Properties

Label 350.2.a.e.1.1
Level 350
Weight 2
Character 350.1
Self dual yes
Analytic conductor 2.795
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 350.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -2.00000 q^{18} -7.00000 q^{19} +1.00000 q^{21} +3.00000 q^{22} +1.00000 q^{24} +2.00000 q^{26} -5.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} +3.00000 q^{34} -2.00000 q^{36} +8.00000 q^{37} -7.00000 q^{38} +2.00000 q^{39} -9.00000 q^{41} +1.00000 q^{42} +8.00000 q^{43} +3.00000 q^{44} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +3.00000 q^{51} +2.00000 q^{52} -12.0000 q^{53} -5.00000 q^{54} +1.00000 q^{56} -7.00000 q^{57} -6.00000 q^{58} +12.0000 q^{59} -10.0000 q^{61} -4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -7.00000 q^{67} +3.00000 q^{68} +6.00000 q^{71} -2.00000 q^{72} +5.00000 q^{73} +8.00000 q^{74} -7.00000 q^{76} +3.00000 q^{77} +2.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} -9.00000 q^{82} -9.00000 q^{83} +1.00000 q^{84} +8.00000 q^{86} -6.00000 q^{87} +3.00000 q^{88} -15.0000 q^{89} +2.00000 q^{91} -4.00000 q^{93} -6.00000 q^{94} +1.00000 q^{96} -10.0000 q^{97} +1.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 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
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 3.00000 0.639602
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) −5.00000 −0.962250
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.00000 0.522233
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −7.00000 −1.13555
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 1.00000 0.154303
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 2.00000 0.277350
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −7.00000 −0.927173
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −4.00000 −0.508001
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 3.00000 0.363803
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −2.00000 −0.235702
\(73\) 5.00000 0.585206 0.292603 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −7.00000 −0.802955
\(77\) 3.00000 0.341882
\(78\) 2.00000 0.226455
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −9.00000 −0.993884
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) −6.00000 −0.643268
\(88\) 3.00000 0.319801
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 1.00000 0.101015
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 3.00000 0.297044
\(103\) 20.0000 1.97066 0.985329 0.170664i \(-0.0545913\pi\)
0.985329 + 0.170664i \(0.0545913\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) −5.00000 −0.481125
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 1.00000 0.0944911
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) −7.00000 −0.655610
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) −4.00000 −0.369800
\(118\) 12.0000 1.10469
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −10.0000 −0.905357
\(123\) −9.00000 −0.811503
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 3.00000 0.261116
\(133\) −7.00000 −0.606977
\(134\) −7.00000 −0.604708
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 21.0000 1.79415 0.897076 0.441877i \(-0.145687\pi\)
0.897076 + 0.441877i \(0.145687\pi\)
\(138\) 0 0
\(139\) −7.00000 −0.593732 −0.296866 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 6.00000 0.503509
\(143\) 6.00000 0.501745
\(144\) −2.00000 −0.166667
\(145\) 0 0
\(146\) 5.00000 0.413803
\(147\) 1.00000 0.0824786
\(148\) 8.00000 0.657596
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −7.00000 −0.567775
\(153\) −6.00000 −0.485071
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 20.0000 1.59617 0.798087 0.602542i \(-0.205846\pi\)
0.798087 + 0.602542i \(0.205846\pi\)
\(158\) 14.0000 1.11378
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 5.00000 0.391630 0.195815 0.980641i \(-0.437265\pi\)
0.195815 + 0.980641i \(0.437265\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 14.0000 1.07061
\(172\) 8.00000 0.609994
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 12.0000 0.901975
\(178\) −15.0000 −1.12430
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 2.00000 0.148250
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 9.00000 0.658145
\(188\) −6.00000 −0.437595
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 1.00000 0.0721688
\(193\) 5.00000 0.359908 0.179954 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −6.00000 −0.426401
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) 0 0
\(203\) −6.00000 −0.421117
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) 20.0000 1.39347
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −21.0000 −1.45260
\(210\) 0 0
\(211\) 17.0000 1.17033 0.585164 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(212\) −12.0000 −0.824163
\(213\) 6.00000 0.411113
\(214\) 3.00000 0.205076
\(215\) 0 0
\(216\) −5.00000 −0.340207
\(217\) −4.00000 −0.271538
\(218\) 14.0000 0.948200
\(219\) 5.00000 0.337869
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 8.00000 0.536925
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −9.00000 −0.598671
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −7.00000 −0.463586
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) −6.00000 −0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 14.0000 0.909398
\(238\) 3.00000 0.194461
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −25.0000 −1.61039 −0.805196 0.593009i \(-0.797940\pi\)
−0.805196 + 0.593009i \(0.797940\pi\)
\(242\) −2.00000 −0.128565
\(243\) 16.0000 1.02640
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) −9.00000 −0.573819
\(247\) −14.0000 −0.890799
\(248\) −4.00000 −0.254000
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) 15.0000 0.946792 0.473396 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 8.00000 0.498058
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) −7.00000 −0.429198
\(267\) −15.0000 −0.917985
\(268\) −7.00000 −0.427593
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 3.00000 0.181902
\(273\) 2.00000 0.121046
\(274\) 21.0000 1.26866
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −7.00000 −0.419832
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −6.00000 −0.357295
\(283\) −1.00000 −0.0594438 −0.0297219 0.999558i \(-0.509462\pi\)
−0.0297219 + 0.999558i \(0.509462\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) −9.00000 −0.531253
\(288\) −2.00000 −0.117851
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 5.00000 0.292603
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) −15.0000 −0.870388
\(298\) 12.0000 0.695141
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) −7.00000 −0.401478
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 3.00000 0.170941
\(309\) 20.0000 1.13776
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 2.00000 0.113228
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 20.0000 1.12867
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) −12.0000 −0.672927
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) 3.00000 0.167444
\(322\) 0 0
\(323\) −21.0000 −1.16847
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 5.00000 0.276924
\(327\) 14.0000 0.774202
\(328\) −9.00000 −0.496942
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) −9.00000 −0.493939
\(333\) −16.0000 −0.876795
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) −9.00000 −0.489535
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 14.0000 0.757033
\(343\) 1.00000 0.0539949
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −21.0000 −1.12734 −0.563670 0.826000i \(-0.690611\pi\)
−0.563670 + 0.826000i \(0.690611\pi\)
\(348\) −6.00000 −0.321634
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) −10.0000 −0.533761
\(352\) 3.00000 0.159901
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −15.0000 −0.794998
\(357\) 3.00000 0.158777
\(358\) 3.00000 0.158555
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 2.00000 0.105118
\(363\) −2.00000 −0.104973
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) −4.00000 −0.207390
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −12.0000 −0.618031
\(378\) −5.00000 −0.257172
\(379\) 17.0000 0.873231 0.436616 0.899648i \(-0.356177\pi\)
0.436616 + 0.899648i \(0.356177\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) −6.00000 −0.306987
\(383\) −30.0000 −1.53293 −0.766464 0.642287i \(-0.777986\pi\)
−0.766464 + 0.642287i \(0.777986\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 5.00000 0.254493
\(387\) −16.0000 −0.813326
\(388\) −10.0000 −0.507673
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 14.0000 0.701757
\(399\) −7.00000 −0.350438
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) −7.00000 −0.349128
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 24.0000 1.18964
\(408\) 3.00000 0.148522
\(409\) −25.0000 −1.23617 −0.618085 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 0 0
\(411\) 21.0000 1.03585
\(412\) 20.0000 0.985329
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −7.00000 −0.342791
\(418\) −21.0000 −1.02714
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 17.0000 0.827547
\(423\) 12.0000 0.583460
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 6.00000 0.290701
\(427\) −10.0000 −0.483934
\(428\) 3.00000 0.145010
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) −5.00000 −0.240563
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) 5.00000 0.238909
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 6.00000 0.285391
\(443\) 21.0000 0.997740 0.498870 0.866677i \(-0.333748\pi\)
0.498870 + 0.866677i \(0.333748\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) 14.0000 0.662919
\(447\) 12.0000 0.567581
\(448\) 1.00000 0.0472456
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) −27.0000 −1.27138
\(452\) −9.00000 −0.423324
\(453\) 8.00000 0.375873
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −7.00000 −0.327805
\(457\) 17.0000 0.795226 0.397613 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(458\) 26.0000 1.21490
\(459\) −15.0000 −0.700140
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 3.00000 0.139573
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) −4.00000 −0.184900
\(469\) −7.00000 −0.323230
\(470\) 0 0
\(471\) 20.0000 0.921551
\(472\) 12.0000 0.552345
\(473\) 24.0000 1.10352
\(474\) 14.0000 0.643041
\(475\) 0 0
\(476\) 3.00000 0.137505
\(477\) 24.0000 1.09888
\(478\) −12.0000 −0.548867
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) −25.0000 −1.13872
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 16.0000 0.725775
\(487\) −34.0000 −1.54069 −0.770344 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) −10.0000 −0.452679
\(489\) 5.00000 0.226108
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −9.00000 −0.405751
\(493\) −18.0000 −0.810679
\(494\) −14.0000 −0.629890
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 6.00000 0.269137
\(498\) −9.00000 −0.403300
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 15.0000 0.669483
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 2.00000 0.0887357
\(509\) 42.0000 1.86162 0.930809 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 0 0
\(511\) 5.00000 0.221187
\(512\) 1.00000 0.0441942
\(513\) 35.0000 1.54529
\(514\) −30.0000 −1.32324
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) −18.0000 −0.791639
\(518\) 8.00000 0.351500
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 39.0000 1.70862 0.854311 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) 12.0000 0.525226
\(523\) −7.00000 −0.306089 −0.153044 0.988219i \(-0.548908\pi\)
−0.153044 + 0.988219i \(0.548908\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) −12.0000 −0.522728
\(528\) 3.00000 0.130558
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −24.0000 −1.04151
\(532\) −7.00000 −0.303488
\(533\) −18.0000 −0.779667
\(534\) −15.0000 −0.649113
\(535\) 0 0
\(536\) −7.00000 −0.302354
\(537\) 3.00000 0.129460
\(538\) −6.00000 −0.258678
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 2.00000 0.0859074
\(543\) 2.00000 0.0858282
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) 35.0000 1.49649 0.748246 0.663421i \(-0.230896\pi\)
0.748246 + 0.663421i \(0.230896\pi\)
\(548\) 21.0000 0.897076
\(549\) 20.0000 0.853579
\(550\) 0 0
\(551\) 42.0000 1.78926
\(552\) 0 0
\(553\) 14.0000 0.595341
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −7.00000 −0.296866
\(557\) 36.0000 1.52537 0.762684 0.646771i \(-0.223881\pi\)
0.762684 + 0.646771i \(0.223881\pi\)
\(558\) 8.00000 0.338667
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 9.00000 0.379980
\(562\) −18.0000 −0.759284
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) −1.00000 −0.0420331
\(567\) 1.00000 0.0419961
\(568\) 6.00000 0.251754
\(569\) −27.0000 −1.13190 −0.565949 0.824440i \(-0.691490\pi\)
−0.565949 + 0.824440i \(0.691490\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 6.00000 0.250873
\(573\) −6.00000 −0.250654
\(574\) −9.00000 −0.375653
\(575\) 0 0
\(576\) −2.00000 −0.0833333
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) −8.00000 −0.332756
\(579\) 5.00000 0.207793
\(580\) 0 0
\(581\) −9.00000 −0.373383
\(582\) −10.0000 −0.414513
\(583\) −36.0000 −1.49097
\(584\) 5.00000 0.206901
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 39.0000 1.60970 0.804851 0.593477i \(-0.202245\pi\)
0.804851 + 0.593477i \(0.202245\pi\)
\(588\) 1.00000 0.0412393
\(589\) 28.0000 1.15372
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 8.00000 0.328798
\(593\) −27.0000 −1.10876 −0.554379 0.832265i \(-0.687044\pi\)
−0.554379 + 0.832265i \(0.687044\pi\)
\(594\) −15.0000 −0.615457
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) 14.0000 0.572982
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 8.00000 0.326056
\(603\) 14.0000 0.570124
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) 44.0000 1.78590 0.892952 0.450151i \(-0.148630\pi\)
0.892952 + 0.450151i \(0.148630\pi\)
\(608\) −7.00000 −0.283887
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) −6.00000 −0.242536
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −7.00000 −0.282497
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 20.0000 0.804518
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) −15.0000 −0.600962
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) −21.0000 −0.838659
\(628\) 20.0000 0.798087
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 14.0000 0.556890
\(633\) 17.0000 0.675689
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) −12.0000 −0.475831
\(637\) 2.00000 0.0792429
\(638\) −18.0000 −0.712627
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 3.00000 0.118401
\(643\) −40.0000 −1.57745 −0.788723 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −21.0000 −0.826234
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 5.00000 0.195815
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) −9.00000 −0.351391
\(657\) −10.0000 −0.390137
\(658\) −6.00000 −0.233904
\(659\) 9.00000 0.350590 0.175295 0.984516i \(-0.443912\pi\)
0.175295 + 0.984516i \(0.443912\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −25.0000 −0.971653
\(663\) 6.00000 0.233021
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) −16.0000 −0.619987
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) −30.0000 −1.15814
\(672\) 1.00000 0.0385758
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −13.0000 −0.500741
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) −9.00000 −0.345643
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) −12.0000 −0.459504
\(683\) 3.00000 0.114792 0.0573959 0.998351i \(-0.481720\pi\)
0.0573959 + 0.998351i \(0.481720\pi\)
\(684\) 14.0000 0.535303
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 26.0000 0.991962
\(688\) 8.00000 0.304997
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) −19.0000 −0.722794 −0.361397 0.932412i \(-0.617700\pi\)
−0.361397 + 0.932412i \(0.617700\pi\)
\(692\) −6.00000 −0.228086
\(693\) −6.00000 −0.227921
\(694\) −21.0000 −0.797149
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −27.0000 −1.02270
\(698\) 8.00000 0.302804
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) −10.0000 −0.377426
\(703\) −56.0000 −2.11208
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) 0 0
\(711\) −28.0000 −1.05008
\(712\) −15.0000 −0.562149
\(713\) 0 0
\(714\) 3.00000 0.112272
\(715\) 0 0
\(716\) 3.00000 0.112115
\(717\) −12.0000 −0.448148
\(718\) −6.00000 −0.223918
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 20.0000 0.744839
\(722\) 30.0000 1.11648
\(723\) −25.0000 −0.929760
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) −34.0000 −1.26099 −0.630495 0.776193i \(-0.717148\pi\)
−0.630495 + 0.776193i \(0.717148\pi\)
\(728\) 2.00000 0.0741249
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) −10.0000 −0.369611
\(733\) −40.0000 −1.47743 −0.738717 0.674016i \(-0.764568\pi\)
−0.738717 + 0.674016i \(0.764568\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) −21.0000 −0.773545
\(738\) 18.0000 0.662589
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −14.0000 −0.514303
\(742\) −12.0000 −0.440534
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) −4.00000 −0.146647
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) 18.0000 0.658586
\(748\) 9.00000 0.329073
\(749\) 3.00000 0.109618
\(750\) 0 0
\(751\) −46.0000 −1.67856 −0.839282 0.543696i \(-0.817024\pi\)
−0.839282 + 0.543696i \(0.817024\pi\)
\(752\) −6.00000 −0.218797
\(753\) 15.0000 0.546630
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) −5.00000 −0.181848
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 17.0000 0.617468
\(759\) 0 0
\(760\) 0 0
\(761\) −9.00000 −0.326250 −0.163125 0.986605i \(-0.552157\pi\)
−0.163125 + 0.986605i \(0.552157\pi\)
\(762\) 2.00000 0.0724524
\(763\) 14.0000 0.506834
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) −30.0000 −1.08394
\(767\) 24.0000 0.866590
\(768\) 1.00000 0.0360844
\(769\) 23.0000 0.829401 0.414701 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) 5.00000 0.179954
\(773\) −12.0000 −0.431610 −0.215805 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) −16.0000 −0.575108
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 8.00000 0.286998
\(778\) −24.0000 −0.860442
\(779\) 63.0000 2.25721
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 0 0
\(783\) 30.0000 1.07211
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −6.00000 −0.213741
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) −9.00000 −0.320003
\(792\) −6.00000 −0.213201
\(793\) −20.0000 −0.710221
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) 14.0000 0.496217
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) −7.00000 −0.247797
\(799\) −18.0000 −0.636794
\(800\) 0 0
\(801\) 30.0000 1.06000
\(802\) −27.0000 −0.953403
\(803\) 15.0000 0.529339
\(804\) −7.00000 −0.246871
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) −6.00000 −0.210559
\(813\) 2.00000 0.0701431
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) −56.0000 −1.95919
\(818\) −25.0000 −0.874105
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 21.0000 0.732459
\(823\) 26.0000 0.906303 0.453152 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) 20.0000 0.696733
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 9.00000 0.312961 0.156480 0.987681i \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(828\) 0 0
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 2.00000 0.0693375
\(833\) 3.00000 0.103944
\(834\) −7.00000 −0.242390
\(835\) 0 0
\(836\) −21.0000 −0.726300
\(837\) 20.0000 0.691301
\(838\) −3.00000 −0.103633
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 20.0000 0.689246
\(843\) −18.0000 −0.619953
\(844\) 17.0000 0.585164
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) −2.00000 −0.0687208
\(848\) −12.0000 −0.412082
\(849\) −1.00000 −0.0343199
\(850\) 0 0
\(851\) 0 0
\(852\) 6.00000 0.205557
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) 15.0000 0.512390 0.256195 0.966625i \(-0.417531\pi\)
0.256195 + 0.966625i \(0.417531\pi\)
\(858\) 6.00000 0.204837
\(859\) −31.0000 −1.05771 −0.528853 0.848713i \(-0.677378\pi\)
−0.528853 + 0.848713i \(0.677378\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) −36.0000 −1.22616
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) 11.0000 0.373795
\(867\) −8.00000 −0.271694
\(868\) −4.00000 −0.135769
\(869\) 42.0000 1.42475
\(870\) 0 0
\(871\) −14.0000 −0.474372
\(872\) 14.0000 0.474100
\(873\) 20.0000 0.676897
\(874\) 0 0
\(875\) 0 0
\(876\) 5.00000 0.168934
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −4.00000 −0.134993
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) −2.00000 −0.0673435
\(883\) 47.0000 1.58168 0.790838 0.612026i \(-0.209645\pi\)
0.790838 + 0.612026i \(0.209645\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 21.0000 0.705509
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 8.00000 0.268462
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 14.0000 0.468755
\(893\) 42.0000 1.40548
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −15.0000 −0.500556
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) −27.0000 −0.899002
\(903\) 8.00000 0.266223
\(904\) −9.00000 −0.299336
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) −7.00000 −0.231793
\(913\) −27.0000 −0.893570
\(914\) 17.0000 0.562310
\(915\) 0 0
\(916\) 26.0000 0.859064
\(917\) 0 0
\(918\) −15.0000 −0.495074
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 18.0000 0.592798
\(923\) 12.0000 0.394985
\(924\) 3.00000 0.0986928
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) −40.0000 −1.31377
\(928\) −6.00000 −0.196960
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −7.00000 −0.229416
\(932\) 6.00000 0.196537
\(933\) −18.0000 −0.589294
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) 29.0000 0.947389 0.473694 0.880689i \(-0.342920\pi\)
0.473694 + 0.880689i \(0.342920\pi\)
\(938\) −7.00000 −0.228558
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 20.0000 0.651635
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 24.0000 0.780307
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 14.0000 0.454699
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 3.00000 0.0972306
\(953\) 57.0000 1.84641 0.923206 0.384307i \(-0.125559\pi\)
0.923206 + 0.384307i \(0.125559\pi\)
\(954\) 24.0000 0.777029
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) −18.0000 −0.581857
\(958\) 18.0000 0.581554
\(959\) 21.0000 0.678125
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 16.0000 0.515861
\(963\) −6.00000 −0.193347
\(964\) −25.0000 −0.805196
\(965\) 0 0
\(966\) 0 0
\(967\) −34.0000 −1.09337 −0.546683 0.837340i \(-0.684110\pi\)
−0.546683 + 0.837340i \(0.684110\pi\)
\(968\) −2.00000 −0.0642824
\(969\) −21.0000 −0.674617
\(970\) 0 0
\(971\) 9.00000 0.288824 0.144412 0.989518i \(-0.453871\pi\)
0.144412 + 0.989518i \(0.453871\pi\)
\(972\) 16.0000 0.513200
\(973\) −7.00000 −0.224410
\(974\) −34.0000 −1.08943
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −3.00000 −0.0959785 −0.0479893 0.998848i \(-0.515281\pi\)
−0.0479893 + 0.998848i \(0.515281\pi\)
\(978\) 5.00000 0.159882
\(979\) −45.0000 −1.43821
\(980\) 0 0
\(981\) −28.0000 −0.893971
\(982\) −12.0000 −0.382935
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) −9.00000 −0.286910
\(985\) 0 0
\(986\) −18.0000 −0.573237
\(987\) −6.00000 −0.190982
\(988\) −14.0000 −0.445399
\(989\) 0 0
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −4.00000 −0.127000
\(993\) −25.0000 −0.793351
\(994\) 6.00000 0.190308
\(995\) 0 0
\(996\) −9.00000 −0.285176
\(997\) 62.0000 1.96356 0.981780 0.190022i \(-0.0608559\pi\)
0.981780 + 0.190022i \(0.0608559\pi\)
\(998\) −28.0000 −0.886325
\(999\) −40.0000 −1.26554
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 350.2.a.e.1.1 yes 1
3.2 odd 2 3150.2.a.m.1.1 1
4.3 odd 2 2800.2.a.h.1.1 1
5.2 odd 4 350.2.c.c.99.2 2
5.3 odd 4 350.2.c.c.99.1 2
5.4 even 2 350.2.a.a.1.1 1
7.6 odd 2 2450.2.a.x.1.1 1
15.2 even 4 3150.2.g.f.2899.1 2
15.8 even 4 3150.2.g.f.2899.2 2
15.14 odd 2 3150.2.a.x.1.1 1
20.3 even 4 2800.2.g.i.449.1 2
20.7 even 4 2800.2.g.i.449.2 2
20.19 odd 2 2800.2.a.x.1.1 1
35.13 even 4 2450.2.c.h.99.1 2
35.27 even 4 2450.2.c.h.99.2 2
35.34 odd 2 2450.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.a.1.1 1 5.4 even 2
350.2.a.e.1.1 yes 1 1.1 even 1 trivial
350.2.c.c.99.1 2 5.3 odd 4
350.2.c.c.99.2 2 5.2 odd 4
2450.2.a.m.1.1 1 35.34 odd 2
2450.2.a.x.1.1 1 7.6 odd 2
2450.2.c.h.99.1 2 35.13 even 4
2450.2.c.h.99.2 2 35.27 even 4
2800.2.a.h.1.1 1 4.3 odd 2
2800.2.a.x.1.1 1 20.19 odd 2
2800.2.g.i.449.1 2 20.3 even 4
2800.2.g.i.449.2 2 20.7 even 4
3150.2.a.m.1.1 1 3.2 odd 2
3150.2.a.x.1.1 1 15.14 odd 2
3150.2.g.f.2899.1 2 15.2 even 4
3150.2.g.f.2899.2 2 15.8 even 4