Properties

Label 6864.2.a.ca.1.2
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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: \(4\)
Coefficient field: 4.4.70164.1
Defining polynomial: \(x^{4} - x^{3} - 10 x^{2} + 10 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.279954\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.279954 q^{5} +4.36642 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.279954 q^{5} +4.36642 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.279954 q^{15} +2.64638 q^{17} -4.58412 q^{19} +4.36642 q^{21} -1.80651 q^{23} -4.92163 q^{25} +1.00000 q^{27} +3.30887 q^{29} +1.50235 q^{31} -1.00000 q^{33} +1.22240 q^{35} -0.559909 q^{37} -1.00000 q^{39} +6.36642 q^{41} +6.97918 q^{43} +0.279954 q^{45} -2.73285 q^{47} +12.0656 q^{49} +2.64638 q^{51} +2.99529 q^{53} -0.279954 q^{55} -4.58412 q^{57} +8.39063 q^{59} -1.10729 q^{61} +4.36642 q^{63} -0.279954 q^{65} +4.39507 q^{67} -1.80651 q^{69} +2.99529 q^{71} +13.6498 q^{73} -4.92163 q^{75} -4.36642 q^{77} +14.9344 q^{79} +1.00000 q^{81} +13.7281 q^{83} +0.740865 q^{85} +3.30887 q^{87} +2.49765 q^{89} -4.36642 q^{91} +1.50235 q^{93} -1.28334 q^{95} -16.0162 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + q^{5} + q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + q^{5} + q^{7} + 4q^{9} - 4q^{11} - 4q^{13} + q^{15} - 6q^{17} + q^{21} + 9q^{23} + q^{25} + 4q^{27} - q^{29} + 8q^{31} - 4q^{33} + 7q^{35} - 2q^{37} - 4q^{39} + 9q^{41} + 5q^{43} + q^{45} + 22q^{47} + 9q^{49} - 6q^{51} + 8q^{53} - q^{55} - q^{59} - 11q^{61} + q^{63} - q^{65} + 13q^{67} + 9q^{69} + 8q^{71} - 3q^{73} + q^{75} - q^{77} + 6q^{79} + 4q^{81} + 18q^{83} + 26q^{85} - q^{87} + 8q^{89} - q^{91} + 8q^{93} + 36q^{95} + 10q^{97} - 4q^{99} + O(q^{100}) \)

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
\(5\) 0.279954 0.125199 0.0625997 0.998039i \(-0.480061\pi\)
0.0625997 + 0.998039i \(0.480061\pi\)
\(6\) 0 0
\(7\) 4.36642 1.65035 0.825176 0.564875i \(-0.191076\pi\)
0.825176 + 0.564875i \(0.191076\pi\)
\(8\) 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
\(15\) 0.279954 0.0722839
\(16\) 0 0
\(17\) 2.64638 0.641841 0.320920 0.947106i \(-0.396008\pi\)
0.320920 + 0.947106i \(0.396008\pi\)
\(18\) 0 0
\(19\) −4.58412 −1.05167 −0.525834 0.850587i \(-0.676247\pi\)
−0.525834 + 0.850587i \(0.676247\pi\)
\(20\) 0 0
\(21\) 4.36642 0.952832
\(22\) 0 0
\(23\) −1.80651 −0.376684 −0.188342 0.982103i \(-0.560311\pi\)
−0.188342 + 0.982103i \(0.560311\pi\)
\(24\) 0 0
\(25\) −4.92163 −0.984325
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.30887 0.614441 0.307221 0.951638i \(-0.400601\pi\)
0.307221 + 0.951638i \(0.400601\pi\)
\(30\) 0 0
\(31\) 1.50235 0.269831 0.134915 0.990857i \(-0.456924\pi\)
0.134915 + 0.990857i \(0.456924\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 1.22240 0.206623
\(36\) 0 0
\(37\) −0.559909 −0.0920484 −0.0460242 0.998940i \(-0.514655\pi\)
−0.0460242 + 0.998940i \(0.514655\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 6.36642 0.994268 0.497134 0.867674i \(-0.334386\pi\)
0.497134 + 0.867674i \(0.334386\pi\)
\(42\) 0 0
\(43\) 6.97918 1.06432 0.532158 0.846645i \(-0.321381\pi\)
0.532158 + 0.846645i \(0.321381\pi\)
\(44\) 0 0
\(45\) 0.279954 0.0417331
\(46\) 0 0
\(47\) −2.73285 −0.398627 −0.199313 0.979936i \(-0.563871\pi\)
−0.199313 + 0.979936i \(0.563871\pi\)
\(48\) 0 0
\(49\) 12.0656 1.72366
\(50\) 0 0
\(51\) 2.64638 0.370567
\(52\) 0 0
\(53\) 2.99529 0.411435 0.205718 0.978611i \(-0.434047\pi\)
0.205718 + 0.978611i \(0.434047\pi\)
\(54\) 0 0
\(55\) −0.279954 −0.0377490
\(56\) 0 0
\(57\) −4.58412 −0.607181
\(58\) 0 0
\(59\) 8.39063 1.09237 0.546184 0.837666i \(-0.316080\pi\)
0.546184 + 0.837666i \(0.316080\pi\)
\(60\) 0 0
\(61\) −1.10729 −0.141774 −0.0708868 0.997484i \(-0.522583\pi\)
−0.0708868 + 0.997484i \(0.522583\pi\)
\(62\) 0 0
\(63\) 4.36642 0.550118
\(64\) 0 0
\(65\) −0.279954 −0.0347241
\(66\) 0 0
\(67\) 4.39507 0.536943 0.268471 0.963288i \(-0.413482\pi\)
0.268471 + 0.963288i \(0.413482\pi\)
\(68\) 0 0
\(69\) −1.80651 −0.217479
\(70\) 0 0
\(71\) 2.99529 0.355476 0.177738 0.984078i \(-0.443122\pi\)
0.177738 + 0.984078i \(0.443122\pi\)
\(72\) 0 0
\(73\) 13.6498 1.59758 0.798792 0.601607i \(-0.205473\pi\)
0.798792 + 0.601607i \(0.205473\pi\)
\(74\) 0 0
\(75\) −4.92163 −0.568300
\(76\) 0 0
\(77\) −4.36642 −0.497600
\(78\) 0 0
\(79\) 14.9344 1.68025 0.840127 0.542390i \(-0.182480\pi\)
0.840127 + 0.542390i \(0.182480\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.7281 1.50686 0.753430 0.657529i \(-0.228398\pi\)
0.753430 + 0.657529i \(0.228398\pi\)
\(84\) 0 0
\(85\) 0.740865 0.0803580
\(86\) 0 0
\(87\) 3.30887 0.354748
\(88\) 0 0
\(89\) 2.49765 0.264750 0.132375 0.991200i \(-0.457740\pi\)
0.132375 + 0.991200i \(0.457740\pi\)
\(90\) 0 0
\(91\) −4.36642 −0.457726
\(92\) 0 0
\(93\) 1.50235 0.155787
\(94\) 0 0
\(95\) −1.28334 −0.131668
\(96\) 0 0
\(97\) −16.0162 −1.62620 −0.813099 0.582126i \(-0.802221\pi\)
−0.813099 + 0.582126i \(0.802221\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −10.4896 −1.04376 −0.521879 0.853020i \(-0.674769\pi\)
−0.521879 + 0.853020i \(0.674769\pi\)
\(102\) 0 0
\(103\) 5.10729 0.503236 0.251618 0.967827i \(-0.419037\pi\)
0.251618 + 0.967827i \(0.419037\pi\)
\(104\) 0 0
\(105\) 1.22240 0.119294
\(106\) 0 0
\(107\) 10.8354 1.04750 0.523750 0.851872i \(-0.324533\pi\)
0.523750 + 0.851872i \(0.324533\pi\)
\(108\) 0 0
\(109\) −9.58882 −0.918443 −0.459221 0.888322i \(-0.651871\pi\)
−0.459221 + 0.888322i \(0.651871\pi\)
\(110\) 0 0
\(111\) −0.559909 −0.0531442
\(112\) 0 0
\(113\) −2.77760 −0.261295 −0.130647 0.991429i \(-0.541706\pi\)
−0.130647 + 0.991429i \(0.541706\pi\)
\(114\) 0 0
\(115\) −0.505741 −0.0471606
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 11.5552 1.05926
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.36642 0.574041
\(124\) 0 0
\(125\) −2.77760 −0.248436
\(126\) 0 0
\(127\) 7.08176 0.628405 0.314202 0.949356i \(-0.398263\pi\)
0.314202 + 0.949356i \(0.398263\pi\)
\(128\) 0 0
\(129\) 6.97918 0.614483
\(130\) 0 0
\(131\) 2.22710 0.194583 0.0972915 0.995256i \(-0.468982\pi\)
0.0972915 + 0.995256i \(0.468982\pi\)
\(132\) 0 0
\(133\) −20.0162 −1.73562
\(134\) 0 0
\(135\) 0.279954 0.0240946
\(136\) 0 0
\(137\) 18.6384 1.59238 0.796191 0.605045i \(-0.206845\pi\)
0.796191 + 0.605045i \(0.206845\pi\)
\(138\) 0 0
\(139\) −12.5313 −1.06289 −0.531444 0.847093i \(-0.678350\pi\)
−0.531444 + 0.847093i \(0.678350\pi\)
\(140\) 0 0
\(141\) −2.73285 −0.230147
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) 0.926332 0.0769276
\(146\) 0 0
\(147\) 12.0656 0.995158
\(148\) 0 0
\(149\) 7.70393 0.631131 0.315565 0.948904i \(-0.397806\pi\)
0.315565 + 0.948904i \(0.397806\pi\)
\(150\) 0 0
\(151\) −4.14873 −0.337619 −0.168809 0.985649i \(-0.553992\pi\)
−0.168809 + 0.985649i \(0.553992\pi\)
\(152\) 0 0
\(153\) 2.64638 0.213947
\(154\) 0 0
\(155\) 0.420590 0.0337826
\(156\) 0 0
\(157\) −2.59353 −0.206986 −0.103493 0.994630i \(-0.533002\pi\)
−0.103493 + 0.994630i \(0.533002\pi\)
\(158\) 0 0
\(159\) 2.99529 0.237542
\(160\) 0 0
\(161\) −7.88801 −0.621662
\(162\) 0 0
\(163\) −6.00809 −0.470590 −0.235295 0.971924i \(-0.575606\pi\)
−0.235295 + 0.971924i \(0.575606\pi\)
\(164\) 0 0
\(165\) −0.279954 −0.0217944
\(166\) 0 0
\(167\) 16.2177 1.25496 0.627481 0.778632i \(-0.284086\pi\)
0.627481 + 0.778632i \(0.284086\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −4.58412 −0.350556
\(172\) 0 0
\(173\) 3.42398 0.260320 0.130160 0.991493i \(-0.458451\pi\)
0.130160 + 0.991493i \(0.458451\pi\)
\(174\) 0 0
\(175\) −21.4899 −1.62448
\(176\) 0 0
\(177\) 8.39063 0.630678
\(178\) 0 0
\(179\) −16.0498 −1.19962 −0.599809 0.800143i \(-0.704757\pi\)
−0.599809 + 0.800143i \(0.704757\pi\)
\(180\) 0 0
\(181\) 1.70393 0.126652 0.0633262 0.997993i \(-0.479829\pi\)
0.0633262 + 0.997993i \(0.479829\pi\)
\(182\) 0 0
\(183\) −1.10729 −0.0818531
\(184\) 0 0
\(185\) −0.156749 −0.0115244
\(186\) 0 0
\(187\) −2.64638 −0.193522
\(188\) 0 0
\(189\) 4.36642 0.317611
\(190\) 0 0
\(191\) 19.5683 1.41591 0.707955 0.706257i \(-0.249618\pi\)
0.707955 + 0.706257i \(0.249618\pi\)
\(192\) 0 0
\(193\) −26.8305 −1.93130 −0.965652 0.259840i \(-0.916330\pi\)
−0.965652 + 0.259840i \(0.916330\pi\)
\(194\) 0 0
\(195\) −0.279954 −0.0200479
\(196\) 0 0
\(197\) −19.6466 −1.39977 −0.699883 0.714258i \(-0.746764\pi\)
−0.699883 + 0.714258i \(0.746764\pi\)
\(198\) 0 0
\(199\) 7.55208 0.535353 0.267677 0.963509i \(-0.413744\pi\)
0.267677 + 0.963509i \(0.413744\pi\)
\(200\) 0 0
\(201\) 4.39507 0.310004
\(202\) 0 0
\(203\) 14.4479 1.01404
\(204\) 0 0
\(205\) 1.78231 0.124482
\(206\) 0 0
\(207\) −1.80651 −0.125561
\(208\) 0 0
\(209\) 4.58412 0.317090
\(210\) 0 0
\(211\) −17.4849 −1.20371 −0.601856 0.798605i \(-0.705572\pi\)
−0.601856 + 0.798605i \(0.705572\pi\)
\(212\) 0 0
\(213\) 2.99529 0.205234
\(214\) 0 0
\(215\) 1.95385 0.133252
\(216\) 0 0
\(217\) 6.55991 0.445316
\(218\) 0 0
\(219\) 13.6498 0.922366
\(220\) 0 0
\(221\) −2.64638 −0.178015
\(222\) 0 0
\(223\) 7.38724 0.494686 0.247343 0.968928i \(-0.420443\pi\)
0.247343 + 0.968928i \(0.420443\pi\)
\(224\) 0 0
\(225\) −4.92163 −0.328108
\(226\) 0 0
\(227\) −7.02891 −0.466525 −0.233263 0.972414i \(-0.574940\pi\)
−0.233263 + 0.972414i \(0.574940\pi\)
\(228\) 0 0
\(229\) −13.6834 −0.904224 −0.452112 0.891961i \(-0.649329\pi\)
−0.452112 + 0.891961i \(0.649329\pi\)
\(230\) 0 0
\(231\) −4.36642 −0.287290
\(232\) 0 0
\(233\) −16.6464 −1.09054 −0.545270 0.838260i \(-0.683573\pi\)
−0.545270 + 0.838260i \(0.683573\pi\)
\(234\) 0 0
\(235\) −0.765072 −0.0499078
\(236\) 0 0
\(237\) 14.9344 0.970095
\(238\) 0 0
\(239\) 11.9795 0.774886 0.387443 0.921894i \(-0.373358\pi\)
0.387443 + 0.921894i \(0.373358\pi\)
\(240\) 0 0
\(241\) 17.5808 1.13248 0.566240 0.824241i \(-0.308398\pi\)
0.566240 + 0.824241i \(0.308398\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.37783 0.215802
\(246\) 0 0
\(247\) 4.58412 0.291680
\(248\) 0 0
\(249\) 13.7281 0.869986
\(250\) 0 0
\(251\) 18.8480 1.18967 0.594836 0.803847i \(-0.297217\pi\)
0.594836 + 0.803847i \(0.297217\pi\)
\(252\) 0 0
\(253\) 1.80651 0.113575
\(254\) 0 0
\(255\) 0.740865 0.0463947
\(256\) 0 0
\(257\) −17.3953 −1.08509 −0.542546 0.840026i \(-0.682539\pi\)
−0.542546 + 0.840026i \(0.682539\pi\)
\(258\) 0 0
\(259\) −2.44480 −0.151912
\(260\) 0 0
\(261\) 3.30887 0.204814
\(262\) 0 0
\(263\) 10.1151 0.623724 0.311862 0.950127i \(-0.399047\pi\)
0.311862 + 0.950127i \(0.399047\pi\)
\(264\) 0 0
\(265\) 0.838545 0.0515114
\(266\) 0 0
\(267\) 2.49765 0.152854
\(268\) 0 0
\(269\) −10.8385 −0.660838 −0.330419 0.943834i \(-0.607190\pi\)
−0.330419 + 0.943834i \(0.607190\pi\)
\(270\) 0 0
\(271\) −2.03362 −0.123534 −0.0617668 0.998091i \(-0.519674\pi\)
−0.0617668 + 0.998091i \(0.519674\pi\)
\(272\) 0 0
\(273\) −4.36642 −0.264268
\(274\) 0 0
\(275\) 4.92163 0.296785
\(276\) 0 0
\(277\) −1.55208 −0.0932557 −0.0466279 0.998912i \(-0.514847\pi\)
−0.0466279 + 0.998912i \(0.514847\pi\)
\(278\) 0 0
\(279\) 1.50235 0.0899435
\(280\) 0 0
\(281\) 10.9747 0.654698 0.327349 0.944903i \(-0.393845\pi\)
0.327349 + 0.944903i \(0.393845\pi\)
\(282\) 0 0
\(283\) −23.1615 −1.37681 −0.688405 0.725326i \(-0.741689\pi\)
−0.688405 + 0.725326i \(0.741689\pi\)
\(284\) 0 0
\(285\) −1.28334 −0.0760187
\(286\) 0 0
\(287\) 27.7985 1.64089
\(288\) 0 0
\(289\) −9.99669 −0.588040
\(290\) 0 0
\(291\) −16.0162 −0.938886
\(292\) 0 0
\(293\) 20.6097 1.20403 0.602016 0.798484i \(-0.294364\pi\)
0.602016 + 0.798484i \(0.294364\pi\)
\(294\) 0 0
\(295\) 2.34899 0.136764
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 1.80651 0.104473
\(300\) 0 0
\(301\) 30.4741 1.75650
\(302\) 0 0
\(303\) −10.4896 −0.602613
\(304\) 0 0
\(305\) −0.309990 −0.0177500
\(306\) 0 0
\(307\) −17.1118 −0.976622 −0.488311 0.872670i \(-0.662387\pi\)
−0.488311 + 0.872670i \(0.662387\pi\)
\(308\) 0 0
\(309\) 5.10729 0.290543
\(310\) 0 0
\(311\) 18.9873 1.07667 0.538335 0.842731i \(-0.319054\pi\)
0.538335 + 0.842731i \(0.319054\pi\)
\(312\) 0 0
\(313\) −21.2050 −1.19858 −0.599288 0.800534i \(-0.704549\pi\)
−0.599288 + 0.800534i \(0.704549\pi\)
\(314\) 0 0
\(315\) 1.22240 0.0688744
\(316\) 0 0
\(317\) −25.4738 −1.43075 −0.715375 0.698741i \(-0.753744\pi\)
−0.715375 + 0.698741i \(0.753744\pi\)
\(318\) 0 0
\(319\) −3.30887 −0.185261
\(320\) 0 0
\(321\) 10.8354 0.604775
\(322\) 0 0
\(323\) −12.1313 −0.675004
\(324\) 0 0
\(325\) 4.92163 0.273003
\(326\) 0 0
\(327\) −9.58882 −0.530263
\(328\) 0 0
\(329\) −11.9328 −0.657874
\(330\) 0 0
\(331\) −3.03561 −0.166852 −0.0834262 0.996514i \(-0.526586\pi\)
−0.0834262 + 0.996514i \(0.526586\pi\)
\(332\) 0 0
\(333\) −0.559909 −0.0306828
\(334\) 0 0
\(335\) 1.23042 0.0672249
\(336\) 0 0
\(337\) −5.58080 −0.304006 −0.152003 0.988380i \(-0.548572\pi\)
−0.152003 + 0.988380i \(0.548572\pi\)
\(338\) 0 0
\(339\) −2.77760 −0.150859
\(340\) 0 0
\(341\) −1.50235 −0.0813570
\(342\) 0 0
\(343\) 22.1188 1.19430
\(344\) 0 0
\(345\) −0.505741 −0.0272282
\(346\) 0 0
\(347\) 9.78597 0.525338 0.262669 0.964886i \(-0.415397\pi\)
0.262669 + 0.964886i \(0.415397\pi\)
\(348\) 0 0
\(349\) 8.55050 0.457698 0.228849 0.973462i \(-0.426504\pi\)
0.228849 + 0.973462i \(0.426504\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −30.2608 −1.61062 −0.805310 0.592854i \(-0.798001\pi\)
−0.805310 + 0.592854i \(0.798001\pi\)
\(354\) 0 0
\(355\) 0.838545 0.0445054
\(356\) 0 0
\(357\) 11.5552 0.611566
\(358\) 0 0
\(359\) −24.5555 −1.29599 −0.647996 0.761644i \(-0.724393\pi\)
−0.647996 + 0.761644i \(0.724393\pi\)
\(360\) 0 0
\(361\) 2.01412 0.106006
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 3.82131 0.200017
\(366\) 0 0
\(367\) 28.7006 1.49816 0.749080 0.662479i \(-0.230496\pi\)
0.749080 + 0.662479i \(0.230496\pi\)
\(368\) 0 0
\(369\) 6.36642 0.331423
\(370\) 0 0
\(371\) 13.0787 0.679013
\(372\) 0 0
\(373\) 4.63026 0.239746 0.119873 0.992789i \(-0.461751\pi\)
0.119873 + 0.992789i \(0.461751\pi\)
\(374\) 0 0
\(375\) −2.77760 −0.143435
\(376\) 0 0
\(377\) −3.30887 −0.170415
\(378\) 0 0
\(379\) −25.5091 −1.31032 −0.655158 0.755492i \(-0.727398\pi\)
−0.655158 + 0.755492i \(0.727398\pi\)
\(380\) 0 0
\(381\) 7.08176 0.362810
\(382\) 0 0
\(383\) −9.95836 −0.508849 −0.254424 0.967093i \(-0.581886\pi\)
−0.254424 + 0.967093i \(0.581886\pi\)
\(384\) 0 0
\(385\) −1.22240 −0.0622992
\(386\) 0 0
\(387\) 6.97918 0.354772
\(388\) 0 0
\(389\) −7.34117 −0.372212 −0.186106 0.982530i \(-0.559587\pi\)
−0.186106 + 0.982530i \(0.559587\pi\)
\(390\) 0 0
\(391\) −4.78072 −0.241771
\(392\) 0 0
\(393\) 2.22710 0.112343
\(394\) 0 0
\(395\) 4.18096 0.210367
\(396\) 0 0
\(397\) 14.8354 0.744569 0.372284 0.928119i \(-0.378575\pi\)
0.372284 + 0.928119i \(0.378575\pi\)
\(398\) 0 0
\(399\) −20.0162 −1.00206
\(400\) 0 0
\(401\) 23.3940 1.16824 0.584121 0.811667i \(-0.301439\pi\)
0.584121 + 0.811667i \(0.301439\pi\)
\(402\) 0 0
\(403\) −1.50235 −0.0748375
\(404\) 0 0
\(405\) 0.279954 0.0139110
\(406\) 0 0
\(407\) 0.559909 0.0277536
\(408\) 0 0
\(409\) 25.8549 1.27844 0.639222 0.769022i \(-0.279257\pi\)
0.639222 + 0.769022i \(0.279257\pi\)
\(410\) 0 0
\(411\) 18.6384 0.919362
\(412\) 0 0
\(413\) 36.6370 1.80279
\(414\) 0 0
\(415\) 3.84325 0.188658
\(416\) 0 0
\(417\) −12.5313 −0.613659
\(418\) 0 0
\(419\) 21.1024 1.03092 0.515460 0.856914i \(-0.327621\pi\)
0.515460 + 0.856914i \(0.327621\pi\)
\(420\) 0 0
\(421\) −12.3678 −0.602770 −0.301385 0.953502i \(-0.597449\pi\)
−0.301385 + 0.953502i \(0.597449\pi\)
\(422\) 0 0
\(423\) −2.73285 −0.132876
\(424\) 0 0
\(425\) −13.0245 −0.631780
\(426\) 0 0
\(427\) −4.83489 −0.233977
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 34.2880 1.65160 0.825799 0.563965i \(-0.190725\pi\)
0.825799 + 0.563965i \(0.190725\pi\)
\(432\) 0 0
\(433\) 9.08184 0.436445 0.218223 0.975899i \(-0.429974\pi\)
0.218223 + 0.975899i \(0.429974\pi\)
\(434\) 0 0
\(435\) 0.926332 0.0444142
\(436\) 0 0
\(437\) 8.28127 0.396147
\(438\) 0 0
\(439\) −33.1330 −1.58135 −0.790675 0.612236i \(-0.790270\pi\)
−0.790675 + 0.612236i \(0.790270\pi\)
\(440\) 0 0
\(441\) 12.0656 0.574555
\(442\) 0 0
\(443\) 5.11982 0.243250 0.121625 0.992576i \(-0.461189\pi\)
0.121625 + 0.992576i \(0.461189\pi\)
\(444\) 0 0
\(445\) 0.699227 0.0331465
\(446\) 0 0
\(447\) 7.70393 0.364384
\(448\) 0 0
\(449\) 15.8320 0.747160 0.373580 0.927598i \(-0.378130\pi\)
0.373580 + 0.927598i \(0.378130\pi\)
\(450\) 0 0
\(451\) −6.36642 −0.299783
\(452\) 0 0
\(453\) −4.14873 −0.194924
\(454\) 0 0
\(455\) −1.22240 −0.0573069
\(456\) 0 0
\(457\) 6.01253 0.281254 0.140627 0.990063i \(-0.455088\pi\)
0.140627 + 0.990063i \(0.455088\pi\)
\(458\) 0 0
\(459\) 2.64638 0.123522
\(460\) 0 0
\(461\) −21.3998 −0.996690 −0.498345 0.866979i \(-0.666059\pi\)
−0.498345 + 0.866979i \(0.666059\pi\)
\(462\) 0 0
\(463\) 38.2420 1.77726 0.888628 0.458629i \(-0.151659\pi\)
0.888628 + 0.458629i \(0.151659\pi\)
\(464\) 0 0
\(465\) 0.420590 0.0195044
\(466\) 0 0
\(467\) 13.8190 0.639469 0.319735 0.947507i \(-0.396406\pi\)
0.319735 + 0.947507i \(0.396406\pi\)
\(468\) 0 0
\(469\) 19.1907 0.886145
\(470\) 0 0
\(471\) −2.59353 −0.119503
\(472\) 0 0
\(473\) −6.97918 −0.320903
\(474\) 0 0
\(475\) 22.5613 1.03518
\(476\) 0 0
\(477\) 2.99529 0.137145
\(478\) 0 0
\(479\) −35.4546 −1.61996 −0.809980 0.586457i \(-0.800522\pi\)
−0.809980 + 0.586457i \(0.800522\pi\)
\(480\) 0 0
\(481\) 0.559909 0.0255296
\(482\) 0 0
\(483\) −7.88801 −0.358917
\(484\) 0 0
\(485\) −4.48380 −0.203599
\(486\) 0 0
\(487\) −29.6175 −1.34210 −0.671048 0.741414i \(-0.734155\pi\)
−0.671048 + 0.741414i \(0.734155\pi\)
\(488\) 0 0
\(489\) −6.00809 −0.271695
\(490\) 0 0
\(491\) 20.5892 0.929176 0.464588 0.885527i \(-0.346202\pi\)
0.464588 + 0.885527i \(0.346202\pi\)
\(492\) 0 0
\(493\) 8.75651 0.394373
\(494\) 0 0
\(495\) −0.279954 −0.0125830
\(496\) 0 0
\(497\) 13.0787 0.586661
\(498\) 0 0
\(499\) −1.45760 −0.0652510 −0.0326255 0.999468i \(-0.510387\pi\)
−0.0326255 + 0.999468i \(0.510387\pi\)
\(500\) 0 0
\(501\) 16.2177 0.724553
\(502\) 0 0
\(503\) −8.10570 −0.361415 −0.180708 0.983537i \(-0.557839\pi\)
−0.180708 + 0.983537i \(0.557839\pi\)
\(504\) 0 0
\(505\) −2.93662 −0.130678
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −31.2145 −1.38356 −0.691778 0.722110i \(-0.743172\pi\)
−0.691778 + 0.722110i \(0.743172\pi\)
\(510\) 0 0
\(511\) 59.6007 2.63658
\(512\) 0 0
\(513\) −4.58412 −0.202394
\(514\) 0 0
\(515\) 1.42981 0.0630048
\(516\) 0 0
\(517\) 2.73285 0.120190
\(518\) 0 0
\(519\) 3.42398 0.150296
\(520\) 0 0
\(521\) −35.9620 −1.57552 −0.787762 0.615979i \(-0.788761\pi\)
−0.787762 + 0.615979i \(0.788761\pi\)
\(522\) 0 0
\(523\) 17.6579 0.772124 0.386062 0.922473i \(-0.373835\pi\)
0.386062 + 0.922473i \(0.373835\pi\)
\(524\) 0 0
\(525\) −21.4899 −0.937896
\(526\) 0 0
\(527\) 3.97579 0.173188
\(528\) 0 0
\(529\) −19.7365 −0.858109
\(530\) 0 0
\(531\) 8.39063 0.364122
\(532\) 0 0
\(533\) −6.36642 −0.275760
\(534\) 0 0
\(535\) 3.03342 0.131146
\(536\) 0 0
\(537\) −16.0498 −0.692600
\(538\) 0 0
\(539\) −12.0656 −0.519704
\(540\) 0 0
\(541\) −23.6771 −1.01796 −0.508979 0.860779i \(-0.669977\pi\)
−0.508979 + 0.860779i \(0.669977\pi\)
\(542\) 0 0
\(543\) 1.70393 0.0731228
\(544\) 0 0
\(545\) −2.68443 −0.114988
\(546\) 0 0
\(547\) 3.24217 0.138625 0.0693126 0.997595i \(-0.477919\pi\)
0.0693126 + 0.997595i \(0.477919\pi\)
\(548\) 0 0
\(549\) −1.10729 −0.0472579
\(550\) 0 0
\(551\) −15.1682 −0.646188
\(552\) 0 0
\(553\) 65.2100 2.77301
\(554\) 0 0
\(555\) −0.156749 −0.00665361
\(556\) 0 0
\(557\) 38.9550 1.65058 0.825289 0.564710i \(-0.191012\pi\)
0.825289 + 0.564710i \(0.191012\pi\)
\(558\) 0 0
\(559\) −6.97918 −0.295188
\(560\) 0 0
\(561\) −2.64638 −0.111730
\(562\) 0 0
\(563\) 0.246259 0.0103786 0.00518930 0.999987i \(-0.498348\pi\)
0.00518930 + 0.999987i \(0.498348\pi\)
\(564\) 0 0
\(565\) −0.777601 −0.0327139
\(566\) 0 0
\(567\) 4.36642 0.183373
\(568\) 0 0
\(569\) 3.13905 0.131596 0.0657979 0.997833i \(-0.479041\pi\)
0.0657979 + 0.997833i \(0.479041\pi\)
\(570\) 0 0
\(571\) −44.8318 −1.87615 −0.938077 0.346426i \(-0.887395\pi\)
−0.938077 + 0.346426i \(0.887395\pi\)
\(572\) 0 0
\(573\) 19.5683 0.817476
\(574\) 0 0
\(575\) 8.89099 0.370780
\(576\) 0 0
\(577\) −23.1933 −0.965549 −0.482775 0.875745i \(-0.660371\pi\)
−0.482775 + 0.875745i \(0.660371\pi\)
\(578\) 0 0
\(579\) −26.8305 −1.11504
\(580\) 0 0
\(581\) 59.9429 2.48685
\(582\) 0 0
\(583\) −2.99529 −0.124052
\(584\) 0 0
\(585\) −0.279954 −0.0115747
\(586\) 0 0
\(587\) −3.27081 −0.135001 −0.0675005 0.997719i \(-0.521502\pi\)
−0.0675005 + 0.997719i \(0.521502\pi\)
\(588\) 0 0
\(589\) −6.88696 −0.283772
\(590\) 0 0
\(591\) −19.6466 −0.808155
\(592\) 0 0
\(593\) −4.85598 −0.199411 −0.0997055 0.995017i \(-0.531790\pi\)
−0.0997055 + 0.995017i \(0.531790\pi\)
\(594\) 0 0
\(595\) 3.23493 0.132619
\(596\) 0 0
\(597\) 7.55208 0.309086
\(598\) 0 0
\(599\) −10.9089 −0.445726 −0.222863 0.974850i \(-0.571540\pi\)
−0.222863 + 0.974850i \(0.571540\pi\)
\(600\) 0 0
\(601\) 23.9011 0.974945 0.487473 0.873138i \(-0.337919\pi\)
0.487473 + 0.873138i \(0.337919\pi\)
\(602\) 0 0
\(603\) 4.39507 0.178981
\(604\) 0 0
\(605\) 0.279954 0.0113818
\(606\) 0 0
\(607\) −7.13905 −0.289765 −0.144883 0.989449i \(-0.546280\pi\)
−0.144883 + 0.989449i \(0.546280\pi\)
\(608\) 0 0
\(609\) 14.4479 0.585459
\(610\) 0 0
\(611\) 2.73285 0.110559
\(612\) 0 0
\(613\) 35.6306 1.43911 0.719553 0.694437i \(-0.244347\pi\)
0.719553 + 0.694437i \(0.244347\pi\)
\(614\) 0 0
\(615\) 1.78231 0.0718696
\(616\) 0 0
\(617\) 24.6133 0.990894 0.495447 0.868638i \(-0.335004\pi\)
0.495447 + 0.868638i \(0.335004\pi\)
\(618\) 0 0
\(619\) −17.6300 −0.708609 −0.354305 0.935130i \(-0.615282\pi\)
−0.354305 + 0.935130i \(0.615282\pi\)
\(620\) 0 0
\(621\) −1.80651 −0.0724929
\(622\) 0 0
\(623\) 10.9058 0.436931
\(624\) 0 0
\(625\) 23.8305 0.953221
\(626\) 0 0
\(627\) 4.58412 0.183072
\(628\) 0 0
\(629\) −1.48173 −0.0590804
\(630\) 0 0
\(631\) 20.8919 0.831695 0.415847 0.909434i \(-0.363485\pi\)
0.415847 + 0.909434i \(0.363485\pi\)
\(632\) 0 0
\(633\) −17.4849 −0.694963
\(634\) 0 0
\(635\) 1.98257 0.0786759
\(636\) 0 0
\(637\) −12.0656 −0.478058
\(638\) 0 0
\(639\) 2.99529 0.118492
\(640\) 0 0
\(641\) 11.4594 0.452619 0.226310 0.974055i \(-0.427334\pi\)
0.226310 + 0.974055i \(0.427334\pi\)
\(642\) 0 0
\(643\) 18.8275 0.742484 0.371242 0.928536i \(-0.378932\pi\)
0.371242 + 0.928536i \(0.378932\pi\)
\(644\) 0 0
\(645\) 1.95385 0.0769328
\(646\) 0 0
\(647\) 17.2739 0.679108 0.339554 0.940587i \(-0.389724\pi\)
0.339554 + 0.940587i \(0.389724\pi\)
\(648\) 0 0
\(649\) −8.39063 −0.329361
\(650\) 0 0
\(651\) 6.55991 0.257103
\(652\) 0 0
\(653\) 14.0512 0.549866 0.274933 0.961463i \(-0.411344\pi\)
0.274933 + 0.961463i \(0.411344\pi\)
\(654\) 0 0
\(655\) 0.623487 0.0243617
\(656\) 0 0
\(657\) 13.6498 0.532528
\(658\) 0 0
\(659\) −20.5120 −0.799035 −0.399518 0.916725i \(-0.630822\pi\)
−0.399518 + 0.916725i \(0.630822\pi\)
\(660\) 0 0
\(661\) 19.5230 0.759356 0.379678 0.925119i \(-0.376035\pi\)
0.379678 + 0.925119i \(0.376035\pi\)
\(662\) 0 0
\(663\) −2.64638 −0.102777
\(664\) 0 0
\(665\) −5.60362 −0.217299
\(666\) 0 0
\(667\) −5.97752 −0.231450
\(668\) 0 0
\(669\) 7.38724 0.285607
\(670\) 0 0
\(671\) 1.10729 0.0427464
\(672\) 0 0
\(673\) −1.65467 −0.0637827 −0.0318914 0.999491i \(-0.510153\pi\)
−0.0318914 + 0.999491i \(0.510153\pi\)
\(674\) 0 0
\(675\) −4.92163 −0.189433
\(676\) 0 0
\(677\) −26.6209 −1.02313 −0.511563 0.859246i \(-0.670933\pi\)
−0.511563 + 0.859246i \(0.670933\pi\)
\(678\) 0 0
\(679\) −69.9335 −2.68380
\(680\) 0 0
\(681\) −7.02891 −0.269349
\(682\) 0 0
\(683\) −23.8563 −0.912837 −0.456418 0.889765i \(-0.650868\pi\)
−0.456418 + 0.889765i \(0.650868\pi\)
\(684\) 0 0
\(685\) 5.21789 0.199365
\(686\) 0 0
\(687\) −13.6834 −0.522054
\(688\) 0 0
\(689\) −2.99529 −0.114112
\(690\) 0 0
\(691\) −16.5811 −0.630774 −0.315387 0.948963i \(-0.602134\pi\)
−0.315387 + 0.948963i \(0.602134\pi\)
\(692\) 0 0
\(693\) −4.36642 −0.165867
\(694\) 0 0
\(695\) −3.50818 −0.133073
\(696\) 0 0
\(697\) 16.8480 0.638162
\(698\) 0 0
\(699\) −16.6464 −0.629624
\(700\) 0 0
\(701\) 43.8622 1.65665 0.828325 0.560247i \(-0.189294\pi\)
0.828325 + 0.560247i \(0.189294\pi\)
\(702\) 0 0
\(703\) 2.56669 0.0968044
\(704\) 0 0
\(705\) −0.765072 −0.0288143
\(706\) 0 0
\(707\) −45.8022 −1.72257
\(708\) 0 0
\(709\) −17.1557 −0.644296 −0.322148 0.946689i \(-0.604405\pi\)
−0.322148 + 0.946689i \(0.604405\pi\)
\(710\) 0 0
\(711\) 14.9344 0.560085
\(712\) 0 0
\(713\) −2.71402 −0.101641
\(714\) 0 0
\(715\) 0.279954 0.0104697
\(716\) 0 0
\(717\) 11.9795 0.447381
\(718\) 0 0
\(719\) −16.4256 −0.612573 −0.306287 0.951939i \(-0.599087\pi\)
−0.306287 + 0.951939i \(0.599087\pi\)
\(720\) 0 0
\(721\) 22.3006 0.830517
\(722\) 0 0
\(723\) 17.5808 0.653837
\(724\) 0 0
\(725\) −16.2850 −0.604810
\(726\) 0 0
\(727\) 8.65221 0.320893 0.160446 0.987045i \(-0.448707\pi\)
0.160446 + 0.987045i \(0.448707\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 18.4695 0.683121
\(732\) 0 0
\(733\) −21.9758 −0.811694 −0.405847 0.913941i \(-0.633023\pi\)
−0.405847 + 0.913941i \(0.633023\pi\)
\(734\) 0 0
\(735\) 3.37783 0.124593
\(736\) 0 0
\(737\) −4.39507 −0.161894
\(738\) 0 0
\(739\) −19.9857 −0.735188 −0.367594 0.929986i \(-0.619818\pi\)
−0.367594 + 0.929986i \(0.619818\pi\)
\(740\) 0 0
\(741\) 4.58412 0.168402
\(742\) 0 0
\(743\) −25.4021 −0.931913 −0.465957 0.884808i \(-0.654290\pi\)
−0.465957 + 0.884808i \(0.654290\pi\)
\(744\) 0 0
\(745\) 2.15675 0.0790172
\(746\) 0 0
\(747\) 13.7281 0.502286
\(748\) 0 0
\(749\) 47.3121 1.72875
\(750\) 0 0
\(751\) −17.2158 −0.628212 −0.314106 0.949388i \(-0.601705\pi\)
−0.314106 + 0.949388i \(0.601705\pi\)
\(752\) 0 0
\(753\) 18.8480 0.686858
\(754\) 0 0
\(755\) −1.16145 −0.0422697
\(756\) 0 0
\(757\) 32.1076 1.16697 0.583486 0.812123i \(-0.301688\pi\)
0.583486 + 0.812123i \(0.301688\pi\)
\(758\) 0 0
\(759\) 1.80651 0.0655723
\(760\) 0 0
\(761\) −12.2353 −0.443528 −0.221764 0.975100i \(-0.571182\pi\)
−0.221764 + 0.975100i \(0.571182\pi\)
\(762\) 0 0
\(763\) −41.8689 −1.51575
\(764\) 0 0
\(765\) 0.740865 0.0267860
\(766\) 0 0
\(767\) −8.39063 −0.302968
\(768\) 0 0
\(769\) −28.1027 −1.01341 −0.506705 0.862119i \(-0.669137\pi\)
−0.506705 + 0.862119i \(0.669137\pi\)
\(770\) 0 0
\(771\) −17.3953 −0.626478
\(772\) 0 0
\(773\) 46.8686 1.68575 0.842873 0.538113i \(-0.180863\pi\)
0.842873 + 0.538113i \(0.180863\pi\)
\(774\) 0 0
\(775\) −7.39402 −0.265601
\(776\) 0 0
\(777\) −2.44480 −0.0877066
\(778\) 0 0
\(779\) −29.1844 −1.04564
\(780\) 0 0
\(781\) −2.99529 −0.107180
\(782\) 0 0
\(783\) 3.30887 0.118249
\(784\) 0 0
\(785\) −0.726069 −0.0259145
\(786\) 0 0
\(787\) 41.9415 1.49505 0.747526 0.664233i \(-0.231242\pi\)
0.747526 + 0.664233i \(0.231242\pi\)
\(788\) 0 0
\(789\) 10.1151 0.360107
\(790\) 0 0
\(791\) −12.1282 −0.431229
\(792\) 0 0
\(793\) 1.10729 0.0393209
\(794\) 0 0
\(795\) 0.838545 0.0297401
\(796\) 0 0
\(797\) 9.31836 0.330073 0.165037 0.986287i \(-0.447226\pi\)
0.165037 + 0.986287i \(0.447226\pi\)
\(798\) 0 0
\(799\) −7.23214 −0.255855
\(800\) 0 0
\(801\) 2.49765 0.0882500
\(802\) 0 0
\(803\) −13.6498 −0.481690
\(804\) 0 0
\(805\) −2.20828 −0.0778317
\(806\) 0 0
\(807\) −10.8385 −0.381535
\(808\) 0 0
\(809\) −26.8839 −0.945188 −0.472594 0.881280i \(-0.656682\pi\)
−0.472594 + 0.881280i \(0.656682\pi\)
\(810\) 0 0
\(811\) −21.9763 −0.771693 −0.385847 0.922563i \(-0.626091\pi\)
−0.385847 + 0.922563i \(0.626091\pi\)
\(812\) 0 0
\(813\) −2.03362 −0.0713221
\(814\) 0 0
\(815\) −1.68199 −0.0589176
\(816\) 0 0
\(817\) −31.9934 −1.11931
\(818\) 0 0
\(819\) −4.36642 −0.152575
\(820\) 0 0
\(821\) −15.7362 −0.549196 −0.274598 0.961559i \(-0.588545\pi\)
−0.274598 + 0.961559i \(0.588545\pi\)
\(822\) 0 0
\(823\) −24.0932 −0.839835 −0.419918 0.907562i \(-0.637941\pi\)
−0.419918 + 0.907562i \(0.637941\pi\)
\(824\) 0 0
\(825\) 4.92163 0.171349
\(826\) 0 0
\(827\) 0.491819 0.0171022 0.00855111 0.999963i \(-0.497278\pi\)
0.00855111 + 0.999963i \(0.497278\pi\)
\(828\) 0 0
\(829\) −5.35945 −0.186141 −0.0930707 0.995660i \(-0.529668\pi\)
−0.0930707 + 0.995660i \(0.529668\pi\)
\(830\) 0 0
\(831\) −1.55208 −0.0538412
\(832\) 0 0
\(833\) 31.9303 1.10632
\(834\) 0 0
\(835\) 4.54021 0.157121
\(836\) 0 0
\(837\) 1.50235 0.0519289
\(838\) 0 0
\(839\) 20.9442 0.723076 0.361538 0.932357i \(-0.382252\pi\)
0.361538 + 0.932357i \(0.382252\pi\)
\(840\) 0 0
\(841\) −18.0514 −0.622462
\(842\) 0 0
\(843\) 10.9747 0.377990
\(844\) 0 0
\(845\) 0.279954 0.00963072
\(846\) 0 0
\(847\) 4.36642 0.150032
\(848\) 0 0
\(849\) −23.1615 −0.794902
\(850\) 0 0
\(851\) 1.01148 0.0346732
\(852\) 0 0
\(853\) −52.4288 −1.79513 −0.897563 0.440886i \(-0.854664\pi\)
−0.897563 + 0.440886i \(0.854664\pi\)
\(854\) 0 0
\(855\) −1.28334 −0.0438894
\(856\) 0 0
\(857\) −27.5616 −0.941485 −0.470743 0.882271i \(-0.656014\pi\)
−0.470743 + 0.882271i \(0.656014\pi\)
\(858\) 0 0
\(859\) −14.6428 −0.499606 −0.249803 0.968297i \(-0.580366\pi\)
−0.249803 + 0.968297i \(0.580366\pi\)
\(860\) 0 0
\(861\) 27.7985 0.947370
\(862\) 0 0
\(863\) 54.7396 1.86336 0.931679 0.363282i \(-0.118344\pi\)
0.931679 + 0.363282i \(0.118344\pi\)
\(864\) 0 0
\(865\) 0.958557 0.0325919
\(866\) 0 0
\(867\) −9.99669 −0.339505
\(868\) 0 0
\(869\) −14.9344 −0.506616
\(870\) 0 0
\(871\) −4.39507 −0.148921
\(872\) 0 0
\(873\) −16.0162 −0.542066
\(874\) 0 0
\(875\) −12.1282 −0.410007
\(876\) 0 0
\(877\) −9.14542 −0.308819 −0.154409 0.988007i \(-0.549347\pi\)
−0.154409 + 0.988007i \(0.549347\pi\)
\(878\) 0 0
\(879\) 20.6097 0.695148
\(880\) 0 0
\(881\) −45.8631 −1.54517 −0.772584 0.634913i \(-0.781036\pi\)
−0.772584 + 0.634913i \(0.781036\pi\)
\(882\) 0 0
\(883\) −26.3131 −0.885507 −0.442753 0.896643i \(-0.645998\pi\)
−0.442753 + 0.896643i \(0.645998\pi\)
\(884\) 0 0
\(885\) 2.34899 0.0789605
\(886\) 0 0
\(887\) 2.09230 0.0702525 0.0351262 0.999383i \(-0.488817\pi\)
0.0351262 + 0.999383i \(0.488817\pi\)
\(888\) 0 0
\(889\) 30.9220 1.03709
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 12.5277 0.419223
\(894\) 0 0
\(895\) −4.49321 −0.150192
\(896\) 0 0
\(897\) 1.80651 0.0603178
\(898\) 0 0
\(899\) 4.97109 0.165795
\(900\) 0 0
\(901\) 7.92668 0.264076
\(902\) 0 0
\(903\) 30.4741 1.01411
\(904\) 0 0
\(905\) 0.477023 0.0158568
\(906\) 0 0
\(907\) 11.6641 0.387299 0.193650 0.981071i \(-0.437967\pi\)
0.193650 + 0.981071i \(0.437967\pi\)
\(908\) 0 0
\(909\) −10.4896 −0.347919
\(910\) 0 0
\(911\) 10.1635 0.336733 0.168366 0.985724i \(-0.446151\pi\)
0.168366 + 0.985724i \(0.446151\pi\)
\(912\) 0 0
\(913\) −13.7281 −0.454335
\(914\) 0 0
\(915\) −0.309990 −0.0102479
\(916\) 0 0
\(917\) 9.72448 0.321131
\(918\) 0 0
\(919\) −50.6215 −1.66985 −0.834924 0.550365i \(-0.814489\pi\)
−0.834924 + 0.550365i \(0.814489\pi\)
\(920\) 0 0
\(921\) −17.1118 −0.563853
\(922\) 0 0
\(923\) −2.99529 −0.0985913
\(924\) 0 0
\(925\) 2.75566 0.0906055
\(926\) 0 0
\(927\) 5.10729 0.167745
\(928\) 0 0
\(929\) 23.3524 0.766167 0.383083 0.923714i \(-0.374862\pi\)
0.383083 + 0.923714i \(0.374862\pi\)
\(930\) 0 0
\(931\) −55.3103 −1.81272
\(932\) 0 0
\(933\) 18.9873 0.621615
\(934\) 0 0
\(935\) −0.740865 −0.0242289
\(936\) 0 0
\(937\) 30.8553 1.00800 0.503999 0.863704i \(-0.331862\pi\)
0.503999 + 0.863704i \(0.331862\pi\)
\(938\) 0 0
\(939\) −21.2050 −0.691998
\(940\) 0 0
\(941\) −16.8237 −0.548439 −0.274219 0.961667i \(-0.588419\pi\)
−0.274219 + 0.961667i \(0.588419\pi\)
\(942\) 0 0
\(943\) −11.5010 −0.374525
\(944\) 0 0
\(945\) 1.22240 0.0397646
\(946\) 0 0
\(947\) −45.3991 −1.47527 −0.737637 0.675197i \(-0.764058\pi\)
−0.737637 + 0.675197i \(0.764058\pi\)
\(948\) 0 0
\(949\) −13.6498 −0.443090
\(950\) 0 0
\(951\) −25.4738 −0.826044
\(952\) 0 0
\(953\) −49.2871 −1.59657 −0.798283 0.602283i \(-0.794258\pi\)
−0.798283 + 0.602283i \(0.794258\pi\)
\(954\) 0 0
\(955\) 5.47822 0.177271
\(956\) 0 0
\(957\) −3.30887 −0.106960
\(958\) 0 0
\(959\) 81.3830 2.62799
\(960\) 0 0
\(961\) −28.7429 −0.927191
\(962\) 0 0
\(963\) 10.8354 0.349167
\(964\) 0 0
\(965\) −7.51132 −0.241798
\(966\) 0 0
\(967\) 36.9587 1.18851 0.594256 0.804276i \(-0.297447\pi\)
0.594256 + 0.804276i \(0.297447\pi\)
\(968\) 0 0
\(969\) −12.1313 −0.389713
\(970\) 0 0
\(971\) −40.2638 −1.29213 −0.646064 0.763283i \(-0.723586\pi\)
−0.646064 + 0.763283i \(0.723586\pi\)
\(972\) 0 0
\(973\) −54.7168 −1.75414
\(974\) 0 0
\(975\) 4.92163 0.157618
\(976\) 0 0
\(977\) −21.7237 −0.695003 −0.347501 0.937679i \(-0.612970\pi\)
−0.347501 + 0.937679i \(0.612970\pi\)
\(978\) 0 0
\(979\) −2.49765 −0.0798251
\(980\) 0 0
\(981\) −9.58882 −0.306148
\(982\) 0 0
\(983\) −7.72868 −0.246507 −0.123253 0.992375i \(-0.539333\pi\)
−0.123253 + 0.992375i \(0.539333\pi\)
\(984\) 0 0
\(985\) −5.50016 −0.175250
\(986\) 0 0
\(987\) −11.9328 −0.379824
\(988\) 0 0
\(989\) −12.6080 −0.400911
\(990\) 0 0
\(991\) 29.8563 0.948417 0.474209 0.880412i \(-0.342734\pi\)
0.474209 + 0.880412i \(0.342734\pi\)
\(992\) 0 0
\(993\) −3.03561 −0.0963323
\(994\) 0 0
\(995\) 2.11424 0.0670259
\(996\) 0 0
\(997\) −26.9856 −0.854641 −0.427320 0.904100i \(-0.640542\pi\)
−0.427320 + 0.904100i \(0.640542\pi\)
\(998\) 0 0
\(999\) −0.559909 −0.0177147
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.ca.1.2 4
4.3 odd 2 3432.2.a.r.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.r.1.2 4 4.3 odd 2
6864.2.a.ca.1.2 4 1.1 even 1 trivial