Properties

Label 6864.2.a.cf.1.5
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.8093159474\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.2172244.1
Defining polynomial: \(x^{5} - 2 x^{4} - 8 x^{3} + 16 x^{2} + 5 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.60211\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.52882 q^{5} +3.24835 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.52882 q^{5} +3.24835 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.52882 q^{15} -5.48469 q^{17} +2.44934 q^{19} +3.24835 q^{21} -2.29362 q^{23} +7.45258 q^{25} +1.00000 q^{27} +9.32897 q^{29} -2.32783 q^{31} +1.00000 q^{33} +11.4629 q^{35} -11.4537 q^{37} +1.00000 q^{39} -10.0588 q^{41} +7.09186 q^{43} +3.52882 q^{45} +2.48433 q^{47} +3.55179 q^{49} -5.48469 q^{51} +13.6449 q^{53} +3.52882 q^{55} +2.44934 q^{57} +5.35995 q^{59} -7.71007 q^{61} +3.24835 q^{63} +3.52882 q^{65} +12.2750 q^{67} -2.29362 q^{69} -1.76516 q^{71} +1.80929 q^{73} +7.45258 q^{75} +3.24835 q^{77} -3.26150 q^{79} +1.00000 q^{81} +13.1482 q^{83} -19.3545 q^{85} +9.32897 q^{87} -2.06824 q^{89} +3.24835 q^{91} -2.32783 q^{93} +8.64329 q^{95} -5.04526 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} + q^{5} + 5q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} + q^{5} + 5q^{7} + 5q^{9} + 5q^{11} + 5q^{13} + q^{15} + 4q^{19} + 5q^{21} + 5q^{23} + 4q^{25} + 5q^{27} + 11q^{29} + 8q^{31} + 5q^{33} + 5q^{35} - 8q^{37} + 5q^{39} - q^{41} - q^{43} + q^{45} + 18q^{47} + 10q^{49} + 2q^{53} + q^{55} + 4q^{57} + 13q^{59} + 9q^{61} + 5q^{63} + q^{65} - 5q^{67} + 5q^{69} + 24q^{71} - 13q^{73} + 4q^{75} + 5q^{77} + 6q^{79} + 5q^{81} + 22q^{83} - 22q^{85} + 11q^{87} - 14q^{89} + 5q^{91} + 8q^{93} + 32q^{95} - 20q^{97} + 5q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 3.52882 1.57814 0.789068 0.614305i \(-0.210564\pi\)
0.789068 + 0.614305i \(0.210564\pi\)
\(6\) 0 0
\(7\) 3.24835 1.22776 0.613881 0.789399i \(-0.289607\pi\)
0.613881 + 0.789399i \(0.289607\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 3.52882 0.911138
\(16\) 0 0
\(17\) −5.48469 −1.33023 −0.665117 0.746740i \(-0.731618\pi\)
−0.665117 + 0.746740i \(0.731618\pi\)
\(18\) 0 0
\(19\) 2.44934 0.561917 0.280959 0.959720i \(-0.409348\pi\)
0.280959 + 0.959720i \(0.409348\pi\)
\(20\) 0 0
\(21\) 3.24835 0.708849
\(22\) 0 0
\(23\) −2.29362 −0.478252 −0.239126 0.970989i \(-0.576861\pi\)
−0.239126 + 0.970989i \(0.576861\pi\)
\(24\) 0 0
\(25\) 7.45258 1.49052
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 9.32897 1.73235 0.866173 0.499745i \(-0.166573\pi\)
0.866173 + 0.499745i \(0.166573\pi\)
\(30\) 0 0
\(31\) −2.32783 −0.418091 −0.209045 0.977906i \(-0.567036\pi\)
−0.209045 + 0.977906i \(0.567036\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 11.4629 1.93758
\(36\) 0 0
\(37\) −11.4537 −1.88298 −0.941489 0.337043i \(-0.890573\pi\)
−0.941489 + 0.337043i \(0.890573\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −10.0588 −1.57092 −0.785459 0.618914i \(-0.787573\pi\)
−0.785459 + 0.618914i \(0.787573\pi\)
\(42\) 0 0
\(43\) 7.09186 1.08150 0.540749 0.841184i \(-0.318141\pi\)
0.540749 + 0.841184i \(0.318141\pi\)
\(44\) 0 0
\(45\) 3.52882 0.526046
\(46\) 0 0
\(47\) 2.48433 0.362376 0.181188 0.983448i \(-0.442006\pi\)
0.181188 + 0.983448i \(0.442006\pi\)
\(48\) 0 0
\(49\) 3.55179 0.507399
\(50\) 0 0
\(51\) −5.48469 −0.768010
\(52\) 0 0
\(53\) 13.6449 1.87427 0.937134 0.348970i \(-0.113468\pi\)
0.937134 + 0.348970i \(0.113468\pi\)
\(54\) 0 0
\(55\) 3.52882 0.475826
\(56\) 0 0
\(57\) 2.44934 0.324423
\(58\) 0 0
\(59\) 5.35995 0.697806 0.348903 0.937159i \(-0.386554\pi\)
0.348903 + 0.937159i \(0.386554\pi\)
\(60\) 0 0
\(61\) −7.71007 −0.987173 −0.493587 0.869697i \(-0.664314\pi\)
−0.493587 + 0.869697i \(0.664314\pi\)
\(62\) 0 0
\(63\) 3.24835 0.409254
\(64\) 0 0
\(65\) 3.52882 0.437696
\(66\) 0 0
\(67\) 12.2750 1.49963 0.749816 0.661647i \(-0.230142\pi\)
0.749816 + 0.661647i \(0.230142\pi\)
\(68\) 0 0
\(69\) −2.29362 −0.276119
\(70\) 0 0
\(71\) −1.76516 −0.209486 −0.104743 0.994499i \(-0.533402\pi\)
−0.104743 + 0.994499i \(0.533402\pi\)
\(72\) 0 0
\(73\) 1.80929 0.211761 0.105881 0.994379i \(-0.466234\pi\)
0.105881 + 0.994379i \(0.466234\pi\)
\(74\) 0 0
\(75\) 7.45258 0.860549
\(76\) 0 0
\(77\) 3.24835 0.370184
\(78\) 0 0
\(79\) −3.26150 −0.366948 −0.183474 0.983025i \(-0.558734\pi\)
−0.183474 + 0.983025i \(0.558734\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.1482 1.44320 0.721600 0.692311i \(-0.243407\pi\)
0.721600 + 0.692311i \(0.243407\pi\)
\(84\) 0 0
\(85\) −19.3545 −2.09929
\(86\) 0 0
\(87\) 9.32897 1.00017
\(88\) 0 0
\(89\) −2.06824 −0.219233 −0.109616 0.993974i \(-0.534962\pi\)
−0.109616 + 0.993974i \(0.534962\pi\)
\(90\) 0 0
\(91\) 3.24835 0.340520
\(92\) 0 0
\(93\) −2.32783 −0.241385
\(94\) 0 0
\(95\) 8.64329 0.886783
\(96\) 0 0
\(97\) −5.04526 −0.512269 −0.256135 0.966641i \(-0.582449\pi\)
−0.256135 + 0.966641i \(0.582449\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 0.897457 0.0893003 0.0446502 0.999003i \(-0.485783\pi\)
0.0446502 + 0.999003i \(0.485783\pi\)
\(102\) 0 0
\(103\) −1.69769 −0.167279 −0.0836394 0.996496i \(-0.526654\pi\)
−0.0836394 + 0.996496i \(0.526654\pi\)
\(104\) 0 0
\(105\) 11.4629 1.11866
\(106\) 0 0
\(107\) −8.44389 −0.816301 −0.408151 0.912915i \(-0.633826\pi\)
−0.408151 + 0.912915i \(0.633826\pi\)
\(108\) 0 0
\(109\) 5.30344 0.507978 0.253989 0.967207i \(-0.418257\pi\)
0.253989 + 0.967207i \(0.418257\pi\)
\(110\) 0 0
\(111\) −11.4537 −1.08714
\(112\) 0 0
\(113\) 13.4569 1.26592 0.632961 0.774183i \(-0.281839\pi\)
0.632961 + 0.774183i \(0.281839\pi\)
\(114\) 0 0
\(115\) −8.09376 −0.754747
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −17.8162 −1.63321
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −10.0588 −0.906969
\(124\) 0 0
\(125\) 8.65470 0.774100
\(126\) 0 0
\(127\) 9.11492 0.808818 0.404409 0.914578i \(-0.367477\pi\)
0.404409 + 0.914578i \(0.367477\pi\)
\(128\) 0 0
\(129\) 7.09186 0.624403
\(130\) 0 0
\(131\) 6.88954 0.601942 0.300971 0.953633i \(-0.402689\pi\)
0.300971 + 0.953633i \(0.402689\pi\)
\(132\) 0 0
\(133\) 7.95632 0.689901
\(134\) 0 0
\(135\) 3.52882 0.303713
\(136\) 0 0
\(137\) −9.28257 −0.793063 −0.396532 0.918021i \(-0.629786\pi\)
−0.396532 + 0.918021i \(0.629786\pi\)
\(138\) 0 0
\(139\) 16.3512 1.38689 0.693444 0.720511i \(-0.256092\pi\)
0.693444 + 0.720511i \(0.256092\pi\)
\(140\) 0 0
\(141\) 2.48433 0.209218
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) 32.9203 2.73388
\(146\) 0 0
\(147\) 3.55179 0.292947
\(148\) 0 0
\(149\) 6.85779 0.561812 0.280906 0.959735i \(-0.409365\pi\)
0.280906 + 0.959735i \(0.409365\pi\)
\(150\) 0 0
\(151\) −2.36108 −0.192142 −0.0960711 0.995374i \(-0.530628\pi\)
−0.0960711 + 0.995374i \(0.530628\pi\)
\(152\) 0 0
\(153\) −5.48469 −0.443411
\(154\) 0 0
\(155\) −8.21450 −0.659805
\(156\) 0 0
\(157\) −17.2539 −1.37701 −0.688504 0.725233i \(-0.741732\pi\)
−0.688504 + 0.725233i \(0.741732\pi\)
\(158\) 0 0
\(159\) 13.6449 1.08211
\(160\) 0 0
\(161\) −7.45048 −0.587180
\(162\) 0 0
\(163\) −8.66881 −0.678994 −0.339497 0.940607i \(-0.610257\pi\)
−0.339497 + 0.940607i \(0.610257\pi\)
\(164\) 0 0
\(165\) 3.52882 0.274718
\(166\) 0 0
\(167\) 13.6094 1.05313 0.526565 0.850135i \(-0.323480\pi\)
0.526565 + 0.850135i \(0.323480\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.44934 0.187306
\(172\) 0 0
\(173\) −12.2378 −0.930419 −0.465210 0.885201i \(-0.654021\pi\)
−0.465210 + 0.885201i \(0.654021\pi\)
\(174\) 0 0
\(175\) 24.2086 1.83000
\(176\) 0 0
\(177\) 5.35995 0.402878
\(178\) 0 0
\(179\) 10.7549 0.803858 0.401929 0.915671i \(-0.368340\pi\)
0.401929 + 0.915671i \(0.368340\pi\)
\(180\) 0 0
\(181\) −8.68491 −0.645544 −0.322772 0.946477i \(-0.604615\pi\)
−0.322772 + 0.946477i \(0.604615\pi\)
\(182\) 0 0
\(183\) −7.71007 −0.569945
\(184\) 0 0
\(185\) −40.4181 −2.97160
\(186\) 0 0
\(187\) −5.48469 −0.401080
\(188\) 0 0
\(189\) 3.24835 0.236283
\(190\) 0 0
\(191\) −14.8523 −1.07468 −0.537339 0.843367i \(-0.680570\pi\)
−0.537339 + 0.843367i \(0.680570\pi\)
\(192\) 0 0
\(193\) −7.50698 −0.540364 −0.270182 0.962809i \(-0.587084\pi\)
−0.270182 + 0.962809i \(0.587084\pi\)
\(194\) 0 0
\(195\) 3.52882 0.252704
\(196\) 0 0
\(197\) −3.19553 −0.227672 −0.113836 0.993500i \(-0.536314\pi\)
−0.113836 + 0.993500i \(0.536314\pi\)
\(198\) 0 0
\(199\) −19.3205 −1.36959 −0.684796 0.728735i \(-0.740109\pi\)
−0.684796 + 0.728735i \(0.740109\pi\)
\(200\) 0 0
\(201\) 12.2750 0.865813
\(202\) 0 0
\(203\) 30.3038 2.12691
\(204\) 0 0
\(205\) −35.4956 −2.47912
\(206\) 0 0
\(207\) −2.29362 −0.159417
\(208\) 0 0
\(209\) 2.44934 0.169424
\(210\) 0 0
\(211\) −4.58374 −0.315558 −0.157779 0.987474i \(-0.550433\pi\)
−0.157779 + 0.987474i \(0.550433\pi\)
\(212\) 0 0
\(213\) −1.76516 −0.120947
\(214\) 0 0
\(215\) 25.0259 1.70675
\(216\) 0 0
\(217\) −7.56162 −0.513316
\(218\) 0 0
\(219\) 1.80929 0.122260
\(220\) 0 0
\(221\) −5.48469 −0.368940
\(222\) 0 0
\(223\) 15.3212 1.02599 0.512993 0.858393i \(-0.328537\pi\)
0.512993 + 0.858393i \(0.328537\pi\)
\(224\) 0 0
\(225\) 7.45258 0.496838
\(226\) 0 0
\(227\) −5.66821 −0.376212 −0.188106 0.982149i \(-0.560235\pi\)
−0.188106 + 0.982149i \(0.560235\pi\)
\(228\) 0 0
\(229\) 6.49994 0.429528 0.214764 0.976666i \(-0.431102\pi\)
0.214764 + 0.976666i \(0.431102\pi\)
\(230\) 0 0
\(231\) 3.24835 0.213726
\(232\) 0 0
\(233\) −11.5430 −0.756208 −0.378104 0.925763i \(-0.623424\pi\)
−0.378104 + 0.925763i \(0.623424\pi\)
\(234\) 0 0
\(235\) 8.76675 0.571880
\(236\) 0 0
\(237\) −3.26150 −0.211857
\(238\) 0 0
\(239\) 7.78299 0.503440 0.251720 0.967800i \(-0.419004\pi\)
0.251720 + 0.967800i \(0.419004\pi\)
\(240\) 0 0
\(241\) −26.6383 −1.71592 −0.857961 0.513714i \(-0.828269\pi\)
−0.857961 + 0.513714i \(0.828269\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 12.5336 0.800745
\(246\) 0 0
\(247\) 2.44934 0.155848
\(248\) 0 0
\(249\) 13.1482 0.833231
\(250\) 0 0
\(251\) −18.5880 −1.17326 −0.586631 0.809854i \(-0.699546\pi\)
−0.586631 + 0.809854i \(0.699546\pi\)
\(252\) 0 0
\(253\) −2.29362 −0.144198
\(254\) 0 0
\(255\) −19.3545 −1.21203
\(256\) 0 0
\(257\) −4.23233 −0.264006 −0.132003 0.991249i \(-0.542141\pi\)
−0.132003 + 0.991249i \(0.542141\pi\)
\(258\) 0 0
\(259\) −37.2057 −2.31185
\(260\) 0 0
\(261\) 9.32897 0.577449
\(262\) 0 0
\(263\) −27.2699 −1.68154 −0.840768 0.541396i \(-0.817896\pi\)
−0.840768 + 0.541396i \(0.817896\pi\)
\(264\) 0 0
\(265\) 48.1503 2.95785
\(266\) 0 0
\(267\) −2.06824 −0.126574
\(268\) 0 0
\(269\) −17.3334 −1.05684 −0.528419 0.848984i \(-0.677215\pi\)
−0.528419 + 0.848984i \(0.677215\pi\)
\(270\) 0 0
\(271\) −19.9402 −1.21128 −0.605640 0.795739i \(-0.707083\pi\)
−0.605640 + 0.795739i \(0.707083\pi\)
\(272\) 0 0
\(273\) 3.24835 0.196599
\(274\) 0 0
\(275\) 7.45258 0.449407
\(276\) 0 0
\(277\) 8.21405 0.493534 0.246767 0.969075i \(-0.420632\pi\)
0.246767 + 0.969075i \(0.420632\pi\)
\(278\) 0 0
\(279\) −2.32783 −0.139364
\(280\) 0 0
\(281\) −15.8135 −0.943354 −0.471677 0.881772i \(-0.656351\pi\)
−0.471677 + 0.881772i \(0.656351\pi\)
\(282\) 0 0
\(283\) −11.2407 −0.668191 −0.334095 0.942539i \(-0.608431\pi\)
−0.334095 + 0.942539i \(0.608431\pi\)
\(284\) 0 0
\(285\) 8.64329 0.511984
\(286\) 0 0
\(287\) −32.6745 −1.92871
\(288\) 0 0
\(289\) 13.0818 0.769520
\(290\) 0 0
\(291\) −5.04526 −0.295759
\(292\) 0 0
\(293\) −16.4756 −0.962517 −0.481259 0.876579i \(-0.659820\pi\)
−0.481259 + 0.876579i \(0.659820\pi\)
\(294\) 0 0
\(295\) 18.9143 1.10123
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) −2.29362 −0.132643
\(300\) 0 0
\(301\) 23.0368 1.32782
\(302\) 0 0
\(303\) 0.897457 0.0515576
\(304\) 0 0
\(305\) −27.2075 −1.55789
\(306\) 0 0
\(307\) 30.0991 1.71785 0.858924 0.512102i \(-0.171133\pi\)
0.858924 + 0.512102i \(0.171133\pi\)
\(308\) 0 0
\(309\) −1.69769 −0.0965784
\(310\) 0 0
\(311\) 18.6929 1.05998 0.529989 0.848004i \(-0.322196\pi\)
0.529989 + 0.848004i \(0.322196\pi\)
\(312\) 0 0
\(313\) 7.73114 0.436990 0.218495 0.975838i \(-0.429885\pi\)
0.218495 + 0.975838i \(0.429885\pi\)
\(314\) 0 0
\(315\) 11.4629 0.645859
\(316\) 0 0
\(317\) −24.2897 −1.36424 −0.682122 0.731239i \(-0.738943\pi\)
−0.682122 + 0.731239i \(0.738943\pi\)
\(318\) 0 0
\(319\) 9.32897 0.522322
\(320\) 0 0
\(321\) −8.44389 −0.471292
\(322\) 0 0
\(323\) −13.4339 −0.747481
\(324\) 0 0
\(325\) 7.45258 0.413395
\(326\) 0 0
\(327\) 5.30344 0.293281
\(328\) 0 0
\(329\) 8.06997 0.444912
\(330\) 0 0
\(331\) 2.62003 0.144010 0.0720050 0.997404i \(-0.477060\pi\)
0.0720050 + 0.997404i \(0.477060\pi\)
\(332\) 0 0
\(333\) −11.4537 −0.627659
\(334\) 0 0
\(335\) 43.3163 2.36662
\(336\) 0 0
\(337\) −14.3625 −0.782375 −0.391188 0.920311i \(-0.627936\pi\)
−0.391188 + 0.920311i \(0.627936\pi\)
\(338\) 0 0
\(339\) 13.4569 0.730881
\(340\) 0 0
\(341\) −2.32783 −0.126059
\(342\) 0 0
\(343\) −11.2010 −0.604797
\(344\) 0 0
\(345\) −8.09376 −0.435754
\(346\) 0 0
\(347\) 34.3771 1.84546 0.922731 0.385445i \(-0.125952\pi\)
0.922731 + 0.385445i \(0.125952\pi\)
\(348\) 0 0
\(349\) 14.4553 0.773774 0.386887 0.922127i \(-0.373550\pi\)
0.386887 + 0.922127i \(0.373550\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 14.3564 0.764114 0.382057 0.924139i \(-0.375216\pi\)
0.382057 + 0.924139i \(0.375216\pi\)
\(354\) 0 0
\(355\) −6.22893 −0.330597
\(356\) 0 0
\(357\) −17.8162 −0.942934
\(358\) 0 0
\(359\) 9.33888 0.492887 0.246444 0.969157i \(-0.420738\pi\)
0.246444 + 0.969157i \(0.420738\pi\)
\(360\) 0 0
\(361\) −13.0007 −0.684249
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 6.38466 0.334188
\(366\) 0 0
\(367\) −27.8507 −1.45379 −0.726897 0.686746i \(-0.759038\pi\)
−0.726897 + 0.686746i \(0.759038\pi\)
\(368\) 0 0
\(369\) −10.0588 −0.523639
\(370\) 0 0
\(371\) 44.3234 2.30115
\(372\) 0 0
\(373\) 25.1455 1.30198 0.650992 0.759084i \(-0.274353\pi\)
0.650992 + 0.759084i \(0.274353\pi\)
\(374\) 0 0
\(375\) 8.65470 0.446927
\(376\) 0 0
\(377\) 9.32897 0.480466
\(378\) 0 0
\(379\) −33.4424 −1.71782 −0.858911 0.512125i \(-0.828859\pi\)
−0.858911 + 0.512125i \(0.828859\pi\)
\(380\) 0 0
\(381\) 9.11492 0.466971
\(382\) 0 0
\(383\) 28.2495 1.44348 0.721741 0.692164i \(-0.243342\pi\)
0.721741 + 0.692164i \(0.243342\pi\)
\(384\) 0 0
\(385\) 11.4629 0.584201
\(386\) 0 0
\(387\) 7.09186 0.360499
\(388\) 0 0
\(389\) −15.8301 −0.802620 −0.401310 0.915942i \(-0.631445\pi\)
−0.401310 + 0.915942i \(0.631445\pi\)
\(390\) 0 0
\(391\) 12.5798 0.636187
\(392\) 0 0
\(393\) 6.88954 0.347531
\(394\) 0 0
\(395\) −11.5093 −0.579093
\(396\) 0 0
\(397\) −4.96388 −0.249130 −0.124565 0.992211i \(-0.539754\pi\)
−0.124565 + 0.992211i \(0.539754\pi\)
\(398\) 0 0
\(399\) 7.95632 0.398314
\(400\) 0 0
\(401\) −28.8698 −1.44169 −0.720846 0.693096i \(-0.756246\pi\)
−0.720846 + 0.693096i \(0.756246\pi\)
\(402\) 0 0
\(403\) −2.32783 −0.115958
\(404\) 0 0
\(405\) 3.52882 0.175349
\(406\) 0 0
\(407\) −11.4537 −0.567739
\(408\) 0 0
\(409\) 26.6445 1.31748 0.658742 0.752369i \(-0.271089\pi\)
0.658742 + 0.752369i \(0.271089\pi\)
\(410\) 0 0
\(411\) −9.28257 −0.457875
\(412\) 0 0
\(413\) 17.4110 0.856739
\(414\) 0 0
\(415\) 46.3975 2.27757
\(416\) 0 0
\(417\) 16.3512 0.800720
\(418\) 0 0
\(419\) 5.83882 0.285245 0.142623 0.989777i \(-0.454447\pi\)
0.142623 + 0.989777i \(0.454447\pi\)
\(420\) 0 0
\(421\) 37.3214 1.81893 0.909466 0.415778i \(-0.136491\pi\)
0.909466 + 0.415778i \(0.136491\pi\)
\(422\) 0 0
\(423\) 2.48433 0.120792
\(424\) 0 0
\(425\) −40.8751 −1.98273
\(426\) 0 0
\(427\) −25.0450 −1.21201
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) −20.7222 −0.998152 −0.499076 0.866558i \(-0.666327\pi\)
−0.499076 + 0.866558i \(0.666327\pi\)
\(432\) 0 0
\(433\) −24.7204 −1.18799 −0.593993 0.804470i \(-0.702449\pi\)
−0.593993 + 0.804470i \(0.702449\pi\)
\(434\) 0 0
\(435\) 32.9203 1.57841
\(436\) 0 0
\(437\) −5.61785 −0.268738
\(438\) 0 0
\(439\) −1.77211 −0.0845783 −0.0422892 0.999105i \(-0.513465\pi\)
−0.0422892 + 0.999105i \(0.513465\pi\)
\(440\) 0 0
\(441\) 3.55179 0.169133
\(442\) 0 0
\(443\) −6.43320 −0.305651 −0.152825 0.988253i \(-0.548837\pi\)
−0.152825 + 0.988253i \(0.548837\pi\)
\(444\) 0 0
\(445\) −7.29844 −0.345979
\(446\) 0 0
\(447\) 6.85779 0.324362
\(448\) 0 0
\(449\) 29.4607 1.39034 0.695169 0.718847i \(-0.255330\pi\)
0.695169 + 0.718847i \(0.255330\pi\)
\(450\) 0 0
\(451\) −10.0588 −0.473649
\(452\) 0 0
\(453\) −2.36108 −0.110933
\(454\) 0 0
\(455\) 11.4629 0.537387
\(456\) 0 0
\(457\) −40.3142 −1.88582 −0.942910 0.333048i \(-0.891923\pi\)
−0.942910 + 0.333048i \(0.891923\pi\)
\(458\) 0 0
\(459\) −5.48469 −0.256003
\(460\) 0 0
\(461\) 18.1182 0.843851 0.421925 0.906631i \(-0.361354\pi\)
0.421925 + 0.906631i \(0.361354\pi\)
\(462\) 0 0
\(463\) −24.9917 −1.16146 −0.580731 0.814096i \(-0.697233\pi\)
−0.580731 + 0.814096i \(0.697233\pi\)
\(464\) 0 0
\(465\) −8.21450 −0.380938
\(466\) 0 0
\(467\) −20.7835 −0.961743 −0.480872 0.876791i \(-0.659680\pi\)
−0.480872 + 0.876791i \(0.659680\pi\)
\(468\) 0 0
\(469\) 39.8736 1.84119
\(470\) 0 0
\(471\) −17.2539 −0.795016
\(472\) 0 0
\(473\) 7.09186 0.326084
\(474\) 0 0
\(475\) 18.2539 0.837546
\(476\) 0 0
\(477\) 13.6449 0.624756
\(478\) 0 0
\(479\) 36.5844 1.67159 0.835793 0.549045i \(-0.185008\pi\)
0.835793 + 0.549045i \(0.185008\pi\)
\(480\) 0 0
\(481\) −11.4537 −0.522244
\(482\) 0 0
\(483\) −7.45048 −0.339008
\(484\) 0 0
\(485\) −17.8038 −0.808430
\(486\) 0 0
\(487\) −21.6052 −0.979025 −0.489513 0.871996i \(-0.662825\pi\)
−0.489513 + 0.871996i \(0.662825\pi\)
\(488\) 0 0
\(489\) −8.66881 −0.392017
\(490\) 0 0
\(491\) 17.8732 0.806607 0.403304 0.915066i \(-0.367862\pi\)
0.403304 + 0.915066i \(0.367862\pi\)
\(492\) 0 0
\(493\) −51.1665 −2.30442
\(494\) 0 0
\(495\) 3.52882 0.158609
\(496\) 0 0
\(497\) −5.73386 −0.257199
\(498\) 0 0
\(499\) −26.1343 −1.16993 −0.584967 0.811057i \(-0.698892\pi\)
−0.584967 + 0.811057i \(0.698892\pi\)
\(500\) 0 0
\(501\) 13.6094 0.608025
\(502\) 0 0
\(503\) 44.3670 1.97823 0.989114 0.147149i \(-0.0470098\pi\)
0.989114 + 0.147149i \(0.0470098\pi\)
\(504\) 0 0
\(505\) 3.16697 0.140928
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −0.450386 −0.0199630 −0.00998151 0.999950i \(-0.503177\pi\)
−0.00998151 + 0.999950i \(0.503177\pi\)
\(510\) 0 0
\(511\) 5.87721 0.259992
\(512\) 0 0
\(513\) 2.44934 0.108141
\(514\) 0 0
\(515\) −5.99086 −0.263989
\(516\) 0 0
\(517\) 2.48433 0.109261
\(518\) 0 0
\(519\) −12.2378 −0.537178
\(520\) 0 0
\(521\) 38.0188 1.66563 0.832817 0.553549i \(-0.186727\pi\)
0.832817 + 0.553549i \(0.186727\pi\)
\(522\) 0 0
\(523\) −19.7961 −0.865625 −0.432812 0.901484i \(-0.642479\pi\)
−0.432812 + 0.901484i \(0.642479\pi\)
\(524\) 0 0
\(525\) 24.2086 1.05655
\(526\) 0 0
\(527\) 12.7674 0.556158
\(528\) 0 0
\(529\) −17.7393 −0.771275
\(530\) 0 0
\(531\) 5.35995 0.232602
\(532\) 0 0
\(533\) −10.0588 −0.435694
\(534\) 0 0
\(535\) −29.7970 −1.28824
\(536\) 0 0
\(537\) 10.7549 0.464107
\(538\) 0 0
\(539\) 3.55179 0.152987
\(540\) 0 0
\(541\) 3.16993 0.136286 0.0681429 0.997676i \(-0.478293\pi\)
0.0681429 + 0.997676i \(0.478293\pi\)
\(542\) 0 0
\(543\) −8.68491 −0.372705
\(544\) 0 0
\(545\) 18.7149 0.801658
\(546\) 0 0
\(547\) −16.1394 −0.690071 −0.345035 0.938590i \(-0.612133\pi\)
−0.345035 + 0.938590i \(0.612133\pi\)
\(548\) 0 0
\(549\) −7.71007 −0.329058
\(550\) 0 0
\(551\) 22.8498 0.973435
\(552\) 0 0
\(553\) −10.5945 −0.450524
\(554\) 0 0
\(555\) −40.4181 −1.71565
\(556\) 0 0
\(557\) 39.5561 1.67605 0.838023 0.545635i \(-0.183711\pi\)
0.838023 + 0.545635i \(0.183711\pi\)
\(558\) 0 0
\(559\) 7.09186 0.299954
\(560\) 0 0
\(561\) −5.48469 −0.231564
\(562\) 0 0
\(563\) 7.62505 0.321357 0.160679 0.987007i \(-0.448632\pi\)
0.160679 + 0.987007i \(0.448632\pi\)
\(564\) 0 0
\(565\) 47.4871 1.99780
\(566\) 0 0
\(567\) 3.24835 0.136418
\(568\) 0 0
\(569\) −14.2286 −0.596495 −0.298247 0.954489i \(-0.596402\pi\)
−0.298247 + 0.954489i \(0.596402\pi\)
\(570\) 0 0
\(571\) −0.707256 −0.0295978 −0.0147989 0.999890i \(-0.504711\pi\)
−0.0147989 + 0.999890i \(0.504711\pi\)
\(572\) 0 0
\(573\) −14.8523 −0.620465
\(574\) 0 0
\(575\) −17.0934 −0.712842
\(576\) 0 0
\(577\) 29.3254 1.22083 0.610416 0.792081i \(-0.291002\pi\)
0.610416 + 0.792081i \(0.291002\pi\)
\(578\) 0 0
\(579\) −7.50698 −0.311980
\(580\) 0 0
\(581\) 42.7099 1.77190
\(582\) 0 0
\(583\) 13.6449 0.565113
\(584\) 0 0
\(585\) 3.52882 0.145899
\(586\) 0 0
\(587\) −30.0849 −1.24174 −0.620869 0.783914i \(-0.713220\pi\)
−0.620869 + 0.783914i \(0.713220\pi\)
\(588\) 0 0
\(589\) −5.70165 −0.234933
\(590\) 0 0
\(591\) −3.19553 −0.131447
\(592\) 0 0
\(593\) −21.5857 −0.886418 −0.443209 0.896418i \(-0.646160\pi\)
−0.443209 + 0.896418i \(0.646160\pi\)
\(594\) 0 0
\(595\) −62.8702 −2.57743
\(596\) 0 0
\(597\) −19.3205 −0.790734
\(598\) 0 0
\(599\) −31.7749 −1.29829 −0.649143 0.760667i \(-0.724872\pi\)
−0.649143 + 0.760667i \(0.724872\pi\)
\(600\) 0 0
\(601\) −8.09700 −0.330283 −0.165142 0.986270i \(-0.552808\pi\)
−0.165142 + 0.986270i \(0.552808\pi\)
\(602\) 0 0
\(603\) 12.2750 0.499877
\(604\) 0 0
\(605\) 3.52882 0.143467
\(606\) 0 0
\(607\) 9.11492 0.369963 0.184982 0.982742i \(-0.440777\pi\)
0.184982 + 0.982742i \(0.440777\pi\)
\(608\) 0 0
\(609\) 30.3038 1.22797
\(610\) 0 0
\(611\) 2.48433 0.100505
\(612\) 0 0
\(613\) −2.60183 −0.105087 −0.0525435 0.998619i \(-0.516733\pi\)
−0.0525435 + 0.998619i \(0.516733\pi\)
\(614\) 0 0
\(615\) −35.4956 −1.43132
\(616\) 0 0
\(617\) −20.8466 −0.839254 −0.419627 0.907697i \(-0.637839\pi\)
−0.419627 + 0.907697i \(0.637839\pi\)
\(618\) 0 0
\(619\) 31.8886 1.28171 0.640855 0.767662i \(-0.278580\pi\)
0.640855 + 0.767662i \(0.278580\pi\)
\(620\) 0 0
\(621\) −2.29362 −0.0920397
\(622\) 0 0
\(623\) −6.71837 −0.269166
\(624\) 0 0
\(625\) −6.72199 −0.268880
\(626\) 0 0
\(627\) 2.44934 0.0978173
\(628\) 0 0
\(629\) 62.8201 2.50480
\(630\) 0 0
\(631\) −5.38401 −0.214334 −0.107167 0.994241i \(-0.534178\pi\)
−0.107167 + 0.994241i \(0.534178\pi\)
\(632\) 0 0
\(633\) −4.58374 −0.182187
\(634\) 0 0
\(635\) 32.1649 1.27643
\(636\) 0 0
\(637\) 3.55179 0.140727
\(638\) 0 0
\(639\) −1.76516 −0.0698286
\(640\) 0 0
\(641\) 16.3920 0.647446 0.323723 0.946152i \(-0.395065\pi\)
0.323723 + 0.946152i \(0.395065\pi\)
\(642\) 0 0
\(643\) −45.9975 −1.81396 −0.906981 0.421171i \(-0.861619\pi\)
−0.906981 + 0.421171i \(0.861619\pi\)
\(644\) 0 0
\(645\) 25.0259 0.985393
\(646\) 0 0
\(647\) −15.2406 −0.599171 −0.299585 0.954069i \(-0.596848\pi\)
−0.299585 + 0.954069i \(0.596848\pi\)
\(648\) 0 0
\(649\) 5.35995 0.210396
\(650\) 0 0
\(651\) −7.56162 −0.296363
\(652\) 0 0
\(653\) −3.21660 −0.125875 −0.0629376 0.998017i \(-0.520047\pi\)
−0.0629376 + 0.998017i \(0.520047\pi\)
\(654\) 0 0
\(655\) 24.3120 0.949947
\(656\) 0 0
\(657\) 1.80929 0.0705871
\(658\) 0 0
\(659\) 1.42094 0.0553521 0.0276761 0.999617i \(-0.491189\pi\)
0.0276761 + 0.999617i \(0.491189\pi\)
\(660\) 0 0
\(661\) 20.5478 0.799215 0.399608 0.916686i \(-0.369146\pi\)
0.399608 + 0.916686i \(0.369146\pi\)
\(662\) 0 0
\(663\) −5.48469 −0.213008
\(664\) 0 0
\(665\) 28.0764 1.08876
\(666\) 0 0
\(667\) −21.3971 −0.828498
\(668\) 0 0
\(669\) 15.3212 0.592353
\(670\) 0 0
\(671\) −7.71007 −0.297644
\(672\) 0 0
\(673\) 9.13806 0.352246 0.176123 0.984368i \(-0.443644\pi\)
0.176123 + 0.984368i \(0.443644\pi\)
\(674\) 0 0
\(675\) 7.45258 0.286850
\(676\) 0 0
\(677\) 26.6437 1.02400 0.512000 0.858985i \(-0.328905\pi\)
0.512000 + 0.858985i \(0.328905\pi\)
\(678\) 0 0
\(679\) −16.3888 −0.628944
\(680\) 0 0
\(681\) −5.66821 −0.217206
\(682\) 0 0
\(683\) 22.9749 0.879110 0.439555 0.898216i \(-0.355136\pi\)
0.439555 + 0.898216i \(0.355136\pi\)
\(684\) 0 0
\(685\) −32.7565 −1.25156
\(686\) 0 0
\(687\) 6.49994 0.247988
\(688\) 0 0
\(689\) 13.6449 0.519828
\(690\) 0 0
\(691\) 24.0127 0.913488 0.456744 0.889598i \(-0.349016\pi\)
0.456744 + 0.889598i \(0.349016\pi\)
\(692\) 0 0
\(693\) 3.24835 0.123395
\(694\) 0 0
\(695\) 57.7003 2.18870
\(696\) 0 0
\(697\) 55.1693 2.08969
\(698\) 0 0
\(699\) −11.5430 −0.436597
\(700\) 0 0
\(701\) 4.66740 0.176285 0.0881425 0.996108i \(-0.471907\pi\)
0.0881425 + 0.996108i \(0.471907\pi\)
\(702\) 0 0
\(703\) −28.0540 −1.05808
\(704\) 0 0
\(705\) 8.76675 0.330175
\(706\) 0 0
\(707\) 2.91526 0.109640
\(708\) 0 0
\(709\) −13.9375 −0.523432 −0.261716 0.965145i \(-0.584288\pi\)
−0.261716 + 0.965145i \(0.584288\pi\)
\(710\) 0 0
\(711\) −3.26150 −0.122316
\(712\) 0 0
\(713\) 5.33915 0.199953
\(714\) 0 0
\(715\) 3.52882 0.131970
\(716\) 0 0
\(717\) 7.78299 0.290661
\(718\) 0 0
\(719\) 25.7087 0.958773 0.479386 0.877604i \(-0.340859\pi\)
0.479386 + 0.877604i \(0.340859\pi\)
\(720\) 0 0
\(721\) −5.51471 −0.205378
\(722\) 0 0
\(723\) −26.6383 −0.990688
\(724\) 0 0
\(725\) 69.5248 2.58209
\(726\) 0 0
\(727\) 7.75698 0.287690 0.143845 0.989600i \(-0.454053\pi\)
0.143845 + 0.989600i \(0.454053\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −38.8966 −1.43864
\(732\) 0 0
\(733\) 51.9270 1.91797 0.958984 0.283461i \(-0.0914826\pi\)
0.958984 + 0.283461i \(0.0914826\pi\)
\(734\) 0 0
\(735\) 12.5336 0.462310
\(736\) 0 0
\(737\) 12.2750 0.452156
\(738\) 0 0
\(739\) −3.00728 −0.110625 −0.0553123 0.998469i \(-0.517615\pi\)
−0.0553123 + 0.998469i \(0.517615\pi\)
\(740\) 0 0
\(741\) 2.44934 0.0899788
\(742\) 0 0
\(743\) −42.4774 −1.55834 −0.779172 0.626810i \(-0.784360\pi\)
−0.779172 + 0.626810i \(0.784360\pi\)
\(744\) 0 0
\(745\) 24.1999 0.886616
\(746\) 0 0
\(747\) 13.1482 0.481066
\(748\) 0 0
\(749\) −27.4287 −1.00222
\(750\) 0 0
\(751\) −4.12506 −0.150526 −0.0752628 0.997164i \(-0.523980\pi\)
−0.0752628 + 0.997164i \(0.523980\pi\)
\(752\) 0 0
\(753\) −18.5880 −0.677383
\(754\) 0 0
\(755\) −8.33184 −0.303227
\(756\) 0 0
\(757\) −24.6428 −0.895657 −0.447828 0.894119i \(-0.647802\pi\)
−0.447828 + 0.894119i \(0.647802\pi\)
\(758\) 0 0
\(759\) −2.29362 −0.0832530
\(760\) 0 0
\(761\) 51.6672 1.87293 0.936467 0.350756i \(-0.114075\pi\)
0.936467 + 0.350756i \(0.114075\pi\)
\(762\) 0 0
\(763\) 17.2274 0.623675
\(764\) 0 0
\(765\) −19.3545 −0.699763
\(766\) 0 0
\(767\) 5.35995 0.193536
\(768\) 0 0
\(769\) −1.60716 −0.0579558 −0.0289779 0.999580i \(-0.509225\pi\)
−0.0289779 + 0.999580i \(0.509225\pi\)
\(770\) 0 0
\(771\) −4.23233 −0.152424
\(772\) 0 0
\(773\) 8.38325 0.301524 0.150762 0.988570i \(-0.451827\pi\)
0.150762 + 0.988570i \(0.451827\pi\)
\(774\) 0 0
\(775\) −17.3483 −0.623171
\(776\) 0 0
\(777\) −37.2057 −1.33475
\(778\) 0 0
\(779\) −24.6374 −0.882726
\(780\) 0 0
\(781\) −1.76516 −0.0631624
\(782\) 0 0
\(783\) 9.32897 0.333390
\(784\) 0 0
\(785\) −60.8858 −2.17311
\(786\) 0 0
\(787\) −37.3531 −1.33150 −0.665748 0.746177i \(-0.731887\pi\)
−0.665748 + 0.746177i \(0.731887\pi\)
\(788\) 0 0
\(789\) −27.2699 −0.970835
\(790\) 0 0
\(791\) 43.7129 1.55425
\(792\) 0 0
\(793\) −7.71007 −0.273793
\(794\) 0 0
\(795\) 48.1503 1.70772
\(796\) 0 0
\(797\) −4.12175 −0.146000 −0.0730000 0.997332i \(-0.523257\pi\)
−0.0730000 + 0.997332i \(0.523257\pi\)
\(798\) 0 0
\(799\) −13.6258 −0.482045
\(800\) 0 0
\(801\) −2.06824 −0.0730776
\(802\) 0 0
\(803\) 1.80929 0.0638484
\(804\) 0 0
\(805\) −26.2914 −0.926650
\(806\) 0 0
\(807\) −17.3334 −0.610166
\(808\) 0 0
\(809\) 43.6064 1.53312 0.766560 0.642172i \(-0.221967\pi\)
0.766560 + 0.642172i \(0.221967\pi\)
\(810\) 0 0
\(811\) 0.946046 0.0332202 0.0166101 0.999862i \(-0.494713\pi\)
0.0166101 + 0.999862i \(0.494713\pi\)
\(812\) 0 0
\(813\) −19.9402 −0.699333
\(814\) 0 0
\(815\) −30.5907 −1.07154
\(816\) 0 0
\(817\) 17.3704 0.607713
\(818\) 0 0
\(819\) 3.24835 0.113507
\(820\) 0 0
\(821\) −13.2226 −0.461470 −0.230735 0.973017i \(-0.574113\pi\)
−0.230735 + 0.973017i \(0.574113\pi\)
\(822\) 0 0
\(823\) 4.96024 0.172903 0.0864516 0.996256i \(-0.472447\pi\)
0.0864516 + 0.996256i \(0.472447\pi\)
\(824\) 0 0
\(825\) 7.45258 0.259465
\(826\) 0 0
\(827\) −36.0721 −1.25435 −0.627175 0.778878i \(-0.715789\pi\)
−0.627175 + 0.778878i \(0.715789\pi\)
\(828\) 0 0
\(829\) −11.8500 −0.411566 −0.205783 0.978598i \(-0.565974\pi\)
−0.205783 + 0.978598i \(0.565974\pi\)
\(830\) 0 0
\(831\) 8.21405 0.284942
\(832\) 0 0
\(833\) −19.4805 −0.674959
\(834\) 0 0
\(835\) 48.0253 1.66198
\(836\) 0 0
\(837\) −2.32783 −0.0804616
\(838\) 0 0
\(839\) −43.2541 −1.49330 −0.746648 0.665219i \(-0.768338\pi\)
−0.746648 + 0.665219i \(0.768338\pi\)
\(840\) 0 0
\(841\) 58.0296 2.00102
\(842\) 0 0
\(843\) −15.8135 −0.544645
\(844\) 0 0
\(845\) 3.52882 0.121395
\(846\) 0 0
\(847\) 3.24835 0.111615
\(848\) 0 0
\(849\) −11.2407 −0.385780
\(850\) 0 0
\(851\) 26.2704 0.900539
\(852\) 0 0
\(853\) −0.307251 −0.0105201 −0.00526003 0.999986i \(-0.501674\pi\)
−0.00526003 + 0.999986i \(0.501674\pi\)
\(854\) 0 0
\(855\) 8.64329 0.295594
\(856\) 0 0
\(857\) 49.7992 1.70111 0.850553 0.525889i \(-0.176267\pi\)
0.850553 + 0.525889i \(0.176267\pi\)
\(858\) 0 0
\(859\) −49.1125 −1.67570 −0.837849 0.545903i \(-0.816187\pi\)
−0.837849 + 0.545903i \(0.816187\pi\)
\(860\) 0 0
\(861\) −32.6745 −1.11354
\(862\) 0 0
\(863\) −27.7615 −0.945013 −0.472507 0.881327i \(-0.656651\pi\)
−0.472507 + 0.881327i \(0.656651\pi\)
\(864\) 0 0
\(865\) −43.1848 −1.46833
\(866\) 0 0
\(867\) 13.0818 0.444283
\(868\) 0 0
\(869\) −3.26150 −0.110639
\(870\) 0 0
\(871\) 12.2750 0.415923
\(872\) 0 0
\(873\) −5.04526 −0.170756
\(874\) 0 0
\(875\) 28.1135 0.950410
\(876\) 0 0
\(877\) −1.79857 −0.0607333 −0.0303666 0.999539i \(-0.509667\pi\)
−0.0303666 + 0.999539i \(0.509667\pi\)
\(878\) 0 0
\(879\) −16.4756 −0.555710
\(880\) 0 0
\(881\) 16.1505 0.544126 0.272063 0.962279i \(-0.412294\pi\)
0.272063 + 0.962279i \(0.412294\pi\)
\(882\) 0 0
\(883\) −27.0031 −0.908725 −0.454363 0.890817i \(-0.650133\pi\)
−0.454363 + 0.890817i \(0.650133\pi\)
\(884\) 0 0
\(885\) 18.9143 0.635797
\(886\) 0 0
\(887\) 49.9740 1.67796 0.838981 0.544161i \(-0.183152\pi\)
0.838981 + 0.544161i \(0.183152\pi\)
\(888\) 0 0
\(889\) 29.6085 0.993036
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 6.08497 0.203626
\(894\) 0 0
\(895\) 37.9520 1.26860
\(896\) 0 0
\(897\) −2.29362 −0.0765816
\(898\) 0 0
\(899\) −21.7163 −0.724278
\(900\) 0 0
\(901\) −74.8379 −2.49321
\(902\) 0 0
\(903\) 23.0368 0.766618
\(904\) 0 0
\(905\) −30.6475 −1.01876
\(906\) 0 0
\(907\) 50.6063 1.68036 0.840178 0.542311i \(-0.182450\pi\)
0.840178 + 0.542311i \(0.182450\pi\)
\(908\) 0 0
\(909\) 0.897457 0.0297668
\(910\) 0 0
\(911\) 46.7755 1.54974 0.774871 0.632120i \(-0.217815\pi\)
0.774871 + 0.632120i \(0.217815\pi\)
\(912\) 0 0
\(913\) 13.1482 0.435141
\(914\) 0 0
\(915\) −27.2075 −0.899451
\(916\) 0 0
\(917\) 22.3797 0.739041
\(918\) 0 0
\(919\) −34.6524 −1.14308 −0.571539 0.820575i \(-0.693653\pi\)
−0.571539 + 0.820575i \(0.693653\pi\)
\(920\) 0 0
\(921\) 30.0991 0.991800
\(922\) 0 0
\(923\) −1.76516 −0.0581009
\(924\) 0 0
\(925\) −85.3596 −2.80661
\(926\) 0 0
\(927\) −1.69769 −0.0557596
\(928\) 0 0
\(929\) −48.8020 −1.60114 −0.800570 0.599239i \(-0.795470\pi\)
−0.800570 + 0.599239i \(0.795470\pi\)
\(930\) 0 0
\(931\) 8.69956 0.285116
\(932\) 0 0
\(933\) 18.6929 0.611979
\(934\) 0 0
\(935\) −19.3545 −0.632960
\(936\) 0 0
\(937\) −7.99427 −0.261161 −0.130581 0.991438i \(-0.541684\pi\)
−0.130581 + 0.991438i \(0.541684\pi\)
\(938\) 0 0
\(939\) 7.73114 0.252296
\(940\) 0 0
\(941\) 58.2642 1.89936 0.949679 0.313225i \(-0.101409\pi\)
0.949679 + 0.313225i \(0.101409\pi\)
\(942\) 0 0
\(943\) 23.0710 0.751295
\(944\) 0 0
\(945\) 11.4629 0.372887
\(946\) 0 0
\(947\) −31.8587 −1.03527 −0.517634 0.855602i \(-0.673187\pi\)
−0.517634 + 0.855602i \(0.673187\pi\)
\(948\) 0 0
\(949\) 1.80929 0.0587320
\(950\) 0 0
\(951\) −24.2897 −0.787646
\(952\) 0 0
\(953\) 15.4081 0.499117 0.249558 0.968360i \(-0.419715\pi\)
0.249558 + 0.968360i \(0.419715\pi\)
\(954\) 0 0
\(955\) −52.4112 −1.69599
\(956\) 0 0
\(957\) 9.32897 0.301563
\(958\) 0 0
\(959\) −30.1531 −0.973693
\(960\) 0 0
\(961\) −25.5812 −0.825200
\(962\) 0 0
\(963\) −8.44389 −0.272100
\(964\) 0 0
\(965\) −26.4908 −0.852769
\(966\) 0 0
\(967\) 37.6657 1.21125 0.605624 0.795751i \(-0.292924\pi\)
0.605624 + 0.795751i \(0.292924\pi\)
\(968\) 0 0
\(969\) −13.4339 −0.431558
\(970\) 0 0
\(971\) 21.2714 0.682632 0.341316 0.939949i \(-0.389127\pi\)
0.341316 + 0.939949i \(0.389127\pi\)
\(972\) 0 0
\(973\) 53.1144 1.70277
\(974\) 0 0
\(975\) 7.45258 0.238673
\(976\) 0 0
\(977\) −40.9806 −1.31109 −0.655543 0.755158i \(-0.727560\pi\)
−0.655543 + 0.755158i \(0.727560\pi\)
\(978\) 0 0
\(979\) −2.06824 −0.0661012
\(980\) 0 0
\(981\) 5.30344 0.169326
\(982\) 0 0
\(983\) −21.0862 −0.672546 −0.336273 0.941765i \(-0.609166\pi\)
−0.336273 + 0.941765i \(0.609166\pi\)
\(984\) 0 0
\(985\) −11.2765 −0.359298
\(986\) 0 0
\(987\) 8.06997 0.256870
\(988\) 0 0
\(989\) −16.2660 −0.517229
\(990\) 0 0
\(991\) −33.8837 −1.07635 −0.538175 0.842833i \(-0.680886\pi\)
−0.538175 + 0.842833i \(0.680886\pi\)
\(992\) 0 0
\(993\) 2.62003 0.0831442
\(994\) 0 0
\(995\) −68.1785 −2.16140
\(996\) 0 0
\(997\) 30.9741 0.980959 0.490480 0.871453i \(-0.336822\pi\)
0.490480 + 0.871453i \(0.336822\pi\)
\(998\) 0 0
\(999\) −11.4537 −0.362379
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 6864.2.a.cf.1.5 5
4.3 odd 2 3432.2.a.w.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.w.1.5 5 4.3 odd 2
6864.2.a.cf.1.5 5 1.1 even 1 trivial