Properties

Label 5550.2.a.bd.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.3169731218\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +6.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -4.00000 q^{21} +6.00000 q^{22} -1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} -6.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} -8.00000 q^{43} +6.00000 q^{44} -6.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -6.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{56} -2.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} +8.00000 q^{61} +8.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} +4.00000 q^{67} +6.00000 q^{68} +1.00000 q^{72} -14.0000 q^{73} -1.00000 q^{74} +2.00000 q^{76} +24.0000 q^{77} +2.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{84} -8.00000 q^{86} -6.00000 q^{87} +6.00000 q^{88} +6.00000 q^{89} -8.00000 q^{91} -8.00000 q^{93} -6.00000 q^{94} -1.00000 q^{96} -8.00000 q^{97} +9.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 6.00000 1.27920
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −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
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.00000 −1.04447
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 2.00000 0.324443
\(39\) 2.00000 0.320256
\(40\) 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
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) −2.00000 −0.264906
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 8.00000 1.01600
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 24.0000 2.73505
\(78\) 2.00000 0.226455
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) −6.00000 −0.643268
\(88\) 6.00000 0.639602
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 9.00000 0.909137
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −6.00000 −0.594089
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 4.00000 0.377964
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) −2.00000 −0.184900
\(118\) −12.0000 −1.10469
\(119\) 24.0000 2.20008
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 8.00000 0.724286
\(123\) 6.00000 0.541002
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −6.00000 −0.522233
\(133\) 8.00000 0.693688
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) −9.00000 −0.742307
\(148\) −1.00000 −0.0821995
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 2.00000 0.162221
\(153\) 6.00000 0.485071
\(154\) 24.0000 1.93398
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −16.0000 −1.27289
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −4.00000 −0.308607
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) −8.00000 −0.609994
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) 12.0000 0.901975
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −8.00000 −0.592999
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 36.0000 2.63258
\(188\) −6.00000 −0.437595
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 6.00000 0.426401
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 24.0000 1.68447
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 32.0000 2.17230
\(218\) 20.0000 1.35457
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 1.00000 0.0671156
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −2.00000 −0.132453
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) −24.0000 −1.57908
\(232\) 6.00000 0.393919
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 16.0000 1.03931
\(238\) 24.0000 1.55569
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 25.0000 1.60706
\(243\) −1.00000 −0.0641500
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) −4.00000 −0.254514
\(248\) 8.00000 0.508001
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 8.00000 0.498058
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) −12.0000 −0.741362
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) −6.00000 −0.367194
\(268\) 4.00000 0.244339
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 6.00000 0.363803
\(273\) 8.00000 0.484182
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −4.00000 −0.239904
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 6.00000 0.357295
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) −24.0000 −1.41668
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) −14.0000 −0.819288
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) −6.00000 −0.348155
\(298\) −12.0000 −0.695141
\(299\) 0 0
\(300\) 0 0
\(301\) −32.0000 −1.84445
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 24.0000 1.36753
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 2.00000 0.113228
\(313\) 16.0000 0.904373 0.452187 0.891923i \(-0.350644\pi\)
0.452187 + 0.891923i \(0.350644\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 6.00000 0.336463
\(319\) 36.0000 2.01561
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) −20.0000 −1.10600
\(328\) −6.00000 −0.331295
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 14.0000 0.769510 0.384755 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) 12.0000 0.658586
\(333\) −1.00000 −0.0547997
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) −9.00000 −0.489535
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 48.0000 2.59935
\(342\) 2.00000 0.108148
\(343\) 8.00000 0.431959
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −6.00000 −0.321634
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 6.00000 0.319801
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) −24.0000 −1.27021
\(358\) −12.0000 −0.634220
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −10.0000 −0.525588
\(363\) −25.0000 −1.31216
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) −8.00000 −0.418167
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) −8.00000 −0.414781
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 36.0000 1.86152
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −12.0000 −0.618031
\(378\) −4.00000 −0.205738
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 16.0000 0.814379
\(387\) −8.00000 −0.406663
\(388\) −8.00000 −0.406138
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) 12.0000 0.605320
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −4.00000 −0.200502
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −4.00000 −0.199502
\(403\) −16.0000 −0.797017
\(404\) 0 0
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) −6.00000 −0.297409
\(408\) −6.00000 −0.297044
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) −14.0000 −0.689730
\(413\) −48.0000 −2.36193
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 4.00000 0.195881
\(418\) 12.0000 0.586939
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) −16.0000 −0.778868
\(423\) −6.00000 −0.291730
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 32.0000 1.54859
\(428\) 0 0
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −38.0000 −1.82616 −0.913082 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) 20.0000 0.957826
\(437\) 0 0
\(438\) 14.0000 0.668946
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −12.0000 −0.570782
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 1.00000 0.0474579
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) 12.0000 0.567581
\(448\) 4.00000 0.188982
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) −6.00000 −0.282216
\(453\) 16.0000 0.751746
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 26.0000 1.21490
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) −24.0000 −1.11658
\(463\) 34.0000 1.58011 0.790057 0.613033i \(-0.210051\pi\)
0.790057 + 0.613033i \(0.210051\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) −12.0000 −0.552345
\(473\) −48.0000 −2.20704
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) 24.0000 1.10004
\(477\) −6.00000 −0.274721
\(478\) 24.0000 1.09773
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 8.00000 0.362143
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 6.00000 0.270501
\(493\) 36.0000 1.62136
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 4.00000 0.177471
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −56.0000 −2.47729
\(512\) 1.00000 0.0441942
\(513\) −2.00000 −0.0883022
\(514\) −30.0000 −1.32324
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) −36.0000 −1.58328
\(518\) −4.00000 −0.175750
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 6.00000 0.262613
\(523\) −32.0000 −1.39926 −0.699631 0.714504i \(-0.746652\pi\)
−0.699631 + 0.714504i \(0.746652\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) 48.0000 2.09091
\(528\) −6.00000 −0.261116
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 8.00000 0.346844
\(533\) 12.0000 0.519778
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 12.0000 0.517838
\(538\) −24.0000 −1.03471
\(539\) 54.0000 2.32594
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 8.00000 0.343629
\(543\) 10.0000 0.429141
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 8.00000 0.342368
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 12.0000 0.512615
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) −64.0000 −2.72156
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 8.00000 0.338667
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) −36.0000 −1.51992
\(562\) −30.0000 −1.26547
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 4.00000 0.167984
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) −12.0000 −0.501745
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −20.0000 −0.832611 −0.416305 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(578\) 19.0000 0.790296
\(579\) −16.0000 −0.664937
\(580\) 0 0
\(581\) 48.0000 1.99138
\(582\) 8.00000 0.331611
\(583\) −36.0000 −1.49097
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) −9.00000 −0.371154
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) −1.00000 −0.0410997
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) 4.00000 0.163709
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −32.0000 −1.30422
\(603\) 4.00000 0.162893
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 0 0
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 2.00000 0.0811107
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 6.00000 0.242536
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 24.0000 0.966988
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 14.0000 0.563163
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 16.0000 0.639489
\(627\) −12.0000 −0.479234
\(628\) −14.0000 −0.558661
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) −16.0000 −0.636446
\(633\) 16.0000 0.635943
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) −18.0000 −0.713186
\(638\) 36.0000 1.42525
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) −72.0000 −2.82625
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 16.0000 0.626608
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −20.0000 −0.782062
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) −14.0000 −0.546192
\(658\) −24.0000 −0.935617
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) 20.0000 0.777910 0.388955 0.921257i \(-0.372836\pi\)
0.388955 + 0.921257i \(0.372836\pi\)
\(662\) 14.0000 0.544125
\(663\) 12.0000 0.466041
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) −4.00000 −0.154649
\(670\) 0 0
\(671\) 48.0000 1.85302
\(672\) −4.00000 −0.154303
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 10.0000 0.385186
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 6.00000 0.230429
\(679\) −32.0000 −1.22805
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 48.0000 1.83801
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) −26.0000 −0.991962
\(688\) −8.00000 −0.304997
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) −6.00000 −0.228086
\(693\) 24.0000 0.911685
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −36.0000 −1.36360
\(698\) 26.0000 0.984115
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 2.00000 0.0754851
\(703\) −2.00000 −0.0754314
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) 44.0000 1.65245 0.826227 0.563337i \(-0.190483\pi\)
0.826227 + 0.563337i \(0.190483\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) −24.0000 −0.898177
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) −24.0000 −0.896296
\(718\) −24.0000 −0.895672
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) −56.0000 −2.08555
\(722\) −15.0000 −0.558242
\(723\) −2.00000 −0.0743808
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) −25.0000 −0.927837
\(727\) −38.0000 −1.40934 −0.704671 0.709534i \(-0.748905\pi\)
−0.704671 + 0.709534i \(0.748905\pi\)
\(728\) −8.00000 −0.296500
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) −8.00000 −0.295689
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) 0 0
\(737\) 24.0000 0.884051
\(738\) −6.00000 −0.220863
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) −24.0000 −0.881068
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) 12.0000 0.439057
\(748\) 36.0000 1.31629
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) −4.00000 −0.144905
\(763\) 80.0000 2.89619
\(764\) 0 0
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) 24.0000 0.866590
\(768\) −1.00000 −0.0360844
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 16.0000 0.575853
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) −8.00000 −0.287183
\(777\) 4.00000 0.143499
\(778\) −30.0000 −1.07555
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −18.0000 −0.641223
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 6.00000 0.213201
\(793\) −16.0000 −0.568177
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) −8.00000 −0.283197
\(799\) −36.0000 −1.27359
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 18.0000 0.635602
\(803\) −84.0000 −2.96430
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 24.0000 0.844840
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 24.0000 0.842235
\(813\) −8.00000 −0.280572
\(814\) −6.00000 −0.210300
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) −16.0000 −0.559769
\(818\) −10.0000 −0.349642
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) −12.0000 −0.418548
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −28.0000 −0.972480 −0.486240 0.873825i \(-0.661632\pi\)
−0.486240 + 0.873825i \(0.661632\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) −2.00000 −0.0693375
\(833\) 54.0000 1.87099
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) −8.00000 −0.276520
\(838\) 6.00000 0.207267
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 8.00000 0.275698
\(843\) 30.0000 1.03325
\(844\) −16.0000 −0.550743
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) 100.000 3.43604
\(848\) −6.00000 −0.206041
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 32.0000 1.09502
\(855\) 0 0
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 12.0000 0.409673
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) 0 0
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −38.0000 −1.29129
\(867\) −19.0000 −0.645274
\(868\) 32.0000 1.08615
\(869\) −96.0000 −3.25658
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 20.0000 0.677285
\(873\) −8.00000 −0.270759
\(874\) 0 0
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 20.0000 0.674967
\(879\) −30.0000 −1.01187
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 9.00000 0.303046
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 1.00000 0.0335578
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 6.00000 0.201008
\(892\) 4.00000 0.133930
\(893\) −12.0000 −0.401565
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) −36.0000 −1.19867
\(903\) 32.0000 1.06489
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 40.0000 1.32818 0.664089 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 72.0000 2.38285
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) 26.0000 0.859064
\(917\) −48.0000 −1.58510
\(918\) −6.00000 −0.198030
\(919\) −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
\(922\) −6.00000 −0.197599
\(923\) 0 0
\(924\) −24.0000 −0.789542
\(925\) 0 0
\(926\) 34.0000 1.11731
\(927\) −14.0000 −0.459820
\(928\) 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
\(931\) 18.0000 0.589926
\(932\) 24.0000 0.786146
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 16.0000 0.522419
\(939\) −16.0000 −0.522140
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) 14.0000 0.456145
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 16.0000 0.519656
\(949\) 28.0000 0.908918
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 24.0000 0.777844
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) −36.0000 −1.16371
\(958\) −24.0000 −0.775405
\(959\) 48.0000 1.55000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 2.00000 0.0644826
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) 46.0000 1.47926 0.739630 0.673014i \(-0.235000\pi\)
0.739630 + 0.673014i \(0.235000\pi\)
\(968\) 25.0000 0.803530
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) −54.0000 −1.73294 −0.866471 0.499227i \(-0.833617\pi\)
−0.866471 + 0.499227i \(0.833617\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −16.0000 −0.512936
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −16.0000 −0.511624
\(979\) 36.0000 1.15056
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) −6.00000 −0.191468
\(983\) 30.0000 0.956851 0.478426 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) 24.0000 0.763928
\(988\) −4.00000 −0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 8.00000 0.254000
\(993\) −14.0000 −0.444277
\(994\) 0 0
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 14.0000 0.443162
\(999\) 1.00000 0.0316386
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 5550.2.a.bd.1.1 1
5.4 even 2 1110.2.a.h.1.1 1
15.14 odd 2 3330.2.a.m.1.1 1
20.19 odd 2 8880.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.h.1.1 1 5.4 even 2
3330.2.a.m.1.1 1 15.14 odd 2
5550.2.a.bd.1.1 1 1.1 even 1 trivial
8880.2.a.n.1.1 1 20.19 odd 2