Properties

Label 4368.2.a.be.1.2
Level $4368$
Weight $2$
Character 4368.1
Self dual yes
Analytic conductor $34.879$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 4368.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.56155 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.56155 q^{11} +1.00000 q^{13} -3.56155 q^{15} -6.68466 q^{17} +4.68466 q^{19} -1.00000 q^{21} +5.56155 q^{23} +7.68466 q^{25} -1.00000 q^{27} +6.68466 q^{29} -6.24621 q^{31} -1.56155 q^{33} +3.56155 q^{35} -7.56155 q^{37} -1.00000 q^{39} -1.12311 q^{41} +6.43845 q^{43} +3.56155 q^{45} +1.00000 q^{49} +6.68466 q^{51} +12.2462 q^{53} +5.56155 q^{55} -4.68466 q^{57} -2.24621 q^{59} +6.68466 q^{61} +1.00000 q^{63} +3.56155 q^{65} +7.12311 q^{67} -5.56155 q^{69} -8.00000 q^{71} -3.56155 q^{73} -7.68466 q^{75} +1.56155 q^{77} +11.1231 q^{79} +1.00000 q^{81} -8.87689 q^{83} -23.8078 q^{85} -6.68466 q^{87} +10.0000 q^{89} +1.00000 q^{91} +6.24621 q^{93} +16.6847 q^{95} +14.4924 q^{97} +1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 3q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 3q^{5} + 2q^{7} + 2q^{9} - q^{11} + 2q^{13} - 3q^{15} - q^{17} - 3q^{19} - 2q^{21} + 7q^{23} + 3q^{25} - 2q^{27} + q^{29} + 4q^{31} + q^{33} + 3q^{35} - 11q^{37} - 2q^{39} + 6q^{41} + 17q^{43} + 3q^{45} + 2q^{49} + q^{51} + 8q^{53} + 7q^{55} + 3q^{57} + 12q^{59} + q^{61} + 2q^{63} + 3q^{65} + 6q^{67} - 7q^{69} - 16q^{71} - 3q^{73} - 3q^{75} - q^{77} + 14q^{79} + 2q^{81} - 26q^{83} - 27q^{85} - q^{87} + 20q^{89} + 2q^{91} - 4q^{93} + 21q^{95} - 4q^{97} - q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.56155 1.59277 0.796387 0.604787i \(-0.206742\pi\)
0.796387 + 0.604787i \(0.206742\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.56155 0.470826 0.235413 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −3.56155 −0.919589
\(16\) 0 0
\(17\) −6.68466 −1.62127 −0.810634 0.585553i \(-0.800877\pi\)
−0.810634 + 0.585553i \(0.800877\pi\)
\(18\) 0 0
\(19\) 4.68466 1.07473 0.537367 0.843348i \(-0.319419\pi\)
0.537367 + 0.843348i \(0.319419\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 5.56155 1.15966 0.579832 0.814736i \(-0.303118\pi\)
0.579832 + 0.814736i \(0.303118\pi\)
\(24\) 0 0
\(25\) 7.68466 1.53693
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.68466 1.24131 0.620655 0.784084i \(-0.286867\pi\)
0.620655 + 0.784084i \(0.286867\pi\)
\(30\) 0 0
\(31\) −6.24621 −1.12185 −0.560926 0.827866i \(-0.689555\pi\)
−0.560926 + 0.827866i \(0.689555\pi\)
\(32\) 0 0
\(33\) −1.56155 −0.271831
\(34\) 0 0
\(35\) 3.56155 0.602012
\(36\) 0 0
\(37\) −7.56155 −1.24311 −0.621556 0.783370i \(-0.713499\pi\)
−0.621556 + 0.783370i \(0.713499\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −1.12311 −0.175400 −0.0876998 0.996147i \(-0.527952\pi\)
−0.0876998 + 0.996147i \(0.527952\pi\)
\(42\) 0 0
\(43\) 6.43845 0.981854 0.490927 0.871201i \(-0.336658\pi\)
0.490927 + 0.871201i \(0.336658\pi\)
\(44\) 0 0
\(45\) 3.56155 0.530925
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.68466 0.936039
\(52\) 0 0
\(53\) 12.2462 1.68215 0.841073 0.540921i \(-0.181924\pi\)
0.841073 + 0.540921i \(0.181924\pi\)
\(54\) 0 0
\(55\) 5.56155 0.749920
\(56\) 0 0
\(57\) −4.68466 −0.620498
\(58\) 0 0
\(59\) −2.24621 −0.292432 −0.146216 0.989253i \(-0.546709\pi\)
−0.146216 + 0.989253i \(0.546709\pi\)
\(60\) 0 0
\(61\) 6.68466 0.855883 0.427941 0.903806i \(-0.359239\pi\)
0.427941 + 0.903806i \(0.359239\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 3.56155 0.441756
\(66\) 0 0
\(67\) 7.12311 0.870226 0.435113 0.900376i \(-0.356708\pi\)
0.435113 + 0.900376i \(0.356708\pi\)
\(68\) 0 0
\(69\) −5.56155 −0.669532
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −3.56155 −0.416848 −0.208424 0.978039i \(-0.566833\pi\)
−0.208424 + 0.978039i \(0.566833\pi\)
\(74\) 0 0
\(75\) −7.68466 −0.887348
\(76\) 0 0
\(77\) 1.56155 0.177955
\(78\) 0 0
\(79\) 11.1231 1.25145 0.625724 0.780045i \(-0.284804\pi\)
0.625724 + 0.780045i \(0.284804\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.87689 −0.974366 −0.487183 0.873300i \(-0.661975\pi\)
−0.487183 + 0.873300i \(0.661975\pi\)
\(84\) 0 0
\(85\) −23.8078 −2.58231
\(86\) 0 0
\(87\) −6.68466 −0.716671
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 6.24621 0.647702
\(94\) 0 0
\(95\) 16.6847 1.71181
\(96\) 0 0
\(97\) 14.4924 1.47148 0.735741 0.677263i \(-0.236834\pi\)
0.735741 + 0.677263i \(0.236834\pi\)
\(98\) 0 0
\(99\) 1.56155 0.156942
\(100\) 0 0
\(101\) −5.12311 −0.509768 −0.254884 0.966972i \(-0.582037\pi\)
−0.254884 + 0.966972i \(0.582037\pi\)
\(102\) 0 0
\(103\) 0.684658 0.0674614 0.0337307 0.999431i \(-0.489261\pi\)
0.0337307 + 0.999431i \(0.489261\pi\)
\(104\) 0 0
\(105\) −3.56155 −0.347572
\(106\) 0 0
\(107\) 8.87689 0.858162 0.429081 0.903266i \(-0.358838\pi\)
0.429081 + 0.903266i \(0.358838\pi\)
\(108\) 0 0
\(109\) −16.9309 −1.62168 −0.810842 0.585266i \(-0.800990\pi\)
−0.810842 + 0.585266i \(0.800990\pi\)
\(110\) 0 0
\(111\) 7.56155 0.717711
\(112\) 0 0
\(113\) −4.24621 −0.399450 −0.199725 0.979852i \(-0.564005\pi\)
−0.199725 + 0.979852i \(0.564005\pi\)
\(114\) 0 0
\(115\) 19.8078 1.84708
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −6.68466 −0.612782
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 0 0
\(123\) 1.12311 0.101267
\(124\) 0 0
\(125\) 9.56155 0.855211
\(126\) 0 0
\(127\) 4.87689 0.432754 0.216377 0.976310i \(-0.430576\pi\)
0.216377 + 0.976310i \(0.430576\pi\)
\(128\) 0 0
\(129\) −6.43845 −0.566874
\(130\) 0 0
\(131\) 9.56155 0.835397 0.417698 0.908586i \(-0.362837\pi\)
0.417698 + 0.908586i \(0.362837\pi\)
\(132\) 0 0
\(133\) 4.68466 0.406211
\(134\) 0 0
\(135\) −3.56155 −0.306530
\(136\) 0 0
\(137\) −3.56155 −0.304284 −0.152142 0.988359i \(-0.548617\pi\)
−0.152142 + 0.988359i \(0.548617\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.56155 0.130584
\(144\) 0 0
\(145\) 23.8078 1.97713
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −17.6155 −1.44312 −0.721560 0.692352i \(-0.756575\pi\)
−0.721560 + 0.692352i \(0.756575\pi\)
\(150\) 0 0
\(151\) −11.8078 −0.960902 −0.480451 0.877022i \(-0.659527\pi\)
−0.480451 + 0.877022i \(0.659527\pi\)
\(152\) 0 0
\(153\) −6.68466 −0.540423
\(154\) 0 0
\(155\) −22.2462 −1.78686
\(156\) 0 0
\(157\) −15.5616 −1.24195 −0.620974 0.783832i \(-0.713263\pi\)
−0.620974 + 0.783832i \(0.713263\pi\)
\(158\) 0 0
\(159\) −12.2462 −0.971188
\(160\) 0 0
\(161\) 5.56155 0.438312
\(162\) 0 0
\(163\) 16.8769 1.32190 0.660950 0.750430i \(-0.270153\pi\)
0.660950 + 0.750430i \(0.270153\pi\)
\(164\) 0 0
\(165\) −5.56155 −0.432966
\(166\) 0 0
\(167\) −22.9309 −1.77444 −0.887222 0.461343i \(-0.847368\pi\)
−0.887222 + 0.461343i \(0.847368\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 4.68466 0.358245
\(172\) 0 0
\(173\) 20.2462 1.53929 0.769645 0.638471i \(-0.220433\pi\)
0.769645 + 0.638471i \(0.220433\pi\)
\(174\) 0 0
\(175\) 7.68466 0.580906
\(176\) 0 0
\(177\) 2.24621 0.168836
\(178\) 0 0
\(179\) −16.4924 −1.23270 −0.616351 0.787472i \(-0.711390\pi\)
−0.616351 + 0.787472i \(0.711390\pi\)
\(180\) 0 0
\(181\) −0.246211 −0.0183007 −0.00915037 0.999958i \(-0.502913\pi\)
−0.00915037 + 0.999958i \(0.502913\pi\)
\(182\) 0 0
\(183\) −6.68466 −0.494144
\(184\) 0 0
\(185\) −26.9309 −1.98000
\(186\) 0 0
\(187\) −10.4384 −0.763335
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −14.9309 −1.08036 −0.540180 0.841550i \(-0.681644\pi\)
−0.540180 + 0.841550i \(0.681644\pi\)
\(192\) 0 0
\(193\) 19.3693 1.39423 0.697117 0.716957i \(-0.254466\pi\)
0.697117 + 0.716957i \(0.254466\pi\)
\(194\) 0 0
\(195\) −3.56155 −0.255048
\(196\) 0 0
\(197\) 10.8769 0.774947 0.387473 0.921881i \(-0.373348\pi\)
0.387473 + 0.921881i \(0.373348\pi\)
\(198\) 0 0
\(199\) −6.93087 −0.491316 −0.245658 0.969357i \(-0.579004\pi\)
−0.245658 + 0.969357i \(0.579004\pi\)
\(200\) 0 0
\(201\) −7.12311 −0.502425
\(202\) 0 0
\(203\) 6.68466 0.469171
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) 5.56155 0.386555
\(208\) 0 0
\(209\) 7.31534 0.506013
\(210\) 0 0
\(211\) 15.8078 1.08825 0.544126 0.839004i \(-0.316861\pi\)
0.544126 + 0.839004i \(0.316861\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) 22.9309 1.56387
\(216\) 0 0
\(217\) −6.24621 −0.424020
\(218\) 0 0
\(219\) 3.56155 0.240667
\(220\) 0 0
\(221\) −6.68466 −0.449659
\(222\) 0 0
\(223\) 23.6155 1.58141 0.790706 0.612196i \(-0.209713\pi\)
0.790706 + 0.612196i \(0.209713\pi\)
\(224\) 0 0
\(225\) 7.68466 0.512311
\(226\) 0 0
\(227\) 7.12311 0.472777 0.236389 0.971659i \(-0.424036\pi\)
0.236389 + 0.971659i \(0.424036\pi\)
\(228\) 0 0
\(229\) 28.2462 1.86656 0.933281 0.359147i \(-0.116932\pi\)
0.933281 + 0.359147i \(0.116932\pi\)
\(230\) 0 0
\(231\) −1.56155 −0.102743
\(232\) 0 0
\(233\) 27.3693 1.79302 0.896512 0.443020i \(-0.146093\pi\)
0.896512 + 0.443020i \(0.146093\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −11.1231 −0.722523
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.56155 0.227539
\(246\) 0 0
\(247\) 4.68466 0.298078
\(248\) 0 0
\(249\) 8.87689 0.562550
\(250\) 0 0
\(251\) 7.80776 0.492822 0.246411 0.969165i \(-0.420749\pi\)
0.246411 + 0.969165i \(0.420749\pi\)
\(252\) 0 0
\(253\) 8.68466 0.546000
\(254\) 0 0
\(255\) 23.8078 1.49090
\(256\) 0 0
\(257\) −4.24621 −0.264871 −0.132436 0.991192i \(-0.542280\pi\)
−0.132436 + 0.991192i \(0.542280\pi\)
\(258\) 0 0
\(259\) −7.56155 −0.469852
\(260\) 0 0
\(261\) 6.68466 0.413770
\(262\) 0 0
\(263\) 30.2462 1.86506 0.932531 0.361091i \(-0.117596\pi\)
0.932531 + 0.361091i \(0.117596\pi\)
\(264\) 0 0
\(265\) 43.6155 2.67928
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 0 0
\(269\) 20.2462 1.23443 0.617217 0.786793i \(-0.288260\pi\)
0.617217 + 0.786793i \(0.288260\pi\)
\(270\) 0 0
\(271\) −4.87689 −0.296250 −0.148125 0.988969i \(-0.547324\pi\)
−0.148125 + 0.988969i \(0.547324\pi\)
\(272\) 0 0
\(273\) −1.00000 −0.0605228
\(274\) 0 0
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) −22.4924 −1.35144 −0.675719 0.737159i \(-0.736167\pi\)
−0.675719 + 0.737159i \(0.736167\pi\)
\(278\) 0 0
\(279\) −6.24621 −0.373951
\(280\) 0 0
\(281\) 16.2462 0.969168 0.484584 0.874745i \(-0.338971\pi\)
0.484584 + 0.874745i \(0.338971\pi\)
\(282\) 0 0
\(283\) −18.2462 −1.08462 −0.542312 0.840177i \(-0.682451\pi\)
−0.542312 + 0.840177i \(0.682451\pi\)
\(284\) 0 0
\(285\) −16.6847 −0.988314
\(286\) 0 0
\(287\) −1.12311 −0.0662948
\(288\) 0 0
\(289\) 27.6847 1.62851
\(290\) 0 0
\(291\) −14.4924 −0.849561
\(292\) 0 0
\(293\) 24.7386 1.44525 0.722623 0.691242i \(-0.242936\pi\)
0.722623 + 0.691242i \(0.242936\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) −1.56155 −0.0906105
\(298\) 0 0
\(299\) 5.56155 0.321633
\(300\) 0 0
\(301\) 6.43845 0.371106
\(302\) 0 0
\(303\) 5.12311 0.294315
\(304\) 0 0
\(305\) 23.8078 1.36323
\(306\) 0 0
\(307\) −26.2462 −1.49795 −0.748975 0.662598i \(-0.769454\pi\)
−0.748975 + 0.662598i \(0.769454\pi\)
\(308\) 0 0
\(309\) −0.684658 −0.0389489
\(310\) 0 0
\(311\) 12.8769 0.730182 0.365091 0.930972i \(-0.381038\pi\)
0.365091 + 0.930972i \(0.381038\pi\)
\(312\) 0 0
\(313\) −13.6155 −0.769595 −0.384798 0.923001i \(-0.625729\pi\)
−0.384798 + 0.923001i \(0.625729\pi\)
\(314\) 0 0
\(315\) 3.56155 0.200671
\(316\) 0 0
\(317\) 2.87689 0.161582 0.0807912 0.996731i \(-0.474255\pi\)
0.0807912 + 0.996731i \(0.474255\pi\)
\(318\) 0 0
\(319\) 10.4384 0.584441
\(320\) 0 0
\(321\) −8.87689 −0.495460
\(322\) 0 0
\(323\) −31.3153 −1.74243
\(324\) 0 0
\(325\) 7.68466 0.426268
\(326\) 0 0
\(327\) 16.9309 0.936279
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −13.3693 −0.734844 −0.367422 0.930054i \(-0.619760\pi\)
−0.367422 + 0.930054i \(0.619760\pi\)
\(332\) 0 0
\(333\) −7.56155 −0.414371
\(334\) 0 0
\(335\) 25.3693 1.38607
\(336\) 0 0
\(337\) 4.43845 0.241778 0.120889 0.992666i \(-0.461426\pi\)
0.120889 + 0.992666i \(0.461426\pi\)
\(338\) 0 0
\(339\) 4.24621 0.230623
\(340\) 0 0
\(341\) −9.75379 −0.528197
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −19.8078 −1.06641
\(346\) 0 0
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) 7.75379 0.415051 0.207525 0.978230i \(-0.433459\pi\)
0.207525 + 0.978230i \(0.433459\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 30.4924 1.62295 0.811474 0.584389i \(-0.198666\pi\)
0.811474 + 0.584389i \(0.198666\pi\)
\(354\) 0 0
\(355\) −28.4924 −1.51222
\(356\) 0 0
\(357\) 6.68466 0.353790
\(358\) 0 0
\(359\) 26.7386 1.41121 0.705606 0.708605i \(-0.250675\pi\)
0.705606 + 0.708605i \(0.250675\pi\)
\(360\) 0 0
\(361\) 2.94602 0.155054
\(362\) 0 0
\(363\) 8.56155 0.449365
\(364\) 0 0
\(365\) −12.6847 −0.663945
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) −1.12311 −0.0584665
\(370\) 0 0
\(371\) 12.2462 0.635792
\(372\) 0 0
\(373\) −17.6155 −0.912097 −0.456049 0.889955i \(-0.650736\pi\)
−0.456049 + 0.889955i \(0.650736\pi\)
\(374\) 0 0
\(375\) −9.56155 −0.493756
\(376\) 0 0
\(377\) 6.68466 0.344277
\(378\) 0 0
\(379\) −34.2462 −1.75911 −0.879555 0.475797i \(-0.842160\pi\)
−0.879555 + 0.475797i \(0.842160\pi\)
\(380\) 0 0
\(381\) −4.87689 −0.249851
\(382\) 0 0
\(383\) −27.4233 −1.40126 −0.700632 0.713522i \(-0.747099\pi\)
−0.700632 + 0.713522i \(0.747099\pi\)
\(384\) 0 0
\(385\) 5.56155 0.283443
\(386\) 0 0
\(387\) 6.43845 0.327285
\(388\) 0 0
\(389\) −28.7386 −1.45711 −0.728553 0.684989i \(-0.759807\pi\)
−0.728553 + 0.684989i \(0.759807\pi\)
\(390\) 0 0
\(391\) −37.1771 −1.88013
\(392\) 0 0
\(393\) −9.56155 −0.482317
\(394\) 0 0
\(395\) 39.6155 1.99327
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 0 0
\(399\) −4.68466 −0.234526
\(400\) 0 0
\(401\) 8.24621 0.411796 0.205898 0.978573i \(-0.433988\pi\)
0.205898 + 0.978573i \(0.433988\pi\)
\(402\) 0 0
\(403\) −6.24621 −0.311146
\(404\) 0 0
\(405\) 3.56155 0.176975
\(406\) 0 0
\(407\) −11.8078 −0.585289
\(408\) 0 0
\(409\) −4.93087 −0.243816 −0.121908 0.992541i \(-0.538901\pi\)
−0.121908 + 0.992541i \(0.538901\pi\)
\(410\) 0 0
\(411\) 3.56155 0.175678
\(412\) 0 0
\(413\) −2.24621 −0.110529
\(414\) 0 0
\(415\) −31.6155 −1.55195
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 36.6847 1.79216 0.896081 0.443890i \(-0.146402\pi\)
0.896081 + 0.443890i \(0.146402\pi\)
\(420\) 0 0
\(421\) −3.75379 −0.182948 −0.0914742 0.995807i \(-0.529158\pi\)
−0.0914742 + 0.995807i \(0.529158\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −51.3693 −2.49178
\(426\) 0 0
\(427\) 6.68466 0.323493
\(428\) 0 0
\(429\) −1.56155 −0.0753925
\(430\) 0 0
\(431\) 33.3693 1.60734 0.803672 0.595073i \(-0.202877\pi\)
0.803672 + 0.595073i \(0.202877\pi\)
\(432\) 0 0
\(433\) −31.3693 −1.50751 −0.753757 0.657154i \(-0.771760\pi\)
−0.753757 + 0.657154i \(0.771760\pi\)
\(434\) 0 0
\(435\) −23.8078 −1.14149
\(436\) 0 0
\(437\) 26.0540 1.24633
\(438\) 0 0
\(439\) 6.93087 0.330792 0.165396 0.986227i \(-0.447110\pi\)
0.165396 + 0.986227i \(0.447110\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 5.36932 0.255104 0.127552 0.991832i \(-0.459288\pi\)
0.127552 + 0.991832i \(0.459288\pi\)
\(444\) 0 0
\(445\) 35.6155 1.68834
\(446\) 0 0
\(447\) 17.6155 0.833186
\(448\) 0 0
\(449\) 10.6847 0.504240 0.252120 0.967696i \(-0.418872\pi\)
0.252120 + 0.967696i \(0.418872\pi\)
\(450\) 0 0
\(451\) −1.75379 −0.0825827
\(452\) 0 0
\(453\) 11.8078 0.554777
\(454\) 0 0
\(455\) 3.56155 0.166968
\(456\) 0 0
\(457\) 27.3693 1.28028 0.640141 0.768257i \(-0.278876\pi\)
0.640141 + 0.768257i \(0.278876\pi\)
\(458\) 0 0
\(459\) 6.68466 0.312013
\(460\) 0 0
\(461\) −28.0540 −1.30660 −0.653302 0.757097i \(-0.726617\pi\)
−0.653302 + 0.757097i \(0.726617\pi\)
\(462\) 0 0
\(463\) 8.68466 0.403610 0.201805 0.979426i \(-0.435319\pi\)
0.201805 + 0.979426i \(0.435319\pi\)
\(464\) 0 0
\(465\) 22.2462 1.03164
\(466\) 0 0
\(467\) 3.31534 0.153416 0.0767079 0.997054i \(-0.475559\pi\)
0.0767079 + 0.997054i \(0.475559\pi\)
\(468\) 0 0
\(469\) 7.12311 0.328914
\(470\) 0 0
\(471\) 15.5616 0.717039
\(472\) 0 0
\(473\) 10.0540 0.462282
\(474\) 0 0
\(475\) 36.0000 1.65179
\(476\) 0 0
\(477\) 12.2462 0.560715
\(478\) 0 0
\(479\) −32.3002 −1.47583 −0.737917 0.674892i \(-0.764190\pi\)
−0.737917 + 0.674892i \(0.764190\pi\)
\(480\) 0 0
\(481\) −7.56155 −0.344777
\(482\) 0 0
\(483\) −5.56155 −0.253059
\(484\) 0 0
\(485\) 51.6155 2.34374
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) −16.8769 −0.763200
\(490\) 0 0
\(491\) 8.87689 0.400609 0.200304 0.979734i \(-0.435807\pi\)
0.200304 + 0.979734i \(0.435807\pi\)
\(492\) 0 0
\(493\) −44.6847 −2.01250
\(494\) 0 0
\(495\) 5.56155 0.249973
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) 22.9309 1.02448
\(502\) 0 0
\(503\) 3.12311 0.139252 0.0696262 0.997573i \(-0.477819\pi\)
0.0696262 + 0.997573i \(0.477819\pi\)
\(504\) 0 0
\(505\) −18.2462 −0.811946
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −12.0540 −0.534283 −0.267142 0.963657i \(-0.586079\pi\)
−0.267142 + 0.963657i \(0.586079\pi\)
\(510\) 0 0
\(511\) −3.56155 −0.157554
\(512\) 0 0
\(513\) −4.68466 −0.206833
\(514\) 0 0
\(515\) 2.43845 0.107451
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −20.2462 −0.888710
\(520\) 0 0
\(521\) 2.68466 0.117617 0.0588085 0.998269i \(-0.481270\pi\)
0.0588085 + 0.998269i \(0.481270\pi\)
\(522\) 0 0
\(523\) −21.7538 −0.951227 −0.475613 0.879654i \(-0.657774\pi\)
−0.475613 + 0.879654i \(0.657774\pi\)
\(524\) 0 0
\(525\) −7.68466 −0.335386
\(526\) 0 0
\(527\) 41.7538 1.81882
\(528\) 0 0
\(529\) 7.93087 0.344820
\(530\) 0 0
\(531\) −2.24621 −0.0974773
\(532\) 0 0
\(533\) −1.12311 −0.0486471
\(534\) 0 0
\(535\) 31.6155 1.36686
\(536\) 0 0
\(537\) 16.4924 0.711701
\(538\) 0 0
\(539\) 1.56155 0.0672608
\(540\) 0 0
\(541\) −32.9309 −1.41581 −0.707904 0.706308i \(-0.750359\pi\)
−0.707904 + 0.706308i \(0.750359\pi\)
\(542\) 0 0
\(543\) 0.246211 0.0105659
\(544\) 0 0
\(545\) −60.3002 −2.58298
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) 6.68466 0.285294
\(550\) 0 0
\(551\) 31.3153 1.33408
\(552\) 0 0
\(553\) 11.1231 0.473003
\(554\) 0 0
\(555\) 26.9309 1.14315
\(556\) 0 0
\(557\) −3.36932 −0.142763 −0.0713813 0.997449i \(-0.522741\pi\)
−0.0713813 + 0.997449i \(0.522741\pi\)
\(558\) 0 0
\(559\) 6.43845 0.272317
\(560\) 0 0
\(561\) 10.4384 0.440712
\(562\) 0 0
\(563\) 6.05398 0.255145 0.127572 0.991829i \(-0.459282\pi\)
0.127572 + 0.991829i \(0.459282\pi\)
\(564\) 0 0
\(565\) −15.1231 −0.636234
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 41.2311 1.72850 0.864248 0.503066i \(-0.167795\pi\)
0.864248 + 0.503066i \(0.167795\pi\)
\(570\) 0 0
\(571\) −18.2462 −0.763580 −0.381790 0.924249i \(-0.624692\pi\)
−0.381790 + 0.924249i \(0.624692\pi\)
\(572\) 0 0
\(573\) 14.9309 0.623746
\(574\) 0 0
\(575\) 42.7386 1.78232
\(576\) 0 0
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) 0 0
\(579\) −19.3693 −0.804961
\(580\) 0 0
\(581\) −8.87689 −0.368276
\(582\) 0 0
\(583\) 19.1231 0.791998
\(584\) 0 0
\(585\) 3.56155 0.147252
\(586\) 0 0
\(587\) −13.3693 −0.551811 −0.275905 0.961185i \(-0.588978\pi\)
−0.275905 + 0.961185i \(0.588978\pi\)
\(588\) 0 0
\(589\) −29.2614 −1.20569
\(590\) 0 0
\(591\) −10.8769 −0.447416
\(592\) 0 0
\(593\) −20.2462 −0.831412 −0.415706 0.909499i \(-0.636466\pi\)
−0.415706 + 0.909499i \(0.636466\pi\)
\(594\) 0 0
\(595\) −23.8078 −0.976023
\(596\) 0 0
\(597\) 6.93087 0.283662
\(598\) 0 0
\(599\) −21.5616 −0.880981 −0.440491 0.897757i \(-0.645195\pi\)
−0.440491 + 0.897757i \(0.645195\pi\)
\(600\) 0 0
\(601\) −10.8769 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(602\) 0 0
\(603\) 7.12311 0.290075
\(604\) 0 0
\(605\) −30.4924 −1.23969
\(606\) 0 0
\(607\) 18.4384 0.748393 0.374197 0.927349i \(-0.377918\pi\)
0.374197 + 0.927349i \(0.377918\pi\)
\(608\) 0 0
\(609\) −6.68466 −0.270876
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 44.5464 1.79921 0.899606 0.436702i \(-0.143854\pi\)
0.899606 + 0.436702i \(0.143854\pi\)
\(614\) 0 0
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) −33.4233 −1.34557 −0.672786 0.739838i \(-0.734902\pi\)
−0.672786 + 0.739838i \(0.734902\pi\)
\(618\) 0 0
\(619\) −19.3153 −0.776349 −0.388175 0.921586i \(-0.626894\pi\)
−0.388175 + 0.921586i \(0.626894\pi\)
\(620\) 0 0
\(621\) −5.56155 −0.223177
\(622\) 0 0
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) 0 0
\(627\) −7.31534 −0.292147
\(628\) 0 0
\(629\) 50.5464 2.01542
\(630\) 0 0
\(631\) 22.9309 0.912864 0.456432 0.889758i \(-0.349127\pi\)
0.456432 + 0.889758i \(0.349127\pi\)
\(632\) 0 0
\(633\) −15.8078 −0.628302
\(634\) 0 0
\(635\) 17.3693 0.689280
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −20.2462 −0.799677 −0.399839 0.916586i \(-0.630934\pi\)
−0.399839 + 0.916586i \(0.630934\pi\)
\(642\) 0 0
\(643\) −16.1922 −0.638559 −0.319280 0.947661i \(-0.603441\pi\)
−0.319280 + 0.947661i \(0.603441\pi\)
\(644\) 0 0
\(645\) −22.9309 −0.902902
\(646\) 0 0
\(647\) 14.2462 0.560076 0.280038 0.959989i \(-0.409653\pi\)
0.280038 + 0.959989i \(0.409653\pi\)
\(648\) 0 0
\(649\) −3.50758 −0.137684
\(650\) 0 0
\(651\) 6.24621 0.244808
\(652\) 0 0
\(653\) −16.9309 −0.662556 −0.331278 0.943533i \(-0.607480\pi\)
−0.331278 + 0.943533i \(0.607480\pi\)
\(654\) 0 0
\(655\) 34.0540 1.33060
\(656\) 0 0
\(657\) −3.56155 −0.138949
\(658\) 0 0
\(659\) −21.3693 −0.832430 −0.416215 0.909266i \(-0.636644\pi\)
−0.416215 + 0.909266i \(0.636644\pi\)
\(660\) 0 0
\(661\) −0.246211 −0.00957651 −0.00478825 0.999989i \(-0.501524\pi\)
−0.00478825 + 0.999989i \(0.501524\pi\)
\(662\) 0 0
\(663\) 6.68466 0.259611
\(664\) 0 0
\(665\) 16.6847 0.647003
\(666\) 0 0
\(667\) 37.1771 1.43950
\(668\) 0 0
\(669\) −23.6155 −0.913029
\(670\) 0 0
\(671\) 10.4384 0.402972
\(672\) 0 0
\(673\) −33.8078 −1.30319 −0.651597 0.758566i \(-0.725901\pi\)
−0.651597 + 0.758566i \(0.725901\pi\)
\(674\) 0 0
\(675\) −7.68466 −0.295783
\(676\) 0 0
\(677\) 2.49242 0.0957916 0.0478958 0.998852i \(-0.484748\pi\)
0.0478958 + 0.998852i \(0.484748\pi\)
\(678\) 0 0
\(679\) 14.4924 0.556168
\(680\) 0 0
\(681\) −7.12311 −0.272958
\(682\) 0 0
\(683\) −35.3153 −1.35130 −0.675652 0.737221i \(-0.736138\pi\)
−0.675652 + 0.737221i \(0.736138\pi\)
\(684\) 0 0
\(685\) −12.6847 −0.484656
\(686\) 0 0
\(687\) −28.2462 −1.07766
\(688\) 0 0
\(689\) 12.2462 0.466543
\(690\) 0 0
\(691\) −16.4924 −0.627401 −0.313701 0.949522i \(-0.601569\pi\)
−0.313701 + 0.949522i \(0.601569\pi\)
\(692\) 0 0
\(693\) 1.56155 0.0593185
\(694\) 0 0
\(695\) −42.7386 −1.62117
\(696\) 0 0
\(697\) 7.50758 0.284370
\(698\) 0 0
\(699\) −27.3693 −1.03520
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) −35.4233 −1.33601
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5.12311 −0.192674
\(708\) 0 0
\(709\) −16.2462 −0.610139 −0.305070 0.952330i \(-0.598680\pi\)
−0.305070 + 0.952330i \(0.598680\pi\)
\(710\) 0 0
\(711\) 11.1231 0.417149
\(712\) 0 0
\(713\) −34.7386 −1.30097
\(714\) 0 0
\(715\) 5.56155 0.207990
\(716\) 0 0
\(717\) 16.0000 0.597531
\(718\) 0 0
\(719\) −43.1231 −1.60822 −0.804110 0.594480i \(-0.797358\pi\)
−0.804110 + 0.594480i \(0.797358\pi\)
\(720\) 0 0
\(721\) 0.684658 0.0254980
\(722\) 0 0
\(723\) 14.0000 0.520666
\(724\) 0 0
\(725\) 51.3693 1.90781
\(726\) 0 0
\(727\) 6.93087 0.257052 0.128526 0.991706i \(-0.458975\pi\)
0.128526 + 0.991706i \(0.458975\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −43.0388 −1.59185
\(732\) 0 0
\(733\) −41.6155 −1.53710 −0.768552 0.639787i \(-0.779023\pi\)
−0.768552 + 0.639787i \(0.779023\pi\)
\(734\) 0 0
\(735\) −3.56155 −0.131370
\(736\) 0 0
\(737\) 11.1231 0.409725
\(738\) 0 0
\(739\) 32.1080 1.18111 0.590555 0.806997i \(-0.298909\pi\)
0.590555 + 0.806997i \(0.298909\pi\)
\(740\) 0 0
\(741\) −4.68466 −0.172095
\(742\) 0 0
\(743\) −28.1080 −1.03118 −0.515590 0.856835i \(-0.672427\pi\)
−0.515590 + 0.856835i \(0.672427\pi\)
\(744\) 0 0
\(745\) −62.7386 −2.29857
\(746\) 0 0
\(747\) −8.87689 −0.324789
\(748\) 0 0
\(749\) 8.87689 0.324355
\(750\) 0 0
\(751\) 6.24621 0.227927 0.113964 0.993485i \(-0.463645\pi\)
0.113964 + 0.993485i \(0.463645\pi\)
\(752\) 0 0
\(753\) −7.80776 −0.284531
\(754\) 0 0
\(755\) −42.0540 −1.53050
\(756\) 0 0
\(757\) 34.4924 1.25365 0.626824 0.779161i \(-0.284354\pi\)
0.626824 + 0.779161i \(0.284354\pi\)
\(758\) 0 0
\(759\) −8.68466 −0.315233
\(760\) 0 0
\(761\) 5.12311 0.185712 0.0928562 0.995680i \(-0.470400\pi\)
0.0928562 + 0.995680i \(0.470400\pi\)
\(762\) 0 0
\(763\) −16.9309 −0.612939
\(764\) 0 0
\(765\) −23.8078 −0.860772
\(766\) 0 0
\(767\) −2.24621 −0.0811060
\(768\) 0 0
\(769\) −16.4384 −0.592786 −0.296393 0.955066i \(-0.595784\pi\)
−0.296393 + 0.955066i \(0.595784\pi\)
\(770\) 0 0
\(771\) 4.24621 0.152924
\(772\) 0 0
\(773\) 52.9309 1.90379 0.951896 0.306423i \(-0.0991321\pi\)
0.951896 + 0.306423i \(0.0991321\pi\)
\(774\) 0 0
\(775\) −48.0000 −1.72421
\(776\) 0 0
\(777\) 7.56155 0.271269
\(778\) 0 0
\(779\) −5.26137 −0.188508
\(780\) 0 0
\(781\) −12.4924 −0.447014
\(782\) 0 0
\(783\) −6.68466 −0.238890
\(784\) 0 0
\(785\) −55.4233 −1.97814
\(786\) 0 0
\(787\) −20.3002 −0.723624 −0.361812 0.932251i \(-0.617842\pi\)
−0.361812 + 0.932251i \(0.617842\pi\)
\(788\) 0 0
\(789\) −30.2462 −1.07679
\(790\) 0 0
\(791\) −4.24621 −0.150978
\(792\) 0 0
\(793\) 6.68466 0.237379
\(794\) 0 0
\(795\) −43.6155 −1.54688
\(796\) 0 0
\(797\) 31.3693 1.11116 0.555579 0.831464i \(-0.312497\pi\)
0.555579 + 0.831464i \(0.312497\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) −5.56155 −0.196263
\(804\) 0 0
\(805\) 19.8078 0.698132
\(806\) 0 0
\(807\) −20.2462 −0.712700
\(808\) 0 0
\(809\) −2.49242 −0.0876289 −0.0438145 0.999040i \(-0.513951\pi\)
−0.0438145 + 0.999040i \(0.513951\pi\)
\(810\) 0 0
\(811\) −41.5616 −1.45942 −0.729712 0.683755i \(-0.760346\pi\)
−0.729712 + 0.683755i \(0.760346\pi\)
\(812\) 0 0
\(813\) 4.87689 0.171040
\(814\) 0 0
\(815\) 60.1080 2.10549
\(816\) 0 0
\(817\) 30.1619 1.05523
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) 14.3845 0.502022 0.251011 0.967984i \(-0.419237\pi\)
0.251011 + 0.967984i \(0.419237\pi\)
\(822\) 0 0
\(823\) −4.49242 −0.156596 −0.0782980 0.996930i \(-0.524949\pi\)
−0.0782980 + 0.996930i \(0.524949\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) −26.9309 −0.936478 −0.468239 0.883602i \(-0.655111\pi\)
−0.468239 + 0.883602i \(0.655111\pi\)
\(828\) 0 0
\(829\) 38.6847 1.34357 0.671787 0.740744i \(-0.265527\pi\)
0.671787 + 0.740744i \(0.265527\pi\)
\(830\) 0 0
\(831\) 22.4924 0.780253
\(832\) 0 0
\(833\) −6.68466 −0.231610
\(834\) 0 0
\(835\) −81.6695 −2.82629
\(836\) 0 0
\(837\) 6.24621 0.215901
\(838\) 0 0
\(839\) 10.7386 0.370739 0.185369 0.982669i \(-0.440652\pi\)
0.185369 + 0.982669i \(0.440652\pi\)
\(840\) 0 0
\(841\) 15.6847 0.540850
\(842\) 0 0
\(843\) −16.2462 −0.559549
\(844\) 0 0
\(845\) 3.56155 0.122521
\(846\) 0 0
\(847\) −8.56155 −0.294178
\(848\) 0 0
\(849\) 18.2462 0.626208
\(850\) 0 0
\(851\) −42.0540 −1.44159
\(852\) 0 0
\(853\) −27.3693 −0.937108 −0.468554 0.883435i \(-0.655225\pi\)
−0.468554 + 0.883435i \(0.655225\pi\)
\(854\) 0 0
\(855\) 16.6847 0.570603
\(856\) 0 0
\(857\) −34.4924 −1.17824 −0.589119 0.808046i \(-0.700525\pi\)
−0.589119 + 0.808046i \(0.700525\pi\)
\(858\) 0 0
\(859\) −5.75379 −0.196317 −0.0981584 0.995171i \(-0.531295\pi\)
−0.0981584 + 0.995171i \(0.531295\pi\)
\(860\) 0 0
\(861\) 1.12311 0.0382753
\(862\) 0 0
\(863\) 17.3693 0.591258 0.295629 0.955303i \(-0.404471\pi\)
0.295629 + 0.955303i \(0.404471\pi\)
\(864\) 0 0
\(865\) 72.1080 2.45174
\(866\) 0 0
\(867\) −27.6847 −0.940220
\(868\) 0 0
\(869\) 17.3693 0.589214
\(870\) 0 0
\(871\) 7.12311 0.241357
\(872\) 0 0
\(873\) 14.4924 0.490494
\(874\) 0 0
\(875\) 9.56155 0.323239
\(876\) 0 0
\(877\) −40.2462 −1.35902 −0.679509 0.733667i \(-0.737807\pi\)
−0.679509 + 0.733667i \(0.737807\pi\)
\(878\) 0 0
\(879\) −24.7386 −0.834413
\(880\) 0 0
\(881\) −19.1771 −0.646092 −0.323046 0.946383i \(-0.604707\pi\)
−0.323046 + 0.946383i \(0.604707\pi\)
\(882\) 0 0
\(883\) −1.56155 −0.0525504 −0.0262752 0.999655i \(-0.508365\pi\)
−0.0262752 + 0.999655i \(0.508365\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) 52.4924 1.76252 0.881262 0.472629i \(-0.156695\pi\)
0.881262 + 0.472629i \(0.156695\pi\)
\(888\) 0 0
\(889\) 4.87689 0.163566
\(890\) 0 0
\(891\) 1.56155 0.0523140
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −58.7386 −1.96342
\(896\) 0 0
\(897\) −5.56155 −0.185695
\(898\) 0 0
\(899\) −41.7538 −1.39257
\(900\) 0 0
\(901\) −81.8617 −2.72721
\(902\) 0 0
\(903\) −6.43845 −0.214258
\(904\) 0 0
\(905\) −0.876894 −0.0291490
\(906\) 0 0
\(907\) 7.50758 0.249285 0.124643 0.992202i \(-0.460222\pi\)
0.124643 + 0.992202i \(0.460222\pi\)
\(908\) 0 0
\(909\) −5.12311 −0.169923
\(910\) 0 0
\(911\) 11.4233 0.378471 0.189235 0.981932i \(-0.439399\pi\)
0.189235 + 0.981932i \(0.439399\pi\)
\(912\) 0 0
\(913\) −13.8617 −0.458757
\(914\) 0 0
\(915\) −23.8078 −0.787060
\(916\) 0 0
\(917\) 9.56155 0.315750
\(918\) 0 0
\(919\) −19.1231 −0.630813 −0.315407 0.948957i \(-0.602141\pi\)
−0.315407 + 0.948957i \(0.602141\pi\)
\(920\) 0 0
\(921\) 26.2462 0.864842
\(922\) 0 0
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) −58.1080 −1.91058
\(926\) 0 0
\(927\) 0.684658 0.0224871
\(928\) 0 0
\(929\) 25.6155 0.840418 0.420209 0.907427i \(-0.361957\pi\)
0.420209 + 0.907427i \(0.361957\pi\)
\(930\) 0 0
\(931\) 4.68466 0.153533
\(932\) 0 0
\(933\) −12.8769 −0.421571
\(934\) 0 0
\(935\) −37.1771 −1.21582
\(936\) 0 0
\(937\) −46.9848 −1.53493 −0.767464 0.641092i \(-0.778482\pi\)
−0.767464 + 0.641092i \(0.778482\pi\)
\(938\) 0 0
\(939\) 13.6155 0.444326
\(940\) 0 0
\(941\) −34.0000 −1.10837 −0.554184 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(942\) 0 0
\(943\) −6.24621 −0.203405
\(944\) 0 0
\(945\) −3.56155 −0.115857
\(946\) 0 0
\(947\) 40.7926 1.32558 0.662791 0.748805i \(-0.269372\pi\)
0.662791 + 0.748805i \(0.269372\pi\)
\(948\) 0 0
\(949\) −3.56155 −0.115613
\(950\) 0 0
\(951\) −2.87689 −0.0932897
\(952\) 0 0
\(953\) 11.3693 0.368288 0.184144 0.982899i \(-0.441049\pi\)
0.184144 + 0.982899i \(0.441049\pi\)
\(954\) 0 0
\(955\) −53.1771 −1.72077
\(956\) 0 0
\(957\) −10.4384 −0.337427
\(958\) 0 0
\(959\) −3.56155 −0.115009
\(960\) 0 0
\(961\) 8.01515 0.258553
\(962\) 0 0
\(963\) 8.87689 0.286054
\(964\) 0 0
\(965\) 68.9848 2.22070
\(966\) 0 0
\(967\) 21.5616 0.693373 0.346686 0.937981i \(-0.387307\pi\)
0.346686 + 0.937981i \(0.387307\pi\)
\(968\) 0 0
\(969\) 31.3153 1.00599
\(970\) 0 0
\(971\) −2.24621 −0.0720843 −0.0360422 0.999350i \(-0.511475\pi\)
−0.0360422 + 0.999350i \(0.511475\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) 0 0
\(975\) −7.68466 −0.246106
\(976\) 0 0
\(977\) −22.6847 −0.725747 −0.362873 0.931839i \(-0.618204\pi\)
−0.362873 + 0.931839i \(0.618204\pi\)
\(978\) 0 0
\(979\) 15.6155 0.499074
\(980\) 0 0
\(981\) −16.9309 −0.540561
\(982\) 0 0
\(983\) 13.1771 0.420284 0.210142 0.977671i \(-0.432607\pi\)
0.210142 + 0.977671i \(0.432607\pi\)
\(984\) 0 0
\(985\) 38.7386 1.23432
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 35.8078 1.13862
\(990\) 0 0
\(991\) 7.61553 0.241915 0.120958 0.992658i \(-0.461403\pi\)
0.120958 + 0.992658i \(0.461403\pi\)
\(992\) 0 0
\(993\) 13.3693 0.424262
\(994\) 0 0
\(995\) −24.6847 −0.782556
\(996\) 0 0
\(997\) 40.7386 1.29021 0.645103 0.764096i \(-0.276815\pi\)
0.645103 + 0.764096i \(0.276815\pi\)
\(998\) 0 0
\(999\) 7.56155 0.239237
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 4368.2.a.be.1.2 2
4.3 odd 2 546.2.a.j.1.2 2
12.11 even 2 1638.2.a.u.1.1 2
28.27 even 2 3822.2.a.bo.1.1 2
52.51 odd 2 7098.2.a.bl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.j.1.2 2 4.3 odd 2
1638.2.a.u.1.1 2 12.11 even 2
3822.2.a.bo.1.1 2 28.27 even 2
4368.2.a.be.1.2 2 1.1 even 1 trivial
7098.2.a.bl.1.1 2 52.51 odd 2