Properties

Label 3150.2.d.e.3149.3
Level 3150
Weight 2
Character 3150.3149
Analytic conductor 25.153
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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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.d (of order \(2\), degree \(1\), not 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: \( 2^{6}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3149.3
Root \(-1.14412 - 1.14412i\)
Character \(\chi\) = 3150.3149
Dual form 3150.2.d.e.3149.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +(-2.23607 + 1.41421i) q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +(-2.23607 + 1.41421i) q^{7} +1.00000 q^{8} -1.41421i q^{11} -5.39835 q^{13} +(-2.23607 + 1.41421i) q^{14} +1.00000 q^{16} +2.23607i q^{17} +1.30986i q^{19} -1.41421i q^{22} +1.00000 q^{23} -5.39835 q^{26} +(-2.23607 + 1.41421i) q^{28} -9.24264i q^{29} -8.56062i q^{31} +1.00000 q^{32} +2.23607i q^{34} -2.82843i q^{37} +1.30986i q^{38} +4.08849 q^{41} -6.41421i q^{43} -1.41421i q^{44} +1.00000 q^{46} -7.63441i q^{47} +(3.00000 - 6.32456i) q^{49} -5.39835 q^{52} -6.07107 q^{53} +(-2.23607 + 1.41421i) q^{56} -9.24264i q^{58} -11.7229 q^{59} -5.39835i q^{61} -8.56062i q^{62} +1.00000 q^{64} +2.23607i q^{68} +4.34315i q^{71} -5.01470 q^{73} -2.82843i q^{74} +1.30986i q^{76} +(2.00000 + 3.16228i) q^{77} -1.07107 q^{79} +4.08849 q^{82} +8.01806i q^{83} -6.41421i q^{86} -1.41421i q^{88} -15.2688 q^{89} +(12.0711 - 7.63441i) q^{91} +1.00000 q^{92} -7.63441i q^{94} +18.4311 q^{97} +(3.00000 - 6.32456i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{4} + 8q^{8} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{4} + 8q^{8} + 8q^{16} + 8q^{23} + 8q^{32} + 8q^{46} + 24q^{49} + 8q^{53} + 8q^{64} + 16q^{77} + 48q^{79} + 40q^{91} + 8q^{92} + 24q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −2.23607 + 1.41421i −0.845154 + 0.534522i
\(8\) 1.00000 0.353553
\(9\) 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
\(13\) −5.39835 −1.49723 −0.748616 0.663004i \(-0.769281\pi\)
−0.748616 + 0.663004i \(0.769281\pi\)
\(14\) −2.23607 + 1.41421i −0.597614 + 0.377964i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.23607i 0.542326i 0.962533 + 0.271163i \(0.0874083\pi\)
−0.962533 + 0.271163i \(0.912592\pi\)
\(18\) 0 0
\(19\) 1.30986i 0.300502i 0.988648 + 0.150251i \(0.0480082\pi\)
−0.988648 + 0.150251i \(0.951992\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.41421i 0.301511i
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.39835 −1.05870
\(27\) 0 0
\(28\) −2.23607 + 1.41421i −0.422577 + 0.267261i
\(29\) 9.24264i 1.71632i −0.513386 0.858158i \(-0.671609\pi\)
0.513386 0.858158i \(-0.328391\pi\)
\(30\) 0 0
\(31\) 8.56062i 1.53753i −0.639529 0.768767i \(-0.720871\pi\)
0.639529 0.768767i \(-0.279129\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.23607i 0.383482i
\(35\) 0 0
\(36\) 0 0
\(37\) 2.82843i 0.464991i −0.972598 0.232495i \(-0.925311\pi\)
0.972598 0.232495i \(-0.0746890\pi\)
\(38\) 1.30986i 0.212487i
\(39\) 0 0
\(40\) 0 0
\(41\) 4.08849 0.638514 0.319257 0.947668i \(-0.396567\pi\)
0.319257 + 0.947668i \(0.396567\pi\)
\(42\) 0 0
\(43\) 6.41421i 0.978158i −0.872239 0.489079i \(-0.837333\pi\)
0.872239 0.489079i \(-0.162667\pi\)
\(44\) 1.41421i 0.213201i
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 7.63441i 1.11359i −0.830649 0.556797i \(-0.812030\pi\)
0.830649 0.556797i \(-0.187970\pi\)
\(48\) 0 0
\(49\) 3.00000 6.32456i 0.428571 0.903508i
\(50\) 0 0
\(51\) 0 0
\(52\) −5.39835 −0.748616
\(53\) −6.07107 −0.833925 −0.416963 0.908924i \(-0.636905\pi\)
−0.416963 + 0.908924i \(0.636905\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.23607 + 1.41421i −0.298807 + 0.188982i
\(57\) 0 0
\(58\) 9.24264i 1.21362i
\(59\) −11.7229 −1.52619 −0.763096 0.646285i \(-0.776322\pi\)
−0.763096 + 0.646285i \(0.776322\pi\)
\(60\) 0 0
\(61\) 5.39835i 0.691187i −0.938384 0.345594i \(-0.887678\pi\)
0.938384 0.345594i \(-0.112322\pi\)
\(62\) 8.56062i 1.08720i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 2.23607i 0.271163i
\(69\) 0 0
\(70\) 0 0
\(71\) 4.34315i 0.515437i 0.966220 + 0.257718i \(0.0829706\pi\)
−0.966220 + 0.257718i \(0.917029\pi\)
\(72\) 0 0
\(73\) −5.01470 −0.586926 −0.293463 0.955970i \(-0.594808\pi\)
−0.293463 + 0.955970i \(0.594808\pi\)
\(74\) 2.82843i 0.328798i
\(75\) 0 0
\(76\) 1.30986i 0.150251i
\(77\) 2.00000 + 3.16228i 0.227921 + 0.360375i
\(78\) 0 0
\(79\) −1.07107 −0.120505 −0.0602523 0.998183i \(-0.519191\pi\)
−0.0602523 + 0.998183i \(0.519191\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.08849 0.451498
\(83\) 8.01806i 0.880097i 0.897974 + 0.440048i \(0.145039\pi\)
−0.897974 + 0.440048i \(0.854961\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.41421i 0.691662i
\(87\) 0 0
\(88\) 1.41421i 0.150756i
\(89\) −15.2688 −1.61849 −0.809246 0.587470i \(-0.800124\pi\)
−0.809246 + 0.587470i \(0.800124\pi\)
\(90\) 0 0
\(91\) 12.0711 7.63441i 1.26539 0.800304i
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 7.63441i 0.787430i
\(95\) 0 0
\(96\) 0 0
\(97\) 18.4311 1.87140 0.935698 0.352803i \(-0.114771\pi\)
0.935698 + 0.352803i \(0.114771\pi\)
\(98\) 3.00000 6.32456i 0.303046 0.638877i
\(99\) 0 0
\(100\) 0 0
\(101\) −5.78199 −0.575330 −0.287665 0.957731i \(-0.592879\pi\)
−0.287665 + 0.957731i \(0.592879\pi\)
\(102\) 0 0
\(103\) −14.8852 −1.46668 −0.733340 0.679862i \(-0.762040\pi\)
−0.733340 + 0.679862i \(0.762040\pi\)
\(104\) −5.39835 −0.529351
\(105\) 0 0
\(106\) −6.07107 −0.589674
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 0 0
\(109\) −7.07107 −0.677285 −0.338643 0.940915i \(-0.609968\pi\)
−0.338643 + 0.940915i \(0.609968\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.23607 + 1.41421i −0.211289 + 0.133631i
\(113\) −13.0711 −1.22962 −0.614811 0.788674i \(-0.710768\pi\)
−0.614811 + 0.788674i \(0.710768\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 9.24264i 0.858158i
\(117\) 0 0
\(118\) −11.7229 −1.07918
\(119\) −3.16228 5.00000i −0.289886 0.458349i
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 5.39835i 0.488743i
\(123\) 0 0
\(124\) 8.56062i 0.768767i
\(125\) 0 0
\(126\) 0 0
\(127\) 10.0000i 0.887357i 0.896186 + 0.443678i \(0.146327\pi\)
−0.896186 + 0.443678i \(0.853673\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 12.6491 1.10516 0.552579 0.833461i \(-0.313644\pi\)
0.552579 + 0.833461i \(0.313644\pi\)
\(132\) 0 0
\(133\) −1.85242 2.92893i −0.160625 0.253971i
\(134\) 0 0
\(135\) 0 0
\(136\) 2.23607i 0.191741i
\(137\) 16.1421 1.37912 0.689558 0.724231i \(-0.257805\pi\)
0.689558 + 0.724231i \(0.257805\pi\)
\(138\) 0 0
\(139\) 1.85242i 0.157120i −0.996909 0.0785601i \(-0.974968\pi\)
0.996909 0.0785601i \(-0.0250322\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.34315i 0.364469i
\(143\) 7.63441i 0.638422i
\(144\) 0 0
\(145\) 0 0
\(146\) −5.01470 −0.415019
\(147\) 0 0
\(148\) 2.82843i 0.232495i
\(149\) 9.24264i 0.757187i 0.925563 + 0.378593i \(0.123592\pi\)
−0.925563 + 0.378593i \(0.876408\pi\)
\(150\) 0 0
\(151\) 14.1421 1.15087 0.575435 0.817847i \(-0.304833\pi\)
0.575435 + 0.817847i \(0.304833\pi\)
\(152\) 1.30986i 0.106244i
\(153\) 0 0
\(154\) 2.00000 + 3.16228i 0.161165 + 0.254824i
\(155\) 0 0
\(156\) 0 0
\(157\) 1.85242 0.147839 0.0739196 0.997264i \(-0.476449\pi\)
0.0739196 + 0.997264i \(0.476449\pi\)
\(158\) −1.07107 −0.0852096
\(159\) 0 0
\(160\) 0 0
\(161\) −2.23607 + 1.41421i −0.176227 + 0.111456i
\(162\) 0 0
\(163\) 22.0711i 1.72874i −0.502857 0.864370i \(-0.667718\pi\)
0.502857 0.864370i \(-0.332282\pi\)
\(164\) 4.08849 0.319257
\(165\) 0 0
\(166\) 8.01806i 0.622322i
\(167\) 5.78199i 0.447424i −0.974655 0.223712i \(-0.928182\pi\)
0.974655 0.223712i \(-0.0718175\pi\)
\(168\) 0 0
\(169\) 16.1421 1.24170
\(170\) 0 0
\(171\) 0 0
\(172\) 6.41421i 0.489079i
\(173\) 16.5787i 1.26045i 0.776411 + 0.630227i \(0.217038\pi\)
−0.776411 + 0.630227i \(0.782962\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.41421i 0.106600i
\(177\) 0 0
\(178\) −15.2688 −1.14445
\(179\) 11.3137i 0.845626i −0.906217 0.422813i \(-0.861043\pi\)
0.906217 0.422813i \(-0.138957\pi\)
\(180\) 0 0
\(181\) 15.2688i 1.13492i −0.823400 0.567461i \(-0.807926\pi\)
0.823400 0.567461i \(-0.192074\pi\)
\(182\) 12.0711 7.63441i 0.894767 0.565900i
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 3.16228 0.231249
\(188\) 7.63441i 0.556797i
\(189\) 0 0
\(190\) 0 0
\(191\) 10.6569i 0.771103i 0.922686 + 0.385551i \(0.125989\pi\)
−0.922686 + 0.385551i \(0.874011\pi\)
\(192\) 0 0
\(193\) 8.48528i 0.610784i 0.952227 + 0.305392i \(0.0987875\pi\)
−0.952227 + 0.305392i \(0.901213\pi\)
\(194\) 18.4311 1.32328
\(195\) 0 0
\(196\) 3.00000 6.32456i 0.214286 0.451754i
\(197\) −20.0711 −1.43000 −0.715002 0.699122i \(-0.753574\pi\)
−0.715002 + 0.699122i \(0.753574\pi\)
\(198\) 0 0
\(199\) 3.70484i 0.262629i −0.991341 0.131315i \(-0.958080\pi\)
0.991341 0.131315i \(-0.0419198\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −5.78199 −0.406820
\(203\) 13.0711 + 20.6672i 0.917409 + 1.45055i
\(204\) 0 0
\(205\) 0 0
\(206\) −14.8852 −1.03710
\(207\) 0 0
\(208\) −5.39835 −0.374308
\(209\) 1.85242 0.128135
\(210\) 0 0
\(211\) −26.2132 −1.80459 −0.902296 0.431118i \(-0.858119\pi\)
−0.902296 + 0.431118i \(0.858119\pi\)
\(212\) −6.07107 −0.416963
\(213\) 0 0
\(214\) −2.00000 −0.136717
\(215\) 0 0
\(216\) 0 0
\(217\) 12.1065 + 19.1421i 0.821846 + 1.29945i
\(218\) −7.07107 −0.478913
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0711i 0.811988i
\(222\) 0 0
\(223\) 0.383649 0.0256910 0.0128455 0.999917i \(-0.495911\pi\)
0.0128455 + 0.999917i \(0.495911\pi\)
\(224\) −2.23607 + 1.41421i −0.149404 + 0.0944911i
\(225\) 0 0
\(226\) −13.0711 −0.869474
\(227\) 6.16564i 0.409228i −0.978843 0.204614i \(-0.934406\pi\)
0.978843 0.204614i \(-0.0655939\pi\)
\(228\) 0 0
\(229\) 1.85242i 0.122411i 0.998125 + 0.0612057i \(0.0194946\pi\)
−0.998125 + 0.0612057i \(0.980505\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 9.24264i 0.606809i
\(233\) −8.92893 −0.584954 −0.292477 0.956273i \(-0.594479\pi\)
−0.292477 + 0.956273i \(0.594479\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −11.7229 −0.763096
\(237\) 0 0
\(238\) −3.16228 5.00000i −0.204980 0.324102i
\(239\) 12.8284i 0.829802i −0.909867 0.414901i \(-0.863816\pi\)
0.909867 0.414901i \(-0.136184\pi\)
\(240\) 0 0
\(241\) 4.47214i 0.288076i −0.989572 0.144038i \(-0.953991\pi\)
0.989572 0.144038i \(-0.0460087\pi\)
\(242\) 9.00000 0.578542
\(243\) 0 0
\(244\) 5.39835i 0.345594i
\(245\) 0 0
\(246\) 0 0
\(247\) 7.07107i 0.449921i
\(248\) 8.56062i 0.543600i
\(249\) 0 0
\(250\) 0 0
\(251\) −12.4902 −0.788374 −0.394187 0.919030i \(-0.628974\pi\)
−0.394187 + 0.919030i \(0.628974\pi\)
\(252\) 0 0
\(253\) 1.41421i 0.0889108i
\(254\) 10.0000i 0.627456i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 16.7376i 1.04406i 0.852926 + 0.522031i \(0.174826\pi\)
−0.852926 + 0.522031i \(0.825174\pi\)
\(258\) 0 0
\(259\) 4.00000 + 6.32456i 0.248548 + 0.392989i
\(260\) 0 0
\(261\) 0 0
\(262\) 12.6491 0.781465
\(263\) −27.2843 −1.68242 −0.841210 0.540708i \(-0.818156\pi\)
−0.841210 + 0.540708i \(0.818156\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.85242 2.92893i −0.113579 0.179584i
\(267\) 0 0
\(268\) 0 0
\(269\) 10.0294 0.611503 0.305751 0.952111i \(-0.401092\pi\)
0.305751 + 0.952111i \(0.401092\pi\)
\(270\) 0 0
\(271\) 19.7410i 1.19918i −0.800308 0.599589i \(-0.795331\pi\)
0.800308 0.599589i \(-0.204669\pi\)
\(272\) 2.23607i 0.135582i
\(273\) 0 0
\(274\) 16.1421 0.975182
\(275\) 0 0
\(276\) 0 0
\(277\) 21.2132i 1.27458i 0.770625 + 0.637289i \(0.219944\pi\)
−0.770625 + 0.637289i \(0.780056\pi\)
\(278\) 1.85242i 0.111101i
\(279\) 0 0
\(280\) 0 0
\(281\) 11.5147i 0.686911i 0.939169 + 0.343455i \(0.111597\pi\)
−0.939169 + 0.343455i \(0.888403\pi\)
\(282\) 0 0
\(283\) −1.30986 −0.0778630 −0.0389315 0.999242i \(-0.512395\pi\)
−0.0389315 + 0.999242i \(0.512395\pi\)
\(284\) 4.34315i 0.257718i
\(285\) 0 0
\(286\) 7.63441i 0.451432i
\(287\) −9.14214 + 5.78199i −0.539643 + 0.341300i
\(288\) 0 0
\(289\) 12.0000 0.705882
\(290\) 0 0
\(291\) 0 0
\(292\) −5.01470 −0.293463
\(293\) 25.2982i 1.47794i 0.673740 + 0.738969i \(0.264687\pi\)
−0.673740 + 0.738969i \(0.735313\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.82843i 0.164399i
\(297\) 0 0
\(298\) 9.24264i 0.535412i
\(299\) −5.39835 −0.312194
\(300\) 0 0
\(301\) 9.07107 + 14.3426i 0.522848 + 0.826695i
\(302\) 14.1421 0.813788
\(303\) 0 0
\(304\) 1.30986i 0.0751255i
\(305\) 0 0
\(306\) 0 0
\(307\) −21.5934 −1.23240 −0.616200 0.787590i \(-0.711329\pi\)
−0.616200 + 0.787590i \(0.711329\pi\)
\(308\) 2.00000 + 3.16228i 0.113961 + 0.180187i
\(309\) 0 0
\(310\) 0 0
\(311\) 30.5377 1.73163 0.865816 0.500363i \(-0.166800\pi\)
0.865816 + 0.500363i \(0.166800\pi\)
\(312\) 0 0
\(313\) 10.2541 0.579598 0.289799 0.957088i \(-0.406412\pi\)
0.289799 + 0.957088i \(0.406412\pi\)
\(314\) 1.85242 0.104538
\(315\) 0 0
\(316\) −1.07107 −0.0602523
\(317\) −15.9289 −0.894658 −0.447329 0.894369i \(-0.647625\pi\)
−0.447329 + 0.894369i \(0.647625\pi\)
\(318\) 0 0
\(319\) −13.0711 −0.731839
\(320\) 0 0
\(321\) 0 0
\(322\) −2.23607 + 1.41421i −0.124611 + 0.0788110i
\(323\) −2.92893 −0.162970
\(324\) 0 0
\(325\) 0 0
\(326\) 22.0711i 1.22240i
\(327\) 0 0
\(328\) 4.08849 0.225749
\(329\) 10.7967 + 17.0711i 0.595241 + 0.941158i
\(330\) 0 0
\(331\) 22.0711 1.21314 0.606568 0.795032i \(-0.292546\pi\)
0.606568 + 0.795032i \(0.292546\pi\)
\(332\) 8.01806i 0.440048i
\(333\) 0 0
\(334\) 5.78199i 0.316377i
\(335\) 0 0
\(336\) 0 0
\(337\) 21.9706i 1.19681i −0.801193 0.598406i \(-0.795801\pi\)
0.801193 0.598406i \(-0.204199\pi\)
\(338\) 16.1421 0.878016
\(339\) 0 0
\(340\) 0 0
\(341\) −12.1065 −0.655606
\(342\) 0 0
\(343\) 2.23607 + 18.3848i 0.120736 + 0.992685i
\(344\) 6.41421i 0.345831i
\(345\) 0 0
\(346\) 16.5787i 0.891276i
\(347\) 5.07107 0.272229 0.136115 0.990693i \(-0.456538\pi\)
0.136115 + 0.990693i \(0.456538\pi\)
\(348\) 0 0
\(349\) 25.1393i 1.34568i −0.739790 0.672838i \(-0.765075\pi\)
0.739790 0.672838i \(-0.234925\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.41421i 0.0753778i
\(353\) 18.9737i 1.00987i 0.863158 + 0.504933i \(0.168483\pi\)
−0.863158 + 0.504933i \(0.831517\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −15.2688 −0.809246
\(357\) 0 0
\(358\) 11.3137i 0.597948i
\(359\) 6.31371i 0.333225i 0.986022 + 0.166612i \(0.0532829\pi\)
−0.986022 + 0.166612i \(0.946717\pi\)
\(360\) 0 0
\(361\) 17.2843 0.909698
\(362\) 15.2688i 0.802512i
\(363\) 0 0
\(364\) 12.0711 7.63441i 0.632696 0.400152i
\(365\) 0 0
\(366\) 0 0
\(367\) 24.5967 1.28394 0.641970 0.766730i \(-0.278117\pi\)
0.641970 + 0.766730i \(0.278117\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 0 0
\(371\) 13.5753 8.58579i 0.704796 0.445752i
\(372\) 0 0
\(373\) 7.07107i 0.366126i −0.983101 0.183063i \(-0.941399\pi\)
0.983101 0.183063i \(-0.0586012\pi\)
\(374\) 3.16228 0.163517
\(375\) 0 0
\(376\) 7.63441i 0.393715i
\(377\) 49.8950i 2.56972i
\(378\) 0 0
\(379\) −7.92893 −0.407282 −0.203641 0.979046i \(-0.565278\pi\)
−0.203641 + 0.979046i \(0.565278\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 10.6569i 0.545252i
\(383\) 28.6852i 1.46575i −0.680365 0.732874i \(-0.738179\pi\)
0.680365 0.732874i \(-0.261821\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.48528i 0.431889i
\(387\) 0 0
\(388\) 18.4311 0.935698
\(389\) 17.1716i 0.870633i 0.900277 + 0.435317i \(0.143364\pi\)
−0.900277 + 0.435317i \(0.856636\pi\)
\(390\) 0 0
\(391\) 2.23607i 0.113083i
\(392\) 3.00000 6.32456i 0.151523 0.319438i
\(393\) 0 0
\(394\) −20.0711 −1.01117
\(395\) 0 0
\(396\) 0 0
\(397\) 10.6378 0.533895 0.266947 0.963711i \(-0.413985\pi\)
0.266947 + 0.963711i \(0.413985\pi\)
\(398\) 3.70484i 0.185707i
\(399\) 0 0
\(400\) 0 0
\(401\) 29.7990i 1.48809i 0.668129 + 0.744045i \(0.267095\pi\)
−0.668129 + 0.744045i \(0.732905\pi\)
\(402\) 0 0
\(403\) 46.2132i 2.30204i
\(404\) −5.78199 −0.287665
\(405\) 0 0
\(406\) 13.0711 + 20.6672i 0.648706 + 1.02569i
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) 3.92957i 0.194305i −0.995270 0.0971525i \(-0.969027\pi\)
0.995270 0.0971525i \(-0.0309735\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −14.8852 −0.733340
\(413\) 26.2132 16.5787i 1.28987 0.815784i
\(414\) 0 0
\(415\) 0 0
\(416\) −5.39835 −0.264676
\(417\) 0 0
\(418\) 1.85242 0.0906048
\(419\) −15.4277 −0.753694 −0.376847 0.926275i \(-0.622992\pi\)
−0.376847 + 0.926275i \(0.622992\pi\)
\(420\) 0 0
\(421\) −37.3553 −1.82059 −0.910294 0.413963i \(-0.864144\pi\)
−0.910294 + 0.413963i \(0.864144\pi\)
\(422\) −26.2132 −1.27604
\(423\) 0 0
\(424\) −6.07107 −0.294837
\(425\) 0 0
\(426\) 0 0
\(427\) 7.63441 + 12.0711i 0.369455 + 0.584160i
\(428\) −2.00000 −0.0966736
\(429\) 0 0
\(430\) 0 0
\(431\) 34.7990i 1.67621i −0.545510 0.838104i \(-0.683664\pi\)
0.545510 0.838104i \(-0.316336\pi\)
\(432\) 0 0
\(433\) 35.0098 1.68246 0.841232 0.540675i \(-0.181831\pi\)
0.841232 + 0.540675i \(0.181831\pi\)
\(434\) 12.1065 + 19.1421i 0.581133 + 0.918852i
\(435\) 0 0
\(436\) −7.07107 −0.338643
\(437\) 1.30986i 0.0626590i
\(438\) 0 0
\(439\) 8.56062i 0.408576i −0.978911 0.204288i \(-0.934512\pi\)
0.978911 0.204288i \(-0.0654879\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 12.0711i 0.574162i
\(443\) −8.92893 −0.424226 −0.212113 0.977245i \(-0.568035\pi\)
−0.212113 + 0.977245i \(0.568035\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0.383649 0.0181663
\(447\) 0 0
\(448\) −2.23607 + 1.41421i −0.105644 + 0.0668153i
\(449\) 18.3848i 0.867631i −0.901002 0.433816i \(-0.857167\pi\)
0.901002 0.433816i \(-0.142833\pi\)
\(450\) 0 0
\(451\) 5.78199i 0.272263i
\(452\) −13.0711 −0.614811
\(453\) 0 0
\(454\) 6.16564i 0.289368i
\(455\) 0 0
\(456\) 0 0
\(457\) 12.1716i 0.569362i −0.958622 0.284681i \(-0.908112\pi\)
0.958622 0.284681i \(-0.0918877\pi\)
\(458\) 1.85242i 0.0865579i
\(459\) 0 0
\(460\) 0 0
\(461\) 32.6148 1.51902 0.759512 0.650494i \(-0.225438\pi\)
0.759512 + 0.650494i \(0.225438\pi\)
\(462\) 0 0
\(463\) 18.2843i 0.849742i −0.905254 0.424871i \(-0.860319\pi\)
0.905254 0.424871i \(-0.139681\pi\)
\(464\) 9.24264i 0.429079i
\(465\) 0 0
\(466\) −8.92893 −0.413625
\(467\) 10.6378i 0.492258i 0.969237 + 0.246129i \(0.0791586\pi\)
−0.969237 + 0.246129i \(0.920841\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −11.7229 −0.539590
\(473\) −9.07107 −0.417088
\(474\) 0 0
\(475\) 0 0
\(476\) −3.16228 5.00000i −0.144943 0.229175i
\(477\) 0 0
\(478\) 12.8284i 0.586759i
\(479\) 2.07716 0.0949077 0.0474538 0.998873i \(-0.484889\pi\)
0.0474538 + 0.998873i \(0.484889\pi\)
\(480\) 0 0
\(481\) 15.2688i 0.696199i
\(482\) 4.47214i 0.203700i
\(483\) 0 0
\(484\) 9.00000 0.409091
\(485\) 0 0
\(486\) 0 0
\(487\) 10.1005i 0.457698i −0.973462 0.228849i \(-0.926504\pi\)
0.973462 0.228849i \(-0.0734961\pi\)
\(488\) 5.39835i 0.244372i
\(489\) 0 0
\(490\) 0 0
\(491\) 38.4853i 1.73682i 0.495850 + 0.868408i \(0.334857\pi\)
−0.495850 + 0.868408i \(0.665143\pi\)
\(492\) 0 0
\(493\) 20.6672 0.930803
\(494\) 7.07107i 0.318142i
\(495\) 0 0
\(496\) 8.56062i 0.384383i
\(497\) −6.14214 9.71157i −0.275512 0.435623i
\(498\) 0 0
\(499\) 7.92893 0.354948 0.177474 0.984126i \(-0.443207\pi\)
0.177474 + 0.984126i \(0.443207\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −12.4902 −0.557465
\(503\) 40.7918i 1.81882i −0.415905 0.909408i \(-0.636535\pi\)
0.415905 0.909408i \(-0.363465\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.41421i 0.0628695i
\(507\) 0 0
\(508\) 10.0000i 0.443678i
\(509\) 44.4966 1.97228 0.986139 0.165921i \(-0.0530596\pi\)
0.986139 + 0.165921i \(0.0530596\pi\)
\(510\) 0 0
\(511\) 11.2132 7.09185i 0.496043 0.313725i
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 16.7376i 0.738264i
\(515\) 0 0
\(516\) 0 0
\(517\) −10.7967 −0.474838
\(518\) 4.00000 + 6.32456i 0.175750 + 0.277885i
\(519\) 0 0
\(520\) 0 0
\(521\) −4.85578 −0.212736 −0.106368 0.994327i \(-0.533922\pi\)
−0.106368 + 0.994327i \(0.533922\pi\)
\(522\) 0 0
\(523\) 28.4605 1.24449 0.622245 0.782822i \(-0.286221\pi\)
0.622245 + 0.782822i \(0.286221\pi\)
\(524\) 12.6491 0.552579
\(525\) 0 0
\(526\) −27.2843 −1.18965
\(527\) 19.1421 0.833845
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) −1.85242 2.92893i −0.0803126 0.126985i
\(533\) −22.0711 −0.956004
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 10.0294 0.432398
\(539\) −8.94427 4.24264i −0.385257 0.182743i
\(540\) 0 0
\(541\) −9.07107 −0.389996 −0.194998 0.980804i \(-0.562470\pi\)
−0.194998 + 0.980804i \(0.562470\pi\)
\(542\) 19.7410i 0.847947i
\(543\) 0 0
\(544\) 2.23607i 0.0958706i
\(545\) 0 0
\(546\) 0 0
\(547\) 26.2132i 1.12080i −0.828224 0.560398i \(-0.810648\pi\)
0.828224 0.560398i \(-0.189352\pi\)
\(548\) 16.1421 0.689558
\(549\) 0 0
\(550\) 0 0
\(551\) 12.1065 0.515756
\(552\) 0 0
\(553\) 2.39498 1.51472i 0.101845 0.0644124i
\(554\) 21.2132i 0.901263i
\(555\) 0 0
\(556\) 1.85242i 0.0785601i
\(557\) −2.14214 −0.0907652 −0.0453826 0.998970i \(-0.514451\pi\)
−0.0453826 + 0.998970i \(0.514451\pi\)
\(558\) 0 0
\(559\) 34.6261i 1.46453i
\(560\) 0 0
\(561\) 0 0
\(562\) 11.5147i 0.485719i
\(563\) 19.8999i 0.838680i −0.907829 0.419340i \(-0.862262\pi\)
0.907829 0.419340i \(-0.137738\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.30986 −0.0550575
\(567\) 0 0
\(568\) 4.34315i 0.182234i
\(569\) 42.5269i 1.78282i 0.453197 + 0.891410i \(0.350283\pi\)
−0.453197 + 0.891410i \(0.649717\pi\)
\(570\) 0 0
\(571\) 4.21320 0.176317 0.0881585 0.996106i \(-0.471902\pi\)
0.0881585 + 0.996106i \(0.471902\pi\)
\(572\) 7.63441i 0.319211i
\(573\) 0 0
\(574\) −9.14214 + 5.78199i −0.381585 + 0.241336i
\(575\) 0 0
\(576\) 0 0
\(577\) −14.5015 −0.603707 −0.301853 0.953354i \(-0.597605\pi\)
−0.301853 + 0.953354i \(0.597605\pi\)
\(578\) 12.0000 0.499134
\(579\) 0 0
\(580\) 0 0
\(581\) −11.3393 17.9289i −0.470431 0.743817i
\(582\) 0 0
\(583\) 8.58579i 0.355587i
\(584\) −5.01470 −0.207510
\(585\) 0 0
\(586\) 25.2982i 1.04506i
\(587\) 28.8441i 1.19053i −0.803531 0.595263i \(-0.797048\pi\)
0.803531 0.595263i \(-0.202952\pi\)
\(588\) 0 0
\(589\) 11.2132 0.462032
\(590\) 0 0
\(591\) 0 0
\(592\) 2.82843i 0.116248i
\(593\) 29.0031i 1.19101i 0.803351 + 0.595506i \(0.203049\pi\)
−0.803351 + 0.595506i \(0.796951\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 9.24264i 0.378593i
\(597\) 0 0
\(598\) −5.39835 −0.220755
\(599\) 26.3137i 1.07515i −0.843216 0.537574i \(-0.819341\pi\)
0.843216 0.537574i \(-0.180659\pi\)
\(600\) 0 0
\(601\) 5.78199i 0.235852i −0.993022 0.117926i \(-0.962375\pi\)
0.993022 0.117926i \(-0.0376246\pi\)
\(602\) 9.07107 + 14.3426i 0.369709 + 0.584561i
\(603\) 0 0
\(604\) 14.1421 0.575435
\(605\) 0 0
\(606\) 0 0
\(607\) −18.6558 −0.757217 −0.378609 0.925557i \(-0.623597\pi\)
−0.378609 + 0.925557i \(0.623597\pi\)
\(608\) 1.30986i 0.0531218i
\(609\) 0 0
\(610\) 0 0
\(611\) 41.2132i 1.66731i
\(612\) 0 0
\(613\) 44.1421i 1.78288i 0.453135 + 0.891442i \(0.350306\pi\)
−0.453135 + 0.891442i \(0.649694\pi\)
\(614\) −21.5934 −0.871438
\(615\) 0 0
\(616\) 2.00000 + 3.16228i 0.0805823 + 0.127412i
\(617\) −32.0000 −1.28827 −0.644136 0.764911i \(-0.722783\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) 0 0
\(619\) 45.0392i 1.81028i 0.425116 + 0.905139i \(0.360233\pi\)
−0.425116 + 0.905139i \(0.639767\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 30.5377 1.22445
\(623\) 34.1421 21.5934i 1.36788 0.865121i
\(624\) 0 0
\(625\) 0 0
\(626\) 10.2541 0.409837
\(627\) 0 0
\(628\) 1.85242 0.0739196
\(629\) 6.32456 0.252177
\(630\) 0 0
\(631\) −46.2843 −1.84255 −0.921274 0.388914i \(-0.872850\pi\)
−0.921274 + 0.388914i \(0.872850\pi\)
\(632\) −1.07107 −0.0426048
\(633\) 0 0
\(634\) −15.9289 −0.632619
\(635\) 0 0
\(636\) 0 0
\(637\) −16.1950 + 34.1421i −0.641671 + 1.35276i
\(638\) −13.0711 −0.517489
\(639\) 0 0
\(640\) 0 0
\(641\) 11.5147i 0.454804i −0.973801 0.227402i \(-0.926977\pi\)
0.973801 0.227402i \(-0.0730231\pi\)
\(642\) 0 0
\(643\) −23.6705 −0.933475 −0.466737 0.884396i \(-0.654571\pi\)
−0.466737 + 0.884396i \(0.654571\pi\)
\(644\) −2.23607 + 1.41421i −0.0881134 + 0.0557278i
\(645\) 0 0
\(646\) −2.92893 −0.115237
\(647\) 43.4115i 1.70668i 0.521353 + 0.853341i \(0.325427\pi\)
−0.521353 + 0.853341i \(0.674573\pi\)
\(648\) 0 0
\(649\) 16.5787i 0.650770i
\(650\) 0 0
\(651\) 0 0
\(652\) 22.0711i 0.864370i
\(653\) 20.1421 0.788223 0.394111 0.919063i \(-0.371052\pi\)
0.394111 + 0.919063i \(0.371052\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4.08849 0.159629
\(657\) 0 0
\(658\) 10.7967 + 17.0711i 0.420899 + 0.665500i
\(659\) 11.6152i 0.452465i −0.974073 0.226232i \(-0.927359\pi\)
0.974073 0.226232i \(-0.0726409\pi\)
\(660\) 0 0
\(661\) 45.0392i 1.75182i 0.482473 + 0.875911i \(0.339739\pi\)
−0.482473 + 0.875911i \(0.660261\pi\)
\(662\) 22.0711 0.857816
\(663\) 0 0
\(664\) 8.01806i 0.311161i
\(665\) 0 0
\(666\) 0 0
\(667\) 9.24264i 0.357876i
\(668\) 5.78199i 0.223712i
\(669\) 0 0
\(670\) 0 0
\(671\) −7.63441 −0.294723
\(672\) 0 0
\(673\) 4.79899i 0.184987i −0.995713 0.0924937i \(-0.970516\pi\)
0.995713 0.0924937i \(-0.0294838\pi\)
\(674\) 21.9706i 0.846274i
\(675\) 0 0
\(676\) 16.1421 0.620851
\(677\) 3.92957i 0.151026i −0.997145 0.0755129i \(-0.975941\pi\)
0.997145 0.0755129i \(-0.0240594\pi\)
\(678\) 0 0
\(679\) −41.2132 + 26.0655i −1.58162 + 1.00030i
\(680\) 0 0
\(681\) 0 0
\(682\) −12.1065 −0.463584
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 2.23607 + 18.3848i 0.0853735 + 0.701934i
\(687\) 0 0
\(688\) 6.41421i 0.244540i
\(689\) 32.7737 1.24858
\(690\) 0 0
\(691\) 32.9326i 1.25282i 0.779495 + 0.626408i \(0.215476\pi\)
−0.779495 + 0.626408i \(0.784524\pi\)
\(692\) 16.5787i 0.630227i
\(693\) 0 0
\(694\) 5.07107 0.192495
\(695\) 0 0
\(696\) 0 0
\(697\) 9.14214i 0.346283i
\(698\) 25.1393i 0.951537i
\(699\) 0 0
\(700\) 0 0
\(701\) 11.8701i 0.448326i −0.974552 0.224163i \(-0.928035\pi\)
0.974552 0.224163i \(-0.0719648\pi\)
\(702\) 0 0
\(703\) 3.70484 0.139731
\(704\) 1.41421i 0.0533002i
\(705\) 0 0
\(706\) 18.9737i 0.714083i
\(707\) 12.9289 8.17697i 0.486243 0.307527i
\(708\) 0 0
\(709\) 4.14214 0.155561 0.0777806 0.996971i \(-0.475217\pi\)
0.0777806 + 0.996971i \(0.475217\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −15.2688 −0.572223
\(713\) 8.56062i 0.320598i
\(714\) 0 0
\(715\) 0 0
\(716\) 11.3137i 0.422813i
\(717\) 0 0
\(718\) 6.31371i 0.235626i
\(719\) −32.9326 −1.22818 −0.614090 0.789236i \(-0.710477\pi\)
−0.614090 + 0.789236i \(0.710477\pi\)
\(720\) 0 0
\(721\) 33.2843 21.0508i 1.23957 0.783974i
\(722\) 17.2843 0.643254
\(723\) 0 0
\(724\) 15.2688i 0.567461i
\(725\) 0 0
\(726\) 0 0
\(727\) 6.70820 0.248794 0.124397 0.992233i \(-0.460300\pi\)
0.124397 + 0.992233i \(0.460300\pi\)
\(728\) 12.0711 7.63441i 0.447384 0.282950i
\(729\) 0 0
\(730\) 0 0
\(731\) 14.3426 0.530481
\(732\) 0 0
\(733\) 17.2802 0.638257 0.319129 0.947711i \(-0.396610\pi\)
0.319129 + 0.947711i \(0.396610\pi\)
\(734\) 24.5967 0.907883
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) −44.3553 −1.63164 −0.815819 0.578308i \(-0.803713\pi\)
−0.815819 + 0.578308i \(0.803713\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 13.5753 8.58579i 0.498366 0.315194i
\(743\) −6.85786 −0.251591 −0.125795 0.992056i \(-0.540148\pi\)
−0.125795 + 0.992056i \(0.540148\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 7.07107i 0.258890i
\(747\) 0 0
\(748\) 3.16228 0.115624
\(749\) 4.47214 2.82843i 0.163408 0.103348i
\(750\) 0 0
\(751\) −11.2132 −0.409176 −0.204588 0.978848i \(-0.565585\pi\)
−0.204588 + 0.978848i \(0.565585\pi\)
\(752\) 7.63441i 0.278398i
\(753\) 0 0
\(754\) 49.8950i 1.81707i
\(755\) 0 0
\(756\) 0 0
\(757\) 4.04163i 0.146896i −0.997299 0.0734478i \(-0.976600\pi\)
0.997299 0.0734478i \(-0.0234002\pi\)
\(758\) −7.92893 −0.287992
\(759\) 0 0
\(760\) 0 0
\(761\) 31.3050 1.13480 0.567402 0.823441i \(-0.307949\pi\)
0.567402 + 0.823441i \(0.307949\pi\)
\(762\) 0 0
\(763\) 15.8114 10.0000i 0.572411 0.362024i
\(764\) 10.6569i 0.385551i
\(765\) 0 0
\(766\) 28.6852i 1.03644i
\(767\) 63.2843 2.28506
\(768\) 0 0
\(769\) 41.8769i 1.51012i 0.655656 + 0.755060i \(0.272392\pi\)
−0.655656 + 0.755060i \(0.727608\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.48528i 0.305392i
\(773\) 36.3196i 1.30633i −0.757217 0.653163i \(-0.773441\pi\)
0.757217 0.653163i \(-0.226559\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 18.4311 0.661638
\(777\) 0 0
\(778\) 17.1716i 0.615631i
\(779\) 5.35534i 0.191875i
\(780\) 0 0
\(781\) 6.14214 0.219783
\(782\) 2.23607i 0.0799616i
\(783\) 0 0
\(784\) 3.00000 6.32456i 0.107143 0.225877i
\(785\) 0 0
\(786\) 0 0
\(787\) −46.5738 −1.66018 −0.830088 0.557632i \(-0.811710\pi\)
−0.830088 + 0.557632i \(0.811710\pi\)
\(788\) −20.0711 −0.715002
\(789\) 0 0
\(790\) 0 0
\(791\) 29.2278 18.4853i 1.03922 0.657261i
\(792\) 0 0
\(793\) 29.1421i 1.03487i
\(794\) 10.6378 0.377521
\(795\) 0 0
\(796\) 3.70484i 0.131315i
\(797\) 42.6442i 1.51054i −0.655417 0.755268i \(-0.727507\pi\)
0.655417 0.755268i \(-0.272493\pi\)
\(798\) 0 0
\(799\) 17.0711 0.603931
\(800\) 0 0
\(801\) 0 0
\(802\) 29.7990i 1.05224i
\(803\) 7.09185i 0.250266i
\(804\) 0 0
\(805\)