Properties

Label 462.2.a.d.1.1
Level $462$
Weight $2$
Character 462.1
Self dual yes
Analytic conductor $3.689$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.68908857338\)
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 \(\chi\) \(=\) 462.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} -6.00000 q^{38} +6.00000 q^{39} -4.00000 q^{41} +1.00000 q^{42} +8.00000 q^{43} -1.00000 q^{44} +4.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +5.00000 q^{50} +4.00000 q^{51} +6.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +1.00000 q^{56} +6.00000 q^{57} -6.00000 q^{58} -2.00000 q^{61} +2.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} -4.00000 q^{67} +4.00000 q^{68} -4.00000 q^{69} +16.0000 q^{71} -1.00000 q^{72} +12.0000 q^{73} -10.0000 q^{74} -5.00000 q^{75} +6.00000 q^{76} +1.00000 q^{77} -6.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} +4.00000 q^{82} -2.00000 q^{83} -1.00000 q^{84} -8.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} -6.00000 q^{91} -4.00000 q^{92} -2.00000 q^{93} +6.00000 q^{94} -1.00000 q^{96} -6.00000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 1.00000 0.213201
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) −6.00000 −0.973329
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 1.00000 0.154303
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 5.00000 0.707107
\(51\) 4.00000 0.560112
\(52\) 6.00000 0.832050
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 6.00000 0.794719
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 2.00000 0.254000
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 4.00000 0.485071
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) −10.0000 −1.16248
\(75\) −5.00000 −0.577350
\(76\) 6.00000 0.688247
\(77\) 1.00000 0.113961
\(78\) −6.00000 −0.679366
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) −2.00000 −0.219529 −0.109764 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 6.00000 0.643268
\(88\) 1.00000 0.106600
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) −4.00000 −0.417029
\(93\) −2.00000 −0.207390
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.00000 −0.100504
\(100\) −5.00000 −0.500000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −4.00000 −0.396059
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 1.00000 0.0962250
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) −1.00000 −0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) −6.00000 −0.561951
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 6.00000 0.554700
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) −4.00000 −0.360668
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −22.0000 −1.92215 −0.961074 0.276289i \(-0.910895\pi\)
−0.961074 + 0.276289i \(0.910895\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −6.00000 −0.520266
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 22.0000 1.87959 0.939793 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) 4.00000 0.340503
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) −16.0000 −1.34269
\(143\) −6.00000 −0.501745
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) 1.00000 0.0824786
\(148\) 10.0000 0.821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 5.00000 0.408248
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −6.00000 −0.486664
\(153\) 4.00000 0.323381
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) 12.0000 0.957704 0.478852 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 16.0000 1.27289
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) −1.00000 −0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 1.00000 0.0771517
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 8.00000 0.609994
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −6.00000 −0.454859
\(175\) 5.00000 0.377964
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 6.00000 0.444750
\(183\) −2.00000 −0.147844
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 2.00000 0.146647
\(187\) −4.00000 −0.292509
\(188\) −6.00000 −0.437595
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 1.00000 0.0710669
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 5.00000 0.353553
\(201\) −4.00000 −0.282138
\(202\) 14.0000 0.985037
\(203\) −6.00000 −0.421117
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) −4.00000 −0.278019
\(208\) 6.00000 0.416025
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −10.0000 −0.686803
\(213\) 16.0000 1.09630
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 2.00000 0.135769
\(218\) 14.0000 0.948200
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) −10.0000 −0.671156
\(223\) −26.0000 −1.74109 −0.870544 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 1.00000 0.0668153
\(225\) −5.00000 −0.333333
\(226\) 2.00000 0.133038
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 6.00000 0.397360
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) −6.00000 −0.393919
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) 4.00000 0.259281
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 4.00000 0.255031
\(247\) 36.0000 2.29063
\(248\) 2.00000 0.127000
\(249\) −2.00000 −0.126745
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 4.00000 0.251478
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.0000 0.623783 0.311891 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) −8.00000 −0.498058
\(259\) −10.0000 −0.621370
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 22.0000 1.35916
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 1.00000 0.0615457
\(265\) 0 0
\(266\) 6.00000 0.367884
\(267\) −6.00000 −0.367194
\(268\) −4.00000 −0.244339
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 4.00000 0.242536
\(273\) −6.00000 −0.363137
\(274\) −22.0000 −1.32907
\(275\) 5.00000 0.301511
\(276\) −4.00000 −0.240772
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 14.0000 0.839664
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) 6.00000 0.357295
\(283\) 18.0000 1.06999 0.534994 0.844856i \(-0.320314\pi\)
0.534994 + 0.844856i \(0.320314\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 4.00000 0.236113
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 12.0000 0.702247
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) −1.00000 −0.0580259
\(298\) 6.00000 0.347571
\(299\) −24.0000 −1.38796
\(300\) −5.00000 −0.288675
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) −14.0000 −0.804279
\(304\) 6.00000 0.344124
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) −30.0000 −1.71219 −0.856095 0.516818i \(-0.827116\pi\)
−0.856095 + 0.516818i \(0.827116\pi\)
\(308\) 1.00000 0.0569803
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −6.00000 −0.339683
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −12.0000 −0.677199
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 10.0000 0.560772
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) −4.00000 −0.222911
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) −30.0000 −1.66410
\(326\) 4.00000 0.221540
\(327\) −14.0000 −0.774202
\(328\) 4.00000 0.220863
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −2.00000 −0.109764
\(333\) 10.0000 0.547997
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) −23.0000 −1.25104
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) −6.00000 −0.324443
\(343\) −1.00000 −0.0539949
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) 6.00000 0.321634
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) −5.00000 −0.267261
\(351\) 6.00000 0.320256
\(352\) 1.00000 0.0533002
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −4.00000 −0.211702
\(358\) −4.00000 −0.211407
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −8.00000 −0.420471
\(363\) 1.00000 0.0524864
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −2.00000 −0.104399 −0.0521996 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) −4.00000 −0.208514
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 10.0000 0.519174
\(372\) −2.00000 −0.103695
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 36.0000 1.85409
\(378\) 1.00000 0.0514344
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 12.0000 0.613973
\(383\) −34.0000 −1.73732 −0.868659 0.495410i \(-0.835018\pi\)
−0.868659 + 0.495410i \(0.835018\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) 8.00000 0.406663
\(388\) −6.00000 −0.304604
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) −1.00000 −0.0505076
\(393\) −22.0000 −1.10975
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) −4.00000 −0.200754 −0.100377 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) 14.0000 0.701757
\(399\) −6.00000 −0.300376
\(400\) −5.00000 −0.250000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 4.00000 0.199502
\(403\) −12.0000 −0.597763
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) −10.0000 −0.495682
\(408\) −4.00000 −0.198030
\(409\) 16.0000 0.791149 0.395575 0.918434i \(-0.370545\pi\)
0.395575 + 0.918434i \(0.370545\pi\)
\(410\) 0 0
\(411\) 22.0000 1.08518
\(412\) −14.0000 −0.689730
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) −14.0000 −0.685583
\(418\) 6.00000 0.293470
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −8.00000 −0.389434
\(423\) −6.00000 −0.291730
\(424\) 10.0000 0.485643
\(425\) −20.0000 −0.970143
\(426\) −16.0000 −0.775203
\(427\) 2.00000 0.0967868
\(428\) 8.00000 0.386695
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) 1.00000 0.0481125
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) −24.0000 −1.14808
\(438\) −12.0000 −0.573382
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −24.0000 −1.14156
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) 26.0000 1.23114
\(447\) −6.00000 −0.283790
\(448\) −1.00000 −0.0472456
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 5.00000 0.235702
\(451\) 4.00000 0.188353
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) −6.00000 −0.280976
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −8.00000 −0.373815
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) −1.00000 −0.0465242
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 6.00000 0.277350
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 12.0000 0.552931
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 16.0000 0.734904
\(475\) −30.0000 −1.37649
\(476\) −4.00000 −0.183340
\(477\) −10.0000 −0.457869
\(478\) −8.00000 −0.365911
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 60.0000 2.73576
\(482\) 20.0000 0.910975
\(483\) 4.00000 0.182006
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 2.00000 0.0905357
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −4.00000 −0.180334
\(493\) 24.0000 1.08091
\(494\) −36.0000 −1.61972
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −16.0000 −0.717698
\(498\) 2.00000 0.0896221
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) −4.00000 −0.177822
\(507\) 23.0000 1.02147
\(508\) −16.0000 −0.709885
\(509\) −16.0000 −0.709188 −0.354594 0.935020i \(-0.615381\pi\)
−0.354594 + 0.935020i \(0.615381\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) −1.00000 −0.0441942
\(513\) 6.00000 0.264906
\(514\) −10.0000 −0.441081
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 6.00000 0.263880
\(518\) 10.0000 0.439375
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 2.00000 0.0876216 0.0438108 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) −6.00000 −0.262613
\(523\) −6.00000 −0.262362 −0.131181 0.991358i \(-0.541877\pi\)
−0.131181 + 0.991358i \(0.541877\pi\)
\(524\) −22.0000 −0.961074
\(525\) 5.00000 0.218218
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) −1.00000 −0.0435194
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) −6.00000 −0.260133
\(533\) −24.0000 −1.03956
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 4.00000 0.172613
\(538\) 24.0000 1.03471
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) −12.0000 −0.515444
\(543\) 8.00000 0.343313
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) 6.00000 0.256776
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 22.0000 0.939793
\(549\) −2.00000 −0.0853579
\(550\) −5.00000 −0.213201
\(551\) 36.0000 1.53365
\(552\) 4.00000 0.170251
\(553\) 16.0000 0.680389
\(554\) 6.00000 0.254916
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 2.00000 0.0846668
\(559\) 48.0000 2.03018
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) −14.0000 −0.590554
\(563\) −22.0000 −0.927189 −0.463595 0.886047i \(-0.653441\pi\)
−0.463595 + 0.886047i \(0.653441\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) −18.0000 −0.756596
\(567\) −1.00000 −0.0419961
\(568\) −16.0000 −0.671345
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −6.00000 −0.250873
\(573\) −12.0000 −0.501307
\(574\) −4.00000 −0.166957
\(575\) 20.0000 0.834058
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 1.00000 0.0415945
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) 2.00000 0.0829740
\(582\) 6.00000 0.248708
\(583\) 10.0000 0.414158
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 1.00000 0.0412393
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 10.0000 0.410997
\(593\) 20.0000 0.821302 0.410651 0.911793i \(-0.365302\pi\)
0.410651 + 0.911793i \(0.365302\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −14.0000 −0.572982
\(598\) 24.0000 0.981433
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 5.00000 0.204124
\(601\) −32.0000 −1.30531 −0.652654 0.757656i \(-0.726344\pi\)
−0.652654 + 0.757656i \(0.726344\pi\)
\(602\) 8.00000 0.326056
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) 14.0000 0.568711
\(607\) 24.0000 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) −6.00000 −0.243332
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) −36.0000 −1.45640
\(612\) 4.00000 0.161690
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 30.0000 1.21070
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 14.0000 0.563163
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 18.0000 0.721734
\(623\) 6.00000 0.240385
\(624\) 6.00000 0.240192
\(625\) 25.0000 1.00000
\(626\) −6.00000 −0.239808
\(627\) −6.00000 −0.239617
\(628\) 12.0000 0.478852
\(629\) 40.0000 1.59490
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 16.0000 0.636446
\(633\) 8.00000 0.317971
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) 6.00000 0.237729
\(638\) 6.00000 0.237542
\(639\) 16.0000 0.632950
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) −8.00000 −0.315735
\(643\) −24.0000 −0.946468 −0.473234 0.880937i \(-0.656913\pi\)
−0.473234 + 0.880937i \(0.656913\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) −34.0000 −1.33668 −0.668339 0.743857i \(-0.732994\pi\)
−0.668339 + 0.743857i \(0.732994\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 30.0000 1.17670
\(651\) 2.00000 0.0783862
\(652\) −4.00000 −0.156652
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) −4.00000 −0.156174
\(657\) 12.0000 0.468165
\(658\) −6.00000 −0.233904
\(659\) 16.0000 0.623272 0.311636 0.950202i \(-0.399123\pi\)
0.311636 + 0.950202i \(0.399123\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 4.00000 0.155464
\(663\) 24.0000 0.932083
\(664\) 2.00000 0.0776151
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) −24.0000 −0.929284
\(668\) 12.0000 0.464294
\(669\) −26.0000 −1.00522
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 1.00000 0.0385758
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) −22.0000 −0.847408
\(675\) −5.00000 −0.192450
\(676\) 23.0000 0.884615
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 2.00000 0.0768095
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) −2.00000 −0.0765840
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 8.00000 0.305219
\(688\) 8.00000 0.304997
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 6.00000 0.228086
\(693\) 1.00000 0.0379869
\(694\) −16.0000 −0.607352
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −16.0000 −0.606043
\(698\) −26.0000 −0.984115
\(699\) 22.0000 0.832116
\(700\) 5.00000 0.188982
\(701\) −46.0000 −1.73740 −0.868698 0.495342i \(-0.835043\pi\)
−0.868698 + 0.495342i \(0.835043\pi\)
\(702\) −6.00000 −0.226455
\(703\) 60.0000 2.26294
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 14.0000 0.526524
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 6.00000 0.224860
\(713\) 8.00000 0.299602
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 8.00000 0.298765
\(718\) −8.00000 −0.298557
\(719\) −50.0000 −1.86469 −0.932343 0.361576i \(-0.882239\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(720\) 0 0
\(721\) 14.0000 0.521387
\(722\) −17.0000 −0.632674
\(723\) −20.0000 −0.743808
\(724\) 8.00000 0.297318
\(725\) −30.0000 −1.11417
\(726\) −1.00000 −0.0371135
\(727\) 22.0000 0.815935 0.407967 0.912996i \(-0.366238\pi\)
0.407967 + 0.912996i \(0.366238\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 32.0000 1.18356
\(732\) −2.00000 −0.0739221
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 2.00000 0.0738213
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 4.00000 0.147342
\(738\) 4.00000 0.147242
\(739\) −52.0000 −1.91285 −0.956425 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 0 0
\(741\) 36.0000 1.32249
\(742\) −10.0000 −0.367112
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 2.00000 0.0733236
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) −2.00000 −0.0731762
\(748\) −4.00000 −0.146254
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) −14.0000 −0.508839 −0.254419 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) 20.0000 0.726433
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 8.00000 0.290000 0.145000 0.989432i \(-0.453682\pi\)
0.145000 + 0.989432i \(0.453682\pi\)
\(762\) 16.0000 0.579619
\(763\) 14.0000 0.506834
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 34.0000 1.22847
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 16.0000 0.576975 0.288487 0.957484i \(-0.406848\pi\)
0.288487 + 0.957484i \(0.406848\pi\)
\(770\) 0 0
\(771\) 10.0000 0.360141
\(772\) −6.00000 −0.215945
\(773\) 20.0000 0.719350 0.359675 0.933078i \(-0.382888\pi\)
0.359675 + 0.933078i \(0.382888\pi\)
\(774\) −8.00000 −0.287554
\(775\) 10.0000 0.359211
\(776\) 6.00000 0.215387
\(777\) −10.0000 −0.358748
\(778\) 30.0000 1.07555
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 16.0000 0.572159
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 22.0000 0.784714
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 0 0
\(791\) 2.00000 0.0711118
\(792\) 1.00000 0.0355335
\(793\) −12.0000 −0.426132
\(794\) 4.00000 0.141955
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) −20.0000 −0.708436 −0.354218 0.935163i \(-0.615253\pi\)
−0.354218 + 0.935163i \(0.615253\pi\)
\(798\) 6.00000 0.212398
\(799\) −24.0000 −0.849059
\(800\) 5.00000 0.176777
\(801\) −6.00000 −0.212000
\(802\) −30.0000 −1.05934
\(803\) −12.0000 −0.423471
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 12.0000 0.422682
\(807\) −24.0000 −0.844840
\(808\) 14.0000 0.492518
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 18.0000 0.632065 0.316033 0.948748i \(-0.397649\pi\)
0.316033 + 0.948748i \(0.397649\pi\)
\(812\) −6.00000 −0.210559
\(813\) 12.0000 0.420858
\(814\) 10.0000 0.350500
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 48.0000 1.67931
\(818\) −16.0000 −0.559427
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) −22.0000 −0.767338
\(823\) 28.0000 0.976019 0.488009 0.872838i \(-0.337723\pi\)
0.488009 + 0.872838i \(0.337723\pi\)
\(824\) 14.0000 0.487713
\(825\) 5.00000 0.174078
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −4.00000 −0.139010
\(829\) 24.0000 0.833554 0.416777 0.909009i \(-0.363160\pi\)
0.416777 + 0.909009i \(0.363160\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) 6.00000 0.208013
\(833\) 4.00000 0.138592
\(834\) 14.0000 0.484780
\(835\) 0 0
\(836\) −6.00000 −0.207514
\(837\) −2.00000 −0.0691301
\(838\) −4.00000 −0.138178
\(839\) −42.0000 −1.45000 −0.725001 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) 14.0000 0.482186
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) −1.00000 −0.0343604
\(848\) −10.0000 −0.343401
\(849\) 18.0000 0.617758
\(850\) 20.0000 0.685994
\(851\) −40.0000 −1.37118
\(852\) 16.0000 0.548151
\(853\) −42.0000 −1.43805 −0.719026 0.694983i \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) 24.0000 0.819824 0.409912 0.912125i \(-0.365559\pi\)
0.409912 + 0.912125i \(0.365559\pi\)
\(858\) 6.00000 0.204837
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 4.00000 0.136320
\(862\) −40.0000 −1.36241
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −26.0000 −0.883516
\(867\) −1.00000 −0.0339618
\(868\) 2.00000 0.0678844
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 14.0000 0.474100
\(873\) −6.00000 −0.203069
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 20.0000 0.674967
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −50.0000 −1.68454 −0.842271 0.539054i \(-0.818782\pi\)
−0.842271 + 0.539054i \(0.818782\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) −10.0000 −0.335578
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −26.0000 −0.870544
\(893\) −36.0000 −1.20469
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) −24.0000 −0.801337
\(898\) 34.0000 1.13459
\(899\) −12.0000 −0.400222
\(900\) −5.00000 −0.166667
\(901\) −40.0000 −1.33259
\(902\) −4.00000 −0.133185
\(903\) −8.00000 −0.266223
\(904\) 2.00000 0.0665190
\(905\) 0 0
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 18.0000 0.597351
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 6.00000 0.198680
\(913\) 2.00000 0.0661903
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 8.00000 0.264327
\(917\) 22.0000 0.726504
\(918\) −4.00000 −0.132020
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −30.0000 −0.988534
\(922\) −2.00000 −0.0658665
\(923\) 96.0000 3.15988
\(924\) 1.00000 0.0328976
\(925\) −50.0000 −1.64399
\(926\) −16.0000 −0.525793
\(927\) −14.0000 −0.459820
\(928\) −6.00000 −0.196960
\(929\) 10.0000 0.328089 0.164045 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 22.0000 0.720634
\(933\) −18.0000 −0.589294
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) −12.0000 −0.392023 −0.196011 0.980602i \(-0.562799\pi\)
−0.196011 + 0.980602i \(0.562799\pi\)
\(938\) −4.00000 −0.130605
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) −12.0000 −0.390981
\(943\) 16.0000 0.521032
\(944\) 0 0
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −16.0000 −0.519656
\(949\) 72.0000 2.33722
\(950\) 30.0000 0.973329
\(951\) −18.0000 −0.583690
\(952\) 4.00000 0.129641
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) −6.00000 −0.193952
\(958\) 24.0000 0.775405
\(959\) −22.0000 −0.710417
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −60.0000 −1.93448
\(963\) 8.00000 0.257796
\(964\) −20.0000 −0.644157
\(965\) 0 0
\(966\) −4.00000 −0.128698
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 1.00000 0.0320750
\(973\) 14.0000 0.448819
\(974\) −20.0000 −0.640841
\(975\) −30.0000 −0.960769
\(976\) −2.00000 −0.0640184
\(977\) −46.0000 −1.47167 −0.735835 0.677161i \(-0.763210\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) 4.00000 0.127906
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) −12.0000 −0.382935
\(983\) 10.0000 0.318950 0.159475 0.987202i \(-0.449020\pi\)
0.159475 + 0.987202i \(0.449020\pi\)
\(984\) 4.00000 0.127515
\(985\) 0 0
\(986\) −24.0000 −0.764316
\(987\) 6.00000 0.190982
\(988\) 36.0000 1.14531
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 2.00000 0.0635001
\(993\) −4.00000 −0.126936
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) −2.00000 −0.0633724
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) −36.0000 −1.13956
\(999\) 10.0000 0.316386
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 462.2.a.d.1.1 1
3.2 odd 2 1386.2.a.j.1.1 1
4.3 odd 2 3696.2.a.j.1.1 1
7.6 odd 2 3234.2.a.b.1.1 1
11.10 odd 2 5082.2.a.ba.1.1 1
21.20 even 2 9702.2.a.bp.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.d.1.1 1 1.1 even 1 trivial
1386.2.a.j.1.1 1 3.2 odd 2
3234.2.a.b.1.1 1 7.6 odd 2
3696.2.a.j.1.1 1 4.3 odd 2
5082.2.a.ba.1.1 1 11.10 odd 2
9702.2.a.bp.1.1 1 21.20 even 2