093\pi\)
0.920346 0.391106i \(-0.127907\pi\)
\(524\) 17.8885 0.781465
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 20.0000i 0.871214i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 7.07107 4.47214i 0.306570 0.193892i
\(533\) 20.0000i 0.866296i
\(534\) 0 0
\(535\) 0 0
\(536\) 7.07107i 0.305424i
\(537\) 0 0
\(538\) 17.8885i 0.771230i
\(539\) 35.7771 + 16.9706i 1.54103 + 0.730974i
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 6.32456 0.271663
\(543\) 0 0
\(544\) 3.16228i 0.135582i
\(545\) 0 0
\(546\) 0 0
\(547\) −21.2132 −0.907011 −0.453506 0.891253i \(-0.649827\pi\)
−0.453506 + 0.891253i \(0.649827\pi\)
\(548\) 2.00000i 0.0854358i
\(549\) 0 0
\(550\) 0 0
\(551\) 8.94427 0.381039
\(552\) 0 0
\(553\) 8.48528 + 13.4164i 0.360831 + 0.570524i
\(554\) 21.2132i 0.901263i
\(555\) 0 0
\(556\) 15.8114i 0.670552i
\(557\) 38.0000i 1.61011i 0.593199 + 0.805056i \(0.297865\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) 6.32456i 0.267500i
\(560\) 0 0
\(561\) 0 0
\(562\) −12.7279 −0.536895
\(563\) 18.9737 0.799645 0.399822 0.916593i \(-0.369072\pi\)
0.399822 + 0.916593i \(0.369072\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 1.41421 0.0593391
\(569\) 24.0416i 1.00788i −0.863739 0.503939i \(-0.831884\pi\)
0.863739 0.503939i \(-0.168116\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 25.2982 1.05777
\(573\) 0 0
\(574\) −10.0000 + 6.32456i −0.417392 + 0.263982i
\(575\) 0 0
\(576\) 0 0
\(577\) 22.3607i 0.930887i −0.885078 0.465444i \(-0.845895\pi\)
0.885078 0.465444i \(-0.154105\pi\)
\(578\) 7.00000i 0.291162i
\(579\) 0 0
\(580\) 0 0
\(581\) 17.8885 + 28.2843i 0.742142 + 1.17343i
\(582\) 0 0
\(583\) −22.6274 −0.937132
\(584\) 13.4164 0.555175
\(585\) 0 0
\(586\) 6.32456i 0.261265i
\(587\) −31.6228 −1.30521 −0.652606 0.757698i \(-0.726324\pi\)
−0.652606 + 0.757698i \(0.726324\pi\)
\(588\) 0 0
\(589\) 20.0000 0.824086
\(590\) 0 0
\(591\) 0 0
\(592\) 9.89949 0.406867
\(593\) −9.48683 −0.389578 −0.194789 0.980845i \(-0.562402\pi\)
−0.194789 + 0.980845i \(0.562402\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 16.9706i 0.695141i
\(597\) 0 0
\(598\) 17.8885i 0.731517i
\(599\) 32.5269i 1.32901i −0.747282 0.664507i \(-0.768642\pi\)
0.747282 0.664507i \(-0.231358\pi\)
\(600\) 0 0
\(601\) 25.2982i 1.03194i −0.856608 0.515968i \(-0.827432\pi\)
0.856608 0.515968i \(-0.172568\pi\)
\(602\) 3.16228 2.00000i 0.128885 0.0815139i
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) 8.94427i 0.363037i 0.983388 + 0.181518i \(0.0581012\pi\)
−0.983388 + 0.181518i \(0.941899\pi\)
\(608\) −3.16228 −0.128247
\(609\) 0 0
\(610\) 0 0
\(611\) 42.4264i 1.71639i
\(612\) 0 0
\(613\) −35.3553 −1.42799 −0.713994 0.700151i \(-0.753116\pi\)
−0.713994 + 0.700151i \(0.753116\pi\)
\(614\) −26.8328 −1.08288
\(615\) 0 0
\(616\) −8.00000 12.6491i −0.322329 0.509647i
\(617\) 38.0000i 1.52982i −0.644136 0.764911i \(-0.722783\pi\)
0.644136 0.764911i \(-0.277217\pi\)
\(618\) 0 0
\(619\) 28.4605i 1.14392i 0.820280 + 0.571962i \(0.193818\pi\)
−0.820280 + 0.571962i \(0.806182\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 8.94427i 0.358633i
\(623\) −6.32456 10.0000i −0.253388 0.400642i
\(624\) 0 0
\(625\) 0 0
\(626\) 13.4164 0.536228
\(627\) 0 0
\(628\) 4.47214i 0.178458i
\(629\) 31.3050 1.24821
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 6.00000i 0.238667i
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 0 0
\(637\) 28.2843 + 13.4164i 1.12066 + 0.531577i
\(638\) 16.0000i 0.633446i
\(639\) 0 0
\(640\) 0 0
\(641\) 1.41421i 0.0558581i −0.999610 0.0279290i \(-0.991109\pi\)
0.999610 0.0279290i \(-0.00889125\pi\)
\(642\) 0 0
\(643\) 44.7214i 1.76364i −0.471588 0.881819i \(-0.656319\pi\)
0.471588 0.881819i \(-0.343681\pi\)
\(644\) 8.94427 5.65685i 0.352454 0.222911i
\(645\) 0 0
\(646\) −10.0000 −0.393445
\(647\) −34.7851 −1.36754 −0.683771 0.729697i \(-0.739661\pi\)
−0.683771 + 0.729697i \(0.739661\pi\)
\(648\) 0 0
\(649\) 25.2982i 0.993042i
\(650\) 0 0
\(651\) 0 0
\(652\) −21.2132 −0.830773
\(653\) 14.0000i 0.547862i −0.961749 0.273931i \(-0.911676\pi\)
0.961749 0.273931i \(-0.0883240\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4.47214 0.174608
\(657\) 0 0
\(658\) 21.2132 13.4164i 0.826977 0.523026i
\(659\) 16.9706i 0.661079i −0.943792 0.330540i \(-0.892769\pi\)
0.943792 0.330540i \(-0.107231\pi\)
\(660\) 0 0
\(661\) 3.16228i 0.122998i 0.998107 + 0.0614992i \(0.0195882\pi\)
−0.998107 + 0.0614992i \(0.980412\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 0 0
\(664\) 12.6491i 0.490881i
\(665\) 0 0
\(666\) 0 0
\(667\) 11.3137 0.438069
\(668\) 9.48683 0.367057
\(669\) 0 0
\(670\) 0 0
\(671\) 53.6656 2.07174
\(672\) 0 0
\(673\) −19.7990 −0.763195 −0.381597 0.924329i \(-0.624626\pi\)
−0.381597 + 0.924329i \(0.624626\pi\)
\(674\) 25.4558i 0.980522i
\(675\) 0 0
\(676\) 7.00000 0.269231
\(677\) 37.9473 1.45843 0.729217 0.684282i \(-0.239884\pi\)
0.729217 + 0.684282i \(0.239884\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 35.7771i 1.36998i
\(683\) 14.0000i 0.535695i −0.963461 0.267848i \(-0.913688\pi\)
0.963461 0.267848i \(-0.0863124\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.23607 18.3848i −0.0853735 0.701934i
\(687\) 0 0
\(688\) −1.41421 −0.0539164
\(689\) −17.8885 −0.681499
\(690\) 0 0
\(691\) 28.4605i 1.08269i −0.840801 0.541344i \(-0.817916\pi\)
0.840801 0.541344i \(-0.182084\pi\)
\(692\) −25.2982 −0.961694
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 14.1421 0.535672
\(698\) 15.8114 0.598470
\(699\) 0 0
\(700\) 0 0
\(701\) 19.7990i 0.747798i 0.927470 + 0.373899i \(0.121979\pi\)
−0.927470 + 0.373899i \(0.878021\pi\)
\(702\) 0 0
\(703\) 31.3050i 1.18069i
\(704\) 5.65685i 0.213201i
\(705\) 0 0
\(706\) 3.16228i 0.119014i
\(707\) 25.2982 + 40.0000i 0.951438 + 1.50435i
\(708\) 0 0
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 4.47214i 0.167600i
\(713\) 25.2982 0.947426
\(714\) 0 0
\(715\) 0 0
\(716\) 2.82843i 0.105703i
\(717\) 0 0
\(718\) 18.3848 0.686114
\(719\) 44.7214 1.66783 0.833913 0.551896i \(-0.186096\pi\)
0.833913 + 0.551896i \(0.186096\pi\)
\(720\) 0 0
\(721\) 20.0000 12.6491i 0.744839 0.471077i
\(722\) 9.00000i 0.334945i
\(723\) 0 0
\(724\) 3.16228i 0.117525i
\(725\) 0 0
\(726\) 0 0
\(727\) 26.8328i 0.995174i −0.867414 0.497587i \(-0.834220\pi\)
0.867414 0.497587i \(-0.165780\pi\)
\(728\) −6.32456 10.0000i −0.234404 0.370625i
\(729\) 0 0
\(730\) 0 0
\(731\) −4.47214 −0.165408
\(732\) 0 0
\(733\) 40.2492i 1.48664i −0.668937 0.743319i \(-0.733250\pi\)
0.668937 0.743319i \(-0.266750\pi\)
\(734\) 13.4164 0.495209
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 40.0000i 1.47342i
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 5.65685 + 8.94427i 0.207670 + 0.328355i
\(743\) 36.0000i 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 21.2132i 0.776671i
\(747\) 0 0
\(748\) 17.8885i 0.654070i
\(749\) −40.2492 + 25.4558i −1.47067 + 0.930136i
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) −9.48683 −0.345949
\(753\) 0 0
\(754\) 12.6491i 0.460653i
\(755\) 0 0
\(756\) 0 0
\(757\) −32.5269 −1.18221 −0.591105 0.806594i \(-0.701308\pi\)
−0.591105 + 0.806594i \(0.701308\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −31.3050 −1.13480 −0.567402 0.823441i \(-0.692051\pi\)
−0.567402 + 0.823441i \(0.692051\pi\)
\(762\) 0 0
\(763\) −14.1421 22.3607i −0.511980 0.809511i
\(764\) 1.41421i 0.0511645i
\(765\) 0 0
\(766\) 9.48683i 0.342773i
\(767\) 20.0000i 0.722158i
\(768\) 0 0
\(769\) 31.6228i 1.14035i −0.821524 0.570173i \(-0.806876\pi\)
0.821524 0.570173i \(-0.193124\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8.48528 −0.305392
\(773\) 31.6228 1.13739 0.568696 0.822548i \(-0.307448\pi\)
0.568696 + 0.822548i \(0.307448\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −16.9706 −0.608424
\(779\) 14.1421i 0.506695i
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) −12.6491 −0.452331
\(783\) 0 0
\(784\) −3.00000 + 6.32456i −0.107143 + 0.225877i
\(785\) 0 0
\(786\) 0 0
\(787\) 26.8328i 0.956487i −0.878227 0.478243i \(-0.841274\pi\)
0.878227 0.478243i \(-0.158726\pi\)
\(788\) 12.0000i 0.427482i
\(789\) 0 0
\(790\) 0 0
\(791\) −13.4164 + 8.48528i −0.477033 + 0.301702i
\(792\) 0 0
\(793\) 42.4264 1.50661
\(794\) −4.47214 −0.158710
\(795\) 0 0
\(796\) 0 0
\(797\) 25.2982 0.896109 0.448054 0.894006i \(-0.352117\pi\)
0.448054 + 0.894006i \(0.352117\pi\)
\(798\) 0 0
\(799\) −30.0000 −1.06132
\(800\) 0 0
\(801\) 0 0
\(802\) 1.41421 0.0499376
\(803\) −75.8947 −2.67826
\(804\) 0 0