Properties

Label 5520.2.a.bm.1.1
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 345)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -1.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -1.82843 q^{7} +1.00000 q^{9} +2.58579 q^{11} +3.41421 q^{13} -1.00000 q^{15} -6.07107 q^{17} +6.24264 q^{19} -1.82843 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.58579 q^{29} +4.17157 q^{31} +2.58579 q^{33} +1.82843 q^{35} +3.00000 q^{37} +3.41421 q^{39} -10.4142 q^{41} -6.00000 q^{43} -1.00000 q^{45} +8.58579 q^{47} -3.65685 q^{49} -6.07107 q^{51} +7.24264 q^{53} -2.58579 q^{55} +6.24264 q^{57} +6.89949 q^{59} +5.41421 q^{61} -1.82843 q^{63} -3.41421 q^{65} +11.4853 q^{67} +1.00000 q^{69} -8.89949 q^{71} -10.2426 q^{73} +1.00000 q^{75} -4.72792 q^{77} +15.3137 q^{79} +1.00000 q^{81} -14.0711 q^{83} +6.07107 q^{85} -3.58579 q^{87} +10.1421 q^{89} -6.24264 q^{91} +4.17157 q^{93} -6.24264 q^{95} -1.17157 q^{97} +2.58579 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 8 q^{11} + 4 q^{13} - 2 q^{15} + 2 q^{17} + 4 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} + 14 q^{31} + 8 q^{33} - 2 q^{35} + 6 q^{37} + 4 q^{39} - 18 q^{41} - 12 q^{43} - 2 q^{45} + 20 q^{47} + 4 q^{49} + 2 q^{51} + 6 q^{53} - 8 q^{55} + 4 q^{57} - 6 q^{59} + 8 q^{61} + 2 q^{63} - 4 q^{65} + 6 q^{67} + 2 q^{69} + 2 q^{71} - 12 q^{73} + 2 q^{75} + 16 q^{77} + 8 q^{79} + 2 q^{81} - 14 q^{83} - 2 q^{85} - 10 q^{87} - 8 q^{89} - 4 q^{91} + 14 q^{93} - 4 q^{95} - 8 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.82843 −0.691080 −0.345540 0.938404i \(-0.612304\pi\)
−0.345540 + 0.938404i \(0.612304\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.58579 0.779644 0.389822 0.920890i \(-0.372537\pi\)
0.389822 + 0.920890i \(0.372537\pi\)
\(12\) 0 0
\(13\) 3.41421 0.946932 0.473466 0.880812i \(-0.343003\pi\)
0.473466 + 0.880812i \(0.343003\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −6.07107 −1.47245 −0.736225 0.676737i \(-0.763394\pi\)
−0.736225 + 0.676737i \(0.763394\pi\)
\(18\) 0 0
\(19\) 6.24264 1.43216 0.716080 0.698018i \(-0.245935\pi\)
0.716080 + 0.698018i \(0.245935\pi\)
\(20\) 0 0
\(21\) −1.82843 −0.398996
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.58579 −0.665864 −0.332932 0.942951i \(-0.608038\pi\)
−0.332932 + 0.942951i \(0.608038\pi\)
\(30\) 0 0
\(31\) 4.17157 0.749237 0.374618 0.927179i \(-0.377774\pi\)
0.374618 + 0.927179i \(0.377774\pi\)
\(32\) 0 0
\(33\) 2.58579 0.450128
\(34\) 0 0
\(35\) 1.82843 0.309061
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 0 0
\(39\) 3.41421 0.546712
\(40\) 0 0
\(41\) −10.4142 −1.62643 −0.813213 0.581966i \(-0.802284\pi\)
−0.813213 + 0.581966i \(0.802284\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 8.58579 1.25237 0.626183 0.779676i \(-0.284616\pi\)
0.626183 + 0.779676i \(0.284616\pi\)
\(48\) 0 0
\(49\) −3.65685 −0.522408
\(50\) 0 0
\(51\) −6.07107 −0.850120
\(52\) 0 0
\(53\) 7.24264 0.994853 0.497427 0.867506i \(-0.334278\pi\)
0.497427 + 0.867506i \(0.334278\pi\)
\(54\) 0 0
\(55\) −2.58579 −0.348667
\(56\) 0 0
\(57\) 6.24264 0.826858
\(58\) 0 0
\(59\) 6.89949 0.898238 0.449119 0.893472i \(-0.351738\pi\)
0.449119 + 0.893472i \(0.351738\pi\)
\(60\) 0 0
\(61\) 5.41421 0.693219 0.346610 0.938010i \(-0.387333\pi\)
0.346610 + 0.938010i \(0.387333\pi\)
\(62\) 0 0
\(63\) −1.82843 −0.230360
\(64\) 0 0
\(65\) −3.41421 −0.423481
\(66\) 0 0
\(67\) 11.4853 1.40315 0.701575 0.712595i \(-0.252480\pi\)
0.701575 + 0.712595i \(0.252480\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −8.89949 −1.05618 −0.528088 0.849190i \(-0.677091\pi\)
−0.528088 + 0.849190i \(0.677091\pi\)
\(72\) 0 0
\(73\) −10.2426 −1.19881 −0.599405 0.800446i \(-0.704596\pi\)
−0.599405 + 0.800446i \(0.704596\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −4.72792 −0.538797
\(78\) 0 0
\(79\) 15.3137 1.72293 0.861463 0.507820i \(-0.169548\pi\)
0.861463 + 0.507820i \(0.169548\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.0711 −1.54450 −0.772250 0.635319i \(-0.780869\pi\)
−0.772250 + 0.635319i \(0.780869\pi\)
\(84\) 0 0
\(85\) 6.07107 0.658500
\(86\) 0 0
\(87\) −3.58579 −0.384437
\(88\) 0 0
\(89\) 10.1421 1.07506 0.537532 0.843243i \(-0.319357\pi\)
0.537532 + 0.843243i \(0.319357\pi\)
\(90\) 0 0
\(91\) −6.24264 −0.654407
\(92\) 0 0
\(93\) 4.17157 0.432572
\(94\) 0 0
\(95\) −6.24264 −0.640481
\(96\) 0 0
\(97\) −1.17157 −0.118955 −0.0594776 0.998230i \(-0.518943\pi\)
−0.0594776 + 0.998230i \(0.518943\pi\)
\(98\) 0 0
\(99\) 2.58579 0.259881
\(100\) 0 0
\(101\) 12.5563 1.24940 0.624702 0.780863i \(-0.285221\pi\)
0.624702 + 0.780863i \(0.285221\pi\)
\(102\) 0 0
\(103\) −0.828427 −0.0816274 −0.0408137 0.999167i \(-0.512995\pi\)
−0.0408137 + 0.999167i \(0.512995\pi\)
\(104\) 0 0
\(105\) 1.82843 0.178436
\(106\) 0 0
\(107\) 4.41421 0.426738 0.213369 0.976972i \(-0.431556\pi\)
0.213369 + 0.976972i \(0.431556\pi\)
\(108\) 0 0
\(109\) 14.2426 1.36420 0.682099 0.731260i \(-0.261067\pi\)
0.682099 + 0.731260i \(0.261067\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) −10.4142 −0.979687 −0.489843 0.871810i \(-0.662946\pi\)
−0.489843 + 0.871810i \(0.662946\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 3.41421 0.315644
\(118\) 0 0
\(119\) 11.1005 1.01758
\(120\) 0 0
\(121\) −4.31371 −0.392155
\(122\) 0 0
\(123\) −10.4142 −0.939018
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0.242641 0.0215309 0.0107654 0.999942i \(-0.496573\pi\)
0.0107654 + 0.999942i \(0.496573\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 13.6569 1.19320 0.596602 0.802537i \(-0.296517\pi\)
0.596602 + 0.802537i \(0.296517\pi\)
\(132\) 0 0
\(133\) −11.4142 −0.989738
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 2.82843 0.241649 0.120824 0.992674i \(-0.461446\pi\)
0.120824 + 0.992674i \(0.461446\pi\)
\(138\) 0 0
\(139\) 11.0000 0.933008 0.466504 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) 0 0
\(141\) 8.58579 0.723054
\(142\) 0 0
\(143\) 8.82843 0.738270
\(144\) 0 0
\(145\) 3.58579 0.297783
\(146\) 0 0
\(147\) −3.65685 −0.301612
\(148\) 0 0
\(149\) 9.07107 0.743131 0.371565 0.928407i \(-0.378821\pi\)
0.371565 + 0.928407i \(0.378821\pi\)
\(150\) 0 0
\(151\) 9.31371 0.757939 0.378969 0.925409i \(-0.376279\pi\)
0.378969 + 0.925409i \(0.376279\pi\)
\(152\) 0 0
\(153\) −6.07107 −0.490817
\(154\) 0 0
\(155\) −4.17157 −0.335069
\(156\) 0 0
\(157\) −1.48528 −0.118538 −0.0592692 0.998242i \(-0.518877\pi\)
−0.0592692 + 0.998242i \(0.518877\pi\)
\(158\) 0 0
\(159\) 7.24264 0.574379
\(160\) 0 0
\(161\) −1.82843 −0.144100
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 0 0
\(165\) −2.58579 −0.201303
\(166\) 0 0
\(167\) 6.72792 0.520622 0.260311 0.965525i \(-0.416175\pi\)
0.260311 + 0.965525i \(0.416175\pi\)
\(168\) 0 0
\(169\) −1.34315 −0.103319
\(170\) 0 0
\(171\) 6.24264 0.477387
\(172\) 0 0
\(173\) 13.6569 1.03831 0.519156 0.854680i \(-0.326246\pi\)
0.519156 + 0.854680i \(0.326246\pi\)
\(174\) 0 0
\(175\) −1.82843 −0.138216
\(176\) 0 0
\(177\) 6.89949 0.518598
\(178\) 0 0
\(179\) 7.65685 0.572300 0.286150 0.958185i \(-0.407624\pi\)
0.286150 + 0.958185i \(0.407624\pi\)
\(180\) 0 0
\(181\) −6.82843 −0.507553 −0.253776 0.967263i \(-0.581673\pi\)
−0.253776 + 0.967263i \(0.581673\pi\)
\(182\) 0 0
\(183\) 5.41421 0.400230
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) −15.6985 −1.14799
\(188\) 0 0
\(189\) −1.82843 −0.132999
\(190\) 0 0
\(191\) −10.2426 −0.741131 −0.370566 0.928806i \(-0.620836\pi\)
−0.370566 + 0.928806i \(0.620836\pi\)
\(192\) 0 0
\(193\) −8.48528 −0.610784 −0.305392 0.952227i \(-0.598787\pi\)
−0.305392 + 0.952227i \(0.598787\pi\)
\(194\) 0 0
\(195\) −3.41421 −0.244497
\(196\) 0 0
\(197\) 16.8284 1.19898 0.599488 0.800384i \(-0.295371\pi\)
0.599488 + 0.800384i \(0.295371\pi\)
\(198\) 0 0
\(199\) −10.4853 −0.743282 −0.371641 0.928377i \(-0.621205\pi\)
−0.371641 + 0.928377i \(0.621205\pi\)
\(200\) 0 0
\(201\) 11.4853 0.810109
\(202\) 0 0
\(203\) 6.55635 0.460166
\(204\) 0 0
\(205\) 10.4142 0.727360
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 16.1421 1.11657
\(210\) 0 0
\(211\) −0.656854 −0.0452197 −0.0226099 0.999744i \(-0.507198\pi\)
−0.0226099 + 0.999744i \(0.507198\pi\)
\(212\) 0 0
\(213\) −8.89949 −0.609783
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) −7.62742 −0.517783
\(218\) 0 0
\(219\) −10.2426 −0.692134
\(220\) 0 0
\(221\) −20.7279 −1.39431
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −17.6569 −1.17193 −0.585963 0.810338i \(-0.699284\pi\)
−0.585963 + 0.810338i \(0.699284\pi\)
\(228\) 0 0
\(229\) 23.7990 1.57268 0.786341 0.617793i \(-0.211973\pi\)
0.786341 + 0.617793i \(0.211973\pi\)
\(230\) 0 0
\(231\) −4.72792 −0.311074
\(232\) 0 0
\(233\) 9.65685 0.632642 0.316321 0.948652i \(-0.397552\pi\)
0.316321 + 0.948652i \(0.397552\pi\)
\(234\) 0 0
\(235\) −8.58579 −0.560075
\(236\) 0 0
\(237\) 15.3137 0.994732
\(238\) 0 0
\(239\) 1.58579 0.102576 0.0512880 0.998684i \(-0.483667\pi\)
0.0512880 + 0.998684i \(0.483667\pi\)
\(240\) 0 0
\(241\) 23.4142 1.50824 0.754121 0.656735i \(-0.228063\pi\)
0.754121 + 0.656735i \(0.228063\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.65685 0.233628
\(246\) 0 0
\(247\) 21.3137 1.35616
\(248\) 0 0
\(249\) −14.0711 −0.891718
\(250\) 0 0
\(251\) 20.6274 1.30199 0.650996 0.759082i \(-0.274352\pi\)
0.650996 + 0.759082i \(0.274352\pi\)
\(252\) 0 0
\(253\) 2.58579 0.162567
\(254\) 0 0
\(255\) 6.07107 0.380185
\(256\) 0 0
\(257\) 30.0416 1.87395 0.936973 0.349403i \(-0.113615\pi\)
0.936973 + 0.349403i \(0.113615\pi\)
\(258\) 0 0
\(259\) −5.48528 −0.340839
\(260\) 0 0
\(261\) −3.58579 −0.221955
\(262\) 0 0
\(263\) −3.72792 −0.229874 −0.114937 0.993373i \(-0.536667\pi\)
−0.114937 + 0.993373i \(0.536667\pi\)
\(264\) 0 0
\(265\) −7.24264 −0.444912
\(266\) 0 0
\(267\) 10.1421 0.620689
\(268\) 0 0
\(269\) −26.5563 −1.61917 −0.809585 0.587003i \(-0.800308\pi\)
−0.809585 + 0.587003i \(0.800308\pi\)
\(270\) 0 0
\(271\) 6.51472 0.395741 0.197870 0.980228i \(-0.436597\pi\)
0.197870 + 0.980228i \(0.436597\pi\)
\(272\) 0 0
\(273\) −6.24264 −0.377822
\(274\) 0 0
\(275\) 2.58579 0.155929
\(276\) 0 0
\(277\) 25.3137 1.52095 0.760477 0.649365i \(-0.224965\pi\)
0.760477 + 0.649365i \(0.224965\pi\)
\(278\) 0 0
\(279\) 4.17157 0.249746
\(280\) 0 0
\(281\) 17.4142 1.03884 0.519422 0.854518i \(-0.326147\pi\)
0.519422 + 0.854518i \(0.326147\pi\)
\(282\) 0 0
\(283\) 28.6569 1.70347 0.851737 0.523970i \(-0.175550\pi\)
0.851737 + 0.523970i \(0.175550\pi\)
\(284\) 0 0
\(285\) −6.24264 −0.369782
\(286\) 0 0
\(287\) 19.0416 1.12399
\(288\) 0 0
\(289\) 19.8579 1.16811
\(290\) 0 0
\(291\) −1.17157 −0.0686788
\(292\) 0 0
\(293\) −3.10051 −0.181133 −0.0905667 0.995890i \(-0.528868\pi\)
−0.0905667 + 0.995890i \(0.528868\pi\)
\(294\) 0 0
\(295\) −6.89949 −0.401704
\(296\) 0 0
\(297\) 2.58579 0.150043
\(298\) 0 0
\(299\) 3.41421 0.197449
\(300\) 0 0
\(301\) 10.9706 0.632333
\(302\) 0 0
\(303\) 12.5563 0.721343
\(304\) 0 0
\(305\) −5.41421 −0.310017
\(306\) 0 0
\(307\) −27.2132 −1.55314 −0.776570 0.630031i \(-0.783042\pi\)
−0.776570 + 0.630031i \(0.783042\pi\)
\(308\) 0 0
\(309\) −0.828427 −0.0471276
\(310\) 0 0
\(311\) −17.6569 −1.00123 −0.500614 0.865671i \(-0.666892\pi\)
−0.500614 + 0.865671i \(0.666892\pi\)
\(312\) 0 0
\(313\) 25.9706 1.46794 0.733971 0.679180i \(-0.237665\pi\)
0.733971 + 0.679180i \(0.237665\pi\)
\(314\) 0 0
\(315\) 1.82843 0.103020
\(316\) 0 0
\(317\) −29.5563 −1.66005 −0.830025 0.557726i \(-0.811674\pi\)
−0.830025 + 0.557726i \(0.811674\pi\)
\(318\) 0 0
\(319\) −9.27208 −0.519137
\(320\) 0 0
\(321\) 4.41421 0.246377
\(322\) 0 0
\(323\) −37.8995 −2.10878
\(324\) 0 0
\(325\) 3.41421 0.189386
\(326\) 0 0
\(327\) 14.2426 0.787620
\(328\) 0 0
\(329\) −15.6985 −0.865485
\(330\) 0 0
\(331\) −1.00000 −0.0549650 −0.0274825 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) 0 0
\(333\) 3.00000 0.164399
\(334\) 0 0
\(335\) −11.4853 −0.627508
\(336\) 0 0
\(337\) 30.9706 1.68707 0.843537 0.537071i \(-0.180469\pi\)
0.843537 + 0.537071i \(0.180469\pi\)
\(338\) 0 0
\(339\) −10.4142 −0.565622
\(340\) 0 0
\(341\) 10.7868 0.584138
\(342\) 0 0
\(343\) 19.4853 1.05211
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −13.1716 −0.707087 −0.353544 0.935418i \(-0.615023\pi\)
−0.353544 + 0.935418i \(0.615023\pi\)
\(348\) 0 0
\(349\) −32.1127 −1.71895 −0.859477 0.511175i \(-0.829210\pi\)
−0.859477 + 0.511175i \(0.829210\pi\)
\(350\) 0 0
\(351\) 3.41421 0.182237
\(352\) 0 0
\(353\) −34.5269 −1.83768 −0.918841 0.394628i \(-0.870874\pi\)
−0.918841 + 0.394628i \(0.870874\pi\)
\(354\) 0 0
\(355\) 8.89949 0.472336
\(356\) 0 0
\(357\) 11.1005 0.587501
\(358\) 0 0
\(359\) 10.9289 0.576807 0.288403 0.957509i \(-0.406876\pi\)
0.288403 + 0.957509i \(0.406876\pi\)
\(360\) 0 0
\(361\) 19.9706 1.05108
\(362\) 0 0
\(363\) −4.31371 −0.226411
\(364\) 0 0
\(365\) 10.2426 0.536124
\(366\) 0 0
\(367\) 13.9706 0.729257 0.364629 0.931153i \(-0.381196\pi\)
0.364629 + 0.931153i \(0.381196\pi\)
\(368\) 0 0
\(369\) −10.4142 −0.542142
\(370\) 0 0
\(371\) −13.2426 −0.687524
\(372\) 0 0
\(373\) −12.9706 −0.671590 −0.335795 0.941935i \(-0.609005\pi\)
−0.335795 + 0.941935i \(0.609005\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −12.2426 −0.630528
\(378\) 0 0
\(379\) −28.9706 −1.48812 −0.744059 0.668114i \(-0.767102\pi\)
−0.744059 + 0.668114i \(0.767102\pi\)
\(380\) 0 0
\(381\) 0.242641 0.0124309
\(382\) 0 0
\(383\) −14.4142 −0.736532 −0.368266 0.929720i \(-0.620048\pi\)
−0.368266 + 0.929720i \(0.620048\pi\)
\(384\) 0 0
\(385\) 4.72792 0.240957
\(386\) 0 0
\(387\) −6.00000 −0.304997
\(388\) 0 0
\(389\) 14.4853 0.734433 0.367216 0.930136i \(-0.380311\pi\)
0.367216 + 0.930136i \(0.380311\pi\)
\(390\) 0 0
\(391\) −6.07107 −0.307027
\(392\) 0 0
\(393\) 13.6569 0.688897
\(394\) 0 0
\(395\) −15.3137 −0.770516
\(396\) 0 0
\(397\) −15.3137 −0.768573 −0.384286 0.923214i \(-0.625553\pi\)
−0.384286 + 0.923214i \(0.625553\pi\)
\(398\) 0 0
\(399\) −11.4142 −0.571425
\(400\) 0 0
\(401\) −32.1421 −1.60510 −0.802551 0.596584i \(-0.796524\pi\)
−0.802551 + 0.596584i \(0.796524\pi\)
\(402\) 0 0
\(403\) 14.2426 0.709476
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 7.75736 0.384518
\(408\) 0 0
\(409\) 1.48528 0.0734424 0.0367212 0.999326i \(-0.488309\pi\)
0.0367212 + 0.999326i \(0.488309\pi\)
\(410\) 0 0
\(411\) 2.82843 0.139516
\(412\) 0 0
\(413\) −12.6152 −0.620755
\(414\) 0 0
\(415\) 14.0711 0.690722
\(416\) 0 0
\(417\) 11.0000 0.538672
\(418\) 0 0
\(419\) 24.0416 1.17451 0.587255 0.809402i \(-0.300208\pi\)
0.587255 + 0.809402i \(0.300208\pi\)
\(420\) 0 0
\(421\) −1.75736 −0.0856485 −0.0428242 0.999083i \(-0.513636\pi\)
−0.0428242 + 0.999083i \(0.513636\pi\)
\(422\) 0 0
\(423\) 8.58579 0.417455
\(424\) 0 0
\(425\) −6.07107 −0.294490
\(426\) 0 0
\(427\) −9.89949 −0.479070
\(428\) 0 0
\(429\) 8.82843 0.426240
\(430\) 0 0
\(431\) 27.7990 1.33903 0.669515 0.742798i \(-0.266502\pi\)
0.669515 + 0.742798i \(0.266502\pi\)
\(432\) 0 0
\(433\) −1.82843 −0.0878686 −0.0439343 0.999034i \(-0.513989\pi\)
−0.0439343 + 0.999034i \(0.513989\pi\)
\(434\) 0 0
\(435\) 3.58579 0.171925
\(436\) 0 0
\(437\) 6.24264 0.298626
\(438\) 0 0
\(439\) 18.4853 0.882254 0.441127 0.897445i \(-0.354579\pi\)
0.441127 + 0.897445i \(0.354579\pi\)
\(440\) 0 0
\(441\) −3.65685 −0.174136
\(442\) 0 0
\(443\) −30.5269 −1.45038 −0.725189 0.688550i \(-0.758247\pi\)
−0.725189 + 0.688550i \(0.758247\pi\)
\(444\) 0 0
\(445\) −10.1421 −0.480783
\(446\) 0 0
\(447\) 9.07107 0.429047
\(448\) 0 0
\(449\) −13.2426 −0.624959 −0.312479 0.949925i \(-0.601160\pi\)
−0.312479 + 0.949925i \(0.601160\pi\)
\(450\) 0 0
\(451\) −26.9289 −1.26803
\(452\) 0 0
\(453\) 9.31371 0.437596
\(454\) 0 0
\(455\) 6.24264 0.292660
\(456\) 0 0
\(457\) −31.0000 −1.45012 −0.725059 0.688686i \(-0.758188\pi\)
−0.725059 + 0.688686i \(0.758188\pi\)
\(458\) 0 0
\(459\) −6.07107 −0.283373
\(460\) 0 0
\(461\) −9.51472 −0.443145 −0.221572 0.975144i \(-0.571119\pi\)
−0.221572 + 0.975144i \(0.571119\pi\)
\(462\) 0 0
\(463\) −17.0711 −0.793360 −0.396680 0.917957i \(-0.629838\pi\)
−0.396680 + 0.917957i \(0.629838\pi\)
\(464\) 0 0
\(465\) −4.17157 −0.193452
\(466\) 0 0
\(467\) −3.58579 −0.165930 −0.0829652 0.996552i \(-0.526439\pi\)
−0.0829652 + 0.996552i \(0.526439\pi\)
\(468\) 0 0
\(469\) −21.0000 −0.969690
\(470\) 0 0
\(471\) −1.48528 −0.0684382
\(472\) 0 0
\(473\) −15.5147 −0.713368
\(474\) 0 0
\(475\) 6.24264 0.286432
\(476\) 0 0
\(477\) 7.24264 0.331618
\(478\) 0 0
\(479\) 18.7279 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(480\) 0 0
\(481\) 10.2426 0.467024
\(482\) 0 0
\(483\) −1.82843 −0.0831963
\(484\) 0 0
\(485\) 1.17157 0.0531984
\(486\) 0 0
\(487\) −10.3848 −0.470579 −0.235290 0.971925i \(-0.575604\pi\)
−0.235290 + 0.971925i \(0.575604\pi\)
\(488\) 0 0
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) −2.07107 −0.0934660 −0.0467330 0.998907i \(-0.514881\pi\)
−0.0467330 + 0.998907i \(0.514881\pi\)
\(492\) 0 0
\(493\) 21.7696 0.980451
\(494\) 0 0
\(495\) −2.58579 −0.116222
\(496\) 0 0
\(497\) 16.2721 0.729902
\(498\) 0 0
\(499\) 13.0000 0.581960 0.290980 0.956729i \(-0.406019\pi\)
0.290980 + 0.956729i \(0.406019\pi\)
\(500\) 0 0
\(501\) 6.72792 0.300581
\(502\) 0 0
\(503\) 14.2721 0.636361 0.318180 0.948030i \(-0.396928\pi\)
0.318180 + 0.948030i \(0.396928\pi\)
\(504\) 0 0
\(505\) −12.5563 −0.558750
\(506\) 0 0
\(507\) −1.34315 −0.0596512
\(508\) 0 0
\(509\) −31.7990 −1.40947 −0.704733 0.709473i \(-0.748933\pi\)
−0.704733 + 0.709473i \(0.748933\pi\)
\(510\) 0 0
\(511\) 18.7279 0.828474
\(512\) 0 0
\(513\) 6.24264 0.275619
\(514\) 0 0
\(515\) 0.828427 0.0365049
\(516\) 0 0
\(517\) 22.2010 0.976399
\(518\) 0 0
\(519\) 13.6569 0.599469
\(520\) 0 0
\(521\) −0.443651 −0.0194367 −0.00971835 0.999953i \(-0.503093\pi\)
−0.00971835 + 0.999953i \(0.503093\pi\)
\(522\) 0 0
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 0 0
\(525\) −1.82843 −0.0797991
\(526\) 0 0
\(527\) −25.3259 −1.10321
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 6.89949 0.299413
\(532\) 0 0
\(533\) −35.5563 −1.54012
\(534\) 0 0
\(535\) −4.41421 −0.190843
\(536\) 0 0
\(537\) 7.65685 0.330418
\(538\) 0 0
\(539\) −9.45584 −0.407292
\(540\) 0 0
\(541\) 34.1421 1.46789 0.733943 0.679212i \(-0.237678\pi\)
0.733943 + 0.679212i \(0.237678\pi\)
\(542\) 0 0
\(543\) −6.82843 −0.293036
\(544\) 0 0
\(545\) −14.2426 −0.610088
\(546\) 0 0
\(547\) 20.9706 0.896637 0.448318 0.893874i \(-0.352023\pi\)
0.448318 + 0.893874i \(0.352023\pi\)
\(548\) 0 0
\(549\) 5.41421 0.231073
\(550\) 0 0
\(551\) −22.3848 −0.953624
\(552\) 0 0
\(553\) −28.0000 −1.19068
\(554\) 0 0
\(555\) −3.00000 −0.127343
\(556\) 0 0
\(557\) 12.8995 0.546569 0.273285 0.961933i \(-0.411890\pi\)
0.273285 + 0.961933i \(0.411890\pi\)
\(558\) 0 0
\(559\) −20.4853 −0.866435
\(560\) 0 0
\(561\) −15.6985 −0.662791
\(562\) 0 0
\(563\) −1.10051 −0.0463808 −0.0231904 0.999731i \(-0.507382\pi\)
−0.0231904 + 0.999731i \(0.507382\pi\)
\(564\) 0 0
\(565\) 10.4142 0.438129
\(566\) 0 0
\(567\) −1.82843 −0.0767867
\(568\) 0 0
\(569\) −16.1421 −0.676714 −0.338357 0.941018i \(-0.609871\pi\)
−0.338357 + 0.941018i \(0.609871\pi\)
\(570\) 0 0
\(571\) −3.27208 −0.136932 −0.0684661 0.997653i \(-0.521810\pi\)
−0.0684661 + 0.997653i \(0.521810\pi\)
\(572\) 0 0
\(573\) −10.2426 −0.427892
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −36.9706 −1.53910 −0.769552 0.638584i \(-0.779521\pi\)
−0.769552 + 0.638584i \(0.779521\pi\)
\(578\) 0 0
\(579\) −8.48528 −0.352636
\(580\) 0 0
\(581\) 25.7279 1.06737
\(582\) 0 0
\(583\) 18.7279 0.775631
\(584\) 0 0
\(585\) −3.41421 −0.141160
\(586\) 0 0
\(587\) −40.6274 −1.67687 −0.838436 0.544999i \(-0.816530\pi\)
−0.838436 + 0.544999i \(0.816530\pi\)
\(588\) 0 0
\(589\) 26.0416 1.07303
\(590\) 0 0
\(591\) 16.8284 0.692229
\(592\) 0 0
\(593\) −2.44365 −0.100349 −0.0501744 0.998740i \(-0.515978\pi\)
−0.0501744 + 0.998740i \(0.515978\pi\)
\(594\) 0 0
\(595\) −11.1005 −0.455076
\(596\) 0 0
\(597\) −10.4853 −0.429134
\(598\) 0 0
\(599\) −33.1716 −1.35535 −0.677677 0.735360i \(-0.737013\pi\)
−0.677677 + 0.735360i \(0.737013\pi\)
\(600\) 0 0
\(601\) −18.3137 −0.747032 −0.373516 0.927624i \(-0.621848\pi\)
−0.373516 + 0.927624i \(0.621848\pi\)
\(602\) 0 0
\(603\) 11.4853 0.467717
\(604\) 0 0
\(605\) 4.31371 0.175377
\(606\) 0 0
\(607\) 38.3848 1.55799 0.778995 0.627030i \(-0.215730\pi\)
0.778995 + 0.627030i \(0.215730\pi\)
\(608\) 0 0
\(609\) 6.55635 0.265677
\(610\) 0 0
\(611\) 29.3137 1.18591
\(612\) 0 0
\(613\) −32.6274 −1.31781 −0.658904 0.752227i \(-0.728980\pi\)
−0.658904 + 0.752227i \(0.728980\pi\)
\(614\) 0 0
\(615\) 10.4142 0.419942
\(616\) 0 0
\(617\) 4.89949 0.197246 0.0986231 0.995125i \(-0.468556\pi\)
0.0986231 + 0.995125i \(0.468556\pi\)
\(618\) 0 0
\(619\) 48.2843 1.94071 0.970354 0.241687i \(-0.0777006\pi\)
0.970354 + 0.241687i \(0.0777006\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −18.5442 −0.742956
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 16.1421 0.644655
\(628\) 0 0
\(629\) −18.2132 −0.726208
\(630\) 0 0
\(631\) −33.2132 −1.32220 −0.661098 0.750299i \(-0.729909\pi\)
−0.661098 + 0.750299i \(0.729909\pi\)
\(632\) 0 0
\(633\) −0.656854 −0.0261076
\(634\) 0 0
\(635\) −0.242641 −0.00962890
\(636\) 0 0
\(637\) −12.4853 −0.494685
\(638\) 0 0
\(639\) −8.89949 −0.352059
\(640\) 0 0
\(641\) 4.92893 0.194681 0.0973406 0.995251i \(-0.468966\pi\)
0.0973406 + 0.995251i \(0.468966\pi\)
\(642\) 0 0
\(643\) −33.0000 −1.30139 −0.650696 0.759338i \(-0.725523\pi\)
−0.650696 + 0.759338i \(0.725523\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) −21.2132 −0.833977 −0.416989 0.908912i \(-0.636914\pi\)
−0.416989 + 0.908912i \(0.636914\pi\)
\(648\) 0 0
\(649\) 17.8406 0.700306
\(650\) 0 0
\(651\) −7.62742 −0.298942
\(652\) 0 0
\(653\) −2.58579 −0.101190 −0.0505948 0.998719i \(-0.516112\pi\)
−0.0505948 + 0.998719i \(0.516112\pi\)
\(654\) 0 0
\(655\) −13.6569 −0.533617
\(656\) 0 0
\(657\) −10.2426 −0.399603
\(658\) 0 0
\(659\) −2.38478 −0.0928977 −0.0464488 0.998921i \(-0.514790\pi\)
−0.0464488 + 0.998921i \(0.514790\pi\)
\(660\) 0 0
\(661\) 11.4558 0.445581 0.222790 0.974866i \(-0.428483\pi\)
0.222790 + 0.974866i \(0.428483\pi\)
\(662\) 0 0
\(663\) −20.7279 −0.805006
\(664\) 0 0
\(665\) 11.4142 0.442624
\(666\) 0 0
\(667\) −3.58579 −0.138842
\(668\) 0 0
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 14.0000 0.540464
\(672\) 0 0
\(673\) 41.6985 1.60736 0.803679 0.595063i \(-0.202873\pi\)
0.803679 + 0.595063i \(0.202873\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −19.2426 −0.739555 −0.369777 0.929120i \(-0.620566\pi\)
−0.369777 + 0.929120i \(0.620566\pi\)
\(678\) 0 0
\(679\) 2.14214 0.0822076
\(680\) 0 0
\(681\) −17.6569 −0.676612
\(682\) 0 0
\(683\) 32.1838 1.23148 0.615739 0.787950i \(-0.288858\pi\)
0.615739 + 0.787950i \(0.288858\pi\)
\(684\) 0 0
\(685\) −2.82843 −0.108069
\(686\) 0 0
\(687\) 23.7990 0.907988
\(688\) 0 0
\(689\) 24.7279 0.942059
\(690\) 0 0
\(691\) 15.6569 0.595615 0.297807 0.954626i \(-0.403745\pi\)
0.297807 + 0.954626i \(0.403745\pi\)
\(692\) 0 0
\(693\) −4.72792 −0.179599
\(694\) 0 0
\(695\) −11.0000 −0.417254
\(696\) 0 0
\(697\) 63.2254 2.39483
\(698\) 0 0
\(699\) 9.65685 0.365256
\(700\) 0 0
\(701\) −36.3848 −1.37423 −0.687117 0.726547i \(-0.741124\pi\)
−0.687117 + 0.726547i \(0.741124\pi\)
\(702\) 0 0
\(703\) 18.7279 0.706337
\(704\) 0 0
\(705\) −8.58579 −0.323359
\(706\) 0 0
\(707\) −22.9584 −0.863438
\(708\) 0 0
\(709\) 16.7279 0.628230 0.314115 0.949385i \(-0.398292\pi\)
0.314115 + 0.949385i \(0.398292\pi\)
\(710\) 0 0
\(711\) 15.3137 0.574309
\(712\) 0 0
\(713\) 4.17157 0.156227
\(714\) 0 0
\(715\) −8.82843 −0.330164
\(716\) 0 0
\(717\) 1.58579 0.0592223
\(718\) 0 0
\(719\) −36.2132 −1.35052 −0.675262 0.737578i \(-0.735970\pi\)
−0.675262 + 0.737578i \(0.735970\pi\)
\(720\) 0 0
\(721\) 1.51472 0.0564111
\(722\) 0 0
\(723\) 23.4142 0.870784
\(724\) 0 0
\(725\) −3.58579 −0.133173
\(726\) 0 0
\(727\) 23.0000 0.853023 0.426511 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 36.4264 1.34728
\(732\) 0 0
\(733\) −17.4853 −0.645834 −0.322917 0.946427i \(-0.604663\pi\)
−0.322917 + 0.946427i \(0.604663\pi\)
\(734\) 0 0
\(735\) 3.65685 0.134885
\(736\) 0 0
\(737\) 29.6985 1.09396
\(738\) 0 0
\(739\) 53.6274 1.97272 0.986358 0.164613i \(-0.0526376\pi\)
0.986358 + 0.164613i \(0.0526376\pi\)
\(740\) 0 0
\(741\) 21.3137 0.782979
\(742\) 0 0
\(743\) 25.6569 0.941259 0.470629 0.882331i \(-0.344027\pi\)
0.470629 + 0.882331i \(0.344027\pi\)
\(744\) 0 0
\(745\) −9.07107 −0.332338
\(746\) 0 0
\(747\) −14.0711 −0.514833
\(748\) 0 0
\(749\) −8.07107 −0.294910
\(750\) 0 0
\(751\) 9.27208 0.338343 0.169171 0.985587i \(-0.445891\pi\)
0.169171 + 0.985587i \(0.445891\pi\)
\(752\) 0 0
\(753\) 20.6274 0.751705
\(754\) 0 0
\(755\) −9.31371 −0.338961
\(756\) 0 0
\(757\) 27.0000 0.981332 0.490666 0.871348i \(-0.336754\pi\)
0.490666 + 0.871348i \(0.336754\pi\)
\(758\) 0 0
\(759\) 2.58579 0.0938581
\(760\) 0 0
\(761\) −19.9289 −0.722423 −0.361212 0.932484i \(-0.617637\pi\)
−0.361212 + 0.932484i \(0.617637\pi\)
\(762\) 0 0
\(763\) −26.0416 −0.942770
\(764\) 0 0
\(765\) 6.07107 0.219500
\(766\) 0 0
\(767\) 23.5563 0.850570
\(768\) 0 0
\(769\) 20.0416 0.722720 0.361360 0.932426i \(-0.382313\pi\)
0.361360 + 0.932426i \(0.382313\pi\)
\(770\) 0 0
\(771\) 30.0416 1.08192
\(772\) 0 0
\(773\) −3.17157 −0.114074 −0.0570368 0.998372i \(-0.518165\pi\)
−0.0570368 + 0.998372i \(0.518165\pi\)
\(774\) 0 0
\(775\) 4.17157 0.149847
\(776\) 0 0
\(777\) −5.48528 −0.196783
\(778\) 0 0
\(779\) −65.0122 −2.32930
\(780\) 0 0
\(781\) −23.0122 −0.823441
\(782\) 0 0
\(783\) −3.58579 −0.128146
\(784\) 0 0
\(785\) 1.48528 0.0530120
\(786\) 0 0
\(787\) −3.62742 −0.129303 −0.0646517 0.997908i \(-0.520594\pi\)
−0.0646517 + 0.997908i \(0.520594\pi\)
\(788\) 0 0
\(789\) −3.72792 −0.132718
\(790\) 0 0
\(791\) 19.0416 0.677042
\(792\) 0 0
\(793\) 18.4853 0.656432
\(794\) 0 0
\(795\) −7.24264 −0.256870
\(796\) 0 0
\(797\) 16.7574 0.593576 0.296788 0.954943i \(-0.404085\pi\)
0.296788 + 0.954943i \(0.404085\pi\)
\(798\) 0 0
\(799\) −52.1249 −1.84405
\(800\) 0 0
\(801\) 10.1421 0.358355
\(802\) 0 0
\(803\) −26.4853 −0.934645
\(804\) 0 0
\(805\) 1.82843 0.0644436
\(806\) 0 0
\(807\) −26.5563 −0.934828
\(808\) 0 0
\(809\) −5.87006 −0.206380 −0.103190 0.994662i \(-0.532905\pi\)
−0.103190 + 0.994662i \(0.532905\pi\)
\(810\) 0 0
\(811\) −39.2843 −1.37946 −0.689729 0.724068i \(-0.742270\pi\)
−0.689729 + 0.724068i \(0.742270\pi\)
\(812\) 0 0
\(813\) 6.51472 0.228481
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) 0 0
\(817\) −37.4558 −1.31041
\(818\) 0 0
\(819\) −6.24264 −0.218136
\(820\) 0 0
\(821\) −43.7990 −1.52860 −0.764298 0.644864i \(-0.776914\pi\)
−0.764298 + 0.644864i \(0.776914\pi\)
\(822\) 0 0
\(823\) −45.6569 −1.59150 −0.795749 0.605627i \(-0.792923\pi\)
−0.795749 + 0.605627i \(0.792923\pi\)
\(824\) 0 0
\(825\) 2.58579 0.0900255
\(826\) 0 0
\(827\) 50.0122 1.73909 0.869547 0.493850i \(-0.164411\pi\)
0.869547 + 0.493850i \(0.164411\pi\)
\(828\) 0 0
\(829\) 23.4853 0.815678 0.407839 0.913054i \(-0.366283\pi\)
0.407839 + 0.913054i \(0.366283\pi\)
\(830\) 0 0
\(831\) 25.3137 0.878123
\(832\) 0 0
\(833\) 22.2010 0.769219
\(834\) 0 0
\(835\) −6.72792 −0.232829
\(836\) 0 0
\(837\) 4.17157 0.144191
\(838\) 0 0
\(839\) 4.62742 0.159756 0.0798781 0.996805i \(-0.474547\pi\)
0.0798781 + 0.996805i \(0.474547\pi\)
\(840\) 0 0
\(841\) −16.1421 −0.556625
\(842\) 0 0
\(843\) 17.4142 0.599777
\(844\) 0 0
\(845\) 1.34315 0.0462056
\(846\) 0 0
\(847\) 7.88730 0.271011
\(848\) 0 0
\(849\) 28.6569 0.983501
\(850\) 0 0
\(851\) 3.00000 0.102839
\(852\) 0 0
\(853\) −31.4558 −1.07703 −0.538514 0.842617i \(-0.681014\pi\)
−0.538514 + 0.842617i \(0.681014\pi\)
\(854\) 0 0
\(855\) −6.24264 −0.213494
\(856\) 0 0
\(857\) 21.5980 0.737773 0.368886 0.929474i \(-0.379739\pi\)
0.368886 + 0.929474i \(0.379739\pi\)
\(858\) 0 0
\(859\) 57.0000 1.94481 0.972407 0.233289i \(-0.0749488\pi\)
0.972407 + 0.233289i \(0.0749488\pi\)
\(860\) 0 0
\(861\) 19.0416 0.648937
\(862\) 0 0
\(863\) −11.1716 −0.380285 −0.190142 0.981757i \(-0.560895\pi\)
−0.190142 + 0.981757i \(0.560895\pi\)
\(864\) 0 0
\(865\) −13.6569 −0.464347
\(866\) 0 0
\(867\) 19.8579 0.674408
\(868\) 0 0
\(869\) 39.5980 1.34327
\(870\) 0 0
\(871\) 39.2132 1.32869
\(872\) 0 0
\(873\) −1.17157 −0.0396517
\(874\) 0 0
\(875\) 1.82843 0.0618121
\(876\) 0 0
\(877\) −1.51472 −0.0511484 −0.0255742 0.999673i \(-0.508141\pi\)
−0.0255742 + 0.999673i \(0.508141\pi\)
\(878\) 0 0
\(879\) −3.10051 −0.104577
\(880\) 0 0
\(881\) 19.4142 0.654081 0.327041 0.945010i \(-0.393949\pi\)
0.327041 + 0.945010i \(0.393949\pi\)
\(882\) 0 0
\(883\) 1.89949 0.0639231 0.0319615 0.999489i \(-0.489825\pi\)
0.0319615 + 0.999489i \(0.489825\pi\)
\(884\) 0 0
\(885\) −6.89949 −0.231924
\(886\) 0 0
\(887\) −21.7990 −0.731938 −0.365969 0.930627i \(-0.619262\pi\)
−0.365969 + 0.930627i \(0.619262\pi\)
\(888\) 0 0
\(889\) −0.443651 −0.0148796
\(890\) 0 0
\(891\) 2.58579 0.0866271
\(892\) 0 0
\(893\) 53.5980 1.79359
\(894\) 0 0
\(895\) −7.65685 −0.255940
\(896\) 0 0
\(897\) 3.41421 0.113997
\(898\) 0 0
\(899\) −14.9584 −0.498890
\(900\) 0 0
\(901\) −43.9706 −1.46487
\(902\) 0 0
\(903\) 10.9706 0.365077
\(904\) 0 0
\(905\) 6.82843 0.226985
\(906\) 0 0
\(907\) −31.1421 −1.03406 −0.517029 0.855968i \(-0.672962\pi\)
−0.517029 + 0.855968i \(0.672962\pi\)
\(908\) 0 0
\(909\) 12.5563 0.416468
\(910\) 0 0
\(911\) 5.65685 0.187420 0.0937100 0.995600i \(-0.470127\pi\)
0.0937100 + 0.995600i \(0.470127\pi\)
\(912\) 0 0
\(913\) −36.3848 −1.20416
\(914\) 0 0
\(915\) −5.41421 −0.178988
\(916\) 0 0
\(917\) −24.9706 −0.824601
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −27.2132 −0.896706
\(922\) 0 0
\(923\) −30.3848 −1.00013
\(924\) 0 0
\(925\) 3.00000 0.0986394
\(926\) 0 0
\(927\) −0.828427 −0.0272091
\(928\) 0 0
\(929\) 41.8701 1.37371 0.686856 0.726794i \(-0.258990\pi\)
0.686856 + 0.726794i \(0.258990\pi\)
\(930\) 0 0
\(931\) −22.8284 −0.748171
\(932\) 0 0
\(933\) −17.6569 −0.578059
\(934\) 0 0
\(935\) 15.6985 0.513395
\(936\) 0 0
\(937\) −59.7990 −1.95355 −0.976774 0.214272i \(-0.931262\pi\)
−0.976774 + 0.214272i \(0.931262\pi\)
\(938\) 0 0
\(939\) 25.9706 0.847517
\(940\) 0 0
\(941\) −7.41421 −0.241696 −0.120848 0.992671i \(-0.538561\pi\)
−0.120848 + 0.992671i \(0.538561\pi\)
\(942\) 0 0
\(943\) −10.4142 −0.339133
\(944\) 0 0
\(945\) 1.82843 0.0594787
\(946\) 0 0
\(947\) 3.79899 0.123451 0.0617253 0.998093i \(-0.480340\pi\)
0.0617253 + 0.998093i \(0.480340\pi\)
\(948\) 0 0
\(949\) −34.9706 −1.13519
\(950\) 0 0
\(951\) −29.5563 −0.958430
\(952\) 0 0
\(953\) 7.37258 0.238821 0.119411 0.992845i \(-0.461899\pi\)
0.119411 + 0.992845i \(0.461899\pi\)
\(954\) 0 0
\(955\) 10.2426 0.331444
\(956\) 0 0
\(957\) −9.27208 −0.299724
\(958\) 0 0
\(959\) −5.17157 −0.166999
\(960\) 0 0
\(961\) −13.5980 −0.438645
\(962\) 0 0
\(963\) 4.41421 0.142246
\(964\) 0 0
\(965\) 8.48528 0.273151
\(966\) 0 0
\(967\) −0.443651 −0.0142668 −0.00713342 0.999975i \(-0.502271\pi\)
−0.00713342 + 0.999975i \(0.502271\pi\)
\(968\) 0 0
\(969\) −37.8995 −1.21751
\(970\) 0 0
\(971\) −30.4264 −0.976430 −0.488215 0.872723i \(-0.662352\pi\)
−0.488215 + 0.872723i \(0.662352\pi\)
\(972\) 0 0
\(973\) −20.1127 −0.644784
\(974\) 0 0
\(975\) 3.41421 0.109342
\(976\) 0 0
\(977\) −15.5858 −0.498633 −0.249317 0.968422i \(-0.580206\pi\)
−0.249317 + 0.968422i \(0.580206\pi\)
\(978\) 0 0
\(979\) 26.2254 0.838167
\(980\) 0 0
\(981\) 14.2426 0.454733
\(982\) 0 0
\(983\) −24.0122 −0.765870 −0.382935 0.923775i \(-0.625087\pi\)
−0.382935 + 0.923775i \(0.625087\pi\)
\(984\) 0 0
\(985\) −16.8284 −0.536198
\(986\) 0 0
\(987\) −15.6985 −0.499688
\(988\) 0 0
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) −1.34315 −0.0426664 −0.0213332 0.999772i \(-0.506791\pi\)
−0.0213332 + 0.999772i \(0.506791\pi\)
\(992\) 0 0
\(993\) −1.00000 −0.0317340
\(994\) 0 0
\(995\) 10.4853 0.332406
\(996\) 0 0
\(997\) −34.1421 −1.08129 −0.540646 0.841250i \(-0.681820\pi\)
−0.540646 + 0.841250i \(0.681820\pi\)
\(998\) 0 0
\(999\) 3.00000 0.0949158
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 5520.2.a.bm.1.1 2
4.3 odd 2 345.2.a.h.1.1 2
12.11 even 2 1035.2.a.j.1.2 2
20.3 even 4 1725.2.b.s.1174.3 4
20.7 even 4 1725.2.b.s.1174.2 4
20.19 odd 2 1725.2.a.z.1.2 2
60.59 even 2 5175.2.a.bj.1.1 2
92.91 even 2 7935.2.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.a.h.1.1 2 4.3 odd 2
1035.2.a.j.1.2 2 12.11 even 2
1725.2.a.z.1.2 2 20.19 odd 2
1725.2.b.s.1174.2 4 20.7 even 4
1725.2.b.s.1174.3 4 20.3 even 4
5175.2.a.bj.1.1 2 60.59 even 2
5520.2.a.bm.1.1 2 1.1 even 1 trivial
7935.2.a.q.1.1 2 92.91 even 2