Properties

Label 966.2.a.a.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} -4.00000 q^{20} -1.00000 q^{21} -2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{30} -1.00000 q^{32} -2.00000 q^{33} -2.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} +4.00000 q^{37} +2.00000 q^{38} -2.00000 q^{39} +4.00000 q^{40} -10.0000 q^{41} +1.00000 q^{42} -10.0000 q^{43} +2.00000 q^{44} -4.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -11.0000 q^{50} -2.00000 q^{51} +2.00000 q^{52} +1.00000 q^{54} -8.00000 q^{55} -1.00000 q^{56} +2.00000 q^{57} +6.00000 q^{58} +4.00000 q^{60} -4.00000 q^{61} +1.00000 q^{63} +1.00000 q^{64} -8.00000 q^{65} +2.00000 q^{66} +2.00000 q^{67} +2.00000 q^{68} -1.00000 q^{69} +4.00000 q^{70} +8.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} -4.00000 q^{74} -11.0000 q^{75} -2.00000 q^{76} +2.00000 q^{77} +2.00000 q^{78} -8.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -6.00000 q^{83} -1.00000 q^{84} -8.00000 q^{85} +10.0000 q^{86} +6.00000 q^{87} -2.00000 q^{88} +6.00000 q^{89} +4.00000 q^{90} +2.00000 q^{91} +1.00000 q^{92} +8.00000 q^{94} +8.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 4.00000 1.26491
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) 4.00000 1.03280
\(16\) 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
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −4.00000 −0.894427
\(21\) −1.00000 −0.218218
\(22\) −2.00000 −0.426401
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 11.0000 2.20000
\(26\) −2.00000 −0.392232
\(27\) −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 \(0.688081\pi\)
\(30\) −4.00000 −0.730297
\(31\) 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
\(35\) −4.00000 −0.676123
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 2.00000 0.324443
\(39\) −2.00000 −0.320256
\(40\) 4.00000 0.632456
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 1.00000 0.154303
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 2.00000 0.301511
\(45\) −4.00000 −0.596285
\(46\) −1.00000 −0.147442
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −11.0000 −1.55563
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 1.00000 0.136083
\(55\) −8.00000 −1.07872
\(56\) −1.00000 −0.133631
\(57\) 2.00000 0.264906
\(58\) 6.00000 0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 4.00000 0.516398
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −8.00000 −0.992278
\(66\) 2.00000 0.246183
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 2.00000 0.242536
\(69\) −1.00000 −0.120386
\(70\) 4.00000 0.478091
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −4.00000 −0.464991
\(75\) −11.0000 −1.27017
\(76\) −2.00000 −0.229416
\(77\) 2.00000 0.227921
\(78\) 2.00000 0.226455
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −1.00000 −0.109109
\(85\) −8.00000 −0.867722
\(86\) 10.0000 1.07833
\(87\) 6.00000 0.643268
\(88\) −2.00000 −0.213201
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 4.00000 0.421637
\(91\) 2.00000 0.209657
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 8.00000 0.820783
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) 2.00000 0.201008
\(100\) 11.0000 1.10000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 2.00000 0.198030
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) −2.00000 −0.196116
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 8.00000 0.762770
\(111\) −4.00000 −0.379663
\(112\) 1.00000 0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −2.00000 −0.187317
\(115\) −4.00000 −0.373002
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) −4.00000 −0.365148
\(121\) −7.00000 −0.636364
\(122\) 4.00000 0.362143
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) −24.0000 −2.14663
\(126\) −1.00000 −0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 10.0000 0.880451
\(130\) 8.00000 0.701646
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −2.00000 −0.174078
\(133\) −2.00000 −0.173422
\(134\) −2.00000 −0.172774
\(135\) 4.00000 0.344265
\(136\) −2.00000 −0.171499
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 1.00000 0.0851257
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) −4.00000 −0.338062
\(141\) 8.00000 0.673722
\(142\) −8.00000 −0.671345
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 24.0000 1.99309
\(146\) 2.00000 0.165521
\(147\) −1.00000 −0.0824786
\(148\) 4.00000 0.328798
\(149\) −24.0000 −1.96616 −0.983078 0.183186i \(-0.941359\pi\)
−0.983078 + 0.183186i \(0.941359\pi\)
\(150\) 11.0000 0.898146
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 2.00000 0.162221
\(153\) 2.00000 0.161690
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 4.00000 0.316228
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) 24.0000 1.87983 0.939913 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) −10.0000 −0.780869
\(165\) 8.00000 0.622799
\(166\) 6.00000 0.465690
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) 8.00000 0.613572
\(171\) −2.00000 −0.152944
\(172\) −10.0000 −0.762493
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −6.00000 −0.454859
\(175\) 11.0000 0.831522
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) −4.00000 −0.298142
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) −2.00000 −0.148250
\(183\) 4.00000 0.295689
\(184\) −1.00000 −0.0737210
\(185\) −16.0000 −1.17634
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) −8.00000 −0.583460
\(189\) −1.00000 −0.0727393
\(190\) −8.00000 −0.580381
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 2.00000 0.143592
\(195\) 8.00000 0.572892
\(196\) 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
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −11.0000 −0.777817
\(201\) −2.00000 −0.141069
\(202\) 14.0000 0.985037
\(203\) −6.00000 −0.421117
\(204\) −2.00000 −0.140028
\(205\) 40.0000 2.79372
\(206\) 12.0000 0.836080
\(207\) 1.00000 0.0695048
\(208\) 2.00000 0.138675
\(209\) −4.00000 −0.276686
\(210\) −4.00000 −0.276026
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) −8.00000 −0.548151
\(214\) 2.00000 0.136717
\(215\) 40.0000 2.72798
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 16.0000 1.08366
\(219\) 2.00000 0.135147
\(220\) −8.00000 −0.539360
\(221\) 4.00000 0.269069
\(222\) 4.00000 0.268462
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 11.0000 0.733333
\(226\) −6.00000 −0.399114
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 2.00000 0.132453
\(229\) 12.0000 0.792982 0.396491 0.918039i \(-0.370228\pi\)
0.396491 + 0.918039i \(0.370228\pi\)
\(230\) 4.00000 0.263752
\(231\) −2.00000 −0.131590
\(232\) 6.00000 0.393919
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) 32.0000 2.08745
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) −2.00000 −0.129641
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 4.00000 0.258199
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) −4.00000 −0.256074
\(245\) −4.00000 −0.255551
\(246\) −10.0000 −0.637577
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 24.0000 1.51789
\(251\) 26.0000 1.64111 0.820553 0.571571i \(-0.193666\pi\)
0.820553 + 0.571571i \(0.193666\pi\)
\(252\) 1.00000 0.0629941
\(253\) 2.00000 0.125739
\(254\) −8.00000 −0.501965
\(255\) 8.00000 0.500979
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) −10.0000 −0.622573
\(259\) 4.00000 0.248548
\(260\) −8.00000 −0.496139
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) −6.00000 −0.367194
\(268\) 2.00000 0.122169
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −4.00000 −0.243432
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 2.00000 0.121268
\(273\) −2.00000 −0.121046
\(274\) −18.0000 −1.08742
\(275\) 22.0000 1.32665
\(276\) −1.00000 −0.0601929
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 16.0000 0.959616
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) −8.00000 −0.476393
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) 8.00000 0.474713
\(285\) −8.00000 −0.473879
\(286\) −4.00000 −0.236525
\(287\) −10.0000 −0.590281
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −24.0000 −1.40933
\(291\) 2.00000 0.117242
\(292\) −2.00000 −0.117041
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) −2.00000 −0.116052
\(298\) 24.0000 1.39028
\(299\) 2.00000 0.115663
\(300\) −11.0000 −0.635085
\(301\) −10.0000 −0.576390
\(302\) −16.0000 −0.920697
\(303\) 14.0000 0.804279
\(304\) −2.00000 −0.114708
\(305\) 16.0000 0.916157
\(306\) −2.00000 −0.114332
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 2.00000 0.113961
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 2.00000 0.113228
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 8.00000 0.451466
\(315\) −4.00000 −0.225374
\(316\) −8.00000 −0.450035
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) −4.00000 −0.223607
\(321\) 2.00000 0.111629
\(322\) −1.00000 −0.0557278
\(323\) −4.00000 −0.222566
\(324\) 1.00000 0.0555556
\(325\) 22.0000 1.22034
\(326\) −24.0000 −1.32924
\(327\) 16.0000 0.884802
\(328\) 10.0000 0.552158
\(329\) −8.00000 −0.441054
\(330\) −8.00000 −0.440386
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −6.00000 −0.329293
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) −1.00000 −0.0545545
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 9.00000 0.489535
\(339\) −6.00000 −0.325875
\(340\) −8.00000 −0.433861
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) 1.00000 0.0539949
\(344\) 10.0000 0.539164
\(345\) 4.00000 0.215353
\(346\) 2.00000 0.107521
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) 6.00000 0.321634
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) −11.0000 −0.587975
\(351\) −2.00000 −0.106752
\(352\) −2.00000 −0.106600
\(353\) 34.0000 1.80964 0.904819 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) 0 0
\(355\) −32.0000 −1.69838
\(356\) 6.00000 0.317999
\(357\) −2.00000 −0.105851
\(358\) −16.0000 −0.845626
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 4.00000 0.210819
\(361\) −15.0000 −0.789474
\(362\) 20.0000 1.05118
\(363\) 7.00000 0.367405
\(364\) 2.00000 0.104828
\(365\) 8.00000 0.418739
\(366\) −4.00000 −0.209083
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 1.00000 0.0521286
\(369\) −10.0000 −0.520579
\(370\) 16.0000 0.831800
\(371\) 0 0
\(372\) 0 0
\(373\) −36.0000 −1.86401 −0.932005 0.362446i \(-0.881942\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) −4.00000 −0.206835
\(375\) 24.0000 1.23935
\(376\) 8.00000 0.412568
\(377\) −12.0000 −0.618031
\(378\) 1.00000 0.0514344
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 8.00000 0.410391
\(381\) −8.00000 −0.409852
\(382\) 20.0000 1.02329
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 1.00000 0.0510310
\(385\) −8.00000 −0.407718
\(386\) 6.00000 0.305392
\(387\) −10.0000 −0.508329
\(388\) −2.00000 −0.101535
\(389\) −36.0000 −1.82527 −0.912636 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) −8.00000 −0.405096
\(391\) 2.00000 0.101144
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −10.0000 −0.503793
\(395\) 32.0000 1.61009
\(396\) 2.00000 0.100504
\(397\) −38.0000 −1.90717 −0.953583 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) 0 0
\(399\) 2.00000 0.100125
\(400\) 11.0000 0.550000
\(401\) −38.0000 −1.89763 −0.948815 0.315833i \(-0.897716\pi\)
−0.948815 + 0.315833i \(0.897716\pi\)
\(402\) 2.00000 0.0997509
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) −4.00000 −0.198762
\(406\) 6.00000 0.297775
\(407\) 8.00000 0.396545
\(408\) 2.00000 0.0990148
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −40.0000 −1.97546
\(411\) −18.0000 −0.887875
\(412\) −12.0000 −0.591198
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 24.0000 1.17811
\(416\) −2.00000 −0.0980581
\(417\) 16.0000 0.783523
\(418\) 4.00000 0.195646
\(419\) 2.00000 0.0977064 0.0488532 0.998806i \(-0.484443\pi\)
0.0488532 + 0.998806i \(0.484443\pi\)
\(420\) 4.00000 0.195180
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 20.0000 0.973585
\(423\) −8.00000 −0.388973
\(424\) 0 0
\(425\) 22.0000 1.06716
\(426\) 8.00000 0.387601
\(427\) −4.00000 −0.193574
\(428\) −2.00000 −0.0966736
\(429\) −4.00000 −0.193122
\(430\) −40.0000 −1.92897
\(431\) −28.0000 −1.34871 −0.674356 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) −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
\(435\) −24.0000 −1.15071
\(436\) −16.0000 −0.766261
\(437\) −2.00000 −0.0956730
\(438\) −2.00000 −0.0955637
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 8.00000 0.381385
\(441\) 1.00000 0.0476190
\(442\) −4.00000 −0.190261
\(443\) 8.00000 0.380091 0.190046 0.981775i \(-0.439136\pi\)
0.190046 + 0.981775i \(0.439136\pi\)
\(444\) −4.00000 −0.189832
\(445\) −24.0000 −1.13771
\(446\) −16.0000 −0.757622
\(447\) 24.0000 1.13516
\(448\) 1.00000 0.0472456
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) −11.0000 −0.518545
\(451\) −20.0000 −0.941763
\(452\) 6.00000 0.282216
\(453\) −16.0000 −0.751746
\(454\) −14.0000 −0.657053
\(455\) −8.00000 −0.375046
\(456\) −2.00000 −0.0936586
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −12.0000 −0.560723
\(459\) −2.00000 −0.0933520
\(460\) −4.00000 −0.186501
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 2.00000 0.0930484
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 34.0000 1.57333 0.786666 0.617379i \(-0.211805\pi\)
0.786666 + 0.617379i \(0.211805\pi\)
\(468\) 2.00000 0.0924500
\(469\) 2.00000 0.0923514
\(470\) −32.0000 −1.47605
\(471\) 8.00000 0.368621
\(472\) 0 0
\(473\) −20.0000 −0.919601
\(474\) −8.00000 −0.367452
\(475\) −22.0000 −1.00943
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) 40.0000 1.82765 0.913823 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) −4.00000 −0.182574
\(481\) 8.00000 0.364769
\(482\) 18.0000 0.819878
\(483\) −1.00000 −0.0455016
\(484\) −7.00000 −0.318182
\(485\) 8.00000 0.363261
\(486\) 1.00000 0.0453609
\(487\) 40.0000 1.81257 0.906287 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) 4.00000 0.181071
\(489\) −24.0000 −1.08532
\(490\) 4.00000 0.180702
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 10.0000 0.450835
\(493\) −12.0000 −0.540453
\(494\) 4.00000 0.179969
\(495\) −8.00000 −0.359573
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) −6.00000 −0.268866
\(499\) −8.00000 −0.358129 −0.179065 0.983837i \(-0.557307\pi\)
−0.179065 + 0.983837i \(0.557307\pi\)
\(500\) −24.0000 −1.07331
\(501\) 0 0
\(502\) −26.0000 −1.16044
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 56.0000 2.49197
\(506\) −2.00000 −0.0889108
\(507\) 9.00000 0.399704
\(508\) 8.00000 0.354943
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) −8.00000 −0.354246
\(511\) −2.00000 −0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) −14.0000 −0.617514
\(515\) 48.0000 2.11513
\(516\) 10.0000 0.440225
\(517\) −16.0000 −0.703679
\(518\) −4.00000 −0.175750
\(519\) 2.00000 0.0877903
\(520\) 8.00000 0.350823
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 6.00000 0.262613
\(523\) −10.0000 −0.437269 −0.218635 0.975807i \(-0.570160\pi\)
−0.218635 + 0.975807i \(0.570160\pi\)
\(524\) 0 0
\(525\) −11.0000 −0.480079
\(526\) 8.00000 0.348817
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) −20.0000 −0.866296
\(534\) 6.00000 0.259645
\(535\) 8.00000 0.345870
\(536\) −2.00000 −0.0863868
\(537\) −16.0000 −0.690451
\(538\) 18.0000 0.776035
\(539\) 2.00000 0.0861461
\(540\) 4.00000 0.172133
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) −16.0000 −0.687259
\(543\) 20.0000 0.858282
\(544\) −2.00000 −0.0857493
\(545\) 64.0000 2.74146
\(546\) 2.00000 0.0855921
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 18.0000 0.768922
\(549\) −4.00000 −0.170716
\(550\) −22.0000 −0.938083
\(551\) 12.0000 0.511217
\(552\) 1.00000 0.0425628
\(553\) −8.00000 −0.340195
\(554\) −2.00000 −0.0849719
\(555\) 16.0000 0.679162
\(556\) −16.0000 −0.678551
\(557\) −28.0000 −1.18640 −0.593199 0.805056i \(-0.702135\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) −20.0000 −0.845910
\(560\) −4.00000 −0.169031
\(561\) −4.00000 −0.168880
\(562\) 10.0000 0.421825
\(563\) −34.0000 −1.43293 −0.716465 0.697623i \(-0.754241\pi\)
−0.716465 + 0.697623i \(0.754241\pi\)
\(564\) 8.00000 0.336861
\(565\) −24.0000 −1.00969
\(566\) 6.00000 0.252199
\(567\) 1.00000 0.0419961
\(568\) −8.00000 −0.335673
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 8.00000 0.335083
\(571\) 2.00000 0.0836974 0.0418487 0.999124i \(-0.486675\pi\)
0.0418487 + 0.999124i \(0.486675\pi\)
\(572\) 4.00000 0.167248
\(573\) 20.0000 0.835512
\(574\) 10.0000 0.417392
\(575\) 11.0000 0.458732
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 13.0000 0.540729
\(579\) 6.00000 0.249351
\(580\) 24.0000 0.996546
\(581\) −6.00000 −0.248922
\(582\) −2.00000 −0.0829027
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) −8.00000 −0.330759
\(586\) 12.0000 0.495715
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 4.00000 0.164399
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 2.00000 0.0820610
\(595\) −8.00000 −0.327968
\(596\) −24.0000 −0.983078
\(597\) 0 0
\(598\) −2.00000 −0.0817861
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 11.0000 0.449073
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 10.0000 0.407570
\(603\) 2.00000 0.0814463
\(604\) 16.0000 0.651031
\(605\) 28.0000 1.13836
\(606\) −14.0000 −0.568711
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 2.00000 0.0811107
\(609\) 6.00000 0.243132
\(610\) −16.0000 −0.647821
\(611\) −16.0000 −0.647291
\(612\) 2.00000 0.0808452
\(613\) −8.00000 −0.323117 −0.161558 0.986863i \(-0.551652\pi\)
−0.161558 + 0.986863i \(0.551652\pi\)
\(614\) 12.0000 0.484281
\(615\) −40.0000 −1.61296
\(616\) −2.00000 −0.0805823
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) −12.0000 −0.482711
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −24.0000 −0.962312
\(623\) 6.00000 0.240385
\(624\) −2.00000 −0.0800641
\(625\) 41.0000 1.64000
\(626\) −22.0000 −0.879297
\(627\) 4.00000 0.159745
\(628\) −8.00000 −0.319235
\(629\) 8.00000 0.318981
\(630\) 4.00000 0.159364
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 8.00000 0.318223
\(633\) 20.0000 0.794929
\(634\) 30.0000 1.19145
\(635\) −32.0000 −1.26988
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 12.0000 0.475085
\(639\) 8.00000 0.316475
\(640\) 4.00000 0.158114
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) −2.00000 −0.0789337
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 1.00000 0.0394055
\(645\) −40.0000 −1.57500
\(646\) 4.00000 0.157378
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −22.0000 −0.862911
\(651\) 0 0
\(652\) 24.0000 0.939913
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −16.0000 −0.625650
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) −2.00000 −0.0780274
\(658\) 8.00000 0.311872
\(659\) −26.0000 −1.01282 −0.506408 0.862294i \(-0.669027\pi\)
−0.506408 + 0.862294i \(0.669027\pi\)
\(660\) 8.00000 0.311400
\(661\) 28.0000 1.08907 0.544537 0.838737i \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) −20.0000 −0.777322
\(663\) −4.00000 −0.155347
\(664\) 6.00000 0.232845
\(665\) 8.00000 0.310227
\(666\) −4.00000 −0.154997
\(667\) −6.00000 −0.232321
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 8.00000 0.309067
\(671\) −8.00000 −0.308837
\(672\) 1.00000 0.0385758
\(673\) −18.0000 −0.693849 −0.346925 0.937893i \(-0.612774\pi\)
−0.346925 + 0.937893i \(0.612774\pi\)
\(674\) −26.0000 −1.00148
\(675\) −11.0000 −0.423390
\(676\) −9.00000 −0.346154
\(677\) 36.0000 1.38359 0.691796 0.722093i \(-0.256820\pi\)
0.691796 + 0.722093i \(0.256820\pi\)
\(678\) 6.00000 0.230429
\(679\) −2.00000 −0.0767530
\(680\) 8.00000 0.306786
\(681\) −14.0000 −0.536481
\(682\) 0 0
\(683\) 28.0000 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −72.0000 −2.75098
\(686\) −1.00000 −0.0381802
\(687\) −12.0000 −0.457829
\(688\) −10.0000 −0.381246
\(689\) 0 0
\(690\) −4.00000 −0.152277
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 2.00000 0.0759737
\(694\) −16.0000 −0.607352
\(695\) 64.0000 2.42766
\(696\) −6.00000 −0.227429
\(697\) −20.0000 −0.757554
\(698\) −14.0000 −0.529908
\(699\) 6.00000 0.226941
\(700\) 11.0000 0.415761
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 2.00000 0.0754851
\(703\) −8.00000 −0.301726
\(704\) 2.00000 0.0753778
\(705\) −32.0000 −1.20519
\(706\) −34.0000 −1.27961
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) −12.0000 −0.450669 −0.225335 0.974281i \(-0.572348\pi\)
−0.225335 + 0.974281i \(0.572348\pi\)
\(710\) 32.0000 1.20094
\(711\) −8.00000 −0.300023
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 2.00000 0.0748481
\(715\) −16.0000 −0.598366
\(716\) 16.0000 0.597948
\(717\) −8.00000 −0.298765
\(718\) 20.0000 0.746393
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) −4.00000 −0.149071
\(721\) −12.0000 −0.446903
\(722\) 15.0000 0.558242
\(723\) 18.0000 0.669427
\(724\) −20.0000 −0.743294
\(725\) −66.0000 −2.45118
\(726\) −7.00000 −0.259794
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) −8.00000 −0.296093
\(731\) −20.0000 −0.739727
\(732\) 4.00000 0.147844
\(733\) −32.0000 −1.18195 −0.590973 0.806691i \(-0.701256\pi\)
−0.590973 + 0.806691i \(0.701256\pi\)
\(734\) 32.0000 1.18114
\(735\) 4.00000 0.147542
\(736\) −1.00000 −0.0368605
\(737\) 4.00000 0.147342
\(738\) 10.0000 0.368105
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) −16.0000 −0.588172
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 96.0000 3.51717
\(746\) 36.0000 1.31805
\(747\) −6.00000 −0.219529
\(748\) 4.00000 0.146254
\(749\) −2.00000 −0.0730784
\(750\) −24.0000 −0.876356
\(751\) 12.0000 0.437886 0.218943 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) −8.00000 −0.291730
\(753\) −26.0000 −0.947493
\(754\) 12.0000 0.437014
\(755\) −64.0000 −2.32920
\(756\) −1.00000 −0.0363696
\(757\) 48.0000 1.74459 0.872295 0.488980i \(-0.162631\pi\)
0.872295 + 0.488980i \(0.162631\pi\)
\(758\) −2.00000 −0.0726433
\(759\) −2.00000 −0.0725954
\(760\) −8.00000 −0.290191
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 8.00000 0.289809
\(763\) −16.0000 −0.579239
\(764\) −20.0000 −0.723575
\(765\) −8.00000 −0.289241
\(766\) 16.0000 0.578103
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 8.00000 0.288300
\(771\) −14.0000 −0.504198
\(772\) −6.00000 −0.215945
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 10.0000 0.359443
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) −4.00000 −0.143499
\(778\) 36.0000 1.29066
\(779\) 20.0000 0.716574
\(780\) 8.00000 0.286446
\(781\) 16.0000 0.572525
\(782\) −2.00000 −0.0715199
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) 32.0000 1.14213
\(786\) 0 0
\(787\) 18.0000 0.641631 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) 10.0000 0.356235
\(789\) 8.00000 0.284808
\(790\) −32.0000 −1.13851
\(791\) 6.00000 0.213335
\(792\) −2.00000 −0.0710669
\(793\) −8.00000 −0.284088
\(794\) 38.0000 1.34857
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) −2.00000 −0.0707992
\(799\) −16.0000 −0.566039
\(800\) −11.0000 −0.388909
\(801\) 6.00000 0.212000
\(802\) 38.0000 1.34183
\(803\) −4.00000 −0.141157
\(804\) −2.00000 −0.0705346
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 14.0000 0.492518
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 4.00000 0.140546
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) −6.00000 −0.210559
\(813\) −16.0000 −0.561144
\(814\) −8.00000 −0.280400
\(815\) −96.0000 −3.36273
\(816\) −2.00000 −0.0700140
\(817\) 20.0000 0.699711
\(818\) 6.00000 0.209785
\(819\) 2.00000 0.0698857
\(820\) 40.0000 1.39686
\(821\) −50.0000 −1.74501 −0.872506 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 18.0000 0.627822
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 12.0000 0.418040
\(825\) −22.0000 −0.765942
\(826\) 0 0
\(827\) 54.0000 1.87776 0.938882 0.344239i \(-0.111863\pi\)
0.938882 + 0.344239i \(0.111863\pi\)
\(828\) 1.00000 0.0347524
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) −24.0000 −0.833052
\(831\) −2.00000 −0.0693792
\(832\) 2.00000 0.0693375
\(833\) 2.00000 0.0692959
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) −2.00000 −0.0690889
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −4.00000 −0.138013
\(841\) 7.00000 0.241379
\(842\) −4.00000 −0.137849
\(843\) 10.0000 0.344418
\(844\) −20.0000 −0.688428
\(845\) 36.0000 1.23844
\(846\) 8.00000 0.275046
\(847\) −7.00000 −0.240523
\(848\) 0 0
\(849\) 6.00000 0.205919
\(850\) −22.0000 −0.754594
\(851\) 4.00000 0.137118
\(852\) −8.00000 −0.274075
\(853\) 42.0000 1.43805 0.719026 0.694983i \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) 4.00000 0.136877
\(855\) 8.00000 0.273594
\(856\) 2.00000 0.0683586
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 4.00000 0.136558
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 40.0000 1.36399
\(861\) 10.0000 0.340799
\(862\) 28.0000 0.953684
\(863\) −40.0000 −1.36162 −0.680808 0.732462i \(-0.738371\pi\)
−0.680808 + 0.732462i \(0.738371\pi\)
\(864\) 1.00000 0.0340207
\(865\) 8.00000 0.272008
\(866\) 34.0000 1.15537
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 24.0000 0.813676
\(871\) 4.00000 0.135535
\(872\) 16.0000 0.541828
\(873\) −2.00000 −0.0676897
\(874\) 2.00000 0.0676510
\(875\) −24.0000 −0.811348
\(876\) 2.00000 0.0675737
\(877\) 58.0000 1.95852 0.979260 0.202606i \(-0.0649409\pi\)
0.979260 + 0.202606i \(0.0649409\pi\)
\(878\) 8.00000 0.269987
\(879\) 12.0000 0.404750
\(880\) −8.00000 −0.269680
\(881\) 38.0000 1.28025 0.640126 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −8.00000 −0.268765
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 4.00000 0.134231
\(889\) 8.00000 0.268311
\(890\) 24.0000 0.804482
\(891\) 2.00000 0.0670025
\(892\) 16.0000 0.535720
\(893\) 16.0000 0.535420
\(894\) −24.0000 −0.802680
\(895\) −64.0000 −2.13928
\(896\) −1.00000 −0.0334077
\(897\) −2.00000 −0.0667781
\(898\) 10.0000 0.333704
\(899\) 0 0
\(900\) 11.0000 0.366667
\(901\) 0 0
\(902\) 20.0000 0.665927
\(903\) 10.0000 0.332779
\(904\) −6.00000 −0.199557
\(905\) 80.0000 2.65929
\(906\) 16.0000 0.531564
\(907\) 38.0000 1.26177 0.630885 0.775877i \(-0.282692\pi\)
0.630885 + 0.775877i \(0.282692\pi\)
\(908\) 14.0000 0.464606
\(909\) −14.0000 −0.464351
\(910\) 8.00000 0.265197
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 2.00000 0.0662266
\(913\) −12.0000 −0.397142
\(914\) −6.00000 −0.198462
\(915\) −16.0000 −0.528944
\(916\) 12.0000 0.396491
\(917\) 0 0
\(918\) 2.00000 0.0660098
\(919\) 48.0000 1.58337 0.791687 0.610927i \(-0.209203\pi\)
0.791687 + 0.610927i \(0.209203\pi\)
\(920\) 4.00000 0.131876
\(921\) 12.0000 0.395413
\(922\) −30.0000 −0.987997
\(923\) 16.0000 0.526646
\(924\) −2.00000 −0.0657952
\(925\) 44.0000 1.44671
\(926\) 32.0000 1.05159
\(927\) −12.0000 −0.394132
\(928\) 6.00000 0.196960
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) −6.00000 −0.196537
\(933\) −24.0000 −0.785725
\(934\) −34.0000 −1.11251
\(935\) −16.0000 −0.523256
\(936\) −2.00000 −0.0653720
\(937\) 34.0000 1.11073 0.555366 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) −2.00000 −0.0653023
\(939\) −22.0000 −0.717943
\(940\) 32.0000 1.04372
\(941\) −20.0000 −0.651981 −0.325991 0.945373i \(-0.605698\pi\)
−0.325991 + 0.945373i \(0.605698\pi\)
\(942\) −8.00000 −0.260654
\(943\) −10.0000 −0.325645
\(944\) 0 0
\(945\) 4.00000 0.130120
\(946\) 20.0000 0.650256
\(947\) −56.0000 −1.81976 −0.909878 0.414876i \(-0.863825\pi\)
−0.909878 + 0.414876i \(0.863825\pi\)
\(948\) 8.00000 0.259828
\(949\) −4.00000 −0.129845
\(950\) 22.0000 0.713774
\(951\) 30.0000 0.972817
\(952\) −2.00000 −0.0648204
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 0 0
\(955\) 80.0000 2.58874
\(956\) 8.00000 0.258738
\(957\) 12.0000 0.387905
\(958\) −40.0000 −1.29234
\(959\) 18.0000 0.581250
\(960\) 4.00000 0.129099
\(961\) −31.0000 −1.00000
\(962\) −8.00000 −0.257930
\(963\) −2.00000 −0.0644491
\(964\) −18.0000 −0.579741
\(965\) 24.0000 0.772587
\(966\) 1.00000 0.0321745
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) 7.00000 0.224989
\(969\) 4.00000 0.128499
\(970\) −8.00000 −0.256865
\(971\) 30.0000 0.962746 0.481373 0.876516i \(-0.340138\pi\)
0.481373 + 0.876516i \(0.340138\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −16.0000 −0.512936
\(974\) −40.0000 −1.28168
\(975\) −22.0000 −0.704564
\(976\) −4.00000 −0.128037
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 24.0000 0.767435
\(979\) 12.0000 0.383522
\(980\) −4.00000 −0.127775
\(981\) −16.0000 −0.510841
\(982\) −8.00000 −0.255290
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) −10.0000 −0.318788
\(985\) −40.0000 −1.27451
\(986\) 12.0000 0.382158
\(987\) 8.00000 0.254643
\(988\) −4.00000 −0.127257
\(989\) −10.0000 −0.317982
\(990\) 8.00000 0.254257
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) 8.00000 0.253236
\(999\) −4.00000 −0.126554
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.a.1.1 1
3.2 odd 2 2898.2.a.u.1.1 1
4.3 odd 2 7728.2.a.m.1.1 1
7.6 odd 2 6762.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.a.1.1 1 1.1 even 1 trivial
2898.2.a.u.1.1 1 3.2 odd 2
6762.2.a.v.1.1 1 7.6 odd 2
7728.2.a.m.1.1 1 4.3 odd 2