Properties

Label 7728.2.a.bg.1.2
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.381966 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.381966 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.23607 q^{11} -0.145898 q^{13} -0.381966 q^{15} -5.47214 q^{17} +8.23607 q^{19} -1.00000 q^{21} -1.00000 q^{23} -4.85410 q^{25} +1.00000 q^{27} -2.70820 q^{29} +5.00000 q^{31} -2.23607 q^{33} +0.381966 q^{35} -0.527864 q^{37} -0.145898 q^{39} +8.70820 q^{41} +8.32624 q^{43} -0.381966 q^{45} -5.23607 q^{47} +1.00000 q^{49} -5.47214 q^{51} -3.61803 q^{53} +0.854102 q^{55} +8.23607 q^{57} +4.85410 q^{59} -9.09017 q^{61} -1.00000 q^{63} +0.0557281 q^{65} +6.85410 q^{67} -1.00000 q^{69} -9.38197 q^{71} -3.47214 q^{73} -4.85410 q^{75} +2.23607 q^{77} -5.94427 q^{79} +1.00000 q^{81} -7.94427 q^{83} +2.09017 q^{85} -2.70820 q^{87} -16.3262 q^{89} +0.145898 q^{91} +5.00000 q^{93} -3.14590 q^{95} -17.1803 q^{97} -2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 3 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 3 q^{5} - 2 q^{7} + 2 q^{9} - 7 q^{13} - 3 q^{15} - 2 q^{17} + 12 q^{19} - 2 q^{21} - 2 q^{23} - 3 q^{25} + 2 q^{27} + 8 q^{29} + 10 q^{31} + 3 q^{35} - 10 q^{37} - 7 q^{39} + 4 q^{41} + q^{43} - 3 q^{45} - 6 q^{47} + 2 q^{49} - 2 q^{51} - 5 q^{53} - 5 q^{55} + 12 q^{57} + 3 q^{59} - 7 q^{61} - 2 q^{63} + 18 q^{65} + 7 q^{67} - 2 q^{69} - 21 q^{71} + 2 q^{73} - 3 q^{75} + 6 q^{79} + 2 q^{81} + 2 q^{83} - 7 q^{85} + 8 q^{87} - 17 q^{89} + 7 q^{91} + 10 q^{93} - 13 q^{95} - 12 q^{97} + O(q^{100}) \)

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
\(5\) −0.381966 −0.170820 −0.0854102 0.996346i \(-0.527220\pi\)
−0.0854102 + 0.996346i \(0.527220\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.23607 −0.674200 −0.337100 0.941469i \(-0.609446\pi\)
−0.337100 + 0.941469i \(0.609446\pi\)
\(12\) 0 0
\(13\) −0.145898 −0.0404648 −0.0202324 0.999795i \(-0.506441\pi\)
−0.0202324 + 0.999795i \(0.506441\pi\)
\(14\) 0 0
\(15\) −0.381966 −0.0986232
\(16\) 0 0
\(17\) −5.47214 −1.32719 −0.663594 0.748093i \(-0.730970\pi\)
−0.663594 + 0.748093i \(0.730970\pi\)
\(18\) 0 0
\(19\) 8.23607 1.88948 0.944742 0.327815i \(-0.106312\pi\)
0.944742 + 0.327815i \(0.106312\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.85410 −0.970820
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.70820 −0.502901 −0.251450 0.967870i \(-0.580908\pi\)
−0.251450 + 0.967870i \(0.580908\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) −2.23607 −0.389249
\(34\) 0 0
\(35\) 0.381966 0.0645640
\(36\) 0 0
\(37\) −0.527864 −0.0867803 −0.0433902 0.999058i \(-0.513816\pi\)
−0.0433902 + 0.999058i \(0.513816\pi\)
\(38\) 0 0
\(39\) −0.145898 −0.0233624
\(40\) 0 0
\(41\) 8.70820 1.35999 0.679996 0.733215i \(-0.261981\pi\)
0.679996 + 0.733215i \(0.261981\pi\)
\(42\) 0 0
\(43\) 8.32624 1.26974 0.634870 0.772619i \(-0.281054\pi\)
0.634870 + 0.772619i \(0.281054\pi\)
\(44\) 0 0
\(45\) −0.381966 −0.0569401
\(46\) 0 0
\(47\) −5.23607 −0.763759 −0.381880 0.924212i \(-0.624723\pi\)
−0.381880 + 0.924212i \(0.624723\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.47214 −0.766252
\(52\) 0 0
\(53\) −3.61803 −0.496975 −0.248488 0.968635i \(-0.579934\pi\)
−0.248488 + 0.968635i \(0.579934\pi\)
\(54\) 0 0
\(55\) 0.854102 0.115167
\(56\) 0 0
\(57\) 8.23607 1.09089
\(58\) 0 0
\(59\) 4.85410 0.631950 0.315975 0.948767i \(-0.397668\pi\)
0.315975 + 0.948767i \(0.397668\pi\)
\(60\) 0 0
\(61\) −9.09017 −1.16388 −0.581938 0.813233i \(-0.697706\pi\)
−0.581938 + 0.813233i \(0.697706\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0.0557281 0.00691222
\(66\) 0 0
\(67\) 6.85410 0.837362 0.418681 0.908133i \(-0.362493\pi\)
0.418681 + 0.908133i \(0.362493\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −9.38197 −1.11343 −0.556717 0.830702i \(-0.687939\pi\)
−0.556717 + 0.830702i \(0.687939\pi\)
\(72\) 0 0
\(73\) −3.47214 −0.406383 −0.203191 0.979139i \(-0.565131\pi\)
−0.203191 + 0.979139i \(0.565131\pi\)
\(74\) 0 0
\(75\) −4.85410 −0.560503
\(76\) 0 0
\(77\) 2.23607 0.254824
\(78\) 0 0
\(79\) −5.94427 −0.668783 −0.334391 0.942434i \(-0.608531\pi\)
−0.334391 + 0.942434i \(0.608531\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −7.94427 −0.871997 −0.435999 0.899947i \(-0.643605\pi\)
−0.435999 + 0.899947i \(0.643605\pi\)
\(84\) 0 0
\(85\) 2.09017 0.226711
\(86\) 0 0
\(87\) −2.70820 −0.290350
\(88\) 0 0
\(89\) −16.3262 −1.73058 −0.865289 0.501274i \(-0.832865\pi\)
−0.865289 + 0.501274i \(0.832865\pi\)
\(90\) 0 0
\(91\) 0.145898 0.0152943
\(92\) 0 0
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) −3.14590 −0.322762
\(96\) 0 0
\(97\) −17.1803 −1.74440 −0.872200 0.489150i \(-0.837307\pi\)
−0.872200 + 0.489150i \(0.837307\pi\)
\(98\) 0 0
\(99\) −2.23607 −0.224733
\(100\) 0 0
\(101\) −16.0902 −1.60103 −0.800516 0.599312i \(-0.795441\pi\)
−0.800516 + 0.599312i \(0.795441\pi\)
\(102\) 0 0
\(103\) −2.47214 −0.243587 −0.121793 0.992555i \(-0.538865\pi\)
−0.121793 + 0.992555i \(0.538865\pi\)
\(104\) 0 0
\(105\) 0.381966 0.0372761
\(106\) 0 0
\(107\) −5.85410 −0.565937 −0.282969 0.959129i \(-0.591319\pi\)
−0.282969 + 0.959129i \(0.591319\pi\)
\(108\) 0 0
\(109\) 15.0344 1.44004 0.720019 0.693954i \(-0.244133\pi\)
0.720019 + 0.693954i \(0.244133\pi\)
\(110\) 0 0
\(111\) −0.527864 −0.0501026
\(112\) 0 0
\(113\) 4.09017 0.384771 0.192385 0.981319i \(-0.438378\pi\)
0.192385 + 0.981319i \(0.438378\pi\)
\(114\) 0 0
\(115\) 0.381966 0.0356185
\(116\) 0 0
\(117\) −0.145898 −0.0134883
\(118\) 0 0
\(119\) 5.47214 0.501630
\(120\) 0 0
\(121\) −6.00000 −0.545455
\(122\) 0 0
\(123\) 8.70820 0.785192
\(124\) 0 0
\(125\) 3.76393 0.336656
\(126\) 0 0
\(127\) 18.3262 1.62619 0.813095 0.582131i \(-0.197781\pi\)
0.813095 + 0.582131i \(0.197781\pi\)
\(128\) 0 0
\(129\) 8.32624 0.733084
\(130\) 0 0
\(131\) −14.4164 −1.25957 −0.629784 0.776771i \(-0.716856\pi\)
−0.629784 + 0.776771i \(0.716856\pi\)
\(132\) 0 0
\(133\) −8.23607 −0.714158
\(134\) 0 0
\(135\) −0.381966 −0.0328744
\(136\) 0 0
\(137\) −2.05573 −0.175633 −0.0878164 0.996137i \(-0.527989\pi\)
−0.0878164 + 0.996137i \(0.527989\pi\)
\(138\) 0 0
\(139\) −13.1459 −1.11502 −0.557510 0.830170i \(-0.688243\pi\)
−0.557510 + 0.830170i \(0.688243\pi\)
\(140\) 0 0
\(141\) −5.23607 −0.440956
\(142\) 0 0
\(143\) 0.326238 0.0272814
\(144\) 0 0
\(145\) 1.03444 0.0859057
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 16.6525 1.36422 0.682112 0.731248i \(-0.261062\pi\)
0.682112 + 0.731248i \(0.261062\pi\)
\(150\) 0 0
\(151\) −11.7082 −0.952800 −0.476400 0.879229i \(-0.658059\pi\)
−0.476400 + 0.879229i \(0.658059\pi\)
\(152\) 0 0
\(153\) −5.47214 −0.442396
\(154\) 0 0
\(155\) −1.90983 −0.153401
\(156\) 0 0
\(157\) 8.18034 0.652862 0.326431 0.945221i \(-0.394154\pi\)
0.326431 + 0.945221i \(0.394154\pi\)
\(158\) 0 0
\(159\) −3.61803 −0.286929
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −15.5066 −1.21457 −0.607284 0.794484i \(-0.707741\pi\)
−0.607284 + 0.794484i \(0.707741\pi\)
\(164\) 0 0
\(165\) 0.854102 0.0664917
\(166\) 0 0
\(167\) −2.52786 −0.195612 −0.0978060 0.995205i \(-0.531182\pi\)
−0.0978060 + 0.995205i \(0.531182\pi\)
\(168\) 0 0
\(169\) −12.9787 −0.998363
\(170\) 0 0
\(171\) 8.23607 0.629828
\(172\) 0 0
\(173\) −24.7082 −1.87853 −0.939265 0.343193i \(-0.888492\pi\)
−0.939265 + 0.343193i \(0.888492\pi\)
\(174\) 0 0
\(175\) 4.85410 0.366936
\(176\) 0 0
\(177\) 4.85410 0.364857
\(178\) 0 0
\(179\) 9.03444 0.675266 0.337633 0.941278i \(-0.390374\pi\)
0.337633 + 0.941278i \(0.390374\pi\)
\(180\) 0 0
\(181\) 7.18034 0.533710 0.266855 0.963737i \(-0.414015\pi\)
0.266855 + 0.963737i \(0.414015\pi\)
\(182\) 0 0
\(183\) −9.09017 −0.671965
\(184\) 0 0
\(185\) 0.201626 0.0148238
\(186\) 0 0
\(187\) 12.2361 0.894790
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −8.29180 −0.599973 −0.299987 0.953943i \(-0.596982\pi\)
−0.299987 + 0.953943i \(0.596982\pi\)
\(192\) 0 0
\(193\) 4.18034 0.300907 0.150454 0.988617i \(-0.451927\pi\)
0.150454 + 0.988617i \(0.451927\pi\)
\(194\) 0 0
\(195\) 0.0557281 0.00399077
\(196\) 0 0
\(197\) −9.67376 −0.689227 −0.344614 0.938745i \(-0.611990\pi\)
−0.344614 + 0.938745i \(0.611990\pi\)
\(198\) 0 0
\(199\) −10.5623 −0.748742 −0.374371 0.927279i \(-0.622141\pi\)
−0.374371 + 0.927279i \(0.622141\pi\)
\(200\) 0 0
\(201\) 6.85410 0.483451
\(202\) 0 0
\(203\) 2.70820 0.190079
\(204\) 0 0
\(205\) −3.32624 −0.232315
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −18.4164 −1.27389
\(210\) 0 0
\(211\) −15.0000 −1.03264 −0.516321 0.856395i \(-0.672699\pi\)
−0.516321 + 0.856395i \(0.672699\pi\)
\(212\) 0 0
\(213\) −9.38197 −0.642842
\(214\) 0 0
\(215\) −3.18034 −0.216897
\(216\) 0 0
\(217\) −5.00000 −0.339422
\(218\) 0 0
\(219\) −3.47214 −0.234625
\(220\) 0 0
\(221\) 0.798374 0.0537044
\(222\) 0 0
\(223\) −2.67376 −0.179048 −0.0895242 0.995985i \(-0.528535\pi\)
−0.0895242 + 0.995985i \(0.528535\pi\)
\(224\) 0 0
\(225\) −4.85410 −0.323607
\(226\) 0 0
\(227\) 18.2705 1.21266 0.606328 0.795215i \(-0.292642\pi\)
0.606328 + 0.795215i \(0.292642\pi\)
\(228\) 0 0
\(229\) −21.3262 −1.40928 −0.704639 0.709566i \(-0.748891\pi\)
−0.704639 + 0.709566i \(0.748891\pi\)
\(230\) 0 0
\(231\) 2.23607 0.147122
\(232\) 0 0
\(233\) −4.09017 −0.267956 −0.133978 0.990984i \(-0.542775\pi\)
−0.133978 + 0.990984i \(0.542775\pi\)
\(234\) 0 0
\(235\) 2.00000 0.130466
\(236\) 0 0
\(237\) −5.94427 −0.386122
\(238\) 0 0
\(239\) 4.38197 0.283446 0.141723 0.989906i \(-0.454736\pi\)
0.141723 + 0.989906i \(0.454736\pi\)
\(240\) 0 0
\(241\) 15.2918 0.985031 0.492516 0.870304i \(-0.336077\pi\)
0.492516 + 0.870304i \(0.336077\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.381966 −0.0244029
\(246\) 0 0
\(247\) −1.20163 −0.0764576
\(248\) 0 0
\(249\) −7.94427 −0.503448
\(250\) 0 0
\(251\) 25.7082 1.62269 0.811344 0.584569i \(-0.198737\pi\)
0.811344 + 0.584569i \(0.198737\pi\)
\(252\) 0 0
\(253\) 2.23607 0.140580
\(254\) 0 0
\(255\) 2.09017 0.130892
\(256\) 0 0
\(257\) 16.2918 1.01625 0.508127 0.861282i \(-0.330338\pi\)
0.508127 + 0.861282i \(0.330338\pi\)
\(258\) 0 0
\(259\) 0.527864 0.0327999
\(260\) 0 0
\(261\) −2.70820 −0.167634
\(262\) 0 0
\(263\) −5.00000 −0.308313 −0.154157 0.988046i \(-0.549266\pi\)
−0.154157 + 0.988046i \(0.549266\pi\)
\(264\) 0 0
\(265\) 1.38197 0.0848935
\(266\) 0 0
\(267\) −16.3262 −0.999150
\(268\) 0 0
\(269\) 3.85410 0.234989 0.117494 0.993074i \(-0.462514\pi\)
0.117494 + 0.993074i \(0.462514\pi\)
\(270\) 0 0
\(271\) −5.18034 −0.314683 −0.157342 0.987544i \(-0.550292\pi\)
−0.157342 + 0.987544i \(0.550292\pi\)
\(272\) 0 0
\(273\) 0.145898 0.00883015
\(274\) 0 0
\(275\) 10.8541 0.654527
\(276\) 0 0
\(277\) −30.2148 −1.81543 −0.907715 0.419587i \(-0.862175\pi\)
−0.907715 + 0.419587i \(0.862175\pi\)
\(278\) 0 0
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) 27.5967 1.64628 0.823142 0.567836i \(-0.192219\pi\)
0.823142 + 0.567836i \(0.192219\pi\)
\(282\) 0 0
\(283\) 1.32624 0.0788367 0.0394183 0.999223i \(-0.487450\pi\)
0.0394183 + 0.999223i \(0.487450\pi\)
\(284\) 0 0
\(285\) −3.14590 −0.186347
\(286\) 0 0
\(287\) −8.70820 −0.514029
\(288\) 0 0
\(289\) 12.9443 0.761428
\(290\) 0 0
\(291\) −17.1803 −1.00713
\(292\) 0 0
\(293\) −18.4721 −1.07915 −0.539577 0.841936i \(-0.681416\pi\)
−0.539577 + 0.841936i \(0.681416\pi\)
\(294\) 0 0
\(295\) −1.85410 −0.107950
\(296\) 0 0
\(297\) −2.23607 −0.129750
\(298\) 0 0
\(299\) 0.145898 0.00843750
\(300\) 0 0
\(301\) −8.32624 −0.479916
\(302\) 0 0
\(303\) −16.0902 −0.924356
\(304\) 0 0
\(305\) 3.47214 0.198814
\(306\) 0 0
\(307\) −8.05573 −0.459765 −0.229882 0.973218i \(-0.573834\pi\)
−0.229882 + 0.973218i \(0.573834\pi\)
\(308\) 0 0
\(309\) −2.47214 −0.140635
\(310\) 0 0
\(311\) −5.38197 −0.305183 −0.152592 0.988289i \(-0.548762\pi\)
−0.152592 + 0.988289i \(0.548762\pi\)
\(312\) 0 0
\(313\) −10.4721 −0.591920 −0.295960 0.955200i \(-0.595640\pi\)
−0.295960 + 0.955200i \(0.595640\pi\)
\(314\) 0 0
\(315\) 0.381966 0.0215213
\(316\) 0 0
\(317\) −28.8541 −1.62061 −0.810304 0.586010i \(-0.800698\pi\)
−0.810304 + 0.586010i \(0.800698\pi\)
\(318\) 0 0
\(319\) 6.05573 0.339056
\(320\) 0 0
\(321\) −5.85410 −0.326744
\(322\) 0 0
\(323\) −45.0689 −2.50770
\(324\) 0 0
\(325\) 0.708204 0.0392841
\(326\) 0 0
\(327\) 15.0344 0.831407
\(328\) 0 0
\(329\) 5.23607 0.288674
\(330\) 0 0
\(331\) 6.94427 0.381692 0.190846 0.981620i \(-0.438877\pi\)
0.190846 + 0.981620i \(0.438877\pi\)
\(332\) 0 0
\(333\) −0.527864 −0.0289268
\(334\) 0 0
\(335\) −2.61803 −0.143038
\(336\) 0 0
\(337\) 1.56231 0.0851042 0.0425521 0.999094i \(-0.486451\pi\)
0.0425521 + 0.999094i \(0.486451\pi\)
\(338\) 0 0
\(339\) 4.09017 0.222148
\(340\) 0 0
\(341\) −11.1803 −0.605449
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0.381966 0.0205644
\(346\) 0 0
\(347\) −27.6525 −1.48446 −0.742231 0.670144i \(-0.766232\pi\)
−0.742231 + 0.670144i \(0.766232\pi\)
\(348\) 0 0
\(349\) 31.5066 1.68651 0.843254 0.537515i \(-0.180637\pi\)
0.843254 + 0.537515i \(0.180637\pi\)
\(350\) 0 0
\(351\) −0.145898 −0.00778746
\(352\) 0 0
\(353\) 22.5967 1.20270 0.601352 0.798984i \(-0.294629\pi\)
0.601352 + 0.798984i \(0.294629\pi\)
\(354\) 0 0
\(355\) 3.58359 0.190197
\(356\) 0 0
\(357\) 5.47214 0.289616
\(358\) 0 0
\(359\) 1.79837 0.0949145 0.0474573 0.998873i \(-0.484888\pi\)
0.0474573 + 0.998873i \(0.484888\pi\)
\(360\) 0 0
\(361\) 48.8328 2.57015
\(362\) 0 0
\(363\) −6.00000 −0.314918
\(364\) 0 0
\(365\) 1.32624 0.0694185
\(366\) 0 0
\(367\) −23.4508 −1.22412 −0.612062 0.790810i \(-0.709660\pi\)
−0.612062 + 0.790810i \(0.709660\pi\)
\(368\) 0 0
\(369\) 8.70820 0.453331
\(370\) 0 0
\(371\) 3.61803 0.187839
\(372\) 0 0
\(373\) −32.8885 −1.70290 −0.851452 0.524432i \(-0.824278\pi\)
−0.851452 + 0.524432i \(0.824278\pi\)
\(374\) 0 0
\(375\) 3.76393 0.194369
\(376\) 0 0
\(377\) 0.395122 0.0203498
\(378\) 0 0
\(379\) 3.05573 0.156962 0.0784811 0.996916i \(-0.474993\pi\)
0.0784811 + 0.996916i \(0.474993\pi\)
\(380\) 0 0
\(381\) 18.3262 0.938882
\(382\) 0 0
\(383\) 13.4721 0.688394 0.344197 0.938897i \(-0.388151\pi\)
0.344197 + 0.938897i \(0.388151\pi\)
\(384\) 0 0
\(385\) −0.854102 −0.0435291
\(386\) 0 0
\(387\) 8.32624 0.423246
\(388\) 0 0
\(389\) −18.1246 −0.918954 −0.459477 0.888190i \(-0.651963\pi\)
−0.459477 + 0.888190i \(0.651963\pi\)
\(390\) 0 0
\(391\) 5.47214 0.276738
\(392\) 0 0
\(393\) −14.4164 −0.727212
\(394\) 0 0
\(395\) 2.27051 0.114242
\(396\) 0 0
\(397\) 10.7639 0.540226 0.270113 0.962829i \(-0.412939\pi\)
0.270113 + 0.962829i \(0.412939\pi\)
\(398\) 0 0
\(399\) −8.23607 −0.412319
\(400\) 0 0
\(401\) 29.1803 1.45720 0.728598 0.684941i \(-0.240172\pi\)
0.728598 + 0.684941i \(0.240172\pi\)
\(402\) 0 0
\(403\) −0.729490 −0.0363385
\(404\) 0 0
\(405\) −0.381966 −0.0189800
\(406\) 0 0
\(407\) 1.18034 0.0585073
\(408\) 0 0
\(409\) −7.65248 −0.378391 −0.189195 0.981939i \(-0.560588\pi\)
−0.189195 + 0.981939i \(0.560588\pi\)
\(410\) 0 0
\(411\) −2.05573 −0.101402
\(412\) 0 0
\(413\) −4.85410 −0.238855
\(414\) 0 0
\(415\) 3.03444 0.148955
\(416\) 0 0
\(417\) −13.1459 −0.643757
\(418\) 0 0
\(419\) 0.562306 0.0274704 0.0137352 0.999906i \(-0.495628\pi\)
0.0137352 + 0.999906i \(0.495628\pi\)
\(420\) 0 0
\(421\) 26.3262 1.28306 0.641531 0.767097i \(-0.278300\pi\)
0.641531 + 0.767097i \(0.278300\pi\)
\(422\) 0 0
\(423\) −5.23607 −0.254586
\(424\) 0 0
\(425\) 26.5623 1.28846
\(426\) 0 0
\(427\) 9.09017 0.439904
\(428\) 0 0
\(429\) 0.326238 0.0157509
\(430\) 0 0
\(431\) −31.0902 −1.49756 −0.748780 0.662818i \(-0.769360\pi\)
−0.748780 + 0.662818i \(0.769360\pi\)
\(432\) 0 0
\(433\) −17.1246 −0.822956 −0.411478 0.911420i \(-0.634987\pi\)
−0.411478 + 0.911420i \(0.634987\pi\)
\(434\) 0 0
\(435\) 1.03444 0.0495977
\(436\) 0 0
\(437\) −8.23607 −0.393985
\(438\) 0 0
\(439\) 0.0557281 0.00265976 0.00132988 0.999999i \(-0.499577\pi\)
0.00132988 + 0.999999i \(0.499577\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 0 0
\(445\) 6.23607 0.295618
\(446\) 0 0
\(447\) 16.6525 0.787635
\(448\) 0 0
\(449\) 24.0902 1.13689 0.568443 0.822723i \(-0.307546\pi\)
0.568443 + 0.822723i \(0.307546\pi\)
\(450\) 0 0
\(451\) −19.4721 −0.916907
\(452\) 0 0
\(453\) −11.7082 −0.550099
\(454\) 0 0
\(455\) −0.0557281 −0.00261257
\(456\) 0 0
\(457\) −8.20163 −0.383656 −0.191828 0.981429i \(-0.561442\pi\)
−0.191828 + 0.981429i \(0.561442\pi\)
\(458\) 0 0
\(459\) −5.47214 −0.255417
\(460\) 0 0
\(461\) 12.8541 0.598675 0.299338 0.954147i \(-0.403234\pi\)
0.299338 + 0.954147i \(0.403234\pi\)
\(462\) 0 0
\(463\) 34.5967 1.60785 0.803924 0.594733i \(-0.202742\pi\)
0.803924 + 0.594733i \(0.202742\pi\)
\(464\) 0 0
\(465\) −1.90983 −0.0885662
\(466\) 0 0
\(467\) 16.8885 0.781509 0.390754 0.920495i \(-0.372214\pi\)
0.390754 + 0.920495i \(0.372214\pi\)
\(468\) 0 0
\(469\) −6.85410 −0.316493
\(470\) 0 0
\(471\) 8.18034 0.376930
\(472\) 0 0
\(473\) −18.6180 −0.856058
\(474\) 0 0
\(475\) −39.9787 −1.83435
\(476\) 0 0
\(477\) −3.61803 −0.165658
\(478\) 0 0
\(479\) −25.6525 −1.17209 −0.586046 0.810278i \(-0.699316\pi\)
−0.586046 + 0.810278i \(0.699316\pi\)
\(480\) 0 0
\(481\) 0.0770143 0.00351155
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 6.56231 0.297979
\(486\) 0 0
\(487\) −22.2361 −1.00761 −0.503806 0.863817i \(-0.668067\pi\)
−0.503806 + 0.863817i \(0.668067\pi\)
\(488\) 0 0
\(489\) −15.5066 −0.701232
\(490\) 0 0
\(491\) −16.9098 −0.763130 −0.381565 0.924342i \(-0.624615\pi\)
−0.381565 + 0.924342i \(0.624615\pi\)
\(492\) 0 0
\(493\) 14.8197 0.667444
\(494\) 0 0
\(495\) 0.854102 0.0383890
\(496\) 0 0
\(497\) 9.38197 0.420839
\(498\) 0 0
\(499\) 11.7984 0.528168 0.264084 0.964500i \(-0.414930\pi\)
0.264084 + 0.964500i \(0.414930\pi\)
\(500\) 0 0
\(501\) −2.52786 −0.112937
\(502\) 0 0
\(503\) 9.90983 0.441857 0.220929 0.975290i \(-0.429091\pi\)
0.220929 + 0.975290i \(0.429091\pi\)
\(504\) 0 0
\(505\) 6.14590 0.273489
\(506\) 0 0
\(507\) −12.9787 −0.576405
\(508\) 0 0
\(509\) 12.2918 0.544824 0.272412 0.962181i \(-0.412179\pi\)
0.272412 + 0.962181i \(0.412179\pi\)
\(510\) 0 0
\(511\) 3.47214 0.153598
\(512\) 0 0
\(513\) 8.23607 0.363631
\(514\) 0 0
\(515\) 0.944272 0.0416096
\(516\) 0 0
\(517\) 11.7082 0.514926
\(518\) 0 0
\(519\) −24.7082 −1.08457
\(520\) 0 0
\(521\) −12.4721 −0.546414 −0.273207 0.961955i \(-0.588084\pi\)
−0.273207 + 0.961955i \(0.588084\pi\)
\(522\) 0 0
\(523\) 20.8328 0.910955 0.455478 0.890247i \(-0.349468\pi\)
0.455478 + 0.890247i \(0.349468\pi\)
\(524\) 0 0
\(525\) 4.85410 0.211850
\(526\) 0 0
\(527\) −27.3607 −1.19185
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 4.85410 0.210650
\(532\) 0 0
\(533\) −1.27051 −0.0550319
\(534\) 0 0
\(535\) 2.23607 0.0966736
\(536\) 0 0
\(537\) 9.03444 0.389865
\(538\) 0 0
\(539\) −2.23607 −0.0963143
\(540\) 0 0
\(541\) −18.7639 −0.806724 −0.403362 0.915040i \(-0.632159\pi\)
−0.403362 + 0.915040i \(0.632159\pi\)
\(542\) 0 0
\(543\) 7.18034 0.308138
\(544\) 0 0
\(545\) −5.74265 −0.245988
\(546\) 0 0
\(547\) 24.3820 1.04250 0.521249 0.853405i \(-0.325466\pi\)
0.521249 + 0.853405i \(0.325466\pi\)
\(548\) 0 0
\(549\) −9.09017 −0.387959
\(550\) 0 0
\(551\) −22.3050 −0.950223
\(552\) 0 0
\(553\) 5.94427 0.252776
\(554\) 0 0
\(555\) 0.201626 0.00855855
\(556\) 0 0
\(557\) −12.7639 −0.540825 −0.270413 0.962745i \(-0.587160\pi\)
−0.270413 + 0.962745i \(0.587160\pi\)
\(558\) 0 0
\(559\) −1.21478 −0.0513798
\(560\) 0 0
\(561\) 12.2361 0.516607
\(562\) 0 0
\(563\) 11.5623 0.487293 0.243647 0.969864i \(-0.421656\pi\)
0.243647 + 0.969864i \(0.421656\pi\)
\(564\) 0 0
\(565\) −1.56231 −0.0657267
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −43.7771 −1.83523 −0.917615 0.397469i \(-0.869889\pi\)
−0.917615 + 0.397469i \(0.869889\pi\)
\(570\) 0 0
\(571\) −26.7771 −1.12059 −0.560293 0.828294i \(-0.689312\pi\)
−0.560293 + 0.828294i \(0.689312\pi\)
\(572\) 0 0
\(573\) −8.29180 −0.346395
\(574\) 0 0
\(575\) 4.85410 0.202430
\(576\) 0 0
\(577\) 9.41641 0.392010 0.196005 0.980603i \(-0.437203\pi\)
0.196005 + 0.980603i \(0.437203\pi\)
\(578\) 0 0
\(579\) 4.18034 0.173729
\(580\) 0 0
\(581\) 7.94427 0.329584
\(582\) 0 0
\(583\) 8.09017 0.335061
\(584\) 0 0
\(585\) 0.0557281 0.00230407
\(586\) 0 0
\(587\) 17.5623 0.724874 0.362437 0.932008i \(-0.381945\pi\)
0.362437 + 0.932008i \(0.381945\pi\)
\(588\) 0 0
\(589\) 41.1803 1.69681
\(590\) 0 0
\(591\) −9.67376 −0.397925
\(592\) 0 0
\(593\) −37.7082 −1.54849 −0.774245 0.632886i \(-0.781870\pi\)
−0.774245 + 0.632886i \(0.781870\pi\)
\(594\) 0 0
\(595\) −2.09017 −0.0856886
\(596\) 0 0
\(597\) −10.5623 −0.432286
\(598\) 0 0
\(599\) −41.2705 −1.68627 −0.843134 0.537704i \(-0.819292\pi\)
−0.843134 + 0.537704i \(0.819292\pi\)
\(600\) 0 0
\(601\) −25.5066 −1.04044 −0.520218 0.854034i \(-0.674149\pi\)
−0.520218 + 0.854034i \(0.674149\pi\)
\(602\) 0 0
\(603\) 6.85410 0.279121
\(604\) 0 0
\(605\) 2.29180 0.0931748
\(606\) 0 0
\(607\) 10.6180 0.430973 0.215486 0.976507i \(-0.430866\pi\)
0.215486 + 0.976507i \(0.430866\pi\)
\(608\) 0 0
\(609\) 2.70820 0.109742
\(610\) 0 0
\(611\) 0.763932 0.0309054
\(612\) 0 0
\(613\) −37.6525 −1.52077 −0.760385 0.649473i \(-0.774990\pi\)
−0.760385 + 0.649473i \(0.774990\pi\)
\(614\) 0 0
\(615\) −3.32624 −0.134127
\(616\) 0 0
\(617\) 37.7426 1.51946 0.759731 0.650238i \(-0.225331\pi\)
0.759731 + 0.650238i \(0.225331\pi\)
\(618\) 0 0
\(619\) 33.2705 1.33725 0.668627 0.743598i \(-0.266882\pi\)
0.668627 + 0.743598i \(0.266882\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 16.3262 0.654097
\(624\) 0 0
\(625\) 22.8328 0.913313
\(626\) 0 0
\(627\) −18.4164 −0.735480
\(628\) 0 0
\(629\) 2.88854 0.115174
\(630\) 0 0
\(631\) 38.2361 1.52215 0.761077 0.648662i \(-0.224671\pi\)
0.761077 + 0.648662i \(0.224671\pi\)
\(632\) 0 0
\(633\) −15.0000 −0.596196
\(634\) 0 0
\(635\) −7.00000 −0.277787
\(636\) 0 0
\(637\) −0.145898 −0.00578069
\(638\) 0 0
\(639\) −9.38197 −0.371145
\(640\) 0 0
\(641\) −28.6180 −1.13034 −0.565172 0.824973i \(-0.691190\pi\)
−0.565172 + 0.824973i \(0.691190\pi\)
\(642\) 0 0
\(643\) −2.50658 −0.0988498 −0.0494249 0.998778i \(-0.515739\pi\)
−0.0494249 + 0.998778i \(0.515739\pi\)
\(644\) 0 0
\(645\) −3.18034 −0.125226
\(646\) 0 0
\(647\) −2.09017 −0.0821731 −0.0410865 0.999156i \(-0.513082\pi\)
−0.0410865 + 0.999156i \(0.513082\pi\)
\(648\) 0 0
\(649\) −10.8541 −0.426061
\(650\) 0 0
\(651\) −5.00000 −0.195965
\(652\) 0 0
\(653\) −32.0902 −1.25579 −0.627893 0.778300i \(-0.716082\pi\)
−0.627893 + 0.778300i \(0.716082\pi\)
\(654\) 0 0
\(655\) 5.50658 0.215160
\(656\) 0 0
\(657\) −3.47214 −0.135461
\(658\) 0 0
\(659\) 28.5967 1.11397 0.556986 0.830522i \(-0.311958\pi\)
0.556986 + 0.830522i \(0.311958\pi\)
\(660\) 0 0
\(661\) 6.81966 0.265254 0.132627 0.991166i \(-0.457659\pi\)
0.132627 + 0.991166i \(0.457659\pi\)
\(662\) 0 0
\(663\) 0.798374 0.0310063
\(664\) 0 0
\(665\) 3.14590 0.121993
\(666\) 0 0
\(667\) 2.70820 0.104862
\(668\) 0 0
\(669\) −2.67376 −0.103374
\(670\) 0 0
\(671\) 20.3262 0.784686
\(672\) 0 0
\(673\) 21.9443 0.845890 0.422945 0.906155i \(-0.360996\pi\)
0.422945 + 0.906155i \(0.360996\pi\)
\(674\) 0 0
\(675\) −4.85410 −0.186834
\(676\) 0 0
\(677\) 0.798374 0.0306840 0.0153420 0.999882i \(-0.495116\pi\)
0.0153420 + 0.999882i \(0.495116\pi\)
\(678\) 0 0
\(679\) 17.1803 0.659321
\(680\) 0 0
\(681\) 18.2705 0.700127
\(682\) 0 0
\(683\) −40.7082 −1.55766 −0.778828 0.627237i \(-0.784186\pi\)
−0.778828 + 0.627237i \(0.784186\pi\)
\(684\) 0 0
\(685\) 0.785218 0.0300016
\(686\) 0 0
\(687\) −21.3262 −0.813647
\(688\) 0 0
\(689\) 0.527864 0.0201100
\(690\) 0 0
\(691\) 20.7984 0.791207 0.395604 0.918421i \(-0.370535\pi\)
0.395604 + 0.918421i \(0.370535\pi\)
\(692\) 0 0
\(693\) 2.23607 0.0849412
\(694\) 0 0
\(695\) 5.02129 0.190468
\(696\) 0 0
\(697\) −47.6525 −1.80497
\(698\) 0 0
\(699\) −4.09017 −0.154704
\(700\) 0 0
\(701\) −31.7984 −1.20101 −0.600504 0.799622i \(-0.705033\pi\)
−0.600504 + 0.799622i \(0.705033\pi\)
\(702\) 0 0
\(703\) −4.34752 −0.163970
\(704\) 0 0
\(705\) 2.00000 0.0753244
\(706\) 0 0
\(707\) 16.0902 0.605133
\(708\) 0 0
\(709\) −12.3262 −0.462922 −0.231461 0.972844i \(-0.574350\pi\)
−0.231461 + 0.972844i \(0.574350\pi\)
\(710\) 0 0
\(711\) −5.94427 −0.222928
\(712\) 0 0
\(713\) −5.00000 −0.187251
\(714\) 0 0
\(715\) −0.124612 −0.00466022
\(716\) 0 0
\(717\) 4.38197 0.163648
\(718\) 0 0
\(719\) −9.29180 −0.346526 −0.173263 0.984876i \(-0.555431\pi\)
−0.173263 + 0.984876i \(0.555431\pi\)
\(720\) 0 0
\(721\) 2.47214 0.0920672
\(722\) 0 0
\(723\) 15.2918 0.568708
\(724\) 0 0
\(725\) 13.1459 0.488226
\(726\) 0 0
\(727\) 30.2361 1.12139 0.560697 0.828021i \(-0.310533\pi\)
0.560697 + 0.828021i \(0.310533\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −45.5623 −1.68518
\(732\) 0 0
\(733\) 28.5410 1.05419 0.527093 0.849807i \(-0.323282\pi\)
0.527093 + 0.849807i \(0.323282\pi\)
\(734\) 0 0
\(735\) −0.381966 −0.0140890
\(736\) 0 0
\(737\) −15.3262 −0.564549
\(738\) 0 0
\(739\) 40.4164 1.48674 0.743371 0.668880i \(-0.233226\pi\)
0.743371 + 0.668880i \(0.233226\pi\)
\(740\) 0 0
\(741\) −1.20163 −0.0441428
\(742\) 0 0
\(743\) −25.3951 −0.931657 −0.465828 0.884875i \(-0.654244\pi\)
−0.465828 + 0.884875i \(0.654244\pi\)
\(744\) 0 0
\(745\) −6.36068 −0.233037
\(746\) 0 0
\(747\) −7.94427 −0.290666
\(748\) 0 0
\(749\) 5.85410 0.213904
\(750\) 0 0
\(751\) 32.0902 1.17099 0.585493 0.810677i \(-0.300901\pi\)
0.585493 + 0.810677i \(0.300901\pi\)
\(752\) 0 0
\(753\) 25.7082 0.936859
\(754\) 0 0
\(755\) 4.47214 0.162758
\(756\) 0 0
\(757\) −32.4164 −1.17819 −0.589097 0.808062i \(-0.700516\pi\)
−0.589097 + 0.808062i \(0.700516\pi\)
\(758\) 0 0
\(759\) 2.23607 0.0811641
\(760\) 0 0
\(761\) 53.8885 1.95346 0.976729 0.214477i \(-0.0688046\pi\)
0.976729 + 0.214477i \(0.0688046\pi\)
\(762\) 0 0
\(763\) −15.0344 −0.544283
\(764\) 0 0
\(765\) 2.09017 0.0755703
\(766\) 0 0
\(767\) −0.708204 −0.0255718
\(768\) 0 0
\(769\) 47.9574 1.72939 0.864695 0.502298i \(-0.167512\pi\)
0.864695 + 0.502298i \(0.167512\pi\)
\(770\) 0 0
\(771\) 16.2918 0.586735
\(772\) 0 0
\(773\) −36.0132 −1.29530 −0.647652 0.761937i \(-0.724249\pi\)
−0.647652 + 0.761937i \(0.724249\pi\)
\(774\) 0 0
\(775\) −24.2705 −0.871822
\(776\) 0 0
\(777\) 0.527864 0.0189370
\(778\) 0 0
\(779\) 71.7214 2.56968
\(780\) 0 0
\(781\) 20.9787 0.750677
\(782\) 0 0
\(783\) −2.70820 −0.0967833
\(784\) 0 0
\(785\) −3.12461 −0.111522
\(786\) 0 0
\(787\) 1.79837 0.0641051 0.0320526 0.999486i \(-0.489796\pi\)
0.0320526 + 0.999486i \(0.489796\pi\)
\(788\) 0 0
\(789\) −5.00000 −0.178005
\(790\) 0 0
\(791\) −4.09017 −0.145430
\(792\) 0 0
\(793\) 1.32624 0.0470961
\(794\) 0 0
\(795\) 1.38197 0.0490133
\(796\) 0 0
\(797\) −3.70820 −0.131351 −0.0656757 0.997841i \(-0.520920\pi\)
−0.0656757 + 0.997841i \(0.520920\pi\)
\(798\) 0 0
\(799\) 28.6525 1.01365
\(800\) 0 0
\(801\) −16.3262 −0.576859
\(802\) 0 0
\(803\) 7.76393 0.273983
\(804\) 0 0
\(805\) −0.381966 −0.0134625
\(806\) 0 0
\(807\) 3.85410 0.135671
\(808\) 0 0
\(809\) −2.43769 −0.0857048 −0.0428524 0.999081i \(-0.513645\pi\)
−0.0428524 + 0.999081i \(0.513645\pi\)
\(810\) 0 0
\(811\) 25.5410 0.896867 0.448433 0.893816i \(-0.351982\pi\)
0.448433 + 0.893816i \(0.351982\pi\)
\(812\) 0 0
\(813\) −5.18034 −0.181682
\(814\) 0 0
\(815\) 5.92299 0.207473
\(816\) 0 0
\(817\) 68.5755 2.39915
\(818\) 0 0
\(819\) 0.145898 0.00509809
\(820\) 0 0
\(821\) 12.2918 0.428987 0.214493 0.976725i \(-0.431190\pi\)
0.214493 + 0.976725i \(0.431190\pi\)
\(822\) 0 0
\(823\) −8.02129 −0.279604 −0.139802 0.990179i \(-0.544647\pi\)
−0.139802 + 0.990179i \(0.544647\pi\)
\(824\) 0 0
\(825\) 10.8541 0.377891
\(826\) 0 0
\(827\) 22.8541 0.794715 0.397357 0.917664i \(-0.369927\pi\)
0.397357 + 0.917664i \(0.369927\pi\)
\(828\) 0 0
\(829\) 35.0689 1.21799 0.608996 0.793173i \(-0.291572\pi\)
0.608996 + 0.793173i \(0.291572\pi\)
\(830\) 0 0
\(831\) −30.2148 −1.04814
\(832\) 0 0
\(833\) −5.47214 −0.189598
\(834\) 0 0
\(835\) 0.965558 0.0334145
\(836\) 0 0
\(837\) 5.00000 0.172825
\(838\) 0 0
\(839\) 2.72949 0.0942325 0.0471162 0.998889i \(-0.484997\pi\)
0.0471162 + 0.998889i \(0.484997\pi\)
\(840\) 0 0
\(841\) −21.6656 −0.747091
\(842\) 0 0
\(843\) 27.5967 0.950482
\(844\) 0 0
\(845\) 4.95743 0.170541
\(846\) 0 0
\(847\) 6.00000 0.206162
\(848\) 0 0
\(849\) 1.32624 0.0455164
\(850\) 0 0
\(851\) 0.527864 0.0180949
\(852\) 0 0
\(853\) −42.6525 −1.46039 −0.730196 0.683237i \(-0.760571\pi\)
−0.730196 + 0.683237i \(0.760571\pi\)
\(854\) 0 0
\(855\) −3.14590 −0.107587
\(856\) 0 0
\(857\) 43.9574 1.50156 0.750779 0.660554i \(-0.229679\pi\)
0.750779 + 0.660554i \(0.229679\pi\)
\(858\) 0 0
\(859\) −1.41641 −0.0483272 −0.0241636 0.999708i \(-0.507692\pi\)
−0.0241636 + 0.999708i \(0.507692\pi\)
\(860\) 0 0
\(861\) −8.70820 −0.296775
\(862\) 0 0
\(863\) −31.5967 −1.07557 −0.537783 0.843083i \(-0.680738\pi\)
−0.537783 + 0.843083i \(0.680738\pi\)
\(864\) 0 0
\(865\) 9.43769 0.320891
\(866\) 0 0
\(867\) 12.9443 0.439611
\(868\) 0 0
\(869\) 13.2918 0.450893
\(870\) 0 0
\(871\) −1.00000 −0.0338837
\(872\) 0 0
\(873\) −17.1803 −0.581466
\(874\) 0 0
\(875\) −3.76393 −0.127244
\(876\) 0 0
\(877\) −26.4721 −0.893901 −0.446950 0.894559i \(-0.647490\pi\)
−0.446950 + 0.894559i \(0.647490\pi\)
\(878\) 0 0
\(879\) −18.4721 −0.623050
\(880\) 0 0
\(881\) −40.5410 −1.36586 −0.682931 0.730483i \(-0.739295\pi\)
−0.682931 + 0.730483i \(0.739295\pi\)
\(882\) 0 0
\(883\) −7.74265 −0.260561 −0.130280 0.991477i \(-0.541588\pi\)
−0.130280 + 0.991477i \(0.541588\pi\)
\(884\) 0 0
\(885\) −1.85410 −0.0623250
\(886\) 0 0
\(887\) −22.3262 −0.749642 −0.374821 0.927097i \(-0.622296\pi\)
−0.374821 + 0.927097i \(0.622296\pi\)
\(888\) 0 0
\(889\) −18.3262 −0.614642
\(890\) 0 0
\(891\) −2.23607 −0.0749111
\(892\) 0 0
\(893\) −43.1246 −1.44311
\(894\) 0 0
\(895\) −3.45085 −0.115349
\(896\) 0 0
\(897\) 0.145898 0.00487139
\(898\) 0 0
\(899\) −13.5410 −0.451618
\(900\) 0 0
\(901\) 19.7984 0.659579
\(902\) 0 0
\(903\) −8.32624 −0.277080
\(904\) 0 0
\(905\) −2.74265 −0.0911686
\(906\) 0 0
\(907\) 0.673762 0.0223719 0.0111860 0.999937i \(-0.496439\pi\)
0.0111860 + 0.999937i \(0.496439\pi\)
\(908\) 0 0
\(909\) −16.0902 −0.533677
\(910\) 0 0
\(911\) 31.7082 1.05054 0.525270 0.850936i \(-0.323964\pi\)
0.525270 + 0.850936i \(0.323964\pi\)
\(912\) 0 0
\(913\) 17.7639 0.587900
\(914\) 0 0
\(915\) 3.47214 0.114785
\(916\) 0 0
\(917\) 14.4164 0.476072
\(918\) 0 0
\(919\) 48.4853 1.59938 0.799691 0.600412i \(-0.204997\pi\)
0.799691 + 0.600412i \(0.204997\pi\)
\(920\) 0 0
\(921\) −8.05573 −0.265445
\(922\) 0 0
\(923\) 1.36881 0.0450549
\(924\) 0 0
\(925\) 2.56231 0.0842481
\(926\) 0 0
\(927\) −2.47214 −0.0811956
\(928\) 0 0
\(929\) −8.74265 −0.286837 −0.143418 0.989662i \(-0.545809\pi\)
−0.143418 + 0.989662i \(0.545809\pi\)
\(930\) 0 0
\(931\) 8.23607 0.269926
\(932\) 0 0
\(933\) −5.38197 −0.176198
\(934\) 0 0
\(935\) −4.67376 −0.152848
\(936\) 0 0
\(937\) 11.3607 0.371137 0.185569 0.982631i \(-0.440587\pi\)
0.185569 + 0.982631i \(0.440587\pi\)
\(938\) 0 0
\(939\) −10.4721 −0.341745
\(940\) 0 0
\(941\) 18.8328 0.613932 0.306966 0.951720i \(-0.400686\pi\)
0.306966 + 0.951720i \(0.400686\pi\)
\(942\) 0 0
\(943\) −8.70820 −0.283578
\(944\) 0 0
\(945\) 0.381966 0.0124254
\(946\) 0 0
\(947\) −42.7214 −1.38826 −0.694129 0.719851i \(-0.744210\pi\)
−0.694129 + 0.719851i \(0.744210\pi\)
\(948\) 0 0
\(949\) 0.506578 0.0164442
\(950\) 0 0
\(951\) −28.8541 −0.935658
\(952\) 0 0
\(953\) −15.9230 −0.515796 −0.257898 0.966172i \(-0.583030\pi\)
−0.257898 + 0.966172i \(0.583030\pi\)
\(954\) 0 0
\(955\) 3.16718 0.102488
\(956\) 0 0
\(957\) 6.05573 0.195754
\(958\) 0 0
\(959\) 2.05573 0.0663829
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) −5.85410 −0.188646
\(964\) 0 0
\(965\) −1.59675 −0.0514011
\(966\) 0 0
\(967\) 7.52786 0.242080 0.121040 0.992648i \(-0.461377\pi\)
0.121040 + 0.992648i \(0.461377\pi\)
\(968\) 0 0
\(969\) −45.0689 −1.44782
\(970\) 0 0
\(971\) 2.90983 0.0933809 0.0466904 0.998909i \(-0.485133\pi\)
0.0466904 + 0.998909i \(0.485133\pi\)
\(972\) 0 0
\(973\) 13.1459 0.421438
\(974\) 0 0
\(975\) 0.708204 0.0226807
\(976\) 0 0
\(977\) 7.02129 0.224631 0.112315 0.993673i \(-0.464173\pi\)
0.112315 + 0.993673i \(0.464173\pi\)
\(978\) 0 0
\(979\) 36.5066 1.16676
\(980\) 0 0
\(981\) 15.0344 0.480013
\(982\) 0 0
\(983\) 14.6393 0.466922 0.233461 0.972366i \(-0.424995\pi\)
0.233461 + 0.972366i \(0.424995\pi\)
\(984\) 0 0
\(985\) 3.69505 0.117734
\(986\) 0 0
\(987\) 5.23607 0.166666
\(988\) 0 0
\(989\) −8.32624 −0.264759
\(990\) 0 0
\(991\) −18.3262 −0.582152 −0.291076 0.956700i \(-0.594013\pi\)
−0.291076 + 0.956700i \(0.594013\pi\)
\(992\) 0 0
\(993\) 6.94427 0.220370
\(994\) 0 0
\(995\) 4.03444 0.127900
\(996\) 0 0
\(997\) 41.5410 1.31562 0.657809 0.753185i \(-0.271484\pi\)
0.657809 + 0.753185i \(0.271484\pi\)
\(998\) 0 0
\(999\) −0.527864 −0.0167009
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.bg.1.2 2
4.3 odd 2 483.2.a.g.1.1 2
12.11 even 2 1449.2.a.f.1.2 2
28.27 even 2 3381.2.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.g.1.1 2 4.3 odd 2
1449.2.a.f.1.2 2 12.11 even 2
3381.2.a.u.1.1 2 28.27 even 2
7728.2.a.bg.1.2 2 1.1 even 1 trivial