Properties

Label 2166.2.a.h.1.1
Level $2166$
Weight $2$
Character 2166.1
Self dual yes
Analytic conductor $17.296$
Analytic rank $0$
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: \(0\)
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 \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 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 \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 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 \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 0 0
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 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 \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 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 \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) 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 \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 4.00000 0.485071
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 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 \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 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 \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 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 \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −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 \(-0.452781\pi\)
0.147799 + 0.989017i \(0.452781\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 1.00000 0.0962250
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 5.00000 0.474579
\(112\) 1.00000 0.0944911
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 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 \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 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 \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 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 \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 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 \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) 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 \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 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 \(-0.345018\pi\)
0.467880 + 0.883792i \(0.345018\pi\)
\(194\) −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 \(-0.829871\pi\)
−0.860535 + 0.509390i \(0.829871\pi\)
\(212\) 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 \(-0.597435\pi\)
−0.301342 + 0.953516i \(0.597435\pi\)
\(224\) 1.00000 0.0668153
\(225\) −5.00000 −0.333333
\(226\) −10.0000 −0.665190
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 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 \(-0.797940\pi\)
−0.805196 + 0.593009i \(0.797940\pi\)
\(242\) −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 \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) −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 \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 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 \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 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 \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −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 \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −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 \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 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 \(-0.635253\pi\)
−0.412237 + 0.911077i \(0.635253\pi\)
\(332\) 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 \(-0.326864\pi\)
0.517498 + 0.855684i \(0.326864\pi\)
\(338\) −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 \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −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 \(-0.391641\pi\)
0.333883 + 0.942615i \(0.391641\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) −8.00000 −0.409316
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 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 \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −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 \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 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 \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −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 \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) 1.00000 0.0481125
\(433\) 9.00000 0.432512 0.216256 0.976337i \(-0.430615\pi\)
0.216256 + 0.976337i \(0.430615\pi\)
\(434\) 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 \(-0.649793\pi\)
−0.453410 + 0.891302i \(0.649793\pi\)
\(440\) 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 \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) −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 \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 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 \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 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 \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 0 0
\(523\) −37.0000 −1.61790 −0.808949 0.587879i \(-0.799963\pi\)
−0.808949 + 0.587879i \(0.799963\pi\)
\(524\) −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 \(-0.575562\pi\)
−0.235163 + 0.971956i \(0.575562\pi\)
\(548\) −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 \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) 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 \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 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 \(-0.111053\pi\)
0.939755 + 0.341850i \(0.111053\pi\)
\(600\) −5.00000 −0.204124
\(601\) −27.0000 −1.10135 −0.550676 0.834719i \(-0.685630\pi\)
−0.550676 + 0.834719i \(0.685630\pi\)
\(602\) −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 \(-0.532355\pi\)
−0.101472 + 0.994838i \(0.532355\pi\)
\(608\) 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 \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) 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 \(-0.412090\pi\)
0.272681 + 0.962104i \(0.412090\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −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 \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 19.0000 0.731853
\(675\) −5.00000 −0.192450
\(676\) −4.00000 −0.153846
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −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 \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 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 \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 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 \(-0.576226\pi\)
−0.237188 + 0.971464i \(0.576226\pi\)
\(752\) 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 \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) −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 \(-0.183909\pi\)
0.837685 + 0.546154i \(0.183909\pi\)
\(788\) −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 \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 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 \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 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 \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 4.00000 0.139010
\(829\) 31.0000 1.07667 0.538337 0.842729i \(-0.319053\pi\)
0.538337 + 0.842729i \(0.319053\pi\)
\(830\) 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 \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 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 \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) −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 \(-0.377833\pi\)
0.374444 + 0.927249i \(0.377833\pi\)
\(864\) 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 \(-0.361295\pi\)
0.422095 + 0.906552i \(0.361295\pi\)
\(878\) −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 \(-0.597718\pi\)
−0.302190 + 0.953248i \(0.597718\pi\)
\(888\) 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 \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) −8.00000 −0.265489
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 60.0000 1.98789 0.993944 0.109885i \(-0.0350482\pi\)
0.993944 + 0.109885i \(0.0350482\pi\)
\(912\) 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 \(-0.325341\pi\)
0.521585 + 0.853199i \(0.325341\pi\)
\(942\) −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 \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 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 \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 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 \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 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 \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) −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 \(-0.412973\pi\)
0.270011 + 0.962857i \(0.412973\pi\)
\(992\) 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.h.1.1 1
3.2 odd 2 6498.2.a.g.1.1 1
19.8 odd 6 114.2.e.b.7.1 2
19.12 odd 6 114.2.e.b.49.1 yes 2
19.18 odd 2 2166.2.a.b.1.1 1
57.8 even 6 342.2.g.c.235.1 2
57.50 even 6 342.2.g.c.163.1 2
57.56 even 2 6498.2.a.u.1.1 1
76.27 even 6 912.2.q.b.577.1 2
76.31 even 6 912.2.q.b.49.1 2
228.107 odd 6 2736.2.s.k.1873.1 2
228.179 odd 6 2736.2.s.k.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.e.b.7.1 2 19.8 odd 6
114.2.e.b.49.1 yes 2 19.12 odd 6
342.2.g.c.163.1 2 57.50 even 6
342.2.g.c.235.1 2 57.8 even 6
912.2.q.b.49.1 2 76.31 even 6
912.2.q.b.577.1 2 76.27 even 6
2166.2.a.b.1.1 1 19.18 odd 2
2166.2.a.h.1.1 1 1.1 even 1 trivial
2736.2.s.k.577.1 2 228.179 odd 6
2736.2.s.k.1873.1 2 228.107 odd 6
6498.2.a.g.1.1 1 3.2 odd 2
6498.2.a.u.1.1 1 57.56 even 2