Properties

Label 6762.2.a.v.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6762.1

$q$-expansion

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

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 \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −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 \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) −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
\(28\) 0 0
\(29\) −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
\(35\) 0 0
\(36\) 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 \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −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 \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −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
\(56\) 0 0
\(57\) 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 \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 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
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −4.00000 −0.464991
\(75\) 11.0000 1.27017
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 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 \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −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 \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −4.00000 −0.421637
\(91\) 0 0
\(92\) 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 \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 11.0000 1.10000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 2.00000 0.198030
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 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
\(112\) 0 0
\(113\) 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
\(119\) 0 0
\(120\) −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
\(126\) 0 0
\(127\) 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
\(133\) 0 0
\(134\) −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 \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 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
\(147\) 0 0
\(148\) 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
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) −4.00000 −0.316228
\(161\) 0 0
\(162\) −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 \(0.5\pi\)
\(168\) 0 0
\(169\) −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 \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 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 \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) 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
\(189\) 0 0
\(190\) −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
\(196\) 0 0
\(197\) 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
\(203\) 0 0
\(204\) −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
\(210\) 0 0
\(211\) −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 \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) −6.00000 −0.399114
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 2.00000 0.132453
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) 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
\(238\) 0 0
\(239\) 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 \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) −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 \(-0.806334\pi\)
−0.820553 + 0.571571i \(0.806334\pi\)
\(252\) 0 0
\(253\) 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 \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 10.0000 0.622573
\(259\) 0 0
\(260\) −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
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 2.00000 0.122169
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −4.00000 −0.243432
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −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
\(280\) 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
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 8.00000 0.474713
\(285\) 8.00000 0.473879
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) −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 \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 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
\(301\) 0 0
\(302\) −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 \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 2.00000 0.113228
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −8.00000 −0.451466
\(315\) 0 0
\(316\) −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
\(322\) 0 0
\(323\) −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
\(329\) 0 0
\(330\) −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
\(336\) 0 0
\(337\) 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
\(343\) 0 0
\(344\) 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 \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −2.00000 −0.106600
\(353\) −34.0000 −1.80964 −0.904819 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 0 0
\(355\) 32.0000 1.69838
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −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
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) −4.00000 −0.209083
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 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
\(378\) 0 0
\(379\) 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 \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 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
\(392\) 0 0
\(393\) 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 \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 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
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 2.00000 0.0990148
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) −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 \(-0.515557\pi\)
−0.0488532 + 0.998806i \(0.515557\pi\)
\(420\) 0 0
\(421\) 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
\(427\) 0 0
\(428\) −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 \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 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 \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) −8.00000 −0.381385
\(441\) 0 0
\(442\) −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
\(448\) 0 0
\(449\) −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
\(455\) 0 0
\(456\) −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 \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −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 \(-0.788195\pi\)
−0.786666 + 0.617379i \(0.788195\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) −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
\(476\) 0 0
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) −40.0000 −1.82765 −0.913823 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) −4.00000 −0.182574
\(481\) −8.00000 −0.364769
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) −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
\(490\) 0 0
\(491\) 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
\(497\) 0 0
\(498\) −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 \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 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 \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 8.00000 0.354246
\(511\) 0 0
\(512\) −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
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 8.00000 0.350823
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 6.00000 0.262613
\(523\) 10.0000 0.437269 0.218635 0.975807i \(-0.429840\pi\)
0.218635 + 0.975807i \(0.429840\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −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
\(539\) 0 0
\(540\) 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
\(546\) 0 0
\(547\) −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
\(553\) 0 0
\(554\) −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
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 10.0000 0.421825
\(563\) 34.0000 1.43293 0.716465 0.697623i \(-0.245759\pi\)
0.716465 + 0.697623i \(0.245759\pi\)
\(564\) 8.00000 0.336861
\(565\) 24.0000 1.00969
\(566\) −6.00000 −0.252199
\(567\) 0 0
\(568\) −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
\(574\) 0 0
\(575\) 11.0000 0.458732
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 13.0000 0.540729
\(579\) −6.00000 −0.249351
\(580\) −24.0000 −0.996546
\(581\) 0 0
\(582\) −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 \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 4.00000 0.164399
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −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 \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 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 \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) −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
\(616\) 0 0
\(617\) 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 \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 24.0000 0.962312
\(623\) 0 0
\(624\) −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
\(630\) 0 0
\(631\) −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
\(637\) 0 0
\(638\) 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 \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) −40.0000 −1.57500
\(646\) 4.00000 0.157378
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) −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
\(658\) 0 0
\(659\) −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 \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) −20.0000 −0.777322
\(663\) 4.00000 0.155347
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) −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
\(672\) 0 0
\(673\) −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 \(-0.743180\pi\)
−0.691796 + 0.722093i \(0.743180\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 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
\(686\) 0 0
\(687\) −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 \(-0.224789\pi\)
0.760836 + 0.648944i \(0.224789\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) −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
\(700\) 0 0
\(701\) 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
\(707\) 0 0
\(708\) 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
\(714\) 0 0
\(715\) −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 \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 4.00000 0.149071
\(721\) 0 0
\(722\) 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 \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 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 \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) −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
\(749\) 0 0
\(750\) −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
\(756\) 0 0
\(757\) 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 \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) −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 \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −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
\(777\) 0 0
\(778\) 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
\(784\) 0 0
\(785\) 32.0000 1.14213
\(786\) 0 0
\(787\) −18.0000 −0.641631 −0.320815 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(788\) 10.0000 0.356235
\(789\) −8.00000 −0.284808
\(790\) 32.0000 1.13851
\(791\) 0 0
\(792\) −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 \(0.5\pi\)
\(798\) 0 0
\(799\) −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
\(805\) 0 0
\(806\) 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 \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) −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
\(819\) 0 0
\(820\) 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 \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) −24.0000 −0.833052
\(831\) 2.00000 0.0693792
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) −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 \(0.5\pi\)
\(840\) 0 0
\(841\) 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
\(847\) 0 0
\(848\) 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 \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 2.00000 0.0683586
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 4.00000 0.136558
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) −40.0000 −1.36399
\(861\) 0 0
\(862\) 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
\(875\) 0 0
\(876\) 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 \(-0.721118\pi\)
−0.640126 + 0.768270i \(0.721118\pi\)
\(882\) 0 0
\(883\) 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 \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) −4.00000 −0.134231
\(889\) 0 0
\(890\) 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
\(896\) 0 0
\(897\) −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
\(903\) 0 0
\(904\) −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
\(910\) 0 0
\(911\) 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
\(924\) 0 0
\(925\) 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 \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −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 \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) −22.0000 −0.717943
\(940\) 32.0000 1.04372
\(941\) 20.0000 0.651981 0.325991 0.945373i \(-0.394302\pi\)
0.325991 + 0.945373i \(0.394302\pi\)
\(942\) −8.00000 −0.260654
\(943\) 10.0000 0.325645
\(944\) 0 0
\(945\) 0 0
\(946\) 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
\(952\) 0 0
\(953\) −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
\(959\) 0 0
\(960\) 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
\(966\) 0 0
\(967\) 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 \(-0.659862\pi\)
−0.481373 + 0.876516i \(0.659862\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −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
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) −8.00000 −0.255290
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) −10.0000 −0.318788
\(985\) 40.0000 1.27451
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) −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
\(994\) 0 0
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) 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 6762.2.a.v.1.1 1
7.6 odd 2 966.2.a.a.1.1 1
21.20 even 2 2898.2.a.u.1.1 1
28.27 even 2 7728.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.a.1.1 1 7.6 odd 2
2898.2.a.u.1.1 1 21.20 even 2
6762.2.a.v.1.1 1 1.1 even 1 trivial
7728.2.a.m.1.1 1 28.27 even 2