Properties

Label 6384.2.a.ce.1.4
Level $6384$
Weight $2$
Character 6384.1
Self dual yes
Analytic conductor $50.976$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6384.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.9764966504\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.135076.1
Defining polynomial: \( x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 3192)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.26848\) of defining polynomial
Character \(\chi\) \(=\) 6384.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.53696 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.53696 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.91239 q^{11} -0.165497 q^{13} +2.53696 q^{15} +6.84768 q^{17} -1.00000 q^{19} -1.00000 q^{21} -2.16550 q^{23} +1.43617 q^{25} +1.00000 q^{27} +4.70246 q^{29} +7.07788 q^{31} -4.91239 q^{33} -2.53696 q^{35} -0.746889 q^{37} -0.165497 q^{39} +3.07392 q^{41} +2.53696 q^{45} -3.88551 q^{47} +1.00000 q^{49} +6.84768 q^{51} -0.702457 q^{53} -12.4625 q^{55} -1.00000 q^{57} +13.3349 q^{59} +6.00000 q^{61} -1.00000 q^{63} -0.419858 q^{65} +13.0439 q^{67} -2.16550 q^{69} +8.26629 q^{71} -5.05761 q^{73} +1.43617 q^{75} +4.91239 q^{77} -9.22311 q^{79} +1.00000 q^{81} -8.19623 q^{83} +17.3723 q^{85} +4.70246 q^{87} -9.38861 q^{89} +0.165497 q^{91} +7.07788 q^{93} -2.53696 q^{95} -3.65531 q^{97} -4.91239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9} - 2 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 5 q^{19} - 5 q^{21} - 2 q^{23} + 19 q^{25} + 5 q^{27} + 4 q^{29} + 4 q^{31} - 2 q^{33} - 2 q^{35} + 10 q^{37} + 8 q^{39} - 6 q^{41} + 2 q^{45} + 2 q^{47} + 5 q^{49} - 2 q^{51} + 16 q^{53} + 16 q^{55} - 5 q^{57} + 12 q^{59} + 30 q^{61} - 5 q^{63} + 4 q^{65} - 18 q^{67} - 2 q^{69} + 10 q^{71} + 14 q^{73} + 19 q^{75} + 2 q^{77} + 2 q^{79} + 5 q^{81} + 6 q^{83} + 12 q^{85} + 4 q^{87} + 10 q^{89} - 8 q^{91} + 4 q^{93} - 2 q^{95} + 8 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(5\) 2.53696 1.13456 0.567281 0.823524i \(-0.307995\pi\)
0.567281 + 0.823524i \(0.307995\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.91239 −1.48114 −0.740570 0.671979i \(-0.765444\pi\)
−0.740570 + 0.671979i \(0.765444\pi\)
\(12\) 0 0
\(13\) −0.165497 −0.0459005 −0.0229502 0.999737i \(-0.507306\pi\)
−0.0229502 + 0.999737i \(0.507306\pi\)
\(14\) 0 0
\(15\) 2.53696 0.655040
\(16\) 0 0
\(17\) 6.84768 1.66081 0.830404 0.557162i \(-0.188110\pi\)
0.830404 + 0.557162i \(0.188110\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −2.16550 −0.451537 −0.225769 0.974181i \(-0.572489\pi\)
−0.225769 + 0.974181i \(0.572489\pi\)
\(24\) 0 0
\(25\) 1.43617 0.287233
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.70246 0.873224 0.436612 0.899650i \(-0.356178\pi\)
0.436612 + 0.899650i \(0.356178\pi\)
\(30\) 0 0
\(31\) 7.07788 1.27123 0.635613 0.772008i \(-0.280748\pi\)
0.635613 + 0.772008i \(0.280748\pi\)
\(32\) 0 0
\(33\) −4.91239 −0.855137
\(34\) 0 0
\(35\) −2.53696 −0.428824
\(36\) 0 0
\(37\) −0.746889 −0.122788 −0.0613939 0.998114i \(-0.519555\pi\)
−0.0613939 + 0.998114i \(0.519555\pi\)
\(38\) 0 0
\(39\) −0.165497 −0.0265007
\(40\) 0 0
\(41\) 3.07392 0.480066 0.240033 0.970765i \(-0.422842\pi\)
0.240033 + 0.970765i \(0.422842\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 2.53696 0.378188
\(46\) 0 0
\(47\) −3.88551 −0.566760 −0.283380 0.959008i \(-0.591456\pi\)
−0.283380 + 0.959008i \(0.591456\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.84768 0.958867
\(52\) 0 0
\(53\) −0.702457 −0.0964898 −0.0482449 0.998836i \(-0.515363\pi\)
−0.0482449 + 0.998836i \(0.515363\pi\)
\(54\) 0 0
\(55\) −12.4625 −1.68045
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 13.3349 1.73605 0.868025 0.496520i \(-0.165389\pi\)
0.868025 + 0.496520i \(0.165389\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −0.419858 −0.0520770
\(66\) 0 0
\(67\) 13.0439 1.59357 0.796784 0.604264i \(-0.206533\pi\)
0.796784 + 0.604264i \(0.206533\pi\)
\(68\) 0 0
\(69\) −2.16550 −0.260695
\(70\) 0 0
\(71\) 8.26629 0.981028 0.490514 0.871433i \(-0.336809\pi\)
0.490514 + 0.871433i \(0.336809\pi\)
\(72\) 0 0
\(73\) −5.05761 −0.591949 −0.295974 0.955196i \(-0.595644\pi\)
−0.295974 + 0.955196i \(0.595644\pi\)
\(74\) 0 0
\(75\) 1.43617 0.165834
\(76\) 0 0
\(77\) 4.91239 0.559818
\(78\) 0 0
\(79\) −9.22311 −1.03768 −0.518840 0.854871i \(-0.673636\pi\)
−0.518840 + 0.854871i \(0.673636\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.19623 −0.899654 −0.449827 0.893116i \(-0.648514\pi\)
−0.449827 + 0.893116i \(0.648514\pi\)
\(84\) 0 0
\(85\) 17.3723 1.88429
\(86\) 0 0
\(87\) 4.70246 0.504156
\(88\) 0 0
\(89\) −9.38861 −0.995190 −0.497595 0.867409i \(-0.665783\pi\)
−0.497595 + 0.867409i \(0.665783\pi\)
\(90\) 0 0
\(91\) 0.165497 0.0173488
\(92\) 0 0
\(93\) 7.07788 0.733942
\(94\) 0 0
\(95\) −2.53696 −0.260287
\(96\) 0 0
\(97\) −3.65531 −0.371141 −0.185570 0.982631i \(-0.559413\pi\)
−0.185570 + 0.982631i \(0.559413\pi\)
\(98\) 0 0
\(99\) −4.91239 −0.493713
\(100\) 0 0
\(101\) 0.389120 0.0387189 0.0193595 0.999813i \(-0.493837\pi\)
0.0193595 + 0.999813i \(0.493837\pi\)
\(102\) 0 0
\(103\) 0.440128 0.0433671 0.0216835 0.999765i \(-0.493097\pi\)
0.0216835 + 0.999765i \(0.493097\pi\)
\(104\) 0 0
\(105\) −2.53696 −0.247582
\(106\) 0 0
\(107\) 5.46962 0.528768 0.264384 0.964418i \(-0.414831\pi\)
0.264384 + 0.964418i \(0.414831\pi\)
\(108\) 0 0
\(109\) 8.00396 0.766641 0.383320 0.923615i \(-0.374781\pi\)
0.383320 + 0.923615i \(0.374781\pi\)
\(110\) 0 0
\(111\) −0.746889 −0.0708916
\(112\) 0 0
\(113\) 16.0210 1.50713 0.753565 0.657374i \(-0.228333\pi\)
0.753565 + 0.657374i \(0.228333\pi\)
\(114\) 0 0
\(115\) −5.49378 −0.512297
\(116\) 0 0
\(117\) −0.165497 −0.0153002
\(118\) 0 0
\(119\) −6.84768 −0.627726
\(120\) 0 0
\(121\) 13.1315 1.19378
\(122\) 0 0
\(123\) 3.07392 0.277166
\(124\) 0 0
\(125\) −9.04130 −0.808679
\(126\) 0 0
\(127\) 13.9660 1.23929 0.619643 0.784884i \(-0.287278\pi\)
0.619643 + 0.784884i \(0.287278\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 18.0210 1.57450 0.787251 0.616632i \(-0.211503\pi\)
0.787251 + 0.616632i \(0.211503\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) 2.53696 0.218347
\(136\) 0 0
\(137\) 5.66901 0.484336 0.242168 0.970234i \(-0.422141\pi\)
0.242168 + 0.970234i \(0.422141\pi\)
\(138\) 0 0
\(139\) −0.0700563 −0.00594210 −0.00297105 0.999996i \(-0.500946\pi\)
−0.00297105 + 0.999996i \(0.500946\pi\)
\(140\) 0 0
\(141\) −3.88551 −0.327219
\(142\) 0 0
\(143\) 0.812983 0.0679851
\(144\) 0 0
\(145\) 11.9299 0.990728
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 13.8208 1.13224 0.566122 0.824321i \(-0.308443\pi\)
0.566122 + 0.824321i \(0.308443\pi\)
\(150\) 0 0
\(151\) −3.81820 −0.310720 −0.155360 0.987858i \(-0.549654\pi\)
−0.155360 + 0.987858i \(0.549654\pi\)
\(152\) 0 0
\(153\) 6.84768 0.553602
\(154\) 0 0
\(155\) 17.9563 1.44228
\(156\) 0 0
\(157\) −5.97261 −0.476666 −0.238333 0.971183i \(-0.576601\pi\)
−0.238333 + 0.971183i \(0.576601\pi\)
\(158\) 0 0
\(159\) −0.702457 −0.0557084
\(160\) 0 0
\(161\) 2.16550 0.170665
\(162\) 0 0
\(163\) 9.95630 0.779838 0.389919 0.920849i \(-0.372503\pi\)
0.389919 + 0.920849i \(0.372503\pi\)
\(164\) 0 0
\(165\) −12.4625 −0.970206
\(166\) 0 0
\(167\) −1.56383 −0.121013 −0.0605066 0.998168i \(-0.519272\pi\)
−0.0605066 + 0.998168i \(0.519272\pi\)
\(168\) 0 0
\(169\) −12.9726 −0.997893
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 23.5444 1.79005 0.895023 0.446021i \(-0.147159\pi\)
0.895023 + 0.446021i \(0.147159\pi\)
\(174\) 0 0
\(175\) −1.43617 −0.108564
\(176\) 0 0
\(177\) 13.3349 1.00231
\(178\) 0 0
\(179\) −22.7639 −1.70146 −0.850728 0.525606i \(-0.823839\pi\)
−0.850728 + 0.525606i \(0.823839\pi\)
\(180\) 0 0
\(181\) 8.42643 0.626332 0.313166 0.949698i \(-0.398610\pi\)
0.313166 + 0.949698i \(0.398610\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 0 0
\(185\) −1.89483 −0.139310
\(186\) 0 0
\(187\) −33.6385 −2.45989
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 11.9497 0.864652 0.432326 0.901717i \(-0.357693\pi\)
0.432326 + 0.901717i \(0.357693\pi\)
\(192\) 0 0
\(193\) 18.6214 1.34040 0.670201 0.742180i \(-0.266208\pi\)
0.670201 + 0.742180i \(0.266208\pi\)
\(194\) 0 0
\(195\) −0.419858 −0.0300667
\(196\) 0 0
\(197\) −20.7195 −1.47620 −0.738102 0.674690i \(-0.764278\pi\)
−0.738102 + 0.674690i \(0.764278\pi\)
\(198\) 0 0
\(199\) 7.05761 0.500301 0.250150 0.968207i \(-0.419520\pi\)
0.250150 + 0.968207i \(0.419520\pi\)
\(200\) 0 0
\(201\) 13.0439 0.920047
\(202\) 0 0
\(203\) −4.70246 −0.330048
\(204\) 0 0
\(205\) 7.79841 0.544665
\(206\) 0 0
\(207\) −2.16550 −0.150512
\(208\) 0 0
\(209\) 4.91239 0.339797
\(210\) 0 0
\(211\) −8.84233 −0.608731 −0.304366 0.952555i \(-0.598444\pi\)
−0.304366 + 0.952555i \(0.598444\pi\)
\(212\) 0 0
\(213\) 8.26629 0.566397
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −7.07788 −0.480478
\(218\) 0 0
\(219\) −5.05761 −0.341762
\(220\) 0 0
\(221\) −1.13327 −0.0762319
\(222\) 0 0
\(223\) −8.28121 −0.554551 −0.277275 0.960791i \(-0.589431\pi\)
−0.277275 + 0.960791i \(0.589431\pi\)
\(224\) 0 0
\(225\) 1.43617 0.0957444
\(226\) 0 0
\(227\) −3.09815 −0.205632 −0.102816 0.994700i \(-0.532785\pi\)
−0.102816 + 0.994700i \(0.532785\pi\)
\(228\) 0 0
\(229\) 5.24915 0.346874 0.173437 0.984845i \(-0.444513\pi\)
0.173437 + 0.984845i \(0.444513\pi\)
\(230\) 0 0
\(231\) 4.91239 0.323211
\(232\) 0 0
\(233\) −3.08184 −0.201898 −0.100949 0.994892i \(-0.532188\pi\)
−0.100949 + 0.994892i \(0.532188\pi\)
\(234\) 0 0
\(235\) −9.85739 −0.643025
\(236\) 0 0
\(237\) −9.22311 −0.599105
\(238\) 0 0
\(239\) 13.6055 0.880068 0.440034 0.897981i \(-0.354966\pi\)
0.440034 + 0.897981i \(0.354966\pi\)
\(240\) 0 0
\(241\) 19.4450 1.25256 0.626280 0.779598i \(-0.284577\pi\)
0.626280 + 0.779598i \(0.284577\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.53696 0.162080
\(246\) 0 0
\(247\) 0.165497 0.0105303
\(248\) 0 0
\(249\) −8.19623 −0.519415
\(250\) 0 0
\(251\) 3.90578 0.246531 0.123265 0.992374i \(-0.460663\pi\)
0.123265 + 0.992374i \(0.460663\pi\)
\(252\) 0 0
\(253\) 10.6378 0.668790
\(254\) 0 0
\(255\) 17.3723 1.08790
\(256\) 0 0
\(257\) −23.9563 −1.49435 −0.747177 0.664626i \(-0.768591\pi\)
−0.747177 + 0.664626i \(0.768591\pi\)
\(258\) 0 0
\(259\) 0.746889 0.0464094
\(260\) 0 0
\(261\) 4.70246 0.291075
\(262\) 0 0
\(263\) −22.3671 −1.37921 −0.689607 0.724184i \(-0.742217\pi\)
−0.689607 + 0.724184i \(0.742217\pi\)
\(264\) 0 0
\(265\) −1.78210 −0.109474
\(266\) 0 0
\(267\) −9.38861 −0.574573
\(268\) 0 0
\(269\) 3.98369 0.242890 0.121445 0.992598i \(-0.461247\pi\)
0.121445 + 0.992598i \(0.461247\pi\)
\(270\) 0 0
\(271\) 5.53432 0.336186 0.168093 0.985771i \(-0.446239\pi\)
0.168093 + 0.985771i \(0.446239\pi\)
\(272\) 0 0
\(273\) 0.165497 0.0100163
\(274\) 0 0
\(275\) −7.05500 −0.425432
\(276\) 0 0
\(277\) 11.2408 0.675392 0.337696 0.941255i \(-0.390352\pi\)
0.337696 + 0.941255i \(0.390352\pi\)
\(278\) 0 0
\(279\) 7.07788 0.423742
\(280\) 0 0
\(281\) −13.7764 −0.821830 −0.410915 0.911674i \(-0.634791\pi\)
−0.410915 + 0.911674i \(0.634791\pi\)
\(282\) 0 0
\(283\) −23.3533 −1.38821 −0.694105 0.719874i \(-0.744199\pi\)
−0.694105 + 0.719874i \(0.744199\pi\)
\(284\) 0 0
\(285\) −2.53696 −0.150277
\(286\) 0 0
\(287\) −3.07392 −0.181448
\(288\) 0 0
\(289\) 29.8908 1.75828
\(290\) 0 0
\(291\) −3.65531 −0.214278
\(292\) 0 0
\(293\) 0.726619 0.0424495 0.0212248 0.999775i \(-0.493243\pi\)
0.0212248 + 0.999775i \(0.493243\pi\)
\(294\) 0 0
\(295\) 33.8300 1.96966
\(296\) 0 0
\(297\) −4.91239 −0.285046
\(298\) 0 0
\(299\) 0.358382 0.0207258
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0.389120 0.0223544
\(304\) 0 0
\(305\) 15.2218 0.871595
\(306\) 0 0
\(307\) −17.3349 −0.989353 −0.494676 0.869077i \(-0.664713\pi\)
−0.494676 + 0.869077i \(0.664713\pi\)
\(308\) 0 0
\(309\) 0.440128 0.0250380
\(310\) 0 0
\(311\) −3.56244 −0.202008 −0.101004 0.994886i \(-0.532205\pi\)
−0.101004 + 0.994886i \(0.532205\pi\)
\(312\) 0 0
\(313\) −17.4827 −0.988180 −0.494090 0.869411i \(-0.664499\pi\)
−0.494090 + 0.869411i \(0.664499\pi\)
\(314\) 0 0
\(315\) −2.53696 −0.142941
\(316\) 0 0
\(317\) 2.02416 0.113688 0.0568442 0.998383i \(-0.481896\pi\)
0.0568442 + 0.998383i \(0.481896\pi\)
\(318\) 0 0
\(319\) −23.1003 −1.29337
\(320\) 0 0
\(321\) 5.46962 0.305284
\(322\) 0 0
\(323\) −6.84768 −0.381015
\(324\) 0 0
\(325\) −0.237681 −0.0131841
\(326\) 0 0
\(327\) 8.00396 0.442620
\(328\) 0 0
\(329\) 3.88551 0.214215
\(330\) 0 0
\(331\) −1.39438 −0.0766418 −0.0383209 0.999265i \(-0.512201\pi\)
−0.0383209 + 0.999265i \(0.512201\pi\)
\(332\) 0 0
\(333\) −0.746889 −0.0409293
\(334\) 0 0
\(335\) 33.0919 1.80800
\(336\) 0 0
\(337\) −5.54753 −0.302193 −0.151097 0.988519i \(-0.548280\pi\)
−0.151097 + 0.988519i \(0.548280\pi\)
\(338\) 0 0
\(339\) 16.0210 0.870142
\(340\) 0 0
\(341\) −34.7693 −1.88286
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −5.49378 −0.295775
\(346\) 0 0
\(347\) −14.0268 −0.753000 −0.376500 0.926417i \(-0.622873\pi\)
−0.376500 + 0.926417i \(0.622873\pi\)
\(348\) 0 0
\(349\) −17.3591 −0.929211 −0.464605 0.885518i \(-0.653804\pi\)
−0.464605 + 0.885518i \(0.653804\pi\)
\(350\) 0 0
\(351\) −0.165497 −0.00883355
\(352\) 0 0
\(353\) −14.0694 −0.748841 −0.374420 0.927259i \(-0.622158\pi\)
−0.374420 + 0.927259i \(0.622158\pi\)
\(354\) 0 0
\(355\) 20.9712 1.11304
\(356\) 0 0
\(357\) −6.84768 −0.362418
\(358\) 0 0
\(359\) −14.1381 −0.746181 −0.373090 0.927795i \(-0.621702\pi\)
−0.373090 + 0.927795i \(0.621702\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 13.1315 0.689227
\(364\) 0 0
\(365\) −12.8310 −0.671603
\(366\) 0 0
\(367\) −16.3925 −0.855680 −0.427840 0.903855i \(-0.640725\pi\)
−0.427840 + 0.903855i \(0.640725\pi\)
\(368\) 0 0
\(369\) 3.07392 0.160022
\(370\) 0 0
\(371\) 0.702457 0.0364697
\(372\) 0 0
\(373\) 25.0779 1.29848 0.649242 0.760582i \(-0.275086\pi\)
0.649242 + 0.760582i \(0.275086\pi\)
\(374\) 0 0
\(375\) −9.04130 −0.466891
\(376\) 0 0
\(377\) −0.778240 −0.0400814
\(378\) 0 0
\(379\) 9.73329 0.499966 0.249983 0.968250i \(-0.419575\pi\)
0.249983 + 0.968250i \(0.419575\pi\)
\(380\) 0 0
\(381\) 13.9660 0.715502
\(382\) 0 0
\(383\) −36.0420 −1.84166 −0.920830 0.389963i \(-0.872488\pi\)
−0.920830 + 0.389963i \(0.872488\pi\)
\(384\) 0 0
\(385\) 12.4625 0.635149
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 10.9915 0.557292 0.278646 0.960394i \(-0.410114\pi\)
0.278646 + 0.960394i \(0.410114\pi\)
\(390\) 0 0
\(391\) −14.8286 −0.749916
\(392\) 0 0
\(393\) 18.0210 0.909039
\(394\) 0 0
\(395\) −23.3987 −1.17731
\(396\) 0 0
\(397\) 13.3185 0.668439 0.334219 0.942495i \(-0.391527\pi\)
0.334219 + 0.942495i \(0.391527\pi\)
\(398\) 0 0
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) 6.27422 0.313319 0.156660 0.987653i \(-0.449927\pi\)
0.156660 + 0.987653i \(0.449927\pi\)
\(402\) 0 0
\(403\) −1.17137 −0.0583499
\(404\) 0 0
\(405\) 2.53696 0.126063
\(406\) 0 0
\(407\) 3.66901 0.181866
\(408\) 0 0
\(409\) −38.0315 −1.88054 −0.940268 0.340436i \(-0.889425\pi\)
−0.940268 + 0.340436i \(0.889425\pi\)
\(410\) 0 0
\(411\) 5.66901 0.279631
\(412\) 0 0
\(413\) −13.3349 −0.656165
\(414\) 0 0
\(415\) −20.7935 −1.02071
\(416\) 0 0
\(417\) −0.0700563 −0.00343067
\(418\) 0 0
\(419\) 29.0598 1.41966 0.709832 0.704371i \(-0.248771\pi\)
0.709832 + 0.704371i \(0.248771\pi\)
\(420\) 0 0
\(421\) −27.1042 −1.32098 −0.660490 0.750835i \(-0.729651\pi\)
−0.660490 + 0.750835i \(0.729651\pi\)
\(422\) 0 0
\(423\) −3.88551 −0.188920
\(424\) 0 0
\(425\) 9.83441 0.477039
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) 0 0
\(429\) 0.812983 0.0392512
\(430\) 0 0
\(431\) −13.0092 −0.626632 −0.313316 0.949649i \(-0.601440\pi\)
−0.313316 + 0.949649i \(0.601440\pi\)
\(432\) 0 0
\(433\) −14.4651 −0.695150 −0.347575 0.937652i \(-0.612995\pi\)
−0.347575 + 0.937652i \(0.612995\pi\)
\(434\) 0 0
\(435\) 11.9299 0.571997
\(436\) 0 0
\(437\) 2.16550 0.103590
\(438\) 0 0
\(439\) 28.8753 1.37814 0.689071 0.724694i \(-0.258019\pi\)
0.689071 + 0.724694i \(0.258019\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 16.7556 0.796082 0.398041 0.917368i \(-0.369690\pi\)
0.398041 + 0.917368i \(0.369690\pi\)
\(444\) 0 0
\(445\) −23.8185 −1.12911
\(446\) 0 0
\(447\) 13.8208 0.653702
\(448\) 0 0
\(449\) 10.2468 0.483578 0.241789 0.970329i \(-0.422266\pi\)
0.241789 + 0.970329i \(0.422266\pi\)
\(450\) 0 0
\(451\) −15.1003 −0.711044
\(452\) 0 0
\(453\) −3.81820 −0.179394
\(454\) 0 0
\(455\) 0.419858 0.0196833
\(456\) 0 0
\(457\) 32.5403 1.52217 0.761086 0.648651i \(-0.224667\pi\)
0.761086 + 0.648651i \(0.224667\pi\)
\(458\) 0 0
\(459\) 6.84768 0.319622
\(460\) 0 0
\(461\) −35.2199 −1.64035 −0.820177 0.572110i \(-0.806125\pi\)
−0.820177 + 0.572110i \(0.806125\pi\)
\(462\) 0 0
\(463\) 29.2755 1.36055 0.680274 0.732958i \(-0.261861\pi\)
0.680274 + 0.732958i \(0.261861\pi\)
\(464\) 0 0
\(465\) 17.9563 0.832704
\(466\) 0 0
\(467\) 39.6196 1.83338 0.916688 0.399604i \(-0.130852\pi\)
0.916688 + 0.399604i \(0.130852\pi\)
\(468\) 0 0
\(469\) −13.0439 −0.602312
\(470\) 0 0
\(471\) −5.97261 −0.275203
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1.43617 −0.0658958
\(476\) 0 0
\(477\) −0.702457 −0.0321633
\(478\) 0 0
\(479\) 16.1760 0.739099 0.369549 0.929211i \(-0.379512\pi\)
0.369549 + 0.929211i \(0.379512\pi\)
\(480\) 0 0
\(481\) 0.123608 0.00563602
\(482\) 0 0
\(483\) 2.16550 0.0985335
\(484\) 0 0
\(485\) −9.27338 −0.421083
\(486\) 0 0
\(487\) −6.41851 −0.290850 −0.145425 0.989369i \(-0.546455\pi\)
−0.145425 + 0.989369i \(0.546455\pi\)
\(488\) 0 0
\(489\) 9.95630 0.450240
\(490\) 0 0
\(491\) −35.1217 −1.58502 −0.792510 0.609859i \(-0.791226\pi\)
−0.792510 + 0.609859i \(0.791226\pi\)
\(492\) 0 0
\(493\) 32.2009 1.45026
\(494\) 0 0
\(495\) −12.4625 −0.560149
\(496\) 0 0
\(497\) −8.26629 −0.370794
\(498\) 0 0
\(499\) −12.9712 −0.580673 −0.290336 0.956925i \(-0.593767\pi\)
−0.290336 + 0.956925i \(0.593767\pi\)
\(500\) 0 0
\(501\) −1.56383 −0.0698670
\(502\) 0 0
\(503\) 29.2225 1.30297 0.651483 0.758663i \(-0.274147\pi\)
0.651483 + 0.758663i \(0.274147\pi\)
\(504\) 0 0
\(505\) 0.987182 0.0439290
\(506\) 0 0
\(507\) −12.9726 −0.576134
\(508\) 0 0
\(509\) −36.9414 −1.63740 −0.818698 0.574224i \(-0.805304\pi\)
−0.818698 + 0.574224i \(0.805304\pi\)
\(510\) 0 0
\(511\) 5.05761 0.223736
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 1.11659 0.0492027
\(516\) 0 0
\(517\) 19.0871 0.839451
\(518\) 0 0
\(519\) 23.5444 1.03348
\(520\) 0 0
\(521\) −12.6298 −0.553323 −0.276661 0.960967i \(-0.589228\pi\)
−0.276661 + 0.960967i \(0.589228\pi\)
\(522\) 0 0
\(523\) −26.5778 −1.16216 −0.581082 0.813845i \(-0.697370\pi\)
−0.581082 + 0.813845i \(0.697370\pi\)
\(524\) 0 0
\(525\) −1.43617 −0.0626794
\(526\) 0 0
\(527\) 48.4671 2.11126
\(528\) 0 0
\(529\) −18.3106 −0.796114
\(530\) 0 0
\(531\) 13.3349 0.578683
\(532\) 0 0
\(533\) −0.508723 −0.0220352
\(534\) 0 0
\(535\) 13.8762 0.599920
\(536\) 0 0
\(537\) −22.7639 −0.982336
\(538\) 0 0
\(539\) −4.91239 −0.211591
\(540\) 0 0
\(541\) 38.4757 1.65420 0.827099 0.562056i \(-0.189989\pi\)
0.827099 + 0.562056i \(0.189989\pi\)
\(542\) 0 0
\(543\) 8.42643 0.361613
\(544\) 0 0
\(545\) 20.3057 0.869802
\(546\) 0 0
\(547\) 22.7993 0.974827 0.487414 0.873171i \(-0.337940\pi\)
0.487414 + 0.873171i \(0.337940\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) −4.70246 −0.200331
\(552\) 0 0
\(553\) 9.22311 0.392206
\(554\) 0 0
\(555\) −1.89483 −0.0804309
\(556\) 0 0
\(557\) −7.45470 −0.315866 −0.157933 0.987450i \(-0.550483\pi\)
−0.157933 + 0.987450i \(0.550483\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −33.6385 −1.42022
\(562\) 0 0
\(563\) 27.0576 1.14034 0.570171 0.821526i \(-0.306877\pi\)
0.570171 + 0.821526i \(0.306877\pi\)
\(564\) 0 0
\(565\) 40.6446 1.70993
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −26.3256 −1.10363 −0.551814 0.833967i \(-0.686064\pi\)
−0.551814 + 0.833967i \(0.686064\pi\)
\(570\) 0 0
\(571\) 2.96875 0.124238 0.0621191 0.998069i \(-0.480214\pi\)
0.0621191 + 0.998069i \(0.480214\pi\)
\(572\) 0 0
\(573\) 11.9497 0.499207
\(574\) 0 0
\(575\) −3.11001 −0.129696
\(576\) 0 0
\(577\) −24.1637 −1.00595 −0.502974 0.864302i \(-0.667761\pi\)
−0.502974 + 0.864302i \(0.667761\pi\)
\(578\) 0 0
\(579\) 18.6214 0.773881
\(580\) 0 0
\(581\) 8.19623 0.340037
\(582\) 0 0
\(583\) 3.45074 0.142915
\(584\) 0 0
\(585\) −0.419858 −0.0173590
\(586\) 0 0
\(587\) −22.2226 −0.917225 −0.458612 0.888636i \(-0.651653\pi\)
−0.458612 + 0.888636i \(0.651653\pi\)
\(588\) 0 0
\(589\) −7.07788 −0.291639
\(590\) 0 0
\(591\) −20.7195 −0.852286
\(592\) 0 0
\(593\) 22.9576 0.942757 0.471378 0.881931i \(-0.343757\pi\)
0.471378 + 0.881931i \(0.343757\pi\)
\(594\) 0 0
\(595\) −17.3723 −0.712195
\(596\) 0 0
\(597\) 7.05761 0.288849
\(598\) 0 0
\(599\) 26.2847 1.07396 0.536982 0.843593i \(-0.319564\pi\)
0.536982 + 0.843593i \(0.319564\pi\)
\(600\) 0 0
\(601\) −12.9229 −0.527136 −0.263568 0.964641i \(-0.584899\pi\)
−0.263568 + 0.964641i \(0.584899\pi\)
\(602\) 0 0
\(603\) 13.0439 0.531189
\(604\) 0 0
\(605\) 33.3142 1.35441
\(606\) 0 0
\(607\) 36.4127 1.47795 0.738974 0.673734i \(-0.235311\pi\)
0.738974 + 0.673734i \(0.235311\pi\)
\(608\) 0 0
\(609\) −4.70246 −0.190553
\(610\) 0 0
\(611\) 0.643039 0.0260146
\(612\) 0 0
\(613\) 35.5444 1.43562 0.717812 0.696237i \(-0.245144\pi\)
0.717812 + 0.696237i \(0.245144\pi\)
\(614\) 0 0
\(615\) 7.79841 0.314462
\(616\) 0 0
\(617\) −14.3231 −0.576625 −0.288313 0.957536i \(-0.593094\pi\)
−0.288313 + 0.957536i \(0.593094\pi\)
\(618\) 0 0
\(619\) −2.83934 −0.114123 −0.0570614 0.998371i \(-0.518173\pi\)
−0.0570614 + 0.998371i \(0.518173\pi\)
\(620\) 0 0
\(621\) −2.16550 −0.0868984
\(622\) 0 0
\(623\) 9.38861 0.376147
\(624\) 0 0
\(625\) −30.1183 −1.20473
\(626\) 0 0
\(627\) 4.91239 0.196182
\(628\) 0 0
\(629\) −5.11446 −0.203927
\(630\) 0 0
\(631\) 15.4827 0.616356 0.308178 0.951329i \(-0.400281\pi\)
0.308178 + 0.951329i \(0.400281\pi\)
\(632\) 0 0
\(633\) −8.84233 −0.351451
\(634\) 0 0
\(635\) 35.4313 1.40605
\(636\) 0 0
\(637\) −0.165497 −0.00655721
\(638\) 0 0
\(639\) 8.26629 0.327009
\(640\) 0 0
\(641\) −14.0321 −0.554234 −0.277117 0.960836i \(-0.589379\pi\)
−0.277117 + 0.960836i \(0.589379\pi\)
\(642\) 0 0
\(643\) 2.02636 0.0799118 0.0399559 0.999201i \(-0.487278\pi\)
0.0399559 + 0.999201i \(0.487278\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 37.9250 1.49099 0.745493 0.666513i \(-0.232214\pi\)
0.745493 + 0.666513i \(0.232214\pi\)
\(648\) 0 0
\(649\) −65.5060 −2.57133
\(650\) 0 0
\(651\) −7.07788 −0.277404
\(652\) 0 0
\(653\) −41.0100 −1.60484 −0.802422 0.596757i \(-0.796455\pi\)
−0.802422 + 0.596757i \(0.796455\pi\)
\(654\) 0 0
\(655\) 45.7186 1.78637
\(656\) 0 0
\(657\) −5.05761 −0.197316
\(658\) 0 0
\(659\) −0.389898 −0.0151883 −0.00759414 0.999971i \(-0.502417\pi\)
−0.00759414 + 0.999971i \(0.502417\pi\)
\(660\) 0 0
\(661\) 24.1734 0.940237 0.470119 0.882603i \(-0.344211\pi\)
0.470119 + 0.882603i \(0.344211\pi\)
\(662\) 0 0
\(663\) −1.13327 −0.0440125
\(664\) 0 0
\(665\) 2.53696 0.0983791
\(666\) 0 0
\(667\) −10.1832 −0.394293
\(668\) 0 0
\(669\) −8.28121 −0.320170
\(670\) 0 0
\(671\) −29.4743 −1.13784
\(672\) 0 0
\(673\) 16.3387 0.629811 0.314906 0.949123i \(-0.398027\pi\)
0.314906 + 0.949123i \(0.398027\pi\)
\(674\) 0 0
\(675\) 1.43617 0.0552780
\(676\) 0 0
\(677\) −11.6932 −0.449408 −0.224704 0.974427i \(-0.572141\pi\)
−0.224704 + 0.974427i \(0.572141\pi\)
\(678\) 0 0
\(679\) 3.65531 0.140278
\(680\) 0 0
\(681\) −3.09815 −0.118721
\(682\) 0 0
\(683\) 14.3115 0.547613 0.273806 0.961785i \(-0.411717\pi\)
0.273806 + 0.961785i \(0.411717\pi\)
\(684\) 0 0
\(685\) 14.3820 0.549510
\(686\) 0 0
\(687\) 5.24915 0.200268
\(688\) 0 0
\(689\) 0.116254 0.00442893
\(690\) 0 0
\(691\) 46.4109 1.76555 0.882777 0.469792i \(-0.155671\pi\)
0.882777 + 0.469792i \(0.155671\pi\)
\(692\) 0 0
\(693\) 4.91239 0.186606
\(694\) 0 0
\(695\) −0.177730 −0.00674169
\(696\) 0 0
\(697\) 21.0492 0.797296
\(698\) 0 0
\(699\) −3.08184 −0.116566
\(700\) 0 0
\(701\) −23.8825 −0.902029 −0.451014 0.892517i \(-0.648938\pi\)
−0.451014 + 0.892517i \(0.648938\pi\)
\(702\) 0 0
\(703\) 0.746889 0.0281695
\(704\) 0 0
\(705\) −9.85739 −0.371251
\(706\) 0 0
\(707\) −0.389120 −0.0146344
\(708\) 0 0
\(709\) 42.8362 1.60875 0.804373 0.594124i \(-0.202501\pi\)
0.804373 + 0.594124i \(0.202501\pi\)
\(710\) 0 0
\(711\) −9.22311 −0.345894
\(712\) 0 0
\(713\) −15.3271 −0.574006
\(714\) 0 0
\(715\) 2.06251 0.0771333
\(716\) 0 0
\(717\) 13.6055 0.508108
\(718\) 0 0
\(719\) −4.07918 −0.152128 −0.0760638 0.997103i \(-0.524235\pi\)
−0.0760638 + 0.997103i \(0.524235\pi\)
\(720\) 0 0
\(721\) −0.440128 −0.0163912
\(722\) 0 0
\(723\) 19.4450 0.723166
\(724\) 0 0
\(725\) 6.75351 0.250819
\(726\) 0 0
\(727\) 50.0525 1.85635 0.928173 0.372150i \(-0.121379\pi\)
0.928173 + 0.372150i \(0.121379\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 19.5879 0.723494 0.361747 0.932276i \(-0.382180\pi\)
0.361747 + 0.932276i \(0.382180\pi\)
\(734\) 0 0
\(735\) 2.53696 0.0935772
\(736\) 0 0
\(737\) −64.0768 −2.36030
\(738\) 0 0
\(739\) 0.530454 0.0195131 0.00975653 0.999952i \(-0.496894\pi\)
0.00975653 + 0.999952i \(0.496894\pi\)
\(740\) 0 0
\(741\) 0.165497 0.00607967
\(742\) 0 0
\(743\) −41.3402 −1.51663 −0.758313 0.651891i \(-0.773976\pi\)
−0.758313 + 0.651891i \(0.773976\pi\)
\(744\) 0 0
\(745\) 35.0628 1.28460
\(746\) 0 0
\(747\) −8.19623 −0.299885
\(748\) 0 0
\(749\) −5.46962 −0.199855
\(750\) 0 0
\(751\) 47.7555 1.74262 0.871311 0.490731i \(-0.163270\pi\)
0.871311 + 0.490731i \(0.163270\pi\)
\(752\) 0 0
\(753\) 3.90578 0.142335
\(754\) 0 0
\(755\) −9.68661 −0.352532
\(756\) 0 0
\(757\) −7.75259 −0.281773 −0.140886 0.990026i \(-0.544995\pi\)
−0.140886 + 0.990026i \(0.544995\pi\)
\(758\) 0 0
\(759\) 10.6378 0.386126
\(760\) 0 0
\(761\) −17.3618 −0.629366 −0.314683 0.949197i \(-0.601898\pi\)
−0.314683 + 0.949197i \(0.601898\pi\)
\(762\) 0 0
\(763\) −8.00396 −0.289763
\(764\) 0 0
\(765\) 17.3723 0.628097
\(766\) 0 0
\(767\) −2.20687 −0.0796856
\(768\) 0 0
\(769\) −18.6026 −0.670828 −0.335414 0.942071i \(-0.608876\pi\)
−0.335414 + 0.942071i \(0.608876\pi\)
\(770\) 0 0
\(771\) −23.9563 −0.862765
\(772\) 0 0
\(773\) 3.59122 0.129167 0.0645837 0.997912i \(-0.479428\pi\)
0.0645837 + 0.997912i \(0.479428\pi\)
\(774\) 0 0
\(775\) 10.1650 0.365138
\(776\) 0 0
\(777\) 0.746889 0.0267945
\(778\) 0 0
\(779\) −3.07392 −0.110135
\(780\) 0 0
\(781\) −40.6072 −1.45304
\(782\) 0 0
\(783\) 4.70246 0.168052
\(784\) 0 0
\(785\) −15.1523 −0.540808
\(786\) 0 0
\(787\) −40.4157 −1.44066 −0.720332 0.693630i \(-0.756010\pi\)
−0.720332 + 0.693630i \(0.756010\pi\)
\(788\) 0 0
\(789\) −22.3671 −0.796289
\(790\) 0 0
\(791\) −16.0210 −0.569641
\(792\) 0 0
\(793\) −0.992979 −0.0352617
\(794\) 0 0
\(795\) −1.78210 −0.0632047
\(796\) 0 0
\(797\) 16.7124 0.591983 0.295991 0.955191i \(-0.404350\pi\)
0.295991 + 0.955191i \(0.404350\pi\)
\(798\) 0 0
\(799\) −26.6067 −0.941279
\(800\) 0 0
\(801\) −9.38861 −0.331730
\(802\) 0 0
\(803\) 24.8449 0.876759
\(804\) 0 0
\(805\) 5.49378 0.193630
\(806\) 0 0
\(807\) 3.98369 0.140233
\(808\) 0 0
\(809\) −33.0324 −1.16136 −0.580678 0.814133i \(-0.697212\pi\)
−0.580678 + 0.814133i \(0.697212\pi\)
\(810\) 0 0
\(811\) −9.55804 −0.335628 −0.167814 0.985819i \(-0.553671\pi\)
−0.167814 + 0.985819i \(0.553671\pi\)
\(812\) 0 0
\(813\) 5.53432 0.194097
\(814\) 0 0
\(815\) 25.2587 0.884775
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0.165497 0.00578292
\(820\) 0 0
\(821\) 32.8489 1.14643 0.573217 0.819403i \(-0.305695\pi\)
0.573217 + 0.819403i \(0.305695\pi\)
\(822\) 0 0
\(823\) −12.2716 −0.427763 −0.213881 0.976860i \(-0.568611\pi\)
−0.213881 + 0.976860i \(0.568611\pi\)
\(824\) 0 0
\(825\) −7.05500 −0.245624
\(826\) 0 0
\(827\) −34.9302 −1.21464 −0.607321 0.794456i \(-0.707756\pi\)
−0.607321 + 0.794456i \(0.707756\pi\)
\(828\) 0 0
\(829\) −13.7099 −0.476163 −0.238082 0.971245i \(-0.576519\pi\)
−0.238082 + 0.971245i \(0.576519\pi\)
\(830\) 0 0
\(831\) 11.2408 0.389938
\(832\) 0 0
\(833\) 6.84768 0.237258
\(834\) 0 0
\(835\) −3.96739 −0.137297
\(836\) 0 0
\(837\) 7.07788 0.244647
\(838\) 0 0
\(839\) −48.2116 −1.66445 −0.832225 0.554438i \(-0.812933\pi\)
−0.832225 + 0.554438i \(0.812933\pi\)
\(840\) 0 0
\(841\) −6.88690 −0.237479
\(842\) 0 0
\(843\) −13.7764 −0.474484
\(844\) 0 0
\(845\) −32.9110 −1.13217
\(846\) 0 0
\(847\) −13.1315 −0.451205
\(848\) 0 0
\(849\) −23.3533 −0.801483
\(850\) 0 0
\(851\) 1.61739 0.0554433
\(852\) 0 0
\(853\) 1.38648 0.0474721 0.0237360 0.999718i \(-0.492444\pi\)
0.0237360 + 0.999718i \(0.492444\pi\)
\(854\) 0 0
\(855\) −2.53696 −0.0867622
\(856\) 0 0
\(857\) −12.1984 −0.416690 −0.208345 0.978055i \(-0.566808\pi\)
−0.208345 + 0.978055i \(0.566808\pi\)
\(858\) 0 0
\(859\) −20.7293 −0.707273 −0.353637 0.935383i \(-0.615055\pi\)
−0.353637 + 0.935383i \(0.615055\pi\)
\(860\) 0 0
\(861\) −3.07392 −0.104759
\(862\) 0 0
\(863\) −46.1843 −1.57213 −0.786066 0.618142i \(-0.787886\pi\)
−0.786066 + 0.618142i \(0.787886\pi\)
\(864\) 0 0
\(865\) 59.7311 2.03092
\(866\) 0 0
\(867\) 29.8908 1.01514
\(868\) 0 0
\(869\) 45.3075 1.53695
\(870\) 0 0
\(871\) −2.15872 −0.0731456
\(872\) 0 0
\(873\) −3.65531 −0.123714
\(874\) 0 0
\(875\) 9.04130 0.305652
\(876\) 0 0
\(877\) 0.232940 0.00786582 0.00393291 0.999992i \(-0.498748\pi\)
0.00393291 + 0.999992i \(0.498748\pi\)
\(878\) 0 0
\(879\) 0.726619 0.0245083
\(880\) 0 0
\(881\) −57.8591 −1.94932 −0.974661 0.223686i \(-0.928191\pi\)
−0.974661 + 0.223686i \(0.928191\pi\)
\(882\) 0 0
\(883\) −46.6825 −1.57099 −0.785496 0.618866i \(-0.787592\pi\)
−0.785496 + 0.618866i \(0.787592\pi\)
\(884\) 0 0
\(885\) 33.8300 1.13718
\(886\) 0 0
\(887\) −53.0508 −1.78127 −0.890636 0.454718i \(-0.849740\pi\)
−0.890636 + 0.454718i \(0.849740\pi\)
\(888\) 0 0
\(889\) −13.9660 −0.468406
\(890\) 0 0
\(891\) −4.91239 −0.164571
\(892\) 0 0
\(893\) 3.88551 0.130024
\(894\) 0 0
\(895\) −57.7512 −1.93041
\(896\) 0 0
\(897\) 0.358382 0.0119660
\(898\) 0 0
\(899\) 33.2834 1.11006
\(900\) 0 0
\(901\) −4.81020 −0.160251
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 21.3775 0.710613
\(906\) 0 0
\(907\) −4.33361 −0.143895 −0.0719475 0.997408i \(-0.522921\pi\)
−0.0719475 + 0.997408i \(0.522921\pi\)
\(908\) 0 0
\(909\) 0.389120 0.0129063
\(910\) 0 0
\(911\) −58.7959 −1.94799 −0.973997 0.226559i \(-0.927252\pi\)
−0.973997 + 0.226559i \(0.927252\pi\)
\(912\) 0 0
\(913\) 40.2631 1.33251
\(914\) 0 0
\(915\) 15.2218 0.503216
\(916\) 0 0
\(917\) −18.0210 −0.595106
\(918\) 0 0
\(919\) −4.31153 −0.142224 −0.0711121 0.997468i \(-0.522655\pi\)
−0.0711121 + 0.997468i \(0.522655\pi\)
\(920\) 0 0
\(921\) −17.3349 −0.571203
\(922\) 0 0
\(923\) −1.36804 −0.0450297
\(924\) 0 0
\(925\) −1.07266 −0.0352687
\(926\) 0 0
\(927\) 0.440128 0.0144557
\(928\) 0 0
\(929\) −51.2454 −1.68131 −0.840653 0.541574i \(-0.817829\pi\)
−0.840653 + 0.541574i \(0.817829\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) −3.56244 −0.116629
\(934\) 0 0
\(935\) −85.3394 −2.79090
\(936\) 0 0
\(937\) −7.97577 −0.260557 −0.130278 0.991477i \(-0.541587\pi\)
−0.130278 + 0.991477i \(0.541587\pi\)
\(938\) 0 0
\(939\) −17.4827 −0.570526
\(940\) 0 0
\(941\) 36.9087 1.20319 0.601595 0.798801i \(-0.294532\pi\)
0.601595 + 0.798801i \(0.294532\pi\)
\(942\) 0 0
\(943\) −6.65656 −0.216768
\(944\) 0 0
\(945\) −2.53696 −0.0825273
\(946\) 0 0
\(947\) −39.3586 −1.27898 −0.639491 0.768798i \(-0.720855\pi\)
−0.639491 + 0.768798i \(0.720855\pi\)
\(948\) 0 0
\(949\) 0.837018 0.0271707
\(950\) 0 0
\(951\) 2.02416 0.0656380
\(952\) 0 0
\(953\) −39.4075 −1.27653 −0.638267 0.769815i \(-0.720348\pi\)
−0.638267 + 0.769815i \(0.720348\pi\)
\(954\) 0 0
\(955\) 30.3160 0.981002
\(956\) 0 0
\(957\) −23.1003 −0.746726
\(958\) 0 0
\(959\) −5.66901 −0.183062
\(960\) 0 0
\(961\) 19.0964 0.616013
\(962\) 0 0
\(963\) 5.46962 0.176256
\(964\) 0 0
\(965\) 47.2419 1.52077
\(966\) 0 0
\(967\) 48.0636 1.54562 0.772811 0.634637i \(-0.218850\pi\)
0.772811 + 0.634637i \(0.218850\pi\)
\(968\) 0 0
\(969\) −6.84768 −0.219979
\(970\) 0 0
\(971\) −24.1041 −0.773539 −0.386769 0.922176i \(-0.626409\pi\)
−0.386769 + 0.922176i \(0.626409\pi\)
\(972\) 0 0
\(973\) 0.0700563 0.00224590
\(974\) 0 0
\(975\) −0.237681 −0.00761187
\(976\) 0 0
\(977\) 33.0840 1.05845 0.529226 0.848481i \(-0.322482\pi\)
0.529226 + 0.848481i \(0.322482\pi\)
\(978\) 0 0
\(979\) 46.1205 1.47402
\(980\) 0 0
\(981\) 8.00396 0.255547
\(982\) 0 0
\(983\) 15.2724 0.487112 0.243556 0.969887i \(-0.421686\pi\)
0.243556 + 0.969887i \(0.421686\pi\)
\(984\) 0 0
\(985\) −52.5645 −1.67485
\(986\) 0 0
\(987\) 3.88551 0.123677
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 13.5946 0.431848 0.215924 0.976410i \(-0.430724\pi\)
0.215924 + 0.976410i \(0.430724\pi\)
\(992\) 0 0
\(993\) −1.39438 −0.0442492
\(994\) 0 0
\(995\) 17.9049 0.567623
\(996\) 0 0
\(997\) 5.15506 0.163262 0.0816312 0.996663i \(-0.473987\pi\)
0.0816312 + 0.996663i \(0.473987\pi\)
\(998\) 0 0
\(999\) −0.746889 −0.0236305
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 6384.2.a.ce.1.4 5
4.3 odd 2 3192.2.a.ba.1.4 5
12.11 even 2 9576.2.a.co.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.ba.1.4 5 4.3 odd 2
6384.2.a.ce.1.4 5 1.1 even 1 trivial
9576.2.a.co.1.2 5 12.11 even 2