Properties

Label 3234.2.a.b.1.1
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3234.1

$q$-expansion

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

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
\(7\) 0 0
\(8\) −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 \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) −1.00000 −0.235702
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 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
\(28\) 0 0
\(29\) 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 \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −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 \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 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 \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 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
\(56\) 0 0
\(57\) 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 \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 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 \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) −10.0000 −1.16248
\(75\) 5.00000 0.577350
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) −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 \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 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 \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −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 \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −5.00000 −0.500000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −4.00000 −0.396059
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 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
\(112\) 0 0
\(113\) −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
\(119\) 0 0
\(120\) 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
\(126\) 0 0
\(127\) −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 \(-0.0891049\pi\)
0.961074 + 0.276289i \(0.0891049\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 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 \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 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
\(147\) 0 0
\(148\) 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
\(154\) 0 0
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) −12.0000 −0.957704 −0.478852 0.877896i \(-0.658947\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 16.0000 1.27289
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) −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 \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 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 \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −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 \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 0 0
\(183\) −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
\(189\) 0 0
\(190\) 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
\(196\) 0 0
\(197\) 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 \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 5.00000 0.353553
\(201\) 4.00000 0.282138
\(202\) −14.0000 −0.985037
\(203\) 0 0
\(204\) 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
\(217\) 0 0
\(218\) 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 \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 2.00000 0.133038
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 6.00000 0.397360
\(229\) −8.00000 −0.528655 −0.264327 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −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
\(238\) 0 0
\(239\) 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 \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) −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 \(0.5\pi\)
\(252\) 0 0
\(253\) 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 \(-0.600963\pi\)
−0.311891 + 0.950118i \(0.600963\pi\)
\(258\) 8.00000 0.498058
\(259\) 0 0
\(260\) 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
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) −4.00000 −0.244339
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) −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 \(-0.679686\pi\)
−0.534994 + 0.844856i \(0.679686\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 0 0
\(288\) −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 \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 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
\(301\) 0 0
\(302\) 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 \(-0.172884\pi\)
0.856095 + 0.516818i \(0.172884\pi\)
\(308\) 0 0
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) −6.00000 −0.339683
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 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
\(322\) 0 0
\(323\) 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
\(329\) 0 0
\(330\) 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
\(336\) 0 0
\(337\) 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
\(343\) 0 0
\(344\) −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 \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 1.00000 0.0533002
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −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
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 2.00000 0.104399 0.0521996 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) −4.00000 −0.208514
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) 0 0
\(372\) −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
\(378\) 0 0
\(379\) −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 \(-0.164982\pi\)
0.868659 + 0.495410i \(0.164982\pi\)
\(384\) 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
\(392\) 0 0
\(393\) −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 \(-0.467995\pi\)
0.100377 + 0.994949i \(0.467995\pi\)
\(398\) −14.0000 −0.701757
\(399\) 0 0
\(400\) −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
\(406\) 0 0
\(407\) −10.0000 −0.495682
\(408\) −4.00000 −0.198030
\(409\) −16.0000 −0.791149 −0.395575 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 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 \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 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
\(427\) 0 0
\(428\) 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 \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 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 \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −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
\(448\) 0 0
\(449\) −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 \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 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 \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) −6.00000 −0.277350
\(469\) 0 0
\(470\) 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
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) −8.00000 −0.365911
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −60.0000 −2.73576
\(482\) −20.0000 −0.910975
\(483\) 0 0
\(484\) 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
\(497\) 0 0
\(498\) 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 \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 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 \(-0.384619\pi\)
0.354594 + 0.935020i \(0.384619\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −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
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) −6.00000 −0.262613
\(523\) 6.00000 0.262362 0.131181 0.991358i \(-0.458123\pi\)
0.131181 + 0.991358i \(0.458123\pi\)
\(524\) 22.0000 0.961074
\(525\) 0 0
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) 1.00000 0.0435194
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −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
\(539\) 0 0
\(540\) 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
\(546\) 0 0
\(547\) 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
\(553\) 0 0
\(554\) 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 \(-0.346559\pi\)
0.463595 + 0.886047i \(0.346559\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) 18.0000 0.756596
\(567\) 0 0
\(568\) −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
\(574\) 0 0
\(575\) 20.0000 0.834058
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 1.00000 0.0415945
\(579\) 6.00000 0.249351
\(580\) 0 0
\(581\) 0 0
\(582\) 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 \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) −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 \(-0.634698\pi\)
−0.410651 + 0.911793i \(0.634698\pi\)
\(594\) −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 \(-0.273656\pi\)
0.652654 + 0.757656i \(0.273656\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) 14.0000 0.568711
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) 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
\(616\) 0 0
\(617\) −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 \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −18.0000 −0.721734
\(623\) 0 0
\(624\) 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
\(637\) 0 0
\(638\) 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 \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 34.0000 1.33668 0.668339 0.743857i \(-0.267006\pi\)
0.668339 + 0.743857i \(0.267006\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −30.0000 −1.17670
\(651\) 0 0
\(652\) −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
\(658\) 0 0
\(659\) 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 \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 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
\(672\) 0 0
\(673\) −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 \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) −2.00000 −0.0768095
\(679\) 0 0
\(680\) 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
\(686\) 0 0
\(687\) 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
\(693\) 0 0
\(694\) −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
\(700\) 0 0
\(701\) −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
\(707\) 0 0
\(708\) 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
\(714\) 0 0
\(715\) 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 \(-0.117761\pi\)
0.932343 + 0.361576i \(0.117761\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −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 \(-0.633762\pi\)
−0.407967 + 0.912996i \(0.633762\pi\)
\(728\) 0 0
\(729\) 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 \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) −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
\(742\) 0 0
\(743\) 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
\(749\) 0 0
\(750\) 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
\(756\) 0 0
\(757\) −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 \(-0.546318\pi\)
−0.145000 + 0.989432i \(0.546318\pi\)
\(762\) −16.0000 −0.579619
\(763\) 0 0
\(764\) −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 \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 10.0000 0.360141
\(772\) −6.00000 −0.215945
\(773\) −20.0000 −0.719350 −0.359675 0.933078i \(-0.617112\pi\)
−0.359675 + 0.933078i \(0.617112\pi\)
\(774\) −8.00000 −0.287554
\(775\) −10.0000 −0.359211
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) 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
\(784\) 0 0
\(785\) 0 0
\(786\) 22.0000 0.784714
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 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 \(-0.384747\pi\)
0.354218 + 0.935163i \(0.384747\pi\)
\(798\) 0 0
\(799\) −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 \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(812\) 0 0
\(813\) 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
\(819\) 0 0
\(820\) 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 \(-0.636840\pi\)
−0.416777 + 0.909009i \(0.636840\pi\)
\(830\) 0 0
\(831\) 6.00000 0.208138
\(832\) −6.00000 −0.208013
\(833\) 0 0
\(834\) 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 \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 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
\(847\) 0 0
\(848\) −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 \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) −24.0000 −0.819824 −0.409912 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 6.00000 0.204837
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −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
\(868\) 0 0
\(869\) 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 \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) 0 0
\(883\) −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 \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 10.0000 0.335578
\(889\) 0 0
\(890\) 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
\(896\) 0 0
\(897\) −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
\(903\) 0 0
\(904\) 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
\(917\) 0 0
\(918\) −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
\(924\) 0 0
\(925\) −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 \(-0.552454\pi\)
−0.164045 + 0.986453i \(0.552454\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 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 \(-0.437201\pi\)
0.196011 + 0.980602i \(0.437201\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −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
\(952\) 0 0
\(953\) 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
\(959\) 0 0
\(960\) 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
\(966\) 0 0
\(967\) 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 \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −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 \(-0.550980\pi\)
−0.159475 + 0.987202i \(0.550980\pi\)
\(984\) 4.00000 0.127515
\(985\) 0 0
\(986\) 24.0000 0.764316
\(987\) 0 0
\(988\) 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
\(994\) 0 0
\(995\) 0 0
\(996\) −2.00000 −0.0633724
\(997\) 46.0000 1.45683 0.728417 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) −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 3234.2.a.b.1.1 1
3.2 odd 2 9702.2.a.bp.1.1 1
7.6 odd 2 462.2.a.d.1.1 1
21.20 even 2 1386.2.a.j.1.1 1
28.27 even 2 3696.2.a.j.1.1 1
77.76 even 2 5082.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.d.1.1 1 7.6 odd 2
1386.2.a.j.1.1 1 21.20 even 2
3234.2.a.b.1.1 1 1.1 even 1 trivial
3696.2.a.j.1.1 1 28.27 even 2
5082.2.a.ba.1.1 1 77.76 even 2
9702.2.a.bp.1.1 1 3.2 odd 2