Properties

Label 483.2.a.a.1.1
Level $483$
Weight $2$
Character 483.1
Self dual yes
Analytic conductor $3.857$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 483.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +2.00000 q^{12} +2.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} +4.00000 q^{17} +2.00000 q^{18} -3.00000 q^{19} +1.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} -5.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -6.00000 q^{29} -2.00000 q^{31} -8.00000 q^{32} +1.00000 q^{33} +8.00000 q^{34} +2.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} +2.00000 q^{39} +1.00000 q^{41} +2.00000 q^{42} -8.00000 q^{43} +2.00000 q^{44} +2.00000 q^{46} -5.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} -10.0000 q^{50} +4.00000 q^{51} +4.00000 q^{52} +3.00000 q^{53} +2.00000 q^{54} -3.00000 q^{57} -12.0000 q^{58} +5.00000 q^{59} +13.0000 q^{61} -4.00000 q^{62} +1.00000 q^{63} -8.00000 q^{64} +2.00000 q^{66} +8.00000 q^{68} +1.00000 q^{69} -16.0000 q^{73} -4.00000 q^{74} -5.00000 q^{75} -6.00000 q^{76} +1.00000 q^{77} +4.00000 q^{78} -2.00000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +6.00000 q^{83} +2.00000 q^{84} -16.0000 q^{86} -6.00000 q^{87} +6.00000 q^{89} +2.00000 q^{91} +2.00000 q^{92} -2.00000 q^{93} -10.0000 q^{94} -8.00000 q^{96} +10.0000 q^{97} +2.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 2.00000 0.816497
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 2.00000 0.577350
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 2.00000 0.471405
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −8.00000 −1.41421
\(33\) 1.00000 0.174078
\(34\) 8.00000 1.37199
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −6.00000 −0.973329
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 2.00000 0.308607
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) −5.00000 −0.729325 −0.364662 0.931140i \(-0.618816\pi\)
−0.364662 + 0.931140i \(0.618816\pi\)
\(48\) −4.00000 −0.577350
\(49\) 1.00000 0.142857
\(50\) −10.0000 −1.41421
\(51\) 4.00000 0.560112
\(52\) 4.00000 0.554700
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 2.00000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) −12.0000 −1.57568
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 8.00000 0.970143
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −16.0000 −1.87266 −0.936329 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −4.00000 −0.464991
\(75\) −5.00000 −0.577350
\(76\) −6.00000 −0.688247
\(77\) 1.00000 0.113961
\(78\) 4.00000 0.452911
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −16.0000 −1.72532
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 2.00000 0.208514
\(93\) −2.00000 −0.207390
\(94\) −10.0000 −1.03142
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 2.00000 0.202031
\(99\) 1.00000 0.100504
\(100\) −10.0000 −1.00000
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) 8.00000 0.792118
\(103\) 11.0000 1.08386 0.541931 0.840423i \(-0.317693\pi\)
0.541931 + 0.840423i \(0.317693\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 2.00000 0.192450
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −4.00000 −0.377964
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −6.00000 −0.561951
\(115\) 0 0
\(116\) −12.0000 −1.11417
\(117\) 2.00000 0.184900
\(118\) 10.0000 0.920575
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 26.0000 2.35393
\(123\) 1.00000 0.0901670
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 2.00000 0.174078
\(133\) −3.00000 −0.260133
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.0000 1.62328 0.811640 0.584158i \(-0.198575\pi\)
0.811640 + 0.584158i \(0.198575\pi\)
\(138\) 2.00000 0.170251
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) −5.00000 −0.421076
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) −32.0000 −2.64834
\(147\) 1.00000 0.0824786
\(148\) −4.00000 −0.328798
\(149\) 17.0000 1.39269 0.696347 0.717705i \(-0.254807\pi\)
0.696347 + 0.717705i \(0.254807\pi\)
\(150\) −10.0000 −0.816497
\(151\) 13.0000 1.05792 0.528962 0.848645i \(-0.322581\pi\)
0.528962 + 0.848645i \(0.322581\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 1.00000 0.0798087 0.0399043 0.999204i \(-0.487295\pi\)
0.0399043 + 0.999204i \(0.487295\pi\)
\(158\) −4.00000 −0.318223
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 2.00000 0.157135
\(163\) −11.0000 −0.861586 −0.430793 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −1.00000 −0.0773823 −0.0386912 0.999251i \(-0.512319\pi\)
−0.0386912 + 0.999251i \(0.512319\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) −16.0000 −1.21999
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) −12.0000 −0.909718
\(175\) −5.00000 −0.377964
\(176\) −4.00000 −0.301511
\(177\) 5.00000 0.375823
\(178\) 12.0000 0.899438
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 4.00000 0.296500
\(183\) 13.0000 0.960988
\(184\) 0 0
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 4.00000 0.292509
\(188\) −10.0000 −0.729325
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) −8.00000 −0.577350
\(193\) 13.0000 0.935760 0.467880 0.883792i \(-0.345018\pi\)
0.467880 + 0.883792i \(0.345018\pi\)
\(194\) 20.0000 1.43592
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 2.00000 0.142134
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 18.0000 1.26648
\(203\) −6.00000 −0.421117
\(204\) 8.00000 0.560112
\(205\) 0 0
\(206\) 22.0000 1.53281
\(207\) 1.00000 0.0695048
\(208\) −8.00000 −0.554700
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −19.0000 −1.30801 −0.654007 0.756489i \(-0.726913\pi\)
−0.654007 + 0.756489i \(0.726913\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) −8.00000 −0.541828
\(219\) −16.0000 −1.08118
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) −4.00000 −0.268462
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −8.00000 −0.534522
\(225\) −5.00000 −0.333333
\(226\) −28.0000 −1.86253
\(227\) −2.00000 −0.132745 −0.0663723 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) −6.00000 −0.397360
\(229\) 13.0000 0.859064 0.429532 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 10.0000 0.650945
\(237\) −2.00000 −0.129914
\(238\) 8.00000 0.518563
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) 15.0000 0.966235 0.483117 0.875556i \(-0.339504\pi\)
0.483117 + 0.875556i \(0.339504\pi\)
\(242\) −20.0000 −1.28565
\(243\) 1.00000 0.0641500
\(244\) 26.0000 1.66448
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) −14.0000 −0.883672 −0.441836 0.897096i \(-0.645673\pi\)
−0.441836 + 0.897096i \(0.645673\pi\)
\(252\) 2.00000 0.125988
\(253\) 1.00000 0.0628695
\(254\) −14.0000 −0.878438
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −3.00000 −0.187135 −0.0935674 0.995613i \(-0.529827\pi\)
−0.0935674 + 0.995613i \(0.529827\pi\)
\(258\) −16.0000 −0.996116
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 30.0000 1.85341
\(263\) −19.0000 −1.17159 −0.585795 0.810459i \(-0.699218\pi\)
−0.585795 + 0.810459i \(0.699218\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.00000 −0.367884
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) 22.0000 1.34136 0.670682 0.741745i \(-0.266002\pi\)
0.670682 + 0.741745i \(0.266002\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −16.0000 −0.970143
\(273\) 2.00000 0.121046
\(274\) 38.0000 2.29566
\(275\) −5.00000 −0.301511
\(276\) 2.00000 0.120386
\(277\) 31.0000 1.86261 0.931305 0.364241i \(-0.118672\pi\)
0.931305 + 0.364241i \(0.118672\pi\)
\(278\) −32.0000 −1.91923
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −10.0000 −0.595491
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 1.00000 0.0590281
\(288\) −8.00000 −0.471405
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) −32.0000 −1.87266
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 34.0000 1.96957
\(299\) 2.00000 0.115663
\(300\) −10.0000 −0.577350
\(301\) −8.00000 −0.461112
\(302\) 26.0000 1.49613
\(303\) 9.00000 0.517036
\(304\) 12.0000 0.688247
\(305\) 0 0
\(306\) 8.00000 0.457330
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 2.00000 0.113961
\(309\) 11.0000 0.625768
\(310\) 0 0
\(311\) 21.0000 1.19080 0.595400 0.803429i \(-0.296993\pi\)
0.595400 + 0.803429i \(0.296993\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 6.00000 0.336463
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 2.00000 0.111456
\(323\) −12.0000 −0.667698
\(324\) 2.00000 0.111111
\(325\) −10.0000 −0.554700
\(326\) −22.0000 −1.21847
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) −5.00000 −0.275659
\(330\) 0 0
\(331\) 5.00000 0.274825 0.137412 0.990514i \(-0.456121\pi\)
0.137412 + 0.990514i \(0.456121\pi\)
\(332\) 12.0000 0.658586
\(333\) −2.00000 −0.109599
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) −18.0000 −0.979071
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) −6.00000 −0.324443
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 4.00000 0.215041
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) −12.0000 −0.643268
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) −10.0000 −0.534522
\(351\) 2.00000 0.106752
\(352\) −8.00000 −0.426401
\(353\) −22.0000 −1.17094 −0.585471 0.810693i \(-0.699090\pi\)
−0.585471 + 0.810693i \(0.699090\pi\)
\(354\) 10.0000 0.531494
\(355\) 0 0
\(356\) 12.0000 0.635999
\(357\) 4.00000 0.211702
\(358\) 32.0000 1.69125
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) −4.00000 −0.210235
\(363\) −10.0000 −0.524864
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 26.0000 1.35904
\(367\) 29.0000 1.51379 0.756894 0.653538i \(-0.226716\pi\)
0.756894 + 0.653538i \(0.226716\pi\)
\(368\) −4.00000 −0.208514
\(369\) 1.00000 0.0520579
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) −4.00000 −0.207390
\(373\) 34.0000 1.76045 0.880227 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 2.00000 0.102869
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) −7.00000 −0.358621
\(382\) 6.00000 0.306987
\(383\) −30.0000 −1.53293 −0.766464 0.642287i \(-0.777986\pi\)
−0.766464 + 0.642287i \(0.777986\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 26.0000 1.32337
\(387\) −8.00000 −0.406663
\(388\) 20.0000 1.01535
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 15.0000 0.756650
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) −22.0000 −1.10276
\(399\) −3.00000 −0.150188
\(400\) 20.0000 1.00000
\(401\) 25.0000 1.24844 0.624220 0.781248i \(-0.285417\pi\)
0.624220 + 0.781248i \(0.285417\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) 19.0000 0.937201
\(412\) 22.0000 1.08386
\(413\) 5.00000 0.246034
\(414\) 2.00000 0.0982946
\(415\) 0 0
\(416\) −16.0000 −0.784465
\(417\) −16.0000 −0.783523
\(418\) −6.00000 −0.293470
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) −38.0000 −1.84981
\(423\) −5.00000 −0.243108
\(424\) 0 0
\(425\) −20.0000 −0.970143
\(426\) 0 0
\(427\) 13.0000 0.629114
\(428\) 8.00000 0.386695
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −23.0000 −1.10787 −0.553936 0.832560i \(-0.686875\pi\)
−0.553936 + 0.832560i \(0.686875\pi\)
\(432\) −4.00000 −0.192450
\(433\) −37.0000 −1.77811 −0.889053 0.457804i \(-0.848636\pi\)
−0.889053 + 0.457804i \(0.848636\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) −3.00000 −0.143509
\(438\) −32.0000 −1.52902
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 16.0000 0.761042
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) 17.0000 0.804072
\(448\) −8.00000 −0.377964
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) −10.0000 −0.471405
\(451\) 1.00000 0.0470882
\(452\) −28.0000 −1.31701
\(453\) 13.0000 0.610793
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) −14.0000 −0.654892 −0.327446 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) 26.0000 1.21490
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 2.00000 0.0930484
\(463\) −11.0000 −0.511213 −0.255607 0.966781i \(-0.582275\pi\)
−0.255607 + 0.966781i \(0.582275\pi\)
\(464\) 24.0000 1.11417
\(465\) 0 0
\(466\) 48.0000 2.22356
\(467\) −10.0000 −0.462745 −0.231372 0.972865i \(-0.574322\pi\)
−0.231372 + 0.972865i \(0.574322\pi\)
\(468\) 4.00000 0.184900
\(469\) 0 0
\(470\) 0 0
\(471\) 1.00000 0.0460776
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) −4.00000 −0.183726
\(475\) 15.0000 0.688247
\(476\) 8.00000 0.366679
\(477\) 3.00000 0.137361
\(478\) −8.00000 −0.365911
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 30.0000 1.36646
\(483\) 1.00000 0.0455016
\(484\) −20.0000 −0.909091
\(485\) 0 0
\(486\) 2.00000 0.0907218
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) −11.0000 −0.497437
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 2.00000 0.0901670
\(493\) −24.0000 −1.08091
\(494\) −12.0000 −0.539906
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) −28.0000 −1.24970
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) −9.00000 −0.399704
\(508\) −14.0000 −0.621150
\(509\) 33.0000 1.46270 0.731350 0.682003i \(-0.238891\pi\)
0.731350 + 0.682003i \(0.238891\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 32.0000 1.41421
\(513\) −3.00000 −0.132453
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) −16.0000 −0.704361
\(517\) −5.00000 −0.219900
\(518\) −4.00000 −0.175750
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) −12.0000 −0.525226
\(523\) −19.0000 −0.830812 −0.415406 0.909636i \(-0.636360\pi\)
−0.415406 + 0.909636i \(0.636360\pi\)
\(524\) 30.0000 1.31056
\(525\) −5.00000 −0.218218
\(526\) −38.0000 −1.65688
\(527\) −8.00000 −0.348485
\(528\) −4.00000 −0.174078
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 5.00000 0.216982
\(532\) −6.00000 −0.260133
\(533\) 2.00000 0.0866296
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) 0 0
\(537\) 16.0000 0.690451
\(538\) 44.0000 1.89697
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) −32.0000 −1.37452
\(543\) −2.00000 −0.0858282
\(544\) −32.0000 −1.37199
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 38.0000 1.62328
\(549\) 13.0000 0.554826
\(550\) −10.0000 −0.426401
\(551\) 18.0000 0.766826
\(552\) 0 0
\(553\) −2.00000 −0.0850487
\(554\) 62.0000 2.63413
\(555\) 0 0
\(556\) −32.0000 −1.35710
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) −4.00000 −0.169334
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) 12.0000 0.506189
\(563\) −32.0000 −1.34864 −0.674320 0.738440i \(-0.735563\pi\)
−0.674320 + 0.738440i \(0.735563\pi\)
\(564\) −10.0000 −0.421076
\(565\) 0 0
\(566\) −40.0000 −1.68133
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −27.0000 −1.13190 −0.565949 0.824440i \(-0.691490\pi\)
−0.565949 + 0.824440i \(0.691490\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 4.00000 0.167248
\(573\) 3.00000 0.125327
\(574\) 2.00000 0.0834784
\(575\) −5.00000 −0.208514
\(576\) −8.00000 −0.333333
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) −2.00000 −0.0831890
\(579\) 13.0000 0.540262
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 20.0000 0.829027
\(583\) 3.00000 0.124247
\(584\) 0 0
\(585\) 0 0
\(586\) −32.0000 −1.32191
\(587\) 35.0000 1.44460 0.722302 0.691577i \(-0.243084\pi\)
0.722302 + 0.691577i \(0.243084\pi\)
\(588\) 2.00000 0.0824786
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 8.00000 0.328798
\(593\) −10.0000 −0.410651 −0.205325 0.978694i \(-0.565825\pi\)
−0.205325 + 0.978694i \(0.565825\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 34.0000 1.39269
\(597\) −11.0000 −0.450200
\(598\) 4.00000 0.163572
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) 24.0000 0.978980 0.489490 0.872009i \(-0.337183\pi\)
0.489490 + 0.872009i \(0.337183\pi\)
\(602\) −16.0000 −0.652111
\(603\) 0 0
\(604\) 26.0000 1.05792
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 24.0000 0.973329
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) −10.0000 −0.404557
\(612\) 8.00000 0.323381
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 32.0000 1.29141
\(615\) 0 0
\(616\) 0 0
\(617\) 46.0000 1.85189 0.925945 0.377658i \(-0.123271\pi\)
0.925945 + 0.377658i \(0.123271\pi\)
\(618\) 22.0000 0.884970
\(619\) −12.0000 −0.482321 −0.241160 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 42.0000 1.68405
\(623\) 6.00000 0.240385
\(624\) −8.00000 −0.320256
\(625\) 25.0000 1.00000
\(626\) 2.00000 0.0799361
\(627\) −3.00000 −0.119808
\(628\) 2.00000 0.0798087
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) −50.0000 −1.99047 −0.995234 0.0975126i \(-0.968911\pi\)
−0.995234 + 0.0975126i \(0.968911\pi\)
\(632\) 0 0
\(633\) −19.0000 −0.755182
\(634\) −4.00000 −0.158860
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 2.00000 0.0792429
\(638\) −12.0000 −0.475085
\(639\) 0 0
\(640\) 0 0
\(641\) 45.0000 1.77739 0.888697 0.458496i \(-0.151612\pi\)
0.888697 + 0.458496i \(0.151612\pi\)
\(642\) 8.00000 0.315735
\(643\) −9.00000 −0.354925 −0.177463 0.984128i \(-0.556789\pi\)
−0.177463 + 0.984128i \(0.556789\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) −4.00000 −0.157256 −0.0786281 0.996904i \(-0.525054\pi\)
−0.0786281 + 0.996904i \(0.525054\pi\)
\(648\) 0 0
\(649\) 5.00000 0.196267
\(650\) −20.0000 −0.784465
\(651\) −2.00000 −0.0783862
\(652\) −22.0000 −0.861586
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) −8.00000 −0.312825
\(655\) 0 0
\(656\) −4.00000 −0.156174
\(657\) −16.0000 −0.624219
\(658\) −10.0000 −0.389841
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) 13.0000 0.505641 0.252821 0.967513i \(-0.418642\pi\)
0.252821 + 0.967513i \(0.418642\pi\)
\(662\) 10.0000 0.388661
\(663\) 8.00000 0.310694
\(664\) 0 0
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) −6.00000 −0.232321
\(668\) −2.00000 −0.0773823
\(669\) −2.00000 −0.0773245
\(670\) 0 0
\(671\) 13.0000 0.501859
\(672\) −8.00000 −0.308607
\(673\) −31.0000 −1.19496 −0.597481 0.801883i \(-0.703832\pi\)
−0.597481 + 0.801883i \(0.703832\pi\)
\(674\) 36.0000 1.38667
\(675\) −5.00000 −0.192450
\(676\) −18.0000 −0.692308
\(677\) 4.00000 0.153732 0.0768662 0.997041i \(-0.475509\pi\)
0.0768662 + 0.997041i \(0.475509\pi\)
\(678\) −28.0000 −1.07533
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −2.00000 −0.0766402
\(682\) −4.00000 −0.153168
\(683\) −16.0000 −0.612223 −0.306111 0.951996i \(-0.599028\pi\)
−0.306111 + 0.951996i \(0.599028\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) 2.00000 0.0763604
\(687\) 13.0000 0.495981
\(688\) 32.0000 1.21999
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 4.00000 0.152057
\(693\) 1.00000 0.0379869
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 20.0000 0.757011
\(699\) 24.0000 0.907763
\(700\) −10.0000 −0.377964
\(701\) −45.0000 −1.69963 −0.849813 0.527084i \(-0.823285\pi\)
−0.849813 + 0.527084i \(0.823285\pi\)
\(702\) 4.00000 0.150970
\(703\) 6.00000 0.226294
\(704\) −8.00000 −0.301511
\(705\) 0 0
\(706\) −44.0000 −1.65596
\(707\) 9.00000 0.338480
\(708\) 10.0000 0.375823
\(709\) 18.0000 0.676004 0.338002 0.941145i \(-0.390249\pi\)
0.338002 + 0.941145i \(0.390249\pi\)
\(710\) 0 0
\(711\) −2.00000 −0.0750059
\(712\) 0 0
\(713\) −2.00000 −0.0749006
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) 32.0000 1.19590
\(717\) −4.00000 −0.149383
\(718\) 32.0000 1.19423
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 11.0000 0.409661
\(722\) −20.0000 −0.744323
\(723\) 15.0000 0.557856
\(724\) −4.00000 −0.148659
\(725\) 30.0000 1.11417
\(726\) −20.0000 −0.742270
\(727\) 23.0000 0.853023 0.426511 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −32.0000 −1.18356
\(732\) 26.0000 0.960988
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 58.0000 2.14082
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) 0 0
\(738\) 2.00000 0.0736210
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) 6.00000 0.220267
\(743\) −3.00000 −0.110059 −0.0550297 0.998485i \(-0.517525\pi\)
−0.0550297 + 0.998485i \(0.517525\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 68.0000 2.48966
\(747\) 6.00000 0.219529
\(748\) 8.00000 0.292509
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 20.0000 0.729325
\(753\) −14.0000 −0.510188
\(754\) −24.0000 −0.874028
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) −12.0000 −0.436147 −0.218074 0.975932i \(-0.569977\pi\)
−0.218074 + 0.975932i \(0.569977\pi\)
\(758\) −20.0000 −0.726433
\(759\) 1.00000 0.0362977
\(760\) 0 0
\(761\) 39.0000 1.41375 0.706874 0.707339i \(-0.250105\pi\)
0.706874 + 0.707339i \(0.250105\pi\)
\(762\) −14.0000 −0.507166
\(763\) −4.00000 −0.144810
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) −60.0000 −2.16789
\(767\) 10.0000 0.361079
\(768\) 16.0000 0.577350
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) −3.00000 −0.108042
\(772\) 26.0000 0.935760
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) −16.0000 −0.575108
\(775\) 10.0000 0.359211
\(776\) 0 0
\(777\) −2.00000 −0.0717496
\(778\) −12.0000 −0.430221
\(779\) −3.00000 −0.107486
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) −6.00000 −0.214423
\(784\) −4.00000 −0.142857
\(785\) 0 0
\(786\) 30.0000 1.07006
\(787\) 39.0000 1.39020 0.695100 0.718913i \(-0.255360\pi\)
0.695100 + 0.718913i \(0.255360\pi\)
\(788\) −12.0000 −0.427482
\(789\) −19.0000 −0.676418
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 26.0000 0.923287
\(794\) 12.0000 0.425864
\(795\) 0 0
\(796\) −22.0000 −0.779769
\(797\) 22.0000 0.779280 0.389640 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) −6.00000 −0.212398
\(799\) −20.0000 −0.707549
\(800\) 40.0000 1.41421
\(801\) 6.00000 0.212000
\(802\) 50.0000 1.76556
\(803\) −16.0000 −0.564628
\(804\) 0 0
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 22.0000 0.774437
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −6.00000 −0.210688 −0.105344 0.994436i \(-0.533594\pi\)
−0.105344 + 0.994436i \(0.533594\pi\)
\(812\) −12.0000 −0.421117
\(813\) −16.0000 −0.561144
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) −16.0000 −0.560112
\(817\) 24.0000 0.839654
\(818\) −4.00000 −0.139857
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −12.0000 −0.418803 −0.209401 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) 38.0000 1.32540
\(823\) −5.00000 −0.174289 −0.0871445 0.996196i \(-0.527774\pi\)
−0.0871445 + 0.996196i \(0.527774\pi\)
\(824\) 0 0
\(825\) −5.00000 −0.174078
\(826\) 10.0000 0.347945
\(827\) −15.0000 −0.521601 −0.260801 0.965393i \(-0.583986\pi\)
−0.260801 + 0.965393i \(0.583986\pi\)
\(828\) 2.00000 0.0695048
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 0 0
\(831\) 31.0000 1.07538
\(832\) −16.0000 −0.554700
\(833\) 4.00000 0.138592
\(834\) −32.0000 −1.10807
\(835\) 0 0
\(836\) −6.00000 −0.207514
\(837\) −2.00000 −0.0691301
\(838\) −36.0000 −1.24360
\(839\) 50.0000 1.72619 0.863096 0.505040i \(-0.168522\pi\)
0.863096 + 0.505040i \(0.168522\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 16.0000 0.551396
\(843\) 6.00000 0.206651
\(844\) −38.0000 −1.30801
\(845\) 0 0
\(846\) −10.0000 −0.343807
\(847\) −10.0000 −0.343604
\(848\) −12.0000 −0.412082
\(849\) −20.0000 −0.686398
\(850\) −40.0000 −1.37199
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) 28.0000 0.958702 0.479351 0.877623i \(-0.340872\pi\)
0.479351 + 0.877623i \(0.340872\pi\)
\(854\) 26.0000 0.889702
\(855\) 0 0
\(856\) 0 0
\(857\) −19.0000 −0.649028 −0.324514 0.945881i \(-0.605201\pi\)
−0.324514 + 0.945881i \(0.605201\pi\)
\(858\) 4.00000 0.136558
\(859\) 56.0000 1.91070 0.955348 0.295484i \(-0.0954809\pi\)
0.955348 + 0.295484i \(0.0954809\pi\)
\(860\) 0 0
\(861\) 1.00000 0.0340799
\(862\) −46.0000 −1.56677
\(863\) 54.0000 1.83818 0.919091 0.394046i \(-0.128925\pi\)
0.919091 + 0.394046i \(0.128925\pi\)
\(864\) −8.00000 −0.272166
\(865\) 0 0
\(866\) −74.0000 −2.51462
\(867\) −1.00000 −0.0339618
\(868\) −4.00000 −0.135769
\(869\) −2.00000 −0.0678454
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) −6.00000 −0.202953
\(875\) 0 0
\(876\) −32.0000 −1.08118
\(877\) 55.0000 1.85722 0.928609 0.371060i \(-0.121005\pi\)
0.928609 + 0.371060i \(0.121005\pi\)
\(878\) −64.0000 −2.15990
\(879\) −16.0000 −0.539667
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 2.00000 0.0673435
\(883\) 48.0000 1.61533 0.807664 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) 16.0000 0.538138
\(885\) 0 0
\(886\) −40.0000 −1.34383
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 0 0
\(889\) −7.00000 −0.234772
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) −4.00000 −0.133930
\(893\) 15.0000 0.501956
\(894\) 34.0000 1.13713
\(895\) 0 0
\(896\) 0 0
\(897\) 2.00000 0.0667781
\(898\) −68.0000 −2.26919
\(899\) 12.0000 0.400222
\(900\) −10.0000 −0.333333
\(901\) 12.0000 0.399778
\(902\) 2.00000 0.0665927
\(903\) −8.00000 −0.266223
\(904\) 0 0
\(905\) 0 0
\(906\) 26.0000 0.863792
\(907\) −48.0000 −1.59381 −0.796907 0.604102i \(-0.793532\pi\)
−0.796907 + 0.604102i \(0.793532\pi\)
\(908\) −4.00000 −0.132745
\(909\) 9.00000 0.298511
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 12.0000 0.397360
\(913\) 6.00000 0.198571
\(914\) −28.0000 −0.926158
\(915\) 0 0
\(916\) 26.0000 0.859064
\(917\) 15.0000 0.495344
\(918\) 8.00000 0.264039
\(919\) 2.00000 0.0659739 0.0329870 0.999456i \(-0.489498\pi\)
0.0329870 + 0.999456i \(0.489498\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) −60.0000 −1.97599
\(923\) 0 0
\(924\) 2.00000 0.0657952
\(925\) 10.0000 0.328798
\(926\) −22.0000 −0.722965
\(927\) 11.0000 0.361287
\(928\) 48.0000 1.57568
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 48.0000 1.57229
\(933\) 21.0000 0.687509
\(934\) −20.0000 −0.654420
\(935\) 0 0
\(936\) 0 0
\(937\) 19.0000 0.620703 0.310351 0.950622i \(-0.399553\pi\)
0.310351 + 0.950622i \(0.399553\pi\)
\(938\) 0 0
\(939\) 1.00000 0.0326338
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) 2.00000 0.0651635
\(943\) 1.00000 0.0325645
\(944\) −20.0000 −0.650945
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 20.0000 0.649913 0.324956 0.945729i \(-0.394650\pi\)
0.324956 + 0.945729i \(0.394650\pi\)
\(948\) −4.00000 −0.129914
\(949\) −32.0000 −1.03876
\(950\) 30.0000 0.973329
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) −53.0000 −1.71684 −0.858419 0.512949i \(-0.828553\pi\)
−0.858419 + 0.512949i \(0.828553\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) −6.00000 −0.193952
\(958\) 8.00000 0.258468
\(959\) 19.0000 0.613542
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −8.00000 −0.257930
\(963\) 4.00000 0.128898
\(964\) 30.0000 0.966235
\(965\) 0 0
\(966\) 2.00000 0.0643489
\(967\) −20.0000 −0.643157 −0.321578 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 0 0
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) −22.0000 −0.706014 −0.353007 0.935621i \(-0.614841\pi\)
−0.353007 + 0.935621i \(0.614841\pi\)
\(972\) 2.00000 0.0641500
\(973\) −16.0000 −0.512936
\(974\) −40.0000 −1.28168
\(975\) −10.0000 −0.320256
\(976\) −52.0000 −1.66448
\(977\) −9.00000 −0.287936 −0.143968 0.989582i \(-0.545986\pi\)
−0.143968 + 0.989582i \(0.545986\pi\)
\(978\) −22.0000 −0.703482
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) −60.0000 −1.91468
\(983\) 10.0000 0.318950 0.159475 0.987202i \(-0.449020\pi\)
0.159475 + 0.987202i \(0.449020\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −48.0000 −1.52863
\(987\) −5.00000 −0.159152
\(988\) −12.0000 −0.381771
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) 13.0000 0.412959 0.206479 0.978451i \(-0.433799\pi\)
0.206479 + 0.978451i \(0.433799\pi\)
\(992\) 16.0000 0.508001
\(993\) 5.00000 0.158670
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 24.0000 0.760088 0.380044 0.924968i \(-0.375909\pi\)
0.380044 + 0.924968i \(0.375909\pi\)
\(998\) −72.0000 −2.27912
\(999\) −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 483.2.a.a.1.1 1
3.2 odd 2 1449.2.a.c.1.1 1
4.3 odd 2 7728.2.a.e.1.1 1
7.6 odd 2 3381.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.a.1.1 1 1.1 even 1 trivial
1449.2.a.c.1.1 1 3.2 odd 2
3381.2.a.m.1.1 1 7.6 odd 2
7728.2.a.e.1.1 1 4.3 odd 2