Properties

Label 690.2.a.e.1.1
Level $690$
Weight $2$
Character 690.1
Self dual yes
Analytic conductor $5.510$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.50967773947\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 690.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -6.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} +4.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} -6.00000 q^{52} -14.0000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +4.00000 q^{57} +6.00000 q^{58} -1.00000 q^{60} +10.0000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} -6.00000 q^{68} +1.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +6.00000 q^{78} -12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -16.0000 q^{83} +6.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} +4.00000 q^{88} -2.00000 q^{89} +1.00000 q^{90} +1.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} -14.0000 q^{97} +7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 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
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 1.00000 0.182574
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) −6.00000 −0.960769
\(40\) 1.00000 0.158114
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) −1.00000 −0.149071
\(46\) −1.00000 −0.147442
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) −6.00000 −0.832050
\(53\) −14.0000 −1.92305 −0.961524 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 6.00000 0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 4.00000 0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −6.00000 −0.727607
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 6.00000 0.679366
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) −4.00000 −0.431331
\(87\) −6.00000 −0.643268
\(88\) 4.00000 0.426401
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −8.00000 −0.829561
\(94\) 8.00000 0.825137
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 7.00000 0.707107
\(99\) −4.00000 −0.402015
\(100\) 1.00000 0.100000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 6.00000 0.594089
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 14.0000 1.35980
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −4.00000 −0.381385
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −4.00000 −0.374634
\(115\) −1.00000 −0.0932505
\(116\) −6.00000 −0.557086
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 10.0000 0.901670
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) −6.00000 −0.526235
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) −1.00000 −0.0860663
\(136\) 6.00000 0.514496
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) −8.00000 −0.671345
\(143\) 24.0000 2.00698
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) −2.00000 −0.165521
\(147\) −7.00000 −0.577350
\(148\) 6.00000 0.493197
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −4.00000 −0.324443
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) −6.00000 −0.480384
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 12.0000 0.954669
\(159\) −14.0000 −1.11027
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 10.0000 0.780869
\(165\) 4.00000 0.311400
\(166\) 16.0000 1.24184
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −6.00000 −0.460179
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) −1.00000 −0.0737210
\(185\) −6.00000 −0.441129
\(186\) 8.00000 0.586588
\(187\) 24.0000 1.75505
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 14.0000 1.00514
\(195\) 6.00000 0.429669
\(196\) −7.00000 −0.500000
\(197\) 26.0000 1.85242 0.926212 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) 4.00000 0.284268
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.00000 0.282138
\(202\) −18.0000 −1.26648
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) −10.0000 −0.698430
\(206\) 16.0000 1.11477
\(207\) 1.00000 0.0695048
\(208\) −6.00000 −0.416025
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −14.0000 −0.961524
\(213\) 8.00000 0.548151
\(214\) −8.00000 −0.546869
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 2.00000 0.135147
\(220\) 4.00000 0.269680
\(221\) 36.0000 2.42162
\(222\) −6.00000 −0.402694
\(223\) −20.0000 −1.33930 −0.669650 0.742677i \(-0.733556\pi\)
−0.669650 + 0.742677i \(0.733556\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 14.0000 0.931266
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 4.00000 0.264906
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 1.00000 0.0659380
\(231\) 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
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) −12.0000 −0.779484
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) 7.00000 0.447214
\(246\) −10.0000 −0.637577
\(247\) −24.0000 −1.52708
\(248\) 8.00000 0.508001
\(249\) −16.0000 −1.01396
\(250\) 1.00000 0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) −12.0000 −0.752947
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 6.00000 0.372104
\(261\) −6.00000 −0.371391
\(262\) −8.00000 −0.494242
\(263\) −32.0000 −1.97320 −0.986602 0.163144i \(-0.947836\pi\)
−0.986602 + 0.163144i \(0.947836\pi\)
\(264\) 4.00000 0.246183
\(265\) 14.0000 0.860013
\(266\) 0 0
\(267\) −2.00000 −0.122398
\(268\) 4.00000 0.244339
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 1.00000 0.0608581
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) −4.00000 −0.241209
\(276\) 1.00000 0.0601929
\(277\) −30.0000 −1.80253 −0.901263 0.433273i \(-0.857359\pi\)
−0.901263 + 0.433273i \(0.857359\pi\)
\(278\) −12.0000 −0.719712
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 8.00000 0.476393
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 8.00000 0.474713
\(285\) −4.00000 −0.236940
\(286\) −24.0000 −1.41915
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) −14.0000 −0.820695
\(292\) 2.00000 0.117041
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) −4.00000 −0.232104
\(298\) −10.0000 −0.579284
\(299\) −6.00000 −0.346989
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) 18.0000 1.03407
\(304\) 4.00000 0.229416
\(305\) −10.0000 −0.572598
\(306\) 6.00000 0.342997
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) −8.00000 −0.454369
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 6.00000 0.339683
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 14.0000 0.785081
\(319\) 24.0000 1.34374
\(320\) −1.00000 −0.0559017
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) −6.00000 −0.332820
\(326\) −4.00000 −0.221540
\(327\) 2.00000 0.110600
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −16.0000 −0.878114
\(333\) 6.00000 0.328798
\(334\) 16.0000 0.875481
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −23.0000 −1.25104
\(339\) −14.0000 −0.760376
\(340\) 6.00000 0.325396
\(341\) 32.0000 1.73290
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) −1.00000 −0.0538382
\(346\) −18.0000 −0.967686
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) −6.00000 −0.321634
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 4.00000 0.213201
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 6.00000 0.315353
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) −10.0000 −0.522708
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 1.00000 0.0521286
\(369\) 10.0000 0.520579
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) −18.0000 −0.932005 −0.466002 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) −24.0000 −1.24101
\(375\) −1.00000 −0.0516398
\(376\) 8.00000 0.412568
\(377\) 36.0000 1.85409
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) −4.00000 −0.205196
\(381\) 12.0000 0.614779
\(382\) 8.00000 0.409316
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 4.00000 0.203331
\(388\) −14.0000 −0.710742
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) −6.00000 −0.303822
\(391\) −6.00000 −0.303433
\(392\) 7.00000 0.353553
\(393\) 8.00000 0.403547
\(394\) −26.0000 −1.30986
\(395\) 12.0000 0.603786
\(396\) −4.00000 −0.201008
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 12.0000 0.601506
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) −4.00000 −0.199502
\(403\) 48.0000 2.39105
\(404\) 18.0000 0.895533
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 6.00000 0.297044
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 10.0000 0.493865
\(411\) 2.00000 0.0986527
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 16.0000 0.785409
\(416\) 6.00000 0.294174
\(417\) 12.0000 0.587643
\(418\) 16.0000 0.782586
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −38.0000 −1.85201 −0.926003 0.377515i \(-0.876779\pi\)
−0.926003 + 0.377515i \(0.876779\pi\)
\(422\) 20.0000 0.973585
\(423\) −8.00000 −0.388973
\(424\) 14.0000 0.679900
\(425\) −6.00000 −0.291043
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) 24.0000 1.15873
\(430\) 4.00000 0.192897
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 2.00000 0.0957826
\(437\) 4.00000 0.191346
\(438\) −2.00000 −0.0955637
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) −4.00000 −0.190693
\(441\) −7.00000 −0.333333
\(442\) −36.0000 −1.71235
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 6.00000 0.284747
\(445\) 2.00000 0.0948091
\(446\) 20.0000 0.947027
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −40.0000 −1.88353
\(452\) −14.0000 −0.658505
\(453\) 16.0000 0.751746
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 14.0000 0.654177
\(459\) −6.00000 −0.280056
\(460\) −1.00000 −0.0466252
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) −6.00000 −0.278543
\(465\) 8.00000 0.370991
\(466\) −22.0000 −1.01913
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) −6.00000 −0.277350
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) 6.00000 0.276465
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 12.0000 0.551178
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −14.0000 −0.641016
\(478\) 8.00000 0.365911
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 1.00000 0.0456435
\(481\) −36.0000 −1.64146
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 14.0000 0.635707
\(486\) −1.00000 −0.0453609
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) −10.0000 −0.452679
\(489\) 4.00000 0.180886
\(490\) −7.00000 −0.316228
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 10.0000 0.450835
\(493\) 36.0000 1.62136
\(494\) 24.0000 1.07981
\(495\) 4.00000 0.179787
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 16.0000 0.716977
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −16.0000 −0.714827
\(502\) 12.0000 0.535586
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 4.00000 0.177822
\(507\) 23.0000 1.02147
\(508\) 12.0000 0.532414
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) −6.00000 −0.265684
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 4.00000 0.176604
\(514\) −6.00000 −0.264649
\(515\) 16.0000 0.705044
\(516\) 4.00000 0.176090
\(517\) 32.0000 1.40736
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) −6.00000 −0.263117
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 6.00000 0.262613
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) 32.0000 1.39527
\(527\) 48.0000 2.09091
\(528\) −4.00000 −0.174078
\(529\) 1.00000 0.0434783
\(530\) −14.0000 −0.608121
\(531\) 0 0
\(532\) 0 0
\(533\) −60.0000 −2.59889
\(534\) 2.00000 0.0865485
\(535\) −8.00000 −0.345870
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) −10.0000 −0.431131
\(539\) 28.0000 1.20605
\(540\) −1.00000 −0.0430331
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 8.00000 0.343629
\(543\) −6.00000 −0.257485
\(544\) 6.00000 0.257248
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 2.00000 0.0854358
\(549\) 10.0000 0.426790
\(550\) 4.00000 0.170561
\(551\) −24.0000 −1.02243
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) 30.0000 1.27458
\(555\) −6.00000 −0.254686
\(556\) 12.0000 0.508913
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 8.00000 0.338667
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) −30.0000 −1.26547
\(563\) −8.00000 −0.337160 −0.168580 0.985688i \(-0.553918\pi\)
−0.168580 + 0.985688i \(0.553918\pi\)
\(564\) −8.00000 −0.336861
\(565\) 14.0000 0.588984
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 4.00000 0.167542
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 24.0000 1.00349
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) −19.0000 −0.790296
\(579\) 10.0000 0.415586
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 14.0000 0.580319
\(583\) 56.0000 2.31928
\(584\) −2.00000 −0.0827606
\(585\) 6.00000 0.248069
\(586\) 14.0000 0.578335
\(587\) −4.00000 −0.165098 −0.0825488 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) −7.00000 −0.288675
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 26.0000 1.06950
\(592\) 6.00000 0.246598
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −12.0000 −0.491127
\(598\) 6.00000 0.245358
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 16.0000 0.651031
\(605\) −5.00000 −0.203279
\(606\) −18.0000 −0.731200
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) 48.0000 1.94187
\(612\) −6.00000 −0.242536
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −12.0000 −0.484281
\(615\) −10.0000 −0.403239
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 16.0000 0.643614
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 8.00000 0.321288
\(621\) 1.00000 0.0401286
\(622\) 8.00000 0.320771
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) −16.0000 −0.638978
\(628\) 6.00000 0.239426
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 12.0000 0.477334
\(633\) −20.0000 −0.794929
\(634\) −18.0000 −0.714871
\(635\) −12.0000 −0.476205
\(636\) −14.0000 −0.555136
\(637\) 42.0000 1.66410
\(638\) −24.0000 −0.950169
\(639\) 8.00000 0.316475
\(640\) 1.00000 0.0395285
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) −8.00000 −0.315735
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) 24.0000 0.944267
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −8.00000 −0.312586
\(656\) 10.0000 0.390434
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 4.00000 0.155700
\(661\) 34.0000 1.32245 0.661223 0.750189i \(-0.270038\pi\)
0.661223 + 0.750189i \(0.270038\pi\)
\(662\) 20.0000 0.777322
\(663\) 36.0000 1.39812
\(664\) 16.0000 0.620920
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) −6.00000 −0.232321
\(668\) −16.0000 −0.619059
\(669\) −20.0000 −0.773245
\(670\) 4.00000 0.154533
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 14.0000 0.539260
\(675\) 1.00000 0.0384900
\(676\) 23.0000 0.884615
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 14.0000 0.537667
\(679\) 0 0
\(680\) −6.00000 −0.230089
\(681\) 8.00000 0.306561
\(682\) −32.0000 −1.22534
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 4.00000 0.152944
\(685\) −2.00000 −0.0764161
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 4.00000 0.152499
\(689\) 84.0000 3.20015
\(690\) 1.00000 0.0380693
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) −12.0000 −0.455186
\(696\) 6.00000 0.227429
\(697\) −60.0000 −2.27266
\(698\) 34.0000 1.28692
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 6.00000 0.226455
\(703\) 24.0000 0.905177
\(704\) −4.00000 −0.150756
\(705\) 8.00000 0.301297
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 8.00000 0.300235
\(711\) −12.0000 −0.450035
\(712\) 2.00000 0.0749532
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) −24.0000 −0.897549
\(716\) 0 0
\(717\) −8.00000 −0.298765
\(718\) 8.00000 0.298557
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 2.00000 0.0743808
\(724\) −6.00000 −0.222988
\(725\) −6.00000 −0.222834
\(726\) −5.00000 −0.185567
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) −24.0000 −0.887672
\(732\) 10.0000 0.369611
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 32.0000 1.18114
\(735\) 7.00000 0.258199
\(736\) −1.00000 −0.0368605
\(737\) −16.0000 −0.589368
\(738\) −10.0000 −0.368105
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) −6.00000 −0.220564
\(741\) −24.0000 −0.881662
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 8.00000 0.293294
\(745\) −10.0000 −0.366372
\(746\) 18.0000 0.659027
\(747\) −16.0000 −0.585409
\(748\) 24.0000 0.877527
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) −8.00000 −0.291730
\(753\) −12.0000 −0.437304
\(754\) −36.0000 −1.31104
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) 54.0000 1.96266 0.981332 0.192323i \(-0.0616021\pi\)
0.981332 + 0.192323i \(0.0616021\pi\)
\(758\) −4.00000 −0.145287
\(759\) −4.00000 −0.145191
\(760\) 4.00000 0.145095
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) −12.0000 −0.434714
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 6.00000 0.216930
\(766\) −16.0000 −0.578103
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −6.00000 −0.216366 −0.108183 0.994131i \(-0.534503\pi\)
−0.108183 + 0.994131i \(0.534503\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 10.0000 0.359908
\(773\) 2.00000 0.0719350 0.0359675 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) −4.00000 −0.143777
\(775\) −8.00000 −0.287368
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 40.0000 1.43315
\(780\) 6.00000 0.214834
\(781\) −32.0000 −1.14505
\(782\) 6.00000 0.214560
\(783\) −6.00000 −0.214423
\(784\) −7.00000 −0.250000
\(785\) −6.00000 −0.214149
\(786\) −8.00000 −0.285351
\(787\) 36.0000 1.28326 0.641631 0.767014i \(-0.278258\pi\)
0.641631 + 0.767014i \(0.278258\pi\)
\(788\) 26.0000 0.926212
\(789\) −32.0000 −1.13923
\(790\) −12.0000 −0.426941
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) −60.0000 −2.13066
\(794\) −2.00000 −0.0709773
\(795\) 14.0000 0.496529
\(796\) −12.0000 −0.425329
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) −1.00000 −0.0353553
\(801\) −2.00000 −0.0706665
\(802\) −6.00000 −0.211867
\(803\) −8.00000 −0.282314
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) 10.0000 0.352017
\(808\) −18.0000 −0.633238
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 1.00000 0.0351364
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 24.0000 0.841200
\(815\) −4.00000 −0.140114
\(816\) −6.00000 −0.210042
\(817\) 16.0000 0.559769
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) −10.0000 −0.349215
\(821\) −46.0000 −1.60541 −0.802706 0.596376i \(-0.796607\pi\)
−0.802706 + 0.596376i \(0.796607\pi\)
\(822\) −2.00000 −0.0697580
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) 16.0000 0.557386
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) 1.00000 0.0347524
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) −16.0000 −0.555368
\(831\) −30.0000 −1.04069
\(832\) −6.00000 −0.208013
\(833\) 42.0000 1.45521
\(834\) −12.0000 −0.415526
\(835\) 16.0000 0.553703
\(836\) −16.0000 −0.553372
\(837\) −8.00000 −0.276520
\(838\) 12.0000 0.414533
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 38.0000 1.30957
\(843\) 30.0000 1.03325
\(844\) −20.0000 −0.688428
\(845\) −23.0000 −0.791224
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) −14.0000 −0.480762
\(849\) 4.00000 0.137280
\(850\) 6.00000 0.205798
\(851\) 6.00000 0.205677
\(852\) 8.00000 0.274075
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) −8.00000 −0.273434
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) −24.0000 −0.819346
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −8.00000 −0.272481
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −18.0000 −0.612018
\(866\) −2.00000 −0.0679628
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 48.0000 1.62829
\(870\) −6.00000 −0.203419
\(871\) −24.0000 −0.813209
\(872\) −2.00000 −0.0677285
\(873\) −14.0000 −0.473828
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 24.0000 0.809961
\(879\) −14.0000 −0.472208
\(880\) 4.00000 0.134840
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 7.00000 0.235702
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 36.0000 1.21081
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) −2.00000 −0.0670402
\(891\) −4.00000 −0.134005
\(892\) −20.0000 −0.669650
\(893\) −32.0000 −1.07084
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) −10.0000 −0.333704
\(899\) 48.0000 1.60089
\(900\) 1.00000 0.0333333
\(901\) 84.0000 2.79845
\(902\) 40.0000 1.33185
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 6.00000 0.199447
\(906\) −16.0000 −0.531564
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 8.00000 0.265489
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 4.00000 0.132453
\(913\) 64.0000 2.11809
\(914\) −10.0000 −0.330771
\(915\) −10.0000 −0.330590
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) 12.0000 0.395843 0.197922 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) 1.00000 0.0329690
\(921\) 12.0000 0.395413
\(922\) −2.00000 −0.0658665
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 20.0000 0.657241
\(927\) −16.0000 −0.525509
\(928\) 6.00000 0.196960
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) −8.00000 −0.262330
\(931\) −28.0000 −0.917663
\(932\) 22.0000 0.720634
\(933\) −8.00000 −0.261908
\(934\) −8.00000 −0.261768
\(935\) −24.0000 −0.784884
\(936\) 6.00000 0.196116
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 8.00000 0.260931
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) −6.00000 −0.195491
\(943\) 10.0000 0.325645
\(944\) 0 0
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) −20.0000 −0.649913 −0.324956 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) −12.0000 −0.389742
\(949\) −12.0000 −0.389536
\(950\) −4.00000 −0.129777
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 14.0000 0.453267
\(955\) 8.00000 0.258874
\(956\) −8.00000 −0.258738
\(957\) 24.0000 0.775810
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) 36.0000 1.16069
\(963\) 8.00000 0.257796
\(964\) 2.00000 0.0644157
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) 28.0000 0.900419 0.450210 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) −5.00000 −0.160706
\(969\) −24.0000 −0.770991
\(970\) −14.0000 −0.449513
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −20.0000 −0.640841
\(975\) −6.00000 −0.192154
\(976\) 10.0000 0.320092
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) −4.00000 −0.127906
\(979\) 8.00000 0.255681
\(980\) 7.00000 0.223607
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −10.0000 −0.318788
\(985\) −26.0000 −0.828429
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) 4.00000 0.127193
\(990\) −4.00000 −0.127128
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 8.00000 0.254000
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 12.0000 0.380426
\(996\) −16.0000 −0.506979
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) −12.0000 −0.379853
\(999\) 6.00000 0.189832
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 690.2.a.e.1.1 1
3.2 odd 2 2070.2.a.r.1.1 1
4.3 odd 2 5520.2.a.f.1.1 1
5.2 odd 4 3450.2.d.a.2899.1 2
5.3 odd 4 3450.2.d.a.2899.2 2
5.4 even 2 3450.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.e.1.1 1 1.1 even 1 trivial
2070.2.a.r.1.1 1 3.2 odd 2
3450.2.a.p.1.1 1 5.4 even 2
3450.2.d.a.2899.1 2 5.2 odd 4
3450.2.d.a.2899.2 2 5.3 odd 4
5520.2.a.f.1.1 1 4.3 odd 2