Properties

Label 8470.2.a.ci.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.14637 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14637 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.68585 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.14637 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14637 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.68585 q^{9} -1.00000 q^{10} +1.14637 q^{12} -4.68585 q^{13} -1.00000 q^{14} +1.14637 q^{15} +1.00000 q^{16} +0.292731 q^{17} +1.68585 q^{18} -6.51806 q^{19} +1.00000 q^{20} +1.14637 q^{21} +2.85363 q^{23} -1.14637 q^{24} +1.00000 q^{25} +4.68585 q^{26} -5.37169 q^{27} +1.00000 q^{28} +1.43910 q^{29} -1.14637 q^{30} +0.978577 q^{31} -1.00000 q^{32} -0.292731 q^{34} +1.00000 q^{35} -1.68585 q^{36} +0.853635 q^{37} +6.51806 q^{38} -5.37169 q^{39} -1.00000 q^{40} +6.22533 q^{41} -1.14637 q^{42} +10.3503 q^{43} -1.68585 q^{45} -2.85363 q^{46} -9.95715 q^{47} +1.14637 q^{48} +1.00000 q^{49} -1.00000 q^{50} +0.335577 q^{51} -4.68585 q^{52} +5.43910 q^{53} +5.37169 q^{54} -1.00000 q^{56} -7.47208 q^{57} -1.43910 q^{58} -9.37169 q^{59} +1.14637 q^{60} +11.9572 q^{61} -0.978577 q^{62} -1.68585 q^{63} +1.00000 q^{64} -4.68585 q^{65} -0.585462 q^{67} +0.292731 q^{68} +3.27131 q^{69} -1.00000 q^{70} -0.335577 q^{71} +1.68585 q^{72} +3.70727 q^{73} -0.853635 q^{74} +1.14637 q^{75} -6.51806 q^{76} +5.37169 q^{78} +2.51806 q^{79} +1.00000 q^{80} -1.10038 q^{81} -6.22533 q^{82} +1.70727 q^{83} +1.14637 q^{84} +0.292731 q^{85} -10.3503 q^{86} +1.64973 q^{87} +13.0790 q^{89} +1.68585 q^{90} -4.68585 q^{91} +2.85363 q^{92} +1.12181 q^{93} +9.95715 q^{94} -6.51806 q^{95} -1.14637 q^{96} +9.10352 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 2q^{3} + 3q^{4} + 3q^{5} - 2q^{6} + 3q^{7} - 3q^{8} + 7q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 2q^{3} + 3q^{4} + 3q^{5} - 2q^{6} + 3q^{7} - 3q^{8} + 7q^{9} - 3q^{10} + 2q^{12} - 2q^{13} - 3q^{14} + 2q^{15} + 3q^{16} - 2q^{17} - 7q^{18} + 6q^{19} + 3q^{20} + 2q^{21} + 10q^{23} - 2q^{24} + 3q^{25} + 2q^{26} + 8q^{27} + 3q^{28} - 2q^{30} - 12q^{31} - 3q^{32} + 2q^{34} + 3q^{35} + 7q^{36} + 4q^{37} - 6q^{38} + 8q^{39} - 3q^{40} - 4q^{41} - 2q^{42} - 8q^{43} + 7q^{45} - 10q^{46} + 2q^{48} + 3q^{49} - 3q^{50} + 28q^{51} - 2q^{52} + 12q^{53} - 8q^{54} - 3q^{56} + 8q^{57} - 4q^{59} + 2q^{60} + 6q^{61} + 12q^{62} + 7q^{63} + 3q^{64} - 2q^{65} + 4q^{67} - 2q^{68} - 8q^{69} - 3q^{70} - 28q^{71} - 7q^{72} + 14q^{73} - 4q^{74} + 2q^{75} + 6q^{76} - 8q^{78} - 18q^{79} + 3q^{80} + 3q^{81} + 4q^{82} + 8q^{83} + 2q^{84} - 2q^{85} + 8q^{86} + 44q^{87} + 18q^{89} - 7q^{90} - 2q^{91} + 10q^{92} + 12q^{93} + 6q^{95} - 2q^{96} - 4q^{97} - 3q^{98} + O(q^{100}) \)

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
\(3\) 1.14637 0.661854 0.330927 0.943656i \(-0.392639\pi\)
0.330927 + 0.943656i \(0.392639\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.14637 −0.468002
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −1.68585 −0.561949
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 1.14637 0.330927
\(13\) −4.68585 −1.29962 −0.649810 0.760097i \(-0.725152\pi\)
−0.649810 + 0.760097i \(0.725152\pi\)
\(14\) −1.00000 −0.267261
\(15\) 1.14637 0.295990
\(16\) 1.00000 0.250000
\(17\) 0.292731 0.0709977 0.0354988 0.999370i \(-0.488698\pi\)
0.0354988 + 0.999370i \(0.488698\pi\)
\(18\) 1.68585 0.397358
\(19\) −6.51806 −1.49535 −0.747673 0.664068i \(-0.768829\pi\)
−0.747673 + 0.664068i \(0.768829\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.14637 0.250157
\(22\) 0 0
\(23\) 2.85363 0.595024 0.297512 0.954718i \(-0.403843\pi\)
0.297512 + 0.954718i \(0.403843\pi\)
\(24\) −1.14637 −0.234001
\(25\) 1.00000 0.200000
\(26\) 4.68585 0.918970
\(27\) −5.37169 −1.03378
\(28\) 1.00000 0.188982
\(29\) 1.43910 0.267234 0.133617 0.991033i \(-0.457341\pi\)
0.133617 + 0.991033i \(0.457341\pi\)
\(30\) −1.14637 −0.209297
\(31\) 0.978577 0.175758 0.0878788 0.996131i \(-0.471991\pi\)
0.0878788 + 0.996131i \(0.471991\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −0.292731 −0.0502029
\(35\) 1.00000 0.169031
\(36\) −1.68585 −0.280974
\(37\) 0.853635 0.140337 0.0701683 0.997535i \(-0.477646\pi\)
0.0701683 + 0.997535i \(0.477646\pi\)
\(38\) 6.51806 1.05737
\(39\) −5.37169 −0.860159
\(40\) −1.00000 −0.158114
\(41\) 6.22533 0.972233 0.486116 0.873894i \(-0.338413\pi\)
0.486116 + 0.873894i \(0.338413\pi\)
\(42\) −1.14637 −0.176888
\(43\) 10.3503 1.57840 0.789201 0.614135i \(-0.210495\pi\)
0.789201 + 0.614135i \(0.210495\pi\)
\(44\) 0 0
\(45\) −1.68585 −0.251311
\(46\) −2.85363 −0.420745
\(47\) −9.95715 −1.45240 −0.726200 0.687483i \(-0.758715\pi\)
−0.726200 + 0.687483i \(0.758715\pi\)
\(48\) 1.14637 0.165464
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0.335577 0.0469901
\(52\) −4.68585 −0.649810
\(53\) 5.43910 0.747117 0.373559 0.927607i \(-0.378137\pi\)
0.373559 + 0.927607i \(0.378137\pi\)
\(54\) 5.37169 0.730995
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −7.47208 −0.989701
\(58\) −1.43910 −0.188963
\(59\) −9.37169 −1.22009 −0.610045 0.792367i \(-0.708849\pi\)
−0.610045 + 0.792367i \(0.708849\pi\)
\(60\) 1.14637 0.147995
\(61\) 11.9572 1.53096 0.765478 0.643462i \(-0.222502\pi\)
0.765478 + 0.643462i \(0.222502\pi\)
\(62\) −0.978577 −0.124279
\(63\) −1.68585 −0.212397
\(64\) 1.00000 0.125000
\(65\) −4.68585 −0.581208
\(66\) 0 0
\(67\) −0.585462 −0.0715256 −0.0357628 0.999360i \(-0.511386\pi\)
−0.0357628 + 0.999360i \(0.511386\pi\)
\(68\) 0.292731 0.0354988
\(69\) 3.27131 0.393819
\(70\) −1.00000 −0.119523
\(71\) −0.335577 −0.0398256 −0.0199128 0.999802i \(-0.506339\pi\)
−0.0199128 + 0.999802i \(0.506339\pi\)
\(72\) 1.68585 0.198679
\(73\) 3.70727 0.433903 0.216952 0.976182i \(-0.430389\pi\)
0.216952 + 0.976182i \(0.430389\pi\)
\(74\) −0.853635 −0.0992330
\(75\) 1.14637 0.132371
\(76\) −6.51806 −0.747673
\(77\) 0 0
\(78\) 5.37169 0.608224
\(79\) 2.51806 0.283304 0.141652 0.989917i \(-0.454759\pi\)
0.141652 + 0.989917i \(0.454759\pi\)
\(80\) 1.00000 0.111803
\(81\) −1.10038 −0.122265
\(82\) −6.22533 −0.687472
\(83\) 1.70727 0.187397 0.0936986 0.995601i \(-0.470131\pi\)
0.0936986 + 0.995601i \(0.470131\pi\)
\(84\) 1.14637 0.125079
\(85\) 0.292731 0.0317511
\(86\) −10.3503 −1.11610
\(87\) 1.64973 0.176870
\(88\) 0 0
\(89\) 13.0790 1.38637 0.693184 0.720761i \(-0.256208\pi\)
0.693184 + 0.720761i \(0.256208\pi\)
\(90\) 1.68585 0.177704
\(91\) −4.68585 −0.491210
\(92\) 2.85363 0.297512
\(93\) 1.12181 0.116326
\(94\) 9.95715 1.02700
\(95\) −6.51806 −0.668739
\(96\) −1.14637 −0.117000
\(97\) 9.10352 0.924322 0.462161 0.886796i \(-0.347074\pi\)
0.462161 + 0.886796i \(0.347074\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.7862 1.07327 0.536635 0.843814i \(-0.319695\pi\)
0.536635 + 0.843814i \(0.319695\pi\)
\(102\) −0.335577 −0.0332270
\(103\) 7.66442 0.755198 0.377599 0.925969i \(-0.376750\pi\)
0.377599 + 0.925969i \(0.376750\pi\)
\(104\) 4.68585 0.459485
\(105\) 1.14637 0.111874
\(106\) −5.43910 −0.528292
\(107\) −9.56404 −0.924591 −0.462295 0.886726i \(-0.652974\pi\)
−0.462295 + 0.886726i \(0.652974\pi\)
\(108\) −5.37169 −0.516891
\(109\) −3.14637 −0.301367 −0.150684 0.988582i \(-0.548147\pi\)
−0.150684 + 0.988582i \(0.548147\pi\)
\(110\) 0 0
\(111\) 0.978577 0.0928824
\(112\) 1.00000 0.0944911
\(113\) −2.58546 −0.243220 −0.121610 0.992578i \(-0.538806\pi\)
−0.121610 + 0.992578i \(0.538806\pi\)
\(114\) 7.47208 0.699824
\(115\) 2.85363 0.266103
\(116\) 1.43910 0.133617
\(117\) 7.89962 0.730320
\(118\) 9.37169 0.862734
\(119\) 0.292731 0.0268346
\(120\) −1.14637 −0.104648
\(121\) 0 0
\(122\) −11.9572 −1.08255
\(123\) 7.13650 0.643477
\(124\) 0.978577 0.0878788
\(125\) 1.00000 0.0894427
\(126\) 1.68585 0.150187
\(127\) 13.3717 1.18655 0.593273 0.805001i \(-0.297836\pi\)
0.593273 + 0.805001i \(0.297836\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.8652 1.04467
\(130\) 4.68585 0.410976
\(131\) 11.4391 0.999438 0.499719 0.866187i \(-0.333437\pi\)
0.499719 + 0.866187i \(0.333437\pi\)
\(132\) 0 0
\(133\) −6.51806 −0.565187
\(134\) 0.585462 0.0505762
\(135\) −5.37169 −0.462322
\(136\) −0.292731 −0.0251015
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −3.27131 −0.278472
\(139\) 14.1825 1.20294 0.601471 0.798895i \(-0.294581\pi\)
0.601471 + 0.798895i \(0.294581\pi\)
\(140\) 1.00000 0.0845154
\(141\) −11.4145 −0.961278
\(142\) 0.335577 0.0281610
\(143\) 0 0
\(144\) −1.68585 −0.140487
\(145\) 1.43910 0.119510
\(146\) −3.70727 −0.306816
\(147\) 1.14637 0.0945506
\(148\) 0.853635 0.0701683
\(149\) −11.5970 −0.950065 −0.475032 0.879968i \(-0.657564\pi\)
−0.475032 + 0.879968i \(0.657564\pi\)
\(150\) −1.14637 −0.0936004
\(151\) −1.73183 −0.140934 −0.0704671 0.997514i \(-0.522449\pi\)
−0.0704671 + 0.997514i \(0.522449\pi\)
\(152\) 6.51806 0.528684
\(153\) −0.493499 −0.0398971
\(154\) 0 0
\(155\) 0.978577 0.0786012
\(156\) −5.37169 −0.430080
\(157\) −12.2927 −0.981067 −0.490533 0.871422i \(-0.663198\pi\)
−0.490533 + 0.871422i \(0.663198\pi\)
\(158\) −2.51806 −0.200326
\(159\) 6.23519 0.494483
\(160\) −1.00000 −0.0790569
\(161\) 2.85363 0.224898
\(162\) 1.10038 0.0864543
\(163\) 23.9143 1.87311 0.936557 0.350516i \(-0.113994\pi\)
0.936557 + 0.350516i \(0.113994\pi\)
\(164\) 6.22533 0.486116
\(165\) 0 0
\(166\) −1.70727 −0.132510
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −1.14637 −0.0884440
\(169\) 8.95715 0.689012
\(170\) −0.292731 −0.0224514
\(171\) 10.9884 0.840307
\(172\) 10.3503 0.789201
\(173\) 7.37169 0.560459 0.280230 0.959933i \(-0.409589\pi\)
0.280230 + 0.959933i \(0.409589\pi\)
\(174\) −1.64973 −0.125066
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −10.7434 −0.807522
\(178\) −13.0790 −0.980310
\(179\) 1.56404 0.116902 0.0584509 0.998290i \(-0.481384\pi\)
0.0584509 + 0.998290i \(0.481384\pi\)
\(180\) −1.68585 −0.125656
\(181\) 14.7862 1.09905 0.549526 0.835477i \(-0.314808\pi\)
0.549526 + 0.835477i \(0.314808\pi\)
\(182\) 4.68585 0.347338
\(183\) 13.7073 1.01327
\(184\) −2.85363 −0.210373
\(185\) 0.853635 0.0627605
\(186\) −1.12181 −0.0822549
\(187\) 0 0
\(188\) −9.95715 −0.726200
\(189\) −5.37169 −0.390733
\(190\) 6.51806 0.472870
\(191\) −17.9572 −1.29933 −0.649667 0.760219i \(-0.725092\pi\)
−0.649667 + 0.760219i \(0.725092\pi\)
\(192\) 1.14637 0.0827318
\(193\) −18.6430 −1.34195 −0.670976 0.741479i \(-0.734125\pi\)
−0.670976 + 0.741479i \(0.734125\pi\)
\(194\) −9.10352 −0.653595
\(195\) −5.37169 −0.384675
\(196\) 1.00000 0.0714286
\(197\) 15.0361 1.07128 0.535639 0.844447i \(-0.320071\pi\)
0.535639 + 0.844447i \(0.320071\pi\)
\(198\) 0 0
\(199\) 15.3288 1.08663 0.543317 0.839528i \(-0.317168\pi\)
0.543317 + 0.839528i \(0.317168\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −0.671153 −0.0473395
\(202\) −10.7862 −0.758917
\(203\) 1.43910 0.101005
\(204\) 0.335577 0.0234951
\(205\) 6.22533 0.434796
\(206\) −7.66442 −0.534006
\(207\) −4.81079 −0.334373
\(208\) −4.68585 −0.324905
\(209\) 0 0
\(210\) −1.14637 −0.0791067
\(211\) −2.04285 −0.140635 −0.0703176 0.997525i \(-0.522401\pi\)
−0.0703176 + 0.997525i \(0.522401\pi\)
\(212\) 5.43910 0.373559
\(213\) −0.384694 −0.0263588
\(214\) 9.56404 0.653784
\(215\) 10.3503 0.705883
\(216\) 5.37169 0.365497
\(217\) 0.978577 0.0664301
\(218\) 3.14637 0.213099
\(219\) 4.24989 0.287181
\(220\) 0 0
\(221\) −1.37169 −0.0922700
\(222\) −0.978577 −0.0656778
\(223\) 11.0790 0.741902 0.370951 0.928652i \(-0.379032\pi\)
0.370951 + 0.928652i \(0.379032\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.68585 −0.112390
\(226\) 2.58546 0.171982
\(227\) 18.5426 1.23072 0.615358 0.788248i \(-0.289011\pi\)
0.615358 + 0.788248i \(0.289011\pi\)
\(228\) −7.47208 −0.494850
\(229\) −8.01469 −0.529626 −0.264813 0.964300i \(-0.585310\pi\)
−0.264813 + 0.964300i \(0.585310\pi\)
\(230\) −2.85363 −0.188163
\(231\) 0 0
\(232\) −1.43910 −0.0944813
\(233\) −27.9572 −1.83153 −0.915767 0.401710i \(-0.868416\pi\)
−0.915767 + 0.401710i \(0.868416\pi\)
\(234\) −7.89962 −0.516414
\(235\) −9.95715 −0.649533
\(236\) −9.37169 −0.610045
\(237\) 2.88661 0.187506
\(238\) −0.292731 −0.0189749
\(239\) 28.8108 1.86362 0.931808 0.362953i \(-0.118231\pi\)
0.931808 + 0.362953i \(0.118231\pi\)
\(240\) 1.14637 0.0739976
\(241\) 8.51806 0.548696 0.274348 0.961630i \(-0.411538\pi\)
0.274348 + 0.961630i \(0.411538\pi\)
\(242\) 0 0
\(243\) 14.8536 0.952861
\(244\) 11.9572 0.765478
\(245\) 1.00000 0.0638877
\(246\) −7.13650 −0.455007
\(247\) 30.5426 1.94338
\(248\) −0.978577 −0.0621397
\(249\) 1.95715 0.124030
\(250\) −1.00000 −0.0632456
\(251\) −23.1281 −1.45983 −0.729916 0.683537i \(-0.760441\pi\)
−0.729916 + 0.683537i \(0.760441\pi\)
\(252\) −1.68585 −0.106198
\(253\) 0 0
\(254\) −13.3717 −0.839015
\(255\) 0.335577 0.0210146
\(256\) 1.00000 0.0625000
\(257\) 14.8108 0.923872 0.461936 0.886913i \(-0.347155\pi\)
0.461936 + 0.886913i \(0.347155\pi\)
\(258\) −11.8652 −0.738695
\(259\) 0.853635 0.0530423
\(260\) −4.68585 −0.290604
\(261\) −2.42610 −0.150172
\(262\) −11.4391 −0.706710
\(263\) −18.7434 −1.15577 −0.577883 0.816119i \(-0.696121\pi\)
−0.577883 + 0.816119i \(0.696121\pi\)
\(264\) 0 0
\(265\) 5.43910 0.334121
\(266\) 6.51806 0.399648
\(267\) 14.9933 0.917573
\(268\) −0.585462 −0.0357628
\(269\) 12.6858 0.773470 0.386735 0.922191i \(-0.373603\pi\)
0.386735 + 0.922191i \(0.373603\pi\)
\(270\) 5.37169 0.326911
\(271\) −1.12181 −0.0681449 −0.0340725 0.999419i \(-0.510848\pi\)
−0.0340725 + 0.999419i \(0.510848\pi\)
\(272\) 0.292731 0.0177494
\(273\) −5.37169 −0.325110
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) 3.27131 0.196910
\(277\) −30.1151 −1.80944 −0.904720 0.426007i \(-0.859920\pi\)
−0.904720 + 0.426007i \(0.859920\pi\)
\(278\) −14.1825 −0.850609
\(279\) −1.64973 −0.0987668
\(280\) −1.00000 −0.0597614
\(281\) 14.3356 0.855189 0.427594 0.903971i \(-0.359361\pi\)
0.427594 + 0.903971i \(0.359361\pi\)
\(282\) 11.4145 0.679726
\(283\) −3.21377 −0.191039 −0.0955194 0.995428i \(-0.530451\pi\)
−0.0955194 + 0.995428i \(0.530451\pi\)
\(284\) −0.335577 −0.0199128
\(285\) −7.47208 −0.442608
\(286\) 0 0
\(287\) 6.22533 0.367469
\(288\) 1.68585 0.0993394
\(289\) −16.9143 −0.994959
\(290\) −1.43910 −0.0845067
\(291\) 10.4360 0.611767
\(292\) 3.70727 0.216952
\(293\) 16.6858 0.974798 0.487399 0.873179i \(-0.337946\pi\)
0.487399 + 0.873179i \(0.337946\pi\)
\(294\) −1.14637 −0.0668574
\(295\) −9.37169 −0.545641
\(296\) −0.853635 −0.0496165
\(297\) 0 0
\(298\) 11.5970 0.671797
\(299\) −13.3717 −0.773305
\(300\) 1.14637 0.0661854
\(301\) 10.3503 0.596580
\(302\) 1.73183 0.0996555
\(303\) 12.3650 0.710349
\(304\) −6.51806 −0.373836
\(305\) 11.9572 0.684665
\(306\) 0.493499 0.0282115
\(307\) 6.29273 0.359145 0.179573 0.983745i \(-0.442529\pi\)
0.179573 + 0.983745i \(0.442529\pi\)
\(308\) 0 0
\(309\) 8.78623 0.499831
\(310\) −0.978577 −0.0555794
\(311\) −20.3931 −1.15639 −0.578194 0.815900i \(-0.696242\pi\)
−0.578194 + 0.815900i \(0.696242\pi\)
\(312\) 5.37169 0.304112
\(313\) 12.0674 0.682090 0.341045 0.940047i \(-0.389219\pi\)
0.341045 + 0.940047i \(0.389219\pi\)
\(314\) 12.2927 0.693719
\(315\) −1.68585 −0.0949867
\(316\) 2.51806 0.141652
\(317\) 28.7679 1.61577 0.807884 0.589341i \(-0.200613\pi\)
0.807884 + 0.589341i \(0.200613\pi\)
\(318\) −6.23519 −0.349652
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −10.9639 −0.611944
\(322\) −2.85363 −0.159027
\(323\) −1.90804 −0.106166
\(324\) −1.10038 −0.0611325
\(325\) −4.68585 −0.259924
\(326\) −23.9143 −1.32449
\(327\) −3.60688 −0.199461
\(328\) −6.22533 −0.343736
\(329\) −9.95715 −0.548956
\(330\) 0 0
\(331\) −3.80765 −0.209288 −0.104644 0.994510i \(-0.533370\pi\)
−0.104644 + 0.994510i \(0.533370\pi\)
\(332\) 1.70727 0.0936986
\(333\) −1.43910 −0.0788620
\(334\) −8.00000 −0.437741
\(335\) −0.585462 −0.0319872
\(336\) 1.14637 0.0625394
\(337\) −15.3717 −0.837349 −0.418675 0.908136i \(-0.637505\pi\)
−0.418675 + 0.908136i \(0.637505\pi\)
\(338\) −8.95715 −0.487205
\(339\) −2.96388 −0.160976
\(340\) 0.292731 0.0158756
\(341\) 0 0
\(342\) −10.9884 −0.594187
\(343\) 1.00000 0.0539949
\(344\) −10.3503 −0.558049
\(345\) 3.27131 0.176121
\(346\) −7.37169 −0.396305
\(347\) −3.02142 −0.162198 −0.0810992 0.996706i \(-0.525843\pi\)
−0.0810992 + 0.996706i \(0.525843\pi\)
\(348\) 1.64973 0.0884348
\(349\) −11.0361 −0.590750 −0.295375 0.955381i \(-0.595445\pi\)
−0.295375 + 0.955381i \(0.595445\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 25.1709 1.34352
\(352\) 0 0
\(353\) 23.9817 1.27642 0.638209 0.769863i \(-0.279676\pi\)
0.638209 + 0.769863i \(0.279676\pi\)
\(354\) 10.7434 0.571004
\(355\) −0.335577 −0.0178106
\(356\) 13.0790 0.693184
\(357\) 0.335577 0.0177606
\(358\) −1.56404 −0.0826620
\(359\) 10.5181 0.555122 0.277561 0.960708i \(-0.410474\pi\)
0.277561 + 0.960708i \(0.410474\pi\)
\(360\) 1.68585 0.0888519
\(361\) 23.4851 1.23606
\(362\) −14.7862 −0.777147
\(363\) 0 0
\(364\) −4.68585 −0.245605
\(365\) 3.70727 0.194047
\(366\) −13.7073 −0.716490
\(367\) −35.5787 −1.85719 −0.928597 0.371089i \(-0.878985\pi\)
−0.928597 + 0.371089i \(0.878985\pi\)
\(368\) 2.85363 0.148756
\(369\) −10.4949 −0.546345
\(370\) −0.853635 −0.0443783
\(371\) 5.43910 0.282384
\(372\) 1.12181 0.0581630
\(373\) −7.50650 −0.388672 −0.194336 0.980935i \(-0.562255\pi\)
−0.194336 + 0.980935i \(0.562255\pi\)
\(374\) 0 0
\(375\) 1.14637 0.0591981
\(376\) 9.95715 0.513501
\(377\) −6.74338 −0.347302
\(378\) 5.37169 0.276290
\(379\) 33.4868 1.72010 0.860050 0.510210i \(-0.170432\pi\)
0.860050 + 0.510210i \(0.170432\pi\)
\(380\) −6.51806 −0.334369
\(381\) 15.3288 0.785321
\(382\) 17.9572 0.918768
\(383\) 15.6644 0.800415 0.400207 0.916425i \(-0.368938\pi\)
0.400207 + 0.916425i \(0.368938\pi\)
\(384\) −1.14637 −0.0585002
\(385\) 0 0
\(386\) 18.6430 0.948904
\(387\) −17.4490 −0.886981
\(388\) 9.10352 0.462161
\(389\) −19.6216 −0.994853 −0.497427 0.867506i \(-0.665722\pi\)
−0.497427 + 0.867506i \(0.665722\pi\)
\(390\) 5.37169 0.272006
\(391\) 0.835347 0.0422453
\(392\) −1.00000 −0.0505076
\(393\) 13.1134 0.661483
\(394\) −15.0361 −0.757509
\(395\) 2.51806 0.126697
\(396\) 0 0
\(397\) 10.2499 0.514427 0.257213 0.966355i \(-0.417196\pi\)
0.257213 + 0.966355i \(0.417196\pi\)
\(398\) −15.3288 −0.768366
\(399\) −7.47208 −0.374072
\(400\) 1.00000 0.0500000
\(401\) −16.8866 −0.843277 −0.421639 0.906764i \(-0.638545\pi\)
−0.421639 + 0.906764i \(0.638545\pi\)
\(402\) 0.671153 0.0334741
\(403\) −4.58546 −0.228418
\(404\) 10.7862 0.536635
\(405\) −1.10038 −0.0546785
\(406\) −1.43910 −0.0714212
\(407\) 0 0
\(408\) −0.335577 −0.0166135
\(409\) −18.2744 −0.903613 −0.451807 0.892116i \(-0.649220\pi\)
−0.451807 + 0.892116i \(0.649220\pi\)
\(410\) −6.22533 −0.307447
\(411\) 11.4637 0.565460
\(412\) 7.66442 0.377599
\(413\) −9.37169 −0.461151
\(414\) 4.81079 0.236437
\(415\) 1.70727 0.0838065
\(416\) 4.68585 0.229743
\(417\) 16.2583 0.796173
\(418\) 0 0
\(419\) −37.2860 −1.82154 −0.910770 0.412914i \(-0.864511\pi\)
−0.910770 + 0.412914i \(0.864511\pi\)
\(420\) 1.14637 0.0559369
\(421\) −2.78623 −0.135793 −0.0678963 0.997692i \(-0.521629\pi\)
−0.0678963 + 0.997692i \(0.521629\pi\)
\(422\) 2.04285 0.0994442
\(423\) 16.7862 0.816174
\(424\) −5.43910 −0.264146
\(425\) 0.292731 0.0141995
\(426\) 0.384694 0.0186385
\(427\) 11.9572 0.578647
\(428\) −9.56404 −0.462295
\(429\) 0 0
\(430\) −10.3503 −0.499134
\(431\) −38.0477 −1.83269 −0.916346 0.400387i \(-0.868876\pi\)
−0.916346 + 0.400387i \(0.868876\pi\)
\(432\) −5.37169 −0.258446
\(433\) −10.8108 −0.519533 −0.259767 0.965671i \(-0.583646\pi\)
−0.259767 + 0.965671i \(0.583646\pi\)
\(434\) −0.978577 −0.0469732
\(435\) 1.64973 0.0790985
\(436\) −3.14637 −0.150684
\(437\) −18.6002 −0.889766
\(438\) −4.24989 −0.203067
\(439\) −30.9933 −1.47923 −0.739614 0.673031i \(-0.764992\pi\)
−0.739614 + 0.673031i \(0.764992\pi\)
\(440\) 0 0
\(441\) −1.68585 −0.0802784
\(442\) 1.37169 0.0652448
\(443\) 25.3717 1.20545 0.602723 0.797951i \(-0.294083\pi\)
0.602723 + 0.797951i \(0.294083\pi\)
\(444\) 0.978577 0.0464412
\(445\) 13.0790 0.620002
\(446\) −11.0790 −0.524604
\(447\) −13.2944 −0.628805
\(448\) 1.00000 0.0472456
\(449\) −0.886615 −0.0418419 −0.0209210 0.999781i \(-0.506660\pi\)
−0.0209210 + 0.999781i \(0.506660\pi\)
\(450\) 1.68585 0.0794716
\(451\) 0 0
\(452\) −2.58546 −0.121610
\(453\) −1.98531 −0.0932779
\(454\) −18.5426 −0.870248
\(455\) −4.68585 −0.219676
\(456\) 7.47208 0.349912
\(457\) −29.9143 −1.39933 −0.699666 0.714470i \(-0.746668\pi\)
−0.699666 + 0.714470i \(0.746668\pi\)
\(458\) 8.01469 0.374502
\(459\) −1.57246 −0.0733962
\(460\) 2.85363 0.133051
\(461\) 2.33558 0.108779 0.0543893 0.998520i \(-0.482679\pi\)
0.0543893 + 0.998520i \(0.482679\pi\)
\(462\) 0 0
\(463\) 0.110250 0.00512374 0.00256187 0.999997i \(-0.499185\pi\)
0.00256187 + 0.999997i \(0.499185\pi\)
\(464\) 1.43910 0.0668084
\(465\) 1.12181 0.0520226
\(466\) 27.9572 1.29509
\(467\) 36.6760 1.69716 0.848581 0.529066i \(-0.177457\pi\)
0.848581 + 0.529066i \(0.177457\pi\)
\(468\) 7.89962 0.365160
\(469\) −0.585462 −0.0270341
\(470\) 9.95715 0.459289
\(471\) −14.0920 −0.649323
\(472\) 9.37169 0.431367
\(473\) 0 0
\(474\) −2.88661 −0.132587
\(475\) −6.51806 −0.299069
\(476\) 0.292731 0.0134173
\(477\) −9.16948 −0.419842
\(478\) −28.8108 −1.31777
\(479\) 42.7434 1.95300 0.976498 0.215529i \(-0.0691474\pi\)
0.976498 + 0.215529i \(0.0691474\pi\)
\(480\) −1.14637 −0.0523242
\(481\) −4.00000 −0.182384
\(482\) −8.51806 −0.387987
\(483\) 3.27131 0.148850
\(484\) 0 0
\(485\) 9.10352 0.413370
\(486\) −14.8536 −0.673775
\(487\) −36.1396 −1.63764 −0.818822 0.574048i \(-0.805372\pi\)
−0.818822 + 0.574048i \(0.805372\pi\)
\(488\) −11.9572 −0.541275
\(489\) 27.4145 1.23973
\(490\) −1.00000 −0.0451754
\(491\) −2.54262 −0.114747 −0.0573733 0.998353i \(-0.518273\pi\)
−0.0573733 + 0.998353i \(0.518273\pi\)
\(492\) 7.13650 0.321738
\(493\) 0.421268 0.0189730
\(494\) −30.5426 −1.37418
\(495\) 0 0
\(496\) 0.978577 0.0439394
\(497\) −0.335577 −0.0150527
\(498\) −1.95715 −0.0877022
\(499\) 3.13650 0.140409 0.0702045 0.997533i \(-0.477635\pi\)
0.0702045 + 0.997533i \(0.477635\pi\)
\(500\) 1.00000 0.0447214
\(501\) 9.17092 0.409727
\(502\) 23.1281 1.03226
\(503\) 28.5855 1.27456 0.637281 0.770631i \(-0.280059\pi\)
0.637281 + 0.770631i \(0.280059\pi\)
\(504\) 1.68585 0.0750936
\(505\) 10.7862 0.479981
\(506\) 0 0
\(507\) 10.2682 0.456026
\(508\) 13.3717 0.593273
\(509\) 2.20077 0.0975473 0.0487737 0.998810i \(-0.484469\pi\)
0.0487737 + 0.998810i \(0.484469\pi\)
\(510\) −0.335577 −0.0148596
\(511\) 3.70727 0.164000
\(512\) −1.00000 −0.0441942
\(513\) 35.0130 1.54586
\(514\) −14.8108 −0.653276
\(515\) 7.66442 0.337735
\(516\) 11.8652 0.522336
\(517\) 0 0
\(518\) −0.853635 −0.0375065
\(519\) 8.45065 0.370943
\(520\) 4.68585 0.205488
\(521\) 37.1940 1.62950 0.814750 0.579812i \(-0.196874\pi\)
0.814750 + 0.579812i \(0.196874\pi\)
\(522\) 2.42610 0.106187
\(523\) −30.4078 −1.32964 −0.664820 0.747003i \(-0.731492\pi\)
−0.664820 + 0.747003i \(0.731492\pi\)
\(524\) 11.4391 0.499719
\(525\) 1.14637 0.0500315
\(526\) 18.7434 0.817250
\(527\) 0.286460 0.0124784
\(528\) 0 0
\(529\) −14.8568 −0.645947
\(530\) −5.43910 −0.236259
\(531\) 15.7992 0.685628
\(532\) −6.51806 −0.282594
\(533\) −29.1709 −1.26353
\(534\) −14.9933 −0.648822
\(535\) −9.56404 −0.413489
\(536\) 0.585462 0.0252881
\(537\) 1.79296 0.0773720
\(538\) −12.6858 −0.546926
\(539\) 0 0
\(540\) −5.37169 −0.231161
\(541\) −39.0607 −1.67935 −0.839675 0.543090i \(-0.817254\pi\)
−0.839675 + 0.543090i \(0.817254\pi\)
\(542\) 1.12181 0.0481857
\(543\) 16.9504 0.727412
\(544\) −0.292731 −0.0125507
\(545\) −3.14637 −0.134775
\(546\) 5.37169 0.229887
\(547\) 0.585462 0.0250325 0.0125163 0.999922i \(-0.496016\pi\)
0.0125163 + 0.999922i \(0.496016\pi\)
\(548\) 10.0000 0.427179
\(549\) −20.1579 −0.860319
\(550\) 0 0
\(551\) −9.38011 −0.399606
\(552\) −3.27131 −0.139236
\(553\) 2.51806 0.107079
\(554\) 30.1151 1.27947
\(555\) 0.978577 0.0415383
\(556\) 14.1825 0.601471
\(557\) −17.4637 −0.739959 −0.369979 0.929040i \(-0.620635\pi\)
−0.369979 + 0.929040i \(0.620635\pi\)
\(558\) 1.64973 0.0698387
\(559\) −48.4998 −2.05132
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −14.3356 −0.604710
\(563\) 18.1579 0.765265 0.382633 0.923901i \(-0.375018\pi\)
0.382633 + 0.923901i \(0.375018\pi\)
\(564\) −11.4145 −0.480639
\(565\) −2.58546 −0.108771
\(566\) 3.21377 0.135085
\(567\) −1.10038 −0.0462118
\(568\) 0.335577 0.0140805
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 7.47208 0.312971
\(571\) 9.03612 0.378150 0.189075 0.981963i \(-0.439451\pi\)
0.189075 + 0.981963i \(0.439451\pi\)
\(572\) 0 0
\(573\) −20.5855 −0.859970
\(574\) −6.22533 −0.259840
\(575\) 2.85363 0.119005
\(576\) −1.68585 −0.0702436
\(577\) 19.3963 0.807476 0.403738 0.914875i \(-0.367711\pi\)
0.403738 + 0.914875i \(0.367711\pi\)
\(578\) 16.9143 0.703542
\(579\) −21.3717 −0.888177
\(580\) 1.43910 0.0597552
\(581\) 1.70727 0.0708295
\(582\) −10.4360 −0.432584
\(583\) 0 0
\(584\) −3.70727 −0.153408
\(585\) 7.89962 0.326609
\(586\) −16.6858 −0.689286
\(587\) 17.2614 0.712456 0.356228 0.934399i \(-0.384063\pi\)
0.356228 + 0.934399i \(0.384063\pi\)
\(588\) 1.14637 0.0472753
\(589\) −6.37842 −0.262818
\(590\) 9.37169 0.385826
\(591\) 17.2369 0.709031
\(592\) 0.853635 0.0350842
\(593\) 11.0361 0.453199 0.226599 0.973988i \(-0.427239\pi\)
0.226599 + 0.973988i \(0.427239\pi\)
\(594\) 0 0
\(595\) 0.292731 0.0120008
\(596\) −11.5970 −0.475032
\(597\) 17.5725 0.719193
\(598\) 13.3717 0.546809
\(599\) −41.7367 −1.70531 −0.852657 0.522472i \(-0.825010\pi\)
−0.852657 + 0.522472i \(0.825010\pi\)
\(600\) −1.14637 −0.0468002
\(601\) −39.3106 −1.60351 −0.801756 0.597652i \(-0.796100\pi\)
−0.801756 + 0.597652i \(0.796100\pi\)
\(602\) −10.3503 −0.421845
\(603\) 0.986999 0.0401937
\(604\) −1.73183 −0.0704671
\(605\) 0 0
\(606\) −12.3650 −0.502292
\(607\) 6.35027 0.257749 0.128875 0.991661i \(-0.458863\pi\)
0.128875 + 0.991661i \(0.458863\pi\)
\(608\) 6.51806 0.264342
\(609\) 1.64973 0.0668505
\(610\) −11.9572 −0.484131
\(611\) 46.6577 1.88757
\(612\) −0.493499 −0.0199485
\(613\) 42.4507 1.71457 0.857283 0.514846i \(-0.172151\pi\)
0.857283 + 0.514846i \(0.172151\pi\)
\(614\) −6.29273 −0.253954
\(615\) 7.13650 0.287771
\(616\) 0 0
\(617\) −0.677425 −0.0272721 −0.0136360 0.999907i \(-0.504341\pi\)
−0.0136360 + 0.999907i \(0.504341\pi\)
\(618\) −8.78623 −0.353434
\(619\) −18.5426 −0.745291 −0.372645 0.927974i \(-0.621549\pi\)
−0.372645 + 0.927974i \(0.621549\pi\)
\(620\) 0.978577 0.0393006
\(621\) −15.3288 −0.615125
\(622\) 20.3931 0.817689
\(623\) 13.0790 0.523998
\(624\) −5.37169 −0.215040
\(625\) 1.00000 0.0400000
\(626\) −12.0674 −0.482310
\(627\) 0 0
\(628\) −12.2927 −0.490533
\(629\) 0.249885 0.00996358
\(630\) 1.68585 0.0671657
\(631\) 21.3717 0.850794 0.425397 0.905007i \(-0.360135\pi\)
0.425397 + 0.905007i \(0.360135\pi\)
\(632\) −2.51806 −0.100163
\(633\) −2.34185 −0.0930801
\(634\) −28.7679 −1.14252
\(635\) 13.3717 0.530639
\(636\) 6.23519 0.247241
\(637\) −4.68585 −0.185660
\(638\) 0 0
\(639\) 0.565731 0.0223800
\(640\) −1.00000 −0.0395285
\(641\) 23.8715 0.942866 0.471433 0.881902i \(-0.343737\pi\)
0.471433 + 0.881902i \(0.343737\pi\)
\(642\) 10.9639 0.432710
\(643\) 0.311018 0.0122654 0.00613268 0.999981i \(-0.498048\pi\)
0.00613268 + 0.999981i \(0.498048\pi\)
\(644\) 2.85363 0.112449
\(645\) 11.8652 0.467191
\(646\) 1.90804 0.0750707
\(647\) −9.62158 −0.378263 −0.189132 0.981952i \(-0.560567\pi\)
−0.189132 + 0.981952i \(0.560567\pi\)
\(648\) 1.10038 0.0432272
\(649\) 0 0
\(650\) 4.68585 0.183794
\(651\) 1.12181 0.0439671
\(652\) 23.9143 0.936557
\(653\) −27.0607 −1.05897 −0.529483 0.848321i \(-0.677614\pi\)
−0.529483 + 0.848321i \(0.677614\pi\)
\(654\) 3.60688 0.141040
\(655\) 11.4391 0.446962
\(656\) 6.22533 0.243058
\(657\) −6.24989 −0.243831
\(658\) 9.95715 0.388170
\(659\) 26.4935 1.03204 0.516020 0.856576i \(-0.327413\pi\)
0.516020 + 0.856576i \(0.327413\pi\)
\(660\) 0 0
\(661\) 35.0852 1.36466 0.682329 0.731046i \(-0.260967\pi\)
0.682329 + 0.731046i \(0.260967\pi\)
\(662\) 3.80765 0.147989
\(663\) −1.57246 −0.0610693
\(664\) −1.70727 −0.0662549
\(665\) −6.51806 −0.252759
\(666\) 1.43910 0.0557639
\(667\) 4.10666 0.159010
\(668\) 8.00000 0.309529
\(669\) 12.7005 0.491031
\(670\) 0.585462 0.0226184
\(671\) 0 0
\(672\) −1.14637 −0.0442220
\(673\) −32.6577 −1.25886 −0.629431 0.777057i \(-0.716712\pi\)
−0.629431 + 0.777057i \(0.716712\pi\)
\(674\) 15.3717 0.592095
\(675\) −5.37169 −0.206757
\(676\) 8.95715 0.344506
\(677\) −9.12808 −0.350821 −0.175410 0.984495i \(-0.556125\pi\)
−0.175410 + 0.984495i \(0.556125\pi\)
\(678\) 2.96388 0.113827
\(679\) 9.10352 0.349361
\(680\) −0.292731 −0.0112257
\(681\) 21.2566 0.814555
\(682\) 0 0
\(683\) 10.4935 0.401523 0.200761 0.979640i \(-0.435658\pi\)
0.200761 + 0.979640i \(0.435658\pi\)
\(684\) 10.9884 0.420154
\(685\) 10.0000 0.382080
\(686\) −1.00000 −0.0381802
\(687\) −9.18777 −0.350535
\(688\) 10.3503 0.394600
\(689\) −25.4868 −0.970969
\(690\) −3.27131 −0.124537
\(691\) 1.70727 0.0649476 0.0324738 0.999473i \(-0.489661\pi\)
0.0324738 + 0.999473i \(0.489661\pi\)
\(692\) 7.37169 0.280230
\(693\) 0 0
\(694\) 3.02142 0.114692
\(695\) 14.1825 0.537972
\(696\) −1.64973 −0.0625329
\(697\) 1.82235 0.0690263
\(698\) 11.0361 0.417723
\(699\) −32.0491 −1.21221
\(700\) 1.00000 0.0377964
\(701\) 37.3534 1.41082 0.705409 0.708800i \(-0.250763\pi\)
0.705409 + 0.708800i \(0.250763\pi\)
\(702\) −25.1709 −0.950015
\(703\) −5.56404 −0.209852
\(704\) 0 0
\(705\) −11.4145 −0.429896
\(706\) −23.9817 −0.902564
\(707\) 10.7862 0.405658
\(708\) −10.7434 −0.403761
\(709\) −8.20704 −0.308222 −0.154111 0.988054i \(-0.549251\pi\)
−0.154111 + 0.988054i \(0.549251\pi\)
\(710\) 0.335577 0.0125940
\(711\) −4.24506 −0.159202
\(712\) −13.0790 −0.490155
\(713\) 2.79250 0.104580
\(714\) −0.335577 −0.0125586
\(715\) 0 0
\(716\) 1.56404 0.0584509
\(717\) 33.0277 1.23344
\(718\) −10.5181 −0.392530
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) −1.68585 −0.0628278
\(721\) 7.66442 0.285438
\(722\) −23.4851 −0.874024
\(723\) 9.76481 0.363157
\(724\) 14.7862 0.549526
\(725\) 1.43910 0.0534467
\(726\) 0 0
\(727\) −46.9442 −1.74106 −0.870531 0.492113i \(-0.836225\pi\)
−0.870531 + 0.492113i \(0.836225\pi\)
\(728\) 4.68585 0.173669
\(729\) 20.3288 0.752920
\(730\) −3.70727 −0.137212
\(731\) 3.02984 0.112063
\(732\) 13.7073 0.506635
\(733\) 42.0000 1.55131 0.775653 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 35.5787 1.31323
\(735\) 1.14637 0.0422843
\(736\) −2.85363 −0.105186
\(737\) 0 0
\(738\) 10.4949 0.386324
\(739\) 0.871922 0.0320742 0.0160371 0.999871i \(-0.494895\pi\)
0.0160371 + 0.999871i \(0.494895\pi\)
\(740\) 0.853635 0.0313802
\(741\) 35.0130 1.28623
\(742\) −5.43910 −0.199676
\(743\) 24.6712 0.905097 0.452548 0.891740i \(-0.350515\pi\)
0.452548 + 0.891740i \(0.350515\pi\)
\(744\) −1.12181 −0.0411274
\(745\) −11.5970 −0.424882
\(746\) 7.50650 0.274833
\(747\) −2.87819 −0.105308
\(748\) 0 0
\(749\) −9.56404 −0.349462
\(750\) −1.14637 −0.0418593
\(751\) −3.75011 −0.136844 −0.0684218 0.997656i \(-0.521796\pi\)
−0.0684218 + 0.997656i \(0.521796\pi\)
\(752\) −9.95715 −0.363100
\(753\) −26.5132 −0.966196
\(754\) 6.74338 0.245580
\(755\) −1.73183 −0.0630277
\(756\) −5.37169 −0.195367
\(757\) 15.7318 0.571783 0.285891 0.958262i \(-0.407710\pi\)
0.285891 + 0.958262i \(0.407710\pi\)
\(758\) −33.4868 −1.21629
\(759\) 0 0
\(760\) 6.51806 0.236435
\(761\) −15.2614 −0.553227 −0.276613 0.960981i \(-0.589212\pi\)
−0.276613 + 0.960981i \(0.589212\pi\)
\(762\) −15.3288 −0.555306
\(763\) −3.14637 −0.113906
\(764\) −17.9572 −0.649667
\(765\) −0.493499 −0.0178425
\(766\) −15.6644 −0.565979
\(767\) 43.9143 1.58565
\(768\) 1.14637 0.0413659
\(769\) 9.63986 0.347622 0.173811 0.984779i \(-0.444392\pi\)
0.173811 + 0.984779i \(0.444392\pi\)
\(770\) 0 0
\(771\) 16.9786 0.611469
\(772\) −18.6430 −0.670976
\(773\) 12.8291 0.461430 0.230715 0.973021i \(-0.425894\pi\)
0.230715 + 0.973021i \(0.425894\pi\)
\(774\) 17.4490 0.627190
\(775\) 0.978577 0.0351515
\(776\) −9.10352 −0.326797
\(777\) 0.978577 0.0351063
\(778\) 19.6216 0.703468
\(779\) −40.5770 −1.45382
\(780\) −5.37169 −0.192337
\(781\) 0 0
\(782\) −0.835347 −0.0298720
\(783\) −7.73038 −0.276261
\(784\) 1.00000 0.0357143
\(785\) −12.2927 −0.438746
\(786\) −13.1134 −0.467739
\(787\) −10.8291 −0.386015 −0.193007 0.981197i \(-0.561824\pi\)
−0.193007 + 0.981197i \(0.561824\pi\)
\(788\) 15.0361 0.535639
\(789\) −21.4868 −0.764949
\(790\) −2.51806 −0.0895885
\(791\) −2.58546 −0.0919284
\(792\) 0 0
\(793\) −56.0294 −1.98966
\(794\) −10.2499 −0.363755
\(795\) 6.23519 0.221139
\(796\) 15.3288 0.543317
\(797\) −21.0790 −0.746655 −0.373328 0.927700i \(-0.621783\pi\)
−0.373328 + 0.927700i \(0.621783\pi\)
\(798\) 7.47208 0.264509
\(799\) −2.91477 −0.103117
\(800\) −1.00000 −0.0353553
\(801\) −22.0491 −0.779067
\(802\) 16.8866 0.596287
\(803\) 0 0
\(804\) −0.671153 −0.0236698
\(805\) 2.85363 0.100577
\(806\) 4.58546 0.161516
\(807\) 14.5426 0.511924
\(808\) −10.7862 −0.379458
\(809\) −3.62158 −0.127328 −0.0636639 0.997971i \(-0.520279\pi\)
−0.0636639 + 0.997971i \(0.520279\pi\)
\(810\) 1.10038 0.0386636
\(811\) −34.7188 −1.21914 −0.609571 0.792731i \(-0.708658\pi\)
−0.609571 + 0.792731i \(0.708658\pi\)
\(812\) 1.43910 0.0505024
\(813\) −1.28600 −0.0451020
\(814\) 0 0
\(815\) 23.9143 0.837682
\(816\) 0.335577 0.0117475
\(817\) −67.4637 −2.36025
\(818\) 18.2744 0.638951
\(819\) 7.89962 0.276035
\(820\) 6.22533 0.217398
\(821\) 37.7549 1.31766 0.658828 0.752293i \(-0.271052\pi\)
0.658828 + 0.752293i \(0.271052\pi\)
\(822\) −11.4637 −0.399841
\(823\) 11.1892 0.390031 0.195016 0.980800i \(-0.437524\pi\)
0.195016 + 0.980800i \(0.437524\pi\)
\(824\) −7.66442 −0.267003
\(825\) 0 0
\(826\) 9.37169 0.326083
\(827\) 7.91431 0.275207 0.137604 0.990487i \(-0.456060\pi\)
0.137604 + 0.990487i \(0.456060\pi\)
\(828\) −4.81079 −0.167186
\(829\) −44.7152 −1.55302 −0.776512 0.630102i \(-0.783013\pi\)
−0.776512 + 0.630102i \(0.783013\pi\)
\(830\) −1.70727 −0.0592602
\(831\) −34.5229 −1.19759
\(832\) −4.68585 −0.162452
\(833\) 0.292731 0.0101425
\(834\) −16.2583 −0.562979
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) −5.25662 −0.181695
\(838\) 37.2860 1.28802
\(839\) 21.6791 0.748446 0.374223 0.927339i \(-0.377909\pi\)
0.374223 + 0.927339i \(0.377909\pi\)
\(840\) −1.14637 −0.0395534
\(841\) −26.9290 −0.928586
\(842\) 2.78623 0.0960198
\(843\) 16.4338 0.566010
\(844\) −2.04285 −0.0703176
\(845\) 8.95715 0.308135
\(846\) −16.7862 −0.577122
\(847\) 0 0
\(848\) 5.43910 0.186779
\(849\) −3.68415 −0.126440
\(850\) −0.292731 −0.0100406
\(851\) 2.43596 0.0835037
\(852\) −0.384694 −0.0131794
\(853\) −9.41454 −0.322348 −0.161174 0.986926i \(-0.551528\pi\)
−0.161174 + 0.986926i \(0.551528\pi\)
\(854\) −11.9572 −0.409165
\(855\) 10.9884 0.375797
\(856\) 9.56404 0.326892
\(857\) 25.8652 0.883538 0.441769 0.897129i \(-0.354351\pi\)
0.441769 + 0.897129i \(0.354351\pi\)
\(858\) 0 0
\(859\) −29.2860 −0.999225 −0.499613 0.866249i \(-0.666524\pi\)
−0.499613 + 0.866249i \(0.666524\pi\)
\(860\) 10.3503 0.352941
\(861\) 7.13650 0.243211
\(862\) 38.0477 1.29591
\(863\) −20.3110 −0.691395 −0.345698 0.938346i \(-0.612358\pi\)
−0.345698 + 0.938346i \(0.612358\pi\)
\(864\) 5.37169 0.182749
\(865\) 7.37169 0.250645
\(866\) 10.8108 0.367366
\(867\) −19.3900 −0.658518
\(868\) 0.978577 0.0332151
\(869\) 0 0
\(870\) −1.64973 −0.0559311
\(871\) 2.74338 0.0929560
\(872\) 3.14637 0.106549
\(873\) −15.3471 −0.519422
\(874\) 18.6002 0.629160
\(875\) 1.00000 0.0338062
\(876\) 4.24989 0.143590
\(877\) 38.3650 1.29549 0.647746 0.761856i \(-0.275712\pi\)
0.647746 + 0.761856i \(0.275712\pi\)
\(878\) 30.9933 1.04597
\(879\) 19.1281 0.645174
\(880\) 0 0
\(881\) 40.5426 1.36592 0.682958 0.730458i \(-0.260693\pi\)
0.682958 + 0.730458i \(0.260693\pi\)
\(882\) 1.68585 0.0567654
\(883\) 28.0491 0.943928 0.471964 0.881618i \(-0.343545\pi\)
0.471964 + 0.881618i \(0.343545\pi\)
\(884\) −1.37169 −0.0461350
\(885\) −10.7434 −0.361135
\(886\) −25.3717 −0.852379
\(887\) 6.65769 0.223543 0.111772 0.993734i \(-0.464347\pi\)
0.111772 + 0.993734i \(0.464347\pi\)
\(888\) −0.978577 −0.0328389
\(889\) 13.3717 0.448472
\(890\) −13.0790 −0.438408
\(891\) 0 0
\(892\) 11.0790 0.370951
\(893\) 64.9013 2.17184
\(894\) 13.2944 0.444632
\(895\) 1.56404 0.0522801
\(896\) −1.00000 −0.0334077
\(897\) −15.3288 −0.511815
\(898\) 0.886615 0.0295867
\(899\) 1.40827 0.0469683
\(900\) −1.68585 −0.0561949
\(901\) 1.59219 0.0530436
\(902\) 0 0
\(903\) 11.8652 0.394849
\(904\) 2.58546 0.0859912
\(905\) 14.7862 0.491511
\(906\) 1.98531 0.0659574
\(907\) 39.5787 1.31419 0.657095 0.753808i \(-0.271785\pi\)
0.657095 + 0.753808i \(0.271785\pi\)
\(908\) 18.5426 0.615358
\(909\) −18.1839 −0.603123
\(910\) 4.68585 0.155334
\(911\) −24.3356 −0.806274 −0.403137 0.915140i \(-0.632080\pi\)
−0.403137 + 0.915140i \(0.632080\pi\)
\(912\) −7.47208 −0.247425
\(913\) 0 0
\(914\) 29.9143 0.989477
\(915\) 13.7073 0.453148
\(916\) −8.01469 −0.264813
\(917\) 11.4391 0.377752
\(918\) 1.57246 0.0518989
\(919\) −50.6331 −1.67023 −0.835117 0.550073i \(-0.814600\pi\)
−0.835117 + 0.550073i \(0.814600\pi\)
\(920\) −2.85363 −0.0940815
\(921\) 7.21377 0.237702
\(922\) −2.33558 −0.0769181
\(923\) 1.57246 0.0517582
\(924\) 0 0
\(925\) 0.853635 0.0280673
\(926\) −0.110250 −0.00362303
\(927\) −12.9210 −0.424383
\(928\) −1.43910 −0.0472407
\(929\) 47.1512 1.54698 0.773490 0.633808i \(-0.218509\pi\)
0.773490 + 0.633808i \(0.218509\pi\)
\(930\) −1.12181 −0.0367855
\(931\) −6.51806 −0.213621
\(932\) −27.9572 −0.915767
\(933\) −23.3780 −0.765360
\(934\) −36.6760 −1.20007
\(935\) 0 0
\(936\) −7.89962 −0.258207
\(937\) 42.0294 1.37304 0.686520 0.727111i \(-0.259137\pi\)
0.686520 + 0.727111i \(0.259137\pi\)
\(938\) 0.585462 0.0191160
\(939\) 13.8337 0.451444
\(940\) −9.95715 −0.324767
\(941\) 36.7434 1.19780 0.598900 0.800824i \(-0.295605\pi\)
0.598900 + 0.800824i \(0.295605\pi\)
\(942\) 14.0920 0.459141
\(943\) 17.7648 0.578502
\(944\) −9.37169 −0.305023
\(945\) −5.37169 −0.174741
\(946\) 0 0
\(947\) 18.8782 0.613459 0.306729 0.951797i \(-0.400765\pi\)
0.306729 + 0.951797i \(0.400765\pi\)
\(948\) 2.88661 0.0937529
\(949\) −17.3717 −0.563909
\(950\) 6.51806 0.211474
\(951\) 32.9786 1.06940
\(952\) −0.292731 −0.00948747
\(953\) 43.2285 1.40031 0.700154 0.713992i \(-0.253115\pi\)
0.700154 + 0.713992i \(0.253115\pi\)
\(954\) 9.16948 0.296873
\(955\) −17.9572 −0.581080
\(956\) 28.8108 0.931808
\(957\) 0 0
\(958\) −42.7434 −1.38098
\(959\) 10.0000 0.322917
\(960\) 1.14637 0.0369988
\(961\) −30.0424 −0.969109
\(962\) 4.00000 0.128965
\(963\) 16.1235 0.519572
\(964\) 8.51806 0.274348
\(965\) −18.6430 −0.600139
\(966\) −3.27131 −0.105253
\(967\) −48.7299 −1.56705 −0.783524 0.621361i \(-0.786580\pi\)
−0.783524 + 0.621361i \(0.786580\pi\)
\(968\) 0 0
\(969\) −2.18731 −0.0702665
\(970\) −9.10352 −0.292296
\(971\) −25.9865 −0.833948 −0.416974 0.908918i \(-0.636909\pi\)
−0.416974 + 0.908918i \(0.636909\pi\)
\(972\) 14.8536 0.476431
\(973\) 14.1825 0.454669
\(974\) 36.1396 1.15799
\(975\) −5.37169 −0.172032
\(976\) 11.9572 0.382739
\(977\) 2.67115 0.0854578 0.0427289 0.999087i \(-0.486395\pi\)
0.0427289 + 0.999087i \(0.486395\pi\)
\(978\) −27.4145 −0.876620
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 5.30429 0.169353
\(982\) 2.54262 0.0811381
\(983\) −8.33558 −0.265864 −0.132932 0.991125i \(-0.542439\pi\)
−0.132932 + 0.991125i \(0.542439\pi\)
\(984\) −7.13650 −0.227503
\(985\) 15.0361 0.479090
\(986\) −0.421268 −0.0134159
\(987\) −11.4145 −0.363329
\(988\) 30.5426 0.971690
\(989\) 29.5359 0.939187
\(990\) 0 0
\(991\) 46.2730 1.46991 0.734955 0.678116i \(-0.237203\pi\)
0.734955 + 0.678116i \(0.237203\pi\)
\(992\) −0.978577 −0.0310699
\(993\) −4.36496 −0.138518
\(994\) 0.335577 0.0106438
\(995\) 15.3288 0.485957
\(996\) 1.95715 0.0620148
\(997\) −35.8715 −1.13606 −0.568030 0.823008i \(-0.692294\pi\)
−0.568030 + 0.823008i \(0.692294\pi\)
\(998\) −3.13650 −0.0992842
\(999\) −4.58546 −0.145078
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.ci.1.2 3
11.10 odd 2 770.2.a.m.1.2 3
33.32 even 2 6930.2.a.ce.1.3 3
44.43 even 2 6160.2.a.bf.1.2 3
55.32 even 4 3850.2.c.ba.1849.5 6
55.43 even 4 3850.2.c.ba.1849.2 6
55.54 odd 2 3850.2.a.bt.1.2 3
77.76 even 2 5390.2.a.ca.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.m.1.2 3 11.10 odd 2
3850.2.a.bt.1.2 3 55.54 odd 2
3850.2.c.ba.1849.2 6 55.43 even 4
3850.2.c.ba.1849.5 6 55.32 even 4
5390.2.a.ca.1.2 3 77.76 even 2
6160.2.a.bf.1.2 3 44.43 even 2
6930.2.a.ce.1.3 3 33.32 even 2
8470.2.a.ci.1.2 3 1.1 even 1 trivial