Properties

Label 1650.2.a.k.1.1
Level $1650$
Weight $2$
Character 1650.1
Self dual yes
Analytic conductor $13.175$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1650.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.1753163335\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1650.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +4.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -6.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} -1.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} +6.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +4.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -2.00000 q^{51} +6.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -4.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} +12.0000 q^{59} -14.0000 q^{61} +4.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} -12.0000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +6.00000 q^{74} +4.00000 q^{76} -4.00000 q^{77} -6.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} +4.00000 q^{84} +4.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} +10.0000 q^{89} +24.0000 q^{91} -4.00000 q^{92} -12.0000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -9.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(6\) −1.00000 −0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 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
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 1.00000 0.213201
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −4.00000 −0.648886
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −4.00000 −0.617213
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 6.00000 0.832050
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 4.00000 0.529813
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.00000 −0.242536
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) −4.00000 −0.455842
\(78\) −6.00000 −0.679366
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 6.00000 0.643268
\(88\) 1.00000 0.106600
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −9.00000 −0.909137
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 2.00000 0.198030
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 4.00000 0.377964
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 6.00000 0.554700
\(118\) −12.0000 −1.10469
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 14.0000 1.26750
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 16.0000 1.38738
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 4.00000 0.340503
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 12.0000 1.00702
\(143\) −6.00000 −0.501745
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) 9.00000 0.742307
\(148\) −6.00000 −0.493197
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) −4.00000 −0.324443
\(153\) −2.00000 −0.161690
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 4.00000 0.318223
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) −1.00000 −0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −4.00000 −0.308607
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 12.0000 0.901975
\(178\) −10.0000 −0.749532
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −24.0000 −1.77900
\(183\) −14.0000 −1.03491
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 12.0000 0.875190
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 1.00000 0.0710669
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) −14.0000 −0.985037
\(203\) 24.0000 1.68447
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 6.00000 0.416025
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −2.00000 −0.137361
\(213\) −12.0000 −0.822226
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 6.00000 0.406371
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 6.00000 0.402694
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 4.00000 0.264906
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) −6.00000 −0.393919
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) −4.00000 −0.259828
\(238\) 8.00000 0.518563
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 24.0000 1.52708
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 4.00000 0.251976
\(253\) 4.00000 0.251478
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 4.00000 0.249029
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) −4.00000 −0.247121
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 1.00000 0.0615457
\(265\) 0 0
\(266\) −16.0000 −0.981023
\(267\) 10.0000 0.611990
\(268\) −4.00000 −0.244339
\(269\) 26.0000 1.58525 0.792624 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −2.00000 −0.121268
\(273\) 24.0000 1.45255
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) −12.0000 −0.714590
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) −24.0000 −1.41668
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 6.00000 0.351123
\(293\) −22.0000 −1.28525 −0.642627 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) −1.00000 −0.0580259
\(298\) 10.0000 0.579284
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) −4.00000 −0.230174
\(303\) 14.0000 0.804279
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) −6.00000 −0.339683
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 2.00000 0.112154
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 16.0000 0.891645
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) −6.00000 −0.331801
\(328\) 6.00000 0.331295
\(329\) 48.0000 2.64633
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −4.00000 −0.219529
\(333\) −6.00000 −0.328798
\(334\) 0 0
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) −23.0000 −1.25104
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 8.00000 0.431959
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 6.00000 0.321634
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 1.00000 0.0533002
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) −8.00000 −0.423405
\(358\) −20.0000 −1.05703
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) 1.00000 0.0524864
\(364\) 24.0000 1.25794
\(365\) 0 0
\(366\) 14.0000 0.731792
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −4.00000 −0.208514
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −8.00000 −0.415339
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 36.0000 1.85409
\(378\) −4.00000 −0.205738
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 12.0000 0.613973
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) −4.00000 −0.203331
\(388\) 14.0000 0.710742
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) −9.00000 −0.454569
\(393\) 4.00000 0.201773
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 16.0000 0.802008
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) −38.0000 −1.89763 −0.948815 0.315833i \(-0.897716\pi\)
−0.948815 + 0.315833i \(0.897716\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) −24.0000 −1.19110
\(407\) 6.00000 0.297409
\(408\) 2.00000 0.0990148
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) 48.0000 2.36193
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) −4.00000 −0.195881
\(418\) 4.00000 0.195646
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.00000 0.194717
\(423\) 12.0000 0.583460
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) −56.0000 −2.71003
\(428\) −4.00000 −0.193347
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) −16.0000 −0.765384
\(438\) −6.00000 −0.286691
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 12.0000 0.570782
\(443\) −28.0000 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) 0 0
\(447\) −10.0000 −0.472984
\(448\) 4.00000 0.188982
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) −2.00000 −0.0940721
\(453\) 4.00000 0.187936
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −34.0000 −1.59045 −0.795226 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) −14.0000 −0.654177
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 4.00000 0.186097
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 6.00000 0.277350
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) −12.0000 −0.552345
\(473\) 4.00000 0.183920
\(474\) 4.00000 0.183726
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) −2.00000 −0.0915737
\(478\) −8.00000 −0.365911
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −36.0000 −1.64146
\(482\) −10.0000 −0.455488
\(483\) −16.0000 −0.728025
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 14.0000 0.633750
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) −6.00000 −0.270501
\(493\) −12.0000 −0.540453
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) 0 0
\(497\) −48.0000 −2.15309
\(498\) 4.00000 0.179244
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) −32.0000 −1.42681 −0.713405 0.700752i \(-0.752848\pi\)
−0.713405 + 0.700752i \(0.752848\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) −4.00000 −0.177822
\(507\) 23.0000 1.02147
\(508\) −12.0000 −0.532414
\(509\) −22.0000 −0.975133 −0.487566 0.873086i \(-0.662115\pi\)
−0.487566 + 0.873086i \(0.662115\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) −1.00000 −0.0441942
\(513\) 4.00000 0.176604
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) −12.0000 −0.527759
\(518\) 24.0000 1.05450
\(519\) 10.0000 0.438951
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −6.00000 −0.262613
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) −1.00000 −0.0435194
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 16.0000 0.693688
\(533\) −36.0000 −1.55933
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 20.0000 0.863064
\(538\) −26.0000 −1.12094
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) −20.0000 −0.859074
\(543\) −2.00000 −0.0858282
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) −24.0000 −1.02711
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 4.00000 0.170251
\(553\) −16.0000 −0.680389
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) 22.0000 0.928014
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) 4.00000 0.167984
\(568\) 12.0000 0.503509
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −6.00000 −0.250873
\(573\) −12.0000 −0.501307
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 46.0000 1.91501 0.957503 0.288425i \(-0.0931316\pi\)
0.957503 + 0.288425i \(0.0931316\pi\)
\(578\) 13.0000 0.540729
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) −14.0000 −0.580319
\(583\) 2.00000 0.0828315
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 9.00000 0.371154
\(589\) 0 0
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) −6.00000 −0.246598
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) −16.0000 −0.654836
\(598\) 24.0000 0.981433
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 16.0000 0.652111
\(603\) −4.00000 −0.162893
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) −14.0000 −0.568711
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) −4.00000 −0.162221
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) 72.0000 2.91281
\(612\) −2.00000 −0.0808452
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −4.00000 −0.160385
\(623\) 40.0000 1.60257
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) 26.0000 1.03917
\(627\) −4.00000 −0.159745
\(628\) 10.0000 0.399043
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 4.00000 0.159111
\(633\) −4.00000 −0.158986
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) 54.0000 2.13956
\(638\) 6.00000 0.237542
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 4.00000 0.157867
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 6.00000 0.234619
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 6.00000 0.234082
\(658\) −48.0000 −1.87123
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) −20.0000 −0.777322
\(663\) −12.0000 −0.466041
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 14.0000 0.540464
\(672\) −4.00000 −0.154303
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 18.0000 0.693334
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −46.0000 −1.76792 −0.883962 0.467559i \(-0.845134\pi\)
−0.883962 + 0.467559i \(0.845134\pi\)
\(678\) 2.00000 0.0768095
\(679\) 56.0000 2.14908
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 14.0000 0.534133
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 10.0000 0.380143
\(693\) −4.00000 −0.151947
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) 12.0000 0.454532
\(698\) 6.00000 0.227103
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −6.00000 −0.226455
\(703\) −24.0000 −0.905177
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 56.0000 2.10610
\(708\) 12.0000 0.450988
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) −10.0000 −0.374766
\(713\) 0 0
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) 8.00000 0.298765
\(718\) 0 0
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 10.0000 0.371904
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −1.00000 −0.0371135
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) −24.0000 −0.889499
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) −14.0000 −0.517455
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 4.00000 0.147342
\(738\) 6.00000 0.220863
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) 0 0
\(741\) 24.0000 0.881662
\(742\) 8.00000 0.293689
\(743\) 32.0000 1.17397 0.586983 0.809599i \(-0.300316\pi\)
0.586983 + 0.809599i \(0.300316\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) −4.00000 −0.146352
\(748\) 2.00000 0.0731272
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 12.0000 0.437595
\(753\) −4.00000 −0.145768
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 28.0000 1.01701
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 12.0000 0.434714
\(763\) −24.0000 −0.868858
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) 72.0000 2.59977
\(768\) 1.00000 0.0360844
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) −10.0000 −0.359908
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) −24.0000 −0.860995
\(778\) 30.0000 1.07555
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) −8.00000 −0.286079
\(783\) 6.00000 0.214423
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) 2.00000 0.0712470
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) −8.00000 −0.284447
\(792\) 1.00000 0.0355335
\(793\) −84.0000 −2.98293
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 54.0000 1.91278 0.956389 0.292096i \(-0.0943526\pi\)
0.956389 + 0.292096i \(0.0943526\pi\)
\(798\) −16.0000 −0.566394
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 38.0000 1.34183
\(803\) −6.00000 −0.211735
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) 26.0000 0.915243
\(808\) −14.0000 −0.492518
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 24.0000 0.842235
\(813\) 20.0000 0.701431
\(814\) −6.00000 −0.210300
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) −16.0000 −0.559769
\(818\) 14.0000 0.489499
\(819\) 24.0000 0.838628
\(820\) 0 0
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 2.00000 0.0697580
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) 52.0000 1.80822 0.904109 0.427303i \(-0.140536\pi\)
0.904109 + 0.427303i \(0.140536\pi\)
\(828\) −4.00000 −0.139010
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) −26.0000 −0.901930
\(832\) 6.00000 0.208013
\(833\) −18.0000 −0.623663
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) −22.0000 −0.757720
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 4.00000 0.137442
\(848\) −2.00000 −0.0686803
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) −12.0000 −0.411113
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 56.0000 1.91628
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) −26.0000 −0.888143 −0.444072 0.895991i \(-0.646466\pi\)
−0.444072 + 0.895991i \(0.646466\pi\)
\(858\) 6.00000 0.204837
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) −20.0000 −0.680808 −0.340404 0.940279i \(-0.610564\pi\)
−0.340404 + 0.940279i \(0.610564\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 6.00000 0.203186
\(873\) 14.0000 0.473828
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) −42.0000 −1.41824 −0.709120 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) −4.00000 −0.134993
\(879\) −22.0000 −0.742042
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) −9.00000 −0.303046
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 6.00000 0.201347
\(889\) −48.0000 −1.60987
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 48.0000 1.60626
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) −24.0000 −0.801337
\(898\) 22.0000 0.734150
\(899\) 0 0
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) −6.00000 −0.199778
\(903\) −16.0000 −0.532447
\(904\) 2.00000 0.0665190
\(905\) 0 0
\(906\) −4.00000 −0.132891
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −12.0000 −0.398234
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 4.00000 0.132453
\(913\) 4.00000 0.132381
\(914\) 34.0000 1.12462
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 16.0000 0.528367
\(918\) 2.00000 0.0660098
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) −14.0000 −0.461065
\(923\) −72.0000 −2.36991
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) −10.0000 −0.327561
\(933\) 4.00000 0.130954
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 16.0000 0.522419
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) −10.0000 −0.325818
\(943\) 24.0000 0.781548
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −4.00000 −0.129914
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 8.00000 0.259281
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) −6.00000 −0.193952
\(958\) 0 0
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 36.0000 1.16069
\(963\) −4.00000 −0.128898
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 16.0000 0.514792
\(967\) 44.0000 1.41494 0.707472 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000 0.0320750
\(973\) −16.0000 −0.512936
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) −26.0000 −0.831814 −0.415907 0.909407i \(-0.636536\pi\)
−0.415907 + 0.909407i \(0.636536\pi\)
\(978\) −20.0000 −0.639529
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 28.0000 0.893516
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 48.0000 1.52786
\(988\) 24.0000 0.763542
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) 48.0000 1.52247
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 4.00000 0.126618
\(999\) −6.00000 −0.189832
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 1650.2.a.k.1.1 1
3.2 odd 2 4950.2.a.bu.1.1 1
5.2 odd 4 1650.2.c.e.199.1 2
5.3 odd 4 1650.2.c.e.199.2 2
5.4 even 2 66.2.a.b.1.1 1
15.2 even 4 4950.2.c.p.199.2 2
15.8 even 4 4950.2.c.p.199.1 2
15.14 odd 2 198.2.a.a.1.1 1
20.19 odd 2 528.2.a.j.1.1 1
35.34 odd 2 3234.2.a.t.1.1 1
40.19 odd 2 2112.2.a.e.1.1 1
40.29 even 2 2112.2.a.r.1.1 1
45.4 even 6 1782.2.e.e.1189.1 2
45.14 odd 6 1782.2.e.v.1189.1 2
45.29 odd 6 1782.2.e.v.595.1 2
45.34 even 6 1782.2.e.e.595.1 2
55.4 even 10 726.2.e.g.511.1 4
55.9 even 10 726.2.e.g.565.1 4
55.14 even 10 726.2.e.g.493.1 4
55.19 odd 10 726.2.e.o.493.1 4
55.24 odd 10 726.2.e.o.565.1 4
55.29 odd 10 726.2.e.o.511.1 4
55.39 odd 10 726.2.e.o.487.1 4
55.49 even 10 726.2.e.g.487.1 4
55.54 odd 2 726.2.a.c.1.1 1
60.59 even 2 1584.2.a.f.1.1 1
105.104 even 2 9702.2.a.x.1.1 1
120.29 odd 2 6336.2.a.bw.1.1 1
120.59 even 2 6336.2.a.cj.1.1 1
165.164 even 2 2178.2.a.g.1.1 1
220.219 even 2 5808.2.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.b.1.1 1 5.4 even 2
198.2.a.a.1.1 1 15.14 odd 2
528.2.a.j.1.1 1 20.19 odd 2
726.2.a.c.1.1 1 55.54 odd 2
726.2.e.g.487.1 4 55.49 even 10
726.2.e.g.493.1 4 55.14 even 10
726.2.e.g.511.1 4 55.4 even 10
726.2.e.g.565.1 4 55.9 even 10
726.2.e.o.487.1 4 55.39 odd 10
726.2.e.o.493.1 4 55.19 odd 10
726.2.e.o.511.1 4 55.29 odd 10
726.2.e.o.565.1 4 55.24 odd 10
1584.2.a.f.1.1 1 60.59 even 2
1650.2.a.k.1.1 1 1.1 even 1 trivial
1650.2.c.e.199.1 2 5.2 odd 4
1650.2.c.e.199.2 2 5.3 odd 4
1782.2.e.e.595.1 2 45.34 even 6
1782.2.e.e.1189.1 2 45.4 even 6
1782.2.e.v.595.1 2 45.29 odd 6
1782.2.e.v.1189.1 2 45.14 odd 6
2112.2.a.e.1.1 1 40.19 odd 2
2112.2.a.r.1.1 1 40.29 even 2
2178.2.a.g.1.1 1 165.164 even 2
3234.2.a.t.1.1 1 35.34 odd 2
4950.2.a.bu.1.1 1 3.2 odd 2
4950.2.c.p.199.1 2 15.8 even 4
4950.2.c.p.199.2 2 15.2 even 4
5808.2.a.bc.1.1 1 220.219 even 2
6336.2.a.bw.1.1 1 120.29 odd 2
6336.2.a.cj.1.1 1 120.59 even 2
9702.2.a.x.1.1 1 105.104 even 2