Properties

Label 546.2.a.e.1.1
Level $546$
Weight $2$
Character 546.1
Self dual yes
Analytic conductor $4.360$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.35983195036\)
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
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} +7.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +3.00000 q^{20} +1.00000 q^{21} +1.00000 q^{22} -7.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +3.00000 q^{29} -3.00000 q^{30} +1.00000 q^{32} -1.00000 q^{33} +7.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -5.00000 q^{37} +1.00000 q^{38} +1.00000 q^{39} +3.00000 q^{40} +4.00000 q^{41} +1.00000 q^{42} +11.0000 q^{43} +1.00000 q^{44} +3.00000 q^{45} -7.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{50} -7.00000 q^{51} -1.00000 q^{52} -14.0000 q^{53} -1.00000 q^{54} +3.00000 q^{55} -1.00000 q^{56} -1.00000 q^{57} +3.00000 q^{58} +4.00000 q^{59} -3.00000 q^{60} +1.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -3.00000 q^{65} -1.00000 q^{66} -6.00000 q^{67} +7.00000 q^{68} +7.00000 q^{69} -3.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} +5.00000 q^{73} -5.00000 q^{74} -4.00000 q^{75} +1.00000 q^{76} -1.00000 q^{77} +1.00000 q^{78} -10.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} -14.0000 q^{83} +1.00000 q^{84} +21.0000 q^{85} +11.0000 q^{86} -3.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} +3.00000 q^{90} +1.00000 q^{91} -7.00000 q^{92} +3.00000 q^{95} -1.00000 q^{96} +6.00000 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) −1.00000 −0.267261
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 3.00000 0.670820
\(21\) 1.00000 0.218218
\(22\) 1.00000 0.213201
\(23\) −7.00000 −1.45960 −0.729800 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) −3.00000 −0.547723
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) 7.00000 1.20049
\(35\) −3.00000 −0.507093
\(36\) 1.00000 0.166667
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 1.00000 0.162221
\(39\) 1.00000 0.160128
\(40\) 3.00000 0.474342
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 1.00000 0.154303
\(43\) 11.0000 1.67748 0.838742 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) 1.00000 0.150756
\(45\) 3.00000 0.447214
\(46\) −7.00000 −1.03209
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) −7.00000 −0.980196
\(52\) −1.00000 −0.138675
\(53\) −14.0000 −1.92305 −0.961524 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) −1.00000 −0.136083
\(55\) 3.00000 0.404520
\(56\) −1.00000 −0.133631
\(57\) −1.00000 −0.132453
\(58\) 3.00000 0.393919
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −3.00000 −0.387298
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −3.00000 −0.372104
\(66\) −1.00000 −0.123091
\(67\) −6.00000 −0.733017 −0.366508 0.930415i \(-0.619447\pi\)
−0.366508 + 0.930415i \(0.619447\pi\)
\(68\) 7.00000 0.848875
\(69\) 7.00000 0.842701
\(70\) −3.00000 −0.358569
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 5.00000 0.585206 0.292603 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) −5.00000 −0.581238
\(75\) −4.00000 −0.461880
\(76\) 1.00000 0.114708
\(77\) −1.00000 −0.113961
\(78\) 1.00000 0.113228
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 1.00000 0.109109
\(85\) 21.0000 2.27777
\(86\) 11.0000 1.18616
\(87\) −3.00000 −0.321634
\(88\) 1.00000 0.106600
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 3.00000 0.316228
\(91\) 1.00000 0.104828
\(92\) −7.00000 −0.729800
\(93\) 0 0
\(94\) 0 0
\(95\) 3.00000 0.307794
\(96\) −1.00000 −0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.00000 0.100504
\(100\) 4.00000 0.400000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) −7.00000 −0.693103
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 3.00000 0.292770
\(106\) −14.0000 −1.35980
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 3.00000 0.286039
\(111\) 5.00000 0.474579
\(112\) −1.00000 −0.0944911
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −1.00000 −0.0936586
\(115\) −21.0000 −1.95826
\(116\) 3.00000 0.278543
\(117\) −1.00000 −0.0924500
\(118\) 4.00000 0.368230
\(119\) −7.00000 −0.641689
\(120\) −3.00000 −0.273861
\(121\) −10.0000 −0.909091
\(122\) 1.00000 0.0905357
\(123\) −4.00000 −0.360668
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) −1.00000 −0.0890871
\(127\) 10.0000 0.887357 0.443678 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(128\) 1.00000 0.0883883
\(129\) −11.0000 −0.968496
\(130\) −3.00000 −0.263117
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −1.00000 −0.0867110
\(134\) −6.00000 −0.518321
\(135\) −3.00000 −0.258199
\(136\) 7.00000 0.600245
\(137\) 15.0000 1.28154 0.640768 0.767734i \(-0.278616\pi\)
0.640768 + 0.767734i \(0.278616\pi\)
\(138\) 7.00000 0.595880
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −3.00000 −0.253546
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) −1.00000 −0.0836242
\(144\) 1.00000 0.0833333
\(145\) 9.00000 0.747409
\(146\) 5.00000 0.413803
\(147\) −1.00000 −0.0824786
\(148\) −5.00000 −0.410997
\(149\) 16.0000 1.31077 0.655386 0.755295i \(-0.272506\pi\)
0.655386 + 0.755295i \(0.272506\pi\)
\(150\) −4.00000 −0.326599
\(151\) 7.00000 0.569652 0.284826 0.958579i \(-0.408064\pi\)
0.284826 + 0.958579i \(0.408064\pi\)
\(152\) 1.00000 0.0811107
\(153\) 7.00000 0.565916
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) 5.00000 0.399043 0.199522 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) −10.0000 −0.795557
\(159\) 14.0000 1.11027
\(160\) 3.00000 0.237171
\(161\) 7.00000 0.551677
\(162\) 1.00000 0.0785674
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 4.00000 0.312348
\(165\) −3.00000 −0.233550
\(166\) −14.0000 −1.08661
\(167\) 7.00000 0.541676 0.270838 0.962625i \(-0.412699\pi\)
0.270838 + 0.962625i \(0.412699\pi\)
\(168\) 1.00000 0.0771517
\(169\) 1.00000 0.0769231
\(170\) 21.0000 1.61063
\(171\) 1.00000 0.0764719
\(172\) 11.0000 0.838742
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) −3.00000 −0.227429
\(175\) −4.00000 −0.302372
\(176\) 1.00000 0.0753778
\(177\) −4.00000 −0.300658
\(178\) −6.00000 −0.449719
\(179\) 8.00000 0.597948 0.298974 0.954261i \(-0.403356\pi\)
0.298974 + 0.954261i \(0.403356\pi\)
\(180\) 3.00000 0.223607
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 1.00000 0.0741249
\(183\) −1.00000 −0.0739221
\(184\) −7.00000 −0.516047
\(185\) −15.0000 −1.10282
\(186\) 0 0
\(187\) 7.00000 0.511891
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 3.00000 0.217643
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −8.00000 −0.575853 −0.287926 0.957653i \(-0.592966\pi\)
−0.287926 + 0.957653i \(0.592966\pi\)
\(194\) 6.00000 0.430775
\(195\) 3.00000 0.214834
\(196\) 1.00000 0.0714286
\(197\) −20.0000 −1.42494 −0.712470 0.701702i \(-0.752424\pi\)
−0.712470 + 0.701702i \(0.752424\pi\)
\(198\) 1.00000 0.0710669
\(199\) 3.00000 0.212664 0.106332 0.994331i \(-0.466089\pi\)
0.106332 + 0.994331i \(0.466089\pi\)
\(200\) 4.00000 0.282843
\(201\) 6.00000 0.423207
\(202\) −12.0000 −0.844317
\(203\) −3.00000 −0.210559
\(204\) −7.00000 −0.490098
\(205\) 12.0000 0.838116
\(206\) −7.00000 −0.487713
\(207\) −7.00000 −0.486534
\(208\) −1.00000 −0.0693375
\(209\) 1.00000 0.0691714
\(210\) 3.00000 0.207020
\(211\) 25.0000 1.72107 0.860535 0.509390i \(-0.170129\pi\)
0.860535 + 0.509390i \(0.170129\pi\)
\(212\) −14.0000 −0.961524
\(213\) 12.0000 0.822226
\(214\) −18.0000 −1.23045
\(215\) 33.0000 2.25058
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −7.00000 −0.474100
\(219\) −5.00000 −0.337869
\(220\) 3.00000 0.202260
\(221\) −7.00000 −0.470871
\(222\) 5.00000 0.335578
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 4.00000 0.266667
\(226\) −14.0000 −0.931266
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) −21.0000 −1.38470
\(231\) 1.00000 0.0657952
\(232\) 3.00000 0.196960
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 10.0000 0.649570
\(238\) −7.00000 −0.453743
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) −3.00000 −0.193649
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −10.0000 −0.642824
\(243\) −1.00000 −0.0641500
\(244\) 1.00000 0.0640184
\(245\) 3.00000 0.191663
\(246\) −4.00000 −0.255031
\(247\) −1.00000 −0.0636285
\(248\) 0 0
\(249\) 14.0000 0.887214
\(250\) −3.00000 −0.189737
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −7.00000 −0.440086
\(254\) 10.0000 0.627456
\(255\) −21.0000 −1.31507
\(256\) 1.00000 0.0625000
\(257\) 10.0000 0.623783 0.311891 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) −11.0000 −0.684830
\(259\) 5.00000 0.310685
\(260\) −3.00000 −0.186052
\(261\) 3.00000 0.185695
\(262\) 15.0000 0.926703
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −42.0000 −2.58004
\(266\) −1.00000 −0.0613139
\(267\) 6.00000 0.367194
\(268\) −6.00000 −0.366508
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) −3.00000 −0.182574
\(271\) −30.0000 −1.82237 −0.911185 0.411997i \(-0.864831\pi\)
−0.911185 + 0.411997i \(0.864831\pi\)
\(272\) 7.00000 0.424437
\(273\) −1.00000 −0.0605228
\(274\) 15.0000 0.906183
\(275\) 4.00000 0.241209
\(276\) 7.00000 0.421350
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) −12.0000 −0.712069
\(285\) −3.00000 −0.177705
\(286\) −1.00000 −0.0591312
\(287\) −4.00000 −0.236113
\(288\) 1.00000 0.0589256
\(289\) 32.0000 1.88235
\(290\) 9.00000 0.528498
\(291\) −6.00000 −0.351726
\(292\) 5.00000 0.292603
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 12.0000 0.698667
\(296\) −5.00000 −0.290619
\(297\) −1.00000 −0.0580259
\(298\) 16.0000 0.926855
\(299\) 7.00000 0.404820
\(300\) −4.00000 −0.230940
\(301\) −11.0000 −0.634029
\(302\) 7.00000 0.402805
\(303\) 12.0000 0.689382
\(304\) 1.00000 0.0573539
\(305\) 3.00000 0.171780
\(306\) 7.00000 0.400163
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 7.00000 0.398216
\(310\) 0 0
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) 1.00000 0.0566139
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 5.00000 0.282166
\(315\) −3.00000 −0.169031
\(316\) −10.0000 −0.562544
\(317\) −28.0000 −1.57264 −0.786318 0.617822i \(-0.788015\pi\)
−0.786318 + 0.617822i \(0.788015\pi\)
\(318\) 14.0000 0.785081
\(319\) 3.00000 0.167968
\(320\) 3.00000 0.167705
\(321\) 18.0000 1.00466
\(322\) 7.00000 0.390095
\(323\) 7.00000 0.389490
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 10.0000 0.553849
\(327\) 7.00000 0.387101
\(328\) 4.00000 0.220863
\(329\) 0 0
\(330\) −3.00000 −0.165145
\(331\) −14.0000 −0.769510 −0.384755 0.923019i \(-0.625714\pi\)
−0.384755 + 0.923019i \(0.625714\pi\)
\(332\) −14.0000 −0.768350
\(333\) −5.00000 −0.273998
\(334\) 7.00000 0.383023
\(335\) −18.0000 −0.983445
\(336\) 1.00000 0.0545545
\(337\) 3.00000 0.163420 0.0817102 0.996656i \(-0.473962\pi\)
0.0817102 + 0.996656i \(0.473962\pi\)
\(338\) 1.00000 0.0543928
\(339\) 14.0000 0.760376
\(340\) 21.0000 1.13888
\(341\) 0 0
\(342\) 1.00000 0.0540738
\(343\) −1.00000 −0.0539949
\(344\) 11.0000 0.593080
\(345\) 21.0000 1.13060
\(346\) 2.00000 0.107521
\(347\) 32.0000 1.71785 0.858925 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(348\) −3.00000 −0.160817
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) −4.00000 −0.213809
\(351\) 1.00000 0.0533761
\(352\) 1.00000 0.0533002
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) −4.00000 −0.212598
\(355\) −36.0000 −1.91068
\(356\) −6.00000 −0.317999
\(357\) 7.00000 0.370479
\(358\) 8.00000 0.422813
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 3.00000 0.158114
\(361\) −18.0000 −0.947368
\(362\) 22.0000 1.15629
\(363\) 10.0000 0.524864
\(364\) 1.00000 0.0524142
\(365\) 15.0000 0.785136
\(366\) −1.00000 −0.0522708
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) −7.00000 −0.364900
\(369\) 4.00000 0.208232
\(370\) −15.0000 −0.779813
\(371\) 14.0000 0.726844
\(372\) 0 0
\(373\) 12.0000 0.621336 0.310668 0.950518i \(-0.399447\pi\)
0.310668 + 0.950518i \(0.399447\pi\)
\(374\) 7.00000 0.361961
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) −3.00000 −0.154508
\(378\) 1.00000 0.0514344
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 3.00000 0.153897
\(381\) −10.0000 −0.512316
\(382\) −15.0000 −0.767467
\(383\) −1.00000 −0.0510976 −0.0255488 0.999674i \(-0.508133\pi\)
−0.0255488 + 0.999674i \(0.508133\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −3.00000 −0.152894
\(386\) −8.00000 −0.407189
\(387\) 11.0000 0.559161
\(388\) 6.00000 0.304604
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) 3.00000 0.151911
\(391\) −49.0000 −2.47804
\(392\) 1.00000 0.0505076
\(393\) −15.0000 −0.756650
\(394\) −20.0000 −1.00759
\(395\) −30.0000 −1.50946
\(396\) 1.00000 0.0502519
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 3.00000 0.150376
\(399\) 1.00000 0.0500626
\(400\) 4.00000 0.200000
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 6.00000 0.299253
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) 3.00000 0.149071
\(406\) −3.00000 −0.148888
\(407\) −5.00000 −0.247841
\(408\) −7.00000 −0.346552
\(409\) 23.0000 1.13728 0.568638 0.822588i \(-0.307470\pi\)
0.568638 + 0.822588i \(0.307470\pi\)
\(410\) 12.0000 0.592638
\(411\) −15.0000 −0.739895
\(412\) −7.00000 −0.344865
\(413\) −4.00000 −0.196827
\(414\) −7.00000 −0.344031
\(415\) −42.0000 −2.06170
\(416\) −1.00000 −0.0490290
\(417\) −4.00000 −0.195881
\(418\) 1.00000 0.0489116
\(419\) 29.0000 1.41674 0.708371 0.705840i \(-0.249430\pi\)
0.708371 + 0.705840i \(0.249430\pi\)
\(420\) 3.00000 0.146385
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 25.0000 1.21698
\(423\) 0 0
\(424\) −14.0000 −0.679900
\(425\) 28.0000 1.35820
\(426\) 12.0000 0.581402
\(427\) −1.00000 −0.0483934
\(428\) −18.0000 −0.870063
\(429\) 1.00000 0.0482805
\(430\) 33.0000 1.59140
\(431\) −10.0000 −0.481683 −0.240842 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −8.00000 −0.384455 −0.192228 0.981350i \(-0.561571\pi\)
−0.192228 + 0.981350i \(0.561571\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) −7.00000 −0.335239
\(437\) −7.00000 −0.334855
\(438\) −5.00000 −0.238909
\(439\) −19.0000 −0.906821 −0.453410 0.891302i \(-0.649793\pi\)
−0.453410 + 0.891302i \(0.649793\pi\)
\(440\) 3.00000 0.143019
\(441\) 1.00000 0.0476190
\(442\) −7.00000 −0.332956
\(443\) 26.0000 1.23530 0.617649 0.786454i \(-0.288085\pi\)
0.617649 + 0.786454i \(0.288085\pi\)
\(444\) 5.00000 0.237289
\(445\) −18.0000 −0.853282
\(446\) 14.0000 0.662919
\(447\) −16.0000 −0.756774
\(448\) −1.00000 −0.0472456
\(449\) 19.0000 0.896665 0.448333 0.893867i \(-0.352018\pi\)
0.448333 + 0.893867i \(0.352018\pi\)
\(450\) 4.00000 0.188562
\(451\) 4.00000 0.188353
\(452\) −14.0000 −0.658505
\(453\) −7.00000 −0.328889
\(454\) −18.0000 −0.844782
\(455\) 3.00000 0.140642
\(456\) −1.00000 −0.0468293
\(457\) −40.0000 −1.87112 −0.935561 0.353166i \(-0.885105\pi\)
−0.935561 + 0.353166i \(0.885105\pi\)
\(458\) 18.0000 0.841085
\(459\) −7.00000 −0.326732
\(460\) −21.0000 −0.979130
\(461\) −23.0000 −1.07122 −0.535608 0.844466i \(-0.679918\pi\)
−0.535608 + 0.844466i \(0.679918\pi\)
\(462\) 1.00000 0.0465242
\(463\) 23.0000 1.06890 0.534450 0.845200i \(-0.320519\pi\)
0.534450 + 0.845200i \(0.320519\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) 3.00000 0.138823 0.0694117 0.997588i \(-0.477888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 6.00000 0.277054
\(470\) 0 0
\(471\) −5.00000 −0.230388
\(472\) 4.00000 0.184115
\(473\) 11.0000 0.505781
\(474\) 10.0000 0.459315
\(475\) 4.00000 0.183533
\(476\) −7.00000 −0.320844
\(477\) −14.0000 −0.641016
\(478\) 20.0000 0.914779
\(479\) −11.0000 −0.502603 −0.251301 0.967909i \(-0.580859\pi\)
−0.251301 + 0.967909i \(0.580859\pi\)
\(480\) −3.00000 −0.136931
\(481\) 5.00000 0.227980
\(482\) −18.0000 −0.819878
\(483\) −7.00000 −0.318511
\(484\) −10.0000 −0.454545
\(485\) 18.0000 0.817338
\(486\) −1.00000 −0.0453609
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 1.00000 0.0452679
\(489\) −10.0000 −0.452216
\(490\) 3.00000 0.135526
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) −4.00000 −0.180334
\(493\) 21.0000 0.945792
\(494\) −1.00000 −0.0449921
\(495\) 3.00000 0.134840
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 14.0000 0.627355
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) −3.00000 −0.134164
\(501\) −7.00000 −0.312737
\(502\) 3.00000 0.133897
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −36.0000 −1.60198
\(506\) −7.00000 −0.311188
\(507\) −1.00000 −0.0444116
\(508\) 10.0000 0.443678
\(509\) −39.0000 −1.72864 −0.864322 0.502938i \(-0.832252\pi\)
−0.864322 + 0.502938i \(0.832252\pi\)
\(510\) −21.0000 −0.929896
\(511\) −5.00000 −0.221187
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 10.0000 0.441081
\(515\) −21.0000 −0.925371
\(516\) −11.0000 −0.484248
\(517\) 0 0
\(518\) 5.00000 0.219687
\(519\) −2.00000 −0.0877903
\(520\) −3.00000 −0.131559
\(521\) −11.0000 −0.481919 −0.240959 0.970535i \(-0.577462\pi\)
−0.240959 + 0.970535i \(0.577462\pi\)
\(522\) 3.00000 0.131306
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 15.0000 0.655278
\(525\) 4.00000 0.174574
\(526\) −4.00000 −0.174408
\(527\) 0 0
\(528\) −1.00000 −0.0435194
\(529\) 26.0000 1.13043
\(530\) −42.0000 −1.82436
\(531\) 4.00000 0.173585
\(532\) −1.00000 −0.0433555
\(533\) −4.00000 −0.173259
\(534\) 6.00000 0.259645
\(535\) −54.0000 −2.33462
\(536\) −6.00000 −0.259161
\(537\) −8.00000 −0.345225
\(538\) 2.00000 0.0862261
\(539\) 1.00000 0.0430730
\(540\) −3.00000 −0.129099
\(541\) 25.0000 1.07483 0.537417 0.843317i \(-0.319400\pi\)
0.537417 + 0.843317i \(0.319400\pi\)
\(542\) −30.0000 −1.28861
\(543\) −22.0000 −0.944110
\(544\) 7.00000 0.300123
\(545\) −21.0000 −0.899541
\(546\) −1.00000 −0.0427960
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 15.0000 0.640768
\(549\) 1.00000 0.0426790
\(550\) 4.00000 0.170561
\(551\) 3.00000 0.127804
\(552\) 7.00000 0.297940
\(553\) 10.0000 0.425243
\(554\) 10.0000 0.424859
\(555\) 15.0000 0.636715
\(556\) 4.00000 0.169638
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) −11.0000 −0.465250
\(560\) −3.00000 −0.126773
\(561\) −7.00000 −0.295540
\(562\) 6.00000 0.253095
\(563\) 39.0000 1.64365 0.821827 0.569737i \(-0.192955\pi\)
0.821827 + 0.569737i \(0.192955\pi\)
\(564\) 0 0
\(565\) −42.0000 −1.76695
\(566\) −16.0000 −0.672530
\(567\) −1.00000 −0.0419961
\(568\) −12.0000 −0.503509
\(569\) 14.0000 0.586911 0.293455 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(570\) −3.00000 −0.125656
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) −1.00000 −0.0418121
\(573\) 15.0000 0.626634
\(574\) −4.00000 −0.166957
\(575\) −28.0000 −1.16768
\(576\) 1.00000 0.0416667
\(577\) −26.0000 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) 32.0000 1.33102
\(579\) 8.00000 0.332469
\(580\) 9.00000 0.373705
\(581\) 14.0000 0.580818
\(582\) −6.00000 −0.248708
\(583\) −14.0000 −0.579821
\(584\) 5.00000 0.206901
\(585\) −3.00000 −0.124035
\(586\) 30.0000 1.23929
\(587\) 42.0000 1.73353 0.866763 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) 12.0000 0.494032
\(591\) 20.0000 0.822690
\(592\) −5.00000 −0.205499
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) −1.00000 −0.0410305
\(595\) −21.0000 −0.860916
\(596\) 16.0000 0.655386
\(597\) −3.00000 −0.122782
\(598\) 7.00000 0.286251
\(599\) −1.00000 −0.0408589 −0.0204294 0.999791i \(-0.506503\pi\)
−0.0204294 + 0.999791i \(0.506503\pi\)
\(600\) −4.00000 −0.163299
\(601\) 16.0000 0.652654 0.326327 0.945257i \(-0.394189\pi\)
0.326327 + 0.945257i \(0.394189\pi\)
\(602\) −11.0000 −0.448327
\(603\) −6.00000 −0.244339
\(604\) 7.00000 0.284826
\(605\) −30.0000 −1.21967
\(606\) 12.0000 0.487467
\(607\) 13.0000 0.527654 0.263827 0.964570i \(-0.415015\pi\)
0.263827 + 0.964570i \(0.415015\pi\)
\(608\) 1.00000 0.0405554
\(609\) 3.00000 0.121566
\(610\) 3.00000 0.121466
\(611\) 0 0
\(612\) 7.00000 0.282958
\(613\) −7.00000 −0.282727 −0.141364 0.989958i \(-0.545149\pi\)
−0.141364 + 0.989958i \(0.545149\pi\)
\(614\) −32.0000 −1.29141
\(615\) −12.0000 −0.483887
\(616\) −1.00000 −0.0402911
\(617\) −27.0000 −1.08698 −0.543490 0.839416i \(-0.682897\pi\)
−0.543490 + 0.839416i \(0.682897\pi\)
\(618\) 7.00000 0.281581
\(619\) 1.00000 0.0401934 0.0200967 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) 0 0
\(621\) 7.00000 0.280900
\(622\) 2.00000 0.0801927
\(623\) 6.00000 0.240385
\(624\) 1.00000 0.0400320
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) −1.00000 −0.0399362
\(628\) 5.00000 0.199522
\(629\) −35.0000 −1.39554
\(630\) −3.00000 −0.119523
\(631\) −37.0000 −1.47295 −0.736473 0.676467i \(-0.763510\pi\)
−0.736473 + 0.676467i \(0.763510\pi\)
\(632\) −10.0000 −0.397779
\(633\) −25.0000 −0.993661
\(634\) −28.0000 −1.11202
\(635\) 30.0000 1.19051
\(636\) 14.0000 0.555136
\(637\) −1.00000 −0.0396214
\(638\) 3.00000 0.118771
\(639\) −12.0000 −0.474713
\(640\) 3.00000 0.118585
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 18.0000 0.710403
\(643\) −9.00000 −0.354925 −0.177463 0.984128i \(-0.556789\pi\)
−0.177463 + 0.984128i \(0.556789\pi\)
\(644\) 7.00000 0.275839
\(645\) −33.0000 −1.29937
\(646\) 7.00000 0.275411
\(647\) −36.0000 −1.41531 −0.707653 0.706560i \(-0.750246\pi\)
−0.707653 + 0.706560i \(0.750246\pi\)
\(648\) 1.00000 0.0392837
\(649\) 4.00000 0.157014
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) −43.0000 −1.68272 −0.841360 0.540475i \(-0.818245\pi\)
−0.841360 + 0.540475i \(0.818245\pi\)
\(654\) 7.00000 0.273722
\(655\) 45.0000 1.75830
\(656\) 4.00000 0.156174
\(657\) 5.00000 0.195069
\(658\) 0 0
\(659\) −18.0000 −0.701180 −0.350590 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) −3.00000 −0.116775
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) −14.0000 −0.544125
\(663\) 7.00000 0.271857
\(664\) −14.0000 −0.543305
\(665\) −3.00000 −0.116335
\(666\) −5.00000 −0.193746
\(667\) −21.0000 −0.813123
\(668\) 7.00000 0.270838
\(669\) −14.0000 −0.541271
\(670\) −18.0000 −0.695401
\(671\) 1.00000 0.0386046
\(672\) 1.00000 0.0385758
\(673\) 7.00000 0.269830 0.134915 0.990857i \(-0.456924\pi\)
0.134915 + 0.990857i \(0.456924\pi\)
\(674\) 3.00000 0.115556
\(675\) −4.00000 −0.153960
\(676\) 1.00000 0.0384615
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 14.0000 0.537667
\(679\) −6.00000 −0.230259
\(680\) 21.0000 0.805313
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) −13.0000 −0.497431 −0.248716 0.968577i \(-0.580008\pi\)
−0.248716 + 0.968577i \(0.580008\pi\)
\(684\) 1.00000 0.0382360
\(685\) 45.0000 1.71936
\(686\) −1.00000 −0.0381802
\(687\) −18.0000 −0.686743
\(688\) 11.0000 0.419371
\(689\) 14.0000 0.533358
\(690\) 21.0000 0.799456
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 2.00000 0.0760286
\(693\) −1.00000 −0.0379869
\(694\) 32.0000 1.21470
\(695\) 12.0000 0.455186
\(696\) −3.00000 −0.113715
\(697\) 28.0000 1.06058
\(698\) 2.00000 0.0757011
\(699\) 12.0000 0.453882
\(700\) −4.00000 −0.151186
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 1.00000 0.0377426
\(703\) −5.00000 −0.188579
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −10.0000 −0.376355
\(707\) 12.0000 0.451306
\(708\) −4.00000 −0.150329
\(709\) 50.0000 1.87779 0.938895 0.344204i \(-0.111851\pi\)
0.938895 + 0.344204i \(0.111851\pi\)
\(710\) −36.0000 −1.35106
\(711\) −10.0000 −0.375029
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 7.00000 0.261968
\(715\) −3.00000 −0.112194
\(716\) 8.00000 0.298974
\(717\) −20.0000 −0.746914
\(718\) 12.0000 0.447836
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 3.00000 0.111803
\(721\) 7.00000 0.260694
\(722\) −18.0000 −0.669891
\(723\) 18.0000 0.669427
\(724\) 22.0000 0.817624
\(725\) 12.0000 0.445669
\(726\) 10.0000 0.371135
\(727\) 37.0000 1.37225 0.686127 0.727482i \(-0.259309\pi\)
0.686127 + 0.727482i \(0.259309\pi\)
\(728\) 1.00000 0.0370625
\(729\) 1.00000 0.0370370
\(730\) 15.0000 0.555175
\(731\) 77.0000 2.84795
\(732\) −1.00000 −0.0369611
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 32.0000 1.18114
\(735\) −3.00000 −0.110657
\(736\) −7.00000 −0.258023
\(737\) −6.00000 −0.221013
\(738\) 4.00000 0.147242
\(739\) 38.0000 1.39785 0.698926 0.715194i \(-0.253662\pi\)
0.698926 + 0.715194i \(0.253662\pi\)
\(740\) −15.0000 −0.551411
\(741\) 1.00000 0.0367359
\(742\) 14.0000 0.513956
\(743\) −26.0000 −0.953847 −0.476924 0.878945i \(-0.658248\pi\)
−0.476924 + 0.878945i \(0.658248\pi\)
\(744\) 0 0
\(745\) 48.0000 1.75858
\(746\) 12.0000 0.439351
\(747\) −14.0000 −0.512233
\(748\) 7.00000 0.255945
\(749\) 18.0000 0.657706
\(750\) 3.00000 0.109545
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) −3.00000 −0.109326
\(754\) −3.00000 −0.109254
\(755\) 21.0000 0.764268
\(756\) 1.00000 0.0363696
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 20.0000 0.726433
\(759\) 7.00000 0.254084
\(760\) 3.00000 0.108821
\(761\) −32.0000 −1.16000 −0.580000 0.814617i \(-0.696947\pi\)
−0.580000 + 0.814617i \(0.696947\pi\)
\(762\) −10.0000 −0.362262
\(763\) 7.00000 0.253417
\(764\) −15.0000 −0.542681
\(765\) 21.0000 0.759257
\(766\) −1.00000 −0.0361315
\(767\) −4.00000 −0.144432
\(768\) −1.00000 −0.0360844
\(769\) −49.0000 −1.76699 −0.883493 0.468445i \(-0.844814\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) −3.00000 −0.108112
\(771\) −10.0000 −0.360141
\(772\) −8.00000 −0.287926
\(773\) −7.00000 −0.251773 −0.125886 0.992045i \(-0.540177\pi\)
−0.125886 + 0.992045i \(0.540177\pi\)
\(774\) 11.0000 0.395387
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) −5.00000 −0.179374
\(778\) 34.0000 1.21896
\(779\) 4.00000 0.143315
\(780\) 3.00000 0.107417
\(781\) −12.0000 −0.429394
\(782\) −49.0000 −1.75224
\(783\) −3.00000 −0.107211
\(784\) 1.00000 0.0357143
\(785\) 15.0000 0.535373
\(786\) −15.0000 −0.535032
\(787\) −31.0000 −1.10503 −0.552515 0.833503i \(-0.686332\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(788\) −20.0000 −0.712470
\(789\) 4.00000 0.142404
\(790\) −30.0000 −1.06735
\(791\) 14.0000 0.497783
\(792\) 1.00000 0.0355335
\(793\) −1.00000 −0.0355110
\(794\) −14.0000 −0.496841
\(795\) 42.0000 1.48959
\(796\) 3.00000 0.106332
\(797\) 36.0000 1.27519 0.637593 0.770374i \(-0.279930\pi\)
0.637593 + 0.770374i \(0.279930\pi\)
\(798\) 1.00000 0.0353996
\(799\) 0 0
\(800\) 4.00000 0.141421
\(801\) −6.00000 −0.212000
\(802\) −2.00000 −0.0706225
\(803\) 5.00000 0.176446
\(804\) 6.00000 0.211604
\(805\) 21.0000 0.740153
\(806\) 0 0
\(807\) −2.00000 −0.0704033
\(808\) −12.0000 −0.422159
\(809\) −50.0000 −1.75791 −0.878953 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) 3.00000 0.105409
\(811\) 21.0000 0.737410 0.368705 0.929547i \(-0.379801\pi\)
0.368705 + 0.929547i \(0.379801\pi\)
\(812\) −3.00000 −0.105279
\(813\) 30.0000 1.05215
\(814\) −5.00000 −0.175250
\(815\) 30.0000 1.05085
\(816\) −7.00000 −0.245049
\(817\) 11.0000 0.384841
\(818\) 23.0000 0.804176
\(819\) 1.00000 0.0349428
\(820\) 12.0000 0.419058
\(821\) −32.0000 −1.11681 −0.558404 0.829569i \(-0.688586\pi\)
−0.558404 + 0.829569i \(0.688586\pi\)
\(822\) −15.0000 −0.523185
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −7.00000 −0.243857
\(825\) −4.00000 −0.139262
\(826\) −4.00000 −0.139178
\(827\) 9.00000 0.312961 0.156480 0.987681i \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(828\) −7.00000 −0.243267
\(829\) 49.0000 1.70184 0.850920 0.525295i \(-0.176045\pi\)
0.850920 + 0.525295i \(0.176045\pi\)
\(830\) −42.0000 −1.45784
\(831\) −10.0000 −0.346896
\(832\) −1.00000 −0.0346688
\(833\) 7.00000 0.242536
\(834\) −4.00000 −0.138509
\(835\) 21.0000 0.726735
\(836\) 1.00000 0.0345857
\(837\) 0 0
\(838\) 29.0000 1.00179
\(839\) −4.00000 −0.138095 −0.0690477 0.997613i \(-0.521996\pi\)
−0.0690477 + 0.997613i \(0.521996\pi\)
\(840\) 3.00000 0.103510
\(841\) −20.0000 −0.689655
\(842\) 26.0000 0.896019
\(843\) −6.00000 −0.206651
\(844\) 25.0000 0.860535
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) −14.0000 −0.480762
\(849\) 16.0000 0.549119
\(850\) 28.0000 0.960392
\(851\) 35.0000 1.19978
\(852\) 12.0000 0.411113
\(853\) 44.0000 1.50653 0.753266 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) −1.00000 −0.0342193
\(855\) 3.00000 0.102598
\(856\) −18.0000 −0.615227
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 1.00000 0.0341394
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 33.0000 1.12529
\(861\) 4.00000 0.136320
\(862\) −10.0000 −0.340601
\(863\) −42.0000 −1.42970 −0.714848 0.699280i \(-0.753504\pi\)
−0.714848 + 0.699280i \(0.753504\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) −8.00000 −0.271851
\(867\) −32.0000 −1.08678
\(868\) 0 0
\(869\) −10.0000 −0.339227
\(870\) −9.00000 −0.305129
\(871\) 6.00000 0.203302
\(872\) −7.00000 −0.237050
\(873\) 6.00000 0.203069
\(874\) −7.00000 −0.236779
\(875\) 3.00000 0.101419
\(876\) −5.00000 −0.168934
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) −19.0000 −0.641219
\(879\) −30.0000 −1.01187
\(880\) 3.00000 0.101130
\(881\) −17.0000 −0.572745 −0.286372 0.958118i \(-0.592449\pi\)
−0.286372 + 0.958118i \(0.592449\pi\)
\(882\) 1.00000 0.0336718
\(883\) 35.0000 1.17784 0.588922 0.808190i \(-0.299553\pi\)
0.588922 + 0.808190i \(0.299553\pi\)
\(884\) −7.00000 −0.235435
\(885\) −12.0000 −0.403376
\(886\) 26.0000 0.873487
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 5.00000 0.167789
\(889\) −10.0000 −0.335389
\(890\) −18.0000 −0.603361
\(891\) 1.00000 0.0335013
\(892\) 14.0000 0.468755
\(893\) 0 0
\(894\) −16.0000 −0.535120
\(895\) 24.0000 0.802232
\(896\) −1.00000 −0.0334077
\(897\) −7.00000 −0.233723
\(898\) 19.0000 0.634038
\(899\) 0 0
\(900\) 4.00000 0.133333
\(901\) −98.0000 −3.26485
\(902\) 4.00000 0.133185
\(903\) 11.0000 0.366057
\(904\) −14.0000 −0.465633
\(905\) 66.0000 2.19391
\(906\) −7.00000 −0.232559
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −18.0000 −0.597351
\(909\) −12.0000 −0.398015
\(910\) 3.00000 0.0994490
\(911\) 15.0000 0.496972 0.248486 0.968635i \(-0.420067\pi\)
0.248486 + 0.968635i \(0.420067\pi\)
\(912\) −1.00000 −0.0331133
\(913\) −14.0000 −0.463332
\(914\) −40.0000 −1.32308
\(915\) −3.00000 −0.0991769
\(916\) 18.0000 0.594737
\(917\) −15.0000 −0.495344
\(918\) −7.00000 −0.231034
\(919\) −50.0000 −1.64935 −0.824674 0.565608i \(-0.808641\pi\)
−0.824674 + 0.565608i \(0.808641\pi\)
\(920\) −21.0000 −0.692349
\(921\) 32.0000 1.05444
\(922\) −23.0000 −0.757465
\(923\) 12.0000 0.394985
\(924\) 1.00000 0.0328976
\(925\) −20.0000 −0.657596
\(926\) 23.0000 0.755827
\(927\) −7.00000 −0.229910
\(928\) 3.00000 0.0984798
\(929\) 48.0000 1.57483 0.787414 0.616424i \(-0.211419\pi\)
0.787414 + 0.616424i \(0.211419\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) −12.0000 −0.393073
\(933\) −2.00000 −0.0654771
\(934\) 3.00000 0.0981630
\(935\) 21.0000 0.686773
\(936\) −1.00000 −0.0326860
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 6.00000 0.195907
\(939\) 0 0
\(940\) 0 0
\(941\) 34.0000 1.10837 0.554184 0.832394i \(-0.313030\pi\)
0.554184 + 0.832394i \(0.313030\pi\)
\(942\) −5.00000 −0.162909
\(943\) −28.0000 −0.911805
\(944\) 4.00000 0.130189
\(945\) 3.00000 0.0975900
\(946\) 11.0000 0.357641
\(947\) −11.0000 −0.357452 −0.178726 0.983899i \(-0.557198\pi\)
−0.178726 + 0.983899i \(0.557198\pi\)
\(948\) 10.0000 0.324785
\(949\) −5.00000 −0.162307
\(950\) 4.00000 0.129777
\(951\) 28.0000 0.907962
\(952\) −7.00000 −0.226871
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) −14.0000 −0.453267
\(955\) −45.0000 −1.45617
\(956\) 20.0000 0.646846
\(957\) −3.00000 −0.0969762
\(958\) −11.0000 −0.355394
\(959\) −15.0000 −0.484375
\(960\) −3.00000 −0.0968246
\(961\) −31.0000 −1.00000
\(962\) 5.00000 0.161206
\(963\) −18.0000 −0.580042
\(964\) −18.0000 −0.579741
\(965\) −24.0000 −0.772587
\(966\) −7.00000 −0.225221
\(967\) 13.0000 0.418052 0.209026 0.977910i \(-0.432971\pi\)
0.209026 + 0.977910i \(0.432971\pi\)
\(968\) −10.0000 −0.321412
\(969\) −7.00000 −0.224872
\(970\) 18.0000 0.577945
\(971\) 48.0000 1.54039 0.770197 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −4.00000 −0.128234
\(974\) −32.0000 −1.02535
\(975\) 4.00000 0.128103
\(976\) 1.00000 0.0320092
\(977\) −7.00000 −0.223950 −0.111975 0.993711i \(-0.535718\pi\)
−0.111975 + 0.993711i \(0.535718\pi\)
\(978\) −10.0000 −0.319765
\(979\) −6.00000 −0.191761
\(980\) 3.00000 0.0958315
\(981\) −7.00000 −0.223493
\(982\) 2.00000 0.0638226
\(983\) −19.0000 −0.606006 −0.303003 0.952990i \(-0.597989\pi\)
−0.303003 + 0.952990i \(0.597989\pi\)
\(984\) −4.00000 −0.127515
\(985\) −60.0000 −1.91176
\(986\) 21.0000 0.668776
\(987\) 0 0
\(988\) −1.00000 −0.0318142
\(989\) −77.0000 −2.44846
\(990\) 3.00000 0.0953463
\(991\) 34.0000 1.08005 0.540023 0.841650i \(-0.318416\pi\)
0.540023 + 0.841650i \(0.318416\pi\)
\(992\) 0 0
\(993\) 14.0000 0.444277
\(994\) 12.0000 0.380617
\(995\) 9.00000 0.285319
\(996\) 14.0000 0.443607
\(997\) 54.0000 1.71020 0.855099 0.518465i \(-0.173497\pi\)
0.855099 + 0.518465i \(0.173497\pi\)
\(998\) 16.0000 0.506471
\(999\) 5.00000 0.158193
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 546.2.a.e.1.1 1
3.2 odd 2 1638.2.a.a.1.1 1
4.3 odd 2 4368.2.a.z.1.1 1
7.6 odd 2 3822.2.a.bc.1.1 1
13.12 even 2 7098.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.e.1.1 1 1.1 even 1 trivial
1638.2.a.a.1.1 1 3.2 odd 2
3822.2.a.bc.1.1 1 7.6 odd 2
4368.2.a.z.1.1 1 4.3 odd 2
7098.2.a.b.1.1 1 13.12 even 2