Properties

Label 3450.2.a.i.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(1\)
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
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{21} +3.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -9.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +1.00000 q^{38} -1.00000 q^{39} +9.00000 q^{41} +1.00000 q^{42} -1.00000 q^{43} -3.00000 q^{44} +1.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +6.00000 q^{51} -1.00000 q^{52} +12.0000 q^{53} -1.00000 q^{54} +1.00000 q^{56} -1.00000 q^{57} +9.00000 q^{58} -6.00000 q^{59} -4.00000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} -1.00000 q^{69} -6.00000 q^{71} -1.00000 q^{72} -7.00000 q^{73} +4.00000 q^{74} -1.00000 q^{76} +3.00000 q^{77} +1.00000 q^{78} -1.00000 q^{79} +1.00000 q^{81} -9.00000 q^{82} -15.0000 q^{83} -1.00000 q^{84} +1.00000 q^{86} -9.00000 q^{87} +3.00000 q^{88} -18.0000 q^{89} +1.00000 q^{91} -1.00000 q^{92} -4.00000 q^{93} -6.00000 q^{94} -1.00000 q^{96} -4.00000 q^{97} +6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 1.00000 0.288675
\(13\) −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
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 3.00000 0.639602
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 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
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 1.00000 0.162221
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 1.00000 0.154303
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) −1.00000 −0.138675
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −1.00000 −0.132453
\(58\) 9.00000 1.18176
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 4.00000 0.508001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −1.00000 −0.117851
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 3.00000 0.341882
\(78\) 1.00000 0.113228
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −9.00000 −0.993884
\(83\) −15.0000 −1.64646 −0.823232 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) −9.00000 −0.964901
\(88\) 3.00000 0.319801
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) −1.00000 −0.104257
\(93\) −4.00000 −0.414781
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 6.00000 0.606092
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −6.00000 −0.594089
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) −1.00000 −0.0944911
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −9.00000 −0.835629
\(117\) −1.00000 −0.0924500
\(118\) 6.00000 0.552345
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 4.00000 0.362143
\(123\) 9.00000 0.811503
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 14.0000 1.24230 0.621150 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −3.00000 −0.261116
\(133\) 1.00000 0.0867110
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 1.00000 0.0851257
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 6.00000 0.503509
\(143\) 3.00000 0.250873
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 7.00000 0.579324
\(147\) −6.00000 −0.494872
\(148\) −4.00000 −0.328798
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 1.00000 0.0811107
\(153\) 6.00000 0.485071
\(154\) −3.00000 −0.241747
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) 1.00000 0.0795557
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 9.00000 0.702782
\(165\) 0 0
\(166\) 15.0000 1.16423
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 1.00000 0.0771517
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) −1.00000 −0.0762493
\(173\) 15.0000 1.14043 0.570214 0.821496i \(-0.306860\pi\)
0.570214 + 0.821496i \(0.306860\pi\)
\(174\) 9.00000 0.682288
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) −6.00000 −0.450988
\(178\) 18.0000 1.34916
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −1.00000 −0.0741249
\(183\) −4.00000 −0.295689
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) −18.0000 −1.31629
\(188\) 6.00000 0.437595
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 15.0000 1.08536 0.542681 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 4.00000 0.287183
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −9.00000 −0.641223 −0.320612 0.947211i \(-0.603888\pi\)
−0.320612 + 0.947211i \(0.603888\pi\)
\(198\) 3.00000 0.213201
\(199\) −13.0000 −0.921546 −0.460773 0.887518i \(-0.652428\pi\)
−0.460773 + 0.887518i \(0.652428\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) −6.00000 −0.422159
\(203\) 9.00000 0.631676
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 7.00000 0.487713
\(207\) −1.00000 −0.0695048
\(208\) −1.00000 −0.0693375
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 12.0000 0.824163
\(213\) −6.00000 −0.411113
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 4.00000 0.271538
\(218\) 4.00000 0.270914
\(219\) −7.00000 −0.473016
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 4.00000 0.268462
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 9.00000 0.590879
\(233\) 9.00000 0.589610 0.294805 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) −1.00000 −0.0649570
\(238\) 6.00000 0.388922
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) −9.00000 −0.573819
\(247\) 1.00000 0.0636285
\(248\) 4.00000 0.254000
\(249\) −15.0000 −0.950586
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 3.00000 0.188608
\(254\) −14.0000 −0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 1.00000 0.0622573
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) −1.00000 −0.0613139
\(267\) −18.0000 −1.10158
\(268\) −4.00000 −0.244339
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 6.00000 0.363803
\(273\) 1.00000 0.0605228
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −19.0000 −1.14160 −0.570800 0.821089i \(-0.693367\pi\)
−0.570800 + 0.821089i \(0.693367\pi\)
\(278\) 10.0000 0.599760
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) −6.00000 −0.357295
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) −3.00000 −0.177394
\(287\) −9.00000 −0.531253
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) −7.00000 −0.409644
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) −3.00000 −0.174078
\(298\) 0 0
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 10.0000 0.575435
\(303\) 6.00000 0.344691
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 3.00000 0.170941
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 1.00000 0.0566139
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −8.00000 −0.451466
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) −12.0000 −0.672927
\(319\) 27.0000 1.51171
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) −1.00000 −0.0557278
\(323\) −6.00000 −0.333849
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) −4.00000 −0.221201
\(328\) −9.00000 −0.496942
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −15.0000 −0.823232
\(333\) −4.00000 −0.219199
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 20.0000 1.08947 0.544735 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 12.0000 0.652714
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 12.0000 0.649836
\(342\) 1.00000 0.0540738
\(343\) 13.0000 0.701934
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) −15.0000 −0.806405
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −9.00000 −0.482451
\(349\) 5.00000 0.267644 0.133822 0.991005i \(-0.457275\pi\)
0.133822 + 0.991005i \(0.457275\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 3.00000 0.159901
\(353\) −9.00000 −0.479022 −0.239511 0.970894i \(-0.576987\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) −6.00000 −0.317554
\(358\) 18.0000 0.951330
\(359\) −3.00000 −0.158334 −0.0791670 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −14.0000 −0.735824
\(363\) −2.00000 −0.104973
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) −37.0000 −1.93138 −0.965692 0.259690i \(-0.916380\pi\)
−0.965692 + 0.259690i \(0.916380\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 9.00000 0.468521
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) −4.00000 −0.207390
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) 9.00000 0.463524
\(378\) 1.00000 0.0514344
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 14.0000 0.717242
\(382\) −15.0000 −0.767467
\(383\) −27.0000 −1.37964 −0.689818 0.723983i \(-0.742309\pi\)
−0.689818 + 0.723983i \(0.742309\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −1.00000 −0.0508329
\(388\) −4.00000 −0.203069
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 6.00000 0.303046
\(393\) 0 0
\(394\) 9.00000 0.453413
\(395\) 0 0
\(396\) −3.00000 −0.150756
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 13.0000 0.651631
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 4.00000 0.199502
\(403\) 4.00000 0.199254
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) −9.00000 −0.446663
\(407\) 12.0000 0.594818
\(408\) −6.00000 −0.297044
\(409\) −7.00000 −0.346128 −0.173064 0.984911i \(-0.555367\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) −7.00000 −0.344865
\(413\) 6.00000 0.295241
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) −10.0000 −0.489702
\(418\) −3.00000 −0.146735
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 22.0000 1.07094
\(423\) 6.00000 0.291730
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 6.00000 0.290701
\(427\) 4.00000 0.193574
\(428\) −12.0000 −0.580042
\(429\) 3.00000 0.144841
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000 0.0481125
\(433\) −28.0000 −1.34559 −0.672797 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 1.00000 0.0478365
\(438\) 7.00000 0.334473
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 6.00000 0.285391
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) 10.0000 0.473514
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −27.0000 −1.27138
\(452\) 18.0000 0.846649
\(453\) −10.0000 −0.469841
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 4.00000 0.186908
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −21.0000 −0.978068 −0.489034 0.872265i \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) −3.00000 −0.139573
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −9.00000 −0.417815
\(465\) 0 0
\(466\) −9.00000 −0.416917
\(467\) −21.0000 −0.971764 −0.485882 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 8.00000 0.368621
\(472\) 6.00000 0.276172
\(473\) 3.00000 0.137940
\(474\) 1.00000 0.0459315
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) 12.0000 0.549442
\(478\) −12.0000 −0.548867
\(479\) 3.00000 0.137073 0.0685367 0.997649i \(-0.478167\pi\)
0.0685367 + 0.997649i \(0.478167\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 10.0000 0.455488
\(483\) 1.00000 0.0455016
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 14.0000 0.634401 0.317200 0.948359i \(-0.397257\pi\)
0.317200 + 0.948359i \(0.397257\pi\)
\(488\) 4.00000 0.181071
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 9.00000 0.405751
\(493\) −54.0000 −2.43204
\(494\) −1.00000 −0.0449921
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 6.00000 0.269137
\(498\) 15.0000 0.672166
\(499\) 26.0000 1.16392 0.581960 0.813217i \(-0.302286\pi\)
0.581960 + 0.813217i \(0.302286\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) −12.0000 −0.535586
\(503\) −9.00000 −0.401290 −0.200645 0.979664i \(-0.564304\pi\)
−0.200645 + 0.979664i \(0.564304\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) −3.00000 −0.133366
\(507\) −12.0000 −0.532939
\(508\) 14.0000 0.621150
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −1.00000 −0.0440225
\(517\) −18.0000 −0.791639
\(518\) −4.00000 −0.175750
\(519\) 15.0000 0.658427
\(520\) 0 0
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 9.00000 0.393919
\(523\) −31.0000 −1.35554 −0.677768 0.735276i \(-0.737052\pi\)
−0.677768 + 0.735276i \(0.737052\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) −24.0000 −1.04546
\(528\) −3.00000 −0.130558
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 1.00000 0.0433555
\(533\) −9.00000 −0.389833
\(534\) 18.0000 0.778936
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) −18.0000 −0.776757
\(538\) 9.00000 0.388018
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 29.0000 1.24681 0.623404 0.781900i \(-0.285749\pi\)
0.623404 + 0.781900i \(0.285749\pi\)
\(542\) −8.00000 −0.343629
\(543\) 14.0000 0.600798
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) −1.00000 −0.0427960
\(547\) 26.0000 1.11168 0.555840 0.831289i \(-0.312397\pi\)
0.555840 + 0.831289i \(0.312397\pi\)
\(548\) −12.0000 −0.512615
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) 9.00000 0.383413
\(552\) 1.00000 0.0425628
\(553\) 1.00000 0.0425243
\(554\) 19.0000 0.807233
\(555\) 0 0
\(556\) −10.0000 −0.424094
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 4.00000 0.169334
\(559\) 1.00000 0.0422955
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) 6.00000 0.253095
\(563\) 21.0000 0.885044 0.442522 0.896758i \(-0.354084\pi\)
0.442522 + 0.896758i \(0.354084\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) −1.00000 −0.0419961
\(568\) 6.00000 0.251754
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 3.00000 0.125436
\(573\) 15.0000 0.626634
\(574\) 9.00000 0.375653
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −31.0000 −1.29055 −0.645273 0.763952i \(-0.723257\pi\)
−0.645273 + 0.763952i \(0.723257\pi\)
\(578\) −19.0000 −0.790296
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) 15.0000 0.622305
\(582\) 4.00000 0.165805
\(583\) −36.0000 −1.49097
\(584\) 7.00000 0.289662
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) 42.0000 1.73353 0.866763 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) −6.00000 −0.247436
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) −9.00000 −0.370211
\(592\) −4.00000 −0.164399
\(593\) 21.0000 0.862367 0.431183 0.902264i \(-0.358096\pi\)
0.431183 + 0.902264i \(0.358096\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) 0 0
\(597\) −13.0000 −0.532055
\(598\) −1.00000 −0.0408930
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −46.0000 −1.87638 −0.938190 0.346122i \(-0.887498\pi\)
−0.938190 + 0.346122i \(0.887498\pi\)
\(602\) −1.00000 −0.0407570
\(603\) −4.00000 −0.162893
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) −34.0000 −1.38002 −0.690009 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) 1.00000 0.0405554
\(609\) 9.00000 0.364698
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 6.00000 0.242536
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 7.00000 0.281581
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −12.0000 −0.481156
\(623\) 18.0000 0.721155
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) 3.00000 0.119808
\(628\) 8.00000 0.319235
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 17.0000 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) 1.00000 0.0397779
\(633\) −22.0000 −0.874421
\(634\) −3.00000 −0.119145
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) 6.00000 0.237729
\(638\) −27.0000 −1.06894
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 36.0000 1.42191 0.710957 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) 12.0000 0.473602
\(643\) −49.0000 −1.93237 −0.966186 0.257847i \(-0.916987\pi\)
−0.966186 + 0.257847i \(0.916987\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 18.0000 0.706562
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) −16.0000 −0.626608
\(653\) −39.0000 −1.52619 −0.763094 0.646288i \(-0.776321\pi\)
−0.763094 + 0.646288i \(0.776321\pi\)
\(654\) 4.00000 0.156412
\(655\) 0 0
\(656\) 9.00000 0.351391
\(657\) −7.00000 −0.273096
\(658\) 6.00000 0.233904
\(659\) −45.0000 −1.75295 −0.876476 0.481446i \(-0.840112\pi\)
−0.876476 + 0.481446i \(0.840112\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) −8.00000 −0.310929
\(663\) −6.00000 −0.233021
\(664\) 15.0000 0.582113
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 9.00000 0.348481
\(668\) 24.0000 0.928588
\(669\) −10.0000 −0.386622
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 1.00000 0.0385758
\(673\) 35.0000 1.34915 0.674575 0.738206i \(-0.264327\pi\)
0.674575 + 0.738206i \(0.264327\pi\)
\(674\) −20.0000 −0.770371
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) −18.0000 −0.691286
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) −24.0000 −0.919682
\(682\) −12.0000 −0.459504
\(683\) 18.0000 0.688751 0.344375 0.938832i \(-0.388091\pi\)
0.344375 + 0.938832i \(0.388091\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) −4.00000 −0.152610
\(688\) −1.00000 −0.0381246
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 50.0000 1.90209 0.951045 0.309053i \(-0.100012\pi\)
0.951045 + 0.309053i \(0.100012\pi\)
\(692\) 15.0000 0.570214
\(693\) 3.00000 0.113961
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 9.00000 0.341144
\(697\) 54.0000 2.04540
\(698\) −5.00000 −0.189253
\(699\) 9.00000 0.340411
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 1.00000 0.0377426
\(703\) 4.00000 0.150863
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 9.00000 0.338719
\(707\) −6.00000 −0.225653
\(708\) −6.00000 −0.225494
\(709\) 8.00000 0.300446 0.150223 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) 18.0000 0.674579
\(713\) 4.00000 0.149801
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) −18.0000 −0.672692
\(717\) 12.0000 0.448148
\(718\) 3.00000 0.111959
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) 7.00000 0.260694
\(722\) 18.0000 0.669891
\(723\) −10.0000 −0.371904
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.00000 −0.221918
\(732\) −4.00000 −0.147844
\(733\) 44.0000 1.62518 0.812589 0.582838i \(-0.198058\pi\)
0.812589 + 0.582838i \(0.198058\pi\)
\(734\) 37.0000 1.36569
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 12.0000 0.442026
\(738\) −9.00000 −0.331295
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 1.00000 0.0367359
\(742\) 12.0000 0.440534
\(743\) 9.00000 0.330178 0.165089 0.986279i \(-0.447209\pi\)
0.165089 + 0.986279i \(0.447209\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) −15.0000 −0.548821
\(748\) −18.0000 −0.658145
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −31.0000 −1.13121 −0.565603 0.824678i \(-0.691357\pi\)
−0.565603 + 0.824678i \(0.691357\pi\)
\(752\) 6.00000 0.218797
\(753\) 12.0000 0.437304
\(754\) −9.00000 −0.327761
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) −4.00000 −0.145382 −0.0726912 0.997354i \(-0.523159\pi\)
−0.0726912 + 0.997354i \(0.523159\pi\)
\(758\) −20.0000 −0.726433
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) −14.0000 −0.507166
\(763\) 4.00000 0.144810
\(764\) 15.0000 0.542681
\(765\) 0 0
\(766\) 27.0000 0.975550
\(767\) 6.00000 0.216647
\(768\) 1.00000 0.0360844
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 2.00000 0.0719816
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) 1.00000 0.0359443
\(775\) 0 0
\(776\) 4.00000 0.143592
\(777\) 4.00000 0.143499
\(778\) 30.0000 1.07555
\(779\) −9.00000 −0.322458
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 6.00000 0.214560
\(783\) −9.00000 −0.321634
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) 17.0000 0.605985 0.302992 0.952993i \(-0.402014\pi\)
0.302992 + 0.952993i \(0.402014\pi\)
\(788\) −9.00000 −0.320612
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 3.00000 0.106600
\(793\) 4.00000 0.142044
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −13.0000 −0.460773
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) −1.00000 −0.0353996
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) −12.0000 −0.423735
\(803\) 21.0000 0.741074
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) −9.00000 −0.316815
\(808\) −6.00000 −0.211079
\(809\) 51.0000 1.79306 0.896532 0.442978i \(-0.146078\pi\)
0.896532 + 0.442978i \(0.146078\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 9.00000 0.315838
\(813\) 8.00000 0.280572
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 1.00000 0.0349856
\(818\) 7.00000 0.244749
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) 45.0000 1.57051 0.785255 0.619172i \(-0.212532\pi\)
0.785255 + 0.619172i \(0.212532\pi\)
\(822\) 12.0000 0.418548
\(823\) 2.00000 0.0697156 0.0348578 0.999392i \(-0.488902\pi\)
0.0348578 + 0.999392i \(0.488902\pi\)
\(824\) 7.00000 0.243857
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 27.0000 0.938882 0.469441 0.882964i \(-0.344455\pi\)
0.469441 + 0.882964i \(0.344455\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 29.0000 1.00721 0.503606 0.863934i \(-0.332006\pi\)
0.503606 + 0.863934i \(0.332006\pi\)
\(830\) 0 0
\(831\) −19.0000 −0.659103
\(832\) −1.00000 −0.0346688
\(833\) −36.0000 −1.24733
\(834\) 10.0000 0.346272
\(835\) 0 0
\(836\) 3.00000 0.103757
\(837\) −4.00000 −0.138260
\(838\) 9.00000 0.310900
\(839\) 3.00000 0.103572 0.0517858 0.998658i \(-0.483509\pi\)
0.0517858 + 0.998658i \(0.483509\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −14.0000 −0.482472
\(843\) −6.00000 −0.206651
\(844\) −22.0000 −0.757271
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) 2.00000 0.0687208
\(848\) 12.0000 0.412082
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) −6.00000 −0.205557
\(853\) −37.0000 −1.26686 −0.633428 0.773802i \(-0.718353\pi\)
−0.633428 + 0.773802i \(0.718353\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) −3.00000 −0.102418
\(859\) 38.0000 1.29654 0.648272 0.761409i \(-0.275492\pi\)
0.648272 + 0.761409i \(0.275492\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) −24.0000 −0.817443
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 28.0000 0.951479
\(867\) 19.0000 0.645274
\(868\) 4.00000 0.135769
\(869\) 3.00000 0.101768
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 4.00000 0.135457
\(873\) −4.00000 −0.135379
\(874\) −1.00000 −0.0338255
\(875\) 0 0
\(876\) −7.00000 −0.236508
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −20.0000 −0.674967
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 6.00000 0.202031
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 4.00000 0.134231
\(889\) −14.0000 −0.469545
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) −10.0000 −0.334825
\(893\) −6.00000 −0.200782
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 1.00000 0.0333890
\(898\) 18.0000 0.600668
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) 72.0000 2.39867
\(902\) 27.0000 0.899002
\(903\) 1.00000 0.0332779
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) 10.0000 0.332228
\(907\) −19.0000 −0.630885 −0.315442 0.948945i \(-0.602153\pi\)
−0.315442 + 0.948945i \(0.602153\pi\)
\(908\) −24.0000 −0.796468
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −27.0000 −0.894550 −0.447275 0.894397i \(-0.647605\pi\)
−0.447275 + 0.894397i \(0.647605\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 45.0000 1.48928
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) 0 0
\(918\) −6.00000 −0.198030
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 21.0000 0.691598
\(923\) 6.00000 0.197492
\(924\) 3.00000 0.0986928
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) −7.00000 −0.229910
\(928\) 9.00000 0.295439
\(929\) −27.0000 −0.885841 −0.442921 0.896561i \(-0.646058\pi\)
−0.442921 + 0.896561i \(0.646058\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 9.00000 0.294805
\(933\) 12.0000 0.392862
\(934\) 21.0000 0.687141
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) −16.0000 −0.522697 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) −4.00000 −0.130605
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) −60.0000 −1.95594 −0.977972 0.208736i \(-0.933065\pi\)
−0.977972 + 0.208736i \(0.933065\pi\)
\(942\) −8.00000 −0.260654
\(943\) −9.00000 −0.293080
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) −3.00000 −0.0975384
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −1.00000 −0.0324785
\(949\) 7.00000 0.227230
\(950\) 0 0
\(951\) 3.00000 0.0972817
\(952\) 6.00000 0.194461
\(953\) 12.0000 0.388718 0.194359 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 27.0000 0.872786
\(958\) −3.00000 −0.0969256
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −4.00000 −0.128965
\(963\) −12.0000 −0.386695
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) −1.00000 −0.0321745
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 2.00000 0.0642824
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) −9.00000 −0.288824 −0.144412 0.989518i \(-0.546129\pi\)
−0.144412 + 0.989518i \(0.546129\pi\)
\(972\) 1.00000 0.0320750
\(973\) 10.0000 0.320585
\(974\) −14.0000 −0.448589
\(975\) 0 0
\(976\) −4.00000 −0.128037
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 16.0000 0.511624
\(979\) 54.0000 1.72585
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) −45.0000 −1.43528 −0.717639 0.696416i \(-0.754777\pi\)
−0.717639 + 0.696416i \(0.754777\pi\)
\(984\) −9.00000 −0.286910
\(985\) 0 0
\(986\) 54.0000 1.71971
\(987\) −6.00000 −0.190982
\(988\) 1.00000 0.0318142
\(989\) 1.00000 0.0317982
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 4.00000 0.127000
\(993\) 8.00000 0.253872
\(994\) −6.00000 −0.190308
\(995\) 0 0
\(996\) −15.0000 −0.475293
\(997\) 53.0000 1.67853 0.839263 0.543725i \(-0.182987\pi\)
0.839263 + 0.543725i \(0.182987\pi\)
\(998\) −26.0000 −0.823016
\(999\) −4.00000 −0.126554
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 3450.2.a.i.1.1 1
5.2 odd 4 3450.2.d.c.2899.1 2
5.3 odd 4 3450.2.d.c.2899.2 2
5.4 even 2 3450.2.a.r.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.i.1.1 1 1.1 even 1 trivial
3450.2.a.r.1.1 yes 1 5.4 even 2
3450.2.d.c.2899.1 2 5.2 odd 4
3450.2.d.c.2899.2 2 5.3 odd 4