Properties

Label 8470.2.a.df.1.1
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 1.1
Root \(0.0816388\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

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

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 \(-0.704001\pi\)
−0.597908 + 0.801565i \(0.704001\pi\)
\(14\) 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
\(19\) −6.54735 −1.50207 −0.751033 0.660265i \(-0.770444\pi\)
−0.751033 + 0.660265i \(0.770444\pi\)
\(20\) 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 \(-0.533515\pi\)
−0.105097 + 0.994462i \(0.533515\pi\)
\(30\) −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 \(-0.487035\pi\)
0.0407193 + 0.999171i \(0.487035\pi\)
\(42\) −1.34292 −0.207218
\(43\) −4.96783 −0.757587 −0.378793 0.925481i \(-0.623661\pi\)
−0.378793 + 0.925481i \(0.623661\pi\)
\(44\) 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 \(-0.378266\pi\)
0.373185 + 0.927757i \(0.378266\pi\)
\(62\) 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 \(-0.248843\pi\)
0.709672 + 0.704533i \(0.248843\pi\)
\(74\) 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 \(-0.0887885\pi\)
0.961349 + 0.275334i \(0.0887885\pi\)
\(80\) 1.00000 0.111803
\(81\) −3.97858 −0.442064
\(82\) 0.521462 0.0575858
\(83\) 9.06142 0.994620 0.497310 0.867573i \(-0.334321\pi\)
0.497310 + 0.867573i \(0.334321\pi\)
\(84\) −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 \(-0.430241\pi\)
0.217404 + 0.976082i \(0.430241\pi\)
\(102\) 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 \(-0.574219\pi\)
−0.231059 + 0.972940i \(0.574219\pi\)
\(108\) 5.63565 0.542291
\(109\) 2.58240 0.247349 0.123675 0.992323i \(-0.460532\pi\)
0.123675 + 0.992323i \(0.460532\pi\)
\(110\) 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 \(-0.337967\pi\)
0.487340 + 0.873212i \(0.337967\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.67141 0.587385
\(130\) −4.31157 −0.378150
\(131\) −11.9717 −1.04597 −0.522986 0.852341i \(-0.675182\pi\)
−0.522986 + 0.852341i \(0.675182\pi\)
\(132\) 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 \(-0.343223\pi\)
0.472857 + 0.881139i \(0.343223\pi\)
\(140\) 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 \(-0.427409\pi\)
0.226079 + 0.974109i \(0.427409\pi\)
\(150\) −1.34292 −0.109649
\(151\) 9.66388 0.786435 0.393218 0.919445i \(-0.371362\pi\)
0.393218 + 0.919445i \(0.371362\pi\)
\(152\) −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 \(-0.604002\pi\)
−0.320948 + 0.947097i \(0.604002\pi\)
\(168\) −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 \(-0.813152\pi\)
−0.832606 + 0.553866i \(0.813152\pi\)
\(174\) 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 \(-0.660562\pi\)
−0.483300 + 0.875455i \(0.660562\pi\)
\(194\) 9.85645 0.707652
\(195\) 5.79011 0.414639
\(196\) 1.00000 0.0714286
\(197\) 11.8667 0.845466 0.422733 0.906254i \(-0.361071\pi\)
0.422733 + 0.906254i \(0.361071\pi\)
\(198\) 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 \(-0.530695\pi\)
−0.0962823 + 0.995354i \(0.530695\pi\)
\(212\) 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 \(-0.418444\pi\)
0.253422 + 0.967356i \(0.418444\pi\)
\(228\) 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 \(-0.760533\pi\)
−0.730114 + 0.683325i \(0.760533\pi\)
\(234\) 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 \(-0.561330\pi\)
−0.191485 + 0.981496i \(0.561330\pi\)
\(240\) −1.34292 −0.0866853
\(241\) −26.9422 −1.73550 −0.867749 0.497002i \(-0.834434\pi\)
−0.867749 + 0.497002i \(0.834434\pi\)
\(242\) 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 \(-0.626861\pi\)
−0.388077 + 0.921627i \(0.626861\pi\)
\(264\) 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 \(-0.574088\pi\)
−0.230657 + 0.973035i \(0.574088\pi\)
\(272\) −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 \(-0.330678\pi\)
0.507208 + 0.861824i \(0.330678\pi\)
\(278\) 11.1498 0.668721
\(279\) −1.37169 −0.0821211
\(280\) 1.00000 0.0597614
\(281\) 9.62695 0.574296 0.287148 0.957886i \(-0.407293\pi\)
0.287148 + 0.957886i \(0.407293\pi\)
\(282\) −15.8585 −0.944361
\(283\) 22.5755 1.34198 0.670988 0.741468i \(-0.265870\pi\)
0.670988 + 0.741468i \(0.265870\pi\)
\(284\) −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 \(-0.717618\pi\)
−0.631640 + 0.775262i \(0.717618\pi\)
\(294\) −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 \(-0.414865\pi\)
0.264282 + 0.964445i \(0.414865\pi\)
\(308\) 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 \(-0.573344\pi\)
−0.228384 + 0.973571i \(0.573344\pi\)
\(338\) 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 \(-0.282148\pi\)
0.632210 + 0.774797i \(0.282148\pi\)
\(348\) 1.52009 0.0814856
\(349\) −25.2540 −1.35182 −0.675909 0.736985i \(-0.736249\pi\)
−0.675909 + 0.736985i \(0.736249\pi\)
\(350\) 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 \(-0.260372\pi\)
0.683696 + 0.729767i \(0.260372\pi\)
\(360\) −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 \(-0.584697\pi\)
−0.262956 + 0.964808i \(0.584697\pi\)
\(374\) 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 \(-0.484841\pi\)
0.0476057 + 0.998866i \(0.484841\pi\)
\(410\) 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 \(-0.380213\pi\)
0.367502 + 0.930023i \(0.380213\pi\)
\(432\) 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 \(-0.225878\pi\)
0.758612 + 0.651543i \(0.225878\pi\)
\(440\) 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 \(-0.477527\pi\)
0.0705418 + 0.997509i \(0.477527\pi\)
\(458\) −1.56937 −0.0733320
\(459\) −14.8536 −0.693308
\(460\) −3.16521 −0.147579
\(461\) −16.8872 −0.786513 −0.393257 0.919429i \(-0.628652\pi\)
−0.393257 + 0.919429i \(0.628652\pi\)
\(462\) 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 \(-0.511305\pi\)
−0.0355083 + 0.999369i \(0.511305\pi\)
\(480\) −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 \(-0.516927\pi\)
−0.0531518 + 0.998586i \(0.516927\pi\)
\(492\) −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 \(-0.522986\pi\)
−0.0721510 + 0.997394i \(0.522986\pi\)
\(504\) −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 \(-0.665904\pi\)
−0.497923 + 0.867221i \(0.665904\pi\)
\(524\) −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 \(-0.215752\pi\)
0.778952 + 0.627084i \(0.215752\pi\)
\(542\) −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 \(-0.380243\pi\)
0.367416 + 0.930057i \(0.380243\pi\)
\(548\) −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 \(-0.733850\pi\)
−0.670335 + 0.742058i \(0.733850\pi\)
\(558\) −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 \(-0.294743\pi\)
0.601066 + 0.799199i \(0.294743\pi\)
\(564\) −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 \(-0.620903\pi\)
−0.370760 + 0.928729i \(0.620903\pi\)
\(570\) 8.79259 0.368281
\(571\) 11.2551 0.471012 0.235506 0.971873i \(-0.424325\pi\)
0.235506 + 0.971873i \(0.424325\pi\)
\(572\) 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 \(-0.473696\pi\)
0.0825439 + 0.996587i \(0.473696\pi\)
\(594\) 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 \(-0.634112\pi\)
−0.408970 + 0.912548i \(0.634112\pi\)
\(602\) −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 \(-0.387857\pi\)
0.345064 + 0.938579i \(0.387857\pi\)
\(608\) −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 \(-0.613368\pi\)
−0.348673 + 0.937244i \(0.613368\pi\)
\(614\) 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 \(-0.761349\pi\)
−0.731863 + 0.681451i \(0.761349\pi\)
\(660\) 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 \(-0.765479\pi\)
−0.740643 + 0.671899i \(0.765479\pi\)
\(674\) −8.38514 −0.322983
\(675\) 5.63565 0.216916
\(676\) 5.58967 0.214987
\(677\) 47.8223 1.83796 0.918980 0.394304i \(-0.129014\pi\)
0.918980 + 0.394304i \(0.129014\pi\)
\(678\) −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 \(-0.600406\pi\)
−0.310230 + 0.950661i \(0.600406\pi\)
\(702\) −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 \(-0.317560\pi\)
0.542282 + 0.840197i \(0.317560\pi\)
\(734\) −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 \(-0.604428\pi\)
−0.322216 + 0.946666i \(0.604428\pi\)
\(740\) 3.13193 0.115132
\(741\) −37.9099 −1.39265
\(742\) 2.59396 0.0952273
\(743\) −21.7292 −0.797169 −0.398584 0.917132i \(-0.630498\pi\)
−0.398584 + 0.917132i \(0.630498\pi\)
\(744\) −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 \(-0.273065\pi\)
0.654058 + 0.756445i \(0.273065\pi\)
\(762\) −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 \(-0.674340\pi\)
−0.520730 + 0.853722i \(0.674340\pi\)
\(770\) 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 \(-0.576416\pi\)
−0.237768 + 0.971322i \(0.576416\pi\)
\(788\) 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 \(-0.657618\pi\)
−0.475182 + 0.879888i \(0.657618\pi\)
\(810\) −3.97858 −0.139793
\(811\) −18.8842 −0.663112 −0.331556 0.943435i \(-0.607574\pi\)
−0.331556 + 0.943435i \(0.607574\pi\)
\(812\) −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 \(-0.607553\pi\)
−0.331496 + 0.943457i \(0.607553\pi\)
\(822\) 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 \(-0.557801\pi\)
−0.180592 + 0.983558i \(0.557801\pi\)
\(828\) 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 \(-0.738883\pi\)
−0.681984 + 0.731367i \(0.738883\pi\)
\(854\) 5.82933 0.199476
\(855\) 7.83428 0.267927
\(856\) −4.78019 −0.163383
\(857\) 50.5306 1.72609 0.863047 0.505124i \(-0.168553\pi\)
0.863047 + 0.505124i \(0.168553\pi\)
\(858\) 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 \(-0.758370\pi\)
−0.725454 + 0.688270i \(0.758370\pi\)
\(878\) 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 \(-0.674637\pi\)
−0.521525 + 0.853236i \(0.674637\pi\)
\(888\) −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 \(-0.490794\pi\)
0.0289184 + 0.999582i \(0.490794\pi\)
\(920\) −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 \(-0.673245\pi\)
−0.517789 + 0.855508i \(0.673245\pi\)
\(938\) −9.72249 −0.317450
\(939\) 2.59279 0.0846124
\(940\) 11.8090 0.385166
\(941\) 15.4489 0.503619 0.251810 0.967777i \(-0.418974\pi\)
0.251810 + 0.967777i \(0.418974\pi\)
\(942\) −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 \(-0.732968\pi\)
−0.668278 + 0.743912i \(0.732968\pi\)
\(954\) −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 \(-0.598248\pi\)
−0.303778 + 0.952743i \(0.598248\pi\)
\(968\) 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 \(-0.458335\pi\)
0.130520 + 0.991446i \(0.458335\pi\)
\(998\) 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.df.1.1 yes 6
11.10 odd 2 8470.2.a.cz.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cz.1.2 6 11.10 odd 2
8470.2.a.df.1.1 yes 6 1.1 even 1 trivial