Properties

Label 6864.2.a.bn.1.1
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(54.8093159474\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Defining polynomial: \(x^{3} - x^{2} - 5 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.86620\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.34889 q^{5} +2.51730 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.34889 q^{5} +2.51730 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +3.34889 q^{15} +7.59859 q^{17} +1.48270 q^{19} -2.51730 q^{21} +1.55191 q^{23} +6.21509 q^{25} -1.00000 q^{27} +1.86620 q^{29} +1.61650 q^{31} +1.00000 q^{33} -8.43018 q^{35} +6.43018 q^{37} -1.00000 q^{39} -10.9475 q^{41} +5.59859 q^{43} -3.34889 q^{45} -0.697788 q^{47} -0.663180 q^{49} -7.59859 q^{51} -9.73240 q^{53} +3.34889 q^{55} -1.48270 q^{57} +4.76700 q^{59} -1.66318 q^{61} +2.51730 q^{63} -3.34889 q^{65} -4.81369 q^{67} -1.55191 q^{69} +0.965392 q^{71} -3.21509 q^{73} -6.21509 q^{75} -2.51730 q^{77} +8.83159 q^{79} +1.00000 q^{81} +0.767005 q^{83} -25.4469 q^{85} -1.86620 q^{87} +3.41811 q^{89} +2.51730 q^{91} -1.61650 q^{93} -4.96539 q^{95} -6.96539 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 4q^{5} + 6q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 4q^{5} + 6q^{7} + 3q^{9} - 3q^{11} + 3q^{13} + 4q^{15} + 6q^{19} - 6q^{21} + 5q^{25} - 3q^{27} - 2q^{29} + 14q^{31} + 3q^{33} + 2q^{35} - 8q^{37} - 3q^{39} - 4q^{41} - 6q^{43} - 4q^{45} + 10q^{47} + 7q^{49} - 14q^{53} + 4q^{55} - 6q^{57} - 4q^{59} + 4q^{61} + 6q^{63} - 4q^{65} + 22q^{67} + 6q^{71} + 4q^{73} - 5q^{75} - 6q^{77} + 22q^{79} + 3q^{81} - 16q^{83} - 14q^{85} + 2q^{87} - 2q^{89} + 6q^{91} - 14q^{93} - 18q^{95} - 24q^{97} - 3q^{99} + O(q^{100}) \)

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
\(5\) −3.34889 −1.49767 −0.748836 0.662756i \(-0.769387\pi\)
−0.748836 + 0.662756i \(0.769387\pi\)
\(6\) 0 0
\(7\) 2.51730 0.951451 0.475726 0.879594i \(-0.342185\pi\)
0.475726 + 0.879594i \(0.342185\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 3.34889 0.864681
\(16\) 0 0
\(17\) 7.59859 1.84293 0.921465 0.388461i \(-0.126993\pi\)
0.921465 + 0.388461i \(0.126993\pi\)
\(18\) 0 0
\(19\) 1.48270 0.340154 0.170077 0.985431i \(-0.445598\pi\)
0.170077 + 0.985431i \(0.445598\pi\)
\(20\) 0 0
\(21\) −2.51730 −0.549321
\(22\) 0 0
\(23\) 1.55191 0.323596 0.161798 0.986824i \(-0.448271\pi\)
0.161798 + 0.986824i \(0.448271\pi\)
\(24\) 0 0
\(25\) 6.21509 1.24302
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.86620 0.346544 0.173272 0.984874i \(-0.444566\pi\)
0.173272 + 0.984874i \(0.444566\pi\)
\(30\) 0 0
\(31\) 1.61650 0.290332 0.145166 0.989407i \(-0.453628\pi\)
0.145166 + 0.989407i \(0.453628\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) −8.43018 −1.42496
\(36\) 0 0
\(37\) 6.43018 1.05712 0.528558 0.848897i \(-0.322733\pi\)
0.528558 + 0.848897i \(0.322733\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −10.9475 −1.70971 −0.854855 0.518867i \(-0.826354\pi\)
−0.854855 + 0.518867i \(0.826354\pi\)
\(42\) 0 0
\(43\) 5.59859 0.853778 0.426889 0.904304i \(-0.359610\pi\)
0.426889 + 0.904304i \(0.359610\pi\)
\(44\) 0 0
\(45\) −3.34889 −0.499224
\(46\) 0 0
\(47\) −0.697788 −0.101783 −0.0508915 0.998704i \(-0.516206\pi\)
−0.0508915 + 0.998704i \(0.516206\pi\)
\(48\) 0 0
\(49\) −0.663180 −0.0947400
\(50\) 0 0
\(51\) −7.59859 −1.06402
\(52\) 0 0
\(53\) −9.73240 −1.33685 −0.668424 0.743781i \(-0.733031\pi\)
−0.668424 + 0.743781i \(0.733031\pi\)
\(54\) 0 0
\(55\) 3.34889 0.451565
\(56\) 0 0
\(57\) −1.48270 −0.196388
\(58\) 0 0
\(59\) 4.76700 0.620611 0.310306 0.950637i \(-0.399569\pi\)
0.310306 + 0.950637i \(0.399569\pi\)
\(60\) 0 0
\(61\) −1.66318 −0.212948 −0.106474 0.994315i \(-0.533956\pi\)
−0.106474 + 0.994315i \(0.533956\pi\)
\(62\) 0 0
\(63\) 2.51730 0.317150
\(64\) 0 0
\(65\) −3.34889 −0.415379
\(66\) 0 0
\(67\) −4.81369 −0.588085 −0.294043 0.955792i \(-0.595001\pi\)
−0.294043 + 0.955792i \(0.595001\pi\)
\(68\) 0 0
\(69\) −1.55191 −0.186828
\(70\) 0 0
\(71\) 0.965392 0.114571 0.0572855 0.998358i \(-0.481755\pi\)
0.0572855 + 0.998358i \(0.481755\pi\)
\(72\) 0 0
\(73\) −3.21509 −0.376298 −0.188149 0.982140i \(-0.560249\pi\)
−0.188149 + 0.982140i \(0.560249\pi\)
\(74\) 0 0
\(75\) −6.21509 −0.717657
\(76\) 0 0
\(77\) −2.51730 −0.286873
\(78\) 0 0
\(79\) 8.83159 0.993632 0.496816 0.867856i \(-0.334502\pi\)
0.496816 + 0.867856i \(0.334502\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.767005 0.0841897 0.0420948 0.999114i \(-0.486597\pi\)
0.0420948 + 0.999114i \(0.486597\pi\)
\(84\) 0 0
\(85\) −25.4469 −2.76010
\(86\) 0 0
\(87\) −1.86620 −0.200077
\(88\) 0 0
\(89\) 3.41811 0.362319 0.181159 0.983454i \(-0.442015\pi\)
0.181159 + 0.983454i \(0.442015\pi\)
\(90\) 0 0
\(91\) 2.51730 0.263885
\(92\) 0 0
\(93\) −1.61650 −0.167623
\(94\) 0 0
\(95\) −4.96539 −0.509438
\(96\) 0 0
\(97\) −6.96539 −0.707228 −0.353614 0.935391i \(-0.615047\pi\)
−0.353614 + 0.935391i \(0.615047\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 6.29638 0.626514 0.313257 0.949668i \(-0.398580\pi\)
0.313257 + 0.949668i \(0.398580\pi\)
\(102\) 0 0
\(103\) 15.4648 1.52379 0.761896 0.647700i \(-0.224269\pi\)
0.761896 + 0.647700i \(0.224269\pi\)
\(104\) 0 0
\(105\) 8.43018 0.822702
\(106\) 0 0
\(107\) −12.0934 −1.16911 −0.584555 0.811354i \(-0.698731\pi\)
−0.584555 + 0.811354i \(0.698731\pi\)
\(108\) 0 0
\(109\) −17.6453 −1.69011 −0.845056 0.534679i \(-0.820433\pi\)
−0.845056 + 0.534679i \(0.820433\pi\)
\(110\) 0 0
\(111\) −6.43018 −0.610326
\(112\) 0 0
\(113\) −2.76700 −0.260298 −0.130149 0.991494i \(-0.541546\pi\)
−0.130149 + 0.991494i \(0.541546\pi\)
\(114\) 0 0
\(115\) −5.19719 −0.484640
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 19.1280 1.75346
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 10.9475 0.987102
\(124\) 0 0
\(125\) −4.06922 −0.363962
\(126\) 0 0
\(127\) 12.6332 1.12102 0.560508 0.828149i \(-0.310606\pi\)
0.560508 + 0.828149i \(0.310606\pi\)
\(128\) 0 0
\(129\) −5.59859 −0.492929
\(130\) 0 0
\(131\) −3.10382 −0.271182 −0.135591 0.990765i \(-0.543293\pi\)
−0.135591 + 0.990765i \(0.543293\pi\)
\(132\) 0 0
\(133\) 3.73240 0.323640
\(134\) 0 0
\(135\) 3.34889 0.288227
\(136\) 0 0
\(137\) 0.813687 0.0695180 0.0347590 0.999396i \(-0.488934\pi\)
0.0347590 + 0.999396i \(0.488934\pi\)
\(138\) 0 0
\(139\) 0.133802 0.0113489 0.00567446 0.999984i \(-0.498194\pi\)
0.00567446 + 0.999984i \(0.498194\pi\)
\(140\) 0 0
\(141\) 0.697788 0.0587644
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −6.24970 −0.519009
\(146\) 0 0
\(147\) 0.663180 0.0546982
\(148\) 0 0
\(149\) −17.0167 −1.39406 −0.697031 0.717041i \(-0.745496\pi\)
−0.697031 + 0.717041i \(0.745496\pi\)
\(150\) 0 0
\(151\) 22.8783 1.86181 0.930904 0.365265i \(-0.119022\pi\)
0.930904 + 0.365265i \(0.119022\pi\)
\(152\) 0 0
\(153\) 7.59859 0.614310
\(154\) 0 0
\(155\) −5.41348 −0.434821
\(156\) 0 0
\(157\) −5.21509 −0.416210 −0.208105 0.978107i \(-0.566730\pi\)
−0.208105 + 0.978107i \(0.566730\pi\)
\(158\) 0 0
\(159\) 9.73240 0.771829
\(160\) 0 0
\(161\) 3.90663 0.307886
\(162\) 0 0
\(163\) 10.7203 0.839680 0.419840 0.907598i \(-0.362086\pi\)
0.419840 + 0.907598i \(0.362086\pi\)
\(164\) 0 0
\(165\) −3.34889 −0.260711
\(166\) 0 0
\(167\) 23.8258 1.84369 0.921846 0.387555i \(-0.126680\pi\)
0.921846 + 0.387555i \(0.126680\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 1.48270 0.113385
\(172\) 0 0
\(173\) 22.7958 1.73313 0.866566 0.499063i \(-0.166322\pi\)
0.866566 + 0.499063i \(0.166322\pi\)
\(174\) 0 0
\(175\) 15.6453 1.18267
\(176\) 0 0
\(177\) −4.76700 −0.358310
\(178\) 0 0
\(179\) −10.6799 −0.798252 −0.399126 0.916896i \(-0.630686\pi\)
−0.399126 + 0.916896i \(0.630686\pi\)
\(180\) 0 0
\(181\) −4.18048 −0.310733 −0.155366 0.987857i \(-0.549656\pi\)
−0.155366 + 0.987857i \(0.549656\pi\)
\(182\) 0 0
\(183\) 1.66318 0.122946
\(184\) 0 0
\(185\) −21.5340 −1.58321
\(186\) 0 0
\(187\) −7.59859 −0.555664
\(188\) 0 0
\(189\) −2.51730 −0.183107
\(190\) 0 0
\(191\) −10.1626 −0.735339 −0.367669 0.929957i \(-0.619844\pi\)
−0.367669 + 0.929957i \(0.619844\pi\)
\(192\) 0 0
\(193\) 20.7491 1.49355 0.746777 0.665075i \(-0.231600\pi\)
0.746777 + 0.665075i \(0.231600\pi\)
\(194\) 0 0
\(195\) 3.34889 0.239819
\(196\) 0 0
\(197\) −11.2151 −0.799042 −0.399521 0.916724i \(-0.630824\pi\)
−0.399521 + 0.916724i \(0.630824\pi\)
\(198\) 0 0
\(199\) −1.93078 −0.136870 −0.0684348 0.997656i \(-0.521801\pi\)
−0.0684348 + 0.997656i \(0.521801\pi\)
\(200\) 0 0
\(201\) 4.81369 0.339531
\(202\) 0 0
\(203\) 4.69779 0.329720
\(204\) 0 0
\(205\) 36.6620 2.56058
\(206\) 0 0
\(207\) 1.55191 0.107865
\(208\) 0 0
\(209\) −1.48270 −0.102560
\(210\) 0 0
\(211\) 17.3310 1.19311 0.596557 0.802570i \(-0.296535\pi\)
0.596557 + 0.802570i \(0.296535\pi\)
\(212\) 0 0
\(213\) −0.965392 −0.0661476
\(214\) 0 0
\(215\) −18.7491 −1.27868
\(216\) 0 0
\(217\) 4.06922 0.276236
\(218\) 0 0
\(219\) 3.21509 0.217256
\(220\) 0 0
\(221\) 7.59859 0.511137
\(222\) 0 0
\(223\) 17.7099 1.18594 0.592970 0.805224i \(-0.297955\pi\)
0.592970 + 0.805224i \(0.297955\pi\)
\(224\) 0 0
\(225\) 6.21509 0.414339
\(226\) 0 0
\(227\) −12.5865 −0.835397 −0.417698 0.908586i \(-0.637163\pi\)
−0.417698 + 0.908586i \(0.637163\pi\)
\(228\) 0 0
\(229\) −18.6978 −1.23558 −0.617792 0.786341i \(-0.711973\pi\)
−0.617792 + 0.786341i \(0.711973\pi\)
\(230\) 0 0
\(231\) 2.51730 0.165626
\(232\) 0 0
\(233\) 26.6332 1.74480 0.872400 0.488793i \(-0.162563\pi\)
0.872400 + 0.488793i \(0.162563\pi\)
\(234\) 0 0
\(235\) 2.33682 0.152437
\(236\) 0 0
\(237\) −8.83159 −0.573673
\(238\) 0 0
\(239\) −26.4815 −1.71295 −0.856473 0.516192i \(-0.827349\pi\)
−0.856473 + 0.516192i \(0.827349\pi\)
\(240\) 0 0
\(241\) −2.40604 −0.154986 −0.0774932 0.996993i \(-0.524692\pi\)
−0.0774932 + 0.996993i \(0.524692\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.22092 0.141889
\(246\) 0 0
\(247\) 1.48270 0.0943417
\(248\) 0 0
\(249\) −0.767005 −0.0486069
\(250\) 0 0
\(251\) −11.3714 −0.717758 −0.358879 0.933384i \(-0.616841\pi\)
−0.358879 + 0.933384i \(0.616841\pi\)
\(252\) 0 0
\(253\) −1.55191 −0.0975679
\(254\) 0 0
\(255\) 25.4469 1.59355
\(256\) 0 0
\(257\) −13.9642 −0.871063 −0.435531 0.900174i \(-0.643439\pi\)
−0.435531 + 0.900174i \(0.643439\pi\)
\(258\) 0 0
\(259\) 16.1867 1.00579
\(260\) 0 0
\(261\) 1.86620 0.115515
\(262\) 0 0
\(263\) −1.44064 −0.0888339 −0.0444170 0.999013i \(-0.514143\pi\)
−0.0444170 + 0.999013i \(0.514143\pi\)
\(264\) 0 0
\(265\) 32.5928 2.00216
\(266\) 0 0
\(267\) −3.41811 −0.209185
\(268\) 0 0
\(269\) 13.7324 0.837279 0.418639 0.908153i \(-0.362507\pi\)
0.418639 + 0.908153i \(0.362507\pi\)
\(270\) 0 0
\(271\) 19.2843 1.17144 0.585719 0.810514i \(-0.300812\pi\)
0.585719 + 0.810514i \(0.300812\pi\)
\(272\) 0 0
\(273\) −2.51730 −0.152354
\(274\) 0 0
\(275\) −6.21509 −0.374784
\(276\) 0 0
\(277\) −1.10382 −0.0663224 −0.0331612 0.999450i \(-0.510557\pi\)
−0.0331612 + 0.999450i \(0.510557\pi\)
\(278\) 0 0
\(279\) 1.61650 0.0967772
\(280\) 0 0
\(281\) −17.1101 −1.02070 −0.510351 0.859966i \(-0.670484\pi\)
−0.510351 + 0.859966i \(0.670484\pi\)
\(282\) 0 0
\(283\) 9.03924 0.537327 0.268663 0.963234i \(-0.413418\pi\)
0.268663 + 0.963234i \(0.413418\pi\)
\(284\) 0 0
\(285\) 4.96539 0.294124
\(286\) 0 0
\(287\) −27.5582 −1.62671
\(288\) 0 0
\(289\) 40.7386 2.39639
\(290\) 0 0
\(291\) 6.96539 0.408319
\(292\) 0 0
\(293\) 18.1805 1.06212 0.531058 0.847336i \(-0.321795\pi\)
0.531058 + 0.847336i \(0.321795\pi\)
\(294\) 0 0
\(295\) −15.9642 −0.929471
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 1.55191 0.0897494
\(300\) 0 0
\(301\) 14.0934 0.812328
\(302\) 0 0
\(303\) −6.29638 −0.361718
\(304\) 0 0
\(305\) 5.56982 0.318927
\(306\) 0 0
\(307\) −0.947489 −0.0540761 −0.0270380 0.999634i \(-0.508608\pi\)
−0.0270380 + 0.999634i \(0.508608\pi\)
\(308\) 0 0
\(309\) −15.4648 −0.879761
\(310\) 0 0
\(311\) 21.2392 1.20437 0.602183 0.798358i \(-0.294298\pi\)
0.602183 + 0.798358i \(0.294298\pi\)
\(312\) 0 0
\(313\) −17.8079 −1.00656 −0.503280 0.864123i \(-0.667874\pi\)
−0.503280 + 0.864123i \(0.667874\pi\)
\(314\) 0 0
\(315\) −8.43018 −0.474987
\(316\) 0 0
\(317\) −9.58069 −0.538105 −0.269053 0.963125i \(-0.586711\pi\)
−0.269053 + 0.963125i \(0.586711\pi\)
\(318\) 0 0
\(319\) −1.86620 −0.104487
\(320\) 0 0
\(321\) 12.0934 0.674986
\(322\) 0 0
\(323\) 11.2664 0.626880
\(324\) 0 0
\(325\) 6.21509 0.344751
\(326\) 0 0
\(327\) 17.6453 0.975786
\(328\) 0 0
\(329\) −1.75655 −0.0968415
\(330\) 0 0
\(331\) 8.61530 0.473540 0.236770 0.971566i \(-0.423911\pi\)
0.236770 + 0.971566i \(0.423911\pi\)
\(332\) 0 0
\(333\) 6.43018 0.352372
\(334\) 0 0
\(335\) 16.1205 0.880759
\(336\) 0 0
\(337\) 32.1626 1.75201 0.876004 0.482304i \(-0.160200\pi\)
0.876004 + 0.482304i \(0.160200\pi\)
\(338\) 0 0
\(339\) 2.76700 0.150283
\(340\) 0 0
\(341\) −1.61650 −0.0875383
\(342\) 0 0
\(343\) −19.2906 −1.04159
\(344\) 0 0
\(345\) 5.19719 0.279807
\(346\) 0 0
\(347\) 26.9988 1.44937 0.724686 0.689079i \(-0.241985\pi\)
0.724686 + 0.689079i \(0.241985\pi\)
\(348\) 0 0
\(349\) 15.6632 0.838431 0.419215 0.907887i \(-0.362305\pi\)
0.419215 + 0.907887i \(0.362305\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 8.74447 0.465421 0.232711 0.972546i \(-0.425241\pi\)
0.232711 + 0.972546i \(0.425241\pi\)
\(354\) 0 0
\(355\) −3.23300 −0.171590
\(356\) 0 0
\(357\) −19.1280 −1.01236
\(358\) 0 0
\(359\) −15.7744 −0.832544 −0.416272 0.909240i \(-0.636664\pi\)
−0.416272 + 0.909240i \(0.636664\pi\)
\(360\) 0 0
\(361\) −16.8016 −0.884295
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 10.7670 0.563571
\(366\) 0 0
\(367\) 9.66318 0.504414 0.252207 0.967673i \(-0.418844\pi\)
0.252207 + 0.967673i \(0.418844\pi\)
\(368\) 0 0
\(369\) −10.9475 −0.569904
\(370\) 0 0
\(371\) −24.4994 −1.27195
\(372\) 0 0
\(373\) 13.5582 0.702015 0.351007 0.936373i \(-0.385839\pi\)
0.351007 + 0.936373i \(0.385839\pi\)
\(374\) 0 0
\(375\) 4.06922 0.210133
\(376\) 0 0
\(377\) 1.86620 0.0961141
\(378\) 0 0
\(379\) 0.419308 0.0215384 0.0107692 0.999942i \(-0.496572\pi\)
0.0107692 + 0.999942i \(0.496572\pi\)
\(380\) 0 0
\(381\) −12.6332 −0.647219
\(382\) 0 0
\(383\) −3.19719 −0.163369 −0.0816844 0.996658i \(-0.526030\pi\)
−0.0816844 + 0.996658i \(0.526030\pi\)
\(384\) 0 0
\(385\) 8.43018 0.429642
\(386\) 0 0
\(387\) 5.59859 0.284593
\(388\) 0 0
\(389\) 31.6274 1.60357 0.801786 0.597612i \(-0.203883\pi\)
0.801786 + 0.597612i \(0.203883\pi\)
\(390\) 0 0
\(391\) 11.7924 0.596365
\(392\) 0 0
\(393\) 3.10382 0.156567
\(394\) 0 0
\(395\) −29.5761 −1.48813
\(396\) 0 0
\(397\) 6.43018 0.322722 0.161361 0.986895i \(-0.448412\pi\)
0.161361 + 0.986895i \(0.448412\pi\)
\(398\) 0 0
\(399\) −3.73240 −0.186854
\(400\) 0 0
\(401\) −12.4769 −0.623065 −0.311533 0.950235i \(-0.600842\pi\)
−0.311533 + 0.950235i \(0.600842\pi\)
\(402\) 0 0
\(403\) 1.61650 0.0805235
\(404\) 0 0
\(405\) −3.34889 −0.166408
\(406\) 0 0
\(407\) −6.43018 −0.318732
\(408\) 0 0
\(409\) −5.28431 −0.261292 −0.130646 0.991429i \(-0.541705\pi\)
−0.130646 + 0.991429i \(0.541705\pi\)
\(410\) 0 0
\(411\) −0.813687 −0.0401362
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −2.56862 −0.126088
\(416\) 0 0
\(417\) −0.133802 −0.00655230
\(418\) 0 0
\(419\) −1.81952 −0.0888892 −0.0444446 0.999012i \(-0.514152\pi\)
−0.0444446 + 0.999012i \(0.514152\pi\)
\(420\) 0 0
\(421\) 23.0346 1.12264 0.561319 0.827599i \(-0.310294\pi\)
0.561319 + 0.827599i \(0.310294\pi\)
\(422\) 0 0
\(423\) −0.697788 −0.0339276
\(424\) 0 0
\(425\) 47.2260 2.29080
\(426\) 0 0
\(427\) −4.18673 −0.202610
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 23.1038 1.11287 0.556436 0.830890i \(-0.312168\pi\)
0.556436 + 0.830890i \(0.312168\pi\)
\(432\) 0 0
\(433\) −16.7912 −0.806931 −0.403466 0.914995i \(-0.632195\pi\)
−0.403466 + 0.914995i \(0.632195\pi\)
\(434\) 0 0
\(435\) 6.24970 0.299650
\(436\) 0 0
\(437\) 2.30101 0.110072
\(438\) 0 0
\(439\) −40.2606 −1.92153 −0.960766 0.277359i \(-0.910541\pi\)
−0.960766 + 0.277359i \(0.910541\pi\)
\(440\) 0 0
\(441\) −0.663180 −0.0315800
\(442\) 0 0
\(443\) 16.8604 0.801060 0.400530 0.916284i \(-0.368826\pi\)
0.400530 + 0.916284i \(0.368826\pi\)
\(444\) 0 0
\(445\) −11.4469 −0.542635
\(446\) 0 0
\(447\) 17.0167 0.804863
\(448\) 0 0
\(449\) −6.98793 −0.329781 −0.164890 0.986312i \(-0.552727\pi\)
−0.164890 + 0.986312i \(0.552727\pi\)
\(450\) 0 0
\(451\) 10.9475 0.515497
\(452\) 0 0
\(453\) −22.8783 −1.07492
\(454\) 0 0
\(455\) −8.43018 −0.395213
\(456\) 0 0
\(457\) 27.0346 1.26463 0.632313 0.774713i \(-0.282106\pi\)
0.632313 + 0.774713i \(0.282106\pi\)
\(458\) 0 0
\(459\) −7.59859 −0.354672
\(460\) 0 0
\(461\) 4.65574 0.216839 0.108420 0.994105i \(-0.465421\pi\)
0.108420 + 0.994105i \(0.465421\pi\)
\(462\) 0 0
\(463\) 31.9417 1.48445 0.742227 0.670148i \(-0.233769\pi\)
0.742227 + 0.670148i \(0.233769\pi\)
\(464\) 0 0
\(465\) 5.41348 0.251044
\(466\) 0 0
\(467\) 30.2738 1.40091 0.700453 0.713698i \(-0.252981\pi\)
0.700453 + 0.713698i \(0.252981\pi\)
\(468\) 0 0
\(469\) −12.1175 −0.559535
\(470\) 0 0
\(471\) 5.21509 0.240299
\(472\) 0 0
\(473\) −5.59859 −0.257424
\(474\) 0 0
\(475\) 9.21509 0.422817
\(476\) 0 0
\(477\) −9.73240 −0.445616
\(478\) 0 0
\(479\) 30.8783 1.41086 0.705432 0.708777i \(-0.250753\pi\)
0.705432 + 0.708777i \(0.250753\pi\)
\(480\) 0 0
\(481\) 6.43018 0.293191
\(482\) 0 0
\(483\) −3.90663 −0.177758
\(484\) 0 0
\(485\) 23.3264 1.05920
\(486\) 0 0
\(487\) −20.8378 −0.944252 −0.472126 0.881531i \(-0.656513\pi\)
−0.472126 + 0.881531i \(0.656513\pi\)
\(488\) 0 0
\(489\) −10.7203 −0.484790
\(490\) 0 0
\(491\) −20.4302 −0.922001 −0.461001 0.887400i \(-0.652509\pi\)
−0.461001 + 0.887400i \(0.652509\pi\)
\(492\) 0 0
\(493\) 14.1805 0.638657
\(494\) 0 0
\(495\) 3.34889 0.150522
\(496\) 0 0
\(497\) 2.43018 0.109009
\(498\) 0 0
\(499\) −27.6406 −1.23737 −0.618683 0.785641i \(-0.712333\pi\)
−0.618683 + 0.785641i \(0.712333\pi\)
\(500\) 0 0
\(501\) −23.8258 −1.06446
\(502\) 0 0
\(503\) 41.9525 1.87057 0.935286 0.353894i \(-0.115143\pi\)
0.935286 + 0.353894i \(0.115143\pi\)
\(504\) 0 0
\(505\) −21.0859 −0.938311
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −7.41811 −0.328802 −0.164401 0.986394i \(-0.552569\pi\)
−0.164401 + 0.986394i \(0.552569\pi\)
\(510\) 0 0
\(511\) −8.09337 −0.358029
\(512\) 0 0
\(513\) −1.48270 −0.0654626
\(514\) 0 0
\(515\) −51.7900 −2.28214
\(516\) 0 0
\(517\) 0.697788 0.0306887
\(518\) 0 0
\(519\) −22.7958 −1.00062
\(520\) 0 0
\(521\) −30.1867 −1.32250 −0.661252 0.750164i \(-0.729975\pi\)
−0.661252 + 0.750164i \(0.729975\pi\)
\(522\) 0 0
\(523\) 11.2618 0.492443 0.246222 0.969214i \(-0.420811\pi\)
0.246222 + 0.969214i \(0.420811\pi\)
\(524\) 0 0
\(525\) −15.6453 −0.682816
\(526\) 0 0
\(527\) 12.2831 0.535061
\(528\) 0 0
\(529\) −20.5916 −0.895286
\(530\) 0 0
\(531\) 4.76700 0.206870
\(532\) 0 0
\(533\) −10.9475 −0.474188
\(534\) 0 0
\(535\) 40.4994 1.75094
\(536\) 0 0
\(537\) 10.6799 0.460871
\(538\) 0 0
\(539\) 0.663180 0.0285652
\(540\) 0 0
\(541\) −5.13963 −0.220970 −0.110485 0.993878i \(-0.535240\pi\)
−0.110485 + 0.993878i \(0.535240\pi\)
\(542\) 0 0
\(543\) 4.18048 0.179402
\(544\) 0 0
\(545\) 59.0922 2.53123
\(546\) 0 0
\(547\) 43.1209 1.84372 0.921859 0.387525i \(-0.126670\pi\)
0.921859 + 0.387525i \(0.126670\pi\)
\(548\) 0 0
\(549\) −1.66318 −0.0709828
\(550\) 0 0
\(551\) 2.76700 0.117878
\(552\) 0 0
\(553\) 22.2318 0.945392
\(554\) 0 0
\(555\) 21.5340 0.914068
\(556\) 0 0
\(557\) 27.3085 1.15710 0.578548 0.815648i \(-0.303619\pi\)
0.578548 + 0.815648i \(0.303619\pi\)
\(558\) 0 0
\(559\) 5.59859 0.236795
\(560\) 0 0
\(561\) 7.59859 0.320813
\(562\) 0 0
\(563\) −5.03461 −0.212183 −0.106092 0.994356i \(-0.533834\pi\)
−0.106092 + 0.994356i \(0.533834\pi\)
\(564\) 0 0
\(565\) 9.26641 0.389841
\(566\) 0 0
\(567\) 2.51730 0.105717
\(568\) 0 0
\(569\) 12.4014 0.519894 0.259947 0.965623i \(-0.416295\pi\)
0.259947 + 0.965623i \(0.416295\pi\)
\(570\) 0 0
\(571\) 15.5744 0.651770 0.325885 0.945409i \(-0.394338\pi\)
0.325885 + 0.945409i \(0.394338\pi\)
\(572\) 0 0
\(573\) 10.1626 0.424548
\(574\) 0 0
\(575\) 9.64528 0.402236
\(576\) 0 0
\(577\) −2.89618 −0.120569 −0.0602847 0.998181i \(-0.519201\pi\)
−0.0602847 + 0.998181i \(0.519201\pi\)
\(578\) 0 0
\(579\) −20.7491 −0.862303
\(580\) 0 0
\(581\) 1.93078 0.0801024
\(582\) 0 0
\(583\) 9.73240 0.403075
\(584\) 0 0
\(585\) −3.34889 −0.138460
\(586\) 0 0
\(587\) 15.3714 0.634447 0.317223 0.948351i \(-0.397250\pi\)
0.317223 + 0.948351i \(0.397250\pi\)
\(588\) 0 0
\(589\) 2.39677 0.0987574
\(590\) 0 0
\(591\) 11.2151 0.461327
\(592\) 0 0
\(593\) 8.38813 0.344459 0.172230 0.985057i \(-0.444903\pi\)
0.172230 + 0.985057i \(0.444903\pi\)
\(594\) 0 0
\(595\) −64.0576 −2.62610
\(596\) 0 0
\(597\) 1.93078 0.0790217
\(598\) 0 0
\(599\) 20.2318 0.826649 0.413324 0.910584i \(-0.364368\pi\)
0.413324 + 0.910584i \(0.364368\pi\)
\(600\) 0 0
\(601\) 18.7219 0.763684 0.381842 0.924228i \(-0.375290\pi\)
0.381842 + 0.924228i \(0.375290\pi\)
\(602\) 0 0
\(603\) −4.81369 −0.196028
\(604\) 0 0
\(605\) −3.34889 −0.136152
\(606\) 0 0
\(607\) 1.56518 0.0635289 0.0317644 0.999495i \(-0.489887\pi\)
0.0317644 + 0.999495i \(0.489887\pi\)
\(608\) 0 0
\(609\) −4.69779 −0.190364
\(610\) 0 0
\(611\) −0.697788 −0.0282295
\(612\) 0 0
\(613\) −25.4710 −1.02877 −0.514383 0.857561i \(-0.671979\pi\)
−0.514383 + 0.857561i \(0.671979\pi\)
\(614\) 0 0
\(615\) −36.6620 −1.47835
\(616\) 0 0
\(617\) −35.1747 −1.41608 −0.708039 0.706173i \(-0.750420\pi\)
−0.708039 + 0.706173i \(0.750420\pi\)
\(618\) 0 0
\(619\) −27.2197 −1.09405 −0.547027 0.837115i \(-0.684240\pi\)
−0.547027 + 0.837115i \(0.684240\pi\)
\(620\) 0 0
\(621\) −1.55191 −0.0622761
\(622\) 0 0
\(623\) 8.60442 0.344729
\(624\) 0 0
\(625\) −17.4481 −0.697924
\(626\) 0 0
\(627\) 1.48270 0.0592132
\(628\) 0 0
\(629\) 48.8604 1.94819
\(630\) 0 0
\(631\) 3.61650 0.143970 0.0719852 0.997406i \(-0.477067\pi\)
0.0719852 + 0.997406i \(0.477067\pi\)
\(632\) 0 0
\(633\) −17.3310 −0.688845
\(634\) 0 0
\(635\) −42.3073 −1.67891
\(636\) 0 0
\(637\) −0.663180 −0.0262762
\(638\) 0 0
\(639\) 0.965392 0.0381903
\(640\) 0 0
\(641\) −2.49940 −0.0987204 −0.0493602 0.998781i \(-0.515718\pi\)
−0.0493602 + 0.998781i \(0.515718\pi\)
\(642\) 0 0
\(643\) 8.48852 0.334755 0.167377 0.985893i \(-0.446470\pi\)
0.167377 + 0.985893i \(0.446470\pi\)
\(644\) 0 0
\(645\) 18.7491 0.738245
\(646\) 0 0
\(647\) 48.0218 1.88793 0.943965 0.330046i \(-0.107064\pi\)
0.943965 + 0.330046i \(0.107064\pi\)
\(648\) 0 0
\(649\) −4.76700 −0.187121
\(650\) 0 0
\(651\) −4.06922 −0.159485
\(652\) 0 0
\(653\) 6.49940 0.254341 0.127171 0.991881i \(-0.459410\pi\)
0.127171 + 0.991881i \(0.459410\pi\)
\(654\) 0 0
\(655\) 10.3944 0.406142
\(656\) 0 0
\(657\) −3.21509 −0.125433
\(658\) 0 0
\(659\) 3.12797 0.121849 0.0609243 0.998142i \(-0.480595\pi\)
0.0609243 + 0.998142i \(0.480595\pi\)
\(660\) 0 0
\(661\) −31.5916 −1.22877 −0.614385 0.789007i \(-0.710596\pi\)
−0.614385 + 0.789007i \(0.710596\pi\)
\(662\) 0 0
\(663\) −7.59859 −0.295105
\(664\) 0 0
\(665\) −12.4994 −0.484706
\(666\) 0 0
\(667\) 2.89618 0.112140
\(668\) 0 0
\(669\) −17.7099 −0.684703
\(670\) 0 0
\(671\) 1.66318 0.0642064
\(672\) 0 0
\(673\) 2.60442 0.100393 0.0501966 0.998739i \(-0.484015\pi\)
0.0501966 + 0.998739i \(0.484015\pi\)
\(674\) 0 0
\(675\) −6.21509 −0.239219
\(676\) 0 0
\(677\) −27.5869 −1.06025 −0.530126 0.847919i \(-0.677855\pi\)
−0.530126 + 0.847919i \(0.677855\pi\)
\(678\) 0 0
\(679\) −17.5340 −0.672894
\(680\) 0 0
\(681\) 12.5865 0.482317
\(682\) 0 0
\(683\) −35.8950 −1.37348 −0.686742 0.726902i \(-0.740960\pi\)
−0.686742 + 0.726902i \(0.740960\pi\)
\(684\) 0 0
\(685\) −2.72495 −0.104115
\(686\) 0 0
\(687\) 18.6978 0.713365
\(688\) 0 0
\(689\) −9.73240 −0.370775
\(690\) 0 0
\(691\) 46.3360 1.76271 0.881353 0.472458i \(-0.156633\pi\)
0.881353 + 0.472458i \(0.156633\pi\)
\(692\) 0 0
\(693\) −2.51730 −0.0956245
\(694\) 0 0
\(695\) −0.448088 −0.0169969
\(696\) 0 0
\(697\) −83.1855 −3.15088
\(698\) 0 0
\(699\) −26.6332 −1.00736
\(700\) 0 0
\(701\) 39.0183 1.47370 0.736851 0.676055i \(-0.236312\pi\)
0.736851 + 0.676055i \(0.236312\pi\)
\(702\) 0 0
\(703\) 9.53401 0.359582
\(704\) 0 0
\(705\) −2.33682 −0.0880097
\(706\) 0 0
\(707\) 15.8499 0.596097
\(708\) 0 0
\(709\) 7.87083 0.295595 0.147798 0.989018i \(-0.452782\pi\)
0.147798 + 0.989018i \(0.452782\pi\)
\(710\) 0 0
\(711\) 8.83159 0.331211
\(712\) 0 0
\(713\) 2.50866 0.0939501
\(714\) 0 0
\(715\) 3.34889 0.125242
\(716\) 0 0
\(717\) 26.4815 0.988970
\(718\) 0 0
\(719\) −40.9988 −1.52900 −0.764499 0.644625i \(-0.777013\pi\)
−0.764499 + 0.644625i \(0.777013\pi\)
\(720\) 0 0
\(721\) 38.9296 1.44981
\(722\) 0 0
\(723\) 2.40604 0.0894814
\(724\) 0 0
\(725\) 11.5986 0.430761
\(726\) 0 0
\(727\) −11.3356 −0.420415 −0.210207 0.977657i \(-0.567414\pi\)
−0.210207 + 0.977657i \(0.567414\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 42.5415 1.57345
\(732\) 0 0
\(733\) 32.3881 1.19628 0.598141 0.801391i \(-0.295906\pi\)
0.598141 + 0.801391i \(0.295906\pi\)
\(734\) 0 0
\(735\) −2.22092 −0.0819199
\(736\) 0 0
\(737\) 4.81369 0.177314
\(738\) 0 0
\(739\) 43.0650 1.58417 0.792086 0.610410i \(-0.208995\pi\)
0.792086 + 0.610410i \(0.208995\pi\)
\(740\) 0 0
\(741\) −1.48270 −0.0544682
\(742\) 0 0
\(743\) −5.80161 −0.212841 −0.106420 0.994321i \(-0.533939\pi\)
−0.106420 + 0.994321i \(0.533939\pi\)
\(744\) 0 0
\(745\) 56.9871 2.08785
\(746\) 0 0
\(747\) 0.767005 0.0280632
\(748\) 0 0
\(749\) −30.4427 −1.11235
\(750\) 0 0
\(751\) −6.73359 −0.245712 −0.122856 0.992424i \(-0.539205\pi\)
−0.122856 + 0.992424i \(0.539205\pi\)
\(752\) 0 0
\(753\) 11.3714 0.414398
\(754\) 0 0
\(755\) −76.6169 −2.78838
\(756\) 0 0
\(757\) −18.7491 −0.681448 −0.340724 0.940163i \(-0.610672\pi\)
−0.340724 + 0.940163i \(0.610672\pi\)
\(758\) 0 0
\(759\) 1.55191 0.0563308
\(760\) 0 0
\(761\) −9.78371 −0.354659 −0.177330 0.984152i \(-0.556746\pi\)
−0.177330 + 0.984152i \(0.556746\pi\)
\(762\) 0 0
\(763\) −44.4185 −1.60806
\(764\) 0 0
\(765\) −25.4469 −0.920034
\(766\) 0 0
\(767\) 4.76700 0.172127
\(768\) 0 0
\(769\) 47.3240 1.70655 0.853273 0.521465i \(-0.174614\pi\)
0.853273 + 0.521465i \(0.174614\pi\)
\(770\) 0 0
\(771\) 13.9642 0.502908
\(772\) 0 0
\(773\) −14.7895 −0.531943 −0.265971 0.963981i \(-0.585693\pi\)
−0.265971 + 0.963981i \(0.585693\pi\)
\(774\) 0 0
\(775\) 10.0467 0.360887
\(776\) 0 0
\(777\) −16.1867 −0.580696
\(778\) 0 0
\(779\) −16.2318 −0.581565
\(780\) 0 0
\(781\) −0.965392 −0.0345444
\(782\) 0 0
\(783\) −1.86620 −0.0666925
\(784\) 0 0
\(785\) 17.4648 0.623345
\(786\) 0 0
\(787\) −20.7733 −0.740486 −0.370243 0.928935i \(-0.620726\pi\)
−0.370243 + 0.928935i \(0.620726\pi\)
\(788\) 0 0
\(789\) 1.44064 0.0512883
\(790\) 0 0
\(791\) −6.96539 −0.247661
\(792\) 0 0
\(793\) −1.66318 −0.0590613
\(794\) 0 0
\(795\) −32.5928 −1.15595
\(796\) 0 0
\(797\) 29.6515 1.05031 0.525155 0.851006i \(-0.324007\pi\)
0.525155 + 0.851006i \(0.324007\pi\)
\(798\) 0 0
\(799\) −5.30221 −0.187579
\(800\) 0 0
\(801\) 3.41811 0.120773
\(802\) 0 0
\(803\) 3.21509 0.113458
\(804\) 0 0
\(805\) −13.0829 −0.461112
\(806\) 0 0
\(807\) −13.7324 −0.483403
\(808\) 0 0
\(809\) −6.34145 −0.222954 −0.111477 0.993767i \(-0.535558\pi\)
−0.111477 + 0.993767i \(0.535558\pi\)
\(810\) 0 0
\(811\) −26.2046 −0.920169 −0.460085 0.887875i \(-0.652181\pi\)
−0.460085 + 0.887875i \(0.652181\pi\)
\(812\) 0 0
\(813\) −19.2843 −0.676330
\(814\) 0 0
\(815\) −35.9012 −1.25757
\(816\) 0 0
\(817\) 8.30101 0.290416
\(818\) 0 0
\(819\) 2.51730 0.0879617
\(820\) 0 0
\(821\) 10.0062 0.349220 0.174610 0.984638i \(-0.444133\pi\)
0.174610 + 0.984638i \(0.444133\pi\)
\(822\) 0 0
\(823\) −8.40604 −0.293016 −0.146508 0.989209i \(-0.546803\pi\)
−0.146508 + 0.989209i \(0.546803\pi\)
\(824\) 0 0
\(825\) 6.21509 0.216382
\(826\) 0 0
\(827\) −46.7040 −1.62406 −0.812029 0.583617i \(-0.801637\pi\)
−0.812029 + 0.583617i \(0.801637\pi\)
\(828\) 0 0
\(829\) 33.6032 1.16709 0.583544 0.812081i \(-0.301666\pi\)
0.583544 + 0.812081i \(0.301666\pi\)
\(830\) 0 0
\(831\) 1.10382 0.0382912
\(832\) 0 0
\(833\) −5.03924 −0.174599
\(834\) 0 0
\(835\) −79.7900 −2.76125
\(836\) 0 0
\(837\) −1.61650 −0.0558743
\(838\) 0 0
\(839\) 11.8258 0.408271 0.204135 0.978943i \(-0.434562\pi\)
0.204135 + 0.978943i \(0.434562\pi\)
\(840\) 0 0
\(841\) −25.5173 −0.879907
\(842\) 0 0
\(843\) 17.1101 0.589302
\(844\) 0 0
\(845\) −3.34889 −0.115205
\(846\) 0 0
\(847\) 2.51730 0.0864956
\(848\) 0 0
\(849\) −9.03924 −0.310226
\(850\) 0 0
\(851\) 9.97908 0.342079
\(852\) 0 0
\(853\) 25.6873 0.879517 0.439759 0.898116i \(-0.355064\pi\)
0.439759 + 0.898116i \(0.355064\pi\)
\(854\) 0 0
\(855\) −4.96539 −0.169813
\(856\) 0 0
\(857\) −43.4578 −1.48449 −0.742245 0.670129i \(-0.766239\pi\)
−0.742245 + 0.670129i \(0.766239\pi\)
\(858\) 0 0
\(859\) −28.0093 −0.955664 −0.477832 0.878451i \(-0.658577\pi\)
−0.477832 + 0.878451i \(0.658577\pi\)
\(860\) 0 0
\(861\) 27.5582 0.939180
\(862\) 0 0
\(863\) −12.5928 −0.428663 −0.214331 0.976761i \(-0.568757\pi\)
−0.214331 + 0.976761i \(0.568757\pi\)
\(864\) 0 0
\(865\) −76.3407 −2.59566
\(866\) 0 0
\(867\) −40.7386 −1.38356
\(868\) 0 0
\(869\) −8.83159 −0.299591
\(870\) 0 0
\(871\) −4.81369 −0.163106
\(872\) 0 0
\(873\) −6.96539 −0.235743
\(874\) 0 0
\(875\) −10.2435 −0.346292
\(876\) 0 0
\(877\) 30.7312 1.03772 0.518859 0.854860i \(-0.326357\pi\)
0.518859 + 0.854860i \(0.326357\pi\)
\(878\) 0 0
\(879\) −18.1805 −0.613213
\(880\) 0 0
\(881\) 52.7195 1.77617 0.888083 0.459683i \(-0.152037\pi\)
0.888083 + 0.459683i \(0.152037\pi\)
\(882\) 0 0
\(883\) −20.8604 −0.702007 −0.351004 0.936374i \(-0.614160\pi\)
−0.351004 + 0.936374i \(0.614160\pi\)
\(884\) 0 0
\(885\) 15.9642 0.536631
\(886\) 0 0
\(887\) 26.0093 0.873306 0.436653 0.899630i \(-0.356164\pi\)
0.436653 + 0.899630i \(0.356164\pi\)
\(888\) 0 0
\(889\) 31.8016 1.06659
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) −1.03461 −0.0346218
\(894\) 0 0
\(895\) 35.7658 1.19552
\(896\) 0 0
\(897\) −1.55191 −0.0518168
\(898\) 0 0
\(899\) 3.01671 0.100613
\(900\) 0 0
\(901\) −73.9525 −2.46372
\(902\) 0 0
\(903\) −14.0934 −0.468998
\(904\) 0 0
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) 6.38512 0.212014 0.106007 0.994365i \(-0.466193\pi\)
0.106007 + 0.994365i \(0.466193\pi\)
\(908\) 0 0
\(909\) 6.29638 0.208838
\(910\) 0 0
\(911\) −46.8845 −1.55335 −0.776677 0.629899i \(-0.783096\pi\)
−0.776677 + 0.629899i \(0.783096\pi\)
\(912\) 0 0
\(913\) −0.767005 −0.0253841
\(914\) 0 0
\(915\) −5.56982 −0.184132
\(916\) 0 0
\(917\) −7.81327 −0.258017
\(918\) 0 0
\(919\) −44.3656 −1.46349 −0.731743 0.681581i \(-0.761293\pi\)
−0.731743 + 0.681581i \(0.761293\pi\)
\(920\) 0 0
\(921\) 0.947489 0.0312208
\(922\) 0 0
\(923\) 0.965392 0.0317763
\(924\) 0 0
\(925\) 39.9642 1.31401
\(926\) 0 0
\(927\) 15.4648 0.507930
\(928\) 0 0
\(929\) −4.35935 −0.143026 −0.0715129 0.997440i \(-0.522783\pi\)
−0.0715129 + 0.997440i \(0.522783\pi\)
\(930\) 0 0
\(931\) −0.983295 −0.0322262
\(932\) 0 0
\(933\) −21.2392 −0.695342
\(934\) 0 0
\(935\) 25.4469 0.832202
\(936\) 0 0
\(937\) −38.5686 −1.25998 −0.629991 0.776603i \(-0.716941\pi\)
−0.629991 + 0.776603i \(0.716941\pi\)
\(938\) 0 0
\(939\) 17.8079 0.581138
\(940\) 0 0
\(941\) 51.0501 1.66419 0.832093 0.554636i \(-0.187142\pi\)
0.832093 + 0.554636i \(0.187142\pi\)
\(942\) 0 0
\(943\) −16.9895 −0.553256
\(944\) 0 0
\(945\) 8.43018 0.274234
\(946\) 0 0
\(947\) −38.5928 −1.25410 −0.627048 0.778980i \(-0.715737\pi\)
−0.627048 + 0.778980i \(0.715737\pi\)
\(948\) 0 0
\(949\) −3.21509 −0.104366
\(950\) 0 0
\(951\) 9.58069 0.310675
\(952\) 0 0
\(953\) −22.4232 −0.726357 −0.363179 0.931720i \(-0.618309\pi\)
−0.363179 + 0.931720i \(0.618309\pi\)
\(954\) 0 0
\(955\) 34.0334 1.10130
\(956\) 0 0
\(957\) 1.86620 0.0603256
\(958\) 0 0
\(959\) 2.04830 0.0661430
\(960\) 0 0
\(961\) −28.3869 −0.915708
\(962\) 0 0
\(963\) −12.0934 −0.389703
\(964\) 0 0
\(965\) −69.4865 −2.23685
\(966\) 0 0
\(967\) 11.0618 0.355723 0.177861 0.984056i \(-0.443082\pi\)
0.177861 + 0.984056i \(0.443082\pi\)
\(968\) 0 0
\(969\) −11.2664 −0.361929
\(970\) 0 0
\(971\) 52.3073 1.67862 0.839310 0.543653i \(-0.182959\pi\)
0.839310 + 0.543653i \(0.182959\pi\)
\(972\) 0 0
\(973\) 0.336820 0.0107979
\(974\) 0 0
\(975\) −6.21509 −0.199042
\(976\) 0 0
\(977\) −60.1400 −1.92405 −0.962025 0.272960i \(-0.911997\pi\)
−0.962025 + 0.272960i \(0.911997\pi\)
\(978\) 0 0
\(979\) −3.41811 −0.109243
\(980\) 0 0
\(981\) −17.6453 −0.563370
\(982\) 0 0
\(983\) 15.9642 0.509179 0.254589 0.967049i \(-0.418060\pi\)
0.254589 + 0.967049i \(0.418060\pi\)
\(984\) 0 0
\(985\) 37.5582 1.19670
\(986\) 0 0
\(987\) 1.75655 0.0559115
\(988\) 0 0
\(989\) 8.68853 0.276279
\(990\) 0 0
\(991\) −23.0230 −0.731348 −0.365674 0.930743i \(-0.619162\pi\)
−0.365674 + 0.930743i \(0.619162\pi\)
\(992\) 0 0
\(993\) −8.61530 −0.273398
\(994\) 0 0
\(995\) 6.46599 0.204986
\(996\) 0 0
\(997\) −11.9759 −0.379279 −0.189640 0.981854i \(-0.560732\pi\)
−0.189640 + 0.981854i \(0.560732\pi\)
\(998\) 0 0
\(999\) −6.43018 −0.203442
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6864.2.a.bn.1.1 3
4.3 odd 2 3432.2.a.n.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.n.1.1 3 4.3 odd 2
6864.2.a.bn.1.1 3 1.1 even 1 trivial