Properties

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

Related objects

Downloads

Learn more

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: \(1\)
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

\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} -1.00000 q^{10} -3.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} -6.00000 q^{18} +7.00000 q^{19} +1.00000 q^{20} +3.00000 q^{21} +1.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} -9.00000 q^{27} -1.00000 q^{28} -8.00000 q^{29} +3.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{35} +6.00000 q^{36} +2.00000 q^{37} -7.00000 q^{38} +3.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} -3.00000 q^{42} +6.00000 q^{43} +6.00000 q^{45} -1.00000 q^{46} -12.0000 q^{47} -3.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} -12.0000 q^{53} +9.00000 q^{54} +1.00000 q^{56} -21.0000 q^{57} +8.00000 q^{58} +3.00000 q^{59} -3.00000 q^{60} +6.00000 q^{61} +4.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +8.00000 q^{67} -3.00000 q^{69} +1.00000 q^{70} -8.00000 q^{71} -6.00000 q^{72} -16.0000 q^{73} -2.00000 q^{74} -3.00000 q^{75} +7.00000 q^{76} -3.00000 q^{78} +9.00000 q^{79} +1.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} +13.0000 q^{83} +3.00000 q^{84} -6.00000 q^{86} +24.0000 q^{87} +6.00000 q^{89} -6.00000 q^{90} +1.00000 q^{91} +1.00000 q^{92} +12.0000 q^{93} +12.0000 q^{94} +7.00000 q^{95} +3.00000 q^{96} -8.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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 \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 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
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 1.00000 0.267261
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −6.00000 −1.41421
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 3.00000 0.612372
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) −9.00000 −1.73205
\(28\) −1.00000 −0.188982
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 3.00000 0.547723
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 6.00000 1.00000
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −7.00000 −1.13555
\(39\) 3.00000 0.480384
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −3.00000 −0.462910
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) −1.00000 −0.147442
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −3.00000 −0.433013
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −21.0000 −2.78152
\(58\) 8.00000 1.05045
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) −3.00000 −0.387298
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 4.00000 0.508001
\(63\) −6.00000 −0.755929
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) −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
\(73\) −16.0000 −1.87266 −0.936329 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −2.00000 −0.232495
\(75\) −3.00000 −0.346410
\(76\) 7.00000 0.802955
\(77\) 0 0
\(78\) −3.00000 −0.339683
\(79\) 9.00000 1.01258 0.506290 0.862364i \(-0.331017\pi\)
0.506290 + 0.862364i \(0.331017\pi\)
\(80\) 1.00000 0.111803
\(81\) 9.00000 1.00000
\(82\) −6.00000 −0.662589
\(83\) 13.0000 1.42694 0.713468 0.700688i \(-0.247124\pi\)
0.713468 + 0.700688i \(0.247124\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 24.0000 2.57307
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −6.00000 −0.632456
\(91\) 1.00000 0.104828
\(92\) 1.00000 0.104257
\(93\) 12.0000 1.24434
\(94\) 12.0000 1.23771
\(95\) 7.00000 0.718185
\(96\) 3.00000 0.306186
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −15.0000 −1.49256 −0.746278 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 1.00000 0.0980581
\(105\) 3.00000 0.292770
\(106\) 12.0000 1.16554
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) −9.00000 −0.866025
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) −1.00000 −0.0944911
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 21.0000 1.96683
\(115\) 1.00000 0.0932505
\(116\) −8.00000 −0.742781
\(117\) −6.00000 −0.554700
\(118\) −3.00000 −0.276172
\(119\) 0 0
\(120\) 3.00000 0.273861
\(121\) 0 0
\(122\) −6.00000 −0.543214
\(123\) −18.0000 −1.62301
\(124\) −4.00000 −0.359211
\(125\) 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
\(129\) −18.0000 −1.58481
\(130\) 1.00000 0.0877058
\(131\) 11.0000 0.961074 0.480537 0.876974i \(-0.340442\pi\)
0.480537 + 0.876974i \(0.340442\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) −8.00000 −0.691095
\(135\) −9.00000 −0.774597
\(136\) 0 0
\(137\) −5.00000 −0.427179 −0.213589 0.976924i \(-0.568515\pi\)
−0.213589 + 0.976924i \(0.568515\pi\)
\(138\) 3.00000 0.255377
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 36.0000 3.03175
\(142\) 8.00000 0.671345
\(143\) 0 0
\(144\) 6.00000 0.500000
\(145\) −8.00000 −0.664364
\(146\) 16.0000 1.32417
\(147\) −3.00000 −0.247436
\(148\) 2.00000 0.164399
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 3.00000 0.244949
\(151\) 7.00000 0.569652 0.284826 0.958579i \(-0.408064\pi\)
0.284826 + 0.958579i \(0.408064\pi\)
\(152\) −7.00000 −0.567775
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 3.00000 0.240192
\(157\) −21.0000 −1.67598 −0.837991 0.545684i \(-0.816270\pi\)
−0.837991 + 0.545684i \(0.816270\pi\)
\(158\) −9.00000 −0.716002
\(159\) 36.0000 2.85499
\(160\) −1.00000 −0.0790569
\(161\) −1.00000 −0.0788110
\(162\) −9.00000 −0.707107
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −13.0000 −1.00900
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) −3.00000 −0.231455
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 42.0000 3.21182
\(172\) 6.00000 0.457496
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −24.0000 −1.81944
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −9.00000 −0.676481
\(178\) −6.00000 −0.449719
\(179\) 22.0000 1.64436 0.822179 0.569230i \(-0.192758\pi\)
0.822179 + 0.569230i \(0.192758\pi\)
\(180\) 6.00000 0.447214
\(181\) 11.0000 0.817624 0.408812 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) −1.00000 −0.0741249
\(183\) −18.0000 −1.33060
\(184\) −1.00000 −0.0737210
\(185\) 2.00000 0.147043
\(186\) −12.0000 −0.879883
\(187\) 0 0
\(188\) −12.0000 −0.875190
\(189\) 9.00000 0.654654
\(190\) −7.00000 −0.507833
\(191\) 9.00000 0.651217 0.325609 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) −3.00000 −0.216506
\(193\) 13.0000 0.935760 0.467880 0.883792i \(-0.345018\pi\)
0.467880 + 0.883792i \(0.345018\pi\)
\(194\) 8.00000 0.574367
\(195\) 3.00000 0.214834
\(196\) 1.00000 0.0714286
\(197\) 4.00000 0.284988 0.142494 0.989796i \(-0.454488\pi\)
0.142494 + 0.989796i \(0.454488\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −24.0000 −1.69283
\(202\) 15.0000 1.05540
\(203\) 8.00000 0.561490
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) −6.00000 −0.418040
\(207\) 6.00000 0.417029
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) −3.00000 −0.207020
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −12.0000 −0.824163
\(213\) 24.0000 1.64445
\(214\) 2.00000 0.136717
\(215\) 6.00000 0.409197
\(216\) 9.00000 0.612372
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) 48.0000 3.24354
\(220\) 0 0
\(221\) 0 0
\(222\) 6.00000 0.402694
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 1.00000 0.0668153
\(225\) 6.00000 0.400000
\(226\) 9.00000 0.598671
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −21.0000 −1.39076
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) 8.00000 0.525226
\(233\) 29.0000 1.89985 0.949927 0.312473i \(-0.101157\pi\)
0.949927 + 0.312473i \(0.101157\pi\)
\(234\) 6.00000 0.392232
\(235\) −12.0000 −0.782794
\(236\) 3.00000 0.195283
\(237\) −27.0000 −1.75384
\(238\) 0 0
\(239\) −17.0000 −1.09964 −0.549819 0.835284i \(-0.685303\pi\)
−0.549819 + 0.835284i \(0.685303\pi\)
\(240\) −3.00000 −0.193649
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) 1.00000 0.0638877
\(246\) 18.0000 1.14764
\(247\) −7.00000 −0.445399
\(248\) 4.00000 0.254000
\(249\) −39.0000 −2.47152
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −6.00000 −0.377964
\(253\) 0 0
\(254\) 7.00000 0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 18.0000 1.12063
\(259\) −2.00000 −0.124274
\(260\) −1.00000 −0.0620174
\(261\) −48.0000 −2.97113
\(262\) −11.0000 −0.679582
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 7.00000 0.429198
\(267\) −18.0000 −1.10158
\(268\) 8.00000 0.488678
\(269\) 3.00000 0.182913 0.0914566 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 9.00000 0.547723
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) −3.00000 −0.181568
\(274\) 5.00000 0.302061
\(275\) 0 0
\(276\) −3.00000 −0.180579
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 5.00000 0.299880
\(279\) −24.0000 −1.43684
\(280\) 1.00000 0.0597614
\(281\) −3.00000 −0.178965 −0.0894825 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) −36.0000 −2.14377
\(283\) −7.00000 −0.416107 −0.208053 0.978117i \(-0.566713\pi\)
−0.208053 + 0.978117i \(0.566713\pi\)
\(284\) −8.00000 −0.474713
\(285\) −21.0000 −1.24393
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) −6.00000 −0.353553
\(289\) −17.0000 −1.00000
\(290\) 8.00000 0.469776
\(291\) 24.0000 1.40690
\(292\) −16.0000 −0.936329
\(293\) −19.0000 −1.10999 −0.554996 0.831853i \(-0.687280\pi\)
−0.554996 + 0.831853i \(0.687280\pi\)
\(294\) 3.00000 0.174964
\(295\) 3.00000 0.174667
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) −1.00000 −0.0578315
\(300\) −3.00000 −0.173205
\(301\) −6.00000 −0.345834
\(302\) −7.00000 −0.402805
\(303\) 45.0000 2.58518
\(304\) 7.00000 0.401478
\(305\) 6.00000 0.343559
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) −18.0000 −1.02398
\(310\) 4.00000 0.227185
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) −3.00000 −0.169842
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) 21.0000 1.18510
\(315\) −6.00000 −0.338062
\(316\) 9.00000 0.506290
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) −36.0000 −2.01878
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 6.00000 0.334887
\(322\) 1.00000 0.0557278
\(323\) 0 0
\(324\) 9.00000 0.500000
\(325\) −1.00000 −0.0554700
\(326\) −6.00000 −0.332309
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −22.0000 −1.20923 −0.604615 0.796518i \(-0.706673\pi\)
−0.604615 + 0.796518i \(0.706673\pi\)
\(332\) 13.0000 0.713468
\(333\) 12.0000 0.657596
\(334\) −2.00000 −0.109435
\(335\) 8.00000 0.437087
\(336\) 3.00000 0.163663
\(337\) 25.0000 1.36184 0.680918 0.732359i \(-0.261581\pi\)
0.680918 + 0.732359i \(0.261581\pi\)
\(338\) 12.0000 0.652714
\(339\) 27.0000 1.46644
\(340\) 0 0
\(341\) 0 0
\(342\) −42.0000 −2.27110
\(343\) −1.00000 −0.0539949
\(344\) −6.00000 −0.323498
\(345\) −3.00000 −0.161515
\(346\) −14.0000 −0.752645
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 24.0000 1.28654
\(349\) 23.0000 1.23116 0.615581 0.788074i \(-0.288921\pi\)
0.615581 + 0.788074i \(0.288921\pi\)
\(350\) 1.00000 0.0534522
\(351\) 9.00000 0.480384
\(352\) 0 0
\(353\) 36.0000 1.91609 0.958043 0.286623i \(-0.0925328\pi\)
0.958043 + 0.286623i \(0.0925328\pi\)
\(354\) 9.00000 0.478345
\(355\) −8.00000 −0.424596
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −22.0000 −1.16274
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −6.00000 −0.316228
\(361\) 30.0000 1.57895
\(362\) −11.0000 −0.578147
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) −16.0000 −0.837478
\(366\) 18.0000 0.940875
\(367\) −30.0000 −1.56599 −0.782994 0.622030i \(-0.786308\pi\)
−0.782994 + 0.622030i \(0.786308\pi\)
\(368\) 1.00000 0.0521286
\(369\) 36.0000 1.87409
\(370\) −2.00000 −0.103975
\(371\) 12.0000 0.623009
\(372\) 12.0000 0.622171
\(373\) −18.0000 −0.932005 −0.466002 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 12.0000 0.618853
\(377\) 8.00000 0.412021
\(378\) −9.00000 −0.462910
\(379\) −14.0000 −0.719132 −0.359566 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) 7.00000 0.359092
\(381\) 21.0000 1.07586
\(382\) −9.00000 −0.460480
\(383\) −38.0000 −1.94171 −0.970855 0.239669i \(-0.922961\pi\)
−0.970855 + 0.239669i \(0.922961\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −13.0000 −0.661683
\(387\) 36.0000 1.82998
\(388\) −8.00000 −0.406138
\(389\) −28.0000 −1.41966 −0.709828 0.704375i \(-0.751227\pi\)
−0.709828 + 0.704375i \(0.751227\pi\)
\(390\) −3.00000 −0.151911
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) −33.0000 −1.66463
\(394\) −4.00000 −0.201517
\(395\) 9.00000 0.452839
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −4.00000 −0.200502
\(399\) 21.0000 1.05131
\(400\) 1.00000 0.0500000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 24.0000 1.19701
\(403\) 4.00000 0.199254
\(404\) −15.0000 −0.746278
\(405\) 9.00000 0.447214
\(406\) −8.00000 −0.397033
\(407\) 0 0
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) −6.00000 −0.296319
\(411\) 15.0000 0.739895
\(412\) 6.00000 0.295599
\(413\) −3.00000 −0.147620
\(414\) −6.00000 −0.294884
\(415\) 13.0000 0.638145
\(416\) 1.00000 0.0490290
\(417\) 15.0000 0.734553
\(418\) 0 0
\(419\) 3.00000 0.146560 0.0732798 0.997311i \(-0.476653\pi\)
0.0732798 + 0.997311i \(0.476653\pi\)
\(420\) 3.00000 0.146385
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) 16.0000 0.778868
\(423\) −72.0000 −3.50076
\(424\) 12.0000 0.582772
\(425\) 0 0
\(426\) −24.0000 −1.16280
\(427\) −6.00000 −0.290360
\(428\) −2.00000 −0.0966736
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) 7.00000 0.337178 0.168589 0.985686i \(-0.446079\pi\)
0.168589 + 0.985686i \(0.446079\pi\)
\(432\) −9.00000 −0.433013
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) −4.00000 −0.192006
\(435\) 24.0000 1.15071
\(436\) 0 0
\(437\) 7.00000 0.334855
\(438\) −48.0000 −2.29353
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) −28.0000 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) −6.00000 −0.284747
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) 54.0000 2.55411
\(448\) −1.00000 −0.0472456
\(449\) −7.00000 −0.330350 −0.165175 0.986264i \(-0.552819\pi\)
−0.165175 + 0.986264i \(0.552819\pi\)
\(450\) −6.00000 −0.282843
\(451\) 0 0
\(452\) −9.00000 −0.423324
\(453\) −21.0000 −0.986666
\(454\) −12.0000 −0.563188
\(455\) 1.00000 0.0468807
\(456\) 21.0000 0.983415
\(457\) 25.0000 1.16945 0.584725 0.811231i \(-0.301202\pi\)
0.584725 + 0.811231i \(0.301202\pi\)
\(458\) −2.00000 −0.0934539
\(459\) 0 0
\(460\) 1.00000 0.0466252
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −23.0000 −1.06890 −0.534450 0.845200i \(-0.679481\pi\)
−0.534450 + 0.845200i \(0.679481\pi\)
\(464\) −8.00000 −0.371391
\(465\) 12.0000 0.556487
\(466\) −29.0000 −1.34340
\(467\) −27.0000 −1.24941 −0.624705 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) −6.00000 −0.277350
\(469\) −8.00000 −0.369406
\(470\) 12.0000 0.553519
\(471\) 63.0000 2.90289
\(472\) −3.00000 −0.138086
\(473\) 0 0
\(474\) 27.0000 1.24015
\(475\) 7.00000 0.321182
\(476\) 0 0
\(477\) −72.0000 −3.29665
\(478\) 17.0000 0.777562
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 3.00000 0.136931
\(481\) −2.00000 −0.0911922
\(482\) −20.0000 −0.910975
\(483\) 3.00000 0.136505
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) 3.00000 0.135943 0.0679715 0.997687i \(-0.478347\pi\)
0.0679715 + 0.997687i \(0.478347\pi\)
\(488\) −6.00000 −0.271607
\(489\) −18.0000 −0.813988
\(490\) −1.00000 −0.0451754
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) −18.0000 −0.811503
\(493\) 0 0
\(494\) 7.00000 0.314945
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 8.00000 0.358849
\(498\) 39.0000 1.74763
\(499\) −18.0000 −0.805791 −0.402895 0.915246i \(-0.631996\pi\)
−0.402895 + 0.915246i \(0.631996\pi\)
\(500\) 1.00000 0.0447214
\(501\) −6.00000 −0.268060
\(502\) 12.0000 0.535586
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 6.00000 0.267261
\(505\) −15.0000 −0.667491
\(506\) 0 0
\(507\) 36.0000 1.59882
\(508\) −7.00000 −0.310575
\(509\) −45.0000 −1.99459 −0.997295 0.0735034i \(-0.976582\pi\)
−0.997295 + 0.0735034i \(0.976582\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) −1.00000 −0.0441942
\(513\) −63.0000 −2.78152
\(514\) −12.0000 −0.529297
\(515\) 6.00000 0.264392
\(516\) −18.0000 −0.792406
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) −42.0000 −1.84360
\(520\) 1.00000 0.0438529
\(521\) 40.0000 1.75243 0.876216 0.481919i \(-0.160060\pi\)
0.876216 + 0.481919i \(0.160060\pi\)
\(522\) 48.0000 2.10090
\(523\) −23.0000 −1.00572 −0.502860 0.864368i \(-0.667719\pi\)
−0.502860 + 0.864368i \(0.667719\pi\)
\(524\) 11.0000 0.480537
\(525\) 3.00000 0.130931
\(526\) 9.00000 0.392419
\(527\) 0 0
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 12.0000 0.521247
\(531\) 18.0000 0.781133
\(532\) −7.00000 −0.303488
\(533\) −6.00000 −0.259889
\(534\) 18.0000 0.778936
\(535\) −2.00000 −0.0864675
\(536\) −8.00000 −0.345547
\(537\) −66.0000 −2.84811
\(538\) −3.00000 −0.129339
\(539\) 0 0
\(540\) −9.00000 −0.387298
\(541\) −26.0000 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) −24.0000 −1.03089
\(543\) −33.0000 −1.41617
\(544\) 0 0
\(545\) 0 0
\(546\) 3.00000 0.128388
\(547\) 42.0000 1.79579 0.897895 0.440209i \(-0.145096\pi\)
0.897895 + 0.440209i \(0.145096\pi\)
\(548\) −5.00000 −0.213589
\(549\) 36.0000 1.53644
\(550\) 0 0
\(551\) −56.0000 −2.38568
\(552\) 3.00000 0.127688
\(553\) −9.00000 −0.382719
\(554\) 2.00000 0.0849719
\(555\) −6.00000 −0.254686
\(556\) −5.00000 −0.212047
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 24.0000 1.01600
\(559\) −6.00000 −0.253773
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 3.00000 0.126547
\(563\) −7.00000 −0.295015 −0.147507 0.989061i \(-0.547125\pi\)
−0.147507 + 0.989061i \(0.547125\pi\)
\(564\) 36.0000 1.51587
\(565\) −9.00000 −0.378633
\(566\) 7.00000 0.294232
\(567\) −9.00000 −0.377964
\(568\) 8.00000 0.335673
\(569\) −1.00000 −0.0419222 −0.0209611 0.999780i \(-0.506673\pi\)
−0.0209611 + 0.999780i \(0.506673\pi\)
\(570\) 21.0000 0.879593
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 0 0
\(573\) −27.0000 −1.12794
\(574\) 6.00000 0.250435
\(575\) 1.00000 0.0417029
\(576\) 6.00000 0.250000
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 17.0000 0.707107
\(579\) −39.0000 −1.62078
\(580\) −8.00000 −0.332182
\(581\) −13.0000 −0.539331
\(582\) −24.0000 −0.994832
\(583\) 0 0
\(584\) 16.0000 0.662085
\(585\) −6.00000 −0.248069
\(586\) 19.0000 0.784883
\(587\) −37.0000 −1.52715 −0.763577 0.645717i \(-0.776559\pi\)
−0.763577 + 0.645717i \(0.776559\pi\)
\(588\) −3.00000 −0.123718
\(589\) −28.0000 −1.15372
\(590\) −3.00000 −0.123508
\(591\) −12.0000 −0.493614
\(592\) 2.00000 0.0821995
\(593\) 36.0000 1.47834 0.739171 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −12.0000 −0.491127
\(598\) 1.00000 0.0408930
\(599\) −15.0000 −0.612883 −0.306442 0.951889i \(-0.599138\pi\)
−0.306442 + 0.951889i \(0.599138\pi\)
\(600\) 3.00000 0.122474
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 6.00000 0.244542
\(603\) 48.0000 1.95471
\(604\) 7.00000 0.284826
\(605\) 0 0
\(606\) −45.0000 −1.82800
\(607\) 2.00000 0.0811775 0.0405887 0.999176i \(-0.487077\pi\)
0.0405887 + 0.999176i \(0.487077\pi\)
\(608\) −7.00000 −0.283887
\(609\) −24.0000 −0.972529
\(610\) −6.00000 −0.242933
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −28.0000 −1.12999
\(615\) −18.0000 −0.725830
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 18.0000 0.724066
\(619\) 1.00000 0.0401934 0.0200967 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) −4.00000 −0.160644
\(621\) −9.00000 −0.361158
\(622\) −30.0000 −1.20289
\(623\) −6.00000 −0.240385
\(624\) 3.00000 0.120096
\(625\) 1.00000 0.0400000
\(626\) 18.0000 0.719425
\(627\) 0 0
\(628\) −21.0000 −0.837991
\(629\) 0 0
\(630\) 6.00000 0.239046
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −9.00000 −0.358001
\(633\) 48.0000 1.90783
\(634\) 12.0000 0.476581
\(635\) −7.00000 −0.277787
\(636\) 36.0000 1.42749
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −48.0000 −1.89885
\(640\) −1.00000 −0.0395285
\(641\) −41.0000 −1.61940 −0.809701 0.586842i \(-0.800371\pi\)
−0.809701 + 0.586842i \(0.800371\pi\)
\(642\) −6.00000 −0.236801
\(643\) −40.0000 −1.57745 −0.788723 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −18.0000 −0.708749
\(646\) 0 0
\(647\) 2.00000 0.0786281 0.0393141 0.999227i \(-0.487483\pi\)
0.0393141 + 0.999227i \(0.487483\pi\)
\(648\) −9.00000 −0.353553
\(649\) 0 0
\(650\) 1.00000 0.0392232
\(651\) −12.0000 −0.470317
\(652\) 6.00000 0.234978
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 0 0
\(655\) 11.0000 0.429806
\(656\) 6.00000 0.234261
\(657\) −96.0000 −3.74532
\(658\) −12.0000 −0.467809
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) 29.0000 1.12797 0.563985 0.825785i \(-0.309268\pi\)
0.563985 + 0.825785i \(0.309268\pi\)
\(662\) 22.0000 0.855054
\(663\) 0 0
\(664\) −13.0000 −0.504498
\(665\) −7.00000 −0.271448
\(666\) −12.0000 −0.464991
\(667\) −8.00000 −0.309761
\(668\) 2.00000 0.0773823
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 0 0
\(672\) −3.00000 −0.115728
\(673\) −17.0000 −0.655302 −0.327651 0.944799i \(-0.606257\pi\)
−0.327651 + 0.944799i \(0.606257\pi\)
\(674\) −25.0000 −0.962964
\(675\) −9.00000 −0.346410
\(676\) −12.0000 −0.461538
\(677\) 23.0000 0.883962 0.441981 0.897024i \(-0.354276\pi\)
0.441981 + 0.897024i \(0.354276\pi\)
\(678\) −27.0000 −1.03693
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) −36.0000 −1.37952
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 42.0000 1.60591
\(685\) −5.00000 −0.191040
\(686\) 1.00000 0.0381802
\(687\) −6.00000 −0.228914
\(688\) 6.00000 0.228748
\(689\) 12.0000 0.457164
\(690\) 3.00000 0.114208
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) 18.0000 0.683271
\(695\) −5.00000 −0.189661
\(696\) −24.0000 −0.909718
\(697\) 0 0
\(698\) −23.0000 −0.870563
\(699\) −87.0000 −3.29064
\(700\) −1.00000 −0.0377964
\(701\) −36.0000 −1.35970 −0.679851 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) −9.00000 −0.339683
\(703\) 14.0000 0.528020
\(704\) 0 0
\(705\) 36.0000 1.35584
\(706\) −36.0000 −1.35488
\(707\) 15.0000 0.564133
\(708\) −9.00000 −0.338241
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 8.00000 0.300235
\(711\) 54.0000 2.02516
\(712\) −6.00000 −0.224860
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) 22.0000 0.822179
\(717\) 51.0000 1.90463
\(718\) −24.0000 −0.895672
\(719\) −28.0000 −1.04422 −0.522112 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) 6.00000 0.223607
\(721\) −6.00000 −0.223452
\(722\) −30.0000 −1.11648
\(723\) −60.0000 −2.23142
\(724\) 11.0000 0.408812
\(725\) −8.00000 −0.297113
\(726\) 0 0
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) −1.00000 −0.0370625
\(729\) −27.0000 −1.00000
\(730\) 16.0000 0.592187
\(731\) 0 0
\(732\) −18.0000 −0.665299
\(733\) 17.0000 0.627909 0.313955 0.949438i \(-0.398346\pi\)
0.313955 + 0.949438i \(0.398346\pi\)
\(734\) 30.0000 1.10732
\(735\) −3.00000 −0.110657
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) −36.0000 −1.32518
\(739\) −24.0000 −0.882854 −0.441427 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) 2.00000 0.0735215
\(741\) 21.0000 0.771454
\(742\) −12.0000 −0.440534
\(743\) −20.0000 −0.733729 −0.366864 0.930274i \(-0.619569\pi\)
−0.366864 + 0.930274i \(0.619569\pi\)
\(744\) −12.0000 −0.439941
\(745\) −18.0000 −0.659469
\(746\) 18.0000 0.659027
\(747\) 78.0000 2.85387
\(748\) 0 0
\(749\) 2.00000 0.0730784
\(750\) 3.00000 0.109545
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) −12.0000 −0.437595
\(753\) 36.0000 1.31191
\(754\) −8.00000 −0.291343
\(755\) 7.00000 0.254756
\(756\) 9.00000 0.327327
\(757\) −14.0000 −0.508839 −0.254419 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) 14.0000 0.508503
\(759\) 0 0
\(760\) −7.00000 −0.253917
\(761\) −52.0000 −1.88500 −0.942499 0.334208i \(-0.891531\pi\)
−0.942499 + 0.334208i \(0.891531\pi\)
\(762\) −21.0000 −0.760750
\(763\) 0 0
\(764\) 9.00000 0.325609
\(765\) 0 0
\(766\) 38.0000 1.37300
\(767\) −3.00000 −0.108324
\(768\) −3.00000 −0.108253
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) −36.0000 −1.29651
\(772\) 13.0000 0.467880
\(773\) −39.0000 −1.40273 −0.701366 0.712801i \(-0.747426\pi\)
−0.701366 + 0.712801i \(0.747426\pi\)
\(774\) −36.0000 −1.29399
\(775\) −4.00000 −0.143684
\(776\) 8.00000 0.287183
\(777\) 6.00000 0.215249
\(778\) 28.0000 1.00385
\(779\) 42.0000 1.50481
\(780\) 3.00000 0.107417
\(781\) 0 0
\(782\) 0 0
\(783\) 72.0000 2.57307
\(784\) 1.00000 0.0357143
\(785\) −21.0000 −0.749522
\(786\) 33.0000 1.17707
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 4.00000 0.142494
\(789\) 27.0000 0.961225
\(790\) −9.00000 −0.320206
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) −6.00000 −0.213066
\(794\) 22.0000 0.780751
\(795\) 36.0000 1.27679
\(796\) 4.00000 0.141776
\(797\) 23.0000 0.814702 0.407351 0.913272i \(-0.366453\pi\)
0.407351 + 0.913272i \(0.366453\pi\)
\(798\) −21.0000 −0.743392
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 36.0000 1.27200
\(802\) 10.0000 0.353112
\(803\) 0 0
\(804\) −24.0000 −0.846415
\(805\) −1.00000 −0.0352454
\(806\) −4.00000 −0.140894
\(807\) −9.00000 −0.316815
\(808\) 15.0000 0.527698
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) −9.00000 −0.316228
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 8.00000 0.280745
\(813\) −72.0000 −2.52515
\(814\) 0 0
\(815\) 6.00000 0.210171
\(816\) 0 0
\(817\) 42.0000 1.46939
\(818\) −4.00000 −0.139857
\(819\) 6.00000 0.209657
\(820\) 6.00000 0.209529
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) −15.0000 −0.523185
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −6.00000 −0.209020
\(825\) 0 0
\(826\) 3.00000 0.104383
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 6.00000 0.208514
\(829\) 33.0000 1.14614 0.573069 0.819507i \(-0.305753\pi\)
0.573069 + 0.819507i \(0.305753\pi\)
\(830\) −13.0000 −0.451237
\(831\) 6.00000 0.208138
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −15.0000 −0.519408
\(835\) 2.00000 0.0692129
\(836\) 0 0
\(837\) 36.0000 1.24434
\(838\) −3.00000 −0.103633
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) −3.00000 −0.103510
\(841\) 35.0000 1.20690
\(842\) −32.0000 −1.10279
\(843\) 9.00000 0.309976
\(844\) −16.0000 −0.550743
\(845\) −12.0000 −0.412813
\(846\) 72.0000 2.47541
\(847\) 0 0
\(848\) −12.0000 −0.412082
\(849\) 21.0000 0.720718
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) 24.0000 0.822226
\(853\) −49.0000 −1.67773 −0.838864 0.544341i \(-0.816780\pi\)
−0.838864 + 0.544341i \(0.816780\pi\)
\(854\) 6.00000 0.205316
\(855\) 42.0000 1.43637
\(856\) 2.00000 0.0683586
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 6.00000 0.204598
\(861\) 18.0000 0.613438
\(862\) −7.00000 −0.238421
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 9.00000 0.306186
\(865\) 14.0000 0.476014
\(866\) −14.0000 −0.475739
\(867\) 51.0000 1.73205
\(868\) 4.00000 0.135769
\(869\) 0 0
\(870\) −24.0000 −0.813676
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) −48.0000 −1.62455
\(874\) −7.00000 −0.236779
\(875\) −1.00000 −0.0338062
\(876\) 48.0000 1.62177
\(877\) −42.0000 −1.41824 −0.709120 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) 10.0000 0.337484
\(879\) 57.0000 1.92256
\(880\) 0 0
\(881\) 38.0000 1.28025 0.640126 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) −6.00000 −0.202031
\(883\) 54.0000 1.81724 0.908622 0.417619i \(-0.137135\pi\)
0.908622 + 0.417619i \(0.137135\pi\)
\(884\) 0 0
\(885\) −9.00000 −0.302532
\(886\) 28.0000 0.940678
\(887\) 22.0000 0.738688 0.369344 0.929293i \(-0.379582\pi\)
0.369344 + 0.929293i \(0.379582\pi\)
\(888\) 6.00000 0.201347
\(889\) 7.00000 0.234772
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) 0 0
\(893\) −84.0000 −2.81095
\(894\) −54.0000 −1.80603
\(895\) 22.0000 0.735379
\(896\) 1.00000 0.0334077
\(897\) 3.00000 0.100167
\(898\) 7.00000 0.233593
\(899\) 32.0000 1.06726
\(900\) 6.00000 0.200000
\(901\) 0 0
\(902\) 0 0
\(903\) 18.0000 0.599002
\(904\) 9.00000 0.299336
\(905\) 11.0000 0.365652
\(906\) 21.0000 0.697678
\(907\) 40.0000 1.32818 0.664089 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) 12.0000 0.398234
\(909\) −90.0000 −2.98511
\(910\) −1.00000 −0.0331497
\(911\) −21.0000 −0.695761 −0.347881 0.937539i \(-0.613099\pi\)
−0.347881 + 0.937539i \(0.613099\pi\)
\(912\) −21.0000 −0.695379
\(913\) 0 0
\(914\) −25.0000 −0.826927
\(915\) −18.0000 −0.595062
\(916\) 2.00000 0.0660819
\(917\) −11.0000 −0.363252
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) −1.00000 −0.0329690
\(921\) −84.0000 −2.76789
\(922\) 30.0000 0.987997
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 23.0000 0.755827
\(927\) 36.0000 1.18240
\(928\) 8.00000 0.262613
\(929\) 12.0000 0.393707 0.196854 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) −12.0000 −0.393496
\(931\) 7.00000 0.229416
\(932\) 29.0000 0.949927
\(933\) −90.0000 −2.94647
\(934\) 27.0000 0.883467
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 28.0000 0.914720 0.457360 0.889282i \(-0.348795\pi\)
0.457360 + 0.889282i \(0.348795\pi\)
\(938\) 8.00000 0.261209
\(939\) 54.0000 1.76222
\(940\) −12.0000 −0.391397
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) −63.0000 −2.05265
\(943\) 6.00000 0.195387
\(944\) 3.00000 0.0976417
\(945\) 9.00000 0.292770
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −27.0000 −0.876919
\(949\) 16.0000 0.519382
\(950\) −7.00000 −0.227110
\(951\) 36.0000 1.16738
\(952\) 0 0
\(953\) 41.0000 1.32812 0.664060 0.747679i \(-0.268832\pi\)
0.664060 + 0.747679i \(0.268832\pi\)
\(954\) 72.0000 2.33109
\(955\) 9.00000 0.291233
\(956\) −17.0000 −0.549819
\(957\) 0 0
\(958\) −18.0000 −0.581554
\(959\) 5.00000 0.161458
\(960\) −3.00000 −0.0968246
\(961\) −15.0000 −0.483871
\(962\) 2.00000 0.0644826
\(963\) −12.0000 −0.386695
\(964\) 20.0000 0.644157
\(965\) 13.0000 0.418485
\(966\) −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
\(969\) 0 0
\(970\) 8.00000 0.256865
\(971\) −39.0000 −1.25157 −0.625785 0.779996i \(-0.715221\pi\)
−0.625785 + 0.779996i \(0.715221\pi\)
\(972\) 0 0
\(973\) 5.00000 0.160293
\(974\) −3.00000 −0.0961262
\(975\) 3.00000 0.0960769
\(976\) 6.00000 0.192055
\(977\) −57.0000 −1.82359 −0.911796 0.410644i \(-0.865304\pi\)
−0.911796 + 0.410644i \(0.865304\pi\)
\(978\) 18.0000 0.575577
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 8.00000 0.255290
\(983\) 30.0000 0.956851 0.478426 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 18.0000 0.573819
\(985\) 4.00000 0.127451
\(986\) 0 0
\(987\) −36.0000 −1.14589
\(988\) −7.00000 −0.222700
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 9.00000 0.285894 0.142947 0.989730i \(-0.454342\pi\)
0.142947 + 0.989730i \(0.454342\pi\)
\(992\) 4.00000 0.127000
\(993\) 66.0000 2.09445
\(994\) −8.00000 −0.253745
\(995\) 4.00000 0.126809
\(996\) −39.0000 −1.23576
\(997\) 57.0000 1.80521 0.902604 0.430472i \(-0.141653\pi\)
0.902604 + 0.430472i \(0.141653\pi\)
\(998\) 18.0000 0.569780
\(999\) −18.0000 −0.569495
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.a.1.1 1
11.10 odd 2 8470.2.a.q.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.a.1.1 1 1.1 even 1 trivial
8470.2.a.q.1.1 yes 1 11.10 odd 2