Properties

Label 8470.2.a.df.1.2
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: 6.6.10784448.1
Defining polynomial: \(x^{6} - 11 x^{4} - 4 x^{3} + 31 x^{2} + 22 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.40765\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.34292 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.34292 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.19656 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.34292 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.34292 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.19656 q^{9} +1.00000 q^{10} -1.34292 q^{12} +4.31157 q^{13} +1.00000 q^{14} -1.34292 q^{15} +1.00000 q^{16} -2.63565 q^{17} -1.19656 q^{18} +1.56877 q^{19} +1.00000 q^{20} -1.34292 q^{21} +5.45794 q^{23} -1.34292 q^{24} +1.00000 q^{25} +4.31157 q^{26} +5.63565 q^{27} +1.00000 q^{28} +2.83920 q^{29} -1.34292 q^{30} +1.14637 q^{31} +1.00000 q^{32} -2.63565 q^{34} +1.00000 q^{35} -1.19656 q^{36} -0.839198 q^{37} +1.56877 q^{38} -5.79011 q^{39} +1.00000 q^{40} +3.47854 q^{41} -1.34292 q^{42} -2.01075 q^{43} -1.19656 q^{45} +5.45794 q^{46} -8.39442 q^{47} -1.34292 q^{48} +1.00000 q^{49} +1.00000 q^{50} +3.53948 q^{51} +4.31157 q^{52} +1.40604 q^{53} +5.63565 q^{54} +1.00000 q^{56} -2.10674 q^{57} +2.83920 q^{58} +2.32408 q^{59} -1.34292 q^{60} -5.24387 q^{61} +1.14637 q^{62} -1.19656 q^{63} +1.00000 q^{64} +4.31157 q^{65} -1.60636 q^{67} -2.63565 q^{68} -7.32959 q^{69} +1.00000 q^{70} -13.8057 q^{71} -1.19656 q^{72} -1.14830 q^{73} -0.839198 q^{74} -1.34292 q^{75} +1.56877 q^{76} -5.79011 q^{78} +9.65408 q^{79} +1.00000 q^{80} -3.97858 q^{81} +3.47854 q^{82} -2.51880 q^{83} -1.34292 q^{84} -2.63565 q^{85} -2.01075 q^{86} -3.81282 q^{87} +2.19326 q^{89} -1.19656 q^{90} +4.31157 q^{91} +5.45794 q^{92} -1.53948 q^{93} -8.39442 q^{94} +1.56877 q^{95} -1.34292 q^{96} -7.56372 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 4q^{3} + 6q^{4} + 6q^{5} + 4q^{6} + 6q^{7} + 6q^{8} + 2q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + 4q^{3} + 6q^{4} + 6q^{5} + 4q^{6} + 6q^{7} + 6q^{8} + 2q^{9} + 6q^{10} + 4q^{12} + 6q^{14} + 4q^{15} + 6q^{16} + 2q^{17} + 2q^{18} + 6q^{20} + 4q^{21} + 4q^{23} + 4q^{24} + 6q^{25} + 16q^{27} + 6q^{28} + 8q^{29} + 4q^{30} + 4q^{31} + 6q^{32} + 2q^{34} + 6q^{35} + 2q^{36} + 4q^{37} + 6q^{40} + 12q^{41} + 4q^{42} - 6q^{43} + 2q^{45} + 4q^{46} + 16q^{47} + 4q^{48} + 6q^{49} + 6q^{50} + 12q^{53} + 16q^{54} + 6q^{56} + 8q^{57} + 8q^{58} + 22q^{59} + 4q^{60} - 4q^{61} + 4q^{62} + 2q^{63} + 6q^{64} + 20q^{67} + 2q^{68} + 12q^{69} + 6q^{70} + 14q^{71} + 2q^{72} + 18q^{73} + 4q^{74} + 4q^{75} + 32q^{79} + 6q^{80} + 6q^{81} + 12q^{82} - 16q^{83} + 4q^{84} + 2q^{85} - 6q^{86} - 4q^{87} + 4q^{89} + 2q^{90} + 4q^{92} + 12q^{93} + 16q^{94} + 4q^{96} + 4q^{97} + 6q^{98} + O(q^{100}) \)

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
\(3\) −1.34292 −0.775337 −0.387669 0.921799i \(-0.626719\pi\)
−0.387669 + 0.921799i \(0.626719\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.34292 −0.548246
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −1.19656 −0.398853
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −1.34292 −0.387669
\(13\) 4.31157 1.19582 0.597908 0.801565i \(-0.295999\pi\)
0.597908 + 0.801565i \(0.295999\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.34292 −0.346741
\(16\) 1.00000 0.250000
\(17\) −2.63565 −0.639240 −0.319620 0.947546i \(-0.603555\pi\)
−0.319620 + 0.947546i \(0.603555\pi\)
\(18\) −1.19656 −0.282031
\(19\) 1.56877 0.359901 0.179951 0.983676i \(-0.442406\pi\)
0.179951 + 0.983676i \(0.442406\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.34292 −0.293050
\(22\) 0 0
\(23\) 5.45794 1.13806 0.569030 0.822317i \(-0.307319\pi\)
0.569030 + 0.822317i \(0.307319\pi\)
\(24\) −1.34292 −0.274123
\(25\) 1.00000 0.200000
\(26\) 4.31157 0.845569
\(27\) 5.63565 1.08458
\(28\) 1.00000 0.188982
\(29\) 2.83920 0.527226 0.263613 0.964629i \(-0.415086\pi\)
0.263613 + 0.964629i \(0.415086\pi\)
\(30\) −1.34292 −0.245183
\(31\) 1.14637 0.205893 0.102947 0.994687i \(-0.467173\pi\)
0.102947 + 0.994687i \(0.467173\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.63565 −0.452011
\(35\) 1.00000 0.169031
\(36\) −1.19656 −0.199426
\(37\) −0.839198 −0.137963 −0.0689816 0.997618i \(-0.521975\pi\)
−0.0689816 + 0.997618i \(0.521975\pi\)
\(38\) 1.56877 0.254489
\(39\) −5.79011 −0.927160
\(40\) 1.00000 0.158114
\(41\) 3.47854 0.543256 0.271628 0.962402i \(-0.412438\pi\)
0.271628 + 0.962402i \(0.412438\pi\)
\(42\) −1.34292 −0.207218
\(43\) −2.01075 −0.306637 −0.153318 0.988177i \(-0.548996\pi\)
−0.153318 + 0.988177i \(0.548996\pi\)
\(44\) 0 0
\(45\) −1.19656 −0.178372
\(46\) 5.45794 0.804729
\(47\) −8.39442 −1.22445 −0.612226 0.790683i \(-0.709726\pi\)
−0.612226 + 0.790683i \(0.709726\pi\)
\(48\) −1.34292 −0.193834
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 3.53948 0.495626
\(52\) 4.31157 0.597908
\(53\) 1.40604 0.193134 0.0965672 0.995326i \(-0.469214\pi\)
0.0965672 + 0.995326i \(0.469214\pi\)
\(54\) 5.63565 0.766915
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −2.10674 −0.279045
\(58\) 2.83920 0.372805
\(59\) 2.32408 0.302569 0.151285 0.988490i \(-0.451659\pi\)
0.151285 + 0.988490i \(0.451659\pi\)
\(60\) −1.34292 −0.173371
\(61\) −5.24387 −0.671409 −0.335704 0.941967i \(-0.608974\pi\)
−0.335704 + 0.941967i \(0.608974\pi\)
\(62\) 1.14637 0.145589
\(63\) −1.19656 −0.150752
\(64\) 1.00000 0.125000
\(65\) 4.31157 0.534785
\(66\) 0 0
\(67\) −1.60636 −0.196248 −0.0981241 0.995174i \(-0.531284\pi\)
−0.0981241 + 0.995174i \(0.531284\pi\)
\(68\) −2.63565 −0.319620
\(69\) −7.32959 −0.882379
\(70\) 1.00000 0.119523
\(71\) −13.8057 −1.63844 −0.819220 0.573480i \(-0.805593\pi\)
−0.819220 + 0.573480i \(0.805593\pi\)
\(72\) −1.19656 −0.141016
\(73\) −1.14830 −0.134398 −0.0671990 0.997740i \(-0.521406\pi\)
−0.0671990 + 0.997740i \(0.521406\pi\)
\(74\) −0.839198 −0.0975548
\(75\) −1.34292 −0.155067
\(76\) 1.56877 0.179951
\(77\) 0 0
\(78\) −5.79011 −0.655601
\(79\) 9.65408 1.08617 0.543084 0.839678i \(-0.317256\pi\)
0.543084 + 0.839678i \(0.317256\pi\)
\(80\) 1.00000 0.111803
\(81\) −3.97858 −0.442064
\(82\) 3.47854 0.384140
\(83\) −2.51880 −0.276475 −0.138237 0.990399i \(-0.544144\pi\)
−0.138237 + 0.990399i \(0.544144\pi\)
\(84\) −1.34292 −0.146525
\(85\) −2.63565 −0.285877
\(86\) −2.01075 −0.216825
\(87\) −3.81282 −0.408778
\(88\) 0 0
\(89\) 2.19326 0.232485 0.116242 0.993221i \(-0.462915\pi\)
0.116242 + 0.993221i \(0.462915\pi\)
\(90\) −1.19656 −0.126128
\(91\) 4.31157 0.451976
\(92\) 5.45794 0.569030
\(93\) −1.53948 −0.159637
\(94\) −8.39442 −0.865818
\(95\) 1.56877 0.160953
\(96\) −1.34292 −0.137062
\(97\) −7.56372 −0.767979 −0.383990 0.923337i \(-0.625450\pi\)
−0.383990 + 0.923337i \(0.625450\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 17.6449 1.75574 0.877868 0.478902i \(-0.158965\pi\)
0.877868 + 0.478902i \(0.158965\pi\)
\(102\) 3.53948 0.350461
\(103\) 10.1528 1.00039 0.500194 0.865913i \(-0.333262\pi\)
0.500194 + 0.865913i \(0.333262\pi\)
\(104\) 4.31157 0.422785
\(105\) −1.34292 −0.131056
\(106\) 1.40604 0.136567
\(107\) 15.4232 1.49102 0.745508 0.666497i \(-0.232207\pi\)
0.745508 + 0.666497i \(0.232207\pi\)
\(108\) 5.63565 0.542291
\(109\) 8.49656 0.813823 0.406911 0.913468i \(-0.366606\pi\)
0.406911 + 0.913468i \(0.366606\pi\)
\(110\) 0 0
\(111\) 1.12698 0.106968
\(112\) 1.00000 0.0944911
\(113\) 2.72299 0.256158 0.128079 0.991764i \(-0.459119\pi\)
0.128079 + 0.991764i \(0.459119\pi\)
\(114\) −2.10674 −0.197314
\(115\) 5.45794 0.508956
\(116\) 2.83920 0.263613
\(117\) −5.15905 −0.476954
\(118\) 2.32408 0.213949
\(119\) −2.63565 −0.241610
\(120\) −1.34292 −0.122592
\(121\) 0 0
\(122\) −5.24387 −0.474758
\(123\) −4.67141 −0.421207
\(124\) 1.14637 0.102947
\(125\) 1.00000 0.0894427
\(126\) −1.19656 −0.106598
\(127\) −9.21928 −0.818079 −0.409040 0.912517i \(-0.634136\pi\)
−0.409040 + 0.912517i \(0.634136\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.70028 0.237747
\(130\) 4.31157 0.378150
\(131\) 17.5357 1.53210 0.766052 0.642779i \(-0.222219\pi\)
0.766052 + 0.642779i \(0.222219\pi\)
\(132\) 0 0
\(133\) 1.56877 0.136030
\(134\) −1.60636 −0.138768
\(135\) 5.63565 0.485040
\(136\) −2.63565 −0.226005
\(137\) 9.12927 0.779966 0.389983 0.920822i \(-0.372481\pi\)
0.389983 + 0.920822i \(0.372481\pi\)
\(138\) −7.32959 −0.623936
\(139\) 0.657844 0.0557976 0.0278988 0.999611i \(-0.491118\pi\)
0.0278988 + 0.999611i \(0.491118\pi\)
\(140\) 1.00000 0.0845154
\(141\) 11.2731 0.949363
\(142\) −13.8057 −1.15855
\(143\) 0 0
\(144\) −1.19656 −0.0997131
\(145\) 2.83920 0.235783
\(146\) −1.14830 −0.0950337
\(147\) −1.34292 −0.110762
\(148\) −0.839198 −0.0689816
\(149\) −20.5982 −1.68747 −0.843737 0.536757i \(-0.819649\pi\)
−0.843737 + 0.536757i \(0.819649\pi\)
\(150\) −1.34292 −0.109649
\(151\) −4.29219 −0.349293 −0.174647 0.984631i \(-0.555878\pi\)
−0.174647 + 0.984631i \(0.555878\pi\)
\(152\) 1.56877 0.127244
\(153\) 3.15371 0.254962
\(154\) 0 0
\(155\) 1.14637 0.0920783
\(156\) −5.79011 −0.463580
\(157\) 10.9039 0.870226 0.435113 0.900376i \(-0.356708\pi\)
0.435113 + 0.900376i \(0.356708\pi\)
\(158\) 9.65408 0.768037
\(159\) −1.88820 −0.149744
\(160\) 1.00000 0.0790569
\(161\) 5.45794 0.430146
\(162\) −3.97858 −0.312587
\(163\) 12.3525 0.967526 0.483763 0.875199i \(-0.339270\pi\)
0.483763 + 0.875199i \(0.339270\pi\)
\(164\) 3.47854 0.271628
\(165\) 0 0
\(166\) −2.51880 −0.195497
\(167\) −6.93334 −0.536518 −0.268259 0.963347i \(-0.586448\pi\)
−0.268259 + 0.963347i \(0.586448\pi\)
\(168\) −1.34292 −0.103609
\(169\) 5.58967 0.429975
\(170\) −2.63565 −0.202145
\(171\) −1.87713 −0.143548
\(172\) −2.01075 −0.153318
\(173\) −11.5843 −0.880741 −0.440370 0.897816i \(-0.645153\pi\)
−0.440370 + 0.897816i \(0.645153\pi\)
\(174\) −3.81282 −0.289049
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −3.12106 −0.234593
\(178\) 2.19326 0.164392
\(179\) 20.4825 1.53093 0.765466 0.643476i \(-0.222508\pi\)
0.765466 + 0.643476i \(0.222508\pi\)
\(180\) −1.19656 −0.0891861
\(181\) −8.01175 −0.595509 −0.297754 0.954642i \(-0.596238\pi\)
−0.297754 + 0.954642i \(0.596238\pi\)
\(182\) 4.31157 0.319595
\(183\) 7.04211 0.520568
\(184\) 5.45794 0.402365
\(185\) −0.839198 −0.0616990
\(186\) −1.53948 −0.112880
\(187\) 0 0
\(188\) −8.39442 −0.612226
\(189\) 5.63565 0.409934
\(190\) 1.56877 0.113811
\(191\) 7.72371 0.558868 0.279434 0.960165i \(-0.409853\pi\)
0.279434 + 0.960165i \(0.409853\pi\)
\(192\) −1.34292 −0.0969171
\(193\) −16.3855 −1.17945 −0.589727 0.807603i \(-0.700765\pi\)
−0.589727 + 0.807603i \(0.700765\pi\)
\(194\) −7.56372 −0.543043
\(195\) −5.79011 −0.414639
\(196\) 1.00000 0.0714286
\(197\) 5.51971 0.393263 0.196631 0.980477i \(-0.437000\pi\)
0.196631 + 0.980477i \(0.437000\pi\)
\(198\) 0 0
\(199\) 4.27443 0.303006 0.151503 0.988457i \(-0.451589\pi\)
0.151503 + 0.988457i \(0.451589\pi\)
\(200\) 1.00000 0.0707107
\(201\) 2.15722 0.152159
\(202\) 17.6449 1.24149
\(203\) 2.83920 0.199273
\(204\) 3.53948 0.247813
\(205\) 3.47854 0.242952
\(206\) 10.1528 0.707381
\(207\) −6.53074 −0.453918
\(208\) 4.31157 0.298954
\(209\) 0 0
\(210\) −1.34292 −0.0926705
\(211\) −19.6106 −1.35005 −0.675026 0.737794i \(-0.735868\pi\)
−0.675026 + 0.737794i \(0.735868\pi\)
\(212\) 1.40604 0.0965672
\(213\) 18.5400 1.27034
\(214\) 15.4232 1.05431
\(215\) −2.01075 −0.137132
\(216\) 5.63565 0.383458
\(217\) 1.14637 0.0778204
\(218\) 8.49656 0.575459
\(219\) 1.54207 0.104204
\(220\) 0 0
\(221\) −11.3638 −0.764413
\(222\) 1.12698 0.0756378
\(223\) 4.01943 0.269161 0.134580 0.990903i \(-0.457031\pi\)
0.134580 + 0.990903i \(0.457031\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.19656 −0.0797705
\(226\) 2.72299 0.181131
\(227\) 20.4787 1.35922 0.679610 0.733574i \(-0.262149\pi\)
0.679610 + 0.733574i \(0.262149\pi\)
\(228\) −2.10674 −0.139522
\(229\) 8.74872 0.578132 0.289066 0.957309i \(-0.406655\pi\)
0.289066 + 0.957309i \(0.406655\pi\)
\(230\) 5.45794 0.359886
\(231\) 0 0
\(232\) 2.83920 0.186402
\(233\) 14.1462 0.926748 0.463374 0.886163i \(-0.346639\pi\)
0.463374 + 0.886163i \(0.346639\pi\)
\(234\) −5.15905 −0.337257
\(235\) −8.39442 −0.547591
\(236\) 2.32408 0.151285
\(237\) −12.9647 −0.842147
\(238\) −2.63565 −0.170844
\(239\) 2.12134 0.137218 0.0686091 0.997644i \(-0.478144\pi\)
0.0686091 + 0.997644i \(0.478144\pi\)
\(240\) −1.34292 −0.0866853
\(241\) 17.4357 1.12313 0.561566 0.827432i \(-0.310199\pi\)
0.561566 + 0.827432i \(0.310199\pi\)
\(242\) 0 0
\(243\) −11.5640 −0.741833
\(244\) −5.24387 −0.335704
\(245\) 1.00000 0.0638877
\(246\) −4.67141 −0.297838
\(247\) 6.76388 0.430376
\(248\) 1.14637 0.0727943
\(249\) 3.38256 0.214361
\(250\) 1.00000 0.0632456
\(251\) 0.936840 0.0591328 0.0295664 0.999563i \(-0.490587\pi\)
0.0295664 + 0.999563i \(0.490587\pi\)
\(252\) −1.19656 −0.0753760
\(253\) 0 0
\(254\) −9.21928 −0.578469
\(255\) 3.53948 0.221651
\(256\) 1.00000 0.0625000
\(257\) 13.7242 0.856095 0.428047 0.903756i \(-0.359202\pi\)
0.428047 + 0.903756i \(0.359202\pi\)
\(258\) 2.70028 0.168112
\(259\) −0.839198 −0.0521452
\(260\) 4.31157 0.267392
\(261\) −3.39726 −0.210285
\(262\) 17.5357 1.08336
\(263\) −18.9341 −1.16753 −0.583763 0.811924i \(-0.698420\pi\)
−0.583763 + 0.811924i \(0.698420\pi\)
\(264\) 0 0
\(265\) 1.40604 0.0863723
\(266\) 1.56877 0.0961877
\(267\) −2.94538 −0.180254
\(268\) −1.60636 −0.0981241
\(269\) 12.9639 0.790422 0.395211 0.918590i \(-0.370672\pi\)
0.395211 + 0.918590i \(0.370672\pi\)
\(270\) 5.63565 0.342975
\(271\) 23.5085 1.42804 0.714020 0.700125i \(-0.246873\pi\)
0.714020 + 0.700125i \(0.246873\pi\)
\(272\) −2.63565 −0.159810
\(273\) −5.79011 −0.350434
\(274\) 9.12927 0.551519
\(275\) 0 0
\(276\) −7.32959 −0.441190
\(277\) 2.44560 0.146942 0.0734710 0.997297i \(-0.476592\pi\)
0.0734710 + 0.997297i \(0.476592\pi\)
\(278\) 0.657844 0.0394549
\(279\) −1.37169 −0.0821211
\(280\) 1.00000 0.0597614
\(281\) −18.0116 −1.07448 −0.537242 0.843428i \(-0.680534\pi\)
−0.537242 + 0.843428i \(0.680534\pi\)
\(282\) 11.2731 0.671301
\(283\) −9.78930 −0.581914 −0.290957 0.956736i \(-0.593974\pi\)
−0.290957 + 0.956736i \(0.593974\pi\)
\(284\) −13.8057 −0.819220
\(285\) −2.10674 −0.124793
\(286\) 0 0
\(287\) 3.47854 0.205332
\(288\) −1.19656 −0.0705078
\(289\) −10.0533 −0.591372
\(290\) 2.83920 0.166723
\(291\) 10.1575 0.595443
\(292\) −1.14830 −0.0671990
\(293\) −8.44836 −0.493558 −0.246779 0.969072i \(-0.579372\pi\)
−0.246779 + 0.969072i \(0.579372\pi\)
\(294\) −1.34292 −0.0783209
\(295\) 2.32408 0.135313
\(296\) −0.839198 −0.0487774
\(297\) 0 0
\(298\) −20.5982 −1.19322
\(299\) 23.5323 1.36091
\(300\) −1.34292 −0.0775337
\(301\) −2.01075 −0.115898
\(302\) −4.29219 −0.246988
\(303\) −23.6958 −1.36129
\(304\) 1.56877 0.0899753
\(305\) −5.24387 −0.300263
\(306\) 3.15371 0.180286
\(307\) −9.34689 −0.533455 −0.266728 0.963772i \(-0.585942\pi\)
−0.266728 + 0.963772i \(0.585942\pi\)
\(308\) 0 0
\(309\) −13.6345 −0.775638
\(310\) 1.14637 0.0651092
\(311\) −27.6729 −1.56919 −0.784593 0.620011i \(-0.787128\pi\)
−0.784593 + 0.620011i \(0.787128\pi\)
\(312\) −5.79011 −0.327801
\(313\) 5.34524 0.302131 0.151065 0.988524i \(-0.451730\pi\)
0.151065 + 0.988524i \(0.451730\pi\)
\(314\) 10.9039 0.615343
\(315\) −1.19656 −0.0674184
\(316\) 9.65408 0.543084
\(317\) 7.13917 0.400976 0.200488 0.979696i \(-0.435747\pi\)
0.200488 + 0.979696i \(0.435747\pi\)
\(318\) −1.88820 −0.105885
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −20.7122 −1.15604
\(322\) 5.45794 0.304159
\(323\) −4.13474 −0.230063
\(324\) −3.97858 −0.221032
\(325\) 4.31157 0.239163
\(326\) 12.3525 0.684144
\(327\) −11.4102 −0.630987
\(328\) 3.47854 0.192070
\(329\) −8.39442 −0.462799
\(330\) 0 0
\(331\) 5.83207 0.320559 0.160280 0.987072i \(-0.448760\pi\)
0.160280 + 0.987072i \(0.448760\pi\)
\(332\) −2.51880 −0.138237
\(333\) 1.00415 0.0550270
\(334\) −6.93334 −0.379375
\(335\) −1.60636 −0.0877649
\(336\) −1.34292 −0.0732625
\(337\) 22.2848 1.21393 0.606964 0.794729i \(-0.292387\pi\)
0.606964 + 0.794729i \(0.292387\pi\)
\(338\) 5.58967 0.304038
\(339\) −3.65677 −0.198609
\(340\) −2.63565 −0.142938
\(341\) 0 0
\(342\) −1.87713 −0.101503
\(343\) 1.00000 0.0539949
\(344\) −2.01075 −0.108412
\(345\) −7.32959 −0.394612
\(346\) −11.5843 −0.622778
\(347\) 16.6253 0.892495 0.446248 0.894910i \(-0.352760\pi\)
0.446248 + 0.894910i \(0.352760\pi\)
\(348\) −3.81282 −0.204389
\(349\) 3.23934 0.173398 0.0866989 0.996235i \(-0.472368\pi\)
0.0866989 + 0.996235i \(0.472368\pi\)
\(350\) 1.00000 0.0534522
\(351\) 24.2985 1.29696
\(352\) 0 0
\(353\) −24.6580 −1.31241 −0.656205 0.754582i \(-0.727839\pi\)
−0.656205 + 0.754582i \(0.727839\pi\)
\(354\) −3.12106 −0.165882
\(355\) −13.8057 −0.732732
\(356\) 2.19326 0.116242
\(357\) 3.53948 0.187329
\(358\) 20.4825 1.08253
\(359\) 20.7493 1.09511 0.547554 0.836771i \(-0.315559\pi\)
0.547554 + 0.836771i \(0.315559\pi\)
\(360\) −1.19656 −0.0630641
\(361\) −16.5390 −0.870471
\(362\) −8.01175 −0.421088
\(363\) 0 0
\(364\) 4.31157 0.225988
\(365\) −1.14830 −0.0601046
\(366\) 7.04211 0.368097
\(367\) 2.12160 0.110746 0.0553732 0.998466i \(-0.482365\pi\)
0.0553732 + 0.998466i \(0.482365\pi\)
\(368\) 5.45794 0.284515
\(369\) −4.16227 −0.216679
\(370\) −0.839198 −0.0436278
\(371\) 1.40604 0.0729979
\(372\) −1.53948 −0.0798184
\(373\) −7.19996 −0.372800 −0.186400 0.982474i \(-0.559682\pi\)
−0.186400 + 0.982474i \(0.559682\pi\)
\(374\) 0 0
\(375\) −1.34292 −0.0693482
\(376\) −8.39442 −0.432909
\(377\) 12.2414 0.630465
\(378\) 5.63565 0.289867
\(379\) −23.1300 −1.18811 −0.594055 0.804424i \(-0.702474\pi\)
−0.594055 + 0.804424i \(0.702474\pi\)
\(380\) 1.56877 0.0804764
\(381\) 12.3808 0.634287
\(382\) 7.72371 0.395180
\(383\) 31.4479 1.60691 0.803455 0.595365i \(-0.202993\pi\)
0.803455 + 0.595365i \(0.202993\pi\)
\(384\) −1.34292 −0.0685308
\(385\) 0 0
\(386\) −16.3855 −0.834000
\(387\) 2.40598 0.122303
\(388\) −7.56372 −0.383990
\(389\) 17.5618 0.890420 0.445210 0.895426i \(-0.353129\pi\)
0.445210 + 0.895426i \(0.353129\pi\)
\(390\) −5.79011 −0.293194
\(391\) −14.3852 −0.727493
\(392\) 1.00000 0.0505076
\(393\) −23.5491 −1.18790
\(394\) 5.51971 0.278079
\(395\) 9.65408 0.485749
\(396\) 0 0
\(397\) 32.7839 1.64538 0.822689 0.568492i \(-0.192473\pi\)
0.822689 + 0.568492i \(0.192473\pi\)
\(398\) 4.27443 0.214258
\(399\) −2.10674 −0.105469
\(400\) 1.00000 0.0500000
\(401\) 9.25146 0.461996 0.230998 0.972954i \(-0.425801\pi\)
0.230998 + 0.972954i \(0.425801\pi\)
\(402\) 2.15722 0.107592
\(403\) 4.94264 0.246210
\(404\) 17.6449 0.877868
\(405\) −3.97858 −0.197697
\(406\) 2.83920 0.140907
\(407\) 0 0
\(408\) 3.53948 0.175230
\(409\) −0.0174947 −0.000865057 0 −0.000432529 1.00000i \(-0.500138\pi\)
−0.000432529 1.00000i \(0.500138\pi\)
\(410\) 3.47854 0.171793
\(411\) −12.2599 −0.604737
\(412\) 10.1528 0.500194
\(413\) 2.32408 0.114360
\(414\) −6.53074 −0.320968
\(415\) −2.51880 −0.123643
\(416\) 4.31157 0.211392
\(417\) −0.883434 −0.0432620
\(418\) 0 0
\(419\) 32.4579 1.58567 0.792837 0.609433i \(-0.208603\pi\)
0.792837 + 0.609433i \(0.208603\pi\)
\(420\) −1.34292 −0.0655279
\(421\) 9.15329 0.446104 0.223052 0.974807i \(-0.428398\pi\)
0.223052 + 0.974807i \(0.428398\pi\)
\(422\) −19.6106 −0.954631
\(423\) 10.0444 0.488376
\(424\) 1.40604 0.0682833
\(425\) −2.63565 −0.127848
\(426\) 18.5400 0.898268
\(427\) −5.24387 −0.253769
\(428\) 15.4232 0.745508
\(429\) 0 0
\(430\) −2.01075 −0.0969670
\(431\) −26.0024 −1.25249 −0.626247 0.779625i \(-0.715410\pi\)
−0.626247 + 0.779625i \(0.715410\pi\)
\(432\) 5.63565 0.271146
\(433\) 23.6006 1.13417 0.567085 0.823659i \(-0.308071\pi\)
0.567085 + 0.823659i \(0.308071\pi\)
\(434\) 1.14637 0.0550273
\(435\) −3.81282 −0.182811
\(436\) 8.49656 0.406911
\(437\) 8.56227 0.409589
\(438\) 1.54207 0.0736832
\(439\) 31.7894 1.51722 0.758612 0.651543i \(-0.225878\pi\)
0.758612 + 0.651543i \(0.225878\pi\)
\(440\) 0 0
\(441\) −1.19656 −0.0569789
\(442\) −11.3638 −0.540522
\(443\) 36.1071 1.71550 0.857750 0.514067i \(-0.171862\pi\)
0.857750 + 0.514067i \(0.171862\pi\)
\(444\) 1.12698 0.0534840
\(445\) 2.19326 0.103970
\(446\) 4.01943 0.190326
\(447\) 27.6619 1.30836
\(448\) 1.00000 0.0472456
\(449\) −4.52269 −0.213439 −0.106719 0.994289i \(-0.534035\pi\)
−0.106719 + 0.994289i \(0.534035\pi\)
\(450\) −1.19656 −0.0564063
\(451\) 0 0
\(452\) 2.72299 0.128079
\(453\) 5.76408 0.270820
\(454\) 20.4787 0.961114
\(455\) 4.31157 0.202130
\(456\) −2.10674 −0.0986572
\(457\) −15.1593 −0.709120 −0.354560 0.935033i \(-0.615369\pi\)
−0.354560 + 0.935033i \(0.615369\pi\)
\(458\) 8.74872 0.408801
\(459\) −14.8536 −0.693308
\(460\) 5.45794 0.254478
\(461\) −27.6126 −1.28605 −0.643024 0.765846i \(-0.722320\pi\)
−0.643024 + 0.765846i \(0.722320\pi\)
\(462\) 0 0
\(463\) 13.3557 0.620692 0.310346 0.950624i \(-0.399555\pi\)
0.310346 + 0.950624i \(0.399555\pi\)
\(464\) 2.83920 0.131806
\(465\) −1.53948 −0.0713917
\(466\) 14.1462 0.655310
\(467\) 26.3287 1.21835 0.609173 0.793038i \(-0.291502\pi\)
0.609173 + 0.793038i \(0.291502\pi\)
\(468\) −5.15905 −0.238477
\(469\) −1.60636 −0.0741749
\(470\) −8.39442 −0.387206
\(471\) −14.6431 −0.674719
\(472\) 2.32408 0.106974
\(473\) 0 0
\(474\) −12.9647 −0.595488
\(475\) 1.56877 0.0719803
\(476\) −2.63565 −0.120805
\(477\) −1.68241 −0.0770322
\(478\) 2.12134 0.0970279
\(479\) 8.18258 0.373872 0.186936 0.982372i \(-0.440144\pi\)
0.186936 + 0.982372i \(0.440144\pi\)
\(480\) −1.34292 −0.0612958
\(481\) −3.61826 −0.164979
\(482\) 17.4357 0.794174
\(483\) −7.32959 −0.333508
\(484\) 0 0
\(485\) −7.56372 −0.343451
\(486\) −11.5640 −0.524555
\(487\) −9.09993 −0.412357 −0.206179 0.978514i \(-0.566103\pi\)
−0.206179 + 0.978514i \(0.566103\pi\)
\(488\) −5.24387 −0.237379
\(489\) −16.5885 −0.750159
\(490\) 1.00000 0.0451754
\(491\) −3.80239 −0.171600 −0.0857998 0.996312i \(-0.527345\pi\)
−0.0857998 + 0.996312i \(0.527345\pi\)
\(492\) −4.67141 −0.210603
\(493\) −7.48314 −0.337024
\(494\) 6.76388 0.304321
\(495\) 0 0
\(496\) 1.14637 0.0514733
\(497\) −13.8057 −0.619272
\(498\) 3.38256 0.151576
\(499\) −22.9856 −1.02898 −0.514488 0.857497i \(-0.672018\pi\)
−0.514488 + 0.857497i \(0.672018\pi\)
\(500\) 1.00000 0.0447214
\(501\) 9.31094 0.415982
\(502\) 0.936840 0.0418132
\(503\) 13.4287 0.598756 0.299378 0.954135i \(-0.403221\pi\)
0.299378 + 0.954135i \(0.403221\pi\)
\(504\) −1.19656 −0.0532989
\(505\) 17.6449 0.785189
\(506\) 0 0
\(507\) −7.50650 −0.333375
\(508\) −9.21928 −0.409040
\(509\) 36.9453 1.63757 0.818785 0.574100i \(-0.194648\pi\)
0.818785 + 0.574100i \(0.194648\pi\)
\(510\) 3.53948 0.156731
\(511\) −1.14830 −0.0507977
\(512\) 1.00000 0.0441942
\(513\) 8.84106 0.390342
\(514\) 13.7242 0.605350
\(515\) 10.1528 0.447387
\(516\) 2.70028 0.118873
\(517\) 0 0
\(518\) −0.839198 −0.0368722
\(519\) 15.5569 0.682871
\(520\) 4.31157 0.189075
\(521\) 12.8367 0.562388 0.281194 0.959651i \(-0.409270\pi\)
0.281194 + 0.959651i \(0.409270\pi\)
\(522\) −3.39726 −0.148694
\(523\) −8.66972 −0.379100 −0.189550 0.981871i \(-0.560703\pi\)
−0.189550 + 0.981871i \(0.560703\pi\)
\(524\) 17.5357 0.766052
\(525\) −1.34292 −0.0586100
\(526\) −18.9341 −0.825565
\(527\) −3.02142 −0.131615
\(528\) 0 0
\(529\) 6.78911 0.295179
\(530\) 1.40604 0.0610745
\(531\) −2.78090 −0.120681
\(532\) 1.56877 0.0680149
\(533\) 14.9980 0.649634
\(534\) −2.94538 −0.127459
\(535\) 15.4232 0.666803
\(536\) −1.60636 −0.0693842
\(537\) −27.5064 −1.18699
\(538\) 12.9639 0.558913
\(539\) 0 0
\(540\) 5.63565 0.242520
\(541\) −38.6634 −1.66227 −0.831136 0.556070i \(-0.812309\pi\)
−0.831136 + 0.556070i \(0.812309\pi\)
\(542\) 23.5085 1.00978
\(543\) 10.7592 0.461720
\(544\) −2.63565 −0.113003
\(545\) 8.49656 0.363953
\(546\) −5.79011 −0.247794
\(547\) −6.20768 −0.265421 −0.132711 0.991155i \(-0.542368\pi\)
−0.132711 + 0.991155i \(0.542368\pi\)
\(548\) 9.12927 0.389983
\(549\) 6.27459 0.267793
\(550\) 0 0
\(551\) 4.45406 0.189749
\(552\) −7.32959 −0.311968
\(553\) 9.65408 0.410533
\(554\) 2.44560 0.103904
\(555\) 1.12698 0.0478376
\(556\) 0.657844 0.0278988
\(557\) −36.9739 −1.56663 −0.783317 0.621623i \(-0.786474\pi\)
−0.783317 + 0.621623i \(0.786474\pi\)
\(558\) −1.37169 −0.0580684
\(559\) −8.66950 −0.366681
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −18.0116 −0.759776
\(563\) 26.0483 1.09780 0.548901 0.835887i \(-0.315046\pi\)
0.548901 + 0.835887i \(0.315046\pi\)
\(564\) 11.2731 0.474681
\(565\) 2.72299 0.114557
\(566\) −9.78930 −0.411475
\(567\) −3.97858 −0.167085
\(568\) −13.8057 −0.579276
\(569\) −18.6278 −0.780920 −0.390460 0.920620i \(-0.627684\pi\)
−0.390460 + 0.920620i \(0.627684\pi\)
\(570\) −2.10674 −0.0882417
\(571\) 34.4156 1.44025 0.720123 0.693846i \(-0.244085\pi\)
0.720123 + 0.693846i \(0.244085\pi\)
\(572\) 0 0
\(573\) −10.3724 −0.433311
\(574\) 3.47854 0.145191
\(575\) 5.45794 0.227612
\(576\) −1.19656 −0.0498566
\(577\) 24.5630 1.02257 0.511285 0.859411i \(-0.329170\pi\)
0.511285 + 0.859411i \(0.329170\pi\)
\(578\) −10.0533 −0.418163
\(579\) 22.0045 0.914474
\(580\) 2.83920 0.117891
\(581\) −2.51880 −0.104498
\(582\) 10.1575 0.421042
\(583\) 0 0
\(584\) −1.14830 −0.0475169
\(585\) −5.15905 −0.213300
\(586\) −8.44836 −0.348999
\(587\) 18.3415 0.757037 0.378518 0.925594i \(-0.376434\pi\)
0.378518 + 0.925594i \(0.376434\pi\)
\(588\) −1.34292 −0.0553812
\(589\) 1.79839 0.0741013
\(590\) 2.32408 0.0956809
\(591\) −7.41254 −0.304911
\(592\) −0.839198 −0.0344908
\(593\) −10.9987 −0.451664 −0.225832 0.974166i \(-0.572510\pi\)
−0.225832 + 0.974166i \(0.572510\pi\)
\(594\) 0 0
\(595\) −2.63565 −0.108051
\(596\) −20.5982 −0.843737
\(597\) −5.74023 −0.234932
\(598\) 23.5323 0.962308
\(599\) 4.76034 0.194502 0.0972511 0.995260i \(-0.468995\pi\)
0.0972511 + 0.995260i \(0.468995\pi\)
\(600\) −1.34292 −0.0548246
\(601\) −16.5137 −0.673608 −0.336804 0.941575i \(-0.609346\pi\)
−0.336804 + 0.941575i \(0.609346\pi\)
\(602\) −2.01075 −0.0819521
\(603\) 1.92210 0.0782741
\(604\) −4.29219 −0.174647
\(605\) 0 0
\(606\) −23.6958 −0.962576
\(607\) 3.14655 0.127714 0.0638572 0.997959i \(-0.479660\pi\)
0.0638572 + 0.997959i \(0.479660\pi\)
\(608\) 1.56877 0.0636222
\(609\) −3.81282 −0.154503
\(610\) −5.24387 −0.212318
\(611\) −36.1932 −1.46422
\(612\) 3.15371 0.127481
\(613\) −45.9924 −1.85761 −0.928807 0.370564i \(-0.879164\pi\)
−0.928807 + 0.370564i \(0.879164\pi\)
\(614\) −9.34689 −0.377210
\(615\) −4.67141 −0.188369
\(616\) 0 0
\(617\) 45.5056 1.83198 0.915992 0.401196i \(-0.131405\pi\)
0.915992 + 0.401196i \(0.131405\pi\)
\(618\) −13.6345 −0.548459
\(619\) −22.5870 −0.907849 −0.453925 0.891040i \(-0.649976\pi\)
−0.453925 + 0.891040i \(0.649976\pi\)
\(620\) 1.14637 0.0460391
\(621\) 30.7591 1.23432
\(622\) −27.6729 −1.10958
\(623\) 2.19326 0.0878710
\(624\) −5.79011 −0.231790
\(625\) 1.00000 0.0400000
\(626\) 5.34524 0.213639
\(627\) 0 0
\(628\) 10.9039 0.435113
\(629\) 2.21184 0.0881916
\(630\) −1.19656 −0.0476720
\(631\) 14.8009 0.589213 0.294606 0.955619i \(-0.404811\pi\)
0.294606 + 0.955619i \(0.404811\pi\)
\(632\) 9.65408 0.384019
\(633\) 26.3356 1.04675
\(634\) 7.13917 0.283533
\(635\) −9.21928 −0.365856
\(636\) −1.88820 −0.0748721
\(637\) 4.31157 0.170831
\(638\) 0 0
\(639\) 16.5194 0.653496
\(640\) 1.00000 0.0395285
\(641\) −34.5333 −1.36398 −0.681992 0.731360i \(-0.738886\pi\)
−0.681992 + 0.731360i \(0.738886\pi\)
\(642\) −20.7122 −0.817444
\(643\) −16.2144 −0.639434 −0.319717 0.947513i \(-0.603588\pi\)
−0.319717 + 0.947513i \(0.603588\pi\)
\(644\) 5.45794 0.215073
\(645\) 2.70028 0.106324
\(646\) −4.13474 −0.162679
\(647\) −43.3962 −1.70608 −0.853041 0.521844i \(-0.825244\pi\)
−0.853041 + 0.521844i \(0.825244\pi\)
\(648\) −3.97858 −0.156293
\(649\) 0 0
\(650\) 4.31157 0.169114
\(651\) −1.53948 −0.0603370
\(652\) 12.3525 0.483763
\(653\) 7.75135 0.303334 0.151667 0.988432i \(-0.451536\pi\)
0.151667 + 0.988432i \(0.451536\pi\)
\(654\) −11.4102 −0.446175
\(655\) 17.5357 0.685178
\(656\) 3.47854 0.135814
\(657\) 1.37400 0.0536050
\(658\) −8.39442 −0.327248
\(659\) 6.80255 0.264990 0.132495 0.991184i \(-0.457701\pi\)
0.132495 + 0.991184i \(0.457701\pi\)
\(660\) 0 0
\(661\) −17.3537 −0.674980 −0.337490 0.941329i \(-0.609578\pi\)
−0.337490 + 0.941329i \(0.609578\pi\)
\(662\) 5.83207 0.226670
\(663\) 15.2607 0.592678
\(664\) −2.51880 −0.0977486
\(665\) 1.56877 0.0608344
\(666\) 1.00415 0.0389100
\(667\) 15.4962 0.600014
\(668\) −6.93334 −0.268259
\(669\) −5.39779 −0.208690
\(670\) −1.60636 −0.0620592
\(671\) 0 0
\(672\) −1.34292 −0.0518044
\(673\) −47.8170 −1.84321 −0.921605 0.388130i \(-0.873121\pi\)
−0.921605 + 0.388130i \(0.873121\pi\)
\(674\) 22.2848 0.858377
\(675\) 5.63565 0.216916
\(676\) 5.58967 0.214987
\(677\) −7.12176 −0.273711 −0.136856 0.990591i \(-0.543700\pi\)
−0.136856 + 0.990591i \(0.543700\pi\)
\(678\) −3.65677 −0.140437
\(679\) −7.56372 −0.290269
\(680\) −2.63565 −0.101073
\(681\) −27.5013 −1.05385
\(682\) 0 0
\(683\) 17.8302 0.682252 0.341126 0.940018i \(-0.389192\pi\)
0.341126 + 0.940018i \(0.389192\pi\)
\(684\) −1.87713 −0.0717738
\(685\) 9.12927 0.348812
\(686\) 1.00000 0.0381802
\(687\) −11.7489 −0.448247
\(688\) −2.01075 −0.0766591
\(689\) 6.06225 0.230953
\(690\) −7.32959 −0.279033
\(691\) −15.7399 −0.598772 −0.299386 0.954132i \(-0.596782\pi\)
−0.299386 + 0.954132i \(0.596782\pi\)
\(692\) −11.5843 −0.440370
\(693\) 0 0
\(694\) 16.6253 0.631089
\(695\) 0.657844 0.0249535
\(696\) −3.81282 −0.144525
\(697\) −9.16822 −0.347271
\(698\) 3.23934 0.122611
\(699\) −18.9973 −0.718542
\(700\) 1.00000 0.0377964
\(701\) −19.1512 −0.723329 −0.361665 0.932308i \(-0.617791\pi\)
−0.361665 + 0.932308i \(0.617791\pi\)
\(702\) 24.2985 0.917089
\(703\) −1.31651 −0.0496532
\(704\) 0 0
\(705\) 11.2731 0.424568
\(706\) −24.6580 −0.928014
\(707\) 17.6449 0.663606
\(708\) −3.12106 −0.117297
\(709\) 6.08007 0.228342 0.114171 0.993461i \(-0.463579\pi\)
0.114171 + 0.993461i \(0.463579\pi\)
\(710\) −13.8057 −0.518120
\(711\) −11.5517 −0.433221
\(712\) 2.19326 0.0821958
\(713\) 6.25679 0.234319
\(714\) 3.53948 0.132462
\(715\) 0 0
\(716\) 20.4825 0.765466
\(717\) −2.84880 −0.106390
\(718\) 20.7493 0.774358
\(719\) 27.3905 1.02149 0.510746 0.859732i \(-0.329369\pi\)
0.510746 + 0.859732i \(0.329369\pi\)
\(720\) −1.19656 −0.0445931
\(721\) 10.1528 0.378111
\(722\) −16.5390 −0.615516
\(723\) −23.4148 −0.870805
\(724\) −8.01175 −0.297754
\(725\) 2.83920 0.105445
\(726\) 0 0
\(727\) 28.3641 1.05197 0.525984 0.850495i \(-0.323697\pi\)
0.525984 + 0.850495i \(0.323697\pi\)
\(728\) 4.31157 0.159798
\(729\) 27.4653 1.01724
\(730\) −1.14830 −0.0425004
\(731\) 5.29964 0.196014
\(732\) 7.04211 0.260284
\(733\) −48.3933 −1.78745 −0.893724 0.448618i \(-0.851916\pi\)
−0.893724 + 0.448618i \(0.851916\pi\)
\(734\) 2.12160 0.0783096
\(735\) −1.34292 −0.0495345
\(736\) 5.45794 0.201182
\(737\) 0 0
\(738\) −4.16227 −0.153215
\(739\) 6.07467 0.223460 0.111730 0.993739i \(-0.464361\pi\)
0.111730 + 0.993739i \(0.464361\pi\)
\(740\) −0.839198 −0.0308495
\(741\) −9.08337 −0.333686
\(742\) 1.40604 0.0516173
\(743\) 32.1862 1.18080 0.590398 0.807112i \(-0.298971\pi\)
0.590398 + 0.807112i \(0.298971\pi\)
\(744\) −1.53948 −0.0564401
\(745\) −20.5982 −0.754661
\(746\) −7.19996 −0.263609
\(747\) 3.01389 0.110273
\(748\) 0 0
\(749\) 15.4232 0.563551
\(750\) −1.34292 −0.0490366
\(751\) −24.5881 −0.897234 −0.448617 0.893724i \(-0.648083\pi\)
−0.448617 + 0.893724i \(0.648083\pi\)
\(752\) −8.39442 −0.306113
\(753\) −1.25810 −0.0458479
\(754\) 12.2414 0.445806
\(755\) −4.29219 −0.156209
\(756\) 5.63565 0.204967
\(757\) −25.5181 −0.927470 −0.463735 0.885974i \(-0.653491\pi\)
−0.463735 + 0.885974i \(0.653491\pi\)
\(758\) −23.1300 −0.840121
\(759\) 0 0
\(760\) 1.56877 0.0569054
\(761\) −13.0436 −0.472830 −0.236415 0.971652i \(-0.575972\pi\)
−0.236415 + 0.971652i \(0.575972\pi\)
\(762\) 12.3808 0.448509
\(763\) 8.49656 0.307596
\(764\) 7.72371 0.279434
\(765\) 3.15371 0.114023
\(766\) 31.4479 1.13626
\(767\) 10.0204 0.361817
\(768\) −1.34292 −0.0484586
\(769\) −48.0699 −1.73344 −0.866722 0.498792i \(-0.833777\pi\)
−0.866722 + 0.498792i \(0.833777\pi\)
\(770\) 0 0
\(771\) −18.4306 −0.663762
\(772\) −16.3855 −0.589727
\(773\) −40.5127 −1.45714 −0.728570 0.684972i \(-0.759815\pi\)
−0.728570 + 0.684972i \(0.759815\pi\)
\(774\) 2.40598 0.0864811
\(775\) 1.14637 0.0411787
\(776\) −7.56372 −0.271522
\(777\) 1.12698 0.0404301
\(778\) 17.5618 0.629622
\(779\) 5.45704 0.195519
\(780\) −5.79011 −0.207319
\(781\) 0 0
\(782\) −14.3852 −0.514415
\(783\) 16.0007 0.571820
\(784\) 1.00000 0.0357143
\(785\) 10.9039 0.389177
\(786\) −23.5491 −0.839970
\(787\) 36.3703 1.29646 0.648231 0.761444i \(-0.275509\pi\)
0.648231 + 0.761444i \(0.275509\pi\)
\(788\) 5.51971 0.196631
\(789\) 25.4270 0.905226
\(790\) 9.65408 0.343477
\(791\) 2.72299 0.0968185
\(792\) 0 0
\(793\) −22.6093 −0.802881
\(794\) 32.7839 1.16346
\(795\) −1.88820 −0.0669677
\(796\) 4.27443 0.151503
\(797\) 30.7299 1.08851 0.544254 0.838920i \(-0.316813\pi\)
0.544254 + 0.838920i \(0.316813\pi\)
\(798\) −2.10674 −0.0745779
\(799\) 22.1248 0.782719
\(800\) 1.00000 0.0353553
\(801\) −2.62436 −0.0927272
\(802\) 9.25146 0.326680
\(803\) 0 0
\(804\) 2.15722 0.0760793
\(805\) 5.45794 0.192367
\(806\) 4.94264 0.174097
\(807\) −17.4095 −0.612843
\(808\) 17.6449 0.620747
\(809\) −45.6988 −1.60668 −0.803342 0.595517i \(-0.796947\pi\)
−0.803342 + 0.595517i \(0.796947\pi\)
\(810\) −3.97858 −0.139793
\(811\) −9.43169 −0.331191 −0.165596 0.986194i \(-0.552955\pi\)
−0.165596 + 0.986194i \(0.552955\pi\)
\(812\) 2.83920 0.0996363
\(813\) −31.5701 −1.10721
\(814\) 0 0
\(815\) 12.3525 0.432691
\(816\) 3.53948 0.123907
\(817\) −3.15441 −0.110359
\(818\) −0.0174947 −0.000611688 0
\(819\) −5.15905 −0.180272
\(820\) 3.47854 0.121476
\(821\) −12.2170 −0.426376 −0.213188 0.977011i \(-0.568385\pi\)
−0.213188 + 0.977011i \(0.568385\pi\)
\(822\) −12.2599 −0.427613
\(823\) 8.03161 0.279964 0.139982 0.990154i \(-0.455295\pi\)
0.139982 + 0.990154i \(0.455295\pi\)
\(824\) 10.1528 0.353691
\(825\) 0 0
\(826\) 2.32408 0.0808651
\(827\) 9.90168 0.344315 0.172158 0.985069i \(-0.444926\pi\)
0.172158 + 0.985069i \(0.444926\pi\)
\(828\) −6.53074 −0.226959
\(829\) −20.2243 −0.702420 −0.351210 0.936297i \(-0.614230\pi\)
−0.351210 + 0.936297i \(0.614230\pi\)
\(830\) −2.51880 −0.0874290
\(831\) −3.28426 −0.113930
\(832\) 4.31157 0.149477
\(833\) −2.63565 −0.0913200
\(834\) −0.883434 −0.0305908
\(835\) −6.93334 −0.239938
\(836\) 0 0
\(837\) 6.46052 0.223308
\(838\) 32.4579 1.12124
\(839\) 35.7528 1.23432 0.617162 0.786836i \(-0.288282\pi\)
0.617162 + 0.786836i \(0.288282\pi\)
\(840\) −1.34292 −0.0463352
\(841\) −20.9390 −0.722033
\(842\) 9.15329 0.315443
\(843\) 24.1883 0.833088
\(844\) −19.6106 −0.675026
\(845\) 5.58967 0.192291
\(846\) 10.0444 0.345334
\(847\) 0 0
\(848\) 1.40604 0.0482836
\(849\) 13.1463 0.451179
\(850\) −2.63565 −0.0904022
\(851\) −4.58029 −0.157010
\(852\) 18.5400 0.635171
\(853\) 23.0500 0.789218 0.394609 0.918849i \(-0.370880\pi\)
0.394609 + 0.918849i \(0.370880\pi\)
\(854\) −5.24387 −0.179442
\(855\) −1.87713 −0.0641964
\(856\) 15.4232 0.527154
\(857\) 27.3702 0.934948 0.467474 0.884007i \(-0.345164\pi\)
0.467474 + 0.884007i \(0.345164\pi\)
\(858\) 0 0
\(859\) 43.4968 1.48409 0.742045 0.670350i \(-0.233856\pi\)
0.742045 + 0.670350i \(0.233856\pi\)
\(860\) −2.01075 −0.0685660
\(861\) −4.67141 −0.159201
\(862\) −26.0024 −0.885647
\(863\) −17.9594 −0.611345 −0.305672 0.952137i \(-0.598881\pi\)
−0.305672 + 0.952137i \(0.598881\pi\)
\(864\) 5.63565 0.191729
\(865\) −11.5843 −0.393879
\(866\) 23.6006 0.801980
\(867\) 13.5008 0.458513
\(868\) 1.14637 0.0389102
\(869\) 0 0
\(870\) −3.81282 −0.129267
\(871\) −6.92595 −0.234677
\(872\) 8.49656 0.287730
\(873\) 9.05042 0.306310
\(874\) 8.56227 0.289623
\(875\) 1.00000 0.0338062
\(876\) 1.54207 0.0521019
\(877\) 32.8608 1.10963 0.554816 0.831973i \(-0.312789\pi\)
0.554816 + 0.831973i \(0.312789\pi\)
\(878\) 31.7894 1.07284
\(879\) 11.3455 0.382674
\(880\) 0 0
\(881\) −24.9270 −0.839812 −0.419906 0.907568i \(-0.637937\pi\)
−0.419906 + 0.907568i \(0.637937\pi\)
\(882\) −1.19656 −0.0402902
\(883\) −18.1957 −0.612334 −0.306167 0.951978i \(-0.599047\pi\)
−0.306167 + 0.951978i \(0.599047\pi\)
\(884\) −11.3638 −0.382207
\(885\) −3.12106 −0.104913
\(886\) 36.1071 1.21304
\(887\) −38.9219 −1.30687 −0.653434 0.756983i \(-0.726672\pi\)
−0.653434 + 0.756983i \(0.726672\pi\)
\(888\) 1.12698 0.0378189
\(889\) −9.21928 −0.309205
\(890\) 2.19326 0.0735182
\(891\) 0 0
\(892\) 4.01943 0.134580
\(893\) −13.1689 −0.440682
\(894\) 27.6619 0.925151
\(895\) 20.4825 0.684654
\(896\) 1.00000 0.0334077
\(897\) −31.6021 −1.05516
\(898\) −4.52269 −0.150924
\(899\) 3.25476 0.108552
\(900\) −1.19656 −0.0398853
\(901\) −3.70583 −0.123459
\(902\) 0 0
\(903\) 2.70028 0.0898598
\(904\) 2.72299 0.0905655
\(905\) −8.01175 −0.266320
\(906\) 5.76408 0.191499
\(907\) −47.4377 −1.57514 −0.787572 0.616223i \(-0.788662\pi\)
−0.787572 + 0.616223i \(0.788662\pi\)
\(908\) 20.4787 0.679610
\(909\) −21.1132 −0.700280
\(910\) 4.31157 0.142927
\(911\) −10.2118 −0.338333 −0.169167 0.985587i \(-0.554108\pi\)
−0.169167 + 0.985587i \(0.554108\pi\)
\(912\) −2.10674 −0.0697612
\(913\) 0 0
\(914\) −15.1593 −0.501423
\(915\) 7.04211 0.232805
\(916\) 8.74872 0.289066
\(917\) 17.5357 0.579081
\(918\) −14.8536 −0.490243
\(919\) −33.7533 −1.11342 −0.556710 0.830707i \(-0.687936\pi\)
−0.556710 + 0.830707i \(0.687936\pi\)
\(920\) 5.45794 0.179943
\(921\) 12.5522 0.413608
\(922\) −27.6126 −0.909373
\(923\) −59.5244 −1.95927
\(924\) 0 0
\(925\) −0.839198 −0.0275927
\(926\) 13.3557 0.438895
\(927\) −12.1484 −0.399007
\(928\) 2.83920 0.0932012
\(929\) 41.2273 1.35262 0.676312 0.736615i \(-0.263577\pi\)
0.676312 + 0.736615i \(0.263577\pi\)
\(930\) −1.53948 −0.0504816
\(931\) 1.56877 0.0514145
\(932\) 14.1462 0.463374
\(933\) 37.1626 1.21665
\(934\) 26.3287 0.861500
\(935\) 0 0
\(936\) −5.15905 −0.168629
\(937\) −32.2808 −1.05457 −0.527283 0.849690i \(-0.676789\pi\)
−0.527283 + 0.849690i \(0.676789\pi\)
\(938\) −1.60636 −0.0524496
\(939\) −7.17825 −0.234253
\(940\) −8.39442 −0.273796
\(941\) −47.3632 −1.54400 −0.771998 0.635625i \(-0.780743\pi\)
−0.771998 + 0.635625i \(0.780743\pi\)
\(942\) −14.6431 −0.477098
\(943\) 18.9857 0.618258
\(944\) 2.32408 0.0756424
\(945\) 5.63565 0.183328
\(946\) 0 0
\(947\) 25.4907 0.828335 0.414167 0.910201i \(-0.364073\pi\)
0.414167 + 0.910201i \(0.364073\pi\)
\(948\) −12.9647 −0.421073
\(949\) −4.95097 −0.160715
\(950\) 1.56877 0.0508977
\(951\) −9.58736 −0.310891
\(952\) −2.63565 −0.0854220
\(953\) −39.8135 −1.28969 −0.644843 0.764315i \(-0.723077\pi\)
−0.644843 + 0.764315i \(0.723077\pi\)
\(954\) −1.68241 −0.0544700
\(955\) 7.72371 0.249934
\(956\) 2.12134 0.0686091
\(957\) 0 0
\(958\) 8.18258 0.264367
\(959\) 9.12927 0.294800
\(960\) −1.34292 −0.0433427
\(961\) −29.6858 −0.957608
\(962\) −3.61826 −0.116657
\(963\) −18.4547 −0.594695
\(964\) 17.4357 0.561566
\(965\) −16.3855 −0.527468
\(966\) −7.32959 −0.235826
\(967\) 29.9522 0.963198 0.481599 0.876392i \(-0.340056\pi\)
0.481599 + 0.876392i \(0.340056\pi\)
\(968\) 0 0
\(969\) 5.55264 0.178377
\(970\) −7.56372 −0.242856
\(971\) 8.45499 0.271333 0.135667 0.990755i \(-0.456682\pi\)
0.135667 + 0.990755i \(0.456682\pi\)
\(972\) −11.5640 −0.370917
\(973\) 0.657844 0.0210895
\(974\) −9.09993 −0.291581
\(975\) −5.79011 −0.185432
\(976\) −5.24387 −0.167852
\(977\) −4.10946 −0.131473 −0.0657366 0.997837i \(-0.520940\pi\)
−0.0657366 + 0.997837i \(0.520940\pi\)
\(978\) −16.5885 −0.530442
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −10.1666 −0.324595
\(982\) −3.80239 −0.121339
\(983\) 20.8447 0.664842 0.332421 0.943131i \(-0.392135\pi\)
0.332421 + 0.943131i \(0.392135\pi\)
\(984\) −4.67141 −0.148919
\(985\) 5.51971 0.175873
\(986\) −7.48314 −0.238312
\(987\) 11.2731 0.358825
\(988\) 6.76388 0.215188
\(989\) −10.9746 −0.348971
\(990\) 0 0
\(991\) 29.5690 0.939290 0.469645 0.882855i \(-0.344382\pi\)
0.469645 + 0.882855i \(0.344382\pi\)
\(992\) 1.14637 0.0363971
\(993\) −7.83202 −0.248542
\(994\) −13.8057 −0.437891
\(995\) 4.27443 0.135508
\(996\) 3.38256 0.107181
\(997\) 57.0730 1.80752 0.903760 0.428040i \(-0.140796\pi\)
0.903760 + 0.428040i \(0.140796\pi\)
\(998\) −22.9856 −0.727596
\(999\) −4.72943 −0.149632
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.df.1.2 yes 6
11.10 odd 2 8470.2.a.cz.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cz.1.1 6 11.10 odd 2
8470.2.a.df.1.2 yes 6 1.1 even 1 trivial