Properties

Label 8470.2.a.f.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} -6.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} +2.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +1.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} +2.00000 q^{42} -7.00000 q^{43} +1.00000 q^{45} +6.00000 q^{46} -2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} +9.00000 q^{53} -4.00000 q^{54} -1.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} +3.00000 q^{59} -2.00000 q^{60} -7.00000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +11.0000 q^{67} +3.00000 q^{68} +12.0000 q^{69} -1.00000 q^{70} -3.00000 q^{71} -1.00000 q^{72} +11.0000 q^{73} -2.00000 q^{74} -2.00000 q^{75} -1.00000 q^{76} +4.00000 q^{78} +11.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -12.0000 q^{83} -2.00000 q^{84} +3.00000 q^{85} +7.00000 q^{86} -12.0000 q^{87} -1.00000 q^{90} +2.00000 q^{91} -6.00000 q^{92} +8.00000 q^{93} -1.00000 q^{95} +2.00000 q^{96} -1.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.00000 0.816497
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −2.00000 −0.577350
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 2.00000 0.408248
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 4.00000 0.769800
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 2.00000 0.365148
\(31\) −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
\(34\) −3.00000 −0.514496
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 1.00000 0.162221
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 2.00000 0.308607
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 6.00000 0.884652
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) 2.00000 0.277350
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 2.00000 0.264906
\(58\) −6.00000 −0.787839
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) −2.00000 −0.258199
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 4.00000 0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 11.0000 1.34386 0.671932 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) 3.00000 0.363803
\(69\) 12.0000 1.44463
\(70\) −1.00000 −0.119523
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) −1.00000 −0.117851
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) −2.00000 −0.232495
\(75\) −2.00000 −0.230940
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −2.00000 −0.218218
\(85\) 3.00000 0.325396
\(86\) 7.00000 0.754829
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −1.00000 −0.105409
\(91\) 2.00000 0.209657
\(92\) −6.00000 −0.625543
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 2.00000 0.204124
\(97\) −1.00000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 6.00000 0.594089
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) −2.00000 −0.196116
\(105\) −2.00000 −0.195180
\(106\) −9.00000 −0.874157
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 4.00000 0.384900
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 1.00000 0.0944911
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) −2.00000 −0.187317
\(115\) −6.00000 −0.559503
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) −3.00000 −0.276172
\(119\) 3.00000 0.275010
\(120\) 2.00000 0.182574
\(121\) 0 0
\(122\) 7.00000 0.633750
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 14.0000 1.23263
\(130\) −2.00000 −0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) −11.0000 −0.950255
\(135\) 4.00000 0.344265
\(136\) −3.00000 −0.257248
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) −12.0000 −1.02151
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 3.00000 0.251754
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) −11.0000 −0.910366
\(147\) −2.00000 −0.164957
\(148\) 2.00000 0.164399
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 2.00000 0.163299
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 1.00000 0.0811107
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) −4.00000 −0.320256
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) −11.0000 −0.875113
\(159\) −18.0000 −1.42749
\(160\) −1.00000 −0.0790569
\(161\) −6.00000 −0.472866
\(162\) 11.0000 0.864242
\(163\) 5.00000 0.391630 0.195815 0.980641i \(-0.437265\pi\)
0.195815 + 0.980641i \(0.437265\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 2.00000 0.154303
\(169\) −9.00000 −0.692308
\(170\) −3.00000 −0.230089
\(171\) −1.00000 −0.0764719
\(172\) −7.00000 −0.533745
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 12.0000 0.909718
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 0 0
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 1.00000 0.0745356
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) −2.00000 −0.148250
\(183\) 14.0000 1.03491
\(184\) 6.00000 0.442326
\(185\) 2.00000 0.147043
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 1.00000 0.0725476
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −2.00000 −0.144338
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 1.00000 0.0717958
\(195\) −4.00000 −0.286446
\(196\) 1.00000 0.0714286
\(197\) −9.00000 −0.641223 −0.320612 0.947211i \(-0.603888\pi\)
−0.320612 + 0.947211i \(0.603888\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −22.0000 −1.55176
\(202\) −6.00000 −0.422159
\(203\) 6.00000 0.421117
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 1.00000 0.0696733
\(207\) −6.00000 −0.417029
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 2.00000 0.138013
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 9.00000 0.618123
\(213\) 6.00000 0.411113
\(214\) 9.00000 0.615227
\(215\) −7.00000 −0.477396
\(216\) −4.00000 −0.272166
\(217\) −4.00000 −0.271538
\(218\) 16.0000 1.08366
\(219\) −22.0000 −1.48662
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 4.00000 0.268462
\(223\) 11.0000 0.736614 0.368307 0.929704i \(-0.379937\pi\)
0.368307 + 0.929704i \(0.379937\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) −12.0000 −0.798228
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 2.00000 0.132453
\(229\) −25.0000 −1.65205 −0.826023 0.563636i \(-0.809402\pi\)
−0.826023 + 0.563636i \(0.809402\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 3.00000 0.195283
\(237\) −22.0000 −1.42905
\(238\) −3.00000 −0.194461
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) −2.00000 −0.129099
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) −7.00000 −0.448129
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 4.00000 0.254000
\(249\) 24.0000 1.52094
\(250\) −1.00000 −0.0632456
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) −2.00000 −0.125491
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) 21.0000 1.30994 0.654972 0.755653i \(-0.272680\pi\)
0.654972 + 0.755653i \(0.272680\pi\)
\(258\) −14.0000 −0.871602
\(259\) 2.00000 0.124274
\(260\) 2.00000 0.124035
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 1.00000 0.0613139
\(267\) 0 0
\(268\) 11.0000 0.671932
\(269\) −15.0000 −0.914566 −0.457283 0.889321i \(-0.651177\pi\)
−0.457283 + 0.889321i \(0.651177\pi\)
\(270\) −4.00000 −0.243432
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 3.00000 0.181902
\(273\) −4.00000 −0.242091
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −5.00000 −0.299880
\(279\) −4.00000 −0.239474
\(280\) −1.00000 −0.0597614
\(281\) −21.0000 −1.25275 −0.626377 0.779520i \(-0.715463\pi\)
−0.626377 + 0.779520i \(0.715463\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) −3.00000 −0.178017
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) −6.00000 −0.352332
\(291\) 2.00000 0.117242
\(292\) 11.0000 0.643726
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 2.00000 0.116642
\(295\) 3.00000 0.174667
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) −2.00000 −0.115470
\(301\) −7.00000 −0.403473
\(302\) −8.00000 −0.460348
\(303\) −12.0000 −0.689382
\(304\) −1.00000 −0.0573539
\(305\) −7.00000 −0.400819
\(306\) −3.00000 −0.171499
\(307\) −34.0000 −1.94048 −0.970241 0.242140i \(-0.922151\pi\)
−0.970241 + 0.242140i \(0.922151\pi\)
\(308\) 0 0
\(309\) 2.00000 0.113776
\(310\) 4.00000 0.227185
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 4.00000 0.226455
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −8.00000 −0.451466
\(315\) 1.00000 0.0563436
\(316\) 11.0000 0.618798
\(317\) 33.0000 1.85346 0.926732 0.375722i \(-0.122605\pi\)
0.926732 + 0.375722i \(0.122605\pi\)
\(318\) 18.0000 1.00939
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 18.0000 1.00466
\(322\) 6.00000 0.334367
\(323\) −3.00000 −0.166924
\(324\) −11.0000 −0.611111
\(325\) 2.00000 0.110940
\(326\) −5.00000 −0.276924
\(327\) 32.0000 1.76960
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 26.0000 1.42909 0.714545 0.699590i \(-0.246634\pi\)
0.714545 + 0.699590i \(0.246634\pi\)
\(332\) −12.0000 −0.658586
\(333\) 2.00000 0.109599
\(334\) −9.00000 −0.492458
\(335\) 11.0000 0.600994
\(336\) −2.00000 −0.109109
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 9.00000 0.489535
\(339\) −24.0000 −1.30350
\(340\) 3.00000 0.162698
\(341\) 0 0
\(342\) 1.00000 0.0540738
\(343\) 1.00000 0.0539949
\(344\) 7.00000 0.377415
\(345\) 12.0000 0.646058
\(346\) −12.0000 −0.645124
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) −12.0000 −0.643268
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) 21.0000 1.11772 0.558859 0.829263i \(-0.311239\pi\)
0.558859 + 0.829263i \(0.311239\pi\)
\(354\) 6.00000 0.318896
\(355\) −3.00000 −0.159223
\(356\) 0 0
\(357\) −6.00000 −0.317554
\(358\) −24.0000 −1.26844
\(359\) −3.00000 −0.158334 −0.0791670 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −18.0000 −0.947368
\(362\) 7.00000 0.367912
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 11.0000 0.575766
\(366\) −14.0000 −0.731792
\(367\) 29.0000 1.51379 0.756894 0.653538i \(-0.226716\pi\)
0.756894 + 0.653538i \(0.226716\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) 9.00000 0.467257
\(372\) 8.00000 0.414781
\(373\) 23.0000 1.19089 0.595447 0.803394i \(-0.296975\pi\)
0.595447 + 0.803394i \(0.296975\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) −4.00000 −0.205738
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) −1.00000 −0.0512989
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 21.0000 1.07305 0.536525 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) −7.00000 −0.355830
\(388\) −1.00000 −0.0507673
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 4.00000 0.202548
\(391\) −18.0000 −0.910299
\(392\) −1.00000 −0.0505076
\(393\) 24.0000 1.21064
\(394\) 9.00000 0.453413
\(395\) 11.0000 0.553470
\(396\) 0 0
\(397\) −16.0000 −0.803017 −0.401508 0.915855i \(-0.631514\pi\)
−0.401508 + 0.915855i \(0.631514\pi\)
\(398\) −2.00000 −0.100251
\(399\) 2.00000 0.100125
\(400\) 1.00000 0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 22.0000 1.09726
\(403\) −8.00000 −0.398508
\(404\) 6.00000 0.298511
\(405\) −11.0000 −0.546594
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) −24.0000 −1.18383
\(412\) −1.00000 −0.0492665
\(413\) 3.00000 0.147620
\(414\) 6.00000 0.294884
\(415\) −12.0000 −0.589057
\(416\) −2.00000 −0.0980581
\(417\) −10.0000 −0.489702
\(418\) 0 0
\(419\) 9.00000 0.439679 0.219839 0.975536i \(-0.429447\pi\)
0.219839 + 0.975536i \(0.429447\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 3.00000 0.145521
\(426\) −6.00000 −0.290701
\(427\) −7.00000 −0.338754
\(428\) −9.00000 −0.435031
\(429\) 0 0
\(430\) 7.00000 0.337570
\(431\) 3.00000 0.144505 0.0722525 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) 4.00000 0.192450
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 4.00000 0.192006
\(435\) −12.0000 −0.575356
\(436\) −16.0000 −0.766261
\(437\) 6.00000 0.287019
\(438\) 22.0000 1.05120
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −6.00000 −0.285391
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −11.0000 −0.520865
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 21.0000 0.991051 0.495526 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) −16.0000 −0.751746
\(454\) 0 0
\(455\) 2.00000 0.0937614
\(456\) −2.00000 −0.0936586
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 25.0000 1.16817
\(459\) 12.0000 0.560112
\(460\) −6.00000 −0.279751
\(461\) 9.00000 0.419172 0.209586 0.977790i \(-0.432788\pi\)
0.209586 + 0.977790i \(0.432788\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 6.00000 0.278543
\(465\) 8.00000 0.370991
\(466\) −24.0000 −1.11178
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 2.00000 0.0924500
\(469\) 11.0000 0.507933
\(470\) 0 0
\(471\) −16.0000 −0.737241
\(472\) −3.00000 −0.138086
\(473\) 0 0
\(474\) 22.0000 1.01049
\(475\) −1.00000 −0.0458831
\(476\) 3.00000 0.137505
\(477\) 9.00000 0.412082
\(478\) 24.0000 1.09773
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 2.00000 0.0912871
\(481\) 4.00000 0.182384
\(482\) −8.00000 −0.364390
\(483\) 12.0000 0.546019
\(484\) 0 0
\(485\) −1.00000 −0.0454077
\(486\) −10.0000 −0.453609
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 7.00000 0.316875
\(489\) −10.0000 −0.452216
\(490\) −1.00000 −0.0451754
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −3.00000 −0.134568
\(498\) −24.0000 −1.07547
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 1.00000 0.0447214
\(501\) −18.0000 −0.804181
\(502\) 15.0000 0.669483
\(503\) 21.0000 0.936344 0.468172 0.883637i \(-0.344913\pi\)
0.468172 + 0.883637i \(0.344913\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 18.0000 0.799408
\(508\) 2.00000 0.0887357
\(509\) −21.0000 −0.930809 −0.465404 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 6.00000 0.265684
\(511\) 11.0000 0.486611
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) −21.0000 −0.926270
\(515\) −1.00000 −0.0440653
\(516\) 14.0000 0.616316
\(517\) 0 0
\(518\) −2.00000 −0.0878750
\(519\) −24.0000 −1.05348
\(520\) −2.00000 −0.0877058
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) −6.00000 −0.262613
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −12.0000 −0.524222
\(525\) −2.00000 −0.0872872
\(526\) 18.0000 0.784837
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −9.00000 −0.390935
\(531\) 3.00000 0.130189
\(532\) −1.00000 −0.0433555
\(533\) 0 0
\(534\) 0 0
\(535\) −9.00000 −0.389104
\(536\) −11.0000 −0.475128
\(537\) −48.0000 −2.07135
\(538\) 15.0000 0.646696
\(539\) 0 0
\(540\) 4.00000 0.172133
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) −2.00000 −0.0859074
\(543\) 14.0000 0.600798
\(544\) −3.00000 −0.128624
\(545\) −16.0000 −0.685365
\(546\) 4.00000 0.171184
\(547\) 35.0000 1.49649 0.748246 0.663421i \(-0.230896\pi\)
0.748246 + 0.663421i \(0.230896\pi\)
\(548\) 12.0000 0.512615
\(549\) −7.00000 −0.298753
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) −12.0000 −0.510754
\(553\) 11.0000 0.467768
\(554\) 22.0000 0.934690
\(555\) −4.00000 −0.169791
\(556\) 5.00000 0.212047
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 4.00000 0.169334
\(559\) −14.0000 −0.592137
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 21.0000 0.885832
\(563\) 30.0000 1.26435 0.632175 0.774826i \(-0.282163\pi\)
0.632175 + 0.774826i \(0.282163\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 16.0000 0.672530
\(567\) −11.0000 −0.461957
\(568\) 3.00000 0.125877
\(569\) −9.00000 −0.377300 −0.188650 0.982044i \(-0.560411\pi\)
−0.188650 + 0.982044i \(0.560411\pi\)
\(570\) −2.00000 −0.0837708
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 8.00000 0.332756
\(579\) 8.00000 0.332469
\(580\) 6.00000 0.249136
\(581\) −12.0000 −0.497844
\(582\) −2.00000 −0.0829027
\(583\) 0 0
\(584\) −11.0000 −0.455183
\(585\) 2.00000 0.0826898
\(586\) −12.0000 −0.495715
\(587\) 30.0000 1.23823 0.619116 0.785299i \(-0.287491\pi\)
0.619116 + 0.785299i \(0.287491\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 4.00000 0.164817
\(590\) −3.00000 −0.123508
\(591\) 18.0000 0.740421
\(592\) 2.00000 0.0821995
\(593\) 39.0000 1.60154 0.800769 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 12.0000 0.490716
\(599\) 27.0000 1.10319 0.551595 0.834112i \(-0.314019\pi\)
0.551595 + 0.834112i \(0.314019\pi\)
\(600\) 2.00000 0.0816497
\(601\) −40.0000 −1.63163 −0.815817 0.578310i \(-0.803712\pi\)
−0.815817 + 0.578310i \(0.803712\pi\)
\(602\) 7.00000 0.285299
\(603\) 11.0000 0.447955
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) 5.00000 0.202944 0.101472 0.994838i \(-0.467645\pi\)
0.101472 + 0.994838i \(0.467645\pi\)
\(608\) 1.00000 0.0405554
\(609\) −12.0000 −0.486265
\(610\) 7.00000 0.283422
\(611\) 0 0
\(612\) 3.00000 0.121268
\(613\) −1.00000 −0.0403896 −0.0201948 0.999796i \(-0.506429\pi\)
−0.0201948 + 0.999796i \(0.506429\pi\)
\(614\) 34.0000 1.37213
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −2.00000 −0.0804518
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) −4.00000 −0.160644
\(621\) −24.0000 −0.963087
\(622\) 0 0
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) −2.00000 −0.0799361
\(627\) 0 0
\(628\) 8.00000 0.319235
\(629\) 6.00000 0.239236
\(630\) −1.00000 −0.0398410
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) −11.0000 −0.437557
\(633\) 8.00000 0.317971
\(634\) −33.0000 −1.31060
\(635\) 2.00000 0.0793676
\(636\) −18.0000 −0.713746
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) −3.00000 −0.118678
\(640\) −1.00000 −0.0395285
\(641\) 21.0000 0.829450 0.414725 0.909947i \(-0.363878\pi\)
0.414725 + 0.909947i \(0.363878\pi\)
\(642\) −18.0000 −0.710403
\(643\) 32.0000 1.26196 0.630978 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) −6.00000 −0.236433
\(645\) 14.0000 0.551249
\(646\) 3.00000 0.118033
\(647\) 45.0000 1.76913 0.884566 0.466415i \(-0.154454\pi\)
0.884566 + 0.466415i \(0.154454\pi\)
\(648\) 11.0000 0.432121
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) 8.00000 0.313545
\(652\) 5.00000 0.195815
\(653\) 21.0000 0.821794 0.410897 0.911682i \(-0.365216\pi\)
0.410897 + 0.911682i \(0.365216\pi\)
\(654\) −32.0000 −1.25130
\(655\) −12.0000 −0.468879
\(656\) 0 0
\(657\) 11.0000 0.429151
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −26.0000 −1.01052
\(663\) −12.0000 −0.466041
\(664\) 12.0000 0.465690
\(665\) −1.00000 −0.0387783
\(666\) −2.00000 −0.0774984
\(667\) −36.0000 −1.39393
\(668\) 9.00000 0.348220
\(669\) −22.0000 −0.850569
\(670\) −11.0000 −0.424967
\(671\) 0 0
\(672\) 2.00000 0.0771517
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 22.0000 0.847408
\(675\) 4.00000 0.153960
\(676\) −9.00000 −0.346154
\(677\) 48.0000 1.84479 0.922395 0.386248i \(-0.126229\pi\)
0.922395 + 0.386248i \(0.126229\pi\)
\(678\) 24.0000 0.921714
\(679\) −1.00000 −0.0383765
\(680\) −3.00000 −0.115045
\(681\) 0 0
\(682\) 0 0
\(683\) 15.0000 0.573959 0.286980 0.957937i \(-0.407349\pi\)
0.286980 + 0.957937i \(0.407349\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 12.0000 0.458496
\(686\) −1.00000 −0.0381802
\(687\) 50.0000 1.90762
\(688\) −7.00000 −0.266872
\(689\) 18.0000 0.685745
\(690\) −12.0000 −0.456832
\(691\) −43.0000 −1.63580 −0.817899 0.575362i \(-0.804861\pi\)
−0.817899 + 0.575362i \(0.804861\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) −3.00000 −0.113878
\(695\) 5.00000 0.189661
\(696\) 12.0000 0.454859
\(697\) 0 0
\(698\) −14.0000 −0.529908
\(699\) −48.0000 −1.81553
\(700\) 1.00000 0.0377964
\(701\) 48.0000 1.81293 0.906467 0.422276i \(-0.138769\pi\)
0.906467 + 0.422276i \(0.138769\pi\)
\(702\) −8.00000 −0.301941
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) 0 0
\(706\) −21.0000 −0.790345
\(707\) 6.00000 0.225653
\(708\) −6.00000 −0.225494
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 3.00000 0.112588
\(711\) 11.0000 0.412532
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) 48.0000 1.79259
\(718\) 3.00000 0.111959
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 1.00000 0.0372678
\(721\) −1.00000 −0.0372419
\(722\) 18.0000 0.669891
\(723\) −16.0000 −0.595046
\(724\) −7.00000 −0.260153
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) −4.00000 −0.148352 −0.0741759 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 13.0000 0.481481
\(730\) −11.0000 −0.407128
\(731\) −21.0000 −0.776713
\(732\) 14.0000 0.517455
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) −29.0000 −1.07041
\(735\) −2.00000 −0.0737711
\(736\) 6.00000 0.221163
\(737\) 0 0
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 2.00000 0.0735215
\(741\) 4.00000 0.146944
\(742\) −9.00000 −0.330400
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) −23.0000 −0.842090
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) −9.00000 −0.328853
\(750\) 2.00000 0.0730297
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 30.0000 1.09326
\(754\) −12.0000 −0.437014
\(755\) 8.00000 0.291150
\(756\) 4.00000 0.145479
\(757\) 47.0000 1.70824 0.854122 0.520073i \(-0.174095\pi\)
0.854122 + 0.520073i \(0.174095\pi\)
\(758\) −8.00000 −0.290573
\(759\) 0 0
\(760\) 1.00000 0.0362738
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) 4.00000 0.144905
\(763\) −16.0000 −0.579239
\(764\) 0 0
\(765\) 3.00000 0.108465
\(766\) −21.0000 −0.758761
\(767\) 6.00000 0.216647
\(768\) −2.00000 −0.0721688
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) −42.0000 −1.51259
\(772\) −4.00000 −0.143963
\(773\) −12.0000 −0.431610 −0.215805 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) 7.00000 0.251610
\(775\) −4.00000 −0.143684
\(776\) 1.00000 0.0358979
\(777\) −4.00000 −0.143499
\(778\) −12.0000 −0.430221
\(779\) 0 0
\(780\) −4.00000 −0.143223
\(781\) 0 0
\(782\) 18.0000 0.643679
\(783\) 24.0000 0.857690
\(784\) 1.00000 0.0357143
\(785\) 8.00000 0.285532
\(786\) −24.0000 −0.856052
\(787\) 50.0000 1.78231 0.891154 0.453701i \(-0.149897\pi\)
0.891154 + 0.453701i \(0.149897\pi\)
\(788\) −9.00000 −0.320612
\(789\) 36.0000 1.28163
\(790\) −11.0000 −0.391362
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −14.0000 −0.497155
\(794\) 16.0000 0.567819
\(795\) −18.0000 −0.638394
\(796\) 2.00000 0.0708881
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) −2.00000 −0.0707992
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) −22.0000 −0.775880
\(805\) −6.00000 −0.211472
\(806\) 8.00000 0.281788
\(807\) 30.0000 1.05605
\(808\) −6.00000 −0.211079
\(809\) 3.00000 0.105474 0.0527372 0.998608i \(-0.483205\pi\)
0.0527372 + 0.998608i \(0.483205\pi\)
\(810\) 11.0000 0.386501
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 6.00000 0.210559
\(813\) −4.00000 −0.140286
\(814\) 0 0
\(815\) 5.00000 0.175142
\(816\) −6.00000 −0.210042
\(817\) 7.00000 0.244899
\(818\) 4.00000 0.139857
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 24.0000 0.837096
\(823\) −22.0000 −0.766872 −0.383436 0.923567i \(-0.625259\pi\)
−0.383436 + 0.923567i \(0.625259\pi\)
\(824\) 1.00000 0.0348367
\(825\) 0 0
\(826\) −3.00000 −0.104383
\(827\) −51.0000 −1.77344 −0.886722 0.462303i \(-0.847023\pi\)
−0.886722 + 0.462303i \(0.847023\pi\)
\(828\) −6.00000 −0.208514
\(829\) 50.0000 1.73657 0.868286 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) 12.0000 0.416526
\(831\) 44.0000 1.52634
\(832\) 2.00000 0.0693375
\(833\) 3.00000 0.103944
\(834\) 10.0000 0.346272
\(835\) 9.00000 0.311458
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) −9.00000 −0.310900
\(839\) 6.00000 0.207143 0.103572 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) 2.00000 0.0690066
\(841\) 7.00000 0.241379
\(842\) −8.00000 −0.275698
\(843\) 42.0000 1.44656
\(844\) −4.00000 −0.137686
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 9.00000 0.309061
\(849\) 32.0000 1.09824
\(850\) −3.00000 −0.102899
\(851\) −12.0000 −0.411355
\(852\) 6.00000 0.205557
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 7.00000 0.239535
\(855\) −1.00000 −0.0341993
\(856\) 9.00000 0.307614
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 47.0000 1.60362 0.801810 0.597580i \(-0.203871\pi\)
0.801810 + 0.597580i \(0.203871\pi\)
\(860\) −7.00000 −0.238698
\(861\) 0 0
\(862\) −3.00000 −0.102180
\(863\) 30.0000 1.02121 0.510606 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) −4.00000 −0.136083
\(865\) 12.0000 0.408012
\(866\) −2.00000 −0.0679628
\(867\) 16.0000 0.543388
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) 12.0000 0.406838
\(871\) 22.0000 0.745442
\(872\) 16.0000 0.541828
\(873\) −1.00000 −0.0338449
\(874\) −6.00000 −0.202953
\(875\) 1.00000 0.0338062
\(876\) −22.0000 −0.743311
\(877\) 17.0000 0.574049 0.287025 0.957923i \(-0.407334\pi\)
0.287025 + 0.957923i \(0.407334\pi\)
\(878\) 10.0000 0.337484
\(879\) −24.0000 −0.809500
\(880\) 0 0
\(881\) −12.0000 −0.404290 −0.202145 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 29.0000 0.975928 0.487964 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) 6.00000 0.201802
\(885\) −6.00000 −0.201688
\(886\) 12.0000 0.403148
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 4.00000 0.134231
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) 0 0
\(892\) 11.0000 0.368307
\(893\) 0 0
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) −1.00000 −0.0334077
\(897\) 24.0000 0.801337
\(898\) −21.0000 −0.700779
\(899\) −24.0000 −0.800445
\(900\) 1.00000 0.0333333
\(901\) 27.0000 0.899500
\(902\) 0 0
\(903\) 14.0000 0.465891
\(904\) −12.0000 −0.399114
\(905\) −7.00000 −0.232688
\(906\) 16.0000 0.531564
\(907\) −1.00000 −0.0332045 −0.0166022 0.999862i \(-0.505285\pi\)
−0.0166022 + 0.999862i \(0.505285\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) −2.00000 −0.0662994
\(911\) −21.0000 −0.695761 −0.347881 0.937539i \(-0.613099\pi\)
−0.347881 + 0.937539i \(0.613099\pi\)
\(912\) 2.00000 0.0662266
\(913\) 0 0
\(914\) −8.00000 −0.264616
\(915\) 14.0000 0.462826
\(916\) −25.0000 −0.826023
\(917\) −12.0000 −0.396275
\(918\) −12.0000 −0.396059
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 6.00000 0.197814
\(921\) 68.0000 2.24068
\(922\) −9.00000 −0.296399
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 22.0000 0.722965
\(927\) −1.00000 −0.0328443
\(928\) −6.00000 −0.196960
\(929\) 12.0000 0.393707 0.196854 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) −8.00000 −0.262330
\(931\) −1.00000 −0.0327737
\(932\) 24.0000 0.786146
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −11.0000 −0.359163
\(939\) −4.00000 −0.130535
\(940\) 0 0
\(941\) 39.0000 1.27136 0.635682 0.771951i \(-0.280719\pi\)
0.635682 + 0.771951i \(0.280719\pi\)
\(942\) 16.0000 0.521308
\(943\) 0 0
\(944\) 3.00000 0.0976417
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −22.0000 −0.714527
\(949\) 22.0000 0.714150
\(950\) 1.00000 0.0324443
\(951\) −66.0000 −2.14020
\(952\) −3.00000 −0.0972306
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −9.00000 −0.291386
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) −36.0000 −1.16311
\(959\) 12.0000 0.387500
\(960\) −2.00000 −0.0645497
\(961\) −15.0000 −0.483871
\(962\) −4.00000 −0.128965
\(963\) −9.00000 −0.290021
\(964\) 8.00000 0.257663
\(965\) −4.00000 −0.128765
\(966\) −12.0000 −0.386094
\(967\) −22.0000 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) 1.00000 0.0321081
\(971\) 21.0000 0.673922 0.336961 0.941519i \(-0.390601\pi\)
0.336961 + 0.941519i \(0.390601\pi\)
\(972\) 10.0000 0.320750
\(973\) 5.00000 0.160293
\(974\) 4.00000 0.128168
\(975\) −4.00000 −0.128103
\(976\) −7.00000 −0.224065
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 10.0000 0.319765
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −16.0000 −0.510841
\(982\) 12.0000 0.382935
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) −9.00000 −0.286764
\(986\) −18.0000 −0.573237
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) 42.0000 1.33552
\(990\) 0 0
\(991\) 5.00000 0.158830 0.0794151 0.996842i \(-0.474695\pi\)
0.0794151 + 0.996842i \(0.474695\pi\)
\(992\) 4.00000 0.127000
\(993\) −52.0000 −1.65017
\(994\) 3.00000 0.0951542
\(995\) 2.00000 0.0634043
\(996\) 24.0000 0.760469
\(997\) 20.0000 0.633406 0.316703 0.948525i \(-0.397424\pi\)
0.316703 + 0.948525i \(0.397424\pi\)
\(998\) −32.0000 −1.01294
\(999\) 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 8470.2.a.f.1.1 1
11.10 odd 2 8470.2.a.u.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.f.1.1 1 1.1 even 1 trivial
8470.2.a.u.1.1 yes 1 11.10 odd 2