Properties

Label 114.2.a.a.1.1
Level $114$
Weight $2$
Character 114.1
Self dual yes
Analytic conductor $0.910$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 114.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.910294583043\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 114.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} -1.00000 q^{12} -4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -4.00000 q^{21} -4.00000 q^{22} -2.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{27} +4.00000 q^{28} -6.00000 q^{29} +6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} -1.00000 q^{38} +10.0000 q^{41} +4.00000 q^{42} -12.0000 q^{43} +4.00000 q^{44} +2.00000 q^{46} +10.0000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +5.00000 q^{50} +2.00000 q^{51} +2.00000 q^{53} +1.00000 q^{54} -4.00000 q^{56} -1.00000 q^{57} +6.00000 q^{58} +4.00000 q^{59} -10.0000 q^{61} -6.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} -2.00000 q^{68} +2.00000 q^{69} -16.0000 q^{71} -1.00000 q^{72} -2.00000 q^{73} +8.00000 q^{74} +5.00000 q^{75} +1.00000 q^{76} +16.0000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -10.0000 q^{82} -16.0000 q^{83} -4.00000 q^{84} +12.0000 q^{86} +6.00000 q^{87} -4.00000 q^{88} -2.00000 q^{89} -2.00000 q^{92} -6.00000 q^{93} -10.0000 q^{94} +1.00000 q^{96} -10.0000 q^{97} -9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(7\) 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
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −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
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) −4.00000 −0.852803
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 4.00000 0.617213
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 5.00000 0.707107
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) −1.00000 −0.132453
\(58\) 6.00000 0.787839
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −6.00000 −0.762001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −2.00000 −0.242536
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 8.00000 0.929981
\(75\) 5.00000 0.577350
\(76\) 1.00000 0.114708
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 6.00000 0.643268
\(88\) −4.00000 −0.426401
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) −6.00000 −0.622171
\(94\) −10.0000 −1.03142
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −9.00000 −0.909137
\(99\) 4.00000 0.402015
\(100\) −5.00000 −0.500000
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) −2.00000 −0.198030
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) 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
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 4.00000 0.377964
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) −10.0000 −0.901670
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) 22.0000 1.95218 0.976092 0.217357i \(-0.0697436\pi\)
0.976092 + 0.217357i \(0.0697436\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −4.00000 −0.348155
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −2.00000 −0.170251
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 16.0000 1.34269
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) −9.00000 −0.742307
\(148\) −8.00000 −0.657596
\(149\) 20.0000 1.63846 0.819232 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) −5.00000 −0.408248
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −2.00000 −0.161690
\(154\) −16.0000 −1.28932
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −10.0000 −0.795557
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) −1.00000 −0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 4.00000 0.308607
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) −12.0000 −0.914991
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −6.00000 −0.454859
\(175\) −20.0000 −1.51186
\(176\) 4.00000 0.301511
\(177\) −4.00000 −0.300658
\(178\) 2.00000 0.149906
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 2.00000 0.147442
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) −8.00000 −0.585018
\(188\) 10.0000 0.729325
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −2.00000 −0.144715 −0.0723575 0.997379i \(-0.523052\pi\)
−0.0723575 + 0.997379i \(0.523052\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −20.0000 −1.42494 −0.712470 0.701702i \(-0.752424\pi\)
−0.712470 + 0.701702i \(0.752424\pi\)
\(198\) −4.00000 −0.284268
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 5.00000 0.353553
\(201\) 0 0
\(202\) −8.00000 −0.562878
\(203\) −24.0000 −1.68447
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 6.00000 0.418040
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 2.00000 0.137361
\(213\) 16.0000 1.09630
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 24.0000 1.62923
\(218\) 4.00000 0.270914
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) −8.00000 −0.536925
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) −4.00000 −0.267261
\(225\) −5.00000 −0.333333
\(226\) 14.0000 0.931266
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) 6.00000 0.393919
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) −10.0000 −0.649570
\(238\) 8.00000 0.518563
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 0 0
\(248\) −6.00000 −0.381000
\(249\) 16.0000 1.01396
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 4.00000 0.251976
\(253\) −8.00000 −0.502956
\(254\) −22.0000 −1.38040
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −12.0000 −0.747087
\(259\) −32.0000 −1.98838
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) 2.00000 0.122398
\(268\) 0 0
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) −20.0000 −1.20605
\(276\) 2.00000 0.120386
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 4.00000 0.239904
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 10.0000 0.595491
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) 0 0
\(287\) 40.0000 2.36113
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) −2.00000 −0.117041
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) −4.00000 −0.232104
\(298\) −20.0000 −1.15857
\(299\) 0 0
\(300\) 5.00000 0.288675
\(301\) −48.0000 −2.76667
\(302\) −10.0000 −0.575435
\(303\) −8.00000 −0.459588
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 16.0000 0.911685
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) 34.0000 1.92796 0.963982 0.265969i \(-0.0856919\pi\)
0.963982 + 0.265969i \(0.0856919\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 2.00000 0.112154
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 8.00000 0.445823
\(323\) −2.00000 −0.111283
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) 4.00000 0.221201
\(328\) −10.0000 −0.552158
\(329\) 40.0000 2.20527
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) −16.0000 −0.878114
\(333\) −8.00000 −0.438397
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 13.0000 0.707107
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) −1.00000 −0.0540738
\(343\) 8.00000 0.431959
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 6.00000 0.321634
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 20.0000 1.06904
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 8.00000 0.423405
\(358\) 20.0000 1.05703
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −12.0000 −0.630706
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) −2.00000 −0.104257
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) −6.00000 −0.311086
\(373\) 8.00000 0.414224 0.207112 0.978317i \(-0.433593\pi\)
0.207112 + 0.978317i \(0.433593\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) −10.0000 −0.515711
\(377\) 0 0
\(378\) 4.00000 0.205738
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −22.0000 −1.12709
\(382\) 2.00000 0.102329
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −12.0000 −0.609994
\(388\) −10.0000 −0.507673
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) 20.0000 1.00759
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) 4.00000 0.200502
\(399\) −4.00000 −0.200250
\(400\) −5.00000 −0.250000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) −32.0000 −1.58618
\(408\) −2.00000 −0.0990148
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) −6.00000 −0.295599
\(413\) 16.0000 0.787309
\(414\) 2.00000 0.0982946
\(415\) 0 0
\(416\) 0 0
\(417\) 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
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) 20.0000 0.973585
\(423\) 10.0000 0.486217
\(424\) −2.00000 −0.0971286
\(425\) 10.0000 0.485071
\(426\) −16.0000 −0.775203
\(427\) −40.0000 −1.93574
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) 0 0
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −18.0000 −0.865025 −0.432512 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) −24.0000 −1.15204
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −2.00000 −0.0956730
\(438\) −2.00000 −0.0955637
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) −6.00000 −0.284108
\(447\) −20.0000 −0.945968
\(448\) 4.00000 0.188982
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 5.00000 0.235702
\(451\) 40.0000 1.88353
\(452\) −14.0000 −0.658505
\(453\) −10.0000 −0.469841
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 2.00000 0.0934539
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 16.0000 0.744387
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) −4.00000 −0.184115
\(473\) −48.0000 −2.20704
\(474\) 10.0000 0.459315
\(475\) −5.00000 −0.229416
\(476\) −8.00000 −0.366679
\(477\) 2.00000 0.0915737
\(478\) 6.00000 0.274434
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 2.00000 0.0910975
\(483\) 8.00000 0.364013
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 10.0000 0.452679
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) −10.0000 −0.450835
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) −64.0000 −2.87079
\(498\) −16.0000 −0.716977
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 24.0000 1.07117
\(503\) 34.0000 1.51599 0.757993 0.652263i \(-0.226180\pi\)
0.757993 + 0.652263i \(0.226180\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) 13.0000 0.577350
\(508\) 22.0000 0.976092
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) 40.0000 1.75920
\(518\) 32.0000 1.40600
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 6.00000 0.262613
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 20.0000 0.872872
\(526\) 6.00000 0.261612
\(527\) −12.0000 −0.522728
\(528\) −4.00000 −0.174078
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 4.00000 0.173422
\(533\) 0 0
\(534\) −2.00000 −0.0865485
\(535\) 0 0
\(536\) 0 0
\(537\) 20.0000 0.863064
\(538\) −14.0000 −0.603583
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 4.00000 0.171815
\(543\) −12.0000 −0.514969
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 6.00000 0.256307
\(549\) −10.0000 −0.426790
\(550\) 20.0000 0.852803
\(551\) −6.00000 −0.255609
\(552\) −2.00000 −0.0851257
\(553\) 40.0000 1.70097
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 4.00000 0.169485 0.0847427 0.996403i \(-0.472993\pi\)
0.0847427 + 0.996403i \(0.472993\pi\)
\(558\) −6.00000 −0.254000
\(559\) 0 0
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −10.0000 −0.421825
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) −10.0000 −0.421076
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) 4.00000 0.167984
\(568\) 16.0000 0.671345
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 2.00000 0.0835512
\(574\) −40.0000 −1.66957
\(575\) 10.0000 0.417029
\(576\) 1.00000 0.0416667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 13.0000 0.540729
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) −64.0000 −2.65517
\(582\) −10.0000 −0.414513
\(583\) 8.00000 0.331326
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) −9.00000 −0.371154
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) 20.0000 0.822690
\(592\) −8.00000 −0.328798
\(593\) 10.0000 0.410651 0.205325 0.978694i \(-0.434175\pi\)
0.205325 + 0.978694i \(0.434175\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 20.0000 0.819232
\(597\) 4.00000 0.163709
\(598\) 0 0
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) −5.00000 −0.204124
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 48.0000 1.95633
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 8.00000 0.324978
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −16.0000 −0.644658
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −6.00000 −0.241355
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 2.00000 0.0802572
\(622\) −34.0000 −1.36328
\(623\) −8.00000 −0.320513
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) −6.00000 −0.239808
\(627\) −4.00000 −0.159745
\(628\) 18.0000 0.718278
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −10.0000 −0.397779
\(633\) 20.0000 0.794929
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) 0 0
\(638\) 24.0000 0.950169
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) −4.00000 −0.157867
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) −24.0000 −0.940634
\(652\) 20.0000 0.783260
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) −2.00000 −0.0780274
\(658\) −40.0000 −1.55936
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) −8.00000 −0.311164 −0.155582 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) −24.0000 −0.932786
\(663\) 0 0
\(664\) 16.0000 0.620920
\(665\) 0 0
\(666\) 8.00000 0.309994
\(667\) 12.0000 0.464642
\(668\) 12.0000 0.464294
\(669\) −6.00000 −0.231973
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) 4.00000 0.154303
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −26.0000 −1.00148
\(675\) 5.00000 0.192450
\(676\) −13.0000 −0.500000
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) −14.0000 −0.537667
\(679\) −40.0000 −1.53506
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) −24.0000 −0.919007
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 2.00000 0.0763048
\(688\) −12.0000 −0.457496
\(689\) 0 0
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 6.00000 0.228086
\(693\) 16.0000 0.607790
\(694\) −28.0000 −1.06287
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −20.0000 −0.757554
\(698\) −10.0000 −0.378506
\(699\) 6.00000 0.226941
\(700\) −20.0000 −0.755929
\(701\) −8.00000 −0.302156 −0.151078 0.988522i \(-0.548274\pi\)
−0.151078 + 0.988522i \(0.548274\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 32.0000 1.20348
\(708\) −4.00000 −0.150329
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 2.00000 0.0749532
\(713\) −12.0000 −0.449404
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 6.00000 0.224074
\(718\) −6.00000 −0.223918
\(719\) 34.0000 1.26799 0.633993 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) −1.00000 −0.0372161
\(723\) 2.00000 0.0743808
\(724\) 12.0000 0.445976
\(725\) 30.0000 1.11417
\(726\) 5.00000 0.185567
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 10.0000 0.369611
\(733\) −38.0000 −1.40356 −0.701781 0.712393i \(-0.747612\pi\)
−0.701781 + 0.712393i \(0.747612\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) 0 0
\(738\) −10.0000 −0.368105
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −8.00000 −0.293689
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) −8.00000 −0.292901
\(747\) −16.0000 −0.585409
\(748\) −8.00000 −0.292509
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 10.0000 0.364905 0.182453 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(752\) 10.0000 0.364662
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −8.00000 −0.290573
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) 22.0000 0.796976
\(763\) −16.0000 −0.579239
\(764\) −2.00000 −0.0723575
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 14.0000 0.503871
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 12.0000 0.431331
\(775\) −30.0000 −1.07763
\(776\) 10.0000 0.358979
\(777\) 32.0000 1.14799
\(778\) −8.00000 −0.286814
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) −64.0000 −2.29010
\(782\) −4.00000 −0.143040
\(783\) 6.00000 0.214423
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −20.0000 −0.712470
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) −56.0000 −1.99113
\(792\) −4.00000 −0.142134
\(793\) 0 0
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 4.00000 0.141598
\(799\) −20.0000 −0.707549
\(800\) 5.00000 0.176777
\(801\) −2.00000 −0.0706665
\(802\) 30.0000 1.05934
\(803\) −8.00000 −0.282314
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) −8.00000 −0.281439
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) −24.0000 −0.842235
\(813\) 4.00000 0.140286
\(814\) 32.0000 1.12160
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) −12.0000 −0.419827
\(818\) 18.0000 0.629355
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 6.00000 0.209274
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 6.00000 0.209020
\(825\) 20.0000 0.696311
\(826\) −16.0000 −0.556711
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −2.00000 −0.0695048
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) −6.00000 −0.207390
\(838\) 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
\(842\) 8.00000 0.275698
\(843\) −10.0000 −0.344418
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) −10.0000 −0.343807
\(847\) 20.0000 0.687208
\(848\) 2.00000 0.0686803
\(849\) −28.0000 −0.960958
\(850\) −10.0000 −0.342997
\(851\) 16.0000 0.548473
\(852\) 16.0000 0.548151
\(853\) 54.0000 1.84892 0.924462 0.381273i \(-0.124514\pi\)
0.924462 + 0.381273i \(0.124514\pi\)
\(854\) 40.0000 1.36877
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) −40.0000 −1.36320
\(862\) 4.00000 0.136241
\(863\) −28.0000 −0.953131 −0.476566 0.879139i \(-0.658119\pi\)
−0.476566 + 0.879139i \(0.658119\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 18.0000 0.611665
\(867\) 13.0000 0.441503
\(868\) 24.0000 0.814613
\(869\) 40.0000 1.35691
\(870\) 0 0
\(871\) 0 0
\(872\) 4.00000 0.135457
\(873\) −10.0000 −0.338449
\(874\) 2.00000 0.0676510
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 8.00000 0.270141 0.135070 0.990836i \(-0.456874\pi\)
0.135070 + 0.990836i \(0.456874\pi\)
\(878\) −22.0000 −0.742464
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) −9.00000 −0.303046
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) −8.00000 −0.268462
\(889\) 88.0000 2.95143
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 6.00000 0.200895
\(893\) 10.0000 0.334637
\(894\) 20.0000 0.668900
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) −36.0000 −1.20067
\(900\) −5.00000 −0.166667
\(901\) −4.00000 −0.133259
\(902\) −40.0000 −1.33185
\(903\) 48.0000 1.59734
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 10.0000 0.332228
\(907\) −32.0000 −1.06254 −0.531271 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) 20.0000 0.663723
\(909\) 8.00000 0.265343
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) −1.00000 −0.0331133
\(913\) −64.0000 −2.11809
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −2.00000 −0.0660819
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 12.0000 0.395199
\(923\) 0 0
\(924\) −16.0000 −0.526361
\(925\) 40.0000 1.31519
\(926\) −36.0000 −1.18303
\(927\) −6.00000 −0.197066
\(928\) 6.00000 0.196960
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) 9.00000 0.294963
\(932\) −6.00000 −0.196537
\(933\) −34.0000 −1.11311
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 18.0000 0.586472
\(943\) −20.0000 −0.651290
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) −8.00000 −0.259965 −0.129983 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) −10.0000 −0.324785
\(949\) 0 0
\(950\) 5.00000 0.162221
\(951\) −6.00000 −0.194563
\(952\) 8.00000 0.259281
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −6.00000 −0.194054
\(957\) 24.0000 0.775810
\(958\) −6.00000 −0.193851
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) 12.0000 0.385894 0.192947 0.981209i \(-0.438195\pi\)
0.192947 + 0.981209i \(0.438195\pi\)
\(968\) −5.00000 −0.160706
\(969\) 2.00000 0.0642493
\(970\) 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
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 20.0000 0.639529
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) −24.0000 −0.765871
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) −40.0000 −1.27321
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −50.0000 −1.58830 −0.794151 0.607720i \(-0.792084\pi\)
−0.794151 + 0.607720i \(0.792084\pi\)
\(992\) −6.00000 −0.190500
\(993\) −24.0000 −0.761617
\(994\) 64.0000 2.02996
\(995\) 0 0
\(996\) 16.0000 0.506979
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) 4.00000 0.126618
\(999\) 8.00000 0.253109
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 114.2.a.a.1.1 1
3.2 odd 2 342.2.a.f.1.1 1
4.3 odd 2 912.2.a.h.1.1 1
5.2 odd 4 2850.2.d.s.799.1 2
5.3 odd 4 2850.2.d.s.799.2 2
5.4 even 2 2850.2.a.x.1.1 1
7.6 odd 2 5586.2.a.p.1.1 1
8.3 odd 2 3648.2.a.j.1.1 1
8.5 even 2 3648.2.a.bb.1.1 1
12.11 even 2 2736.2.a.j.1.1 1
15.14 odd 2 8550.2.a.a.1.1 1
19.18 odd 2 2166.2.a.i.1.1 1
57.56 even 2 6498.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.a.a.1.1 1 1.1 even 1 trivial
342.2.a.f.1.1 1 3.2 odd 2
912.2.a.h.1.1 1 4.3 odd 2
2166.2.a.i.1.1 1 19.18 odd 2
2736.2.a.j.1.1 1 12.11 even 2
2850.2.a.x.1.1 1 5.4 even 2
2850.2.d.s.799.1 2 5.2 odd 4
2850.2.d.s.799.2 2 5.3 odd 4
3648.2.a.j.1.1 1 8.3 odd 2
3648.2.a.bb.1.1 1 8.5 even 2
5586.2.a.p.1.1 1 7.6 odd 2
6498.2.a.h.1.1 1 57.56 even 2
8550.2.a.a.1.1 1 15.14 odd 2