Properties

Label 1638.2.a.u.1.1
Level $1638$
Weight $2$
Character 1638.1
Self dual yes
Analytic conductor $13.079$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.0794958511\)
Analytic rank: \(1\)
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 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 1638.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.56155 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.56155 q^{5} -1.00000 q^{7} -1.00000 q^{8} +3.56155 q^{10} +1.56155 q^{11} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.68466 q^{17} -4.68466 q^{19} -3.56155 q^{20} -1.56155 q^{22} +5.56155 q^{23} +7.68466 q^{25} -1.00000 q^{26} -1.00000 q^{28} -6.68466 q^{29} +6.24621 q^{31} -1.00000 q^{32} -6.68466 q^{34} +3.56155 q^{35} -7.56155 q^{37} +4.68466 q^{38} +3.56155 q^{40} +1.12311 q^{41} -6.43845 q^{43} +1.56155 q^{44} -5.56155 q^{46} +1.00000 q^{49} -7.68466 q^{50} +1.00000 q^{52} -12.2462 q^{53} -5.56155 q^{55} +1.00000 q^{56} +6.68466 q^{58} -2.24621 q^{59} +6.68466 q^{61} -6.24621 q^{62} +1.00000 q^{64} -3.56155 q^{65} -7.12311 q^{67} +6.68466 q^{68} -3.56155 q^{70} -8.00000 q^{71} -3.56155 q^{73} +7.56155 q^{74} -4.68466 q^{76} -1.56155 q^{77} -11.1231 q^{79} -3.56155 q^{80} -1.12311 q^{82} -8.87689 q^{83} -23.8078 q^{85} +6.43845 q^{86} -1.56155 q^{88} -10.0000 q^{89} -1.00000 q^{91} +5.56155 q^{92} +16.6847 q^{95} +14.4924 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 3q^{5} - 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 3q^{5} - 2q^{7} - 2q^{8} + 3q^{10} - q^{11} + 2q^{13} + 2q^{14} + 2q^{16} + q^{17} + 3q^{19} - 3q^{20} + q^{22} + 7q^{23} + 3q^{25} - 2q^{26} - 2q^{28} - q^{29} - 4q^{31} - 2q^{32} - q^{34} + 3q^{35} - 11q^{37} - 3q^{38} + 3q^{40} - 6q^{41} - 17q^{43} - q^{44} - 7q^{46} + 2q^{49} - 3q^{50} + 2q^{52} - 8q^{53} - 7q^{55} + 2q^{56} + q^{58} + 12q^{59} + q^{61} + 4q^{62} + 2q^{64} - 3q^{65} - 6q^{67} + q^{68} - 3q^{70} - 16q^{71} - 3q^{73} + 11q^{74} + 3q^{76} + q^{77} - 14q^{79} - 3q^{80} + 6q^{82} - 26q^{83} - 27q^{85} + 17q^{86} + q^{88} - 20q^{89} - 2q^{91} + 7q^{92} + 21q^{95} - 4q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.56155 −1.59277 −0.796387 0.604787i \(-0.793258\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.56155 1.12626
\(11\) 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
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.68466 1.62127 0.810634 0.585553i \(-0.199123\pi\)
0.810634 + 0.585553i \(0.199123\pi\)
\(18\) 0 0
\(19\) −4.68466 −1.07473 −0.537367 0.843348i \(-0.680581\pi\)
−0.537367 + 0.843348i \(0.680581\pi\)
\(20\) −3.56155 −0.796387
\(21\) 0 0
\(22\) −1.56155 −0.332924
\(23\) 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
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −6.68466 −1.24131 −0.620655 0.784084i \(-0.713133\pi\)
−0.620655 + 0.784084i \(0.713133\pi\)
\(30\) 0 0
\(31\) 6.24621 1.12185 0.560926 0.827866i \(-0.310445\pi\)
0.560926 + 0.827866i \(0.310445\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.68466 −1.14641
\(35\) 3.56155 0.602012
\(36\) 0 0
\(37\) −7.56155 −1.24311 −0.621556 0.783370i \(-0.713499\pi\)
−0.621556 + 0.783370i \(0.713499\pi\)
\(38\) 4.68466 0.759952
\(39\) 0 0
\(40\) 3.56155 0.563131
\(41\) 1.12311 0.175400 0.0876998 0.996147i \(-0.472048\pi\)
0.0876998 + 0.996147i \(0.472048\pi\)
\(42\) 0 0
\(43\) −6.43845 −0.981854 −0.490927 0.871201i \(-0.663342\pi\)
−0.490927 + 0.871201i \(0.663342\pi\)
\(44\) 1.56155 0.235413
\(45\) 0 0
\(46\) −5.56155 −0.820006
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −7.68466 −1.08677
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −12.2462 −1.68215 −0.841073 0.540921i \(-0.818076\pi\)
−0.841073 + 0.540921i \(0.818076\pi\)
\(54\) 0 0
\(55\) −5.56155 −0.749920
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 6.68466 0.877739
\(59\) −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 \(0.359239\pi\)
\(62\) −6.24621 −0.793270
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.56155 −0.441756
\(66\) 0 0
\(67\) −7.12311 −0.870226 −0.435113 0.900376i \(-0.643292\pi\)
−0.435113 + 0.900376i \(0.643292\pi\)
\(68\) 6.68466 0.810634
\(69\) 0 0
\(70\) −3.56155 −0.425687
\(71\) −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 \(0.566833\pi\)
\(74\) 7.56155 0.879013
\(75\) 0 0
\(76\) −4.68466 −0.537367
\(77\) −1.56155 −0.177955
\(78\) 0 0
\(79\) −11.1231 −1.25145 −0.625724 0.780045i \(-0.715196\pi\)
−0.625724 + 0.780045i \(0.715196\pi\)
\(80\) −3.56155 −0.398194
\(81\) 0 0
\(82\) −1.12311 −0.124026
\(83\) −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
\(86\) 6.43845 0.694276
\(87\) 0 0
\(88\) −1.56155 −0.166462
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 5.56155 0.579832
\(93\) 0 0
\(94\) 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 \(0.236834\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 7.68466 0.768466
\(101\) 5.12311 0.509768 0.254884 0.966972i \(-0.417963\pi\)
0.254884 + 0.966972i \(0.417963\pi\)
\(102\) 0 0
\(103\) −0.684658 −0.0674614 −0.0337307 0.999431i \(-0.510739\pi\)
−0.0337307 + 0.999431i \(0.510739\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 12.2462 1.18946
\(107\) 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 \(0.800990\pi\)
\(110\) 5.56155 0.530273
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 4.24621 0.399450 0.199725 0.979852i \(-0.435995\pi\)
0.199725 + 0.979852i \(0.435995\pi\)
\(114\) 0 0
\(115\) −19.8078 −1.84708
\(116\) −6.68466 −0.620655
\(117\) 0 0
\(118\) 2.24621 0.206781
\(119\) −6.68466 −0.612782
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) −6.68466 −0.605201
\(123\) 0 0
\(124\) 6.24621 0.560926
\(125\) −9.56155 −0.855211
\(126\) 0 0
\(127\) −4.87689 −0.432754 −0.216377 0.976310i \(-0.569424\pi\)
−0.216377 + 0.976310i \(0.569424\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 3.56155 0.312369
\(131\) 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
\(134\) 7.12311 0.615343
\(135\) 0 0
\(136\) −6.68466 −0.573205
\(137\) 3.56155 0.304284 0.152142 0.988359i \(-0.451383\pi\)
0.152142 + 0.988359i \(0.451383\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 3.56155 0.301006
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 1.56155 0.130584
\(144\) 0 0
\(145\) 23.8078 1.97713
\(146\) 3.56155 0.294756
\(147\) 0 0
\(148\) −7.56155 −0.621556
\(149\) 17.6155 1.44312 0.721560 0.692352i \(-0.243425\pi\)
0.721560 + 0.692352i \(0.243425\pi\)
\(150\) 0 0
\(151\) 11.8078 0.960902 0.480451 0.877022i \(-0.340473\pi\)
0.480451 + 0.877022i \(0.340473\pi\)
\(152\) 4.68466 0.379976
\(153\) 0 0
\(154\) 1.56155 0.125834
\(155\) −22.2462 −1.78686
\(156\) 0 0
\(157\) −15.5616 −1.24195 −0.620974 0.783832i \(-0.713263\pi\)
−0.620974 + 0.783832i \(0.713263\pi\)
\(158\) 11.1231 0.884907
\(159\) 0 0
\(160\) 3.56155 0.281565
\(161\) −5.56155 −0.438312
\(162\) 0 0
\(163\) −16.8769 −1.32190 −0.660950 0.750430i \(-0.729847\pi\)
−0.660950 + 0.750430i \(0.729847\pi\)
\(164\) 1.12311 0.0876998
\(165\) 0 0
\(166\) 8.87689 0.688981
\(167\) −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
\(170\) 23.8078 1.82597
\(171\) 0 0
\(172\) −6.43845 −0.490927
\(173\) −20.2462 −1.53929 −0.769645 0.638471i \(-0.779567\pi\)
−0.769645 + 0.638471i \(0.779567\pi\)
\(174\) 0 0
\(175\) −7.68466 −0.580906
\(176\) 1.56155 0.117706
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) −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 \(0.502913\pi\)
\(182\) 1.00000 0.0741249
\(183\) 0 0
\(184\) −5.56155 −0.410003
\(185\) 26.9309 1.98000
\(186\) 0 0
\(187\) 10.4384 0.763335
\(188\) 0 0
\(189\) 0 0
\(190\) −16.6847 −1.21043
\(191\) −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 \(0.254466\pi\)
\(194\) −14.4924 −1.04050
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −10.8769 −0.774947 −0.387473 0.921881i \(-0.626652\pi\)
−0.387473 + 0.921881i \(0.626652\pi\)
\(198\) 0 0
\(199\) 6.93087 0.491316 0.245658 0.969357i \(-0.420996\pi\)
0.245658 + 0.969357i \(0.420996\pi\)
\(200\) −7.68466 −0.543387
\(201\) 0 0
\(202\) −5.12311 −0.360460
\(203\) 6.68466 0.469171
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0.684658 0.0477024
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) −7.31534 −0.506013
\(210\) 0 0
\(211\) −15.8078 −1.08825 −0.544126 0.839004i \(-0.683139\pi\)
−0.544126 + 0.839004i \(0.683139\pi\)
\(212\) −12.2462 −0.841073
\(213\) 0 0
\(214\) −8.87689 −0.606812
\(215\) 22.9309 1.56387
\(216\) 0 0
\(217\) −6.24621 −0.424020
\(218\) 16.9309 1.14670
\(219\) 0 0
\(220\) −5.56155 −0.374960
\(221\) 6.68466 0.449659
\(222\) 0 0
\(223\) −23.6155 −1.58141 −0.790706 0.612196i \(-0.790287\pi\)
−0.790706 + 0.612196i \(0.790287\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −4.24621 −0.282454
\(227\) 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 \(0.116932\pi\)
\(230\) 19.8078 1.30609
\(231\) 0 0
\(232\) 6.68466 0.438869
\(233\) −27.3693 −1.79302 −0.896512 0.443020i \(-0.853907\pi\)
−0.896512 + 0.443020i \(0.853907\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2.24621 −0.146216
\(237\) 0 0
\(238\) 6.68466 0.433302
\(239\) −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 \(0.648900\pi\)
\(242\) 8.56155 0.550357
\(243\) 0 0
\(244\) 6.68466 0.427941
\(245\) −3.56155 −0.227539
\(246\) 0 0
\(247\) −4.68466 −0.298078
\(248\) −6.24621 −0.396635
\(249\) 0 0
\(250\) 9.56155 0.604726
\(251\) 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
\(254\) 4.87689 0.306004
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.24621 0.264871 0.132436 0.991192i \(-0.457720\pi\)
0.132436 + 0.991192i \(0.457720\pi\)
\(258\) 0 0
\(259\) 7.56155 0.469852
\(260\) −3.56155 −0.220878
\(261\) 0 0
\(262\) −9.56155 −0.590715
\(263\) 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
\(266\) −4.68466 −0.287235
\(267\) 0 0
\(268\) −7.12311 −0.435113
\(269\) −20.2462 −1.23443 −0.617217 0.786793i \(-0.711740\pi\)
−0.617217 + 0.786793i \(0.711740\pi\)
\(270\) 0 0
\(271\) 4.87689 0.296250 0.148125 0.988969i \(-0.452676\pi\)
0.148125 + 0.988969i \(0.452676\pi\)
\(272\) 6.68466 0.405317
\(273\) 0 0
\(274\) −3.56155 −0.215161
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) −22.4924 −1.35144 −0.675719 0.737159i \(-0.736167\pi\)
−0.675719 + 0.737159i \(0.736167\pi\)
\(278\) −12.0000 −0.719712
\(279\) 0 0
\(280\) −3.56155 −0.212843
\(281\) −16.2462 −0.969168 −0.484584 0.874745i \(-0.661029\pi\)
−0.484584 + 0.874745i \(0.661029\pi\)
\(282\) 0 0
\(283\) 18.2462 1.08462 0.542312 0.840177i \(-0.317549\pi\)
0.542312 + 0.840177i \(0.317549\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −1.56155 −0.0923366
\(287\) −1.12311 −0.0662948
\(288\) 0 0
\(289\) 27.6847 1.62851
\(290\) −23.8078 −1.39804
\(291\) 0 0
\(292\) −3.56155 −0.208424
\(293\) −24.7386 −1.44525 −0.722623 0.691242i \(-0.757064\pi\)
−0.722623 + 0.691242i \(0.757064\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 7.56155 0.439506
\(297\) 0 0
\(298\) −17.6155 −1.02044
\(299\) 5.56155 0.321633
\(300\) 0 0
\(301\) 6.43845 0.371106
\(302\) −11.8078 −0.679460
\(303\) 0 0
\(304\) −4.68466 −0.268684
\(305\) −23.8078 −1.36323
\(306\) 0 0
\(307\) 26.2462 1.49795 0.748975 0.662598i \(-0.230546\pi\)
0.748975 + 0.662598i \(0.230546\pi\)
\(308\) −1.56155 −0.0889777
\(309\) 0 0
\(310\) 22.2462 1.26350
\(311\) 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 \(0.625729\pi\)
\(314\) 15.5616 0.878189
\(315\) 0 0
\(316\) −11.1231 −0.625724
\(317\) −2.87689 −0.161582 −0.0807912 0.996731i \(-0.525745\pi\)
−0.0807912 + 0.996731i \(0.525745\pi\)
\(318\) 0 0
\(319\) −10.4384 −0.584441
\(320\) −3.56155 −0.199097
\(321\) 0 0
\(322\) 5.56155 0.309933
\(323\) −31.3153 −1.74243
\(324\) 0 0
\(325\) 7.68466 0.426268
\(326\) 16.8769 0.934725
\(327\) 0 0
\(328\) −1.12311 −0.0620131
\(329\) 0 0
\(330\) 0 0
\(331\) 13.3693 0.734844 0.367422 0.930054i \(-0.380240\pi\)
0.367422 + 0.930054i \(0.380240\pi\)
\(332\) −8.87689 −0.487183
\(333\) 0 0
\(334\) 22.9309 1.25472
\(335\) 25.3693 1.38607
\(336\) 0 0
\(337\) 4.43845 0.241778 0.120889 0.992666i \(-0.461426\pi\)
0.120889 + 0.992666i \(0.461426\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) −23.8078 −1.29116
\(341\) 9.75379 0.528197
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 6.43845 0.347138
\(345\) 0 0
\(346\) 20.2462 1.08844
\(347\) 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 \(0.433459\pi\)
\(350\) 7.68466 0.410762
\(351\) 0 0
\(352\) −1.56155 −0.0832310
\(353\) −30.4924 −1.62295 −0.811474 0.584389i \(-0.801334\pi\)
−0.811474 + 0.584389i \(0.801334\pi\)
\(354\) 0 0
\(355\) 28.4924 1.51222
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 16.4924 0.871652
\(359\) 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
\(362\) 0.246211 0.0129406
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) 12.6847 0.663945
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 5.56155 0.289916
\(369\) 0 0
\(370\) −26.9309 −1.40007
\(371\) 12.2462 0.635792
\(372\) 0 0
\(373\) −17.6155 −0.912097 −0.456049 0.889955i \(-0.650736\pi\)
−0.456049 + 0.889955i \(0.650736\pi\)
\(374\) −10.4384 −0.539759
\(375\) 0 0
\(376\) 0 0
\(377\) −6.68466 −0.344277
\(378\) 0 0
\(379\) 34.2462 1.75911 0.879555 0.475797i \(-0.157840\pi\)
0.879555 + 0.475797i \(0.157840\pi\)
\(380\) 16.6847 0.855905
\(381\) 0 0
\(382\) 14.9309 0.763930
\(383\) −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
\(386\) −19.3693 −0.985872
\(387\) 0 0
\(388\) 14.4924 0.735741
\(389\) 28.7386 1.45711 0.728553 0.684989i \(-0.240193\pi\)
0.728553 + 0.684989i \(0.240193\pi\)
\(390\) 0 0
\(391\) 37.1771 1.88013
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 10.8769 0.547970
\(395\) 39.6155 1.99327
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) −6.93087 −0.347413
\(399\) 0 0
\(400\) 7.68466 0.384233
\(401\) −8.24621 −0.411796 −0.205898 0.978573i \(-0.566012\pi\)
−0.205898 + 0.978573i \(0.566012\pi\)
\(402\) 0 0
\(403\) 6.24621 0.311146
\(404\) 5.12311 0.254884
\(405\) 0 0
\(406\) −6.68466 −0.331754
\(407\) −11.8078 −0.585289
\(408\) 0 0
\(409\) −4.93087 −0.243816 −0.121908 0.992541i \(-0.538901\pi\)
−0.121908 + 0.992541i \(0.538901\pi\)
\(410\) 4.00000 0.197546
\(411\) 0 0
\(412\) −0.684658 −0.0337307
\(413\) 2.24621 0.110529
\(414\) 0 0
\(415\) 31.6155 1.55195
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 7.31534 0.357805
\(419\) 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 \(0.529158\pi\)
\(422\) 15.8078 0.769510
\(423\) 0 0
\(424\) 12.2462 0.594729
\(425\) 51.3693 2.49178
\(426\) 0 0
\(427\) −6.68466 −0.323493
\(428\) 8.87689 0.429081
\(429\) 0 0
\(430\) −22.9309 −1.10582
\(431\) 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 \(0.771760\pi\)
\(434\) 6.24621 0.299828
\(435\) 0 0
\(436\) −16.9309 −0.810842
\(437\) −26.0540 −1.24633
\(438\) 0 0
\(439\) −6.93087 −0.330792 −0.165396 0.986227i \(-0.552890\pi\)
−0.165396 + 0.986227i \(0.552890\pi\)
\(440\) 5.56155 0.265137
\(441\) 0 0
\(442\) −6.68466 −0.317957
\(443\) 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
\(446\) 23.6155 1.11823
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −10.6847 −0.504240 −0.252120 0.967696i \(-0.581128\pi\)
−0.252120 + 0.967696i \(0.581128\pi\)
\(450\) 0 0
\(451\) 1.75379 0.0825827
\(452\) 4.24621 0.199725
\(453\) 0 0
\(454\) −7.12311 −0.334304
\(455\) 3.56155 0.166968
\(456\) 0 0
\(457\) 27.3693 1.28028 0.640141 0.768257i \(-0.278876\pi\)
0.640141 + 0.768257i \(0.278876\pi\)
\(458\) −28.2462 −1.31986
\(459\) 0 0
\(460\) −19.8078 −0.923542
\(461\) 28.0540 1.30660 0.653302 0.757097i \(-0.273383\pi\)
0.653302 + 0.757097i \(0.273383\pi\)
\(462\) 0 0
\(463\) −8.68466 −0.403610 −0.201805 0.979426i \(-0.564681\pi\)
−0.201805 + 0.979426i \(0.564681\pi\)
\(464\) −6.68466 −0.310327
\(465\) 0 0
\(466\) 27.3693 1.26786
\(467\) 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
\(471\) 0 0
\(472\) 2.24621 0.103390
\(473\) −10.0540 −0.462282
\(474\) 0 0
\(475\) −36.0000 −1.65179
\(476\) −6.68466 −0.306391
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) −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
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) −8.56155 −0.389161
\(485\) −51.6155 −2.34374
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −6.68466 −0.302600
\(489\) 0 0
\(490\) 3.56155 0.160895
\(491\) 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
\(494\) 4.68466 0.210773
\(495\) 0 0
\(496\) 6.24621 0.280463
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) −9.56155 −0.427606
\(501\) 0 0
\(502\) −7.80776 −0.348478
\(503\) 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
\(506\) −8.68466 −0.386080
\(507\) 0 0
\(508\) −4.87689 −0.216377
\(509\) 12.0540 0.534283 0.267142 0.963657i \(-0.413921\pi\)
0.267142 + 0.963657i \(0.413921\pi\)
\(510\) 0 0
\(511\) 3.56155 0.157554
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −4.24621 −0.187292
\(515\) 2.43845 0.107451
\(516\) 0 0
\(517\) 0 0
\(518\) −7.56155 −0.332236
\(519\) 0 0
\(520\) 3.56155 0.156184
\(521\) −2.68466 −0.117617 −0.0588085 0.998269i \(-0.518730\pi\)
−0.0588085 + 0.998269i \(0.518730\pi\)
\(522\) 0 0
\(523\) 21.7538 0.951227 0.475613 0.879654i \(-0.342226\pi\)
0.475613 + 0.879654i \(0.342226\pi\)
\(524\) 9.56155 0.417698
\(525\) 0 0
\(526\) −30.2462 −1.31880
\(527\) 41.7538 1.81882
\(528\) 0 0
\(529\) 7.93087 0.344820
\(530\) −43.6155 −1.89454
\(531\) 0 0
\(532\) 4.68466 0.203106
\(533\) 1.12311 0.0486471
\(534\) 0 0
\(535\) −31.6155 −1.36686
\(536\) 7.12311 0.307671
\(537\) 0 0
\(538\) 20.2462 0.872876
\(539\) 1.56155 0.0672608
\(540\) 0 0
\(541\) −32.9309 −1.41581 −0.707904 0.706308i \(-0.750359\pi\)
−0.707904 + 0.706308i \(0.750359\pi\)
\(542\) −4.87689 −0.209481
\(543\) 0 0
\(544\) −6.68466 −0.286602
\(545\) 60.3002 2.58298
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 3.56155 0.152142
\(549\) 0 0
\(550\) −12.0000 −0.511682
\(551\) 31.3153 1.33408
\(552\) 0 0
\(553\) 11.1231 0.473003
\(554\) 22.4924 0.955611
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) 3.36932 0.142763 0.0713813 0.997449i \(-0.477259\pi\)
0.0713813 + 0.997449i \(0.477259\pi\)
\(558\) 0 0
\(559\) −6.43845 −0.272317
\(560\) 3.56155 0.150503
\(561\) 0 0
\(562\) 16.2462 0.685305
\(563\) 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
\(566\) −18.2462 −0.766945
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −41.2311 −1.72850 −0.864248 0.503066i \(-0.832205\pi\)
−0.864248 + 0.503066i \(0.832205\pi\)
\(570\) 0 0
\(571\) 18.2462 0.763580 0.381790 0.924249i \(-0.375308\pi\)
0.381790 + 0.924249i \(0.375308\pi\)
\(572\) 1.56155 0.0652918
\(573\) 0 0
\(574\) 1.12311 0.0468775
\(575\) 42.7386 1.78232
\(576\) 0 0
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) −27.6847 −1.15153
\(579\) 0 0
\(580\) 23.8078 0.988564
\(581\) 8.87689 0.368276
\(582\) 0 0
\(583\) −19.1231 −0.791998
\(584\) 3.56155 0.147378
\(585\) 0 0
\(586\) 24.7386 1.02194
\(587\) −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
\(590\) −8.00000 −0.329355
\(591\) 0 0
\(592\) −7.56155 −0.310778
\(593\) 20.2462 0.831412 0.415706 0.909499i \(-0.363534\pi\)
0.415706 + 0.909499i \(0.363534\pi\)
\(594\) 0 0
\(595\) 23.8078 0.976023
\(596\) 17.6155 0.721560
\(597\) 0 0
\(598\) −5.56155 −0.227429
\(599\) −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 \(0.571206\pi\)
\(602\) −6.43845 −0.262412
\(603\) 0 0
\(604\) 11.8078 0.480451
\(605\) 30.4924 1.23969
\(606\) 0 0
\(607\) −18.4384 −0.748393 −0.374197 0.927349i \(-0.622082\pi\)
−0.374197 + 0.927349i \(0.622082\pi\)
\(608\) 4.68466 0.189988
\(609\) 0 0
\(610\) 23.8078 0.963948
\(611\) 0 0
\(612\) 0 0
\(613\) 44.5464 1.79921 0.899606 0.436702i \(-0.143854\pi\)
0.899606 + 0.436702i \(0.143854\pi\)
\(614\) −26.2462 −1.05921
\(615\) 0 0
\(616\) 1.56155 0.0629168
\(617\) 33.4233 1.34557 0.672786 0.739838i \(-0.265098\pi\)
0.672786 + 0.739838i \(0.265098\pi\)
\(618\) 0 0
\(619\) 19.3153 0.776349 0.388175 0.921586i \(-0.373106\pi\)
0.388175 + 0.921586i \(0.373106\pi\)
\(620\) −22.2462 −0.893429
\(621\) 0 0
\(622\) −12.8769 −0.516316
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) 13.6155 0.544186
\(627\) 0 0
\(628\) −15.5616 −0.620974
\(629\) −50.5464 −2.01542
\(630\) 0 0
\(631\) −22.9309 −0.912864 −0.456432 0.889758i \(-0.650873\pi\)
−0.456432 + 0.889758i \(0.650873\pi\)
\(632\) 11.1231 0.442453
\(633\) 0 0
\(634\) 2.87689 0.114256
\(635\) 17.3693 0.689280
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 10.4384 0.413262
\(639\) 0 0
\(640\) 3.56155 0.140783
\(641\) 20.2462 0.799677 0.399839 0.916586i \(-0.369066\pi\)
0.399839 + 0.916586i \(0.369066\pi\)
\(642\) 0 0
\(643\) 16.1922 0.638559 0.319280 0.947661i \(-0.396559\pi\)
0.319280 + 0.947661i \(0.396559\pi\)
\(644\) −5.56155 −0.219156
\(645\) 0 0
\(646\) 31.3153 1.23209
\(647\) 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
\(650\) −7.68466 −0.301417
\(651\) 0 0
\(652\) −16.8769 −0.660950
\(653\) 16.9309 0.662556 0.331278 0.943533i \(-0.392520\pi\)
0.331278 + 0.943533i \(0.392520\pi\)
\(654\) 0 0
\(655\) −34.0540 −1.33060
\(656\) 1.12311 0.0438499
\(657\) 0 0
\(658\) 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 \(0.501524\pi\)
\(662\) −13.3693 −0.519613
\(663\) 0 0
\(664\) 8.87689 0.344490
\(665\) −16.6847 −0.647003
\(666\) 0 0
\(667\) −37.1771 −1.43950
\(668\) −22.9309 −0.887222
\(669\) 0 0
\(670\) −25.3693 −0.980102
\(671\) 10.4384 0.402972
\(672\) 0 0
\(673\) −33.8078 −1.30319 −0.651597 0.758566i \(-0.725901\pi\)
−0.651597 + 0.758566i \(0.725901\pi\)
\(674\) −4.43845 −0.170963
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −2.49242 −0.0957916 −0.0478958 0.998852i \(-0.515252\pi\)
−0.0478958 + 0.998852i \(0.515252\pi\)
\(678\) 0 0
\(679\) −14.4924 −0.556168
\(680\) 23.8078 0.912986
\(681\) 0 0
\(682\) −9.75379 −0.373492
\(683\) −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
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −6.43845 −0.245463
\(689\) −12.2462 −0.466543
\(690\) 0 0
\(691\) 16.4924 0.627401 0.313701 0.949522i \(-0.398431\pi\)
0.313701 + 0.949522i \(0.398431\pi\)
\(692\) −20.2462 −0.769645
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) −42.7386 −1.62117
\(696\) 0 0
\(697\) 7.50758 0.284370
\(698\) −7.75379 −0.293485
\(699\) 0 0
\(700\) −7.68466 −0.290453
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 35.4233 1.33601
\(704\) 1.56155 0.0588532
\(705\) 0 0
\(706\) 30.4924 1.14760
\(707\) −5.12311 −0.192674
\(708\) 0 0
\(709\) −16.2462 −0.610139 −0.305070 0.952330i \(-0.598680\pi\)
−0.305070 + 0.952330i \(0.598680\pi\)
\(710\) −28.4924 −1.06930
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) 34.7386 1.30097
\(714\) 0 0
\(715\) −5.56155 −0.207990
\(716\) −16.4924 −0.616351
\(717\) 0 0
\(718\) −26.7386 −0.997877
\(719\) −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
\(722\) −2.94602 −0.109640
\(723\) 0 0
\(724\) −0.246211 −0.00915037
\(725\) −51.3693 −1.90781
\(726\) 0 0
\(727\) −6.93087 −0.257052 −0.128526 0.991706i \(-0.541025\pi\)
−0.128526 + 0.991706i \(0.541025\pi\)
\(728\) 1.00000 0.0370625
\(729\) 0 0
\(730\) −12.6847 −0.469480
\(731\) −43.0388 −1.59185
\(732\) 0 0
\(733\) −41.6155 −1.53710 −0.768552 0.639787i \(-0.779023\pi\)
−0.768552 + 0.639787i \(0.779023\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) −5.56155 −0.205002
\(737\) −11.1231 −0.409725
\(738\) 0 0
\(739\) −32.1080 −1.18111 −0.590555 0.806997i \(-0.701091\pi\)
−0.590555 + 0.806997i \(0.701091\pi\)
\(740\) 26.9309 0.989998
\(741\) 0 0
\(742\) −12.2462 −0.449573
\(743\) −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
\(746\) 17.6155 0.644950
\(747\) 0 0
\(748\) 10.4384 0.381667
\(749\) −8.87689 −0.324355
\(750\) 0 0
\(751\) −6.24621 −0.227927 −0.113964 0.993485i \(-0.536355\pi\)
−0.113964 + 0.993485i \(0.536355\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 6.68466 0.243441
\(755\) −42.0540 −1.53050
\(756\) 0 0
\(757\) 34.4924 1.25365 0.626824 0.779161i \(-0.284354\pi\)
0.626824 + 0.779161i \(0.284354\pi\)
\(758\) −34.2462 −1.24388
\(759\) 0 0
\(760\) −16.6847 −0.605216
\(761\) −5.12311 −0.185712 −0.0928562 0.995680i \(-0.529600\pi\)
−0.0928562 + 0.995680i \(0.529600\pi\)
\(762\) 0 0
\(763\) 16.9309 0.612939
\(764\) −14.9309 −0.540180
\(765\) 0 0
\(766\) 27.4233 0.990844
\(767\) −2.24621 −0.0811060
\(768\) 0 0
\(769\) −16.4384 −0.592786 −0.296393 0.955066i \(-0.595784\pi\)
−0.296393 + 0.955066i \(0.595784\pi\)
\(770\) −5.56155 −0.200424
\(771\) 0 0
\(772\) 19.3693 0.697117
\(773\) −52.9309 −1.90379 −0.951896 0.306423i \(-0.900868\pi\)
−0.951896 + 0.306423i \(0.900868\pi\)
\(774\) 0 0
\(775\) 48.0000 1.72421
\(776\) −14.4924 −0.520248
\(777\) 0 0
\(778\) −28.7386 −1.03033
\(779\) −5.26137 −0.188508
\(780\) 0 0
\(781\) −12.4924 −0.447014
\(782\) −37.1771 −1.32945
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 55.4233 1.97814
\(786\) 0 0
\(787\) 20.3002 0.723624 0.361812 0.932251i \(-0.382158\pi\)
0.361812 + 0.932251i \(0.382158\pi\)
\(788\) −10.8769 −0.387473
\(789\) 0 0
\(790\) −39.6155 −1.40946
\(791\) −4.24621 −0.150978
\(792\) 0 0
\(793\) 6.68466 0.237379
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) 6.93087 0.245658
\(797\) −31.3693 −1.11116 −0.555579 0.831464i \(-0.687503\pi\)
−0.555579 + 0.831464i \(0.687503\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −7.68466 −0.271694
\(801\) 0 0
\(802\) 8.24621 0.291184
\(803\) −5.56155 −0.196263
\(804\) 0 0
\(805\) 19.8078 0.698132
\(806\) −6.24621 −0.220013
\(807\) 0 0
\(808\) −5.12311 −0.180230
\(809\) 2.49242 0.0876289 0.0438145 0.999040i \(-0.486049\pi\)
0.0438145 + 0.999040i \(0.486049\pi\)
\(810\) 0 0
\(811\) 41.5616 1.45942 0.729712 0.683755i \(-0.239654\pi\)
0.729712 + 0.683755i \(0.239654\pi\)
\(812\) 6.68466 0.234586
\(813\) 0 0
\(814\) 11.8078 0.413862
\(815\) 60.1080 2.10549
\(816\) 0 0
\(817\) 30.1619 1.05523
\(818\) 4.93087 0.172404
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) −14.3845 −0.502022 −0.251011 0.967984i \(-0.580763\pi\)
−0.251011 + 0.967984i \(0.580763\pi\)
\(822\) 0 0
\(823\) 4.49242 0.156596 0.0782980 0.996930i \(-0.475051\pi\)
0.0782980 + 0.996930i \(0.475051\pi\)
\(824\) 0.684658 0.0238512
\(825\) 0 0
\(826\) −2.24621 −0.0781557
\(827\) −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 \(0.265527\pi\)
\(830\) −31.6155 −1.09739
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) 6.68466 0.231610
\(834\) 0 0
\(835\) 81.6695 2.82629
\(836\) −7.31534 −0.253006
\(837\) 0 0
\(838\) −36.6847 −1.26725
\(839\) 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
\(842\) 3.75379 0.129364
\(843\) 0 0
\(844\) −15.8078 −0.544126
\(845\) −3.56155 −0.122521
\(846\) 0 0
\(847\) 8.56155 0.294178
\(848\) −12.2462 −0.420537
\(849\) 0 0
\(850\) −51.3693 −1.76195
\(851\) −42.0540 −1.44159
\(852\) 0 0
\(853\) −27.3693 −0.937108 −0.468554 0.883435i \(-0.655225\pi\)
−0.468554 + 0.883435i \(0.655225\pi\)
\(854\) 6.68466 0.228744
\(855\) 0 0
\(856\) −8.87689 −0.303406
\(857\) 34.4924 1.17824 0.589119 0.808046i \(-0.299475\pi\)
0.589119 + 0.808046i \(0.299475\pi\)
\(858\) 0 0
\(859\) 5.75379 0.196317 0.0981584 0.995171i \(-0.468705\pi\)
0.0981584 + 0.995171i \(0.468705\pi\)
\(860\) 22.9309 0.781936
\(861\) 0 0
\(862\) −33.3693 −1.13656
\(863\) 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
\(866\) 31.3693 1.06597
\(867\) 0 0
\(868\) −6.24621 −0.212010
\(869\) −17.3693 −0.589214
\(870\) 0 0
\(871\) −7.12311 −0.241357
\(872\) 16.9309 0.573352
\(873\) 0 0
\(874\) 26.0540 0.881289
\(875\) 9.56155 0.323239
\(876\) 0 0
\(877\) −40.2462 −1.35902 −0.679509 0.733667i \(-0.737807\pi\)
−0.679509 + 0.733667i \(0.737807\pi\)
\(878\) 6.93087 0.233906
\(879\) 0 0
\(880\) −5.56155 −0.187480
\(881\) 19.1771 0.646092 0.323046 0.946383i \(-0.395293\pi\)
0.323046 + 0.946383i \(0.395293\pi\)
\(882\) 0 0
\(883\) 1.56155 0.0525504 0.0262752 0.999655i \(-0.491635\pi\)
0.0262752 + 0.999655i \(0.491635\pi\)
\(884\) 6.68466 0.224829
\(885\) 0 0
\(886\) −5.36932 −0.180386
\(887\) 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
\(890\) −35.6155 −1.19384
\(891\) 0 0
\(892\) −23.6155 −0.790706
\(893\) 0 0
\(894\) 0 0
\(895\) 58.7386 1.96342
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 10.6847 0.356552
\(899\) −41.7538 −1.39257
\(900\) 0 0
\(901\) −81.8617 −2.72721
\(902\) −1.75379 −0.0583948
\(903\) 0 0
\(904\) −4.24621 −0.141227
\(905\) 0.876894 0.0291490
\(906\) 0 0
\(907\) −7.50758 −0.249285 −0.124643 0.992202i \(-0.539778\pi\)
−0.124643 + 0.992202i \(0.539778\pi\)
\(908\) 7.12311 0.236389
\(909\) 0 0
\(910\) −3.56155 −0.118064
\(911\) 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
\(914\) −27.3693 −0.905297
\(915\) 0 0
\(916\) 28.2462 0.933281
\(917\) −9.56155 −0.315750
\(918\) 0 0
\(919\) 19.1231 0.630813 0.315407 0.948957i \(-0.397859\pi\)
0.315407 + 0.948957i \(0.397859\pi\)
\(920\) 19.8078 0.653043
\(921\) 0 0
\(922\) −28.0540 −0.923908
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) −58.1080 −1.91058
\(926\) 8.68466 0.285396
\(927\) 0 0
\(928\) 6.68466 0.219435
\(929\) −25.6155 −0.840418 −0.420209 0.907427i \(-0.638043\pi\)
−0.420209 + 0.907427i \(0.638043\pi\)
\(930\) 0 0
\(931\) −4.68466 −0.153533
\(932\) −27.3693 −0.896512
\(933\) 0 0
\(934\) −3.31534 −0.108481
\(935\) −37.1771 −1.21582
\(936\) 0 0
\(937\) −46.9848 −1.53493 −0.767464 0.641092i \(-0.778482\pi\)
−0.767464 + 0.641092i \(0.778482\pi\)
\(938\) −7.12311 −0.232578
\(939\) 0 0
\(940\) 0 0
\(941\) 34.0000 1.10837 0.554184 0.832394i \(-0.313030\pi\)
0.554184 + 0.832394i \(0.313030\pi\)
\(942\) 0 0
\(943\) 6.24621 0.203405
\(944\) −2.24621 −0.0731079
\(945\) 0 0
\(946\) 10.0540 0.326883
\(947\) 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
\(950\) 36.0000 1.16799
\(951\) 0 0
\(952\) 6.68466 0.216651
\(953\) −11.3693 −0.368288 −0.184144 0.982899i \(-0.558951\pi\)
−0.184144 + 0.982899i \(0.558951\pi\)
\(954\) 0 0
\(955\) 53.1771 1.72077
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) 32.3002 1.04357
\(959\) −3.56155 −0.115009
\(960\) 0 0
\(961\) 8.01515 0.258553
\(962\) 7.56155 0.243794
\(963\) 0 0
\(964\) −14.0000 −0.450910
\(965\) −68.9848 −2.22070
\(966\) 0 0
\(967\) −21.5616 −0.693373 −0.346686 0.937981i \(-0.612693\pi\)
−0.346686 + 0.937981i \(0.612693\pi\)
\(968\) 8.56155 0.275179
\(969\) 0 0
\(970\) 51.6155 1.65727
\(971\) −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
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 6.68466 0.213971
\(977\) 22.6847 0.725747 0.362873 0.931839i \(-0.381796\pi\)
0.362873 + 0.931839i \(0.381796\pi\)
\(978\) 0 0
\(979\) −15.6155 −0.499074
\(980\) −3.56155 −0.113770
\(981\) 0 0
\(982\) −8.87689 −0.283273
\(983\) 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
\(986\) 44.6847 1.42305
\(987\) 0 0
\(988\) −4.68466 −0.149039
\(989\) −35.8078 −1.13862
\(990\) 0 0
\(991\) −7.61553 −0.241915 −0.120958 0.992658i \(-0.538597\pi\)
−0.120958 + 0.992658i \(0.538597\pi\)
\(992\) −6.24621 −0.198317
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) −24.6847 −0.782556
\(996\) 0 0
\(997\) 40.7386 1.29021 0.645103 0.764096i \(-0.276815\pi\)
0.645103 + 0.764096i \(0.276815\pi\)
\(998\) −36.0000 −1.13956
\(999\) 0 0
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 1638.2.a.u.1.1 2
3.2 odd 2 546.2.a.j.1.2 2
12.11 even 2 4368.2.a.be.1.2 2
21.20 even 2 3822.2.a.bo.1.1 2
39.38 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 3.2 odd 2
1638.2.a.u.1.1 2 1.1 even 1 trivial
3822.2.a.bo.1.1 2 21.20 even 2
4368.2.a.be.1.2 2 12.11 even 2
7098.2.a.bl.1.1 2 39.38 odd 2