Properties

Label 1950.2.a.w.1.1
Level $1950$
Weight $2$
Character 1950.1
Self dual yes
Analytic conductor $15.571$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.5708283941\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1950.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} -4.00000 q^{21} -4.00000 q^{22} +1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -8.00000 q^{38} -1.00000 q^{39} -10.0000 q^{41} -4.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} -8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -2.00000 q^{51} -1.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} -4.00000 q^{56} -8.00000 q^{57} +6.00000 q^{58} +4.00000 q^{59} -2.00000 q^{61} -4.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} +16.0000 q^{67} -2.00000 q^{68} -8.00000 q^{71} +1.00000 q^{72} -2.00000 q^{73} +2.00000 q^{74} -8.00000 q^{76} +16.0000 q^{77} -1.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -10.0000 q^{82} -12.0000 q^{83} -4.00000 q^{84} -4.00000 q^{86} +6.00000 q^{87} -4.00000 q^{88} +14.0000 q^{89} +4.00000 q^{91} -4.00000 q^{93} -8.00000 q^{94} +1.00000 q^{96} -10.0000 q^{97} +9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) −4.00000 −0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −8.00000 −1.29777
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) −4.00000 −0.617213
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) −1.00000 −0.138675
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) −8.00000 −1.05963
\(58\) 6.00000 0.787839
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −4.00000 −0.508001
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 16.0000 1.95471 0.977356 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 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
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 16.0000 1.82337
\(78\) −1.00000 −0.113228
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 6.00000 0.643268
\(88\) −4.00000 −0.426401
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 9.00000 0.909137
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −2.00000 −0.198030
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −4.00000 −0.377964
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) −1.00000 −0.0924500
\(118\) 4.00000 0.368230
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) −10.0000 −0.901670
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −4.00000 −0.348155
\(133\) 32.0000 2.77475
\(134\) 16.0000 1.38219
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) −8.00000 −0.671345
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 9.00000 0.742307
\(148\) 2.00000 0.164399
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) −8.00000 −0.648886
\(153\) −2.00000 −0.161690
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 8.00000 0.636446
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −4.00000 −0.308607
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) −4.00000 −0.304997
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 4.00000 0.300658
\(178\) 14.0000 1.04934
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 4.00000 0.296500
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 8.00000 0.585018
\(188\) −8.00000 −0.583460
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) −4.00000 −0.284268
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) −2.00000 −0.140720
\(203\) −24.0000 −1.68447
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 32.0000 2.21349
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 10.0000 0.686803
\(213\) −8.00000 −0.548151
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) −2.00000 −0.135457
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) 2.00000 0.134231
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) −8.00000 −0.529813
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 6.00000 0.393919
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 8.00000 0.519656
\(238\) 8.00000 0.518563
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) 8.00000 0.509028
\(248\) −4.00000 −0.254000
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) −4.00000 −0.251976
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −4.00000 −0.249029
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 4.00000 0.247121
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) 32.0000 1.96205
\(267\) 14.0000 0.856786
\(268\) 16.0000 0.977356
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) −2.00000 −0.121268
\(273\) 4.00000 0.242091
\(274\) 10.0000 0.604122
\(275\) 0 0
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 12.0000 0.719712
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) −8.00000 −0.476393
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 40.0000 2.36113
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) −2.00000 −0.117041
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) −4.00000 −0.232104
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 12.0000 0.690522
\(303\) −2.00000 −0.114897
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 16.0000 0.911685
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 10.0000 0.560772
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 16.0000 0.890264
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) −2.00000 −0.110600
\(328\) −10.0000 −0.552158
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −12.0000 −0.658586
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 1.00000 0.0543928
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) −8.00000 −0.432590
\(343\) −8.00000 −0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 10.0000 0.537603
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 6.00000 0.321634
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −4.00000 −0.213201
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 8.00000 0.423405
\(358\) −12.0000 −0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −10.0000 −0.525588
\(363\) 5.00000 0.262432
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −40.0000 −2.07670
\(372\) −4.00000 −0.207390
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) −6.00000 −0.309016
\(378\) −4.00000 −0.205738
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) −4.00000 −0.203331
\(388\) −10.0000 −0.507673
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) 4.00000 0.201773
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) −8.00000 −0.401004
\(399\) 32.0000 1.60200
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 16.0000 0.798007
\(403\) 4.00000 0.199254
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) −24.0000 −1.19110
\(407\) −8.00000 −0.396545
\(408\) −2.00000 −0.0990148
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) −16.0000 −0.788263
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 12.0000 0.587643
\(418\) 32.0000 1.56517
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 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
\(423\) −8.00000 −0.388973
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 8.00000 0.387147
\(428\) −12.0000 −0.580042
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 1.00000 0.0481125
\(433\) 30.0000 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) −2.00000 −0.0955637
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 2.00000 0.0951303
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) −6.00000 −0.283790
\(448\) −4.00000 −0.188982
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) 6.00000 0.282216
\(453\) 12.0000 0.563809
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) −8.00000 −0.374634
\(457\) 30.0000 1.40334 0.701670 0.712502i \(-0.252438\pi\)
0.701670 + 0.712502i \(0.252438\pi\)
\(458\) 22.0000 1.02799
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 16.0000 0.744387
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −64.0000 −2.95525
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 4.00000 0.184115
\(473\) 16.0000 0.735681
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) 10.0000 0.457869
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) −10.0000 −0.450835
\(493\) −12.0000 −0.540453
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 32.0000 1.43540
\(498\) −12.0000 −0.537733
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) −40.0000 −1.78351 −0.891756 0.452517i \(-0.850526\pi\)
−0.891756 + 0.452517i \(0.850526\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 42.0000 1.86162 0.930809 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 1.00000 0.0441942
\(513\) −8.00000 −0.353209
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 32.0000 1.40736
\(518\) −8.00000 −0.351500
\(519\) 10.0000 0.438951
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 6.00000 0.262613
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 8.00000 0.348485
\(528\) −4.00000 −0.174078
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 32.0000 1.38738
\(533\) 10.0000 0.433148
\(534\) 14.0000 0.605839
\(535\) 0 0
\(536\) 16.0000 0.691095
\(537\) −12.0000 −0.517838
\(538\) −26.0000 −1.12094
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −4.00000 −0.171815
\(543\) −10.0000 −0.429141
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 10.0000 0.427179
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −48.0000 −2.04487
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −4.00000 −0.169334
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −26.0000 −1.09674
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) −4.00000 −0.167984
\(568\) −8.00000 −0.335673
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 4.00000 0.167248
\(573\) −8.00000 −0.334205
\(574\) 40.0000 1.66957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) −13.0000 −0.540729
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 48.0000 1.99138
\(582\) −10.0000 −0.414513
\(583\) −40.0000 −1.65663
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) −4.00000 −0.165098 −0.0825488 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) 9.00000 0.371154
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 2.00000 0.0821995
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 16.0000 0.652111
\(603\) 16.0000 0.651570
\(604\) 12.0000 0.488273
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) −8.00000 −0.324443
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) −2.00000 −0.0808452
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 8.00000 0.322854
\(615\) 0 0
\(616\) 16.0000 0.644658
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −16.0000 −0.643614
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −56.0000 −2.24359
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) 32.0000 1.27796
\(628\) −14.0000 −0.558661
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −36.0000 −1.43314 −0.716569 0.697517i \(-0.754288\pi\)
−0.716569 + 0.697517i \(0.754288\pi\)
\(632\) 8.00000 0.318223
\(633\) 12.0000 0.476957
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) −9.00000 −0.356593
\(638\) −24.0000 −0.950169
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −12.0000 −0.473602
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 16.0000 0.629512
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 16.0000 0.626608
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) −2.00000 −0.0780274
\(658\) 32.0000 1.24749
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 8.00000 0.310929
\(663\) 2.00000 0.0776736
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) 0 0
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) −4.00000 −0.154303
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 6.00000 0.230429
\(679\) 40.0000 1.53506
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 16.0000 0.612672
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 22.0000 0.839352
\(688\) −4.00000 −0.152499
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 10.0000 0.380143
\(693\) 16.0000 0.607790
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 20.0000 0.757554
\(698\) 6.00000 0.227103
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −16.0000 −0.603451
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 8.00000 0.300871
\(708\) 4.00000 0.150329
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 14.0000 0.524672
\(713\) 0 0
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 64.0000 2.38348
\(722\) 45.0000 1.67473
\(723\) 10.0000 0.371904
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) −2.00000 −0.0739221
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 0 0
\(737\) −64.0000 −2.35747
\(738\) −10.0000 −0.368105
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) −40.0000 −1.46845
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) −4.00000 −0.146647
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) −12.0000 −0.439057
\(748\) 8.00000 0.292509
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) −8.00000 −0.291730
\(753\) 4.00000 0.145768
\(754\) −6.00000 −0.218507
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) −54.0000 −1.96266 −0.981332 0.192323i \(-0.938398\pi\)
−0.981332 + 0.192323i \(0.938398\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −26.0000 −0.942499 −0.471250 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −4.00000 −0.144432
\(768\) 1.00000 0.0360844
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 14.0000 0.503871
\(773\) 54.0000 1.94225 0.971123 0.238581i \(-0.0766824\pi\)
0.971123 + 0.238581i \(0.0766824\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) −8.00000 −0.286998
\(778\) −26.0000 −0.932145
\(779\) 80.0000 2.86630
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 4.00000 0.142675
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) −18.0000 −0.641223
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) −4.00000 −0.142134
\(793\) 2.00000 0.0710221
\(794\) −6.00000 −0.212932
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 32.0000 1.13279
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) 6.00000 0.211867
\(803\) 8.00000 0.282314
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) −26.0000 −0.915243
\(808\) −2.00000 −0.0703598
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) −24.0000 −0.842235
\(813\) −4.00000 −0.140286
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 32.0000 1.11954
\(818\) 2.00000 0.0699284
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 10.0000 0.348790
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) −1.00000 −0.0346688
\(833\) −18.0000 −0.623663
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 32.0000 1.10674
\(837\) −4.00000 −0.138260
\(838\) 4.00000 0.138178
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 22.0000 0.758170
\(843\) −26.0000 −0.895488
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) −20.0000 −0.687208
\(848\) 10.0000 0.343401
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 0 0
\(852\) −8.00000 −0.274075
\(853\) 2.00000 0.0684787 0.0342393 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 4.00000 0.136558
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 40.0000 1.36320
\(862\) −8.00000 −0.272481
\(863\) −40.0000 −1.36162 −0.680808 0.732462i \(-0.738371\pi\)
−0.680808 + 0.732462i \(0.738371\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 30.0000 1.01944
\(867\) −13.0000 −0.441503
\(868\) 16.0000 0.543075
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) −2.00000 −0.0677285
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 16.0000 0.539974
\(879\) −26.0000 −0.876958
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 9.00000 0.303046
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 2.00000 0.0672673
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 4.00000 0.133930
\(893\) 64.0000 2.14168
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) 40.0000 1.33185
\(903\) 16.0000 0.532447
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 12.0000 0.398673
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −20.0000 −0.663723
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) −8.00000 −0.264906
\(913\) 48.0000 1.58857
\(914\) 30.0000 0.992312
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) −16.0000 −0.528367
\(918\) −2.00000 −0.0660098
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) −6.00000 −0.197599
\(923\) 8.00000 0.263323
\(924\) 16.0000 0.526361
\(925\) 0 0
\(926\) 20.0000 0.657241
\(927\) −16.0000 −0.525509
\(928\) 6.00000 0.196960
\(929\) 46.0000 1.50921 0.754606 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 0 0
\(931\) −72.0000 −2.35970
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) 4.00000 0.130884
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) −64.0000 −2.08967
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −46.0000 −1.49956 −0.749779 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) −14.0000 −0.456145
\(943\) 0 0
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 8.00000 0.259828
\(949\) 2.00000 0.0649227
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 8.00000 0.259281
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 0 0
\(957\) −24.0000 −0.775810
\(958\) −16.0000 −0.516937
\(959\) −40.0000 −1.29167
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −2.00000 −0.0644826
\(963\) −12.0000 −0.386695
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 5.00000 0.160706
\(969\) 16.0000 0.513994
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 1.00000 0.0320750
\(973\) −48.0000 −1.53881
\(974\) −4.00000 −0.128168
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 16.0000 0.511624
\(979\) −56.0000 −1.78977
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 36.0000 1.14881
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −10.0000 −0.318788
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) 32.0000 1.01857
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) 0 0
\(991\) −48.0000 −1.52477 −0.762385 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) −4.00000 −0.127000
\(993\) 8.00000 0.253872
\(994\) 32.0000 1.01498
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 0 0
\(999\) 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 1950.2.a.w.1.1 1
3.2 odd 2 5850.2.a.d.1.1 1
5.2 odd 4 1950.2.e.i.1249.2 2
5.3 odd 4 1950.2.e.i.1249.1 2
5.4 even 2 78.2.a.a.1.1 1
15.2 even 4 5850.2.e.bb.5149.1 2
15.8 even 4 5850.2.e.bb.5149.2 2
15.14 odd 2 234.2.a.c.1.1 1
20.19 odd 2 624.2.a.h.1.1 1
35.34 odd 2 3822.2.a.j.1.1 1
40.19 odd 2 2496.2.a.b.1.1 1
40.29 even 2 2496.2.a.t.1.1 1
45.4 even 6 2106.2.e.q.703.1 2
45.14 odd 6 2106.2.e.j.703.1 2
45.29 odd 6 2106.2.e.j.1405.1 2
45.34 even 6 2106.2.e.q.1405.1 2
55.54 odd 2 9438.2.a.t.1.1 1
60.59 even 2 1872.2.a.c.1.1 1
65.4 even 6 1014.2.e.c.991.1 2
65.9 even 6 1014.2.e.f.991.1 2
65.19 odd 12 1014.2.i.d.361.2 4
65.24 odd 12 1014.2.i.d.823.2 4
65.29 even 6 1014.2.e.f.529.1 2
65.34 odd 4 1014.2.b.b.337.1 2
65.44 odd 4 1014.2.b.b.337.2 2
65.49 even 6 1014.2.e.c.529.1 2
65.54 odd 12 1014.2.i.d.823.1 4
65.59 odd 12 1014.2.i.d.361.1 4
65.64 even 2 1014.2.a.d.1.1 1
120.29 odd 2 7488.2.a.bz.1.1 1
120.59 even 2 7488.2.a.bk.1.1 1
195.44 even 4 3042.2.b.g.1351.1 2
195.164 even 4 3042.2.b.g.1351.2 2
195.194 odd 2 3042.2.a.f.1.1 1
260.259 odd 2 8112.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.a.a.1.1 1 5.4 even 2
234.2.a.c.1.1 1 15.14 odd 2
624.2.a.h.1.1 1 20.19 odd 2
1014.2.a.d.1.1 1 65.64 even 2
1014.2.b.b.337.1 2 65.34 odd 4
1014.2.b.b.337.2 2 65.44 odd 4
1014.2.e.c.529.1 2 65.49 even 6
1014.2.e.c.991.1 2 65.4 even 6
1014.2.e.f.529.1 2 65.29 even 6
1014.2.e.f.991.1 2 65.9 even 6
1014.2.i.d.361.1 4 65.59 odd 12
1014.2.i.d.361.2 4 65.19 odd 12
1014.2.i.d.823.1 4 65.54 odd 12
1014.2.i.d.823.2 4 65.24 odd 12
1872.2.a.c.1.1 1 60.59 even 2
1950.2.a.w.1.1 1 1.1 even 1 trivial
1950.2.e.i.1249.1 2 5.3 odd 4
1950.2.e.i.1249.2 2 5.2 odd 4
2106.2.e.j.703.1 2 45.14 odd 6
2106.2.e.j.1405.1 2 45.29 odd 6
2106.2.e.q.703.1 2 45.4 even 6
2106.2.e.q.1405.1 2 45.34 even 6
2496.2.a.b.1.1 1 40.19 odd 2
2496.2.a.t.1.1 1 40.29 even 2
3042.2.a.f.1.1 1 195.194 odd 2
3042.2.b.g.1351.1 2 195.44 even 4
3042.2.b.g.1351.2 2 195.164 even 4
3822.2.a.j.1.1 1 35.34 odd 2
5850.2.a.d.1.1 1 3.2 odd 2
5850.2.e.bb.5149.1 2 15.2 even 4
5850.2.e.bb.5149.2 2 15.8 even 4
7488.2.a.bk.1.1 1 120.59 even 2
7488.2.a.bz.1.1 1 120.29 odd 2
8112.2.a.v.1.1 1 260.259 odd 2
9438.2.a.t.1.1 1 55.54 odd 2