Properties

Label 1862.2.a.f.1.1
Level $1862$
Weight $2$
Character 1862.1
Self dual yes
Analytic conductor $14.868$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1862 = 2 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1862.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8681448564\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1862.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} +4.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +4.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} +1.00000 q^{19} +4.00000 q^{20} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} +1.00000 q^{26} -5.00000 q^{27} -5.00000 q^{29} +4.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} -3.00000 q^{34} -2.00000 q^{36} -2.00000 q^{37} +1.00000 q^{38} +1.00000 q^{39} +4.00000 q^{40} +8.00000 q^{41} +4.00000 q^{43} +2.00000 q^{44} -8.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +11.0000 q^{50} -3.00000 q^{51} +1.00000 q^{52} -1.00000 q^{53} -5.00000 q^{54} +8.00000 q^{55} +1.00000 q^{57} -5.00000 q^{58} -15.0000 q^{59} +4.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +2.00000 q^{66} +3.00000 q^{67} -3.00000 q^{68} -1.00000 q^{69} +2.00000 q^{71} -2.00000 q^{72} -9.00000 q^{73} -2.00000 q^{74} +11.0000 q^{75} +1.00000 q^{76} +1.00000 q^{78} -10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +8.00000 q^{82} +6.00000 q^{83} -12.0000 q^{85} +4.00000 q^{86} -5.00000 q^{87} +2.00000 q^{88} -8.00000 q^{90} -1.00000 q^{92} +8.00000 q^{93} -8.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) 4.00000 1.26491
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −2.00000 −0.471405
\(19\) 1.00000 0.229416
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 1.00000 0.204124
\(25\) 11.0000 2.20000
\(26\) 1.00000 0.196116
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 4.00000 0.730297
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 1.00000 0.162221
\(39\) 1.00000 0.160128
\(40\) 4.00000 0.632456
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 2.00000 0.301511
\(45\) −8.00000 −1.19257
\(46\) −1.00000 −0.147442
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 11.0000 1.55563
\(51\) −3.00000 −0.420084
\(52\) 1.00000 0.138675
\(53\) −1.00000 −0.137361 −0.0686803 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) −5.00000 −0.680414
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) −5.00000 −0.656532
\(59\) −15.0000 −1.95283 −0.976417 0.215894i \(-0.930733\pi\)
−0.976417 + 0.215894i \(0.930733\pi\)
\(60\) 4.00000 0.516398
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 2.00000 0.246183
\(67\) 3.00000 0.366508 0.183254 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) −3.00000 −0.363803
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −2.00000 −0.235702
\(73\) −9.00000 −1.05337 −0.526685 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) −2.00000 −0.232495
\(75\) 11.0000 1.27017
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) 8.00000 0.883452
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 4.00000 0.431331
\(87\) −5.00000 −0.536056
\(88\) 2.00000 0.213201
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −8.00000 −0.843274
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 8.00000 0.829561
\(94\) −8.00000 −0.825137
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 11.0000 1.10000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −3.00000 −0.297044
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) −7.00000 −0.676716 −0.338358 0.941018i \(-0.609871\pi\)
−0.338358 + 0.941018i \(0.609871\pi\)
\(108\) −5.00000 −0.481125
\(109\) −15.0000 −1.43674 −0.718370 0.695662i \(-0.755111\pi\)
−0.718370 + 0.695662i \(0.755111\pi\)
\(110\) 8.00000 0.762770
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 1.00000 0.0936586
\(115\) −4.00000 −0.373002
\(116\) −5.00000 −0.464238
\(117\) −2.00000 −0.184900
\(118\) −15.0000 −1.38086
\(119\) 0 0
\(120\) 4.00000 0.365148
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) 8.00000 0.721336
\(124\) 8.00000 0.718421
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) 18.0000 1.59724 0.798621 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 4.00000 0.350823
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 3.00000 0.259161
\(135\) −20.0000 −1.72133
\(136\) −3.00000 −0.257248
\(137\) −17.0000 −1.45241 −0.726204 0.687479i \(-0.758717\pi\)
−0.726204 + 0.687479i \(0.758717\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 2.00000 0.167836
\(143\) 2.00000 0.167248
\(144\) −2.00000 −0.166667
\(145\) −20.0000 −1.66091
\(146\) −9.00000 −0.744845
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 11.0000 0.898146
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 1.00000 0.0811107
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 32.0000 2.57030
\(156\) 1.00000 0.0800641
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −10.0000 −0.795557
\(159\) −1.00000 −0.0793052
\(160\) 4.00000 0.316228
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 8.00000 0.624695
\(165\) 8.00000 0.622799
\(166\) 6.00000 0.465690
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −12.0000 −0.920358
\(171\) −2.00000 −0.152944
\(172\) 4.00000 0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −5.00000 −0.379049
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) −15.0000 −1.12747
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −8.00000 −0.596285
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −1.00000 −0.0737210
\(185\) −8.00000 −0.588172
\(186\) 8.00000 0.586588
\(187\) −6.00000 −0.438763
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 7.00000 0.506502 0.253251 0.967401i \(-0.418500\pi\)
0.253251 + 0.967401i \(0.418500\pi\)
\(192\) 1.00000 0.0721688
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 2.00000 0.143592
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) −4.00000 −0.284268
\(199\) 25.0000 1.77220 0.886102 0.463491i \(-0.153403\pi\)
0.886102 + 0.463491i \(0.153403\pi\)
\(200\) 11.0000 0.777817
\(201\) 3.00000 0.211604
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) −3.00000 −0.210042
\(205\) 32.0000 2.23498
\(206\) 6.00000 0.418040
\(207\) 2.00000 0.139010
\(208\) 1.00000 0.0693375
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) 27.0000 1.85876 0.929378 0.369129i \(-0.120344\pi\)
0.929378 + 0.369129i \(0.120344\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 2.00000 0.137038
\(214\) −7.00000 −0.478510
\(215\) 16.0000 1.09119
\(216\) −5.00000 −0.340207
\(217\) 0 0
\(218\) −15.0000 −1.01593
\(219\) −9.00000 −0.608164
\(220\) 8.00000 0.539360
\(221\) −3.00000 −0.201802
\(222\) −2.00000 −0.134231
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) −22.0000 −1.46667
\(226\) 14.0000 0.931266
\(227\) 17.0000 1.12833 0.564165 0.825662i \(-0.309198\pi\)
0.564165 + 0.825662i \(0.309198\pi\)
\(228\) 1.00000 0.0662266
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −5.00000 −0.328266
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) −32.0000 −2.08745
\(236\) −15.0000 −0.976417
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) 15.0000 0.970269 0.485135 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 4.00000 0.258199
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) −7.00000 −0.449977
\(243\) 16.0000 1.02640
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 8.00000 0.510061
\(247\) 1.00000 0.0636285
\(248\) 8.00000 0.508001
\(249\) 6.00000 0.380235
\(250\) 24.0000 1.51789
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 18.0000 1.12942
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) 10.0000 0.618984
\(262\) −12.0000 −0.741362
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 2.00000 0.123091
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 0 0
\(268\) 3.00000 0.183254
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) −20.0000 −1.21716
\(271\) −7.00000 −0.425220 −0.212610 0.977137i \(-0.568196\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) −17.0000 −1.02701
\(275\) 22.0000 1.32665
\(276\) −1.00000 −0.0601929
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 0 0
\(279\) −16.0000 −0.957895
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) −8.00000 −0.476393
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 2.00000 0.118678
\(285\) 4.00000 0.236940
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) −2.00000 −0.117851
\(289\) −8.00000 −0.470588
\(290\) −20.0000 −1.17444
\(291\) 2.00000 0.117242
\(292\) −9.00000 −0.526685
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) −60.0000 −3.49334
\(296\) −2.00000 −0.116248
\(297\) −10.0000 −0.580259
\(298\) 0 0
\(299\) −1.00000 −0.0578315
\(300\) 11.0000 0.635085
\(301\) 0 0
\(302\) 2.00000 0.115087
\(303\) −2.00000 −0.114897
\(304\) 1.00000 0.0573539
\(305\) −8.00000 −0.458079
\(306\) 6.00000 0.342997
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 6.00000 0.341328
\(310\) 32.0000 1.81748
\(311\) −7.00000 −0.396934 −0.198467 0.980108i \(-0.563596\pi\)
−0.198467 + 0.980108i \(0.563596\pi\)
\(312\) 1.00000 0.0566139
\(313\) −29.0000 −1.63918 −0.819588 0.572953i \(-0.805798\pi\)
−0.819588 + 0.572953i \(0.805798\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −27.0000 −1.51647 −0.758236 0.651981i \(-0.773938\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) −1.00000 −0.0560772
\(319\) −10.0000 −0.559893
\(320\) 4.00000 0.223607
\(321\) −7.00000 −0.390702
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 1.00000 0.0555556
\(325\) 11.0000 0.610170
\(326\) −16.0000 −0.886158
\(327\) −15.0000 −0.829502
\(328\) 8.00000 0.441726
\(329\) 0 0
\(330\) 8.00000 0.440386
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 6.00000 0.329293
\(333\) 4.00000 0.219199
\(334\) 12.0000 0.656611
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) −12.0000 −0.652714
\(339\) 14.0000 0.760376
\(340\) −12.0000 −0.650791
\(341\) 16.0000 0.866449
\(342\) −2.00000 −0.108148
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) −4.00000 −0.215353
\(346\) 6.00000 0.322562
\(347\) −2.00000 −0.107366 −0.0536828 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) −5.00000 −0.268028
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 2.00000 0.106600
\(353\) −9.00000 −0.479022 −0.239511 0.970894i \(-0.576987\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) −15.0000 −0.797241
\(355\) 8.00000 0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) −8.00000 −0.421637
\(361\) 1.00000 0.0526316
\(362\) −22.0000 −1.15629
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) −36.0000 −1.88433
\(366\) −2.00000 −0.104542
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −16.0000 −0.832927
\(370\) −8.00000 −0.415900
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) 29.0000 1.50156 0.750782 0.660551i \(-0.229677\pi\)
0.750782 + 0.660551i \(0.229677\pi\)
\(374\) −6.00000 −0.310253
\(375\) 24.0000 1.23935
\(376\) −8.00000 −0.412568
\(377\) −5.00000 −0.257513
\(378\) 0 0
\(379\) 15.0000 0.770498 0.385249 0.922813i \(-0.374116\pi\)
0.385249 + 0.922813i \(0.374116\pi\)
\(380\) 4.00000 0.205196
\(381\) 18.0000 0.922168
\(382\) 7.00000 0.358151
\(383\) 26.0000 1.32854 0.664269 0.747494i \(-0.268743\pi\)
0.664269 + 0.747494i \(0.268743\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) −8.00000 −0.406663
\(388\) 2.00000 0.101535
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 4.00000 0.202548
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 8.00000 0.403034
\(395\) −40.0000 −2.01262
\(396\) −4.00000 −0.201008
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) 25.0000 1.25314
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 3.00000 0.149626
\(403\) 8.00000 0.398508
\(404\) −2.00000 −0.0995037
\(405\) 4.00000 0.198762
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) −3.00000 −0.148522
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 32.0000 1.58037
\(411\) −17.0000 −0.838548
\(412\) 6.00000 0.295599
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) 24.0000 1.17811
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 2.00000 0.0978232
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) 27.0000 1.31434
\(423\) 16.0000 0.777947
\(424\) −1.00000 −0.0485643
\(425\) −33.0000 −1.60074
\(426\) 2.00000 0.0969003
\(427\) 0 0
\(428\) −7.00000 −0.338358
\(429\) 2.00000 0.0965609
\(430\) 16.0000 0.771589
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) −5.00000 −0.240563
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −20.0000 −0.958927
\(436\) −15.0000 −0.718370
\(437\) −1.00000 −0.0478365
\(438\) −9.00000 −0.430037
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 8.00000 0.381385
\(441\) 0 0
\(442\) −3.00000 −0.142695
\(443\) −26.0000 −1.23530 −0.617649 0.786454i \(-0.711915\pi\)
−0.617649 + 0.786454i \(0.711915\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) 0 0
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) −22.0000 −1.03709
\(451\) 16.0000 0.753411
\(452\) 14.0000 0.658505
\(453\) 2.00000 0.0939682
\(454\) 17.0000 0.797850
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) −7.00000 −0.327446 −0.163723 0.986506i \(-0.552350\pi\)
−0.163723 + 0.986506i \(0.552350\pi\)
\(458\) 10.0000 0.467269
\(459\) 15.0000 0.700140
\(460\) −4.00000 −0.186501
\(461\) 28.0000 1.30409 0.652045 0.758180i \(-0.273911\pi\)
0.652045 + 0.758180i \(0.273911\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −5.00000 −0.232119
\(465\) 32.0000 1.48396
\(466\) −6.00000 −0.277945
\(467\) 2.00000 0.0925490 0.0462745 0.998929i \(-0.485265\pi\)
0.0462745 + 0.998929i \(0.485265\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) −32.0000 −1.47605
\(471\) 2.00000 0.0921551
\(472\) −15.0000 −0.690431
\(473\) 8.00000 0.367840
\(474\) −10.0000 −0.459315
\(475\) 11.0000 0.504715
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 15.0000 0.686084
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 4.00000 0.182574
\(481\) −2.00000 −0.0911922
\(482\) 8.00000 0.364390
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 8.00000 0.363261
\(486\) 16.0000 0.725775
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 8.00000 0.360668
\(493\) 15.0000 0.675566
\(494\) 1.00000 0.0449921
\(495\) −16.0000 −0.719147
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 6.00000 0.268866
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) 24.0000 1.07331
\(501\) 12.0000 0.536120
\(502\) −2.00000 −0.0892644
\(503\) −39.0000 −1.73892 −0.869462 0.494000i \(-0.835534\pi\)
−0.869462 + 0.494000i \(0.835534\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) −2.00000 −0.0889108
\(507\) −12.0000 −0.532939
\(508\) 18.0000 0.798621
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) −12.0000 −0.531369
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −5.00000 −0.220755
\(514\) −8.00000 −0.352865
\(515\) 24.0000 1.05757
\(516\) 4.00000 0.176090
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 4.00000 0.175412
\(521\) 28.0000 1.22670 0.613351 0.789810i \(-0.289821\pi\)
0.613351 + 0.789810i \(0.289821\pi\)
\(522\) 10.0000 0.437688
\(523\) −29.0000 −1.26808 −0.634041 0.773300i \(-0.718605\pi\)
−0.634041 + 0.773300i \(0.718605\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −24.0000 −1.04546
\(528\) 2.00000 0.0870388
\(529\) −22.0000 −0.956522
\(530\) −4.00000 −0.173749
\(531\) 30.0000 1.30189
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) −28.0000 −1.21055
\(536\) 3.00000 0.129580
\(537\) 0 0
\(538\) −30.0000 −1.29339
\(539\) 0 0
\(540\) −20.0000 −0.860663
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) −7.00000 −0.300676
\(543\) −22.0000 −0.944110
\(544\) −3.00000 −0.128624
\(545\) −60.0000 −2.57012
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −17.0000 −0.726204
\(549\) 4.00000 0.170716
\(550\) 22.0000 0.938083
\(551\) −5.00000 −0.213007
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) 28.0000 1.18961
\(555\) −8.00000 −0.339581
\(556\) 0 0
\(557\) 28.0000 1.18640 0.593199 0.805056i \(-0.297865\pi\)
0.593199 + 0.805056i \(0.297865\pi\)
\(558\) −16.0000 −0.677334
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −6.00000 −0.253320
\(562\) −8.00000 −0.337460
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) −8.00000 −0.336861
\(565\) 56.0000 2.35594
\(566\) 6.00000 0.252199
\(567\) 0 0
\(568\) 2.00000 0.0839181
\(569\) 40.0000 1.67689 0.838444 0.544988i \(-0.183466\pi\)
0.838444 + 0.544988i \(0.183466\pi\)
\(570\) 4.00000 0.167542
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 2.00000 0.0836242
\(573\) 7.00000 0.292429
\(574\) 0 0
\(575\) −11.0000 −0.458732
\(576\) −2.00000 −0.0833333
\(577\) 37.0000 1.54033 0.770165 0.637845i \(-0.220174\pi\)
0.770165 + 0.637845i \(0.220174\pi\)
\(578\) −8.00000 −0.332756
\(579\) −6.00000 −0.249351
\(580\) −20.0000 −0.830455
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) −2.00000 −0.0828315
\(584\) −9.00000 −0.372423
\(585\) −8.00000 −0.330759
\(586\) −9.00000 −0.371787
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) −60.0000 −2.47016
\(591\) 8.00000 0.329076
\(592\) −2.00000 −0.0821995
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) −10.0000 −0.410305
\(595\) 0 0
\(596\) 0 0
\(597\) 25.0000 1.02318
\(598\) −1.00000 −0.0408930
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 11.0000 0.449073
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 0 0
\(603\) −6.00000 −0.244339
\(604\) 2.00000 0.0813788
\(605\) −28.0000 −1.13836
\(606\) −2.00000 −0.0812444
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) −8.00000 −0.323911
\(611\) −8.00000 −0.323645
\(612\) 6.00000 0.242536
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 12.0000 0.484281
\(615\) 32.0000 1.29036
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 6.00000 0.241355
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 32.0000 1.28515
\(621\) 5.00000 0.200643
\(622\) −7.00000 −0.280674
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 41.0000 1.64000
\(626\) −29.0000 −1.15907
\(627\) 2.00000 0.0798723
\(628\) 2.00000 0.0798087
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) −10.0000 −0.397779
\(633\) 27.0000 1.07315
\(634\) −27.0000 −1.07231
\(635\) 72.0000 2.85723
\(636\) −1.00000 −0.0396526
\(637\) 0 0
\(638\) −10.0000 −0.395904
\(639\) −4.00000 −0.158238
\(640\) 4.00000 0.158114
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) −7.00000 −0.276268
\(643\) 26.0000 1.02534 0.512670 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) 0 0
\(645\) 16.0000 0.629999
\(646\) −3.00000 −0.118033
\(647\) −23.0000 −0.904223 −0.452112 0.891961i \(-0.649329\pi\)
−0.452112 + 0.891961i \(0.649329\pi\)
\(648\) 1.00000 0.0392837
\(649\) −30.0000 −1.17760
\(650\) 11.0000 0.431455
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) −15.0000 −0.586546
\(655\) −48.0000 −1.87552
\(656\) 8.00000 0.312348
\(657\) 18.0000 0.702247
\(658\) 0 0
\(659\) 5.00000 0.194772 0.0973862 0.995247i \(-0.468952\pi\)
0.0973862 + 0.995247i \(0.468952\pi\)
\(660\) 8.00000 0.311400
\(661\) 23.0000 0.894596 0.447298 0.894385i \(-0.352386\pi\)
0.447298 + 0.894385i \(0.352386\pi\)
\(662\) 17.0000 0.660724
\(663\) −3.00000 −0.116510
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 5.00000 0.193601
\(668\) 12.0000 0.464294
\(669\) −14.0000 −0.541271
\(670\) 12.0000 0.463600
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 44.0000 1.69608 0.848038 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) −32.0000 −1.23259
\(675\) −55.0000 −2.11695
\(676\) −12.0000 −0.461538
\(677\) −13.0000 −0.499631 −0.249815 0.968294i \(-0.580370\pi\)
−0.249815 + 0.968294i \(0.580370\pi\)
\(678\) 14.0000 0.537667
\(679\) 0 0
\(680\) −12.0000 −0.460179
\(681\) 17.0000 0.651441
\(682\) 16.0000 0.612672
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −68.0000 −2.59815
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 4.00000 0.152499
\(689\) −1.00000 −0.0380970
\(690\) −4.00000 −0.152277
\(691\) −42.0000 −1.59776 −0.798878 0.601494i \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −2.00000 −0.0759190
\(695\) 0 0
\(696\) −5.00000 −0.189525
\(697\) −24.0000 −0.909065
\(698\) −10.0000 −0.378506
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −28.0000 −1.05755 −0.528773 0.848763i \(-0.677348\pi\)
−0.528773 + 0.848763i \(0.677348\pi\)
\(702\) −5.00000 −0.188713
\(703\) −2.00000 −0.0754314
\(704\) 2.00000 0.0753778
\(705\) −32.0000 −1.20519
\(706\) −9.00000 −0.338719
\(707\) 0 0
\(708\) −15.0000 −0.563735
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 8.00000 0.300235
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 0 0
\(717\) 15.0000 0.560185
\(718\) −15.0000 −0.559795
\(719\) 5.00000 0.186469 0.0932343 0.995644i \(-0.470279\pi\)
0.0932343 + 0.995644i \(0.470279\pi\)
\(720\) −8.00000 −0.298142
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) 8.00000 0.297523
\(724\) −22.0000 −0.817624
\(725\) −55.0000 −2.04265
\(726\) −7.00000 −0.259794
\(727\) 17.0000 0.630495 0.315248 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) −36.0000 −1.33242
\(731\) −12.0000 −0.443836
\(732\) −2.00000 −0.0739221
\(733\) 36.0000 1.32969 0.664845 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 6.00000 0.221013
\(738\) −16.0000 −0.588968
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) −8.00000 −0.294086
\(741\) 1.00000 0.0367359
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) 29.0000 1.06177
\(747\) −12.0000 −0.439057
\(748\) −6.00000 −0.219382
\(749\) 0 0
\(750\) 24.0000 0.876356
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −8.00000 −0.291730
\(753\) −2.00000 −0.0728841
\(754\) −5.00000 −0.182089
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 15.0000 0.544825
\(759\) −2.00000 −0.0725954
\(760\) 4.00000 0.145095
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 18.0000 0.652071
\(763\) 0 0
\(764\) 7.00000 0.253251
\(765\) 24.0000 0.867722
\(766\) 26.0000 0.939418
\(767\) −15.0000 −0.541619
\(768\) 1.00000 0.0360844
\(769\) 35.0000 1.26213 0.631066 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(770\) 0 0
\(771\) −8.00000 −0.288113
\(772\) −6.00000 −0.215945
\(773\) −9.00000 −0.323708 −0.161854 0.986815i \(-0.551747\pi\)
−0.161854 + 0.986815i \(0.551747\pi\)
\(774\) −8.00000 −0.287554
\(775\) 88.0000 3.16105
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) 8.00000 0.286630
\(780\) 4.00000 0.143223
\(781\) 4.00000 0.143131
\(782\) 3.00000 0.107280
\(783\) 25.0000 0.893427
\(784\) 0 0
\(785\) 8.00000 0.285532
\(786\) −12.0000 −0.428026
\(787\) 17.0000 0.605985 0.302992 0.952993i \(-0.402014\pi\)
0.302992 + 0.952993i \(0.402014\pi\)
\(788\) 8.00000 0.284988
\(789\) 24.0000 0.854423
\(790\) −40.0000 −1.42314
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) −2.00000 −0.0710221
\(794\) −8.00000 −0.283909
\(795\) −4.00000 −0.141865
\(796\) 25.0000 0.886102
\(797\) −3.00000 −0.106265 −0.0531327 0.998587i \(-0.516921\pi\)
−0.0531327 + 0.998587i \(0.516921\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 11.0000 0.388909
\(801\) 0 0
\(802\) −8.00000 −0.282490
\(803\) −18.0000 −0.635206
\(804\) 3.00000 0.105802
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) −30.0000 −1.05605
\(808\) −2.00000 −0.0703598
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 4.00000 0.140546
\(811\) 3.00000 0.105344 0.0526721 0.998612i \(-0.483226\pi\)
0.0526721 + 0.998612i \(0.483226\pi\)
\(812\) 0 0
\(813\) −7.00000 −0.245501
\(814\) −4.00000 −0.140200
\(815\) −64.0000 −2.24182
\(816\) −3.00000 −0.105021
\(817\) 4.00000 0.139942
\(818\) 20.0000 0.699284
\(819\) 0 0
\(820\) 32.0000 1.11749
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) −17.0000 −0.592943
\(823\) 29.0000 1.01088 0.505438 0.862863i \(-0.331331\pi\)
0.505438 + 0.862863i \(0.331331\pi\)
\(824\) 6.00000 0.209020
\(825\) 22.0000 0.765942
\(826\) 0 0
\(827\) 23.0000 0.799788 0.399894 0.916561i \(-0.369047\pi\)
0.399894 + 0.916561i \(0.369047\pi\)
\(828\) 2.00000 0.0695048
\(829\) 15.0000 0.520972 0.260486 0.965478i \(-0.416117\pi\)
0.260486 + 0.965478i \(0.416117\pi\)
\(830\) 24.0000 0.833052
\(831\) 28.0000 0.971309
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) 0 0
\(835\) 48.0000 1.66111
\(836\) 2.00000 0.0691714
\(837\) −40.0000 −1.38260
\(838\) 0 0
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −13.0000 −0.448010
\(843\) −8.00000 −0.275535
\(844\) 27.0000 0.929378
\(845\) −48.0000 −1.65125
\(846\) 16.0000 0.550091
\(847\) 0 0
\(848\) −1.00000 −0.0343401
\(849\) 6.00000 0.205919
\(850\) −33.0000 −1.13189
\(851\) 2.00000 0.0685591
\(852\) 2.00000 0.0685189
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) −7.00000 −0.239255
\(857\) 12.0000 0.409912 0.204956 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(858\) 2.00000 0.0682789
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 16.0000 0.545595
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) 54.0000 1.83818 0.919091 0.394046i \(-0.128925\pi\)
0.919091 + 0.394046i \(0.128925\pi\)
\(864\) −5.00000 −0.170103
\(865\) 24.0000 0.816024
\(866\) −14.0000 −0.475739
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −20.0000 −0.678454
\(870\) −20.0000 −0.678064
\(871\) 3.00000 0.101651
\(872\) −15.0000 −0.507964
\(873\) −4.00000 −0.135379
\(874\) −1.00000 −0.0338255
\(875\) 0 0
\(876\) −9.00000 −0.304082
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) −20.0000 −0.674967
\(879\) −9.00000 −0.303562
\(880\) 8.00000 0.269680
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 34.0000 1.14419 0.572096 0.820187i \(-0.306131\pi\)
0.572096 + 0.820187i \(0.306131\pi\)
\(884\) −3.00000 −0.100901
\(885\) −60.0000 −2.01688
\(886\) −26.0000 −0.873487
\(887\) 2.00000 0.0671534 0.0335767 0.999436i \(-0.489310\pi\)
0.0335767 + 0.999436i \(0.489310\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −14.0000 −0.468755
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1.00000 −0.0333890
\(898\) 10.0000 0.333704
\(899\) −40.0000 −1.33407
\(900\) −22.0000 −0.733333
\(901\) 3.00000 0.0999445
\(902\) 16.0000 0.532742
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) −88.0000 −2.92522
\(906\) 2.00000 0.0664455
\(907\) 53.0000 1.75984 0.879918 0.475125i \(-0.157597\pi\)
0.879918 + 0.475125i \(0.157597\pi\)
\(908\) 17.0000 0.564165
\(909\) 4.00000 0.132672
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 1.00000 0.0331133
\(913\) 12.0000 0.397142
\(914\) −7.00000 −0.231539
\(915\) −8.00000 −0.264472
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 15.0000 0.495074
\(919\) 5.00000 0.164935 0.0824674 0.996594i \(-0.473720\pi\)
0.0824674 + 0.996594i \(0.473720\pi\)
\(920\) −4.00000 −0.131876
\(921\) 12.0000 0.395413
\(922\) 28.0000 0.922131
\(923\) 2.00000 0.0658308
\(924\) 0 0
\(925\) −22.0000 −0.723356
\(926\) 4.00000 0.131448
\(927\) −12.0000 −0.394132
\(928\) −5.00000 −0.164133
\(929\) 55.0000 1.80449 0.902246 0.431222i \(-0.141918\pi\)
0.902246 + 0.431222i \(0.141918\pi\)
\(930\) 32.0000 1.04932
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) −7.00000 −0.229170
\(934\) 2.00000 0.0654420
\(935\) −24.0000 −0.784884
\(936\) −2.00000 −0.0653720
\(937\) 7.00000 0.228680 0.114340 0.993442i \(-0.463525\pi\)
0.114340 + 0.993442i \(0.463525\pi\)
\(938\) 0 0
\(939\) −29.0000 −0.946379
\(940\) −32.0000 −1.04372
\(941\) −7.00000 −0.228193 −0.114097 0.993470i \(-0.536397\pi\)
−0.114097 + 0.993470i \(0.536397\pi\)
\(942\) 2.00000 0.0651635
\(943\) −8.00000 −0.260516
\(944\) −15.0000 −0.488208
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −10.0000 −0.324785
\(949\) −9.00000 −0.292152
\(950\) 11.0000 0.356887
\(951\) −27.0000 −0.875535
\(952\) 0 0
\(953\) −46.0000 −1.49009 −0.745043 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) 2.00000 0.0647524
\(955\) 28.0000 0.906059
\(956\) 15.0000 0.485135
\(957\) −10.0000 −0.323254
\(958\) 20.0000 0.646171
\(959\) 0 0
\(960\) 4.00000 0.129099
\(961\) 33.0000 1.06452
\(962\) −2.00000 −0.0644826
\(963\) 14.0000 0.451144
\(964\) 8.00000 0.257663
\(965\) −24.0000 −0.772587
\(966\) 0 0
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) −7.00000 −0.224989
\(969\) −3.00000 −0.0963739
\(970\) 8.00000 0.256865
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 16.0000 0.513200
\(973\) 0 0
\(974\) −2.00000 −0.0640841
\(975\) 11.0000 0.352282
\(976\) −2.00000 −0.0640184
\(977\) 8.00000 0.255943 0.127971 0.991778i \(-0.459153\pi\)
0.127971 + 0.991778i \(0.459153\pi\)
\(978\) −16.0000 −0.511624
\(979\) 0 0
\(980\) 0 0
\(981\) 30.0000 0.957826
\(982\) −28.0000 −0.893516
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) 8.00000 0.255031
\(985\) 32.0000 1.01960
\(986\) 15.0000 0.477697
\(987\) 0 0
\(988\) 1.00000 0.0318142
\(989\) −4.00000 −0.127193
\(990\) −16.0000 −0.508513
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 8.00000 0.254000
\(993\) 17.0000 0.539479
\(994\) 0 0
\(995\) 100.000 3.17021
\(996\) 6.00000 0.190117
\(997\) −28.0000 −0.886769 −0.443384 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) 40.0000 1.26618
\(999\) 10.0000 0.316386
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 1862.2.a.f.1.1 1
7.6 odd 2 38.2.a.b.1.1 1
21.20 even 2 342.2.a.d.1.1 1
28.27 even 2 304.2.a.d.1.1 1
35.13 even 4 950.2.b.c.799.1 2
35.27 even 4 950.2.b.c.799.2 2
35.34 odd 2 950.2.a.b.1.1 1
56.13 odd 2 1216.2.a.n.1.1 1
56.27 even 2 1216.2.a.g.1.1 1
77.76 even 2 4598.2.a.a.1.1 1
84.83 odd 2 2736.2.a.w.1.1 1
91.90 odd 2 6422.2.a.b.1.1 1
105.104 even 2 8550.2.a.u.1.1 1
133.6 odd 18 722.2.e.c.245.1 6
133.13 even 18 722.2.e.d.245.1 6
133.27 even 6 722.2.c.f.653.1 2
133.34 even 18 722.2.e.d.415.1 6
133.41 even 18 722.2.e.d.389.1 6
133.48 even 18 722.2.e.d.423.1 6
133.55 odd 18 722.2.e.c.99.1 6
133.62 odd 18 722.2.e.c.595.1 6
133.69 even 6 722.2.c.f.429.1 2
133.83 odd 6 722.2.c.d.429.1 2
133.90 even 18 722.2.e.d.595.1 6
133.97 even 18 722.2.e.d.99.1 6
133.104 odd 18 722.2.e.c.423.1 6
133.111 odd 18 722.2.e.c.389.1 6
133.118 odd 18 722.2.e.c.415.1 6
133.125 odd 6 722.2.c.d.653.1 2
133.132 even 2 722.2.a.b.1.1 1
140.139 even 2 7600.2.a.h.1.1 1
399.398 odd 2 6498.2.a.y.1.1 1
532.531 odd 2 5776.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.b.1.1 1 7.6 odd 2
304.2.a.d.1.1 1 28.27 even 2
342.2.a.d.1.1 1 21.20 even 2
722.2.a.b.1.1 1 133.132 even 2
722.2.c.d.429.1 2 133.83 odd 6
722.2.c.d.653.1 2 133.125 odd 6
722.2.c.f.429.1 2 133.69 even 6
722.2.c.f.653.1 2 133.27 even 6
722.2.e.c.99.1 6 133.55 odd 18
722.2.e.c.245.1 6 133.6 odd 18
722.2.e.c.389.1 6 133.111 odd 18
722.2.e.c.415.1 6 133.118 odd 18
722.2.e.c.423.1 6 133.104 odd 18
722.2.e.c.595.1 6 133.62 odd 18
722.2.e.d.99.1 6 133.97 even 18
722.2.e.d.245.1 6 133.13 even 18
722.2.e.d.389.1 6 133.41 even 18
722.2.e.d.415.1 6 133.34 even 18
722.2.e.d.423.1 6 133.48 even 18
722.2.e.d.595.1 6 133.90 even 18
950.2.a.b.1.1 1 35.34 odd 2
950.2.b.c.799.1 2 35.13 even 4
950.2.b.c.799.2 2 35.27 even 4
1216.2.a.g.1.1 1 56.27 even 2
1216.2.a.n.1.1 1 56.13 odd 2
1862.2.a.f.1.1 1 1.1 even 1 trivial
2736.2.a.w.1.1 1 84.83 odd 2
4598.2.a.a.1.1 1 77.76 even 2
5776.2.a.d.1.1 1 532.531 odd 2
6422.2.a.b.1.1 1 91.90 odd 2
6498.2.a.y.1.1 1 399.398 odd 2
7600.2.a.h.1.1 1 140.139 even 2
8550.2.a.u.1.1 1 105.104 even 2