Properties

Label 6422.2.a.b.1.1
Level 6422
Weight 2
Character 6422.1
Self dual yes
Analytic conductor 51.280
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.2799281781\)
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 \(\chi\) \(=\) 6422.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -4.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} +3.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} +2.00000 q^{18} +1.00000 q^{19} +4.00000 q^{20} +3.00000 q^{21} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} +5.00000 q^{27} -3.00000 q^{28} -5.00000 q^{29} +4.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} -3.00000 q^{34} -12.0000 q^{35} -2.00000 q^{36} +2.00000 q^{37} -1.00000 q^{38} -4.00000 q^{40} +8.00000 q^{41} -3.00000 q^{42} +4.00000 q^{43} -2.00000 q^{44} -8.00000 q^{45} +1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -11.0000 q^{50} -3.00000 q^{51} -1.00000 q^{53} -5.00000 q^{54} -8.00000 q^{55} +3.00000 q^{56} -1.00000 q^{57} +5.00000 q^{58} -15.0000 q^{59} -4.00000 q^{60} +2.00000 q^{61} -8.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} -3.00000 q^{67} +3.00000 q^{68} +1.00000 q^{69} +12.0000 q^{70} -2.00000 q^{71} +2.00000 q^{72} -9.00000 q^{73} -2.00000 q^{74} -11.0000 q^{75} +1.00000 q^{76} +6.00000 q^{77} -10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} +6.00000 q^{83} +3.00000 q^{84} +12.0000 q^{85} -4.00000 q^{86} +5.00000 q^{87} +2.00000 q^{88} +8.00000 q^{90} -1.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} -2.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 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
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) −4.00000 −1.26491
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 3.00000 0.801784
\(15\) −4.00000 −1.03280
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 2.00000 0.471405
\(19\) 1.00000 0.229416
\(20\) 4.00000 0.894427
\(21\) 3.00000 0.654654
\(22\) 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
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) −3.00000 −0.566947
\(29\) −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
\(35\) −12.0000 −2.02837
\(36\) −2.00000 −0.333333
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −4.00000 −0.632456
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) −3.00000 −0.462910
\(43\) 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
\(49\) 2.00000 0.285714
\(50\) −11.0000 −1.55563
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) −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
\(56\) 3.00000 0.400892
\(57\) −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 \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −8.00000 −1.01600
\(63\) 6.00000 0.755929
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 3.00000 0.363803
\(69\) 1.00000 0.120386
\(70\) 12.0000 1.43427
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 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
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) −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 \(0.393190\pi\)
\(84\) 3.00000 0.327327
\(85\) 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 \(0.467625\pi\)
\(98\) −2.00000 −0.202031
\(99\) 4.00000 0.402015
\(100\) 11.0000 1.10000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 3.00000 0.297044
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) 12.0000 1.17108
\(106\) 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 \(-0.244889\pi\)
0.718370 + 0.695662i \(0.244889\pi\)
\(110\) 8.00000 0.762770
\(111\) −2.00000 −0.189832
\(112\) −3.00000 −0.283473
\(113\) 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
\(117\) 0 0
\(118\) 15.0000 1.38086
\(119\) −9.00000 −0.825029
\(120\) 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
\(126\) −6.00000 −0.534522
\(127\) 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
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 2.00000 0.174078
\(133\) −3.00000 −0.260133
\(134\) 3.00000 0.259161
\(135\) 20.0000 1.72133
\(136\) −3.00000 −0.257248
\(137\) 17.0000 1.45241 0.726204 0.687479i \(-0.241283\pi\)
0.726204 + 0.687479i \(0.241283\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −12.0000 −1.01419
\(141\) 8.00000 0.673722
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) −20.0000 −1.66091
\(146\) 9.00000 0.744845
\(147\) −2.00000 −0.164957
\(148\) 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 \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −6.00000 −0.485071
\(154\) −6.00000 −0.483494
\(155\) 32.0000 2.57030
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 10.0000 0.795557
\(159\) 1.00000 0.0793052
\(160\) −4.00000 −0.316228
\(161\) 3.00000 0.236433
\(162\) −1.00000 −0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 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 \(0.346308\pi\)
\(168\) −3.00000 −0.231455
\(169\) 0 0
\(170\) −12.0000 −0.920358
\(171\) −2.00000 −0.152944
\(172\) 4.00000 0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −5.00000 −0.379049
\(175\) −33.0000 −2.49457
\(176\) −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 \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 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
\(189\) −15.0000 −1.09109
\(190\) −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 \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) −4.00000 −0.284268
\(199\) −25.0000 −1.77220 −0.886102 0.463491i \(-0.846597\pi\)
−0.886102 + 0.463491i \(0.846597\pi\)
\(200\) −11.0000 −0.777817
\(201\) 3.00000 0.211604
\(202\) −2.00000 −0.140720
\(203\) 15.0000 1.05279
\(204\) −3.00000 −0.210042
\(205\) 32.0000 2.23498
\(206\) 6.00000 0.418040
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) −2.00000 −0.138343
\(210\) −12.0000 −0.828079
\(211\) 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
\(217\) −24.0000 −1.62923
\(218\) −15.0000 −1.01593
\(219\) 9.00000 0.608164
\(220\) −8.00000 −0.539360
\(221\) 0 0
\(222\) 2.00000 0.134231
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 3.00000 0.200446
\(225\) −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
\(231\) −6.00000 −0.394771
\(232\) 5.00000 0.328266
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −32.0000 −2.08745
\(236\) −15.0000 −0.976417
\(237\) 10.0000 0.649570
\(238\) 9.00000 0.583383
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) −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
\(245\) 8.00000 0.511101
\(246\) 8.00000 0.510061
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) −6.00000 −0.380235
\(250\) −24.0000 −1.51789
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 6.00000 0.377964
\(253\) 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 \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) 4.00000 0.249029
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 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
\(266\) 3.00000 0.183942
\(267\) 0 0
\(268\) −3.00000 −0.183254
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) −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
\(280\) 12.0000 0.717137
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) −8.00000 −0.476393
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) −2.00000 −0.118678
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) 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 \(0.584677\pi\)
\(294\) 2.00000 0.116642
\(295\) −60.0000 −3.49334
\(296\) −2.00000 −0.116248
\(297\) −10.0000 −0.580259
\(298\) 0 0
\(299\) 0 0
\(300\) −11.0000 −0.635085
\(301\) −12.0000 −0.691669
\(302\) 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 \(0.388747\pi\)
\(308\) 6.00000 0.341882
\(309\) 6.00000 0.341328
\(310\) −32.0000 −1.81748
\(311\) 7.00000 0.396934 0.198467 0.980108i \(-0.436404\pi\)
0.198467 + 0.980108i \(0.436404\pi\)
\(312\) 0 0
\(313\) 29.0000 1.63918 0.819588 0.572953i \(-0.194202\pi\)
0.819588 + 0.572953i \(0.194202\pi\)
\(314\) 2.00000 0.112867
\(315\) 24.0000 1.35225
\(316\) −10.0000 −0.562544
\(317\) 27.0000 1.51647 0.758236 0.651981i \(-0.226062\pi\)
0.758236 + 0.651981i \(0.226062\pi\)
\(318\) −1.00000 −0.0560772
\(319\) 10.0000 0.559893
\(320\) 4.00000 0.223607
\(321\) 7.00000 0.390702
\(322\) −3.00000 −0.167183
\(323\) 3.00000 0.166924
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) −15.0000 −0.829502
\(328\) −8.00000 −0.441726
\(329\) 24.0000 1.32316
\(330\) −8.00000 −0.440386
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) 6.00000 0.329293
\(333\) −4.00000 −0.219199
\(334\) −12.0000 −0.656611
\(335\) −12.0000 −0.655630
\(336\) 3.00000 0.163663
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) −14.0000 −0.760376
\(340\) 12.0000 0.650791
\(341\) −16.0000 −0.866449
\(342\) 2.00000 0.108148
\(343\) 15.0000 0.809924
\(344\) −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 \(0.586245\pi\)
\(350\) 33.0000 1.76392
\(351\) 0 0
\(352\) 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
\(357\) 9.00000 0.476331
\(358\) 0 0
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 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 \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −16.0000 −0.832927
\(370\) −8.00000 −0.415900
\(371\) 3.00000 0.155752
\(372\) −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
\(377\) 0 0
\(378\) 15.0000 0.771517
\(379\) −15.0000 −0.770498 −0.385249 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 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
\(385\) 24.0000 1.22315
\(386\) −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 \(0.775061\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) −2.00000 −0.101015
\(393\) −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
\(399\) 3.00000 0.150188
\(400\) 11.0000 0.550000
\(401\) 8.00000 0.399501 0.199750 0.979847i \(-0.435987\pi\)
0.199750 + 0.979847i \(0.435987\pi\)
\(402\) −3.00000 −0.149626
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 4.00000 0.198762
\(406\) −15.0000 −0.744438
\(407\) −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
\(413\) 45.0000 2.21431
\(414\) −2.00000 −0.0982946
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) 0 0
\(418\) 2.00000 0.0978232
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 12.0000 0.585540
\(421\) 13.0000 0.633581 0.316791 0.948495i \(-0.397395\pi\)
0.316791 + 0.948495i \(0.397395\pi\)
\(422\) −27.0000 −1.31434
\(423\) 16.0000 0.777947
\(424\) 1.00000 0.0485643
\(425\) 33.0000 1.60074
\(426\) −2.00000 −0.0969003
\(427\) −6.00000 −0.290360
\(428\) −7.00000 −0.338358
\(429\) 0 0
\(430\) −16.0000 −0.771589
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 5.00000 0.240563
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 24.0000 1.15204
\(435\) 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 \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 8.00000 0.381385
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) −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
\(448\) −3.00000 −0.141737
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 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 \(-0.447650\pi\)
0.163723 + 0.986506i \(0.447650\pi\)
\(458\) −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 \(0.273911\pi\)
\(462\) 6.00000 0.279145
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −5.00000 −0.232119
\(465\) −32.0000 −1.48396
\(466\) 6.00000 0.277945
\(467\) −2.00000 −0.0925490 −0.0462745 0.998929i \(-0.514735\pi\)
−0.0462745 + 0.998929i \(0.514735\pi\)
\(468\) 0 0
\(469\) 9.00000 0.415581
\(470\) 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
\(476\) −9.00000 −0.412514
\(477\) 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
\(481\) 0 0
\(482\) −8.00000 −0.364390
\(483\) −3.00000 −0.136505
\(484\) −7.00000 −0.318182
\(485\) 8.00000 0.363261
\(486\) 16.0000 0.725775
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −16.0000 −0.723545
\(490\) −8.00000 −0.361403
\(491\) −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
\(494\) 0 0
\(495\) 16.0000 0.719147
\(496\) 8.00000 0.359211
\(497\) 6.00000 0.269137
\(498\) 6.00000 0.268866
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 24.0000 1.07331
\(501\) −12.0000 −0.536120
\(502\) −2.00000 −0.0892644
\(503\) 39.0000 1.73892 0.869462 0.494000i \(-0.164466\pi\)
0.869462 + 0.494000i \(0.164466\pi\)
\(504\) −6.00000 −0.267261
\(505\) 8.00000 0.355995
\(506\) −2.00000 −0.0889108
\(507\) 0 0
\(508\) 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
\(511\) 27.0000 1.19441
\(512\) −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
\(518\) 6.00000 0.263625
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) −10.0000 −0.437688
\(523\) 29.0000 1.26808 0.634041 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 12.0000 0.524222
\(525\) 33.0000 1.44024
\(526\) −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
\(532\) −3.00000 −0.130066
\(533\) 0 0
\(534\) 0 0
\(535\) −28.0000 −1.21055
\(536\) 3.00000 0.129580
\(537\) 0 0
\(538\) −30.0000 −1.29339
\(539\) −4.00000 −0.172292
\(540\) 20.0000 0.860663
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 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
\(553\) 30.0000 1.27573
\(554\) −28.0000 −1.18961
\(555\) −8.00000 −0.339581
\(556\) 0 0
\(557\) −28.0000 −1.18640 −0.593199 0.805056i \(-0.702135\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 16.0000 0.677334
\(559\) 0 0
\(560\) −12.0000 −0.507093
\(561\) 6.00000 0.253320
\(562\) −8.00000 −0.337460
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 8.00000 0.336861
\(565\) 56.0000 2.35594
\(566\) 6.00000 0.252199
\(567\) −3.00000 −0.125988
\(568\) 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 \(0.699252\pi\)
\(572\) 0 0
\(573\) −7.00000 −0.292429
\(574\) 24.0000 1.00174
\(575\) −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
\(581\) −18.0000 −0.746766
\(582\) 2.00000 0.0829027
\(583\) 2.00000 0.0828315
\(584\) 9.00000 0.372423
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 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
\(595\) −36.0000 −1.47586
\(596\) 0 0
\(597\) 25.0000 1.02318
\(598\) 0 0
\(599\) 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 \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 12.0000 0.489083
\(603\) 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 \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −15.0000 −0.607831
\(610\) −8.00000 −0.323911
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) −12.0000 −0.484281
\(615\) −32.0000 −1.29036
\(616\) −6.00000 −0.241747
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −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
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) −29.0000 −1.15907
\(627\) 2.00000 0.0798723
\(628\) −2.00000 −0.0798087
\(629\) 6.00000 0.239236
\(630\) −24.0000 −0.956183
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 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 \(0.328656\pi\)
\(644\) 3.00000 0.118217
\(645\) −16.0000 −0.629999
\(646\) −3.00000 −0.118033
\(647\) 23.0000 0.904223 0.452112 0.891961i \(-0.350671\pi\)
0.452112 + 0.891961i \(0.350671\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) 24.0000 0.940634
\(652\) 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
\(658\) −24.0000 −0.935617
\(659\) 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
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) −12.0000 −0.465340
\(666\) 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
\(672\) −3.00000 −0.115728
\(673\) 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
\(676\) 0 0
\(677\) 13.0000 0.499631 0.249815 0.968294i \(-0.419630\pi\)
0.249815 + 0.968294i \(0.419630\pi\)
\(678\) 14.0000 0.537667
\(679\) −6.00000 −0.230259
\(680\) −12.0000 −0.460179
\(681\) −17.0000 −0.651441
\(682\) 16.0000 0.612672
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 68.0000 2.59815
\(686\) −15.0000 −0.572703
\(687\) −10.0000 −0.381524
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) −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
\(693\) −12.0000 −0.455842
\(694\) 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
\(700\) −33.0000 −1.24728
\(701\) −28.0000 −1.05755 −0.528773 0.848763i \(-0.677348\pi\)
−0.528773 + 0.848763i \(0.677348\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) −2.00000 −0.0753778
\(705\) 32.0000 1.20519
\(706\) 9.00000 0.338719
\(707\) −6.00000 −0.225653
\(708\) 15.0000 0.563735
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 8.00000 0.300235
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) −9.00000 −0.336817
\(715\) 0 0
\(716\) 0 0
\(717\) 15.0000 0.560185
\(718\) −15.0000 −0.559795
\(719\) −5.00000 −0.186469 −0.0932343 0.995644i \(-0.529721\pi\)
−0.0932343 + 0.995644i \(0.529721\pi\)
\(720\) −8.00000 −0.298142
\(721\) 18.0000 0.670355
\(722\) −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 \(-0.602088\pi\)
−0.315248 + 0.949009i \(0.602088\pi\)
\(728\) 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
\(735\) −8.00000 −0.295084
\(736\) 1.00000 0.0368605
\(737\) 6.00000 0.221013
\(738\) 16.0000 0.588968
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) −3.00000 −0.110133
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) −29.0000 −1.06177
\(747\) −12.0000 −0.439057
\(748\) −6.00000 −0.219382
\(749\) 21.0000 0.767323
\(750\) 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
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) −15.0000 −0.545545
\(757\) −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
\(763\) −45.0000 −1.62911
\(764\) 7.00000 0.253251
\(765\) −24.0000 −0.867722
\(766\) −26.0000 −0.939418
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 35.0000 1.26213 0.631066 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(770\) −24.0000 −0.864900
\(771\) −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
\(777\) 6.00000 0.215249
\(778\) 30.0000 1.07555
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 3.00000 0.107280
\(783\) −25.0000 −0.893427
\(784\) 2.00000 0.0714286
\(785\) −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
\(791\) −42.0000 −1.49335
\(792\) −4.00000 −0.142134
\(793\) 0 0
\(794\) 8.00000 0.283909
\(795\) 4.00000 0.141865
\(796\) −25.0000 −0.886102
\(797\) 3.00000 0.106265 0.0531327 0.998587i \(-0.483079\pi\)
0.0531327 + 0.998587i \(0.483079\pi\)
\(798\) −3.00000 −0.106199
\(799\) −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
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) −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 \(0.483226\pi\)
\(812\) 15.0000 0.526397
\(813\) 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 \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) 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
\(826\) −45.0000 −1.56575
\(827\) −23.0000 −0.799788 −0.399894 0.916561i \(-0.630953\pi\)
−0.399894 + 0.916561i \(0.630953\pi\)
\(828\) 2.00000 0.0695048
\(829\) −15.0000 −0.520972 −0.260486 0.965478i \(-0.583883\pi\)
−0.260486 + 0.965478i \(0.583883\pi\)
\(830\) −24.0000 −0.833052
\(831\) −28.0000 −0.971309
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 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 \(0.612202\pi\)
\(840\) −12.0000 −0.414039
\(841\) −4.00000 −0.137931
\(842\) −13.0000 −0.448010
\(843\) −8.00000 −0.275535
\(844\) 27.0000 0.929378
\(845\) 0 0
\(846\) −16.0000 −0.550091
\(847\) 21.0000 0.721569
\(848\) −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 \(0.467246\pi\)
\(854\) 6.00000 0.205316
\(855\) −8.00000 −0.273594
\(856\) 7.00000 0.239255
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 0 0
\(859\) −50.0000 −1.70598 −0.852989 0.521929i \(-0.825213\pi\)
−0.852989 + 0.521929i \(0.825213\pi\)
\(860\) 16.0000 0.545595
\(861\) 24.0000 0.817918
\(862\) −18.0000 −0.613082
\(863\) −54.0000 −1.83818 −0.919091 0.394046i \(-0.871075\pi\)
−0.919091 + 0.394046i \(0.871075\pi\)
\(864\) −5.00000 −0.170103
\(865\) −24.0000 −0.816024
\(866\) −14.0000 −0.475739
\(867\) 8.00000 0.271694
\(868\) −24.0000 −0.814613
\(869\) 20.0000 0.678454
\(870\) −20.0000 −0.678064
\(871\) 0 0
\(872\) −15.0000 −0.507964
\(873\) −4.00000 −0.135379
\(874\) 1.00000 0.0338255
\(875\) −72.0000 −2.43404
\(876\) 9.00000 0.304082
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) −20.0000 −0.674967
\(879\) 9.00000 0.303562
\(880\) −8.00000 −0.269680
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 4.00000 0.134687
\(883\) 34.0000 1.14419 0.572096 0.820187i \(-0.306131\pi\)
0.572096 + 0.820187i \(0.306131\pi\)
\(884\) 0 0
\(885\) 60.0000 2.01688
\(886\) 26.0000 0.873487
\(887\) −2.00000 −0.0671534 −0.0335767 0.999436i \(-0.510690\pi\)
−0.0335767 + 0.999436i \(0.510690\pi\)
\(888\) 2.00000 0.0671156
\(889\) −54.0000 −1.81110
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) −14.0000 −0.468755
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) 10.0000 0.333704
\(899\) −40.0000 −1.33407
\(900\) −22.0000 −0.733333
\(901\) −3.00000 −0.0999445
\(902\) 16.0000 0.532742
\(903\) 12.0000 0.399335
\(904\) −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
\(917\) −36.0000 −1.18882
\(918\) −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
\(923\) 0 0
\(924\) −6.00000 −0.197386
\(925\) 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
\(931\) 2.00000 0.0655474
\(932\) −6.00000 −0.196537
\(933\) −7.00000 −0.229170
\(934\) 2.00000 0.0654420
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) −7.00000 −0.228680 −0.114340 0.993442i \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(938\) −9.00000 −0.293860
\(939\) −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
\(945\) −60.0000 −1.95180
\(946\) 8.00000 0.260102
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 10.0000 0.324785
\(949\) 0 0
\(950\) −11.0000 −0.356887
\(951\) −27.0000 −0.875535
\(952\) 9.00000 0.291692
\(953\) −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
\(959\) −51.0000 −1.64688
\(960\) −4.00000 −0.129099
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 14.0000 0.451144
\(964\) 8.00000 0.257663
\(965\) 24.0000 0.772587
\(966\) 3.00000 0.0965234
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 7.00000 0.224989
\(969\) −3.00000 −0.0963739
\(970\) −8.00000 −0.256865
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) −16.0000 −0.513200
\(973\) 0 0
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −8.00000 −0.255943 −0.127971 0.991778i \(-0.540847\pi\)
−0.127971 + 0.991778i \(0.540847\pi\)
\(978\) 16.0000 0.511624
\(979\) 0 0
\(980\) 8.00000 0.255551
\(981\) −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
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) −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
\(994\) −6.00000 −0.190308
\(995\) −100.000 −3.17021
\(996\) −6.00000 −0.190117
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) 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 6422.2.a.b.1.1 1
13.12 even 2 38.2.a.b.1.1 1
39.38 odd 2 342.2.a.d.1.1 1
52.51 odd 2 304.2.a.d.1.1 1
65.12 odd 4 950.2.b.c.799.2 2
65.38 odd 4 950.2.b.c.799.1 2
65.64 even 2 950.2.a.b.1.1 1
91.90 odd 2 1862.2.a.f.1.1 1
104.51 odd 2 1216.2.a.g.1.1 1
104.77 even 2 1216.2.a.n.1.1 1
143.142 odd 2 4598.2.a.a.1.1 1
156.155 even 2 2736.2.a.w.1.1 1
195.194 odd 2 8550.2.a.u.1.1 1
247.12 odd 6 722.2.c.f.429.1 2
247.25 even 18 722.2.e.c.245.1 6
247.51 odd 18 722.2.e.d.245.1 6
247.64 even 6 722.2.c.d.429.1 2
247.90 odd 18 722.2.e.d.595.1 6
247.103 odd 6 722.2.c.f.653.1 2
247.116 odd 18 722.2.e.d.99.1 6
247.129 odd 18 722.2.e.d.415.1 6
247.142 even 18 722.2.e.c.423.1 6
247.155 odd 18 722.2.e.d.389.1 6
247.168 even 18 722.2.e.c.389.1 6
247.181 odd 18 722.2.e.d.423.1 6
247.194 even 18 722.2.e.c.415.1 6
247.207 even 18 722.2.e.c.99.1 6
247.220 even 6 722.2.c.d.653.1 2
247.233 even 18 722.2.e.c.595.1 6
247.246 odd 2 722.2.a.b.1.1 1
260.259 odd 2 7600.2.a.h.1.1 1
741.740 even 2 6498.2.a.y.1.1 1
988.987 even 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 13.12 even 2
304.2.a.d.1.1 1 52.51 odd 2
342.2.a.d.1.1 1 39.38 odd 2
722.2.a.b.1.1 1 247.246 odd 2
722.2.c.d.429.1 2 247.64 even 6
722.2.c.d.653.1 2 247.220 even 6
722.2.c.f.429.1 2 247.12 odd 6
722.2.c.f.653.1 2 247.103 odd 6
722.2.e.c.99.1 6 247.207 even 18
722.2.e.c.245.1 6 247.25 even 18
722.2.e.c.389.1 6 247.168 even 18
722.2.e.c.415.1 6 247.194 even 18
722.2.e.c.423.1 6 247.142 even 18
722.2.e.c.595.1 6 247.233 even 18
722.2.e.d.99.1 6 247.116 odd 18
722.2.e.d.245.1 6 247.51 odd 18
722.2.e.d.389.1 6 247.155 odd 18
722.2.e.d.415.1 6 247.129 odd 18
722.2.e.d.423.1 6 247.181 odd 18
722.2.e.d.595.1 6 247.90 odd 18
950.2.a.b.1.1 1 65.64 even 2
950.2.b.c.799.1 2 65.38 odd 4
950.2.b.c.799.2 2 65.12 odd 4
1216.2.a.g.1.1 1 104.51 odd 2
1216.2.a.n.1.1 1 104.77 even 2
1862.2.a.f.1.1 1 91.90 odd 2
2736.2.a.w.1.1 1 156.155 even 2
4598.2.a.a.1.1 1 143.142 odd 2
5776.2.a.d.1.1 1 988.987 even 2
6422.2.a.b.1.1 1 1.1 even 1 trivial
6498.2.a.y.1.1 1 741.740 even 2
7600.2.a.h.1.1 1 260.259 odd 2
8550.2.a.u.1.1 1 195.194 odd 2