Properties

Label 8470.2.a.bz.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.73205 q^{12} -5.73205 q^{13} +1.00000 q^{14} -1.73205 q^{15} +1.00000 q^{16} -0.732051 q^{17} -0.267949 q^{19} -1.00000 q^{20} +1.73205 q^{21} -5.00000 q^{23} +1.73205 q^{24} +1.00000 q^{25} -5.73205 q^{26} -5.19615 q^{27} +1.00000 q^{28} +8.00000 q^{29} -1.73205 q^{30} -2.73205 q^{31} +1.00000 q^{32} -0.732051 q^{34} -1.00000 q^{35} +6.00000 q^{37} -0.267949 q^{38} -9.92820 q^{39} -1.00000 q^{40} -6.19615 q^{41} +1.73205 q^{42} -5.26795 q^{43} -5.00000 q^{46} -4.73205 q^{47} +1.73205 q^{48} +1.00000 q^{49} +1.00000 q^{50} -1.26795 q^{51} -5.73205 q^{52} -6.19615 q^{53} -5.19615 q^{54} +1.00000 q^{56} -0.464102 q^{57} +8.00000 q^{58} -5.53590 q^{59} -1.73205 q^{60} -2.92820 q^{61} -2.73205 q^{62} +1.00000 q^{64} +5.73205 q^{65} -9.66025 q^{67} -0.732051 q^{68} -8.66025 q^{69} -1.00000 q^{70} +6.92820 q^{71} +9.12436 q^{73} +6.00000 q^{74} +1.73205 q^{75} -0.267949 q^{76} -9.92820 q^{78} -7.00000 q^{79} -1.00000 q^{80} -9.00000 q^{81} -6.19615 q^{82} +5.00000 q^{83} +1.73205 q^{84} +0.732051 q^{85} -5.26795 q^{86} +13.8564 q^{87} -13.6603 q^{89} -5.73205 q^{91} -5.00000 q^{92} -4.73205 q^{93} -4.73205 q^{94} +0.267949 q^{95} +1.73205 q^{96} +10.9282 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} - 8 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{19} - 2 q^{20} - 10 q^{23} + 2 q^{25} - 8 q^{26} + 2 q^{28} + 16 q^{29} - 2 q^{31} + 2 q^{32} + 2 q^{34} - 2 q^{35} + 12 q^{37} - 4 q^{38} - 6 q^{39} - 2 q^{40} - 2 q^{41} - 14 q^{43} - 10 q^{46} - 6 q^{47} + 2 q^{49} + 2 q^{50} - 6 q^{51} - 8 q^{52} - 2 q^{53} + 2 q^{56} + 6 q^{57} + 16 q^{58} - 18 q^{59} + 8 q^{61} - 2 q^{62} + 2 q^{64} + 8 q^{65} - 2 q^{67} + 2 q^{68} - 2 q^{70} - 6 q^{73} + 12 q^{74} - 4 q^{76} - 6 q^{78} - 14 q^{79} - 2 q^{80} - 18 q^{81} - 2 q^{82} + 10 q^{83} - 2 q^{85} - 14 q^{86} - 10 q^{89} - 8 q^{91} - 10 q^{92} - 6 q^{93} - 6 q^{94} + 4 q^{95} + 8 q^{97} + 2 q^{98} + O(q^{100}) \)

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
\(3\) 1.73205 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.73205 0.707107
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 1.73205 0.500000
\(13\) −5.73205 −1.58978 −0.794892 0.606750i \(-0.792473\pi\)
−0.794892 + 0.606750i \(0.792473\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.73205 −0.447214
\(16\) 1.00000 0.250000
\(17\) −0.732051 −0.177548 −0.0887742 0.996052i \(-0.528295\pi\)
−0.0887742 + 0.996052i \(0.528295\pi\)
\(18\) 0 0
\(19\) −0.267949 −0.0614718 −0.0307359 0.999528i \(-0.509785\pi\)
−0.0307359 + 0.999528i \(0.509785\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.73205 0.377964
\(22\) 0 0
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) 1.73205 0.353553
\(25\) 1.00000 0.200000
\(26\) −5.73205 −1.12415
\(27\) −5.19615 −1.00000
\(28\) 1.00000 0.188982
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) −1.73205 −0.316228
\(31\) −2.73205 −0.490691 −0.245345 0.969436i \(-0.578901\pi\)
−0.245345 + 0.969436i \(0.578901\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.732051 −0.125546
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −0.267949 −0.0434671
\(39\) −9.92820 −1.58978
\(40\) −1.00000 −0.158114
\(41\) −6.19615 −0.967676 −0.483838 0.875157i \(-0.660758\pi\)
−0.483838 + 0.875157i \(0.660758\pi\)
\(42\) 1.73205 0.267261
\(43\) −5.26795 −0.803355 −0.401677 0.915781i \(-0.631573\pi\)
−0.401677 + 0.915781i \(0.631573\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −5.00000 −0.737210
\(47\) −4.73205 −0.690241 −0.345120 0.938558i \(-0.612162\pi\)
−0.345120 + 0.938558i \(0.612162\pi\)
\(48\) 1.73205 0.250000
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −1.26795 −0.177548
\(52\) −5.73205 −0.794892
\(53\) −6.19615 −0.851107 −0.425553 0.904933i \(-0.639921\pi\)
−0.425553 + 0.904933i \(0.639921\pi\)
\(54\) −5.19615 −0.707107
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −0.464102 −0.0614718
\(58\) 8.00000 1.05045
\(59\) −5.53590 −0.720713 −0.360356 0.932815i \(-0.617345\pi\)
−0.360356 + 0.932815i \(0.617345\pi\)
\(60\) −1.73205 −0.223607
\(61\) −2.92820 −0.374918 −0.187459 0.982272i \(-0.560025\pi\)
−0.187459 + 0.982272i \(0.560025\pi\)
\(62\) −2.73205 −0.346971
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 5.73205 0.710973
\(66\) 0 0
\(67\) −9.66025 −1.18019 −0.590094 0.807335i \(-0.700909\pi\)
−0.590094 + 0.807335i \(0.700909\pi\)
\(68\) −0.732051 −0.0887742
\(69\) −8.66025 −1.04257
\(70\) −1.00000 −0.119523
\(71\) 6.92820 0.822226 0.411113 0.911584i \(-0.365140\pi\)
0.411113 + 0.911584i \(0.365140\pi\)
\(72\) 0 0
\(73\) 9.12436 1.06793 0.533963 0.845508i \(-0.320702\pi\)
0.533963 + 0.845508i \(0.320702\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.73205 0.200000
\(76\) −0.267949 −0.0307359
\(77\) 0 0
\(78\) −9.92820 −1.12415
\(79\) −7.00000 −0.787562 −0.393781 0.919204i \(-0.628833\pi\)
−0.393781 + 0.919204i \(0.628833\pi\)
\(80\) −1.00000 −0.111803
\(81\) −9.00000 −1.00000
\(82\) −6.19615 −0.684251
\(83\) 5.00000 0.548821 0.274411 0.961613i \(-0.411517\pi\)
0.274411 + 0.961613i \(0.411517\pi\)
\(84\) 1.73205 0.188982
\(85\) 0.732051 0.0794021
\(86\) −5.26795 −0.568058
\(87\) 13.8564 1.48556
\(88\) 0 0
\(89\) −13.6603 −1.44798 −0.723992 0.689808i \(-0.757695\pi\)
−0.723992 + 0.689808i \(0.757695\pi\)
\(90\) 0 0
\(91\) −5.73205 −0.600882
\(92\) −5.00000 −0.521286
\(93\) −4.73205 −0.490691
\(94\) −4.73205 −0.488074
\(95\) 0.267949 0.0274910
\(96\) 1.73205 0.176777
\(97\) 10.9282 1.10959 0.554795 0.831987i \(-0.312797\pi\)
0.554795 + 0.831987i \(0.312797\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −11.5359 −1.14786 −0.573932 0.818903i \(-0.694583\pi\)
−0.573932 + 0.818903i \(0.694583\pi\)
\(102\) −1.26795 −0.125546
\(103\) −10.7321 −1.05746 −0.528730 0.848790i \(-0.677332\pi\)
−0.528730 + 0.848790i \(0.677332\pi\)
\(104\) −5.73205 −0.562074
\(105\) −1.73205 −0.169031
\(106\) −6.19615 −0.601824
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) −5.19615 −0.500000
\(109\) 20.0526 1.92069 0.960343 0.278820i \(-0.0899433\pi\)
0.960343 + 0.278820i \(0.0899433\pi\)
\(110\) 0 0
\(111\) 10.3923 0.986394
\(112\) 1.00000 0.0944911
\(113\) −14.1244 −1.32871 −0.664354 0.747418i \(-0.731293\pi\)
−0.664354 + 0.747418i \(0.731293\pi\)
\(114\) −0.464102 −0.0434671
\(115\) 5.00000 0.466252
\(116\) 8.00000 0.742781
\(117\) 0 0
\(118\) −5.53590 −0.509621
\(119\) −0.732051 −0.0671070
\(120\) −1.73205 −0.158114
\(121\) 0 0
\(122\) −2.92820 −0.265107
\(123\) −10.7321 −0.967676
\(124\) −2.73205 −0.245345
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.12436 −0.543449 −0.271724 0.962375i \(-0.587594\pi\)
−0.271724 + 0.962375i \(0.587594\pi\)
\(128\) 1.00000 0.0883883
\(129\) −9.12436 −0.803355
\(130\) 5.73205 0.502734
\(131\) −13.1962 −1.15295 −0.576477 0.817114i \(-0.695573\pi\)
−0.576477 + 0.817114i \(0.695573\pi\)
\(132\) 0 0
\(133\) −0.267949 −0.0232341
\(134\) −9.66025 −0.834519
\(135\) 5.19615 0.447214
\(136\) −0.732051 −0.0627728
\(137\) 14.6603 1.25251 0.626255 0.779618i \(-0.284587\pi\)
0.626255 + 0.779618i \(0.284587\pi\)
\(138\) −8.66025 −0.737210
\(139\) −17.1962 −1.45856 −0.729279 0.684216i \(-0.760144\pi\)
−0.729279 + 0.684216i \(0.760144\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −8.19615 −0.690241
\(142\) 6.92820 0.581402
\(143\) 0 0
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 9.12436 0.755137
\(147\) 1.73205 0.142857
\(148\) 6.00000 0.493197
\(149\) 19.4641 1.59456 0.797281 0.603609i \(-0.206271\pi\)
0.797281 + 0.603609i \(0.206271\pi\)
\(150\) 1.73205 0.141421
\(151\) −18.8564 −1.53451 −0.767256 0.641341i \(-0.778379\pi\)
−0.767256 + 0.641341i \(0.778379\pi\)
\(152\) −0.267949 −0.0217335
\(153\) 0 0
\(154\) 0 0
\(155\) 2.73205 0.219444
\(156\) −9.92820 −0.794892
\(157\) −5.39230 −0.430353 −0.215176 0.976575i \(-0.569033\pi\)
−0.215176 + 0.976575i \(0.569033\pi\)
\(158\) −7.00000 −0.556890
\(159\) −10.7321 −0.851107
\(160\) −1.00000 −0.0790569
\(161\) −5.00000 −0.394055
\(162\) −9.00000 −0.707107
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) −6.19615 −0.483838
\(165\) 0 0
\(166\) 5.00000 0.388075
\(167\) −0.339746 −0.0262903 −0.0131452 0.999914i \(-0.504184\pi\)
−0.0131452 + 0.999914i \(0.504184\pi\)
\(168\) 1.73205 0.133631
\(169\) 19.8564 1.52742
\(170\) 0.732051 0.0561457
\(171\) 0 0
\(172\) −5.26795 −0.401677
\(173\) 7.85641 0.597312 0.298656 0.954361i \(-0.403462\pi\)
0.298656 + 0.954361i \(0.403462\pi\)
\(174\) 13.8564 1.05045
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −9.58846 −0.720713
\(178\) −13.6603 −1.02388
\(179\) −16.1962 −1.21056 −0.605279 0.796014i \(-0.706938\pi\)
−0.605279 + 0.796014i \(0.706938\pi\)
\(180\) 0 0
\(181\) −22.6603 −1.68432 −0.842162 0.539225i \(-0.818717\pi\)
−0.842162 + 0.539225i \(0.818717\pi\)
\(182\) −5.73205 −0.424888
\(183\) −5.07180 −0.374918
\(184\) −5.00000 −0.368605
\(185\) −6.00000 −0.441129
\(186\) −4.73205 −0.346971
\(187\) 0 0
\(188\) −4.73205 −0.345120
\(189\) −5.19615 −0.377964
\(190\) 0.267949 0.0194391
\(191\) −8.66025 −0.626634 −0.313317 0.949649i \(-0.601440\pi\)
−0.313317 + 0.949649i \(0.601440\pi\)
\(192\) 1.73205 0.125000
\(193\) 11.3923 0.820036 0.410018 0.912077i \(-0.365522\pi\)
0.410018 + 0.912077i \(0.365522\pi\)
\(194\) 10.9282 0.784599
\(195\) 9.92820 0.710973
\(196\) 1.00000 0.0714286
\(197\) −9.46410 −0.674289 −0.337145 0.941453i \(-0.609461\pi\)
−0.337145 + 0.941453i \(0.609461\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 1.00000 0.0707107
\(201\) −16.7321 −1.18019
\(202\) −11.5359 −0.811663
\(203\) 8.00000 0.561490
\(204\) −1.26795 −0.0887742
\(205\) 6.19615 0.432758
\(206\) −10.7321 −0.747737
\(207\) 0 0
\(208\) −5.73205 −0.397446
\(209\) 0 0
\(210\) −1.73205 −0.119523
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −6.19615 −0.425553
\(213\) 12.0000 0.822226
\(214\) 2.00000 0.136717
\(215\) 5.26795 0.359271
\(216\) −5.19615 −0.353553
\(217\) −2.73205 −0.185464
\(218\) 20.0526 1.35813
\(219\) 15.8038 1.06793
\(220\) 0 0
\(221\) 4.19615 0.282264
\(222\) 10.3923 0.697486
\(223\) −11.2679 −0.754558 −0.377279 0.926100i \(-0.623140\pi\)
−0.377279 + 0.926100i \(0.623140\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −14.1244 −0.939538
\(227\) 17.3205 1.14960 0.574801 0.818293i \(-0.305079\pi\)
0.574801 + 0.818293i \(0.305079\pi\)
\(228\) −0.464102 −0.0307359
\(229\) 5.85641 0.387002 0.193501 0.981100i \(-0.438016\pi\)
0.193501 + 0.981100i \(0.438016\pi\)
\(230\) 5.00000 0.329690
\(231\) 0 0
\(232\) 8.00000 0.525226
\(233\) 5.53590 0.362669 0.181334 0.983421i \(-0.441958\pi\)
0.181334 + 0.983421i \(0.441958\pi\)
\(234\) 0 0
\(235\) 4.73205 0.308685
\(236\) −5.53590 −0.360356
\(237\) −12.1244 −0.787562
\(238\) −0.732051 −0.0474518
\(239\) −4.46410 −0.288759 −0.144379 0.989522i \(-0.546119\pi\)
−0.144379 + 0.989522i \(0.546119\pi\)
\(240\) −1.73205 −0.111803
\(241\) 17.4641 1.12496 0.562481 0.826810i \(-0.309847\pi\)
0.562481 + 0.826810i \(0.309847\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −2.92820 −0.187459
\(245\) −1.00000 −0.0638877
\(246\) −10.7321 −0.684251
\(247\) 1.53590 0.0977269
\(248\) −2.73205 −0.173485
\(249\) 8.66025 0.548821
\(250\) −1.00000 −0.0632456
\(251\) −6.39230 −0.403479 −0.201739 0.979439i \(-0.564659\pi\)
−0.201739 + 0.979439i \(0.564659\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −6.12436 −0.384276
\(255\) 1.26795 0.0794021
\(256\) 1.00000 0.0625000
\(257\) 8.19615 0.511262 0.255631 0.966774i \(-0.417717\pi\)
0.255631 + 0.966774i \(0.417717\pi\)
\(258\) −9.12436 −0.568058
\(259\) 6.00000 0.372822
\(260\) 5.73205 0.355487
\(261\) 0 0
\(262\) −13.1962 −0.815261
\(263\) 10.1244 0.624295 0.312147 0.950034i \(-0.398952\pi\)
0.312147 + 0.950034i \(0.398952\pi\)
\(264\) 0 0
\(265\) 6.19615 0.380627
\(266\) −0.267949 −0.0164290
\(267\) −23.6603 −1.44798
\(268\) −9.66025 −0.590094
\(269\) −21.7321 −1.32503 −0.662513 0.749050i \(-0.730510\pi\)
−0.662513 + 0.749050i \(0.730510\pi\)
\(270\) 5.19615 0.316228
\(271\) 21.4641 1.30385 0.651926 0.758283i \(-0.273961\pi\)
0.651926 + 0.758283i \(0.273961\pi\)
\(272\) −0.732051 −0.0443871
\(273\) −9.92820 −0.600882
\(274\) 14.6603 0.885658
\(275\) 0 0
\(276\) −8.66025 −0.521286
\(277\) 1.80385 0.108383 0.0541913 0.998531i \(-0.482742\pi\)
0.0541913 + 0.998531i \(0.482742\pi\)
\(278\) −17.1962 −1.03136
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 7.73205 0.461255 0.230628 0.973042i \(-0.425922\pi\)
0.230628 + 0.973042i \(0.425922\pi\)
\(282\) −8.19615 −0.488074
\(283\) 25.7846 1.53274 0.766368 0.642402i \(-0.222062\pi\)
0.766368 + 0.642402i \(0.222062\pi\)
\(284\) 6.92820 0.411113
\(285\) 0.464102 0.0274910
\(286\) 0 0
\(287\) −6.19615 −0.365747
\(288\) 0 0
\(289\) −16.4641 −0.968477
\(290\) −8.00000 −0.469776
\(291\) 18.9282 1.10959
\(292\) 9.12436 0.533963
\(293\) 11.7321 0.685394 0.342697 0.939446i \(-0.388660\pi\)
0.342697 + 0.939446i \(0.388660\pi\)
\(294\) 1.73205 0.101015
\(295\) 5.53590 0.322312
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 19.4641 1.12753
\(299\) 28.6603 1.65747
\(300\) 1.73205 0.100000
\(301\) −5.26795 −0.303640
\(302\) −18.8564 −1.08506
\(303\) −19.9808 −1.14786
\(304\) −0.267949 −0.0153679
\(305\) 2.92820 0.167668
\(306\) 0 0
\(307\) −7.07180 −0.403609 −0.201804 0.979426i \(-0.564681\pi\)
−0.201804 + 0.979426i \(0.564681\pi\)
\(308\) 0 0
\(309\) −18.5885 −1.05746
\(310\) 2.73205 0.155170
\(311\) 12.7321 0.721968 0.360984 0.932572i \(-0.382441\pi\)
0.360984 + 0.932572i \(0.382441\pi\)
\(312\) −9.92820 −0.562074
\(313\) −5.60770 −0.316966 −0.158483 0.987362i \(-0.550660\pi\)
−0.158483 + 0.987362i \(0.550660\pi\)
\(314\) −5.39230 −0.304305
\(315\) 0 0
\(316\) −7.00000 −0.393781
\(317\) 5.12436 0.287812 0.143906 0.989591i \(-0.454034\pi\)
0.143906 + 0.989591i \(0.454034\pi\)
\(318\) −10.7321 −0.601824
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 3.46410 0.193347
\(322\) −5.00000 −0.278639
\(323\) 0.196152 0.0109142
\(324\) −9.00000 −0.500000
\(325\) −5.73205 −0.317957
\(326\) 10.0000 0.553849
\(327\) 34.7321 1.92069
\(328\) −6.19615 −0.342125
\(329\) −4.73205 −0.260886
\(330\) 0 0
\(331\) 34.0526 1.87170 0.935849 0.352401i \(-0.114635\pi\)
0.935849 + 0.352401i \(0.114635\pi\)
\(332\) 5.00000 0.274411
\(333\) 0 0
\(334\) −0.339746 −0.0185901
\(335\) 9.66025 0.527796
\(336\) 1.73205 0.0944911
\(337\) −9.39230 −0.511631 −0.255816 0.966726i \(-0.582344\pi\)
−0.255816 + 0.966726i \(0.582344\pi\)
\(338\) 19.8564 1.08005
\(339\) −24.4641 −1.32871
\(340\) 0.732051 0.0397010
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −5.26795 −0.284029
\(345\) 8.66025 0.466252
\(346\) 7.85641 0.422363
\(347\) −15.4641 −0.830156 −0.415078 0.909786i \(-0.636246\pi\)
−0.415078 + 0.909786i \(0.636246\pi\)
\(348\) 13.8564 0.742781
\(349\) 0.464102 0.0248428 0.0124214 0.999923i \(-0.496046\pi\)
0.0124214 + 0.999923i \(0.496046\pi\)
\(350\) 1.00000 0.0534522
\(351\) 29.7846 1.58978
\(352\) 0 0
\(353\) 12.1962 0.649136 0.324568 0.945862i \(-0.394781\pi\)
0.324568 + 0.945862i \(0.394781\pi\)
\(354\) −9.58846 −0.509621
\(355\) −6.92820 −0.367711
\(356\) −13.6603 −0.723992
\(357\) −1.26795 −0.0671070
\(358\) −16.1962 −0.855993
\(359\) 21.7128 1.14596 0.572979 0.819570i \(-0.305788\pi\)
0.572979 + 0.819570i \(0.305788\pi\)
\(360\) 0 0
\(361\) −18.9282 −0.996221
\(362\) −22.6603 −1.19100
\(363\) 0 0
\(364\) −5.73205 −0.300441
\(365\) −9.12436 −0.477591
\(366\) −5.07180 −0.265107
\(367\) −17.2679 −0.901380 −0.450690 0.892681i \(-0.648822\pi\)
−0.450690 + 0.892681i \(0.648822\pi\)
\(368\) −5.00000 −0.260643
\(369\) 0 0
\(370\) −6.00000 −0.311925
\(371\) −6.19615 −0.321688
\(372\) −4.73205 −0.245345
\(373\) 21.3205 1.10393 0.551967 0.833866i \(-0.313877\pi\)
0.551967 + 0.833866i \(0.313877\pi\)
\(374\) 0 0
\(375\) −1.73205 −0.0894427
\(376\) −4.73205 −0.244037
\(377\) −45.8564 −2.36173
\(378\) −5.19615 −0.267261
\(379\) −6.78461 −0.348502 −0.174251 0.984701i \(-0.555750\pi\)
−0.174251 + 0.984701i \(0.555750\pi\)
\(380\) 0.267949 0.0137455
\(381\) −10.6077 −0.543449
\(382\) −8.66025 −0.443097
\(383\) 25.6603 1.31118 0.655589 0.755118i \(-0.272420\pi\)
0.655589 + 0.755118i \(0.272420\pi\)
\(384\) 1.73205 0.0883883
\(385\) 0 0
\(386\) 11.3923 0.579853
\(387\) 0 0
\(388\) 10.9282 0.554795
\(389\) −23.1244 −1.17245 −0.586226 0.810148i \(-0.699387\pi\)
−0.586226 + 0.810148i \(0.699387\pi\)
\(390\) 9.92820 0.502734
\(391\) 3.66025 0.185107
\(392\) 1.00000 0.0505076
\(393\) −22.8564 −1.15295
\(394\) −9.46410 −0.476795
\(395\) 7.00000 0.352208
\(396\) 0 0
\(397\) 12.7846 0.641641 0.320821 0.947140i \(-0.396041\pi\)
0.320821 + 0.947140i \(0.396041\pi\)
\(398\) −4.00000 −0.200502
\(399\) −0.464102 −0.0232341
\(400\) 1.00000 0.0500000
\(401\) −18.5359 −0.925639 −0.462819 0.886453i \(-0.653162\pi\)
−0.462819 + 0.886453i \(0.653162\pi\)
\(402\) −16.7321 −0.834519
\(403\) 15.6603 0.780093
\(404\) −11.5359 −0.573932
\(405\) 9.00000 0.447214
\(406\) 8.00000 0.397033
\(407\) 0 0
\(408\) −1.26795 −0.0627728
\(409\) 35.7128 1.76588 0.882942 0.469481i \(-0.155559\pi\)
0.882942 + 0.469481i \(0.155559\pi\)
\(410\) 6.19615 0.306006
\(411\) 25.3923 1.25251
\(412\) −10.7321 −0.528730
\(413\) −5.53590 −0.272404
\(414\) 0 0
\(415\) −5.00000 −0.245440
\(416\) −5.73205 −0.281037
\(417\) −29.7846 −1.45856
\(418\) 0 0
\(419\) 15.9282 0.778144 0.389072 0.921207i \(-0.372796\pi\)
0.389072 + 0.921207i \(0.372796\pi\)
\(420\) −1.73205 −0.0845154
\(421\) −7.60770 −0.370776 −0.185388 0.982665i \(-0.559354\pi\)
−0.185388 + 0.982665i \(0.559354\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) −6.19615 −0.300912
\(425\) −0.732051 −0.0355097
\(426\) 12.0000 0.581402
\(427\) −2.92820 −0.141706
\(428\) 2.00000 0.0966736
\(429\) 0 0
\(430\) 5.26795 0.254043
\(431\) −34.4641 −1.66008 −0.830039 0.557706i \(-0.811682\pi\)
−0.830039 + 0.557706i \(0.811682\pi\)
\(432\) −5.19615 −0.250000
\(433\) −20.0526 −0.963664 −0.481832 0.876263i \(-0.660029\pi\)
−0.481832 + 0.876263i \(0.660029\pi\)
\(434\) −2.73205 −0.131143
\(435\) −13.8564 −0.664364
\(436\) 20.0526 0.960343
\(437\) 1.33975 0.0640887
\(438\) 15.8038 0.755137
\(439\) −14.7321 −0.703122 −0.351561 0.936165i \(-0.614349\pi\)
−0.351561 + 0.936165i \(0.614349\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 4.19615 0.199591
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 10.3923 0.493197
\(445\) 13.6603 0.647558
\(446\) −11.2679 −0.533553
\(447\) 33.7128 1.59456
\(448\) 1.00000 0.0472456
\(449\) 31.3923 1.48149 0.740747 0.671784i \(-0.234472\pi\)
0.740747 + 0.671784i \(0.234472\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −14.1244 −0.664354
\(453\) −32.6603 −1.53451
\(454\) 17.3205 0.812892
\(455\) 5.73205 0.268723
\(456\) −0.464102 −0.0217335
\(457\) −3.39230 −0.158685 −0.0793427 0.996847i \(-0.525282\pi\)
−0.0793427 + 0.996847i \(0.525282\pi\)
\(458\) 5.85641 0.273652
\(459\) 3.80385 0.177548
\(460\) 5.00000 0.233126
\(461\) 10.5359 0.490706 0.245353 0.969434i \(-0.421096\pi\)
0.245353 + 0.969434i \(0.421096\pi\)
\(462\) 0 0
\(463\) 26.1769 1.21654 0.608272 0.793729i \(-0.291863\pi\)
0.608272 + 0.793729i \(0.291863\pi\)
\(464\) 8.00000 0.371391
\(465\) 4.73205 0.219444
\(466\) 5.53590 0.256446
\(467\) 13.5885 0.628799 0.314399 0.949291i \(-0.398197\pi\)
0.314399 + 0.949291i \(0.398197\pi\)
\(468\) 0 0
\(469\) −9.66025 −0.446069
\(470\) 4.73205 0.218273
\(471\) −9.33975 −0.430353
\(472\) −5.53590 −0.254810
\(473\) 0 0
\(474\) −12.1244 −0.556890
\(475\) −0.267949 −0.0122944
\(476\) −0.732051 −0.0335535
\(477\) 0 0
\(478\) −4.46410 −0.204183
\(479\) 28.9808 1.32416 0.662082 0.749431i \(-0.269673\pi\)
0.662082 + 0.749431i \(0.269673\pi\)
\(480\) −1.73205 −0.0790569
\(481\) −34.3923 −1.56815
\(482\) 17.4641 0.795468
\(483\) −8.66025 −0.394055
\(484\) 0 0
\(485\) −10.9282 −0.496224
\(486\) 0 0
\(487\) 17.3923 0.788121 0.394060 0.919085i \(-0.371070\pi\)
0.394060 + 0.919085i \(0.371070\pi\)
\(488\) −2.92820 −0.132554
\(489\) 17.3205 0.783260
\(490\) −1.00000 −0.0451754
\(491\) 9.66025 0.435961 0.217981 0.975953i \(-0.430053\pi\)
0.217981 + 0.975953i \(0.430053\pi\)
\(492\) −10.7321 −0.483838
\(493\) −5.85641 −0.263759
\(494\) 1.53590 0.0691033
\(495\) 0 0
\(496\) −2.73205 −0.122673
\(497\) 6.92820 0.310772
\(498\) 8.66025 0.388075
\(499\) −22.0000 −0.984855 −0.492428 0.870353i \(-0.663890\pi\)
−0.492428 + 0.870353i \(0.663890\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −0.588457 −0.0262903
\(502\) −6.39230 −0.285303
\(503\) 18.1962 0.811326 0.405663 0.914023i \(-0.367041\pi\)
0.405663 + 0.914023i \(0.367041\pi\)
\(504\) 0 0
\(505\) 11.5359 0.513341
\(506\) 0 0
\(507\) 34.3923 1.52742
\(508\) −6.12436 −0.271724
\(509\) −32.6603 −1.44764 −0.723820 0.689989i \(-0.757615\pi\)
−0.723820 + 0.689989i \(0.757615\pi\)
\(510\) 1.26795 0.0561457
\(511\) 9.12436 0.403638
\(512\) 1.00000 0.0441942
\(513\) 1.39230 0.0614718
\(514\) 8.19615 0.361517
\(515\) 10.7321 0.472911
\(516\) −9.12436 −0.401677
\(517\) 0 0
\(518\) 6.00000 0.263625
\(519\) 13.6077 0.597312
\(520\) 5.73205 0.251367
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 0 0
\(523\) −23.3923 −1.02287 −0.511437 0.859321i \(-0.670887\pi\)
−0.511437 + 0.859321i \(0.670887\pi\)
\(524\) −13.1962 −0.576477
\(525\) 1.73205 0.0755929
\(526\) 10.1244 0.441443
\(527\) 2.00000 0.0871214
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 6.19615 0.269144
\(531\) 0 0
\(532\) −0.267949 −0.0116171
\(533\) 35.5167 1.53840
\(534\) −23.6603 −1.02388
\(535\) −2.00000 −0.0864675
\(536\) −9.66025 −0.417259
\(537\) −28.0526 −1.21056
\(538\) −21.7321 −0.936935
\(539\) 0 0
\(540\) 5.19615 0.223607
\(541\) 30.3923 1.30667 0.653334 0.757070i \(-0.273370\pi\)
0.653334 + 0.757070i \(0.273370\pi\)
\(542\) 21.4641 0.921962
\(543\) −39.2487 −1.68432
\(544\) −0.732051 −0.0313864
\(545\) −20.0526 −0.858957
\(546\) −9.92820 −0.424888
\(547\) −7.51666 −0.321389 −0.160695 0.987004i \(-0.551373\pi\)
−0.160695 + 0.987004i \(0.551373\pi\)
\(548\) 14.6603 0.626255
\(549\) 0 0
\(550\) 0 0
\(551\) −2.14359 −0.0913202
\(552\) −8.66025 −0.368605
\(553\) −7.00000 −0.297670
\(554\) 1.80385 0.0766381
\(555\) −10.3923 −0.441129
\(556\) −17.1962 −0.729279
\(557\) −45.3731 −1.92252 −0.961259 0.275646i \(-0.911108\pi\)
−0.961259 + 0.275646i \(0.911108\pi\)
\(558\) 0 0
\(559\) 30.1962 1.27716
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 7.73205 0.326157
\(563\) −8.46410 −0.356719 −0.178360 0.983965i \(-0.557079\pi\)
−0.178360 + 0.983965i \(0.557079\pi\)
\(564\) −8.19615 −0.345120
\(565\) 14.1244 0.594216
\(566\) 25.7846 1.08381
\(567\) −9.00000 −0.377964
\(568\) 6.92820 0.290701
\(569\) 13.8756 0.581697 0.290849 0.956769i \(-0.406062\pi\)
0.290849 + 0.956769i \(0.406062\pi\)
\(570\) 0.464102 0.0194391
\(571\) 6.53590 0.273519 0.136759 0.990604i \(-0.456331\pi\)
0.136759 + 0.990604i \(0.456331\pi\)
\(572\) 0 0
\(573\) −15.0000 −0.626634
\(574\) −6.19615 −0.258622
\(575\) −5.00000 −0.208514
\(576\) 0 0
\(577\) −40.4449 −1.68374 −0.841871 0.539679i \(-0.818546\pi\)
−0.841871 + 0.539679i \(0.818546\pi\)
\(578\) −16.4641 −0.684816
\(579\) 19.7321 0.820036
\(580\) −8.00000 −0.332182
\(581\) 5.00000 0.207435
\(582\) 18.9282 0.784599
\(583\) 0 0
\(584\) 9.12436 0.377569
\(585\) 0 0
\(586\) 11.7321 0.484647
\(587\) −30.5167 −1.25956 −0.629779 0.776775i \(-0.716854\pi\)
−0.629779 + 0.776775i \(0.716854\pi\)
\(588\) 1.73205 0.0714286
\(589\) 0.732051 0.0301636
\(590\) 5.53590 0.227909
\(591\) −16.3923 −0.674289
\(592\) 6.00000 0.246598
\(593\) −2.14359 −0.0880268 −0.0440134 0.999031i \(-0.514014\pi\)
−0.0440134 + 0.999031i \(0.514014\pi\)
\(594\) 0 0
\(595\) 0.732051 0.0300112
\(596\) 19.4641 0.797281
\(597\) −6.92820 −0.283552
\(598\) 28.6603 1.17200
\(599\) 19.9808 0.816392 0.408196 0.912894i \(-0.366158\pi\)
0.408196 + 0.912894i \(0.366158\pi\)
\(600\) 1.73205 0.0707107
\(601\) −13.6603 −0.557213 −0.278607 0.960405i \(-0.589873\pi\)
−0.278607 + 0.960405i \(0.589873\pi\)
\(602\) −5.26795 −0.214706
\(603\) 0 0
\(604\) −18.8564 −0.767256
\(605\) 0 0
\(606\) −19.9808 −0.811663
\(607\) 48.2487 1.95835 0.979177 0.203006i \(-0.0650713\pi\)
0.979177 + 0.203006i \(0.0650713\pi\)
\(608\) −0.267949 −0.0108668
\(609\) 13.8564 0.561490
\(610\) 2.92820 0.118559
\(611\) 27.1244 1.09733
\(612\) 0 0
\(613\) 40.7321 1.64515 0.822576 0.568655i \(-0.192536\pi\)
0.822576 + 0.568655i \(0.192536\pi\)
\(614\) −7.07180 −0.285394
\(615\) 10.7321 0.432758
\(616\) 0 0
\(617\) 34.3923 1.38458 0.692291 0.721618i \(-0.256601\pi\)
0.692291 + 0.721618i \(0.256601\pi\)
\(618\) −18.5885 −0.747737
\(619\) −22.7128 −0.912905 −0.456453 0.889748i \(-0.650880\pi\)
−0.456453 + 0.889748i \(0.650880\pi\)
\(620\) 2.73205 0.109722
\(621\) 25.9808 1.04257
\(622\) 12.7321 0.510509
\(623\) −13.6603 −0.547287
\(624\) −9.92820 −0.397446
\(625\) 1.00000 0.0400000
\(626\) −5.60770 −0.224129
\(627\) 0 0
\(628\) −5.39230 −0.215176
\(629\) −4.39230 −0.175133
\(630\) 0 0
\(631\) −4.67949 −0.186288 −0.0931438 0.995653i \(-0.529692\pi\)
−0.0931438 + 0.995653i \(0.529692\pi\)
\(632\) −7.00000 −0.278445
\(633\) −13.8564 −0.550743
\(634\) 5.12436 0.203514
\(635\) 6.12436 0.243038
\(636\) −10.7321 −0.425553
\(637\) −5.73205 −0.227112
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −46.4641 −1.83522 −0.917611 0.397480i \(-0.869885\pi\)
−0.917611 + 0.397480i \(0.869885\pi\)
\(642\) 3.46410 0.136717
\(643\) −39.1769 −1.54499 −0.772493 0.635023i \(-0.780991\pi\)
−0.772493 + 0.635023i \(0.780991\pi\)
\(644\) −5.00000 −0.197028
\(645\) 9.12436 0.359271
\(646\) 0.196152 0.00771751
\(647\) −32.1962 −1.26576 −0.632881 0.774249i \(-0.718128\pi\)
−0.632881 + 0.774249i \(0.718128\pi\)
\(648\) −9.00000 −0.353553
\(649\) 0 0
\(650\) −5.73205 −0.224830
\(651\) −4.73205 −0.185464
\(652\) 10.0000 0.391630
\(653\) 8.19615 0.320740 0.160370 0.987057i \(-0.448731\pi\)
0.160370 + 0.987057i \(0.448731\pi\)
\(654\) 34.7321 1.35813
\(655\) 13.1962 0.515616
\(656\) −6.19615 −0.241919
\(657\) 0 0
\(658\) −4.73205 −0.184475
\(659\) 26.4449 1.03015 0.515073 0.857146i \(-0.327765\pi\)
0.515073 + 0.857146i \(0.327765\pi\)
\(660\) 0 0
\(661\) 20.6603 0.803591 0.401795 0.915729i \(-0.368386\pi\)
0.401795 + 0.915729i \(0.368386\pi\)
\(662\) 34.0526 1.32349
\(663\) 7.26795 0.282264
\(664\) 5.00000 0.194038
\(665\) 0.267949 0.0103906
\(666\) 0 0
\(667\) −40.0000 −1.54881
\(668\) −0.339746 −0.0131452
\(669\) −19.5167 −0.754558
\(670\) 9.66025 0.373208
\(671\) 0 0
\(672\) 1.73205 0.0668153
\(673\) −29.1051 −1.12192 −0.560960 0.827843i \(-0.689568\pi\)
−0.560960 + 0.827843i \(0.689568\pi\)
\(674\) −9.39230 −0.361778
\(675\) −5.19615 −0.200000
\(676\) 19.8564 0.763708
\(677\) −5.98076 −0.229859 −0.114930 0.993374i \(-0.536664\pi\)
−0.114930 + 0.993374i \(0.536664\pi\)
\(678\) −24.4641 −0.939538
\(679\) 10.9282 0.419386
\(680\) 0.732051 0.0280729
\(681\) 30.0000 1.14960
\(682\) 0 0
\(683\) −25.0718 −0.959346 −0.479673 0.877447i \(-0.659245\pi\)
−0.479673 + 0.877447i \(0.659245\pi\)
\(684\) 0 0
\(685\) −14.6603 −0.560140
\(686\) 1.00000 0.0381802
\(687\) 10.1436 0.387002
\(688\) −5.26795 −0.200839
\(689\) 35.5167 1.35308
\(690\) 8.66025 0.329690
\(691\) −25.8564 −0.983624 −0.491812 0.870701i \(-0.663665\pi\)
−0.491812 + 0.870701i \(0.663665\pi\)
\(692\) 7.85641 0.298656
\(693\) 0 0
\(694\) −15.4641 −0.587009
\(695\) 17.1962 0.652287
\(696\) 13.8564 0.525226
\(697\) 4.53590 0.171809
\(698\) 0.464102 0.0175665
\(699\) 9.58846 0.362669
\(700\) 1.00000 0.0377964
\(701\) −17.9090 −0.676412 −0.338206 0.941072i \(-0.609820\pi\)
−0.338206 + 0.941072i \(0.609820\pi\)
\(702\) 29.7846 1.12415
\(703\) −1.60770 −0.0606354
\(704\) 0 0
\(705\) 8.19615 0.308685
\(706\) 12.1962 0.459008
\(707\) −11.5359 −0.433852
\(708\) −9.58846 −0.360356
\(709\) −33.3205 −1.25138 −0.625689 0.780073i \(-0.715182\pi\)
−0.625689 + 0.780073i \(0.715182\pi\)
\(710\) −6.92820 −0.260011
\(711\) 0 0
\(712\) −13.6603 −0.511940
\(713\) 13.6603 0.511581
\(714\) −1.26795 −0.0474518
\(715\) 0 0
\(716\) −16.1962 −0.605279
\(717\) −7.73205 −0.288759
\(718\) 21.7128 0.810315
\(719\) −21.0718 −0.785845 −0.392923 0.919572i \(-0.628536\pi\)
−0.392923 + 0.919572i \(0.628536\pi\)
\(720\) 0 0
\(721\) −10.7321 −0.399682
\(722\) −18.9282 −0.704435
\(723\) 30.2487 1.12496
\(724\) −22.6603 −0.842162
\(725\) 8.00000 0.297113
\(726\) 0 0
\(727\) 19.3205 0.716558 0.358279 0.933615i \(-0.383364\pi\)
0.358279 + 0.933615i \(0.383364\pi\)
\(728\) −5.73205 −0.212444
\(729\) 27.0000 1.00000
\(730\) −9.12436 −0.337708
\(731\) 3.85641 0.142634
\(732\) −5.07180 −0.187459
\(733\) 7.58846 0.280286 0.140143 0.990131i \(-0.455244\pi\)
0.140143 + 0.990131i \(0.455244\pi\)
\(734\) −17.2679 −0.637372
\(735\) −1.73205 −0.0638877
\(736\) −5.00000 −0.184302
\(737\) 0 0
\(738\) 0 0
\(739\) −24.9808 −0.918932 −0.459466 0.888195i \(-0.651959\pi\)
−0.459466 + 0.888195i \(0.651959\pi\)
\(740\) −6.00000 −0.220564
\(741\) 2.66025 0.0977269
\(742\) −6.19615 −0.227468
\(743\) 19.4641 0.714069 0.357034 0.934091i \(-0.383788\pi\)
0.357034 + 0.934091i \(0.383788\pi\)
\(744\) −4.73205 −0.173485
\(745\) −19.4641 −0.713110
\(746\) 21.3205 0.780599
\(747\) 0 0
\(748\) 0 0
\(749\) 2.00000 0.0730784
\(750\) −1.73205 −0.0632456
\(751\) 30.3731 1.10833 0.554164 0.832407i \(-0.313038\pi\)
0.554164 + 0.832407i \(0.313038\pi\)
\(752\) −4.73205 −0.172560
\(753\) −11.0718 −0.403479
\(754\) −45.8564 −1.66999
\(755\) 18.8564 0.686255
\(756\) −5.19615 −0.188982
\(757\) 39.9090 1.45052 0.725258 0.688477i \(-0.241720\pi\)
0.725258 + 0.688477i \(0.241720\pi\)
\(758\) −6.78461 −0.246428
\(759\) 0 0
\(760\) 0.267949 0.00971954
\(761\) 18.4449 0.668626 0.334313 0.942462i \(-0.391496\pi\)
0.334313 + 0.942462i \(0.391496\pi\)
\(762\) −10.6077 −0.384276
\(763\) 20.0526 0.725951
\(764\) −8.66025 −0.313317
\(765\) 0 0
\(766\) 25.6603 0.927143
\(767\) 31.7321 1.14578
\(768\) 1.73205 0.0625000
\(769\) −6.78461 −0.244659 −0.122330 0.992490i \(-0.539037\pi\)
−0.122330 + 0.992490i \(0.539037\pi\)
\(770\) 0 0
\(771\) 14.1962 0.511262
\(772\) 11.3923 0.410018
\(773\) 34.4641 1.23959 0.619794 0.784765i \(-0.287216\pi\)
0.619794 + 0.784765i \(0.287216\pi\)
\(774\) 0 0
\(775\) −2.73205 −0.0981382
\(776\) 10.9282 0.392300
\(777\) 10.3923 0.372822
\(778\) −23.1244 −0.829048
\(779\) 1.66025 0.0594848
\(780\) 9.92820 0.355487
\(781\) 0 0
\(782\) 3.66025 0.130890
\(783\) −41.5692 −1.48556
\(784\) 1.00000 0.0357143
\(785\) 5.39230 0.192460
\(786\) −22.8564 −0.815261
\(787\) −50.4974 −1.80004 −0.900019 0.435850i \(-0.856448\pi\)
−0.900019 + 0.435850i \(0.856448\pi\)
\(788\) −9.46410 −0.337145
\(789\) 17.5359 0.624295
\(790\) 7.00000 0.249049
\(791\) −14.1244 −0.502204
\(792\) 0 0
\(793\) 16.7846 0.596039
\(794\) 12.7846 0.453709
\(795\) 10.7321 0.380627
\(796\) −4.00000 −0.141776
\(797\) −28.1769 −0.998078 −0.499039 0.866580i \(-0.666313\pi\)
−0.499039 + 0.866580i \(0.666313\pi\)
\(798\) −0.464102 −0.0164290
\(799\) 3.46410 0.122551
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −18.5359 −0.654525
\(803\) 0 0
\(804\) −16.7321 −0.590094
\(805\) 5.00000 0.176227
\(806\) 15.6603 0.551609
\(807\) −37.6410 −1.32503
\(808\) −11.5359 −0.405831
\(809\) −50.7846 −1.78549 −0.892746 0.450561i \(-0.851224\pi\)
−0.892746 + 0.450561i \(0.851224\pi\)
\(810\) 9.00000 0.316228
\(811\) −37.4641 −1.31554 −0.657771 0.753218i \(-0.728501\pi\)
−0.657771 + 0.753218i \(0.728501\pi\)
\(812\) 8.00000 0.280745
\(813\) 37.1769 1.30385
\(814\) 0 0
\(815\) −10.0000 −0.350285
\(816\) −1.26795 −0.0443871
\(817\) 1.41154 0.0493836
\(818\) 35.7128 1.24867
\(819\) 0 0
\(820\) 6.19615 0.216379
\(821\) 16.1436 0.563415 0.281708 0.959500i \(-0.409099\pi\)
0.281708 + 0.959500i \(0.409099\pi\)
\(822\) 25.3923 0.885658
\(823\) 1.21539 0.0423658 0.0211829 0.999776i \(-0.493257\pi\)
0.0211829 + 0.999776i \(0.493257\pi\)
\(824\) −10.7321 −0.373869
\(825\) 0 0
\(826\) −5.53590 −0.192619
\(827\) −45.9090 −1.59641 −0.798206 0.602385i \(-0.794217\pi\)
−0.798206 + 0.602385i \(0.794217\pi\)
\(828\) 0 0
\(829\) 24.1244 0.837874 0.418937 0.908015i \(-0.362403\pi\)
0.418937 + 0.908015i \(0.362403\pi\)
\(830\) −5.00000 −0.173553
\(831\) 3.12436 0.108383
\(832\) −5.73205 −0.198723
\(833\) −0.732051 −0.0253641
\(834\) −29.7846 −1.03136
\(835\) 0.339746 0.0117574
\(836\) 0 0
\(837\) 14.1962 0.490691
\(838\) 15.9282 0.550231
\(839\) 1.80385 0.0622757 0.0311379 0.999515i \(-0.490087\pi\)
0.0311379 + 0.999515i \(0.490087\pi\)
\(840\) −1.73205 −0.0597614
\(841\) 35.0000 1.20690
\(842\) −7.60770 −0.262178
\(843\) 13.3923 0.461255
\(844\) −8.00000 −0.275371
\(845\) −19.8564 −0.683081
\(846\) 0 0
\(847\) 0 0
\(848\) −6.19615 −0.212777
\(849\) 44.6603 1.53274
\(850\) −0.732051 −0.0251091
\(851\) −30.0000 −1.02839
\(852\) 12.0000 0.411113
\(853\) −35.8372 −1.22704 −0.613521 0.789679i \(-0.710247\pi\)
−0.613521 + 0.789679i \(0.710247\pi\)
\(854\) −2.92820 −0.100201
\(855\) 0 0
\(856\) 2.00000 0.0683586
\(857\) −36.3923 −1.24314 −0.621569 0.783360i \(-0.713504\pi\)
−0.621569 + 0.783360i \(0.713504\pi\)
\(858\) 0 0
\(859\) 31.8564 1.08693 0.543463 0.839433i \(-0.317113\pi\)
0.543463 + 0.839433i \(0.317113\pi\)
\(860\) 5.26795 0.179636
\(861\) −10.7321 −0.365747
\(862\) −34.4641 −1.17385
\(863\) −37.8564 −1.28865 −0.644324 0.764753i \(-0.722861\pi\)
−0.644324 + 0.764753i \(0.722861\pi\)
\(864\) −5.19615 −0.176777
\(865\) −7.85641 −0.267126
\(866\) −20.0526 −0.681414
\(867\) −28.5167 −0.968477
\(868\) −2.73205 −0.0927318
\(869\) 0 0
\(870\) −13.8564 −0.469776
\(871\) 55.3731 1.87624
\(872\) 20.0526 0.679065
\(873\) 0 0
\(874\) 1.33975 0.0453176
\(875\) −1.00000 −0.0338062
\(876\) 15.8038 0.533963
\(877\) 48.0526 1.62262 0.811310 0.584616i \(-0.198755\pi\)
0.811310 + 0.584616i \(0.198755\pi\)
\(878\) −14.7321 −0.497183
\(879\) 20.3205 0.685394
\(880\) 0 0
\(881\) 57.4256 1.93472 0.967359 0.253409i \(-0.0815518\pi\)
0.967359 + 0.253409i \(0.0815518\pi\)
\(882\) 0 0
\(883\) 36.6410 1.23307 0.616534 0.787328i \(-0.288536\pi\)
0.616534 + 0.787328i \(0.288536\pi\)
\(884\) 4.19615 0.141132
\(885\) 9.58846 0.322312
\(886\) 4.00000 0.134383
\(887\) 19.6603 0.660127 0.330063 0.943959i \(-0.392930\pi\)
0.330063 + 0.943959i \(0.392930\pi\)
\(888\) 10.3923 0.348743
\(889\) −6.12436 −0.205404
\(890\) 13.6603 0.457893
\(891\) 0 0
\(892\) −11.2679 −0.377279
\(893\) 1.26795 0.0424303
\(894\) 33.7128 1.12753
\(895\) 16.1962 0.541378
\(896\) 1.00000 0.0334077
\(897\) 49.6410 1.65747
\(898\) 31.3923 1.04757
\(899\) −21.8564 −0.728952
\(900\) 0 0
\(901\) 4.53590 0.151113
\(902\) 0 0
\(903\) −9.12436 −0.303640
\(904\) −14.1244 −0.469769
\(905\) 22.6603 0.753252
\(906\) −32.6603 −1.08506
\(907\) 36.0000 1.19536 0.597680 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) 17.3205 0.574801
\(909\) 0 0
\(910\) 5.73205 0.190016
\(911\) −21.9808 −0.728255 −0.364128 0.931349i \(-0.618633\pi\)
−0.364128 + 0.931349i \(0.618633\pi\)
\(912\) −0.464102 −0.0153679
\(913\) 0 0
\(914\) −3.39230 −0.112207
\(915\) 5.07180 0.167668
\(916\) 5.85641 0.193501
\(917\) −13.1962 −0.435775
\(918\) 3.80385 0.125546
\(919\) 39.8564 1.31474 0.657371 0.753567i \(-0.271669\pi\)
0.657371 + 0.753567i \(0.271669\pi\)
\(920\) 5.00000 0.164845
\(921\) −12.2487 −0.403609
\(922\) 10.5359 0.346981
\(923\) −39.7128 −1.30716
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 26.1769 0.860227
\(927\) 0 0
\(928\) 8.00000 0.262613
\(929\) −12.6795 −0.416001 −0.208000 0.978129i \(-0.566696\pi\)
−0.208000 + 0.978129i \(0.566696\pi\)
\(930\) 4.73205 0.155170
\(931\) −0.267949 −0.00878168
\(932\) 5.53590 0.181334
\(933\) 22.0526 0.721968
\(934\) 13.5885 0.444628
\(935\) 0 0
\(936\) 0 0
\(937\) 17.4641 0.570527 0.285264 0.958449i \(-0.407919\pi\)
0.285264 + 0.958449i \(0.407919\pi\)
\(938\) −9.66025 −0.315418
\(939\) −9.71281 −0.316966
\(940\) 4.73205 0.154342
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) −9.33975 −0.304305
\(943\) 30.9808 1.00887
\(944\) −5.53590 −0.180178
\(945\) 5.19615 0.169031
\(946\) 0 0
\(947\) −21.9474 −0.713196 −0.356598 0.934258i \(-0.616063\pi\)
−0.356598 + 0.934258i \(0.616063\pi\)
\(948\) −12.1244 −0.393781
\(949\) −52.3013 −1.69777
\(950\) −0.267949 −0.00869342
\(951\) 8.87564 0.287812
\(952\) −0.732051 −0.0237259
\(953\) −23.3923 −0.757751 −0.378876 0.925448i \(-0.623689\pi\)
−0.378876 + 0.925448i \(0.623689\pi\)
\(954\) 0 0
\(955\) 8.66025 0.280239
\(956\) −4.46410 −0.144379
\(957\) 0 0
\(958\) 28.9808 0.936326
\(959\) 14.6603 0.473404
\(960\) −1.73205 −0.0559017
\(961\) −23.5359 −0.759223
\(962\) −34.3923 −1.10885
\(963\) 0 0
\(964\) 17.4641 0.562481
\(965\) −11.3923 −0.366731
\(966\) −8.66025 −0.278639
\(967\) −28.7846 −0.925651 −0.462825 0.886450i \(-0.653164\pi\)
−0.462825 + 0.886450i \(0.653164\pi\)
\(968\) 0 0
\(969\) 0.339746 0.0109142
\(970\) −10.9282 −0.350883
\(971\) 50.7128 1.62745 0.813726 0.581249i \(-0.197436\pi\)
0.813726 + 0.581249i \(0.197436\pi\)
\(972\) 0 0
\(973\) −17.1962 −0.551283
\(974\) 17.3923 0.557285
\(975\) −9.92820 −0.317957
\(976\) −2.92820 −0.0937295
\(977\) −14.2679 −0.456472 −0.228236 0.973606i \(-0.573296\pi\)
−0.228236 + 0.973606i \(0.573296\pi\)
\(978\) 17.3205 0.553849
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 9.66025 0.308271
\(983\) 37.6077 1.19950 0.599750 0.800188i \(-0.295267\pi\)
0.599750 + 0.800188i \(0.295267\pi\)
\(984\) −10.7321 −0.342125
\(985\) 9.46410 0.301551
\(986\) −5.85641 −0.186506
\(987\) −8.19615 −0.260886
\(988\) 1.53590 0.0488634
\(989\) 26.3397 0.837555
\(990\) 0 0
\(991\) −1.87564 −0.0595818 −0.0297909 0.999556i \(-0.509484\pi\)
−0.0297909 + 0.999556i \(0.509484\pi\)
\(992\) −2.73205 −0.0867427
\(993\) 58.9808 1.87170
\(994\) 6.92820 0.219749
\(995\) 4.00000 0.126809
\(996\) 8.66025 0.274411
\(997\) 38.6603 1.22438 0.612191 0.790710i \(-0.290288\pi\)
0.612191 + 0.790710i \(0.290288\pi\)
\(998\) −22.0000 −0.696398
\(999\) −31.1769 −0.986394
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.bz.1.2 yes 2
11.10 odd 2 8470.2.a.bm.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.bm.1.2 2 11.10 odd 2
8470.2.a.bz.1.2 yes 2 1.1 even 1 trivial