Properties

Label 1950.2.a.z.1.1
Level $1950$
Weight $2$
Character 1950.1
Self dual yes
Analytic conductor $15.571$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1950.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{21} +2.00000 q^{22} +1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} +2.00000 q^{42} +4.00000 q^{43} +2.00000 q^{44} -8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} -1.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} -4.00000 q^{57} +4.00000 q^{58} +10.0000 q^{59} -14.0000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -16.0000 q^{67} +2.00000 q^{68} -4.00000 q^{71} +1.00000 q^{72} +8.00000 q^{73} +6.00000 q^{74} -4.00000 q^{76} +4.00000 q^{77} -1.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +2.00000 q^{84} +4.00000 q^{86} +4.00000 q^{87} +2.00000 q^{88} +6.00000 q^{89} -2.00000 q^{91} +8.00000 q^{93} -8.00000 q^{94} +1.00000 q^{96} +12.0000 q^{97} -3.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 2.00000 0.426401
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 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
\(33\) 2.00000 0.348155
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 2.00000 0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) −1.00000 −0.138675
\(53\) 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
\(56\) 2.00000 0.267261
\(57\) −4.00000 −0.529813
\(58\) 4.00000 0.525226
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 8.00000 1.01600
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −16.0000 −1.95471 −0.977356 0.211604i \(-0.932131\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 1.00000 0.117851
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 4.00000 0.455842
\(78\) −1.00000 −0.113228
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 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 \(0.271155\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 4.00000 0.428845
\(88\) 2.00000 0.213201
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) −3.00000 −0.303046
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 2.00000 0.198030
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 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
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 2.00000 0.188982
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) −1.00000 −0.0924500
\(118\) 10.0000 0.920575
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −14.0000 −1.26750
\(123\) −6.00000 −0.541002
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 2.00000 0.174078
\(133\) −8.00000 −0.693688
\(134\) −16.0000 −1.38219
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) −4.00000 −0.335673
\(143\) −2.00000 −0.167248
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 8.00000 0.662085
\(147\) −3.00000 −0.247436
\(148\) 6.00000 0.493197
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −4.00000 −0.324443
\(153\) 2.00000 0.161690
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −8.00000 −0.636446
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 2.00000 0.154303
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 10.0000 0.751646
\(178\) 6.00000 0.449719
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) −2.00000 −0.148250
\(183\) −14.0000 −1.03491
\(184\) 0 0
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 4.00000 0.292509
\(188\) −8.00000 −0.583460
\(189\) 2.00000 0.145479
\(190\) 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
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 12.0000 0.861550
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 2.00000 0.142134
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 4.00000 0.281439
\(203\) 8.00000 0.561490
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 6.00000 0.418040
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) −8.00000 −0.553372
\(210\) 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
\(213\) −4.00000 −0.274075
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) −18.0000 −1.21911
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 6.00000 0.402694
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −4.00000 −0.264906
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 4.00000 0.262613
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) 10.0000 0.650945
\(237\) −8.00000 −0.519656
\(238\) 4.00000 0.259281
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) 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
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) 6.00000 0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 4.00000 0.249029
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) −6.00000 −0.370681
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) 6.00000 0.367194
\(268\) −16.0000 −0.977356
\(269\) −20.0000 −1.21942 −0.609711 0.792624i \(-0.708714\pi\)
−0.609711 + 0.792624i \(0.708714\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 2.00000 0.121268
\(273\) −2.00000 −0.121046
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 4.00000 0.239904
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −8.00000 −0.476393
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) −12.0000 −0.708338
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) 8.00000 0.468165
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 2.00000 0.116052
\(298\) 12.0000 0.695141
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) −16.0000 −0.920697
\(303\) 4.00000 0.229794
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 4.00000 0.227921
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −20.0000 −1.13047 −0.565233 0.824931i \(-0.691214\pi\)
−0.565233 + 0.824931i \(0.691214\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 2.00000 0.112154
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) −18.0000 −0.995402
\(328\) −6.00000 −0.331295
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 12.0000 0.658586
\(333\) 6.00000 0.328798
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 20.0000 1.08947 0.544735 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 1.00000 0.0543928
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) −4.00000 −0.216295
\(343\) −20.0000 −1.07990
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 4.00000 0.214423
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 2.00000 0.106600
\(353\) −34.0000 −1.80964 −0.904819 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 10.0000 0.531494
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 4.00000 0.211702
\(358\) −18.0000 −0.951330
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −6.00000 −0.315353
\(363\) −7.00000 −0.367405
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) −14.0000 −0.731792
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 8.00000 0.414781
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) −4.00000 −0.206010
\(378\) 2.00000 0.102869
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) −12.0000 −0.613973
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 12.0000 0.609208
\(389\) 16.0000 0.811232 0.405616 0.914044i \(-0.367057\pi\)
0.405616 + 0.914044i \(0.367057\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) −6.00000 −0.302660
\(394\) 10.0000 0.503793
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) −24.0000 −1.20301
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −16.0000 −0.798007
\(403\) −8.00000 −0.398508
\(404\) 4.00000 0.199007
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) 12.0000 0.594818
\(408\) 2.00000 0.0990148
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) −14.0000 −0.690569
\(412\) 6.00000 0.295599
\(413\) 20.0000 0.984136
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 4.00000 0.195881
\(418\) −8.00000 −0.391293
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) −4.00000 −0.194717
\(423\) −8.00000 −0.388973
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) −4.00000 −0.193801
\(427\) −28.0000 −1.35501
\(428\) −4.00000 −0.193347
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 1.00000 0.0481125
\(433\) 12.0000 0.576683 0.288342 0.957528i \(-0.406896\pi\)
0.288342 + 0.957528i \(0.406896\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) 0 0
\(438\) 8.00000 0.382255
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −2.00000 −0.0951303
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) 12.0000 0.567581
\(448\) 2.00000 0.0944911
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) 6.00000 0.282216
\(453\) −16.0000 −0.751746
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) −22.0000 −1.02799
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 4.00000 0.186097
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 10.0000 0.460287
\(473\) 8.00000 0.367840
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 2.00000 0.0915737
\(478\) −12.0000 −0.548867
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) −14.0000 −0.633750
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −22.0000 −0.992846 −0.496423 0.868081i \(-0.665354\pi\)
−0.496423 + 0.868081i \(0.665354\pi\)
\(492\) −6.00000 −0.270501
\(493\) 8.00000 0.360302
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −8.00000 −0.358849
\(498\) 12.0000 0.537733
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 18.0000 0.803379
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 6.00000 0.266207
\(509\) −40.0000 −1.77297 −0.886484 0.462758i \(-0.846860\pi\)
−0.886484 + 0.462758i \(0.846860\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) −16.0000 −0.703679
\(518\) 12.0000 0.527250
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 4.00000 0.175075
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 16.0000 0.696971
\(528\) 2.00000 0.0870388
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) −8.00000 −0.346844
\(533\) 6.00000 0.259889
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) −16.0000 −0.691095
\(537\) −18.0000 −0.776757
\(538\) −20.0000 −0.862261
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 16.0000 0.687259
\(543\) −6.00000 −0.257485
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −14.0000 −0.598050
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) −16.0000 −0.681623
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 8.00000 0.338667
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) −18.0000 −0.759284
\(563\) −44.0000 −1.85438 −0.927189 0.374593i \(-0.877783\pi\)
−0.927189 + 0.374593i \(0.877783\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 2.00000 0.0839921
\(568\) −4.00000 −0.167836
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) −2.00000 −0.0836242
\(573\) −12.0000 −0.501307
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 28.0000 1.16566 0.582828 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 12.0000 0.497416
\(583\) 4.00000 0.165663
\(584\) 8.00000 0.331042
\(585\) 0 0
\(586\) 26.0000 1.07405
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) −3.00000 −0.123718
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 6.00000 0.246598
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −46.0000 −1.87638 −0.938190 0.346122i \(-0.887498\pi\)
−0.938190 + 0.346122i \(0.887498\pi\)
\(602\) 8.00000 0.326056
\(603\) −16.0000 −0.651570
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 4.00000 0.162489
\(607\) 18.0000 0.730597 0.365299 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) −4.00000 −0.162221
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 2.00000 0.0808452
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) 46.0000 1.85189 0.925945 0.377658i \(-0.123271\pi\)
0.925945 + 0.377658i \(0.123271\pi\)
\(618\) 6.00000 0.241355
\(619\) −24.0000 −0.964641 −0.482321 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 20.0000 0.801927
\(623\) 12.0000 0.480770
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) −20.0000 −0.799361
\(627\) −8.00000 −0.319489
\(628\) 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 \(0.550904\pi\)
\(632\) −8.00000 −0.318223
\(633\) −4.00000 −0.158986
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 3.00000 0.118864
\(638\) 8.00000 0.316723
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) −38.0000 −1.50091 −0.750455 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) −4.00000 −0.157867
\(643\) 32.0000 1.26196 0.630978 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 36.0000 1.41531 0.707653 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) 1.00000 0.0392837
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) −8.00000 −0.313304
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) −18.0000 −0.703856
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 8.00000 0.312110
\(658\) −16.0000 −0.623745
\(659\) 10.0000 0.389545 0.194772 0.980848i \(-0.437603\pi\)
0.194772 + 0.980848i \(0.437603\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) −12.0000 −0.466393
\(663\) −2.00000 −0.0776736
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) −28.0000 −1.08093
\(672\) 2.00000 0.0771517
\(673\) −36.0000 −1.38770 −0.693849 0.720121i \(-0.744086\pi\)
−0.693849 + 0.720121i \(0.744086\pi\)
\(674\) 20.0000 0.770371
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −10.0000 −0.384331 −0.192166 0.981363i \(-0.561551\pi\)
−0.192166 + 0.981363i \(0.561551\pi\)
\(678\) 6.00000 0.230429
\(679\) 24.0000 0.921035
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 16.0000 0.612672
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) −22.0000 −0.839352
\(688\) 4.00000 0.152499
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 14.0000 0.532200
\(693\) 4.00000 0.151947
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) −12.0000 −0.454532
\(698\) 14.0000 0.529908
\(699\) −14.0000 −0.529529
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −24.0000 −0.905177
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −34.0000 −1.27961
\(707\) 8.00000 0.300871
\(708\) 10.0000 0.375823
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) −18.0000 −0.672692
\(717\) −12.0000 −0.448148
\(718\) 32.0000 1.19423
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) −3.00000 −0.111648
\(723\) −18.0000 −0.669427
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) 34.0000 1.26099 0.630495 0.776193i \(-0.282852\pi\)
0.630495 + 0.776193i \(0.282852\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) −14.0000 −0.517455
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) 0 0
\(737\) −32.0000 −1.17874
\(738\) −6.00000 −0.220863
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 4.00000 0.146845
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) 12.0000 0.439057
\(748\) 4.00000 0.146254
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) −8.00000 −0.291730
\(753\) 18.0000 0.655956
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 8.00000 0.290573
\(759\) 0 0
\(760\) 0 0
\(761\) 38.0000 1.37750 0.688749 0.724999i \(-0.258160\pi\)
0.688749 + 0.724999i \(0.258160\pi\)
\(762\) 6.00000 0.217357
\(763\) −36.0000 −1.30329
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −10.0000 −0.361079
\(768\) 1.00000 0.0360844
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 0 0
\(773\) 22.0000 0.791285 0.395643 0.918405i \(-0.370522\pi\)
0.395643 + 0.918405i \(0.370522\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 12.0000 0.430775
\(777\) 12.0000 0.430498
\(778\) 16.0000 0.573628
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 4.00000 0.142948
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −6.00000 −0.214013
\(787\) −8.00000 −0.285169 −0.142585 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 10.0000 0.356235
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 2.00000 0.0710669
\(793\) 14.0000 0.497155
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) 14.0000 0.495905 0.247953 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(798\) −8.00000 −0.283197
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) −18.0000 −0.635602
\(803\) 16.0000 0.564628
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) −20.0000 −0.704033
\(808\) 4.00000 0.140720
\(809\) −22.0000 −0.773479 −0.386739 0.922189i \(-0.626399\pi\)
−0.386739 + 0.922189i \(0.626399\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 8.00000 0.280745
\(813\) 16.0000 0.561144
\(814\) 12.0000 0.420600
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) −16.0000 −0.559769
\(818\) −26.0000 −0.909069
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) −14.0000 −0.488306
\(823\) 34.0000 1.18517 0.592583 0.805510i \(-0.298108\pi\)
0.592583 + 0.805510i \(0.298108\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) 20.0000 0.695889
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) −1.00000 −0.0346688
\(833\) −6.00000 −0.207888
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) 8.00000 0.276520
\(838\) −14.0000 −0.483622
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 34.0000 1.17172
\(843\) −18.0000 −0.619953
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) −14.0000 −0.481046
\(848\) 2.00000 0.0686803
\(849\) 16.0000 0.549119
\(850\) 0 0
\(851\) 0 0
\(852\) −4.00000 −0.137038
\(853\) −18.0000 −0.616308 −0.308154 0.951336i \(-0.599711\pi\)
−0.308154 + 0.951336i \(0.599711\pi\)
\(854\) −28.0000 −0.958140
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 34.0000 1.16142 0.580709 0.814111i \(-0.302775\pi\)
0.580709 + 0.814111i \(0.302775\pi\)
\(858\) −2.00000 −0.0682789
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 16.0000 0.544962
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 12.0000 0.407777
\(867\) −13.0000 −0.441503
\(868\) 16.0000 0.543075
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) −18.0000 −0.609557
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) 42.0000 1.41824 0.709120 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) 32.0000 1.07995
\(879\) 26.0000 0.876958
\(880\) 0 0
\(881\) −10.0000 −0.336909 −0.168454 0.985709i \(-0.553878\pi\)
−0.168454 + 0.985709i \(0.553878\pi\)
\(882\) −3.00000 −0.101015
\(883\) 24.0000 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 6.00000 0.201347
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −14.0000 −0.468755
\(893\) 32.0000 1.07084
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) 32.0000 1.06726
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) −12.0000 −0.399556
\(903\) 8.00000 0.266223
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 12.0000 0.398234
\(909\) 4.00000 0.132672
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) −4.00000 −0.132453
\(913\) 24.0000 0.794284
\(914\) −28.0000 −0.926158
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) −12.0000 −0.396275
\(918\) 2.00000 0.0660098
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) 20.0000 0.658665
\(923\) 4.00000 0.131662
\(924\) 4.00000 0.131590
\(925\) 0 0
\(926\) 6.00000 0.197172
\(927\) 6.00000 0.197066
\(928\) 4.00000 0.131306
\(929\) −50.0000 −1.64045 −0.820223 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) −14.0000 −0.458585
\(933\) 20.0000 0.654771
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) 56.0000 1.82944 0.914720 0.404088i \(-0.132411\pi\)
0.914720 + 0.404088i \(0.132411\pi\)
\(938\) −32.0000 −1.04484
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) −44.0000 −1.43436 −0.717180 0.696888i \(-0.754567\pi\)
−0.717180 + 0.696888i \(0.754567\pi\)
\(942\) 10.0000 0.325818
\(943\) 0 0
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −8.00000 −0.259828
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 4.00000 0.129641
\(953\) 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
\(956\) −12.0000 −0.388108
\(957\) 8.00000 0.258603
\(958\) 36.0000 1.16311
\(959\) −28.0000 −0.904167
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −6.00000 −0.193448
\(963\) −4.00000 −0.128898
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) 0 0
\(967\) 58.0000 1.86515 0.932577 0.360971i \(-0.117555\pi\)
0.932577 + 0.360971i \(0.117555\pi\)
\(968\) −7.00000 −0.224989
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) 50.0000 1.60458 0.802288 0.596937i \(-0.203616\pi\)
0.802288 + 0.596937i \(0.203616\pi\)
\(972\) 1.00000 0.0320750
\(973\) 8.00000 0.256468
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) −8.00000 −0.255812
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) −22.0000 −0.702048
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 8.00000 0.254772
\(987\) −16.0000 −0.509286
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 8.00000 0.254000
\(993\) −12.0000 −0.380808
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 36.0000 1.13956
\(999\) 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 1950.2.a.z.1.1 1
3.2 odd 2 5850.2.a.t.1.1 1
5.2 odd 4 390.2.e.c.79.2 yes 2
5.3 odd 4 390.2.e.c.79.1 2
5.4 even 2 1950.2.a.c.1.1 1
15.2 even 4 1170.2.e.a.469.1 2
15.8 even 4 1170.2.e.a.469.2 2
15.14 odd 2 5850.2.a.bj.1.1 1
20.3 even 4 3120.2.l.g.1249.1 2
20.7 even 4 3120.2.l.g.1249.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.e.c.79.1 2 5.3 odd 4
390.2.e.c.79.2 yes 2 5.2 odd 4
1170.2.e.a.469.1 2 15.2 even 4
1170.2.e.a.469.2 2 15.8 even 4
1950.2.a.c.1.1 1 5.4 even 2
1950.2.a.z.1.1 1 1.1 even 1 trivial
3120.2.l.g.1249.1 2 20.3 even 4
3120.2.l.g.1249.2 2 20.7 even 4
5850.2.a.t.1.1 1 3.2 odd 2
5850.2.a.bj.1.1 1 15.14 odd 2