Properties

Label 8470.2.a.bh.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
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 \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +1.00000 q^{10} +3.00000 q^{12} +3.00000 q^{13} -1.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +6.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} -3.00000 q^{21} -1.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} +3.00000 q^{26} +9.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} +3.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} +6.00000 q^{36} -8.00000 q^{37} +1.00000 q^{38} +9.00000 q^{39} +1.00000 q^{40} -3.00000 q^{42} -8.00000 q^{43} +6.00000 q^{45} -1.00000 q^{46} +10.0000 q^{47} +3.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +3.00000 q^{52} +4.00000 q^{53} +9.00000 q^{54} -1.00000 q^{56} +3.00000 q^{57} -6.00000 q^{58} +3.00000 q^{59} +3.00000 q^{60} +2.00000 q^{61} +6.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} +3.00000 q^{65} +16.0000 q^{67} +2.00000 q^{68} -3.00000 q^{69} -1.00000 q^{70} -8.00000 q^{71} +6.00000 q^{72} +4.00000 q^{73} -8.00000 q^{74} +3.00000 q^{75} +1.00000 q^{76} +9.00000 q^{78} -1.00000 q^{79} +1.00000 q^{80} +9.00000 q^{81} -3.00000 q^{83} -3.00000 q^{84} +2.00000 q^{85} -8.00000 q^{86} -18.0000 q^{87} +10.0000 q^{89} +6.00000 q^{90} -3.00000 q^{91} -1.00000 q^{92} +18.0000 q^{93} +10.0000 q^{94} +1.00000 q^{95} +3.00000 q^{96} -6.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 3.00000 1.22474
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 6.00000 2.00000
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 3.00000 0.866025
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) −1.00000 −0.267261
\(15\) 3.00000 0.774597
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 6.00000 1.41421
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 3.00000 0.612372
\(25\) 1.00000 0.200000
\(26\) 3.00000 0.588348
\(27\) 9.00000 1.73205
\(28\) −1.00000 −0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 3.00000 0.547723
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −1.00000 −0.169031
\(36\) 6.00000 1.00000
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 1.00000 0.162221
\(39\) 9.00000 1.44115
\(40\) 1.00000 0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −3.00000 −0.462910
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) −1.00000 −0.147442
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 3.00000 0.433013
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 6.00000 0.840168
\(52\) 3.00000 0.416025
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 3.00000 0.397360
\(58\) −6.00000 −0.787839
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 3.00000 0.387298
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 6.00000 0.762001
\(63\) −6.00000 −0.755929
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 16.0000 1.95471 0.977356 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) 2.00000 0.242536
\(69\) −3.00000 −0.361158
\(70\) −1.00000 −0.119523
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 6.00000 0.707107
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −8.00000 −0.929981
\(75\) 3.00000 0.346410
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 9.00000 1.01905
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 1.00000 0.111803
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) −3.00000 −0.327327
\(85\) 2.00000 0.216930
\(86\) −8.00000 −0.862662
\(87\) −18.0000 −1.92980
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 6.00000 0.632456
\(91\) −3.00000 −0.314485
\(92\) −1.00000 −0.104257
\(93\) 18.0000 1.86651
\(94\) 10.0000 1.03142
\(95\) 1.00000 0.102598
\(96\) 3.00000 0.306186
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 19.0000 1.89057 0.945285 0.326245i \(-0.105783\pi\)
0.945285 + 0.326245i \(0.105783\pi\)
\(102\) 6.00000 0.594089
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 3.00000 0.294174
\(105\) −3.00000 −0.292770
\(106\) 4.00000 0.388514
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 9.00000 0.866025
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −24.0000 −2.27798
\(112\) −1.00000 −0.0944911
\(113\) −3.00000 −0.282216 −0.141108 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 3.00000 0.280976
\(115\) −1.00000 −0.0932505
\(116\) −6.00000 −0.557086
\(117\) 18.0000 1.66410
\(118\) 3.00000 0.276172
\(119\) −2.00000 −0.183340
\(120\) 3.00000 0.273861
\(121\) 0 0
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 6.00000 0.538816
\(125\) 1.00000 0.0894427
\(126\) −6.00000 −0.534522
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 1.00000 0.0883883
\(129\) −24.0000 −2.11308
\(130\) 3.00000 0.263117
\(131\) −3.00000 −0.262111 −0.131056 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 16.0000 1.38219
\(135\) 9.00000 0.774597
\(136\) 2.00000 0.171499
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) −3.00000 −0.255377
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 30.0000 2.52646
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) 6.00000 0.500000
\(145\) −6.00000 −0.498273
\(146\) 4.00000 0.331042
\(147\) 3.00000 0.247436
\(148\) −8.00000 −0.657596
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 3.00000 0.244949
\(151\) −23.0000 −1.87171 −0.935857 0.352381i \(-0.885372\pi\)
−0.935857 + 0.352381i \(0.885372\pi\)
\(152\) 1.00000 0.0811107
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 9.00000 0.720577
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) −1.00000 −0.0795557
\(159\) 12.0000 0.951662
\(160\) 1.00000 0.0790569
\(161\) 1.00000 0.0788110
\(162\) 9.00000 0.707107
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) −3.00000 −0.231455
\(169\) −4.00000 −0.307692
\(170\) 2.00000 0.153393
\(171\) 6.00000 0.458831
\(172\) −8.00000 −0.609994
\(173\) −22.0000 −1.67263 −0.836315 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) −18.0000 −1.36458
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 9.00000 0.676481
\(178\) 10.0000 0.749532
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 6.00000 0.447214
\(181\) 3.00000 0.222988 0.111494 0.993765i \(-0.464436\pi\)
0.111494 + 0.993765i \(0.464436\pi\)
\(182\) −3.00000 −0.222375
\(183\) 6.00000 0.443533
\(184\) −1.00000 −0.0737210
\(185\) −8.00000 −0.588172
\(186\) 18.0000 1.31982
\(187\) 0 0
\(188\) 10.0000 0.729325
\(189\) −9.00000 −0.654654
\(190\) 1.00000 0.0725476
\(191\) 5.00000 0.361787 0.180894 0.983503i \(-0.442101\pi\)
0.180894 + 0.983503i \(0.442101\pi\)
\(192\) 3.00000 0.216506
\(193\) 9.00000 0.647834 0.323917 0.946085i \(-0.395000\pi\)
0.323917 + 0.946085i \(0.395000\pi\)
\(194\) −6.00000 −0.430775
\(195\) 9.00000 0.644503
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 1.00000 0.0707107
\(201\) 48.0000 3.38566
\(202\) 19.0000 1.33684
\(203\) 6.00000 0.421117
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) −6.00000 −0.417029
\(208\) 3.00000 0.208013
\(209\) 0 0
\(210\) −3.00000 −0.207020
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 4.00000 0.274721
\(213\) −24.0000 −1.64445
\(214\) −16.0000 −1.09374
\(215\) −8.00000 −0.545595
\(216\) 9.00000 0.612372
\(217\) −6.00000 −0.407307
\(218\) −14.0000 −0.948200
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) −24.0000 −1.61077
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 6.00000 0.400000
\(226\) −3.00000 −0.199557
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 3.00000 0.198680
\(229\) −30.0000 −1.98246 −0.991228 0.132164i \(-0.957808\pi\)
−0.991228 + 0.132164i \(0.957808\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 1.00000 0.0655122 0.0327561 0.999463i \(-0.489572\pi\)
0.0327561 + 0.999463i \(0.489572\pi\)
\(234\) 18.0000 1.17670
\(235\) 10.0000 0.652328
\(236\) 3.00000 0.195283
\(237\) −3.00000 −0.194871
\(238\) −2.00000 −0.129641
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 3.00000 0.193649
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 3.00000 0.190885
\(248\) 6.00000 0.381000
\(249\) −9.00000 −0.570352
\(250\) 1.00000 0.0632456
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) −6.00000 −0.377964
\(253\) 0 0
\(254\) −7.00000 −0.439219
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) −24.0000 −1.49417
\(259\) 8.00000 0.497096
\(260\) 3.00000 0.186052
\(261\) −36.0000 −2.22834
\(262\) −3.00000 −0.185341
\(263\) 23.0000 1.41824 0.709120 0.705087i \(-0.249092\pi\)
0.709120 + 0.705087i \(0.249092\pi\)
\(264\) 0 0
\(265\) 4.00000 0.245718
\(266\) −1.00000 −0.0613139
\(267\) 30.0000 1.83597
\(268\) 16.0000 0.977356
\(269\) −5.00000 −0.304855 −0.152428 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(270\) 9.00000 0.547723
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 2.00000 0.121268
\(273\) −9.00000 −0.544705
\(274\) −3.00000 −0.181237
\(275\) 0 0
\(276\) −3.00000 −0.180579
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 5.00000 0.299880
\(279\) 36.0000 2.15526
\(280\) −1.00000 −0.0597614
\(281\) −9.00000 −0.536895 −0.268447 0.963294i \(-0.586511\pi\)
−0.268447 + 0.963294i \(0.586511\pi\)
\(282\) 30.0000 1.78647
\(283\) −19.0000 −1.12943 −0.564716 0.825285i \(-0.691014\pi\)
−0.564716 + 0.825285i \(0.691014\pi\)
\(284\) −8.00000 −0.474713
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) 0 0
\(288\) 6.00000 0.353553
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) −18.0000 −1.05518
\(292\) 4.00000 0.234082
\(293\) −7.00000 −0.408944 −0.204472 0.978872i \(-0.565548\pi\)
−0.204472 + 0.978872i \(0.565548\pi\)
\(294\) 3.00000 0.174964
\(295\) 3.00000 0.174667
\(296\) −8.00000 −0.464991
\(297\) 0 0
\(298\) 0 0
\(299\) −3.00000 −0.173494
\(300\) 3.00000 0.173205
\(301\) 8.00000 0.461112
\(302\) −23.0000 −1.32350
\(303\) 57.0000 3.27456
\(304\) 1.00000 0.0573539
\(305\) 2.00000 0.114520
\(306\) 12.0000 0.685994
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 42.0000 2.38930
\(310\) 6.00000 0.340777
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 9.00000 0.509525
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 13.0000 0.733632
\(315\) −6.00000 −0.338062
\(316\) −1.00000 −0.0562544
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 12.0000 0.672927
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −48.0000 −2.67910
\(322\) 1.00000 0.0557278
\(323\) 2.00000 0.111283
\(324\) 9.00000 0.500000
\(325\) 3.00000 0.166410
\(326\) −10.0000 −0.553849
\(327\) −42.0000 −2.32261
\(328\) 0 0
\(329\) −10.0000 −0.551318
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −3.00000 −0.164646
\(333\) −48.0000 −2.63038
\(334\) −24.0000 −1.31322
\(335\) 16.0000 0.874173
\(336\) −3.00000 −0.163663
\(337\) 17.0000 0.926049 0.463025 0.886345i \(-0.346764\pi\)
0.463025 + 0.886345i \(0.346764\pi\)
\(338\) −4.00000 −0.217571
\(339\) −9.00000 −0.488813
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) −1.00000 −0.0539949
\(344\) −8.00000 −0.431331
\(345\) −3.00000 −0.161515
\(346\) −22.0000 −1.18273
\(347\) 22.0000 1.18102 0.590511 0.807030i \(-0.298926\pi\)
0.590511 + 0.807030i \(0.298926\pi\)
\(348\) −18.0000 −0.964901
\(349\) 1.00000 0.0535288 0.0267644 0.999642i \(-0.491480\pi\)
0.0267644 + 0.999642i \(0.491480\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 27.0000 1.44115
\(352\) 0 0
\(353\) 16.0000 0.851594 0.425797 0.904819i \(-0.359994\pi\)
0.425797 + 0.904819i \(0.359994\pi\)
\(354\) 9.00000 0.478345
\(355\) −8.00000 −0.424596
\(356\) 10.0000 0.529999
\(357\) −6.00000 −0.317554
\(358\) −16.0000 −0.845626
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 6.00000 0.316228
\(361\) −18.0000 −0.947368
\(362\) 3.00000 0.157676
\(363\) 0 0
\(364\) −3.00000 −0.157243
\(365\) 4.00000 0.209370
\(366\) 6.00000 0.313625
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −8.00000 −0.415900
\(371\) −4.00000 −0.207670
\(372\) 18.0000 0.933257
\(373\) 32.0000 1.65690 0.828449 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 10.0000 0.515711
\(377\) −18.0000 −0.927047
\(378\) −9.00000 −0.462910
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 1.00000 0.0512989
\(381\) −21.0000 −1.07586
\(382\) 5.00000 0.255822
\(383\) 26.0000 1.32854 0.664269 0.747494i \(-0.268743\pi\)
0.664269 + 0.747494i \(0.268743\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 9.00000 0.458088
\(387\) −48.0000 −2.43998
\(388\) −6.00000 −0.304604
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 9.00000 0.455733
\(391\) −2.00000 −0.101144
\(392\) 1.00000 0.0505076
\(393\) −9.00000 −0.453990
\(394\) −6.00000 −0.302276
\(395\) −1.00000 −0.0503155
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −18.0000 −0.902258
\(399\) −3.00000 −0.150188
\(400\) 1.00000 0.0500000
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 48.0000 2.39402
\(403\) 18.0000 0.896644
\(404\) 19.0000 0.945285
\(405\) 9.00000 0.447214
\(406\) 6.00000 0.297775
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 14.0000 0.689730
\(413\) −3.00000 −0.147620
\(414\) −6.00000 −0.294884
\(415\) −3.00000 −0.147264
\(416\) 3.00000 0.147087
\(417\) 15.0000 0.734553
\(418\) 0 0
\(419\) −21.0000 −1.02592 −0.512959 0.858413i \(-0.671451\pi\)
−0.512959 + 0.858413i \(0.671451\pi\)
\(420\) −3.00000 −0.146385
\(421\) −32.0000 −1.55958 −0.779792 0.626038i \(-0.784675\pi\)
−0.779792 + 0.626038i \(0.784675\pi\)
\(422\) 14.0000 0.681509
\(423\) 60.0000 2.91730
\(424\) 4.00000 0.194257
\(425\) 2.00000 0.0970143
\(426\) −24.0000 −1.16280
\(427\) −2.00000 −0.0967868
\(428\) −16.0000 −0.773389
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 37.0000 1.78223 0.891114 0.453780i \(-0.149925\pi\)
0.891114 + 0.453780i \(0.149925\pi\)
\(432\) 9.00000 0.433013
\(433\) −18.0000 −0.865025 −0.432512 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) −6.00000 −0.288009
\(435\) −18.0000 −0.863034
\(436\) −14.0000 −0.670478
\(437\) −1.00000 −0.0478365
\(438\) 12.0000 0.573382
\(439\) 30.0000 1.43182 0.715911 0.698192i \(-0.246012\pi\)
0.715911 + 0.698192i \(0.246012\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 6.00000 0.285391
\(443\) −32.0000 −1.52037 −0.760183 0.649709i \(-0.774891\pi\)
−0.760183 + 0.649709i \(0.774891\pi\)
\(444\) −24.0000 −1.13899
\(445\) 10.0000 0.474045
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 41.0000 1.93491 0.967455 0.253044i \(-0.0814317\pi\)
0.967455 + 0.253044i \(0.0814317\pi\)
\(450\) 6.00000 0.282843
\(451\) 0 0
\(452\) −3.00000 −0.141108
\(453\) −69.0000 −3.24190
\(454\) 20.0000 0.938647
\(455\) −3.00000 −0.140642
\(456\) 3.00000 0.140488
\(457\) 17.0000 0.795226 0.397613 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(458\) −30.0000 −1.40181
\(459\) 18.0000 0.840168
\(460\) −1.00000 −0.0466252
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 23.0000 1.06890 0.534450 0.845200i \(-0.320519\pi\)
0.534450 + 0.845200i \(0.320519\pi\)
\(464\) −6.00000 −0.278543
\(465\) 18.0000 0.834730
\(466\) 1.00000 0.0463241
\(467\) 27.0000 1.24941 0.624705 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) 18.0000 0.832050
\(469\) −16.0000 −0.738811
\(470\) 10.0000 0.461266
\(471\) 39.0000 1.79703
\(472\) 3.00000 0.138086
\(473\) 0 0
\(474\) −3.00000 −0.137795
\(475\) 1.00000 0.0458831
\(476\) −2.00000 −0.0916698
\(477\) 24.0000 1.09888
\(478\) 9.00000 0.411650
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 3.00000 0.136931
\(481\) −24.0000 −1.09431
\(482\) 22.0000 1.00207
\(483\) 3.00000 0.136505
\(484\) 0 0
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) −19.0000 −0.860972 −0.430486 0.902597i \(-0.641658\pi\)
−0.430486 + 0.902597i \(0.641658\pi\)
\(488\) 2.00000 0.0905357
\(489\) −30.0000 −1.35665
\(490\) 1.00000 0.0451754
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 3.00000 0.134976
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 8.00000 0.358849
\(498\) −9.00000 −0.403300
\(499\) −8.00000 −0.358129 −0.179065 0.983837i \(-0.557307\pi\)
−0.179065 + 0.983837i \(0.557307\pi\)
\(500\) 1.00000 0.0447214
\(501\) −72.0000 −3.21672
\(502\) 20.0000 0.892644
\(503\) 14.0000 0.624229 0.312115 0.950044i \(-0.398963\pi\)
0.312115 + 0.950044i \(0.398963\pi\)
\(504\) −6.00000 −0.267261
\(505\) 19.0000 0.845489
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) −7.00000 −0.310575
\(509\) −41.0000 −1.81729 −0.908647 0.417566i \(-0.862883\pi\)
−0.908647 + 0.417566i \(0.862883\pi\)
\(510\) 6.00000 0.265684
\(511\) −4.00000 −0.176950
\(512\) 1.00000 0.0441942
\(513\) 9.00000 0.397360
\(514\) −24.0000 −1.05859
\(515\) 14.0000 0.616914
\(516\) −24.0000 −1.05654
\(517\) 0 0
\(518\) 8.00000 0.351500
\(519\) −66.0000 −2.89708
\(520\) 3.00000 0.131559
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −36.0000 −1.57568
\(523\) 21.0000 0.918266 0.459133 0.888368i \(-0.348160\pi\)
0.459133 + 0.888368i \(0.348160\pi\)
\(524\) −3.00000 −0.131056
\(525\) −3.00000 −0.130931
\(526\) 23.0000 1.00285
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 4.00000 0.173749
\(531\) 18.0000 0.781133
\(532\) −1.00000 −0.0433555
\(533\) 0 0
\(534\) 30.0000 1.29823
\(535\) −16.0000 −0.691740
\(536\) 16.0000 0.691095
\(537\) −48.0000 −2.07135
\(538\) −5.00000 −0.215565
\(539\) 0 0
\(540\) 9.00000 0.387298
\(541\) −6.00000 −0.257960 −0.128980 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) −12.0000 −0.515444
\(543\) 9.00000 0.386227
\(544\) 2.00000 0.0857493
\(545\) −14.0000 −0.599694
\(546\) −9.00000 −0.385164
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) −3.00000 −0.128154
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) −3.00000 −0.127688
\(553\) 1.00000 0.0425243
\(554\) −8.00000 −0.339887
\(555\) −24.0000 −1.01874
\(556\) 5.00000 0.212047
\(557\) −42.0000 −1.77960 −0.889799 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 36.0000 1.52400
\(559\) −24.0000 −1.01509
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −9.00000 −0.379642
\(563\) 25.0000 1.05362 0.526812 0.849982i \(-0.323387\pi\)
0.526812 + 0.849982i \(0.323387\pi\)
\(564\) 30.0000 1.26323
\(565\) −3.00000 −0.126211
\(566\) −19.0000 −0.798630
\(567\) −9.00000 −0.377964
\(568\) −8.00000 −0.335673
\(569\) 9.00000 0.377300 0.188650 0.982044i \(-0.439589\pi\)
0.188650 + 0.982044i \(0.439589\pi\)
\(570\) 3.00000 0.125656
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 15.0000 0.626634
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 6.00000 0.250000
\(577\) 16.0000 0.666089 0.333044 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) −13.0000 −0.540729
\(579\) 27.0000 1.12208
\(580\) −6.00000 −0.249136
\(581\) 3.00000 0.124461
\(582\) −18.0000 −0.746124
\(583\) 0 0
\(584\) 4.00000 0.165521
\(585\) 18.0000 0.744208
\(586\) −7.00000 −0.289167
\(587\) 5.00000 0.206372 0.103186 0.994662i \(-0.467096\pi\)
0.103186 + 0.994662i \(0.467096\pi\)
\(588\) 3.00000 0.123718
\(589\) 6.00000 0.247226
\(590\) 3.00000 0.123508
\(591\) −18.0000 −0.740421
\(592\) −8.00000 −0.328798
\(593\) −24.0000 −0.985562 −0.492781 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) 0 0
\(597\) −54.0000 −2.21007
\(598\) −3.00000 −0.122679
\(599\) −43.0000 −1.75693 −0.878466 0.477805i \(-0.841433\pi\)
−0.878466 + 0.477805i \(0.841433\pi\)
\(600\) 3.00000 0.122474
\(601\) −20.0000 −0.815817 −0.407909 0.913023i \(-0.633742\pi\)
−0.407909 + 0.913023i \(0.633742\pi\)
\(602\) 8.00000 0.326056
\(603\) 96.0000 3.90942
\(604\) −23.0000 −0.935857
\(605\) 0 0
\(606\) 57.0000 2.31547
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) 1.00000 0.0405554
\(609\) 18.0000 0.729397
\(610\) 2.00000 0.0809776
\(611\) 30.0000 1.21367
\(612\) 12.0000 0.485071
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 42.0000 1.68949
\(619\) 25.0000 1.00483 0.502417 0.864625i \(-0.332444\pi\)
0.502417 + 0.864625i \(0.332444\pi\)
\(620\) 6.00000 0.240966
\(621\) −9.00000 −0.361158
\(622\) 30.0000 1.20289
\(623\) −10.0000 −0.400642
\(624\) 9.00000 0.360288
\(625\) 1.00000 0.0400000
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) 13.0000 0.518756
\(629\) −16.0000 −0.637962
\(630\) −6.00000 −0.239046
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −1.00000 −0.0397779
\(633\) 42.0000 1.66935
\(634\) −22.0000 −0.873732
\(635\) −7.00000 −0.277787
\(636\) 12.0000 0.475831
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) −48.0000 −1.89885
\(640\) 1.00000 0.0395285
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) −48.0000 −1.89441
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 1.00000 0.0394055
\(645\) −24.0000 −0.944999
\(646\) 2.00000 0.0786889
\(647\) −30.0000 −1.17942 −0.589711 0.807614i \(-0.700758\pi\)
−0.589711 + 0.807614i \(0.700758\pi\)
\(648\) 9.00000 0.353553
\(649\) 0 0
\(650\) 3.00000 0.117670
\(651\) −18.0000 −0.705476
\(652\) −10.0000 −0.391630
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) −42.0000 −1.64233
\(655\) −3.00000 −0.117220
\(656\) 0 0
\(657\) 24.0000 0.936329
\(658\) −10.0000 −0.389841
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −11.0000 −0.427850 −0.213925 0.976850i \(-0.568625\pi\)
−0.213925 + 0.976850i \(0.568625\pi\)
\(662\) −4.00000 −0.155464
\(663\) 18.0000 0.699062
\(664\) −3.00000 −0.116423
\(665\) −1.00000 −0.0387783
\(666\) −48.0000 −1.85996
\(667\) 6.00000 0.232321
\(668\) −24.0000 −0.928588
\(669\) −12.0000 −0.463947
\(670\) 16.0000 0.618134
\(671\) 0 0
\(672\) −3.00000 −0.115728
\(673\) 7.00000 0.269830 0.134915 0.990857i \(-0.456924\pi\)
0.134915 + 0.990857i \(0.456924\pi\)
\(674\) 17.0000 0.654816
\(675\) 9.00000 0.346410
\(676\) −4.00000 −0.153846
\(677\) −13.0000 −0.499631 −0.249815 0.968294i \(-0.580370\pi\)
−0.249815 + 0.968294i \(0.580370\pi\)
\(678\) −9.00000 −0.345643
\(679\) 6.00000 0.230259
\(680\) 2.00000 0.0766965
\(681\) 60.0000 2.29920
\(682\) 0 0
\(683\) −30.0000 −1.14792 −0.573959 0.818884i \(-0.694593\pi\)
−0.573959 + 0.818884i \(0.694593\pi\)
\(684\) 6.00000 0.229416
\(685\) −3.00000 −0.114624
\(686\) −1.00000 −0.0381802
\(687\) −90.0000 −3.43371
\(688\) −8.00000 −0.304997
\(689\) 12.0000 0.457164
\(690\) −3.00000 −0.114208
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) −22.0000 −0.836315
\(693\) 0 0
\(694\) 22.0000 0.835109
\(695\) 5.00000 0.189661
\(696\) −18.0000 −0.682288
\(697\) 0 0
\(698\) 1.00000 0.0378506
\(699\) 3.00000 0.113470
\(700\) −1.00000 −0.0377964
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 27.0000 1.01905
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 30.0000 1.12987
\(706\) 16.0000 0.602168
\(707\) −19.0000 −0.714569
\(708\) 9.00000 0.338241
\(709\) −44.0000 −1.65245 −0.826227 0.563337i \(-0.809517\pi\)
−0.826227 + 0.563337i \(0.809517\pi\)
\(710\) −8.00000 −0.300235
\(711\) −6.00000 −0.225018
\(712\) 10.0000 0.374766
\(713\) −6.00000 −0.224702
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) −16.0000 −0.597948
\(717\) 27.0000 1.00833
\(718\) −32.0000 −1.19423
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 6.00000 0.223607
\(721\) −14.0000 −0.521387
\(722\) −18.0000 −0.669891
\(723\) 66.0000 2.45457
\(724\) 3.00000 0.111494
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 6.00000 0.222528 0.111264 0.993791i \(-0.464510\pi\)
0.111264 + 0.993791i \(0.464510\pi\)
\(728\) −3.00000 −0.111187
\(729\) −27.0000 −1.00000
\(730\) 4.00000 0.148047
\(731\) −16.0000 −0.591781
\(732\) 6.00000 0.221766
\(733\) 13.0000 0.480166 0.240083 0.970752i \(-0.422825\pi\)
0.240083 + 0.970752i \(0.422825\pi\)
\(734\) −16.0000 −0.590571
\(735\) 3.00000 0.110657
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) −8.00000 −0.294086
\(741\) 9.00000 0.330623
\(742\) −4.00000 −0.146845
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 18.0000 0.659912
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) −18.0000 −0.658586
\(748\) 0 0
\(749\) 16.0000 0.584627
\(750\) 3.00000 0.109545
\(751\) 25.0000 0.912263 0.456131 0.889912i \(-0.349235\pi\)
0.456131 + 0.889912i \(0.349235\pi\)
\(752\) 10.0000 0.364662
\(753\) 60.0000 2.18652
\(754\) −18.0000 −0.655521
\(755\) −23.0000 −0.837056
\(756\) −9.00000 −0.327327
\(757\) −48.0000 −1.74459 −0.872295 0.488980i \(-0.837369\pi\)
−0.872295 + 0.488980i \(0.837369\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) 1.00000 0.0362738
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) −21.0000 −0.760750
\(763\) 14.0000 0.506834
\(764\) 5.00000 0.180894
\(765\) 12.0000 0.433861
\(766\) 26.0000 0.939418
\(767\) 9.00000 0.324971
\(768\) 3.00000 0.108253
\(769\) 32.0000 1.15395 0.576975 0.816762i \(-0.304233\pi\)
0.576975 + 0.816762i \(0.304233\pi\)
\(770\) 0 0
\(771\) −72.0000 −2.59302
\(772\) 9.00000 0.323917
\(773\) 3.00000 0.107903 0.0539513 0.998544i \(-0.482818\pi\)
0.0539513 + 0.998544i \(0.482818\pi\)
\(774\) −48.0000 −1.72532
\(775\) 6.00000 0.215526
\(776\) −6.00000 −0.215387
\(777\) 24.0000 0.860995
\(778\) 12.0000 0.430221
\(779\) 0 0
\(780\) 9.00000 0.322252
\(781\) 0 0
\(782\) −2.00000 −0.0715199
\(783\) −54.0000 −1.92980
\(784\) 1.00000 0.0357143
\(785\) 13.0000 0.463990
\(786\) −9.00000 −0.321019
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) −6.00000 −0.213741
\(789\) 69.0000 2.45647
\(790\) −1.00000 −0.0355784
\(791\) 3.00000 0.106668
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) 14.0000 0.496841
\(795\) 12.0000 0.425596
\(796\) −18.0000 −0.637993
\(797\) −7.00000 −0.247953 −0.123976 0.992285i \(-0.539565\pi\)
−0.123976 + 0.992285i \(0.539565\pi\)
\(798\) −3.00000 −0.106199
\(799\) 20.0000 0.707549
\(800\) 1.00000 0.0353553
\(801\) 60.0000 2.12000
\(802\) 22.0000 0.776847
\(803\) 0 0
\(804\) 48.0000 1.69283
\(805\) 1.00000 0.0352454
\(806\) 18.0000 0.634023
\(807\) −15.0000 −0.528025
\(808\) 19.0000 0.668418
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 9.00000 0.316228
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 6.00000 0.210559
\(813\) −36.0000 −1.26258
\(814\) 0 0
\(815\) −10.0000 −0.350285
\(816\) 6.00000 0.210042
\(817\) −8.00000 −0.279885
\(818\) 14.0000 0.489499
\(819\) −18.0000 −0.628971
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) −9.00000 −0.313911
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) −3.00000 −0.104383
\(827\) −24.0000 −0.834562 −0.417281 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(828\) −6.00000 −0.208514
\(829\) −15.0000 −0.520972 −0.260486 0.965478i \(-0.583883\pi\)
−0.260486 + 0.965478i \(0.583883\pi\)
\(830\) −3.00000 −0.104132
\(831\) −24.0000 −0.832551
\(832\) 3.00000 0.104006
\(833\) 2.00000 0.0692959
\(834\) 15.0000 0.519408
\(835\) −24.0000 −0.830554
\(836\) 0 0
\(837\) 54.0000 1.86651
\(838\) −21.0000 −0.725433
\(839\) 26.0000 0.897620 0.448810 0.893627i \(-0.351848\pi\)
0.448810 + 0.893627i \(0.351848\pi\)
\(840\) −3.00000 −0.103510
\(841\) 7.00000 0.241379
\(842\) −32.0000 −1.10279
\(843\) −27.0000 −0.929929
\(844\) 14.0000 0.481900
\(845\) −4.00000 −0.137604
\(846\) 60.0000 2.06284
\(847\) 0 0
\(848\) 4.00000 0.137361
\(849\) −57.0000 −1.95623
\(850\) 2.00000 0.0685994
\(851\) 8.00000 0.274236
\(852\) −24.0000 −0.822226
\(853\) −21.0000 −0.719026 −0.359513 0.933140i \(-0.617057\pi\)
−0.359513 + 0.933140i \(0.617057\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 6.00000 0.205196
\(856\) −16.0000 −0.546869
\(857\) 12.0000 0.409912 0.204956 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(858\) 0 0
\(859\) 52.0000 1.77422 0.887109 0.461561i \(-0.152710\pi\)
0.887109 + 0.461561i \(0.152710\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 37.0000 1.26023
\(863\) 20.0000 0.680808 0.340404 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(864\) 9.00000 0.306186
\(865\) −22.0000 −0.748022
\(866\) −18.0000 −0.611665
\(867\) −39.0000 −1.32451
\(868\) −6.00000 −0.203653
\(869\) 0 0
\(870\) −18.0000 −0.610257
\(871\) 48.0000 1.62642
\(872\) −14.0000 −0.474100
\(873\) −36.0000 −1.21842
\(874\) −1.00000 −0.0338255
\(875\) −1.00000 −0.0338062
\(876\) 12.0000 0.405442
\(877\) −52.0000 −1.75592 −0.877958 0.478738i \(-0.841094\pi\)
−0.877958 + 0.478738i \(0.841094\pi\)
\(878\) 30.0000 1.01245
\(879\) −21.0000 −0.708312
\(880\) 0 0
\(881\) −52.0000 −1.75192 −0.875962 0.482380i \(-0.839773\pi\)
−0.875962 + 0.482380i \(0.839773\pi\)
\(882\) 6.00000 0.202031
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 6.00000 0.201802
\(885\) 9.00000 0.302532
\(886\) −32.0000 −1.07506
\(887\) −54.0000 −1.81314 −0.906571 0.422053i \(-0.861310\pi\)
−0.906571 + 0.422053i \(0.861310\pi\)
\(888\) −24.0000 −0.805387
\(889\) 7.00000 0.234772
\(890\) 10.0000 0.335201
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 10.0000 0.334637
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) −1.00000 −0.0334077
\(897\) −9.00000 −0.300501
\(898\) 41.0000 1.36819
\(899\) −36.0000 −1.20067
\(900\) 6.00000 0.200000
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) 24.0000 0.798670
\(904\) −3.00000 −0.0997785
\(905\) 3.00000 0.0997234
\(906\) −69.0000 −2.29237
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 20.0000 0.663723
\(909\) 114.000 3.78114
\(910\) −3.00000 −0.0994490
\(911\) −1.00000 −0.0331315 −0.0165657 0.999863i \(-0.505273\pi\)
−0.0165657 + 0.999863i \(0.505273\pi\)
\(912\) 3.00000 0.0993399
\(913\) 0 0
\(914\) 17.0000 0.562310
\(915\) 6.00000 0.198354
\(916\) −30.0000 −0.991228
\(917\) 3.00000 0.0990687
\(918\) 18.0000 0.594089
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) −1.00000 −0.0329690
\(921\) −48.0000 −1.58165
\(922\) 6.00000 0.197599
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) 23.0000 0.755827
\(927\) 84.0000 2.75892
\(928\) −6.00000 −0.196960
\(929\) 12.0000 0.393707 0.196854 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 18.0000 0.590243
\(931\) 1.00000 0.0327737
\(932\) 1.00000 0.0327561
\(933\) 90.0000 2.94647
\(934\) 27.0000 0.883467
\(935\) 0 0
\(936\) 18.0000 0.588348
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) −16.0000 −0.522419
\(939\) 66.0000 2.15383
\(940\) 10.0000 0.326164
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 39.0000 1.27069
\(943\) 0 0
\(944\) 3.00000 0.0976417
\(945\) −9.00000 −0.292770
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −3.00000 −0.0974355
\(949\) 12.0000 0.389536
\(950\) 1.00000 0.0324443
\(951\) −66.0000 −2.14020
\(952\) −2.00000 −0.0648204
\(953\) −31.0000 −1.00419 −0.502094 0.864813i \(-0.667437\pi\)
−0.502094 + 0.864813i \(0.667437\pi\)
\(954\) 24.0000 0.777029
\(955\) 5.00000 0.161796
\(956\) 9.00000 0.291081
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 3.00000 0.0968751
\(960\) 3.00000 0.0968246
\(961\) 5.00000 0.161290
\(962\) −24.0000 −0.773791
\(963\) −96.0000 −3.09356
\(964\) 22.0000 0.708572
\(965\) 9.00000 0.289720
\(966\) 3.00000 0.0965234
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) −6.00000 −0.192648
\(971\) 41.0000 1.31575 0.657876 0.753126i \(-0.271455\pi\)
0.657876 + 0.753126i \(0.271455\pi\)
\(972\) 0 0
\(973\) −5.00000 −0.160293
\(974\) −19.0000 −0.608799
\(975\) 9.00000 0.288231
\(976\) 2.00000 0.0640184
\(977\) −3.00000 −0.0959785 −0.0479893 0.998848i \(-0.515281\pi\)
−0.0479893 + 0.998848i \(0.515281\pi\)
\(978\) −30.0000 −0.959294
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −84.0000 −2.68191
\(982\) −8.00000 −0.255290
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) −12.0000 −0.382158
\(987\) −30.0000 −0.954911
\(988\) 3.00000 0.0954427
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 5.00000 0.158830 0.0794151 0.996842i \(-0.474695\pi\)
0.0794151 + 0.996842i \(0.474695\pi\)
\(992\) 6.00000 0.190500
\(993\) −12.0000 −0.380808
\(994\) 8.00000 0.253745
\(995\) −18.0000 −0.570638
\(996\) −9.00000 −0.285176
\(997\) −55.0000 −1.74187 −0.870934 0.491400i \(-0.836485\pi\)
−0.870934 + 0.491400i \(0.836485\pi\)
\(998\) −8.00000 −0.253236
\(999\) −72.0000 −2.27798
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 8470.2.a.bh.1.1 yes 1
11.10 odd 2 8470.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.p.1.1 1 11.10 odd 2
8470.2.a.bh.1.1 yes 1 1.1 even 1 trivial