Properties

Label 624.2.a.k.1.1
Level $624$
Weight $2$
Character 624.1
Self dual yes
Analytic conductor $4.983$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 624.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} +2.82843 q^{15} +7.65685 q^{17} +2.82843 q^{19} +2.82843 q^{21} +4.00000 q^{23} +3.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +1.17157 q^{31} -2.00000 q^{33} +8.00000 q^{35} -7.65685 q^{37} +1.00000 q^{39} +5.17157 q^{41} +1.65685 q^{43} -2.82843 q^{45} +11.6569 q^{47} +1.00000 q^{49} -7.65685 q^{51} -2.00000 q^{53} -5.65685 q^{55} -2.82843 q^{57} -7.65685 q^{59} +13.3137 q^{61} -2.82843 q^{63} +2.82843 q^{65} -6.82843 q^{67} -4.00000 q^{69} -2.00000 q^{71} +0.343146 q^{73} -3.00000 q^{75} -5.65685 q^{77} +11.3137 q^{79} +1.00000 q^{81} -3.65685 q^{83} -21.6569 q^{85} -2.00000 q^{87} +14.8284 q^{89} +2.82843 q^{91} -1.17157 q^{93} -8.00000 q^{95} +3.65685 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 2 q^{9} + 4 q^{11} - 2 q^{13} + 4 q^{17} + 8 q^{23} + 6 q^{25} - 2 q^{27} + 4 q^{29} + 8 q^{31} - 4 q^{33} + 16 q^{35} - 4 q^{37} + 2 q^{39} + 16 q^{41} - 8 q^{43} + 12 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} - 4 q^{59} + 4 q^{61} - 8 q^{67} - 8 q^{69} - 4 q^{71} + 12 q^{73} - 6 q^{75} + 2 q^{81} + 4 q^{83} - 32 q^{85} - 4 q^{87} + 24 q^{89} - 8 q^{93} - 16 q^{95} - 4 q^{97} + 4 q^{99} + O(q^{100}) \)

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
\(5\) −2.82843 −1.26491 −0.632456 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 2.82843 0.730297
\(16\) 0 0
\(17\) 7.65685 1.85706 0.928530 0.371257i \(-0.121073\pi\)
0.928530 + 0.371257i \(0.121073\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 2.82843 0.617213
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 1.17157 0.210421 0.105210 0.994450i \(-0.466448\pi\)
0.105210 + 0.994450i \(0.466448\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 8.00000 1.35225
\(36\) 0 0
\(37\) −7.65685 −1.25878 −0.629390 0.777090i \(-0.716695\pi\)
−0.629390 + 0.777090i \(0.716695\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 5.17157 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(42\) 0 0
\(43\) 1.65685 0.252668 0.126334 0.991988i \(-0.459679\pi\)
0.126334 + 0.991988i \(0.459679\pi\)
\(44\) 0 0
\(45\) −2.82843 −0.421637
\(46\) 0 0
\(47\) 11.6569 1.70033 0.850163 0.526519i \(-0.176503\pi\)
0.850163 + 0.526519i \(0.176503\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −7.65685 −1.07217
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) −2.82843 −0.374634
\(58\) 0 0
\(59\) −7.65685 −0.996838 −0.498419 0.866936i \(-0.666086\pi\)
−0.498419 + 0.866936i \(0.666086\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0 0
\(63\) −2.82843 −0.356348
\(64\) 0 0
\(65\) 2.82843 0.350823
\(66\) 0 0
\(67\) −6.82843 −0.834225 −0.417113 0.908855i \(-0.636958\pi\)
−0.417113 + 0.908855i \(0.636958\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 0.343146 0.0401622 0.0200811 0.999798i \(-0.493608\pi\)
0.0200811 + 0.999798i \(0.493608\pi\)
\(74\) 0 0
\(75\) −3.00000 −0.346410
\(76\) 0 0
\(77\) −5.65685 −0.644658
\(78\) 0 0
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.65685 −0.401392 −0.200696 0.979654i \(-0.564320\pi\)
−0.200696 + 0.979654i \(0.564320\pi\)
\(84\) 0 0
\(85\) −21.6569 −2.34902
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 14.8284 1.57181 0.785905 0.618347i \(-0.212197\pi\)
0.785905 + 0.618347i \(0.212197\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 0 0
\(93\) −1.17157 −0.121486
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) 3.65685 0.371297 0.185649 0.982616i \(-0.440561\pi\)
0.185649 + 0.982616i \(0.440561\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 7.65685 0.761885 0.380943 0.924599i \(-0.375599\pi\)
0.380943 + 0.924599i \(0.375599\pi\)
\(102\) 0 0
\(103\) −2.34315 −0.230877 −0.115439 0.993315i \(-0.536827\pi\)
−0.115439 + 0.993315i \(0.536827\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) 0 0
\(107\) 11.3137 1.09374 0.546869 0.837218i \(-0.315820\pi\)
0.546869 + 0.837218i \(0.315820\pi\)
\(108\) 0 0
\(109\) 5.31371 0.508961 0.254480 0.967078i \(-0.418096\pi\)
0.254480 + 0.967078i \(0.418096\pi\)
\(110\) 0 0
\(111\) 7.65685 0.726756
\(112\) 0 0
\(113\) −5.31371 −0.499872 −0.249936 0.968262i \(-0.580410\pi\)
−0.249936 + 0.968262i \(0.580410\pi\)
\(114\) 0 0
\(115\) −11.3137 −1.05501
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −21.6569 −1.98528
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −5.17157 −0.466305
\(124\) 0 0
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) −5.65685 −0.501965 −0.250982 0.967992i \(-0.580754\pi\)
−0.250982 + 0.967992i \(0.580754\pi\)
\(128\) 0 0
\(129\) −1.65685 −0.145878
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) 2.82843 0.243432
\(136\) 0 0
\(137\) −10.8284 −0.925135 −0.462567 0.886584i \(-0.653072\pi\)
−0.462567 + 0.886584i \(0.653072\pi\)
\(138\) 0 0
\(139\) 7.31371 0.620341 0.310170 0.950681i \(-0.399614\pi\)
0.310170 + 0.950681i \(0.399614\pi\)
\(140\) 0 0
\(141\) −11.6569 −0.981684
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) −5.65685 −0.469776
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −9.17157 −0.751365 −0.375682 0.926749i \(-0.622592\pi\)
−0.375682 + 0.926749i \(0.622592\pi\)
\(150\) 0 0
\(151\) 3.51472 0.286024 0.143012 0.989721i \(-0.454321\pi\)
0.143012 + 0.989721i \(0.454321\pi\)
\(152\) 0 0
\(153\) 7.65685 0.619020
\(154\) 0 0
\(155\) −3.31371 −0.266163
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) −11.3137 −0.891645
\(162\) 0 0
\(163\) −18.8284 −1.47476 −0.737378 0.675480i \(-0.763936\pi\)
−0.737378 + 0.675480i \(0.763936\pi\)
\(164\) 0 0
\(165\) 5.65685 0.440386
\(166\) 0 0
\(167\) 3.65685 0.282976 0.141488 0.989940i \(-0.454811\pi\)
0.141488 + 0.989940i \(0.454811\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) 0 0
\(173\) −11.6569 −0.886254 −0.443127 0.896459i \(-0.646131\pi\)
−0.443127 + 0.896459i \(0.646131\pi\)
\(174\) 0 0
\(175\) −8.48528 −0.641427
\(176\) 0 0
\(177\) 7.65685 0.575524
\(178\) 0 0
\(179\) 23.3137 1.74255 0.871274 0.490797i \(-0.163294\pi\)
0.871274 + 0.490797i \(0.163294\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) −13.3137 −0.984178
\(184\) 0 0
\(185\) 21.6569 1.59224
\(186\) 0 0
\(187\) 15.3137 1.11985
\(188\) 0 0
\(189\) 2.82843 0.205738
\(190\) 0 0
\(191\) −3.31371 −0.239772 −0.119886 0.992788i \(-0.538253\pi\)
−0.119886 + 0.992788i \(0.538253\pi\)
\(192\) 0 0
\(193\) 5.31371 0.382489 0.191245 0.981542i \(-0.438748\pi\)
0.191245 + 0.981542i \(0.438748\pi\)
\(194\) 0 0
\(195\) −2.82843 −0.202548
\(196\) 0 0
\(197\) 0.485281 0.0345749 0.0172874 0.999851i \(-0.494497\pi\)
0.0172874 + 0.999851i \(0.494497\pi\)
\(198\) 0 0
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) 0 0
\(201\) 6.82843 0.481640
\(202\) 0 0
\(203\) −5.65685 −0.397033
\(204\) 0 0
\(205\) −14.6274 −1.02162
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 5.65685 0.391293
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 2.00000 0.137038
\(214\) 0 0
\(215\) −4.68629 −0.319602
\(216\) 0 0
\(217\) −3.31371 −0.224949
\(218\) 0 0
\(219\) −0.343146 −0.0231876
\(220\) 0 0
\(221\) −7.65685 −0.515056
\(222\) 0 0
\(223\) 12.4853 0.836076 0.418038 0.908429i \(-0.362718\pi\)
0.418038 + 0.908429i \(0.362718\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 0 0
\(227\) 17.3137 1.14915 0.574576 0.818452i \(-0.305167\pi\)
0.574576 + 0.818452i \(0.305167\pi\)
\(228\) 0 0
\(229\) −1.31371 −0.0868123 −0.0434062 0.999058i \(-0.513821\pi\)
−0.0434062 + 0.999058i \(0.513821\pi\)
\(230\) 0 0
\(231\) 5.65685 0.372194
\(232\) 0 0
\(233\) 6.97056 0.456657 0.228328 0.973584i \(-0.426674\pi\)
0.228328 + 0.973584i \(0.426674\pi\)
\(234\) 0 0
\(235\) −32.9706 −2.15076
\(236\) 0 0
\(237\) −11.3137 −0.734904
\(238\) 0 0
\(239\) −2.00000 −0.129369 −0.0646846 0.997906i \(-0.520604\pi\)
−0.0646846 + 0.997906i \(0.520604\pi\)
\(240\) 0 0
\(241\) 0.343146 0.0221040 0.0110520 0.999939i \(-0.496482\pi\)
0.0110520 + 0.999939i \(0.496482\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.82843 −0.180702
\(246\) 0 0
\(247\) −2.82843 −0.179969
\(248\) 0 0
\(249\) 3.65685 0.231744
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) 21.6569 1.35620
\(256\) 0 0
\(257\) −4.34315 −0.270918 −0.135459 0.990783i \(-0.543251\pi\)
−0.135459 + 0.990783i \(0.543251\pi\)
\(258\) 0 0
\(259\) 21.6569 1.34569
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 5.65685 0.347498
\(266\) 0 0
\(267\) −14.8284 −0.907485
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 27.7990 1.68867 0.844334 0.535817i \(-0.179996\pi\)
0.844334 + 0.535817i \(0.179996\pi\)
\(272\) 0 0
\(273\) −2.82843 −0.171184
\(274\) 0 0
\(275\) 6.00000 0.361814
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 1.17157 0.0701402
\(280\) 0 0
\(281\) 21.1716 1.26299 0.631495 0.775380i \(-0.282442\pi\)
0.631495 + 0.775380i \(0.282442\pi\)
\(282\) 0 0
\(283\) −28.9706 −1.72212 −0.861061 0.508502i \(-0.830199\pi\)
−0.861061 + 0.508502i \(0.830199\pi\)
\(284\) 0 0
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) −14.6274 −0.863429
\(288\) 0 0
\(289\) 41.6274 2.44867
\(290\) 0 0
\(291\) −3.65685 −0.214369
\(292\) 0 0
\(293\) 2.14214 0.125145 0.0625724 0.998040i \(-0.480070\pi\)
0.0625724 + 0.998040i \(0.480070\pi\)
\(294\) 0 0
\(295\) 21.6569 1.26091
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) 0 0
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) −4.68629 −0.270113
\(302\) 0 0
\(303\) −7.65685 −0.439875
\(304\) 0 0
\(305\) −37.6569 −2.15623
\(306\) 0 0
\(307\) 22.8284 1.30289 0.651444 0.758697i \(-0.274164\pi\)
0.651444 + 0.758697i \(0.274164\pi\)
\(308\) 0 0
\(309\) 2.34315 0.133297
\(310\) 0 0
\(311\) 10.6274 0.602626 0.301313 0.953525i \(-0.402575\pi\)
0.301313 + 0.953525i \(0.402575\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) 8.00000 0.450749
\(316\) 0 0
\(317\) 8.48528 0.476581 0.238290 0.971194i \(-0.423413\pi\)
0.238290 + 0.971194i \(0.423413\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) −11.3137 −0.631470
\(322\) 0 0
\(323\) 21.6569 1.20502
\(324\) 0 0
\(325\) −3.00000 −0.166410
\(326\) 0 0
\(327\) −5.31371 −0.293849
\(328\) 0 0
\(329\) −32.9706 −1.81773
\(330\) 0 0
\(331\) −26.1421 −1.43690 −0.718451 0.695578i \(-0.755148\pi\)
−0.718451 + 0.695578i \(0.755148\pi\)
\(332\) 0 0
\(333\) −7.65685 −0.419593
\(334\) 0 0
\(335\) 19.3137 1.05522
\(336\) 0 0
\(337\) 9.31371 0.507350 0.253675 0.967290i \(-0.418361\pi\)
0.253675 + 0.967290i \(0.418361\pi\)
\(338\) 0 0
\(339\) 5.31371 0.288601
\(340\) 0 0
\(341\) 2.34315 0.126888
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 11.3137 0.609110
\(346\) 0 0
\(347\) 8.68629 0.466305 0.233152 0.972440i \(-0.425096\pi\)
0.233152 + 0.972440i \(0.425096\pi\)
\(348\) 0 0
\(349\) 3.65685 0.195747 0.0978735 0.995199i \(-0.468796\pi\)
0.0978735 + 0.995199i \(0.468796\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −33.4558 −1.78067 −0.890337 0.455301i \(-0.849532\pi\)
−0.890337 + 0.455301i \(0.849532\pi\)
\(354\) 0 0
\(355\) 5.65685 0.300235
\(356\) 0 0
\(357\) 21.6569 1.14620
\(358\) 0 0
\(359\) −34.9706 −1.84568 −0.922838 0.385189i \(-0.874136\pi\)
−0.922838 + 0.385189i \(0.874136\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) −0.970563 −0.0508016
\(366\) 0 0
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 0 0
\(369\) 5.17157 0.269221
\(370\) 0 0
\(371\) 5.65685 0.293689
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) −5.65685 −0.292119
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −0.485281 −0.0249272 −0.0124636 0.999922i \(-0.503967\pi\)
−0.0124636 + 0.999922i \(0.503967\pi\)
\(380\) 0 0
\(381\) 5.65685 0.289809
\(382\) 0 0
\(383\) −30.9706 −1.58252 −0.791261 0.611479i \(-0.790575\pi\)
−0.791261 + 0.611479i \(0.790575\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) 0 0
\(387\) 1.65685 0.0842226
\(388\) 0 0
\(389\) −26.9706 −1.36746 −0.683731 0.729734i \(-0.739644\pi\)
−0.683731 + 0.729734i \(0.739644\pi\)
\(390\) 0 0
\(391\) 30.6274 1.54890
\(392\) 0 0
\(393\) −8.00000 −0.403547
\(394\) 0 0
\(395\) −32.0000 −1.61009
\(396\) 0 0
\(397\) 30.9706 1.55437 0.777184 0.629273i \(-0.216647\pi\)
0.777184 + 0.629273i \(0.216647\pi\)
\(398\) 0 0
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) 26.1421 1.30548 0.652738 0.757584i \(-0.273620\pi\)
0.652738 + 0.757584i \(0.273620\pi\)
\(402\) 0 0
\(403\) −1.17157 −0.0583602
\(404\) 0 0
\(405\) −2.82843 −0.140546
\(406\) 0 0
\(407\) −15.3137 −0.759072
\(408\) 0 0
\(409\) −34.9706 −1.72918 −0.864592 0.502475i \(-0.832423\pi\)
−0.864592 + 0.502475i \(0.832423\pi\)
\(410\) 0 0
\(411\) 10.8284 0.534127
\(412\) 0 0
\(413\) 21.6569 1.06566
\(414\) 0 0
\(415\) 10.3431 0.507725
\(416\) 0 0
\(417\) −7.31371 −0.358154
\(418\) 0 0
\(419\) −14.6274 −0.714596 −0.357298 0.933990i \(-0.616302\pi\)
−0.357298 + 0.933990i \(0.616302\pi\)
\(420\) 0 0
\(421\) 37.3137 1.81856 0.909279 0.416186i \(-0.136634\pi\)
0.909279 + 0.416186i \(0.136634\pi\)
\(422\) 0 0
\(423\) 11.6569 0.566776
\(424\) 0 0
\(425\) 22.9706 1.11424
\(426\) 0 0
\(427\) −37.6569 −1.82234
\(428\) 0 0
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −8.34315 −0.401875 −0.200938 0.979604i \(-0.564399\pi\)
−0.200938 + 0.979604i \(0.564399\pi\)
\(432\) 0 0
\(433\) −21.3137 −1.02427 −0.512136 0.858905i \(-0.671146\pi\)
−0.512136 + 0.858905i \(0.671146\pi\)
\(434\) 0 0
\(435\) 5.65685 0.271225
\(436\) 0 0
\(437\) 11.3137 0.541208
\(438\) 0 0
\(439\) −16.9706 −0.809961 −0.404980 0.914325i \(-0.632722\pi\)
−0.404980 + 0.914325i \(0.632722\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 25.9411 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(444\) 0 0
\(445\) −41.9411 −1.98820
\(446\) 0 0
\(447\) 9.17157 0.433801
\(448\) 0 0
\(449\) −31.7990 −1.50069 −0.750344 0.661048i \(-0.770112\pi\)
−0.750344 + 0.661048i \(0.770112\pi\)
\(450\) 0 0
\(451\) 10.3431 0.487040
\(452\) 0 0
\(453\) −3.51472 −0.165136
\(454\) 0 0
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −7.65685 −0.358173 −0.179086 0.983833i \(-0.557314\pi\)
−0.179086 + 0.983833i \(0.557314\pi\)
\(458\) 0 0
\(459\) −7.65685 −0.357391
\(460\) 0 0
\(461\) 5.17157 0.240864 0.120432 0.992722i \(-0.461572\pi\)
0.120432 + 0.992722i \(0.461572\pi\)
\(462\) 0 0
\(463\) 24.4853 1.13793 0.568964 0.822363i \(-0.307344\pi\)
0.568964 + 0.822363i \(0.307344\pi\)
\(464\) 0 0
\(465\) 3.31371 0.153670
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 19.3137 0.891824
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) 3.31371 0.152364
\(474\) 0 0
\(475\) 8.48528 0.389331
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) −25.3137 −1.15661 −0.578306 0.815820i \(-0.696286\pi\)
−0.578306 + 0.815820i \(0.696286\pi\)
\(480\) 0 0
\(481\) 7.65685 0.349123
\(482\) 0 0
\(483\) 11.3137 0.514792
\(484\) 0 0
\(485\) −10.3431 −0.469658
\(486\) 0 0
\(487\) 7.79899 0.353406 0.176703 0.984264i \(-0.443457\pi\)
0.176703 + 0.984264i \(0.443457\pi\)
\(488\) 0 0
\(489\) 18.8284 0.851451
\(490\) 0 0
\(491\) −30.6274 −1.38220 −0.691098 0.722761i \(-0.742873\pi\)
−0.691098 + 0.722761i \(0.742873\pi\)
\(492\) 0 0
\(493\) 15.3137 0.689695
\(494\) 0 0
\(495\) −5.65685 −0.254257
\(496\) 0 0
\(497\) 5.65685 0.253745
\(498\) 0 0
\(499\) −26.1421 −1.17028 −0.585141 0.810931i \(-0.698961\pi\)
−0.585141 + 0.810931i \(0.698961\pi\)
\(500\) 0 0
\(501\) −3.65685 −0.163376
\(502\) 0 0
\(503\) 7.31371 0.326102 0.163051 0.986618i \(-0.447866\pi\)
0.163051 + 0.986618i \(0.447866\pi\)
\(504\) 0 0
\(505\) −21.6569 −0.963717
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 11.7990 0.522981 0.261491 0.965206i \(-0.415786\pi\)
0.261491 + 0.965206i \(0.415786\pi\)
\(510\) 0 0
\(511\) −0.970563 −0.0429352
\(512\) 0 0
\(513\) −2.82843 −0.124878
\(514\) 0 0
\(515\) 6.62742 0.292039
\(516\) 0 0
\(517\) 23.3137 1.02534
\(518\) 0 0
\(519\) 11.6569 0.511679
\(520\) 0 0
\(521\) 25.3137 1.10901 0.554507 0.832179i \(-0.312907\pi\)
0.554507 + 0.832179i \(0.312907\pi\)
\(522\) 0 0
\(523\) 15.3137 0.669622 0.334811 0.942285i \(-0.391328\pi\)
0.334811 + 0.942285i \(0.391328\pi\)
\(524\) 0 0
\(525\) 8.48528 0.370328
\(526\) 0 0
\(527\) 8.97056 0.390764
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −7.65685 −0.332279
\(532\) 0 0
\(533\) −5.17157 −0.224006
\(534\) 0 0
\(535\) −32.0000 −1.38348
\(536\) 0 0
\(537\) −23.3137 −1.00606
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) −15.0294 −0.643790
\(546\) 0 0
\(547\) −23.3137 −0.996822 −0.498411 0.866941i \(-0.666083\pi\)
−0.498411 + 0.866941i \(0.666083\pi\)
\(548\) 0 0
\(549\) 13.3137 0.568215
\(550\) 0 0
\(551\) 5.65685 0.240990
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) 0 0
\(555\) −21.6569 −0.919282
\(556\) 0 0
\(557\) −7.79899 −0.330454 −0.165227 0.986256i \(-0.552836\pi\)
−0.165227 + 0.986256i \(0.552836\pi\)
\(558\) 0 0
\(559\) −1.65685 −0.0700775
\(560\) 0 0
\(561\) −15.3137 −0.646545
\(562\) 0 0
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 15.0294 0.632293
\(566\) 0 0
\(567\) −2.82843 −0.118783
\(568\) 0 0
\(569\) −42.9706 −1.80142 −0.900710 0.434421i \(-0.856953\pi\)
−0.900710 + 0.434421i \(0.856953\pi\)
\(570\) 0 0
\(571\) 12.9706 0.542801 0.271401 0.962466i \(-0.412513\pi\)
0.271401 + 0.962466i \(0.412513\pi\)
\(572\) 0 0
\(573\) 3.31371 0.138432
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −31.9411 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(578\) 0 0
\(579\) −5.31371 −0.220830
\(580\) 0 0
\(581\) 10.3431 0.429106
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 0 0
\(585\) 2.82843 0.116941
\(586\) 0 0
\(587\) 10.9706 0.452804 0.226402 0.974034i \(-0.427304\pi\)
0.226402 + 0.974034i \(0.427304\pi\)
\(588\) 0 0
\(589\) 3.31371 0.136539
\(590\) 0 0
\(591\) −0.485281 −0.0199618
\(592\) 0 0
\(593\) −20.4853 −0.841230 −0.420615 0.907239i \(-0.638186\pi\)
−0.420615 + 0.907239i \(0.638186\pi\)
\(594\) 0 0
\(595\) 61.2548 2.51120
\(596\) 0 0
\(597\) 21.6569 0.886356
\(598\) 0 0
\(599\) 23.3137 0.952572 0.476286 0.879290i \(-0.341983\pi\)
0.476286 + 0.879290i \(0.341983\pi\)
\(600\) 0 0
\(601\) −0.627417 −0.0255929 −0.0127964 0.999918i \(-0.504073\pi\)
−0.0127964 + 0.999918i \(0.504073\pi\)
\(602\) 0 0
\(603\) −6.82843 −0.278075
\(604\) 0 0
\(605\) 19.7990 0.804943
\(606\) 0 0
\(607\) −41.9411 −1.70234 −0.851169 0.524892i \(-0.824106\pi\)
−0.851169 + 0.524892i \(0.824106\pi\)
\(608\) 0 0
\(609\) 5.65685 0.229227
\(610\) 0 0
\(611\) −11.6569 −0.471586
\(612\) 0 0
\(613\) −47.6569 −1.92484 −0.962421 0.271561i \(-0.912460\pi\)
−0.962421 + 0.271561i \(0.912460\pi\)
\(614\) 0 0
\(615\) 14.6274 0.589834
\(616\) 0 0
\(617\) −34.8284 −1.40214 −0.701070 0.713093i \(-0.747294\pi\)
−0.701070 + 0.713093i \(0.747294\pi\)
\(618\) 0 0
\(619\) −23.7990 −0.956562 −0.478281 0.878207i \(-0.658740\pi\)
−0.478281 + 0.878207i \(0.658740\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) −41.9411 −1.68034
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) −5.65685 −0.225913
\(628\) 0 0
\(629\) −58.6274 −2.33763
\(630\) 0 0
\(631\) −43.1127 −1.71629 −0.858145 0.513408i \(-0.828383\pi\)
−0.858145 + 0.513408i \(0.828383\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 30.2843 1.19616 0.598078 0.801438i \(-0.295931\pi\)
0.598078 + 0.801438i \(0.295931\pi\)
\(642\) 0 0
\(643\) −22.8284 −0.900265 −0.450133 0.892962i \(-0.648623\pi\)
−0.450133 + 0.892962i \(0.648623\pi\)
\(644\) 0 0
\(645\) 4.68629 0.184523
\(646\) 0 0
\(647\) −11.3137 −0.444788 −0.222394 0.974957i \(-0.571387\pi\)
−0.222394 + 0.974957i \(0.571387\pi\)
\(648\) 0 0
\(649\) −15.3137 −0.601116
\(650\) 0 0
\(651\) 3.31371 0.129874
\(652\) 0 0
\(653\) −25.3137 −0.990602 −0.495301 0.868721i \(-0.664942\pi\)
−0.495301 + 0.868721i \(0.664942\pi\)
\(654\) 0 0
\(655\) −22.6274 −0.884126
\(656\) 0 0
\(657\) 0.343146 0.0133874
\(658\) 0 0
\(659\) 47.3137 1.84308 0.921540 0.388283i \(-0.126932\pi\)
0.921540 + 0.388283i \(0.126932\pi\)
\(660\) 0 0
\(661\) −34.9706 −1.36020 −0.680099 0.733121i \(-0.738063\pi\)
−0.680099 + 0.733121i \(0.738063\pi\)
\(662\) 0 0
\(663\) 7.65685 0.297368
\(664\) 0 0
\(665\) 22.6274 0.877454
\(666\) 0 0
\(667\) 8.00000 0.309761
\(668\) 0 0
\(669\) −12.4853 −0.482709
\(670\) 0 0
\(671\) 26.6274 1.02794
\(672\) 0 0
\(673\) 16.6274 0.640940 0.320470 0.947259i \(-0.396159\pi\)
0.320470 + 0.947259i \(0.396159\pi\)
\(674\) 0 0
\(675\) −3.00000 −0.115470
\(676\) 0 0
\(677\) 26.6863 1.02564 0.512819 0.858497i \(-0.328601\pi\)
0.512819 + 0.858497i \(0.328601\pi\)
\(678\) 0 0
\(679\) −10.3431 −0.396934
\(680\) 0 0
\(681\) −17.3137 −0.663463
\(682\) 0 0
\(683\) −47.9411 −1.83442 −0.917208 0.398408i \(-0.869563\pi\)
−0.917208 + 0.398408i \(0.869563\pi\)
\(684\) 0 0
\(685\) 30.6274 1.17021
\(686\) 0 0
\(687\) 1.31371 0.0501211
\(688\) 0 0
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) 5.85786 0.222844 0.111422 0.993773i \(-0.464460\pi\)
0.111422 + 0.993773i \(0.464460\pi\)
\(692\) 0 0
\(693\) −5.65685 −0.214886
\(694\) 0 0
\(695\) −20.6863 −0.784676
\(696\) 0 0
\(697\) 39.5980 1.49988
\(698\) 0 0
\(699\) −6.97056 −0.263651
\(700\) 0 0
\(701\) 5.02944 0.189959 0.0949796 0.995479i \(-0.469721\pi\)
0.0949796 + 0.995479i \(0.469721\pi\)
\(702\) 0 0
\(703\) −21.6569 −0.816804
\(704\) 0 0
\(705\) 32.9706 1.24174
\(706\) 0 0
\(707\) −21.6569 −0.814490
\(708\) 0 0
\(709\) −4.62742 −0.173786 −0.0868931 0.996218i \(-0.527694\pi\)
−0.0868931 + 0.996218i \(0.527694\pi\)
\(710\) 0 0
\(711\) 11.3137 0.424297
\(712\) 0 0
\(713\) 4.68629 0.175503
\(714\) 0 0
\(715\) 5.65685 0.211554
\(716\) 0 0
\(717\) 2.00000 0.0746914
\(718\) 0 0
\(719\) 29.9411 1.11662 0.558308 0.829634i \(-0.311451\pi\)
0.558308 + 0.829634i \(0.311451\pi\)
\(720\) 0 0
\(721\) 6.62742 0.246818
\(722\) 0 0
\(723\) −0.343146 −0.0127617
\(724\) 0 0
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 10.3431 0.383606 0.191803 0.981433i \(-0.438567\pi\)
0.191803 + 0.981433i \(0.438567\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.6863 0.469219
\(732\) 0 0
\(733\) −36.6274 −1.35286 −0.676432 0.736505i \(-0.736475\pi\)
−0.676432 + 0.736505i \(0.736475\pi\)
\(734\) 0 0
\(735\) 2.82843 0.104328
\(736\) 0 0
\(737\) −13.6569 −0.503057
\(738\) 0 0
\(739\) 18.1421 0.667369 0.333685 0.942685i \(-0.391708\pi\)
0.333685 + 0.942685i \(0.391708\pi\)
\(740\) 0 0
\(741\) 2.82843 0.103905
\(742\) 0 0
\(743\) −2.00000 −0.0733729 −0.0366864 0.999327i \(-0.511680\pi\)
−0.0366864 + 0.999327i \(0.511680\pi\)
\(744\) 0 0
\(745\) 25.9411 0.950409
\(746\) 0 0
\(747\) −3.65685 −0.133797
\(748\) 0 0
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) 0.970563 0.0354163 0.0177082 0.999843i \(-0.494363\pi\)
0.0177082 + 0.999843i \(0.494363\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −9.94113 −0.361795
\(756\) 0 0
\(757\) 51.9411 1.88783 0.943916 0.330185i \(-0.107111\pi\)
0.943916 + 0.330185i \(0.107111\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 32.4853 1.17759 0.588795 0.808282i \(-0.299602\pi\)
0.588795 + 0.808282i \(0.299602\pi\)
\(762\) 0 0
\(763\) −15.0294 −0.544102
\(764\) 0 0
\(765\) −21.6569 −0.783005
\(766\) 0 0
\(767\) 7.65685 0.276473
\(768\) 0 0
\(769\) 42.0000 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 0 0
\(771\) 4.34315 0.156415
\(772\) 0 0
\(773\) 34.1421 1.22801 0.614004 0.789303i \(-0.289558\pi\)
0.614004 + 0.789303i \(0.289558\pi\)
\(774\) 0 0
\(775\) 3.51472 0.126252
\(776\) 0 0
\(777\) −21.6569 −0.776935
\(778\) 0 0
\(779\) 14.6274 0.524082
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 28.2843 1.00951
\(786\) 0 0
\(787\) 40.7696 1.45328 0.726639 0.687020i \(-0.241081\pi\)
0.726639 + 0.687020i \(0.241081\pi\)
\(788\) 0 0
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 15.0294 0.534385
\(792\) 0 0
\(793\) −13.3137 −0.472784
\(794\) 0 0
\(795\) −5.65685 −0.200628
\(796\) 0 0
\(797\) 24.3431 0.862278 0.431139 0.902285i \(-0.358112\pi\)
0.431139 + 0.902285i \(0.358112\pi\)
\(798\) 0 0
\(799\) 89.2548 3.15761
\(800\) 0 0
\(801\) 14.8284 0.523937
\(802\) 0 0
\(803\) 0.686292 0.0242187
\(804\) 0 0
\(805\) 32.0000 1.12785
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) 0 0
\(809\) 18.6863 0.656975 0.328488 0.944508i \(-0.393461\pi\)
0.328488 + 0.944508i \(0.393461\pi\)
\(810\) 0 0
\(811\) 30.1421 1.05843 0.529217 0.848487i \(-0.322486\pi\)
0.529217 + 0.848487i \(0.322486\pi\)
\(812\) 0 0
\(813\) −27.7990 −0.974953
\(814\) 0 0
\(815\) 53.2548 1.86544
\(816\) 0 0
\(817\) 4.68629 0.163953
\(818\) 0 0
\(819\) 2.82843 0.0988332
\(820\) 0 0
\(821\) −23.7990 −0.830590 −0.415295 0.909687i \(-0.636322\pi\)
−0.415295 + 0.909687i \(0.636322\pi\)
\(822\) 0 0
\(823\) −15.0294 −0.523893 −0.261947 0.965082i \(-0.584364\pi\)
−0.261947 + 0.965082i \(0.584364\pi\)
\(824\) 0 0
\(825\) −6.00000 −0.208893
\(826\) 0 0
\(827\) 26.0000 0.904109 0.452054 0.891990i \(-0.350691\pi\)
0.452054 + 0.891990i \(0.350691\pi\)
\(828\) 0 0
\(829\) −17.3137 −0.601330 −0.300665 0.953730i \(-0.597209\pi\)
−0.300665 + 0.953730i \(0.597209\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 7.65685 0.265294
\(834\) 0 0
\(835\) −10.3431 −0.357939
\(836\) 0 0
\(837\) −1.17157 −0.0404955
\(838\) 0 0
\(839\) −43.2548 −1.49332 −0.746661 0.665204i \(-0.768344\pi\)
−0.746661 + 0.665204i \(0.768344\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −21.1716 −0.729188
\(844\) 0 0
\(845\) −2.82843 −0.0973009
\(846\) 0 0
\(847\) 19.7990 0.680301
\(848\) 0 0
\(849\) 28.9706 0.994267
\(850\) 0 0
\(851\) −30.6274 −1.04989
\(852\) 0 0
\(853\) 3.65685 0.125208 0.0626042 0.998038i \(-0.480059\pi\)
0.0626042 + 0.998038i \(0.480059\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) −49.5980 −1.69423 −0.847117 0.531406i \(-0.821664\pi\)
−0.847117 + 0.531406i \(0.821664\pi\)
\(858\) 0 0
\(859\) 0.686292 0.0234160 0.0117080 0.999931i \(-0.496273\pi\)
0.0117080 + 0.999931i \(0.496273\pi\)
\(860\) 0 0
\(861\) 14.6274 0.498501
\(862\) 0 0
\(863\) 28.3431 0.964812 0.482406 0.875948i \(-0.339763\pi\)
0.482406 + 0.875948i \(0.339763\pi\)
\(864\) 0 0
\(865\) 32.9706 1.12103
\(866\) 0 0
\(867\) −41.6274 −1.41374
\(868\) 0 0
\(869\) 22.6274 0.767583
\(870\) 0 0
\(871\) 6.82843 0.231372
\(872\) 0 0
\(873\) 3.65685 0.123766
\(874\) 0 0
\(875\) −16.0000 −0.540899
\(876\) 0 0
\(877\) 42.2843 1.42784 0.713919 0.700228i \(-0.246918\pi\)
0.713919 + 0.700228i \(0.246918\pi\)
\(878\) 0 0
\(879\) −2.14214 −0.0722524
\(880\) 0 0
\(881\) −25.5980 −0.862418 −0.431209 0.902252i \(-0.641913\pi\)
−0.431209 + 0.902252i \(0.641913\pi\)
\(882\) 0 0
\(883\) −27.5980 −0.928746 −0.464373 0.885640i \(-0.653720\pi\)
−0.464373 + 0.885640i \(0.653720\pi\)
\(884\) 0 0
\(885\) −21.6569 −0.727987
\(886\) 0 0
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) 32.9706 1.10332
\(894\) 0 0
\(895\) −65.9411 −2.20417
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) 0 0
\(899\) 2.34315 0.0781483
\(900\) 0 0
\(901\) −15.3137 −0.510174
\(902\) 0 0
\(903\) 4.68629 0.155950
\(904\) 0 0
\(905\) −39.5980 −1.31628
\(906\) 0 0
\(907\) 12.9706 0.430680 0.215340 0.976539i \(-0.430914\pi\)
0.215340 + 0.976539i \(0.430914\pi\)
\(908\) 0 0
\(909\) 7.65685 0.253962
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) −7.31371 −0.242048
\(914\) 0 0
\(915\) 37.6569 1.24490
\(916\) 0 0
\(917\) −22.6274 −0.747223
\(918\) 0 0
\(919\) −3.31371 −0.109309 −0.0546546 0.998505i \(-0.517406\pi\)
−0.0546546 + 0.998505i \(0.517406\pi\)
\(920\) 0 0
\(921\) −22.8284 −0.752222
\(922\) 0 0
\(923\) 2.00000 0.0658308
\(924\) 0 0
\(925\) −22.9706 −0.755267
\(926\) 0 0
\(927\) −2.34315 −0.0769590
\(928\) 0 0
\(929\) 11.7990 0.387112 0.193556 0.981089i \(-0.437998\pi\)
0.193556 + 0.981089i \(0.437998\pi\)
\(930\) 0 0
\(931\) 2.82843 0.0926980
\(932\) 0 0
\(933\) −10.6274 −0.347926
\(934\) 0 0
\(935\) −43.3137 −1.41651
\(936\) 0 0
\(937\) −21.3137 −0.696289 −0.348144 0.937441i \(-0.613188\pi\)
−0.348144 + 0.937441i \(0.613188\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 34.1421 1.11300 0.556501 0.830847i \(-0.312144\pi\)
0.556501 + 0.830847i \(0.312144\pi\)
\(942\) 0 0
\(943\) 20.6863 0.673638
\(944\) 0 0
\(945\) −8.00000 −0.260240
\(946\) 0 0
\(947\) 21.0294 0.683365 0.341682 0.939815i \(-0.389003\pi\)
0.341682 + 0.939815i \(0.389003\pi\)
\(948\) 0 0
\(949\) −0.343146 −0.0111390
\(950\) 0 0
\(951\) −8.48528 −0.275154
\(952\) 0 0
\(953\) 40.3431 1.30684 0.653421 0.756994i \(-0.273333\pi\)
0.653421 + 0.756994i \(0.273333\pi\)
\(954\) 0 0
\(955\) 9.37258 0.303290
\(956\) 0 0
\(957\) −4.00000 −0.129302
\(958\) 0 0
\(959\) 30.6274 0.989011
\(960\) 0 0
\(961\) −29.6274 −0.955723
\(962\) 0 0
\(963\) 11.3137 0.364579
\(964\) 0 0
\(965\) −15.0294 −0.483815
\(966\) 0 0
\(967\) −18.1421 −0.583412 −0.291706 0.956508i \(-0.594223\pi\)
−0.291706 + 0.956508i \(0.594223\pi\)
\(968\) 0 0
\(969\) −21.6569 −0.695718
\(970\) 0 0
\(971\) 15.3137 0.491440 0.245720 0.969341i \(-0.420976\pi\)
0.245720 + 0.969341i \(0.420976\pi\)
\(972\) 0 0
\(973\) −20.6863 −0.663172
\(974\) 0 0
\(975\) 3.00000 0.0960769
\(976\) 0 0
\(977\) 42.1421 1.34825 0.674123 0.738619i \(-0.264522\pi\)
0.674123 + 0.738619i \(0.264522\pi\)
\(978\) 0 0
\(979\) 29.6569 0.947837
\(980\) 0 0
\(981\) 5.31371 0.169654
\(982\) 0 0
\(983\) −25.3137 −0.807382 −0.403691 0.914895i \(-0.632273\pi\)
−0.403691 + 0.914895i \(0.632273\pi\)
\(984\) 0 0
\(985\) −1.37258 −0.0437341
\(986\) 0 0
\(987\) 32.9706 1.04946
\(988\) 0 0
\(989\) 6.62742 0.210740
\(990\) 0 0
\(991\) −4.68629 −0.148865 −0.0744325 0.997226i \(-0.523715\pi\)
−0.0744325 + 0.997226i \(0.523715\pi\)
\(992\) 0 0
\(993\) 26.1421 0.829596
\(994\) 0 0
\(995\) 61.2548 1.94191
\(996\) 0 0
\(997\) −39.2548 −1.24321 −0.621607 0.783330i \(-0.713520\pi\)
−0.621607 + 0.783330i \(0.713520\pi\)
\(998\) 0 0
\(999\) 7.65685 0.242252
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 624.2.a.k.1.1 2
3.2 odd 2 1872.2.a.w.1.2 2
4.3 odd 2 39.2.a.b.1.2 2
8.3 odd 2 2496.2.a.bf.1.2 2
8.5 even 2 2496.2.a.bi.1.2 2
12.11 even 2 117.2.a.c.1.1 2
13.12 even 2 8112.2.a.bm.1.2 2
20.3 even 4 975.2.c.h.274.2 4
20.7 even 4 975.2.c.h.274.3 4
20.19 odd 2 975.2.a.l.1.1 2
24.5 odd 2 7488.2.a.co.1.1 2
24.11 even 2 7488.2.a.cl.1.1 2
28.27 even 2 1911.2.a.h.1.2 2
36.7 odd 6 1053.2.e.m.352.1 4
36.11 even 6 1053.2.e.e.352.2 4
36.23 even 6 1053.2.e.e.703.2 4
36.31 odd 6 1053.2.e.m.703.1 4
44.43 even 2 4719.2.a.p.1.1 2
52.3 odd 6 507.2.e.h.22.1 4
52.7 even 12 507.2.j.f.361.3 8
52.11 even 12 507.2.j.f.316.2 8
52.15 even 12 507.2.j.f.316.3 8
52.19 even 12 507.2.j.f.361.2 8
52.23 odd 6 507.2.e.d.22.2 4
52.31 even 4 507.2.b.e.337.2 4
52.35 odd 6 507.2.e.h.484.1 4
52.43 odd 6 507.2.e.d.484.2 4
52.47 even 4 507.2.b.e.337.3 4
52.51 odd 2 507.2.a.h.1.1 2
60.23 odd 4 2925.2.c.u.2224.3 4
60.47 odd 4 2925.2.c.u.2224.2 4
60.59 even 2 2925.2.a.v.1.2 2
84.83 odd 2 5733.2.a.u.1.1 2
156.47 odd 4 1521.2.b.j.1351.2 4
156.83 odd 4 1521.2.b.j.1351.3 4
156.155 even 2 1521.2.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.b.1.2 2 4.3 odd 2
117.2.a.c.1.1 2 12.11 even 2
507.2.a.h.1.1 2 52.51 odd 2
507.2.b.e.337.2 4 52.31 even 4
507.2.b.e.337.3 4 52.47 even 4
507.2.e.d.22.2 4 52.23 odd 6
507.2.e.d.484.2 4 52.43 odd 6
507.2.e.h.22.1 4 52.3 odd 6
507.2.e.h.484.1 4 52.35 odd 6
507.2.j.f.316.2 8 52.11 even 12
507.2.j.f.316.3 8 52.15 even 12
507.2.j.f.361.2 8 52.19 even 12
507.2.j.f.361.3 8 52.7 even 12
624.2.a.k.1.1 2 1.1 even 1 trivial
975.2.a.l.1.1 2 20.19 odd 2
975.2.c.h.274.2 4 20.3 even 4
975.2.c.h.274.3 4 20.7 even 4
1053.2.e.e.352.2 4 36.11 even 6
1053.2.e.e.703.2 4 36.23 even 6
1053.2.e.m.352.1 4 36.7 odd 6
1053.2.e.m.703.1 4 36.31 odd 6
1521.2.a.f.1.2 2 156.155 even 2
1521.2.b.j.1351.2 4 156.47 odd 4
1521.2.b.j.1351.3 4 156.83 odd 4
1872.2.a.w.1.2 2 3.2 odd 2
1911.2.a.h.1.2 2 28.27 even 2
2496.2.a.bf.1.2 2 8.3 odd 2
2496.2.a.bi.1.2 2 8.5 even 2
2925.2.a.v.1.2 2 60.59 even 2
2925.2.c.u.2224.2 4 60.47 odd 4
2925.2.c.u.2224.3 4 60.23 odd 4
4719.2.a.p.1.1 2 44.43 even 2
5733.2.a.u.1.1 2 84.83 odd 2
7488.2.a.cl.1.1 2 24.11 even 2
7488.2.a.co.1.1 2 24.5 odd 2
8112.2.a.bm.1.2 2 13.12 even 2