Properties

Label 2760.2.a.u.1.3
Level $2760$
Weight $2$
Character 2760.1
Self dual yes
Analytic conductor $22.039$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2760.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.0387109579\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.363328\) of defining polynomial
Character \(\chi\) \(=\) 2760.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +4.86799 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +4.86799 q^{7} +1.00000 q^{9} +4.77801 q^{11} +2.77801 q^{13} +1.00000 q^{15} -0.636672 q^{17} -3.50466 q^{19} +4.86799 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +7.36333 q^{29} -5.15066 q^{31} +4.77801 q^{33} +4.86799 q^{35} +2.86799 q^{37} +2.77801 q^{39} -6.19269 q^{41} -8.28267 q^{43} +1.00000 q^{45} +7.50466 q^{47} +16.6974 q^{49} -0.636672 q^{51} -4.91934 q^{53} +4.77801 q^{55} -3.50466 q^{57} -8.65533 q^{59} -15.0607 q^{61} +4.86799 q^{63} +2.77801 q^{65} -13.4333 q^{67} -1.00000 q^{69} -11.0993 q^{71} -15.2406 q^{73} +1.00000 q^{75} +23.2593 q^{77} -1.45331 q^{79} +1.00000 q^{81} +6.63667 q^{83} -0.636672 q^{85} +7.36333 q^{87} +9.29200 q^{89} +13.5233 q^{91} -5.15066 q^{93} -3.50466 q^{95} -14.7453 q^{97} +4.77801 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9} + 8 q^{11} + 2 q^{13} + 3 q^{15} - 4 q^{17} + 2 q^{21} - 3 q^{23} + 3 q^{25} + 3 q^{27} + 20 q^{29} + 14 q^{31} + 8 q^{33} + 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} - 8 q^{43} + 3 q^{45} + 12 q^{47} + 29 q^{49} - 4 q^{51} + 8 q^{55} + 14 q^{59} - 22 q^{61} + 2 q^{63} + 2 q^{65} + 6 q^{67} - 3 q^{69} - 6 q^{71} - 10 q^{73} + 3 q^{75} - 8 q^{77} + 4 q^{79} + 3 q^{81} + 22 q^{83} - 4 q^{85} + 20 q^{87} - 10 q^{89} - 12 q^{91} + 14 q^{93} + 2 q^{97} + 8 q^{99}+O(q^{100}) \) 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
\(7\) 4.86799 1.83993 0.919964 0.392003i \(-0.128218\pi\)
0.919964 + 0.392003i \(0.128218\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.77801 1.44062 0.720312 0.693650i \(-0.243999\pi\)
0.720312 + 0.693650i \(0.243999\pi\)
\(12\) 0 0
\(13\) 2.77801 0.770481 0.385240 0.922816i \(-0.374119\pi\)
0.385240 + 0.922816i \(0.374119\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −0.636672 −0.154416 −0.0772078 0.997015i \(-0.524600\pi\)
−0.0772078 + 0.997015i \(0.524600\pi\)
\(18\) 0 0
\(19\) −3.50466 −0.804025 −0.402013 0.915634i \(-0.631689\pi\)
−0.402013 + 0.915634i \(0.631689\pi\)
\(20\) 0 0
\(21\) 4.86799 1.06228
\(22\) 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
\(29\) 7.36333 1.36734 0.683668 0.729793i \(-0.260384\pi\)
0.683668 + 0.729793i \(0.260384\pi\)
\(30\) 0 0
\(31\) −5.15066 −0.925087 −0.462543 0.886597i \(-0.653063\pi\)
−0.462543 + 0.886597i \(0.653063\pi\)
\(32\) 0 0
\(33\) 4.77801 0.831744
\(34\) 0 0
\(35\) 4.86799 0.822841
\(36\) 0 0
\(37\) 2.86799 0.471495 0.235748 0.971814i \(-0.424246\pi\)
0.235748 + 0.971814i \(0.424246\pi\)
\(38\) 0 0
\(39\) 2.77801 0.444837
\(40\) 0 0
\(41\) −6.19269 −0.967135 −0.483568 0.875307i \(-0.660659\pi\)
−0.483568 + 0.875307i \(0.660659\pi\)
\(42\) 0 0
\(43\) −8.28267 −1.26310 −0.631548 0.775337i \(-0.717580\pi\)
−0.631548 + 0.775337i \(0.717580\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 7.50466 1.09467 0.547334 0.836914i \(-0.315643\pi\)
0.547334 + 0.836914i \(0.315643\pi\)
\(48\) 0 0
\(49\) 16.6974 2.38534
\(50\) 0 0
\(51\) −0.636672 −0.0891519
\(52\) 0 0
\(53\) −4.91934 −0.675724 −0.337862 0.941196i \(-0.609704\pi\)
−0.337862 + 0.941196i \(0.609704\pi\)
\(54\) 0 0
\(55\) 4.77801 0.644266
\(56\) 0 0
\(57\) −3.50466 −0.464204
\(58\) 0 0
\(59\) −8.65533 −1.12683 −0.563414 0.826175i \(-0.690512\pi\)
−0.563414 + 0.826175i \(0.690512\pi\)
\(60\) 0 0
\(61\) −15.0607 −1.92832 −0.964161 0.265317i \(-0.914523\pi\)
−0.964161 + 0.265317i \(0.914523\pi\)
\(62\) 0 0
\(63\) 4.86799 0.613309
\(64\) 0 0
\(65\) 2.77801 0.344569
\(66\) 0 0
\(67\) −13.4333 −1.64114 −0.820572 0.571544i \(-0.806345\pi\)
−0.820572 + 0.571544i \(0.806345\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −11.0993 −1.31725 −0.658623 0.752473i \(-0.728861\pi\)
−0.658623 + 0.752473i \(0.728861\pi\)
\(72\) 0 0
\(73\) −15.2406 −1.78378 −0.891892 0.452249i \(-0.850622\pi\)
−0.891892 + 0.452249i \(0.850622\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 23.2593 2.65064
\(78\) 0 0
\(79\) −1.45331 −0.163510 −0.0817552 0.996652i \(-0.526053\pi\)
−0.0817552 + 0.996652i \(0.526053\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.63667 0.728469 0.364235 0.931307i \(-0.381331\pi\)
0.364235 + 0.931307i \(0.381331\pi\)
\(84\) 0 0
\(85\) −0.636672 −0.0690567
\(86\) 0 0
\(87\) 7.36333 0.789432
\(88\) 0 0
\(89\) 9.29200 0.984950 0.492475 0.870327i \(-0.336092\pi\)
0.492475 + 0.870327i \(0.336092\pi\)
\(90\) 0 0
\(91\) 13.5233 1.41763
\(92\) 0 0
\(93\) −5.15066 −0.534099
\(94\) 0 0
\(95\) −3.50466 −0.359571
\(96\) 0 0
\(97\) −14.7453 −1.49716 −0.748580 0.663045i \(-0.769264\pi\)
−0.748580 + 0.663045i \(0.769264\pi\)
\(98\) 0 0
\(99\) 4.77801 0.480208
\(100\) 0 0
\(101\) −17.3820 −1.72957 −0.864786 0.502140i \(-0.832546\pi\)
−0.864786 + 0.502140i \(0.832546\pi\)
\(102\) 0 0
\(103\) 15.2920 1.50677 0.753383 0.657582i \(-0.228421\pi\)
0.753383 + 0.657582i \(0.228421\pi\)
\(104\) 0 0
\(105\) 4.86799 0.475067
\(106\) 0 0
\(107\) 15.3820 1.48703 0.743516 0.668718i \(-0.233157\pi\)
0.743516 + 0.668718i \(0.233157\pi\)
\(108\) 0 0
\(109\) −15.2406 −1.45979 −0.729895 0.683560i \(-0.760431\pi\)
−0.729895 + 0.683560i \(0.760431\pi\)
\(110\) 0 0
\(111\) 2.86799 0.272218
\(112\) 0 0
\(113\) 6.19269 0.582559 0.291280 0.956638i \(-0.405919\pi\)
0.291280 + 0.956638i \(0.405919\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 2.77801 0.256827
\(118\) 0 0
\(119\) −3.09931 −0.284114
\(120\) 0 0
\(121\) 11.8294 1.07540
\(122\) 0 0
\(123\) −6.19269 −0.558376
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.77801 −0.423980 −0.211990 0.977272i \(-0.567994\pi\)
−0.211990 + 0.977272i \(0.567994\pi\)
\(128\) 0 0
\(129\) −8.28267 −0.729249
\(130\) 0 0
\(131\) 6.54669 0.571987 0.285993 0.958232i \(-0.407676\pi\)
0.285993 + 0.958232i \(0.407676\pi\)
\(132\) 0 0
\(133\) −17.0607 −1.47935
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −10.7453 −0.918034 −0.459017 0.888427i \(-0.651798\pi\)
−0.459017 + 0.888427i \(0.651798\pi\)
\(138\) 0 0
\(139\) 14.4240 1.22343 0.611714 0.791079i \(-0.290480\pi\)
0.611714 + 0.791079i \(0.290480\pi\)
\(140\) 0 0
\(141\) 7.50466 0.632007
\(142\) 0 0
\(143\) 13.2733 1.10997
\(144\) 0 0
\(145\) 7.36333 0.611491
\(146\) 0 0
\(147\) 16.6974 1.37717
\(148\) 0 0
\(149\) 6.67531 0.546862 0.273431 0.961892i \(-0.411841\pi\)
0.273431 + 0.961892i \(0.411841\pi\)
\(150\) 0 0
\(151\) 18.3013 1.48934 0.744671 0.667432i \(-0.232607\pi\)
0.744671 + 0.667432i \(0.232607\pi\)
\(152\) 0 0
\(153\) −0.636672 −0.0514719
\(154\) 0 0
\(155\) −5.15066 −0.413711
\(156\) 0 0
\(157\) 18.4427 1.47188 0.735942 0.677044i \(-0.236739\pi\)
0.735942 + 0.677044i \(0.236739\pi\)
\(158\) 0 0
\(159\) −4.91934 −0.390129
\(160\) 0 0
\(161\) −4.86799 −0.383652
\(162\) 0 0
\(163\) −0.282672 −0.0221406 −0.0110703 0.999939i \(-0.503524\pi\)
−0.0110703 + 0.999939i \(0.503524\pi\)
\(164\) 0 0
\(165\) 4.77801 0.371967
\(166\) 0 0
\(167\) 6.51399 0.504068 0.252034 0.967718i \(-0.418901\pi\)
0.252034 + 0.967718i \(0.418901\pi\)
\(168\) 0 0
\(169\) −5.28267 −0.406359
\(170\) 0 0
\(171\) −3.50466 −0.268008
\(172\) 0 0
\(173\) 8.30133 0.631138 0.315569 0.948903i \(-0.397805\pi\)
0.315569 + 0.948903i \(0.397805\pi\)
\(174\) 0 0
\(175\) 4.86799 0.367986
\(176\) 0 0
\(177\) −8.65533 −0.650574
\(178\) 0 0
\(179\) 5.17064 0.386472 0.193236 0.981152i \(-0.438102\pi\)
0.193236 + 0.981152i \(0.438102\pi\)
\(180\) 0 0
\(181\) 21.8573 1.62464 0.812322 0.583209i \(-0.198203\pi\)
0.812322 + 0.583209i \(0.198203\pi\)
\(182\) 0 0
\(183\) −15.0607 −1.11332
\(184\) 0 0
\(185\) 2.86799 0.210859
\(186\) 0 0
\(187\) −3.04202 −0.222455
\(188\) 0 0
\(189\) 4.86799 0.354094
\(190\) 0 0
\(191\) −8.49534 −0.614701 −0.307350 0.951596i \(-0.599442\pi\)
−0.307350 + 0.951596i \(0.599442\pi\)
\(192\) 0 0
\(193\) 7.83869 0.564241 0.282121 0.959379i \(-0.408962\pi\)
0.282121 + 0.959379i \(0.408962\pi\)
\(194\) 0 0
\(195\) 2.77801 0.198937
\(196\) 0 0
\(197\) −17.5747 −1.25214 −0.626072 0.779765i \(-0.715338\pi\)
−0.626072 + 0.779765i \(0.715338\pi\)
\(198\) 0 0
\(199\) 4.17997 0.296310 0.148155 0.988964i \(-0.452667\pi\)
0.148155 + 0.988964i \(0.452667\pi\)
\(200\) 0 0
\(201\) −13.4333 −0.947515
\(202\) 0 0
\(203\) 35.8446 2.51580
\(204\) 0 0
\(205\) −6.19269 −0.432516
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −16.7453 −1.15830
\(210\) 0 0
\(211\) −22.4240 −1.54373 −0.771866 0.635785i \(-0.780676\pi\)
−0.771866 + 0.635785i \(0.780676\pi\)
\(212\) 0 0
\(213\) −11.0993 −0.760512
\(214\) 0 0
\(215\) −8.28267 −0.564874
\(216\) 0 0
\(217\) −25.0734 −1.70209
\(218\) 0 0
\(219\) −15.2406 −1.02987
\(220\) 0 0
\(221\) −1.76868 −0.118974
\(222\) 0 0
\(223\) −12.2827 −0.822509 −0.411254 0.911521i \(-0.634909\pi\)
−0.411254 + 0.911521i \(0.634909\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 24.5653 1.63046 0.815230 0.579138i \(-0.196611\pi\)
0.815230 + 0.579138i \(0.196611\pi\)
\(228\) 0 0
\(229\) −5.82003 −0.384598 −0.192299 0.981336i \(-0.561594\pi\)
−0.192299 + 0.981336i \(0.561594\pi\)
\(230\) 0 0
\(231\) 23.2593 1.53035
\(232\) 0 0
\(233\) −2.28267 −0.149543 −0.0747714 0.997201i \(-0.523823\pi\)
−0.0747714 + 0.997201i \(0.523823\pi\)
\(234\) 0 0
\(235\) 7.50466 0.489550
\(236\) 0 0
\(237\) −1.45331 −0.0944028
\(238\) 0 0
\(239\) 2.53397 0.163909 0.0819544 0.996636i \(-0.473884\pi\)
0.0819544 + 0.996636i \(0.473884\pi\)
\(240\) 0 0
\(241\) 2.67531 0.172332 0.0861658 0.996281i \(-0.472539\pi\)
0.0861658 + 0.996281i \(0.472539\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 16.6974 1.06675
\(246\) 0 0
\(247\) −9.73599 −0.619486
\(248\) 0 0
\(249\) 6.63667 0.420582
\(250\) 0 0
\(251\) 12.2827 0.775275 0.387638 0.921812i \(-0.373291\pi\)
0.387638 + 0.921812i \(0.373291\pi\)
\(252\) 0 0
\(253\) −4.77801 −0.300391
\(254\) 0 0
\(255\) −0.636672 −0.0398699
\(256\) 0 0
\(257\) 9.89004 0.616924 0.308462 0.951237i \(-0.400186\pi\)
0.308462 + 0.951237i \(0.400186\pi\)
\(258\) 0 0
\(259\) 13.9614 0.867517
\(260\) 0 0
\(261\) 7.36333 0.455779
\(262\) 0 0
\(263\) −4.26995 −0.263297 −0.131648 0.991296i \(-0.542027\pi\)
−0.131648 + 0.991296i \(0.542027\pi\)
\(264\) 0 0
\(265\) −4.91934 −0.302193
\(266\) 0 0
\(267\) 9.29200 0.568661
\(268\) 0 0
\(269\) 30.5526 1.86283 0.931413 0.363963i \(-0.118577\pi\)
0.931413 + 0.363963i \(0.118577\pi\)
\(270\) 0 0
\(271\) −23.6974 −1.43951 −0.719756 0.694227i \(-0.755746\pi\)
−0.719756 + 0.694227i \(0.755746\pi\)
\(272\) 0 0
\(273\) 13.5233 0.818469
\(274\) 0 0
\(275\) 4.77801 0.288125
\(276\) 0 0
\(277\) 20.5840 1.23677 0.618386 0.785874i \(-0.287787\pi\)
0.618386 + 0.785874i \(0.287787\pi\)
\(278\) 0 0
\(279\) −5.15066 −0.308362
\(280\) 0 0
\(281\) 8.61670 0.514029 0.257014 0.966408i \(-0.417261\pi\)
0.257014 + 0.966408i \(0.417261\pi\)
\(282\) 0 0
\(283\) 7.02930 0.417849 0.208924 0.977932i \(-0.433004\pi\)
0.208924 + 0.977932i \(0.433004\pi\)
\(284\) 0 0
\(285\) −3.50466 −0.207598
\(286\) 0 0
\(287\) −30.1460 −1.77946
\(288\) 0 0
\(289\) −16.5946 −0.976156
\(290\) 0 0
\(291\) −14.7453 −0.864386
\(292\) 0 0
\(293\) −21.2793 −1.24315 −0.621574 0.783355i \(-0.713507\pi\)
−0.621574 + 0.783355i \(0.713507\pi\)
\(294\) 0 0
\(295\) −8.65533 −0.503933
\(296\) 0 0
\(297\) 4.77801 0.277248
\(298\) 0 0
\(299\) −2.77801 −0.160656
\(300\) 0 0
\(301\) −40.3200 −2.32401
\(302\) 0 0
\(303\) −17.3820 −0.998569
\(304\) 0 0
\(305\) −15.0607 −0.862372
\(306\) 0 0
\(307\) 13.0607 0.745412 0.372706 0.927949i \(-0.378430\pi\)
0.372706 + 0.927949i \(0.378430\pi\)
\(308\) 0 0
\(309\) 15.2920 0.869932
\(310\) 0 0
\(311\) 5.45331 0.309229 0.154615 0.987975i \(-0.450586\pi\)
0.154615 + 0.987975i \(0.450586\pi\)
\(312\) 0 0
\(313\) 4.60398 0.260232 0.130116 0.991499i \(-0.458465\pi\)
0.130116 + 0.991499i \(0.458465\pi\)
\(314\) 0 0
\(315\) 4.86799 0.274280
\(316\) 0 0
\(317\) −30.3527 −1.70478 −0.852388 0.522910i \(-0.824847\pi\)
−0.852388 + 0.522910i \(0.824847\pi\)
\(318\) 0 0
\(319\) 35.1820 1.96982
\(320\) 0 0
\(321\) 15.3820 0.858539
\(322\) 0 0
\(323\) 2.23132 0.124154
\(324\) 0 0
\(325\) 2.77801 0.154096
\(326\) 0 0
\(327\) −15.2406 −0.842810
\(328\) 0 0
\(329\) 36.5327 2.01411
\(330\) 0 0
\(331\) 6.42401 0.353095 0.176548 0.984292i \(-0.443507\pi\)
0.176548 + 0.984292i \(0.443507\pi\)
\(332\) 0 0
\(333\) 2.86799 0.157165
\(334\) 0 0
\(335\) −13.4333 −0.733942
\(336\) 0 0
\(337\) −25.1120 −1.36794 −0.683970 0.729510i \(-0.739748\pi\)
−0.683970 + 0.729510i \(0.739748\pi\)
\(338\) 0 0
\(339\) 6.19269 0.336341
\(340\) 0 0
\(341\) −24.6099 −1.33270
\(342\) 0 0
\(343\) 47.2066 2.54892
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) 8.46264 0.454298 0.227149 0.973860i \(-0.427059\pi\)
0.227149 + 0.973860i \(0.427059\pi\)
\(348\) 0 0
\(349\) −6.40535 −0.342871 −0.171435 0.985195i \(-0.554840\pi\)
−0.171435 + 0.985195i \(0.554840\pi\)
\(350\) 0 0
\(351\) 2.77801 0.148279
\(352\) 0 0
\(353\) −25.7873 −1.37252 −0.686261 0.727356i \(-0.740749\pi\)
−0.686261 + 0.727356i \(0.740749\pi\)
\(354\) 0 0
\(355\) −11.0993 −0.589090
\(356\) 0 0
\(357\) −3.09931 −0.164033
\(358\) 0 0
\(359\) 5.94865 0.313958 0.156979 0.987602i \(-0.449825\pi\)
0.156979 + 0.987602i \(0.449825\pi\)
\(360\) 0 0
\(361\) −6.71733 −0.353544
\(362\) 0 0
\(363\) 11.8294 0.620880
\(364\) 0 0
\(365\) −15.2406 −0.797732
\(366\) 0 0
\(367\) −10.5013 −0.548162 −0.274081 0.961707i \(-0.588374\pi\)
−0.274081 + 0.961707i \(0.588374\pi\)
\(368\) 0 0
\(369\) −6.19269 −0.322378
\(370\) 0 0
\(371\) −23.9473 −1.24328
\(372\) 0 0
\(373\) −7.17064 −0.371282 −0.185641 0.982618i \(-0.559436\pi\)
−0.185641 + 0.982618i \(0.559436\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 20.4554 1.05351
\(378\) 0 0
\(379\) −7.11203 −0.365321 −0.182660 0.983176i \(-0.558471\pi\)
−0.182660 + 0.983176i \(0.558471\pi\)
\(380\) 0 0
\(381\) −4.77801 −0.244785
\(382\) 0 0
\(383\) 7.80731 0.398935 0.199468 0.979904i \(-0.436079\pi\)
0.199468 + 0.979904i \(0.436079\pi\)
\(384\) 0 0
\(385\) 23.2593 1.18540
\(386\) 0 0
\(387\) −8.28267 −0.421032
\(388\) 0 0
\(389\) 27.5933 1.39904 0.699519 0.714614i \(-0.253398\pi\)
0.699519 + 0.714614i \(0.253398\pi\)
\(390\) 0 0
\(391\) 0.636672 0.0321979
\(392\) 0 0
\(393\) 6.54669 0.330237
\(394\) 0 0
\(395\) −1.45331 −0.0731241
\(396\) 0 0
\(397\) −14.6426 −0.734892 −0.367446 0.930045i \(-0.619768\pi\)
−0.367446 + 0.930045i \(0.619768\pi\)
\(398\) 0 0
\(399\) −17.0607 −0.854102
\(400\) 0 0
\(401\) −3.55602 −0.177579 −0.0887895 0.996050i \(-0.528300\pi\)
−0.0887895 + 0.996050i \(0.528300\pi\)
\(402\) 0 0
\(403\) −14.3086 −0.712762
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 13.7033 0.679247
\(408\) 0 0
\(409\) 28.9894 1.43343 0.716716 0.697366i \(-0.245645\pi\)
0.716716 + 0.697366i \(0.245645\pi\)
\(410\) 0 0
\(411\) −10.7453 −0.530027
\(412\) 0 0
\(413\) −42.1341 −2.07328
\(414\) 0 0
\(415\) 6.63667 0.325781
\(416\) 0 0
\(417\) 14.4240 0.706347
\(418\) 0 0
\(419\) 28.3527 1.38512 0.692560 0.721361i \(-0.256483\pi\)
0.692560 + 0.721361i \(0.256483\pi\)
\(420\) 0 0
\(421\) 9.32469 0.454458 0.227229 0.973841i \(-0.427033\pi\)
0.227229 + 0.973841i \(0.427033\pi\)
\(422\) 0 0
\(423\) 7.50466 0.364889
\(424\) 0 0
\(425\) −0.636672 −0.0308831
\(426\) 0 0
\(427\) −73.3153 −3.54798
\(428\) 0 0
\(429\) 13.2733 0.640843
\(430\) 0 0
\(431\) −3.89730 −0.187726 −0.0938631 0.995585i \(-0.529922\pi\)
−0.0938631 + 0.995585i \(0.529922\pi\)
\(432\) 0 0
\(433\) −28.4613 −1.36776 −0.683882 0.729593i \(-0.739710\pi\)
−0.683882 + 0.729593i \(0.739710\pi\)
\(434\) 0 0
\(435\) 7.36333 0.353045
\(436\) 0 0
\(437\) 3.50466 0.167651
\(438\) 0 0
\(439\) 19.0866 0.910953 0.455477 0.890248i \(-0.349469\pi\)
0.455477 + 0.890248i \(0.349469\pi\)
\(440\) 0 0
\(441\) 16.6974 0.795112
\(442\) 0 0
\(443\) 6.47668 0.307716 0.153858 0.988093i \(-0.450830\pi\)
0.153858 + 0.988093i \(0.450830\pi\)
\(444\) 0 0
\(445\) 9.29200 0.440483
\(446\) 0 0
\(447\) 6.67531 0.315731
\(448\) 0 0
\(449\) 16.8166 0.793626 0.396813 0.917899i \(-0.370116\pi\)
0.396813 + 0.917899i \(0.370116\pi\)
\(450\) 0 0
\(451\) −29.5887 −1.39328
\(452\) 0 0
\(453\) 18.3013 0.859871
\(454\) 0 0
\(455\) 13.5233 0.633983
\(456\) 0 0
\(457\) −4.52671 −0.211751 −0.105875 0.994379i \(-0.533764\pi\)
−0.105875 + 0.994379i \(0.533764\pi\)
\(458\) 0 0
\(459\) −0.636672 −0.0297173
\(460\) 0 0
\(461\) −8.44398 −0.393276 −0.196638 0.980476i \(-0.563002\pi\)
−0.196638 + 0.980476i \(0.563002\pi\)
\(462\) 0 0
\(463\) −30.5140 −1.41811 −0.709053 0.705155i \(-0.750877\pi\)
−0.709053 + 0.705155i \(0.750877\pi\)
\(464\) 0 0
\(465\) −5.15066 −0.238856
\(466\) 0 0
\(467\) 12.2700 0.567786 0.283893 0.958856i \(-0.408374\pi\)
0.283893 + 0.958856i \(0.408374\pi\)
\(468\) 0 0
\(469\) −65.3934 −3.01959
\(470\) 0 0
\(471\) 18.4427 0.849793
\(472\) 0 0
\(473\) −39.5747 −1.81965
\(474\) 0 0
\(475\) −3.50466 −0.160805
\(476\) 0 0
\(477\) −4.91934 −0.225241
\(478\) 0 0
\(479\) −1.06068 −0.0484637 −0.0242319 0.999706i \(-0.507714\pi\)
−0.0242319 + 0.999706i \(0.507714\pi\)
\(480\) 0 0
\(481\) 7.96731 0.363278
\(482\) 0 0
\(483\) −4.86799 −0.221501
\(484\) 0 0
\(485\) −14.7453 −0.669550
\(486\) 0 0
\(487\) −5.13795 −0.232823 −0.116411 0.993201i \(-0.537139\pi\)
−0.116411 + 0.993201i \(0.537139\pi\)
\(488\) 0 0
\(489\) −0.282672 −0.0127829
\(490\) 0 0
\(491\) 2.81664 0.127113 0.0635566 0.997978i \(-0.479756\pi\)
0.0635566 + 0.997978i \(0.479756\pi\)
\(492\) 0 0
\(493\) −4.68802 −0.211138
\(494\) 0 0
\(495\) 4.77801 0.214755
\(496\) 0 0
\(497\) −54.0314 −2.42364
\(498\) 0 0
\(499\) 13.2534 0.593302 0.296651 0.954986i \(-0.404130\pi\)
0.296651 + 0.954986i \(0.404130\pi\)
\(500\) 0 0
\(501\) 6.51399 0.291024
\(502\) 0 0
\(503\) −28.2300 −1.25871 −0.629357 0.777116i \(-0.716682\pi\)
−0.629357 + 0.777116i \(0.716682\pi\)
\(504\) 0 0
\(505\) −17.3820 −0.773488
\(506\) 0 0
\(507\) −5.28267 −0.234612
\(508\) 0 0
\(509\) 2.70800 0.120030 0.0600150 0.998197i \(-0.480885\pi\)
0.0600150 + 0.998197i \(0.480885\pi\)
\(510\) 0 0
\(511\) −74.1914 −3.28203
\(512\) 0 0
\(513\) −3.50466 −0.154735
\(514\) 0 0
\(515\) 15.2920 0.673846
\(516\) 0 0
\(517\) 35.8573 1.57700
\(518\) 0 0
\(519\) 8.30133 0.364388
\(520\) 0 0
\(521\) 23.4206 1.02608 0.513038 0.858366i \(-0.328520\pi\)
0.513038 + 0.858366i \(0.328520\pi\)
\(522\) 0 0
\(523\) 2.01866 0.0882697 0.0441349 0.999026i \(-0.485947\pi\)
0.0441349 + 0.999026i \(0.485947\pi\)
\(524\) 0 0
\(525\) 4.86799 0.212457
\(526\) 0 0
\(527\) 3.27928 0.142848
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −8.65533 −0.375609
\(532\) 0 0
\(533\) −17.2033 −0.745159
\(534\) 0 0
\(535\) 15.3820 0.665021
\(536\) 0 0
\(537\) 5.17064 0.223130
\(538\) 0 0
\(539\) 79.7801 3.43637
\(540\) 0 0
\(541\) 11.8387 0.508985 0.254492 0.967075i \(-0.418092\pi\)
0.254492 + 0.967075i \(0.418092\pi\)
\(542\) 0 0
\(543\) 21.8573 0.937988
\(544\) 0 0
\(545\) −15.2406 −0.652838
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) −15.0607 −0.642774
\(550\) 0 0
\(551\) −25.8060 −1.09937
\(552\) 0 0
\(553\) −7.07472 −0.300848
\(554\) 0 0
\(555\) 2.86799 0.121740
\(556\) 0 0
\(557\) −36.9193 −1.56432 −0.782161 0.623076i \(-0.785883\pi\)
−0.782161 + 0.623076i \(0.785883\pi\)
\(558\) 0 0
\(559\) −23.0093 −0.973191
\(560\) 0 0
\(561\) −3.04202 −0.128434
\(562\) 0 0
\(563\) 37.7860 1.59249 0.796245 0.604974i \(-0.206816\pi\)
0.796245 + 0.604974i \(0.206816\pi\)
\(564\) 0 0
\(565\) 6.19269 0.260528
\(566\) 0 0
\(567\) 4.86799 0.204436
\(568\) 0 0
\(569\) −25.9346 −1.08724 −0.543618 0.839333i \(-0.682946\pi\)
−0.543618 + 0.839333i \(0.682946\pi\)
\(570\) 0 0
\(571\) 2.69396 0.112739 0.0563694 0.998410i \(-0.482048\pi\)
0.0563694 + 0.998410i \(0.482048\pi\)
\(572\) 0 0
\(573\) −8.49534 −0.354898
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 17.3947 0.724151 0.362075 0.932149i \(-0.382068\pi\)
0.362075 + 0.932149i \(0.382068\pi\)
\(578\) 0 0
\(579\) 7.83869 0.325765
\(580\) 0 0
\(581\) 32.3073 1.34033
\(582\) 0 0
\(583\) −23.5047 −0.973464
\(584\) 0 0
\(585\) 2.77801 0.114856
\(586\) 0 0
\(587\) 25.2080 1.04044 0.520222 0.854031i \(-0.325849\pi\)
0.520222 + 0.854031i \(0.325849\pi\)
\(588\) 0 0
\(589\) 18.0514 0.743793
\(590\) 0 0
\(591\) −17.5747 −0.722925
\(592\) 0 0
\(593\) 44.7313 1.83689 0.918447 0.395545i \(-0.129444\pi\)
0.918447 + 0.395545i \(0.129444\pi\)
\(594\) 0 0
\(595\) −3.09931 −0.127059
\(596\) 0 0
\(597\) 4.17997 0.171075
\(598\) 0 0
\(599\) 23.0093 0.940136 0.470068 0.882630i \(-0.344229\pi\)
0.470068 + 0.882630i \(0.344229\pi\)
\(600\) 0 0
\(601\) 40.8867 1.66780 0.833901 0.551915i \(-0.186103\pi\)
0.833901 + 0.551915i \(0.186103\pi\)
\(602\) 0 0
\(603\) −13.4333 −0.547048
\(604\) 0 0
\(605\) 11.8294 0.480932
\(606\) 0 0
\(607\) −2.65665 −0.107830 −0.0539150 0.998546i \(-0.517170\pi\)
−0.0539150 + 0.998546i \(0.517170\pi\)
\(608\) 0 0
\(609\) 35.8446 1.45250
\(610\) 0 0
\(611\) 20.8480 0.843420
\(612\) 0 0
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 0 0
\(615\) −6.19269 −0.249713
\(616\) 0 0
\(617\) −40.0314 −1.61160 −0.805801 0.592186i \(-0.798265\pi\)
−0.805801 + 0.592186i \(0.798265\pi\)
\(618\) 0 0
\(619\) 22.9066 0.920695 0.460348 0.887739i \(-0.347725\pi\)
0.460348 + 0.887739i \(0.347725\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 45.2334 1.81224
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −16.7453 −0.668743
\(628\) 0 0
\(629\) −1.82597 −0.0728062
\(630\) 0 0
\(631\) −31.1566 −1.24032 −0.620162 0.784473i \(-0.712933\pi\)
−0.620162 + 0.784473i \(0.712933\pi\)
\(632\) 0 0
\(633\) −22.4240 −0.891274
\(634\) 0 0
\(635\) −4.77801 −0.189609
\(636\) 0 0
\(637\) 46.3854 1.83786
\(638\) 0 0
\(639\) −11.0993 −0.439082
\(640\) 0 0
\(641\) −37.7474 −1.49093 −0.745466 0.666544i \(-0.767773\pi\)
−0.745466 + 0.666544i \(0.767773\pi\)
\(642\) 0 0
\(643\) 21.8187 0.860446 0.430223 0.902723i \(-0.358435\pi\)
0.430223 + 0.902723i \(0.358435\pi\)
\(644\) 0 0
\(645\) −8.28267 −0.326130
\(646\) 0 0
\(647\) 32.4554 1.27595 0.637976 0.770056i \(-0.279772\pi\)
0.637976 + 0.770056i \(0.279772\pi\)
\(648\) 0 0
\(649\) −41.3552 −1.62333
\(650\) 0 0
\(651\) −25.0734 −0.982704
\(652\) 0 0
\(653\) 15.0607 0.589370 0.294685 0.955594i \(-0.404785\pi\)
0.294685 + 0.955594i \(0.404785\pi\)
\(654\) 0 0
\(655\) 6.54669 0.255800
\(656\) 0 0
\(657\) −15.2406 −0.594594
\(658\) 0 0
\(659\) 32.2500 1.25628 0.628140 0.778100i \(-0.283816\pi\)
0.628140 + 0.778100i \(0.283816\pi\)
\(660\) 0 0
\(661\) −37.0093 −1.43950 −0.719748 0.694235i \(-0.755743\pi\)
−0.719748 + 0.694235i \(0.755743\pi\)
\(662\) 0 0
\(663\) −1.76868 −0.0686898
\(664\) 0 0
\(665\) −17.0607 −0.661585
\(666\) 0 0
\(667\) −7.36333 −0.285109
\(668\) 0 0
\(669\) −12.2827 −0.474876
\(670\) 0 0
\(671\) −71.9600 −2.77799
\(672\) 0 0
\(673\) −0.411290 −0.0158541 −0.00792704 0.999969i \(-0.502523\pi\)
−0.00792704 + 0.999969i \(0.502523\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −1.98728 −0.0763774 −0.0381887 0.999271i \(-0.512159\pi\)
−0.0381887 + 0.999271i \(0.512159\pi\)
\(678\) 0 0
\(679\) −71.7801 −2.75467
\(680\) 0 0
\(681\) 24.5653 0.941346
\(682\) 0 0
\(683\) 17.3179 0.662652 0.331326 0.943516i \(-0.392504\pi\)
0.331326 + 0.943516i \(0.392504\pi\)
\(684\) 0 0
\(685\) −10.7453 −0.410557
\(686\) 0 0
\(687\) −5.82003 −0.222048
\(688\) 0 0
\(689\) −13.6660 −0.520632
\(690\) 0 0
\(691\) −5.94139 −0.226021 −0.113011 0.993594i \(-0.536049\pi\)
−0.113011 + 0.993594i \(0.536049\pi\)
\(692\) 0 0
\(693\) 23.2593 0.883548
\(694\) 0 0
\(695\) 14.4240 0.547134
\(696\) 0 0
\(697\) 3.94271 0.149341
\(698\) 0 0
\(699\) −2.28267 −0.0863386
\(700\) 0 0
\(701\) 34.2127 1.29219 0.646097 0.763255i \(-0.276400\pi\)
0.646097 + 0.763255i \(0.276400\pi\)
\(702\) 0 0
\(703\) −10.0514 −0.379094
\(704\) 0 0
\(705\) 7.50466 0.282642
\(706\) 0 0
\(707\) −84.6154 −3.18229
\(708\) 0 0
\(709\) 18.6354 0.699865 0.349933 0.936775i \(-0.386204\pi\)
0.349933 + 0.936775i \(0.386204\pi\)
\(710\) 0 0
\(711\) −1.45331 −0.0545035
\(712\) 0 0
\(713\) 5.15066 0.192894
\(714\) 0 0
\(715\) 13.2733 0.496395
\(716\) 0 0
\(717\) 2.53397 0.0946328
\(718\) 0 0
\(719\) 10.5594 0.393799 0.196900 0.980424i \(-0.436913\pi\)
0.196900 + 0.980424i \(0.436913\pi\)
\(720\) 0 0
\(721\) 74.4413 2.77234
\(722\) 0 0
\(723\) 2.67531 0.0994957
\(724\) 0 0
\(725\) 7.36333 0.273467
\(726\) 0 0
\(727\) −47.3493 −1.75609 −0.878044 0.478580i \(-0.841152\pi\)
−0.878044 + 0.478580i \(0.841152\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 5.27334 0.195042
\(732\) 0 0
\(733\) −42.1600 −1.55721 −0.778607 0.627511i \(-0.784074\pi\)
−0.778607 + 0.627511i \(0.784074\pi\)
\(734\) 0 0
\(735\) 16.6974 0.615891
\(736\) 0 0
\(737\) −64.1846 −2.36427
\(738\) 0 0
\(739\) 13.2933 0.489003 0.244501 0.969649i \(-0.421376\pi\)
0.244501 + 0.969649i \(0.421376\pi\)
\(740\) 0 0
\(741\) −9.73599 −0.357660
\(742\) 0 0
\(743\) 3.47197 0.127374 0.0636871 0.997970i \(-0.479714\pi\)
0.0636871 + 0.997970i \(0.479714\pi\)
\(744\) 0 0
\(745\) 6.67531 0.244564
\(746\) 0 0
\(747\) 6.63667 0.242823
\(748\) 0 0
\(749\) 74.8794 2.73603
\(750\) 0 0
\(751\) 22.5513 0.822909 0.411454 0.911430i \(-0.365021\pi\)
0.411454 + 0.911430i \(0.365021\pi\)
\(752\) 0 0
\(753\) 12.2827 0.447605
\(754\) 0 0
\(755\) 18.3013 0.666054
\(756\) 0 0
\(757\) −13.9028 −0.505304 −0.252652 0.967557i \(-0.581303\pi\)
−0.252652 + 0.967557i \(0.581303\pi\)
\(758\) 0 0
\(759\) −4.77801 −0.173431
\(760\) 0 0
\(761\) 29.8701 1.08279 0.541394 0.840769i \(-0.317897\pi\)
0.541394 + 0.840769i \(0.317897\pi\)
\(762\) 0 0
\(763\) −74.1914 −2.68591
\(764\) 0 0
\(765\) −0.636672 −0.0230189
\(766\) 0 0
\(767\) −24.0446 −0.868199
\(768\) 0 0
\(769\) 10.4299 0.376114 0.188057 0.982158i \(-0.439781\pi\)
0.188057 + 0.982158i \(0.439781\pi\)
\(770\) 0 0
\(771\) 9.89004 0.356181
\(772\) 0 0
\(773\) −50.5000 −1.81636 −0.908179 0.418583i \(-0.862527\pi\)
−0.908179 + 0.418583i \(0.862527\pi\)
\(774\) 0 0
\(775\) −5.15066 −0.185017
\(776\) 0 0
\(777\) 13.9614 0.500861
\(778\) 0 0
\(779\) 21.7033 0.777601
\(780\) 0 0
\(781\) −53.0326 −1.89766
\(782\) 0 0
\(783\) 7.36333 0.263144
\(784\) 0 0
\(785\) 18.4427 0.658247
\(786\) 0 0
\(787\) 41.2907 1.47185 0.735927 0.677061i \(-0.236747\pi\)
0.735927 + 0.677061i \(0.236747\pi\)
\(788\) 0 0
\(789\) −4.26995 −0.152014
\(790\) 0 0
\(791\) 30.1460 1.07187
\(792\) 0 0
\(793\) −41.8387 −1.48574
\(794\) 0 0
\(795\) −4.91934 −0.174471
\(796\) 0 0
\(797\) 25.4847 0.902714 0.451357 0.892343i \(-0.350940\pi\)
0.451357 + 0.892343i \(0.350940\pi\)
\(798\) 0 0
\(799\) −4.77801 −0.169034
\(800\) 0 0
\(801\) 9.29200 0.328317
\(802\) 0 0
\(803\) −72.8199 −2.56976
\(804\) 0 0
\(805\) −4.86799 −0.171574
\(806\) 0 0
\(807\) 30.5526 1.07550
\(808\) 0 0
\(809\) −30.9380 −1.08772 −0.543861 0.839175i \(-0.683038\pi\)
−0.543861 + 0.839175i \(0.683038\pi\)
\(810\) 0 0
\(811\) 31.5946 1.10944 0.554719 0.832038i \(-0.312826\pi\)
0.554719 + 0.832038i \(0.312826\pi\)
\(812\) 0 0
\(813\) −23.6974 −0.831102
\(814\) 0 0
\(815\) −0.282672 −0.00990158
\(816\) 0 0
\(817\) 29.0280 1.01556
\(818\) 0 0
\(819\) 13.5233 0.472543
\(820\) 0 0
\(821\) −56.7640 −1.98108 −0.990538 0.137238i \(-0.956178\pi\)
−0.990538 + 0.137238i \(0.956178\pi\)
\(822\) 0 0
\(823\) −10.9066 −0.380181 −0.190091 0.981767i \(-0.560878\pi\)
−0.190091 + 0.981767i \(0.560878\pi\)
\(824\) 0 0
\(825\) 4.77801 0.166349
\(826\) 0 0
\(827\) 8.75803 0.304547 0.152273 0.988338i \(-0.451341\pi\)
0.152273 + 0.988338i \(0.451341\pi\)
\(828\) 0 0
\(829\) 21.2720 0.738808 0.369404 0.929269i \(-0.379562\pi\)
0.369404 + 0.929269i \(0.379562\pi\)
\(830\) 0 0
\(831\) 20.5840 0.714051
\(832\) 0 0
\(833\) −10.6307 −0.368333
\(834\) 0 0
\(835\) 6.51399 0.225426
\(836\) 0 0
\(837\) −5.15066 −0.178033
\(838\) 0 0
\(839\) −14.6240 −0.504875 −0.252437 0.967613i \(-0.581232\pi\)
−0.252437 + 0.967613i \(0.581232\pi\)
\(840\) 0 0
\(841\) 25.2186 0.869607
\(842\) 0 0
\(843\) 8.61670 0.296775
\(844\) 0 0
\(845\) −5.28267 −0.181729
\(846\) 0 0
\(847\) 57.5852 1.97865
\(848\) 0 0
\(849\) 7.02930 0.241245
\(850\) 0 0
\(851\) −2.86799 −0.0983135
\(852\) 0 0
\(853\) −52.6867 −1.80396 −0.901979 0.431779i \(-0.857886\pi\)
−0.901979 + 0.431779i \(0.857886\pi\)
\(854\) 0 0
\(855\) −3.50466 −0.119857
\(856\) 0 0
\(857\) −6.77075 −0.231284 −0.115642 0.993291i \(-0.536893\pi\)
−0.115642 + 0.993291i \(0.536893\pi\)
\(858\) 0 0
\(859\) −47.9146 −1.63483 −0.817413 0.576052i \(-0.804593\pi\)
−0.817413 + 0.576052i \(0.804593\pi\)
\(860\) 0 0
\(861\) −30.1460 −1.02737
\(862\) 0 0
\(863\) −27.8573 −0.948275 −0.474138 0.880451i \(-0.657240\pi\)
−0.474138 + 0.880451i \(0.657240\pi\)
\(864\) 0 0
\(865\) 8.30133 0.282254
\(866\) 0 0
\(867\) −16.5946 −0.563584
\(868\) 0 0
\(869\) −6.94394 −0.235557
\(870\) 0 0
\(871\) −37.3179 −1.26447
\(872\) 0 0
\(873\) −14.7453 −0.499053
\(874\) 0 0
\(875\) 4.86799 0.164568
\(876\) 0 0
\(877\) 12.4440 0.420203 0.210102 0.977680i \(-0.432620\pi\)
0.210102 + 0.977680i \(0.432620\pi\)
\(878\) 0 0
\(879\) −21.2793 −0.717732
\(880\) 0 0
\(881\) −7.70329 −0.259530 −0.129765 0.991545i \(-0.541422\pi\)
−0.129765 + 0.991545i \(0.541422\pi\)
\(882\) 0 0
\(883\) −47.9274 −1.61288 −0.806442 0.591313i \(-0.798610\pi\)
−0.806442 + 0.591313i \(0.798610\pi\)
\(884\) 0 0
\(885\) −8.65533 −0.290946
\(886\) 0 0
\(887\) −7.61462 −0.255674 −0.127837 0.991795i \(-0.540803\pi\)
−0.127837 + 0.991795i \(0.540803\pi\)
\(888\) 0 0
\(889\) −23.2593 −0.780092
\(890\) 0 0
\(891\) 4.77801 0.160069
\(892\) 0 0
\(893\) −26.3013 −0.880140
\(894\) 0 0
\(895\) 5.17064 0.172835
\(896\) 0 0
\(897\) −2.77801 −0.0927550
\(898\) 0 0
\(899\) −37.9260 −1.26490
\(900\) 0 0
\(901\) 3.13201 0.104342
\(902\) 0 0
\(903\) −40.3200 −1.34177
\(904\) 0 0
\(905\) 21.8573 0.726563
\(906\) 0 0
\(907\) 50.0759 1.66274 0.831372 0.555716i \(-0.187556\pi\)
0.831372 + 0.555716i \(0.187556\pi\)
\(908\) 0 0
\(909\) −17.3820 −0.576524
\(910\) 0 0
\(911\) 14.5840 0.483190 0.241595 0.970377i \(-0.422330\pi\)
0.241595 + 0.970377i \(0.422330\pi\)
\(912\) 0 0
\(913\) 31.7101 1.04945
\(914\) 0 0
\(915\) −15.0607 −0.497891
\(916\) 0 0
\(917\) 31.8692 1.05241
\(918\) 0 0
\(919\) 8.03731 0.265127 0.132563 0.991175i \(-0.457679\pi\)
0.132563 + 0.991175i \(0.457679\pi\)
\(920\) 0 0
\(921\) 13.0607 0.430364
\(922\) 0 0
\(923\) −30.8340 −1.01491
\(924\) 0 0
\(925\) 2.86799 0.0942990
\(926\) 0 0
\(927\) 15.2920 0.502255
\(928\) 0 0
\(929\) 14.3353 0.470327 0.235164 0.971956i \(-0.424437\pi\)
0.235164 + 0.971956i \(0.424437\pi\)
\(930\) 0 0
\(931\) −58.5186 −1.91787
\(932\) 0 0
\(933\) 5.45331 0.178533
\(934\) 0 0
\(935\) −3.04202 −0.0994848
\(936\) 0 0
\(937\) 14.7080 0.480489 0.240245 0.970712i \(-0.422772\pi\)
0.240245 + 0.970712i \(0.422772\pi\)
\(938\) 0 0
\(939\) 4.60398 0.150245
\(940\) 0 0
\(941\) 2.77801 0.0905605 0.0452802 0.998974i \(-0.485582\pi\)
0.0452802 + 0.998974i \(0.485582\pi\)
\(942\) 0 0
\(943\) 6.19269 0.201662
\(944\) 0 0
\(945\) 4.86799 0.158356
\(946\) 0 0
\(947\) −32.6680 −1.06157 −0.530784 0.847507i \(-0.678103\pi\)
−0.530784 + 0.847507i \(0.678103\pi\)
\(948\) 0 0
\(949\) −42.3386 −1.37437
\(950\) 0 0
\(951\) −30.3527 −0.984253
\(952\) 0 0
\(953\) −25.4347 −0.823909 −0.411955 0.911204i \(-0.635154\pi\)
−0.411955 + 0.911204i \(0.635154\pi\)
\(954\) 0 0
\(955\) −8.49534 −0.274903
\(956\) 0 0
\(957\) 35.1820 1.13727
\(958\) 0 0
\(959\) −52.3081 −1.68912
\(960\) 0 0
\(961\) −4.47065 −0.144215
\(962\) 0 0
\(963\) 15.3820 0.495678
\(964\) 0 0
\(965\) 7.83869 0.252336
\(966\) 0 0
\(967\) −46.4740 −1.49450 −0.747252 0.664541i \(-0.768627\pi\)
−0.747252 + 0.664541i \(0.768627\pi\)
\(968\) 0 0
\(969\) 2.23132 0.0716804
\(970\) 0 0
\(971\) 59.6120 1.91304 0.956520 0.291667i \(-0.0942100\pi\)
0.956520 + 0.291667i \(0.0942100\pi\)
\(972\) 0 0
\(973\) 70.2160 2.25102
\(974\) 0 0
\(975\) 2.77801 0.0889675
\(976\) 0 0
\(977\) 11.2207 0.358981 0.179491 0.983760i \(-0.442555\pi\)
0.179491 + 0.983760i \(0.442555\pi\)
\(978\) 0 0
\(979\) 44.3973 1.41894
\(980\) 0 0
\(981\) −15.2406 −0.486596
\(982\) 0 0
\(983\) 4.29539 0.137002 0.0685008 0.997651i \(-0.478178\pi\)
0.0685008 + 0.997651i \(0.478178\pi\)
\(984\) 0 0
\(985\) −17.5747 −0.559976
\(986\) 0 0
\(987\) 36.5327 1.16285
\(988\) 0 0
\(989\) 8.28267 0.263374
\(990\) 0 0
\(991\) 26.3841 0.838117 0.419059 0.907959i \(-0.362360\pi\)
0.419059 + 0.907959i \(0.362360\pi\)
\(992\) 0 0
\(993\) 6.42401 0.203860
\(994\) 0 0
\(995\) 4.17997 0.132514
\(996\) 0 0
\(997\) 13.2547 0.419780 0.209890 0.977725i \(-0.432689\pi\)
0.209890 + 0.977725i \(0.432689\pi\)
\(998\) 0 0
\(999\) 2.86799 0.0907393
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 2760.2.a.u.1.3 3
3.2 odd 2 8280.2.a.bi.1.3 3
4.3 odd 2 5520.2.a.bz.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.u.1.3 3 1.1 even 1 trivial
5520.2.a.bz.1.1 3 4.3 odd 2
8280.2.a.bi.1.3 3 3.2 odd 2