Properties

Label 6160.2.a.bf.1.2
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.1878476451\)
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 \(\chi\) \(=\) 6160.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.14637 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.68585 q^{9} +O(q^{10})\) \(q-1.14637 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.68585 q^{9} -1.00000 q^{11} +4.68585 q^{13} -1.14637 q^{15} -0.292731 q^{17} -6.51806 q^{19} -1.14637 q^{21} -2.85363 q^{23} +1.00000 q^{25} +5.37169 q^{27} -1.43910 q^{29} -0.978577 q^{31} +1.14637 q^{33} +1.00000 q^{35} +0.853635 q^{37} -5.37169 q^{39} -6.22533 q^{41} +10.3503 q^{43} -1.68585 q^{45} +9.95715 q^{47} +1.00000 q^{49} +0.335577 q^{51} +5.43910 q^{53} -1.00000 q^{55} +7.47208 q^{57} +9.37169 q^{59} -11.9572 q^{61} -1.68585 q^{63} +4.68585 q^{65} +0.585462 q^{67} +3.27131 q^{69} +0.335577 q^{71} -3.70727 q^{73} -1.14637 q^{75} -1.00000 q^{77} +2.51806 q^{79} -1.10038 q^{81} +1.70727 q^{83} -0.292731 q^{85} +1.64973 q^{87} +13.0790 q^{89} +4.68585 q^{91} +1.12181 q^{93} -6.51806 q^{95} +9.10352 q^{97} +1.68585 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9} + O(q^{10}) \) \( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9} - 3 q^{11} + 2 q^{13} - 2 q^{15} + 2 q^{17} + 6 q^{19} - 2 q^{21} - 10 q^{23} + 3 q^{25} - 8 q^{27} + 12 q^{31} + 2 q^{33} + 3 q^{35} + 4 q^{37} + 8 q^{39} + 4 q^{41} - 8 q^{43} + 7 q^{45} + 3 q^{49} + 28 q^{51} + 12 q^{53} - 3 q^{55} - 8 q^{57} + 4 q^{59} - 6 q^{61} + 7 q^{63} + 2 q^{65} - 4 q^{67} - 8 q^{69} + 28 q^{71} - 14 q^{73} - 2 q^{75} - 3 q^{77} - 18 q^{79} + 3 q^{81} + 8 q^{83} + 2 q^{85} + 44 q^{87} + 18 q^{89} + 2 q^{91} + 12 q^{93} + 6 q^{95} - 4 q^{97} - 7 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.14637 −0.661854 −0.330927 0.943656i \(-0.607361\pi\)
−0.330927 + 0.943656i \(0.607361\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −1.68585 −0.561949
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 4.68585 1.29962 0.649810 0.760097i \(-0.274848\pi\)
0.649810 + 0.760097i \(0.274848\pi\)
\(14\) 0 0
\(15\) −1.14637 −0.295990
\(16\) 0 0
\(17\) −0.292731 −0.0709977 −0.0354988 0.999370i \(-0.511302\pi\)
−0.0354988 + 0.999370i \(0.511302\pi\)
\(18\) 0 0
\(19\) −6.51806 −1.49535 −0.747673 0.664068i \(-0.768829\pi\)
−0.747673 + 0.664068i \(0.768829\pi\)
\(20\) 0 0
\(21\) −1.14637 −0.250157
\(22\) 0 0
\(23\) −2.85363 −0.595024 −0.297512 0.954718i \(-0.596157\pi\)
−0.297512 + 0.954718i \(0.596157\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.37169 1.03378
\(28\) 0 0
\(29\) −1.43910 −0.267234 −0.133617 0.991033i \(-0.542659\pi\)
−0.133617 + 0.991033i \(0.542659\pi\)
\(30\) 0 0
\(31\) −0.978577 −0.175758 −0.0878788 0.996131i \(-0.528009\pi\)
−0.0878788 + 0.996131i \(0.528009\pi\)
\(32\) 0 0
\(33\) 1.14637 0.199557
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 0.853635 0.140337 0.0701683 0.997535i \(-0.477646\pi\)
0.0701683 + 0.997535i \(0.477646\pi\)
\(38\) 0 0
\(39\) −5.37169 −0.860159
\(40\) 0 0
\(41\) −6.22533 −0.972233 −0.486116 0.873894i \(-0.661587\pi\)
−0.486116 + 0.873894i \(0.661587\pi\)
\(42\) 0 0
\(43\) 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
\(46\) 0 0
\(47\) 9.95715 1.45240 0.726200 0.687483i \(-0.241285\pi\)
0.726200 + 0.687483i \(0.241285\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.335577 0.0469901
\(52\) 0 0
\(53\) 5.43910 0.747117 0.373559 0.927607i \(-0.378137\pi\)
0.373559 + 0.927607i \(0.378137\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 7.47208 0.989701
\(58\) 0 0
\(59\) 9.37169 1.22009 0.610045 0.792367i \(-0.291151\pi\)
0.610045 + 0.792367i \(0.291151\pi\)
\(60\) 0 0
\(61\) −11.9572 −1.53096 −0.765478 0.643462i \(-0.777498\pi\)
−0.765478 + 0.643462i \(0.777498\pi\)
\(62\) 0 0
\(63\) −1.68585 −0.212397
\(64\) 0 0
\(65\) 4.68585 0.581208
\(66\) 0 0
\(67\) 0.585462 0.0715256 0.0357628 0.999360i \(-0.488614\pi\)
0.0357628 + 0.999360i \(0.488614\pi\)
\(68\) 0 0
\(69\) 3.27131 0.393819
\(70\) 0 0
\(71\) 0.335577 0.0398256 0.0199128 0.999802i \(-0.493661\pi\)
0.0199128 + 0.999802i \(0.493661\pi\)
\(72\) 0 0
\(73\) −3.70727 −0.433903 −0.216952 0.976182i \(-0.569611\pi\)
−0.216952 + 0.976182i \(0.569611\pi\)
\(74\) 0 0
\(75\) −1.14637 −0.132371
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 2.51806 0.283304 0.141652 0.989917i \(-0.454759\pi\)
0.141652 + 0.989917i \(0.454759\pi\)
\(80\) 0 0
\(81\) −1.10038 −0.122265
\(82\) 0 0
\(83\) 1.70727 0.187397 0.0936986 0.995601i \(-0.470131\pi\)
0.0936986 + 0.995601i \(0.470131\pi\)
\(84\) 0 0
\(85\) −0.292731 −0.0317511
\(86\) 0 0
\(87\) 1.64973 0.176870
\(88\) 0 0
\(89\) 13.0790 1.38637 0.693184 0.720761i \(-0.256208\pi\)
0.693184 + 0.720761i \(0.256208\pi\)
\(90\) 0 0
\(91\) 4.68585 0.491210
\(92\) 0 0
\(93\) 1.12181 0.116326
\(94\) 0 0
\(95\) −6.51806 −0.668739
\(96\) 0 0
\(97\) 9.10352 0.924322 0.462161 0.886796i \(-0.347074\pi\)
0.462161 + 0.886796i \(0.347074\pi\)
\(98\) 0 0
\(99\) 1.68585 0.169434
\(100\) 0 0
\(101\) −10.7862 −1.07327 −0.536635 0.843814i \(-0.680305\pi\)
−0.536635 + 0.843814i \(0.680305\pi\)
\(102\) 0 0
\(103\) −7.66442 −0.755198 −0.377599 0.925969i \(-0.623250\pi\)
−0.377599 + 0.925969i \(0.623250\pi\)
\(104\) 0 0
\(105\) −1.14637 −0.111874
\(106\) 0 0
\(107\) −9.56404 −0.924591 −0.462295 0.886726i \(-0.652974\pi\)
−0.462295 + 0.886726i \(0.652974\pi\)
\(108\) 0 0
\(109\) 3.14637 0.301367 0.150684 0.988582i \(-0.451853\pi\)
0.150684 + 0.988582i \(0.451853\pi\)
\(110\) 0 0
\(111\) −0.978577 −0.0928824
\(112\) 0 0
\(113\) −2.58546 −0.243220 −0.121610 0.992578i \(-0.538806\pi\)
−0.121610 + 0.992578i \(0.538806\pi\)
\(114\) 0 0
\(115\) −2.85363 −0.266103
\(116\) 0 0
\(117\) −7.89962 −0.730320
\(118\) 0 0
\(119\) −0.292731 −0.0268346
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 7.13650 0.643477
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 13.3717 1.18655 0.593273 0.805001i \(-0.297836\pi\)
0.593273 + 0.805001i \(0.297836\pi\)
\(128\) 0 0
\(129\) −11.8652 −1.04467
\(130\) 0 0
\(131\) 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 0
\(135\) 5.37169 0.462322
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 14.1825 1.20294 0.601471 0.798895i \(-0.294581\pi\)
0.601471 + 0.798895i \(0.294581\pi\)
\(140\) 0 0
\(141\) −11.4145 −0.961278
\(142\) 0 0
\(143\) −4.68585 −0.391850
\(144\) 0 0
\(145\) −1.43910 −0.119510
\(146\) 0 0
\(147\) −1.14637 −0.0945506
\(148\) 0 0
\(149\) 11.5970 0.950065 0.475032 0.879968i \(-0.342436\pi\)
0.475032 + 0.879968i \(0.342436\pi\)
\(150\) 0 0
\(151\) −1.73183 −0.140934 −0.0704671 0.997514i \(-0.522449\pi\)
−0.0704671 + 0.997514i \(0.522449\pi\)
\(152\) 0 0
\(153\) 0.493499 0.0398971
\(154\) 0 0
\(155\) −0.978577 −0.0786012
\(156\) 0 0
\(157\) −12.2927 −0.981067 −0.490533 0.871422i \(-0.663198\pi\)
−0.490533 + 0.871422i \(0.663198\pi\)
\(158\) 0 0
\(159\) −6.23519 −0.494483
\(160\) 0 0
\(161\) −2.85363 −0.224898
\(162\) 0 0
\(163\) −23.9143 −1.87311 −0.936557 0.350516i \(-0.886006\pi\)
−0.936557 + 0.350516i \(0.886006\pi\)
\(164\) 0 0
\(165\) 1.14637 0.0892444
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 8.95715 0.689012
\(170\) 0 0
\(171\) 10.9884 0.840307
\(172\) 0 0
\(173\) −7.37169 −0.560459 −0.280230 0.959933i \(-0.590411\pi\)
−0.280230 + 0.959933i \(0.590411\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −10.7434 −0.807522
\(178\) 0 0
\(179\) −1.56404 −0.116902 −0.0584509 0.998290i \(-0.518616\pi\)
−0.0584509 + 0.998290i \(0.518616\pi\)
\(180\) 0 0
\(181\) 14.7862 1.09905 0.549526 0.835477i \(-0.314808\pi\)
0.549526 + 0.835477i \(0.314808\pi\)
\(182\) 0 0
\(183\) 13.7073 1.01327
\(184\) 0 0
\(185\) 0.853635 0.0627605
\(186\) 0 0
\(187\) 0.292731 0.0214066
\(188\) 0 0
\(189\) 5.37169 0.390733
\(190\) 0 0
\(191\) 17.9572 1.29933 0.649667 0.760219i \(-0.274908\pi\)
0.649667 + 0.760219i \(0.274908\pi\)
\(192\) 0 0
\(193\) 18.6430 1.34195 0.670976 0.741479i \(-0.265875\pi\)
0.670976 + 0.741479i \(0.265875\pi\)
\(194\) 0 0
\(195\) −5.37169 −0.384675
\(196\) 0 0
\(197\) −15.0361 −1.07128 −0.535639 0.844447i \(-0.679929\pi\)
−0.535639 + 0.844447i \(0.679929\pi\)
\(198\) 0 0
\(199\) −15.3288 −1.08663 −0.543317 0.839528i \(-0.682832\pi\)
−0.543317 + 0.839528i \(0.682832\pi\)
\(200\) 0 0
\(201\) −0.671153 −0.0473395
\(202\) 0 0
\(203\) −1.43910 −0.101005
\(204\) 0 0
\(205\) −6.22533 −0.434796
\(206\) 0 0
\(207\) 4.81079 0.334373
\(208\) 0 0
\(209\) 6.51806 0.450863
\(210\) 0 0
\(211\) −2.04285 −0.140635 −0.0703176 0.997525i \(-0.522401\pi\)
−0.0703176 + 0.997525i \(0.522401\pi\)
\(212\) 0 0
\(213\) −0.384694 −0.0263588
\(214\) 0 0
\(215\) 10.3503 0.705883
\(216\) 0 0
\(217\) −0.978577 −0.0664301
\(218\) 0 0
\(219\) 4.24989 0.287181
\(220\) 0 0
\(221\) −1.37169 −0.0922700
\(222\) 0 0
\(223\) −11.0790 −0.741902 −0.370951 0.928652i \(-0.620968\pi\)
−0.370951 + 0.928652i \(0.620968\pi\)
\(224\) 0 0
\(225\) −1.68585 −0.112390
\(226\) 0 0
\(227\) 18.5426 1.23072 0.615358 0.788248i \(-0.289011\pi\)
0.615358 + 0.788248i \(0.289011\pi\)
\(228\) 0 0
\(229\) −8.01469 −0.529626 −0.264813 0.964300i \(-0.585310\pi\)
−0.264813 + 0.964300i \(0.585310\pi\)
\(230\) 0 0
\(231\) 1.14637 0.0754253
\(232\) 0 0
\(233\) 27.9572 1.83153 0.915767 0.401710i \(-0.131584\pi\)
0.915767 + 0.401710i \(0.131584\pi\)
\(234\) 0 0
\(235\) 9.95715 0.649533
\(236\) 0 0
\(237\) −2.88661 −0.187506
\(238\) 0 0
\(239\) 28.8108 1.86362 0.931808 0.362953i \(-0.118231\pi\)
0.931808 + 0.362953i \(0.118231\pi\)
\(240\) 0 0
\(241\) −8.51806 −0.548696 −0.274348 0.961630i \(-0.588462\pi\)
−0.274348 + 0.961630i \(0.588462\pi\)
\(242\) 0 0
\(243\) −14.8536 −0.952861
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −30.5426 −1.94338
\(248\) 0 0
\(249\) −1.95715 −0.124030
\(250\) 0 0
\(251\) 23.1281 1.45983 0.729916 0.683537i \(-0.239559\pi\)
0.729916 + 0.683537i \(0.239559\pi\)
\(252\) 0 0
\(253\) 2.85363 0.179406
\(254\) 0 0
\(255\) 0.335577 0.0210146
\(256\) 0 0
\(257\) 14.8108 0.923872 0.461936 0.886913i \(-0.347155\pi\)
0.461936 + 0.886913i \(0.347155\pi\)
\(258\) 0 0
\(259\) 0.853635 0.0530423
\(260\) 0 0
\(261\) 2.42610 0.150172
\(262\) 0 0
\(263\) −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
\(266\) 0 0
\(267\) −14.9933 −0.917573
\(268\) 0 0
\(269\) 12.6858 0.773470 0.386735 0.922191i \(-0.373603\pi\)
0.386735 + 0.922191i \(0.373603\pi\)
\(270\) 0 0
\(271\) −1.12181 −0.0681449 −0.0340725 0.999419i \(-0.510848\pi\)
−0.0340725 + 0.999419i \(0.510848\pi\)
\(272\) 0 0
\(273\) −5.37169 −0.325110
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 30.1151 1.80944 0.904720 0.426007i \(-0.140080\pi\)
0.904720 + 0.426007i \(0.140080\pi\)
\(278\) 0 0
\(279\) 1.64973 0.0987668
\(280\) 0 0
\(281\) −14.3356 −0.855189 −0.427594 0.903971i \(-0.640639\pi\)
−0.427594 + 0.903971i \(0.640639\pi\)
\(282\) 0 0
\(283\) −3.21377 −0.191039 −0.0955194 0.995428i \(-0.530451\pi\)
−0.0955194 + 0.995428i \(0.530451\pi\)
\(284\) 0 0
\(285\) 7.47208 0.442608
\(286\) 0 0
\(287\) −6.22533 −0.367469
\(288\) 0 0
\(289\) −16.9143 −0.994959
\(290\) 0 0
\(291\) −10.4360 −0.611767
\(292\) 0 0
\(293\) −16.6858 −0.974798 −0.487399 0.873179i \(-0.662054\pi\)
−0.487399 + 0.873179i \(0.662054\pi\)
\(294\) 0 0
\(295\) 9.37169 0.545641
\(296\) 0 0
\(297\) −5.37169 −0.311697
\(298\) 0 0
\(299\) −13.3717 −0.773305
\(300\) 0 0
\(301\) 10.3503 0.596580
\(302\) 0 0
\(303\) 12.3650 0.710349
\(304\) 0 0
\(305\) −11.9572 −0.684665
\(306\) 0 0
\(307\) 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 0
\(311\) 20.3931 1.15639 0.578194 0.815900i \(-0.303758\pi\)
0.578194 + 0.815900i \(0.303758\pi\)
\(312\) 0 0
\(313\) 12.0674 0.682090 0.341045 0.940047i \(-0.389219\pi\)
0.341045 + 0.940047i \(0.389219\pi\)
\(314\) 0 0
\(315\) −1.68585 −0.0949867
\(316\) 0 0
\(317\) 28.7679 1.61577 0.807884 0.589341i \(-0.200613\pi\)
0.807884 + 0.589341i \(0.200613\pi\)
\(318\) 0 0
\(319\) 1.43910 0.0805739
\(320\) 0 0
\(321\) 10.9639 0.611944
\(322\) 0 0
\(323\) 1.90804 0.106166
\(324\) 0 0
\(325\) 4.68585 0.259924
\(326\) 0 0
\(327\) −3.60688 −0.199461
\(328\) 0 0
\(329\) 9.95715 0.548956
\(330\) 0 0
\(331\) 3.80765 0.209288 0.104644 0.994510i \(-0.466630\pi\)
0.104644 + 0.994510i \(0.466630\pi\)
\(332\) 0 0
\(333\) −1.43910 −0.0788620
\(334\) 0 0
\(335\) 0.585462 0.0319872
\(336\) 0 0
\(337\) 15.3717 0.837349 0.418675 0.908136i \(-0.362495\pi\)
0.418675 + 0.908136i \(0.362495\pi\)
\(338\) 0 0
\(339\) 2.96388 0.160976
\(340\) 0 0
\(341\) 0.978577 0.0529929
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 3.27131 0.176121
\(346\) 0 0
\(347\) −3.02142 −0.162198 −0.0810992 0.996706i \(-0.525843\pi\)
−0.0810992 + 0.996706i \(0.525843\pi\)
\(348\) 0 0
\(349\) 11.0361 0.590750 0.295375 0.955381i \(-0.404555\pi\)
0.295375 + 0.955381i \(0.404555\pi\)
\(350\) 0 0
\(351\) 25.1709 1.34352
\(352\) 0 0
\(353\) 23.9817 1.27642 0.638209 0.769863i \(-0.279676\pi\)
0.638209 + 0.769863i \(0.279676\pi\)
\(354\) 0 0
\(355\) 0.335577 0.0178106
\(356\) 0 0
\(357\) 0.335577 0.0177606
\(358\) 0 0
\(359\) 10.5181 0.555122 0.277561 0.960708i \(-0.410474\pi\)
0.277561 + 0.960708i \(0.410474\pi\)
\(360\) 0 0
\(361\) 23.4851 1.23606
\(362\) 0 0
\(363\) −1.14637 −0.0601686
\(364\) 0 0
\(365\) −3.70727 −0.194047
\(366\) 0 0
\(367\) 35.5787 1.85719 0.928597 0.371089i \(-0.121015\pi\)
0.928597 + 0.371089i \(0.121015\pi\)
\(368\) 0 0
\(369\) 10.4949 0.546345
\(370\) 0 0
\(371\) 5.43910 0.282384
\(372\) 0 0
\(373\) 7.50650 0.388672 0.194336 0.980935i \(-0.437745\pi\)
0.194336 + 0.980935i \(0.437745\pi\)
\(374\) 0 0
\(375\) −1.14637 −0.0591981
\(376\) 0 0
\(377\) −6.74338 −0.347302
\(378\) 0 0
\(379\) −33.4868 −1.72010 −0.860050 0.510210i \(-0.829568\pi\)
−0.860050 + 0.510210i \(0.829568\pi\)
\(380\) 0 0
\(381\) −15.3288 −0.785321
\(382\) 0 0
\(383\) −15.6644 −0.800415 −0.400207 0.916425i \(-0.631062\pi\)
−0.400207 + 0.916425i \(0.631062\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −17.4490 −0.886981
\(388\) 0 0
\(389\) −19.6216 −0.994853 −0.497427 0.867506i \(-0.665722\pi\)
−0.497427 + 0.867506i \(0.665722\pi\)
\(390\) 0 0
\(391\) 0.835347 0.0422453
\(392\) 0 0
\(393\) −13.1134 −0.661483
\(394\) 0 0
\(395\) 2.51806 0.126697
\(396\) 0 0
\(397\) 10.2499 0.514427 0.257213 0.966355i \(-0.417196\pi\)
0.257213 + 0.966355i \(0.417196\pi\)
\(398\) 0 0
\(399\) 7.47208 0.374072
\(400\) 0 0
\(401\) −16.8866 −0.843277 −0.421639 0.906764i \(-0.638545\pi\)
−0.421639 + 0.906764i \(0.638545\pi\)
\(402\) 0 0
\(403\) −4.58546 −0.228418
\(404\) 0 0
\(405\) −1.10038 −0.0546785
\(406\) 0 0
\(407\) −0.853635 −0.0423131
\(408\) 0 0
\(409\) 18.2744 0.903613 0.451807 0.892116i \(-0.350780\pi\)
0.451807 + 0.892116i \(0.350780\pi\)
\(410\) 0 0
\(411\) −11.4637 −0.565460
\(412\) 0 0
\(413\) 9.37169 0.461151
\(414\) 0 0
\(415\) 1.70727 0.0838065
\(416\) 0 0
\(417\) −16.2583 −0.796173
\(418\) 0 0
\(419\) 37.2860 1.82154 0.910770 0.412914i \(-0.135489\pi\)
0.910770 + 0.412914i \(0.135489\pi\)
\(420\) 0 0
\(421\) −2.78623 −0.135793 −0.0678963 0.997692i \(-0.521629\pi\)
−0.0678963 + 0.997692i \(0.521629\pi\)
\(422\) 0 0
\(423\) −16.7862 −0.816174
\(424\) 0 0
\(425\) −0.292731 −0.0141995
\(426\) 0 0
\(427\) −11.9572 −0.578647
\(428\) 0 0
\(429\) 5.37169 0.259348
\(430\) 0 0
\(431\) −38.0477 −1.83269 −0.916346 0.400387i \(-0.868876\pi\)
−0.916346 + 0.400387i \(0.868876\pi\)
\(432\) 0 0
\(433\) −10.8108 −0.519533 −0.259767 0.965671i \(-0.583646\pi\)
−0.259767 + 0.965671i \(0.583646\pi\)
\(434\) 0 0
\(435\) 1.64973 0.0790985
\(436\) 0 0
\(437\) 18.6002 0.889766
\(438\) 0 0
\(439\) −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
\(442\) 0 0
\(443\) −25.3717 −1.20545 −0.602723 0.797951i \(-0.705917\pi\)
−0.602723 + 0.797951i \(0.705917\pi\)
\(444\) 0 0
\(445\) 13.0790 0.620002
\(446\) 0 0
\(447\) −13.2944 −0.628805
\(448\) 0 0
\(449\) −0.886615 −0.0418419 −0.0209210 0.999781i \(-0.506660\pi\)
−0.0209210 + 0.999781i \(0.506660\pi\)
\(450\) 0 0
\(451\) 6.22533 0.293139
\(452\) 0 0
\(453\) 1.98531 0.0932779
\(454\) 0 0
\(455\) 4.68585 0.219676
\(456\) 0 0
\(457\) 29.9143 1.39933 0.699666 0.714470i \(-0.253332\pi\)
0.699666 + 0.714470i \(0.253332\pi\)
\(458\) 0 0
\(459\) −1.57246 −0.0733962
\(460\) 0 0
\(461\) −2.33558 −0.108779 −0.0543893 0.998520i \(-0.517321\pi\)
−0.0543893 + 0.998520i \(0.517321\pi\)
\(462\) 0 0
\(463\) −0.110250 −0.00512374 −0.00256187 0.999997i \(-0.500815\pi\)
−0.00256187 + 0.999997i \(0.500815\pi\)
\(464\) 0 0
\(465\) 1.12181 0.0520226
\(466\) 0 0
\(467\) −36.6760 −1.69716 −0.848581 0.529066i \(-0.822543\pi\)
−0.848581 + 0.529066i \(0.822543\pi\)
\(468\) 0 0
\(469\) 0.585462 0.0270341
\(470\) 0 0
\(471\) 14.0920 0.649323
\(472\) 0 0
\(473\) −10.3503 −0.475906
\(474\) 0 0
\(475\) −6.51806 −0.299069
\(476\) 0 0
\(477\) −9.16948 −0.419842
\(478\) 0 0
\(479\) 42.7434 1.95300 0.976498 0.215529i \(-0.0691474\pi\)
0.976498 + 0.215529i \(0.0691474\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) 3.27131 0.148850
\(484\) 0 0
\(485\) 9.10352 0.413370
\(486\) 0 0
\(487\) 36.1396 1.63764 0.818822 0.574048i \(-0.194628\pi\)
0.818822 + 0.574048i \(0.194628\pi\)
\(488\) 0 0
\(489\) 27.4145 1.23973
\(490\) 0 0
\(491\) −2.54262 −0.114747 −0.0573733 0.998353i \(-0.518273\pi\)
−0.0573733 + 0.998353i \(0.518273\pi\)
\(492\) 0 0
\(493\) 0.421268 0.0189730
\(494\) 0 0
\(495\) 1.68585 0.0757732
\(496\) 0 0
\(497\) 0.335577 0.0150527
\(498\) 0 0
\(499\) −3.13650 −0.140409 −0.0702045 0.997533i \(-0.522365\pi\)
−0.0702045 + 0.997533i \(0.522365\pi\)
\(500\) 0 0
\(501\) −9.17092 −0.409727
\(502\) 0 0
\(503\) 28.5855 1.27456 0.637281 0.770631i \(-0.280059\pi\)
0.637281 + 0.770631i \(0.280059\pi\)
\(504\) 0 0
\(505\) −10.7862 −0.479981
\(506\) 0 0
\(507\) −10.2682 −0.456026
\(508\) 0 0
\(509\) 2.20077 0.0975473 0.0487737 0.998810i \(-0.484469\pi\)
0.0487737 + 0.998810i \(0.484469\pi\)
\(510\) 0 0
\(511\) −3.70727 −0.164000
\(512\) 0 0
\(513\) −35.0130 −1.54586
\(514\) 0 0
\(515\) −7.66442 −0.337735
\(516\) 0 0
\(517\) −9.95715 −0.437915
\(518\) 0 0
\(519\) 8.45065 0.370943
\(520\) 0 0
\(521\) 37.1940 1.62950 0.814750 0.579812i \(-0.196874\pi\)
0.814750 + 0.579812i \(0.196874\pi\)
\(522\) 0 0
\(523\) −30.4078 −1.32964 −0.664820 0.747003i \(-0.731492\pi\)
−0.664820 + 0.747003i \(0.731492\pi\)
\(524\) 0 0
\(525\) −1.14637 −0.0500315
\(526\) 0 0
\(527\) 0.286460 0.0124784
\(528\) 0 0
\(529\) −14.8568 −0.645947
\(530\) 0 0
\(531\) −15.7992 −0.685628
\(532\) 0 0
\(533\) −29.1709 −1.26353
\(534\) 0 0
\(535\) −9.56404 −0.413489
\(536\) 0 0
\(537\) 1.79296 0.0773720
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 39.0607 1.67935 0.839675 0.543090i \(-0.182746\pi\)
0.839675 + 0.543090i \(0.182746\pi\)
\(542\) 0 0
\(543\) −16.9504 −0.727412
\(544\) 0 0
\(545\) 3.14637 0.134775
\(546\) 0 0
\(547\) 0.585462 0.0250325 0.0125163 0.999922i \(-0.496016\pi\)
0.0125163 + 0.999922i \(0.496016\pi\)
\(548\) 0 0
\(549\) 20.1579 0.860319
\(550\) 0 0
\(551\) 9.38011 0.399606
\(552\) 0 0
\(553\) 2.51806 0.107079
\(554\) 0 0
\(555\) −0.978577 −0.0415383
\(556\) 0 0
\(557\) 17.4637 0.739959 0.369979 0.929040i \(-0.379365\pi\)
0.369979 + 0.929040i \(0.379365\pi\)
\(558\) 0 0
\(559\) 48.4998 2.05132
\(560\) 0 0
\(561\) −0.335577 −0.0141681
\(562\) 0 0
\(563\) 18.1579 0.765265 0.382633 0.923901i \(-0.375018\pi\)
0.382633 + 0.923901i \(0.375018\pi\)
\(564\) 0 0
\(565\) −2.58546 −0.108771
\(566\) 0 0
\(567\) −1.10038 −0.0462118
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) 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
\(574\) 0 0
\(575\) −2.85363 −0.119005
\(576\) 0 0
\(577\) 19.3963 0.807476 0.403738 0.914875i \(-0.367711\pi\)
0.403738 + 0.914875i \(0.367711\pi\)
\(578\) 0 0
\(579\) −21.3717 −0.888177
\(580\) 0 0
\(581\) 1.70727 0.0708295
\(582\) 0 0
\(583\) −5.43910 −0.225264
\(584\) 0 0
\(585\) −7.89962 −0.326609
\(586\) 0 0
\(587\) −17.2614 −0.712456 −0.356228 0.934399i \(-0.615937\pi\)
−0.356228 + 0.934399i \(0.615937\pi\)
\(588\) 0 0
\(589\) 6.37842 0.262818
\(590\) 0 0
\(591\) 17.2369 0.709031
\(592\) 0 0
\(593\) −11.0361 −0.453199 −0.226599 0.973988i \(-0.572761\pi\)
−0.226599 + 0.973988i \(0.572761\pi\)
\(594\) 0 0
\(595\) −0.292731 −0.0120008
\(596\) 0 0
\(597\) 17.5725 0.719193
\(598\) 0 0
\(599\) 41.7367 1.70531 0.852657 0.522472i \(-0.174990\pi\)
0.852657 + 0.522472i \(0.174990\pi\)
\(600\) 0 0
\(601\) 39.3106 1.60351 0.801756 0.597652i \(-0.203900\pi\)
0.801756 + 0.597652i \(0.203900\pi\)
\(602\) 0 0
\(603\) −0.986999 −0.0401937
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 6.35027 0.257749 0.128875 0.991661i \(-0.458863\pi\)
0.128875 + 0.991661i \(0.458863\pi\)
\(608\) 0 0
\(609\) 1.64973 0.0668505
\(610\) 0 0
\(611\) 46.6577 1.88757
\(612\) 0 0
\(613\) −42.4507 −1.71457 −0.857283 0.514846i \(-0.827849\pi\)
−0.857283 + 0.514846i \(0.827849\pi\)
\(614\) 0 0
\(615\) 7.13650 0.287771
\(616\) 0 0
\(617\) −0.677425 −0.0272721 −0.0136360 0.999907i \(-0.504341\pi\)
−0.0136360 + 0.999907i \(0.504341\pi\)
\(618\) 0 0
\(619\) 18.5426 0.745291 0.372645 0.927974i \(-0.378451\pi\)
0.372645 + 0.927974i \(0.378451\pi\)
\(620\) 0 0
\(621\) −15.3288 −0.615125
\(622\) 0 0
\(623\) 13.0790 0.523998
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −7.47208 −0.298406
\(628\) 0 0
\(629\) −0.249885 −0.00996358
\(630\) 0 0
\(631\) −21.3717 −0.850794 −0.425397 0.905007i \(-0.639865\pi\)
−0.425397 + 0.905007i \(0.639865\pi\)
\(632\) 0 0
\(633\) 2.34185 0.0930801
\(634\) 0 0
\(635\) 13.3717 0.530639
\(636\) 0 0
\(637\) 4.68585 0.185660
\(638\) 0 0
\(639\) −0.565731 −0.0223800
\(640\) 0 0
\(641\) 23.8715 0.942866 0.471433 0.881902i \(-0.343737\pi\)
0.471433 + 0.881902i \(0.343737\pi\)
\(642\) 0 0
\(643\) −0.311018 −0.0122654 −0.00613268 0.999981i \(-0.501952\pi\)
−0.00613268 + 0.999981i \(0.501952\pi\)
\(644\) 0 0
\(645\) −11.8652 −0.467191
\(646\) 0 0
\(647\) 9.62158 0.378263 0.189132 0.981952i \(-0.439433\pi\)
0.189132 + 0.981952i \(0.439433\pi\)
\(648\) 0 0
\(649\) −9.37169 −0.367871
\(650\) 0 0
\(651\) 1.12181 0.0439671
\(652\) 0 0
\(653\) −27.0607 −1.05897 −0.529483 0.848321i \(-0.677614\pi\)
−0.529483 + 0.848321i \(0.677614\pi\)
\(654\) 0 0
\(655\) 11.4391 0.446962
\(656\) 0 0
\(657\) 6.24989 0.243831
\(658\) 0 0
\(659\) 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 \(0.260967\pi\)
\(662\) 0 0
\(663\) 1.57246 0.0610693
\(664\) 0 0
\(665\) −6.51806 −0.252759
\(666\) 0 0
\(667\) 4.10666 0.159010
\(668\) 0 0
\(669\) 12.7005 0.491031
\(670\) 0 0
\(671\) 11.9572 0.461601
\(672\) 0 0
\(673\) 32.6577 1.25886 0.629431 0.777057i \(-0.283288\pi\)
0.629431 + 0.777057i \(0.283288\pi\)
\(674\) 0 0
\(675\) 5.37169 0.206757
\(676\) 0 0
\(677\) 9.12808 0.350821 0.175410 0.984495i \(-0.443875\pi\)
0.175410 + 0.984495i \(0.443875\pi\)
\(678\) 0 0
\(679\) 9.10352 0.349361
\(680\) 0 0
\(681\) −21.2566 −0.814555
\(682\) 0 0
\(683\) −10.4935 −0.401523 −0.200761 0.979640i \(-0.564342\pi\)
−0.200761 + 0.979640i \(0.564342\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) 0 0
\(687\) 9.18777 0.350535
\(688\) 0 0
\(689\) 25.4868 0.970969
\(690\) 0 0
\(691\) −1.70727 −0.0649476 −0.0324738 0.999473i \(-0.510339\pi\)
−0.0324738 + 0.999473i \(0.510339\pi\)
\(692\) 0 0
\(693\) 1.68585 0.0640400
\(694\) 0 0
\(695\) 14.1825 0.537972
\(696\) 0 0
\(697\) 1.82235 0.0690263
\(698\) 0 0
\(699\) −32.0491 −1.21221
\(700\) 0 0
\(701\) −37.3534 −1.41082 −0.705409 0.708800i \(-0.749237\pi\)
−0.705409 + 0.708800i \(0.749237\pi\)
\(702\) 0 0
\(703\) −5.56404 −0.209852
\(704\) 0 0
\(705\) −11.4145 −0.429896
\(706\) 0 0
\(707\) −10.7862 −0.405658
\(708\) 0 0
\(709\) −8.20704 −0.308222 −0.154111 0.988054i \(-0.549251\pi\)
−0.154111 + 0.988054i \(0.549251\pi\)
\(710\) 0 0
\(711\) −4.24506 −0.159202
\(712\) 0 0
\(713\) 2.79250 0.104580
\(714\) 0 0
\(715\) −4.68585 −0.175241
\(716\) 0 0
\(717\) −33.0277 −1.23344
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) −7.66442 −0.285438
\(722\) 0 0
\(723\) 9.76481 0.363157
\(724\) 0 0
\(725\) −1.43910 −0.0534467
\(726\) 0 0
\(727\) 46.9442 1.74106 0.870531 0.492113i \(-0.163775\pi\)
0.870531 + 0.492113i \(0.163775\pi\)
\(728\) 0 0
\(729\) 20.3288 0.752920
\(730\) 0 0
\(731\) −3.02984 −0.112063
\(732\) 0 0
\(733\) −42.0000 −1.55131 −0.775653 0.631160i \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) 0 0
\(735\) −1.14637 −0.0422843
\(736\) 0 0
\(737\) −0.585462 −0.0215658
\(738\) 0 0
\(739\) 0.871922 0.0320742 0.0160371 0.999871i \(-0.494895\pi\)
0.0160371 + 0.999871i \(0.494895\pi\)
\(740\) 0 0
\(741\) 35.0130 1.28623
\(742\) 0 0
\(743\) 24.6712 0.905097 0.452548 0.891740i \(-0.350515\pi\)
0.452548 + 0.891740i \(0.350515\pi\)
\(744\) 0 0
\(745\) 11.5970 0.424882
\(746\) 0 0
\(747\) −2.87819 −0.105308
\(748\) 0 0
\(749\) −9.56404 −0.349462
\(750\) 0 0
\(751\) 3.75011 0.136844 0.0684218 0.997656i \(-0.478204\pi\)
0.0684218 + 0.997656i \(0.478204\pi\)
\(752\) 0 0
\(753\) −26.5132 −0.966196
\(754\) 0 0
\(755\) −1.73183 −0.0630277
\(756\) 0 0
\(757\) 15.7318 0.571783 0.285891 0.958262i \(-0.407710\pi\)
0.285891 + 0.958262i \(0.407710\pi\)
\(758\) 0 0
\(759\) −3.27131 −0.118741
\(760\) 0 0
\(761\) 15.2614 0.553227 0.276613 0.960981i \(-0.410788\pi\)
0.276613 + 0.960981i \(0.410788\pi\)
\(762\) 0 0
\(763\) 3.14637 0.113906
\(764\) 0 0
\(765\) 0.493499 0.0178425
\(766\) 0 0
\(767\) 43.9143 1.58565
\(768\) 0 0
\(769\) −9.63986 −0.347622 −0.173811 0.984779i \(-0.555608\pi\)
−0.173811 + 0.984779i \(0.555608\pi\)
\(770\) 0 0
\(771\) −16.9786 −0.611469
\(772\) 0 0
\(773\) 12.8291 0.461430 0.230715 0.973021i \(-0.425894\pi\)
0.230715 + 0.973021i \(0.425894\pi\)
\(774\) 0 0
\(775\) −0.978577 −0.0351515
\(776\) 0 0
\(777\) −0.978577 −0.0351063
\(778\) 0 0
\(779\) 40.5770 1.45382
\(780\) 0 0
\(781\) −0.335577 −0.0120079
\(782\) 0 0
\(783\) −7.73038 −0.276261
\(784\) 0 0
\(785\) −12.2927 −0.438746
\(786\) 0 0
\(787\) −10.8291 −0.386015 −0.193007 0.981197i \(-0.561824\pi\)
−0.193007 + 0.981197i \(0.561824\pi\)
\(788\) 0 0
\(789\) 21.4868 0.764949
\(790\) 0 0
\(791\) −2.58546 −0.0919284
\(792\) 0 0
\(793\) −56.0294 −1.98966
\(794\) 0 0
\(795\) −6.23519 −0.221139
\(796\) 0 0
\(797\) −21.0790 −0.746655 −0.373328 0.927700i \(-0.621783\pi\)
−0.373328 + 0.927700i \(0.621783\pi\)
\(798\) 0 0
\(799\) −2.91477 −0.103117
\(800\) 0 0
\(801\) −22.0491 −0.779067
\(802\) 0 0
\(803\) 3.70727 0.130827
\(804\) 0 0
\(805\) −2.85363 −0.100577
\(806\) 0 0
\(807\) −14.5426 −0.511924
\(808\) 0 0
\(809\) 3.62158 0.127328 0.0636639 0.997971i \(-0.479721\pi\)
0.0636639 + 0.997971i \(0.479721\pi\)
\(810\) 0 0
\(811\) −34.7188 −1.21914 −0.609571 0.792731i \(-0.708658\pi\)
−0.609571 + 0.792731i \(0.708658\pi\)
\(812\) 0 0
\(813\) 1.28600 0.0451020
\(814\) 0 0
\(815\) −23.9143 −0.837682
\(816\) 0 0
\(817\) −67.4637 −2.36025
\(818\) 0 0
\(819\) −7.89962 −0.276035
\(820\) 0 0
\(821\) −37.7549 −1.31766 −0.658828 0.752293i \(-0.728948\pi\)
−0.658828 + 0.752293i \(0.728948\pi\)
\(822\) 0 0
\(823\) −11.1892 −0.390031 −0.195016 0.980800i \(-0.562476\pi\)
−0.195016 + 0.980800i \(0.562476\pi\)
\(824\) 0 0
\(825\) 1.14637 0.0399113
\(826\) 0 0
\(827\) 7.91431 0.275207 0.137604 0.990487i \(-0.456060\pi\)
0.137604 + 0.990487i \(0.456060\pi\)
\(828\) 0 0
\(829\) −44.7152 −1.55302 −0.776512 0.630102i \(-0.783013\pi\)
−0.776512 + 0.630102i \(0.783013\pi\)
\(830\) 0 0
\(831\) −34.5229 −1.19759
\(832\) 0 0
\(833\) −0.292731 −0.0101425
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) −5.25662 −0.181695
\(838\) 0 0
\(839\) −21.6791 −0.748446 −0.374223 0.927339i \(-0.622091\pi\)
−0.374223 + 0.927339i \(0.622091\pi\)
\(840\) 0 0
\(841\) −26.9290 −0.928586
\(842\) 0 0
\(843\) 16.4338 0.566010
\(844\) 0 0
\(845\) 8.95715 0.308135
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) 3.68415 0.126440
\(850\) 0 0
\(851\) −2.43596 −0.0835037
\(852\) 0 0
\(853\) 9.41454 0.322348 0.161174 0.986926i \(-0.448472\pi\)
0.161174 + 0.986926i \(0.448472\pi\)
\(854\) 0 0
\(855\) 10.9884 0.375797
\(856\) 0 0
\(857\) −25.8652 −0.883538 −0.441769 0.897129i \(-0.645649\pi\)
−0.441769 + 0.897129i \(0.645649\pi\)
\(858\) 0 0
\(859\) 29.2860 0.999225 0.499613 0.866249i \(-0.333476\pi\)
0.499613 + 0.866249i \(0.333476\pi\)
\(860\) 0 0
\(861\) 7.13650 0.243211
\(862\) 0 0
\(863\) 20.3110 0.691395 0.345698 0.938346i \(-0.387642\pi\)
0.345698 + 0.938346i \(0.387642\pi\)
\(864\) 0 0
\(865\) −7.37169 −0.250645
\(866\) 0 0
\(867\) 19.3900 0.658518
\(868\) 0 0
\(869\) −2.51806 −0.0854193
\(870\) 0 0
\(871\) 2.74338 0.0929560
\(872\) 0 0
\(873\) −15.3471 −0.519422
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −38.3650 −1.29549 −0.647746 0.761856i \(-0.724288\pi\)
−0.647746 + 0.761856i \(0.724288\pi\)
\(878\) 0 0
\(879\) 19.1281 0.645174
\(880\) 0 0
\(881\) 40.5426 1.36592 0.682958 0.730458i \(-0.260693\pi\)
0.682958 + 0.730458i \(0.260693\pi\)
\(882\) 0 0
\(883\) −28.0491 −0.943928 −0.471964 0.881618i \(-0.656455\pi\)
−0.471964 + 0.881618i \(0.656455\pi\)
\(884\) 0 0
\(885\) −10.7434 −0.361135
\(886\) 0 0
\(887\) 6.65769 0.223543 0.111772 0.993734i \(-0.464347\pi\)
0.111772 + 0.993734i \(0.464347\pi\)
\(888\) 0 0
\(889\) 13.3717 0.448472
\(890\) 0 0
\(891\) 1.10038 0.0368643
\(892\) 0 0
\(893\) −64.9013 −2.17184
\(894\) 0 0
\(895\) −1.56404 −0.0522801
\(896\) 0 0
\(897\) 15.3288 0.511815
\(898\) 0 0
\(899\) 1.40827 0.0469683
\(900\) 0 0
\(901\) −1.59219 −0.0530436
\(902\) 0 0
\(903\) −11.8652 −0.394849
\(904\) 0 0
\(905\) 14.7862 0.491511
\(906\) 0 0
\(907\) −39.5787 −1.31419 −0.657095 0.753808i \(-0.728215\pi\)
−0.657095 + 0.753808i \(0.728215\pi\)
\(908\) 0 0
\(909\) 18.1839 0.603123
\(910\) 0 0
\(911\) 24.3356 0.806274 0.403137 0.915140i \(-0.367920\pi\)
0.403137 + 0.915140i \(0.367920\pi\)
\(912\) 0 0
\(913\) −1.70727 −0.0565024
\(914\) 0 0
\(915\) 13.7073 0.453148
\(916\) 0 0
\(917\) 11.4391 0.377752
\(918\) 0 0
\(919\) −50.6331 −1.67023 −0.835117 0.550073i \(-0.814600\pi\)
−0.835117 + 0.550073i \(0.814600\pi\)
\(920\) 0 0
\(921\) −7.21377 −0.237702
\(922\) 0 0
\(923\) 1.57246 0.0517582
\(924\) 0 0
\(925\) 0.853635 0.0280673
\(926\) 0 0
\(927\) 12.9210 0.424383
\(928\) 0 0
\(929\) 47.1512 1.54698 0.773490 0.633808i \(-0.218509\pi\)
0.773490 + 0.633808i \(0.218509\pi\)
\(930\) 0 0
\(931\) −6.51806 −0.213621
\(932\) 0 0
\(933\) −23.3780 −0.765360
\(934\) 0 0
\(935\) 0.292731 0.00957333
\(936\) 0 0
\(937\) −42.0294 −1.37304 −0.686520 0.727111i \(-0.740863\pi\)
−0.686520 + 0.727111i \(0.740863\pi\)
\(938\) 0 0
\(939\) −13.8337 −0.451444
\(940\) 0 0
\(941\) −36.7434 −1.19780 −0.598900 0.800824i \(-0.704395\pi\)
−0.598900 + 0.800824i \(0.704395\pi\)
\(942\) 0 0
\(943\) 17.7648 0.578502
\(944\) 0 0
\(945\) 5.37169 0.174741
\(946\) 0 0
\(947\) −18.8782 −0.613459 −0.306729 0.951797i \(-0.599235\pi\)
−0.306729 + 0.951797i \(0.599235\pi\)
\(948\) 0 0
\(949\) −17.3717 −0.563909
\(950\) 0 0
\(951\) −32.9786 −1.06940
\(952\) 0 0
\(953\) −43.2285 −1.40031 −0.700154 0.713992i \(-0.746885\pi\)
−0.700154 + 0.713992i \(0.746885\pi\)
\(954\) 0 0
\(955\) 17.9572 0.581080
\(956\) 0 0
\(957\) −1.64973 −0.0533282
\(958\) 0 0
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) −30.0424 −0.969109
\(962\) 0 0
\(963\) 16.1235 0.519572
\(964\) 0 0
\(965\) 18.6430 0.600139
\(966\) 0 0
\(967\) −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
\(970\) 0 0
\(971\) 25.9865 0.833948 0.416974 0.908918i \(-0.363091\pi\)
0.416974 + 0.908918i \(0.363091\pi\)
\(972\) 0 0
\(973\) 14.1825 0.454669
\(974\) 0 0
\(975\) −5.37169 −0.172032
\(976\) 0 0
\(977\) 2.67115 0.0854578 0.0427289 0.999087i \(-0.486395\pi\)
0.0427289 + 0.999087i \(0.486395\pi\)
\(978\) 0 0
\(979\) −13.0790 −0.418005
\(980\) 0 0
\(981\) −5.30429 −0.169353
\(982\) 0 0
\(983\) 8.33558 0.265864 0.132932 0.991125i \(-0.457561\pi\)
0.132932 + 0.991125i \(0.457561\pi\)
\(984\) 0 0
\(985\) −15.0361 −0.479090
\(986\) 0 0
\(987\) −11.4145 −0.363329
\(988\) 0 0
\(989\) −29.5359 −0.939187
\(990\) 0 0
\(991\) −46.2730 −1.46991 −0.734955 0.678116i \(-0.762797\pi\)
−0.734955 + 0.678116i \(0.762797\pi\)
\(992\) 0 0
\(993\) −4.36496 −0.138518
\(994\) 0 0
\(995\) −15.3288 −0.485957
\(996\) 0 0
\(997\) 35.8715 1.13606 0.568030 0.823008i \(-0.307706\pi\)
0.568030 + 0.823008i \(0.307706\pi\)
\(998\) 0 0
\(999\) 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 6160.2.a.bf.1.2 3
4.3 odd 2 770.2.a.m.1.2 3
12.11 even 2 6930.2.a.ce.1.3 3
20.3 even 4 3850.2.c.ba.1849.2 6
20.7 even 4 3850.2.c.ba.1849.5 6
20.19 odd 2 3850.2.a.bt.1.2 3
28.27 even 2 5390.2.a.ca.1.2 3
44.43 even 2 8470.2.a.ci.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.m.1.2 3 4.3 odd 2
3850.2.a.bt.1.2 3 20.19 odd 2
3850.2.c.ba.1849.2 6 20.3 even 4
3850.2.c.ba.1849.5 6 20.7 even 4
5390.2.a.ca.1.2 3 28.27 even 2
6160.2.a.bf.1.2 3 1.1 even 1 trivial
6930.2.a.ce.1.3 3 12.11 even 2
8470.2.a.ci.1.2 3 44.43 even 2