Properties

Label 6240.2.a.bn.1.1
Level $6240$
Weight $2$
Character 6240.1
Self dual yes
Analytic conductor $49.827$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.8266508613\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6240.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} +4.82843 q^{17} +2.82843 q^{19} +2.82843 q^{21} -2.82843 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.82843 q^{29} +9.65685 q^{31} +4.00000 q^{33} -2.82843 q^{35} +7.65685 q^{37} +1.00000 q^{39} +0.343146 q^{41} -9.65685 q^{43} +1.00000 q^{45} +1.65685 q^{47} +1.00000 q^{49} -4.82843 q^{51} +13.3137 q^{53} -4.00000 q^{55} -2.82843 q^{57} -4.00000 q^{59} -2.00000 q^{61} -2.82843 q^{63} -1.00000 q^{65} +2.82843 q^{69} -8.00000 q^{71} +8.82843 q^{73} -1.00000 q^{75} +11.3137 q^{77} +5.65685 q^{79} +1.00000 q^{81} +7.31371 q^{83} +4.82843 q^{85} +4.82843 q^{87} -7.65685 q^{89} +2.82843 q^{91} -9.65685 q^{93} +2.82843 q^{95} -18.4853 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} - 8 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 8 q^{31} + 8 q^{33} + 4 q^{37} + 2 q^{39} + 12 q^{41} - 8 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} - 4 q^{51} + 4 q^{53} - 8 q^{55} - 8 q^{59} - 4 q^{61} - 2 q^{65} - 16 q^{71} + 12 q^{73} - 2 q^{75} + 2 q^{81} - 8 q^{83} + 4 q^{85} + 4 q^{87} - 4 q^{89} - 8 q^{93} - 20 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 4.82843 1.17107 0.585533 0.810649i \(-0.300885\pi\)
0.585533 + 0.810649i \(0.300885\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 2.82843 0.617213
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.82843 −0.896616 −0.448308 0.893879i \(-0.647973\pi\)
−0.448308 + 0.893879i \(0.647973\pi\)
\(30\) 0 0
\(31\) 9.65685 1.73442 0.867211 0.497941i \(-0.165910\pi\)
0.867211 + 0.497941i \(0.165910\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) −2.82843 −0.478091
\(36\) 0 0
\(37\) 7.65685 1.25878 0.629390 0.777090i \(-0.283305\pi\)
0.629390 + 0.777090i \(0.283305\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 0.343146 0.0535904 0.0267952 0.999641i \(-0.491470\pi\)
0.0267952 + 0.999641i \(0.491470\pi\)
\(42\) 0 0
\(43\) −9.65685 −1.47266 −0.736328 0.676625i \(-0.763442\pi\)
−0.736328 + 0.676625i \(0.763442\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 1.65685 0.241677 0.120839 0.992672i \(-0.461442\pi\)
0.120839 + 0.992672i \(0.461442\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.82843 −0.676115
\(52\) 0 0
\(53\) 13.3137 1.82878 0.914389 0.404836i \(-0.132671\pi\)
0.914389 + 0.404836i \(0.132671\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) −2.82843 −0.374634
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −2.82843 −0.356348
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 2.82843 0.340503
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 8.82843 1.03329 0.516645 0.856200i \(-0.327181\pi\)
0.516645 + 0.856200i \(0.327181\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 11.3137 1.28932
\(78\) 0 0
\(79\) 5.65685 0.636446 0.318223 0.948016i \(-0.396914\pi\)
0.318223 + 0.948016i \(0.396914\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.31371 0.802784 0.401392 0.915906i \(-0.368527\pi\)
0.401392 + 0.915906i \(0.368527\pi\)
\(84\) 0 0
\(85\) 4.82843 0.523716
\(86\) 0 0
\(87\) 4.82843 0.517662
\(88\) 0 0
\(89\) −7.65685 −0.811625 −0.405812 0.913956i \(-0.633011\pi\)
−0.405812 + 0.913956i \(0.633011\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 0 0
\(93\) −9.65685 −1.00137
\(94\) 0 0
\(95\) 2.82843 0.290191
\(96\) 0 0
\(97\) −18.4853 −1.87690 −0.938448 0.345421i \(-0.887736\pi\)
−0.938448 + 0.345421i \(0.887736\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −10.4853 −1.04332 −0.521662 0.853152i \(-0.674688\pi\)
−0.521662 + 0.853152i \(0.674688\pi\)
\(102\) 0 0
\(103\) −13.6569 −1.34565 −0.672825 0.739802i \(-0.734919\pi\)
−0.672825 + 0.739802i \(0.734919\pi\)
\(104\) 0 0
\(105\) 2.82843 0.276026
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −14.4853 −1.38744 −0.693719 0.720246i \(-0.744029\pi\)
−0.693719 + 0.720246i \(0.744029\pi\)
\(110\) 0 0
\(111\) −7.65685 −0.726756
\(112\) 0 0
\(113\) 2.48528 0.233796 0.116898 0.993144i \(-0.462705\pi\)
0.116898 + 0.993144i \(0.462705\pi\)
\(114\) 0 0
\(115\) −2.82843 −0.263752
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −13.6569 −1.25192
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −0.343146 −0.0309404
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 13.6569 1.21185 0.605925 0.795522i \(-0.292803\pi\)
0.605925 + 0.795522i \(0.292803\pi\)
\(128\) 0 0
\(129\) 9.65685 0.850239
\(130\) 0 0
\(131\) −5.17157 −0.451842 −0.225921 0.974146i \(-0.572539\pi\)
−0.225921 + 0.974146i \(0.572539\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −9.31371 −0.795724 −0.397862 0.917445i \(-0.630248\pi\)
−0.397862 + 0.917445i \(0.630248\pi\)
\(138\) 0 0
\(139\) −13.6569 −1.15836 −0.579180 0.815200i \(-0.696627\pi\)
−0.579180 + 0.815200i \(0.696627\pi\)
\(140\) 0 0
\(141\) −1.65685 −0.139532
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) −4.82843 −0.400979
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −4.34315 −0.355804 −0.177902 0.984048i \(-0.556931\pi\)
−0.177902 + 0.984048i \(0.556931\pi\)
\(150\) 0 0
\(151\) −12.9706 −1.05553 −0.527765 0.849391i \(-0.676970\pi\)
−0.527765 + 0.849391i \(0.676970\pi\)
\(152\) 0 0
\(153\) 4.82843 0.390355
\(154\) 0 0
\(155\) 9.65685 0.775657
\(156\) 0 0
\(157\) −1.31371 −0.104845 −0.0524227 0.998625i \(-0.516694\pi\)
−0.0524227 + 0.998625i \(0.516694\pi\)
\(158\) 0 0
\(159\) −13.3137 −1.05585
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) −16.9706 −1.32924 −0.664619 0.747183i \(-0.731406\pi\)
−0.664619 + 0.747183i \(0.731406\pi\)
\(164\) 0 0
\(165\) 4.00000 0.311400
\(166\) 0 0
\(167\) 7.31371 0.565952 0.282976 0.959127i \(-0.408678\pi\)
0.282976 + 0.959127i \(0.408678\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) 0 0
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) −2.82843 −0.213809
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) 8.48528 0.634220 0.317110 0.948389i \(-0.397288\pi\)
0.317110 + 0.948389i \(0.397288\pi\)
\(180\) 0 0
\(181\) 8.34315 0.620141 0.310071 0.950714i \(-0.399647\pi\)
0.310071 + 0.950714i \(0.399647\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 7.65685 0.562943
\(186\) 0 0
\(187\) −19.3137 −1.41236
\(188\) 0 0
\(189\) 2.82843 0.205738
\(190\) 0 0
\(191\) 3.31371 0.239772 0.119886 0.992788i \(-0.461747\pi\)
0.119886 + 0.992788i \(0.461747\pi\)
\(192\) 0 0
\(193\) 17.7990 1.28120 0.640600 0.767875i \(-0.278686\pi\)
0.640600 + 0.767875i \(0.278686\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −13.3137 −0.948562 −0.474281 0.880373i \(-0.657292\pi\)
−0.474281 + 0.880373i \(0.657292\pi\)
\(198\) 0 0
\(199\) 16.9706 1.20301 0.601506 0.798869i \(-0.294568\pi\)
0.601506 + 0.798869i \(0.294568\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 13.6569 0.958523
\(204\) 0 0
\(205\) 0.343146 0.0239663
\(206\) 0 0
\(207\) −2.82843 −0.196589
\(208\) 0 0
\(209\) −11.3137 −0.782586
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) −9.65685 −0.658592
\(216\) 0 0
\(217\) −27.3137 −1.85418
\(218\) 0 0
\(219\) −8.82843 −0.596570
\(220\) 0 0
\(221\) −4.82843 −0.324795
\(222\) 0 0
\(223\) 2.82843 0.189405 0.0947027 0.995506i \(-0.469810\pi\)
0.0947027 + 0.995506i \(0.469810\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −17.6569 −1.17193 −0.585963 0.810338i \(-0.699284\pi\)
−0.585963 + 0.810338i \(0.699284\pi\)
\(228\) 0 0
\(229\) −3.17157 −0.209583 −0.104792 0.994494i \(-0.533418\pi\)
−0.104792 + 0.994494i \(0.533418\pi\)
\(230\) 0 0
\(231\) −11.3137 −0.744387
\(232\) 0 0
\(233\) 9.51472 0.623330 0.311665 0.950192i \(-0.399113\pi\)
0.311665 + 0.950192i \(0.399113\pi\)
\(234\) 0 0
\(235\) 1.65685 0.108081
\(236\) 0 0
\(237\) −5.65685 −0.367452
\(238\) 0 0
\(239\) −16.9706 −1.09773 −0.548867 0.835910i \(-0.684941\pi\)
−0.548867 + 0.835910i \(0.684941\pi\)
\(240\) 0 0
\(241\) −1.31371 −0.0846234 −0.0423117 0.999104i \(-0.513472\pi\)
−0.0423117 + 0.999104i \(0.513472\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −2.82843 −0.179969
\(248\) 0 0
\(249\) −7.31371 −0.463487
\(250\) 0 0
\(251\) −10.8284 −0.683484 −0.341742 0.939794i \(-0.611017\pi\)
−0.341742 + 0.939794i \(0.611017\pi\)
\(252\) 0 0
\(253\) 11.3137 0.711287
\(254\) 0 0
\(255\) −4.82843 −0.302368
\(256\) 0 0
\(257\) −8.82843 −0.550702 −0.275351 0.961344i \(-0.588794\pi\)
−0.275351 + 0.961344i \(0.588794\pi\)
\(258\) 0 0
\(259\) −21.6569 −1.34569
\(260\) 0 0
\(261\) −4.82843 −0.298872
\(262\) 0 0
\(263\) 14.1421 0.872041 0.436021 0.899937i \(-0.356387\pi\)
0.436021 + 0.899937i \(0.356387\pi\)
\(264\) 0 0
\(265\) 13.3137 0.817855
\(266\) 0 0
\(267\) 7.65685 0.468592
\(268\) 0 0
\(269\) −18.4853 −1.12707 −0.563534 0.826093i \(-0.690559\pi\)
−0.563534 + 0.826093i \(0.690559\pi\)
\(270\) 0 0
\(271\) 1.65685 0.100647 0.0503234 0.998733i \(-0.483975\pi\)
0.0503234 + 0.998733i \(0.483975\pi\)
\(272\) 0 0
\(273\) −2.82843 −0.171184
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 32.6274 1.96039 0.980196 0.198031i \(-0.0634547\pi\)
0.980196 + 0.198031i \(0.0634547\pi\)
\(278\) 0 0
\(279\) 9.65685 0.578141
\(280\) 0 0
\(281\) 6.97056 0.415829 0.207914 0.978147i \(-0.433332\pi\)
0.207914 + 0.978147i \(0.433332\pi\)
\(282\) 0 0
\(283\) −14.3431 −0.852612 −0.426306 0.904579i \(-0.640185\pi\)
−0.426306 + 0.904579i \(0.640185\pi\)
\(284\) 0 0
\(285\) −2.82843 −0.167542
\(286\) 0 0
\(287\) −0.970563 −0.0572905
\(288\) 0 0
\(289\) 6.31371 0.371395
\(290\) 0 0
\(291\) 18.4853 1.08363
\(292\) 0 0
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) 2.82843 0.163572
\(300\) 0 0
\(301\) 27.3137 1.57434
\(302\) 0 0
\(303\) 10.4853 0.602364
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 13.6569 0.776911
\(310\) 0 0
\(311\) 14.6274 0.829445 0.414722 0.909948i \(-0.363879\pi\)
0.414722 + 0.909948i \(0.363879\pi\)
\(312\) 0 0
\(313\) −22.9706 −1.29837 −0.649186 0.760629i \(-0.724891\pi\)
−0.649186 + 0.760629i \(0.724891\pi\)
\(314\) 0 0
\(315\) −2.82843 −0.159364
\(316\) 0 0
\(317\) −13.3137 −0.747772 −0.373886 0.927475i \(-0.621975\pi\)
−0.373886 + 0.927475i \(0.621975\pi\)
\(318\) 0 0
\(319\) 19.3137 1.08136
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 13.6569 0.759888
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) 14.4853 0.801038
\(328\) 0 0
\(329\) −4.68629 −0.258364
\(330\) 0 0
\(331\) −24.4853 −1.34583 −0.672916 0.739719i \(-0.734959\pi\)
−0.672916 + 0.739719i \(0.734959\pi\)
\(332\) 0 0
\(333\) 7.65685 0.419593
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 12.3431 0.672374 0.336187 0.941795i \(-0.390863\pi\)
0.336187 + 0.941795i \(0.390863\pi\)
\(338\) 0 0
\(339\) −2.48528 −0.134982
\(340\) 0 0
\(341\) −38.6274 −2.09179
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 2.82843 0.152277
\(346\) 0 0
\(347\) −9.65685 −0.518407 −0.259204 0.965823i \(-0.583460\pi\)
−0.259204 + 0.965823i \(0.583460\pi\)
\(348\) 0 0
\(349\) 20.8284 1.11492 0.557460 0.830204i \(-0.311776\pi\)
0.557460 + 0.830204i \(0.311776\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −25.3137 −1.34731 −0.673656 0.739045i \(-0.735277\pi\)
−0.673656 + 0.739045i \(0.735277\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 13.6569 0.722797
\(358\) 0 0
\(359\) −11.3137 −0.597115 −0.298557 0.954392i \(-0.596505\pi\)
−0.298557 + 0.954392i \(0.596505\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 8.82843 0.462101
\(366\) 0 0
\(367\) −13.6569 −0.712882 −0.356441 0.934318i \(-0.616010\pi\)
−0.356441 + 0.934318i \(0.616010\pi\)
\(368\) 0 0
\(369\) 0.343146 0.0178635
\(370\) 0 0
\(371\) −37.6569 −1.95505
\(372\) 0 0
\(373\) −9.31371 −0.482246 −0.241123 0.970495i \(-0.577516\pi\)
−0.241123 + 0.970495i \(0.577516\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 4.82843 0.248677
\(378\) 0 0
\(379\) 19.7990 1.01701 0.508503 0.861060i \(-0.330199\pi\)
0.508503 + 0.861060i \(0.330199\pi\)
\(380\) 0 0
\(381\) −13.6569 −0.699662
\(382\) 0 0
\(383\) −10.6274 −0.543036 −0.271518 0.962433i \(-0.587526\pi\)
−0.271518 + 0.962433i \(0.587526\pi\)
\(384\) 0 0
\(385\) 11.3137 0.576600
\(386\) 0 0
\(387\) −9.65685 −0.490885
\(388\) 0 0
\(389\) −3.85786 −0.195601 −0.0978007 0.995206i \(-0.531181\pi\)
−0.0978007 + 0.995206i \(0.531181\pi\)
\(390\) 0 0
\(391\) −13.6569 −0.690657
\(392\) 0 0
\(393\) 5.17157 0.260871
\(394\) 0 0
\(395\) 5.65685 0.284627
\(396\) 0 0
\(397\) 15.6569 0.785795 0.392897 0.919582i \(-0.371473\pi\)
0.392897 + 0.919582i \(0.371473\pi\)
\(398\) 0 0
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) 26.2843 1.31257 0.656287 0.754511i \(-0.272126\pi\)
0.656287 + 0.754511i \(0.272126\pi\)
\(402\) 0 0
\(403\) −9.65685 −0.481042
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −30.6274 −1.51814
\(408\) 0 0
\(409\) −0.343146 −0.0169675 −0.00848373 0.999964i \(-0.502700\pi\)
−0.00848373 + 0.999964i \(0.502700\pi\)
\(410\) 0 0
\(411\) 9.31371 0.459411
\(412\) 0 0
\(413\) 11.3137 0.556711
\(414\) 0 0
\(415\) 7.31371 0.359016
\(416\) 0 0
\(417\) 13.6569 0.668779
\(418\) 0 0
\(419\) −24.4853 −1.19618 −0.598092 0.801427i \(-0.704074\pi\)
−0.598092 + 0.801427i \(0.704074\pi\)
\(420\) 0 0
\(421\) −0.828427 −0.0403751 −0.0201875 0.999796i \(-0.506426\pi\)
−0.0201875 + 0.999796i \(0.506426\pi\)
\(422\) 0 0
\(423\) 1.65685 0.0805590
\(424\) 0 0
\(425\) 4.82843 0.234213
\(426\) 0 0
\(427\) 5.65685 0.273754
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −24.9706 −1.20279 −0.601395 0.798952i \(-0.705388\pi\)
−0.601395 + 0.798952i \(0.705388\pi\)
\(432\) 0 0
\(433\) −8.34315 −0.400946 −0.200473 0.979699i \(-0.564248\pi\)
−0.200473 + 0.979699i \(0.564248\pi\)
\(434\) 0 0
\(435\) 4.82843 0.231505
\(436\) 0 0
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −20.9706 −0.996342 −0.498171 0.867079i \(-0.665995\pi\)
−0.498171 + 0.867079i \(0.665995\pi\)
\(444\) 0 0
\(445\) −7.65685 −0.362970
\(446\) 0 0
\(447\) 4.34315 0.205424
\(448\) 0 0
\(449\) −4.34315 −0.204966 −0.102483 0.994735i \(-0.532679\pi\)
−0.102483 + 0.994735i \(0.532679\pi\)
\(450\) 0 0
\(451\) −1.37258 −0.0646324
\(452\) 0 0
\(453\) 12.9706 0.609410
\(454\) 0 0
\(455\) 2.82843 0.132599
\(456\) 0 0
\(457\) 28.1421 1.31643 0.658217 0.752828i \(-0.271311\pi\)
0.658217 + 0.752828i \(0.271311\pi\)
\(458\) 0 0
\(459\) −4.82843 −0.225372
\(460\) 0 0
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 0 0
\(463\) 22.1421 1.02903 0.514516 0.857481i \(-0.327972\pi\)
0.514516 + 0.857481i \(0.327972\pi\)
\(464\) 0 0
\(465\) −9.65685 −0.447826
\(466\) 0 0
\(467\) 7.31371 0.338438 0.169219 0.985578i \(-0.445875\pi\)
0.169219 + 0.985578i \(0.445875\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 1.31371 0.0605325
\(472\) 0 0
\(473\) 38.6274 1.77609
\(474\) 0 0
\(475\) 2.82843 0.129777
\(476\) 0 0
\(477\) 13.3137 0.609593
\(478\) 0 0
\(479\) 21.6569 0.989527 0.494763 0.869028i \(-0.335255\pi\)
0.494763 + 0.869028i \(0.335255\pi\)
\(480\) 0 0
\(481\) −7.65685 −0.349123
\(482\) 0 0
\(483\) −8.00000 −0.364013
\(484\) 0 0
\(485\) −18.4853 −0.839373
\(486\) 0 0
\(487\) −10.8284 −0.490683 −0.245341 0.969437i \(-0.578900\pi\)
−0.245341 + 0.969437i \(0.578900\pi\)
\(488\) 0 0
\(489\) 16.9706 0.767435
\(490\) 0 0
\(491\) −24.4853 −1.10501 −0.552503 0.833511i \(-0.686327\pi\)
−0.552503 + 0.833511i \(0.686327\pi\)
\(492\) 0 0
\(493\) −23.3137 −1.05000
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) 22.6274 1.01498
\(498\) 0 0
\(499\) −18.8284 −0.842876 −0.421438 0.906857i \(-0.638475\pi\)
−0.421438 + 0.906857i \(0.638475\pi\)
\(500\) 0 0
\(501\) −7.31371 −0.326752
\(502\) 0 0
\(503\) 15.5147 0.691767 0.345884 0.938277i \(-0.387579\pi\)
0.345884 + 0.938277i \(0.387579\pi\)
\(504\) 0 0
\(505\) −10.4853 −0.466589
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −21.3137 −0.944714 −0.472357 0.881407i \(-0.656597\pi\)
−0.472357 + 0.881407i \(0.656597\pi\)
\(510\) 0 0
\(511\) −24.9706 −1.10463
\(512\) 0 0
\(513\) −2.82843 −0.124878
\(514\) 0 0
\(515\) −13.6569 −0.601793
\(516\) 0 0
\(517\) −6.62742 −0.291473
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −35.6569 −1.56216 −0.781078 0.624434i \(-0.785330\pi\)
−0.781078 + 0.624434i \(0.785330\pi\)
\(522\) 0 0
\(523\) −42.6274 −1.86397 −0.931983 0.362501i \(-0.881923\pi\)
−0.931983 + 0.362501i \(0.881923\pi\)
\(524\) 0 0
\(525\) 2.82843 0.123443
\(526\) 0 0
\(527\) 46.6274 2.03112
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −0.343146 −0.0148633
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) −8.48528 −0.366167
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −39.4558 −1.69634 −0.848170 0.529725i \(-0.822295\pi\)
−0.848170 + 0.529725i \(0.822295\pi\)
\(542\) 0 0
\(543\) −8.34315 −0.358039
\(544\) 0 0
\(545\) −14.4853 −0.620481
\(546\) 0 0
\(547\) 23.3137 0.996822 0.498411 0.866941i \(-0.333917\pi\)
0.498411 + 0.866941i \(0.333917\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −13.6569 −0.581802
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) −7.65685 −0.325015
\(556\) 0 0
\(557\) 39.9411 1.69236 0.846180 0.532897i \(-0.178897\pi\)
0.846180 + 0.532897i \(0.178897\pi\)
\(558\) 0 0
\(559\) 9.65685 0.408441
\(560\) 0 0
\(561\) 19.3137 0.815425
\(562\) 0 0
\(563\) −25.6569 −1.08131 −0.540654 0.841245i \(-0.681823\pi\)
−0.540654 + 0.841245i \(0.681823\pi\)
\(564\) 0 0
\(565\) 2.48528 0.104557
\(566\) 0 0
\(567\) −2.82843 −0.118783
\(568\) 0 0
\(569\) −42.2843 −1.77265 −0.886324 0.463065i \(-0.846750\pi\)
−0.886324 + 0.463065i \(0.846750\pi\)
\(570\) 0 0
\(571\) −40.9706 −1.71456 −0.857282 0.514847i \(-0.827849\pi\)
−0.857282 + 0.514847i \(0.827849\pi\)
\(572\) 0 0
\(573\) −3.31371 −0.138432
\(574\) 0 0
\(575\) −2.82843 −0.117954
\(576\) 0 0
\(577\) 30.4853 1.26912 0.634559 0.772874i \(-0.281182\pi\)
0.634559 + 0.772874i \(0.281182\pi\)
\(578\) 0 0
\(579\) −17.7990 −0.739701
\(580\) 0 0
\(581\) −20.6863 −0.858212
\(582\) 0 0
\(583\) −53.2548 −2.20559
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) −17.6569 −0.728776 −0.364388 0.931247i \(-0.618722\pi\)
−0.364388 + 0.931247i \(0.618722\pi\)
\(588\) 0 0
\(589\) 27.3137 1.12544
\(590\) 0 0
\(591\) 13.3137 0.547653
\(592\) 0 0
\(593\) 15.6569 0.642950 0.321475 0.946918i \(-0.395821\pi\)
0.321475 + 0.946918i \(0.395821\pi\)
\(594\) 0 0
\(595\) −13.6569 −0.559876
\(596\) 0 0
\(597\) −16.9706 −0.694559
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −25.3137 −1.03257 −0.516284 0.856418i \(-0.672685\pi\)
−0.516284 + 0.856418i \(0.672685\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) 5.65685 0.229605 0.114802 0.993388i \(-0.463377\pi\)
0.114802 + 0.993388i \(0.463377\pi\)
\(608\) 0 0
\(609\) −13.6569 −0.553404
\(610\) 0 0
\(611\) −1.65685 −0.0670291
\(612\) 0 0
\(613\) −46.9706 −1.89712 −0.948562 0.316593i \(-0.897461\pi\)
−0.948562 + 0.316593i \(0.897461\pi\)
\(614\) 0 0
\(615\) −0.343146 −0.0138370
\(616\) 0 0
\(617\) −28.6274 −1.15250 −0.576248 0.817275i \(-0.695484\pi\)
−0.576248 + 0.817275i \(0.695484\pi\)
\(618\) 0 0
\(619\) 29.1716 1.17250 0.586252 0.810129i \(-0.300603\pi\)
0.586252 + 0.810129i \(0.300603\pi\)
\(620\) 0 0
\(621\) 2.82843 0.113501
\(622\) 0 0
\(623\) 21.6569 0.867664
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 11.3137 0.451826
\(628\) 0 0
\(629\) 36.9706 1.47411
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) 16.0000 0.635943
\(634\) 0 0
\(635\) 13.6569 0.541956
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 40.6274 1.60469 0.802343 0.596863i \(-0.203586\pi\)
0.802343 + 0.596863i \(0.203586\pi\)
\(642\) 0 0
\(643\) 30.6274 1.20783 0.603914 0.797050i \(-0.293607\pi\)
0.603914 + 0.797050i \(0.293607\pi\)
\(644\) 0 0
\(645\) 9.65685 0.380238
\(646\) 0 0
\(647\) 9.45584 0.371748 0.185874 0.982574i \(-0.440488\pi\)
0.185874 + 0.982574i \(0.440488\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 27.3137 1.07051
\(652\) 0 0
\(653\) −18.2843 −0.715519 −0.357759 0.933814i \(-0.616459\pi\)
−0.357759 + 0.933814i \(0.616459\pi\)
\(654\) 0 0
\(655\) −5.17157 −0.202070
\(656\) 0 0
\(657\) 8.82843 0.344430
\(658\) 0 0
\(659\) −43.7990 −1.70617 −0.853083 0.521775i \(-0.825270\pi\)
−0.853083 + 0.521775i \(0.825270\pi\)
\(660\) 0 0
\(661\) −0.828427 −0.0322221 −0.0161110 0.999870i \(-0.505129\pi\)
−0.0161110 + 0.999870i \(0.505129\pi\)
\(662\) 0 0
\(663\) 4.82843 0.187521
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 13.6569 0.528796
\(668\) 0 0
\(669\) −2.82843 −0.109353
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −50.2843 −1.93258 −0.966291 0.257453i \(-0.917117\pi\)
−0.966291 + 0.257453i \(0.917117\pi\)
\(678\) 0 0
\(679\) 52.2843 2.00649
\(680\) 0 0
\(681\) 17.6569 0.676612
\(682\) 0 0
\(683\) 37.9411 1.45178 0.725888 0.687812i \(-0.241429\pi\)
0.725888 + 0.687812i \(0.241429\pi\)
\(684\) 0 0
\(685\) −9.31371 −0.355859
\(686\) 0 0
\(687\) 3.17157 0.121003
\(688\) 0 0
\(689\) −13.3137 −0.507212
\(690\) 0 0
\(691\) −29.1716 −1.10974 −0.554869 0.831937i \(-0.687232\pi\)
−0.554869 + 0.831937i \(0.687232\pi\)
\(692\) 0 0
\(693\) 11.3137 0.429772
\(694\) 0 0
\(695\) −13.6569 −0.518034
\(696\) 0 0
\(697\) 1.65685 0.0627578
\(698\) 0 0
\(699\) −9.51472 −0.359880
\(700\) 0 0
\(701\) −26.4853 −1.00034 −0.500168 0.865929i \(-0.666728\pi\)
−0.500168 + 0.865929i \(0.666728\pi\)
\(702\) 0 0
\(703\) 21.6569 0.816804
\(704\) 0 0
\(705\) −1.65685 −0.0624007
\(706\) 0 0
\(707\) 29.6569 1.11536
\(708\) 0 0
\(709\) 46.7696 1.75647 0.878234 0.478232i \(-0.158722\pi\)
0.878234 + 0.478232i \(0.158722\pi\)
\(710\) 0 0
\(711\) 5.65685 0.212149
\(712\) 0 0
\(713\) −27.3137 −1.02291
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 0 0
\(717\) 16.9706 0.633777
\(718\) 0 0
\(719\) 26.3431 0.982434 0.491217 0.871037i \(-0.336552\pi\)
0.491217 + 0.871037i \(0.336552\pi\)
\(720\) 0 0
\(721\) 38.6274 1.43856
\(722\) 0 0
\(723\) 1.31371 0.0488573
\(724\) 0 0
\(725\) −4.82843 −0.179323
\(726\) 0 0
\(727\) −30.6274 −1.13591 −0.567954 0.823060i \(-0.692265\pi\)
−0.567954 + 0.823060i \(0.692265\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −46.6274 −1.72458
\(732\) 0 0
\(733\) 34.9706 1.29167 0.645834 0.763478i \(-0.276510\pi\)
0.645834 + 0.763478i \(0.276510\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 39.1127 1.43878 0.719392 0.694604i \(-0.244421\pi\)
0.719392 + 0.694604i \(0.244421\pi\)
\(740\) 0 0
\(741\) 2.82843 0.103905
\(742\) 0 0
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 0 0
\(745\) −4.34315 −0.159121
\(746\) 0 0
\(747\) 7.31371 0.267595
\(748\) 0 0
\(749\) −11.3137 −0.413394
\(750\) 0 0
\(751\) 28.6863 1.04678 0.523389 0.852094i \(-0.324668\pi\)
0.523389 + 0.852094i \(0.324668\pi\)
\(752\) 0 0
\(753\) 10.8284 0.394610
\(754\) 0 0
\(755\) −12.9706 −0.472047
\(756\) 0 0
\(757\) 45.3137 1.64695 0.823477 0.567349i \(-0.192031\pi\)
0.823477 + 0.567349i \(0.192031\pi\)
\(758\) 0 0
\(759\) −11.3137 −0.410662
\(760\) 0 0
\(761\) −18.9706 −0.687682 −0.343841 0.939028i \(-0.611728\pi\)
−0.343841 + 0.939028i \(0.611728\pi\)
\(762\) 0 0
\(763\) 40.9706 1.48323
\(764\) 0 0
\(765\) 4.82843 0.174572
\(766\) 0 0
\(767\) 4.00000 0.144432
\(768\) 0 0
\(769\) −31.9411 −1.15183 −0.575913 0.817511i \(-0.695353\pi\)
−0.575913 + 0.817511i \(0.695353\pi\)
\(770\) 0 0
\(771\) 8.82843 0.317948
\(772\) 0 0
\(773\) −30.6863 −1.10371 −0.551855 0.833940i \(-0.686080\pi\)
−0.551855 + 0.833940i \(0.686080\pi\)
\(774\) 0 0
\(775\) 9.65685 0.346884
\(776\) 0 0
\(777\) 21.6569 0.776935
\(778\) 0 0
\(779\) 0.970563 0.0347740
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) 4.82843 0.172554
\(784\) 0 0
\(785\) −1.31371 −0.0468883
\(786\) 0 0
\(787\) 53.6569 1.91266 0.956330 0.292289i \(-0.0944168\pi\)
0.956330 + 0.292289i \(0.0944168\pi\)
\(788\) 0 0
\(789\) −14.1421 −0.503473
\(790\) 0 0
\(791\) −7.02944 −0.249938
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 0 0
\(795\) −13.3137 −0.472189
\(796\) 0 0
\(797\) 32.6274 1.15572 0.577861 0.816135i \(-0.303887\pi\)
0.577861 + 0.816135i \(0.303887\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) −7.65685 −0.270542
\(802\) 0 0
\(803\) −35.3137 −1.24619
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 0 0
\(807\) 18.4853 0.650713
\(808\) 0 0
\(809\) −2.68629 −0.0944450 −0.0472225 0.998884i \(-0.515037\pi\)
−0.0472225 + 0.998884i \(0.515037\pi\)
\(810\) 0 0
\(811\) −47.1127 −1.65435 −0.827175 0.561944i \(-0.810054\pi\)
−0.827175 + 0.561944i \(0.810054\pi\)
\(812\) 0 0
\(813\) −1.65685 −0.0581084
\(814\) 0 0
\(815\) −16.9706 −0.594453
\(816\) 0 0
\(817\) −27.3137 −0.955586
\(818\) 0 0
\(819\) 2.82843 0.0988332
\(820\) 0 0
\(821\) 42.2843 1.47573 0.737866 0.674948i \(-0.235834\pi\)
0.737866 + 0.674948i \(0.235834\pi\)
\(822\) 0 0
\(823\) 5.65685 0.197186 0.0985928 0.995128i \(-0.468566\pi\)
0.0985928 + 0.995128i \(0.468566\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 8.68629 0.302052 0.151026 0.988530i \(-0.451742\pi\)
0.151026 + 0.988530i \(0.451742\pi\)
\(828\) 0 0
\(829\) 18.6863 0.649002 0.324501 0.945885i \(-0.394804\pi\)
0.324501 + 0.945885i \(0.394804\pi\)
\(830\) 0 0
\(831\) −32.6274 −1.13183
\(832\) 0 0
\(833\) 4.82843 0.167295
\(834\) 0 0
\(835\) 7.31371 0.253101
\(836\) 0 0
\(837\) −9.65685 −0.333790
\(838\) 0 0
\(839\) 24.9706 0.862080 0.431040 0.902333i \(-0.358147\pi\)
0.431040 + 0.902333i \(0.358147\pi\)
\(840\) 0 0
\(841\) −5.68629 −0.196079
\(842\) 0 0
\(843\) −6.97056 −0.240079
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −14.1421 −0.485930
\(848\) 0 0
\(849\) 14.3431 0.492255
\(850\) 0 0
\(851\) −21.6569 −0.742387
\(852\) 0 0
\(853\) 1.02944 0.0352473 0.0176236 0.999845i \(-0.494390\pi\)
0.0176236 + 0.999845i \(0.494390\pi\)
\(854\) 0 0
\(855\) 2.82843 0.0967302
\(856\) 0 0
\(857\) −8.82843 −0.301573 −0.150787 0.988566i \(-0.548181\pi\)
−0.150787 + 0.988566i \(0.548181\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) 0.970563 0.0330767
\(862\) 0 0
\(863\) 20.0000 0.680808 0.340404 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(864\) 0 0
\(865\) −14.0000 −0.476014
\(866\) 0 0
\(867\) −6.31371 −0.214425
\(868\) 0 0
\(869\) −22.6274 −0.767583
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −18.4853 −0.625632
\(874\) 0 0
\(875\) −2.82843 −0.0956183
\(876\) 0 0
\(877\) 53.3137 1.80028 0.900138 0.435605i \(-0.143465\pi\)
0.900138 + 0.435605i \(0.143465\pi\)
\(878\) 0 0
\(879\) 10.0000 0.337292
\(880\) 0 0
\(881\) −22.9706 −0.773898 −0.386949 0.922101i \(-0.626471\pi\)
−0.386949 + 0.922101i \(0.626471\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 4.00000 0.134459
\(886\) 0 0
\(887\) −28.7696 −0.965987 −0.482994 0.875624i \(-0.660451\pi\)
−0.482994 + 0.875624i \(0.660451\pi\)
\(888\) 0 0
\(889\) −38.6274 −1.29552
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 4.68629 0.156821
\(894\) 0 0
\(895\) 8.48528 0.283632
\(896\) 0 0
\(897\) −2.82843 −0.0944384
\(898\) 0 0
\(899\) −46.6274 −1.55511
\(900\) 0 0
\(901\) 64.2843 2.14162
\(902\) 0 0
\(903\) −27.3137 −0.908943
\(904\) 0 0
\(905\) 8.34315 0.277336
\(906\) 0 0
\(907\) 38.3431 1.27316 0.636582 0.771209i \(-0.280348\pi\)
0.636582 + 0.771209i \(0.280348\pi\)
\(908\) 0 0
\(909\) −10.4853 −0.347775
\(910\) 0 0
\(911\) 24.9706 0.827312 0.413656 0.910433i \(-0.364252\pi\)
0.413656 + 0.910433i \(0.364252\pi\)
\(912\) 0 0
\(913\) −29.2548 −0.968194
\(914\) 0 0
\(915\) 2.00000 0.0661180
\(916\) 0 0
\(917\) 14.6274 0.483040
\(918\) 0 0
\(919\) −22.6274 −0.746410 −0.373205 0.927749i \(-0.621741\pi\)
−0.373205 + 0.927749i \(0.621741\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) 0 0
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 7.65685 0.251756
\(926\) 0 0
\(927\) −13.6569 −0.448550
\(928\) 0 0
\(929\) 26.2843 0.862359 0.431179 0.902266i \(-0.358098\pi\)
0.431179 + 0.902266i \(0.358098\pi\)
\(930\) 0 0
\(931\) 2.82843 0.0926980
\(932\) 0 0
\(933\) −14.6274 −0.478880
\(934\) 0 0
\(935\) −19.3137 −0.631626
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 22.9706 0.749616
\(940\) 0 0
\(941\) −39.6569 −1.29278 −0.646388 0.763009i \(-0.723721\pi\)
−0.646388 + 0.763009i \(0.723721\pi\)
\(942\) 0 0
\(943\) −0.970563 −0.0316059
\(944\) 0 0
\(945\) 2.82843 0.0920087
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) −8.82843 −0.286583
\(950\) 0 0
\(951\) 13.3137 0.431727
\(952\) 0 0
\(953\) −59.7401 −1.93517 −0.967586 0.252541i \(-0.918734\pi\)
−0.967586 + 0.252541i \(0.918734\pi\)
\(954\) 0 0
\(955\) 3.31371 0.107229
\(956\) 0 0
\(957\) −19.3137 −0.624324
\(958\) 0 0
\(959\) 26.3431 0.850665
\(960\) 0 0
\(961\) 62.2548 2.00822
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 0 0
\(965\) 17.7990 0.572970
\(966\) 0 0
\(967\) 35.7990 1.15122 0.575609 0.817725i \(-0.304765\pi\)
0.575609 + 0.817725i \(0.304765\pi\)
\(968\) 0 0
\(969\) −13.6569 −0.438721
\(970\) 0 0
\(971\) 5.17157 0.165964 0.0829818 0.996551i \(-0.473556\pi\)
0.0829818 + 0.996551i \(0.473556\pi\)
\(972\) 0 0
\(973\) 38.6274 1.23834
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) −37.5980 −1.20287 −0.601433 0.798923i \(-0.705403\pi\)
−0.601433 + 0.798923i \(0.705403\pi\)
\(978\) 0 0
\(979\) 30.6274 0.978856
\(980\) 0 0
\(981\) −14.4853 −0.462479
\(982\) 0 0
\(983\) 26.6274 0.849283 0.424641 0.905362i \(-0.360400\pi\)
0.424641 + 0.905362i \(0.360400\pi\)
\(984\) 0 0
\(985\) −13.3137 −0.424210
\(986\) 0 0
\(987\) 4.68629 0.149166
\(988\) 0 0
\(989\) 27.3137 0.868525
\(990\) 0 0
\(991\) −2.34315 −0.0744325 −0.0372162 0.999307i \(-0.511849\pi\)
−0.0372162 + 0.999307i \(0.511849\pi\)
\(992\) 0 0
\(993\) 24.4853 0.777017
\(994\) 0 0
\(995\) 16.9706 0.538003
\(996\) 0 0
\(997\) −25.3137 −0.801693 −0.400847 0.916145i \(-0.631284\pi\)
−0.400847 + 0.916145i \(0.631284\pi\)
\(998\) 0 0
\(999\) −7.65685 −0.242252
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 6240.2.a.bn.1.1 2
4.3 odd 2 6240.2.a.bt.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6240.2.a.bn.1.1 2 1.1 even 1 trivial
6240.2.a.bt.1.2 yes 2 4.3 odd 2