Properties

Label 2850.2.a.e.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -2.00000 q^{21} -2.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} -8.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} +1.00000 q^{38} +4.00000 q^{39} -8.00000 q^{41} +2.00000 q^{42} +6.00000 q^{43} +2.00000 q^{44} +4.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -2.00000 q^{51} -4.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -2.00000 q^{56} +1.00000 q^{57} +2.00000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -8.00000 q^{67} +2.00000 q^{68} +4.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} -14.0000 q^{73} +8.00000 q^{74} -1.00000 q^{76} +4.00000 q^{77} -4.00000 q^{78} +1.00000 q^{81} +8.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} -6.00000 q^{86} -2.00000 q^{88} -8.00000 q^{91} -4.00000 q^{92} +8.00000 q^{93} -12.0000 q^{94} +1.00000 q^{96} +12.0000 q^{97} +3.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) −2.00000 −0.426401
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 1.00000 0.162221
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 2.00000 0.308607
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) −4.00000 −0.554700
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 8.00000 1.01600
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 2.00000 0.242536
\(69\) 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
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 4.00000 0.455842
\(78\) −4.00000 −0.452911
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.00000 0.883452
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) −4.00000 −0.417029
\(93\) 8.00000 0.829561
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 3.00000 0.303046
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 2.00000 0.198030
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 2.00000 0.188982
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 0 0
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) 8.00000 0.721336
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) −2.00000 −0.174078
\(133\) −2.00000 −0.173422
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 22.0000 1.87959 0.939793 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) −4.00000 −0.340503
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 8.00000 0.671345
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 3.00000 0.247436
\(148\) −8.00000 −0.657596
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 1.00000 0.0811107
\(153\) 2.00000 0.161690
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) −1.00000 −0.0785674
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) −8.00000 −0.624695
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 2.00000 0.154303
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 6.00000 0.457496
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 8.00000 0.592999
\(183\) −2.00000 −0.147844
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 4.00000 0.292509
\(188\) 12.0000 0.875190
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −24.0000 −1.72756 −0.863779 0.503871i \(-0.831909\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) −2.00000 −0.142134
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) −4.00000 −0.278019
\(208\) −4.00000 −0.277350
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 6.00000 0.412082
\(213\) 8.00000 0.548151
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −16.0000 −1.08615
\(218\) −10.0000 −0.677285
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) −8.00000 −0.536925
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 1.00000 0.0662266
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) 10.0000 0.646846 0.323423 0.946254i \(-0.395166\pi\)
0.323423 + 0.946254i \(0.395166\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −8.00000 −0.510061
\(247\) 4.00000 0.254514
\(248\) 8.00000 0.508001
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 2.00000 0.125988
\(253\) −8.00000 −0.502956
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 6.00000 0.373544
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 0 0
\(262\) 18.0000 1.11204
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 2.00000 0.121268
\(273\) 8.00000 0.484182
\(274\) −22.0000 −1.32907
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 12.0000 0.714590
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) −16.0000 −0.944450
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) −14.0000 −0.819288
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) −2.00000 −0.116052
\(298\) −10.0000 −0.579284
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 8.00000 0.460348
\(303\) −2.00000 −0.114897
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 32.0000 1.82634 0.913168 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) 4.00000 0.227921
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −4.00000 −0.226455
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 8.00000 0.445823
\(323\) −2.00000 −0.111283
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 14.0000 0.775388
\(327\) −10.0000 −0.553001
\(328\) 8.00000 0.441726
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −4.00000 −0.219529
\(333\) −8.00000 −0.438397
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) −28.0000 −1.52526 −0.762629 0.646837i \(-0.776092\pi\)
−0.762629 + 0.646837i \(0.776092\pi\)
\(338\) −3.00000 −0.163178
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 1.00000 0.0540738
\(343\) −20.0000 −1.07990
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) −2.00000 −0.106600
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) −10.0000 −0.527780 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −2.00000 −0.105118
\(363\) 7.00000 0.367405
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) −4.00000 −0.208514
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 8.00000 0.414781
\(373\) 36.0000 1.86401 0.932005 0.362446i \(-0.118058\pi\)
0.932005 + 0.362446i \(0.118058\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 2.00000 0.102869
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 18.0000 0.920960
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 24.0000 1.22157
\(387\) 6.00000 0.304997
\(388\) 12.0000 0.609208
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 3.00000 0.151523
\(393\) 18.0000 0.907980
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 20.0000 1.00251
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) −8.00000 −0.399004
\(403\) 32.0000 1.59403
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) −16.0000 −0.793091
\(408\) 2.00000 0.0990148
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 0 0
\(411\) −22.0000 −1.08518
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) 2.00000 0.0978232
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −12.0000 −0.584151
\(423\) 12.0000 0.583460
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 4.00000 0.193574
\(428\) 12.0000 0.580042
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 4.00000 0.191346
\(438\) −14.0000 −0.668946
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 8.00000 0.380521
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) −10.0000 −0.472984
\(448\) 2.00000 0.0944911
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) −16.0000 −0.753411
\(452\) 6.00000 0.282216
\(453\) 8.00000 0.375873
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) −10.0000 −0.467269
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 4.00000 0.186097
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) −4.00000 −0.184900
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 6.00000 0.274721
\(478\) −10.0000 −0.457389
\(479\) −10.0000 −0.456912 −0.228456 0.973554i \(-0.573368\pi\)
−0.228456 + 0.973554i \(0.573368\pi\)
\(480\) 0 0
\(481\) 32.0000 1.45907
\(482\) −2.00000 −0.0910975
\(483\) 8.00000 0.364013
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 14.0000 0.633102
\(490\) 0 0
\(491\) 22.0000 0.992846 0.496423 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) 8.00000 0.360668
\(493\) 0 0
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −16.0000 −0.717698
\(498\) −4.00000 −0.179244
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) −2.00000 −0.0892644
\(503\) 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
\(506\) 8.00000 0.355643
\(507\) −3.00000 −0.133235
\(508\) −8.00000 −0.354943
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 0 0
\(511\) −28.0000 −1.23865
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) −6.00000 −0.264135
\(517\) 24.0000 1.05552
\(518\) 16.0000 0.703000
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −18.0000 −0.786334
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −16.0000 −0.696971
\(528\) −2.00000 −0.0870388
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) 32.0000 1.38607
\(534\) 0 0
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −12.0000 −0.515444
\(543\) −2.00000 −0.0858282
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) 32.0000 1.36822 0.684111 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) 22.0000 0.939793
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) −4.00000 −0.170251
\(553\) 0 0
\(554\) 18.0000 0.764747
\(555\) 0 0
\(556\) 0 0
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 8.00000 0.338667
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) −12.0000 −0.506189
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) 2.00000 0.0839921
\(568\) 8.00000 0.335673
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) −8.00000 −0.334497
\(573\) 18.0000 0.751961
\(574\) 16.0000 0.667827
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 13.0000 0.540729
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 12.0000 0.497416
\(583\) 12.0000 0.496989
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) −8.00000 −0.330195 −0.165098 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) 3.00000 0.123718
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) −8.00000 −0.328798
\(593\) 46.0000 1.88899 0.944497 0.328521i \(-0.106550\pi\)
0.944497 + 0.328521i \(0.106550\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 20.0000 0.818546
\(598\) −16.0000 −0.654289
\(599\) 20.0000 0.817178 0.408589 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(600\) 0 0
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) −12.0000 −0.489083
\(603\) −8.00000 −0.325785
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 2.00000 0.0812444
\(607\) 12.0000 0.487065 0.243532 0.969893i \(-0.421694\pi\)
0.243532 + 0.969893i \(0.421694\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 2.00000 0.0808452
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) −32.0000 −1.29141
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) −4.00000 −0.160904
\(619\) −40.0000 −1.60774 −0.803868 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 18.0000 0.721734
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) 2.00000 0.0798723
\(628\) −18.0000 −0.718278
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 32.0000 1.26392 0.631962 0.774999i \(-0.282250\pi\)
0.631962 + 0.774999i \(0.282250\pi\)
\(642\) 12.0000 0.473602
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) −14.0000 −0.548282
\(653\) 46.0000 1.80012 0.900060 0.435767i \(-0.143523\pi\)
0.900060 + 0.435767i \(0.143523\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) −8.00000 −0.312348
\(657\) −14.0000 −0.546192
\(658\) −24.0000 −0.935617
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) −12.0000 −0.466393
\(663\) 8.00000 0.310694
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 8.00000 0.309994
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 2.00000 0.0771517
\(673\) −4.00000 −0.154189 −0.0770943 0.997024i \(-0.524564\pi\)
−0.0770943 + 0.997024i \(0.524564\pi\)
\(674\) 28.0000 1.07852
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 6.00000 0.230429
\(679\) 24.0000 0.921035
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 16.0000 0.612672
\(683\) 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
\(687\) −10.0000 −0.381524
\(688\) 6.00000 0.228748
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) −14.0000 −0.532200
\(693\) 4.00000 0.151947
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) 10.0000 0.378506
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −4.00000 −0.150970
\(703\) 8.00000 0.301726
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 32.0000 1.19841
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 0 0
\(717\) −10.0000 −0.373457
\(718\) 10.0000 0.373197
\(719\) 50.0000 1.86469 0.932343 0.361576i \(-0.117761\pi\)
0.932343 + 0.361576i \(0.117761\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) −1.00000 −0.0372161
\(723\) −2.00000 −0.0743808
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) −2.00000 −0.0739221
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 18.0000 0.664392
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) −16.0000 −0.589368
\(738\) 8.00000 0.294484
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) −12.0000 −0.440534
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) −36.0000 −1.31805
\(747\) −4.00000 −0.146352
\(748\) 4.00000 0.146254
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 12.0000 0.437595
\(753\) −2.00000 −0.0728841
\(754\) 0 0
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 20.0000 0.726433
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −8.00000 −0.289809
\(763\) 20.0000 0.724049
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 50.0000 1.80305 0.901523 0.432731i \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) −24.0000 −0.863779
\(773\) −34.0000 −1.22290 −0.611448 0.791285i \(-0.709412\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) −6.00000 −0.215666
\(775\) 0 0
\(776\) −12.0000 −0.430775
\(777\) 16.0000 0.573997
\(778\) −30.0000 −1.07555
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −18.0000 −0.642039
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −18.0000 −0.641223
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) −2.00000 −0.0710669
\(793\) −8.00000 −0.284088
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) −2.00000 −0.0707992
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) −12.0000 −0.423735
\(803\) −28.0000 −0.988099
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −32.0000 −1.12715
\(807\) 0 0
\(808\) −2.00000 −0.0703598
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) −12.0000 −0.420858
\(814\) 16.0000 0.560800
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) −6.00000 −0.209913
\(818\) −30.0000 −1.04893
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 22.0000 0.767338
\(823\) −34.0000 −1.18517 −0.592583 0.805510i \(-0.701892\pi\)
−0.592583 + 0.805510i \(0.701892\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) −4.00000 −0.139010
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) −4.00000 −0.138675
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) 8.00000 0.276520
\(838\) 30.0000 1.03633
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −22.0000 −0.758170
\(843\) −12.0000 −0.413302
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) −14.0000 −0.481046
\(848\) 6.00000 0.206041
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 8.00000 0.274075
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) −8.00000 −0.273115
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 16.0000 0.545279
\(862\) −12.0000 −0.408722
\(863\) 16.0000 0.544646 0.272323 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −16.0000 −0.543702
\(867\) 13.0000 0.441503
\(868\) −16.0000 −0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) −10.0000 −0.338643
\(873\) 12.0000 0.406138
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) 12.0000 0.405211 0.202606 0.979260i \(-0.435059\pi\)
0.202606 + 0.979260i \(0.435059\pi\)
\(878\) 40.0000 1.34993
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 3.00000 0.101015
\(883\) 26.0000 0.874970 0.437485 0.899226i \(-0.355869\pi\)
0.437485 + 0.899226i \(0.355869\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −8.00000 −0.268462
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −4.00000 −0.133930
\(893\) −12.0000 −0.401565
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) −16.0000 −0.534224
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 16.0000 0.532742
\(903\) −12.0000 −0.399335
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 12.0000 0.398234
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 1.00000 0.0331133
\(913\) −8.00000 −0.264761
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) −36.0000 −1.18882
\(918\) 2.00000 0.0660098
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −32.0000 −1.05444
\(922\) −2.00000 −0.0658665
\(923\) 32.0000 1.05329
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) −6.00000 −0.197172
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) 6.00000 0.196537
\(933\) 18.0000 0.589294
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 16.0000 0.522419
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) −18.0000 −0.586472
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 0 0
\(949\) 56.0000 1.81784
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) −4.00000 −0.129641
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 10.0000 0.323423
\(957\) 0 0
\(958\) 10.0000 0.323085
\(959\) 44.0000 1.42083
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −32.0000 −1.03172
\(963\) 12.0000 0.386695
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) −58.0000 −1.86515 −0.932577 0.360971i \(-0.882445\pi\)
−0.932577 + 0.360971i \(0.882445\pi\)
\(968\) 7.00000 0.224989
\(969\) 2.00000 0.0642493
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 28.0000 0.897178
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −38.0000 −1.21573 −0.607864 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) −14.0000 −0.447671
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) −22.0000 −0.702048
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −8.00000 −0.255031
\(985\) 0 0
\(986\) 0 0
\(987\) −24.0000 −0.763928
\(988\) 4.00000 0.127257
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 8.00000 0.254000
\(993\) −12.0000 −0.380808
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 62.0000 1.96356 0.981780 0.190022i \(-0.0608559\pi\)
0.981780 + 0.190022i \(0.0608559\pi\)
\(998\) 40.0000 1.26618
\(999\) 8.00000 0.253109
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.e.1.1 1
3.2 odd 2 8550.2.a.bf.1.1 1
5.2 odd 4 2850.2.d.q.799.1 2
5.3 odd 4 2850.2.d.q.799.2 2
5.4 even 2 570.2.a.l.1.1 1
15.14 odd 2 1710.2.a.b.1.1 1
20.19 odd 2 4560.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.l.1.1 1 5.4 even 2
1710.2.a.b.1.1 1 15.14 odd 2
2850.2.a.e.1.1 1 1.1 even 1 trivial
2850.2.d.q.799.1 2 5.2 odd 4
2850.2.d.q.799.2 2 5.3 odd 4
4560.2.a.o.1.1 1 20.19 odd 2
8550.2.a.bf.1.1 1 3.2 odd 2