Properties

Label 690.2.a.c.1.1
Level $690$
Weight $2$
Character 690.1
Self dual yes
Analytic conductor $5.510$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.50967773947\)
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 \(\chi\) \(=\) 690.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +8.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} -10.0000 q^{29} +1.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -2.00000 q^{35} +1.00000 q^{36} +8.00000 q^{37} -8.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} -2.00000 q^{42} +12.0000 q^{43} +2.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} +2.00000 q^{55} +2.00000 q^{56} -8.00000 q^{57} +10.0000 q^{58} +4.00000 q^{59} -1.00000 q^{60} +12.0000 q^{61} -8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +2.00000 q^{66} -4.00000 q^{67} +1.00000 q^{69} +2.00000 q^{70} +16.0000 q^{71} -1.00000 q^{72} -10.0000 q^{73} -8.00000 q^{74} -1.00000 q^{75} +8.00000 q^{76} -4.00000 q^{77} -2.00000 q^{78} +10.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -10.0000 q^{83} +2.00000 q^{84} -12.0000 q^{86} +10.0000 q^{87} -2.00000 q^{88} -1.00000 q^{90} +4.00000 q^{91} -1.00000 q^{92} -8.00000 q^{93} -8.00000 q^{94} +8.00000 q^{95} +1.00000 q^{96} +10.0000 q^{97} +3.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 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
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.00000 0.436436
\(22\) −2.00000 −0.426401
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 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
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −8.00000 −1.29777
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −2.00000 −0.308607
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 2.00000 0.301511
\(45\) 1.00000 0.149071
\(46\) 1.00000 0.147442
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.00000 0.269680
\(56\) 2.00000 0.267261
\(57\) −8.00000 −1.05963
\(58\) 10.0000 1.31306
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −1.00000 −0.129099
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) −8.00000 −1.01600
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 2.00000 0.246183
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 2.00000 0.239046
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) −1.00000 −0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −8.00000 −0.929981
\(75\) −1.00000 −0.115470
\(76\) 8.00000 0.917663
\(77\) −4.00000 −0.455842
\(78\) −2.00000 −0.226455
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 10.0000 1.07211
\(88\) −2.00000 −0.213201
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −1.00000 −0.105409
\(91\) 4.00000 0.419314
\(92\) −1.00000 −0.104257
\(93\) −8.00000 −0.829561
\(94\) −8.00000 −0.825137
\(95\) 8.00000 0.820783
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 3.00000 0.303046
\(99\) 2.00000 0.201008
\(100\) 1.00000 0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 2.00000 0.196116
\(105\) 2.00000 0.195180
\(106\) −10.0000 −0.971286
\(107\) −10.0000 −0.966736 −0.483368 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) −2.00000 −0.190693
\(111\) −8.00000 −0.759326
\(112\) −2.00000 −0.188982
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 8.00000 0.749269
\(115\) −1.00000 −0.0932505
\(116\) −10.0000 −0.928477
\(117\) −2.00000 −0.184900
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −7.00000 −0.636364
\(122\) −12.0000 −1.08643
\(123\) 6.00000 0.541002
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) 2.00000 0.178174
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −12.0000 −1.05654
\(130\) 2.00000 0.175412
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) −2.00000 −0.174078
\(133\) −16.0000 −1.38738
\(134\) 4.00000 0.345547
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −2.00000 −0.169031
\(141\) −8.00000 −0.673722
\(142\) −16.0000 −1.34269
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) −10.0000 −0.830455
\(146\) 10.0000 0.827606
\(147\) 3.00000 0.247436
\(148\) 8.00000 0.657596
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 1.00000 0.0816497
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −8.00000 −0.648886
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 8.00000 0.642575
\(156\) 2.00000 0.160128
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) −10.0000 −0.795557
\(159\) −10.0000 −0.793052
\(160\) −1.00000 −0.0790569
\(161\) 2.00000 0.157622
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −6.00000 −0.468521
\(165\) −2.00000 −0.155700
\(166\) 10.0000 0.776151
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −2.00000 −0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) 12.0000 0.914991
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −10.0000 −0.758098
\(175\) −2.00000 −0.151186
\(176\) 2.00000 0.150756
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 1.00000 0.0745356
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) −4.00000 −0.296500
\(183\) −12.0000 −0.887066
\(184\) 1.00000 0.0737210
\(185\) 8.00000 0.588172
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) 2.00000 0.145479
\(190\) −8.00000 −0.580381
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −10.0000 −0.717958
\(195\) 2.00000 0.143223
\(196\) −3.00000 −0.214286
\(197\) 26.0000 1.85242 0.926212 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) −2.00000 −0.142134
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.00000 0.282138
\(202\) −10.0000 −0.703598
\(203\) 20.0000 1.40372
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 2.00000 0.139347
\(207\) −1.00000 −0.0695048
\(208\) −2.00000 −0.138675
\(209\) 16.0000 1.10674
\(210\) −2.00000 −0.138013
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 10.0000 0.686803
\(213\) −16.0000 −1.09630
\(214\) 10.0000 0.683586
\(215\) 12.0000 0.818393
\(216\) 1.00000 0.0680414
\(217\) −16.0000 −1.08615
\(218\) 8.00000 0.541828
\(219\) 10.0000 0.675737
\(220\) 2.00000 0.134840
\(221\) 0 0
\(222\) 8.00000 0.536925
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 2.00000 0.133631
\(225\) 1.00000 0.0666667
\(226\) −8.00000 −0.532152
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) −8.00000 −0.529813
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 1.00000 0.0659380
\(231\) 4.00000 0.263181
\(232\) 10.0000 0.656532
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 2.00000 0.130744
\(235\) 8.00000 0.521862
\(236\) 4.00000 0.260378
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) 12.0000 0.768221
\(245\) −3.00000 −0.191663
\(246\) −6.00000 −0.382546
\(247\) −16.0000 −1.01806
\(248\) −8.00000 −0.508001
\(249\) 10.0000 0.633724
\(250\) −1.00000 −0.0632456
\(251\) −14.0000 −0.883672 −0.441836 0.897096i \(-0.645673\pi\)
−0.441836 + 0.897096i \(0.645673\pi\)
\(252\) −2.00000 −0.125988
\(253\) −2.00000 −0.125739
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 12.0000 0.747087
\(259\) −16.0000 −0.994192
\(260\) −2.00000 −0.124035
\(261\) −10.0000 −0.618984
\(262\) 8.00000 0.494242
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 2.00000 0.123091
\(265\) 10.0000 0.614295
\(266\) 16.0000 0.981023
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 1.00000 0.0608581
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) 12.0000 0.724947
\(275\) 2.00000 0.120605
\(276\) 1.00000 0.0601929
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 20.0000 1.19952
\(279\) 8.00000 0.478947
\(280\) 2.00000 0.119523
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 8.00000 0.476393
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 16.0000 0.949425
\(285\) −8.00000 −0.473879
\(286\) 4.00000 0.236525
\(287\) 12.0000 0.708338
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 10.0000 0.587220
\(291\) −10.0000 −0.586210
\(292\) −10.0000 −0.585206
\(293\) 2.00000 0.116841 0.0584206 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) −3.00000 −0.174964
\(295\) 4.00000 0.232889
\(296\) −8.00000 −0.464991
\(297\) −2.00000 −0.116052
\(298\) 10.0000 0.579284
\(299\) 2.00000 0.115663
\(300\) −1.00000 −0.0577350
\(301\) −24.0000 −1.38334
\(302\) −20.0000 −1.15087
\(303\) −10.0000 −0.574485
\(304\) 8.00000 0.458831
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) −4.00000 −0.227921
\(309\) 2.00000 0.113776
\(310\) −8.00000 −0.454369
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) −2.00000 −0.113228
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 8.00000 0.451466
\(315\) −2.00000 −0.112687
\(316\) 10.0000 0.562544
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 10.0000 0.560772
\(319\) −20.0000 −1.11979
\(320\) 1.00000 0.0559017
\(321\) 10.0000 0.558146
\(322\) −2.00000 −0.111456
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −4.00000 −0.221540
\(327\) 8.00000 0.442401
\(328\) 6.00000 0.331295
\(329\) −16.0000 −0.882109
\(330\) 2.00000 0.110096
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −10.0000 −0.548821
\(333\) 8.00000 0.438397
\(334\) 8.00000 0.437741
\(335\) −4.00000 −0.218543
\(336\) 2.00000 0.109109
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 9.00000 0.489535
\(339\) −8.00000 −0.434500
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) −8.00000 −0.432590
\(343\) 20.0000 1.07990
\(344\) −12.0000 −0.646997
\(345\) 1.00000 0.0538382
\(346\) 2.00000 0.107521
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 10.0000 0.536056
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 2.00000 0.106904
\(351\) 2.00000 0.106752
\(352\) −2.00000 −0.106600
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 4.00000 0.212598
\(355\) 16.0000 0.849192
\(356\) 0 0
\(357\) 0 0
\(358\) 16.0000 0.845626
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 45.0000 2.36842
\(362\) −16.0000 −0.840941
\(363\) 7.00000 0.367405
\(364\) 4.00000 0.209657
\(365\) −10.0000 −0.523424
\(366\) 12.0000 0.627250
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −6.00000 −0.312348
\(370\) −8.00000 −0.415900
\(371\) −20.0000 −1.03835
\(372\) −8.00000 −0.414781
\(373\) −24.0000 −1.24267 −0.621336 0.783544i \(-0.713410\pi\)
−0.621336 + 0.783544i \(0.713410\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −8.00000 −0.412568
\(377\) 20.0000 1.03005
\(378\) −2.00000 −0.102869
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 8.00000 0.410391
\(381\) 4.00000 0.204926
\(382\) 16.0000 0.818631
\(383\) 36.0000 1.83951 0.919757 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 1.00000 0.0510310
\(385\) −4.00000 −0.203859
\(386\) −14.0000 −0.712581
\(387\) 12.0000 0.609994
\(388\) 10.0000 0.507673
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) 8.00000 0.403547
\(394\) −26.0000 −1.30986
\(395\) 10.0000 0.503155
\(396\) 2.00000 0.100504
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 14.0000 0.701757
\(399\) 16.0000 0.801002
\(400\) 1.00000 0.0500000
\(401\) −16.0000 −0.799002 −0.399501 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(402\) −4.00000 −0.199502
\(403\) −16.0000 −0.797017
\(404\) 10.0000 0.497519
\(405\) 1.00000 0.0496904
\(406\) −20.0000 −0.992583
\(407\) 16.0000 0.793091
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 6.00000 0.296319
\(411\) 12.0000 0.591916
\(412\) −2.00000 −0.0985329
\(413\) −8.00000 −0.393654
\(414\) 1.00000 0.0491473
\(415\) −10.0000 −0.490881
\(416\) 2.00000 0.0980581
\(417\) 20.0000 0.979404
\(418\) −16.0000 −0.782586
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 2.00000 0.0975900
\(421\) −36.0000 −1.75453 −0.877266 0.480004i \(-0.840635\pi\)
−0.877266 + 0.480004i \(0.840635\pi\)
\(422\) −4.00000 −0.194717
\(423\) 8.00000 0.388973
\(424\) −10.0000 −0.485643
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) −24.0000 −1.16144
\(428\) −10.0000 −0.483368
\(429\) 4.00000 0.193122
\(430\) −12.0000 −0.578691
\(431\) −28.0000 −1.34871 −0.674356 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) −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
\(435\) 10.0000 0.479463
\(436\) −8.00000 −0.383131
\(437\) −8.00000 −0.382692
\(438\) −10.0000 −0.477818
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) −2.00000 −0.0953463
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −28.0000 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) 10.0000 0.472984
\(448\) −2.00000 −0.0944911
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −12.0000 −0.565058
\(452\) 8.00000 0.376288
\(453\) −20.0000 −0.939682
\(454\) −14.0000 −0.657053
\(455\) 4.00000 0.187523
\(456\) 8.00000 0.374634
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 16.0000 0.747631
\(459\) 0 0
\(460\) −1.00000 −0.0466252
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) −4.00000 −0.186097
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −10.0000 −0.464238
\(465\) −8.00000 −0.370991
\(466\) 6.00000 0.277945
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 8.00000 0.369406
\(470\) −8.00000 −0.369012
\(471\) 8.00000 0.368621
\(472\) −4.00000 −0.184115
\(473\) 24.0000 1.10352
\(474\) 10.0000 0.459315
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) 16.0000 0.731823
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 1.00000 0.0456435
\(481\) −16.0000 −0.729537
\(482\) −10.0000 −0.455488
\(483\) −2.00000 −0.0910032
\(484\) −7.00000 −0.318182
\(485\) 10.0000 0.454077
\(486\) 1.00000 0.0453609
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) −12.0000 −0.543214
\(489\) −4.00000 −0.180886
\(490\) 3.00000 0.135526
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000 0.270501
\(493\) 0 0
\(494\) 16.0000 0.719874
\(495\) 2.00000 0.0898933
\(496\) 8.00000 0.359211
\(497\) −32.0000 −1.43540
\(498\) −10.0000 −0.448111
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 1.00000 0.0447214
\(501\) 8.00000 0.357414
\(502\) 14.0000 0.624851
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 2.00000 0.0890871
\(505\) 10.0000 0.444994
\(506\) 2.00000 0.0889108
\(507\) 9.00000 0.399704
\(508\) −4.00000 −0.177471
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) −1.00000 −0.0441942
\(513\) −8.00000 −0.353209
\(514\) 6.00000 0.264649
\(515\) −2.00000 −0.0881305
\(516\) −12.0000 −0.528271
\(517\) 16.0000 0.703679
\(518\) 16.0000 0.703000
\(519\) 2.00000 0.0877903
\(520\) 2.00000 0.0877058
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 10.0000 0.437688
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −8.00000 −0.349482
\(525\) 2.00000 0.0872872
\(526\) −4.00000 −0.174408
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 1.00000 0.0434783
\(530\) −10.0000 −0.434372
\(531\) 4.00000 0.173585
\(532\) −16.0000 −0.693688
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) −10.0000 −0.432338
\(536\) 4.00000 0.172774
\(537\) 16.0000 0.690451
\(538\) −18.0000 −0.776035
\(539\) −6.00000 −0.258438
\(540\) −1.00000 −0.0430331
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) −28.0000 −1.20270
\(543\) −16.0000 −0.686626
\(544\) 0 0
\(545\) −8.00000 −0.342682
\(546\) 4.00000 0.171184
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −12.0000 −0.512615
\(549\) 12.0000 0.512148
\(550\) −2.00000 −0.0852803
\(551\) −80.0000 −3.40811
\(552\) −1.00000 −0.0425628
\(553\) −20.0000 −0.850487
\(554\) 6.00000 0.254916
\(555\) −8.00000 −0.339581
\(556\) −20.0000 −0.848189
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) −8.00000 −0.338667
\(559\) −24.0000 −1.01509
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 16.0000 0.674919
\(563\) 6.00000 0.252870 0.126435 0.991975i \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) −8.00000 −0.336861
\(565\) 8.00000 0.336563
\(566\) 16.0000 0.672530
\(567\) −2.00000 −0.0839921
\(568\) −16.0000 −0.671345
\(569\) −32.0000 −1.34151 −0.670755 0.741679i \(-0.734030\pi\)
−0.670755 + 0.741679i \(0.734030\pi\)
\(570\) 8.00000 0.335083
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −4.00000 −0.167248
\(573\) 16.0000 0.668410
\(574\) −12.0000 −0.500870
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 17.0000 0.707107
\(579\) −14.0000 −0.581820
\(580\) −10.0000 −0.415227
\(581\) 20.0000 0.829740
\(582\) 10.0000 0.414513
\(583\) 20.0000 0.828315
\(584\) 10.0000 0.413803
\(585\) −2.00000 −0.0826898
\(586\) −2.00000 −0.0826192
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 3.00000 0.123718
\(589\) 64.0000 2.63707
\(590\) −4.00000 −0.164677
\(591\) −26.0000 −1.06950
\(592\) 8.00000 0.328798
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 14.0000 0.572982
\(598\) −2.00000 −0.0817861
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 1.00000 0.0408248
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 24.0000 0.978167
\(603\) −4.00000 −0.162893
\(604\) 20.0000 0.813788
\(605\) −7.00000 −0.284590
\(606\) 10.0000 0.406222
\(607\) 12.0000 0.487065 0.243532 0.969893i \(-0.421694\pi\)
0.243532 + 0.969893i \(0.421694\pi\)
\(608\) −8.00000 −0.324443
\(609\) −20.0000 −0.810441
\(610\) −12.0000 −0.485866
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) −44.0000 −1.77714 −0.888572 0.458738i \(-0.848302\pi\)
−0.888572 + 0.458738i \(0.848302\pi\)
\(614\) −28.0000 −1.12999
\(615\) 6.00000 0.241943
\(616\) 4.00000 0.161165
\(617\) 48.0000 1.93241 0.966204 0.257780i \(-0.0829910\pi\)
0.966204 + 0.257780i \(0.0829910\pi\)
\(618\) −2.00000 −0.0804518
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 8.00000 0.321288
\(621\) 1.00000 0.0401286
\(622\) −16.0000 −0.641542
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) −16.0000 −0.638978
\(628\) −8.00000 −0.319235
\(629\) 0 0
\(630\) 2.00000 0.0796819
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) −10.0000 −0.397779
\(633\) −4.00000 −0.158986
\(634\) −6.00000 −0.238290
\(635\) −4.00000 −0.158735
\(636\) −10.0000 −0.396526
\(637\) 6.00000 0.237729
\(638\) 20.0000 0.791808
\(639\) 16.0000 0.632950
\(640\) −1.00000 −0.0395285
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) −10.0000 −0.394669
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 2.00000 0.0788110
\(645\) −12.0000 −0.472500
\(646\) 0 0
\(647\) −40.0000 −1.57256 −0.786281 0.617869i \(-0.787996\pi\)
−0.786281 + 0.617869i \(0.787996\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 8.00000 0.314027
\(650\) 2.00000 0.0784465
\(651\) 16.0000 0.627089
\(652\) 4.00000 0.156652
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) −8.00000 −0.312825
\(655\) −8.00000 −0.312586
\(656\) −6.00000 −0.234261
\(657\) −10.0000 −0.390137
\(658\) 16.0000 0.623745
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) −2.00000 −0.0778499
\(661\) −28.0000 −1.08907 −0.544537 0.838737i \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) 12.0000 0.466393
\(663\) 0 0
\(664\) 10.0000 0.388075
\(665\) −16.0000 −0.620453
\(666\) −8.00000 −0.309994
\(667\) 10.0000 0.387202
\(668\) −8.00000 −0.309529
\(669\) 4.00000 0.154649
\(670\) 4.00000 0.154533
\(671\) 24.0000 0.926510
\(672\) −2.00000 −0.0771517
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −22.0000 −0.847408
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 46.0000 1.76792 0.883962 0.467559i \(-0.154866\pi\)
0.883962 + 0.467559i \(0.154866\pi\)
\(678\) 8.00000 0.307238
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) −14.0000 −0.536481
\(682\) −16.0000 −0.612672
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 8.00000 0.305888
\(685\) −12.0000 −0.458496
\(686\) −20.0000 −0.763604
\(687\) 16.0000 0.610438
\(688\) 12.0000 0.457496
\(689\) −20.0000 −0.761939
\(690\) −1.00000 −0.0380693
\(691\) −52.0000 −1.97817 −0.989087 0.147335i \(-0.952930\pi\)
−0.989087 + 0.147335i \(0.952930\pi\)
\(692\) −2.00000 −0.0760286
\(693\) −4.00000 −0.151947
\(694\) 24.0000 0.911028
\(695\) −20.0000 −0.758643
\(696\) −10.0000 −0.379049
\(697\) 0 0
\(698\) 6.00000 0.227103
\(699\) 6.00000 0.226941
\(700\) −2.00000 −0.0755929
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 64.0000 2.41381
\(704\) 2.00000 0.0753778
\(705\) −8.00000 −0.301297
\(706\) −14.0000 −0.526897
\(707\) −20.0000 −0.752177
\(708\) −4.00000 −0.150329
\(709\) 28.0000 1.05156 0.525781 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) −16.0000 −0.600469
\(711\) 10.0000 0.375029
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) −16.0000 −0.597948
\(717\) 16.0000 0.597531
\(718\) −8.00000 −0.298557
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 1.00000 0.0372678
\(721\) 4.00000 0.148968
\(722\) −45.0000 −1.67473
\(723\) −10.0000 −0.371904
\(724\) 16.0000 0.594635
\(725\) −10.0000 −0.371391
\(726\) −7.00000 −0.259794
\(727\) 14.0000 0.519231 0.259616 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) 0 0
\(732\) −12.0000 −0.443533
\(733\) −52.0000 −1.92066 −0.960332 0.278859i \(-0.910044\pi\)
−0.960332 + 0.278859i \(0.910044\pi\)
\(734\) −18.0000 −0.664392
\(735\) 3.00000 0.110657
\(736\) 1.00000 0.0368605
\(737\) −8.00000 −0.294684
\(738\) 6.00000 0.220863
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) 8.00000 0.294086
\(741\) 16.0000 0.587775
\(742\) 20.0000 0.734223
\(743\) −28.0000 −1.02722 −0.513610 0.858024i \(-0.671692\pi\)
−0.513610 + 0.858024i \(0.671692\pi\)
\(744\) 8.00000 0.293294
\(745\) −10.0000 −0.366372
\(746\) 24.0000 0.878702
\(747\) −10.0000 −0.365881
\(748\) 0 0
\(749\) 20.0000 0.730784
\(750\) 1.00000 0.0365148
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) 8.00000 0.291730
\(753\) 14.0000 0.510188
\(754\) −20.0000 −0.728357
\(755\) 20.0000 0.727875
\(756\) 2.00000 0.0727393
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) −8.00000 −0.290573
\(759\) 2.00000 0.0725954
\(760\) −8.00000 −0.290191
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) −4.00000 −0.144905
\(763\) 16.0000 0.579239
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) −8.00000 −0.288863
\(768\) −1.00000 −0.0360844
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 4.00000 0.144150
\(771\) 6.00000 0.216085
\(772\) 14.0000 0.503871
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) −12.0000 −0.431331
\(775\) 8.00000 0.287368
\(776\) −10.0000 −0.358979
\(777\) 16.0000 0.573997
\(778\) 2.00000 0.0717035
\(779\) −48.0000 −1.71978
\(780\) 2.00000 0.0716115
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) 10.0000 0.357371
\(784\) −3.00000 −0.107143
\(785\) −8.00000 −0.285532
\(786\) −8.00000 −0.285351
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 26.0000 0.926212
\(789\) −4.00000 −0.142404
\(790\) −10.0000 −0.355784
\(791\) −16.0000 −0.568895
\(792\) −2.00000 −0.0710669
\(793\) −24.0000 −0.852265
\(794\) 10.0000 0.354887
\(795\) −10.0000 −0.354663
\(796\) −14.0000 −0.496217
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) −16.0000 −0.566394
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 16.0000 0.564980
\(803\) −20.0000 −0.705785
\(804\) 4.00000 0.141069
\(805\) 2.00000 0.0704907
\(806\) 16.0000 0.563576
\(807\) −18.0000 −0.633630
\(808\) −10.0000 −0.351799
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 20.0000 0.701862
\(813\) −28.0000 −0.982003
\(814\) −16.0000 −0.560800
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 96.0000 3.35861
\(818\) 10.0000 0.349642
\(819\) 4.00000 0.139771
\(820\) −6.00000 −0.209529
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) −12.0000 −0.418548
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 2.00000 0.0696733
\(825\) −2.00000 −0.0696311
\(826\) 8.00000 0.278356
\(827\) −6.00000 −0.208640 −0.104320 0.994544i \(-0.533267\pi\)
−0.104320 + 0.994544i \(0.533267\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 10.0000 0.347105
\(831\) 6.00000 0.208138
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) −20.0000 −0.692543
\(835\) −8.00000 −0.276851
\(836\) 16.0000 0.553372
\(837\) −8.00000 −0.276520
\(838\) −30.0000 −1.03633
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) −2.00000 −0.0690066
\(841\) 71.0000 2.44828
\(842\) 36.0000 1.24064
\(843\) 16.0000 0.551069
\(844\) 4.00000 0.137686
\(845\) −9.00000 −0.309609
\(846\) −8.00000 −0.275046
\(847\) 14.0000 0.481046
\(848\) 10.0000 0.343401
\(849\) 16.0000 0.549119
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) −16.0000 −0.548151
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 24.0000 0.821263
\(855\) 8.00000 0.273594
\(856\) 10.0000 0.341793
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) −4.00000 −0.136558
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 12.0000 0.409197
\(861\) −12.0000 −0.408959
\(862\) 28.0000 0.953684
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 1.00000 0.0340207
\(865\) −2.00000 −0.0680020
\(866\) −14.0000 −0.475739
\(867\) 17.0000 0.577350
\(868\) −16.0000 −0.543075
\(869\) 20.0000 0.678454
\(870\) −10.0000 −0.339032
\(871\) 8.00000 0.271070
\(872\) 8.00000 0.270914
\(873\) 10.0000 0.338449
\(874\) 8.00000 0.270604
\(875\) −2.00000 −0.0676123
\(876\) 10.0000 0.337869
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) −16.0000 −0.539974
\(879\) −2.00000 −0.0674583
\(880\) 2.00000 0.0674200
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 3.00000 0.101015
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) 28.0000 0.940678
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 8.00000 0.268462
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −4.00000 −0.133930
\(893\) 64.0000 2.14168
\(894\) −10.0000 −0.334450
\(895\) −16.0000 −0.534821
\(896\) 2.00000 0.0668153
\(897\) −2.00000 −0.0667781
\(898\) −18.0000 −0.600668
\(899\) −80.0000 −2.66815
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 12.0000 0.399556
\(903\) 24.0000 0.798670
\(904\) −8.00000 −0.266076
\(905\) 16.0000 0.531858
\(906\) 20.0000 0.664455
\(907\) 40.0000 1.32818 0.664089 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) 14.0000 0.464606
\(909\) 10.0000 0.331679
\(910\) −4.00000 −0.132599
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) −8.00000 −0.264906
\(913\) −20.0000 −0.661903
\(914\) −10.0000 −0.330771
\(915\) −12.0000 −0.396708
\(916\) −16.0000 −0.528655
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 1.00000 0.0329690
\(921\) −28.0000 −0.922631
\(922\) 30.0000 0.987997
\(923\) −32.0000 −1.05329
\(924\) 4.00000 0.131590
\(925\) 8.00000 0.263038
\(926\) −8.00000 −0.262896
\(927\) −2.00000 −0.0656886
\(928\) 10.0000 0.328266
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 8.00000 0.262330
\(931\) −24.0000 −0.786568
\(932\) −6.00000 −0.196537
\(933\) −16.0000 −0.523816
\(934\) 6.00000 0.196326
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) −8.00000 −0.261209
\(939\) 10.0000 0.326338
\(940\) 8.00000 0.260931
\(941\) −34.0000 −1.10837 −0.554184 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(942\) −8.00000 −0.260654
\(943\) 6.00000 0.195387
\(944\) 4.00000 0.130189
\(945\) 2.00000 0.0650600
\(946\) −24.0000 −0.780307
\(947\) −8.00000 −0.259965 −0.129983 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) −10.0000 −0.324785
\(949\) 20.0000 0.649227
\(950\) −8.00000 −0.259554
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 8.00000 0.259145 0.129573 0.991570i \(-0.458639\pi\)
0.129573 + 0.991570i \(0.458639\pi\)
\(954\) −10.0000 −0.323762
\(955\) −16.0000 −0.517748
\(956\) −16.0000 −0.517477
\(957\) 20.0000 0.646508
\(958\) −24.0000 −0.775405
\(959\) 24.0000 0.775000
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) 16.0000 0.515861
\(963\) −10.0000 −0.322245
\(964\) 10.0000 0.322078
\(965\) 14.0000 0.450676
\(966\) 2.00000 0.0643489
\(967\) −56.0000 −1.80084 −0.900419 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) −10.0000 −0.321081
\(971\) 30.0000 0.962746 0.481373 0.876516i \(-0.340138\pi\)
0.481373 + 0.876516i \(0.340138\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 40.0000 1.28234
\(974\) 40.0000 1.28168
\(975\) 2.00000 0.0640513
\(976\) 12.0000 0.384111
\(977\) −20.0000 −0.639857 −0.319928 0.947442i \(-0.603659\pi\)
−0.319928 + 0.947442i \(0.603659\pi\)
\(978\) 4.00000 0.127906
\(979\) 0 0
\(980\) −3.00000 −0.0958315
\(981\) −8.00000 −0.255420
\(982\) 0 0
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) −6.00000 −0.191273
\(985\) 26.0000 0.828429
\(986\) 0 0
\(987\) 16.0000 0.509286
\(988\) −16.0000 −0.509028
\(989\) −12.0000 −0.381578
\(990\) −2.00000 −0.0635642
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −8.00000 −0.254000
\(993\) 12.0000 0.380808
\(994\) 32.0000 1.01498
\(995\) −14.0000 −0.443830
\(996\) 10.0000 0.316862
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) 4.00000 0.126618
\(999\) −8.00000 −0.253109
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 690.2.a.c.1.1 1
3.2 odd 2 2070.2.a.k.1.1 1
4.3 odd 2 5520.2.a.bg.1.1 1
5.2 odd 4 3450.2.d.q.2899.1 2
5.3 odd 4 3450.2.d.q.2899.2 2
5.4 even 2 3450.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.c.1.1 1 1.1 even 1 trivial
2070.2.a.k.1.1 1 3.2 odd 2
3450.2.a.z.1.1 1 5.4 even 2
3450.2.d.q.2899.1 2 5.2 odd 4
3450.2.d.q.2899.2 2 5.3 odd 4
5520.2.a.bg.1.1 1 4.3 odd 2