Properties

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

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(34.8786556029\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 4368.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.27492 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.27492 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.27492 q^{11} -1.00000 q^{13} +3.27492 q^{15} -7.27492 q^{17} -5.27492 q^{19} -1.00000 q^{21} +5.27492 q^{23} +5.72508 q^{25} +1.00000 q^{27} +0.725083 q^{29} -8.00000 q^{31} -5.27492 q^{33} -3.27492 q^{35} +3.27492 q^{37} -1.00000 q^{39} -8.54983 q^{41} -5.27492 q^{43} +3.27492 q^{45} +1.00000 q^{49} -7.27492 q^{51} +10.0000 q^{53} -17.2749 q^{55} -5.27492 q^{57} -8.00000 q^{59} -4.72508 q^{61} -1.00000 q^{63} -3.27492 q^{65} +2.54983 q^{67} +5.27492 q^{69} +9.82475 q^{73} +5.72508 q^{75} +5.27492 q^{77} +2.54983 q^{79} +1.00000 q^{81} -10.5498 q^{83} -23.8248 q^{85} +0.725083 q^{87} -14.0000 q^{89} +1.00000 q^{91} -8.00000 q^{93} -17.2749 q^{95} -15.0997 q^{97} -5.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} - 3 q^{11} - 2 q^{13} - q^{15} - 7 q^{17} - 3 q^{19} - 2 q^{21} + 3 q^{23} + 19 q^{25} + 2 q^{27} + 9 q^{29} - 16 q^{31} - 3 q^{33} + q^{35} - q^{37} - 2 q^{39} - 2 q^{41} - 3 q^{43} - q^{45} + 2 q^{49} - 7 q^{51} + 20 q^{53} - 27 q^{55} - 3 q^{57} - 16 q^{59} - 17 q^{61} - 2 q^{63} + q^{65} - 10 q^{67} + 3 q^{69} - 3 q^{73} + 19 q^{75} + 3 q^{77} - 10 q^{79} + 2 q^{81} - 6 q^{83} - 25 q^{85} + 9 q^{87} - 28 q^{89} + 2 q^{91} - 16 q^{93} - 27 q^{95} - 3 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 3.27492 1.46459 0.732294 0.680989i \(-0.238450\pi\)
0.732294 + 0.680989i \(0.238450\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.27492 −1.59045 −0.795224 0.606316i \(-0.792647\pi\)
−0.795224 + 0.606316i \(0.792647\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 3.27492 0.845580
\(16\) 0 0
\(17\) −7.27492 −1.76443 −0.882213 0.470850i \(-0.843947\pi\)
−0.882213 + 0.470850i \(0.843947\pi\)
\(18\) 0 0
\(19\) −5.27492 −1.21015 −0.605075 0.796169i \(-0.706857\pi\)
−0.605075 + 0.796169i \(0.706857\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 5.27492 1.09990 0.549948 0.835199i \(-0.314647\pi\)
0.549948 + 0.835199i \(0.314647\pi\)
\(24\) 0 0
\(25\) 5.72508 1.14502
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.725083 0.134644 0.0673222 0.997731i \(-0.478554\pi\)
0.0673222 + 0.997731i \(0.478554\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) −5.27492 −0.918245
\(34\) 0 0
\(35\) −3.27492 −0.553562
\(36\) 0 0
\(37\) 3.27492 0.538393 0.269197 0.963085i \(-0.413242\pi\)
0.269197 + 0.963085i \(0.413242\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −8.54983 −1.33526 −0.667630 0.744493i \(-0.732691\pi\)
−0.667630 + 0.744493i \(0.732691\pi\)
\(42\) 0 0
\(43\) −5.27492 −0.804417 −0.402209 0.915548i \(-0.631757\pi\)
−0.402209 + 0.915548i \(0.631757\pi\)
\(44\) 0 0
\(45\) 3.27492 0.488196
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −7.27492 −1.01869
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) −17.2749 −2.32935
\(56\) 0 0
\(57\) −5.27492 −0.698680
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −4.72508 −0.604985 −0.302492 0.953152i \(-0.597819\pi\)
−0.302492 + 0.953152i \(0.597819\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −3.27492 −0.406203
\(66\) 0 0
\(67\) 2.54983 0.311512 0.155756 0.987796i \(-0.450219\pi\)
0.155756 + 0.987796i \(0.450219\pi\)
\(68\) 0 0
\(69\) 5.27492 0.635025
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 9.82475 1.14990 0.574950 0.818188i \(-0.305021\pi\)
0.574950 + 0.818188i \(0.305021\pi\)
\(74\) 0 0
\(75\) 5.72508 0.661076
\(76\) 0 0
\(77\) 5.27492 0.601133
\(78\) 0 0
\(79\) 2.54983 0.286879 0.143439 0.989659i \(-0.454184\pi\)
0.143439 + 0.989659i \(0.454184\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.5498 −1.15799 −0.578997 0.815329i \(-0.696556\pi\)
−0.578997 + 0.815329i \(0.696556\pi\)
\(84\) 0 0
\(85\) −23.8248 −2.58416
\(86\) 0 0
\(87\) 0.725083 0.0777370
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) −17.2749 −1.77237
\(96\) 0 0
\(97\) −15.0997 −1.53314 −0.766570 0.642161i \(-0.778038\pi\)
−0.766570 + 0.642161i \(0.778038\pi\)
\(98\) 0 0
\(99\) −5.27492 −0.530149
\(100\) 0 0
\(101\) −12.5498 −1.24876 −0.624378 0.781123i \(-0.714647\pi\)
−0.624378 + 0.781123i \(0.714647\pi\)
\(102\) 0 0
\(103\) −2.72508 −0.268510 −0.134255 0.990947i \(-0.542864\pi\)
−0.134255 + 0.990947i \(0.542864\pi\)
\(104\) 0 0
\(105\) −3.27492 −0.319599
\(106\) 0 0
\(107\) 14.5498 1.40659 0.703293 0.710900i \(-0.251712\pi\)
0.703293 + 0.710900i \(0.251712\pi\)
\(108\) 0 0
\(109\) −7.27492 −0.696811 −0.348405 0.937344i \(-0.613277\pi\)
−0.348405 + 0.937344i \(0.613277\pi\)
\(110\) 0 0
\(111\) 3.27492 0.310841
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 17.2749 1.61089
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 7.27492 0.666891
\(120\) 0 0
\(121\) 16.8248 1.52952
\(122\) 0 0
\(123\) −8.54983 −0.770913
\(124\) 0 0
\(125\) 2.37459 0.212389
\(126\) 0 0
\(127\) 5.45017 0.483624 0.241812 0.970323i \(-0.422258\pi\)
0.241812 + 0.970323i \(0.422258\pi\)
\(128\) 0 0
\(129\) −5.27492 −0.464431
\(130\) 0 0
\(131\) 15.8248 1.38261 0.691307 0.722561i \(-0.257035\pi\)
0.691307 + 0.722561i \(0.257035\pi\)
\(132\) 0 0
\(133\) 5.27492 0.457393
\(134\) 0 0
\(135\) 3.27492 0.281860
\(136\) 0 0
\(137\) 20.3746 1.74072 0.870359 0.492417i \(-0.163887\pi\)
0.870359 + 0.492417i \(0.163887\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.27492 0.441111
\(144\) 0 0
\(145\) 2.37459 0.197199
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 7.45017 0.610341 0.305171 0.952298i \(-0.401286\pi\)
0.305171 + 0.952298i \(0.401286\pi\)
\(150\) 0 0
\(151\) 1.27492 0.103751 0.0518756 0.998654i \(-0.483480\pi\)
0.0518756 + 0.998654i \(0.483480\pi\)
\(152\) 0 0
\(153\) −7.27492 −0.588142
\(154\) 0 0
\(155\) −26.1993 −2.10438
\(156\) 0 0
\(157\) 24.3746 1.94530 0.972652 0.232268i \(-0.0746146\pi\)
0.972652 + 0.232268i \(0.0746146\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) −5.27492 −0.415722
\(162\) 0 0
\(163\) 2.54983 0.199718 0.0998592 0.995002i \(-0.468161\pi\)
0.0998592 + 0.995002i \(0.468161\pi\)
\(164\) 0 0
\(165\) −17.2749 −1.34485
\(166\) 0 0
\(167\) 9.27492 0.717715 0.358857 0.933392i \(-0.383166\pi\)
0.358857 + 0.933392i \(0.383166\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −5.27492 −0.403383
\(172\) 0 0
\(173\) −23.0997 −1.75624 −0.878118 0.478445i \(-0.841201\pi\)
−0.878118 + 0.478445i \(0.841201\pi\)
\(174\) 0 0
\(175\) −5.72508 −0.432776
\(176\) 0 0
\(177\) −8.00000 −0.601317
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −11.0997 −0.825032 −0.412516 0.910950i \(-0.635350\pi\)
−0.412516 + 0.910950i \(0.635350\pi\)
\(182\) 0 0
\(183\) −4.72508 −0.349288
\(184\) 0 0
\(185\) 10.7251 0.788524
\(186\) 0 0
\(187\) 38.3746 2.80623
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 5.27492 0.381680 0.190840 0.981621i \(-0.438879\pi\)
0.190840 + 0.981621i \(0.438879\pi\)
\(192\) 0 0
\(193\) −3.45017 −0.248348 −0.124174 0.992260i \(-0.539628\pi\)
−0.124174 + 0.992260i \(0.539628\pi\)
\(194\) 0 0
\(195\) −3.27492 −0.234522
\(196\) 0 0
\(197\) 12.5498 0.894139 0.447069 0.894499i \(-0.352468\pi\)
0.447069 + 0.894499i \(0.352468\pi\)
\(198\) 0 0
\(199\) 15.8248 1.12179 0.560893 0.827888i \(-0.310458\pi\)
0.560893 + 0.827888i \(0.310458\pi\)
\(200\) 0 0
\(201\) 2.54983 0.179851
\(202\) 0 0
\(203\) −0.725083 −0.0508908
\(204\) 0 0
\(205\) −28.0000 −1.95560
\(206\) 0 0
\(207\) 5.27492 0.366632
\(208\) 0 0
\(209\) 27.8248 1.92468
\(210\) 0 0
\(211\) −23.8248 −1.64016 −0.820082 0.572246i \(-0.806072\pi\)
−0.820082 + 0.572246i \(0.806072\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −17.2749 −1.17814
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) 9.82475 0.663895
\(220\) 0 0
\(221\) 7.27492 0.489364
\(222\) 0 0
\(223\) 23.6495 1.58369 0.791844 0.610723i \(-0.209121\pi\)
0.791844 + 0.610723i \(0.209121\pi\)
\(224\) 0 0
\(225\) 5.72508 0.381672
\(226\) 0 0
\(227\) 5.45017 0.361740 0.180870 0.983507i \(-0.442109\pi\)
0.180870 + 0.983507i \(0.442109\pi\)
\(228\) 0 0
\(229\) −11.0997 −0.733487 −0.366743 0.930322i \(-0.619527\pi\)
−0.366743 + 0.930322i \(0.619527\pi\)
\(230\) 0 0
\(231\) 5.27492 0.347064
\(232\) 0 0
\(233\) 20.5498 1.34626 0.673132 0.739522i \(-0.264948\pi\)
0.673132 + 0.739522i \(0.264948\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2.54983 0.165630
\(238\) 0 0
\(239\) 5.09967 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(240\) 0 0
\(241\) −15.0997 −0.972655 −0.486328 0.873777i \(-0.661664\pi\)
−0.486328 + 0.873777i \(0.661664\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.27492 0.209227
\(246\) 0 0
\(247\) 5.27492 0.335635
\(248\) 0 0
\(249\) −10.5498 −0.668569
\(250\) 0 0
\(251\) 28.9244 1.82569 0.912847 0.408303i \(-0.133879\pi\)
0.912847 + 0.408303i \(0.133879\pi\)
\(252\) 0 0
\(253\) −27.8248 −1.74933
\(254\) 0 0
\(255\) −23.8248 −1.49196
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −3.27492 −0.203493
\(260\) 0 0
\(261\) 0.725083 0.0448815
\(262\) 0 0
\(263\) −28.0000 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) 32.7492 2.01177
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −13.4502 −0.817039 −0.408520 0.912750i \(-0.633955\pi\)
−0.408520 + 0.912750i \(0.633955\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) 0 0
\(275\) −30.1993 −1.82109
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −15.0997 −0.900771 −0.450385 0.892834i \(-0.648713\pi\)
−0.450385 + 0.892834i \(0.648713\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) −17.2749 −1.02328
\(286\) 0 0
\(287\) 8.54983 0.504681
\(288\) 0 0
\(289\) 35.9244 2.11320
\(290\) 0 0
\(291\) −15.0997 −0.885158
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) −26.1993 −1.52538
\(296\) 0 0
\(297\) −5.27492 −0.306082
\(298\) 0 0
\(299\) −5.27492 −0.305056
\(300\) 0 0
\(301\) 5.27492 0.304041
\(302\) 0 0
\(303\) −12.5498 −0.720969
\(304\) 0 0
\(305\) −15.4743 −0.886053
\(306\) 0 0
\(307\) 9.09967 0.519346 0.259673 0.965697i \(-0.416385\pi\)
0.259673 + 0.965697i \(0.416385\pi\)
\(308\) 0 0
\(309\) −2.72508 −0.155025
\(310\) 0 0
\(311\) −18.5498 −1.05186 −0.525932 0.850526i \(-0.676283\pi\)
−0.525932 + 0.850526i \(0.676283\pi\)
\(312\) 0 0
\(313\) 17.6495 0.997609 0.498804 0.866715i \(-0.333773\pi\)
0.498804 + 0.866715i \(0.333773\pi\)
\(314\) 0 0
\(315\) −3.27492 −0.184521
\(316\) 0 0
\(317\) −16.5498 −0.929531 −0.464766 0.885434i \(-0.653861\pi\)
−0.464766 + 0.885434i \(0.653861\pi\)
\(318\) 0 0
\(319\) −3.82475 −0.214145
\(320\) 0 0
\(321\) 14.5498 0.812093
\(322\) 0 0
\(323\) 38.3746 2.13522
\(324\) 0 0
\(325\) −5.72508 −0.317570
\(326\) 0 0
\(327\) −7.27492 −0.402304
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −15.6495 −0.860174 −0.430087 0.902787i \(-0.641517\pi\)
−0.430087 + 0.902787i \(0.641517\pi\)
\(332\) 0 0
\(333\) 3.27492 0.179464
\(334\) 0 0
\(335\) 8.35050 0.456236
\(336\) 0 0
\(337\) −28.3746 −1.54566 −0.772831 0.634612i \(-0.781160\pi\)
−0.772831 + 0.634612i \(0.781160\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 42.1993 2.28522
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 17.2749 0.930050
\(346\) 0 0
\(347\) 25.0997 1.34742 0.673710 0.738995i \(-0.264700\pi\)
0.673710 + 0.738995i \(0.264700\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −3.09967 −0.164979 −0.0824894 0.996592i \(-0.526287\pi\)
−0.0824894 + 0.996592i \(0.526287\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 7.27492 0.385029
\(358\) 0 0
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 8.82475 0.464461
\(362\) 0 0
\(363\) 16.8248 0.883070
\(364\) 0 0
\(365\) 32.1752 1.68413
\(366\) 0 0
\(367\) 33.0997 1.72779 0.863894 0.503673i \(-0.168018\pi\)
0.863894 + 0.503673i \(0.168018\pi\)
\(368\) 0 0
\(369\) −8.54983 −0.445087
\(370\) 0 0
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) 11.4502 0.592867 0.296434 0.955053i \(-0.404203\pi\)
0.296434 + 0.955053i \(0.404203\pi\)
\(374\) 0 0
\(375\) 2.37459 0.122623
\(376\) 0 0
\(377\) −0.725083 −0.0373437
\(378\) 0 0
\(379\) 2.90033 0.148980 0.0744900 0.997222i \(-0.476267\pi\)
0.0744900 + 0.997222i \(0.476267\pi\)
\(380\) 0 0
\(381\) 5.45017 0.279220
\(382\) 0 0
\(383\) −22.7251 −1.16120 −0.580599 0.814190i \(-0.697181\pi\)
−0.580599 + 0.814190i \(0.697181\pi\)
\(384\) 0 0
\(385\) 17.2749 0.880411
\(386\) 0 0
\(387\) −5.27492 −0.268139
\(388\) 0 0
\(389\) −32.1993 −1.63257 −0.816286 0.577649i \(-0.803970\pi\)
−0.816286 + 0.577649i \(0.803970\pi\)
\(390\) 0 0
\(391\) −38.3746 −1.94069
\(392\) 0 0
\(393\) 15.8248 0.798253
\(394\) 0 0
\(395\) 8.35050 0.420159
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 5.27492 0.264076
\(400\) 0 0
\(401\) 27.0997 1.35329 0.676646 0.736308i \(-0.263433\pi\)
0.676646 + 0.736308i \(0.263433\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 0 0
\(405\) 3.27492 0.162732
\(406\) 0 0
\(407\) −17.2749 −0.856286
\(408\) 0 0
\(409\) −29.8248 −1.47474 −0.737370 0.675490i \(-0.763932\pi\)
−0.737370 + 0.675490i \(0.763932\pi\)
\(410\) 0 0
\(411\) 20.3746 1.00500
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) −34.5498 −1.69598
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 5.27492 0.257697 0.128848 0.991664i \(-0.458872\pi\)
0.128848 + 0.991664i \(0.458872\pi\)
\(420\) 0 0
\(421\) −27.0997 −1.32076 −0.660379 0.750933i \(-0.729604\pi\)
−0.660379 + 0.750933i \(0.729604\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −41.6495 −2.02030
\(426\) 0 0
\(427\) 4.72508 0.228663
\(428\) 0 0
\(429\) 5.27492 0.254675
\(430\) 0 0
\(431\) 26.5498 1.27886 0.639430 0.768849i \(-0.279170\pi\)
0.639430 + 0.768849i \(0.279170\pi\)
\(432\) 0 0
\(433\) −21.6495 −1.04041 −0.520204 0.854042i \(-0.674144\pi\)
−0.520204 + 0.854042i \(0.674144\pi\)
\(434\) 0 0
\(435\) 2.37459 0.113853
\(436\) 0 0
\(437\) −27.8248 −1.33104
\(438\) 0 0
\(439\) −7.82475 −0.373455 −0.186728 0.982412i \(-0.559788\pi\)
−0.186728 + 0.982412i \(0.559788\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −22.5498 −1.07137 −0.535687 0.844416i \(-0.679947\pi\)
−0.535687 + 0.844416i \(0.679947\pi\)
\(444\) 0 0
\(445\) −45.8488 −2.17344
\(446\) 0 0
\(447\) 7.45017 0.352381
\(448\) 0 0
\(449\) 7.27492 0.343325 0.171662 0.985156i \(-0.445086\pi\)
0.171662 + 0.985156i \(0.445086\pi\)
\(450\) 0 0
\(451\) 45.0997 2.12366
\(452\) 0 0
\(453\) 1.27492 0.0599008
\(454\) 0 0
\(455\) 3.27492 0.153530
\(456\) 0 0
\(457\) −3.45017 −0.161392 −0.0806960 0.996739i \(-0.525714\pi\)
−0.0806960 + 0.996739i \(0.525714\pi\)
\(458\) 0 0
\(459\) −7.27492 −0.339564
\(460\) 0 0
\(461\) 34.9244 1.62659 0.813296 0.581850i \(-0.197671\pi\)
0.813296 + 0.581850i \(0.197671\pi\)
\(462\) 0 0
\(463\) −22.7251 −1.05612 −0.528062 0.849206i \(-0.677081\pi\)
−0.528062 + 0.849206i \(0.677081\pi\)
\(464\) 0 0
\(465\) −26.1993 −1.21497
\(466\) 0 0
\(467\) 18.7251 0.866493 0.433247 0.901275i \(-0.357368\pi\)
0.433247 + 0.901275i \(0.357368\pi\)
\(468\) 0 0
\(469\) −2.54983 −0.117740
\(470\) 0 0
\(471\) 24.3746 1.12312
\(472\) 0 0
\(473\) 27.8248 1.27938
\(474\) 0 0
\(475\) −30.1993 −1.38564
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) 0 0
\(479\) −14.3746 −0.656792 −0.328396 0.944540i \(-0.606508\pi\)
−0.328396 + 0.944540i \(0.606508\pi\)
\(480\) 0 0
\(481\) −3.27492 −0.149323
\(482\) 0 0
\(483\) −5.27492 −0.240017
\(484\) 0 0
\(485\) −49.4502 −2.24542
\(486\) 0 0
\(487\) −42.1993 −1.91223 −0.956117 0.292984i \(-0.905352\pi\)
−0.956117 + 0.292984i \(0.905352\pi\)
\(488\) 0 0
\(489\) 2.54983 0.115307
\(490\) 0 0
\(491\) 6.54983 0.295590 0.147795 0.989018i \(-0.452782\pi\)
0.147795 + 0.989018i \(0.452782\pi\)
\(492\) 0 0
\(493\) −5.27492 −0.237570
\(494\) 0 0
\(495\) −17.2749 −0.776450
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 5.09967 0.228293 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(500\) 0 0
\(501\) 9.27492 0.414373
\(502\) 0 0
\(503\) 18.5498 0.827096 0.413548 0.910482i \(-0.364289\pi\)
0.413548 + 0.910482i \(0.364289\pi\)
\(504\) 0 0
\(505\) −41.0997 −1.82891
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 42.9244 1.90259 0.951296 0.308280i \(-0.0997533\pi\)
0.951296 + 0.308280i \(0.0997533\pi\)
\(510\) 0 0
\(511\) −9.82475 −0.434621
\(512\) 0 0
\(513\) −5.27492 −0.232893
\(514\) 0 0
\(515\) −8.92442 −0.393257
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −23.0997 −1.01396
\(520\) 0 0
\(521\) 3.27492 0.143477 0.0717384 0.997423i \(-0.477145\pi\)
0.0717384 + 0.997423i \(0.477145\pi\)
\(522\) 0 0
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 0 0
\(525\) −5.72508 −0.249863
\(526\) 0 0
\(527\) 58.1993 2.53520
\(528\) 0 0
\(529\) 4.82475 0.209772
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 8.54983 0.370334
\(534\) 0 0
\(535\) 47.6495 2.06007
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) −5.27492 −0.227207
\(540\) 0 0
\(541\) −25.4743 −1.09522 −0.547612 0.836732i \(-0.684463\pi\)
−0.547612 + 0.836732i \(0.684463\pi\)
\(542\) 0 0
\(543\) −11.0997 −0.476332
\(544\) 0 0
\(545\) −23.8248 −1.02054
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 0 0
\(549\) −4.72508 −0.201662
\(550\) 0 0
\(551\) −3.82475 −0.162940
\(552\) 0 0
\(553\) −2.54983 −0.108430
\(554\) 0 0
\(555\) 10.7251 0.455254
\(556\) 0 0
\(557\) 30.7492 1.30288 0.651442 0.758698i \(-0.274164\pi\)
0.651442 + 0.758698i \(0.274164\pi\)
\(558\) 0 0
\(559\) 5.27492 0.223105
\(560\) 0 0
\(561\) 38.3746 1.62018
\(562\) 0 0
\(563\) −2.37459 −0.100077 −0.0500384 0.998747i \(-0.515934\pi\)
−0.0500384 + 0.998747i \(0.515934\pi\)
\(564\) 0 0
\(565\) −19.6495 −0.826661
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −27.0997 −1.13608 −0.568039 0.823002i \(-0.692298\pi\)
−0.568039 + 0.823002i \(0.692298\pi\)
\(570\) 0 0
\(571\) 6.90033 0.288770 0.144385 0.989522i \(-0.453880\pi\)
0.144385 + 0.989522i \(0.453880\pi\)
\(572\) 0 0
\(573\) 5.27492 0.220363
\(574\) 0 0
\(575\) 30.1993 1.25940
\(576\) 0 0
\(577\) −39.0997 −1.62774 −0.813870 0.581047i \(-0.802643\pi\)
−0.813870 + 0.581047i \(0.802643\pi\)
\(578\) 0 0
\(579\) −3.45017 −0.143384
\(580\) 0 0
\(581\) 10.5498 0.437681
\(582\) 0 0
\(583\) −52.7492 −2.18465
\(584\) 0 0
\(585\) −3.27492 −0.135401
\(586\) 0 0
\(587\) 28.7492 1.18661 0.593303 0.804979i \(-0.297824\pi\)
0.593303 + 0.804979i \(0.297824\pi\)
\(588\) 0 0
\(589\) 42.1993 1.73879
\(590\) 0 0
\(591\) 12.5498 0.516231
\(592\) 0 0
\(593\) 23.0997 0.948590 0.474295 0.880366i \(-0.342703\pi\)
0.474295 + 0.880366i \(0.342703\pi\)
\(594\) 0 0
\(595\) 23.8248 0.976720
\(596\) 0 0
\(597\) 15.8248 0.647664
\(598\) 0 0
\(599\) −29.2749 −1.19614 −0.598070 0.801444i \(-0.704066\pi\)
−0.598070 + 0.801444i \(0.704066\pi\)
\(600\) 0 0
\(601\) −21.6495 −0.883102 −0.441551 0.897236i \(-0.645572\pi\)
−0.441551 + 0.897236i \(0.645572\pi\)
\(602\) 0 0
\(603\) 2.54983 0.103837
\(604\) 0 0
\(605\) 55.0997 2.24012
\(606\) 0 0
\(607\) −31.8248 −1.29173 −0.645863 0.763453i \(-0.723502\pi\)
−0.645863 + 0.763453i \(0.723502\pi\)
\(608\) 0 0
\(609\) −0.725083 −0.0293818
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.72508 0.352403 0.176201 0.984354i \(-0.443619\pi\)
0.176201 + 0.984354i \(0.443619\pi\)
\(614\) 0 0
\(615\) −28.0000 −1.12907
\(616\) 0 0
\(617\) 1.82475 0.0734617 0.0367309 0.999325i \(-0.488306\pi\)
0.0367309 + 0.999325i \(0.488306\pi\)
\(618\) 0 0
\(619\) −37.2749 −1.49821 −0.749103 0.662454i \(-0.769515\pi\)
−0.749103 + 0.662454i \(0.769515\pi\)
\(620\) 0 0
\(621\) 5.27492 0.211675
\(622\) 0 0
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) −20.8488 −0.833954
\(626\) 0 0
\(627\) 27.8248 1.11121
\(628\) 0 0
\(629\) −23.8248 −0.949955
\(630\) 0 0
\(631\) −11.8248 −0.470736 −0.235368 0.971906i \(-0.575630\pi\)
−0.235368 + 0.971906i \(0.575630\pi\)
\(632\) 0 0
\(633\) −23.8248 −0.946949
\(634\) 0 0
\(635\) 17.8488 0.708310
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.0997 0.596401 0.298201 0.954503i \(-0.403614\pi\)
0.298201 + 0.954503i \(0.403614\pi\)
\(642\) 0 0
\(643\) −28.9244 −1.14067 −0.570334 0.821413i \(-0.693186\pi\)
−0.570334 + 0.821413i \(0.693186\pi\)
\(644\) 0 0
\(645\) −17.2749 −0.680199
\(646\) 0 0
\(647\) −42.1993 −1.65903 −0.829514 0.558487i \(-0.811382\pi\)
−0.829514 + 0.558487i \(0.811382\pi\)
\(648\) 0 0
\(649\) 42.1993 1.65647
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 0 0
\(653\) 32.3746 1.26692 0.633458 0.773777i \(-0.281635\pi\)
0.633458 + 0.773777i \(0.281635\pi\)
\(654\) 0 0
\(655\) 51.8248 2.02496
\(656\) 0 0
\(657\) 9.82475 0.383300
\(658\) 0 0
\(659\) 22.5498 0.878417 0.439208 0.898385i \(-0.355259\pi\)
0.439208 + 0.898385i \(0.355259\pi\)
\(660\) 0 0
\(661\) 20.1993 0.785663 0.392832 0.919610i \(-0.371496\pi\)
0.392832 + 0.919610i \(0.371496\pi\)
\(662\) 0 0
\(663\) 7.27492 0.282534
\(664\) 0 0
\(665\) 17.2749 0.669893
\(666\) 0 0
\(667\) 3.82475 0.148095
\(668\) 0 0
\(669\) 23.6495 0.914343
\(670\) 0 0
\(671\) 24.9244 0.962197
\(672\) 0 0
\(673\) 8.72508 0.336327 0.168164 0.985759i \(-0.446216\pi\)
0.168164 + 0.985759i \(0.446216\pi\)
\(674\) 0 0
\(675\) 5.72508 0.220359
\(676\) 0 0
\(677\) −44.1993 −1.69872 −0.849359 0.527815i \(-0.823011\pi\)
−0.849359 + 0.527815i \(0.823011\pi\)
\(678\) 0 0
\(679\) 15.0997 0.579472
\(680\) 0 0
\(681\) 5.45017 0.208851
\(682\) 0 0
\(683\) 34.3746 1.31531 0.657653 0.753321i \(-0.271549\pi\)
0.657653 + 0.753321i \(0.271549\pi\)
\(684\) 0 0
\(685\) 66.7251 2.54943
\(686\) 0 0
\(687\) −11.0997 −0.423479
\(688\) 0 0
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) 5.27492 0.200378
\(694\) 0 0
\(695\) −13.0997 −0.496899
\(696\) 0 0
\(697\) 62.1993 2.35597
\(698\) 0 0
\(699\) 20.5498 0.777266
\(700\) 0 0
\(701\) 15.0997 0.570307 0.285153 0.958482i \(-0.407955\pi\)
0.285153 + 0.958482i \(0.407955\pi\)
\(702\) 0 0
\(703\) −17.2749 −0.651536
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.5498 0.471985
\(708\) 0 0
\(709\) 7.09967 0.266634 0.133317 0.991073i \(-0.457437\pi\)
0.133317 + 0.991073i \(0.457437\pi\)
\(710\) 0 0
\(711\) 2.54983 0.0956263
\(712\) 0 0
\(713\) −42.1993 −1.58038
\(714\) 0 0
\(715\) 17.2749 0.646045
\(716\) 0 0
\(717\) 5.09967 0.190451
\(718\) 0 0
\(719\) −15.6495 −0.583628 −0.291814 0.956475i \(-0.594259\pi\)
−0.291814 + 0.956475i \(0.594259\pi\)
\(720\) 0 0
\(721\) 2.72508 0.101487
\(722\) 0 0
\(723\) −15.0997 −0.561563
\(724\) 0 0
\(725\) 4.15116 0.154170
\(726\) 0 0
\(727\) 2.37459 0.0880685 0.0440343 0.999030i \(-0.485979\pi\)
0.0440343 + 0.999030i \(0.485979\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 38.3746 1.41934
\(732\) 0 0
\(733\) −0.549834 −0.0203086 −0.0101543 0.999948i \(-0.503232\pi\)
−0.0101543 + 0.999948i \(0.503232\pi\)
\(734\) 0 0
\(735\) 3.27492 0.120797
\(736\) 0 0
\(737\) −13.4502 −0.495443
\(738\) 0 0
\(739\) 7.64950 0.281392 0.140696 0.990053i \(-0.455066\pi\)
0.140696 + 0.990053i \(0.455066\pi\)
\(740\) 0 0
\(741\) 5.27492 0.193779
\(742\) 0 0
\(743\) −26.5498 −0.974019 −0.487009 0.873397i \(-0.661912\pi\)
−0.487009 + 0.873397i \(0.661912\pi\)
\(744\) 0 0
\(745\) 24.3987 0.893898
\(746\) 0 0
\(747\) −10.5498 −0.385998
\(748\) 0 0
\(749\) −14.5498 −0.531639
\(750\) 0 0
\(751\) 50.1993 1.83180 0.915900 0.401407i \(-0.131479\pi\)
0.915900 + 0.401407i \(0.131479\pi\)
\(752\) 0 0
\(753\) 28.9244 1.05406
\(754\) 0 0
\(755\) 4.17525 0.151953
\(756\) 0 0
\(757\) −4.90033 −0.178106 −0.0890528 0.996027i \(-0.528384\pi\)
−0.0890528 + 0.996027i \(0.528384\pi\)
\(758\) 0 0
\(759\) −27.8248 −1.00997
\(760\) 0 0
\(761\) 23.4502 0.850068 0.425034 0.905177i \(-0.360262\pi\)
0.425034 + 0.905177i \(0.360262\pi\)
\(762\) 0 0
\(763\) 7.27492 0.263370
\(764\) 0 0
\(765\) −23.8248 −0.861386
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 10.1752 0.366929 0.183464 0.983026i \(-0.441269\pi\)
0.183464 + 0.983026i \(0.441269\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 0 0
\(773\) 42.9244 1.54388 0.771942 0.635693i \(-0.219286\pi\)
0.771942 + 0.635693i \(0.219286\pi\)
\(774\) 0 0
\(775\) −45.8007 −1.64521
\(776\) 0 0
\(777\) −3.27492 −0.117487
\(778\) 0 0
\(779\) 45.0997 1.61586
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0.725083 0.0259123
\(784\) 0 0
\(785\) 79.8248 2.84907
\(786\) 0 0
\(787\) 2.72508 0.0971387 0.0485694 0.998820i \(-0.484534\pi\)
0.0485694 + 0.998820i \(0.484534\pi\)
\(788\) 0 0
\(789\) −28.0000 −0.996826
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 4.72508 0.167793
\(794\) 0 0
\(795\) 32.7492 1.16149
\(796\) 0 0
\(797\) 29.6495 1.05024 0.525120 0.851028i \(-0.324021\pi\)
0.525120 + 0.851028i \(0.324021\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) −51.8248 −1.82886
\(804\) 0 0
\(805\) −17.2749 −0.608861
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) 4.19934 0.147641 0.0738204 0.997272i \(-0.476481\pi\)
0.0738204 + 0.997272i \(0.476481\pi\)
\(810\) 0 0
\(811\) −26.3746 −0.926137 −0.463068 0.886322i \(-0.653252\pi\)
−0.463068 + 0.886322i \(0.653252\pi\)
\(812\) 0 0
\(813\) −13.4502 −0.471718
\(814\) 0 0
\(815\) 8.35050 0.292505
\(816\) 0 0
\(817\) 27.8248 0.973465
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) −13.6495 −0.476371 −0.238185 0.971220i \(-0.576553\pi\)
−0.238185 + 0.971220i \(0.576553\pi\)
\(822\) 0 0
\(823\) −5.09967 −0.177763 −0.0888816 0.996042i \(-0.528329\pi\)
−0.0888816 + 0.996042i \(0.528329\pi\)
\(824\) 0 0
\(825\) −30.1993 −1.05141
\(826\) 0 0
\(827\) 18.7251 0.651135 0.325567 0.945519i \(-0.394445\pi\)
0.325567 + 0.945519i \(0.394445\pi\)
\(828\) 0 0
\(829\) 19.2749 0.669446 0.334723 0.942317i \(-0.391357\pi\)
0.334723 + 0.942317i \(0.391357\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) −7.27492 −0.252061
\(834\) 0 0
\(835\) 30.3746 1.05116
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 0 0
\(839\) −53.0997 −1.83320 −0.916602 0.399801i \(-0.869079\pi\)
−0.916602 + 0.399801i \(0.869079\pi\)
\(840\) 0 0
\(841\) −28.4743 −0.981871
\(842\) 0 0
\(843\) −15.0997 −0.520060
\(844\) 0 0
\(845\) 3.27492 0.112661
\(846\) 0 0
\(847\) −16.8248 −0.578105
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 17.2749 0.592177
\(852\) 0 0
\(853\) 17.6495 0.604307 0.302154 0.953259i \(-0.402294\pi\)
0.302154 + 0.953259i \(0.402294\pi\)
\(854\) 0 0
\(855\) −17.2749 −0.590790
\(856\) 0 0
\(857\) −11.0997 −0.379157 −0.189579 0.981866i \(-0.560712\pi\)
−0.189579 + 0.981866i \(0.560712\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 8.54983 0.291378
\(862\) 0 0
\(863\) 15.6495 0.532715 0.266358 0.963874i \(-0.414180\pi\)
0.266358 + 0.963874i \(0.414180\pi\)
\(864\) 0 0
\(865\) −75.6495 −2.57216
\(866\) 0 0
\(867\) 35.9244 1.22006
\(868\) 0 0
\(869\) −13.4502 −0.456266
\(870\) 0 0
\(871\) −2.54983 −0.0863978
\(872\) 0 0
\(873\) −15.0997 −0.511046
\(874\) 0 0
\(875\) −2.37459 −0.0802757
\(876\) 0 0
\(877\) 20.9003 0.705754 0.352877 0.935670i \(-0.385203\pi\)
0.352877 + 0.935670i \(0.385203\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −39.2749 −1.32321 −0.661603 0.749854i \(-0.730123\pi\)
−0.661603 + 0.749854i \(0.730123\pi\)
\(882\) 0 0
\(883\) 20.9244 0.704163 0.352081 0.935969i \(-0.385474\pi\)
0.352081 + 0.935969i \(0.385474\pi\)
\(884\) 0 0
\(885\) −26.1993 −0.880681
\(886\) 0 0
\(887\) 2.90033 0.0973836 0.0486918 0.998814i \(-0.484495\pi\)
0.0486918 + 0.998814i \(0.484495\pi\)
\(888\) 0 0
\(889\) −5.45017 −0.182793
\(890\) 0 0
\(891\) −5.27492 −0.176716
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −39.2990 −1.31362
\(896\) 0 0
\(897\) −5.27492 −0.176124
\(898\) 0 0
\(899\) −5.80066 −0.193463
\(900\) 0 0
\(901\) −72.7492 −2.42363
\(902\) 0 0
\(903\) 5.27492 0.175538
\(904\) 0 0
\(905\) −36.3505 −1.20833
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 0 0
\(909\) −12.5498 −0.416252
\(910\) 0 0
\(911\) −15.4743 −0.512685 −0.256342 0.966586i \(-0.582517\pi\)
−0.256342 + 0.966586i \(0.582517\pi\)
\(912\) 0 0
\(913\) 55.6495 1.84173
\(914\) 0 0
\(915\) −15.4743 −0.511563
\(916\) 0 0
\(917\) −15.8248 −0.522579
\(918\) 0 0
\(919\) −2.54983 −0.0841113 −0.0420556 0.999115i \(-0.513391\pi\)
−0.0420556 + 0.999115i \(0.513391\pi\)
\(920\) 0 0
\(921\) 9.09967 0.299844
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 18.7492 0.616469
\(926\) 0 0
\(927\) −2.72508 −0.0895035
\(928\) 0 0
\(929\) −0.549834 −0.0180395 −0.00901974 0.999959i \(-0.502871\pi\)
−0.00901974 + 0.999959i \(0.502871\pi\)
\(930\) 0 0
\(931\) −5.27492 −0.172878
\(932\) 0 0
\(933\) −18.5498 −0.607294
\(934\) 0 0
\(935\) 125.674 4.10997
\(936\) 0 0
\(937\) 47.0997 1.53868 0.769340 0.638840i \(-0.220585\pi\)
0.769340 + 0.638840i \(0.220585\pi\)
\(938\) 0 0
\(939\) 17.6495 0.575970
\(940\) 0 0
\(941\) 20.9003 0.681331 0.340666 0.940185i \(-0.389348\pi\)
0.340666 + 0.940185i \(0.389348\pi\)
\(942\) 0 0
\(943\) −45.0997 −1.46865
\(944\) 0 0
\(945\) −3.27492 −0.106533
\(946\) 0 0
\(947\) 39.8248 1.29413 0.647065 0.762435i \(-0.275996\pi\)
0.647065 + 0.762435i \(0.275996\pi\)
\(948\) 0 0
\(949\) −9.82475 −0.318925
\(950\) 0 0
\(951\) −16.5498 −0.536665
\(952\) 0 0
\(953\) −13.6495 −0.442151 −0.221075 0.975257i \(-0.570957\pi\)
−0.221075 + 0.975257i \(0.570957\pi\)
\(954\) 0 0
\(955\) 17.2749 0.559003
\(956\) 0 0
\(957\) −3.82475 −0.123637
\(958\) 0 0
\(959\) −20.3746 −0.657930
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 14.5498 0.468862
\(964\) 0 0
\(965\) −11.2990 −0.363728
\(966\) 0 0
\(967\) −20.1752 −0.648792 −0.324396 0.945921i \(-0.605161\pi\)
−0.324396 + 0.945921i \(0.605161\pi\)
\(968\) 0 0
\(969\) 38.3746 1.23277
\(970\) 0 0
\(971\) −1.09967 −0.0352901 −0.0176450 0.999844i \(-0.505617\pi\)
−0.0176450 + 0.999844i \(0.505617\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) 0 0
\(975\) −5.72508 −0.183349
\(976\) 0 0
\(977\) 38.9244 1.24530 0.622651 0.782499i \(-0.286056\pi\)
0.622651 + 0.782499i \(0.286056\pi\)
\(978\) 0 0
\(979\) 73.8488 2.36022
\(980\) 0 0
\(981\) −7.27492 −0.232270
\(982\) 0 0
\(983\) 22.0241 0.702459 0.351230 0.936289i \(-0.385764\pi\)
0.351230 + 0.936289i \(0.385764\pi\)
\(984\) 0 0
\(985\) 41.0997 1.30954
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −27.8248 −0.884776
\(990\) 0 0
\(991\) −28.7492 −0.913248 −0.456624 0.889660i \(-0.650941\pi\)
−0.456624 + 0.889660i \(0.650941\pi\)
\(992\) 0 0
\(993\) −15.6495 −0.496622
\(994\) 0 0
\(995\) 51.8248 1.64296
\(996\) 0 0
\(997\) −43.0997 −1.36498 −0.682490 0.730895i \(-0.739103\pi\)
−0.682490 + 0.730895i \(0.739103\pi\)
\(998\) 0 0
\(999\) 3.27492 0.103614
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4368.2.a.bh.1.2 2
4.3 odd 2 546.2.a.h.1.2 2
12.11 even 2 1638.2.a.y.1.1 2
28.27 even 2 3822.2.a.bm.1.1 2
52.51 odd 2 7098.2.a.bu.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.h.1.2 2 4.3 odd 2
1638.2.a.y.1.1 2 12.11 even 2
3822.2.a.bm.1.1 2 28.27 even 2
4368.2.a.bh.1.2 2 1.1 even 1 trivial
7098.2.a.bu.1.1 2 52.51 odd 2