Properties

Label 6160.2.a.r.1.1
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 6160.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -4.74456 q^{13} -2.00000 q^{15} +4.74456 q^{17} -4.74456 q^{19} +2.00000 q^{21} -4.74456 q^{23} +1.00000 q^{25} +4.00000 q^{27} -2.74456 q^{29} +6.74456 q^{31} -2.00000 q^{33} -1.00000 q^{35} -10.7446 q^{37} +9.48913 q^{39} -4.00000 q^{41} -4.00000 q^{43} +1.00000 q^{45} +6.74456 q^{47} +1.00000 q^{49} -9.48913 q^{51} -1.25544 q^{53} +1.00000 q^{55} +9.48913 q^{57} -2.74456 q^{59} +12.7446 q^{61} -1.00000 q^{63} -4.74456 q^{65} +4.00000 q^{67} +9.48913 q^{69} +4.00000 q^{71} -0.744563 q^{73} -2.00000 q^{75} -1.00000 q^{77} +4.74456 q^{79} -11.0000 q^{81} -8.00000 q^{83} +4.74456 q^{85} +5.48913 q^{87} -7.48913 q^{89} +4.74456 q^{91} -13.4891 q^{93} -4.74456 q^{95} -5.25544 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} - 2 q^{17} + 2 q^{19} + 4 q^{21} + 2 q^{23} + 2 q^{25} + 8 q^{27} + 6 q^{29} + 2 q^{31} - 4 q^{33} - 2 q^{35} - 10 q^{37} - 4 q^{39} - 8 q^{41} - 8 q^{43} + 2 q^{45} + 2 q^{47} + 2 q^{49} + 4 q^{51} - 14 q^{53} + 2 q^{55} - 4 q^{57} + 6 q^{59} + 14 q^{61} - 2 q^{63} + 2 q^{65} + 8 q^{67} - 4 q^{69} + 8 q^{71} + 10 q^{73} - 4 q^{75} - 2 q^{77} - 2 q^{79} - 22 q^{81} - 16 q^{83} - 2 q^{85} - 12 q^{87} + 8 q^{89} - 2 q^{91} - 4 q^{93} + 2 q^{95} - 22 q^{97} + 2 q^{99} + O(q^{100}) \)

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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −4.74456 −1.31590 −0.657952 0.753059i \(-0.728577\pi\)
−0.657952 + 0.753059i \(0.728577\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) 4.74456 1.15073 0.575363 0.817898i \(-0.304861\pi\)
0.575363 + 0.817898i \(0.304861\pi\)
\(18\) 0 0
\(19\) −4.74456 −1.08848 −0.544239 0.838930i \(-0.683181\pi\)
−0.544239 + 0.838930i \(0.683181\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −4.74456 −0.989310 −0.494655 0.869090i \(-0.664706\pi\)
−0.494655 + 0.869090i \(0.664706\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −2.74456 −0.509652 −0.254826 0.966987i \(-0.582018\pi\)
−0.254826 + 0.966987i \(0.582018\pi\)
\(30\) 0 0
\(31\) 6.74456 1.21136 0.605680 0.795709i \(-0.292901\pi\)
0.605680 + 0.795709i \(0.292901\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −10.7446 −1.76640 −0.883198 0.469001i \(-0.844614\pi\)
−0.883198 + 0.469001i \(0.844614\pi\)
\(38\) 0 0
\(39\) 9.48913 1.51948
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 6.74456 0.983796 0.491898 0.870653i \(-0.336303\pi\)
0.491898 + 0.870653i \(0.336303\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −9.48913 −1.32874
\(52\) 0 0
\(53\) −1.25544 −0.172448 −0.0862238 0.996276i \(-0.527480\pi\)
−0.0862238 + 0.996276i \(0.527480\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 9.48913 1.25687
\(58\) 0 0
\(59\) −2.74456 −0.357312 −0.178656 0.983912i \(-0.557175\pi\)
−0.178656 + 0.983912i \(0.557175\pi\)
\(60\) 0 0
\(61\) 12.7446 1.63177 0.815887 0.578211i \(-0.196249\pi\)
0.815887 + 0.578211i \(0.196249\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −4.74456 −0.588491
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 9.48913 1.14236
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −0.744563 −0.0871445 −0.0435722 0.999050i \(-0.513874\pi\)
−0.0435722 + 0.999050i \(0.513874\pi\)
\(74\) 0 0
\(75\) −2.00000 −0.230940
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 4.74456 0.533805 0.266903 0.963724i \(-0.414000\pi\)
0.266903 + 0.963724i \(0.414000\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 4.74456 0.514620
\(86\) 0 0
\(87\) 5.48913 0.588496
\(88\) 0 0
\(89\) −7.48913 −0.793846 −0.396923 0.917852i \(-0.629922\pi\)
−0.396923 + 0.917852i \(0.629922\pi\)
\(90\) 0 0
\(91\) 4.74456 0.497365
\(92\) 0 0
\(93\) −13.4891 −1.39876
\(94\) 0 0
\(95\) −4.74456 −0.486782
\(96\) 0 0
\(97\) −5.25544 −0.533609 −0.266804 0.963751i \(-0.585968\pi\)
−0.266804 + 0.963751i \(0.585968\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −8.74456 −0.870117 −0.435058 0.900402i \(-0.643272\pi\)
−0.435058 + 0.900402i \(0.643272\pi\)
\(102\) 0 0
\(103\) −10.7446 −1.05869 −0.529347 0.848406i \(-0.677563\pi\)
−0.529347 + 0.848406i \(0.677563\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 16.2337 1.55491 0.777453 0.628941i \(-0.216511\pi\)
0.777453 + 0.628941i \(0.216511\pi\)
\(110\) 0 0
\(111\) 21.4891 2.03966
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) −4.74456 −0.442433
\(116\) 0 0
\(117\) −4.74456 −0.438635
\(118\) 0 0
\(119\) −4.74456 −0.434933
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −8.74456 −0.764016 −0.382008 0.924159i \(-0.624767\pi\)
−0.382008 + 0.924159i \(0.624767\pi\)
\(132\) 0 0
\(133\) 4.74456 0.411406
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) −19.4891 −1.66507 −0.832534 0.553974i \(-0.813111\pi\)
−0.832534 + 0.553974i \(0.813111\pi\)
\(138\) 0 0
\(139\) −3.25544 −0.276123 −0.138061 0.990424i \(-0.544087\pi\)
−0.138061 + 0.990424i \(0.544087\pi\)
\(140\) 0 0
\(141\) −13.4891 −1.13599
\(142\) 0 0
\(143\) −4.74456 −0.396760
\(144\) 0 0
\(145\) −2.74456 −0.227924
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) 10.7446 0.880229 0.440114 0.897942i \(-0.354938\pi\)
0.440114 + 0.897942i \(0.354938\pi\)
\(150\) 0 0
\(151\) 20.7446 1.68817 0.844084 0.536211i \(-0.180145\pi\)
0.844084 + 0.536211i \(0.180145\pi\)
\(152\) 0 0
\(153\) 4.74456 0.383575
\(154\) 0 0
\(155\) 6.74456 0.541736
\(156\) 0 0
\(157\) 23.4891 1.87464 0.937318 0.348475i \(-0.113300\pi\)
0.937318 + 0.348475i \(0.113300\pi\)
\(158\) 0 0
\(159\) 2.51087 0.199125
\(160\) 0 0
\(161\) 4.74456 0.373924
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) −2.00000 −0.155700
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 9.51087 0.731606
\(170\) 0 0
\(171\) −4.74456 −0.362826
\(172\) 0 0
\(173\) −14.2337 −1.08217 −0.541084 0.840969i \(-0.681986\pi\)
−0.541084 + 0.840969i \(0.681986\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 5.48913 0.412588
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 3.48913 0.259345 0.129672 0.991557i \(-0.458607\pi\)
0.129672 + 0.991557i \(0.458607\pi\)
\(182\) 0 0
\(183\) −25.4891 −1.88421
\(184\) 0 0
\(185\) −10.7446 −0.789956
\(186\) 0 0
\(187\) 4.74456 0.346957
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 9.48913 0.679530
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) −14.7446 −1.04521 −0.522607 0.852574i \(-0.675041\pi\)
−0.522607 + 0.852574i \(0.675041\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) 2.74456 0.192631
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) −4.74456 −0.329770
\(208\) 0 0
\(209\) −4.74456 −0.328188
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −6.74456 −0.457851
\(218\) 0 0
\(219\) 1.48913 0.100626
\(220\) 0 0
\(221\) −22.5109 −1.51425
\(222\) 0 0
\(223\) −26.7446 −1.79095 −0.895474 0.445113i \(-0.853163\pi\)
−0.895474 + 0.445113i \(0.853163\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) 20.9783 1.37433 0.687165 0.726501i \(-0.258855\pi\)
0.687165 + 0.726501i \(0.258855\pi\)
\(234\) 0 0
\(235\) 6.74456 0.439967
\(236\) 0 0
\(237\) −9.48913 −0.616385
\(238\) 0 0
\(239\) 3.25544 0.210577 0.105288 0.994442i \(-0.466423\pi\)
0.105288 + 0.994442i \(0.466423\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 22.5109 1.43233
\(248\) 0 0
\(249\) 16.0000 1.01396
\(250\) 0 0
\(251\) −8.23369 −0.519706 −0.259853 0.965648i \(-0.583674\pi\)
−0.259853 + 0.965648i \(0.583674\pi\)
\(252\) 0 0
\(253\) −4.74456 −0.298288
\(254\) 0 0
\(255\) −9.48913 −0.594232
\(256\) 0 0
\(257\) 24.2337 1.51166 0.755828 0.654770i \(-0.227235\pi\)
0.755828 + 0.654770i \(0.227235\pi\)
\(258\) 0 0
\(259\) 10.7446 0.667635
\(260\) 0 0
\(261\) −2.74456 −0.169884
\(262\) 0 0
\(263\) 18.9783 1.17025 0.585125 0.810943i \(-0.301046\pi\)
0.585125 + 0.810943i \(0.301046\pi\)
\(264\) 0 0
\(265\) −1.25544 −0.0771209
\(266\) 0 0
\(267\) 14.9783 0.916654
\(268\) 0 0
\(269\) −24.9783 −1.52295 −0.761475 0.648194i \(-0.775525\pi\)
−0.761475 + 0.648194i \(0.775525\pi\)
\(270\) 0 0
\(271\) 30.9783 1.88179 0.940897 0.338692i \(-0.109984\pi\)
0.940897 + 0.338692i \(0.109984\pi\)
\(272\) 0 0
\(273\) −9.48913 −0.574308
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −7.48913 −0.449978 −0.224989 0.974361i \(-0.572235\pi\)
−0.224989 + 0.974361i \(0.572235\pi\)
\(278\) 0 0
\(279\) 6.74456 0.403786
\(280\) 0 0
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) 0 0
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 0 0
\(285\) 9.48913 0.562087
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) 5.51087 0.324169
\(290\) 0 0
\(291\) 10.5109 0.616158
\(292\) 0 0
\(293\) 24.7446 1.44559 0.722796 0.691061i \(-0.242856\pi\)
0.722796 + 0.691061i \(0.242856\pi\)
\(294\) 0 0
\(295\) −2.74456 −0.159795
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) 22.5109 1.30184
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) 17.4891 1.00472
\(304\) 0 0
\(305\) 12.7446 0.729752
\(306\) 0 0
\(307\) −21.4891 −1.22645 −0.613225 0.789909i \(-0.710128\pi\)
−0.613225 + 0.789909i \(0.710128\pi\)
\(308\) 0 0
\(309\) 21.4891 1.22247
\(310\) 0 0
\(311\) 9.25544 0.524828 0.262414 0.964955i \(-0.415481\pi\)
0.262414 + 0.964955i \(0.415481\pi\)
\(312\) 0 0
\(313\) 32.2337 1.82196 0.910978 0.412455i \(-0.135329\pi\)
0.910978 + 0.412455i \(0.135329\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) −32.2337 −1.81042 −0.905212 0.424960i \(-0.860288\pi\)
−0.905212 + 0.424960i \(0.860288\pi\)
\(318\) 0 0
\(319\) −2.74456 −0.153666
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) −22.5109 −1.25254
\(324\) 0 0
\(325\) −4.74456 −0.263181
\(326\) 0 0
\(327\) −32.4674 −1.79545
\(328\) 0 0
\(329\) −6.74456 −0.371840
\(330\) 0 0
\(331\) 30.9783 1.70272 0.851359 0.524583i \(-0.175779\pi\)
0.851359 + 0.524583i \(0.175779\pi\)
\(332\) 0 0
\(333\) −10.7446 −0.588798
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 0 0
\(339\) −20.0000 −1.08625
\(340\) 0 0
\(341\) 6.74456 0.365239
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 9.48913 0.510877
\(346\) 0 0
\(347\) −22.9783 −1.23354 −0.616769 0.787145i \(-0.711559\pi\)
−0.616769 + 0.787145i \(0.711559\pi\)
\(348\) 0 0
\(349\) 19.2554 1.03072 0.515360 0.856974i \(-0.327658\pi\)
0.515360 + 0.856974i \(0.327658\pi\)
\(350\) 0 0
\(351\) −18.9783 −1.01298
\(352\) 0 0
\(353\) −2.74456 −0.146078 −0.0730392 0.997329i \(-0.523270\pi\)
−0.0730392 + 0.997329i \(0.523270\pi\)
\(354\) 0 0
\(355\) 4.00000 0.212298
\(356\) 0 0
\(357\) 9.48913 0.502218
\(358\) 0 0
\(359\) 12.7446 0.672632 0.336316 0.941749i \(-0.390819\pi\)
0.336316 + 0.941749i \(0.390819\pi\)
\(360\) 0 0
\(361\) 3.51087 0.184783
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) −0.744563 −0.0389722
\(366\) 0 0
\(367\) −2.74456 −0.143265 −0.0716325 0.997431i \(-0.522821\pi\)
−0.0716325 + 0.997431i \(0.522821\pi\)
\(368\) 0 0
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 1.25544 0.0651791
\(372\) 0 0
\(373\) 28.9783 1.50044 0.750218 0.661190i \(-0.229948\pi\)
0.750218 + 0.661190i \(0.229948\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) 0 0
\(377\) 13.0217 0.670654
\(378\) 0 0
\(379\) 14.5109 0.745374 0.372687 0.927957i \(-0.378437\pi\)
0.372687 + 0.927957i \(0.378437\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 37.7228 1.92755 0.963773 0.266724i \(-0.0859413\pi\)
0.963773 + 0.266724i \(0.0859413\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 15.4891 0.785330 0.392665 0.919682i \(-0.371553\pi\)
0.392665 + 0.919682i \(0.371553\pi\)
\(390\) 0 0
\(391\) −22.5109 −1.13842
\(392\) 0 0
\(393\) 17.4891 0.882210
\(394\) 0 0
\(395\) 4.74456 0.238725
\(396\) 0 0
\(397\) 12.5109 0.627903 0.313951 0.949439i \(-0.398347\pi\)
0.313951 + 0.949439i \(0.398347\pi\)
\(398\) 0 0
\(399\) −9.48913 −0.475050
\(400\) 0 0
\(401\) 0.510875 0.0255119 0.0127559 0.999919i \(-0.495940\pi\)
0.0127559 + 0.999919i \(0.495940\pi\)
\(402\) 0 0
\(403\) −32.0000 −1.59403
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) −10.7446 −0.532588
\(408\) 0 0
\(409\) 1.48913 0.0736325 0.0368163 0.999322i \(-0.488278\pi\)
0.0368163 + 0.999322i \(0.488278\pi\)
\(410\) 0 0
\(411\) 38.9783 1.92266
\(412\) 0 0
\(413\) 2.74456 0.135051
\(414\) 0 0
\(415\) −8.00000 −0.392705
\(416\) 0 0
\(417\) 6.51087 0.318839
\(418\) 0 0
\(419\) −10.7446 −0.524906 −0.262453 0.964945i \(-0.584532\pi\)
−0.262453 + 0.964945i \(0.584532\pi\)
\(420\) 0 0
\(421\) 27.4891 1.33974 0.669869 0.742479i \(-0.266350\pi\)
0.669869 + 0.742479i \(0.266350\pi\)
\(422\) 0 0
\(423\) 6.74456 0.327932
\(424\) 0 0
\(425\) 4.74456 0.230145
\(426\) 0 0
\(427\) −12.7446 −0.616753
\(428\) 0 0
\(429\) 9.48913 0.458139
\(430\) 0 0
\(431\) 28.7446 1.38458 0.692288 0.721621i \(-0.256603\pi\)
0.692288 + 0.721621i \(0.256603\pi\)
\(432\) 0 0
\(433\) 25.2554 1.21370 0.606849 0.794817i \(-0.292433\pi\)
0.606849 + 0.794817i \(0.292433\pi\)
\(434\) 0 0
\(435\) 5.48913 0.263183
\(436\) 0 0
\(437\) 22.5109 1.07684
\(438\) 0 0
\(439\) 30.9783 1.47851 0.739256 0.673425i \(-0.235178\pi\)
0.739256 + 0.673425i \(0.235178\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 6.51087 0.309341 0.154670 0.987966i \(-0.450568\pi\)
0.154670 + 0.987966i \(0.450568\pi\)
\(444\) 0 0
\(445\) −7.48913 −0.355019
\(446\) 0 0
\(447\) −21.4891 −1.01640
\(448\) 0 0
\(449\) 40.9783 1.93388 0.966942 0.254998i \(-0.0820748\pi\)
0.966942 + 0.254998i \(0.0820748\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 0 0
\(453\) −41.4891 −1.94933
\(454\) 0 0
\(455\) 4.74456 0.222429
\(456\) 0 0
\(457\) −19.4891 −0.911663 −0.455831 0.890066i \(-0.650658\pi\)
−0.455831 + 0.890066i \(0.650658\pi\)
\(458\) 0 0
\(459\) 18.9783 0.885829
\(460\) 0 0
\(461\) −23.7228 −1.10488 −0.552441 0.833552i \(-0.686303\pi\)
−0.552441 + 0.833552i \(0.686303\pi\)
\(462\) 0 0
\(463\) −12.7446 −0.592290 −0.296145 0.955143i \(-0.595701\pi\)
−0.296145 + 0.955143i \(0.595701\pi\)
\(464\) 0 0
\(465\) −13.4891 −0.625543
\(466\) 0 0
\(467\) 28.9783 1.34095 0.670477 0.741931i \(-0.266090\pi\)
0.670477 + 0.741931i \(0.266090\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −46.9783 −2.16464
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) −4.74456 −0.217695
\(476\) 0 0
\(477\) −1.25544 −0.0574825
\(478\) 0 0
\(479\) 18.5109 0.845783 0.422892 0.906180i \(-0.361015\pi\)
0.422892 + 0.906180i \(0.361015\pi\)
\(480\) 0 0
\(481\) 50.9783 2.32441
\(482\) 0 0
\(483\) −9.48913 −0.431770
\(484\) 0 0
\(485\) −5.25544 −0.238637
\(486\) 0 0
\(487\) −20.7446 −0.940026 −0.470013 0.882660i \(-0.655751\pi\)
−0.470013 + 0.882660i \(0.655751\pi\)
\(488\) 0 0
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) 14.9783 0.675959 0.337979 0.941153i \(-0.390257\pi\)
0.337979 + 0.941153i \(0.390257\pi\)
\(492\) 0 0
\(493\) −13.0217 −0.586470
\(494\) 0 0
\(495\) 1.00000 0.0449467
\(496\) 0 0
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) −9.48913 −0.424792 −0.212396 0.977184i \(-0.568127\pi\)
−0.212396 + 0.977184i \(0.568127\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −29.4891 −1.31486 −0.657428 0.753518i \(-0.728355\pi\)
−0.657428 + 0.753518i \(0.728355\pi\)
\(504\) 0 0
\(505\) −8.74456 −0.389128
\(506\) 0 0
\(507\) −19.0217 −0.844786
\(508\) 0 0
\(509\) −12.5109 −0.554535 −0.277267 0.960793i \(-0.589429\pi\)
−0.277267 + 0.960793i \(0.589429\pi\)
\(510\) 0 0
\(511\) 0.744563 0.0329375
\(512\) 0 0
\(513\) −18.9783 −0.837910
\(514\) 0 0
\(515\) −10.7446 −0.473462
\(516\) 0 0
\(517\) 6.74456 0.296626
\(518\) 0 0
\(519\) 28.4674 1.24958
\(520\) 0 0
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) 0 0
\(523\) −5.48913 −0.240023 −0.120011 0.992773i \(-0.538293\pi\)
−0.120011 + 0.992773i \(0.538293\pi\)
\(524\) 0 0
\(525\) 2.00000 0.0872872
\(526\) 0 0
\(527\) 32.0000 1.39394
\(528\) 0 0
\(529\) −0.489125 −0.0212663
\(530\) 0 0
\(531\) −2.74456 −0.119104
\(532\) 0 0
\(533\) 18.9783 0.822039
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 36.2337 1.55781 0.778904 0.627143i \(-0.215776\pi\)
0.778904 + 0.627143i \(0.215776\pi\)
\(542\) 0 0
\(543\) −6.97825 −0.299465
\(544\) 0 0
\(545\) 16.2337 0.695375
\(546\) 0 0
\(547\) 30.9783 1.32453 0.662267 0.749268i \(-0.269594\pi\)
0.662267 + 0.749268i \(0.269594\pi\)
\(548\) 0 0
\(549\) 12.7446 0.543925
\(550\) 0 0
\(551\) 13.0217 0.554745
\(552\) 0 0
\(553\) −4.74456 −0.201759
\(554\) 0 0
\(555\) 21.4891 0.912163
\(556\) 0 0
\(557\) 44.9783 1.90579 0.952895 0.303301i \(-0.0980887\pi\)
0.952895 + 0.303301i \(0.0980887\pi\)
\(558\) 0 0
\(559\) 18.9783 0.802694
\(560\) 0 0
\(561\) −9.48913 −0.400631
\(562\) 0 0
\(563\) 17.4891 0.737079 0.368539 0.929612i \(-0.379858\pi\)
0.368539 + 0.929612i \(0.379858\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) 0 0
\(567\) 11.0000 0.461957
\(568\) 0 0
\(569\) 39.4891 1.65547 0.827735 0.561119i \(-0.189629\pi\)
0.827735 + 0.561119i \(0.189629\pi\)
\(570\) 0 0
\(571\) −5.48913 −0.229713 −0.114856 0.993382i \(-0.536641\pi\)
−0.114856 + 0.993382i \(0.536641\pi\)
\(572\) 0 0
\(573\) −32.0000 −1.33682
\(574\) 0 0
\(575\) −4.74456 −0.197862
\(576\) 0 0
\(577\) 2.74456 0.114258 0.0571288 0.998367i \(-0.481805\pi\)
0.0571288 + 0.998367i \(0.481805\pi\)
\(578\) 0 0
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) −1.25544 −0.0519949
\(584\) 0 0
\(585\) −4.74456 −0.196164
\(586\) 0 0
\(587\) −40.9783 −1.69135 −0.845677 0.533696i \(-0.820803\pi\)
−0.845677 + 0.533696i \(0.820803\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) −20.0000 −0.822690
\(592\) 0 0
\(593\) −5.76631 −0.236794 −0.118397 0.992966i \(-0.537776\pi\)
−0.118397 + 0.992966i \(0.537776\pi\)
\(594\) 0 0
\(595\) −4.74456 −0.194508
\(596\) 0 0
\(597\) 29.4891 1.20691
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −10.5109 −0.428748 −0.214374 0.976752i \(-0.568771\pi\)
−0.214374 + 0.976752i \(0.568771\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) −5.48913 −0.222431
\(610\) 0 0
\(611\) −32.0000 −1.29458
\(612\) 0 0
\(613\) −11.4891 −0.464041 −0.232021 0.972711i \(-0.574534\pi\)
−0.232021 + 0.972711i \(0.574534\pi\)
\(614\) 0 0
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 0 0
\(619\) 10.7446 0.431860 0.215930 0.976409i \(-0.430722\pi\)
0.215930 + 0.976409i \(0.430722\pi\)
\(620\) 0 0
\(621\) −18.9783 −0.761571
\(622\) 0 0
\(623\) 7.48913 0.300045
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 9.48913 0.378959
\(628\) 0 0
\(629\) −50.9783 −2.03264
\(630\) 0 0
\(631\) −42.9783 −1.71094 −0.855469 0.517855i \(-0.826731\pi\)
−0.855469 + 0.517855i \(0.826731\pi\)
\(632\) 0 0
\(633\) −24.0000 −0.953914
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.74456 −0.187986
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) −11.4891 −0.453793 −0.226897 0.973919i \(-0.572858\pi\)
−0.226897 + 0.973919i \(0.572858\pi\)
\(642\) 0 0
\(643\) 22.4674 0.886027 0.443013 0.896515i \(-0.353909\pi\)
0.443013 + 0.896515i \(0.353909\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) 2.74456 0.107900 0.0539499 0.998544i \(-0.482819\pi\)
0.0539499 + 0.998544i \(0.482819\pi\)
\(648\) 0 0
\(649\) −2.74456 −0.107734
\(650\) 0 0
\(651\) 13.4891 0.528681
\(652\) 0 0
\(653\) −41.7228 −1.63274 −0.816370 0.577529i \(-0.804017\pi\)
−0.816370 + 0.577529i \(0.804017\pi\)
\(654\) 0 0
\(655\) −8.74456 −0.341678
\(656\) 0 0
\(657\) −0.744563 −0.0290482
\(658\) 0 0
\(659\) 18.5109 0.721081 0.360541 0.932744i \(-0.382592\pi\)
0.360541 + 0.932744i \(0.382592\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) 45.0217 1.74850
\(664\) 0 0
\(665\) 4.74456 0.183986
\(666\) 0 0
\(667\) 13.0217 0.504204
\(668\) 0 0
\(669\) 53.4891 2.06801
\(670\) 0 0
\(671\) 12.7446 0.491998
\(672\) 0 0
\(673\) −24.9783 −0.962841 −0.481420 0.876490i \(-0.659879\pi\)
−0.481420 + 0.876490i \(0.659879\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 1.76631 0.0678849 0.0339424 0.999424i \(-0.489194\pi\)
0.0339424 + 0.999424i \(0.489194\pi\)
\(678\) 0 0
\(679\) 5.25544 0.201685
\(680\) 0 0
\(681\) 40.0000 1.53280
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −19.4891 −0.744641
\(686\) 0 0
\(687\) 12.0000 0.457829
\(688\) 0 0
\(689\) 5.95650 0.226925
\(690\) 0 0
\(691\) −36.2337 −1.37839 −0.689197 0.724574i \(-0.742037\pi\)
−0.689197 + 0.724574i \(0.742037\pi\)
\(692\) 0 0
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) −3.25544 −0.123486
\(696\) 0 0
\(697\) −18.9783 −0.718853
\(698\) 0 0
\(699\) −41.9565 −1.58694
\(700\) 0 0
\(701\) −12.2337 −0.462060 −0.231030 0.972947i \(-0.574210\pi\)
−0.231030 + 0.972947i \(0.574210\pi\)
\(702\) 0 0
\(703\) 50.9783 1.92268
\(704\) 0 0
\(705\) −13.4891 −0.508030
\(706\) 0 0
\(707\) 8.74456 0.328873
\(708\) 0 0
\(709\) 23.4891 0.882153 0.441076 0.897470i \(-0.354597\pi\)
0.441076 + 0.897470i \(0.354597\pi\)
\(710\) 0 0
\(711\) 4.74456 0.177935
\(712\) 0 0
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) −4.74456 −0.177437
\(716\) 0 0
\(717\) −6.51087 −0.243153
\(718\) 0 0
\(719\) 49.7228 1.85435 0.927174 0.374631i \(-0.122231\pi\)
0.927174 + 0.374631i \(0.122231\pi\)
\(720\) 0 0
\(721\) 10.7446 0.400148
\(722\) 0 0
\(723\) −40.0000 −1.48762
\(724\) 0 0
\(725\) −2.74456 −0.101930
\(726\) 0 0
\(727\) 20.2337 0.750426 0.375213 0.926939i \(-0.377570\pi\)
0.375213 + 0.926939i \(0.377570\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −18.9783 −0.701936
\(732\) 0 0
\(733\) 18.2337 0.673477 0.336738 0.941598i \(-0.390676\pi\)
0.336738 + 0.941598i \(0.390676\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −14.9783 −0.550984 −0.275492 0.961303i \(-0.588841\pi\)
−0.275492 + 0.961303i \(0.588841\pi\)
\(740\) 0 0
\(741\) −45.0217 −1.65392
\(742\) 0 0
\(743\) 18.9783 0.696244 0.348122 0.937449i \(-0.386819\pi\)
0.348122 + 0.937449i \(0.386819\pi\)
\(744\) 0 0
\(745\) 10.7446 0.393650
\(746\) 0 0
\(747\) −8.00000 −0.292705
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 0 0
\(753\) 16.4674 0.600105
\(754\) 0 0
\(755\) 20.7446 0.754972
\(756\) 0 0
\(757\) −30.7446 −1.11743 −0.558715 0.829360i \(-0.688705\pi\)
−0.558715 + 0.829360i \(0.688705\pi\)
\(758\) 0 0
\(759\) 9.48913 0.344433
\(760\) 0 0
\(761\) −6.51087 −0.236019 −0.118010 0.993012i \(-0.537651\pi\)
−0.118010 + 0.993012i \(0.537651\pi\)
\(762\) 0 0
\(763\) −16.2337 −0.587699
\(764\) 0 0
\(765\) 4.74456 0.171540
\(766\) 0 0
\(767\) 13.0217 0.470188
\(768\) 0 0
\(769\) −8.00000 −0.288487 −0.144244 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) 0 0
\(771\) −48.4674 −1.74551
\(772\) 0 0
\(773\) −28.5109 −1.02546 −0.512732 0.858548i \(-0.671367\pi\)
−0.512732 + 0.858548i \(0.671367\pi\)
\(774\) 0 0
\(775\) 6.74456 0.242272
\(776\) 0 0
\(777\) −21.4891 −0.770918
\(778\) 0 0
\(779\) 18.9783 0.679966
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) −10.9783 −0.392331
\(784\) 0 0
\(785\) 23.4891 0.838363
\(786\) 0 0
\(787\) 17.4891 0.623420 0.311710 0.950177i \(-0.399098\pi\)
0.311710 + 0.950177i \(0.399098\pi\)
\(788\) 0 0
\(789\) −37.9565 −1.35129
\(790\) 0 0
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) −60.4674 −2.14726
\(794\) 0 0
\(795\) 2.51087 0.0890515
\(796\) 0 0
\(797\) 26.4674 0.937523 0.468761 0.883325i \(-0.344700\pi\)
0.468761 + 0.883325i \(0.344700\pi\)
\(798\) 0 0
\(799\) 32.0000 1.13208
\(800\) 0 0
\(801\) −7.48913 −0.264615
\(802\) 0 0
\(803\) −0.744563 −0.0262750
\(804\) 0 0
\(805\) 4.74456 0.167224
\(806\) 0 0
\(807\) 49.9565 1.75855
\(808\) 0 0
\(809\) −34.4674 −1.21181 −0.605904 0.795538i \(-0.707189\pi\)
−0.605904 + 0.795538i \(0.707189\pi\)
\(810\) 0 0
\(811\) 7.25544 0.254773 0.127386 0.991853i \(-0.459341\pi\)
0.127386 + 0.991853i \(0.459341\pi\)
\(812\) 0 0
\(813\) −61.9565 −2.17291
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 18.9783 0.663965
\(818\) 0 0
\(819\) 4.74456 0.165788
\(820\) 0 0
\(821\) −49.7228 −1.73534 −0.867669 0.497142i \(-0.834383\pi\)
−0.867669 + 0.497142i \(0.834383\pi\)
\(822\) 0 0
\(823\) −42.2337 −1.47217 −0.736087 0.676887i \(-0.763329\pi\)
−0.736087 + 0.676887i \(0.763329\pi\)
\(824\) 0 0
\(825\) −2.00000 −0.0696311
\(826\) 0 0
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 0 0
\(829\) −54.4674 −1.89173 −0.945865 0.324560i \(-0.894784\pi\)
−0.945865 + 0.324560i \(0.894784\pi\)
\(830\) 0 0
\(831\) 14.9783 0.519590
\(832\) 0 0
\(833\) 4.74456 0.164389
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 26.9783 0.932505
\(838\) 0 0
\(839\) 14.7446 0.509039 0.254519 0.967068i \(-0.418083\pi\)
0.254519 + 0.967068i \(0.418083\pi\)
\(840\) 0 0
\(841\) −21.4674 −0.740254
\(842\) 0 0
\(843\) −28.0000 −0.964371
\(844\) 0 0
\(845\) 9.51087 0.327184
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) 56.0000 1.92192
\(850\) 0 0
\(851\) 50.9783 1.74751
\(852\) 0 0
\(853\) −11.2554 −0.385379 −0.192689 0.981260i \(-0.561721\pi\)
−0.192689 + 0.981260i \(0.561721\pi\)
\(854\) 0 0
\(855\) −4.74456 −0.162261
\(856\) 0 0
\(857\) 37.2119 1.27114 0.635568 0.772045i \(-0.280766\pi\)
0.635568 + 0.772045i \(0.280766\pi\)
\(858\) 0 0
\(859\) −51.2119 −1.74733 −0.873664 0.486529i \(-0.838263\pi\)
−0.873664 + 0.486529i \(0.838263\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) 5.76631 0.196288 0.0981438 0.995172i \(-0.468709\pi\)
0.0981438 + 0.995172i \(0.468709\pi\)
\(864\) 0 0
\(865\) −14.2337 −0.483960
\(866\) 0 0
\(867\) −11.0217 −0.374318
\(868\) 0 0
\(869\) 4.74456 0.160948
\(870\) 0 0
\(871\) −18.9783 −0.643053
\(872\) 0 0
\(873\) −5.25544 −0.177870
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 42.4674 1.43402 0.717011 0.697062i \(-0.245510\pi\)
0.717011 + 0.697062i \(0.245510\pi\)
\(878\) 0 0
\(879\) −49.4891 −1.66923
\(880\) 0 0
\(881\) −32.5109 −1.09532 −0.547660 0.836701i \(-0.684481\pi\)
−0.547660 + 0.836701i \(0.684481\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) 5.48913 0.184515
\(886\) 0 0
\(887\) 18.5109 0.621534 0.310767 0.950486i \(-0.399414\pi\)
0.310767 + 0.950486i \(0.399414\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −11.0000 −0.368514
\(892\) 0 0
\(893\) −32.0000 −1.07084
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) −45.0217 −1.50323
\(898\) 0 0
\(899\) −18.5109 −0.617372
\(900\) 0 0
\(901\) −5.95650 −0.198440
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) 0 0
\(905\) 3.48913 0.115982
\(906\) 0 0
\(907\) 37.4891 1.24481 0.622403 0.782697i \(-0.286157\pi\)
0.622403 + 0.782697i \(0.286157\pi\)
\(908\) 0 0
\(909\) −8.74456 −0.290039
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) 0 0
\(915\) −25.4891 −0.842644
\(916\) 0 0
\(917\) 8.74456 0.288771
\(918\) 0 0
\(919\) −25.2119 −0.831665 −0.415833 0.909441i \(-0.636510\pi\)
−0.415833 + 0.909441i \(0.636510\pi\)
\(920\) 0 0
\(921\) 42.9783 1.41618
\(922\) 0 0
\(923\) −18.9783 −0.624677
\(924\) 0 0
\(925\) −10.7446 −0.353279
\(926\) 0 0
\(927\) −10.7446 −0.352898
\(928\) 0 0
\(929\) 16.9783 0.557038 0.278519 0.960431i \(-0.410156\pi\)
0.278519 + 0.960431i \(0.410156\pi\)
\(930\) 0 0
\(931\) −4.74456 −0.155497
\(932\) 0 0
\(933\) −18.5109 −0.606019
\(934\) 0 0
\(935\) 4.74456 0.155164
\(936\) 0 0
\(937\) −7.25544 −0.237025 −0.118512 0.992953i \(-0.537813\pi\)
−0.118512 + 0.992953i \(0.537813\pi\)
\(938\) 0 0
\(939\) −64.4674 −2.10381
\(940\) 0 0
\(941\) −22.2337 −0.724798 −0.362399 0.932023i \(-0.618042\pi\)
−0.362399 + 0.932023i \(0.618042\pi\)
\(942\) 0 0
\(943\) 18.9783 0.618017
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 0 0
\(949\) 3.53262 0.114674
\(950\) 0 0
\(951\) 64.4674 2.09050
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) 0 0
\(957\) 5.48913 0.177438
\(958\) 0 0
\(959\) 19.4891 0.629337
\(960\) 0 0
\(961\) 14.4891 0.467391
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) 5.02175 0.161489 0.0807443 0.996735i \(-0.474270\pi\)
0.0807443 + 0.996735i \(0.474270\pi\)
\(968\) 0 0
\(969\) 45.0217 1.44631
\(970\) 0 0
\(971\) 37.7228 1.21058 0.605291 0.796004i \(-0.293057\pi\)
0.605291 + 0.796004i \(0.293057\pi\)
\(972\) 0 0
\(973\) 3.25544 0.104365
\(974\) 0 0
\(975\) 9.48913 0.303895
\(976\) 0 0
\(977\) −32.9783 −1.05507 −0.527534 0.849534i \(-0.676883\pi\)
−0.527534 + 0.849534i \(0.676883\pi\)
\(978\) 0 0
\(979\) −7.48913 −0.239353
\(980\) 0 0
\(981\) 16.2337 0.518302
\(982\) 0 0
\(983\) 32.2337 1.02809 0.514047 0.857762i \(-0.328146\pi\)
0.514047 + 0.857762i \(0.328146\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 0 0
\(987\) 13.4891 0.429364
\(988\) 0 0
\(989\) 18.9783 0.603473
\(990\) 0 0
\(991\) 21.4891 0.682625 0.341312 0.939950i \(-0.389129\pi\)
0.341312 + 0.939950i \(0.389129\pi\)
\(992\) 0 0
\(993\) −61.9565 −1.96613
\(994\) 0 0
\(995\) −14.7446 −0.467434
\(996\) 0 0
\(997\) −41.2119 −1.30520 −0.652598 0.757705i \(-0.726321\pi\)
−0.652598 + 0.757705i \(0.726321\pi\)
\(998\) 0 0
\(999\) −42.9783 −1.35977
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.r.1.1 2
4.3 odd 2 770.2.a.k.1.1 2
12.11 even 2 6930.2.a.bo.1.1 2
20.3 even 4 3850.2.c.y.1849.1 4
20.7 even 4 3850.2.c.y.1849.4 4
20.19 odd 2 3850.2.a.bc.1.2 2
28.27 even 2 5390.2.a.bq.1.2 2
44.43 even 2 8470.2.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.k.1.1 2 4.3 odd 2
3850.2.a.bc.1.2 2 20.19 odd 2
3850.2.c.y.1849.1 4 20.3 even 4
3850.2.c.y.1849.4 4 20.7 even 4
5390.2.a.bq.1.2 2 28.27 even 2
6160.2.a.r.1.1 2 1.1 even 1 trivial
6930.2.a.bo.1.1 2 12.11 even 2
8470.2.a.bu.1.2 2 44.43 even 2