Properties

Label 7728.2.a.bz.1.2
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.39605.1
Defining polynomial: \(x^{4} - 2 x^{3} - 8 x^{2} + 9 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.318315\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.681685 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.681685 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.71255 q^{11} +3.66749 q^{13} +0.681685 q^{15} +2.21699 q^{17} -4.34918 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.53531 q^{25} -1.00000 q^{27} -4.63226 q^{29} -8.21699 q^{31} -3.71255 q^{33} -0.681685 q^{35} +3.93391 q^{37} -3.66749 q^{39} -5.99563 q^{41} -5.16305 q^{43} -0.681685 q^{45} -6.41527 q^{47} +1.00000 q^{49} -2.21699 q^{51} -3.60140 q^{53} -2.53079 q^{55} +4.34918 q^{57} -6.83258 q^{59} -4.71939 q^{61} +1.00000 q^{63} -2.50007 q^{65} +2.96477 q^{67} -1.00000 q^{69} -3.12003 q^{71} +10.2787 q^{73} +4.53531 q^{75} +3.71255 q^{77} +11.6137 q^{79} +1.00000 q^{81} -11.5520 q^{83} -1.51129 q^{85} +4.63226 q^{87} +1.88885 q^{89} +3.66749 q^{91} +8.21699 q^{93} +2.96477 q^{95} +7.33062 q^{97} +3.71255 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 2q^{5} + 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 2q^{5} + 4q^{7} + 4q^{9} + 2q^{13} + 2q^{15} - 10q^{17} - 4q^{19} - 4q^{21} + 4q^{23} - 4q^{27} + 11q^{29} - 14q^{31} - 2q^{35} + 13q^{37} - 2q^{39} + 7q^{41} - 12q^{43} - 2q^{45} - 15q^{47} + 4q^{49} + 10q^{51} + q^{53} - 31q^{55} + 4q^{57} - 5q^{59} + 3q^{61} + 4q^{63} + 25q^{65} - 5q^{67} - 4q^{69} - 5q^{71} - 6q^{73} - 26q^{79} + 4q^{81} - 2q^{83} + 5q^{85} - 11q^{87} + 7q^{89} + 2q^{91} + 14q^{93} - 5q^{95} - 27q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −0.681685 −0.304859 −0.152429 0.988314i \(-0.548710\pi\)
−0.152429 + 0.988314i \(0.548710\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.71255 1.11938 0.559688 0.828704i \(-0.310921\pi\)
0.559688 + 0.828704i \(0.310921\pi\)
\(12\) 0 0
\(13\) 3.66749 1.01718 0.508590 0.861009i \(-0.330167\pi\)
0.508590 + 0.861009i \(0.330167\pi\)
\(14\) 0 0
\(15\) 0.681685 0.176010
\(16\) 0 0
\(17\) 2.21699 0.537699 0.268850 0.963182i \(-0.413357\pi\)
0.268850 + 0.963182i \(0.413357\pi\)
\(18\) 0 0
\(19\) −4.34918 −0.997770 −0.498885 0.866668i \(-0.666257\pi\)
−0.498885 + 0.866668i \(0.666257\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.53531 −0.907061
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.63226 −0.860189 −0.430095 0.902784i \(-0.641520\pi\)
−0.430095 + 0.902784i \(0.641520\pi\)
\(30\) 0 0
\(31\) −8.21699 −1.47582 −0.737908 0.674902i \(-0.764186\pi\)
−0.737908 + 0.674902i \(0.764186\pi\)
\(32\) 0 0
\(33\) −3.71255 −0.646272
\(34\) 0 0
\(35\) −0.681685 −0.115226
\(36\) 0 0
\(37\) 3.93391 0.646730 0.323365 0.946274i \(-0.395186\pi\)
0.323365 + 0.946274i \(0.395186\pi\)
\(38\) 0 0
\(39\) −3.66749 −0.587269
\(40\) 0 0
\(41\) −5.99563 −0.936360 −0.468180 0.883633i \(-0.655090\pi\)
−0.468180 + 0.883633i \(0.655090\pi\)
\(42\) 0 0
\(43\) −5.16305 −0.787358 −0.393679 0.919248i \(-0.628798\pi\)
−0.393679 + 0.919248i \(0.628798\pi\)
\(44\) 0 0
\(45\) −0.681685 −0.101620
\(46\) 0 0
\(47\) −6.41527 −0.935763 −0.467882 0.883791i \(-0.654983\pi\)
−0.467882 + 0.883791i \(0.654983\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.21699 −0.310441
\(52\) 0 0
\(53\) −3.60140 −0.494690 −0.247345 0.968927i \(-0.579558\pi\)
−0.247345 + 0.968927i \(0.579558\pi\)
\(54\) 0 0
\(55\) −2.53079 −0.341251
\(56\) 0 0
\(57\) 4.34918 0.576063
\(58\) 0 0
\(59\) −6.83258 −0.889526 −0.444763 0.895648i \(-0.646712\pi\)
−0.444763 + 0.895648i \(0.646712\pi\)
\(60\) 0 0
\(61\) −4.71939 −0.604256 −0.302128 0.953267i \(-0.597697\pi\)
−0.302128 + 0.953267i \(0.597697\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −2.50007 −0.310096
\(66\) 0 0
\(67\) 2.96477 0.362204 0.181102 0.983464i \(-0.442034\pi\)
0.181102 + 0.983464i \(0.442034\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −3.12003 −0.370280 −0.185140 0.982712i \(-0.559274\pi\)
−0.185140 + 0.982712i \(0.559274\pi\)
\(72\) 0 0
\(73\) 10.2787 1.20303 0.601516 0.798860i \(-0.294563\pi\)
0.601516 + 0.798860i \(0.294563\pi\)
\(74\) 0 0
\(75\) 4.53531 0.523692
\(76\) 0 0
\(77\) 3.71255 0.423084
\(78\) 0 0
\(79\) 11.6137 1.30664 0.653322 0.757080i \(-0.273375\pi\)
0.653322 + 0.757080i \(0.273375\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −11.5520 −1.26799 −0.633997 0.773335i \(-0.718587\pi\)
−0.633997 + 0.773335i \(0.718587\pi\)
\(84\) 0 0
\(85\) −1.51129 −0.163922
\(86\) 0 0
\(87\) 4.63226 0.496631
\(88\) 0 0
\(89\) 1.88885 0.200218 0.100109 0.994976i \(-0.468081\pi\)
0.100109 + 0.994976i \(0.468081\pi\)
\(90\) 0 0
\(91\) 3.66749 0.384458
\(92\) 0 0
\(93\) 8.21699 0.852062
\(94\) 0 0
\(95\) 2.96477 0.304179
\(96\) 0 0
\(97\) 7.33062 0.744311 0.372156 0.928170i \(-0.378619\pi\)
0.372156 + 0.928170i \(0.378619\pi\)
\(98\) 0 0
\(99\) 3.71255 0.373125
\(100\) 0 0
\(101\) 4.59251 0.456972 0.228486 0.973547i \(-0.426622\pi\)
0.228486 + 0.973547i \(0.426622\pi\)
\(102\) 0 0
\(103\) −14.3448 −1.41344 −0.706718 0.707495i \(-0.749825\pi\)
−0.706718 + 0.707495i \(0.749825\pi\)
\(104\) 0 0
\(105\) 0.681685 0.0665256
\(106\) 0 0
\(107\) −1.44613 −0.139803 −0.0699015 0.997554i \(-0.522269\pi\)
−0.0699015 + 0.997554i \(0.522269\pi\)
\(108\) 0 0
\(109\) 9.11567 0.873122 0.436561 0.899675i \(-0.356196\pi\)
0.436561 + 0.899675i \(0.356196\pi\)
\(110\) 0 0
\(111\) −3.93391 −0.373390
\(112\) 0 0
\(113\) 12.8051 1.20461 0.602303 0.798268i \(-0.294250\pi\)
0.602303 + 0.798268i \(0.294250\pi\)
\(114\) 0 0
\(115\) −0.681685 −0.0635675
\(116\) 0 0
\(117\) 3.66749 0.339060
\(118\) 0 0
\(119\) 2.21699 0.203231
\(120\) 0 0
\(121\) 2.78301 0.253001
\(122\) 0 0
\(123\) 5.99563 0.540608
\(124\) 0 0
\(125\) 6.50007 0.581384
\(126\) 0 0
\(127\) −17.5069 −1.55349 −0.776744 0.629816i \(-0.783130\pi\)
−0.776744 + 0.629816i \(0.783130\pi\)
\(128\) 0 0
\(129\) 5.16305 0.454581
\(130\) 0 0
\(131\) −2.08480 −0.182150 −0.0910751 0.995844i \(-0.529030\pi\)
−0.0910751 + 0.995844i \(0.529030\pi\)
\(132\) 0 0
\(133\) −4.34918 −0.377122
\(134\) 0 0
\(135\) 0.681685 0.0586701
\(136\) 0 0
\(137\) −14.9163 −1.27438 −0.637192 0.770705i \(-0.719904\pi\)
−0.637192 + 0.770705i \(0.719904\pi\)
\(138\) 0 0
\(139\) −5.45050 −0.462306 −0.231153 0.972917i \(-0.574250\pi\)
−0.231153 + 0.972917i \(0.574250\pi\)
\(140\) 0 0
\(141\) 6.41527 0.540263
\(142\) 0 0
\(143\) 13.6157 1.13861
\(144\) 0 0
\(145\) 3.15774 0.262236
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 7.09900 0.581572 0.290786 0.956788i \(-0.406083\pi\)
0.290786 + 0.956788i \(0.406083\pi\)
\(150\) 0 0
\(151\) −9.29197 −0.756170 −0.378085 0.925771i \(-0.623417\pi\)
−0.378085 + 0.925771i \(0.623417\pi\)
\(152\) 0 0
\(153\) 2.21699 0.179233
\(154\) 0 0
\(155\) 5.60140 0.449915
\(156\) 0 0
\(157\) −1.33499 −0.106543 −0.0532717 0.998580i \(-0.516965\pi\)
−0.0532717 + 0.998580i \(0.516965\pi\)
\(158\) 0 0
\(159\) 3.60140 0.285610
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −7.49760 −0.587257 −0.293629 0.955920i \(-0.594863\pi\)
−0.293629 + 0.955920i \(0.594863\pi\)
\(164\) 0 0
\(165\) 2.53079 0.197022
\(166\) 0 0
\(167\) −4.34918 −0.336549 −0.168275 0.985740i \(-0.553820\pi\)
−0.168275 + 0.985740i \(0.553820\pi\)
\(168\) 0 0
\(169\) 0.450502 0.0346540
\(170\) 0 0
\(171\) −4.34918 −0.332590
\(172\) 0 0
\(173\) 23.7459 1.80537 0.902683 0.430306i \(-0.141594\pi\)
0.902683 + 0.430306i \(0.141594\pi\)
\(174\) 0 0
\(175\) −4.53531 −0.342837
\(176\) 0 0
\(177\) 6.83258 0.513568
\(178\) 0 0
\(179\) 6.28512 0.469772 0.234886 0.972023i \(-0.424528\pi\)
0.234886 + 0.972023i \(0.424528\pi\)
\(180\) 0 0
\(181\) −1.97692 −0.146943 −0.0734717 0.997297i \(-0.523408\pi\)
−0.0734717 + 0.997297i \(0.523408\pi\)
\(182\) 0 0
\(183\) 4.71939 0.348868
\(184\) 0 0
\(185\) −2.68168 −0.197161
\(186\) 0 0
\(187\) 8.23068 0.601887
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 10.6279 0.769007 0.384504 0.923123i \(-0.374373\pi\)
0.384504 + 0.923123i \(0.374373\pi\)
\(192\) 0 0
\(193\) −16.9531 −1.22031 −0.610154 0.792283i \(-0.708892\pi\)
−0.610154 + 0.792283i \(0.708892\pi\)
\(194\) 0 0
\(195\) 2.50007 0.179034
\(196\) 0 0
\(197\) −6.45596 −0.459968 −0.229984 0.973194i \(-0.573867\pi\)
−0.229984 + 0.973194i \(0.573867\pi\)
\(198\) 0 0
\(199\) −15.0256 −1.06513 −0.532567 0.846388i \(-0.678772\pi\)
−0.532567 + 0.846388i \(0.678772\pi\)
\(200\) 0 0
\(201\) −2.96477 −0.209119
\(202\) 0 0
\(203\) −4.63226 −0.325121
\(204\) 0 0
\(205\) 4.08713 0.285458
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −16.1465 −1.11688
\(210\) 0 0
\(211\) 1.68416 0.115943 0.0579713 0.998318i \(-0.481537\pi\)
0.0579713 + 0.998318i \(0.481537\pi\)
\(212\) 0 0
\(213\) 3.12003 0.213781
\(214\) 0 0
\(215\) 3.51957 0.240033
\(216\) 0 0
\(217\) −8.21699 −0.557806
\(218\) 0 0
\(219\) −10.2787 −0.694571
\(220\) 0 0
\(221\) 8.13080 0.546937
\(222\) 0 0
\(223\) 0.742472 0.0497196 0.0248598 0.999691i \(-0.492086\pi\)
0.0248598 + 0.999691i \(0.492086\pi\)
\(224\) 0 0
\(225\) −4.53531 −0.302354
\(226\) 0 0
\(227\) 20.0500 1.33077 0.665383 0.746502i \(-0.268268\pi\)
0.665383 + 0.746502i \(0.268268\pi\)
\(228\) 0 0
\(229\) −11.7710 −0.777850 −0.388925 0.921269i \(-0.627153\pi\)
−0.388925 + 0.921269i \(0.627153\pi\)
\(230\) 0 0
\(231\) −3.71255 −0.244268
\(232\) 0 0
\(233\) 17.0882 1.11949 0.559743 0.828666i \(-0.310900\pi\)
0.559743 + 0.828666i \(0.310900\pi\)
\(234\) 0 0
\(235\) 4.37319 0.285276
\(236\) 0 0
\(237\) −11.6137 −0.754391
\(238\) 0 0
\(239\) −24.5838 −1.59019 −0.795096 0.606483i \(-0.792580\pi\)
−0.795096 + 0.606483i \(0.792580\pi\)
\(240\) 0 0
\(241\) −12.3963 −0.798514 −0.399257 0.916839i \(-0.630732\pi\)
−0.399257 + 0.916839i \(0.630732\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.681685 −0.0435513
\(246\) 0 0
\(247\) −15.9506 −1.01491
\(248\) 0 0
\(249\) 11.5520 0.732077
\(250\) 0 0
\(251\) −6.40653 −0.404377 −0.202188 0.979347i \(-0.564805\pi\)
−0.202188 + 0.979347i \(0.564805\pi\)
\(252\) 0 0
\(253\) 3.71255 0.233406
\(254\) 0 0
\(255\) 1.51129 0.0946406
\(256\) 0 0
\(257\) −29.3262 −1.82932 −0.914661 0.404223i \(-0.867542\pi\)
−0.914661 + 0.404223i \(0.867542\pi\)
\(258\) 0 0
\(259\) 3.93391 0.244441
\(260\) 0 0
\(261\) −4.63226 −0.286730
\(262\) 0 0
\(263\) 21.0905 1.30050 0.650249 0.759721i \(-0.274664\pi\)
0.650249 + 0.759721i \(0.274664\pi\)
\(264\) 0 0
\(265\) 2.45502 0.150811
\(266\) 0 0
\(267\) −1.88885 −0.115596
\(268\) 0 0
\(269\) 15.7302 0.959085 0.479542 0.877519i \(-0.340803\pi\)
0.479542 + 0.877519i \(0.340803\pi\)
\(270\) 0 0
\(271\) −21.0055 −1.27599 −0.637995 0.770040i \(-0.720236\pi\)
−0.637995 + 0.770040i \(0.720236\pi\)
\(272\) 0 0
\(273\) −3.66749 −0.221967
\(274\) 0 0
\(275\) −16.8375 −1.01534
\(276\) 0 0
\(277\) −0.643976 −0.0386928 −0.0193464 0.999813i \(-0.506159\pi\)
−0.0193464 + 0.999813i \(0.506159\pi\)
\(278\) 0 0
\(279\) −8.21699 −0.491938
\(280\) 0 0
\(281\) −21.8404 −1.30289 −0.651444 0.758697i \(-0.725836\pi\)
−0.651444 + 0.758697i \(0.725836\pi\)
\(282\) 0 0
\(283\) 6.89431 0.409824 0.204912 0.978780i \(-0.434309\pi\)
0.204912 + 0.978780i \(0.434309\pi\)
\(284\) 0 0
\(285\) −2.96477 −0.175618
\(286\) 0 0
\(287\) −5.99563 −0.353911
\(288\) 0 0
\(289\) −12.0850 −0.710880
\(290\) 0 0
\(291\) −7.33062 −0.429728
\(292\) 0 0
\(293\) −5.30164 −0.309725 −0.154863 0.987936i \(-0.549494\pi\)
−0.154863 + 0.987936i \(0.549494\pi\)
\(294\) 0 0
\(295\) 4.65767 0.271180
\(296\) 0 0
\(297\) −3.71255 −0.215424
\(298\) 0 0
\(299\) 3.66749 0.212097
\(300\) 0 0
\(301\) −5.16305 −0.297593
\(302\) 0 0
\(303\) −4.59251 −0.263833
\(304\) 0 0
\(305\) 3.21714 0.184213
\(306\) 0 0
\(307\) −7.87218 −0.449289 −0.224645 0.974441i \(-0.572122\pi\)
−0.224645 + 0.974441i \(0.572122\pi\)
\(308\) 0 0
\(309\) 14.3448 0.816048
\(310\) 0 0
\(311\) 10.7390 0.608955 0.304478 0.952519i \(-0.401518\pi\)
0.304478 + 0.952519i \(0.401518\pi\)
\(312\) 0 0
\(313\) −14.1038 −0.797194 −0.398597 0.917126i \(-0.630503\pi\)
−0.398597 + 0.917126i \(0.630503\pi\)
\(314\) 0 0
\(315\) −0.681685 −0.0384086
\(316\) 0 0
\(317\) 17.3800 0.976160 0.488080 0.872799i \(-0.337697\pi\)
0.488080 + 0.872799i \(0.337697\pi\)
\(318\) 0 0
\(319\) −17.1975 −0.962875
\(320\) 0 0
\(321\) 1.44613 0.0807153
\(322\) 0 0
\(323\) −9.64209 −0.536500
\(324\) 0 0
\(325\) −16.6332 −0.922644
\(326\) 0 0
\(327\) −9.11567 −0.504097
\(328\) 0 0
\(329\) −6.41527 −0.353685
\(330\) 0 0
\(331\) −27.4251 −1.50742 −0.753710 0.657207i \(-0.771738\pi\)
−0.753710 + 0.657207i \(0.771738\pi\)
\(332\) 0 0
\(333\) 3.93391 0.215577
\(334\) 0 0
\(335\) −2.02104 −0.110421
\(336\) 0 0
\(337\) 24.8803 1.35531 0.677657 0.735378i \(-0.262995\pi\)
0.677657 + 0.735378i \(0.262995\pi\)
\(338\) 0 0
\(339\) −12.8051 −0.695479
\(340\) 0 0
\(341\) −30.5060 −1.65199
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0.681685 0.0367007
\(346\) 0 0
\(347\) 5.84380 0.313711 0.156856 0.987622i \(-0.449864\pi\)
0.156856 + 0.987622i \(0.449864\pi\)
\(348\) 0 0
\(349\) −28.5750 −1.52959 −0.764793 0.644276i \(-0.777159\pi\)
−0.764793 + 0.644276i \(0.777159\pi\)
\(350\) 0 0
\(351\) −3.66749 −0.195756
\(352\) 0 0
\(353\) −1.57039 −0.0835833 −0.0417916 0.999126i \(-0.513307\pi\)
−0.0417916 + 0.999126i \(0.513307\pi\)
\(354\) 0 0
\(355\) 2.12688 0.112883
\(356\) 0 0
\(357\) −2.21699 −0.117336
\(358\) 0 0
\(359\) −17.1441 −0.904828 −0.452414 0.891808i \(-0.649437\pi\)
−0.452414 + 0.891808i \(0.649437\pi\)
\(360\) 0 0
\(361\) −0.0846542 −0.00445549
\(362\) 0 0
\(363\) −2.78301 −0.146070
\(364\) 0 0
\(365\) −7.00685 −0.366755
\(366\) 0 0
\(367\) −6.56850 −0.342873 −0.171436 0.985195i \(-0.554841\pi\)
−0.171436 + 0.985195i \(0.554841\pi\)
\(368\) 0 0
\(369\) −5.99563 −0.312120
\(370\) 0 0
\(371\) −3.60140 −0.186975
\(372\) 0 0
\(373\) 14.8449 0.768639 0.384319 0.923200i \(-0.374436\pi\)
0.384319 + 0.923200i \(0.374436\pi\)
\(374\) 0 0
\(375\) −6.50007 −0.335662
\(376\) 0 0
\(377\) −16.9888 −0.874967
\(378\) 0 0
\(379\) 9.81606 0.504217 0.252109 0.967699i \(-0.418876\pi\)
0.252109 + 0.967699i \(0.418876\pi\)
\(380\) 0 0
\(381\) 17.5069 0.896907
\(382\) 0 0
\(383\) 21.4198 1.09450 0.547250 0.836969i \(-0.315675\pi\)
0.547250 + 0.836969i \(0.315675\pi\)
\(384\) 0 0
\(385\) −2.53079 −0.128981
\(386\) 0 0
\(387\) −5.16305 −0.262453
\(388\) 0 0
\(389\) 4.34044 0.220069 0.110035 0.993928i \(-0.464904\pi\)
0.110035 + 0.993928i \(0.464904\pi\)
\(390\) 0 0
\(391\) 2.21699 0.112118
\(392\) 0 0
\(393\) 2.08480 0.105164
\(394\) 0 0
\(395\) −7.91689 −0.398342
\(396\) 0 0
\(397\) 32.8551 1.64895 0.824476 0.565897i \(-0.191470\pi\)
0.824476 + 0.565897i \(0.191470\pi\)
\(398\) 0 0
\(399\) 4.34918 0.217731
\(400\) 0 0
\(401\) 12.9153 0.644962 0.322481 0.946576i \(-0.395483\pi\)
0.322481 + 0.946576i \(0.395483\pi\)
\(402\) 0 0
\(403\) −30.1358 −1.50117
\(404\) 0 0
\(405\) −0.681685 −0.0338732
\(406\) 0 0
\(407\) 14.6048 0.723934
\(408\) 0 0
\(409\) −33.0301 −1.63323 −0.816616 0.577182i \(-0.804152\pi\)
−0.816616 + 0.577182i \(0.804152\pi\)
\(410\) 0 0
\(411\) 14.9163 0.735766
\(412\) 0 0
\(413\) −6.83258 −0.336209
\(414\) 0 0
\(415\) 7.87481 0.386559
\(416\) 0 0
\(417\) 5.45050 0.266912
\(418\) 0 0
\(419\) 0.522051 0.0255039 0.0127519 0.999919i \(-0.495941\pi\)
0.0127519 + 0.999919i \(0.495941\pi\)
\(420\) 0 0
\(421\) −18.9888 −0.925457 −0.462728 0.886500i \(-0.653129\pi\)
−0.462728 + 0.886500i \(0.653129\pi\)
\(422\) 0 0
\(423\) −6.41527 −0.311921
\(424\) 0 0
\(425\) −10.0547 −0.487726
\(426\) 0 0
\(427\) −4.71939 −0.228387
\(428\) 0 0
\(429\) −13.6157 −0.657374
\(430\) 0 0
\(431\) 34.9978 1.68579 0.842893 0.538081i \(-0.180851\pi\)
0.842893 + 0.538081i \(0.180851\pi\)
\(432\) 0 0
\(433\) 8.71707 0.418915 0.209458 0.977818i \(-0.432830\pi\)
0.209458 + 0.977818i \(0.432830\pi\)
\(434\) 0 0
\(435\) −3.15774 −0.151402
\(436\) 0 0
\(437\) −4.34918 −0.208049
\(438\) 0 0
\(439\) −19.3316 −0.922645 −0.461322 0.887233i \(-0.652625\pi\)
−0.461322 + 0.887233i \(0.652625\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 12.7504 0.605790 0.302895 0.953024i \(-0.402047\pi\)
0.302895 + 0.953024i \(0.402047\pi\)
\(444\) 0 0
\(445\) −1.28760 −0.0610382
\(446\) 0 0
\(447\) −7.09900 −0.335771
\(448\) 0 0
\(449\) −12.9080 −0.609166 −0.304583 0.952486i \(-0.598517\pi\)
−0.304583 + 0.952486i \(0.598517\pi\)
\(450\) 0 0
\(451\) −22.2591 −1.04814
\(452\) 0 0
\(453\) 9.29197 0.436575
\(454\) 0 0
\(455\) −2.50007 −0.117205
\(456\) 0 0
\(457\) −14.4516 −0.676017 −0.338008 0.941143i \(-0.609753\pi\)
−0.338008 + 0.941143i \(0.609753\pi\)
\(458\) 0 0
\(459\) −2.21699 −0.103480
\(460\) 0 0
\(461\) −31.5309 −1.46854 −0.734271 0.678856i \(-0.762476\pi\)
−0.734271 + 0.678856i \(0.762476\pi\)
\(462\) 0 0
\(463\) 13.9672 0.649113 0.324557 0.945866i \(-0.394785\pi\)
0.324557 + 0.945866i \(0.394785\pi\)
\(464\) 0 0
\(465\) −5.60140 −0.259759
\(466\) 0 0
\(467\) 8.37662 0.387624 0.193812 0.981039i \(-0.437915\pi\)
0.193812 + 0.981039i \(0.437915\pi\)
\(468\) 0 0
\(469\) 2.96477 0.136900
\(470\) 0 0
\(471\) 1.33499 0.0615129
\(472\) 0 0
\(473\) −19.1681 −0.881349
\(474\) 0 0
\(475\) 19.7248 0.905038
\(476\) 0 0
\(477\) −3.60140 −0.164897
\(478\) 0 0
\(479\) −6.17957 −0.282352 −0.141176 0.989985i \(-0.545088\pi\)
−0.141176 + 0.989985i \(0.545088\pi\)
\(480\) 0 0
\(481\) 14.4276 0.657841
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −4.99717 −0.226910
\(486\) 0 0
\(487\) −5.84567 −0.264893 −0.132446 0.991190i \(-0.542283\pi\)
−0.132446 + 0.991190i \(0.542283\pi\)
\(488\) 0 0
\(489\) 7.49760 0.339053
\(490\) 0 0
\(491\) 30.0877 1.35784 0.678920 0.734212i \(-0.262448\pi\)
0.678920 + 0.734212i \(0.262448\pi\)
\(492\) 0 0
\(493\) −10.2697 −0.462523
\(494\) 0 0
\(495\) −2.53079 −0.113750
\(496\) 0 0
\(497\) −3.12003 −0.139953
\(498\) 0 0
\(499\) −15.3086 −0.685309 −0.342654 0.939462i \(-0.611326\pi\)
−0.342654 + 0.939462i \(0.611326\pi\)
\(500\) 0 0
\(501\) 4.34918 0.194307
\(502\) 0 0
\(503\) −15.0402 −0.670609 −0.335304 0.942110i \(-0.608839\pi\)
−0.335304 + 0.942110i \(0.608839\pi\)
\(504\) 0 0
\(505\) −3.13065 −0.139312
\(506\) 0 0
\(507\) −0.450502 −0.0200075
\(508\) 0 0
\(509\) 30.4870 1.35131 0.675656 0.737217i \(-0.263861\pi\)
0.675656 + 0.737217i \(0.263861\pi\)
\(510\) 0 0
\(511\) 10.2787 0.454704
\(512\) 0 0
\(513\) 4.34918 0.192021
\(514\) 0 0
\(515\) 9.77864 0.430898
\(516\) 0 0
\(517\) −23.8170 −1.04747
\(518\) 0 0
\(519\) −23.7459 −1.04233
\(520\) 0 0
\(521\) −10.1500 −0.444678 −0.222339 0.974969i \(-0.571369\pi\)
−0.222339 + 0.974969i \(0.571369\pi\)
\(522\) 0 0
\(523\) 2.26437 0.0990142 0.0495071 0.998774i \(-0.484235\pi\)
0.0495071 + 0.998774i \(0.484235\pi\)
\(524\) 0 0
\(525\) 4.53531 0.197937
\(526\) 0 0
\(527\) −18.2170 −0.793545
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −6.83258 −0.296509
\(532\) 0 0
\(533\) −21.9889 −0.952446
\(534\) 0 0
\(535\) 0.985808 0.0426202
\(536\) 0 0
\(537\) −6.28512 −0.271223
\(538\) 0 0
\(539\) 3.71255 0.159911
\(540\) 0 0
\(541\) 31.7512 1.36509 0.682545 0.730843i \(-0.260873\pi\)
0.682545 + 0.730843i \(0.260873\pi\)
\(542\) 0 0
\(543\) 1.97692 0.0848378
\(544\) 0 0
\(545\) −6.21401 −0.266179
\(546\) 0 0
\(547\) −25.7710 −1.10189 −0.550944 0.834542i \(-0.685732\pi\)
−0.550944 + 0.834542i \(0.685732\pi\)
\(548\) 0 0
\(549\) −4.71939 −0.201419
\(550\) 0 0
\(551\) 20.1465 0.858271
\(552\) 0 0
\(553\) 11.6137 0.493865
\(554\) 0 0
\(555\) 2.68168 0.113831
\(556\) 0 0
\(557\) −1.65613 −0.0701724 −0.0350862 0.999384i \(-0.511171\pi\)
−0.0350862 + 0.999384i \(0.511171\pi\)
\(558\) 0 0
\(559\) −18.9354 −0.800884
\(560\) 0 0
\(561\) −8.23068 −0.347500
\(562\) 0 0
\(563\) 8.00233 0.337258 0.168629 0.985680i \(-0.446066\pi\)
0.168629 + 0.985680i \(0.446066\pi\)
\(564\) 0 0
\(565\) −8.72907 −0.367235
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −6.58080 −0.275881 −0.137941 0.990440i \(-0.544048\pi\)
−0.137941 + 0.990440i \(0.544048\pi\)
\(570\) 0 0
\(571\) 34.9820 1.46395 0.731976 0.681330i \(-0.238598\pi\)
0.731976 + 0.681330i \(0.238598\pi\)
\(572\) 0 0
\(573\) −10.6279 −0.443987
\(574\) 0 0
\(575\) −4.53531 −0.189135
\(576\) 0 0
\(577\) 11.2558 0.468585 0.234292 0.972166i \(-0.424723\pi\)
0.234292 + 0.972166i \(0.424723\pi\)
\(578\) 0 0
\(579\) 16.9531 0.704545
\(580\) 0 0
\(581\) −11.5520 −0.479257
\(582\) 0 0
\(583\) −13.3704 −0.553744
\(584\) 0 0
\(585\) −2.50007 −0.103365
\(586\) 0 0
\(587\) 16.3805 0.676095 0.338047 0.941129i \(-0.390234\pi\)
0.338047 + 0.941129i \(0.390234\pi\)
\(588\) 0 0
\(589\) 35.7372 1.47252
\(590\) 0 0
\(591\) 6.45596 0.265563
\(592\) 0 0
\(593\) −11.2920 −0.463706 −0.231853 0.972751i \(-0.574479\pi\)
−0.231853 + 0.972751i \(0.574479\pi\)
\(594\) 0 0
\(595\) −1.51129 −0.0619568
\(596\) 0 0
\(597\) 15.0256 0.614955
\(598\) 0 0
\(599\) −14.0456 −0.573889 −0.286945 0.957947i \(-0.592640\pi\)
−0.286945 + 0.957947i \(0.592640\pi\)
\(600\) 0 0
\(601\) 6.71596 0.273950 0.136975 0.990575i \(-0.456262\pi\)
0.136975 + 0.990575i \(0.456262\pi\)
\(602\) 0 0
\(603\) 2.96477 0.120735
\(604\) 0 0
\(605\) −1.89714 −0.0771295
\(606\) 0 0
\(607\) −41.5453 −1.68627 −0.843135 0.537701i \(-0.819293\pi\)
−0.843135 + 0.537701i \(0.819293\pi\)
\(608\) 0 0
\(609\) 4.63226 0.187709
\(610\) 0 0
\(611\) −23.5280 −0.951839
\(612\) 0 0
\(613\) −9.99376 −0.403644 −0.201822 0.979422i \(-0.564686\pi\)
−0.201822 + 0.979422i \(0.564686\pi\)
\(614\) 0 0
\(615\) −4.08713 −0.164809
\(616\) 0 0
\(617\) −15.6157 −0.628666 −0.314333 0.949313i \(-0.601781\pi\)
−0.314333 + 0.949313i \(0.601781\pi\)
\(618\) 0 0
\(619\) −4.23272 −0.170127 −0.0850637 0.996376i \(-0.527109\pi\)
−0.0850637 + 0.996376i \(0.527109\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 1.88885 0.0756752
\(624\) 0 0
\(625\) 18.2455 0.729821
\(626\) 0 0
\(627\) 16.1465 0.644830
\(628\) 0 0
\(629\) 8.72143 0.347746
\(630\) 0 0
\(631\) 42.1468 1.67784 0.838919 0.544256i \(-0.183188\pi\)
0.838919 + 0.544256i \(0.183188\pi\)
\(632\) 0 0
\(633\) −1.68416 −0.0669395
\(634\) 0 0
\(635\) 11.9342 0.473594
\(636\) 0 0
\(637\) 3.66749 0.145311
\(638\) 0 0
\(639\) −3.12003 −0.123427
\(640\) 0 0
\(641\) −9.66749 −0.381843 −0.190922 0.981605i \(-0.561148\pi\)
−0.190922 + 0.981605i \(0.561148\pi\)
\(642\) 0 0
\(643\) −30.8753 −1.21760 −0.608802 0.793322i \(-0.708349\pi\)
−0.608802 + 0.793322i \(0.708349\pi\)
\(644\) 0 0
\(645\) −3.51957 −0.138583
\(646\) 0 0
\(647\) 11.1106 0.436805 0.218402 0.975859i \(-0.429915\pi\)
0.218402 + 0.975859i \(0.429915\pi\)
\(648\) 0 0
\(649\) −25.3663 −0.995714
\(650\) 0 0
\(651\) 8.21699 0.322049
\(652\) 0 0
\(653\) 40.6203 1.58959 0.794797 0.606876i \(-0.207577\pi\)
0.794797 + 0.606876i \(0.207577\pi\)
\(654\) 0 0
\(655\) 1.42118 0.0555301
\(656\) 0 0
\(657\) 10.2787 0.401011
\(658\) 0 0
\(659\) −7.13764 −0.278043 −0.139022 0.990289i \(-0.544396\pi\)
−0.139022 + 0.990289i \(0.544396\pi\)
\(660\) 0 0
\(661\) −10.5980 −0.412214 −0.206107 0.978529i \(-0.566079\pi\)
−0.206107 + 0.978529i \(0.566079\pi\)
\(662\) 0 0
\(663\) −8.13080 −0.315774
\(664\) 0 0
\(665\) 2.96477 0.114969
\(666\) 0 0
\(667\) −4.63226 −0.179362
\(668\) 0 0
\(669\) −0.742472 −0.0287056
\(670\) 0 0
\(671\) −17.5210 −0.676390
\(672\) 0 0
\(673\) −32.8352 −1.26570 −0.632852 0.774273i \(-0.718116\pi\)
−0.632852 + 0.774273i \(0.718116\pi\)
\(674\) 0 0
\(675\) 4.53531 0.174564
\(676\) 0 0
\(677\) 26.2542 1.00903 0.504516 0.863402i \(-0.331671\pi\)
0.504516 + 0.863402i \(0.331671\pi\)
\(678\) 0 0
\(679\) 7.33062 0.281323
\(680\) 0 0
\(681\) −20.0500 −0.768318
\(682\) 0 0
\(683\) 12.8203 0.490554 0.245277 0.969453i \(-0.421121\pi\)
0.245277 + 0.969453i \(0.421121\pi\)
\(684\) 0 0
\(685\) 10.1682 0.388507
\(686\) 0 0
\(687\) 11.7710 0.449092
\(688\) 0 0
\(689\) −13.2081 −0.503189
\(690\) 0 0
\(691\) 47.2581 1.79778 0.898892 0.438171i \(-0.144374\pi\)
0.898892 + 0.438171i \(0.144374\pi\)
\(692\) 0 0
\(693\) 3.71255 0.141028
\(694\) 0 0
\(695\) 3.71553 0.140938
\(696\) 0 0
\(697\) −13.2923 −0.503480
\(698\) 0 0
\(699\) −17.0882 −0.646336
\(700\) 0 0
\(701\) 14.3948 0.543685 0.271842 0.962342i \(-0.412367\pi\)
0.271842 + 0.962342i \(0.412367\pi\)
\(702\) 0 0
\(703\) −17.1093 −0.645288
\(704\) 0 0
\(705\) −4.37319 −0.164704
\(706\) 0 0
\(707\) 4.59251 0.172719
\(708\) 0 0
\(709\) −3.40406 −0.127842 −0.0639210 0.997955i \(-0.520361\pi\)
−0.0639210 + 0.997955i \(0.520361\pi\)
\(710\) 0 0
\(711\) 11.6137 0.435548
\(712\) 0 0
\(713\) −8.21699 −0.307729
\(714\) 0 0
\(715\) −9.28165 −0.347114
\(716\) 0 0
\(717\) 24.5838 0.918098
\(718\) 0 0
\(719\) −38.6326 −1.44075 −0.720376 0.693584i \(-0.756031\pi\)
−0.720376 + 0.693584i \(0.756031\pi\)
\(720\) 0 0
\(721\) −14.3448 −0.534229
\(722\) 0 0
\(723\) 12.3963 0.461022
\(724\) 0 0
\(725\) 21.0087 0.780244
\(726\) 0 0
\(727\) −17.7830 −0.659535 −0.329768 0.944062i \(-0.606970\pi\)
−0.329768 + 0.944062i \(0.606970\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −11.4464 −0.423362
\(732\) 0 0
\(733\) 1.92050 0.0709354 0.0354677 0.999371i \(-0.488708\pi\)
0.0354677 + 0.999371i \(0.488708\pi\)
\(734\) 0 0
\(735\) 0.681685 0.0251443
\(736\) 0 0
\(737\) 11.0068 0.405442
\(738\) 0 0
\(739\) 8.58925 0.315961 0.157980 0.987442i \(-0.449502\pi\)
0.157980 + 0.987442i \(0.449502\pi\)
\(740\) 0 0
\(741\) 15.9506 0.585959
\(742\) 0 0
\(743\) 10.5308 0.386337 0.193169 0.981166i \(-0.438124\pi\)
0.193169 + 0.981166i \(0.438124\pi\)
\(744\) 0 0
\(745\) −4.83928 −0.177297
\(746\) 0 0
\(747\) −11.5520 −0.422665
\(748\) 0 0
\(749\) −1.44613 −0.0528406
\(750\) 0 0
\(751\) 30.8659 1.12631 0.563157 0.826350i \(-0.309587\pi\)
0.563157 + 0.826350i \(0.309587\pi\)
\(752\) 0 0
\(753\) 6.40653 0.233467
\(754\) 0 0
\(755\) 6.33420 0.230525
\(756\) 0 0
\(757\) −6.50567 −0.236453 −0.118226 0.992987i \(-0.537721\pi\)
−0.118226 + 0.992987i \(0.537721\pi\)
\(758\) 0 0
\(759\) −3.71255 −0.134757
\(760\) 0 0
\(761\) −32.7554 −1.18738 −0.593692 0.804693i \(-0.702330\pi\)
−0.593692 + 0.804693i \(0.702330\pi\)
\(762\) 0 0
\(763\) 9.11567 0.330009
\(764\) 0 0
\(765\) −1.51129 −0.0546408
\(766\) 0 0
\(767\) −25.0584 −0.904808
\(768\) 0 0
\(769\) 19.5298 0.704264 0.352132 0.935950i \(-0.385457\pi\)
0.352132 + 0.935950i \(0.385457\pi\)
\(770\) 0 0
\(771\) 29.3262 1.05616
\(772\) 0 0
\(773\) −46.4062 −1.66911 −0.834557 0.550921i \(-0.814276\pi\)
−0.834557 + 0.550921i \(0.814276\pi\)
\(774\) 0 0
\(775\) 37.2666 1.33865
\(776\) 0 0
\(777\) −3.93391 −0.141128
\(778\) 0 0
\(779\) 26.0761 0.934272
\(780\) 0 0
\(781\) −11.5833 −0.414482
\(782\) 0 0
\(783\) 4.63226 0.165544
\(784\) 0 0
\(785\) 0.910039 0.0324807
\(786\) 0 0
\(787\) −30.4937 −1.08698 −0.543491 0.839415i \(-0.682898\pi\)
−0.543491 + 0.839415i \(0.682898\pi\)
\(788\) 0 0
\(789\) −21.0905 −0.750843
\(790\) 0 0
\(791\) 12.8051 0.455298
\(792\) 0 0
\(793\) −17.3083 −0.614637
\(794\) 0 0
\(795\) −2.45502 −0.0870706
\(796\) 0 0
\(797\) −38.9621 −1.38011 −0.690054 0.723758i \(-0.742413\pi\)
−0.690054 + 0.723758i \(0.742413\pi\)
\(798\) 0 0
\(799\) −14.2226 −0.503159
\(800\) 0 0
\(801\) 1.88885 0.0667393
\(802\) 0 0
\(803\) 38.1602 1.34664
\(804\) 0 0
\(805\) −0.681685 −0.0240262
\(806\) 0 0
\(807\) −15.7302 −0.553728
\(808\) 0 0
\(809\) 6.82728 0.240034 0.120017 0.992772i \(-0.461705\pi\)
0.120017 + 0.992772i \(0.461705\pi\)
\(810\) 0 0
\(811\) −48.1986 −1.69248 −0.846241 0.532801i \(-0.821139\pi\)
−0.846241 + 0.532801i \(0.821139\pi\)
\(812\) 0 0
\(813\) 21.0055 0.736693
\(814\) 0 0
\(815\) 5.11100 0.179030
\(816\) 0 0
\(817\) 22.4550 0.785602
\(818\) 0 0
\(819\) 3.66749 0.128153
\(820\) 0 0
\(821\) 56.3174 1.96549 0.982745 0.184967i \(-0.0592178\pi\)
0.982745 + 0.184967i \(0.0592178\pi\)
\(822\) 0 0
\(823\) −7.86484 −0.274151 −0.137075 0.990561i \(-0.543770\pi\)
−0.137075 + 0.990561i \(0.543770\pi\)
\(824\) 0 0
\(825\) 16.8375 0.586208
\(826\) 0 0
\(827\) 17.8131 0.619422 0.309711 0.950831i \(-0.399768\pi\)
0.309711 + 0.950831i \(0.399768\pi\)
\(828\) 0 0
\(829\) −53.7175 −1.86569 −0.932843 0.360283i \(-0.882680\pi\)
−0.932843 + 0.360283i \(0.882680\pi\)
\(830\) 0 0
\(831\) 0.643976 0.0223393
\(832\) 0 0
\(833\) 2.21699 0.0768142
\(834\) 0 0
\(835\) 2.96477 0.102600
\(836\) 0 0
\(837\) 8.21699 0.284021
\(838\) 0 0
\(839\) −3.38033 −0.116702 −0.0583510 0.998296i \(-0.518584\pi\)
−0.0583510 + 0.998296i \(0.518584\pi\)
\(840\) 0 0
\(841\) −7.54215 −0.260074
\(842\) 0 0
\(843\) 21.8404 0.752222
\(844\) 0 0
\(845\) −0.307100 −0.0105646
\(846\) 0 0
\(847\) 2.78301 0.0956253
\(848\) 0 0
\(849\) −6.89431 −0.236612
\(850\) 0 0
\(851\) 3.93391 0.134853
\(852\) 0 0
\(853\) 28.9072 0.989764 0.494882 0.868960i \(-0.335211\pi\)
0.494882 + 0.868960i \(0.335211\pi\)
\(854\) 0 0
\(855\) 2.96477 0.101393
\(856\) 0 0
\(857\) −19.4329 −0.663815 −0.331908 0.943312i \(-0.607692\pi\)
−0.331908 + 0.943312i \(0.607692\pi\)
\(858\) 0 0
\(859\) 12.8159 0.437273 0.218637 0.975806i \(-0.429839\pi\)
0.218637 + 0.975806i \(0.429839\pi\)
\(860\) 0 0
\(861\) 5.99563 0.204331
\(862\) 0 0
\(863\) 3.25957 0.110957 0.0554785 0.998460i \(-0.482332\pi\)
0.0554785 + 0.998460i \(0.482332\pi\)
\(864\) 0 0
\(865\) −16.1872 −0.550382
\(866\) 0 0
\(867\) 12.0850 0.410427
\(868\) 0 0
\(869\) 43.1164 1.46262
\(870\) 0 0
\(871\) 10.8733 0.368427
\(872\) 0 0
\(873\) 7.33062 0.248104
\(874\) 0 0
\(875\) 6.50007 0.219743
\(876\) 0 0
\(877\) −13.9913 −0.472451 −0.236226 0.971698i \(-0.575910\pi\)
−0.236226 + 0.971698i \(0.575910\pi\)
\(878\) 0 0
\(879\) 5.30164 0.178820
\(880\) 0 0
\(881\) −0.174264 −0.00587111 −0.00293555 0.999996i \(-0.500934\pi\)
−0.00293555 + 0.999996i \(0.500934\pi\)
\(882\) 0 0
\(883\) −51.5852 −1.73598 −0.867989 0.496583i \(-0.834588\pi\)
−0.867989 + 0.496583i \(0.834588\pi\)
\(884\) 0 0
\(885\) −4.65767 −0.156566
\(886\) 0 0
\(887\) 42.4004 1.42367 0.711833 0.702348i \(-0.247865\pi\)
0.711833 + 0.702348i \(0.247865\pi\)
\(888\) 0 0
\(889\) −17.5069 −0.587163
\(890\) 0 0
\(891\) 3.71255 0.124375
\(892\) 0 0
\(893\) 27.9012 0.933676
\(894\) 0 0
\(895\) −4.28447 −0.143214
\(896\) 0 0
\(897\) −3.66749 −0.122454
\(898\) 0 0
\(899\) 38.0633 1.26948
\(900\) 0 0
\(901\) −7.98427 −0.265995
\(902\) 0 0
\(903\) 5.16305 0.171816
\(904\) 0 0
\(905\) 1.34764 0.0447970
\(906\) 0 0
\(907\) −10.1863 −0.338230 −0.169115 0.985596i \(-0.554091\pi\)
−0.169115 + 0.985596i \(0.554091\pi\)
\(908\) 0 0
\(909\) 4.59251 0.152324
\(910\) 0 0
\(911\) −29.4173 −0.974639 −0.487319 0.873224i \(-0.662025\pi\)
−0.487319 + 0.873224i \(0.662025\pi\)
\(912\) 0 0
\(913\) −42.8873 −1.41936
\(914\) 0 0
\(915\) −3.21714 −0.106355
\(916\) 0 0
\(917\) −2.08480 −0.0688463
\(918\) 0 0
\(919\) 42.7597 1.41051 0.705257 0.708952i \(-0.250832\pi\)
0.705257 + 0.708952i \(0.250832\pi\)
\(920\) 0 0
\(921\) 7.87218 0.259397
\(922\) 0 0
\(923\) −11.4427 −0.376641
\(924\) 0 0
\(925\) −17.8415 −0.586624
\(926\) 0 0
\(927\) −14.3448 −0.471145
\(928\) 0 0
\(929\) 7.12936 0.233907 0.116953 0.993137i \(-0.462687\pi\)
0.116953 + 0.993137i \(0.462687\pi\)
\(930\) 0 0
\(931\) −4.34918 −0.142539
\(932\) 0 0
\(933\) −10.7390 −0.351580
\(934\) 0 0
\(935\) −5.61073 −0.183491
\(936\) 0 0
\(937\) −51.4494 −1.68078 −0.840389 0.541983i \(-0.817674\pi\)
−0.840389 + 0.541983i \(0.817674\pi\)
\(938\) 0 0
\(939\) 14.1038 0.460260
\(940\) 0 0
\(941\) −21.0999 −0.687838 −0.343919 0.938999i \(-0.611755\pi\)
−0.343919 + 0.938999i \(0.611755\pi\)
\(942\) 0 0
\(943\) −5.99563 −0.195245
\(944\) 0 0
\(945\) 0.681685 0.0221752
\(946\) 0 0
\(947\) −6.17333 −0.200606 −0.100303 0.994957i \(-0.531981\pi\)
−0.100303 + 0.994957i \(0.531981\pi\)
\(948\) 0 0
\(949\) 37.6971 1.22370
\(950\) 0 0
\(951\) −17.3800 −0.563586
\(952\) 0 0
\(953\) −32.1838 −1.04254 −0.521268 0.853393i \(-0.674541\pi\)
−0.521268 + 0.853393i \(0.674541\pi\)
\(954\) 0 0
\(955\) −7.24488 −0.234439
\(956\) 0 0
\(957\) 17.1975 0.555916
\(958\) 0 0
\(959\) −14.9163 −0.481672
\(960\) 0 0
\(961\) 36.5189 1.17803
\(962\) 0 0
\(963\) −1.44613 −0.0466010
\(964\) 0 0
\(965\) 11.5566 0.372021
\(966\) 0 0
\(967\) 8.21822 0.264280 0.132140 0.991231i \(-0.457815\pi\)
0.132140 + 0.991231i \(0.457815\pi\)
\(968\) 0 0
\(969\) 9.64209 0.309748
\(970\) 0 0
\(971\) −0.611810 −0.0196339 −0.00981697 0.999952i \(-0.503125\pi\)
−0.00981697 + 0.999952i \(0.503125\pi\)
\(972\) 0 0
\(973\) −5.45050 −0.174735
\(974\) 0 0
\(975\) 16.6332 0.532689
\(976\) 0 0
\(977\) −53.8950 −1.72426 −0.862128 0.506691i \(-0.830868\pi\)
−0.862128 + 0.506691i \(0.830868\pi\)
\(978\) 0 0
\(979\) 7.01245 0.224119
\(980\) 0 0
\(981\) 9.11567 0.291041
\(982\) 0 0
\(983\) −10.9487 −0.349209 −0.174604 0.984639i \(-0.555865\pi\)
−0.174604 + 0.984639i \(0.555865\pi\)
\(984\) 0 0
\(985\) 4.40093 0.140225
\(986\) 0 0
\(987\) 6.41527 0.204200
\(988\) 0 0
\(989\) −5.16305 −0.164175
\(990\) 0 0
\(991\) −9.22384 −0.293005 −0.146502 0.989210i \(-0.546802\pi\)
−0.146502 + 0.989210i \(0.546802\pi\)
\(992\) 0 0
\(993\) 27.4251 0.870309
\(994\) 0 0
\(995\) 10.2427 0.324715
\(996\) 0 0
\(997\) −33.8410 −1.07175 −0.535877 0.844296i \(-0.680019\pi\)
−0.535877 + 0.844296i \(0.680019\pi\)
\(998\) 0 0
\(999\) −3.93391 −0.124463
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 7728.2.a.bz.1.2 4
4.3 odd 2 3864.2.a.t.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.t.1.2 4 4.3 odd 2
7728.2.a.bz.1.2 4 1.1 even 1 trivial