Properties

Label 4560.2.a.bt.1.3
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.12489\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +3.60975 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +3.60975 q^{7} +1.00000 q^{9} +1.60975 q^{11} +0.640023 q^{13} -1.00000 q^{15} +5.28005 q^{17} -1.00000 q^{19} +3.60975 q^{21} +8.24977 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.88979 q^{29} +7.28005 q^{31} +1.60975 q^{33} -3.60975 q^{35} -9.92007 q^{37} +0.640023 q^{39} +4.57947 q^{41} -7.92007 q^{43} -1.00000 q^{45} -0.249771 q^{47} +6.03028 q^{49} +5.28005 q^{51} +1.75023 q^{53} -1.60975 q^{55} -1.00000 q^{57} -12.3103 q^{59} +11.5298 q^{61} +3.60975 q^{63} -0.640023 q^{65} -12.4995 q^{67} +8.24977 q^{69} -13.7796 q^{71} +3.93945 q^{73} +1.00000 q^{75} +5.81078 q^{77} +3.03028 q^{79} +1.00000 q^{81} +10.4995 q^{83} -5.28005 q^{85} -4.88979 q^{87} +15.9201 q^{89} +2.31032 q^{91} +7.28005 q^{93} +1.00000 q^{95} -9.92007 q^{97} +1.60975 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} - 3 q^{15} - 3 q^{19} + 2 q^{21} + 8 q^{23} + 3 q^{25} + 3 q^{27} + 10 q^{29} + 6 q^{31} - 4 q^{33} - 2 q^{35} - 6 q^{37} - 6 q^{39} + 4 q^{41} - 3 q^{45} + 16 q^{47} + 19 q^{49} + 22 q^{53} + 4 q^{55} - 3 q^{57} - 22 q^{59} + 2 q^{61} + 2 q^{63} + 6 q^{65} - 4 q^{67} + 8 q^{69} + 8 q^{71} + 10 q^{73} + 3 q^{75} + 36 q^{77} + 10 q^{79} + 3 q^{81} - 2 q^{83} + 10 q^{87} + 24 q^{89} - 8 q^{91} + 6 q^{93} + 3 q^{95} - 6 q^{97} - 4 q^{99} + O(q^{100}) \)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.60975 1.36436 0.682178 0.731186i \(-0.261033\pi\)
0.682178 + 0.731186i \(0.261033\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.60975 0.485357 0.242679 0.970107i \(-0.421974\pi\)
0.242679 + 0.970107i \(0.421974\pi\)
\(12\) 0 0
\(13\) 0.640023 0.177511 0.0887553 0.996053i \(-0.471711\pi\)
0.0887553 + 0.996053i \(0.471711\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 5.28005 1.28060 0.640300 0.768125i \(-0.278810\pi\)
0.640300 + 0.768125i \(0.278810\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 3.60975 0.787711
\(22\) 0 0
\(23\) 8.24977 1.72020 0.860098 0.510129i \(-0.170402\pi\)
0.860098 + 0.510129i \(0.170402\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.88979 −0.908012 −0.454006 0.890999i \(-0.650006\pi\)
−0.454006 + 0.890999i \(0.650006\pi\)
\(30\) 0 0
\(31\) 7.28005 1.30754 0.653768 0.756695i \(-0.273187\pi\)
0.653768 + 0.756695i \(0.273187\pi\)
\(32\) 0 0
\(33\) 1.60975 0.280221
\(34\) 0 0
\(35\) −3.60975 −0.610159
\(36\) 0 0
\(37\) −9.92007 −1.63085 −0.815425 0.578863i \(-0.803497\pi\)
−0.815425 + 0.578863i \(0.803497\pi\)
\(38\) 0 0
\(39\) 0.640023 0.102486
\(40\) 0 0
\(41\) 4.57947 0.715193 0.357597 0.933876i \(-0.383596\pi\)
0.357597 + 0.933876i \(0.383596\pi\)
\(42\) 0 0
\(43\) −7.92007 −1.20780 −0.603900 0.797060i \(-0.706387\pi\)
−0.603900 + 0.797060i \(0.706387\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −0.249771 −0.0364328 −0.0182164 0.999834i \(-0.505799\pi\)
−0.0182164 + 0.999834i \(0.505799\pi\)
\(48\) 0 0
\(49\) 6.03028 0.861468
\(50\) 0 0
\(51\) 5.28005 0.739354
\(52\) 0 0
\(53\) 1.75023 0.240412 0.120206 0.992749i \(-0.461644\pi\)
0.120206 + 0.992749i \(0.461644\pi\)
\(54\) 0 0
\(55\) −1.60975 −0.217058
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −12.3103 −1.60267 −0.801334 0.598218i \(-0.795876\pi\)
−0.801334 + 0.598218i \(0.795876\pi\)
\(60\) 0 0
\(61\) 11.5298 1.47624 0.738121 0.674668i \(-0.235713\pi\)
0.738121 + 0.674668i \(0.235713\pi\)
\(62\) 0 0
\(63\) 3.60975 0.454785
\(64\) 0 0
\(65\) −0.640023 −0.0793851
\(66\) 0 0
\(67\) −12.4995 −1.52706 −0.763531 0.645771i \(-0.776536\pi\)
−0.763531 + 0.645771i \(0.776536\pi\)
\(68\) 0 0
\(69\) 8.24977 0.993156
\(70\) 0 0
\(71\) −13.7796 −1.63534 −0.817668 0.575690i \(-0.804734\pi\)
−0.817668 + 0.575690i \(0.804734\pi\)
\(72\) 0 0
\(73\) 3.93945 0.461077 0.230539 0.973063i \(-0.425951\pi\)
0.230539 + 0.973063i \(0.425951\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 5.81078 0.662200
\(78\) 0 0
\(79\) 3.03028 0.340933 0.170466 0.985363i \(-0.445473\pi\)
0.170466 + 0.985363i \(0.445473\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.4995 1.15247 0.576237 0.817282i \(-0.304520\pi\)
0.576237 + 0.817282i \(0.304520\pi\)
\(84\) 0 0
\(85\) −5.28005 −0.572701
\(86\) 0 0
\(87\) −4.88979 −0.524241
\(88\) 0 0
\(89\) 15.9201 1.68752 0.843762 0.536717i \(-0.180336\pi\)
0.843762 + 0.536717i \(0.180336\pi\)
\(90\) 0 0
\(91\) 2.31032 0.242188
\(92\) 0 0
\(93\) 7.28005 0.754906
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −9.92007 −1.00723 −0.503615 0.863928i \(-0.667997\pi\)
−0.503615 + 0.863928i \(0.667997\pi\)
\(98\) 0 0
\(99\) 1.60975 0.161786
\(100\) 0 0
\(101\) 17.2195 1.71340 0.856702 0.515812i \(-0.172510\pi\)
0.856702 + 0.515812i \(0.172510\pi\)
\(102\) 0 0
\(103\) −12.4995 −1.23162 −0.615808 0.787896i \(-0.711170\pi\)
−0.615808 + 0.787896i \(0.711170\pi\)
\(104\) 0 0
\(105\) −3.60975 −0.352275
\(106\) 0 0
\(107\) 10.5601 1.02088 0.510441 0.859913i \(-0.329482\pi\)
0.510441 + 0.859913i \(0.329482\pi\)
\(108\) 0 0
\(109\) 10.3103 0.987550 0.493775 0.869590i \(-0.335617\pi\)
0.493775 + 0.869590i \(0.335617\pi\)
\(110\) 0 0
\(111\) −9.92007 −0.941571
\(112\) 0 0
\(113\) 0.470182 0.0442310 0.0221155 0.999755i \(-0.492960\pi\)
0.0221155 + 0.999755i \(0.492960\pi\)
\(114\) 0 0
\(115\) −8.24977 −0.769295
\(116\) 0 0
\(117\) 0.640023 0.0591702
\(118\) 0 0
\(119\) 19.0596 1.74719
\(120\) 0 0
\(121\) −8.40871 −0.764428
\(122\) 0 0
\(123\) 4.57947 0.412917
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 12.4995 1.10915 0.554577 0.832132i \(-0.312880\pi\)
0.554577 + 0.832132i \(0.312880\pi\)
\(128\) 0 0
\(129\) −7.92007 −0.697323
\(130\) 0 0
\(131\) −4.32970 −0.378288 −0.189144 0.981949i \(-0.560571\pi\)
−0.189144 + 0.981949i \(0.560571\pi\)
\(132\) 0 0
\(133\) −3.60975 −0.313005
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 5.03028 0.429765 0.214883 0.976640i \(-0.431063\pi\)
0.214883 + 0.976640i \(0.431063\pi\)
\(138\) 0 0
\(139\) 9.77959 0.829494 0.414747 0.909937i \(-0.363870\pi\)
0.414747 + 0.909937i \(0.363870\pi\)
\(140\) 0 0
\(141\) −0.249771 −0.0210345
\(142\) 0 0
\(143\) 1.03028 0.0861560
\(144\) 0 0
\(145\) 4.88979 0.406075
\(146\) 0 0
\(147\) 6.03028 0.497369
\(148\) 0 0
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) −1.34060 −0.109096 −0.0545482 0.998511i \(-0.517372\pi\)
−0.0545482 + 0.998511i \(0.517372\pi\)
\(152\) 0 0
\(153\) 5.28005 0.426866
\(154\) 0 0
\(155\) −7.28005 −0.584747
\(156\) 0 0
\(157\) −10.1892 −0.813188 −0.406594 0.913609i \(-0.633284\pi\)
−0.406594 + 0.913609i \(0.633284\pi\)
\(158\) 0 0
\(159\) 1.75023 0.138802
\(160\) 0 0
\(161\) 29.7796 2.34696
\(162\) 0 0
\(163\) −15.7990 −1.23747 −0.618735 0.785600i \(-0.712355\pi\)
−0.618735 + 0.785600i \(0.712355\pi\)
\(164\) 0 0
\(165\) −1.60975 −0.125319
\(166\) 0 0
\(167\) −9.52982 −0.737439 −0.368720 0.929541i \(-0.620204\pi\)
−0.368720 + 0.929541i \(0.620204\pi\)
\(168\) 0 0
\(169\) −12.5904 −0.968490
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 13.4693 1.02405 0.512025 0.858971i \(-0.328896\pi\)
0.512025 + 0.858971i \(0.328896\pi\)
\(174\) 0 0
\(175\) 3.60975 0.272871
\(176\) 0 0
\(177\) −12.3103 −0.925301
\(178\) 0 0
\(179\) −20.1892 −1.50901 −0.754507 0.656293i \(-0.772124\pi\)
−0.754507 + 0.656293i \(0.772124\pi\)
\(180\) 0 0
\(181\) −7.75023 −0.576070 −0.288035 0.957620i \(-0.593002\pi\)
−0.288035 + 0.957620i \(0.593002\pi\)
\(182\) 0 0
\(183\) 11.5298 0.852309
\(184\) 0 0
\(185\) 9.92007 0.729338
\(186\) 0 0
\(187\) 8.49954 0.621548
\(188\) 0 0
\(189\) 3.60975 0.262570
\(190\) 0 0
\(191\) 2.39025 0.172953 0.0864763 0.996254i \(-0.472439\pi\)
0.0864763 + 0.996254i \(0.472439\pi\)
\(192\) 0 0
\(193\) −0.640023 −0.0460699 −0.0230349 0.999735i \(-0.507333\pi\)
−0.0230349 + 0.999735i \(0.507333\pi\)
\(194\) 0 0
\(195\) −0.640023 −0.0458330
\(196\) 0 0
\(197\) −14.7493 −1.05085 −0.525423 0.850841i \(-0.676093\pi\)
−0.525423 + 0.850841i \(0.676093\pi\)
\(198\) 0 0
\(199\) −19.8401 −1.40643 −0.703215 0.710977i \(-0.748253\pi\)
−0.703215 + 0.710977i \(0.748253\pi\)
\(200\) 0 0
\(201\) −12.4995 −0.881650
\(202\) 0 0
\(203\) −17.6509 −1.23885
\(204\) 0 0
\(205\) −4.57947 −0.319844
\(206\) 0 0
\(207\) 8.24977 0.573399
\(208\) 0 0
\(209\) −1.60975 −0.111349
\(210\) 0 0
\(211\) 2.06055 0.141854 0.0709271 0.997481i \(-0.477404\pi\)
0.0709271 + 0.997481i \(0.477404\pi\)
\(212\) 0 0
\(213\) −13.7796 −0.944162
\(214\) 0 0
\(215\) 7.92007 0.540144
\(216\) 0 0
\(217\) 26.2791 1.78394
\(218\) 0 0
\(219\) 3.93945 0.266203
\(220\) 0 0
\(221\) 3.37935 0.227320
\(222\) 0 0
\(223\) 26.5601 1.77860 0.889298 0.457329i \(-0.151194\pi\)
0.889298 + 0.457329i \(0.151194\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 5.03028 0.333871 0.166936 0.985968i \(-0.446613\pi\)
0.166936 + 0.985968i \(0.446613\pi\)
\(228\) 0 0
\(229\) 23.5298 1.55489 0.777447 0.628948i \(-0.216514\pi\)
0.777447 + 0.628948i \(0.216514\pi\)
\(230\) 0 0
\(231\) 5.81078 0.382321
\(232\) 0 0
\(233\) 15.9688 1.04615 0.523076 0.852286i \(-0.324785\pi\)
0.523076 + 0.852286i \(0.324785\pi\)
\(234\) 0 0
\(235\) 0.249771 0.0162933
\(236\) 0 0
\(237\) 3.03028 0.196838
\(238\) 0 0
\(239\) −15.6703 −1.01363 −0.506814 0.862056i \(-0.669177\pi\)
−0.506814 + 0.862056i \(0.669177\pi\)
\(240\) 0 0
\(241\) −28.4390 −1.83192 −0.915958 0.401274i \(-0.868568\pi\)
−0.915958 + 0.401274i \(0.868568\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −6.03028 −0.385260
\(246\) 0 0
\(247\) −0.640023 −0.0407237
\(248\) 0 0
\(249\) 10.4995 0.665381
\(250\) 0 0
\(251\) 24.8292 1.56721 0.783604 0.621261i \(-0.213379\pi\)
0.783604 + 0.621261i \(0.213379\pi\)
\(252\) 0 0
\(253\) 13.2800 0.834909
\(254\) 0 0
\(255\) −5.28005 −0.330649
\(256\) 0 0
\(257\) −9.46927 −0.590677 −0.295338 0.955393i \(-0.595432\pi\)
−0.295338 + 0.955393i \(0.595432\pi\)
\(258\) 0 0
\(259\) −35.8089 −2.22506
\(260\) 0 0
\(261\) −4.88979 −0.302671
\(262\) 0 0
\(263\) 13.5298 0.834284 0.417142 0.908841i \(-0.363032\pi\)
0.417142 + 0.908841i \(0.363032\pi\)
\(264\) 0 0
\(265\) −1.75023 −0.107516
\(266\) 0 0
\(267\) 15.9201 0.974292
\(268\) 0 0
\(269\) 25.8889 1.57847 0.789236 0.614090i \(-0.210477\pi\)
0.789236 + 0.614090i \(0.210477\pi\)
\(270\) 0 0
\(271\) 29.6585 1.80162 0.900812 0.434209i \(-0.142972\pi\)
0.900812 + 0.434209i \(0.142972\pi\)
\(272\) 0 0
\(273\) 2.31032 0.139827
\(274\) 0 0
\(275\) 1.60975 0.0970714
\(276\) 0 0
\(277\) 1.03028 0.0619033 0.0309516 0.999521i \(-0.490146\pi\)
0.0309516 + 0.999521i \(0.490146\pi\)
\(278\) 0 0
\(279\) 7.28005 0.435845
\(280\) 0 0
\(281\) 12.0799 0.720628 0.360314 0.932831i \(-0.382670\pi\)
0.360314 + 0.932831i \(0.382670\pi\)
\(282\) 0 0
\(283\) 13.2001 0.784666 0.392333 0.919823i \(-0.371668\pi\)
0.392333 + 0.919823i \(0.371668\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 16.5307 0.975778
\(288\) 0 0
\(289\) 10.8789 0.639935
\(290\) 0 0
\(291\) −9.92007 −0.581525
\(292\) 0 0
\(293\) 11.0303 0.644396 0.322198 0.946672i \(-0.395578\pi\)
0.322198 + 0.946672i \(0.395578\pi\)
\(294\) 0 0
\(295\) 12.3103 0.716735
\(296\) 0 0
\(297\) 1.60975 0.0934070
\(298\) 0 0
\(299\) 5.28005 0.305353
\(300\) 0 0
\(301\) −28.5895 −1.64787
\(302\) 0 0
\(303\) 17.2195 0.989234
\(304\) 0 0
\(305\) −11.5298 −0.660195
\(306\) 0 0
\(307\) −17.7796 −1.01473 −0.507367 0.861730i \(-0.669381\pi\)
−0.507367 + 0.861730i \(0.669381\pi\)
\(308\) 0 0
\(309\) −12.4995 −0.711074
\(310\) 0 0
\(311\) −2.88979 −0.163865 −0.0819326 0.996638i \(-0.526109\pi\)
−0.0819326 + 0.996638i \(0.526109\pi\)
\(312\) 0 0
\(313\) −3.40115 −0.192244 −0.0961222 0.995370i \(-0.530644\pi\)
−0.0961222 + 0.995370i \(0.530644\pi\)
\(314\) 0 0
\(315\) −3.60975 −0.203386
\(316\) 0 0
\(317\) 4.18922 0.235290 0.117645 0.993056i \(-0.462466\pi\)
0.117645 + 0.993056i \(0.462466\pi\)
\(318\) 0 0
\(319\) −7.87133 −0.440710
\(320\) 0 0
\(321\) 10.5601 0.589407
\(322\) 0 0
\(323\) −5.28005 −0.293790
\(324\) 0 0
\(325\) 0.640023 0.0355021
\(326\) 0 0
\(327\) 10.3103 0.570162
\(328\) 0 0
\(329\) −0.901610 −0.0497073
\(330\) 0 0
\(331\) −10.3103 −0.566707 −0.283353 0.959016i \(-0.591447\pi\)
−0.283353 + 0.959016i \(0.591447\pi\)
\(332\) 0 0
\(333\) −9.92007 −0.543617
\(334\) 0 0
\(335\) 12.4995 0.682923
\(336\) 0 0
\(337\) −16.6400 −0.906440 −0.453220 0.891399i \(-0.649725\pi\)
−0.453220 + 0.891399i \(0.649725\pi\)
\(338\) 0 0
\(339\) 0.470182 0.0255368
\(340\) 0 0
\(341\) 11.7190 0.634621
\(342\) 0 0
\(343\) −3.50046 −0.189007
\(344\) 0 0
\(345\) −8.24977 −0.444153
\(346\) 0 0
\(347\) 16.0606 0.862176 0.431088 0.902310i \(-0.358130\pi\)
0.431088 + 0.902310i \(0.358130\pi\)
\(348\) 0 0
\(349\) −12.0606 −0.645587 −0.322793 0.946469i \(-0.604622\pi\)
−0.322793 + 0.946469i \(0.604622\pi\)
\(350\) 0 0
\(351\) 0.640023 0.0341619
\(352\) 0 0
\(353\) 13.5298 0.720120 0.360060 0.932929i \(-0.382756\pi\)
0.360060 + 0.932929i \(0.382756\pi\)
\(354\) 0 0
\(355\) 13.7796 0.731345
\(356\) 0 0
\(357\) 19.0596 1.00874
\(358\) 0 0
\(359\) −29.6097 −1.56274 −0.781371 0.624066i \(-0.785479\pi\)
−0.781371 + 0.624066i \(0.785479\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −8.40871 −0.441343
\(364\) 0 0
\(365\) −3.93945 −0.206200
\(366\) 0 0
\(367\) 11.4886 0.599702 0.299851 0.953986i \(-0.403063\pi\)
0.299851 + 0.953986i \(0.403063\pi\)
\(368\) 0 0
\(369\) 4.57947 0.238398
\(370\) 0 0
\(371\) 6.31789 0.328008
\(372\) 0 0
\(373\) 7.20012 0.372808 0.186404 0.982473i \(-0.440317\pi\)
0.186404 + 0.982473i \(0.440317\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −3.12958 −0.161182
\(378\) 0 0
\(379\) −9.68968 −0.497725 −0.248863 0.968539i \(-0.580057\pi\)
−0.248863 + 0.968539i \(0.580057\pi\)
\(380\) 0 0
\(381\) 12.4995 0.640371
\(382\) 0 0
\(383\) −22.9385 −1.17210 −0.586052 0.810273i \(-0.699319\pi\)
−0.586052 + 0.810273i \(0.699319\pi\)
\(384\) 0 0
\(385\) −5.81078 −0.296145
\(386\) 0 0
\(387\) −7.92007 −0.402600
\(388\) 0 0
\(389\) −21.5592 −1.09309 −0.546547 0.837429i \(-0.684058\pi\)
−0.546547 + 0.837429i \(0.684058\pi\)
\(390\) 0 0
\(391\) 43.5592 2.20288
\(392\) 0 0
\(393\) −4.32970 −0.218404
\(394\) 0 0
\(395\) −3.03028 −0.152470
\(396\) 0 0
\(397\) 22.0294 1.10562 0.552811 0.833307i \(-0.313555\pi\)
0.552811 + 0.833307i \(0.313555\pi\)
\(398\) 0 0
\(399\) −3.60975 −0.180713
\(400\) 0 0
\(401\) −17.0790 −0.852885 −0.426443 0.904515i \(-0.640233\pi\)
−0.426443 + 0.904515i \(0.640233\pi\)
\(402\) 0 0
\(403\) 4.65940 0.232101
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −15.9688 −0.791544
\(408\) 0 0
\(409\) −23.6585 −1.16984 −0.584918 0.811092i \(-0.698873\pi\)
−0.584918 + 0.811092i \(0.698873\pi\)
\(410\) 0 0
\(411\) 5.03028 0.248125
\(412\) 0 0
\(413\) −44.4372 −2.18661
\(414\) 0 0
\(415\) −10.4995 −0.515402
\(416\) 0 0
\(417\) 9.77959 0.478909
\(418\) 0 0
\(419\) −22.7687 −1.11232 −0.556162 0.831074i \(-0.687726\pi\)
−0.556162 + 0.831074i \(0.687726\pi\)
\(420\) 0 0
\(421\) −2.47018 −0.120389 −0.0601947 0.998187i \(-0.519172\pi\)
−0.0601947 + 0.998187i \(0.519172\pi\)
\(422\) 0 0
\(423\) −0.249771 −0.0121443
\(424\) 0 0
\(425\) 5.28005 0.256120
\(426\) 0 0
\(427\) 41.6197 2.01412
\(428\) 0 0
\(429\) 1.03028 0.0497422
\(430\) 0 0
\(431\) 9.40115 0.452838 0.226419 0.974030i \(-0.427298\pi\)
0.226419 + 0.974030i \(0.427298\pi\)
\(432\) 0 0
\(433\) 36.4802 1.75312 0.876562 0.481288i \(-0.159831\pi\)
0.876562 + 0.481288i \(0.159831\pi\)
\(434\) 0 0
\(435\) 4.88979 0.234448
\(436\) 0 0
\(437\) −8.24977 −0.394640
\(438\) 0 0
\(439\) 30.5895 1.45995 0.729977 0.683471i \(-0.239531\pi\)
0.729977 + 0.683471i \(0.239531\pi\)
\(440\) 0 0
\(441\) 6.03028 0.287156
\(442\) 0 0
\(443\) −21.9612 −1.04341 −0.521705 0.853126i \(-0.674704\pi\)
−0.521705 + 0.853126i \(0.674704\pi\)
\(444\) 0 0
\(445\) −15.9201 −0.754684
\(446\) 0 0
\(447\) 14.0000 0.662177
\(448\) 0 0
\(449\) 13.3212 0.628667 0.314334 0.949313i \(-0.398219\pi\)
0.314334 + 0.949313i \(0.398219\pi\)
\(450\) 0 0
\(451\) 7.37179 0.347124
\(452\) 0 0
\(453\) −1.34060 −0.0629868
\(454\) 0 0
\(455\) −2.31032 −0.108310
\(456\) 0 0
\(457\) −17.9612 −0.840192 −0.420096 0.907480i \(-0.638004\pi\)
−0.420096 + 0.907480i \(0.638004\pi\)
\(458\) 0 0
\(459\) 5.28005 0.246451
\(460\) 0 0
\(461\) 29.5592 1.37671 0.688354 0.725375i \(-0.258334\pi\)
0.688354 + 0.725375i \(0.258334\pi\)
\(462\) 0 0
\(463\) −11.3288 −0.526493 −0.263247 0.964729i \(-0.584793\pi\)
−0.263247 + 0.964729i \(0.584793\pi\)
\(464\) 0 0
\(465\) −7.28005 −0.337604
\(466\) 0 0
\(467\) 1.84014 0.0851516 0.0425758 0.999093i \(-0.486444\pi\)
0.0425758 + 0.999093i \(0.486444\pi\)
\(468\) 0 0
\(469\) −45.1202 −2.08346
\(470\) 0 0
\(471\) −10.1892 −0.469494
\(472\) 0 0
\(473\) −12.7493 −0.586214
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 1.75023 0.0801375
\(478\) 0 0
\(479\) −7.51044 −0.343161 −0.171580 0.985170i \(-0.554887\pi\)
−0.171580 + 0.985170i \(0.554887\pi\)
\(480\) 0 0
\(481\) −6.34908 −0.289493
\(482\) 0 0
\(483\) 29.7796 1.35502
\(484\) 0 0
\(485\) 9.92007 0.450447
\(486\) 0 0
\(487\) −18.2791 −0.828306 −0.414153 0.910207i \(-0.635922\pi\)
−0.414153 + 0.910207i \(0.635922\pi\)
\(488\) 0 0
\(489\) −15.7990 −0.714454
\(490\) 0 0
\(491\) 35.7678 1.61418 0.807089 0.590430i \(-0.201042\pi\)
0.807089 + 0.590430i \(0.201042\pi\)
\(492\) 0 0
\(493\) −25.8183 −1.16280
\(494\) 0 0
\(495\) −1.60975 −0.0723528
\(496\) 0 0
\(497\) −49.7408 −2.23118
\(498\) 0 0
\(499\) 5.65848 0.253309 0.126654 0.991947i \(-0.459576\pi\)
0.126654 + 0.991947i \(0.459576\pi\)
\(500\) 0 0
\(501\) −9.52982 −0.425761
\(502\) 0 0
\(503\) −38.4078 −1.71252 −0.856260 0.516546i \(-0.827218\pi\)
−0.856260 + 0.516546i \(0.827218\pi\)
\(504\) 0 0
\(505\) −17.2195 −0.766257
\(506\) 0 0
\(507\) −12.5904 −0.559158
\(508\) 0 0
\(509\) −16.1093 −0.714032 −0.357016 0.934098i \(-0.616206\pi\)
−0.357016 + 0.934098i \(0.616206\pi\)
\(510\) 0 0
\(511\) 14.2204 0.629074
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 12.4995 0.550796
\(516\) 0 0
\(517\) −0.402068 −0.0176829
\(518\) 0 0
\(519\) 13.4693 0.591235
\(520\) 0 0
\(521\) −41.9182 −1.83647 −0.918236 0.396034i \(-0.870386\pi\)
−0.918236 + 0.396034i \(0.870386\pi\)
\(522\) 0 0
\(523\) −27.5979 −1.20677 −0.603387 0.797449i \(-0.706182\pi\)
−0.603387 + 0.797449i \(0.706182\pi\)
\(524\) 0 0
\(525\) 3.60975 0.157542
\(526\) 0 0
\(527\) 38.4390 1.67443
\(528\) 0 0
\(529\) 45.0587 1.95907
\(530\) 0 0
\(531\) −12.3103 −0.534223
\(532\) 0 0
\(533\) 2.93097 0.126954
\(534\) 0 0
\(535\) −10.5601 −0.456553
\(536\) 0 0
\(537\) −20.1892 −0.871229
\(538\) 0 0
\(539\) 9.70722 0.418120
\(540\) 0 0
\(541\) 38.9991 1.67670 0.838351 0.545131i \(-0.183520\pi\)
0.838351 + 0.545131i \(0.183520\pi\)
\(542\) 0 0
\(543\) −7.75023 −0.332594
\(544\) 0 0
\(545\) −10.3103 −0.441646
\(546\) 0 0
\(547\) 7.21949 0.308683 0.154342 0.988018i \(-0.450674\pi\)
0.154342 + 0.988018i \(0.450674\pi\)
\(548\) 0 0
\(549\) 11.5298 0.492081
\(550\) 0 0
\(551\) 4.88979 0.208312
\(552\) 0 0
\(553\) 10.9385 0.465154
\(554\) 0 0
\(555\) 9.92007 0.421084
\(556\) 0 0
\(557\) 12.3179 0.521926 0.260963 0.965349i \(-0.415960\pi\)
0.260963 + 0.965349i \(0.415960\pi\)
\(558\) 0 0
\(559\) −5.06903 −0.214397
\(560\) 0 0
\(561\) 8.49954 0.358851
\(562\) 0 0
\(563\) 32.2110 1.35753 0.678766 0.734354i \(-0.262515\pi\)
0.678766 + 0.734354i \(0.262515\pi\)
\(564\) 0 0
\(565\) −0.470182 −0.0197807
\(566\) 0 0
\(567\) 3.60975 0.151595
\(568\) 0 0
\(569\) −27.9201 −1.17047 −0.585235 0.810864i \(-0.698998\pi\)
−0.585235 + 0.810864i \(0.698998\pi\)
\(570\) 0 0
\(571\) −40.9603 −1.71414 −0.857068 0.515203i \(-0.827717\pi\)
−0.857068 + 0.515203i \(0.827717\pi\)
\(572\) 0 0
\(573\) 2.39025 0.0998542
\(574\) 0 0
\(575\) 8.24977 0.344039
\(576\) 0 0
\(577\) −39.6585 −1.65100 −0.825502 0.564399i \(-0.809108\pi\)
−0.825502 + 0.564399i \(0.809108\pi\)
\(578\) 0 0
\(579\) −0.640023 −0.0265985
\(580\) 0 0
\(581\) 37.9007 1.57239
\(582\) 0 0
\(583\) 2.81743 0.116686
\(584\) 0 0
\(585\) −0.640023 −0.0264617
\(586\) 0 0
\(587\) −43.3388 −1.78878 −0.894391 0.447286i \(-0.852391\pi\)
−0.894391 + 0.447286i \(0.852391\pi\)
\(588\) 0 0
\(589\) −7.28005 −0.299969
\(590\) 0 0
\(591\) −14.7493 −0.606706
\(592\) 0 0
\(593\) −32.5895 −1.33829 −0.669144 0.743133i \(-0.733339\pi\)
−0.669144 + 0.743133i \(0.733339\pi\)
\(594\) 0 0
\(595\) −19.0596 −0.781369
\(596\) 0 0
\(597\) −19.8401 −0.812003
\(598\) 0 0
\(599\) 27.1807 1.11057 0.555287 0.831658i \(-0.312608\pi\)
0.555287 + 0.831658i \(0.312608\pi\)
\(600\) 0 0
\(601\) −9.71904 −0.396448 −0.198224 0.980157i \(-0.563517\pi\)
−0.198224 + 0.980157i \(0.563517\pi\)
\(602\) 0 0
\(603\) −12.4995 −0.509021
\(604\) 0 0
\(605\) 8.40871 0.341863
\(606\) 0 0
\(607\) 17.7796 0.721651 0.360826 0.932633i \(-0.382495\pi\)
0.360826 + 0.932633i \(0.382495\pi\)
\(608\) 0 0
\(609\) −17.6509 −0.715251
\(610\) 0 0
\(611\) −0.159859 −0.00646721
\(612\) 0 0
\(613\) −16.9915 −0.686281 −0.343141 0.939284i \(-0.611491\pi\)
−0.343141 + 0.939284i \(0.611491\pi\)
\(614\) 0 0
\(615\) −4.57947 −0.184662
\(616\) 0 0
\(617\) 27.3406 1.10069 0.550346 0.834937i \(-0.314496\pi\)
0.550346 + 0.834937i \(0.314496\pi\)
\(618\) 0 0
\(619\) −41.2800 −1.65919 −0.829593 0.558369i \(-0.811427\pi\)
−0.829593 + 0.558369i \(0.811427\pi\)
\(620\) 0 0
\(621\) 8.24977 0.331052
\(622\) 0 0
\(623\) 57.4674 2.30238
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.60975 −0.0642871
\(628\) 0 0
\(629\) −52.3784 −2.08847
\(630\) 0 0
\(631\) 26.5601 1.05734 0.528670 0.848827i \(-0.322691\pi\)
0.528670 + 0.848827i \(0.322691\pi\)
\(632\) 0 0
\(633\) 2.06055 0.0818996
\(634\) 0 0
\(635\) −12.4995 −0.496029
\(636\) 0 0
\(637\) 3.85952 0.152920
\(638\) 0 0
\(639\) −13.7796 −0.545112
\(640\) 0 0
\(641\) −3.01846 −0.119222 −0.0596110 0.998222i \(-0.518986\pi\)
−0.0596110 + 0.998222i \(0.518986\pi\)
\(642\) 0 0
\(643\) 11.7990 0.465306 0.232653 0.972560i \(-0.425259\pi\)
0.232653 + 0.972560i \(0.425259\pi\)
\(644\) 0 0
\(645\) 7.92007 0.311852
\(646\) 0 0
\(647\) 7.46927 0.293647 0.146824 0.989163i \(-0.453095\pi\)
0.146824 + 0.989163i \(0.453095\pi\)
\(648\) 0 0
\(649\) −19.8165 −0.777866
\(650\) 0 0
\(651\) 26.2791 1.02996
\(652\) 0 0
\(653\) 41.8089 1.63611 0.818055 0.575140i \(-0.195052\pi\)
0.818055 + 0.575140i \(0.195052\pi\)
\(654\) 0 0
\(655\) 4.32970 0.169175
\(656\) 0 0
\(657\) 3.93945 0.153692
\(658\) 0 0
\(659\) −29.8477 −1.16270 −0.581351 0.813653i \(-0.697476\pi\)
−0.581351 + 0.813653i \(0.697476\pi\)
\(660\) 0 0
\(661\) 18.5677 0.722198 0.361099 0.932527i \(-0.382402\pi\)
0.361099 + 0.932527i \(0.382402\pi\)
\(662\) 0 0
\(663\) 3.37935 0.131243
\(664\) 0 0
\(665\) 3.60975 0.139980
\(666\) 0 0
\(667\) −40.3397 −1.56196
\(668\) 0 0
\(669\) 26.5601 1.02687
\(670\) 0 0
\(671\) 18.5601 0.716504
\(672\) 0 0
\(673\) −4.64002 −0.178860 −0.0894299 0.995993i \(-0.528504\pi\)
−0.0894299 + 0.995993i \(0.528504\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 18.3709 0.706050 0.353025 0.935614i \(-0.385153\pi\)
0.353025 + 0.935614i \(0.385153\pi\)
\(678\) 0 0
\(679\) −35.8089 −1.37422
\(680\) 0 0
\(681\) 5.03028 0.192761
\(682\) 0 0
\(683\) −18.9385 −0.724663 −0.362331 0.932049i \(-0.618019\pi\)
−0.362331 + 0.932049i \(0.618019\pi\)
\(684\) 0 0
\(685\) −5.03028 −0.192197
\(686\) 0 0
\(687\) 23.5298 0.897719
\(688\) 0 0
\(689\) 1.12019 0.0426758
\(690\) 0 0
\(691\) −38.1580 −1.45160 −0.725800 0.687906i \(-0.758530\pi\)
−0.725800 + 0.687906i \(0.758530\pi\)
\(692\) 0 0
\(693\) 5.81078 0.220733
\(694\) 0 0
\(695\) −9.77959 −0.370961
\(696\) 0 0
\(697\) 24.1798 0.915876
\(698\) 0 0
\(699\) 15.9688 0.603996
\(700\) 0 0
\(701\) 23.9394 0.904180 0.452090 0.891972i \(-0.350678\pi\)
0.452090 + 0.891972i \(0.350678\pi\)
\(702\) 0 0
\(703\) 9.92007 0.374143
\(704\) 0 0
\(705\) 0.249771 0.00940691
\(706\) 0 0
\(707\) 62.1580 2.33769
\(708\) 0 0
\(709\) −13.0908 −0.491636 −0.245818 0.969316i \(-0.579057\pi\)
−0.245818 + 0.969316i \(0.579057\pi\)
\(710\) 0 0
\(711\) 3.03028 0.113644
\(712\) 0 0
\(713\) 60.0587 2.24922
\(714\) 0 0
\(715\) −1.03028 −0.0385301
\(716\) 0 0
\(717\) −15.6703 −0.585218
\(718\) 0 0
\(719\) −22.6088 −0.843167 −0.421584 0.906790i \(-0.638526\pi\)
−0.421584 + 0.906790i \(0.638526\pi\)
\(720\) 0 0
\(721\) −45.1202 −1.68036
\(722\) 0 0
\(723\) −28.4390 −1.05766
\(724\) 0 0
\(725\) −4.88979 −0.181602
\(726\) 0 0
\(727\) −4.10929 −0.152405 −0.0762025 0.997092i \(-0.524280\pi\)
−0.0762025 + 0.997092i \(0.524280\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −41.8183 −1.54671
\(732\) 0 0
\(733\) −6.52890 −0.241150 −0.120575 0.992704i \(-0.538474\pi\)
−0.120575 + 0.992704i \(0.538474\pi\)
\(734\) 0 0
\(735\) −6.03028 −0.222430
\(736\) 0 0
\(737\) −20.1211 −0.741170
\(738\) 0 0
\(739\) −32.8780 −1.20944 −0.604718 0.796440i \(-0.706714\pi\)
−0.604718 + 0.796440i \(0.706714\pi\)
\(740\) 0 0
\(741\) −0.640023 −0.0235118
\(742\) 0 0
\(743\) −2.06055 −0.0755943 −0.0377972 0.999285i \(-0.512034\pi\)
−0.0377972 + 0.999285i \(0.512034\pi\)
\(744\) 0 0
\(745\) −14.0000 −0.512920
\(746\) 0 0
\(747\) 10.4995 0.384158
\(748\) 0 0
\(749\) 38.1193 1.39285
\(750\) 0 0
\(751\) −12.7787 −0.466300 −0.233150 0.972441i \(-0.574903\pi\)
−0.233150 + 0.972441i \(0.574903\pi\)
\(752\) 0 0
\(753\) 24.8292 0.904828
\(754\) 0 0
\(755\) 1.34060 0.0487894
\(756\) 0 0
\(757\) −11.3094 −0.411047 −0.205524 0.978652i \(-0.565890\pi\)
−0.205524 + 0.978652i \(0.565890\pi\)
\(758\) 0 0
\(759\) 13.2800 0.482035
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 37.2177 1.34737
\(764\) 0 0
\(765\) −5.28005 −0.190900
\(766\) 0 0
\(767\) −7.87890 −0.284490
\(768\) 0 0
\(769\) 10.6206 0.382990 0.191495 0.981494i \(-0.438666\pi\)
0.191495 + 0.981494i \(0.438666\pi\)
\(770\) 0 0
\(771\) −9.46927 −0.341027
\(772\) 0 0
\(773\) −27.8695 −1.00240 −0.501198 0.865333i \(-0.667107\pi\)
−0.501198 + 0.865333i \(0.667107\pi\)
\(774\) 0 0
\(775\) 7.28005 0.261507
\(776\) 0 0
\(777\) −35.8089 −1.28464
\(778\) 0 0
\(779\) −4.57947 −0.164077
\(780\) 0 0
\(781\) −22.1817 −0.793722
\(782\) 0 0
\(783\) −4.88979 −0.174747
\(784\) 0 0
\(785\) 10.1892 0.363669
\(786\) 0 0
\(787\) 43.2782 1.54270 0.771351 0.636410i \(-0.219581\pi\)
0.771351 + 0.636410i \(0.219581\pi\)
\(788\) 0 0
\(789\) 13.5298 0.481674
\(790\) 0 0
\(791\) 1.69724 0.0603469
\(792\) 0 0
\(793\) 7.37935 0.262049
\(794\) 0 0
\(795\) −1.75023 −0.0620742
\(796\) 0 0
\(797\) 2.53073 0.0896432 0.0448216 0.998995i \(-0.485728\pi\)
0.0448216 + 0.998995i \(0.485728\pi\)
\(798\) 0 0
\(799\) −1.31880 −0.0466559
\(800\) 0 0
\(801\) 15.9201 0.562508
\(802\) 0 0
\(803\) 6.34152 0.223787
\(804\) 0 0
\(805\) −29.7796 −1.04959
\(806\) 0 0
\(807\) 25.8889 0.911332
\(808\) 0 0
\(809\) −2.99908 −0.105442 −0.0527211 0.998609i \(-0.516789\pi\)
−0.0527211 + 0.998609i \(0.516789\pi\)
\(810\) 0 0
\(811\) −36.3784 −1.27742 −0.638710 0.769448i \(-0.720532\pi\)
−0.638710 + 0.769448i \(0.720532\pi\)
\(812\) 0 0
\(813\) 29.6585 1.04017
\(814\) 0 0
\(815\) 15.7990 0.553414
\(816\) 0 0
\(817\) 7.92007 0.277088
\(818\) 0 0
\(819\) 2.31032 0.0807292
\(820\) 0 0
\(821\) 26.7181 0.932469 0.466234 0.884661i \(-0.345610\pi\)
0.466234 + 0.884661i \(0.345610\pi\)
\(822\) 0 0
\(823\) −10.8292 −0.377484 −0.188742 0.982027i \(-0.560441\pi\)
−0.188742 + 0.982027i \(0.560441\pi\)
\(824\) 0 0
\(825\) 1.60975 0.0560442
\(826\) 0 0
\(827\) 17.0303 0.592201 0.296100 0.955157i \(-0.404314\pi\)
0.296100 + 0.955157i \(0.404314\pi\)
\(828\) 0 0
\(829\) −53.8695 −1.87097 −0.935483 0.353373i \(-0.885035\pi\)
−0.935483 + 0.353373i \(0.885035\pi\)
\(830\) 0 0
\(831\) 1.03028 0.0357399
\(832\) 0 0
\(833\) 31.8401 1.10320
\(834\) 0 0
\(835\) 9.52982 0.329793
\(836\) 0 0
\(837\) 7.28005 0.251635
\(838\) 0 0
\(839\) −36.4608 −1.25877 −0.629383 0.777095i \(-0.716692\pi\)
−0.629383 + 0.777095i \(0.716692\pi\)
\(840\) 0 0
\(841\) −5.08991 −0.175514
\(842\) 0 0
\(843\) 12.0799 0.416055
\(844\) 0 0
\(845\) 12.5904 0.433122
\(846\) 0 0
\(847\) −30.3533 −1.04295
\(848\) 0 0
\(849\) 13.2001 0.453027
\(850\) 0 0
\(851\) −81.8383 −2.80538
\(852\) 0 0
\(853\) −34.6888 −1.18772 −0.593860 0.804568i \(-0.702397\pi\)
−0.593860 + 0.804568i \(0.702397\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) −46.2498 −1.57986 −0.789931 0.613196i \(-0.789884\pi\)
−0.789931 + 0.613196i \(0.789884\pi\)
\(858\) 0 0
\(859\) 23.0596 0.786785 0.393392 0.919371i \(-0.371301\pi\)
0.393392 + 0.919371i \(0.371301\pi\)
\(860\) 0 0
\(861\) 16.5307 0.563366
\(862\) 0 0
\(863\) −22.9385 −0.780837 −0.390418 0.920638i \(-0.627670\pi\)
−0.390418 + 0.920638i \(0.627670\pi\)
\(864\) 0 0
\(865\) −13.4693 −0.457969
\(866\) 0 0
\(867\) 10.8789 0.369467
\(868\) 0 0
\(869\) 4.87798 0.165474
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) −9.92007 −0.335744
\(874\) 0 0
\(875\) −3.60975 −0.122032
\(876\) 0 0
\(877\) 1.42053 0.0479678 0.0239839 0.999712i \(-0.492365\pi\)
0.0239839 + 0.999712i \(0.492365\pi\)
\(878\) 0 0
\(879\) 11.0303 0.372042
\(880\) 0 0
\(881\) −6.90161 −0.232521 −0.116261 0.993219i \(-0.537091\pi\)
−0.116261 + 0.993219i \(0.537091\pi\)
\(882\) 0 0
\(883\) −1.98062 −0.0666533 −0.0333266 0.999445i \(-0.510610\pi\)
−0.0333266 + 0.999445i \(0.510610\pi\)
\(884\) 0 0
\(885\) 12.3103 0.413807
\(886\) 0 0
\(887\) 9.56101 0.321027 0.160514 0.987034i \(-0.448685\pi\)
0.160514 + 0.987034i \(0.448685\pi\)
\(888\) 0 0
\(889\) 45.1202 1.51328
\(890\) 0 0
\(891\) 1.60975 0.0539286
\(892\) 0 0
\(893\) 0.249771 0.00835826
\(894\) 0 0
\(895\) 20.1892 0.674851
\(896\) 0 0
\(897\) 5.28005 0.176296
\(898\) 0 0
\(899\) −35.5979 −1.18726
\(900\) 0 0
\(901\) 9.24129 0.307872
\(902\) 0 0
\(903\) −28.5895 −0.951397
\(904\) 0 0
\(905\) 7.75023 0.257626
\(906\) 0 0
\(907\) 15.2195 0.505355 0.252678 0.967551i \(-0.418689\pi\)
0.252678 + 0.967551i \(0.418689\pi\)
\(908\) 0 0
\(909\) 17.2195 0.571135
\(910\) 0 0
\(911\) 44.9991 1.49089 0.745443 0.666569i \(-0.232238\pi\)
0.745443 + 0.666569i \(0.232238\pi\)
\(912\) 0 0
\(913\) 16.9016 0.559362
\(914\) 0 0
\(915\) −11.5298 −0.381164
\(916\) 0 0
\(917\) −15.6291 −0.516119
\(918\) 0 0
\(919\) −45.3775 −1.49687 −0.748433 0.663210i \(-0.769194\pi\)
−0.748433 + 0.663210i \(0.769194\pi\)
\(920\) 0 0
\(921\) −17.7796 −0.585857
\(922\) 0 0
\(923\) −8.81926 −0.290289
\(924\) 0 0
\(925\) −9.92007 −0.326170
\(926\) 0 0
\(927\) −12.4995 −0.410539
\(928\) 0 0
\(929\) −21.3406 −0.700162 −0.350081 0.936719i \(-0.613846\pi\)
−0.350081 + 0.936719i \(0.613846\pi\)
\(930\) 0 0
\(931\) −6.03028 −0.197634
\(932\) 0 0
\(933\) −2.88979 −0.0946076
\(934\) 0 0
\(935\) −8.49954 −0.277965
\(936\) 0 0
\(937\) 19.9007 0.650127 0.325064 0.945692i \(-0.394614\pi\)
0.325064 + 0.945692i \(0.394614\pi\)
\(938\) 0 0
\(939\) −3.40115 −0.110992
\(940\) 0 0
\(941\) 5.70905 0.186110 0.0930549 0.995661i \(-0.470337\pi\)
0.0930549 + 0.995661i \(0.470337\pi\)
\(942\) 0 0
\(943\) 37.7796 1.23027
\(944\) 0 0
\(945\) −3.60975 −0.117425
\(946\) 0 0
\(947\) 28.3179 0.920208 0.460104 0.887865i \(-0.347812\pi\)
0.460104 + 0.887865i \(0.347812\pi\)
\(948\) 0 0
\(949\) 2.52134 0.0818461
\(950\) 0 0
\(951\) 4.18922 0.135845
\(952\) 0 0
\(953\) 3.97064 0.128622 0.0643108 0.997930i \(-0.479515\pi\)
0.0643108 + 0.997930i \(0.479515\pi\)
\(954\) 0 0
\(955\) −2.39025 −0.0773468
\(956\) 0 0
\(957\) −7.87133 −0.254444
\(958\) 0 0
\(959\) 18.1580 0.586353
\(960\) 0 0
\(961\) 21.9991 0.709648
\(962\) 0 0
\(963\) 10.5601 0.340294
\(964\) 0 0
\(965\) 0.640023 0.0206031
\(966\) 0 0
\(967\) −4.55011 −0.146322 −0.0731609 0.997320i \(-0.523309\pi\)
−0.0731609 + 0.997320i \(0.523309\pi\)
\(968\) 0 0
\(969\) −5.28005 −0.169620
\(970\) 0 0
\(971\) −11.5298 −0.370009 −0.185005 0.982738i \(-0.559230\pi\)
−0.185005 + 0.982738i \(0.559230\pi\)
\(972\) 0 0
\(973\) 35.3018 1.13173
\(974\) 0 0
\(975\) 0.640023 0.0204972
\(976\) 0 0
\(977\) 10.4920 0.335668 0.167834 0.985815i \(-0.446323\pi\)
0.167834 + 0.985815i \(0.446323\pi\)
\(978\) 0 0
\(979\) 25.6273 0.819052
\(980\) 0 0
\(981\) 10.3103 0.329183
\(982\) 0 0
\(983\) 5.40871 0.172511 0.0862556 0.996273i \(-0.472510\pi\)
0.0862556 + 0.996273i \(0.472510\pi\)
\(984\) 0 0
\(985\) 14.7493 0.469952
\(986\) 0 0
\(987\) −0.901610 −0.0286986
\(988\) 0 0
\(989\) −65.3388 −2.07765
\(990\) 0 0
\(991\) 27.0890 0.860510 0.430255 0.902707i \(-0.358424\pi\)
0.430255 + 0.902707i \(0.358424\pi\)
\(992\) 0 0
\(993\) −10.3103 −0.327188
\(994\) 0 0
\(995\) 19.8401 0.628975
\(996\) 0 0
\(997\) 22.8099 0.722396 0.361198 0.932489i \(-0.382368\pi\)
0.361198 + 0.932489i \(0.382368\pi\)
\(998\) 0 0
\(999\) −9.92007 −0.313857
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 4560.2.a.bt.1.3 3
4.3 odd 2 2280.2.a.r.1.1 3
12.11 even 2 6840.2.a.bk.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.r.1.1 3 4.3 odd 2
4560.2.a.bt.1.3 3 1.1 even 1 trivial
6840.2.a.bk.1.1 3 12.11 even 2