Properties

Label 5070.2.a.i.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.00000 q^{11} +1.00000 q^{12} +5.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -8.00000 q^{17} -1.00000 q^{18} -5.00000 q^{19} -1.00000 q^{20} -5.00000 q^{21} +3.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -5.00000 q^{28} -4.00000 q^{29} +1.00000 q^{30} -2.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +8.00000 q^{34} +5.00000 q^{35} +1.00000 q^{36} -7.00000 q^{37} +5.00000 q^{38} +1.00000 q^{40} +6.00000 q^{41} +5.00000 q^{42} +6.00000 q^{43} -3.00000 q^{44} -1.00000 q^{45} +4.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} +18.0000 q^{49} -1.00000 q^{50} -8.00000 q^{51} +1.00000 q^{53} -1.00000 q^{54} +3.00000 q^{55} +5.00000 q^{56} -5.00000 q^{57} +4.00000 q^{58} +12.0000 q^{59} -1.00000 q^{60} +2.00000 q^{61} +2.00000 q^{62} -5.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +8.00000 q^{67} -8.00000 q^{68} -4.00000 q^{69} -5.00000 q^{70} +2.00000 q^{71} -1.00000 q^{72} +7.00000 q^{74} +1.00000 q^{75} -5.00000 q^{76} +15.0000 q^{77} -2.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +8.00000 q^{83} -5.00000 q^{84} +8.00000 q^{85} -6.00000 q^{86} -4.00000 q^{87} +3.00000 q^{88} -11.0000 q^{89} +1.00000 q^{90} -4.00000 q^{92} -2.00000 q^{93} +3.00000 q^{94} +5.00000 q^{95} -1.00000 q^{96} -18.0000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −5.00000 −1.88982 −0.944911 0.327327i \(-0.893852\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 5.00000 1.33631
\(15\) −1.00000 −0.258199
\(16\) 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
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) −1.00000 −0.223607
\(21\) −5.00000 −1.09109
\(22\) 3.00000 0.639602
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −5.00000 −0.944911
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 1.00000 0.182574
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) 8.00000 1.37199
\(35\) 5.00000 0.845154
\(36\) 1.00000 0.166667
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 5.00000 0.811107
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 5.00000 0.771517
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −3.00000 −0.452267
\(45\) −1.00000 −0.149071
\(46\) 4.00000 0.589768
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 1.00000 0.144338
\(49\) 18.0000 2.57143
\(50\) −1.00000 −0.141421
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) −1.00000 −0.136083
\(55\) 3.00000 0.404520
\(56\) 5.00000 0.668153
\(57\) −5.00000 −0.662266
\(58\) 4.00000 0.525226
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 2.00000 0.254000
\(63\) −5.00000 −0.629941
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −8.00000 −0.970143
\(69\) −4.00000 −0.481543
\(70\) −5.00000 −0.597614
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −1.00000 −0.117851
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 7.00000 0.813733
\(75\) 1.00000 0.115470
\(76\) −5.00000 −0.573539
\(77\) 15.0000 1.70941
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −5.00000 −0.545545
\(85\) 8.00000 0.867722
\(86\) −6.00000 −0.646997
\(87\) −4.00000 −0.428845
\(88\) 3.00000 0.319801
\(89\) −11.0000 −1.16600 −0.582999 0.812473i \(-0.698121\pi\)
−0.582999 + 0.812473i \(0.698121\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) −2.00000 −0.207390
\(94\) 3.00000 0.309426
\(95\) 5.00000 0.512989
\(96\) −1.00000 −0.102062
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −18.0000 −1.81827
\(99\) −3.00000 −0.301511
\(100\) 1.00000 0.100000
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 8.00000 0.792118
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 0 0
\(105\) 5.00000 0.487950
\(106\) −1.00000 −0.0971286
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 1.00000 0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) −3.00000 −0.286039
\(111\) −7.00000 −0.664411
\(112\) −5.00000 −0.472456
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 5.00000 0.468293
\(115\) 4.00000 0.373002
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) 40.0000 3.66679
\(120\) 1.00000 0.0912871
\(121\) −2.00000 −0.181818
\(122\) −2.00000 −0.181071
\(123\) 6.00000 0.541002
\(124\) −2.00000 −0.179605
\(125\) −1.00000 −0.0894427
\(126\) 5.00000 0.445435
\(127\) −21.0000 −1.86345 −0.931724 0.363166i \(-0.881696\pi\)
−0.931724 + 0.363166i \(0.881696\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −19.0000 −1.66004 −0.830019 0.557735i \(-0.811670\pi\)
−0.830019 + 0.557735i \(0.811670\pi\)
\(132\) −3.00000 −0.261116
\(133\) 25.0000 2.16777
\(134\) −8.00000 −0.691095
\(135\) −1.00000 −0.0860663
\(136\) 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
\(139\) 7.00000 0.593732 0.296866 0.954919i \(-0.404058\pi\)
0.296866 + 0.954919i \(0.404058\pi\)
\(140\) 5.00000 0.422577
\(141\) −3.00000 −0.252646
\(142\) −2.00000 −0.167836
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) 18.0000 1.48461
\(148\) −7.00000 −0.575396
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 22.0000 1.79033 0.895167 0.445730i \(-0.147056\pi\)
0.895167 + 0.445730i \(0.147056\pi\)
\(152\) 5.00000 0.405554
\(153\) −8.00000 −0.646762
\(154\) −15.0000 −1.20873
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 15.0000 1.19713 0.598565 0.801074i \(-0.295738\pi\)
0.598565 + 0.801074i \(0.295738\pi\)
\(158\) 2.00000 0.159111
\(159\) 1.00000 0.0793052
\(160\) 1.00000 0.0790569
\(161\) 20.0000 1.57622
\(162\) −1.00000 −0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 6.00000 0.468521
\(165\) 3.00000 0.233550
\(166\) −8.00000 −0.620920
\(167\) −23.0000 −1.77979 −0.889897 0.456162i \(-0.849224\pi\)
−0.889897 + 0.456162i \(0.849224\pi\)
\(168\) 5.00000 0.385758
\(169\) 0 0
\(170\) −8.00000 −0.613572
\(171\) −5.00000 −0.382360
\(172\) 6.00000 0.457496
\(173\) 5.00000 0.380143 0.190071 0.981770i \(-0.439128\pi\)
0.190071 + 0.981770i \(0.439128\pi\)
\(174\) 4.00000 0.303239
\(175\) −5.00000 −0.377964
\(176\) −3.00000 −0.226134
\(177\) 12.0000 0.901975
\(178\) 11.0000 0.824485
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 4.00000 0.294884
\(185\) 7.00000 0.514650
\(186\) 2.00000 0.146647
\(187\) 24.0000 1.75505
\(188\) −3.00000 −0.218797
\(189\) −5.00000 −0.363696
\(190\) −5.00000 −0.362738
\(191\) 2.00000 0.144715 0.0723575 0.997379i \(-0.476948\pi\)
0.0723575 + 0.997379i \(0.476948\pi\)
\(192\) 1.00000 0.0721688
\(193\) 24.0000 1.72756 0.863779 0.503871i \(-0.168091\pi\)
0.863779 + 0.503871i \(0.168091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 18.0000 1.28571
\(197\) 3.00000 0.213741 0.106871 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) 3.00000 0.213201
\(199\) 22.0000 1.55954 0.779769 0.626067i \(-0.215336\pi\)
0.779769 + 0.626067i \(0.215336\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) 8.00000 0.562878
\(203\) 20.0000 1.40372
\(204\) −8.00000 −0.560112
\(205\) −6.00000 −0.419058
\(206\) 7.00000 0.487713
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 15.0000 1.03757
\(210\) −5.00000 −0.345033
\(211\) −15.0000 −1.03264 −0.516321 0.856395i \(-0.672699\pi\)
−0.516321 + 0.856395i \(0.672699\pi\)
\(212\) 1.00000 0.0686803
\(213\) 2.00000 0.137038
\(214\) 6.00000 0.410152
\(215\) −6.00000 −0.409197
\(216\) −1.00000 −0.0680414
\(217\) 10.0000 0.678844
\(218\) −14.0000 −0.948200
\(219\) 0 0
\(220\) 3.00000 0.202260
\(221\) 0 0
\(222\) 7.00000 0.469809
\(223\) −3.00000 −0.200895 −0.100447 0.994942i \(-0.532027\pi\)
−0.100447 + 0.994942i \(0.532027\pi\)
\(224\) 5.00000 0.334077
\(225\) 1.00000 0.0666667
\(226\) −8.00000 −0.532152
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) −5.00000 −0.331133
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −4.00000 −0.263752
\(231\) 15.0000 0.986928
\(232\) 4.00000 0.262613
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) 3.00000 0.195698
\(236\) 12.0000 0.781133
\(237\) −2.00000 −0.129914
\(238\) −40.0000 −2.59281
\(239\) −18.0000 −1.16432 −0.582162 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −25.0000 −1.61039 −0.805196 0.593009i \(-0.797940\pi\)
−0.805196 + 0.593009i \(0.797940\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) −18.0000 −1.14998
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) 2.00000 0.127000
\(249\) 8.00000 0.506979
\(250\) 1.00000 0.0632456
\(251\) 15.0000 0.946792 0.473396 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) −5.00000 −0.314970
\(253\) 12.0000 0.754434
\(254\) 21.0000 1.31766
\(255\) 8.00000 0.500979
\(256\) 1.00000 0.0625000
\(257\) 28.0000 1.74659 0.873296 0.487190i \(-0.161978\pi\)
0.873296 + 0.487190i \(0.161978\pi\)
\(258\) −6.00000 −0.373544
\(259\) 35.0000 2.17479
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) 19.0000 1.17382
\(263\) −15.0000 −0.924940 −0.462470 0.886635i \(-0.653037\pi\)
−0.462470 + 0.886635i \(0.653037\pi\)
\(264\) 3.00000 0.184637
\(265\) −1.00000 −0.0614295
\(266\) −25.0000 −1.53285
\(267\) −11.0000 −0.673189
\(268\) 8.00000 0.488678
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 1.00000 0.0608581
\(271\) 4.00000 0.242983 0.121491 0.992592i \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) −3.00000 −0.180907
\(276\) −4.00000 −0.240772
\(277\) −15.0000 −0.901263 −0.450631 0.892710i \(-0.648801\pi\)
−0.450631 + 0.892710i \(0.648801\pi\)
\(278\) −7.00000 −0.419832
\(279\) −2.00000 −0.119737
\(280\) −5.00000 −0.298807
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 3.00000 0.178647
\(283\) −10.0000 −0.594438 −0.297219 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) 2.00000 0.118678
\(285\) 5.00000 0.296174
\(286\) 0 0
\(287\) −30.0000 −1.77084
\(288\) −1.00000 −0.0589256
\(289\) 47.0000 2.76471
\(290\) −4.00000 −0.234888
\(291\) 0 0
\(292\) 0 0
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) −18.0000 −1.04978
\(295\) −12.0000 −0.698667
\(296\) 7.00000 0.406867
\(297\) −3.00000 −0.174078
\(298\) 2.00000 0.115857
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −30.0000 −1.72917
\(302\) −22.0000 −1.26596
\(303\) −8.00000 −0.459588
\(304\) −5.00000 −0.286770
\(305\) −2.00000 −0.114520
\(306\) 8.00000 0.457330
\(307\) −6.00000 −0.342438 −0.171219 0.985233i \(-0.554771\pi\)
−0.171219 + 0.985233i \(0.554771\pi\)
\(308\) 15.0000 0.854704
\(309\) −7.00000 −0.398216
\(310\) −2.00000 −0.113592
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −15.0000 −0.846499
\(315\) 5.00000 0.281718
\(316\) −2.00000 −0.112509
\(317\) −23.0000 −1.29181 −0.645904 0.763418i \(-0.723520\pi\)
−0.645904 + 0.763418i \(0.723520\pi\)
\(318\) −1.00000 −0.0560772
\(319\) 12.0000 0.671871
\(320\) −1.00000 −0.0559017
\(321\) −6.00000 −0.334887
\(322\) −20.0000 −1.11456
\(323\) 40.0000 2.22566
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) 14.0000 0.774202
\(328\) −6.00000 −0.331295
\(329\) 15.0000 0.826977
\(330\) −3.00000 −0.165145
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 8.00000 0.439057
\(333\) −7.00000 −0.383598
\(334\) 23.0000 1.25850
\(335\) −8.00000 −0.437087
\(336\) −5.00000 −0.272772
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 8.00000 0.434500
\(340\) 8.00000 0.433861
\(341\) 6.00000 0.324918
\(342\) 5.00000 0.270369
\(343\) −55.0000 −2.96972
\(344\) −6.00000 −0.323498
\(345\) 4.00000 0.215353
\(346\) −5.00000 −0.268802
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) −4.00000 −0.214423
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 5.00000 0.267261
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) 16.0000 0.851594 0.425797 0.904819i \(-0.359994\pi\)
0.425797 + 0.904819i \(0.359994\pi\)
\(354\) −12.0000 −0.637793
\(355\) −2.00000 −0.106149
\(356\) −11.0000 −0.582999
\(357\) 40.0000 2.11702
\(358\) −4.00000 −0.211407
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 1.00000 0.0527046
\(361\) 6.00000 0.315789
\(362\) 2.00000 0.105118
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −4.00000 −0.208514
\(369\) 6.00000 0.312348
\(370\) −7.00000 −0.363913
\(371\) −5.00000 −0.259587
\(372\) −2.00000 −0.103695
\(373\) 38.0000 1.96757 0.983783 0.179364i \(-0.0574041\pi\)
0.983783 + 0.179364i \(0.0574041\pi\)
\(374\) −24.0000 −1.24101
\(375\) −1.00000 −0.0516398
\(376\) 3.00000 0.154713
\(377\) 0 0
\(378\) 5.00000 0.257172
\(379\) −25.0000 −1.28416 −0.642082 0.766636i \(-0.721929\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(380\) 5.00000 0.256495
\(381\) −21.0000 −1.07586
\(382\) −2.00000 −0.102329
\(383\) −28.0000 −1.43073 −0.715367 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −15.0000 −0.764471
\(386\) −24.0000 −1.22157
\(387\) 6.00000 0.304997
\(388\) 0 0
\(389\) −32.0000 −1.62246 −0.811232 0.584724i \(-0.801203\pi\)
−0.811232 + 0.584724i \(0.801203\pi\)
\(390\) 0 0
\(391\) 32.0000 1.61831
\(392\) −18.0000 −0.909137
\(393\) −19.0000 −0.958423
\(394\) −3.00000 −0.151138
\(395\) 2.00000 0.100631
\(396\) −3.00000 −0.150756
\(397\) 25.0000 1.25471 0.627357 0.778732i \(-0.284137\pi\)
0.627357 + 0.778732i \(0.284137\pi\)
\(398\) −22.0000 −1.10276
\(399\) 25.0000 1.25157
\(400\) 1.00000 0.0500000
\(401\) −19.0000 −0.948815 −0.474407 0.880305i \(-0.657338\pi\)
−0.474407 + 0.880305i \(0.657338\pi\)
\(402\) −8.00000 −0.399004
\(403\) 0 0
\(404\) −8.00000 −0.398015
\(405\) −1.00000 −0.0496904
\(406\) −20.0000 −0.992583
\(407\) 21.0000 1.04093
\(408\) 8.00000 0.396059
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 6.00000 0.296319
\(411\) −12.0000 −0.591916
\(412\) −7.00000 −0.344865
\(413\) −60.0000 −2.95241
\(414\) 4.00000 0.196589
\(415\) −8.00000 −0.392705
\(416\) 0 0
\(417\) 7.00000 0.342791
\(418\) −15.0000 −0.733674
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 5.00000 0.243975
\(421\) 12.0000 0.584844 0.292422 0.956289i \(-0.405539\pi\)
0.292422 + 0.956289i \(0.405539\pi\)
\(422\) 15.0000 0.730189
\(423\) −3.00000 −0.145865
\(424\) −1.00000 −0.0485643
\(425\) −8.00000 −0.388057
\(426\) −2.00000 −0.0969003
\(427\) −10.0000 −0.483934
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) 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 \(0.374389\pi\)
\(434\) −10.0000 −0.480015
\(435\) 4.00000 0.191785
\(436\) 14.0000 0.670478
\(437\) 20.0000 0.956730
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) −3.00000 −0.143019
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) −7.00000 −0.332205
\(445\) 11.0000 0.521450
\(446\) 3.00000 0.142054
\(447\) −2.00000 −0.0945968
\(448\) −5.00000 −0.236228
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −18.0000 −0.847587
\(452\) 8.00000 0.376288
\(453\) 22.0000 1.03365
\(454\) 0 0
\(455\) 0 0
\(456\) 5.00000 0.234146
\(457\) 30.0000 1.40334 0.701670 0.712502i \(-0.252438\pi\)
0.701670 + 0.712502i \(0.252438\pi\)
\(458\) −14.0000 −0.654177
\(459\) −8.00000 −0.373408
\(460\) 4.00000 0.186501
\(461\) 8.00000 0.372597 0.186299 0.982493i \(-0.440351\pi\)
0.186299 + 0.982493i \(0.440351\pi\)
\(462\) −15.0000 −0.697863
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −4.00000 −0.185695
\(465\) 2.00000 0.0927478
\(466\) 14.0000 0.648537
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) −40.0000 −1.84703
\(470\) −3.00000 −0.138380
\(471\) 15.0000 0.691164
\(472\) −12.0000 −0.552345
\(473\) −18.0000 −0.827641
\(474\) 2.00000 0.0918630
\(475\) −5.00000 −0.229416
\(476\) 40.0000 1.83340
\(477\) 1.00000 0.0457869
\(478\) 18.0000 0.823301
\(479\) −28.0000 −1.27935 −0.639676 0.768644i \(-0.720932\pi\)
−0.639676 + 0.768644i \(0.720932\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 25.0000 1.13872
\(483\) 20.0000 0.910032
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 37.0000 1.67663 0.838315 0.545186i \(-0.183541\pi\)
0.838315 + 0.545186i \(0.183541\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −20.0000 −0.904431
\(490\) 18.0000 0.813157
\(491\) 21.0000 0.947717 0.473858 0.880601i \(-0.342861\pi\)
0.473858 + 0.880601i \(0.342861\pi\)
\(492\) 6.00000 0.270501
\(493\) 32.0000 1.44121
\(494\) 0 0
\(495\) 3.00000 0.134840
\(496\) −2.00000 −0.0898027
\(497\) −10.0000 −0.448561
\(498\) −8.00000 −0.358489
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −23.0000 −1.02756
\(502\) −15.0000 −0.669483
\(503\) −11.0000 −0.490466 −0.245233 0.969464i \(-0.578864\pi\)
−0.245233 + 0.969464i \(0.578864\pi\)
\(504\) 5.00000 0.222718
\(505\) 8.00000 0.355995
\(506\) −12.0000 −0.533465
\(507\) 0 0
\(508\) −21.0000 −0.931724
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) −8.00000 −0.354246
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −5.00000 −0.220755
\(514\) −28.0000 −1.23503
\(515\) 7.00000 0.308457
\(516\) 6.00000 0.264135
\(517\) 9.00000 0.395820
\(518\) −35.0000 −1.53781
\(519\) 5.00000 0.219476
\(520\) 0 0
\(521\) 5.00000 0.219054 0.109527 0.993984i \(-0.465066\pi\)
0.109527 + 0.993984i \(0.465066\pi\)
\(522\) 4.00000 0.175075
\(523\) −10.0000 −0.437269 −0.218635 0.975807i \(-0.570160\pi\)
−0.218635 + 0.975807i \(0.570160\pi\)
\(524\) −19.0000 −0.830019
\(525\) −5.00000 −0.218218
\(526\) 15.0000 0.654031
\(527\) 16.0000 0.696971
\(528\) −3.00000 −0.130558
\(529\) −7.00000 −0.304348
\(530\) 1.00000 0.0434372
\(531\) 12.0000 0.520756
\(532\) 25.0000 1.08389
\(533\) 0 0
\(534\) 11.0000 0.476017
\(535\) 6.00000 0.259403
\(536\) −8.00000 −0.345547
\(537\) 4.00000 0.172613
\(538\) −4.00000 −0.172452
\(539\) −54.0000 −2.32594
\(540\) −1.00000 −0.0430331
\(541\) 34.0000 1.46177 0.730887 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) −4.00000 −0.171815
\(543\) −2.00000 −0.0858282
\(544\) 8.00000 0.342997
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) 6.00000 0.256541 0.128271 0.991739i \(-0.459057\pi\)
0.128271 + 0.991739i \(0.459057\pi\)
\(548\) −12.0000 −0.512615
\(549\) 2.00000 0.0853579
\(550\) 3.00000 0.127920
\(551\) 20.0000 0.852029
\(552\) 4.00000 0.170251
\(553\) 10.0000 0.425243
\(554\) 15.0000 0.637289
\(555\) 7.00000 0.297133
\(556\) 7.00000 0.296866
\(557\) 15.0000 0.635570 0.317785 0.948163i \(-0.397061\pi\)
0.317785 + 0.948163i \(0.397061\pi\)
\(558\) 2.00000 0.0846668
\(559\) 0 0
\(560\) 5.00000 0.211289
\(561\) 24.0000 1.01328
\(562\) −18.0000 −0.759284
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) −3.00000 −0.126323
\(565\) −8.00000 −0.336563
\(566\) 10.0000 0.420331
\(567\) −5.00000 −0.209980
\(568\) −2.00000 −0.0839181
\(569\) −15.0000 −0.628833 −0.314416 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) −5.00000 −0.209427
\(571\) −33.0000 −1.38101 −0.690504 0.723329i \(-0.742611\pi\)
−0.690504 + 0.723329i \(0.742611\pi\)
\(572\) 0 0
\(573\) 2.00000 0.0835512
\(574\) 30.0000 1.25218
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −47.0000 −1.95494
\(579\) 24.0000 0.997406
\(580\) 4.00000 0.166091
\(581\) −40.0000 −1.65948
\(582\) 0 0
\(583\) −3.00000 −0.124247
\(584\) 0 0
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 18.0000 0.742307
\(589\) 10.0000 0.412043
\(590\) 12.0000 0.494032
\(591\) 3.00000 0.123404
\(592\) −7.00000 −0.287698
\(593\) 20.0000 0.821302 0.410651 0.911793i \(-0.365302\pi\)
0.410651 + 0.911793i \(0.365302\pi\)
\(594\) 3.00000 0.123091
\(595\) −40.0000 −1.63984
\(596\) −2.00000 −0.0819232
\(597\) 22.0000 0.900400
\(598\) 0 0
\(599\) 34.0000 1.38920 0.694601 0.719395i \(-0.255581\pi\)
0.694601 + 0.719395i \(0.255581\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 37.0000 1.50926 0.754631 0.656150i \(-0.227816\pi\)
0.754631 + 0.656150i \(0.227816\pi\)
\(602\) 30.0000 1.22271
\(603\) 8.00000 0.325785
\(604\) 22.0000 0.895167
\(605\) 2.00000 0.0813116
\(606\) 8.00000 0.324978
\(607\) −29.0000 −1.17707 −0.588537 0.808470i \(-0.700296\pi\)
−0.588537 + 0.808470i \(0.700296\pi\)
\(608\) 5.00000 0.202777
\(609\) 20.0000 0.810441
\(610\) 2.00000 0.0809776
\(611\) 0 0
\(612\) −8.00000 −0.323381
\(613\) 25.0000 1.00974 0.504870 0.863195i \(-0.331540\pi\)
0.504870 + 0.863195i \(0.331540\pi\)
\(614\) 6.00000 0.242140
\(615\) −6.00000 −0.241943
\(616\) −15.0000 −0.604367
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 7.00000 0.281581
\(619\) 17.0000 0.683288 0.341644 0.939829i \(-0.389016\pi\)
0.341644 + 0.939829i \(0.389016\pi\)
\(620\) 2.00000 0.0803219
\(621\) −4.00000 −0.160514
\(622\) 12.0000 0.481156
\(623\) 55.0000 2.20353
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 6.00000 0.239808
\(627\) 15.0000 0.599042
\(628\) 15.0000 0.598565
\(629\) 56.0000 2.23287
\(630\) −5.00000 −0.199205
\(631\) 12.0000 0.477712 0.238856 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) 2.00000 0.0795557
\(633\) −15.0000 −0.596196
\(634\) 23.0000 0.913447
\(635\) 21.0000 0.833360
\(636\) 1.00000 0.0396526
\(637\) 0 0
\(638\) −12.0000 −0.475085
\(639\) 2.00000 0.0791188
\(640\) 1.00000 0.0395285
\(641\) 27.0000 1.06644 0.533218 0.845978i \(-0.320983\pi\)
0.533218 + 0.845978i \(0.320983\pi\)
\(642\) 6.00000 0.236801
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 20.0000 0.788110
\(645\) −6.00000 −0.236250
\(646\) −40.0000 −1.57378
\(647\) 3.00000 0.117942 0.0589711 0.998260i \(-0.481218\pi\)
0.0589711 + 0.998260i \(0.481218\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) −20.0000 −0.783260
\(653\) 27.0000 1.05659 0.528296 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) −14.0000 −0.547443
\(655\) 19.0000 0.742391
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) −15.0000 −0.584761
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 3.00000 0.116775
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) −25.0000 −0.969458
\(666\) 7.00000 0.271244
\(667\) 16.0000 0.619522
\(668\) −23.0000 −0.889897
\(669\) −3.00000 −0.115987
\(670\) 8.00000 0.309067
\(671\) −6.00000 −0.231627
\(672\) 5.00000 0.192879
\(673\) −32.0000 −1.23351 −0.616755 0.787155i \(-0.711553\pi\)
−0.616755 + 0.787155i \(0.711553\pi\)
\(674\) −14.0000 −0.539260
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) −8.00000 −0.307238
\(679\) 0 0
\(680\) −8.00000 −0.306786
\(681\) 0 0
\(682\) −6.00000 −0.229752
\(683\) 30.0000 1.14792 0.573959 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(684\) −5.00000 −0.191180
\(685\) 12.0000 0.458496
\(686\) 55.0000 2.09991
\(687\) 14.0000 0.534133
\(688\) 6.00000 0.228748
\(689\) 0 0
\(690\) −4.00000 −0.152277
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) 5.00000 0.190071
\(693\) 15.0000 0.569803
\(694\) 16.0000 0.607352
\(695\) −7.00000 −0.265525
\(696\) 4.00000 0.151620
\(697\) −48.0000 −1.81813
\(698\) −8.00000 −0.302804
\(699\) −14.0000 −0.529529
\(700\) −5.00000 −0.188982
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 35.0000 1.32005
\(704\) −3.00000 −0.113067
\(705\) 3.00000 0.112987
\(706\) −16.0000 −0.602168
\(707\) 40.0000 1.50435
\(708\) 12.0000 0.450988
\(709\) 32.0000 1.20179 0.600893 0.799330i \(-0.294812\pi\)
0.600893 + 0.799330i \(0.294812\pi\)
\(710\) 2.00000 0.0750587
\(711\) −2.00000 −0.0750059
\(712\) 11.0000 0.412242
\(713\) 8.00000 0.299602
\(714\) −40.0000 −1.49696
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) −18.0000 −0.672222
\(718\) 18.0000 0.671754
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 35.0000 1.30347
\(722\) −6.00000 −0.223297
\(723\) −25.0000 −0.929760
\(724\) −2.00000 −0.0743294
\(725\) −4.00000 −0.148556
\(726\) 2.00000 0.0742270
\(727\) −11.0000 −0.407967 −0.203984 0.978974i \(-0.565389\pi\)
−0.203984 + 0.978974i \(0.565389\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) 2.00000 0.0739221
\(733\) −43.0000 −1.58824 −0.794121 0.607760i \(-0.792068\pi\)
−0.794121 + 0.607760i \(0.792068\pi\)
\(734\) 8.00000 0.295285
\(735\) −18.0000 −0.663940
\(736\) 4.00000 0.147442
\(737\) −24.0000 −0.884051
\(738\) −6.00000 −0.220863
\(739\) 19.0000 0.698926 0.349463 0.936950i \(-0.386364\pi\)
0.349463 + 0.936950i \(0.386364\pi\)
\(740\) 7.00000 0.257325
\(741\) 0 0
\(742\) 5.00000 0.183556
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 2.00000 0.0733236
\(745\) 2.00000 0.0732743
\(746\) −38.0000 −1.39128
\(747\) 8.00000 0.292705
\(748\) 24.0000 0.877527
\(749\) 30.0000 1.09618
\(750\) 1.00000 0.0365148
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) −3.00000 −0.109399
\(753\) 15.0000 0.546630
\(754\) 0 0
\(755\) −22.0000 −0.800662
\(756\) −5.00000 −0.181848
\(757\) 17.0000 0.617876 0.308938 0.951082i \(-0.400027\pi\)
0.308938 + 0.951082i \(0.400027\pi\)
\(758\) 25.0000 0.908041
\(759\) 12.0000 0.435572
\(760\) −5.00000 −0.181369
\(761\) 9.00000 0.326250 0.163125 0.986605i \(-0.447843\pi\)
0.163125 + 0.986605i \(0.447843\pi\)
\(762\) 21.0000 0.760750
\(763\) −70.0000 −2.53417
\(764\) 2.00000 0.0723575
\(765\) 8.00000 0.289241
\(766\) 28.0000 1.01168
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 15.0000 0.540562
\(771\) 28.0000 1.00840
\(772\) 24.0000 0.863779
\(773\) 1.00000 0.0359675 0.0179838 0.999838i \(-0.494275\pi\)
0.0179838 + 0.999838i \(0.494275\pi\)
\(774\) −6.00000 −0.215666
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) 35.0000 1.25562
\(778\) 32.0000 1.14726
\(779\) −30.0000 −1.07486
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) −32.0000 −1.14432
\(783\) −4.00000 −0.142948
\(784\) 18.0000 0.642857
\(785\) −15.0000 −0.535373
\(786\) 19.0000 0.677708
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 3.00000 0.106871
\(789\) −15.0000 −0.534014
\(790\) −2.00000 −0.0711568
\(791\) −40.0000 −1.42224
\(792\) 3.00000 0.106600
\(793\) 0 0
\(794\) −25.0000 −0.887217
\(795\) −1.00000 −0.0354663
\(796\) 22.0000 0.779769
\(797\) −10.0000 −0.354218 −0.177109 0.984191i \(-0.556675\pi\)
−0.177109 + 0.984191i \(0.556675\pi\)
\(798\) −25.0000 −0.884990
\(799\) 24.0000 0.849059
\(800\) −1.00000 −0.0353553
\(801\) −11.0000 −0.388666
\(802\) 19.0000 0.670913
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) −20.0000 −0.704907
\(806\) 0 0
\(807\) 4.00000 0.140807
\(808\) 8.00000 0.281439
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 1.00000 0.0351364
\(811\) 25.0000 0.877869 0.438934 0.898519i \(-0.355356\pi\)
0.438934 + 0.898519i \(0.355356\pi\)
\(812\) 20.0000 0.701862
\(813\) 4.00000 0.140286
\(814\) −21.0000 −0.736050
\(815\) 20.0000 0.700569
\(816\) −8.00000 −0.280056
\(817\) −30.0000 −1.04957
\(818\) −25.0000 −0.874105
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) −22.0000 −0.767805 −0.383903 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) 12.0000 0.418548
\(823\) −35.0000 −1.22002 −0.610012 0.792392i \(-0.708835\pi\)
−0.610012 + 0.792392i \(0.708835\pi\)
\(824\) 7.00000 0.243857
\(825\) −3.00000 −0.104447
\(826\) 60.0000 2.08767
\(827\) 6.00000 0.208640 0.104320 0.994544i \(-0.466733\pi\)
0.104320 + 0.994544i \(0.466733\pi\)
\(828\) −4.00000 −0.139010
\(829\) −16.0000 −0.555703 −0.277851 0.960624i \(-0.589622\pi\)
−0.277851 + 0.960624i \(0.589622\pi\)
\(830\) 8.00000 0.277684
\(831\) −15.0000 −0.520344
\(832\) 0 0
\(833\) −144.000 −4.98930
\(834\) −7.00000 −0.242390
\(835\) 23.0000 0.795948
\(836\) 15.0000 0.518786
\(837\) −2.00000 −0.0691301
\(838\) 12.0000 0.414533
\(839\) −14.0000 −0.483334 −0.241667 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) −5.00000 −0.172516
\(841\) −13.0000 −0.448276
\(842\) −12.0000 −0.413547
\(843\) 18.0000 0.619953
\(844\) −15.0000 −0.516321
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) 10.0000 0.343604
\(848\) 1.00000 0.0343401
\(849\) −10.0000 −0.343199
\(850\) 8.00000 0.274398
\(851\) 28.0000 0.959828
\(852\) 2.00000 0.0685189
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 10.0000 0.342193
\(855\) 5.00000 0.170996
\(856\) 6.00000 0.205076
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) 3.00000 0.102359 0.0511793 0.998689i \(-0.483702\pi\)
0.0511793 + 0.998689i \(0.483702\pi\)
\(860\) −6.00000 −0.204598
\(861\) −30.0000 −1.02240
\(862\) −12.0000 −0.408722
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −5.00000 −0.170005
\(866\) −16.0000 −0.543702
\(867\) 47.0000 1.59620
\(868\) 10.0000 0.339422
\(869\) 6.00000 0.203536
\(870\) −4.00000 −0.135613
\(871\) 0 0
\(872\) −14.0000 −0.474100
\(873\) 0 0
\(874\) −20.0000 −0.676510
\(875\) 5.00000 0.169031
\(876\) 0 0
\(877\) −54.0000 −1.82345 −0.911725 0.410801i \(-0.865249\pi\)
−0.911725 + 0.410801i \(0.865249\pi\)
\(878\) −10.0000 −0.337484
\(879\) −9.00000 −0.303562
\(880\) 3.00000 0.101130
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) −18.0000 −0.606092
\(883\) 30.0000 1.00958 0.504790 0.863242i \(-0.331570\pi\)
0.504790 + 0.863242i \(0.331570\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) −6.00000 −0.201574
\(887\) 41.0000 1.37665 0.688323 0.725405i \(-0.258347\pi\)
0.688323 + 0.725405i \(0.258347\pi\)
\(888\) 7.00000 0.234905
\(889\) 105.000 3.52159
\(890\) −11.0000 −0.368721
\(891\) −3.00000 −0.100504
\(892\) −3.00000 −0.100447
\(893\) 15.0000 0.501956
\(894\) 2.00000 0.0668900
\(895\) −4.00000 −0.133705
\(896\) 5.00000 0.167038
\(897\) 0 0
\(898\) 27.0000 0.901002
\(899\) 8.00000 0.266815
\(900\) 1.00000 0.0333333
\(901\) −8.00000 −0.266519
\(902\) 18.0000 0.599334
\(903\) −30.0000 −0.998337
\(904\) −8.00000 −0.266076
\(905\) 2.00000 0.0664822
\(906\) −22.0000 −0.730901
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) 0 0
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) −5.00000 −0.165567
\(913\) −24.0000 −0.794284
\(914\) −30.0000 −0.992312
\(915\) −2.00000 −0.0661180
\(916\) 14.0000 0.462573
\(917\) 95.0000 3.13718
\(918\) 8.00000 0.264039
\(919\) −2.00000 −0.0659739 −0.0329870 0.999456i \(-0.510502\pi\)
−0.0329870 + 0.999456i \(0.510502\pi\)
\(920\) −4.00000 −0.131876
\(921\) −6.00000 −0.197707
\(922\) −8.00000 −0.263466
\(923\) 0 0
\(924\) 15.0000 0.493464
\(925\) −7.00000 −0.230159
\(926\) −8.00000 −0.262896
\(927\) −7.00000 −0.229910
\(928\) 4.00000 0.131306
\(929\) 2.00000 0.0656179 0.0328089 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) −2.00000 −0.0655826
\(931\) −90.0000 −2.94963
\(932\) −14.0000 −0.458585
\(933\) −12.0000 −0.392862
\(934\) 24.0000 0.785304
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 40.0000 1.30605
\(939\) −6.00000 −0.195803
\(940\) 3.00000 0.0978492
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) −15.0000 −0.488726
\(943\) −24.0000 −0.781548
\(944\) 12.0000 0.390567
\(945\) 5.00000 0.162650
\(946\) 18.0000 0.585230
\(947\) 58.0000 1.88475 0.942373 0.334563i \(-0.108589\pi\)
0.942373 + 0.334563i \(0.108589\pi\)
\(948\) −2.00000 −0.0649570
\(949\) 0 0
\(950\) 5.00000 0.162221
\(951\) −23.0000 −0.745826
\(952\) −40.0000 −1.29641
\(953\) −54.0000 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) −1.00000 −0.0323762
\(955\) −2.00000 −0.0647185
\(956\) −18.0000 −0.582162
\(957\) 12.0000 0.387905
\(958\) 28.0000 0.904639
\(959\) 60.0000 1.93750
\(960\) −1.00000 −0.0322749
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) −25.0000 −0.805196
\(965\) −24.0000 −0.772587
\(966\) −20.0000 −0.643489
\(967\) −31.0000 −0.996893 −0.498446 0.866921i \(-0.666096\pi\)
−0.498446 + 0.866921i \(0.666096\pi\)
\(968\) 2.00000 0.0642824
\(969\) 40.0000 1.28499
\(970\) 0 0
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 1.00000 0.0320750
\(973\) −35.0000 −1.12205
\(974\) −37.0000 −1.18556
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 20.0000 0.639857 0.319928 0.947442i \(-0.396341\pi\)
0.319928 + 0.947442i \(0.396341\pi\)
\(978\) 20.0000 0.639529
\(979\) 33.0000 1.05468
\(980\) −18.0000 −0.574989
\(981\) 14.0000 0.446986
\(982\) −21.0000 −0.670137
\(983\) 23.0000 0.733586 0.366793 0.930303i \(-0.380456\pi\)
0.366793 + 0.930303i \(0.380456\pi\)
\(984\) −6.00000 −0.191273
\(985\) −3.00000 −0.0955879
\(986\) −32.0000 −1.01909
\(987\) 15.0000 0.477455
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) −3.00000 −0.0953463
\(991\) 10.0000 0.317660 0.158830 0.987306i \(-0.449228\pi\)
0.158830 + 0.987306i \(0.449228\pi\)
\(992\) 2.00000 0.0635001
\(993\) 4.00000 0.126936
\(994\) 10.0000 0.317181
\(995\) −22.0000 −0.697447
\(996\) 8.00000 0.253490
\(997\) −1.00000 −0.0316703 −0.0158352 0.999875i \(-0.505041\pi\)
−0.0158352 + 0.999875i \(0.505041\pi\)
\(998\) −4.00000 −0.126618
\(999\) −7.00000 −0.221470
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 5070.2.a.i.1.1 1
13.3 even 3 390.2.i.d.61.1 2
13.5 odd 4 5070.2.b.l.1351.2 2
13.8 odd 4 5070.2.b.l.1351.1 2
13.9 even 3 390.2.i.d.211.1 yes 2
13.12 even 2 5070.2.a.x.1.1 1
39.29 odd 6 1170.2.i.g.451.1 2
39.35 odd 6 1170.2.i.g.991.1 2
65.3 odd 12 1950.2.z.j.1699.1 4
65.9 even 6 1950.2.i.i.601.1 2
65.22 odd 12 1950.2.z.j.1849.1 4
65.29 even 6 1950.2.i.i.451.1 2
65.42 odd 12 1950.2.z.j.1699.2 4
65.48 odd 12 1950.2.z.j.1849.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.d.61.1 2 13.3 even 3
390.2.i.d.211.1 yes 2 13.9 even 3
1170.2.i.g.451.1 2 39.29 odd 6
1170.2.i.g.991.1 2 39.35 odd 6
1950.2.i.i.451.1 2 65.29 even 6
1950.2.i.i.601.1 2 65.9 even 6
1950.2.z.j.1699.1 4 65.3 odd 12
1950.2.z.j.1699.2 4 65.42 odd 12
1950.2.z.j.1849.1 4 65.22 odd 12
1950.2.z.j.1849.2 4 65.48 odd 12
5070.2.a.i.1.1 1 1.1 even 1 trivial
5070.2.a.x.1.1 1 13.12 even 2
5070.2.b.l.1351.1 2 13.8 odd 4
5070.2.b.l.1351.2 2 13.5 odd 4