Properties

Label 2166.2.a.b.1.1
Level $2166$
Weight $2$
Character 2166.1
Self dual yes
Analytic conductor $17.296$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.2955970778\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2166.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} -3.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -1.00000 q^{21} +2.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} +3.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -3.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -5.00000 q^{37} +3.00000 q^{39} +4.00000 q^{41} +1.00000 q^{42} -9.00000 q^{43} -2.00000 q^{44} -4.00000 q^{46} +10.0000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} -4.00000 q^{51} -3.00000 q^{52} -4.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -14.0000 q^{59} +11.0000 q^{61} +3.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +3.00000 q^{67} +4.00000 q^{68} -4.00000 q^{69} +14.0000 q^{71} -1.00000 q^{72} -11.0000 q^{73} +5.00000 q^{74} +5.00000 q^{75} -2.00000 q^{77} -3.00000 q^{78} +1.00000 q^{79} +1.00000 q^{81} -4.00000 q^{82} +8.00000 q^{83} -1.00000 q^{84} +9.00000 q^{86} +2.00000 q^{88} -14.0000 q^{89} -3.00000 q^{91} +4.00000 q^{92} +3.00000 q^{93} -10.0000 q^{94} +1.00000 q^{96} +2.00000 q^{97} +6.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −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
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 2.00000 0.426401
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) 3.00000 0.588348
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 1.00000 0.154303
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.00000 −0.857143
\(50\) 5.00000 0.707107
\(51\) −4.00000 −0.560112
\(52\) −3.00000 −0.416025
\(53\) −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
\(57\) 0 0
\(58\) 0 0
\(59\) −14.0000 −1.82264 −0.911322 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) 3.00000 0.381000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 3.00000 0.366508 0.183254 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) 4.00000 0.485071
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) −1.00000 −0.117851
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 5.00000 0.581238
\(75\) 5.00000 0.577350
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) −3.00000 −0.339683
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −4.00000 −0.441726
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 9.00000 0.970495
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 4.00000 0.417029
\(93\) 3.00000 0.311086
\(94\) −10.0000 −1.03142
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 6.00000 0.606092
\(99\) −2.00000 −0.201008
\(100\) −5.00000 −0.500000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 4.00000 0.396059
\(103\) −3.00000 −0.295599 −0.147799 0.989017i \(-0.547219\pi\)
−0.147799 + 0.989017i \(0.547219\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) −10.0000 −0.966736 −0.483368 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 5.00000 0.474579
\(112\) 1.00000 0.0944911
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.00000 −0.277350
\(118\) 14.0000 1.28880
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −11.0000 −0.995893
\(123\) −4.00000 −0.360668
\(124\) −3.00000 −0.269408
\(125\) 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
\(129\) 9.00000 0.792406
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) −3.00000 −0.259161
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 4.00000 0.340503
\(139\) −19.0000 −1.61156 −0.805779 0.592216i \(-0.798253\pi\)
−0.805779 + 0.592216i \(0.798253\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) −14.0000 −1.17485
\(143\) 6.00000 0.501745
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 11.0000 0.910366
\(147\) 6.00000 0.494872
\(148\) −5.00000 −0.410997
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) −5.00000 −0.408248
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 3.00000 0.240192
\(157\) −21.0000 −1.67598 −0.837991 0.545684i \(-0.816270\pi\)
−0.837991 + 0.545684i \(0.816270\pi\)
\(158\) −1.00000 −0.0795557
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) −1.00000 −0.0785674
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 4.00000 0.312348
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 1.00000 0.0771517
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) −9.00000 −0.686244
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 0 0
\(175\) −5.00000 −0.377964
\(176\) −2.00000 −0.150756
\(177\) 14.0000 1.05230
\(178\) 14.0000 1.04934
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 3.00000 0.222375
\(183\) −11.0000 −0.813143
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −3.00000 −0.219971
\(187\) −8.00000 −0.585018
\(188\) 10.0000 0.729325
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −13.0000 −0.935760 −0.467880 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 2.00000 0.142134
\(199\) 5.00000 0.354441 0.177220 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) 5.00000 0.353553
\(201\) −3.00000 −0.211604
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 3.00000 0.209020
\(207\) 4.00000 0.278019
\(208\) −3.00000 −0.208013
\(209\) 0 0
\(210\) 0 0
\(211\) 25.0000 1.72107 0.860535 0.509390i \(-0.170129\pi\)
0.860535 + 0.509390i \(0.170129\pi\)
\(212\) −4.00000 −0.274721
\(213\) −14.0000 −0.959264
\(214\) 10.0000 0.683586
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −3.00000 −0.203653
\(218\) −14.0000 −0.948200
\(219\) 11.0000 0.743311
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) −5.00000 −0.335578
\(223\) 9.00000 0.602685 0.301342 0.953516i \(-0.402565\pi\)
0.301342 + 0.953516i \(0.402565\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −5.00000 −0.333333
\(226\) −10.0000 −0.665190
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) −11.0000 −0.726900 −0.363450 0.931614i \(-0.618401\pi\)
−0.363450 + 0.931614i \(0.618401\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 3.00000 0.196116
\(235\) 0 0
\(236\) −14.0000 −0.911322
\(237\) −1.00000 −0.0649570
\(238\) −4.00000 −0.259281
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 25.0000 1.61039 0.805196 0.593009i \(-0.202060\pi\)
0.805196 + 0.593009i \(0.202060\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) 11.0000 0.704203
\(245\) 0 0
\(246\) 4.00000 0.255031
\(247\) 0 0
\(248\) 3.00000 0.190500
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 1.00000 0.0629941
\(253\) −8.00000 −0.502956
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) −9.00000 −0.560316
\(259\) −5.00000 −0.310685
\(260\) 0 0
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) 14.0000 0.856786
\(268\) 3.00000 0.183254
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 4.00000 0.242536
\(273\) 3.00000 0.181568
\(274\) 6.00000 0.362473
\(275\) 10.0000 0.603023
\(276\) −4.00000 −0.240772
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 19.0000 1.13954
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 10.0000 0.595491
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 14.0000 0.830747
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 4.00000 0.236113
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) −11.0000 −0.643726
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) −6.00000 −0.349927
\(295\) 0 0
\(296\) 5.00000 0.290619
\(297\) 2.00000 0.116052
\(298\) 22.0000 1.27443
\(299\) −12.0000 −0.693978
\(300\) 5.00000 0.288675
\(301\) −9.00000 −0.518751
\(302\) 20.0000 1.15087
\(303\) 10.0000 0.574485
\(304\) 0 0
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) −2.00000 −0.113961
\(309\) 3.00000 0.170664
\(310\) 0 0
\(311\) 10.0000 0.567048 0.283524 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) −3.00000 −0.169842
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 21.0000 1.18510
\(315\) 0 0
\(316\) 1.00000 0.0562544
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) −4.00000 −0.224309
\(319\) 0 0
\(320\) 0 0
\(321\) 10.0000 0.558146
\(322\) −4.00000 −0.222911
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 15.0000 0.832050
\(326\) −11.0000 −0.609234
\(327\) −14.0000 −0.774202
\(328\) −4.00000 −0.220863
\(329\) 10.0000 0.551318
\(330\) 0 0
\(331\) 15.0000 0.824475 0.412237 0.911077i \(-0.364747\pi\)
0.412237 + 0.911077i \(0.364747\pi\)
\(332\) 8.00000 0.439057
\(333\) −5.00000 −0.273998
\(334\) −6.00000 −0.328305
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −19.0000 −1.03500 −0.517498 0.855684i \(-0.673136\pi\)
−0.517498 + 0.855684i \(0.673136\pi\)
\(338\) 4.00000 0.217571
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 9.00000 0.485247
\(345\) 0 0
\(346\) 24.0000 1.29025
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) 0 0
\(349\) −29.0000 −1.55233 −0.776167 0.630527i \(-0.782839\pi\)
−0.776167 + 0.630527i \(0.782839\pi\)
\(350\) 5.00000 0.267261
\(351\) 3.00000 0.160128
\(352\) 2.00000 0.106600
\(353\) −12.0000 −0.638696 −0.319348 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) −14.0000 −0.744092
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) −4.00000 −0.211702
\(358\) −4.00000 −0.211407
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 18.0000 0.946059
\(363\) 7.00000 0.367405
\(364\) −3.00000 −0.157243
\(365\) 0 0
\(366\) 11.0000 0.574979
\(367\) 23.0000 1.20059 0.600295 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) 4.00000 0.208514
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 3.00000 0.155543
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) −10.0000 −0.515711
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −13.0000 −0.667765 −0.333883 0.942615i \(-0.608359\pi\)
−0.333883 + 0.942615i \(0.608359\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 8.00000 0.409316
\(383\) 14.0000 0.715367 0.357683 0.933843i \(-0.383567\pi\)
0.357683 + 0.933843i \(0.383567\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 13.0000 0.661683
\(387\) −9.00000 −0.457496
\(388\) 2.00000 0.101535
\(389\) −16.0000 −0.811232 −0.405616 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 6.00000 0.303046
\(393\) 6.00000 0.302660
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) 15.0000 0.752828 0.376414 0.926451i \(-0.377157\pi\)
0.376414 + 0.926451i \(0.377157\pi\)
\(398\) −5.00000 −0.250627
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 3.00000 0.149626
\(403\) 9.00000 0.448322
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) 10.0000 0.495682
\(408\) 4.00000 0.198030
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) −3.00000 −0.147799
\(413\) −14.0000 −0.688895
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 3.00000 0.147087
\(417\) 19.0000 0.930434
\(418\) 0 0
\(419\) 18.0000 0.879358 0.439679 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −25.0000 −1.21698
\(423\) 10.0000 0.486217
\(424\) 4.00000 0.194257
\(425\) −20.0000 −0.970143
\(426\) 14.0000 0.678302
\(427\) 11.0000 0.532327
\(428\) −10.0000 −0.483368
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) −10.0000 −0.481683 −0.240842 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −9.00000 −0.432512 −0.216256 0.976337i \(-0.569385\pi\)
−0.216256 + 0.976337i \(0.569385\pi\)
\(434\) 3.00000 0.144005
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) −11.0000 −0.525600
\(439\) 19.0000 0.906821 0.453410 0.891302i \(-0.350207\pi\)
0.453410 + 0.891302i \(0.350207\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 12.0000 0.570782
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 5.00000 0.237289
\(445\) 0 0
\(446\) −9.00000 −0.426162
\(447\) 22.0000 1.04056
\(448\) 1.00000 0.0472456
\(449\) −36.0000 −1.69895 −0.849473 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 5.00000 0.235702
\(451\) −8.00000 −0.376705
\(452\) 10.0000 0.470360
\(453\) 20.0000 0.939682
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) 5.00000 0.233890 0.116945 0.993138i \(-0.462690\pi\)
0.116945 + 0.993138i \(0.462690\pi\)
\(458\) 11.0000 0.513996
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) −2.00000 −0.0930484
\(463\) −9.00000 −0.418265 −0.209133 0.977887i \(-0.567064\pi\)
−0.209133 + 0.977887i \(0.567064\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) −3.00000 −0.138675
\(469\) 3.00000 0.138527
\(470\) 0 0
\(471\) 21.0000 0.967629
\(472\) 14.0000 0.644402
\(473\) 18.0000 0.827641
\(474\) 1.00000 0.0459315
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) −4.00000 −0.183147
\(478\) 12.0000 0.548867
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 15.0000 0.683941
\(482\) −25.0000 −1.13872
\(483\) −4.00000 −0.182006
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −11.0000 −0.497947
\(489\) −11.0000 −0.497437
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) −4.00000 −0.180334
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) 14.0000 0.627986
\(498\) 8.00000 0.358489
\(499\) 5.00000 0.223831 0.111915 0.993718i \(-0.464301\pi\)
0.111915 + 0.993718i \(0.464301\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 18.0000 0.803379
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) 4.00000 0.177646
\(508\) −8.00000 −0.354943
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) 9.00000 0.396203
\(517\) −20.0000 −0.879599
\(518\) 5.00000 0.219687
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) 34.0000 1.48957 0.744784 0.667306i \(-0.232553\pi\)
0.744784 + 0.667306i \(0.232553\pi\)
\(522\) 0 0
\(523\) 37.0000 1.61790 0.808949 0.587879i \(-0.200037\pi\)
0.808949 + 0.587879i \(0.200037\pi\)
\(524\) −6.00000 −0.262111
\(525\) 5.00000 0.218218
\(526\) −18.0000 −0.784837
\(527\) −12.0000 −0.522728
\(528\) 2.00000 0.0870388
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −14.0000 −0.607548
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) −14.0000 −0.605839
\(535\) 0 0
\(536\) −3.00000 −0.129580
\(537\) −4.00000 −0.172613
\(538\) −2.00000 −0.0862261
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) −31.0000 −1.33279 −0.666397 0.745597i \(-0.732164\pi\)
−0.666397 + 0.745597i \(0.732164\pi\)
\(542\) 16.0000 0.687259
\(543\) 18.0000 0.772454
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) −3.00000 −0.128388
\(547\) 11.0000 0.470326 0.235163 0.971956i \(-0.424438\pi\)
0.235163 + 0.971956i \(0.424438\pi\)
\(548\) −6.00000 −0.256307
\(549\) 11.0000 0.469469
\(550\) −10.0000 −0.426401
\(551\) 0 0
\(552\) 4.00000 0.170251
\(553\) 1.00000 0.0425243
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −19.0000 −0.805779
\(557\) 34.0000 1.44063 0.720313 0.693649i \(-0.243998\pi\)
0.720313 + 0.693649i \(0.243998\pi\)
\(558\) 3.00000 0.127000
\(559\) 27.0000 1.14198
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 8.00000 0.337460
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) −10.0000 −0.421076
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 1.00000 0.0419961
\(568\) −14.0000 −0.587427
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) −7.00000 −0.292941 −0.146470 0.989215i \(-0.546791\pi\)
−0.146470 + 0.989215i \(0.546791\pi\)
\(572\) 6.00000 0.250873
\(573\) 8.00000 0.334205
\(574\) −4.00000 −0.166957
\(575\) −20.0000 −0.834058
\(576\) 1.00000 0.0416667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 1.00000 0.0415945
\(579\) 13.0000 0.540262
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 2.00000 0.0829027
\(583\) 8.00000 0.331326
\(584\) 11.0000 0.455183
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) −2.00000 −0.0825488 −0.0412744 0.999148i \(-0.513142\pi\)
−0.0412744 + 0.999148i \(0.513142\pi\)
\(588\) 6.00000 0.247436
\(589\) 0 0
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) −5.00000 −0.205499
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −22.0000 −0.901155
\(597\) −5.00000 −0.204636
\(598\) 12.0000 0.490716
\(599\) −46.0000 −1.87951 −0.939755 0.341850i \(-0.888947\pi\)
−0.939755 + 0.341850i \(0.888947\pi\)
\(600\) −5.00000 −0.204124
\(601\) 27.0000 1.10135 0.550676 0.834719i \(-0.314370\pi\)
0.550676 + 0.834719i \(0.314370\pi\)
\(602\) 9.00000 0.366813
\(603\) 3.00000 0.122169
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) 5.00000 0.202944 0.101472 0.994838i \(-0.467645\pi\)
0.101472 + 0.994838i \(0.467645\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −30.0000 −1.21367
\(612\) 4.00000 0.161690
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 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 \(0.820655\pi\)
\(618\) −3.00000 −0.120678
\(619\) 1.00000 0.0401934 0.0200967 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −10.0000 −0.400963
\(623\) −14.0000 −0.560898
\(624\) 3.00000 0.120096
\(625\) 25.0000 1.00000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) −21.0000 −0.837991
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) 17.0000 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) −1.00000 −0.0397779
\(633\) −25.0000 −0.993661
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) 4.00000 0.158610
\(637\) 18.0000 0.713186
\(638\) 0 0
\(639\) 14.0000 0.553831
\(640\) 0 0
\(641\) −14.0000 −0.552967 −0.276483 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(642\) −10.0000 −0.394669
\(643\) 37.0000 1.45914 0.729569 0.683907i \(-0.239721\pi\)
0.729569 + 0.683907i \(0.239721\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 28.0000 1.09910
\(650\) −15.0000 −0.588348
\(651\) 3.00000 0.117579
\(652\) 11.0000 0.430793
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) 4.00000 0.156174
\(657\) −11.0000 −0.429151
\(658\) −10.0000 −0.389841
\(659\) −14.0000 −0.545363 −0.272681 0.962104i \(-0.587910\pi\)
−0.272681 + 0.962104i \(0.587910\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) −15.0000 −0.582992
\(663\) 12.0000 0.466041
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) 5.00000 0.193746
\(667\) 0 0
\(668\) 6.00000 0.232147
\(669\) −9.00000 −0.347960
\(670\) 0 0
\(671\) −22.0000 −0.849301
\(672\) 1.00000 0.0385758
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 19.0000 0.731853
\(675\) 5.00000 0.192450
\(676\) −4.00000 −0.153846
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 10.0000 0.384048
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) −6.00000 −0.229752
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) 11.0000 0.419676
\(688\) −9.00000 −0.343122
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) −24.0000 −0.912343
\(693\) −2.00000 −0.0759737
\(694\) −16.0000 −0.607352
\(695\) 0 0
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 29.0000 1.09767
\(699\) −18.0000 −0.680823
\(700\) −5.00000 −0.188982
\(701\) −44.0000 −1.66186 −0.830929 0.556379i \(-0.812190\pi\)
−0.830929 + 0.556379i \(0.812190\pi\)
\(702\) −3.00000 −0.113228
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 12.0000 0.451626
\(707\) −10.0000 −0.376089
\(708\) 14.0000 0.526152
\(709\) −31.0000 −1.16423 −0.582115 0.813107i \(-0.697775\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) 0 0
\(711\) 1.00000 0.0375029
\(712\) 14.0000 0.524672
\(713\) −12.0000 −0.449404
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 12.0000 0.448148
\(718\) 36.0000 1.34351
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 0 0
\(721\) −3.00000 −0.111726
\(722\) 0 0
\(723\) −25.0000 −0.929760
\(724\) −18.0000 −0.668965
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) 17.0000 0.630495 0.315248 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) 3.00000 0.111187
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −36.0000 −1.33151
\(732\) −11.0000 −0.406572
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) −23.0000 −0.848945
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −6.00000 −0.221013
\(738\) −4.00000 −0.147242
\(739\) −5.00000 −0.183928 −0.0919640 0.995762i \(-0.529314\pi\)
−0.0919640 + 0.995762i \(0.529314\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.00000 0.146845
\(743\) −18.0000 −0.660356 −0.330178 0.943919i \(-0.607109\pi\)
−0.330178 + 0.943919i \(0.607109\pi\)
\(744\) −3.00000 −0.109985
\(745\) 0 0
\(746\) −26.0000 −0.951928
\(747\) 8.00000 0.292705
\(748\) −8.00000 −0.292509
\(749\) −10.0000 −0.365392
\(750\) 0 0
\(751\) 13.0000 0.474377 0.237188 0.971464i \(-0.423774\pi\)
0.237188 + 0.971464i \(0.423774\pi\)
\(752\) 10.0000 0.364662
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) 5.00000 0.181728 0.0908640 0.995863i \(-0.471037\pi\)
0.0908640 + 0.995863i \(0.471037\pi\)
\(758\) 13.0000 0.472181
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) −34.0000 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(762\) −8.00000 −0.289809
\(763\) 14.0000 0.506834
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −14.0000 −0.505841
\(767\) 42.0000 1.51653
\(768\) −1.00000 −0.0360844
\(769\) 27.0000 0.973645 0.486822 0.873501i \(-0.338156\pi\)
0.486822 + 0.873501i \(0.338156\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) −13.0000 −0.467880
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 9.00000 0.323498
\(775\) 15.0000 0.538816
\(776\) −2.00000 −0.0717958
\(777\) 5.00000 0.179374
\(778\) 16.0000 0.573628
\(779\) 0 0
\(780\) 0 0
\(781\) −28.0000 −1.00192
\(782\) −16.0000 −0.572159
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) −6.00000 −0.214013
\(787\) −47.0000 −1.67537 −0.837685 0.546154i \(-0.816091\pi\)
−0.837685 + 0.546154i \(0.816091\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) 10.0000 0.355559
\(792\) 2.00000 0.0710669
\(793\) −33.0000 −1.17186
\(794\) −15.0000 −0.532330
\(795\) 0 0
\(796\) 5.00000 0.177220
\(797\) 26.0000 0.920967 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) 0 0
\(799\) 40.0000 1.41510
\(800\) 5.00000 0.176777
\(801\) −14.0000 −0.494666
\(802\) −18.0000 −0.635602
\(803\) 22.0000 0.776363
\(804\) −3.00000 −0.105802
\(805\) 0 0
\(806\) −9.00000 −0.317011
\(807\) −2.00000 −0.0704033
\(808\) 10.0000 0.351799
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) −10.0000 −0.350500
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 0 0
\(818\) 30.0000 1.04893
\(819\) −3.00000 −0.104828
\(820\) 0 0
\(821\) 48.0000 1.67521 0.837606 0.546275i \(-0.183955\pi\)
0.837606 + 0.546275i \(0.183955\pi\)
\(822\) −6.00000 −0.209274
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 3.00000 0.104510
\(825\) −10.0000 −0.348155
\(826\) 14.0000 0.487122
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 4.00000 0.139010
\(829\) −31.0000 −1.07667 −0.538337 0.842729i \(-0.680947\pi\)
−0.538337 + 0.842729i \(0.680947\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) −3.00000 −0.104006
\(833\) −24.0000 −0.831551
\(834\) −19.0000 −0.657916
\(835\) 0 0
\(836\) 0 0
\(837\) 3.00000 0.103695
\(838\) −18.0000 −0.621800
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −10.0000 −0.344623
\(843\) 8.00000 0.275535
\(844\) 25.0000 0.860535
\(845\) 0 0
\(846\) −10.0000 −0.343807
\(847\) −7.00000 −0.240523
\(848\) −4.00000 −0.137361
\(849\) 20.0000 0.686398
\(850\) 20.0000 0.685994
\(851\) −20.0000 −0.685591
\(852\) −14.0000 −0.479632
\(853\) 3.00000 0.102718 0.0513590 0.998680i \(-0.483645\pi\)
0.0513590 + 0.998680i \(0.483645\pi\)
\(854\) −11.0000 −0.376412
\(855\) 0 0
\(856\) 10.0000 0.341793
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 6.00000 0.204837
\(859\) 7.00000 0.238837 0.119418 0.992844i \(-0.461897\pi\)
0.119418 + 0.992844i \(0.461897\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 10.0000 0.340601
\(863\) −22.0000 −0.748889 −0.374444 0.927249i \(-0.622167\pi\)
−0.374444 + 0.927249i \(0.622167\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 9.00000 0.305832
\(867\) 1.00000 0.0339618
\(868\) −3.00000 −0.101827
\(869\) −2.00000 −0.0678454
\(870\) 0 0
\(871\) −9.00000 −0.304953
\(872\) −14.0000 −0.474100
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 11.0000 0.371656
\(877\) −25.0000 −0.844190 −0.422095 0.906552i \(-0.638705\pi\)
−0.422095 + 0.906552i \(0.638705\pi\)
\(878\) −19.0000 −0.641219
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) 36.0000 1.21287 0.606435 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) 6.00000 0.202031
\(883\) −9.00000 −0.302874 −0.151437 0.988467i \(-0.548390\pi\)
−0.151437 + 0.988467i \(0.548390\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 18.0000 0.604381 0.302190 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) −5.00000 −0.167789
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 9.00000 0.301342
\(893\) 0 0
\(894\) −22.0000 −0.735790
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 12.0000 0.400668
\(898\) 36.0000 1.20134
\(899\) 0 0
\(900\) −5.00000 −0.166667
\(901\) −16.0000 −0.533037
\(902\) 8.00000 0.266371
\(903\) 9.00000 0.299501
\(904\) −10.0000 −0.332595
\(905\) 0 0
\(906\) −20.0000 −0.664455
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) 8.00000 0.265489
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −60.0000 −1.98789 −0.993944 0.109885i \(-0.964952\pi\)
−0.993944 + 0.109885i \(0.964952\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) −5.00000 −0.165385
\(915\) 0 0
\(916\) −11.0000 −0.363450
\(917\) −6.00000 −0.198137
\(918\) 4.00000 0.132020
\(919\) 55.0000 1.81428 0.907141 0.420826i \(-0.138260\pi\)
0.907141 + 0.420826i \(0.138260\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) −6.00000 −0.197599
\(923\) −42.0000 −1.38245
\(924\) 2.00000 0.0657952
\(925\) 25.0000 0.821995
\(926\) 9.00000 0.295758
\(927\) −3.00000 −0.0985329
\(928\) 0 0
\(929\) 12.0000 0.393707 0.196854 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 18.0000 0.589610
\(933\) −10.0000 −0.327385
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 3.00000 0.0980581
\(937\) −25.0000 −0.816714 −0.408357 0.912822i \(-0.633898\pi\)
−0.408357 + 0.912822i \(0.633898\pi\)
\(938\) −3.00000 −0.0979535
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −32.0000 −1.04317 −0.521585 0.853199i \(-0.674659\pi\)
−0.521585 + 0.853199i \(0.674659\pi\)
\(942\) −21.0000 −0.684217
\(943\) 16.0000 0.521032
\(944\) −14.0000 −0.455661
\(945\) 0 0
\(946\) −18.0000 −0.585230
\(947\) 10.0000 0.324956 0.162478 0.986712i \(-0.448051\pi\)
0.162478 + 0.986712i \(0.448051\pi\)
\(948\) −1.00000 −0.0324785
\(949\) 33.0000 1.07123
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) −4.00000 −0.129641
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 4.00000 0.129505
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) −6.00000 −0.193851
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −15.0000 −0.483619
\(963\) −10.0000 −0.322245
\(964\) 25.0000 0.805196
\(965\) 0 0
\(966\) 4.00000 0.128698
\(967\) 51.0000 1.64005 0.820025 0.572328i \(-0.193960\pi\)
0.820025 + 0.572328i \(0.193960\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 0 0
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −19.0000 −0.609112
\(974\) −32.0000 −1.02535
\(975\) −15.0000 −0.480384
\(976\) 11.0000 0.352101
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 11.0000 0.351741
\(979\) 28.0000 0.894884
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) −30.0000 −0.957338
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 4.00000 0.127515
\(985\) 0 0
\(986\) 0 0
\(987\) −10.0000 −0.318304
\(988\) 0 0
\(989\) −36.0000 −1.14473
\(990\) 0 0
\(991\) −17.0000 −0.540023 −0.270011 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(992\) 3.00000 0.0952501
\(993\) −15.0000 −0.476011
\(994\) −14.0000 −0.444053
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) −49.0000 −1.55185 −0.775923 0.630828i \(-0.782715\pi\)
−0.775923 + 0.630828i \(0.782715\pi\)
\(998\) −5.00000 −0.158272
\(999\) 5.00000 0.158193
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 2166.2.a.b.1.1 1
3.2 odd 2 6498.2.a.u.1.1 1
19.7 even 3 114.2.e.b.49.1 yes 2
19.11 even 3 114.2.e.b.7.1 2
19.18 odd 2 2166.2.a.h.1.1 1
57.11 odd 6 342.2.g.c.235.1 2
57.26 odd 6 342.2.g.c.163.1 2
57.56 even 2 6498.2.a.g.1.1 1
76.7 odd 6 912.2.q.b.49.1 2
76.11 odd 6 912.2.q.b.577.1 2
228.11 even 6 2736.2.s.k.577.1 2
228.83 even 6 2736.2.s.k.1873.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.e.b.7.1 2 19.11 even 3
114.2.e.b.49.1 yes 2 19.7 even 3
342.2.g.c.163.1 2 57.26 odd 6
342.2.g.c.235.1 2 57.11 odd 6
912.2.q.b.49.1 2 76.7 odd 6
912.2.q.b.577.1 2 76.11 odd 6
2166.2.a.b.1.1 1 1.1 even 1 trivial
2166.2.a.h.1.1 1 19.18 odd 2
2736.2.s.k.577.1 2 228.11 even 6
2736.2.s.k.1873.1 2 228.83 even 6
6498.2.a.g.1.1 1 57.56 even 2
6498.2.a.u.1.1 1 3.2 odd 2