Properties

Label 8470.2.a.dd.1.6
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.745749504.1
Defining polynomial: \(x^{6} - 18 x^{4} - 4 x^{3} + 81 x^{2} + 36 x - 44\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-3.24337\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.24337 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.24337 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.51947 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.24337 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.24337 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.51947 q^{9} -1.00000 q^{10} +3.24337 q^{12} +2.14515 q^{13} -1.00000 q^{14} -3.24337 q^{15} +1.00000 q^{16} +1.43942 q^{17} +7.51947 q^{18} -0.633832 q^{19} -1.00000 q^{20} -3.24337 q^{21} -5.61769 q^{23} +3.24337 q^{24} +1.00000 q^{25} +2.14515 q^{26} +14.6583 q^{27} -1.00000 q^{28} -1.19605 q^{29} -3.24337 q^{30} +6.95889 q^{31} +1.00000 q^{32} +1.43942 q^{34} +1.00000 q^{35} +7.51947 q^{36} +4.80395 q^{37} -0.633832 q^{38} +6.95754 q^{39} -1.00000 q^{40} -8.93663 q^{41} -3.24337 q^{42} +9.70747 q^{43} -7.51947 q^{45} -5.61769 q^{46} +8.29874 q^{47} +3.24337 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.66858 q^{51} +2.14515 q^{52} +14.1537 q^{53} +14.6583 q^{54} -1.00000 q^{56} -2.05576 q^{57} -1.19605 q^{58} -1.19248 q^{59} -3.24337 q^{60} -4.19441 q^{61} +6.95889 q^{62} -7.51947 q^{63} +1.00000 q^{64} -2.14515 q^{65} -12.0099 q^{67} +1.43942 q^{68} -18.2203 q^{69} +1.00000 q^{70} -5.71591 q^{71} +7.51947 q^{72} +6.43981 q^{73} +4.80395 q^{74} +3.24337 q^{75} -0.633832 q^{76} +6.95754 q^{78} -7.69133 q^{79} -1.00000 q^{80} +24.9840 q^{81} -8.93663 q^{82} +11.5685 q^{83} -3.24337 q^{84} -1.43942 q^{85} +9.70747 q^{86} -3.87923 q^{87} +16.7855 q^{89} -7.51947 q^{90} -2.14515 q^{91} -5.61769 q^{92} +22.5703 q^{93} +8.29874 q^{94} +0.633832 q^{95} +3.24337 q^{96} -2.96529 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 6q^{4} - 6q^{5} - 6q^{7} + 6q^{8} + 18q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + 6q^{4} - 6q^{5} - 6q^{7} + 6q^{8} + 18q^{9} - 6q^{10} - 6q^{14} + 6q^{16} - 6q^{17} + 18q^{18} - 6q^{20} + 6q^{25} - 12q^{27} - 6q^{28} - 12q^{29} + 6q^{32} - 6q^{34} + 6q^{35} + 18q^{36} + 24q^{37} + 24q^{39} - 6q^{40} - 12q^{41} + 18q^{43} - 18q^{45} + 24q^{47} + 6q^{49} + 6q^{50} + 12q^{51} + 36q^{53} - 12q^{54} - 6q^{56} + 12q^{57} - 12q^{58} + 30q^{59} - 36q^{61} - 18q^{63} + 6q^{64} - 12q^{67} - 6q^{68} + 6q^{70} + 6q^{71} + 18q^{72} + 6q^{73} + 24q^{74} + 24q^{78} + 24q^{79} - 6q^{80} + 54q^{81} - 12q^{82} - 24q^{83} + 6q^{85} + 18q^{86} + 24q^{87} + 36q^{89} - 18q^{90} + 24q^{94} + 6q^{98} + O(q^{100}) \)

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
\(3\) 3.24337 1.87256 0.936281 0.351252i \(-0.114244\pi\)
0.936281 + 0.351252i \(0.114244\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 3.24337 1.32410
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 7.51947 2.50649
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 3.24337 0.936281
\(13\) 2.14515 0.594959 0.297479 0.954728i \(-0.403854\pi\)
0.297479 + 0.954728i \(0.403854\pi\)
\(14\) −1.00000 −0.267261
\(15\) −3.24337 −0.837435
\(16\) 1.00000 0.250000
\(17\) 1.43942 0.349111 0.174555 0.984647i \(-0.444151\pi\)
0.174555 + 0.984647i \(0.444151\pi\)
\(18\) 7.51947 1.77236
\(19\) −0.633832 −0.145411 −0.0727056 0.997353i \(-0.523163\pi\)
−0.0727056 + 0.997353i \(0.523163\pi\)
\(20\) −1.00000 −0.223607
\(21\) −3.24337 −0.707762
\(22\) 0 0
\(23\) −5.61769 −1.17137 −0.585684 0.810539i \(-0.699174\pi\)
−0.585684 + 0.810539i \(0.699174\pi\)
\(24\) 3.24337 0.662051
\(25\) 1.00000 0.200000
\(26\) 2.14515 0.420699
\(27\) 14.6583 2.82100
\(28\) −1.00000 −0.188982
\(29\) −1.19605 −0.222101 −0.111050 0.993815i \(-0.535421\pi\)
−0.111050 + 0.993815i \(0.535421\pi\)
\(30\) −3.24337 −0.592156
\(31\) 6.95889 1.24985 0.624927 0.780683i \(-0.285129\pi\)
0.624927 + 0.780683i \(0.285129\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.43942 0.246859
\(35\) 1.00000 0.169031
\(36\) 7.51947 1.25324
\(37\) 4.80395 0.789765 0.394882 0.918732i \(-0.370785\pi\)
0.394882 + 0.918732i \(0.370785\pi\)
\(38\) −0.633832 −0.102821
\(39\) 6.95754 1.11410
\(40\) −1.00000 −0.158114
\(41\) −8.93663 −1.39567 −0.697834 0.716260i \(-0.745853\pi\)
−0.697834 + 0.716260i \(0.745853\pi\)
\(42\) −3.24337 −0.500463
\(43\) 9.70747 1.48038 0.740188 0.672400i \(-0.234736\pi\)
0.740188 + 0.672400i \(0.234736\pi\)
\(44\) 0 0
\(45\) −7.51947 −1.12094
\(46\) −5.61769 −0.828283
\(47\) 8.29874 1.21050 0.605248 0.796037i \(-0.293074\pi\)
0.605248 + 0.796037i \(0.293074\pi\)
\(48\) 3.24337 0.468141
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 4.66858 0.653732
\(52\) 2.14515 0.297479
\(53\) 14.1537 1.94416 0.972079 0.234652i \(-0.0753952\pi\)
0.972079 + 0.234652i \(0.0753952\pi\)
\(54\) 14.6583 1.99475
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −2.05576 −0.272291
\(58\) −1.19605 −0.157049
\(59\) −1.19248 −0.155248 −0.0776238 0.996983i \(-0.524733\pi\)
−0.0776238 + 0.996983i \(0.524733\pi\)
\(60\) −3.24337 −0.418718
\(61\) −4.19441 −0.537039 −0.268520 0.963274i \(-0.586534\pi\)
−0.268520 + 0.963274i \(0.586534\pi\)
\(62\) 6.95889 0.883780
\(63\) −7.51947 −0.947364
\(64\) 1.00000 0.125000
\(65\) −2.14515 −0.266074
\(66\) 0 0
\(67\) −12.0099 −1.46724 −0.733621 0.679559i \(-0.762171\pi\)
−0.733621 + 0.679559i \(0.762171\pi\)
\(68\) 1.43942 0.174555
\(69\) −18.2203 −2.19346
\(70\) 1.00000 0.119523
\(71\) −5.71591 −0.678353 −0.339177 0.940723i \(-0.610148\pi\)
−0.339177 + 0.940723i \(0.610148\pi\)
\(72\) 7.51947 0.886178
\(73\) 6.43981 0.753723 0.376861 0.926270i \(-0.377003\pi\)
0.376861 + 0.926270i \(0.377003\pi\)
\(74\) 4.80395 0.558448
\(75\) 3.24337 0.374512
\(76\) −0.633832 −0.0727056
\(77\) 0 0
\(78\) 6.95754 0.787786
\(79\) −7.69133 −0.865342 −0.432671 0.901552i \(-0.642429\pi\)
−0.432671 + 0.901552i \(0.642429\pi\)
\(80\) −1.00000 −0.111803
\(81\) 24.9840 2.77600
\(82\) −8.93663 −0.986886
\(83\) 11.5685 1.26981 0.634906 0.772590i \(-0.281039\pi\)
0.634906 + 0.772590i \(0.281039\pi\)
\(84\) −3.24337 −0.353881
\(85\) −1.43942 −0.156127
\(86\) 9.70747 1.04678
\(87\) −3.87923 −0.415897
\(88\) 0 0
\(89\) 16.7855 1.77926 0.889629 0.456684i \(-0.150963\pi\)
0.889629 + 0.456684i \(0.150963\pi\)
\(90\) −7.51947 −0.792622
\(91\) −2.14515 −0.224873
\(92\) −5.61769 −0.585684
\(93\) 22.5703 2.34043
\(94\) 8.29874 0.855950
\(95\) 0.633832 0.0650298
\(96\) 3.24337 0.331025
\(97\) −2.96529 −0.301080 −0.150540 0.988604i \(-0.548101\pi\)
−0.150540 + 0.988604i \(0.548101\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −15.5584 −1.54812 −0.774060 0.633113i \(-0.781777\pi\)
−0.774060 + 0.633113i \(0.781777\pi\)
\(102\) 4.66858 0.462258
\(103\) 0.404673 0.0398736 0.0199368 0.999801i \(-0.493653\pi\)
0.0199368 + 0.999801i \(0.493653\pi\)
\(104\) 2.14515 0.210350
\(105\) 3.24337 0.316521
\(106\) 14.1537 1.37473
\(107\) −8.17962 −0.790754 −0.395377 0.918519i \(-0.629386\pi\)
−0.395377 + 0.918519i \(0.629386\pi\)
\(108\) 14.6583 1.41050
\(109\) −1.34828 −0.129142 −0.0645709 0.997913i \(-0.520568\pi\)
−0.0645709 + 0.997913i \(0.520568\pi\)
\(110\) 0 0
\(111\) 15.5810 1.47888
\(112\) −1.00000 −0.0944911
\(113\) 12.6845 1.19326 0.596630 0.802516i \(-0.296506\pi\)
0.596630 + 0.802516i \(0.296506\pi\)
\(114\) −2.05576 −0.192539
\(115\) 5.61769 0.523852
\(116\) −1.19605 −0.111050
\(117\) 16.1304 1.49126
\(118\) −1.19248 −0.109777
\(119\) −1.43942 −0.131952
\(120\) −3.24337 −0.296078
\(121\) 0 0
\(122\) −4.19441 −0.379744
\(123\) −28.9848 −2.61347
\(124\) 6.95889 0.624927
\(125\) −1.00000 −0.0894427
\(126\) −7.51947 −0.669887
\(127\) 12.6218 1.12000 0.560001 0.828492i \(-0.310801\pi\)
0.560001 + 0.828492i \(0.310801\pi\)
\(128\) 1.00000 0.0883883
\(129\) 31.4850 2.77210
\(130\) −2.14515 −0.188143
\(131\) 4.14932 0.362528 0.181264 0.983434i \(-0.441981\pi\)
0.181264 + 0.983434i \(0.441981\pi\)
\(132\) 0 0
\(133\) 0.633832 0.0549602
\(134\) −12.0099 −1.03750
\(135\) −14.6583 −1.26159
\(136\) 1.43942 0.123429
\(137\) 20.8283 1.77948 0.889742 0.456463i \(-0.150884\pi\)
0.889742 + 0.456463i \(0.150884\pi\)
\(138\) −18.2203 −1.55101
\(139\) −18.4862 −1.56798 −0.783988 0.620776i \(-0.786818\pi\)
−0.783988 + 0.620776i \(0.786818\pi\)
\(140\) 1.00000 0.0845154
\(141\) 26.9159 2.26673
\(142\) −5.71591 −0.479668
\(143\) 0 0
\(144\) 7.51947 0.626622
\(145\) 1.19605 0.0993264
\(146\) 6.43981 0.532962
\(147\) 3.24337 0.267509
\(148\) 4.80395 0.394882
\(149\) −15.4372 −1.26467 −0.632333 0.774697i \(-0.717902\pi\)
−0.632333 + 0.774697i \(0.717902\pi\)
\(150\) 3.24337 0.264820
\(151\) −11.4606 −0.932648 −0.466324 0.884614i \(-0.654422\pi\)
−0.466324 + 0.884614i \(0.654422\pi\)
\(152\) −0.633832 −0.0514106
\(153\) 10.8237 0.875043
\(154\) 0 0
\(155\) −6.95889 −0.558951
\(156\) 6.95754 0.557049
\(157\) −10.2685 −0.819514 −0.409757 0.912195i \(-0.634387\pi\)
−0.409757 + 0.912195i \(0.634387\pi\)
\(158\) −7.69133 −0.611889
\(159\) 45.9057 3.64056
\(160\) −1.00000 −0.0790569
\(161\) 5.61769 0.442736
\(162\) 24.9840 1.96293
\(163\) −2.90989 −0.227920 −0.113960 0.993485i \(-0.536354\pi\)
−0.113960 + 0.993485i \(0.536354\pi\)
\(164\) −8.93663 −0.697834
\(165\) 0 0
\(166\) 11.5685 0.897892
\(167\) 11.7960 0.912798 0.456399 0.889775i \(-0.349139\pi\)
0.456399 + 0.889775i \(0.349139\pi\)
\(168\) −3.24337 −0.250232
\(169\) −8.39831 −0.646024
\(170\) −1.43942 −0.110399
\(171\) −4.76608 −0.364472
\(172\) 9.70747 0.740188
\(173\) 0.146508 0.0111388 0.00556939 0.999984i \(-0.498227\pi\)
0.00556939 + 0.999984i \(0.498227\pi\)
\(174\) −3.87923 −0.294084
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −3.86766 −0.290711
\(178\) 16.7855 1.25813
\(179\) 10.2987 0.769764 0.384882 0.922966i \(-0.374242\pi\)
0.384882 + 0.922966i \(0.374242\pi\)
\(180\) −7.51947 −0.560468
\(181\) −19.8342 −1.47426 −0.737131 0.675750i \(-0.763820\pi\)
−0.737131 + 0.675750i \(0.763820\pi\)
\(182\) −2.14515 −0.159009
\(183\) −13.6040 −1.00564
\(184\) −5.61769 −0.414141
\(185\) −4.80395 −0.353194
\(186\) 22.5703 1.65493
\(187\) 0 0
\(188\) 8.29874 0.605248
\(189\) −14.6583 −1.06624
\(190\) 0.633832 0.0459830
\(191\) 5.95268 0.430720 0.215360 0.976535i \(-0.430907\pi\)
0.215360 + 0.976535i \(0.430907\pi\)
\(192\) 3.24337 0.234070
\(193\) −14.9671 −1.07736 −0.538678 0.842512i \(-0.681076\pi\)
−0.538678 + 0.842512i \(0.681076\pi\)
\(194\) −2.96529 −0.212896
\(195\) −6.95754 −0.498240
\(196\) 1.00000 0.0714286
\(197\) −19.1531 −1.36460 −0.682301 0.731071i \(-0.739021\pi\)
−0.682301 + 0.731071i \(0.739021\pi\)
\(198\) 0 0
\(199\) −0.268053 −0.0190018 −0.00950089 0.999955i \(-0.503024\pi\)
−0.00950089 + 0.999955i \(0.503024\pi\)
\(200\) 1.00000 0.0707107
\(201\) −38.9525 −2.74750
\(202\) −15.5584 −1.09469
\(203\) 1.19605 0.0839461
\(204\) 4.66858 0.326866
\(205\) 8.93663 0.624162
\(206\) 0.404673 0.0281949
\(207\) −42.2420 −2.93602
\(208\) 2.14515 0.148740
\(209\) 0 0
\(210\) 3.24337 0.223814
\(211\) −9.35228 −0.643837 −0.321919 0.946767i \(-0.604328\pi\)
−0.321919 + 0.946767i \(0.604328\pi\)
\(212\) 14.1537 0.972079
\(213\) −18.5388 −1.27026
\(214\) −8.17962 −0.559147
\(215\) −9.70747 −0.662044
\(216\) 14.6583 0.997373
\(217\) −6.95889 −0.472400
\(218\) −1.34828 −0.0913171
\(219\) 20.8867 1.41139
\(220\) 0 0
\(221\) 3.08778 0.207707
\(222\) 15.5810 1.04573
\(223\) −13.8365 −0.926559 −0.463280 0.886212i \(-0.653327\pi\)
−0.463280 + 0.886212i \(0.653327\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 7.51947 0.501298
\(226\) 12.6845 0.843763
\(227\) 25.8686 1.71696 0.858481 0.512846i \(-0.171409\pi\)
0.858481 + 0.512846i \(0.171409\pi\)
\(228\) −2.05576 −0.136146
\(229\) 11.1800 0.738796 0.369398 0.929271i \(-0.379564\pi\)
0.369398 + 0.929271i \(0.379564\pi\)
\(230\) 5.61769 0.370419
\(231\) 0 0
\(232\) −1.19605 −0.0785244
\(233\) 2.87302 0.188218 0.0941088 0.995562i \(-0.470000\pi\)
0.0941088 + 0.995562i \(0.470000\pi\)
\(234\) 16.1304 1.05448
\(235\) −8.29874 −0.541350
\(236\) −1.19248 −0.0776238
\(237\) −24.9458 −1.62041
\(238\) −1.43942 −0.0933038
\(239\) 8.22791 0.532219 0.266110 0.963943i \(-0.414262\pi\)
0.266110 + 0.963943i \(0.414262\pi\)
\(240\) −3.24337 −0.209359
\(241\) −6.54207 −0.421412 −0.210706 0.977550i \(-0.567576\pi\)
−0.210706 + 0.977550i \(0.567576\pi\)
\(242\) 0 0
\(243\) 37.0575 2.37724
\(244\) −4.19441 −0.268520
\(245\) −1.00000 −0.0638877
\(246\) −28.9848 −1.84801
\(247\) −1.35967 −0.0865136
\(248\) 6.95889 0.441890
\(249\) 37.5211 2.37780
\(250\) −1.00000 −0.0632456
\(251\) −11.0809 −0.699422 −0.349711 0.936858i \(-0.613720\pi\)
−0.349711 + 0.936858i \(0.613720\pi\)
\(252\) −7.51947 −0.473682
\(253\) 0 0
\(254\) 12.6218 0.791960
\(255\) −4.66858 −0.292358
\(256\) 1.00000 0.0625000
\(257\) 18.1347 1.13121 0.565606 0.824676i \(-0.308642\pi\)
0.565606 + 0.824676i \(0.308642\pi\)
\(258\) 31.4850 1.96017
\(259\) −4.80395 −0.298503
\(260\) −2.14515 −0.133037
\(261\) −8.99365 −0.556693
\(262\) 4.14932 0.256346
\(263\) −17.9268 −1.10541 −0.552706 0.833376i \(-0.686405\pi\)
−0.552706 + 0.833376i \(0.686405\pi\)
\(264\) 0 0
\(265\) −14.1537 −0.869454
\(266\) 0.633832 0.0388628
\(267\) 54.4416 3.33177
\(268\) −12.0099 −0.733621
\(269\) 6.06753 0.369944 0.184972 0.982744i \(-0.440781\pi\)
0.184972 + 0.982744i \(0.440781\pi\)
\(270\) −14.6583 −0.892077
\(271\) 21.2185 1.28893 0.644466 0.764633i \(-0.277080\pi\)
0.644466 + 0.764633i \(0.277080\pi\)
\(272\) 1.43942 0.0872777
\(273\) −6.95754 −0.421089
\(274\) 20.8283 1.25829
\(275\) 0 0
\(276\) −18.2203 −1.09673
\(277\) −5.01421 −0.301275 −0.150637 0.988589i \(-0.548133\pi\)
−0.150637 + 0.988589i \(0.548133\pi\)
\(278\) −18.4862 −1.10873
\(279\) 52.3272 3.13274
\(280\) 1.00000 0.0597614
\(281\) 4.76751 0.284406 0.142203 0.989838i \(-0.454581\pi\)
0.142203 + 0.989838i \(0.454581\pi\)
\(282\) 26.9159 1.60282
\(283\) 9.65019 0.573644 0.286822 0.957984i \(-0.407401\pi\)
0.286822 + 0.957984i \(0.407401\pi\)
\(284\) −5.71591 −0.339177
\(285\) 2.05576 0.121772
\(286\) 0 0
\(287\) 8.93663 0.527513
\(288\) 7.51947 0.443089
\(289\) −14.9281 −0.878122
\(290\) 1.19605 0.0702344
\(291\) −9.61756 −0.563791
\(292\) 6.43981 0.376861
\(293\) −28.5011 −1.66505 −0.832526 0.553986i \(-0.813106\pi\)
−0.832526 + 0.553986i \(0.813106\pi\)
\(294\) 3.24337 0.189157
\(295\) 1.19248 0.0694288
\(296\) 4.80395 0.279224
\(297\) 0 0
\(298\) −15.4372 −0.894253
\(299\) −12.0508 −0.696916
\(300\) 3.24337 0.187256
\(301\) −9.70747 −0.559530
\(302\) −11.4606 −0.659482
\(303\) −50.4617 −2.89895
\(304\) −0.633832 −0.0363528
\(305\) 4.19441 0.240171
\(306\) 10.8237 0.618749
\(307\) −0.552191 −0.0315152 −0.0157576 0.999876i \(-0.505016\pi\)
−0.0157576 + 0.999876i \(0.505016\pi\)
\(308\) 0 0
\(309\) 1.31251 0.0746659
\(310\) −6.95889 −0.395238
\(311\) −30.4747 −1.72806 −0.864032 0.503437i \(-0.832069\pi\)
−0.864032 + 0.503437i \(0.832069\pi\)
\(312\) 6.95754 0.393893
\(313\) −33.4678 −1.89171 −0.945856 0.324587i \(-0.894775\pi\)
−0.945856 + 0.324587i \(0.894775\pi\)
\(314\) −10.2685 −0.579484
\(315\) 7.51947 0.423674
\(316\) −7.69133 −0.432671
\(317\) −20.2125 −1.13525 −0.567623 0.823288i \(-0.692137\pi\)
−0.567623 + 0.823288i \(0.692137\pi\)
\(318\) 45.9057 2.57426
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −26.5296 −1.48074
\(322\) 5.61769 0.313061
\(323\) −0.912352 −0.0507646
\(324\) 24.9840 1.38800
\(325\) 2.14515 0.118992
\(326\) −2.90989 −0.161164
\(327\) −4.37298 −0.241826
\(328\) −8.93663 −0.493443
\(329\) −8.29874 −0.457524
\(330\) 0 0
\(331\) −19.3667 −1.06449 −0.532244 0.846591i \(-0.678651\pi\)
−0.532244 + 0.846591i \(0.678651\pi\)
\(332\) 11.5685 0.634906
\(333\) 36.1232 1.97954
\(334\) 11.7960 0.645446
\(335\) 12.0099 0.656170
\(336\) −3.24337 −0.176941
\(337\) 0.0549787 0.00299488 0.00149744 0.999999i \(-0.499523\pi\)
0.00149744 + 0.999999i \(0.499523\pi\)
\(338\) −8.39831 −0.456808
\(339\) 41.1407 2.23446
\(340\) −1.43942 −0.0780636
\(341\) 0 0
\(342\) −4.76608 −0.257720
\(343\) −1.00000 −0.0539949
\(344\) 9.70747 0.523392
\(345\) 18.2203 0.980945
\(346\) 0.146508 0.00787631
\(347\) 22.9466 1.23184 0.615920 0.787809i \(-0.288784\pi\)
0.615920 + 0.787809i \(0.288784\pi\)
\(348\) −3.87923 −0.207949
\(349\) −21.7380 −1.16361 −0.581804 0.813329i \(-0.697653\pi\)
−0.581804 + 0.813329i \(0.697653\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 31.4444 1.67838
\(352\) 0 0
\(353\) −10.3596 −0.551384 −0.275692 0.961246i \(-0.588907\pi\)
−0.275692 + 0.961246i \(0.588907\pi\)
\(354\) −3.86766 −0.205564
\(355\) 5.71591 0.303369
\(356\) 16.7855 0.889629
\(357\) −4.66858 −0.247087
\(358\) 10.2987 0.544305
\(359\) 33.6387 1.77538 0.887690 0.460442i \(-0.152309\pi\)
0.887690 + 0.460442i \(0.152309\pi\)
\(360\) −7.51947 −0.396311
\(361\) −18.5983 −0.978856
\(362\) −19.8342 −1.04246
\(363\) 0 0
\(364\) −2.14515 −0.112437
\(365\) −6.43981 −0.337075
\(366\) −13.6040 −0.711095
\(367\) −23.1345 −1.20761 −0.603806 0.797131i \(-0.706350\pi\)
−0.603806 + 0.797131i \(0.706350\pi\)
\(368\) −5.61769 −0.292842
\(369\) −67.1987 −3.49823
\(370\) −4.80395 −0.249746
\(371\) −14.1537 −0.734823
\(372\) 22.5703 1.17021
\(373\) −20.5321 −1.06311 −0.531555 0.847024i \(-0.678392\pi\)
−0.531555 + 0.847024i \(0.678392\pi\)
\(374\) 0 0
\(375\) −3.24337 −0.167487
\(376\) 8.29874 0.427975
\(377\) −2.56571 −0.132141
\(378\) −14.6583 −0.753943
\(379\) 21.0362 1.08056 0.540278 0.841487i \(-0.318319\pi\)
0.540278 + 0.841487i \(0.318319\pi\)
\(380\) 0.633832 0.0325149
\(381\) 40.9371 2.09727
\(382\) 5.95268 0.304565
\(383\) 11.3755 0.581263 0.290632 0.956835i \(-0.406135\pi\)
0.290632 + 0.956835i \(0.406135\pi\)
\(384\) 3.24337 0.165513
\(385\) 0 0
\(386\) −14.9671 −0.761805
\(387\) 72.9951 3.71055
\(388\) −2.96529 −0.150540
\(389\) 25.9812 1.31730 0.658649 0.752450i \(-0.271128\pi\)
0.658649 + 0.752450i \(0.271128\pi\)
\(390\) −6.95754 −0.352309
\(391\) −8.08622 −0.408938
\(392\) 1.00000 0.0505076
\(393\) 13.4578 0.678856
\(394\) −19.1531 −0.964920
\(395\) 7.69133 0.386993
\(396\) 0 0
\(397\) 7.16936 0.359820 0.179910 0.983683i \(-0.442419\pi\)
0.179910 + 0.983683i \(0.442419\pi\)
\(398\) −0.268053 −0.0134363
\(399\) 2.05576 0.102916
\(400\) 1.00000 0.0500000
\(401\) −26.2702 −1.31187 −0.655937 0.754816i \(-0.727726\pi\)
−0.655937 + 0.754816i \(0.727726\pi\)
\(402\) −38.9525 −1.94278
\(403\) 14.9279 0.743611
\(404\) −15.5584 −0.774060
\(405\) −24.9840 −1.24147
\(406\) 1.19605 0.0593589
\(407\) 0 0
\(408\) 4.66858 0.231129
\(409\) 36.5385 1.80671 0.903356 0.428891i \(-0.141096\pi\)
0.903356 + 0.428891i \(0.141096\pi\)
\(410\) 8.93663 0.441349
\(411\) 67.5541 3.33220
\(412\) 0.404673 0.0199368
\(413\) 1.19248 0.0586781
\(414\) −42.2420 −2.07608
\(415\) −11.5685 −0.567877
\(416\) 2.14515 0.105175
\(417\) −59.9575 −2.93613
\(418\) 0 0
\(419\) −17.9411 −0.876480 −0.438240 0.898858i \(-0.644398\pi\)
−0.438240 + 0.898858i \(0.644398\pi\)
\(420\) 3.24337 0.158260
\(421\) 36.8460 1.79576 0.897882 0.440237i \(-0.145106\pi\)
0.897882 + 0.440237i \(0.145106\pi\)
\(422\) −9.35228 −0.455262
\(423\) 62.4021 3.03409
\(424\) 14.1537 0.687364
\(425\) 1.43942 0.0698222
\(426\) −18.5388 −0.898208
\(427\) 4.19441 0.202982
\(428\) −8.17962 −0.395377
\(429\) 0 0
\(430\) −9.70747 −0.468136
\(431\) 31.8543 1.53437 0.767185 0.641426i \(-0.221657\pi\)
0.767185 + 0.641426i \(0.221657\pi\)
\(432\) 14.6583 0.705249
\(433\) 25.0278 1.20276 0.601380 0.798963i \(-0.294618\pi\)
0.601380 + 0.798963i \(0.294618\pi\)
\(434\) −6.95889 −0.334037
\(435\) 3.87923 0.185995
\(436\) −1.34828 −0.0645709
\(437\) 3.56067 0.170330
\(438\) 20.8867 0.998005
\(439\) −40.3537 −1.92598 −0.962988 0.269544i \(-0.913127\pi\)
−0.962988 + 0.269544i \(0.913127\pi\)
\(440\) 0 0
\(441\) 7.51947 0.358070
\(442\) 3.08778 0.146871
\(443\) −0.519900 −0.0247012 −0.0123506 0.999924i \(-0.503931\pi\)
−0.0123506 + 0.999924i \(0.503931\pi\)
\(444\) 15.5810 0.739442
\(445\) −16.7855 −0.795708
\(446\) −13.8365 −0.655176
\(447\) −50.0686 −2.36816
\(448\) −1.00000 −0.0472456
\(449\) 23.0974 1.09003 0.545016 0.838426i \(-0.316524\pi\)
0.545016 + 0.838426i \(0.316524\pi\)
\(450\) 7.51947 0.354471
\(451\) 0 0
\(452\) 12.6845 0.596630
\(453\) −37.1709 −1.74644
\(454\) 25.8686 1.21407
\(455\) 2.14515 0.100566
\(456\) −2.05576 −0.0962696
\(457\) −20.3786 −0.953273 −0.476637 0.879100i \(-0.658144\pi\)
−0.476637 + 0.879100i \(0.658144\pi\)
\(458\) 11.1800 0.522407
\(459\) 21.0995 0.984840
\(460\) 5.61769 0.261926
\(461\) −12.1816 −0.567354 −0.283677 0.958920i \(-0.591554\pi\)
−0.283677 + 0.958920i \(0.591554\pi\)
\(462\) 0 0
\(463\) −11.4172 −0.530601 −0.265301 0.964166i \(-0.585471\pi\)
−0.265301 + 0.964166i \(0.585471\pi\)
\(464\) −1.19605 −0.0555252
\(465\) −22.5703 −1.04667
\(466\) 2.87302 0.133090
\(467\) 22.7698 1.05366 0.526831 0.849970i \(-0.323380\pi\)
0.526831 + 0.849970i \(0.323380\pi\)
\(468\) 16.1304 0.745629
\(469\) 12.0099 0.554565
\(470\) −8.29874 −0.382792
\(471\) −33.3045 −1.53459
\(472\) −1.19248 −0.0548883
\(473\) 0 0
\(474\) −24.9458 −1.14580
\(475\) −0.633832 −0.0290822
\(476\) −1.43942 −0.0659758
\(477\) 106.428 4.87301
\(478\) 8.22791 0.376336
\(479\) −15.9269 −0.727719 −0.363859 0.931454i \(-0.618541\pi\)
−0.363859 + 0.931454i \(0.618541\pi\)
\(480\) −3.24337 −0.148039
\(481\) 10.3052 0.469878
\(482\) −6.54207 −0.297983
\(483\) 18.2203 0.829050
\(484\) 0 0
\(485\) 2.96529 0.134647
\(486\) 37.0575 1.68096
\(487\) 4.57953 0.207518 0.103759 0.994602i \(-0.466913\pi\)
0.103759 + 0.994602i \(0.466913\pi\)
\(488\) −4.19441 −0.189872
\(489\) −9.43784 −0.426794
\(490\) −1.00000 −0.0451754
\(491\) −28.0073 −1.26395 −0.631975 0.774989i \(-0.717756\pi\)
−0.631975 + 0.774989i \(0.717756\pi\)
\(492\) −28.9848 −1.30674
\(493\) −1.72162 −0.0775378
\(494\) −1.35967 −0.0611744
\(495\) 0 0
\(496\) 6.95889 0.312463
\(497\) 5.71591 0.256393
\(498\) 37.5211 1.68136
\(499\) 27.8295 1.24582 0.622910 0.782294i \(-0.285950\pi\)
0.622910 + 0.782294i \(0.285950\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 38.2587 1.70927
\(502\) −11.0809 −0.494566
\(503\) −30.4709 −1.35863 −0.679316 0.733846i \(-0.737723\pi\)
−0.679316 + 0.733846i \(0.737723\pi\)
\(504\) −7.51947 −0.334944
\(505\) 15.5584 0.692340
\(506\) 0 0
\(507\) −27.2389 −1.20972
\(508\) 12.6218 0.560001
\(509\) −25.7729 −1.14236 −0.571181 0.820824i \(-0.693515\pi\)
−0.571181 + 0.820824i \(0.693515\pi\)
\(510\) −4.66858 −0.206728
\(511\) −6.43981 −0.284880
\(512\) 1.00000 0.0441942
\(513\) −9.29092 −0.410204
\(514\) 18.1347 0.799888
\(515\) −0.404673 −0.0178320
\(516\) 31.4850 1.38605
\(517\) 0 0
\(518\) −4.80395 −0.211074
\(519\) 0.475179 0.0208581
\(520\) −2.14515 −0.0940713
\(521\) 3.08204 0.135027 0.0675133 0.997718i \(-0.478493\pi\)
0.0675133 + 0.997718i \(0.478493\pi\)
\(522\) −8.99365 −0.393641
\(523\) −10.0632 −0.440033 −0.220016 0.975496i \(-0.570611\pi\)
−0.220016 + 0.975496i \(0.570611\pi\)
\(524\) 4.14932 0.181264
\(525\) −3.24337 −0.141552
\(526\) −17.9268 −0.781644
\(527\) 10.0168 0.436338
\(528\) 0 0
\(529\) 8.55841 0.372105
\(530\) −14.1537 −0.614797
\(531\) −8.96681 −0.389126
\(532\) 0.633832 0.0274801
\(533\) −19.1705 −0.830365
\(534\) 54.4416 2.35592
\(535\) 8.17962 0.353636
\(536\) −12.0099 −0.518748
\(537\) 33.4027 1.44143
\(538\) 6.06753 0.261590
\(539\) 0 0
\(540\) −14.6583 −0.630794
\(541\) 21.7416 0.934744 0.467372 0.884061i \(-0.345201\pi\)
0.467372 + 0.884061i \(0.345201\pi\)
\(542\) 21.2185 0.911413
\(543\) −64.3296 −2.76065
\(544\) 1.43942 0.0617147
\(545\) 1.34828 0.0577540
\(546\) −6.95754 −0.297755
\(547\) 33.5075 1.43268 0.716338 0.697753i \(-0.245817\pi\)
0.716338 + 0.697753i \(0.245817\pi\)
\(548\) 20.8283 0.889742
\(549\) −31.5397 −1.34608
\(550\) 0 0
\(551\) 0.758094 0.0322959
\(552\) −18.2203 −0.775506
\(553\) 7.69133 0.327069
\(554\) −5.01421 −0.213034
\(555\) −15.5810 −0.661377
\(556\) −18.4862 −0.783988
\(557\) −27.9709 −1.18517 −0.592583 0.805509i \(-0.701892\pi\)
−0.592583 + 0.805509i \(0.701892\pi\)
\(558\) 52.3272 2.21519
\(559\) 20.8240 0.880763
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 4.76751 0.201105
\(563\) 19.1724 0.808019 0.404009 0.914755i \(-0.367616\pi\)
0.404009 + 0.914755i \(0.367616\pi\)
\(564\) 26.9159 1.13336
\(565\) −12.6845 −0.533642
\(566\) 9.65019 0.405628
\(567\) −24.9840 −1.04923
\(568\) −5.71591 −0.239834
\(569\) 39.8416 1.67025 0.835124 0.550062i \(-0.185396\pi\)
0.835124 + 0.550062i \(0.185396\pi\)
\(570\) 2.05576 0.0861061
\(571\) 47.3425 1.98122 0.990610 0.136718i \(-0.0436555\pi\)
0.990610 + 0.136718i \(0.0436555\pi\)
\(572\) 0 0
\(573\) 19.3067 0.806551
\(574\) 8.93663 0.373008
\(575\) −5.61769 −0.234274
\(576\) 7.51947 0.313311
\(577\) −11.3882 −0.474097 −0.237048 0.971498i \(-0.576180\pi\)
−0.237048 + 0.971498i \(0.576180\pi\)
\(578\) −14.9281 −0.620926
\(579\) −48.5439 −2.01741
\(580\) 1.19605 0.0496632
\(581\) −11.5685 −0.479944
\(582\) −9.61756 −0.398661
\(583\) 0 0
\(584\) 6.43981 0.266481
\(585\) −16.1304 −0.666911
\(586\) −28.5011 −1.17737
\(587\) −22.3027 −0.920529 −0.460265 0.887782i \(-0.652245\pi\)
−0.460265 + 0.887782i \(0.652245\pi\)
\(588\) 3.24337 0.133754
\(589\) −4.41077 −0.181743
\(590\) 1.19248 0.0490936
\(591\) −62.1207 −2.55530
\(592\) 4.80395 0.197441
\(593\) −14.0840 −0.578361 −0.289180 0.957275i \(-0.593383\pi\)
−0.289180 + 0.957275i \(0.593383\pi\)
\(594\) 0 0
\(595\) 1.43942 0.0590105
\(596\) −15.4372 −0.632333
\(597\) −0.869397 −0.0355820
\(598\) −12.0508 −0.492794
\(599\) −22.5478 −0.921280 −0.460640 0.887587i \(-0.652380\pi\)
−0.460640 + 0.887587i \(0.652380\pi\)
\(600\) 3.24337 0.132410
\(601\) 38.4433 1.56814 0.784068 0.620675i \(-0.213141\pi\)
0.784068 + 0.620675i \(0.213141\pi\)
\(602\) −9.70747 −0.395647
\(603\) −90.3080 −3.67762
\(604\) −11.4606 −0.466324
\(605\) 0 0
\(606\) −50.4617 −2.04987
\(607\) 24.2061 0.982497 0.491248 0.871020i \(-0.336541\pi\)
0.491248 + 0.871020i \(0.336541\pi\)
\(608\) −0.633832 −0.0257053
\(609\) 3.87923 0.157194
\(610\) 4.19441 0.169827
\(611\) 17.8021 0.720195
\(612\) 10.8237 0.437521
\(613\) −26.3266 −1.06332 −0.531662 0.846957i \(-0.678432\pi\)
−0.531662 + 0.846957i \(0.678432\pi\)
\(614\) −0.552191 −0.0222846
\(615\) 28.9848 1.16878
\(616\) 0 0
\(617\) −35.7839 −1.44061 −0.720303 0.693660i \(-0.755997\pi\)
−0.720303 + 0.693660i \(0.755997\pi\)
\(618\) 1.31251 0.0527968
\(619\) −8.17683 −0.328654 −0.164327 0.986406i \(-0.552545\pi\)
−0.164327 + 0.986406i \(0.552545\pi\)
\(620\) −6.95889 −0.279476
\(621\) −82.3459 −3.30443
\(622\) −30.4747 −1.22193
\(623\) −16.7855 −0.672496
\(624\) 6.95754 0.278524
\(625\) 1.00000 0.0400000
\(626\) −33.4678 −1.33764
\(627\) 0 0
\(628\) −10.2685 −0.409757
\(629\) 6.91491 0.275716
\(630\) 7.51947 0.299583
\(631\) 20.1283 0.801293 0.400647 0.916233i \(-0.368785\pi\)
0.400647 + 0.916233i \(0.368785\pi\)
\(632\) −7.69133 −0.305945
\(633\) −30.3329 −1.20563
\(634\) −20.2125 −0.802741
\(635\) −12.6218 −0.500880
\(636\) 45.9057 1.82028
\(637\) 2.14515 0.0849941
\(638\) 0 0
\(639\) −42.9806 −1.70029
\(640\) −1.00000 −0.0395285
\(641\) 8.58218 0.338976 0.169488 0.985532i \(-0.445789\pi\)
0.169488 + 0.985532i \(0.445789\pi\)
\(642\) −26.5296 −1.04704
\(643\) −12.5152 −0.493550 −0.246775 0.969073i \(-0.579371\pi\)
−0.246775 + 0.969073i \(0.579371\pi\)
\(644\) 5.61769 0.221368
\(645\) −31.4850 −1.23972
\(646\) −0.912352 −0.0358960
\(647\) 16.7526 0.658611 0.329305 0.944223i \(-0.393185\pi\)
0.329305 + 0.944223i \(0.393185\pi\)
\(648\) 24.9840 0.981464
\(649\) 0 0
\(650\) 2.14515 0.0841399
\(651\) −22.5703 −0.884599
\(652\) −2.90989 −0.113960
\(653\) −27.3451 −1.07010 −0.535048 0.844822i \(-0.679707\pi\)
−0.535048 + 0.844822i \(0.679707\pi\)
\(654\) −4.37298 −0.170997
\(655\) −4.14932 −0.162127
\(656\) −8.93663 −0.348917
\(657\) 48.4239 1.88920
\(658\) −8.29874 −0.323519
\(659\) −12.7483 −0.496604 −0.248302 0.968683i \(-0.579873\pi\)
−0.248302 + 0.968683i \(0.579873\pi\)
\(660\) 0 0
\(661\) −0.892400 −0.0347103 −0.0173552 0.999849i \(-0.505525\pi\)
−0.0173552 + 0.999849i \(0.505525\pi\)
\(662\) −19.3667 −0.752707
\(663\) 10.0148 0.388944
\(664\) 11.5685 0.448946
\(665\) −0.633832 −0.0245790
\(666\) 36.1232 1.39974
\(667\) 6.71903 0.260162
\(668\) 11.7960 0.456399
\(669\) −44.8769 −1.73504
\(670\) 12.0099 0.463982
\(671\) 0 0
\(672\) −3.24337 −0.125116
\(673\) −16.8765 −0.650541 −0.325271 0.945621i \(-0.605455\pi\)
−0.325271 + 0.945621i \(0.605455\pi\)
\(674\) 0.0549787 0.00211770
\(675\) 14.6583 0.564199
\(676\) −8.39831 −0.323012
\(677\) 19.6250 0.754248 0.377124 0.926163i \(-0.376913\pi\)
0.377124 + 0.926163i \(0.376913\pi\)
\(678\) 41.1407 1.58000
\(679\) 2.96529 0.113798
\(680\) −1.43942 −0.0551993
\(681\) 83.9016 3.21512
\(682\) 0 0
\(683\) −14.3107 −0.547585 −0.273792 0.961789i \(-0.588278\pi\)
−0.273792 + 0.961789i \(0.588278\pi\)
\(684\) −4.76608 −0.182236
\(685\) −20.8283 −0.795810
\(686\) −1.00000 −0.0381802
\(687\) 36.2609 1.38344
\(688\) 9.70747 0.370094
\(689\) 30.3619 1.15669
\(690\) 18.2203 0.693633
\(691\) −6.52873 −0.248365 −0.124182 0.992259i \(-0.539631\pi\)
−0.124182 + 0.992259i \(0.539631\pi\)
\(692\) 0.146508 0.00556939
\(693\) 0 0
\(694\) 22.9466 0.871042
\(695\) 18.4862 0.701220
\(696\) −3.87923 −0.147042
\(697\) −12.8636 −0.487243
\(698\) −21.7380 −0.822795
\(699\) 9.31826 0.352449
\(700\) −1.00000 −0.0377964
\(701\) −0.193487 −0.00730791 −0.00365395 0.999993i \(-0.501163\pi\)
−0.00365395 + 0.999993i \(0.501163\pi\)
\(702\) 31.4444 1.18679
\(703\) −3.04490 −0.114841
\(704\) 0 0
\(705\) −26.9159 −1.01371
\(706\) −10.3596 −0.389888
\(707\) 15.5584 0.585134
\(708\) −3.86766 −0.145355
\(709\) −19.2772 −0.723970 −0.361985 0.932184i \(-0.617901\pi\)
−0.361985 + 0.932184i \(0.617901\pi\)
\(710\) 5.71591 0.214514
\(711\) −57.8347 −2.16897
\(712\) 16.7855 0.629063
\(713\) −39.0929 −1.46404
\(714\) −4.66858 −0.174717
\(715\) 0 0
\(716\) 10.2987 0.384882
\(717\) 26.6862 0.996613
\(718\) 33.6387 1.25538
\(719\) −41.7245 −1.55606 −0.778030 0.628227i \(-0.783781\pi\)
−0.778030 + 0.628227i \(0.783781\pi\)
\(720\) −7.51947 −0.280234
\(721\) −0.404673 −0.0150708
\(722\) −18.5983 −0.692155
\(723\) −21.2184 −0.789120
\(724\) −19.8342 −0.737131
\(725\) −1.19605 −0.0444201
\(726\) 0 0
\(727\) 2.81130 0.104265 0.0521326 0.998640i \(-0.483398\pi\)
0.0521326 + 0.998640i \(0.483398\pi\)
\(728\) −2.14515 −0.0795047
\(729\) 45.2392 1.67553
\(730\) −6.43981 −0.238348
\(731\) 13.9731 0.516816
\(732\) −13.6040 −0.502820
\(733\) 50.4642 1.86394 0.931969 0.362539i \(-0.118090\pi\)
0.931969 + 0.362539i \(0.118090\pi\)
\(734\) −23.1345 −0.853911
\(735\) −3.24337 −0.119634
\(736\) −5.61769 −0.207071
\(737\) 0 0
\(738\) −67.1987 −2.47362
\(739\) −22.9651 −0.844783 −0.422392 0.906413i \(-0.638809\pi\)
−0.422392 + 0.906413i \(0.638809\pi\)
\(740\) −4.80395 −0.176597
\(741\) −4.40991 −0.162002
\(742\) −14.1537 −0.519598
\(743\) −45.8455 −1.68191 −0.840954 0.541107i \(-0.818006\pi\)
−0.840954 + 0.541107i \(0.818006\pi\)
\(744\) 22.5703 0.827466
\(745\) 15.4372 0.565575
\(746\) −20.5321 −0.751733
\(747\) 86.9892 3.18277
\(748\) 0 0
\(749\) 8.17962 0.298877
\(750\) −3.24337 −0.118431
\(751\) −3.84194 −0.140194 −0.0700972 0.997540i \(-0.522331\pi\)
−0.0700972 + 0.997540i \(0.522331\pi\)
\(752\) 8.29874 0.302624
\(753\) −35.9396 −1.30971
\(754\) −2.56571 −0.0934376
\(755\) 11.4606 0.417093
\(756\) −14.6583 −0.533118
\(757\) −26.9122 −0.978140 −0.489070 0.872245i \(-0.662664\pi\)
−0.489070 + 0.872245i \(0.662664\pi\)
\(758\) 21.0362 0.764068
\(759\) 0 0
\(760\) 0.633832 0.0229915
\(761\) 0.765350 0.0277439 0.0138720 0.999904i \(-0.495584\pi\)
0.0138720 + 0.999904i \(0.495584\pi\)
\(762\) 40.9371 1.48300
\(763\) 1.34828 0.0488110
\(764\) 5.95268 0.215360
\(765\) −10.8237 −0.391331
\(766\) 11.3755 0.411015
\(767\) −2.55805 −0.0923659
\(768\) 3.24337 0.117035
\(769\) −42.2441 −1.52336 −0.761681 0.647952i \(-0.775626\pi\)
−0.761681 + 0.647952i \(0.775626\pi\)
\(770\) 0 0
\(771\) 58.8176 2.11826
\(772\) −14.9671 −0.538678
\(773\) 31.5640 1.13528 0.567639 0.823277i \(-0.307857\pi\)
0.567639 + 0.823277i \(0.307857\pi\)
\(774\) 72.9951 2.62375
\(775\) 6.95889 0.249971
\(776\) −2.96529 −0.106448
\(777\) −15.5810 −0.558966
\(778\) 25.9812 0.931470
\(779\) 5.66433 0.202946
\(780\) −6.95754 −0.249120
\(781\) 0 0
\(782\) −8.08622 −0.289163
\(783\) −17.5321 −0.626545
\(784\) 1.00000 0.0357143
\(785\) 10.2685 0.366498
\(786\) 13.4578 0.480024
\(787\) −20.4895 −0.730370 −0.365185 0.930935i \(-0.618994\pi\)
−0.365185 + 0.930935i \(0.618994\pi\)
\(788\) −19.1531 −0.682301
\(789\) −58.1432 −2.06995
\(790\) 7.69133 0.273645
\(791\) −12.6845 −0.451010
\(792\) 0 0
\(793\) −8.99766 −0.319516
\(794\) 7.16936 0.254431
\(795\) −45.9057 −1.62811
\(796\) −0.268053 −0.00950089
\(797\) −6.13739 −0.217397 −0.108699 0.994075i \(-0.534668\pi\)
−0.108699 + 0.994075i \(0.534668\pi\)
\(798\) 2.05576 0.0727729
\(799\) 11.9454 0.422597
\(800\) 1.00000 0.0353553
\(801\) 126.218 4.45969
\(802\) −26.2702 −0.927634
\(803\) 0 0
\(804\) −38.9525 −1.37375
\(805\) −5.61769 −0.197997
\(806\) 14.9279 0.525813
\(807\) 19.6793 0.692743
\(808\) −15.5584 −0.547343
\(809\) 31.7238 1.11535 0.557674 0.830060i \(-0.311694\pi\)
0.557674 + 0.830060i \(0.311694\pi\)
\(810\) −24.9840 −0.877848
\(811\) −9.91528 −0.348173 −0.174086 0.984730i \(-0.555697\pi\)
−0.174086 + 0.984730i \(0.555697\pi\)
\(812\) 1.19605 0.0419731
\(813\) 68.8196 2.41361
\(814\) 0 0
\(815\) 2.90989 0.101929
\(816\) 4.66858 0.163433
\(817\) −6.15291 −0.215263
\(818\) 36.5385 1.27754
\(819\) −16.1304 −0.563643
\(820\) 8.93663 0.312081
\(821\) 15.9145 0.555419 0.277710 0.960665i \(-0.410425\pi\)
0.277710 + 0.960665i \(0.410425\pi\)
\(822\) 67.5541 2.35622
\(823\) −5.51317 −0.192177 −0.0960885 0.995373i \(-0.530633\pi\)
−0.0960885 + 0.995373i \(0.530633\pi\)
\(824\) 0.404673 0.0140975
\(825\) 0 0
\(826\) 1.19248 0.0414917
\(827\) 18.8463 0.655350 0.327675 0.944790i \(-0.393735\pi\)
0.327675 + 0.944790i \(0.393735\pi\)
\(828\) −42.2420 −1.46801
\(829\) −23.0204 −0.799532 −0.399766 0.916617i \(-0.630909\pi\)
−0.399766 + 0.916617i \(0.630909\pi\)
\(830\) −11.5685 −0.401550
\(831\) −16.2630 −0.564156
\(832\) 2.14515 0.0743699
\(833\) 1.43942 0.0498730
\(834\) −59.9575 −2.07616
\(835\) −11.7960 −0.408216
\(836\) 0 0
\(837\) 102.006 3.52583
\(838\) −17.9411 −0.619765
\(839\) 41.3387 1.42717 0.713585 0.700569i \(-0.247070\pi\)
0.713585 + 0.700569i \(0.247070\pi\)
\(840\) 3.24337 0.111907
\(841\) −27.5695 −0.950671
\(842\) 36.8460 1.26980
\(843\) 15.4628 0.532567
\(844\) −9.35228 −0.321919
\(845\) 8.39831 0.288911
\(846\) 62.4021 2.14543
\(847\) 0 0
\(848\) 14.1537 0.486040
\(849\) 31.2992 1.07418
\(850\) 1.43942 0.0493717
\(851\) −26.9871 −0.925106
\(852\) −18.5388 −0.635129
\(853\) −19.8786 −0.680630 −0.340315 0.940312i \(-0.610534\pi\)
−0.340315 + 0.940312i \(0.610534\pi\)
\(854\) 4.19441 0.143530
\(855\) 4.76608 0.162997
\(856\) −8.17962 −0.279574
\(857\) −30.9589 −1.05753 −0.528767 0.848767i \(-0.677346\pi\)
−0.528767 + 0.848767i \(0.677346\pi\)
\(858\) 0 0
\(859\) −38.3155 −1.30731 −0.653654 0.756793i \(-0.726765\pi\)
−0.653654 + 0.756793i \(0.726765\pi\)
\(860\) −9.70747 −0.331022
\(861\) 28.9848 0.987801
\(862\) 31.8543 1.08496
\(863\) 33.7087 1.14746 0.573729 0.819045i \(-0.305496\pi\)
0.573729 + 0.819045i \(0.305496\pi\)
\(864\) 14.6583 0.498686
\(865\) −0.146508 −0.00498141
\(866\) 25.0278 0.850480
\(867\) −48.4173 −1.64434
\(868\) −6.95889 −0.236200
\(869\) 0 0
\(870\) 3.87923 0.131518
\(871\) −25.7631 −0.872948
\(872\) −1.34828 −0.0456585
\(873\) −22.2974 −0.754654
\(874\) 3.56067 0.120442
\(875\) 1.00000 0.0338062
\(876\) 20.8867 0.705696
\(877\) 32.7548 1.10605 0.553025 0.833165i \(-0.313473\pi\)
0.553025 + 0.833165i \(0.313473\pi\)
\(878\) −40.3537 −1.36187
\(879\) −92.4397 −3.11791
\(880\) 0 0
\(881\) −41.6696 −1.40389 −0.701943 0.712233i \(-0.747684\pi\)
−0.701943 + 0.712233i \(0.747684\pi\)
\(882\) 7.51947 0.253194
\(883\) −19.7448 −0.664467 −0.332234 0.943197i \(-0.607802\pi\)
−0.332234 + 0.943197i \(0.607802\pi\)
\(884\) 3.08778 0.103853
\(885\) 3.86766 0.130010
\(886\) −0.519900 −0.0174664
\(887\) −25.1277 −0.843704 −0.421852 0.906665i \(-0.638620\pi\)
−0.421852 + 0.906665i \(0.638620\pi\)
\(888\) 15.5810 0.522864
\(889\) −12.6218 −0.423321
\(890\) −16.7855 −0.562651
\(891\) 0 0
\(892\) −13.8365 −0.463280
\(893\) −5.26001 −0.176020
\(894\) −50.0686 −1.67455
\(895\) −10.2987 −0.344249
\(896\) −1.00000 −0.0334077
\(897\) −39.0853 −1.30502
\(898\) 23.0974 0.770769
\(899\) −8.32317 −0.277593
\(900\) 7.51947 0.250649
\(901\) 20.3731 0.678727
\(902\) 0 0
\(903\) −31.4850 −1.04775
\(904\) 12.6845 0.421881
\(905\) 19.8342 0.659310
\(906\) −37.1709 −1.23492
\(907\) 15.1970 0.504608 0.252304 0.967648i \(-0.418812\pi\)
0.252304 + 0.967648i \(0.418812\pi\)
\(908\) 25.8686 0.858481
\(909\) −116.991 −3.88034
\(910\) 2.14515 0.0711112
\(911\) −31.4976 −1.04356 −0.521781 0.853080i \(-0.674732\pi\)
−0.521781 + 0.853080i \(0.674732\pi\)
\(912\) −2.05576 −0.0680729
\(913\) 0 0
\(914\) −20.3786 −0.674066
\(915\) 13.6040 0.449736
\(916\) 11.1800 0.369398
\(917\) −4.14932 −0.137023
\(918\) 21.0995 0.696387
\(919\) 3.67799 0.121326 0.0606628 0.998158i \(-0.480679\pi\)
0.0606628 + 0.998158i \(0.480679\pi\)
\(920\) 5.61769 0.185210
\(921\) −1.79096 −0.0590142
\(922\) −12.1816 −0.401180
\(923\) −12.2615 −0.403592
\(924\) 0 0
\(925\) 4.80395 0.157953
\(926\) −11.4172 −0.375192
\(927\) 3.04293 0.0999429
\(928\) −1.19605 −0.0392622
\(929\) 57.4781 1.88579 0.942897 0.333084i \(-0.108089\pi\)
0.942897 + 0.333084i \(0.108089\pi\)
\(930\) −22.5703 −0.740109
\(931\) −0.633832 −0.0207730
\(932\) 2.87302 0.0941088
\(933\) −98.8410 −3.23591
\(934\) 22.7698 0.745052
\(935\) 0 0
\(936\) 16.1304 0.527239
\(937\) −18.0162 −0.588564 −0.294282 0.955719i \(-0.595081\pi\)
−0.294282 + 0.955719i \(0.595081\pi\)
\(938\) 12.0099 0.392137
\(939\) −108.549 −3.54235
\(940\) −8.29874 −0.270675
\(941\) −27.4433 −0.894625 −0.447312 0.894378i \(-0.647619\pi\)
−0.447312 + 0.894378i \(0.647619\pi\)
\(942\) −33.3045 −1.08512
\(943\) 50.2032 1.63484
\(944\) −1.19248 −0.0388119
\(945\) 14.6583 0.476835
\(946\) 0 0
\(947\) 0.524970 0.0170592 0.00852961 0.999964i \(-0.497285\pi\)
0.00852961 + 0.999964i \(0.497285\pi\)
\(948\) −24.9458 −0.810204
\(949\) 13.8144 0.448434
\(950\) −0.633832 −0.0205642
\(951\) −65.5567 −2.12582
\(952\) −1.43942 −0.0466519
\(953\) 47.1112 1.52608 0.763040 0.646351i \(-0.223706\pi\)
0.763040 + 0.646351i \(0.223706\pi\)
\(954\) 106.428 3.44574
\(955\) −5.95268 −0.192624
\(956\) 8.22791 0.266110
\(957\) 0 0
\(958\) −15.9269 −0.514575
\(959\) −20.8283 −0.672582
\(960\) −3.24337 −0.104679
\(961\) 17.4261 0.562134
\(962\) 10.3052 0.332254
\(963\) −61.5064 −1.98202
\(964\) −6.54207 −0.210706
\(965\) 14.9671 0.481808
\(966\) 18.2203 0.586227
\(967\) −30.5133 −0.981243 −0.490621 0.871373i \(-0.663230\pi\)
−0.490621 + 0.871373i \(0.663230\pi\)
\(968\) 0 0
\(969\) −2.95910 −0.0950599
\(970\) 2.96529 0.0952099
\(971\) 39.5470 1.26912 0.634562 0.772872i \(-0.281180\pi\)
0.634562 + 0.772872i \(0.281180\pi\)
\(972\) 37.0575 1.18862
\(973\) 18.4862 0.592639
\(974\) 4.57953 0.146737
\(975\) 6.95754 0.222820
\(976\) −4.19441 −0.134260
\(977\) −31.0366 −0.992950 −0.496475 0.868051i \(-0.665373\pi\)
−0.496475 + 0.868051i \(0.665373\pi\)
\(978\) −9.43784 −0.301789
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −10.1384 −0.323693
\(982\) −28.0073 −0.893748
\(983\) −51.0988 −1.62980 −0.814900 0.579602i \(-0.803208\pi\)
−0.814900 + 0.579602i \(0.803208\pi\)
\(984\) −28.9848 −0.924003
\(985\) 19.1531 0.610269
\(986\) −1.72162 −0.0548275
\(987\) −26.9159 −0.856743
\(988\) −1.35967 −0.0432568
\(989\) −54.5336 −1.73407
\(990\) 0 0
\(991\) −8.89709 −0.282625 −0.141313 0.989965i \(-0.545132\pi\)
−0.141313 + 0.989965i \(0.545132\pi\)
\(992\) 6.95889 0.220945
\(993\) −62.8133 −1.99332
\(994\) 5.71591 0.181298
\(995\) 0.268053 0.00849786
\(996\) 37.5211 1.18890
\(997\) −26.9013 −0.851974 −0.425987 0.904729i \(-0.640073\pi\)
−0.425987 + 0.904729i \(0.640073\pi\)
\(998\) 27.8295 0.880928
\(999\) 70.4179 2.22792
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.dd.1.6 yes 6
11.10 odd 2 8470.2.a.cx.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cx.1.6 6 11.10 odd 2
8470.2.a.dd.1.6 yes 6 1.1 even 1 trivial