Properties

Label 2850.2.a.n.1.1
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.7573645761\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} -6.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -8.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} +2.00000 q^{21} +4.00000 q^{23} -1.00000 q^{24} +6.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +2.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} +8.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -1.00000 q^{38} -6.00000 q^{39} -12.0000 q^{41} -2.00000 q^{42} -4.00000 q^{43} -4.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -8.00000 q^{51} -6.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} -2.00000 q^{56} +1.00000 q^{57} -2.00000 q^{58} +6.00000 q^{59} -14.0000 q^{61} +2.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +12.0000 q^{67} -8.00000 q^{68} +4.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} +1.00000 q^{76} +6.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} +12.0000 q^{82} -2.00000 q^{83} +2.00000 q^{84} +4.00000 q^{86} +2.00000 q^{87} -12.0000 q^{91} +4.00000 q^{92} -2.00000 q^{93} +12.0000 q^{94} -1.00000 q^{96} -2.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −8.00000 −1.94029 −0.970143 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 8.00000 1.37199
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −1.00000 −0.162221
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) −2.00000 −0.308607
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) −6.00000 −0.832050
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 1.00000 0.132453
\(58\) −2.00000 −0.262613
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 2.00000 0.254000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −8.00000 −0.970143
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 6.00000 0.679366
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) −2.00000 −0.219529 −0.109764 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 4.00000 0.417029
\(93\) −2.00000 −0.207390
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 8.00000 0.792118
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 1.00000 0.0962250
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 2.00000 0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) −6.00000 −0.554700
\(118\) −6.00000 −0.552345
\(119\) −16.0000 −1.46672
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 14.0000 1.26750
\(123\) −12.0000 −1.08200
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 8.00000 0.685994
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) −4.00000 −0.340503
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 8.00000 0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) −3.00000 −0.247436
\(148\) 2.00000 0.164399
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −8.00000 −0.646762
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) −14.0000 −1.11378
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) −1.00000 −0.0785674
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −2.00000 −0.154303
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) −4.00000 −0.304997
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 0 0
\(177\) 6.00000 0.450988
\(178\) 0 0
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 12.0000 0.889499
\(183\) −14.0000 −1.03491
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 2.00000 0.146647
\(187\) 0 0
\(188\) −12.0000 −0.875190
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 6.00000 0.422159
\(203\) 4.00000 0.280745
\(204\) −8.00000 −0.560112
\(205\) 0 0
\(206\) −16.0000 −1.11477
\(207\) 4.00000 0.278019
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −10.0000 −0.686803
\(213\) −8.00000 −0.548151
\(214\) −16.0000 −1.09374
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −4.00000 −0.271538
\(218\) 12.0000 0.812743
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 48.0000 3.22883
\(222\) −2.00000 −0.134231
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 1.00000 0.0662266
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 14.0000 0.909398
\(238\) 16.0000 1.03713
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 11.0000 0.707107
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) −6.00000 −0.381771
\(248\) 2.00000 0.127000
\(249\) −2.00000 −0.126745
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 30.0000 1.87135 0.935674 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 4.00000 0.249029
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) −8.00000 −0.485071
\(273\) −12.0000 −0.726273
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) 12.0000 0.719712
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) 12.0000 0.714590
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) −1.00000 −0.0589256
\(289\) 47.0000 2.76471
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 10.0000 0.585206
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) −18.0000 −1.03578
\(303\) −6.00000 −0.344691
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 8.00000 0.457330
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 6.00000 0.339683
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 10.0000 0.560772
\(319\) 0 0
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) −8.00000 −0.445823
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) −12.0000 −0.663602
\(328\) 12.0000 0.662589
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −2.00000 −0.109764
\(333\) 2.00000 0.109599
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −23.0000 −1.25104
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) −1.00000 −0.0540738
\(343\) −20.0000 −1.07990
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) 10.0000 0.536828 0.268414 0.963304i \(-0.413500\pi\)
0.268414 + 0.963304i \(0.413500\pi\)
\(348\) 2.00000 0.107211
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 0 0
\(353\) −20.0000 −1.06449 −0.532246 0.846590i \(-0.678652\pi\)
−0.532246 + 0.846590i \(0.678652\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) 0 0
\(357\) −16.0000 −0.846810
\(358\) −10.0000 −0.528516
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 12.0000 0.630706
\(363\) −11.0000 −0.577350
\(364\) −12.0000 −0.628971
\(365\) 0 0
\(366\) 14.0000 0.731792
\(367\) −34.0000 −1.77479 −0.887393 0.461014i \(-0.847486\pi\)
−0.887393 + 0.461014i \(0.847486\pi\)
\(368\) 4.00000 0.208514
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) −20.0000 −1.03835
\(372\) −2.00000 −0.103695
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) −12.0000 −0.618031
\(378\) −2.00000 −0.102869
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) −2.00000 −0.101535
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −32.0000 −1.61831
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) −4.00000 −0.200502
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) −12.0000 −0.598506
\(403\) 12.0000 0.597763
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) 0 0
\(408\) 8.00000 0.396059
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 16.0000 0.788263
\(413\) 12.0000 0.590481
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 16.0000 0.778868
\(423\) −12.0000 −0.583460
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) −28.0000 −1.35501
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000 0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) −12.0000 −0.574696
\(437\) 4.00000 0.191346
\(438\) −10.0000 −0.477818
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −48.0000 −2.28313
\(443\) 34.0000 1.61539 0.807694 0.589601i \(-0.200715\pi\)
0.807694 + 0.589601i \(0.200715\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) 10.0000 0.472984
\(448\) 2.00000 0.0944911
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 18.0000 0.845714
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −6.00000 −0.280362
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) 22.0000 1.01804 0.509019 0.860755i \(-0.330008\pi\)
0.509019 + 0.860755i \(0.330008\pi\)
\(468\) −6.00000 −0.277350
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) −6.00000 −0.276172
\(473\) 0 0
\(474\) −14.0000 −0.643041
\(475\) 0 0
\(476\) −16.0000 −0.733359
\(477\) −10.0000 −0.457869
\(478\) 8.00000 0.365911
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 14.0000 0.637683
\(483\) 8.00000 0.364013
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 14.0000 0.633750
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −16.0000 −0.722070 −0.361035 0.932552i \(-0.617576\pi\)
−0.361035 + 0.932552i \(0.617576\pi\)
\(492\) −12.0000 −0.541002
\(493\) −16.0000 −0.720604
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −16.0000 −0.717698
\(498\) 2.00000 0.0896221
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) 12.0000 0.535586
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) 23.0000 1.02147
\(508\) 8.00000 0.354943
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) −30.0000 −1.32324
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) −4.00000 −0.175750
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 2.00000 0.0867110
\(533\) 72.0000 3.11867
\(534\) 0 0
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 10.0000 0.431532
\(538\) 30.0000 1.29339
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 4.00000 0.171815
\(543\) −12.0000 −0.514969
\(544\) 8.00000 0.342997
\(545\) 0 0
\(546\) 12.0000 0.513553
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) −12.0000 −0.512615
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) −4.00000 −0.170251
\(553\) 28.0000 1.19068
\(554\) 4.00000 0.169944
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 2.00000 0.0846668
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 20.0000 0.843649
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 2.00000 0.0839921
\(568\) 8.00000 0.335673
\(569\) 4.00000 0.167689 0.0838444 0.996479i \(-0.473280\pi\)
0.0838444 + 0.996479i \(0.473280\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) −47.0000 −1.95494
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 2.00000 0.0829027
\(583\) 0 0
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 26.0000 1.07313 0.536567 0.843857i \(-0.319721\pi\)
0.536567 + 0.843857i \(0.319721\pi\)
\(588\) −3.00000 −0.123718
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 2.00000 0.0821995
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 4.00000 0.163709
\(598\) 24.0000 0.981433
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 8.00000 0.326056
\(603\) 12.0000 0.488678
\(604\) 18.0000 0.732410
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 72.0000 2.91281
\(612\) −8.00000 −0.323381
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) −20.0000 −0.805170 −0.402585 0.915383i \(-0.631888\pi\)
−0.402585 + 0.915383i \(0.631888\pi\)
\(618\) −16.0000 −0.643614
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 4.00000 0.159617
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −14.0000 −0.556890
\(633\) −16.0000 −0.635943
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) 18.0000 0.713186
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −20.0000 −0.789953 −0.394976 0.918691i \(-0.629247\pi\)
−0.394976 + 0.918691i \(0.629247\pi\)
\(642\) −16.0000 −0.631470
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) 16.0000 0.629025 0.314512 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) −8.00000 −0.313304
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 12.0000 0.469237
\(655\) 0 0
\(656\) −12.0000 −0.468521
\(657\) 10.0000 0.390137
\(658\) 24.0000 0.935617
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) −20.0000 −0.777322
\(663\) 48.0000 1.86417
\(664\) 2.00000 0.0776151
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 8.00000 0.309761
\(668\) −8.00000 −0.309529
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 6.00000 0.230429
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 6.00000 0.228914
\(688\) −4.00000 −0.152499
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) −10.0000 −0.379595
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) 96.0000 3.63626
\(698\) −14.0000 −0.529908
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 6.00000 0.226455
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) 0 0
\(706\) 20.0000 0.752710
\(707\) −12.0000 −0.451306
\(708\) 6.00000 0.225494
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 16.0000 0.598785
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) −8.00000 −0.298765
\(718\) 0 0
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) −1.00000 −0.0372161
\(723\) −14.0000 −0.520666
\(724\) −12.0000 −0.445976
\(725\) 0 0
\(726\) 11.0000 0.408248
\(727\) −42.0000 −1.55769 −0.778847 0.627214i \(-0.784195\pi\)
−0.778847 + 0.627214i \(0.784195\pi\)
\(728\) 12.0000 0.444750
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 32.0000 1.18356
\(732\) −14.0000 −0.517455
\(733\) −8.00000 −0.295487 −0.147743 0.989026i \(-0.547201\pi\)
−0.147743 + 0.989026i \(0.547201\pi\)
\(734\) 34.0000 1.25496
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 0 0
\(738\) 12.0000 0.441726
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) 20.0000 0.734223
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 2.00000 0.0733236
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) −2.00000 −0.0731762
\(748\) 0 0
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) −2.00000 −0.0729810 −0.0364905 0.999334i \(-0.511618\pi\)
−0.0364905 + 0.999334i \(0.511618\pi\)
\(752\) −12.0000 −0.437595
\(753\) −12.0000 −0.437304
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) −40.0000 −1.45382 −0.726912 0.686730i \(-0.759045\pi\)
−0.726912 + 0.686730i \(0.759045\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) −8.00000 −0.289809
\(763\) −24.0000 −0.868858
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −36.0000 −1.29988
\(768\) 1.00000 0.0360844
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 2.00000 0.0719816
\(773\) −10.0000 −0.359675 −0.179838 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 4.00000 0.143499
\(778\) −18.0000 −0.645331
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 32.0000 1.14432
\(783\) 2.00000 0.0714742
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) −18.0000 −0.641223
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 84.0000 2.98293
\(794\) 8.00000 0.283909
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) −2.00000 −0.0707992
\(799\) 96.0000 3.39624
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −12.0000 −0.422682
\(807\) −30.0000 −1.05605
\(808\) 6.00000 0.211079
\(809\) 38.0000 1.33601 0.668004 0.744157i \(-0.267149\pi\)
0.668004 + 0.744157i \(0.267149\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 4.00000 0.140372
\(813\) −4.00000 −0.140286
\(814\) 0 0
\(815\) 0 0
\(816\) −8.00000 −0.280056
\(817\) −4.00000 −0.139942
\(818\) 6.00000 0.209785
\(819\) −12.0000 −0.419314
\(820\) 0 0
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 12.0000 0.418548
\(823\) −14.0000 −0.488009 −0.244005 0.969774i \(-0.578461\pi\)
−0.244005 + 0.969774i \(0.578461\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) −32.0000 −1.11275 −0.556375 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) 4.00000 0.139010
\(829\) 28.0000 0.972480 0.486240 0.873825i \(-0.338368\pi\)
0.486240 + 0.873825i \(0.338368\pi\)
\(830\) 0 0
\(831\) −4.00000 −0.138758
\(832\) −6.00000 −0.208013
\(833\) 24.0000 0.831551
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 28.0000 0.967244
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −4.00000 −0.137849
\(843\) −20.0000 −0.688837
\(844\) −16.0000 −0.550743
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) −22.0000 −0.755929
\(848\) −10.0000 −0.343401
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) −8.00000 −0.274075
\(853\) 28.0000 0.958702 0.479351 0.877623i \(-0.340872\pi\)
0.479351 + 0.877623i \(0.340872\pi\)
\(854\) 28.0000 0.958140
\(855\) 0 0
\(856\) −16.0000 −0.546869
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) −24.0000 −0.817443
\(863\) −8.00000 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) 47.0000 1.59620
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) 0 0
\(871\) −72.0000 −2.43963
\(872\) 12.0000 0.406371
\(873\) −2.00000 −0.0676897
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 10.0000 0.337484
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 3.00000 0.101015
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 48.0000 1.61441
\(885\) 0 0
\(886\) −34.0000 −1.14225
\(887\) −32.0000 −1.07445 −0.537227 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 4.00000 0.133930
\(893\) −12.0000 −0.401565
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) −24.0000 −0.801337
\(898\) −36.0000 −1.20134
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) 80.0000 2.66519
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −18.0000 −0.598010
\(907\) 36.0000 1.19536 0.597680 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) −20.0000 −0.663723
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 1.00000 0.0331133
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) 8.00000 0.264039
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) 18.0000 0.592798
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 0 0
\(926\) 22.0000 0.722965
\(927\) 16.0000 0.525509
\(928\) −2.00000 −0.0656532
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) −24.0000 −0.786146
\(933\) 24.0000 0.785725
\(934\) −22.0000 −0.719862
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) −24.0000 −0.783628
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) −4.00000 −0.130327
\(943\) −48.0000 −1.56310
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) −2.00000 −0.0649913 −0.0324956 0.999472i \(-0.510346\pi\)
−0.0324956 + 0.999472i \(0.510346\pi\)
\(948\) 14.0000 0.454699
\(949\) −60.0000 −1.94768
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 16.0000 0.518563
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 0 0
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 12.0000 0.386896
\(963\) 16.0000 0.515593
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) 34.0000 1.09337 0.546683 0.837340i \(-0.315890\pi\)
0.546683 + 0.837340i \(0.315890\pi\)
\(968\) 11.0000 0.353553
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 1.00000 0.0320750
\(973\) −24.0000 −0.769405
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 8.00000 0.255812
\(979\) 0 0
\(980\) 0 0
\(981\) −12.0000 −0.383131
\(982\) 16.0000 0.510581
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 12.0000 0.382546
\(985\) 0 0
\(986\) 16.0000 0.509544
\(987\) −24.0000 −0.763928
\(988\) −6.00000 −0.190885
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) 50.0000 1.58830 0.794151 0.607720i \(-0.207916\pi\)
0.794151 + 0.607720i \(0.207916\pi\)
\(992\) 2.00000 0.0635001
\(993\) 20.0000 0.634681
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) −2.00000 −0.0633724
\(997\) −44.0000 −1.39349 −0.696747 0.717317i \(-0.745370\pi\)
−0.696747 + 0.717317i \(0.745370\pi\)
\(998\) −28.0000 −0.886325
\(999\) 2.00000 0.0632772
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2850.2.a.n.1.1 1
3.2 odd 2 8550.2.a.bh.1.1 1
5.2 odd 4 2850.2.d.e.799.1 2
5.3 odd 4 2850.2.d.e.799.2 2
5.4 even 2 570.2.a.h.1.1 1
15.14 odd 2 1710.2.a.c.1.1 1
20.19 odd 2 4560.2.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.h.1.1 1 5.4 even 2
1710.2.a.c.1.1 1 15.14 odd 2
2850.2.a.n.1.1 1 1.1 even 1 trivial
2850.2.d.e.799.1 2 5.2 odd 4
2850.2.d.e.799.2 2 5.3 odd 4
4560.2.a.bc.1.1 1 20.19 odd 2
8550.2.a.bh.1.1 1 3.2 odd 2