Properties

Label 3150.2.b.d.251.8
Level 3150
Weight 2
Character 3150.251
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.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: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.8
Root \(1.14412 - 1.14412i\)
Character \(\chi\) = 3150.251
Dual form 3150.2.b.d.251.3

$q$-expansion

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

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
\(7\) 1.41421 + 2.23607i 0.534522 + 0.845154i
\(8\) 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
\(13\) 5.39835i 1.49723i −0.663004 0.748616i \(-0.730719\pi\)
0.663004 0.748616i \(-0.269281\pi\)
\(14\) −2.23607 + 1.41421i −0.597614 + 0.377964i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.23607 −0.542326 −0.271163 0.962533i \(-0.587408\pi\)
−0.271163 + 0.962533i \(0.587408\pi\)
\(18\) 0 0
\(19\) 1.30986i 0.300502i −0.988648 0.150251i \(-0.951992\pi\)
0.988648 0.150251i \(-0.0480082\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.41421 −0.301511
\(23\) 1.00000i 0.208514i −0.994550 0.104257i \(-0.966753\pi\)
0.994550 0.104257i \(-0.0332465\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 5.39835 1.05870
\(27\) 0 0
\(28\) −1.41421 2.23607i −0.267261 0.422577i
\(29\) 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 \(-0.279129\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 2.23607i 0.383482i
\(35\) 0 0
\(36\) 0 0
\(37\) −2.82843 −0.464991 −0.232495 0.972598i \(-0.574689\pi\)
−0.232495 + 0.972598i \(0.574689\pi\)
\(38\) 1.30986 0.212487
\(39\) 0 0
\(40\) 0 0
\(41\) −4.08849 −0.638514 −0.319257 0.947668i \(-0.603433\pi\)
−0.319257 + 0.947668i \(0.603433\pi\)
\(42\) 0 0
\(43\) 6.41421 0.978158 0.489079 0.872239i \(-0.337333\pi\)
0.489079 + 0.872239i \(0.337333\pi\)
\(44\) 1.41421i 0.213201i
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 7.63441 1.11359 0.556797 0.830649i \(-0.312030\pi\)
0.556797 + 0.830649i \(0.312030\pi\)
\(48\) 0 0
\(49\) −3.00000 + 6.32456i −0.428571 + 0.903508i
\(50\) 0 0
\(51\) 0 0
\(52\) 5.39835i 0.748616i
\(53\) 6.07107i 0.833925i 0.908924 + 0.416963i \(0.136905\pi\)
−0.908924 + 0.416963i \(0.863095\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.23607 1.41421i 0.298807 0.188982i
\(57\) 0 0
\(58\) 9.24264 1.21362
\(59\) −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 \(-0.112322\pi\)
\(62\) 8.56062 1.08720
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 2.23607 0.271163
\(69\) 0 0
\(70\) 0 0
\(71\) 4.34315i 0.515437i −0.966220 0.257718i \(-0.917029\pi\)
0.966220 0.257718i \(-0.0829706\pi\)
\(72\) 0 0
\(73\) 5.01470i 0.586926i −0.955970 0.293463i \(-0.905192\pi\)
0.955970 0.293463i \(-0.0948077\pi\)
\(74\) 2.82843i 0.328798i
\(75\) 0 0
\(76\) 1.30986i 0.150251i
\(77\) −3.16228 + 2.00000i −0.360375 + 0.227921i
\(78\) 0 0
\(79\) 1.07107 0.120505 0.0602523 0.998183i \(-0.480809\pi\)
0.0602523 + 0.998183i \(0.480809\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.08849i 0.451498i
\(83\) 8.01806 0.880097 0.440048 0.897974i \(-0.354961\pi\)
0.440048 + 0.897974i \(0.354961\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.41421i 0.691662i
\(87\) 0 0
\(88\) 1.41421 0.150756
\(89\) −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
\(92\) 1.00000i 0.104257i
\(93\) 0 0
\(94\) 7.63441i 0.787430i
\(95\) 0 0
\(96\) 0 0
\(97\) 18.4311i 1.87140i −0.352803 0.935698i \(-0.614771\pi\)
0.352803 0.935698i \(-0.385229\pi\)
\(98\) −6.32456 3.00000i −0.638877 0.303046i
\(99\) 0 0
\(100\) 0 0
\(101\) 5.78199 0.575330 0.287665 0.957731i \(-0.407121\pi\)
0.287665 + 0.957731i \(0.407121\pi\)
\(102\) 0 0
\(103\) 14.8852i 1.46668i −0.679862 0.733340i \(-0.737960\pi\)
0.679862 0.733340i \(-0.262040\pi\)
\(104\) −5.39835 −0.529351
\(105\) 0 0
\(106\) −6.07107 −0.589674
\(107\) 2.00000i 0.193347i −0.995316 0.0966736i \(-0.969180\pi\)
0.995316 0.0966736i \(-0.0308203\pi\)
\(108\) 0 0
\(109\) 7.07107 0.677285 0.338643 0.940915i \(-0.390032\pi\)
0.338643 + 0.940915i \(0.390032\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.41421 + 2.23607i 0.133631 + 0.211289i
\(113\) 13.0711i 1.22962i 0.788674 + 0.614811i \(0.210768\pi\)
−0.788674 + 0.614811i \(0.789232\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 9.24264i 0.858158i
\(117\) 0 0
\(118\) 11.7229i 1.07918i
\(119\) −3.16228 5.00000i −0.289886 0.458349i
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 5.39835 0.488743
\(123\) 0 0
\(124\) 8.56062i 0.768767i
\(125\) 0 0
\(126\) 0 0
\(127\) 10.0000 0.887357 0.443678 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −12.6491 −1.10516 −0.552579 0.833461i \(-0.686356\pi\)
−0.552579 + 0.833461i \(0.686356\pi\)
\(132\) 0 0
\(133\) 2.92893 1.85242i 0.253971 0.160625i
\(134\) 0 0
\(135\) 0 0
\(136\) 2.23607i 0.191741i
\(137\) 16.1421i 1.37912i 0.724231 + 0.689558i \(0.242195\pi\)
−0.724231 + 0.689558i \(0.757805\pi\)
\(138\) 0 0
\(139\) 1.85242i 0.157120i 0.996909 + 0.0785601i \(0.0250322\pi\)
−0.996909 + 0.0785601i \(0.974968\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.34315 0.364469
\(143\) 7.63441 0.638422
\(144\) 0 0
\(145\) 0 0
\(146\) 5.01470 0.415019
\(147\) 0 0
\(148\) 2.82843 0.232495
\(149\) 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 \(0.304833\pi\)
\(152\) −1.30986 −0.106244
\(153\) 0 0
\(154\) −2.00000 3.16228i −0.161165 0.254824i
\(155\) 0 0
\(156\) 0 0
\(157\) 1.85242i 0.147839i −0.997264 0.0739196i \(-0.976449\pi\)
0.997264 0.0739196i \(-0.0235508\pi\)
\(158\) 1.07107i 0.0852096i
\(159\) 0 0
\(160\) 0 0
\(161\) 2.23607 1.41421i 0.176227 0.111456i
\(162\) 0 0
\(163\) 22.0711 1.72874 0.864370 0.502857i \(-0.167718\pi\)
0.864370 + 0.502857i \(0.167718\pi\)
\(164\) 4.08849 0.319257
\(165\) 0 0
\(166\) 8.01806i 0.622322i
\(167\) 5.78199 0.447424 0.223712 0.974655i \(-0.428182\pi\)
0.223712 + 0.974655i \(0.428182\pi\)
\(168\) 0 0
\(169\) −16.1421 −1.24170
\(170\) 0 0
\(171\) 0 0
\(172\) −6.41421 −0.489079
\(173\) 16.5787 1.26045 0.630227 0.776411i \(-0.282962\pi\)
0.630227 + 0.776411i \(0.282962\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.41421i 0.106600i
\(177\) 0 0
\(178\) 15.2688i 1.14445i
\(179\) 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 \(-0.192074\pi\)
\(182\) 7.63441 + 12.0711i 0.565900 + 0.894767i
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 3.16228i 0.231249i
\(188\) −7.63441 −0.556797
\(189\) 0 0
\(190\) 0 0
\(191\) 10.6569i 0.771103i −0.922686 0.385551i \(-0.874011\pi\)
0.922686 0.385551i \(-0.125989\pi\)
\(192\) 0 0
\(193\) −8.48528 −0.610784 −0.305392 0.952227i \(-0.598787\pi\)
−0.305392 + 0.952227i \(0.598787\pi\)
\(194\) 18.4311 1.32328
\(195\) 0 0
\(196\) 3.00000 6.32456i 0.214286 0.451754i
\(197\) 20.0711i 1.43000i −0.699122 0.715002i \(-0.746426\pi\)
0.699122 0.715002i \(-0.253574\pi\)
\(198\) 0 0
\(199\) 3.70484i 0.262629i 0.991341 + 0.131315i \(0.0419198\pi\)
−0.991341 + 0.131315i \(0.958080\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 5.78199i 0.406820i
\(203\) 20.6672 13.0711i 1.45055 0.917409i
\(204\) 0 0
\(205\) 0 0
\(206\) 14.8852 1.03710
\(207\) 0 0
\(208\) 5.39835i 0.374308i
\(209\) 1.85242 0.128135
\(210\) 0 0
\(211\) −26.2132 −1.80459 −0.902296 0.431118i \(-0.858119\pi\)
−0.902296 + 0.431118i \(0.858119\pi\)
\(212\) 6.07107i 0.416963i
\(213\) 0 0
\(214\) 2.00000 0.136717
\(215\) 0 0
\(216\) 0 0
\(217\) 19.1421 12.1065i 1.29945 0.821846i
\(218\) 7.07107i 0.478913i
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0711i 0.811988i
\(222\) 0 0
\(223\) 0.383649i 0.0256910i 0.999917 + 0.0128455i \(0.00408896\pi\)
−0.999917 + 0.0128455i \(0.995911\pi\)
\(224\) −2.23607 + 1.41421i −0.149404 + 0.0944911i
\(225\) 0 0
\(226\) −13.0711 −0.869474
\(227\) 6.16564 0.409228 0.204614 0.978843i \(-0.434406\pi\)
0.204614 + 0.978843i \(0.434406\pi\)
\(228\) 0 0
\(229\) 1.85242i 0.122411i −0.998125 0.0612057i \(-0.980505\pi\)
0.998125 0.0612057i \(-0.0194946\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −9.24264 −0.606809
\(233\) 8.92893i 0.584954i 0.956273 + 0.292477i \(0.0944795\pi\)
−0.956273 + 0.292477i \(0.905521\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 11.7229 0.763096
\(237\) 0 0
\(238\) 5.00000 3.16228i 0.324102 0.204980i
\(239\) 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 \(-0.0460087\pi\)
\(242\) 9.00000i 0.578542i
\(243\) 0 0
\(244\) 5.39835i 0.345594i
\(245\) 0 0
\(246\) 0 0
\(247\) −7.07107 −0.449921
\(248\) −8.56062 −0.543600
\(249\) 0 0
\(250\) 0 0
\(251\) 12.4902 0.788374 0.394187 0.919030i \(-0.371026\pi\)
0.394187 + 0.919030i \(0.371026\pi\)
\(252\) 0 0
\(253\) 1.41421 0.0889108
\(254\) 10.0000i 0.627456i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −16.7376 −1.04406 −0.522031 0.852926i \(-0.674826\pi\)
−0.522031 + 0.852926i \(0.674826\pi\)
\(258\) 0 0
\(259\) −4.00000 6.32456i −0.248548 0.392989i
\(260\) 0 0
\(261\) 0 0
\(262\) 12.6491i 0.781465i
\(263\) 27.2843i 1.68242i 0.540708 + 0.841210i \(0.318156\pi\)
−0.540708 + 0.841210i \(0.681844\pi\)
\(264\) 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 \(-0.204669\pi\)
\(272\) −2.23607 −0.135582
\(273\) 0 0
\(274\) −16.1421 −0.975182
\(275\) 0 0
\(276\) 0 0
\(277\) 21.2132 1.27458 0.637289 0.770625i \(-0.280056\pi\)
0.637289 + 0.770625i \(0.280056\pi\)
\(278\) −1.85242 −0.111101
\(279\) 0 0
\(280\) 0 0
\(281\) 11.5147i 0.686911i −0.939169 0.343455i \(-0.888403\pi\)
0.939169 0.343455i \(-0.111597\pi\)
\(282\) 0 0
\(283\) 1.30986i 0.0778630i −0.999242 0.0389315i \(-0.987605\pi\)
0.999242 0.0389315i \(-0.0123954\pi\)
\(284\) 4.34315i 0.257718i
\(285\) 0 0
\(286\) 7.63441i 0.451432i
\(287\) −5.78199 9.14214i −0.341300 0.539643i
\(288\) 0 0
\(289\) −12.0000 −0.705882
\(290\) 0 0
\(291\) 0 0
\(292\) 5.01470i 0.293463i
\(293\) 25.2982 1.47794 0.738969 0.673740i \(-0.235313\pi\)
0.738969 + 0.673740i \(0.235313\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.82843i 0.164399i
\(297\) 0 0
\(298\) −9.24264 −0.535412
\(299\) −5.39835 −0.312194
\(300\) 0 0
\(301\) 9.07107 + 14.3426i 0.522848 + 0.826695i
\(302\) 14.1421i 0.813788i
\(303\) 0 0
\(304\) 1.30986i 0.0751255i
\(305\) 0 0
\(306\) 0 0
\(307\) 21.5934i 1.23240i 0.787590 + 0.616200i \(0.211329\pi\)
−0.787590 + 0.616200i \(0.788671\pi\)
\(308\) 3.16228 2.00000i 0.180187 0.113961i
\(309\) 0 0
\(310\) 0 0
\(311\) −30.5377 −1.73163 −0.865816 0.500363i \(-0.833200\pi\)
−0.865816 + 0.500363i \(0.833200\pi\)
\(312\) 0 0
\(313\) 10.2541i 0.579598i 0.957088 + 0.289799i \(0.0935884\pi\)
−0.957088 + 0.289799i \(0.906412\pi\)
\(314\) 1.85242 0.104538
\(315\) 0 0
\(316\) −1.07107 −0.0602523
\(317\) 15.9289i 0.894658i −0.894369 0.447329i \(-0.852375\pi\)
0.894369 0.447329i \(-0.147625\pi\)
\(318\) 0 0
\(319\) 13.0711 0.731839
\(320\) 0 0
\(321\) 0 0
\(322\) 1.41421 + 2.23607i 0.0788110 + 0.124611i
\(323\) 2.92893i 0.162970i
\(324\) 0 0
\(325\) 0 0
\(326\) 22.0711i 1.22240i
\(327\) 0 0
\(328\) 4.08849i 0.225749i
\(329\) 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 \(0.292546\pi\)
\(332\) −8.01806 −0.440048
\(333\) 0 0
\(334\) 5.78199i 0.316377i
\(335\) 0 0
\(336\) 0 0
\(337\) −21.9706 −1.19681 −0.598406 0.801193i \(-0.704199\pi\)
−0.598406 + 0.801193i \(0.704199\pi\)
\(338\) 16.1421i 0.878016i
\(339\) 0 0
\(340\) 0 0
\(341\) 12.1065 0.655606
\(342\) 0 0
\(343\) −18.3848 + 2.23607i −0.992685 + 0.120736i
\(344\) 6.41421i 0.345831i
\(345\) 0 0
\(346\) 16.5787i 0.891276i
\(347\) 5.07107i 0.272229i 0.990693 + 0.136115i \(0.0434615\pi\)
−0.990693 + 0.136115i \(0.956538\pi\)
\(348\) 0 0
\(349\) 25.1393i 1.34568i 0.739790 + 0.672838i \(0.234925\pi\)
−0.739790 + 0.672838i \(0.765075\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.41421 −0.0753778
\(353\) 18.9737 1.00987 0.504933 0.863158i \(-0.331517\pi\)
0.504933 + 0.863158i \(0.331517\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 15.2688 0.809246
\(357\) 0 0
\(358\) 11.3137 0.597948
\(359\) 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
\(362\) 15.2688 0.802512
\(363\) 0 0
\(364\) −12.0711 + 7.63441i −0.632696 + 0.400152i
\(365\) 0 0
\(366\) 0 0
\(367\) 24.5967i 1.28394i −0.766730 0.641970i \(-0.778117\pi\)
0.766730 0.641970i \(-0.221883\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 0 0
\(370\) 0 0
\(371\) −13.5753 + 8.58579i −0.704796 + 0.445752i
\(372\) 0 0
\(373\) 7.07107 0.366126 0.183063 0.983101i \(-0.441399\pi\)
0.183063 + 0.983101i \(0.441399\pi\)
\(374\) 3.16228 0.163517
\(375\) 0 0
\(376\) 7.63441i 0.393715i
\(377\) −49.8950 −2.56972
\(378\) 0 0
\(379\) 7.92893 0.407282 0.203641 0.979046i \(-0.434722\pi\)
0.203641 + 0.979046i \(0.434722\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 10.6569 0.545252
\(383\) −28.6852 −1.46575 −0.732874 0.680365i \(-0.761821\pi\)
−0.732874 + 0.680365i \(0.761821\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.48528i 0.431889i
\(387\) 0 0
\(388\) 18.4311i 0.935698i
\(389\) 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
\(392\) 6.32456 + 3.00000i 0.319438 + 0.151523i
\(393\) 0 0
\(394\) 20.0711 1.01117
\(395\) 0 0
\(396\) 0 0
\(397\) 10.6378i 0.533895i −0.963711 0.266947i \(-0.913985\pi\)
0.963711 0.266947i \(-0.0860150\pi\)
\(398\) −3.70484 −0.185707
\(399\) 0 0
\(400\) 0 0
\(401\) 29.7990i 1.48809i −0.668129 0.744045i \(-0.732905\pi\)
0.668129 0.744045i \(-0.267095\pi\)
\(402\) 0 0
\(403\) −46.2132 −2.30204
\(404\) −5.78199 −0.287665
\(405\) 0 0
\(406\) 13.0711 + 20.6672i 0.648706 + 1.02569i
\(407\) 4.00000i 0.198273i
\(408\) 0 0
\(409\) 3.92957i 0.194305i 0.995270 + 0.0971525i \(0.0309735\pi\)
−0.995270 + 0.0971525i \(0.969027\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 14.8852i 0.733340i
\(413\) −16.5787 26.2132i −0.815784 1.28987i
\(414\) 0 0
\(415\) 0 0
\(416\) 5.39835 0.264676
\(417\) 0 0
\(418\) 1.85242i 0.0906048i
\(419\) −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 \(0.864144\pi\)
\(422\) 26.2132i 1.27604i
\(423\) 0 0
\(424\) 6.07107 0.294837
\(425\) 0 0
\(426\) 0 0
\(427\) 12.0711 7.63441i 0.584160 0.369455i
\(428\) 2.00000i 0.0966736i
\(429\) 0 0
\(430\) 0 0
\(431\) 34.7990i 1.67621i 0.545510 + 0.838104i \(0.316336\pi\)
−0.545510 + 0.838104i \(0.683664\pi\)
\(432\) 0 0
\(433\) 35.0098i 1.68246i 0.540675 + 0.841232i \(0.318169\pi\)
−0.540675 + 0.841232i \(0.681831\pi\)
\(434\) 12.1065 + 19.1421i 0.581133 + 0.918852i
\(435\) 0 0
\(436\) −7.07107 −0.338643
\(437\) −1.30986 −0.0626590
\(438\) 0 0
\(439\) 8.56062i 0.408576i 0.978911 + 0.204288i \(0.0654879\pi\)
−0.978911 + 0.204288i \(0.934512\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −12.0711 −0.574162
\(443\) 8.92893i 0.424226i 0.977245 + 0.212113i \(0.0680346\pi\)
−0.977245 + 0.212113i \(0.931965\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −0.383649 −0.0181663
\(447\) 0 0
\(448\) −1.41421 2.23607i −0.0668153 0.105644i
\(449\) 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
\(452\) 13.0711i 0.614811i
\(453\) 0 0
\(454\) 6.16564i 0.289368i
\(455\) 0 0
\(456\) 0 0
\(457\) −12.1716 −0.569362 −0.284681 0.958622i \(-0.591888\pi\)
−0.284681 + 0.958622i \(0.591888\pi\)
\(458\) 1.85242 0.0865579
\(459\) 0 0
\(460\) 0 0
\(461\) −32.6148 −1.51902 −0.759512 0.650494i \(-0.774562\pi\)
−0.759512 + 0.650494i \(0.774562\pi\)
\(462\) 0 0
\(463\) 18.2843 0.849742 0.424871 0.905254i \(-0.360319\pi\)
0.424871 + 0.905254i \(0.360319\pi\)
\(464\) 9.24264i 0.429079i
\(465\) 0 0
\(466\) −8.92893 −0.413625
\(467\) −10.6378 −0.492258 −0.246129 0.969237i \(-0.579159\pi\)
−0.246129 + 0.969237i \(0.579159\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 11.7229i 0.539590i
\(473\) 9.07107i 0.417088i
\(474\) 0 0
\(475\) 0 0
\(476\) 3.16228 + 5.00000i 0.144943 + 0.229175i
\(477\) 0 0
\(478\) 12.8284 0.586759
\(479\) 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
\(482\) 4.47214 0.203700
\(483\) 0 0
\(484\) −9.00000 −0.409091
\(485\) 0 0
\(486\) 0 0
\(487\) −10.1005 −0.457698 −0.228849 0.973462i \(-0.573496\pi\)
−0.228849 + 0.973462i \(0.573496\pi\)
\(488\) −5.39835 −0.244372
\(489\) 0 0
\(490\) 0 0
\(491\) 38.4853i 1.73682i −0.495850 0.868408i \(-0.665143\pi\)
0.495850 0.868408i \(-0.334857\pi\)
\(492\) 0 0
\(493\) 20.6672i 0.930803i
\(494\) 7.07107i 0.318142i
\(495\) 0 0
\(496\) 8.56062i 0.384383i
\(497\) 9.71157 6.14214i 0.435623 0.275512i
\(498\) 0 0
\(499\) −7.92893 −0.354948 −0.177474 0.984126i \(-0.556793\pi\)
−0.177474 + 0.984126i \(0.556793\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 12.4902i 0.557465i
\(503\) −40.7918 −1.81882 −0.909408 0.415905i \(-0.863465\pi\)
−0.909408 + 0.415905i \(0.863465\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.41421i 0.0628695i
\(507\) 0 0
\(508\) −10.0000 −0.443678
\(509\) 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
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 16.7376i 0.738264i
\(515\) 0 0
\(516\) 0 0
\(517\) 10.7967i 0.474838i
\(518\) 6.32456 4.00000i 0.277885 0.175750i
\(519\) 0 0
\(520\) 0 0
\(521\) 4.85578 0.212736 0.106368 0.994327i \(-0.466078\pi\)
0.106368 + 0.994327i \(0.466078\pi\)
\(522\) 0 0
\(523\) 28.4605i 1.24449i 0.782822 + 0.622245i \(0.213779\pi\)
−0.782822 + 0.622245i \(0.786221\pi\)
\(524\) 12.6491 0.552579
\(525\) 0 0
\(526\) −27.2843 −1.18965
\(527\) 19.1421i 0.833845i
\(528\) 0 0
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) −2.92893 + 1.85242i −0.126985 + 0.0803126i
\(533\) 22.0711i 0.956004i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 10.0294i 0.432398i
\(539\) −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 \(0.562470\pi\)
\(542\) 19.7410 0.847947
\(543\) 0 0
\(544\) 2.23607i 0.0958706i
\(545\) 0 0
\(546\) 0 0
\(547\) −26.2132 −1.12080 −0.560398 0.828224i \(-0.689352\pi\)
−0.560398 + 0.828224i \(0.689352\pi\)
\(548\) 16.1421i 0.689558i
\(549\) 0 0
\(550\) 0 0
\(551\) −12.1065 −0.515756
\(552\) 0 0
\(553\) 1.51472 + 2.39498i 0.0644124 + 0.101845i
\(554\) 21.2132i 0.901263i
\(555\) 0 0
\(556\) 1.85242i 0.0785601i
\(557\) 2.14214i 0.0907652i −0.998970 0.0453826i \(-0.985549\pi\)
0.998970 0.0453826i \(-0.0144507\pi\)
\(558\) 0 0
\(559\) 34.6261i 1.46453i
\(560\) 0 0
\(561\) 0 0
\(562\) 11.5147 0.485719
\(563\) −19.8999 −0.838680 −0.419340 0.907829i \(-0.637738\pi\)
−0.419340 + 0.907829i \(0.637738\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.30986 0.0550575
\(567\) 0 0
\(568\) −4.34315 −0.182234
\(569\) 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 \(0.471902\pi\)
\(572\) −7.63441 −0.319211
\(573\) 0 0
\(574\) 9.14214 5.78199i 0.381585 0.241336i
\(575\) 0 0
\(576\) 0 0
\(577\) 14.5015i 0.603707i 0.953354 + 0.301853i \(0.0976053\pi\)
−0.953354 + 0.301853i \(0.902395\pi\)
\(578\) 12.0000i 0.499134i
\(579\) 0 0
\(580\) 0 0
\(581\) 11.3393 + 17.9289i 0.470431 + 0.743817i
\(582\) 0 0
\(583\) −8.58579 −0.355587
\(584\) −5.01470 −0.207510
\(585\) 0 0
\(586\) 25.2982i 1.04506i
\(587\) 28.8441 1.19053 0.595263 0.803531i \(-0.297048\pi\)
0.595263 + 0.803531i \(0.297048\pi\)
\(588\) 0 0
\(589\) −11.2132 −0.462032
\(590\) 0 0
\(591\) 0 0
\(592\) −2.82843 −0.116248
\(593\) 29.0031 1.19101 0.595506 0.803351i \(-0.296951\pi\)
0.595506 + 0.803351i \(0.296951\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 9.24264i 0.378593i
\(597\) 0 0
\(598\) 5.39835i 0.220755i
\(599\) 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 \(-0.0376246\pi\)
\(602\) −14.3426 + 9.07107i −0.584561 + 0.369709i
\(603\) 0 0
\(604\) −14.1421 −0.575435
\(605\) 0 0
\(606\) 0 0
\(607\) 18.6558i 0.757217i 0.925557 + 0.378609i \(0.123597\pi\)
−0.925557 + 0.378609i \(0.876403\pi\)
\(608\) 1.30986 0.0531218
\(609\) 0 0
\(610\) 0 0
\(611\) 41.2132i 1.66731i
\(612\) 0 0
\(613\) −44.1421 −1.78288 −0.891442 0.453135i \(-0.850306\pi\)
−0.891442 + 0.453135i \(0.850306\pi\)
\(614\) −21.5934 −0.871438
\(615\) 0 0
\(616\) 2.00000 + 3.16228i 0.0805823 + 0.127412i
\(617\) 32.0000i 1.28827i −0.764911 0.644136i \(-0.777217\pi\)
0.764911 0.644136i \(-0.222783\pi\)
\(618\) 0 0
\(619\) 45.0392i 1.81028i −0.425116 0.905139i \(-0.639767\pi\)
0.425116 0.905139i \(-0.360233\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 30.5377i 1.22445i
\(623\) −21.5934 34.1421i −0.865121 1.36788i
\(624\) 0 0
\(625\) 0 0
\(626\) −10.2541 −0.409837
\(627\) 0 0
\(628\) 1.85242i 0.0739196i
\(629\) 6.32456 0.252177
\(630\) 0 0
\(631\) −46.2843 −1.84255 −0.921274 0.388914i \(-0.872850\pi\)
−0.921274 + 0.388914i \(0.872850\pi\)
\(632\) 1.07107i 0.0426048i
\(633\) 0 0
\(634\) 15.9289 0.632619
\(635\) 0 0
\(636\) 0 0
\(637\) 34.1421 + 16.1950i 1.35276 + 0.641671i
\(638\) 13.0711i 0.517489i
\(639\) 0 0
\(640\) 0 0
\(641\) 11.5147i 0.454804i 0.973801 + 0.227402i \(0.0730231\pi\)
−0.973801 + 0.227402i \(0.926977\pi\)
\(642\) 0 0
\(643\) 23.6705i 0.933475i −0.884396 0.466737i \(-0.845429\pi\)
0.884396 0.466737i \(-0.154571\pi\)
\(644\) −2.23607 + 1.41421i −0.0881134 + 0.0557278i
\(645\) 0 0
\(646\) −2.92893 −0.115237
\(647\) −43.4115 −1.70668 −0.853341 0.521353i \(-0.825427\pi\)
−0.853341 + 0.521353i \(0.825427\pi\)
\(648\) 0 0
\(649\) 16.5787i 0.650770i
\(650\) 0 0
\(651\) 0 0
\(652\) −22.0711 −0.864370
\(653\) 20.1421i 0.788223i −0.919063 0.394111i \(-0.871052\pi\)
0.919063 0.394111i \(-0.128948\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −4.08849 −0.159629
\(657\) 0 0
\(658\) −17.0711 + 10.7967i −0.665500 + 0.420899i
\(659\) 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 \(0.660261\pi\)
\(662\) 22.0711i 0.857816i
\(663\) 0 0
\(664\) 8.01806i 0.311161i
\(665\) 0 0
\(666\) 0 0
\(667\) −9.24264 −0.357876
\(668\) −5.78199 −0.223712
\(669\) 0 0
\(670\) 0 0
\(671\) 7.63441 0.294723
\(672\) 0 0
\(673\) 4.79899 0.184987 0.0924937 0.995713i \(-0.470516\pi\)
0.0924937 + 0.995713i \(0.470516\pi\)
\(674\) 21.9706i 0.846274i
\(675\) 0 0
\(676\) 16.1421 0.620851
\(677\) 3.92957 0.151026 0.0755129 0.997145i \(-0.475941\pi\)
0.0755129 + 0.997145i \(0.475941\pi\)
\(678\) 0 0
\(679\) 41.2132 26.0655i 1.58162 1.00030i
\(680\) 0 0
\(681\) 0 0
\(682\) 12.1065i 0.463584i
\(683\) 36.0000i 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.23607 18.3848i −0.0853735 0.701934i
\(687\) 0 0
\(688\) 6.41421 0.244540
\(689\) 32.7737 1.24858
\(690\) 0 0
\(691\) 32.9326i 1.25282i 0.779495 + 0.626408i \(0.215476\pi\)
−0.779495 + 0.626408i \(0.784524\pi\)
\(692\) −16.5787 −0.630227
\(693\) 0 0
\(694\) −5.07107 −0.192495
\(695\) 0 0
\(696\) 0 0
\(697\) 9.14214 0.346283
\(698\) −25.1393 −0.951537
\(699\) 0 0
\(700\) 0 0
\(701\) 11.8701i 0.448326i 0.974552 + 0.224163i \(0.0719648\pi\)
−0.974552 + 0.224163i \(0.928035\pi\)
\(702\) 0 0
\(703\) 3.70484i 0.139731i
\(704\) 1.41421i 0.0533002i
\(705\) 0 0
\(706\) 18.9737i 0.714083i
\(707\) 8.17697 + 12.9289i 0.307527 + 0.486243i
\(708\) 0 0
\(709\) −4.14214 −0.155561 −0.0777806 0.996971i \(-0.524783\pi\)
−0.0777806 + 0.996971i \(0.524783\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 15.2688i 0.572223i
\(713\) −8.56062 −0.320598
\(714\) 0 0
\(715\) 0 0
\(716\) 11.3137i 0.422813i
\(717\) 0 0
\(718\) −6.31371 −0.235626
\(719\) −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
\(722\) 17.2843i 0.643254i
\(723\) 0 0
\(724\) 15.2688i 0.567461i
\(725\) 0 0
\(726\) 0 0
\(727\) 6.70820i 0.248794i −0.992233 0.124397i \(-0.960300\pi\)
0.992233 0.124397i \(-0.0396996\pi\)
\(728\) −7.63441 12.0711i −0.282950 0.447384i
\(729\) 0 0
\(730\) 0 0
\(731\) −14.3426 −0.530481
\(732\) 0 0
\(733\) 17.2802i 0.638257i 0.947711 + 0.319129i \(0.103390\pi\)
−0.947711 + 0.319129i \(0.896610\pi\)
\(734\) 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 \(-0.196287\pi\)
0.815819 + 0.578308i \(0.196287\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −8.58579 13.5753i −0.315194 0.498366i
\(743\) 6.85786i 0.251591i 0.992056 + 0.125795i \(0.0401483\pi\)
−0.992056 + 0.125795i \(0.959852\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 7.07107i 0.258890i
\(747\) 0 0
\(748\) 3.16228i 0.115624i
\(749\) 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 \(0.565585\pi\)
\(752\) 7.63441 0.278398
\(753\) 0 0
\(754\) 49.8950i 1.81707i
\(755\) 0 0
\(756\) 0 0
\(757\) −4.04163 −0.146896 −0.0734478 0.997299i \(-0.523400\pi\)
−0.0734478 + 0.997299i \(0.523400\pi\)
\(758\) 7.92893i 0.287992i
\(759\) 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
\(763\) 10.0000 + 15.8114i 0.362024 + 0.572411i
\(764\) 10.6569i 0.385551i
\(765\) 0 0
\(766\) 28.6852i 1.03644i
\(767\) 63.2843i 2.28506i
\(768\) 0 0
\(769\) 41.8769i 1.51012i −0.655656 0.755060i \(-0.727608\pi\)
0.655656 0.755060i \(-0.272392\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.48528 0.305392
\(773\) −36.3196 −1.30633 −0.653163 0.757217i \(-0.726559\pi\)
−0.653163 + 0.757217i \(0.726559\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −18.4311 −0.661638
\(777\) 0 0
\(778\) −17.1716 −0.615631
\(779\) 5.35534i 0.191875i
\(780\) 0 0
\(781\) 6.14214 0.219783
\(782\) −2.23607 −0.0799616
\(783\) 0 0
\(784\) −3.00000 + 6.32456i −0.107143 + 0.225877i
\(785\) 0 0
\(786\) 0 0
\(787\) 46.5738i 1.66018i 0.557632 + 0.830088i \(0.311710\pi\)
−0.557632 + 0.830088i \(0.688290\pi\)
\(788\) 20.0711i 0.715002i
\(789\) 0 0
\(790\) 0 0
\(791\) −29.2278 + 18.4853i −1.03922 + 0.657261i
\(792\) 0 0
\(793\) −29.1421 −1.03487
\(794\) 10.6378 0.377521
\(795\) 0 0
\(796\) 3.70484i 0.131315i
\(797\) 42.6442 1.51054 0.755268 0.655417i \(-0.227507\pi\)
0.755268 + 0.655417i \(0.227507\pi\)
\(798\) 0 0
\(799\) −17.0711 −0.603931
\(800\) 0 0
\(801\) 0 0
\(802\) 29.7990 1.05224
\(803\) 7.09185 0.250266
\(804\)