Properties

Label 390.2.a.d.1.1
Level $390$
Weight $2$
Character 390.1
Self dual yes
Analytic conductor $3.114$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.11416567883\)
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 \(\chi\) \(=\) 390.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +1.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +2.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -6.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -1.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{38} +1.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} +1.00000 q^{45} +6.00000 q^{46} +1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +1.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -2.00000 q^{56} +2.00000 q^{57} +1.00000 q^{60} +14.0000 q^{61} -8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} -4.00000 q^{67} -6.00000 q^{69} -2.00000 q^{70} -1.00000 q^{72} -4.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} -1.00000 q^{78} -16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} +2.00000 q^{84} +4.00000 q^{86} -6.00000 q^{89} -1.00000 q^{90} +2.00000 q^{91} -6.00000 q^{92} +8.00000 q^{93} +2.00000 q^{95} -1.00000 q^{96} -4.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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
\(5\) 1.00000 0.447214
\(6\) −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
\(10\) −1.00000 −0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) −2.00000 −0.534522
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −2.00000 −0.324443
\(39\) 1.00000 0.160128
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −2.00000 −0.308607
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 6.00000 0.884652
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −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
\(56\) −2.00000 −0.267261
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000 0.129099
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) −8.00000 −1.01600
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) −2.00000 −0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −1.00000 −0.105409
\(91\) 2.00000 0.209657
\(92\) −6.00000 −0.625543
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) −1.00000 −0.102062
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 2.00000 0.195180
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 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
\(111\) 2.00000 0.189832
\(112\) 2.00000 0.188982
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) −2.00000 −0.187317
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) −11.0000 −1.00000
\(122\) −14.0000 −1.26750
\(123\) 6.00000 0.541002
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) −2.00000 −0.178174
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) −1.00000 −0.0877058
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 4.00000 0.345547
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 6.00000 0.510754
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) −3.00000 −0.247436
\(148\) 2.00000 0.164399
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 1.00000 0.0800641
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 16.0000 1.27289
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) −12.0000 −0.945732
\(162\) −1.00000 −0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 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
\(171\) 2.00000 0.152944
\(172\) −4.00000 −0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 1.00000 0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −2.00000 −0.148250
\(183\) 14.0000 1.03491
\(184\) 6.00000 0.442326
\(185\) 2.00000 0.147043
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) −2.00000 −0.145095
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) 4.00000 0.287183
\(195\) 1.00000 0.0716115
\(196\) −3.00000 −0.214286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.00000 −0.282138
\(202\) 12.0000 0.844317
\(203\) 0 0
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 16.0000 1.11477
\(207\) −6.00000 −0.417029
\(208\) 1.00000 0.0693375
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) 16.0000 1.08615
\(218\) 4.00000 0.270914
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) −2.00000 −0.133631
\(225\) 1.00000 0.0666667
\(226\) −12.0000 −0.798228
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 2.00000 0.132453
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 1.00000 0.0645497
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 11.0000 0.707107
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) −3.00000 −0.191663
\(246\) −6.00000 −0.382546
\(247\) 2.00000 0.127257
\(248\) −8.00000 −0.508001
\(249\) −12.0000 −0.760469
\(250\) −1.00000 −0.0632456
\(251\) −30.0000 −1.89358 −0.946792 0.321847i \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 4.00000 0.249029
\(259\) 4.00000 0.248548
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) 30.0000 1.84988 0.924940 0.380114i \(-0.124115\pi\)
0.924940 + 0.380114i \(0.124115\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −4.00000 −0.245256
\(267\) −6.00000 −0.367194
\(268\) −4.00000 −0.244339
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −8.00000 −0.479808
\(279\) 8.00000 0.478947
\(280\) −2.00000 −0.119523
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) −4.00000 −0.234082
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) −6.00000 −0.346989
\(300\) 1.00000 0.0577350
\(301\) −8.00000 −0.461112
\(302\) −8.00000 −0.460348
\(303\) −12.0000 −0.689382
\(304\) 2.00000 0.114708
\(305\) 14.0000 0.801638
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) −8.00000 −0.454369
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −2.00000 −0.112867
\(315\) 2.00000 0.112687
\(316\) −16.0000 −0.900070
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −12.0000 −0.669775
\(322\) 12.0000 0.668734
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 1.00000 0.0554700
\(326\) 16.0000 0.886158
\(327\) −4.00000 −0.221201
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) −12.0000 −0.658586
\(333\) 2.00000 0.109599
\(334\) −12.0000 −0.656611
\(335\) −4.00000 −0.218543
\(336\) 2.00000 0.109109
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) −2.00000 −0.108148
\(343\) −20.0000 −1.07990
\(344\) 4.00000 0.215666
\(345\) −6.00000 −0.323029
\(346\) −18.0000 −0.967686
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) −2.00000 −0.106904
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 6.00000 0.317110
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −15.0000 −0.789474
\(362\) −2.00000 −0.105118
\(363\) −11.0000 −0.577350
\(364\) 2.00000 0.104828
\(365\) −4.00000 −0.209370
\(366\) −14.0000 −0.731792
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −6.00000 −0.312772
\(369\) 6.00000 0.312348
\(370\) −2.00000 −0.103975
\(371\) −12.0000 −0.623009
\(372\) 8.00000 0.414781
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) 2.00000 0.102598
\(381\) −16.0000 −0.819705
\(382\) 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
\(386\) −8.00000 −0.407189
\(387\) −4.00000 −0.203331
\(388\) −4.00000 −0.203069
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) −1.00000 −0.0506370
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) 6.00000 0.302660
\(394\) −18.0000 −0.906827
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) 26.0000 1.30490 0.652451 0.757831i \(-0.273741\pi\)
0.652451 + 0.757831i \(0.273741\pi\)
\(398\) 16.0000 0.802008
\(399\) 4.00000 0.200250
\(400\) 1.00000 0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 4.00000 0.199502
\(403\) 8.00000 0.398508
\(404\) −12.0000 −0.597022
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −6.00000 −0.296319
\(411\) 6.00000 0.295958
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) −12.0000 −0.589057
\(416\) −1.00000 −0.0490290
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 2.00000 0.0975900
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 28.0000 1.35501
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −12.0000 −0.574038
\(438\) 4.00000 0.191127
\(439\) 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
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 2.00000 0.0949158
\(445\) −6.00000 −0.284427
\(446\) 10.0000 0.473514
\(447\) 18.0000 0.851371
\(448\) 2.00000 0.0944911
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) 8.00000 0.375873
\(454\) −12.0000 −0.563188
\(455\) 2.00000 0.0937614
\(456\) −2.00000 −0.0936586
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 4.00000 0.186908
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) −42.0000 −1.95614 −0.978068 0.208288i \(-0.933211\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 0 0
\(463\) −10.0000 −0.464739 −0.232370 0.972628i \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 0 0
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 1.00000 0.0462250
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) 16.0000 0.734904
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 2.00000 0.0911922
\(482\) −26.0000 −1.18427
\(483\) −12.0000 −0.546019
\(484\) −11.0000 −0.500000
\(485\) −4.00000 −0.181631
\(486\) −1.00000 −0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −14.0000 −0.633750
\(489\) −16.0000 −0.723545
\(490\) 3.00000 0.135526
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 6.00000 0.270501
\(493\) 0 0
\(494\) −2.00000 −0.0899843
\(495\) 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 \(0.398544\pi\)
\(500\) 1.00000 0.0447214
\(501\) 12.0000 0.536120
\(502\) 30.0000 1.33897
\(503\) −18.0000 −0.802580 −0.401290 0.915951i \(-0.631438\pi\)
−0.401290 + 0.915951i \(0.631438\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) −16.0000 −0.709885
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) 12.0000 0.529297
\(515\) −16.0000 −0.705044
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) −4.00000 −0.175750
\(519\) 18.0000 0.790112
\(520\) −1.00000 −0.0438529
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 6.00000 0.262111
\(525\) 2.00000 0.0872872
\(526\) −30.0000 −1.30806
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) 6.00000 0.259889
\(534\) 6.00000 0.259645
\(535\) −12.0000 −0.518805
\(536\) 4.00000 0.172774
\(537\) −6.00000 −0.258919
\(538\) −12.0000 −0.517357
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) −8.00000 −0.343629
\(543\) 2.00000 0.0858282
\(544\) 0 0
\(545\) −4.00000 −0.171341
\(546\) −2.00000 −0.0855921
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 6.00000 0.256307
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) 0 0
\(552\) 6.00000 0.255377
\(553\) −32.0000 −1.36078
\(554\) −26.0000 −1.10463
\(555\) 2.00000 0.0848953
\(556\) 8.00000 0.339276
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) −8.00000 −0.338667
\(559\) −4.00000 −0.169182
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 28.0000 1.17693
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) −2.00000 −0.0837708
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) −16.0000 −0.666089 −0.333044 0.942911i \(-0.608076\pi\)
−0.333044 + 0.942911i \(0.608076\pi\)
\(578\) 17.0000 0.707107
\(579\) 8.00000 0.332469
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) 4.00000 0.165805
\(583\) 0 0
\(584\) 4.00000 0.165521
\(585\) 1.00000 0.0413449
\(586\) 6.00000 0.247858
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −3.00000 −0.123718
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 2.00000 0.0821995
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) −16.0000 −0.654836
\(598\) 6.00000 0.245358
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 8.00000 0.326056
\(603\) −4.00000 −0.162893
\(604\) 8.00000 0.325515
\(605\) −11.0000 −0.447214
\(606\) 12.0000 0.487467
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) −14.0000 −0.566843
\(611\) 0 0
\(612\) 0 0
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 4.00000 0.161427
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 16.0000 0.643614
\(619\) 38.0000 1.52735 0.763674 0.645601i \(-0.223393\pi\)
0.763674 + 0.645601i \(0.223393\pi\)
\(620\) 8.00000 0.321288
\(621\) −6.00000 −0.240772
\(622\) −24.0000 −0.962312
\(623\) −12.0000 −0.480770
\(624\) 1.00000 0.0400320
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 0 0
\(630\) −2.00000 −0.0796819
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 16.0000 0.636446
\(633\) −4.00000 −0.158986
\(634\) 18.0000 0.714871
\(635\) −16.0000 −0.634941
\(636\) −6.00000 −0.237915
\(637\) −3.00000 −0.118864
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 12.0000 0.473602
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −12.0000 −0.472866
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) 6.00000 0.235884 0.117942 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −1.00000 −0.0392232
\(651\) 16.0000 0.627089
\(652\) −16.0000 −0.626608
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 4.00000 0.156412
\(655\) 6.00000 0.234439
\(656\) 6.00000 0.234261
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −18.0000 −0.701180 −0.350590 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) −2.00000 −0.0777322
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 4.00000 0.155113
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) −10.0000 −0.386622
\(670\) 4.00000 0.154533
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −14.0000 −0.539260
\(675\) 1.00000 0.0384900
\(676\) 1.00000 0.0384615
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) −12.0000 −0.460857
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 2.00000 0.0764719
\(685\) 6.00000 0.229248
\(686\) 20.0000 0.763604
\(687\) −4.00000 −0.152610
\(688\) −4.00000 −0.152499
\(689\) −6.00000 −0.228582
\(690\) 6.00000 0.228416
\(691\) 26.0000 0.989087 0.494543 0.869153i \(-0.335335\pi\)
0.494543 + 0.869153i \(0.335335\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) −24.0000 −0.911028
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) 0 0
\(698\) −20.0000 −0.757011
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −24.0000 −0.902613
\(708\) 0 0
\(709\) 20.0000 0.751116 0.375558 0.926799i \(-0.377451\pi\)
0.375558 + 0.926799i \(0.377451\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 6.00000 0.224860
\(713\) −48.0000 −1.79761
\(714\) 0 0
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 1.00000 0.0372678
\(721\) −32.0000 −1.19174
\(722\) 15.0000 0.558242
\(723\) 26.0000 0.966950
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 11.0000 0.408248
\(727\) −4.00000 −0.148352 −0.0741759 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 4.00000 0.148047
\(731\) 0 0
\(732\) 14.0000 0.517455
\(733\) 50.0000 1.84679 0.923396 0.383849i \(-0.125402\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) −8.00000 −0.295285
\(735\) −3.00000 −0.110657
\(736\) 6.00000 0.221163
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) 38.0000 1.39785 0.698926 0.715194i \(-0.253662\pi\)
0.698926 + 0.715194i \(0.253662\pi\)
\(740\) 2.00000 0.0735215
\(741\) 2.00000 0.0734718
\(742\) 12.0000 0.440534
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) −8.00000 −0.293294
\(745\) 18.0000 0.659469
\(746\) −26.0000 −0.951928
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) −24.0000 −0.876941
\(750\) −1.00000 −0.0365148
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) −30.0000 −1.09326
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 2.00000 0.0727393
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 34.0000 1.23494
\(759\) 0 0
\(760\) −2.00000 −0.0725476
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 16.0000 0.579619
\(763\) −8.00000 −0.289619
\(764\) 0 0
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 8.00000 0.287926
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 4.00000 0.143777
\(775\) 8.00000 0.287368
\(776\) 4.00000 0.143592
\(777\) 4.00000 0.143499
\(778\) −36.0000 −1.29066
\(779\) 12.0000 0.429945
\(780\) 1.00000 0.0358057
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 2.00000 0.0713831
\(786\) −6.00000 −0.214013
\(787\) −16.0000 −0.570338 −0.285169 0.958477i \(-0.592050\pi\)
−0.285169 + 0.958477i \(0.592050\pi\)
\(788\) 18.0000 0.641223
\(789\) 30.0000 1.06803
\(790\) 16.0000 0.569254
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) 14.0000 0.497155
\(794\) −26.0000 −0.922705
\(795\) −6.00000 −0.212798
\(796\) −16.0000 −0.567105
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) −12.0000 −0.422944
\(806\) −8.00000 −0.281788
\(807\) 12.0000 0.422420
\(808\) 12.0000 0.422159
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 26.0000 0.912983 0.456492 0.889728i \(-0.349106\pi\)
0.456492 + 0.889728i \(0.349106\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) 0 0
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) −14.0000 −0.489499
\(819\) 2.00000 0.0698857
\(820\) 6.00000 0.209529
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) −6.00000 −0.209274
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −6.00000 −0.208514
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 12.0000 0.416526
\(831\) 26.0000 0.901930
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) −8.00000 −0.277017
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) −6.00000 −0.207267
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −29.0000 −1.00000
\(842\) −32.0000 −1.10279
\(843\) −30.0000 −1.03325
\(844\) −4.00000 −0.137686
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −22.0000 −0.755929
\(848\) −6.00000 −0.206041
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) −28.0000 −0.958140
\(855\) 2.00000 0.0683986
\(856\) 12.0000 0.410152
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −4.00000 −0.136399
\(861\) 12.0000 0.408959
\(862\) −24.0000 −0.817443
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 18.0000 0.612018
\(866\) −14.0000 −0.475739
\(867\) −17.0000 −0.577350
\(868\) 16.0000 0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 4.00000 0.135457
\(873\) −4.00000 −0.135379
\(874\) 12.0000 0.405906
\(875\) 2.00000 0.0676123
\(876\) −4.00000 −0.135147
\(877\) −58.0000 −1.95852 −0.979260 0.202606i \(-0.935059\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) −32.0000 −1.07995
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 3.00000 0.101015
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −30.0000 −1.00730 −0.503651 0.863907i \(-0.668010\pi\)
−0.503651 + 0.863907i \(0.668010\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −32.0000 −1.07325
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −10.0000 −0.334825
\(893\) 0 0
\(894\) −18.0000 −0.602010
\(895\) −6.00000 −0.200558
\(896\) −2.00000 −0.0668153
\(897\) −6.00000 −0.200334
\(898\) 30.0000 1.00111
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) −12.0000 −0.399114
\(905\) 2.00000 0.0664822
\(906\) −8.00000 −0.265782
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 12.0000 0.398234
\(909\) −12.0000 −0.398015
\(910\) −2.00000 −0.0662994
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 2.00000 0.0662266
\(913\) 0 0
\(914\) 28.0000 0.926158
\(915\) 14.0000 0.462826
\(916\) −4.00000 −0.132164
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 6.00000 0.197814
\(921\) −4.00000 −0.131804
\(922\) 42.0000 1.38320
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 10.0000 0.328620
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) −8.00000 −0.262330
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) 24.0000 0.785725
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 8.00000 0.261209
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) −2.00000 −0.0651635
\(943\) −36.0000 −1.17232
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) −16.0000 −0.519656
\(949\) −4.00000 −0.129845
\(950\) −2.00000 −0.0648886
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 12.0000 0.387500
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) −2.00000 −0.0644826
\(963\) −12.0000 −0.386695
\(964\) 26.0000 0.837404
\(965\) 8.00000 0.257529
\(966\) 12.0000 0.386094
\(967\) 50.0000 1.60789 0.803946 0.594703i \(-0.202730\pi\)
0.803946 + 0.594703i \(0.202730\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) −30.0000 −0.962746 −0.481373 0.876516i \(-0.659862\pi\)
−0.481373 + 0.876516i \(0.659862\pi\)
\(972\) 1.00000 0.0320750
\(973\) 16.0000 0.512936
\(974\) −2.00000 −0.0640841
\(975\) 1.00000 0.0320256
\(976\) 14.0000 0.448129
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 16.0000 0.511624
\(979\) 0 0
\(980\) −3.00000 −0.0958315
\(981\) −4.00000 −0.127710
\(982\) −6.00000 −0.191468
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) −6.00000 −0.191273
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) 24.0000 0.763156
\(990\) 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
\(993\) 2.00000 0.0634681
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) −12.0000 −0.380235
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) −14.0000 −0.443162
\(999\) 2.00000 0.0632772
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 390.2.a.d.1.1 1
3.2 odd 2 1170.2.a.k.1.1 1
4.3 odd 2 3120.2.a.j.1.1 1
5.2 odd 4 1950.2.e.d.1249.1 2
5.3 odd 4 1950.2.e.d.1249.2 2
5.4 even 2 1950.2.a.o.1.1 1
12.11 even 2 9360.2.a.g.1.1 1
13.5 odd 4 5070.2.b.m.1351.2 2
13.8 odd 4 5070.2.b.m.1351.1 2
13.12 even 2 5070.2.a.t.1.1 1
15.2 even 4 5850.2.e.o.5149.2 2
15.8 even 4 5850.2.e.o.5149.1 2
15.14 odd 2 5850.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.d.1.1 1 1.1 even 1 trivial
1170.2.a.k.1.1 1 3.2 odd 2
1950.2.a.o.1.1 1 5.4 even 2
1950.2.e.d.1249.1 2 5.2 odd 4
1950.2.e.d.1249.2 2 5.3 odd 4
3120.2.a.j.1.1 1 4.3 odd 2
5070.2.a.t.1.1 1 13.12 even 2
5070.2.b.m.1351.1 2 13.8 odd 4
5070.2.b.m.1351.2 2 13.5 odd 4
5850.2.a.g.1.1 1 15.14 odd 2
5850.2.e.o.5149.1 2 15.8 even 4
5850.2.e.o.5149.2 2 15.2 even 4
9360.2.a.g.1.1 1 12.11 even 2