Properties

Label 507.2.a.a.1.1
Level $507$
Weight $2$
Character 507.1
Self dual yes
Analytic conductor $4.048$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.04841538248\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -4.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +2.00000 q^{20} -4.00000 q^{21} +4.00000 q^{22} -3.00000 q^{24} -1.00000 q^{25} -1.00000 q^{27} -4.00000 q^{28} -10.0000 q^{29} -2.00000 q^{30} -4.00000 q^{31} -5.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} -8.00000 q^{35} -1.00000 q^{36} +2.00000 q^{37} -6.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} -12.0000 q^{43} +4.00000 q^{44} -2.00000 q^{45} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{53} +1.00000 q^{54} +8.00000 q^{55} +12.0000 q^{56} +10.0000 q^{58} -12.0000 q^{59} -2.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} +4.00000 q^{63} +7.00000 q^{64} -4.00000 q^{66} +8.00000 q^{67} -2.00000 q^{68} +8.00000 q^{70} +3.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -16.0000 q^{77} +8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} +4.00000 q^{84} -4.00000 q^{85} +12.0000 q^{86} +10.0000 q^{87} -12.0000 q^{88} +2.00000 q^{89} +2.00000 q^{90} +4.00000 q^{93} +5.00000 q^{96} -10.0000 q^{97} -9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −4.00000 −1.06904
\(15\) 2.00000 0.516398
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) −4.00000 −0.872872
\(22\) 4.00000 0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −3.00000 −0.612372
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −4.00000 −0.755929
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) −2.00000 −0.365148
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −5.00000 −0.883883
\(33\) 4.00000 0.696311
\(34\) −2.00000 −0.342997
\(35\) −8.00000 −1.35225
\(36\) −1.00000 −0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −6.00000 −0.948683
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000 0.617213
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 4.00000 0.603023
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 8.00000 1.07872
\(56\) 12.0000 1.60357
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −2.00000 −0.258199
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.00000 0.508001
\(63\) 4.00000 0.503953
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 8.00000 0.956183
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 3.00000 0.353553
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −16.0000 −1.82337
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 4.00000 0.436436
\(85\) −4.00000 −0.433861
\(86\) 12.0000 1.29399
\(87\) 10.0000 1.07211
\(88\) −12.0000 −1.27920
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −9.00000 −0.909137
\(99\) −4.00000 −0.402015
\(100\) 1.00000 0.100000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 2.00000 0.198030
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 8.00000 0.780720
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −8.00000 −0.762770
\(111\) −2.00000 −0.189832
\(112\) −4.00000 −0.377964
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 8.00000 0.733359
\(120\) 6.00000 0.547723
\(121\) 5.00000 0.454545
\(122\) 2.00000 0.181071
\(123\) 6.00000 0.541002
\(124\) 4.00000 0.359211
\(125\) 12.0000 1.07331
\(126\) −4.00000 −0.356348
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 3.00000 0.265165
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 2.00000 0.172133
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 8.00000 0.676123
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 20.0000 1.66091
\(146\) 2.00000 0.165521
\(147\) −9.00000 −0.742307
\(148\) −2.00000 −0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 16.0000 1.28932
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −8.00000 −0.636446
\(159\) −6.00000 −0.475831
\(160\) 10.0000 0.790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 6.00000 0.468521
\(165\) −8.00000 −0.622799
\(166\) 4.00000 0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −12.0000 −0.925820
\(169\) 0 0
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −10.0000 −0.758098
\(175\) −4.00000 −0.302372
\(176\) 4.00000 0.301511
\(177\) 12.0000 0.901975
\(178\) −2.00000 −0.149906
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 2.00000 0.149071
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) −4.00000 −0.293294
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −7.00000 −0.505181
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 4.00000 0.284268
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −3.00000 −0.212132
\(201\) −8.00000 −0.564276
\(202\) 18.0000 1.26648
\(203\) −40.0000 −2.80745
\(204\) 2.00000 0.140028
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) −8.00000 −0.552052
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 24.0000 1.63679
\(216\) −3.00000 −0.204124
\(217\) −16.0000 −1.08615
\(218\) −2.00000 −0.135457
\(219\) 2.00000 0.135147
\(220\) −8.00000 −0.539360
\(221\) 0 0
\(222\) 2.00000 0.134231
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −20.0000 −1.33631
\(225\) −1.00000 −0.0666667
\(226\) 6.00000 0.399114
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) −30.0000 −1.96960
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) −8.00000 −0.519656
\(238\) −8.00000 −0.518563
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −2.00000 −0.129099
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) −18.0000 −1.14998
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) −12.0000 −0.762001
\(249\) 4.00000 0.253490
\(250\) −12.0000 −0.758947
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −4.00000 −0.251976
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 4.00000 0.250490
\(256\) −17.0000 −1.06250
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) −12.0000 −0.747087
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) −4.00000 −0.247121
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 12.0000 0.738549
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) −2.00000 −0.122398
\(268\) −8.00000 −0.488678
\(269\) 22.0000 1.34136 0.670682 0.741745i \(-0.266002\pi\)
0.670682 + 0.741745i \(0.266002\pi\)
\(270\) −2.00000 −0.121716
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −12.0000 −0.719712
\(279\) −4.00000 −0.239474
\(280\) −24.0000 −1.43427
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) −20.0000 −1.17444
\(291\) 10.0000 0.586210
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 9.00000 0.524891
\(295\) 24.0000 1.39733
\(296\) 6.00000 0.348743
\(297\) 4.00000 0.232104
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −48.0000 −2.76667
\(302\) 4.00000 0.230174
\(303\) 18.0000 1.03407
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) −2.00000 −0.114332
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 16.0000 0.911685
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 18.0000 1.01580
\(315\) −8.00000 −0.450749
\(316\) −8.00000 −0.450035
\(317\) −26.0000 −1.46031 −0.730153 0.683284i \(-0.760551\pi\)
−0.730153 + 0.683284i \(0.760551\pi\)
\(318\) 6.00000 0.336463
\(319\) 40.0000 2.23957
\(320\) −14.0000 −0.782624
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) −2.00000 −0.110600
\(328\) −18.0000 −0.993884
\(329\) 0 0
\(330\) 8.00000 0.440386
\(331\) 16.0000 0.879440 0.439720 0.898135i \(-0.355078\pi\)
0.439720 + 0.898135i \(0.355078\pi\)
\(332\) 4.00000 0.219529
\(333\) 2.00000 0.109599
\(334\) −8.00000 −0.437741
\(335\) −16.0000 −0.874173
\(336\) 4.00000 0.218218
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 4.00000 0.216930
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) −36.0000 −1.94099
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −10.0000 −0.536056
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 20.0000 1.06600
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) −8.00000 −0.423405
\(358\) −4.00000 −0.211407
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −6.00000 −0.316228
\(361\) −19.0000 −1.00000
\(362\) 10.0000 0.525588
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) −2.00000 −0.104542
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 4.00000 0.207950
\(371\) 24.0000 1.24602
\(372\) −4.00000 −0.207390
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 8.00000 0.413670
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 0 0
\(378\) 4.00000 0.205738
\(379\) 24.0000 1.23280 0.616399 0.787434i \(-0.288591\pi\)
0.616399 + 0.787434i \(0.288591\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) −8.00000 −0.409316
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) −3.00000 −0.153093
\(385\) 32.0000 1.63087
\(386\) 18.0000 0.916176
\(387\) −12.0000 −0.609994
\(388\) 10.0000 0.507673
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 27.0000 1.36371
\(393\) −4.00000 −0.201773
\(394\) 18.0000 0.906827
\(395\) −16.0000 −0.805047
\(396\) 4.00000 0.201008
\(397\) −38.0000 −1.90717 −0.953583 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 8.00000 0.399004
\(403\) 0 0
\(404\) 18.0000 0.895533
\(405\) −2.00000 −0.0993808
\(406\) 40.0000 1.98517
\(407\) −8.00000 −0.396545
\(408\) −6.00000 −0.297044
\(409\) −34.0000 −1.68119 −0.840596 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) −12.0000 −0.592638
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) −48.0000 −2.36193
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) −8.00000 −0.390360
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −24.0000 −1.15738
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 16.0000 0.768025
\(435\) −20.0000 −0.958927
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) −2.00000 −0.0955637
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 24.0000 1.14416
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 2.00000 0.0949158
\(445\) −4.00000 −0.189618
\(446\) 4.00000 0.189405
\(447\) −6.00000 −0.283790
\(448\) 28.0000 1.32288
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 1.00000 0.0471405
\(451\) 24.0000 1.13012
\(452\) 6.00000 0.282216
\(453\) 4.00000 0.187936
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −10.0000 −0.467269
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 38.0000 1.76984 0.884918 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(462\) −16.0000 −0.744387
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 10.0000 0.464238
\(465\) −8.00000 −0.370991
\(466\) 14.0000 0.648537
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 32.0000 1.47762
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) −36.0000 −1.65703
\(473\) 48.0000 2.20704
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) 6.00000 0.274721
\(478\) −24.0000 −1.09773
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −10.0000 −0.456435
\(481\) 0 0
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 20.0000 0.908153
\(486\) 1.00000 0.0453609
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) −6.00000 −0.271607
\(489\) 8.00000 0.361773
\(490\) 18.0000 0.813157
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −6.00000 −0.270501
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) −12.0000 −0.536656
\(501\) −8.00000 −0.357414
\(502\) 12.0000 0.535586
\(503\) −8.00000 −0.356702 −0.178351 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) 12.0000 0.534522
\(505\) 36.0000 1.60198
\(506\) 0 0
\(507\) 0 0
\(508\) 16.0000 0.709885
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) −4.00000 −0.177123
\(511\) −8.00000 −0.353899
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −26.0000 −1.14681
\(515\) 0 0
\(516\) −12.0000 −0.528271
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 10.0000 0.437688
\(523\) 44.0000 1.92399 0.961993 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) −4.00000 −0.174741
\(525\) 4.00000 0.174574
\(526\) −24.0000 −1.04645
\(527\) −8.00000 −0.348485
\(528\) −4.00000 −0.174078
\(529\) −23.0000 −1.00000
\(530\) 12.0000 0.521247
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) 2.00000 0.0865485
\(535\) −24.0000 −1.03761
\(536\) 24.0000 1.03664
\(537\) −4.00000 −0.172613
\(538\) −22.0000 −0.948487
\(539\) −36.0000 −1.55063
\(540\) −2.00000 −0.0860663
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) −12.0000 −0.515444
\(543\) 10.0000 0.429141
\(544\) −10.0000 −0.428746
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 6.00000 0.256307
\(549\) −2.00000 −0.0853579
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) 10.0000 0.424859
\(555\) 4.00000 0.169791
\(556\) −12.0000 −0.508913
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 4.00000 0.169334
\(559\) 0 0
\(560\) 8.00000 0.338062
\(561\) 8.00000 0.337760
\(562\) −10.0000 −0.421825
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) −12.0000 −0.504398
\(567\) 4.00000 0.167984
\(568\) 0 0
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) 46.0000 1.91501 0.957503 0.288425i \(-0.0931316\pi\)
0.957503 + 0.288425i \(0.0931316\pi\)
\(578\) 13.0000 0.540729
\(579\) 18.0000 0.748054
\(580\) −20.0000 −0.830455
\(581\) −16.0000 −0.663792
\(582\) −10.0000 −0.414513
\(583\) −24.0000 −0.993978
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 9.00000 0.371154
\(589\) 0 0
\(590\) −24.0000 −0.988064
\(591\) 18.0000 0.740421
\(592\) −2.00000 −0.0821995
\(593\) 26.0000 1.06769 0.533846 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) −4.00000 −0.164122
\(595\) −16.0000 −0.655936
\(596\) −6.00000 −0.245770
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 3.00000 0.122474
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 48.0000 1.95633
\(603\) 8.00000 0.325785
\(604\) 4.00000 0.162758
\(605\) −10.0000 −0.406558
\(606\) −18.0000 −0.731200
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 0 0
\(609\) 40.0000 1.62088
\(610\) −4.00000 −0.161955
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −16.0000 −0.645707
\(615\) −12.0000 −0.483887
\(616\) −48.0000 −1.93398
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) −24.0000 −0.964641 −0.482321 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) 0 0
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) 4.00000 0.159490
\(630\) 8.00000 0.318728
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 24.0000 0.954669
\(633\) 20.0000 0.794929
\(634\) 26.0000 1.03259
\(635\) 32.0000 1.26988
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) −40.0000 −1.58362
\(639\) 0 0
\(640\) −6.00000 −0.237171
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 12.0000 0.473602
\(643\) 40.0000 1.57745 0.788723 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.944999
\(646\) 0 0
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 3.00000 0.117851
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 8.00000 0.313304
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 2.00000 0.0782062
\(655\) −8.00000 −0.312586
\(656\) 6.00000 0.234261
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 8.00000 0.311400
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) −16.0000 −0.621858
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) 4.00000 0.154649
\(670\) 16.0000 0.618134
\(671\) 8.00000 0.308837
\(672\) 20.0000 0.771517
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −18.0000 −0.693334
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) −6.00000 −0.230429
\(679\) −40.0000 −1.53506
\(680\) −12.0000 −0.460179
\(681\) −20.0000 −0.766402
\(682\) −16.0000 −0.612672
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) −8.00000 −0.305441
\(687\) −10.0000 −0.381524
\(688\) 12.0000 0.457496
\(689\) 0 0
\(690\) 0 0
\(691\) 24.0000 0.913003 0.456502 0.889723i \(-0.349102\pi\)
0.456502 + 0.889723i \(0.349102\pi\)
\(692\) −6.00000 −0.228086
\(693\) −16.0000 −0.607790
\(694\) 12.0000 0.455514
\(695\) −24.0000 −0.910372
\(696\) 30.0000 1.13715
\(697\) −12.0000 −0.454532
\(698\) −26.0000 −0.984115
\(699\) 14.0000 0.529529
\(700\) 4.00000 0.151186
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −28.0000 −1.05529
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) −72.0000 −2.70784
\(708\) −12.0000 −0.450988
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) −24.0000 −0.896296
\(718\) 24.0000 0.895672
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) 10.0000 0.371904
\(724\) 10.0000 0.371647
\(725\) 10.0000 0.371391
\(726\) 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
\(730\) −4.00000 −0.148047
\(731\) −24.0000 −0.887672
\(732\) −2.00000 −0.0739221
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) −16.0000 −0.590571
\(735\) 18.0000 0.663940
\(736\) 0 0
\(737\) −32.0000 −1.17874
\(738\) 6.00000 0.220863
\(739\) −32.0000 −1.17714 −0.588570 0.808447i \(-0.700309\pi\)
−0.588570 + 0.808447i \(0.700309\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 12.0000 0.439941
\(745\) −12.0000 −0.439646
\(746\) 26.0000 0.951928
\(747\) −4.00000 −0.146352
\(748\) 8.00000 0.292509
\(749\) 48.0000 1.75388
\(750\) 12.0000 0.438178
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 4.00000 0.145479
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −24.0000 −0.871719
\(759\) 0 0
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) −16.0000 −0.579619
\(763\) 8.00000 0.289619
\(764\) −8.00000 −0.289430
\(765\) −4.00000 −0.144620
\(766\) −16.0000 −0.578103
\(767\) 0 0
\(768\) 17.0000 0.613435
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) −32.0000 −1.15320
\(771\) −26.0000 −0.936367
\(772\) 18.0000 0.647834
\(773\) −10.0000 −0.359675 −0.179838 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(774\) 12.0000 0.431331
\(775\) 4.00000 0.143684
\(776\) −30.0000 −1.07694
\(777\) −8.00000 −0.286998
\(778\) −22.0000 −0.788738
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 10.0000 0.357371
\(784\) −9.00000 −0.321429
\(785\) 36.0000 1.28490
\(786\) 4.00000 0.142675
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 18.0000 0.641223
\(789\) −24.0000 −0.854423
\(790\) 16.0000 0.569254
\(791\) −24.0000 −0.853342
\(792\) −12.0000 −0.426401
\(793\) 0 0
\(794\) 38.0000 1.34857
\(795\) 12.0000 0.425596
\(796\) −8.00000 −0.283552
\(797\) 46.0000 1.62940 0.814702 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) 2.00000 0.0706665
\(802\) 22.0000 0.776847
\(803\) 8.00000 0.282314
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) −22.0000 −0.774437
\(808\) −54.0000 −1.89971
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 2.00000 0.0702728
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) 40.0000 1.40372
\(813\) −12.0000 −0.420858
\(814\) 8.00000 0.280400
\(815\) 16.0000 0.560456
\(816\) 2.00000 0.0700140
\(817\) 0 0
\(818\) 34.0000 1.18878
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) −6.00000 −0.209274
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 48.0000 1.67013
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) −8.00000 −0.277684
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 12.0000 0.415526
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) −4.00000 −0.138178
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 24.0000 0.828079
\(841\) 71.0000 2.44828
\(842\) −10.0000 −0.344623
\(843\) −10.0000 −0.344418
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) 20.0000 0.687208
\(848\) −6.00000 −0.206041
\(849\) −12.0000 −0.411839
\(850\) 2.00000 0.0685994
\(851\) 0 0
\(852\) 0 0
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) −46.0000 −1.57133 −0.785665 0.618652i \(-0.787679\pi\)
−0.785665 + 0.618652i \(0.787679\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) −24.0000 −0.818393
\(861\) 24.0000 0.817918
\(862\) 0 0
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 5.00000 0.170103
\(865\) −12.0000 −0.408012
\(866\) −34.0000 −1.15537
\(867\) 13.0000 0.441503
\(868\) 16.0000 0.543075
\(869\) −32.0000 −1.08553
\(870\) 20.0000 0.678064
\(871\) 0 0
\(872\) 6.00000 0.203186
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 48.0000 1.62270
\(876\) −2.00000 −0.0675737
\(877\) 10.0000 0.337676 0.168838 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) −32.0000 −1.07995
\(879\) −6.00000 −0.202375
\(880\) −8.00000 −0.269680
\(881\) 58.0000 1.95407 0.977035 0.213080i \(-0.0683494\pi\)
0.977035 + 0.213080i \(0.0683494\pi\)
\(882\) −9.00000 −0.303046
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 0 0
\(885\) −24.0000 −0.806751
\(886\) 4.00000 0.134383
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −6.00000 −0.201347
\(889\) −64.0000 −2.14649
\(890\) 4.00000 0.134080
\(891\) −4.00000 −0.134005
\(892\) 4.00000 0.133930
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) −8.00000 −0.267411
\(896\) 12.0000 0.400892
\(897\) 0 0
\(898\) 22.0000 0.734150
\(899\) 40.0000 1.33407
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) −24.0000 −0.799113
\(903\) 48.0000 1.59734
\(904\) −18.0000 −0.598671
\(905\) 20.0000 0.664822
\(906\) −4.00000 −0.132891
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −20.0000 −0.663723
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 2.00000 0.0661541
\(915\) −4.00000 −0.132236
\(916\) −10.0000 −0.330409
\(917\) 16.0000 0.528367
\(918\) 2.00000 0.0660098
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) −38.0000 −1.25146
\(923\) 0 0
\(924\) −16.0000 −0.526361
\(925\) −2.00000 −0.0657596
\(926\) 4.00000 0.131448
\(927\) 0 0
\(928\) 50.0000 1.64133
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 8.00000 0.262330
\(931\) 0 0
\(932\) 14.0000 0.458585
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) −32.0000 −1.04484
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) −18.0000 −0.586472
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 8.00000 0.260240
\(946\) −48.0000 −1.56061
\(947\) 60.0000 1.94974 0.974869 0.222779i \(-0.0715128\pi\)
0.974869 + 0.222779i \(0.0715128\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) 0 0
\(951\) 26.0000 0.843108
\(952\) 24.0000 0.777844
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) −6.00000 −0.194257
\(955\) −16.0000 −0.517748
\(956\) −24.0000 −0.776215
\(957\) −40.0000 −1.29302
\(958\) −24.0000 −0.775405
\(959\) −24.0000 −0.775000
\(960\) 14.0000 0.451848
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 10.0000 0.322078
\(965\) 36.0000 1.15888
\(966\) 0 0
\(967\) 52.0000 1.67221 0.836104 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) 15.0000 0.482118
\(969\) 0 0
\(970\) −20.0000 −0.642161
\(971\) −60.0000 −1.92549 −0.962746 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 1.00000 0.0320750
\(973\) 48.0000 1.53881
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) −8.00000 −0.255812
\(979\) −8.00000 −0.255681
\(980\) 18.0000 0.574989
\(981\) 2.00000 0.0638551
\(982\) 12.0000 0.382935
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 18.0000 0.573819
\(985\) 36.0000 1.14706
\(986\) 20.0000 0.636930
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) −8.00000 −0.254257
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 20.0000 0.635001
\(993\) −16.0000 −0.507745
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) −4.00000 −0.126745
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) −24.0000 −0.759707
\(999\) −2.00000 −0.0632772
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 507.2.a.a.1.1 1
3.2 odd 2 1521.2.a.e.1.1 1
4.3 odd 2 8112.2.a.s.1.1 1
13.2 odd 12 507.2.j.e.316.1 4
13.3 even 3 507.2.e.b.22.1 2
13.4 even 6 507.2.e.a.484.1 2
13.5 odd 4 507.2.b.a.337.2 2
13.6 odd 12 507.2.j.e.361.2 4
13.7 odd 12 507.2.j.e.361.1 4
13.8 odd 4 507.2.b.a.337.1 2
13.9 even 3 507.2.e.b.484.1 2
13.10 even 6 507.2.e.a.22.1 2
13.11 odd 12 507.2.j.e.316.2 4
13.12 even 2 39.2.a.a.1.1 1
39.5 even 4 1521.2.b.b.1351.1 2
39.8 even 4 1521.2.b.b.1351.2 2
39.38 odd 2 117.2.a.a.1.1 1
52.51 odd 2 624.2.a.i.1.1 1
65.12 odd 4 975.2.c.f.274.2 2
65.38 odd 4 975.2.c.f.274.1 2
65.64 even 2 975.2.a.f.1.1 1
91.90 odd 2 1911.2.a.f.1.1 1
104.51 odd 2 2496.2.a.e.1.1 1
104.77 even 2 2496.2.a.q.1.1 1
117.25 even 6 1053.2.e.b.352.1 2
117.38 odd 6 1053.2.e.d.352.1 2
117.77 odd 6 1053.2.e.d.703.1 2
117.103 even 6 1053.2.e.b.703.1 2
143.142 odd 2 4719.2.a.c.1.1 1
156.155 even 2 1872.2.a.h.1.1 1
195.38 even 4 2925.2.c.e.2224.2 2
195.77 even 4 2925.2.c.e.2224.1 2
195.194 odd 2 2925.2.a.p.1.1 1
273.272 even 2 5733.2.a.e.1.1 1
312.77 odd 2 7488.2.a.bl.1.1 1
312.155 even 2 7488.2.a.by.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.a.1.1 1 13.12 even 2
117.2.a.a.1.1 1 39.38 odd 2
507.2.a.a.1.1 1 1.1 even 1 trivial
507.2.b.a.337.1 2 13.8 odd 4
507.2.b.a.337.2 2 13.5 odd 4
507.2.e.a.22.1 2 13.10 even 6
507.2.e.a.484.1 2 13.4 even 6
507.2.e.b.22.1 2 13.3 even 3
507.2.e.b.484.1 2 13.9 even 3
507.2.j.e.316.1 4 13.2 odd 12
507.2.j.e.316.2 4 13.11 odd 12
507.2.j.e.361.1 4 13.7 odd 12
507.2.j.e.361.2 4 13.6 odd 12
624.2.a.i.1.1 1 52.51 odd 2
975.2.a.f.1.1 1 65.64 even 2
975.2.c.f.274.1 2 65.38 odd 4
975.2.c.f.274.2 2 65.12 odd 4
1053.2.e.b.352.1 2 117.25 even 6
1053.2.e.b.703.1 2 117.103 even 6
1053.2.e.d.352.1 2 117.38 odd 6
1053.2.e.d.703.1 2 117.77 odd 6
1521.2.a.e.1.1 1 3.2 odd 2
1521.2.b.b.1351.1 2 39.5 even 4
1521.2.b.b.1351.2 2 39.8 even 4
1872.2.a.h.1.1 1 156.155 even 2
1911.2.a.f.1.1 1 91.90 odd 2
2496.2.a.e.1.1 1 104.51 odd 2
2496.2.a.q.1.1 1 104.77 even 2
2925.2.a.p.1.1 1 195.194 odd 2
2925.2.c.e.2224.1 2 195.77 even 4
2925.2.c.e.2224.2 2 195.38 even 4
4719.2.a.c.1.1 1 143.142 odd 2
5733.2.a.e.1.1 1 273.272 even 2
7488.2.a.bl.1.1 1 312.77 odd 2
7488.2.a.by.1.1 1 312.155 even 2
8112.2.a.s.1.1 1 4.3 odd 2