Properties

Label 966.2.a.e.1.1
Level $966$
Weight $2$
Character 966.1
Self dual yes
Analytic conductor $7.714$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.71354883526\)
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 \(\chi\) \(=\) 966.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} -1.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{32} -2.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +6.00000 q^{38} -6.00000 q^{39} +6.00000 q^{41} +1.00000 q^{42} -6.00000 q^{43} -2.00000 q^{44} -1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +5.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} -4.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} -6.00000 q^{57} +6.00000 q^{58} -8.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -2.00000 q^{67} +2.00000 q^{68} +1.00000 q^{69} -1.00000 q^{72} -2.00000 q^{73} -5.00000 q^{75} -6.00000 q^{76} +2.00000 q^{77} +6.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -2.00000 q^{83} -1.00000 q^{84} +6.00000 q^{86} -6.00000 q^{87} +2.00000 q^{88} -2.00000 q^{89} +6.00000 q^{91} +1.00000 q^{92} -8.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 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
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) 6.00000 1.17670
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 6.00000 0.973329
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 1.00000 0.154303
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 5.00000 0.707107
\(51\) 2.00000 0.280056
\(52\) −6.00000 −0.832050
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −6.00000 −0.794719
\(58\) 6.00000 0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 2.00000 0.242536
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) −5.00000 −0.577350
\(76\) −6.00000 −0.688247
\(77\) 2.00000 0.227921
\(78\) 6.00000 0.679366
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −2.00000 −0.219529 −0.109764 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) −6.00000 −0.643268
\(88\) 2.00000 0.213201
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −1.00000 −0.101015
\(99\) −2.00000 −0.201008
\(100\) −5.00000 −0.500000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −2.00000 −0.198030
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 1.00000 0.0962250
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 6.00000 0.561951
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 8.00000 0.724286
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −2.00000 −0.174078
\(133\) 6.00000 0.520266
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 5.00000 0.408248
\(151\) −24.0000 −1.95309 −0.976546 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) 6.00000 0.486664
\(153\) 2.00000 0.161690
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) −20.0000 −1.59617 −0.798087 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) −8.00000 −0.636446
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) −1.00000 −0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 1.00000 0.0771517
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) −6.00000 −0.457496
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 6.00000 0.454859
\(175\) 5.00000 0.377964
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) −8.00000 −0.597948 −0.298974 0.954261i \(-0.596644\pi\)
−0.298974 + 0.954261i \(0.596644\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) −6.00000 −0.444750
\(183\) −8.00000 −0.591377
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 8.00000 0.583460
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 2.00000 0.142134
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 5.00000 0.353553
\(201\) −2.00000 −0.141069
\(202\) 14.0000 0.985037
\(203\) 6.00000 0.421117
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 1.00000 0.0695048
\(208\) −6.00000 −0.416025
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −4.00000 −0.274721
\(213\) 0 0
\(214\) −2.00000 −0.136717
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 12.0000 0.812743
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 1.00000 0.0668153
\(225\) −5.00000 −0.333333
\(226\) −6.00000 −0.399114
\(227\) 26.0000 1.72568 0.862840 0.505477i \(-0.168683\pi\)
0.862840 + 0.505477i \(0.168683\pi\)
\(228\) −6.00000 −0.397360
\(229\) 24.0000 1.58596 0.792982 0.609245i \(-0.208527\pi\)
0.792982 + 0.609245i \(0.208527\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 6.00000 0.393919
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 2.00000 0.129641
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 36.0000 2.29063
\(248\) 0 0
\(249\) −2.00000 −0.126745
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −2.00000 −0.125739
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 6.00000 0.373544
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) −6.00000 −0.367884
\(267\) −2.00000 −0.122398
\(268\) −2.00000 −0.122169
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 2.00000 0.121268
\(273\) 6.00000 0.363137
\(274\) 6.00000 0.362473
\(275\) 10.0000 0.603023
\(276\) 1.00000 0.0601929
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) −8.00000 −0.476393
\(283\) 30.0000 1.78331 0.891657 0.452711i \(-0.149543\pi\)
0.891657 + 0.452711i \(0.149543\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) −6.00000 −0.354169
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) −2.00000 −0.117041
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) −4.00000 −0.231714
\(299\) −6.00000 −0.346989
\(300\) −5.00000 −0.288675
\(301\) 6.00000 0.345834
\(302\) 24.0000 1.38104
\(303\) −14.0000 −0.804279
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 2.00000 0.113961
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 6.00000 0.339683
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 20.0000 1.12867
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 4.00000 0.224309
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) 2.00000 0.111629
\(322\) 1.00000 0.0557278
\(323\) −12.0000 −0.667698
\(324\) 1.00000 0.0555556
\(325\) 30.0000 1.66410
\(326\) −16.0000 −0.886158
\(327\) −12.0000 −0.663602
\(328\) −6.00000 −0.331295
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −2.00000 −0.109764
\(333\) 0 0
\(334\) −16.0000 −0.875481
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) −23.0000 −1.25104
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) −1.00000 −0.0539949
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 8.00000 0.429463 0.214731 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) −6.00000 −0.321634
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −5.00000 −0.267261
\(351\) −6.00000 −0.320256
\(352\) 2.00000 0.106600
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) −2.00000 −0.105851
\(358\) 8.00000 0.422813
\(359\) −28.0000 −1.47778 −0.738892 0.673824i \(-0.764651\pi\)
−0.738892 + 0.673824i \(0.764651\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 8.00000 0.420471
\(363\) −7.00000 −0.367405
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) 8.00000 0.418167
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 1.00000 0.0521286
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) −8.00000 −0.414224 −0.207112 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 36.0000 1.85409
\(378\) 1.00000 0.0514344
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 12.0000 0.613973
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −6.00000 −0.304997
\(388\) 14.0000 0.710742
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −16.0000 −0.802008
\(399\) 6.00000 0.300376
\(400\) −5.00000 −0.250000
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 2.00000 0.0997509
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) −2.00000 −0.0990148
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) −12.0000 −0.586939
\(419\) 38.0000 1.85642 0.928211 0.372055i \(-0.121347\pi\)
0.928211 + 0.372055i \(0.121347\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) −12.0000 −0.584151
\(423\) 8.00000 0.388973
\(424\) 4.00000 0.194257
\(425\) −10.0000 −0.485071
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) 2.00000 0.0966736
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) 1.00000 0.0481125
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −12.0000 −0.574696
\(437\) −6.00000 −0.287019
\(438\) 2.00000 0.0955637
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) −32.0000 −1.52037 −0.760183 0.649709i \(-0.774891\pi\)
−0.760183 + 0.649709i \(0.774891\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 4.00000 0.189194
\(448\) −1.00000 −0.0472456
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 5.00000 0.235702
\(451\) −12.0000 −0.565058
\(452\) 6.00000 0.282216
\(453\) −24.0000 −1.12762
\(454\) −26.0000 −1.22024
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) −24.0000 −1.12145
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) −2.00000 −0.0930484
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) 22.0000 1.01804 0.509019 0.860755i \(-0.330008\pi\)
0.509019 + 0.860755i \(0.330008\pi\)
\(468\) −6.00000 −0.277350
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) −20.0000 −0.921551
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) −8.00000 −0.367452
\(475\) 30.0000 1.37649
\(476\) −2.00000 −0.0916698
\(477\) −4.00000 −0.183147
\(478\) −24.0000 −1.09773
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 2.00000 0.0910975
\(483\) −1.00000 −0.0455016
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 8.00000 0.362143
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000 0.270501
\(493\) −12.0000 −0.540453
\(494\) −36.0000 −1.61972
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 2.00000 0.0896221
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) 2.00000 0.0892644
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) 23.0000 1.02147
\(508\) −8.00000 −0.354943
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) −1.00000 −0.0441942
\(513\) −6.00000 −0.264906
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) −6.00000 −0.264135
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 6.00000 0.262613
\(523\) 18.0000 0.787085 0.393543 0.919306i \(-0.371249\pi\)
0.393543 + 0.919306i \(0.371249\pi\)
\(524\) 0 0
\(525\) 5.00000 0.218218
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 6.00000 0.260133
\(533\) −36.0000 −1.55933
\(534\) 2.00000 0.0865485
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) −8.00000 −0.345225
\(538\) −6.00000 −0.258678
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) −8.00000 −0.343629
\(543\) −8.00000 −0.343313
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) −6.00000 −0.256776
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −6.00000 −0.256307
\(549\) −8.00000 −0.341432
\(550\) −10.0000 −0.426401
\(551\) 36.0000 1.53365
\(552\) −1.00000 −0.0425628
\(553\) −8.00000 −0.340195
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) 0 0
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) 36.0000 1.52264
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 10.0000 0.421825
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −30.0000 −1.26099
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −46.0000 −1.92842 −0.964210 0.265139i \(-0.914582\pi\)
−0.964210 + 0.265139i \(0.914582\pi\)
\(570\) 0 0
\(571\) −2.00000 −0.0836974 −0.0418487 0.999124i \(-0.513325\pi\)
−0.0418487 + 0.999124i \(0.513325\pi\)
\(572\) 12.0000 0.501745
\(573\) −12.0000 −0.501307
\(574\) 6.00000 0.250435
\(575\) −5.00000 −0.208514
\(576\) 1.00000 0.0416667
\(577\) 6.00000 0.249783 0.124892 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) 13.0000 0.540729
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) 2.00000 0.0829740
\(582\) −14.0000 −0.580319
\(583\) 8.00000 0.331326
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 48.0000 1.98117 0.990586 0.136892i \(-0.0437113\pi\)
0.990586 + 0.136892i \(0.0437113\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 0 0
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) 16.0000 0.654836
\(598\) 6.00000 0.245358
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 5.00000 0.204124
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) −6.00000 −0.244542
\(603\) −2.00000 −0.0814463
\(604\) −24.0000 −0.976546
\(605\) 0 0
\(606\) 14.0000 0.568711
\(607\) 48.0000 1.94826 0.974130 0.225989i \(-0.0725612\pi\)
0.974130 + 0.225989i \(0.0725612\pi\)
\(608\) 6.00000 0.243332
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 2.00000 0.0808452
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 4.00000 0.160904
\(619\) 14.0000 0.562708 0.281354 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 16.0000 0.641542
\(623\) 2.00000 0.0801283
\(624\) −6.00000 −0.240192
\(625\) 25.0000 1.00000
\(626\) 26.0000 1.03917
\(627\) 12.0000 0.479234
\(628\) −20.0000 −0.798087
\(629\) 0 0
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) −8.00000 −0.318223
\(633\) 12.0000 0.476957
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) −4.00000 −0.158610
\(637\) −6.00000 −0.237729
\(638\) −12.0000 −0.475085
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −2.00000 −0.0789337
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −30.0000 −1.17670
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) 22.0000 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) 12.0000 0.469237
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −2.00000 −0.0780274
\(658\) 8.00000 0.311872
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −8.00000 −0.311164 −0.155582 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) −20.0000 −0.777322
\(663\) −12.0000 −0.466041
\(664\) 2.00000 0.0776151
\(665\) 0 0
\(666\) 0 0
\(667\) −6.00000 −0.232321
\(668\) 16.0000 0.619059
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 16.0000 0.617673
\(672\) 1.00000 0.0385758
\(673\) −50.0000 −1.92736 −0.963679 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) 6.00000 0.231111
\(675\) −5.00000 −0.192450
\(676\) 23.0000 0.884615
\(677\) −8.00000 −0.307465 −0.153732 0.988113i \(-0.549129\pi\)
−0.153732 + 0.988113i \(0.549129\pi\)
\(678\) −6.00000 −0.230429
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) 26.0000 0.996322
\(682\) 0 0
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 24.0000 0.915657
\(688\) −6.00000 −0.228748
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 6.00000 0.228086
\(693\) 2.00000 0.0759737
\(694\) −8.00000 −0.303676
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 12.0000 0.454532
\(698\) 2.00000 0.0757011
\(699\) 26.0000 0.983410
\(700\) 5.00000 0.188982
\(701\) 24.0000 0.906467 0.453234 0.891392i \(-0.350270\pi\)
0.453234 + 0.891392i \(0.350270\pi\)
\(702\) 6.00000 0.226455
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) 14.0000 0.526524
\(708\) 0 0
\(709\) 32.0000 1.20179 0.600893 0.799330i \(-0.294812\pi\)
0.600893 + 0.799330i \(0.294812\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 2.00000 0.0749532
\(713\) 0 0
\(714\) 2.00000 0.0748481
\(715\) 0 0
\(716\) −8.00000 −0.298974
\(717\) 24.0000 0.896296
\(718\) 28.0000 1.04495
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) −17.0000 −0.632674
\(723\) −2.00000 −0.0743808
\(724\) −8.00000 −0.297318
\(725\) 30.0000 1.11417
\(726\) 7.00000 0.259794
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) −6.00000 −0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) −8.00000 −0.295689
\(733\) 20.0000 0.738717 0.369358 0.929287i \(-0.379577\pi\)
0.369358 + 0.929287i \(0.379577\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 4.00000 0.147342
\(738\) −6.00000 −0.220863
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 0 0
\(741\) 36.0000 1.32249
\(742\) −4.00000 −0.146845
\(743\) −32.0000 −1.17397 −0.586983 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 8.00000 0.292901
\(747\) −2.00000 −0.0731762
\(748\) −4.00000 −0.146254
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 8.00000 0.291730
\(753\) −2.00000 −0.0728841
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) −28.0000 −1.01768 −0.508839 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(758\) 34.0000 1.23494
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 8.00000 0.289809
\(763\) 12.0000 0.434429
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 10.0000 0.359908
\(773\) 28.0000 1.00709 0.503545 0.863969i \(-0.332029\pi\)
0.503545 + 0.863969i \(0.332029\pi\)
\(774\) 6.00000 0.215666
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) 0 0
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) 0 0
\(782\) −2.00000 −0.0715199
\(783\) −6.00000 −0.214423
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) −10.0000 −0.356462 −0.178231 0.983989i \(-0.557037\pi\)
−0.178231 + 0.983989i \(0.557037\pi\)
\(788\) 10.0000 0.356235
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 2.00000 0.0710669
\(793\) 48.0000 1.70453
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −52.0000 −1.84193 −0.920967 0.389640i \(-0.872599\pi\)
−0.920967 + 0.389640i \(0.872599\pi\)
\(798\) −6.00000 −0.212398
\(799\) 16.0000 0.566039
\(800\) 5.00000 0.176777
\(801\) −2.00000 −0.0706665
\(802\) −2.00000 −0.0706225
\(803\) 4.00000 0.141157
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 14.0000 0.492518
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 0 0
\(811\) −48.0000 −1.68551 −0.842754 0.538299i \(-0.819067\pi\)
−0.842754 + 0.538299i \(0.819067\pi\)
\(812\) 6.00000 0.210559
\(813\) 8.00000 0.280572
\(814\) 0 0
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) 36.0000 1.25948
\(818\) −10.0000 −0.349642
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 6.00000 0.209274
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 4.00000 0.139347
\(825\) 10.0000 0.348155
\(826\) 0 0
\(827\) 42.0000 1.46048 0.730242 0.683189i \(-0.239408\pi\)
0.730242 + 0.683189i \(0.239408\pi\)
\(828\) 1.00000 0.0347524
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) −6.00000 −0.208013
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) 0 0
\(838\) −38.0000 −1.31269
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −10.0000 −0.344418
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 7.00000 0.240523
\(848\) −4.00000 −0.137361
\(849\) 30.0000 1.02960
\(850\) 10.0000 0.342997
\(851\) 0 0
\(852\) 0 0
\(853\) 34.0000 1.16414 0.582069 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) −12.0000 −0.409673
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 36.0000 1.22616
\(863\) −56.0000 −1.90626 −0.953131 0.302558i \(-0.902160\pi\)
−0.953131 + 0.302558i \(0.902160\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 12.0000 0.406371
\(873\) 14.0000 0.473828
\(874\) 6.00000 0.202953
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 16.0000 0.539974
\(879\) −24.0000 −0.809500
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 32.0000 1.07506
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 16.0000 0.535720
\(893\) −48.0000 −1.60626
\(894\) −4.00000 −0.133780
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) −6.00000 −0.200334
\(898\) −6.00000 −0.200223
\(899\) 0 0
\(900\) −5.00000 −0.166667
\(901\) −8.00000 −0.266519
\(902\) 12.0000 0.399556
\(903\) 6.00000 0.199667
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 24.0000 0.797347
\(907\) −38.0000 −1.26177 −0.630885 0.775877i \(-0.717308\pi\)
−0.630885 + 0.775877i \(0.717308\pi\)
\(908\) 26.0000 0.862840
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 44.0000 1.45779 0.728893 0.684628i \(-0.240035\pi\)
0.728893 + 0.684628i \(0.240035\pi\)
\(912\) −6.00000 −0.198680
\(913\) 4.00000 0.132381
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) 24.0000 0.792982
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 10.0000 0.329332
\(923\) 0 0
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) −16.0000 −0.525793
\(927\) −4.00000 −0.131377
\(928\) 6.00000 0.196960
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 26.0000 0.851658
\(933\) −16.0000 −0.523816
\(934\) −22.0000 −0.719862
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) −2.00000 −0.0653023
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) 20.0000 0.651635
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 8.00000 0.259828
\(949\) 12.0000 0.389536
\(950\) −30.0000 −0.973329
\(951\) 2.00000 0.0648544
\(952\) 2.00000 0.0648204
\(953\) 58.0000 1.87880 0.939402 0.342817i \(-0.111381\pi\)
0.939402 + 0.342817i \(0.111381\pi\)
\(954\) 4.00000 0.129505
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 12.0000 0.387905
\(958\) 24.0000 0.775405
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 2.00000 0.0644491
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 1.00000 0.0321745
\(967\) −56.0000 −1.80084 −0.900419 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) 7.00000 0.224989
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) 30.0000 0.960769
\(976\) −8.00000 −0.256074
\(977\) 10.0000 0.319928 0.159964 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) −16.0000 −0.511624
\(979\) 4.00000 0.127841
\(980\) 0 0
\(981\) −12.0000 −0.383131
\(982\) 0 0
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) −8.00000 −0.254643
\(988\) 36.0000 1.14531
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) −2.00000 −0.0633724
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −16.0000 −0.506471
\(999\) 0 0
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 966.2.a.e.1.1 1
3.2 odd 2 2898.2.a.n.1.1 1
4.3 odd 2 7728.2.a.g.1.1 1
7.6 odd 2 6762.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.e.1.1 1 1.1 even 1 trivial
2898.2.a.n.1.1 1 3.2 odd 2
6762.2.a.f.1.1 1 7.6 odd 2
7728.2.a.g.1.1 1 4.3 odd 2