Properties

Label 8470.2.a.cz.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 $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.34292 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.34292 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.19656 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.34292 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.34292 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.19656 q^{9} -1.00000 q^{10} -1.34292 q^{12} +4.31157 q^{13} +1.00000 q^{14} -1.34292 q^{15} +1.00000 q^{16} +2.63565 q^{17} +1.19656 q^{18} +6.54735 q^{19} +1.00000 q^{20} +1.34292 q^{21} -3.16521 q^{23} +1.34292 q^{24} +1.00000 q^{25} -4.31157 q^{26} +5.63565 q^{27} -1.00000 q^{28} +1.13193 q^{29} +1.34292 q^{30} +1.14637 q^{31} -1.00000 q^{32} -2.63565 q^{34} -1.00000 q^{35} -1.19656 q^{36} +3.13193 q^{37} -6.54735 q^{38} -5.79011 q^{39} -1.00000 q^{40} -0.521462 q^{41} -1.34292 q^{42} +4.96783 q^{43} -1.19656 q^{45} +3.16521 q^{46} +11.8090 q^{47} -1.34292 q^{48} +1.00000 q^{49} -1.00000 q^{50} -3.53948 q^{51} +4.31157 q^{52} +2.59396 q^{53} -5.63565 q^{54} +1.00000 q^{56} -8.79259 q^{57} -1.13193 q^{58} +10.9472 q^{59} -1.34292 q^{60} -5.82933 q^{61} -1.14637 q^{62} +1.19656 q^{63} +1.00000 q^{64} +4.31157 q^{65} -9.72249 q^{67} +2.63565 q^{68} +4.25063 q^{69} +1.00000 q^{70} -4.50169 q^{71} +1.19656 q^{72} -12.1269 q^{73} -3.13193 q^{74} -1.34292 q^{75} +6.54735 q^{76} +5.79011 q^{78} -17.0893 q^{79} +1.00000 q^{80} -3.97858 q^{81} +0.521462 q^{82} -9.06142 q^{83} +1.34292 q^{84} +2.63565 q^{85} -4.96783 q^{86} -1.52009 q^{87} +7.17843 q^{89} +1.19656 q^{90} -4.31157 q^{91} -3.16521 q^{92} -1.53948 q^{93} -11.8090 q^{94} +6.54735 q^{95} +1.34292 q^{96} +9.85645 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 4 q^{3} + 6 q^{4} + 6 q^{5} - 4 q^{6} - 6 q^{7} - 6 q^{8} + 2 q^{9} + O(q^{10}) \) \( 6 q - 6 q^{2} + 4 q^{3} + 6 q^{4} + 6 q^{5} - 4 q^{6} - 6 q^{7} - 6 q^{8} + 2 q^{9} - 6 q^{10} + 4 q^{12} + 6 q^{14} + 4 q^{15} + 6 q^{16} - 2 q^{17} - 2 q^{18} + 6 q^{20} - 4 q^{21} + 4 q^{23} - 4 q^{24} + 6 q^{25} + 16 q^{27} - 6 q^{28} - 8 q^{29} - 4 q^{30} + 4 q^{31} - 6 q^{32} + 2 q^{34} - 6 q^{35} + 2 q^{36} + 4 q^{37} - 6 q^{40} - 12 q^{41} + 4 q^{42} + 6 q^{43} + 2 q^{45} - 4 q^{46} + 16 q^{47} + 4 q^{48} + 6 q^{49} - 6 q^{50} + 12 q^{53} - 16 q^{54} + 6 q^{56} - 8 q^{57} + 8 q^{58} + 22 q^{59} + 4 q^{60} + 4 q^{61} - 4 q^{62} - 2 q^{63} + 6 q^{64} + 20 q^{67} - 2 q^{68} + 12 q^{69} + 6 q^{70} + 14 q^{71} - 2 q^{72} - 18 q^{73} - 4 q^{74} + 4 q^{75} - 32 q^{79} + 6 q^{80} + 6 q^{81} + 12 q^{82} + 16 q^{83} - 4 q^{84} - 2 q^{85} - 6 q^{86} + 4 q^{87} + 4 q^{89} - 2 q^{90} + 4 q^{92} + 12 q^{93} - 16 q^{94} - 4 q^{96} + 4 q^{97} - 6 q^{98} + O(q^{100}) \)

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
\(3\) −1.34292 −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 \(-0.396445\pi\)
0.319620 + 0.947546i \(0.396445\pi\)
\(18\) 1.19656 0.282031
\(19\) 6.54735 1.50207 0.751033 0.660265i \(-0.229556\pi\)
0.751033 + 0.660265i \(0.229556\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.34292 0.293050
\(22\) 0 0
\(23\) −3.16521 −0.659992 −0.329996 0.943982i \(-0.607047\pi\)
−0.329996 + 0.943982i \(0.607047\pi\)
\(24\) 1.34292 0.274123
\(25\) 1.00000 0.200000
\(26\) −4.31157 −0.845569
\(27\) 5.63565 1.08458
\(28\) −1.00000 −0.188982
\(29\) 1.13193 0.210194 0.105097 0.994462i \(-0.466485\pi\)
0.105097 + 0.994462i \(0.466485\pi\)
\(30\) 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
\(37\) 3.13193 0.514886 0.257443 0.966293i \(-0.417120\pi\)
0.257443 + 0.966293i \(0.417120\pi\)
\(38\) −6.54735 −1.06212
\(39\) −5.79011 −0.927160
\(40\) −1.00000 −0.158114
\(41\) −0.521462 −0.0814386 −0.0407193 0.999171i \(-0.512965\pi\)
−0.0407193 + 0.999171i \(0.512965\pi\)
\(42\) −1.34292 −0.207218
\(43\) 4.96783 0.757587 0.378793 0.925481i \(-0.376339\pi\)
0.378793 + 0.925481i \(0.376339\pi\)
\(44\) 0 0
\(45\) −1.19656 −0.178372
\(46\) 3.16521 0.466685
\(47\) 11.8090 1.72251 0.861257 0.508170i \(-0.169678\pi\)
0.861257 + 0.508170i \(0.169678\pi\)
\(48\) −1.34292 −0.193834
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −3.53948 −0.495626
\(52\) 4.31157 0.597908
\(53\) 2.59396 0.356308 0.178154 0.984003i \(-0.442988\pi\)
0.178154 + 0.984003i \(0.442988\pi\)
\(54\) −5.63565 −0.766915
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −8.79259 −1.16461
\(58\) −1.13193 −0.148630
\(59\) 10.9472 1.42521 0.712604 0.701567i \(-0.247516\pi\)
0.712604 + 0.701567i \(0.247516\pi\)
\(60\) −1.34292 −0.173371
\(61\) −5.82933 −0.746369 −0.373185 0.927757i \(-0.621734\pi\)
−0.373185 + 0.927757i \(0.621734\pi\)
\(62\) −1.14637 −0.145589
\(63\) 1.19656 0.150752
\(64\) 1.00000 0.125000
\(65\) 4.31157 0.534785
\(66\) 0 0
\(67\) −9.72249 −1.18779 −0.593895 0.804542i \(-0.702411\pi\)
−0.593895 + 0.804542i \(0.702411\pi\)
\(68\) 2.63565 0.319620
\(69\) 4.25063 0.511716
\(70\) 1.00000 0.119523
\(71\) −4.50169 −0.534252 −0.267126 0.963662i \(-0.586074\pi\)
−0.267126 + 0.963662i \(0.586074\pi\)
\(72\) 1.19656 0.141016
\(73\) −12.1269 −1.41934 −0.709672 0.704533i \(-0.751157\pi\)
−0.709672 + 0.704533i \(0.751157\pi\)
\(74\) −3.13193 −0.364079
\(75\) −1.34292 −0.155067
\(76\) 6.54735 0.751033
\(77\) 0 0
\(78\) 5.79011 0.655601
\(79\) −17.0893 −1.92270 −0.961349 0.275334i \(-0.911212\pi\)
−0.961349 + 0.275334i \(0.911212\pi\)
\(80\) 1.00000 0.111803
\(81\) −3.97858 −0.442064
\(82\) 0.521462 0.0575858
\(83\) −9.06142 −0.994620 −0.497310 0.867573i \(-0.665679\pi\)
−0.497310 + 0.867573i \(0.665679\pi\)
\(84\) 1.34292 0.146525
\(85\) 2.63565 0.285877
\(86\) −4.96783 −0.535695
\(87\) −1.52009 −0.162971
\(88\) 0 0
\(89\) 7.17843 0.760913 0.380456 0.924799i \(-0.375767\pi\)
0.380456 + 0.924799i \(0.375767\pi\)
\(90\) 1.19656 0.126128
\(91\) −4.31157 −0.451976
\(92\) −3.16521 −0.329996
\(93\) −1.53948 −0.159637
\(94\) −11.8090 −1.21800
\(95\) 6.54735 0.671744
\(96\) 1.34292 0.137062
\(97\) 9.85645 1.00077 0.500385 0.865803i \(-0.333192\pi\)
0.500385 + 0.865803i \(0.333192\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −4.36976 −0.434807 −0.217404 0.976082i \(-0.569759\pi\)
−0.217404 + 0.976082i \(0.569759\pi\)
\(102\) 3.53948 0.350461
\(103\) 14.2978 1.40881 0.704403 0.709800i \(-0.251215\pi\)
0.704403 + 0.709800i \(0.251215\pi\)
\(104\) −4.31157 −0.422785
\(105\) 1.34292 0.131056
\(106\) −2.59396 −0.251948
\(107\) 4.78019 0.462118 0.231059 0.972940i \(-0.425781\pi\)
0.231059 + 0.972940i \(0.425781\pi\)
\(108\) 5.63565 0.542291
\(109\) −2.58240 −0.247349 −0.123675 0.992323i \(-0.539468\pi\)
−0.123675 + 0.992323i \(0.539468\pi\)
\(110\) 0 0
\(111\) −4.20594 −0.399210
\(112\) −1.00000 −0.0944911
\(113\) 12.9414 1.21743 0.608714 0.793390i \(-0.291686\pi\)
0.608714 + 0.793390i \(0.291686\pi\)
\(114\) 8.79259 0.823501
\(115\) −3.16521 −0.295157
\(116\) 1.13193 0.105097
\(117\) −5.15905 −0.476954
\(118\) −10.9472 −1.00777
\(119\) −2.63565 −0.241610
\(120\) 1.34292 0.122592
\(121\) 0 0
\(122\) 5.82933 0.527763
\(123\) 0.700283 0.0631424
\(124\) 1.14637 0.102947
\(125\) 1.00000 0.0894427
\(126\) −1.19656 −0.106598
\(127\) −10.9841 −0.974680 −0.487340 0.873212i \(-0.662033\pi\)
−0.487340 + 0.873212i \(0.662033\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.67141 −0.587385
\(130\) −4.31157 −0.378150
\(131\) 11.9717 1.04597 0.522986 0.852341i \(-0.324818\pi\)
0.522986 + 0.852341i \(0.324818\pi\)
\(132\) 0 0
\(133\) −6.54735 −0.567727
\(134\) 9.72249 0.839895
\(135\) 5.63565 0.485040
\(136\) −2.63565 −0.226005
\(137\) −9.37916 −0.801315 −0.400658 0.916228i \(-0.631218\pi\)
−0.400658 + 0.916228i \(0.631218\pi\)
\(138\) −4.25063 −0.361838
\(139\) −11.1498 −0.945714 −0.472857 0.881139i \(-0.656777\pi\)
−0.472857 + 0.881139i \(0.656777\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −15.8585 −1.33553
\(142\) 4.50169 0.377773
\(143\) 0 0
\(144\) −1.19656 −0.0997131
\(145\) 1.13193 0.0940016
\(146\) 12.1269 1.00363
\(147\) −1.34292 −0.110762
\(148\) 3.13193 0.257443
\(149\) −5.51928 −0.452157 −0.226079 0.974109i \(-0.572591\pi\)
−0.226079 + 0.974109i \(0.572591\pi\)
\(150\) 1.34292 0.109649
\(151\) −9.66388 −0.786435 −0.393218 0.919445i \(-0.628638\pi\)
−0.393218 + 0.919445i \(0.628638\pi\)
\(152\) −6.54735 −0.531060
\(153\) −3.15371 −0.254962
\(154\) 0 0
\(155\) 1.14637 0.0920783
\(156\) −5.79011 −0.463580
\(157\) 16.8181 1.34223 0.671114 0.741355i \(-0.265816\pi\)
0.671114 + 0.741355i \(0.265816\pi\)
\(158\) 17.0893 1.35955
\(159\) −3.48349 −0.276259
\(160\) −1.00000 −0.0790569
\(161\) 3.16521 0.249453
\(162\) 3.97858 0.312587
\(163\) −22.8952 −1.79329 −0.896644 0.442753i \(-0.854002\pi\)
−0.896644 + 0.442753i \(0.854002\pi\)
\(164\) −0.521462 −0.0407193
\(165\) 0 0
\(166\) 9.06142 0.703303
\(167\) 8.29513 0.641896 0.320948 0.947097i \(-0.395998\pi\)
0.320948 + 0.947097i \(0.395998\pi\)
\(168\) −1.34292 −0.103609
\(169\) 5.58967 0.429975
\(170\) −2.63565 −0.202145
\(171\) −7.83428 −0.599103
\(172\) 4.96783 0.378793
\(173\) 21.9024 1.66521 0.832606 0.553866i \(-0.186848\pi\)
0.832606 + 0.553866i \(0.186848\pi\)
\(174\) 1.52009 0.115238
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −14.7013 −1.10502
\(178\) −7.17843 −0.538046
\(179\) −3.11079 −0.232511 −0.116256 0.993219i \(-0.537089\pi\)
−0.116256 + 0.993219i \(0.537089\pi\)
\(180\) −1.19656 −0.0891861
\(181\) 4.58252 0.340616 0.170308 0.985391i \(-0.445524\pi\)
0.170308 + 0.985391i \(0.445524\pi\)
\(182\) 4.31157 0.319595
\(183\) 7.82834 0.578688
\(184\) 3.16521 0.233342
\(185\) 3.13193 0.230264
\(186\) 1.53948 0.112880
\(187\) 0 0
\(188\) 11.8090 0.861257
\(189\) −5.63565 −0.409934
\(190\) −6.54735 −0.474995
\(191\) 4.33382 0.313584 0.156792 0.987632i \(-0.449885\pi\)
0.156792 + 0.987632i \(0.449885\pi\)
\(192\) −1.34292 −0.0969171
\(193\) 13.4284 0.966599 0.483300 0.875455i \(-0.339438\pi\)
0.483300 + 0.875455i \(0.339438\pi\)
\(194\) −9.85645 −0.707652
\(195\) −5.79011 −0.414639
\(196\) 1.00000 0.0714286
\(197\) −11.8667 −0.845466 −0.422733 0.906254i \(-0.638929\pi\)
−0.422733 + 0.906254i \(0.638929\pi\)
\(198\) 0 0
\(199\) −14.5672 −1.03264 −0.516319 0.856396i \(-0.672698\pi\)
−0.516319 + 0.856396i \(0.672698\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 13.0565 0.920938
\(202\) 4.36976 0.307455
\(203\) −1.13193 −0.0794458
\(204\) −3.53948 −0.247813
\(205\) −0.521462 −0.0364205
\(206\) −14.2978 −0.996177
\(207\) 3.78735 0.263239
\(208\) 4.31157 0.298954
\(209\) 0 0
\(210\) −1.34292 −0.0926705
\(211\) 2.79716 0.192565 0.0962823 0.995354i \(-0.469305\pi\)
0.0962823 + 0.995354i \(0.469305\pi\)
\(212\) 2.59396 0.178154
\(213\) 6.04542 0.414226
\(214\) −4.78019 −0.326767
\(215\) 4.96783 0.338803
\(216\) −5.63565 −0.383458
\(217\) −1.14637 −0.0778204
\(218\) 2.58240 0.174902
\(219\) 16.2855 1.10047
\(220\) 0 0
\(221\) 11.3638 0.764413
\(222\) 4.20594 0.282284
\(223\) 19.6450 1.31553 0.657763 0.753225i \(-0.271503\pi\)
0.657763 + 0.753225i \(0.271503\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.19656 −0.0797705
\(226\) −12.9414 −0.860851
\(227\) −7.63636 −0.506843 −0.253422 0.967356i \(-0.581556\pi\)
−0.253422 + 0.967356i \(0.581556\pi\)
\(228\) −8.79259 −0.582303
\(229\) −1.56937 −0.103707 −0.0518536 0.998655i \(-0.516513\pi\)
−0.0518536 + 0.998655i \(0.516513\pi\)
\(230\) 3.16521 0.208708
\(231\) 0 0
\(232\) −1.13193 −0.0743148
\(233\) 22.2894 1.46023 0.730114 0.683325i \(-0.239467\pi\)
0.730114 + 0.683325i \(0.239467\pi\)
\(234\) 5.15905 0.337257
\(235\) 11.8090 0.770331
\(236\) 10.9472 0.712604
\(237\) 22.9496 1.49074
\(238\) 2.63565 0.170844
\(239\) 5.92057 0.382970 0.191485 0.981496i \(-0.438670\pi\)
0.191485 + 0.981496i \(0.438670\pi\)
\(240\) −1.34292 −0.0866853
\(241\) 26.9422 1.73550 0.867749 0.497002i \(-0.165566\pi\)
0.867749 + 0.497002i \(0.165566\pi\)
\(242\) 0 0
\(243\) −11.5640 −0.741833
\(244\) −5.82933 −0.373185
\(245\) 1.00000 0.0638877
\(246\) −0.700283 −0.0446484
\(247\) 28.2294 1.79619
\(248\) −1.14637 −0.0727943
\(249\) 12.1688 0.771166
\(250\) −1.00000 −0.0632456
\(251\) −10.5437 −0.665514 −0.332757 0.943013i \(-0.607979\pi\)
−0.332757 + 0.943013i \(0.607979\pi\)
\(252\) 1.19656 0.0753760
\(253\) 0 0
\(254\) 10.9841 0.689203
\(255\) −3.53948 −0.221651
\(256\) 1.00000 0.0625000
\(257\) 19.8973 1.24116 0.620581 0.784142i \(-0.286897\pi\)
0.620581 + 0.784142i \(0.286897\pi\)
\(258\) 6.67141 0.415344
\(259\) −3.13193 −0.194609
\(260\) 4.31157 0.267392
\(261\) −1.35442 −0.0838364
\(262\) −11.9717 −0.739613
\(263\) 12.5871 0.776155 0.388077 0.921627i \(-0.373139\pi\)
0.388077 + 0.921627i \(0.373139\pi\)
\(264\) 0 0
\(265\) 2.59396 0.159346
\(266\) 6.54735 0.401444
\(267\) −9.64009 −0.589964
\(268\) −9.72249 −0.593895
\(269\) −6.90634 −0.421087 −0.210544 0.977584i \(-0.567523\pi\)
−0.210544 + 0.977584i \(0.567523\pi\)
\(270\) −5.63565 −0.342975
\(271\) 7.59420 0.461315 0.230657 0.973035i \(-0.425912\pi\)
0.230657 + 0.973035i \(0.425912\pi\)
\(272\) 2.63565 0.159810
\(273\) 5.79011 0.350434
\(274\) 9.37916 0.566616
\(275\) 0 0
\(276\) 4.25063 0.255858
\(277\) −16.8832 −1.01442 −0.507208 0.861824i \(-0.669322\pi\)
−0.507208 + 0.861824i \(0.669322\pi\)
\(278\) 11.1498 0.668721
\(279\) −1.37169 −0.0821211
\(280\) 1.00000 0.0597614
\(281\) −9.62695 −0.574296 −0.287148 0.957886i \(-0.592707\pi\)
−0.287148 + 0.957886i \(0.592707\pi\)
\(282\) 15.8585 0.944361
\(283\) −22.5755 −1.34198 −0.670988 0.741468i \(-0.734130\pi\)
−0.670988 + 0.741468i \(0.734130\pi\)
\(284\) −4.50169 −0.267126
\(285\) −8.79259 −0.520828
\(286\) 0 0
\(287\) 0.521462 0.0307809
\(288\) 1.19656 0.0705078
\(289\) −10.0533 −0.591372
\(290\) −1.13193 −0.0664691
\(291\) −13.2365 −0.775935
\(292\) −12.1269 −0.709672
\(293\) 21.6239 1.26328 0.631640 0.775262i \(-0.282382\pi\)
0.631640 + 0.775262i \(0.282382\pi\)
\(294\) 1.34292 0.0783209
\(295\) 10.9472 0.637372
\(296\) −3.13193 −0.182040
\(297\) 0 0
\(298\) 5.51928 0.319723
\(299\) −13.6470 −0.789228
\(300\) −1.34292 −0.0775337
\(301\) −4.96783 −0.286341
\(302\) 9.66388 0.556094
\(303\) 5.86825 0.337122
\(304\) 6.54735 0.375516
\(305\) −5.82933 −0.333787
\(306\) 3.15371 0.180286
\(307\) −9.26120 −0.528565 −0.264282 0.964445i \(-0.585135\pi\)
−0.264282 + 0.964445i \(0.585135\pi\)
\(308\) 0 0
\(309\) −19.2009 −1.09230
\(310\) −1.14637 −0.0651092
\(311\) 6.38689 0.362167 0.181084 0.983468i \(-0.442040\pi\)
0.181084 + 0.983468i \(0.442040\pi\)
\(312\) 5.79011 0.327801
\(313\) −1.93070 −0.109130 −0.0545649 0.998510i \(-0.517377\pi\)
−0.0545649 + 0.998510i \(0.517377\pi\)
\(314\) −16.8181 −0.949098
\(315\) 1.19656 0.0674184
\(316\) −17.0893 −0.961349
\(317\) −11.8763 −0.667039 −0.333519 0.942743i \(-0.608236\pi\)
−0.333519 + 0.942743i \(0.608236\pi\)
\(318\) 3.48349 0.195344
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −6.41942 −0.358297
\(322\) −3.16521 −0.176390
\(323\) 17.2566 0.960180
\(324\) −3.97858 −0.221032
\(325\) 4.31157 0.239163
\(326\) 22.8952 1.26805
\(327\) 3.46797 0.191779
\(328\) 0.521462 0.0287929
\(329\) −11.8090 −0.651049
\(330\) 0 0
\(331\) −28.5755 −1.57065 −0.785324 0.619084i \(-0.787504\pi\)
−0.785324 + 0.619084i \(0.787504\pi\)
\(332\) −9.06142 −0.497310
\(333\) −3.74753 −0.205364
\(334\) −8.29513 −0.453889
\(335\) −9.72249 −0.531196
\(336\) 1.34292 0.0732625
\(337\) 8.38514 0.456768 0.228384 0.973571i \(-0.426656\pi\)
0.228384 + 0.973571i \(0.426656\pi\)
\(338\) −5.58967 −0.304038
\(339\) −17.3793 −0.943916
\(340\) 2.63565 0.142938
\(341\) 0 0
\(342\) 7.83428 0.423629
\(343\) −1.00000 −0.0539949
\(344\) −4.96783 −0.267847
\(345\) 4.25063 0.228846
\(346\) −21.9024 −1.17748
\(347\) −23.5535 −1.26442 −0.632210 0.774797i \(-0.717852\pi\)
−0.632210 + 0.774797i \(0.717852\pi\)
\(348\) −1.52009 −0.0814856
\(349\) 25.2540 1.35182 0.675909 0.736985i \(-0.263751\pi\)
0.675909 + 0.736985i \(0.263751\pi\)
\(350\) 1.00000 0.0534522
\(351\) 24.2985 1.29696
\(352\) 0 0
\(353\) 20.9078 1.11281 0.556406 0.830911i \(-0.312180\pi\)
0.556406 + 0.830911i \(0.312180\pi\)
\(354\) 14.7013 0.781365
\(355\) −4.50169 −0.238925
\(356\) 7.17843 0.380456
\(357\) 3.53948 0.187329
\(358\) 3.11079 0.164410
\(359\) −25.9084 −1.36739 −0.683696 0.729767i \(-0.739628\pi\)
−0.683696 + 0.729767i \(0.739628\pi\)
\(360\) 1.19656 0.0630641
\(361\) 23.8678 1.25620
\(362\) −4.58252 −0.240852
\(363\) 0 0
\(364\) −4.31157 −0.225988
\(365\) −12.1269 −0.634750
\(366\) −7.82834 −0.409194
\(367\) −20.8650 −1.08914 −0.544572 0.838714i \(-0.683308\pi\)
−0.544572 + 0.838714i \(0.683308\pi\)
\(368\) −3.16521 −0.164998
\(369\) 0.623959 0.0324820
\(370\) −3.13193 −0.162821
\(371\) −2.59396 −0.134672
\(372\) −1.53948 −0.0798184
\(373\) 10.1570 0.525911 0.262956 0.964808i \(-0.415303\pi\)
0.262956 + 0.964808i \(0.415303\pi\)
\(374\) 0 0
\(375\) −1.34292 −0.0693482
\(376\) −11.8090 −0.609000
\(377\) 4.88039 0.251353
\(378\) 5.63565 0.289867
\(379\) 29.5378 1.51726 0.758628 0.651524i \(-0.225870\pi\)
0.758628 + 0.651524i \(0.225870\pi\)
\(380\) 6.54735 0.335872
\(381\) 14.7508 0.755706
\(382\) −4.33382 −0.221738
\(383\) 26.2888 1.34330 0.671648 0.740871i \(-0.265587\pi\)
0.671648 + 0.740871i \(0.265587\pi\)
\(384\) 1.34292 0.0685308
\(385\) 0 0
\(386\) −13.4284 −0.683489
\(387\) −5.94429 −0.302165
\(388\) 9.85645 0.500385
\(389\) −8.64078 −0.438105 −0.219052 0.975713i \(-0.570297\pi\)
−0.219052 + 0.975713i \(0.570297\pi\)
\(390\) 5.79011 0.293194
\(391\) −8.34240 −0.421893
\(392\) −1.00000 −0.0505076
\(393\) −16.0771 −0.810980
\(394\) 11.8667 0.597835
\(395\) −17.0893 −0.859856
\(396\) 0 0
\(397\) 35.1598 1.76462 0.882309 0.470670i \(-0.155988\pi\)
0.882309 + 0.470670i \(0.155988\pi\)
\(398\) 14.5672 0.730186
\(399\) 8.79259 0.440180
\(400\) 1.00000 0.0500000
\(401\) −7.99484 −0.399243 −0.199622 0.979873i \(-0.563971\pi\)
−0.199622 + 0.979873i \(0.563971\pi\)
\(402\) −13.0565 −0.651202
\(403\) 4.94264 0.246210
\(404\) −4.36976 −0.217404
\(405\) −3.97858 −0.197697
\(406\) 1.13193 0.0561767
\(407\) 0 0
\(408\) 3.53948 0.175230
\(409\) −1.92553 −0.0952114 −0.0476057 0.998866i \(-0.515159\pi\)
−0.0476057 + 0.998866i \(0.515159\pi\)
\(410\) 0.521462 0.0257532
\(411\) 12.5955 0.621290
\(412\) 14.2978 0.704403
\(413\) −10.9472 −0.538678
\(414\) −3.78735 −0.186138
\(415\) −9.06142 −0.444808
\(416\) −4.31157 −0.211392
\(417\) 14.9733 0.733247
\(418\) 0 0
\(419\) −22.4861 −1.09852 −0.549259 0.835652i \(-0.685090\pi\)
−0.549259 + 0.835652i \(0.685090\pi\)
\(420\) 1.34292 0.0655279
\(421\) −23.3112 −1.13612 −0.568059 0.822988i \(-0.692306\pi\)
−0.568059 + 0.822988i \(0.692306\pi\)
\(422\) −2.79716 −0.136164
\(423\) −14.1301 −0.687029
\(424\) −2.59396 −0.125974
\(425\) 2.63565 0.127848
\(426\) −6.04542 −0.292902
\(427\) 5.82933 0.282101
\(428\) 4.78019 0.231059
\(429\) 0 0
\(430\) −4.96783 −0.239570
\(431\) −15.2591 −0.735003 −0.367502 0.930023i \(-0.619787\pi\)
−0.367502 + 0.930023i \(0.619787\pi\)
\(432\) 5.63565 0.271146
\(433\) −0.722365 −0.0347146 −0.0173573 0.999849i \(-0.505525\pi\)
−0.0173573 + 0.999849i \(0.505525\pi\)
\(434\) 1.14637 0.0550273
\(435\) −1.52009 −0.0728829
\(436\) −2.58240 −0.123675
\(437\) −20.7237 −0.991350
\(438\) −16.2855 −0.778149
\(439\) −31.7894 −1.51722 −0.758612 0.651543i \(-0.774122\pi\)
−0.758612 + 0.651543i \(0.774122\pi\)
\(440\) 0 0
\(441\) −1.19656 −0.0569789
\(442\) −11.3638 −0.540522
\(443\) 29.1789 1.38633 0.693166 0.720778i \(-0.256215\pi\)
0.693166 + 0.720778i \(0.256215\pi\)
\(444\) −4.20594 −0.199605
\(445\) 7.17843 0.340290
\(446\) −19.6450 −0.930217
\(447\) 7.41197 0.350574
\(448\) −1.00000 −0.0472456
\(449\) 18.6378 0.879570 0.439785 0.898103i \(-0.355055\pi\)
0.439785 + 0.898103i \(0.355055\pi\)
\(450\) 1.19656 0.0564063
\(451\) 0 0
\(452\) 12.9414 0.608714
\(453\) 12.9778 0.609752
\(454\) 7.63636 0.358392
\(455\) −4.31157 −0.202130
\(456\) 8.79259 0.411751
\(457\) −3.01602 −0.141084 −0.0705418 0.997509i \(-0.522473\pi\)
−0.0705418 + 0.997509i \(0.522473\pi\)
\(458\) 1.56937 0.0733320
\(459\) 14.8536 0.693308
\(460\) −3.16521 −0.147579
\(461\) 16.8872 0.786513 0.393257 0.919429i \(-0.371348\pi\)
0.393257 + 0.919429i \(0.371348\pi\)
\(462\) 0 0
\(463\) 0.180188 0.00837403 0.00418701 0.999991i \(-0.498667\pi\)
0.00418701 + 0.999991i \(0.498667\pi\)
\(464\) 1.13193 0.0525485
\(465\) −1.53948 −0.0713917
\(466\) −22.2894 −1.03254
\(467\) 12.4723 0.577147 0.288574 0.957458i \(-0.406819\pi\)
0.288574 + 0.957458i \(0.406819\pi\)
\(468\) −5.15905 −0.238477
\(469\) 9.72249 0.448943
\(470\) −11.8090 −0.544707
\(471\) −22.5854 −1.04068
\(472\) −10.9472 −0.503887
\(473\) 0 0
\(474\) −22.9496 −1.05411
\(475\) 6.54735 0.300413
\(476\) −2.63565 −0.120805
\(477\) −3.10382 −0.142114
\(478\) −5.92057 −0.270801
\(479\) 1.55428 0.0710166 0.0355083 0.999369i \(-0.488695\pi\)
0.0355083 + 0.999369i \(0.488695\pi\)
\(480\) 1.34292 0.0612958
\(481\) 13.5035 0.615709
\(482\) −26.9422 −1.22718
\(483\) −4.25063 −0.193410
\(484\) 0 0
\(485\) 9.85645 0.447558
\(486\) 11.5640 0.524555
\(487\) 29.2641 1.32608 0.663042 0.748582i \(-0.269265\pi\)
0.663042 + 0.748582i \(0.269265\pi\)
\(488\) 5.82933 0.263881
\(489\) 30.7464 1.39040
\(490\) −1.00000 −0.0451754
\(491\) 2.35553 0.106304 0.0531518 0.998586i \(-0.483073\pi\)
0.0531518 + 0.998586i \(0.483073\pi\)
\(492\) 0.700283 0.0315712
\(493\) 2.98337 0.134364
\(494\) −28.2294 −1.27010
\(495\) 0 0
\(496\) 1.14637 0.0514733
\(497\) 4.50169 0.201928
\(498\) −12.1688 −0.545297
\(499\) 25.2783 1.13161 0.565807 0.824538i \(-0.308565\pi\)
0.565807 + 0.824538i \(0.308565\pi\)
\(500\) 1.00000 0.0447214
\(501\) −11.1397 −0.497686
\(502\) 10.5437 0.470589
\(503\) 3.23636 0.144302 0.0721510 0.997394i \(-0.477014\pi\)
0.0721510 + 0.997394i \(0.477014\pi\)
\(504\) −1.19656 −0.0532989
\(505\) −4.36976 −0.194452
\(506\) 0 0
\(507\) −7.50650 −0.333375
\(508\) −10.9841 −0.487340
\(509\) 39.5692 1.75387 0.876937 0.480606i \(-0.159583\pi\)
0.876937 + 0.480606i \(0.159583\pi\)
\(510\) 3.53948 0.156731
\(511\) 12.1269 0.536461
\(512\) −1.00000 −0.0441942
\(513\) 36.8986 1.62911
\(514\) −19.8973 −0.877634
\(515\) 14.2978 0.630037
\(516\) −6.67141 −0.293692
\(517\) 0 0
\(518\) 3.13193 0.137609
\(519\) −29.4133 −1.29110
\(520\) −4.31157 −0.189075
\(521\) −0.586854 −0.0257105 −0.0128553 0.999917i \(-0.504092\pi\)
−0.0128553 + 0.999917i \(0.504092\pi\)
\(522\) 1.35442 0.0592813
\(523\) 22.7742 0.995846 0.497923 0.867221i \(-0.334096\pi\)
0.497923 + 0.867221i \(0.334096\pi\)
\(524\) 11.9717 0.522986
\(525\) 1.34292 0.0586100
\(526\) −12.5871 −0.548824
\(527\) 3.02142 0.131615
\(528\) 0 0
\(529\) −12.9815 −0.564411
\(530\) −2.59396 −0.112674
\(531\) −13.0990 −0.568448
\(532\) −6.54735 −0.283864
\(533\) −2.24832 −0.0973856
\(534\) 9.64009 0.417167
\(535\) 4.78019 0.206665
\(536\) 9.72249 0.419947
\(537\) 4.17755 0.180274
\(538\) 6.90634 0.297753
\(539\) 0 0
\(540\) 5.63565 0.242520
\(541\) −36.2359 −1.55790 −0.778952 0.627084i \(-0.784248\pi\)
−0.778952 + 0.627084i \(0.784248\pi\)
\(542\) −7.59420 −0.326199
\(543\) −6.15397 −0.264092
\(544\) −2.63565 −0.113003
\(545\) −2.58240 −0.110618
\(546\) −5.79011 −0.247794
\(547\) −17.1863 −0.734831 −0.367416 0.930057i \(-0.619757\pi\)
−0.367416 + 0.930057i \(0.619757\pi\)
\(548\) −9.37916 −0.400658
\(549\) 6.97513 0.297691
\(550\) 0 0
\(551\) 7.41113 0.315725
\(552\) −4.25063 −0.180919
\(553\) 17.0893 0.726711
\(554\) 16.8832 0.717300
\(555\) −4.20594 −0.178532
\(556\) −11.1498 −0.472857
\(557\) 31.6410 1.34067 0.670335 0.742058i \(-0.266150\pi\)
0.670335 + 0.742058i \(0.266150\pi\)
\(558\) 1.37169 0.0580684
\(559\) 21.4192 0.905934
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 9.62695 0.406089
\(563\) −28.5238 −1.20213 −0.601066 0.799199i \(-0.705257\pi\)
−0.601066 + 0.799199i \(0.705257\pi\)
\(564\) −15.8585 −0.667764
\(565\) 12.9414 0.544450
\(566\) 22.5755 0.948920
\(567\) 3.97858 0.167085
\(568\) 4.50169 0.188887
\(569\) 17.6880 0.741520 0.370760 0.928729i \(-0.379097\pi\)
0.370760 + 0.928729i \(0.379097\pi\)
\(570\) 8.79259 0.368281
\(571\) −11.2551 −0.471012 −0.235506 0.971873i \(-0.575675\pi\)
−0.235506 + 0.971873i \(0.575675\pi\)
\(572\) 0 0
\(573\) −5.81999 −0.243134
\(574\) −0.521462 −0.0217654
\(575\) −3.16521 −0.131998
\(576\) −1.19656 −0.0498566
\(577\) 3.75289 0.156235 0.0781175 0.996944i \(-0.475109\pi\)
0.0781175 + 0.996944i \(0.475109\pi\)
\(578\) 10.0533 0.418163
\(579\) −18.0333 −0.749440
\(580\) 1.13193 0.0470008
\(581\) 9.06142 0.375931
\(582\) 13.2365 0.548669
\(583\) 0 0
\(584\) 12.1269 0.501814
\(585\) −5.15905 −0.213300
\(586\) −21.6239 −0.893274
\(587\) −31.3692 −1.29475 −0.647374 0.762173i \(-0.724133\pi\)
−0.647374 + 0.762173i \(0.724133\pi\)
\(588\) −1.34292 −0.0553812
\(589\) 7.50566 0.309265
\(590\) −10.9472 −0.450690
\(591\) 15.9360 0.655521
\(592\) 3.13193 0.128721
\(593\) −4.02015 −0.165088 −0.0825439 0.996587i \(-0.526304\pi\)
−0.0825439 + 0.996587i \(0.526304\pi\)
\(594\) 0 0
\(595\) −2.63565 −0.108051
\(596\) −5.51928 −0.226079
\(597\) 19.5626 0.800643
\(598\) 13.6470 0.558069
\(599\) −15.0102 −0.613301 −0.306651 0.951822i \(-0.599208\pi\)
−0.306651 + 0.951822i \(0.599208\pi\)
\(600\) 1.34292 0.0548246
\(601\) 20.0520 0.817939 0.408970 0.912548i \(-0.365888\pi\)
0.408970 + 0.912548i \(0.365888\pi\)
\(602\) 4.96783 0.202474
\(603\) 11.6335 0.473753
\(604\) −9.66388 −0.393218
\(605\) 0 0
\(606\) −5.86825 −0.238381
\(607\) −17.0030 −0.690129 −0.345064 0.938579i \(-0.612143\pi\)
−0.345064 + 0.938579i \(0.612143\pi\)
\(608\) −6.54735 −0.265530
\(609\) 1.52009 0.0615973
\(610\) 5.82933 0.236023
\(611\) 50.9152 2.05981
\(612\) −3.15371 −0.127481
\(613\) 17.2655 0.697347 0.348673 0.937244i \(-0.386632\pi\)
0.348673 + 0.937244i \(0.386632\pi\)
\(614\) 9.26120 0.373752
\(615\) 0.700283 0.0282381
\(616\) 0 0
\(617\) −18.0616 −0.727134 −0.363567 0.931568i \(-0.618441\pi\)
−0.363567 + 0.931568i \(0.618441\pi\)
\(618\) 19.2009 0.772373
\(619\) 32.2086 1.29457 0.647287 0.762247i \(-0.275904\pi\)
0.647287 + 0.762247i \(0.275904\pi\)
\(620\) 1.14637 0.0460391
\(621\) −17.8380 −0.715815
\(622\) −6.38689 −0.256091
\(623\) −7.17843 −0.287598
\(624\) −5.79011 −0.231790
\(625\) 1.00000 0.0400000
\(626\) 1.93070 0.0771664
\(627\) 0 0
\(628\) 16.8181 0.671114
\(629\) 8.25468 0.329136
\(630\) −1.19656 −0.0476720
\(631\) 10.0492 0.400052 0.200026 0.979791i \(-0.435897\pi\)
0.200026 + 0.979791i \(0.435897\pi\)
\(632\) 17.0893 0.679776
\(633\) −3.75637 −0.149302
\(634\) 11.8763 0.471667
\(635\) −10.9841 −0.435890
\(636\) −3.48349 −0.138129
\(637\) 4.31157 0.170831
\(638\) 0 0
\(639\) 5.38653 0.213088
\(640\) −1.00000 −0.0395285
\(641\) −22.0387 −0.870476 −0.435238 0.900315i \(-0.643336\pi\)
−0.435238 + 0.900315i \(0.643336\pi\)
\(642\) 6.41942 0.253354
\(643\) −18.2425 −0.719414 −0.359707 0.933065i \(-0.617123\pi\)
−0.359707 + 0.933065i \(0.617123\pi\)
\(644\) 3.16521 0.124727
\(645\) −6.67141 −0.262686
\(646\) −17.2566 −0.678950
\(647\) −10.5244 −0.413755 −0.206878 0.978367i \(-0.566330\pi\)
−0.206878 + 0.978367i \(0.566330\pi\)
\(648\) 3.97858 0.156293
\(649\) 0 0
\(650\) −4.31157 −0.169114
\(651\) 1.53948 0.0603370
\(652\) −22.8952 −0.896644
\(653\) 30.1567 1.18012 0.590061 0.807358i \(-0.299104\pi\)
0.590061 + 0.807358i \(0.299104\pi\)
\(654\) −3.46797 −0.135608
\(655\) 11.9717 0.467773
\(656\) −0.521462 −0.0203597
\(657\) 14.5105 0.566109
\(658\) 11.8090 0.460361
\(659\) 37.5753 1.46373 0.731863 0.681451i \(-0.238651\pi\)
0.731863 + 0.681451i \(0.238651\pi\)
\(660\) 0 0
\(661\) −24.1822 −0.940579 −0.470290 0.882512i \(-0.655851\pi\)
−0.470290 + 0.882512i \(0.655851\pi\)
\(662\) 28.5755 1.11062
\(663\) −15.2607 −0.592678
\(664\) 9.06142 0.351651
\(665\) −6.54735 −0.253895
\(666\) 3.74753 0.145214
\(667\) −3.58279 −0.138726
\(668\) 8.29513 0.320948
\(669\) −26.3817 −1.01998
\(670\) 9.72249 0.375612
\(671\) 0 0
\(672\) −1.34292 −0.0518044
\(673\) 38.4279 1.48129 0.740643 0.671899i \(-0.234521\pi\)
0.740643 + 0.671899i \(0.234521\pi\)
\(674\) −8.38514 −0.322983
\(675\) 5.63565 0.216916
\(676\) 5.58967 0.214987
\(677\) −47.8223 −1.83796 −0.918980 0.394304i \(-0.870986\pi\)
−0.918980 + 0.394304i \(0.870986\pi\)
\(678\) 17.3793 0.667450
\(679\) −9.85645 −0.378256
\(680\) −2.63565 −0.101073
\(681\) 10.2550 0.392974
\(682\) 0 0
\(683\) 4.96235 0.189879 0.0949395 0.995483i \(-0.469734\pi\)
0.0949395 + 0.995483i \(0.469734\pi\)
\(684\) −7.83428 −0.299551
\(685\) −9.37916 −0.358359
\(686\) 1.00000 0.0381802
\(687\) 2.10755 0.0804080
\(688\) 4.96783 0.189397
\(689\) 11.1841 0.426078
\(690\) −4.25063 −0.161819
\(691\) 28.2052 1.07298 0.536488 0.843908i \(-0.319751\pi\)
0.536488 + 0.843908i \(0.319751\pi\)
\(692\) 21.9024 0.832606
\(693\) 0 0
\(694\) 23.5535 0.894080
\(695\) −11.1498 −0.422936
\(696\) 1.52009 0.0576190
\(697\) −1.37439 −0.0520588
\(698\) −25.2540 −0.955879
\(699\) −29.9330 −1.13217
\(700\) −1.00000 −0.0377964
\(701\) 16.4276 0.620461 0.310230 0.950661i \(-0.399594\pi\)
0.310230 + 0.950661i \(0.399594\pi\)
\(702\) −24.2985 −0.917089
\(703\) 20.5058 0.773392
\(704\) 0 0
\(705\) −15.8585 −0.597266
\(706\) −20.9078 −0.786877
\(707\) 4.36976 0.164342
\(708\) −14.7013 −0.552508
\(709\) 26.0841 0.979610 0.489805 0.871832i \(-0.337068\pi\)
0.489805 + 0.871832i \(0.337068\pi\)
\(710\) 4.50169 0.168945
\(711\) 20.4483 0.766873
\(712\) −7.17843 −0.269023
\(713\) −3.62849 −0.135888
\(714\) −3.53948 −0.132462
\(715\) 0 0
\(716\) −3.11079 −0.116256
\(717\) −7.95087 −0.296931
\(718\) 25.9084 0.966892
\(719\) 16.7246 0.623723 0.311862 0.950128i \(-0.399047\pi\)
0.311862 + 0.950128i \(0.399047\pi\)
\(720\) −1.19656 −0.0445931
\(721\) −14.2978 −0.532479
\(722\) −23.8678 −0.888267
\(723\) −36.1813 −1.34560
\(724\) 4.58252 0.170308
\(725\) 1.13193 0.0420388
\(726\) 0 0
\(727\) 30.1587 1.11853 0.559263 0.828990i \(-0.311084\pi\)
0.559263 + 0.828990i \(0.311084\pi\)
\(728\) 4.31157 0.159798
\(729\) 27.4653 1.01724
\(730\) 12.1269 0.448836
\(731\) 13.0935 0.484280
\(732\) 7.82834 0.289344
\(733\) −29.3635 −1.08456 −0.542282 0.840197i \(-0.682440\pi\)
−0.542282 + 0.840197i \(0.682440\pi\)
\(734\) 20.8650 0.770141
\(735\) −1.34292 −0.0495345
\(736\) 3.16521 0.116671
\(737\) 0 0
\(738\) −0.623959 −0.0229682
\(739\) 17.5186 0.644432 0.322216 0.946666i \(-0.395572\pi\)
0.322216 + 0.946666i \(0.395572\pi\)
\(740\) 3.13193 0.115132
\(741\) −37.9099 −1.39265
\(742\) 2.59396 0.0952273
\(743\) 21.7292 0.797169 0.398584 0.917132i \(-0.369502\pi\)
0.398584 + 0.917132i \(0.369502\pi\)
\(744\) 1.53948 0.0564401
\(745\) −5.51928 −0.202211
\(746\) −10.1570 −0.371876
\(747\) 10.8425 0.396707
\(748\) 0 0
\(749\) −4.78019 −0.174664
\(750\) 1.34292 0.0490366
\(751\) 37.8166 1.37995 0.689974 0.723835i \(-0.257622\pi\)
0.689974 + 0.723835i \(0.257622\pi\)
\(752\) 11.8090 0.430628
\(753\) 14.1594 0.515998
\(754\) −4.88039 −0.177734
\(755\) −9.66388 −0.351705
\(756\) −5.63565 −0.204967
\(757\) 45.5840 1.65678 0.828390 0.560152i \(-0.189257\pi\)
0.828390 + 0.560152i \(0.189257\pi\)
\(758\) −29.5378 −1.07286
\(759\) 0 0
\(760\) −6.54735 −0.237497
\(761\) −36.0860 −1.30812 −0.654058 0.756445i \(-0.726935\pi\)
−0.654058 + 0.756445i \(0.726935\pi\)
\(762\) −14.7508 −0.534365
\(763\) 2.58240 0.0934893
\(764\) 4.33382 0.156792
\(765\) −3.15371 −0.114023
\(766\) −26.2888 −0.949853
\(767\) 47.1998 1.70429
\(768\) −1.34292 −0.0484586
\(769\) 28.8806 1.04146 0.520730 0.853722i \(-0.325660\pi\)
0.520730 + 0.853722i \(0.325660\pi\)
\(770\) 0 0
\(771\) −26.7206 −0.962319
\(772\) 13.4284 0.483300
\(773\) 41.6626 1.49850 0.749250 0.662287i \(-0.230414\pi\)
0.749250 + 0.662287i \(0.230414\pi\)
\(774\) 5.94429 0.213663
\(775\) 1.14637 0.0411787
\(776\) −9.85645 −0.353826
\(777\) 4.20594 0.150887
\(778\) 8.64078 0.309787
\(779\) −3.41419 −0.122326
\(780\) −5.79011 −0.207319
\(781\) 0 0
\(782\) 8.34240 0.298323
\(783\) 6.37916 0.227973
\(784\) 1.00000 0.0357143
\(785\) 16.8181 0.600262
\(786\) 16.0771 0.573450
\(787\) 13.3405 0.475537 0.237768 0.971322i \(-0.423584\pi\)
0.237768 + 0.971322i \(0.423584\pi\)
\(788\) −11.8667 −0.422733
\(789\) −16.9035 −0.601781
\(790\) 17.0893 0.608010
\(791\) −12.9414 −0.460144
\(792\) 0 0
\(793\) −25.1336 −0.892520
\(794\) −35.1598 −1.24777
\(795\) −3.48349 −0.123547
\(796\) −14.5672 −0.516319
\(797\) 32.9064 1.16560 0.582802 0.812614i \(-0.301956\pi\)
0.582802 + 0.812614i \(0.301956\pi\)
\(798\) −8.79259 −0.311254
\(799\) 31.1243 1.10110
\(800\) −1.00000 −0.0353553
\(801\) −8.58941 −0.303492
\(802\) 7.99484 0.282308
\(803\) 0 0
\(804\) 13.0565 0.460469
\(805\) 3.16521 0.111559
\(806\) −4.94264 −0.174097
\(807\) 9.27468 0.326484
\(808\) 4.36976 0.153728
\(809\) 27.0311 0.950363 0.475182 0.879888i \(-0.342382\pi\)
0.475182 + 0.879888i \(0.342382\pi\)
\(810\) 3.97858 0.139793
\(811\) 18.8842 0.663112 0.331556 0.943435i \(-0.392426\pi\)
0.331556 + 0.943435i \(0.392426\pi\)
\(812\) −1.13193 −0.0397229
\(813\) −10.1984 −0.357674
\(814\) 0 0
\(815\) −22.8952 −0.801982
\(816\) −3.53948 −0.123907
\(817\) 32.5261 1.13794
\(818\) 1.92553 0.0673247
\(819\) 5.15905 0.180272
\(820\) −0.521462 −0.0182102
\(821\) 18.9968 0.662992 0.331496 0.943457i \(-0.392447\pi\)
0.331496 + 0.943457i \(0.392447\pi\)
\(822\) −12.5955 −0.439318
\(823\) 32.5539 1.13476 0.567378 0.823457i \(-0.307958\pi\)
0.567378 + 0.823457i \(0.307958\pi\)
\(824\) −14.2978 −0.498088
\(825\) 0 0
\(826\) 10.9472 0.380903
\(827\) 10.3868 0.361183 0.180592 0.983558i \(-0.442199\pi\)
0.180592 + 0.983558i \(0.442199\pi\)
\(828\) 3.78735 0.131620
\(829\) −13.1965 −0.458332 −0.229166 0.973387i \(-0.573600\pi\)
−0.229166 + 0.973387i \(0.573600\pi\)
\(830\) 9.06142 0.314527
\(831\) 22.6729 0.786514
\(832\) 4.31157 0.149477
\(833\) 2.63565 0.0913200
\(834\) −14.9733 −0.518484
\(835\) 8.29513 0.287065
\(836\) 0 0
\(837\) 6.46052 0.223308
\(838\) 22.4861 0.776769
\(839\) −20.4534 −0.706128 −0.353064 0.935599i \(-0.614860\pi\)
−0.353064 + 0.935599i \(0.614860\pi\)
\(840\) −1.34292 −0.0463352
\(841\) −27.7187 −0.955819
\(842\) 23.3112 0.803357
\(843\) 12.9283 0.445273
\(844\) 2.79716 0.0962823
\(845\) 5.58967 0.192291
\(846\) 14.1301 0.485803
\(847\) 0 0
\(848\) 2.59396 0.0890770
\(849\) 30.3172 1.04048
\(850\) −2.63565 −0.0904022
\(851\) −9.91321 −0.339820
\(852\) 6.04542 0.207113
\(853\) 39.8363 1.36397 0.681984 0.731367i \(-0.261117\pi\)
0.681984 + 0.731367i \(0.261117\pi\)
\(854\) −5.82933 −0.199476
\(855\) −7.83428 −0.267927
\(856\) −4.78019 −0.163383
\(857\) −50.5306 −1.72609 −0.863047 0.505124i \(-0.831447\pi\)
−0.863047 + 0.505124i \(0.831447\pi\)
\(858\) 0 0
\(859\) −34.2746 −1.16943 −0.584717 0.811238i \(-0.698794\pi\)
−0.584717 + 0.811238i \(0.698794\pi\)
\(860\) 4.96783 0.169401
\(861\) −0.700283 −0.0238656
\(862\) 15.2591 0.519726
\(863\) 50.1602 1.70747 0.853736 0.520706i \(-0.174331\pi\)
0.853736 + 0.520706i \(0.174331\pi\)
\(864\) −5.63565 −0.191729
\(865\) 21.9024 0.744705
\(866\) 0.722365 0.0245470
\(867\) 13.5008 0.458513
\(868\) −1.14637 −0.0389102
\(869\) 0 0
\(870\) 1.52009 0.0515360
\(871\) −41.9192 −1.42038
\(872\) 2.58240 0.0874512
\(873\) −11.7938 −0.399160
\(874\) 20.7237 0.700991
\(875\) −1.00000 −0.0338062
\(876\) 16.2855 0.550235
\(877\) 42.9675 1.45091 0.725454 0.688270i \(-0.241630\pi\)
0.725454 + 0.688270i \(0.241630\pi\)
\(878\) 31.7894 1.07284
\(879\) −29.0392 −0.979468
\(880\) 0 0
\(881\) 7.10467 0.239362 0.119681 0.992812i \(-0.461813\pi\)
0.119681 + 0.992812i \(0.461813\pi\)
\(882\) 1.19656 0.0402902
\(883\) −41.1823 −1.38589 −0.692947 0.720989i \(-0.743688\pi\)
−0.692947 + 0.720989i \(0.743688\pi\)
\(884\) 11.3638 0.382207
\(885\) −14.7013 −0.494178
\(886\) −29.1789 −0.980284
\(887\) 31.0647 1.04305 0.521525 0.853236i \(-0.325363\pi\)
0.521525 + 0.853236i \(0.325363\pi\)
\(888\) 4.20594 0.141142
\(889\) 10.9841 0.368395
\(890\) −7.17843 −0.240622
\(891\) 0 0
\(892\) 19.6450 0.657763
\(893\) 77.3174 2.58733
\(894\) −7.41197 −0.247893
\(895\) −3.11079 −0.103982
\(896\) 1.00000 0.0334077
\(897\) 18.3269 0.611918
\(898\) −18.6378 −0.621950
\(899\) 1.29760 0.0432775
\(900\) −1.19656 −0.0398853
\(901\) 6.83678 0.227766
\(902\) 0 0
\(903\) 6.67141 0.222011
\(904\) −12.9414 −0.430426
\(905\) 4.58252 0.152328
\(906\) −12.9778 −0.431160
\(907\) 17.7502 0.589386 0.294693 0.955592i \(-0.404783\pi\)
0.294693 + 0.955592i \(0.404783\pi\)
\(908\) −7.63636 −0.253422
\(909\) 5.22867 0.173424
\(910\) 4.31157 0.142927
\(911\) 27.0043 0.894694 0.447347 0.894361i \(-0.352369\pi\)
0.447347 + 0.894361i \(0.352369\pi\)
\(912\) −8.79259 −0.291152
\(913\) 0 0
\(914\) 3.01602 0.0997611
\(915\) 7.82834 0.258797
\(916\) −1.56937 −0.0518536
\(917\) −11.9717 −0.395340
\(918\) −14.8536 −0.490243
\(919\) −1.75332 −0.0578368 −0.0289184 0.999582i \(-0.509206\pi\)
−0.0289184 + 0.999582i \(0.509206\pi\)
\(920\) 3.16521 0.104354
\(921\) 12.4371 0.409816
\(922\) −16.8872 −0.556149
\(923\) −19.4094 −0.638867
\(924\) 0 0
\(925\) 3.13193 0.102977
\(926\) −0.180188 −0.00592133
\(927\) −17.1082 −0.561906
\(928\) −1.13193 −0.0371574
\(929\) 1.51606 0.0497403 0.0248702 0.999691i \(-0.492083\pi\)
0.0248702 + 0.999691i \(0.492083\pi\)
\(930\) 1.53948 0.0504816
\(931\) 6.54735 0.214581
\(932\) 22.2894 0.730114
\(933\) −8.57710 −0.280802
\(934\) −12.4723 −0.408105
\(935\) 0 0
\(936\) 5.15905 0.168629
\(937\) 31.6995 1.03558 0.517789 0.855508i \(-0.326755\pi\)
0.517789 + 0.855508i \(0.326755\pi\)
\(938\) −9.72249 −0.317450
\(939\) 2.59279 0.0846124
\(940\) 11.8090 0.385166
\(941\) −15.4489 −0.503619 −0.251810 0.967777i \(-0.581026\pi\)
−0.251810 + 0.967777i \(0.581026\pi\)
\(942\) 22.5854 0.735871
\(943\) 1.65054 0.0537488
\(944\) 10.9472 0.356302
\(945\) −5.63565 −0.183328
\(946\) 0 0
\(947\) 10.0390 0.326222 0.163111 0.986608i \(-0.447847\pi\)
0.163111 + 0.986608i \(0.447847\pi\)
\(948\) 22.9496 0.745369
\(949\) −52.2859 −1.69727
\(950\) −6.54735 −0.212424
\(951\) 15.9489 0.517180
\(952\) 2.63565 0.0854220
\(953\) 41.2604 1.33656 0.668278 0.743912i \(-0.267032\pi\)
0.668278 + 0.743912i \(0.267032\pi\)
\(954\) 3.10382 0.100490
\(955\) 4.33382 0.140239
\(956\) 5.92057 0.191485
\(957\) 0 0
\(958\) −1.55428 −0.0502164
\(959\) 9.37916 0.302869
\(960\) −1.34292 −0.0433427
\(961\) −29.6858 −0.957608
\(962\) −13.5035 −0.435372
\(963\) −5.71977 −0.184317
\(964\) 26.9422 0.867749
\(965\) 13.4284 0.432276
\(966\) 4.25063 0.136762
\(967\) 18.8930 0.607557 0.303778 0.952743i \(-0.401752\pi\)
0.303778 + 0.952743i \(0.401752\pi\)
\(968\) 0 0
\(969\) −23.1742 −0.744463
\(970\) −9.85645 −0.316472
\(971\) 13.6882 0.439277 0.219638 0.975581i \(-0.429512\pi\)
0.219638 + 0.975581i \(0.429512\pi\)
\(972\) −11.5640 −0.370917
\(973\) 11.1498 0.357446
\(974\) −29.2641 −0.937683
\(975\) −5.79011 −0.185432
\(976\) −5.82933 −0.186592
\(977\) 34.5173 1.10431 0.552153 0.833743i \(-0.313807\pi\)
0.552153 + 0.833743i \(0.313807\pi\)
\(978\) −30.7464 −0.983163
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 3.09000 0.0986560
\(982\) −2.35553 −0.0751680
\(983\) −29.7951 −0.950316 −0.475158 0.879900i \(-0.657609\pi\)
−0.475158 + 0.879900i \(0.657609\pi\)
\(984\) −0.700283 −0.0223242
\(985\) −11.8667 −0.378104
\(986\) −2.98337 −0.0950099
\(987\) 15.8585 0.504782
\(988\) 28.2294 0.898096
\(989\) −15.7242 −0.500001
\(990\) 0 0
\(991\) 39.5393 1.25601 0.628004 0.778210i \(-0.283872\pi\)
0.628004 + 0.778210i \(0.283872\pi\)
\(992\) −1.14637 −0.0363971
\(993\) 38.3746 1.21778
\(994\) −4.50169 −0.142785
\(995\) −14.5672 −0.461810
\(996\) 12.1688 0.385583
\(997\) −8.24240 −0.261040 −0.130520 0.991446i \(-0.541665\pi\)
−0.130520 + 0.991446i \(0.541665\pi\)
\(998\) −25.2783 −0.800172
\(999\) 17.6505 0.558436
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.cz.1.2 6
11.10 odd 2 8470.2.a.df.1.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cz.1.2 6 1.1 even 1 trivial
8470.2.a.df.1.1 yes 6 11.10 odd 2