Properties

Label 1050.2.a.r.1.1
Level 1050
Weight 2
Character 1050.1
Self dual yes
Analytic conductor 8.384
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.38429221223\)
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 \(\chi\) = 1050.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} +1.00000 q^{21} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -5.00000 q^{29} +7.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{39} +7.00000 q^{41} +1.00000 q^{42} -11.0000 q^{43} +2.00000 q^{44} -1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +3.00000 q^{51} -1.00000 q^{52} -1.00000 q^{53} +1.00000 q^{54} +1.00000 q^{56} -5.00000 q^{58} -5.00000 q^{59} -3.00000 q^{61} +7.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -12.0000 q^{67} +3.00000 q^{68} -1.00000 q^{69} +12.0000 q^{71} +1.00000 q^{72} -6.00000 q^{73} -2.00000 q^{74} +2.00000 q^{77} -1.00000 q^{78} +10.0000 q^{79} +1.00000 q^{81} +7.00000 q^{82} -11.0000 q^{83} +1.00000 q^{84} -11.0000 q^{86} -5.00000 q^{87} +2.00000 q^{88} -10.0000 q^{89} -1.00000 q^{91} -1.00000 q^{92} +7.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 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 \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 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 \(0.381478\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 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
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 1.00000 0.154303
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 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 \(0.521879\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −5.00000 −0.656532
\(59\) −5.00000 −0.650945 −0.325472 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) 0 0
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) 7.00000 0.889001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 3.00000 0.363803
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) −1.00000 −0.113228
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 7.00000 0.773021
\(83\) −11.0000 −1.20741 −0.603703 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −11.0000 −1.18616
\(87\) −5.00000 −0.536056
\(88\) 2.00000 0.213201
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) −1.00000 −0.104257
\(93\) 7.00000 0.725866
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 1.00000 0.101015
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 3.00000 0.297044
\(103\) −11.0000 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −5.00000 −0.464238
\(117\) −1.00000 −0.0924500
\(118\) −5.00000 −0.460287
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −3.00000 −0.271607
\(123\) 7.00000 0.631169
\(124\) 7.00000 0.628619
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) −22.0000 −1.95218 −0.976092 0.217357i \(-0.930256\pi\)
−0.976092 + 0.217357i \(0.930256\pi\)
\(128\) 1.00000 0.0883883
\(129\) −11.0000 −0.968496
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 12.0000 1.00702
\(143\) −2.00000 −0.167248
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) 1.00000 0.0824786
\(148\) −2.00000 −0.164399
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) −18.0000 −1.46482 −0.732410 0.680864i \(-0.761604\pi\)
−0.732410 + 0.680864i \(0.761604\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) −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
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 1.00000 0.0785674
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 7.00000 0.546608
\(165\) 0 0
\(166\) −11.0000 −0.853766
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 1.00000 0.0771517
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) −11.0000 −0.838742
\(173\) 24.0000 1.82469 0.912343 0.409426i \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) −5.00000 −0.379049
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) −5.00000 −0.375823
\(178\) −10.0000 −0.749532
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) −1.00000 −0.0741249
\(183\) −3.00000 −0.221766
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 7.00000 0.513265
\(187\) 6.00000 0.438763
\(188\) 8.00000 0.583460
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −17.0000 −1.21120 −0.605600 0.795769i \(-0.707067\pi\)
−0.605600 + 0.795769i \(0.707067\pi\)
\(198\) 2.00000 0.142134
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 12.0000 0.844317
\(203\) −5.00000 −0.350931
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) −11.0000 −0.766406
\(207\) −1.00000 −0.0695048
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 12.0000 0.822226
\(214\) −2.00000 −0.136717
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 7.00000 0.475191
\(218\) 0 0
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) −2.00000 −0.134231
\(223\) 9.00000 0.602685 0.301342 0.953516i \(-0.402565\pi\)
0.301342 + 0.953516i \(0.402565\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −17.0000 −1.12833 −0.564165 0.825662i \(-0.690802\pi\)
−0.564165 + 0.825662i \(0.690802\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) −5.00000 −0.328266
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −5.00000 −0.325472
\(237\) 10.0000 0.649570
\(238\) 3.00000 0.194461
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) −3.00000 −0.192055
\(245\) 0 0
\(246\) 7.00000 0.446304
\(247\) 0 0
\(248\) 7.00000 0.444500
\(249\) −11.0000 −0.697097
\(250\) 0 0
\(251\) 7.00000 0.441836 0.220918 0.975292i \(-0.429095\pi\)
0.220918 + 0.975292i \(0.429095\pi\)
\(252\) 1.00000 0.0629941
\(253\) −2.00000 −0.125739
\(254\) −22.0000 −1.38040
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 23.0000 1.43470 0.717350 0.696713i \(-0.245355\pi\)
0.717350 + 0.696713i \(0.245355\pi\)
\(258\) −11.0000 −0.684830
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) −8.00000 −0.494242
\(263\) −21.0000 −1.29492 −0.647458 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) −12.0000 −0.733017
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 3.00000 0.181902
\(273\) −1.00000 −0.0605228
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 20.0000 1.19952
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) 32.0000 1.90896 0.954480 0.298275i \(-0.0964112\pi\)
0.954480 + 0.298275i \(0.0964112\pi\)
\(282\) 8.00000 0.476393
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 7.00000 0.413197
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) −6.00000 −0.351123
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 2.00000 0.116052
\(298\) 15.0000 0.868927
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) −11.0000 −0.634029
\(302\) −18.0000 −1.03578
\(303\) 12.0000 0.689382
\(304\) 0 0
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 2.00000 0.113961
\(309\) −11.0000 −0.625768
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) −1.00000 −0.0560772
\(319\) −10.0000 −0.559893
\(320\) 0 0
\(321\) −2.00000 −0.111629
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −1.00000 −0.0553849
\(327\) 0 0
\(328\) 7.00000 0.386510
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) −11.0000 −0.603703
\(333\) −2.00000 −0.109599
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) −27.0000 −1.47078 −0.735392 0.677642i \(-0.763002\pi\)
−0.735392 + 0.677642i \(0.763002\pi\)
\(338\) −12.0000 −0.652714
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 14.0000 0.758143
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −11.0000 −0.593080
\(345\) 0 0
\(346\) 24.0000 1.29025
\(347\) −2.00000 −0.107366 −0.0536828 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) −5.00000 −0.268028
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 2.00000 0.106600
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) −5.00000 −0.265747
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 3.00000 0.158777
\(358\) −10.0000 −0.528516
\(359\) 25.0000 1.31945 0.659725 0.751507i \(-0.270673\pi\)
0.659725 + 0.751507i \(0.270673\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −18.0000 −0.946059
\(363\) −7.00000 −0.367405
\(364\) −1.00000 −0.0524142
\(365\) 0 0
\(366\) −3.00000 −0.156813
\(367\) 23.0000 1.20059 0.600295 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 7.00000 0.364405
\(370\) 0 0
\(371\) −1.00000 −0.0519174
\(372\) 7.00000 0.362933
\(373\) 24.0000 1.24267 0.621336 0.783544i \(-0.286590\pi\)
0.621336 + 0.783544i \(0.286590\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 5.00000 0.257513
\(378\) 1.00000 0.0514344
\(379\) −25.0000 −1.28416 −0.642082 0.766636i \(-0.721929\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(380\) 0 0
\(381\) −22.0000 −1.12709
\(382\) −3.00000 −0.153493
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) −11.0000 −0.559161
\(388\) −2.00000 −0.101535
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 1.00000 0.0505076
\(393\) −8.00000 −0.403547
\(394\) −17.0000 −0.856448
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 13.0000 0.652451 0.326226 0.945292i \(-0.394223\pi\)
0.326226 + 0.945292i \(0.394223\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) −12.0000 −0.598506
\(403\) −7.00000 −0.348695
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) −5.00000 −0.248146
\(407\) −4.00000 −0.198273
\(408\) 3.00000 0.148522
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) −11.0000 −0.541931
\(413\) −5.00000 −0.246034
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) 25.0000 1.22133 0.610665 0.791889i \(-0.290902\pi\)
0.610665 + 0.791889i \(0.290902\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −13.0000 −0.632830
\(423\) 8.00000 0.388973
\(424\) −1.00000 −0.0485643
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) −3.00000 −0.145180
\(428\) −2.00000 −0.0966736
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 7.00000 0.336011
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) 35.0000 1.67046 0.835229 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −3.00000 −0.142695
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 9.00000 0.426162
\(447\) 15.0000 0.709476
\(448\) 1.00000 0.0472456
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 14.0000 0.659234
\(452\) −6.00000 −0.282216
\(453\) −18.0000 −0.845714
\(454\) −17.0000 −0.797850
\(455\) 0 0
\(456\) 0 0
\(457\) 33.0000 1.54367 0.771837 0.635820i \(-0.219338\pi\)
0.771837 + 0.635820i \(0.219338\pi\)
\(458\) −10.0000 −0.467269
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 2.00000 0.0930484
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −5.00000 −0.232119
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) 33.0000 1.52706 0.763529 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) −5.00000 −0.230144
\(473\) −22.0000 −1.01156
\(474\) 10.0000 0.459315
\(475\) 0 0
\(476\) 3.00000 0.137505
\(477\) −1.00000 −0.0457869
\(478\) 0 0
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 22.0000 1.00207
\(483\) −1.00000 −0.0455016
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) −3.00000 −0.135804
\(489\) −1.00000 −0.0452216
\(490\) 0 0
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) 7.00000 0.315584
\(493\) −15.0000 −0.675566
\(494\) 0 0
\(495\) 0 0
\(496\) 7.00000 0.314309
\(497\) 12.0000 0.538274
\(498\) −11.0000 −0.492922
\(499\) 25.0000 1.11915 0.559577 0.828778i \(-0.310964\pi\)
0.559577 + 0.828778i \(0.310964\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 7.00000 0.312425
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) −2.00000 −0.0889108
\(507\) −12.0000 −0.532939
\(508\) −22.0000 −0.976092
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 23.0000 1.01449
\(515\) 0 0
\(516\) −11.0000 −0.484248
\(517\) 16.0000 0.703679
\(518\) −2.00000 −0.0878750
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) −5.00000 −0.218844
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) −21.0000 −0.915644
\(527\) 21.0000 0.914774
\(528\) 2.00000 0.0870388
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) −5.00000 −0.216982
\(532\) 0 0
\(533\) −7.00000 −0.303204
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) −10.0000 −0.431532
\(538\) −10.0000 −0.431131
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) −28.0000 −1.20270
\(543\) −18.0000 −0.772454
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) −1.00000 −0.0427960
\(547\) −37.0000 −1.58201 −0.791003 0.611812i \(-0.790441\pi\)
−0.791003 + 0.611812i \(0.790441\pi\)
\(548\) −12.0000 −0.512615
\(549\) −3.00000 −0.128037
\(550\) 0 0
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) 10.0000 0.425243
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 7.00000 0.296334
\(559\) 11.0000 0.465250
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 32.0000 1.34984
\(563\) −31.0000 −1.30649 −0.653247 0.757145i \(-0.726594\pi\)
−0.653247 + 0.757145i \(0.726594\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) 1.00000 0.0419961
\(568\) 12.0000 0.503509
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 7.00000 0.292941 0.146470 0.989215i \(-0.453209\pi\)
0.146470 + 0.989215i \(0.453209\pi\)
\(572\) −2.00000 −0.0836242
\(573\) −3.00000 −0.125327
\(574\) 7.00000 0.292174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −8.00000 −0.332756
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) −11.0000 −0.456357
\(582\) −2.00000 −0.0829027
\(583\) −2.00000 −0.0828315
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) −27.0000 −1.11441 −0.557205 0.830375i \(-0.688126\pi\)
−0.557205 + 0.830375i \(0.688126\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) 0 0
\(591\) −17.0000 −0.699287
\(592\) −2.00000 −0.0821995
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) 0 0
\(598\) 1.00000 0.0408930
\(599\) 5.00000 0.204294 0.102147 0.994769i \(-0.467429\pi\)
0.102147 + 0.994769i \(0.467429\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) −11.0000 −0.448327
\(603\) −12.0000 −0.488678
\(604\) −18.0000 −0.732410
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 0 0
\(609\) −5.00000 −0.202610
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 3.00000 0.121268
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 2.00000 0.0805823
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) −11.0000 −0.442485
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −18.0000 −0.721734
\(623\) −10.0000 −0.400642
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) 10.0000 0.397779
\(633\) −13.0000 −0.516704
\(634\) 3.00000 0.119145
\(635\) 0 0
\(636\) −1.00000 −0.0396526
\(637\) −1.00000 −0.0396214
\(638\) −10.0000 −0.395904
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) −2.00000 −0.0789337
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) −22.0000 −0.864909 −0.432455 0.901656i \(-0.642352\pi\)
−0.432455 + 0.901656i \(0.642352\pi\)
\(648\) 1.00000 0.0392837
\(649\) −10.0000 −0.392534
\(650\) 0 0
\(651\) 7.00000 0.274352
\(652\) −1.00000 −0.0391630
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 7.00000 0.273304
\(657\) −6.00000 −0.234082
\(658\) 8.00000 0.311872
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) 17.0000 0.660724
\(663\) −3.00000 −0.116510
\(664\) −11.0000 −0.426883
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 5.00000 0.193601
\(668\) 18.0000 0.696441
\(669\) 9.00000 0.347960
\(670\) 0 0
\(671\) −6.00000 −0.231627
\(672\) 1.00000 0.0385758
\(673\) 29.0000 1.11787 0.558934 0.829212i \(-0.311211\pi\)
0.558934 + 0.829212i \(0.311211\pi\)
\(674\) −27.0000 −1.04000
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) −32.0000 −1.22986 −0.614930 0.788582i \(-0.710816\pi\)
−0.614930 + 0.788582i \(0.710816\pi\)
\(678\) −6.00000 −0.230429
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) −17.0000 −0.651441
\(682\) 14.0000 0.536088
\(683\) −46.0000 −1.76014 −0.880071 0.474843i \(-0.842505\pi\)
−0.880071 + 0.474843i \(0.842505\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) −10.0000 −0.381524
\(688\) −11.0000 −0.419371
\(689\) 1.00000 0.0380970
\(690\) 0 0
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) 24.0000 0.912343
\(693\) 2.00000 0.0759737
\(694\) −2.00000 −0.0759190
\(695\) 0 0
\(696\) −5.00000 −0.189525
\(697\) 21.0000 0.795432
\(698\) 15.0000 0.567758
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) 27.0000 1.01978 0.509888 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 12.0000 0.451306
\(708\) −5.00000 −0.187912
\(709\) 20.0000 0.751116 0.375558 0.926799i \(-0.377451\pi\)
0.375558 + 0.926799i \(0.377451\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) −10.0000 −0.374766
\(713\) −7.00000 −0.262152
\(714\) 3.00000 0.112272
\(715\) 0 0
\(716\) −10.0000 −0.373718
\(717\) 0 0
\(718\) 25.0000 0.932992
\(719\) −10.0000 −0.372937 −0.186469 0.982461i \(-0.559704\pi\)
−0.186469 + 0.982461i \(0.559704\pi\)
\(720\) 0 0
\(721\) −11.0000 −0.409661
\(722\) −19.0000 −0.707107
\(723\) 22.0000 0.818189
\(724\) −18.0000 −0.668965
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) 13.0000 0.482143 0.241072 0.970507i \(-0.422501\pi\)
0.241072 + 0.970507i \(0.422501\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −33.0000 −1.22055
\(732\) −3.00000 −0.110883
\(733\) −11.0000 −0.406294 −0.203147 0.979148i \(-0.565117\pi\)
−0.203147 + 0.979148i \(0.565117\pi\)
\(734\) 23.0000 0.848945
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −24.0000 −0.884051
\(738\) 7.00000 0.257674
\(739\) −25.0000 −0.919640 −0.459820 0.888012i \(-0.652086\pi\)
−0.459820 + 0.888012i \(0.652086\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.00000 −0.0367112
\(743\) 29.0000 1.06391 0.531953 0.846774i \(-0.321458\pi\)
0.531953 + 0.846774i \(0.321458\pi\)
\(744\) 7.00000 0.256632
\(745\) 0 0
\(746\) 24.0000 0.878702
\(747\) −11.0000 −0.402469
\(748\) 6.00000 0.219382
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 8.00000 0.291730
\(753\) 7.00000 0.255094
\(754\) 5.00000 0.182089
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −25.0000 −0.908041
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −22.0000 −0.796976
\(763\) 0 0
\(764\) −3.00000 −0.108536
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) 5.00000 0.180540
\(768\) 1.00000 0.0360844
\(769\) −20.0000 −0.721218 −0.360609 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(770\) 0 0
\(771\) 23.0000 0.828325
\(772\) 14.0000 0.503871
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) −11.0000 −0.395387
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) −2.00000 −0.0717496
\(778\) 10.0000 0.358517
\(779\) 0 0
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) −3.00000 −0.107280
\(783\) −5.00000 −0.178685
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) −8.00000 −0.285351
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) −17.0000 −0.605600
\(789\) −21.0000 −0.747620
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 2.00000 0.0710669
\(793\) 3.00000 0.106533
\(794\) 13.0000 0.461353
\(795\) 0 0
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) −8.00000 −0.282490
\(803\) −12.0000 −0.423471
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) −7.00000 −0.246564
\(807\) −10.0000 −0.352017
\(808\) 12.0000 0.422159
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −18.0000 −0.632065 −0.316033 0.948748i \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(812\) −5.00000 −0.175466
\(813\) −28.0000 −0.982003
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) 0 0
\(818\) −10.0000 −0.349642
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) −12.0000 −0.418548
\(823\) 54.0000 1.88232 0.941161 0.337959i \(-0.109737\pi\)
0.941161 + 0.337959i \(0.109737\pi\)
\(824\) −11.0000 −0.383203
\(825\) 0 0
\(826\) −5.00000 −0.173972
\(827\) −2.00000 −0.0695468 −0.0347734 0.999395i \(-0.511071\pi\)
−0.0347734 + 0.999395i \(0.511071\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −5.00000 −0.173657 −0.0868286 0.996223i \(-0.527673\pi\)
−0.0868286 + 0.996223i \(0.527673\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) −1.00000 −0.0346688
\(833\) 3.00000 0.103944
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) 7.00000 0.241955
\(838\) 25.0000 0.863611
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 22.0000 0.758170
\(843\) 32.0000 1.10214
\(844\) −13.0000 −0.447478
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) −7.00000 −0.240523
\(848\) −1.00000 −0.0343401
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) 12.0000 0.411113
\(853\) 29.0000 0.992941 0.496471 0.868054i \(-0.334629\pi\)
0.496471 + 0.868054i \(0.334629\pi\)
\(854\) −3.00000 −0.102658
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) −2.00000 −0.0682789
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 7.00000 0.238559
\(862\) −3.00000 −0.102180
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 14.0000 0.475739
\(867\) −8.00000 −0.271694
\(868\) 7.00000 0.237595
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 35.0000 1.18119
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) 7.00000 0.235836 0.117918 0.993023i \(-0.462378\pi\)
0.117918 + 0.993023i \(0.462378\pi\)
\(882\) 1.00000 0.0336718
\(883\) 39.0000 1.31245 0.656227 0.754563i \(-0.272151\pi\)
0.656227 + 0.754563i \(0.272151\pi\)
\(884\) −3.00000 −0.100901
\(885\) 0 0
\(886\) −16.0000 −0.537531
\(887\) 58.0000 1.94745 0.973725 0.227728i \(-0.0731298\pi\)
0.973725 + 0.227728i \(0.0731298\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −22.0000 −0.737856
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 9.00000 0.301342
\(893\) 0 0
\(894\) 15.0000 0.501675
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 1.00000 0.0333890
\(898\) 20.0000 0.667409
\(899\) −35.0000 −1.16732
\(900\) 0 0
\(901\) −3.00000 −0.0999445
\(902\) 14.0000 0.466149
\(903\) −11.0000 −0.366057
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −18.0000 −0.598010
\(907\) −17.0000 −0.564476 −0.282238 0.959344i \(-0.591077\pi\)
−0.282238 + 0.959344i \(0.591077\pi\)
\(908\) −17.0000 −0.564165
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −53.0000 −1.75597 −0.877984 0.478690i \(-0.841112\pi\)
−0.877984 + 0.478690i \(0.841112\pi\)
\(912\) 0 0
\(913\) −22.0000 −0.728094
\(914\) 33.0000 1.09154
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −8.00000 −0.264183
\(918\) 3.00000 0.0990148
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −2.00000 −0.0659022
\(922\) 12.0000 0.395199
\(923\) −12.0000 −0.394985
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) −11.0000 −0.361287
\(928\) −5.00000 −0.164133
\(929\) −45.0000 −1.47640 −0.738201 0.674581i \(-0.764324\pi\)
−0.738201 + 0.674581i \(0.764324\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 24.0000 0.786146
\(933\) −18.0000 −0.589294
\(934\) 33.0000 1.07979
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) −12.0000 −0.391814
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) −2.00000 −0.0651635
\(943\) −7.00000 −0.227951
\(944\) −5.00000 −0.162736
\(945\) 0 0
\(946\) −22.0000 −0.715282
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 10.0000 0.324785
\(949\) 6.00000 0.194768
\(950\) 0 0
\(951\) 3.00000 0.0972817
\(952\) 3.00000 0.0972306
\(953\) −16.0000 −0.518291 −0.259145 0.965838i \(-0.583441\pi\)
−0.259145 + 0.965838i \(0.583441\pi\)
\(954\) −1.00000 −0.0323762
\(955\) 0 0
\(956\) 0 0
\(957\) −10.0000 −0.323254
\(958\) −20.0000 −0.646171
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 2.00000 0.0644826
\(963\) −2.00000 −0.0644491
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) −1.00000 −0.0321745
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000 0.0320750
\(973\) 20.0000 0.641171
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −3.00000 −0.0960277
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −1.00000 −0.0319765
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) 0 0
\(982\) 2.00000 0.0638226
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 7.00000 0.223152
\(985\) 0 0
\(986\) −15.0000 −0.477697
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) 11.0000 0.349780
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 7.00000 0.222250
\(993\) 17.0000 0.539479
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) −11.0000 −0.348548
\(997\) 58.0000 1.83688 0.918439 0.395562i \(-0.129450\pi\)
0.918439 + 0.395562i \(0.129450\pi\)
\(998\) 25.0000 0.791361
\(999\) −2.00000 −0.0632772
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1050.2.a.r.1.1 yes 1
3.2 odd 2 3150.2.a.n.1.1 1
4.3 odd 2 8400.2.a.d.1.1 1
5.2 odd 4 1050.2.g.i.799.2 2
5.3 odd 4 1050.2.g.i.799.1 2
5.4 even 2 1050.2.a.b.1.1 1
7.6 odd 2 7350.2.a.cb.1.1 1
15.2 even 4 3150.2.g.h.2899.1 2
15.8 even 4 3150.2.g.h.2899.2 2
15.14 odd 2 3150.2.a.y.1.1 1
20.19 odd 2 8400.2.a.ck.1.1 1
35.34 odd 2 7350.2.a.bi.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.b.1.1 1 5.4 even 2
1050.2.a.r.1.1 yes 1 1.1 even 1 trivial
1050.2.g.i.799.1 2 5.3 odd 4
1050.2.g.i.799.2 2 5.2 odd 4
3150.2.a.n.1.1 1 3.2 odd 2
3150.2.a.y.1.1 1 15.14 odd 2
3150.2.g.h.2899.1 2 15.2 even 4
3150.2.g.h.2899.2 2 15.8 even 4
7350.2.a.bi.1.1 1 35.34 odd 2
7350.2.a.cb.1.1 1 7.6 odd 2
8400.2.a.d.1.1 1 4.3 odd 2
8400.2.a.ck.1.1 1 20.19 odd 2