Properties

Label 4560.2.a.bi.1.2
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +4.73205 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +4.73205 q^{7} +1.00000 q^{9} -4.19615 q^{11} -1.26795 q^{13} -1.00000 q^{15} -6.92820 q^{17} -1.00000 q^{19} -4.73205 q^{21} +6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -10.1962 q^{29} +2.00000 q^{31} +4.19615 q^{33} +4.73205 q^{35} +2.73205 q^{37} +1.26795 q^{39} +11.6603 q^{41} +4.73205 q^{43} +1.00000 q^{45} +10.0000 q^{47} +15.3923 q^{49} +6.92820 q^{51} -4.19615 q^{55} +1.00000 q^{57} +9.46410 q^{59} +5.46410 q^{61} +4.73205 q^{63} -1.26795 q^{65} +14.9282 q^{67} -6.00000 q^{69} -12.3923 q^{71} +0.928203 q^{73} -1.00000 q^{75} -19.8564 q^{77} +8.00000 q^{79} +1.00000 q^{81} -8.53590 q^{83} -6.92820 q^{85} +10.1962 q^{87} -7.66025 q^{89} -6.00000 q^{91} -2.00000 q^{93} -1.00000 q^{95} +1.66025 q^{97} -4.19615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{11} - 6 q^{13} - 2 q^{15} - 2 q^{19} - 6 q^{21} + 12 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} + 4 q^{31} - 2 q^{33} + 6 q^{35} + 2 q^{37} + 6 q^{39} + 6 q^{41} + 6 q^{43} + 2 q^{45} + 20 q^{47} + 10 q^{49} + 2 q^{55} + 2 q^{57} + 12 q^{59} + 4 q^{61} + 6 q^{63} - 6 q^{65} + 16 q^{67} - 12 q^{69} - 4 q^{71} - 12 q^{73} - 2 q^{75} - 12 q^{77} + 16 q^{79} + 2 q^{81} - 24 q^{83} + 10 q^{87} + 2 q^{89} - 12 q^{91} - 4 q^{93} - 2 q^{95} - 14 q^{97} + 2 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.73205 1.78855 0.894274 0.447521i \(-0.147693\pi\)
0.894274 + 0.447521i \(0.147693\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.19615 −1.26519 −0.632594 0.774484i \(-0.718010\pi\)
−0.632594 + 0.774484i \(0.718010\pi\)
\(12\) 0 0
\(13\) −1.26795 −0.351666 −0.175833 0.984420i \(-0.556262\pi\)
−0.175833 + 0.984420i \(0.556262\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −6.92820 −1.68034 −0.840168 0.542326i \(-0.817544\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −4.73205 −1.03262
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.1962 −1.89338 −0.946689 0.322149i \(-0.895595\pi\)
−0.946689 + 0.322149i \(0.895595\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 4.19615 0.730456
\(34\) 0 0
\(35\) 4.73205 0.799863
\(36\) 0 0
\(37\) 2.73205 0.449146 0.224573 0.974457i \(-0.427901\pi\)
0.224573 + 0.974457i \(0.427901\pi\)
\(38\) 0 0
\(39\) 1.26795 0.203034
\(40\) 0 0
\(41\) 11.6603 1.82103 0.910513 0.413481i \(-0.135687\pi\)
0.910513 + 0.413481i \(0.135687\pi\)
\(42\) 0 0
\(43\) 4.73205 0.721631 0.360815 0.932637i \(-0.382498\pi\)
0.360815 + 0.932637i \(0.382498\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) 15.3923 2.19890
\(50\) 0 0
\(51\) 6.92820 0.970143
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −4.19615 −0.565809
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 9.46410 1.23212 0.616061 0.787699i \(-0.288728\pi\)
0.616061 + 0.787699i \(0.288728\pi\)
\(60\) 0 0
\(61\) 5.46410 0.699607 0.349803 0.936823i \(-0.386248\pi\)
0.349803 + 0.936823i \(0.386248\pi\)
\(62\) 0 0
\(63\) 4.73205 0.596182
\(64\) 0 0
\(65\) −1.26795 −0.157270
\(66\) 0 0
\(67\) 14.9282 1.82377 0.911885 0.410445i \(-0.134627\pi\)
0.911885 + 0.410445i \(0.134627\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −12.3923 −1.47070 −0.735348 0.677690i \(-0.762981\pi\)
−0.735348 + 0.677690i \(0.762981\pi\)
\(72\) 0 0
\(73\) 0.928203 0.108638 0.0543190 0.998524i \(-0.482701\pi\)
0.0543190 + 0.998524i \(0.482701\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −19.8564 −2.26285
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.53590 −0.936937 −0.468468 0.883480i \(-0.655194\pi\)
−0.468468 + 0.883480i \(0.655194\pi\)
\(84\) 0 0
\(85\) −6.92820 −0.751469
\(86\) 0 0
\(87\) 10.1962 1.09314
\(88\) 0 0
\(89\) −7.66025 −0.811985 −0.405993 0.913876i \(-0.633074\pi\)
−0.405993 + 0.913876i \(0.633074\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 1.66025 0.168573 0.0842866 0.996442i \(-0.473139\pi\)
0.0842866 + 0.996442i \(0.473139\pi\)
\(98\) 0 0
\(99\) −4.19615 −0.421729
\(100\) 0 0
\(101\) −3.46410 −0.344691 −0.172345 0.985037i \(-0.555135\pi\)
−0.172345 + 0.985037i \(0.555135\pi\)
\(102\) 0 0
\(103\) 13.8564 1.36531 0.682656 0.730740i \(-0.260825\pi\)
0.682656 + 0.730740i \(0.260825\pi\)
\(104\) 0 0
\(105\) −4.73205 −0.461801
\(106\) 0 0
\(107\) 10.9282 1.05647 0.528235 0.849098i \(-0.322854\pi\)
0.528235 + 0.849098i \(0.322854\pi\)
\(108\) 0 0
\(109\) −10.3923 −0.995402 −0.497701 0.867349i \(-0.665822\pi\)
−0.497701 + 0.867349i \(0.665822\pi\)
\(110\) 0 0
\(111\) −2.73205 −0.259315
\(112\) 0 0
\(113\) 2.53590 0.238557 0.119279 0.992861i \(-0.461942\pi\)
0.119279 + 0.992861i \(0.461942\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 0 0
\(117\) −1.26795 −0.117222
\(118\) 0 0
\(119\) −32.7846 −3.00536
\(120\) 0 0
\(121\) 6.60770 0.600700
\(122\) 0 0
\(123\) −11.6603 −1.05137
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 1.07180 0.0951066 0.0475533 0.998869i \(-0.484858\pi\)
0.0475533 + 0.998869i \(0.484858\pi\)
\(128\) 0 0
\(129\) −4.73205 −0.416634
\(130\) 0 0
\(131\) 20.5885 1.79882 0.899411 0.437104i \(-0.143996\pi\)
0.899411 + 0.437104i \(0.143996\pi\)
\(132\) 0 0
\(133\) −4.73205 −0.410321
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 22.7846 1.94662 0.973310 0.229493i \(-0.0737068\pi\)
0.973310 + 0.229493i \(0.0737068\pi\)
\(138\) 0 0
\(139\) 17.4641 1.48129 0.740643 0.671899i \(-0.234521\pi\)
0.740643 + 0.671899i \(0.234521\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 0 0
\(143\) 5.32051 0.444923
\(144\) 0 0
\(145\) −10.1962 −0.846744
\(146\) 0 0
\(147\) −15.3923 −1.26954
\(148\) 0 0
\(149\) 8.92820 0.731427 0.365713 0.930727i \(-0.380825\pi\)
0.365713 + 0.930727i \(0.380825\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) −6.92820 −0.560112
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) −3.46410 −0.276465 −0.138233 0.990400i \(-0.544142\pi\)
−0.138233 + 0.990400i \(0.544142\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 28.3923 2.23763
\(162\) 0 0
\(163\) −25.1244 −1.96789 −0.983946 0.178468i \(-0.942886\pi\)
−0.983946 + 0.178468i \(0.942886\pi\)
\(164\) 0 0
\(165\) 4.19615 0.326670
\(166\) 0 0
\(167\) −14.3923 −1.11371 −0.556855 0.830610i \(-0.687992\pi\)
−0.556855 + 0.830610i \(0.687992\pi\)
\(168\) 0 0
\(169\) −11.3923 −0.876331
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 12.3923 0.942169 0.471085 0.882088i \(-0.343863\pi\)
0.471085 + 0.882088i \(0.343863\pi\)
\(174\) 0 0
\(175\) 4.73205 0.357709
\(176\) 0 0
\(177\) −9.46410 −0.711365
\(178\) 0 0
\(179\) −21.4641 −1.60430 −0.802151 0.597121i \(-0.796311\pi\)
−0.802151 + 0.597121i \(0.796311\pi\)
\(180\) 0 0
\(181\) 4.53590 0.337151 0.168575 0.985689i \(-0.446083\pi\)
0.168575 + 0.985689i \(0.446083\pi\)
\(182\) 0 0
\(183\) −5.46410 −0.403918
\(184\) 0 0
\(185\) 2.73205 0.200864
\(186\) 0 0
\(187\) 29.0718 2.12594
\(188\) 0 0
\(189\) −4.73205 −0.344206
\(190\) 0 0
\(191\) 13.2679 0.960035 0.480018 0.877259i \(-0.340630\pi\)
0.480018 + 0.877259i \(0.340630\pi\)
\(192\) 0 0
\(193\) −11.8038 −0.849660 −0.424830 0.905273i \(-0.639666\pi\)
−0.424830 + 0.905273i \(0.639666\pi\)
\(194\) 0 0
\(195\) 1.26795 0.0907997
\(196\) 0 0
\(197\) 1.07180 0.0763624 0.0381812 0.999271i \(-0.487844\pi\)
0.0381812 + 0.999271i \(0.487844\pi\)
\(198\) 0 0
\(199\) 2.53590 0.179765 0.0898825 0.995952i \(-0.471351\pi\)
0.0898825 + 0.995952i \(0.471351\pi\)
\(200\) 0 0
\(201\) −14.9282 −1.05295
\(202\) 0 0
\(203\) −48.2487 −3.38640
\(204\) 0 0
\(205\) 11.6603 0.814387
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 4.19615 0.290254
\(210\) 0 0
\(211\) 6.92820 0.476957 0.238479 0.971148i \(-0.423351\pi\)
0.238479 + 0.971148i \(0.423351\pi\)
\(212\) 0 0
\(213\) 12.3923 0.849107
\(214\) 0 0
\(215\) 4.73205 0.322723
\(216\) 0 0
\(217\) 9.46410 0.642465
\(218\) 0 0
\(219\) −0.928203 −0.0627222
\(220\) 0 0
\(221\) 8.78461 0.590917
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −23.4641 −1.55737 −0.778684 0.627417i \(-0.784112\pi\)
−0.778684 + 0.627417i \(0.784112\pi\)
\(228\) 0 0
\(229\) 17.4641 1.15406 0.577030 0.816723i \(-0.304212\pi\)
0.577030 + 0.816723i \(0.304212\pi\)
\(230\) 0 0
\(231\) 19.8564 1.30646
\(232\) 0 0
\(233\) 0.143594 0.00940713 0.00470356 0.999989i \(-0.498503\pi\)
0.00470356 + 0.999989i \(0.498503\pi\)
\(234\) 0 0
\(235\) 10.0000 0.652328
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 22.0526 1.42646 0.713231 0.700929i \(-0.247231\pi\)
0.713231 + 0.700929i \(0.247231\pi\)
\(240\) 0 0
\(241\) −8.92820 −0.575116 −0.287558 0.957763i \(-0.592843\pi\)
−0.287558 + 0.957763i \(0.592843\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 15.3923 0.983378
\(246\) 0 0
\(247\) 1.26795 0.0806777
\(248\) 0 0
\(249\) 8.53590 0.540941
\(250\) 0 0
\(251\) 28.1962 1.77973 0.889863 0.456228i \(-0.150800\pi\)
0.889863 + 0.456228i \(0.150800\pi\)
\(252\) 0 0
\(253\) −25.1769 −1.58286
\(254\) 0 0
\(255\) 6.92820 0.433861
\(256\) 0 0
\(257\) 27.3205 1.70421 0.852103 0.523374i \(-0.175327\pi\)
0.852103 + 0.523374i \(0.175327\pi\)
\(258\) 0 0
\(259\) 12.9282 0.803319
\(260\) 0 0
\(261\) −10.1962 −0.631126
\(262\) 0 0
\(263\) −3.46410 −0.213606 −0.106803 0.994280i \(-0.534061\pi\)
−0.106803 + 0.994280i \(0.534061\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 7.66025 0.468800
\(268\) 0 0
\(269\) −25.5167 −1.55578 −0.777889 0.628402i \(-0.783709\pi\)
−0.777889 + 0.628402i \(0.783709\pi\)
\(270\) 0 0
\(271\) −23.3205 −1.41662 −0.708310 0.705902i \(-0.750542\pi\)
−0.708310 + 0.705902i \(0.750542\pi\)
\(272\) 0 0
\(273\) 6.00000 0.363137
\(274\) 0 0
\(275\) −4.19615 −0.253038
\(276\) 0 0
\(277\) 11.8564 0.712382 0.356191 0.934413i \(-0.384075\pi\)
0.356191 + 0.934413i \(0.384075\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −8.05256 −0.480375 −0.240188 0.970726i \(-0.577209\pi\)
−0.240188 + 0.970726i \(0.577209\pi\)
\(282\) 0 0
\(283\) 13.8038 0.820554 0.410277 0.911961i \(-0.365432\pi\)
0.410277 + 0.911961i \(0.365432\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 55.1769 3.25699
\(288\) 0 0
\(289\) 31.0000 1.82353
\(290\) 0 0
\(291\) −1.66025 −0.0973258
\(292\) 0 0
\(293\) −24.3923 −1.42501 −0.712507 0.701665i \(-0.752440\pi\)
−0.712507 + 0.701665i \(0.752440\pi\)
\(294\) 0 0
\(295\) 9.46410 0.551021
\(296\) 0 0
\(297\) 4.19615 0.243485
\(298\) 0 0
\(299\) −7.60770 −0.439964
\(300\) 0 0
\(301\) 22.3923 1.29067
\(302\) 0 0
\(303\) 3.46410 0.199007
\(304\) 0 0
\(305\) 5.46410 0.312874
\(306\) 0 0
\(307\) −7.32051 −0.417803 −0.208902 0.977937i \(-0.566989\pi\)
−0.208902 + 0.977937i \(0.566989\pi\)
\(308\) 0 0
\(309\) −13.8564 −0.788263
\(310\) 0 0
\(311\) −21.2679 −1.20599 −0.602997 0.797743i \(-0.706027\pi\)
−0.602997 + 0.797743i \(0.706027\pi\)
\(312\) 0 0
\(313\) −13.3205 −0.752920 −0.376460 0.926433i \(-0.622859\pi\)
−0.376460 + 0.926433i \(0.622859\pi\)
\(314\) 0 0
\(315\) 4.73205 0.266621
\(316\) 0 0
\(317\) 1.07180 0.0601981 0.0300991 0.999547i \(-0.490418\pi\)
0.0300991 + 0.999547i \(0.490418\pi\)
\(318\) 0 0
\(319\) 42.7846 2.39548
\(320\) 0 0
\(321\) −10.9282 −0.609953
\(322\) 0 0
\(323\) 6.92820 0.385496
\(324\) 0 0
\(325\) −1.26795 −0.0703332
\(326\) 0 0
\(327\) 10.3923 0.574696
\(328\) 0 0
\(329\) 47.3205 2.60886
\(330\) 0 0
\(331\) 16.9282 0.930458 0.465229 0.885190i \(-0.345972\pi\)
0.465229 + 0.885190i \(0.345972\pi\)
\(332\) 0 0
\(333\) 2.73205 0.149715
\(334\) 0 0
\(335\) 14.9282 0.815615
\(336\) 0 0
\(337\) −2.73205 −0.148824 −0.0744121 0.997228i \(-0.523708\pi\)
−0.0744121 + 0.997228i \(0.523708\pi\)
\(338\) 0 0
\(339\) −2.53590 −0.137731
\(340\) 0 0
\(341\) −8.39230 −0.454469
\(342\) 0 0
\(343\) 39.7128 2.14429
\(344\) 0 0
\(345\) −6.00000 −0.323029
\(346\) 0 0
\(347\) 3.46410 0.185963 0.0929814 0.995668i \(-0.470360\pi\)
0.0929814 + 0.995668i \(0.470360\pi\)
\(348\) 0 0
\(349\) 29.7128 1.59049 0.795245 0.606288i \(-0.207342\pi\)
0.795245 + 0.606288i \(0.207342\pi\)
\(350\) 0 0
\(351\) 1.26795 0.0676781
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 0 0
\(355\) −12.3923 −0.657715
\(356\) 0 0
\(357\) 32.7846 1.73515
\(358\) 0 0
\(359\) −0.588457 −0.0310576 −0.0155288 0.999879i \(-0.504943\pi\)
−0.0155288 + 0.999879i \(0.504943\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −6.60770 −0.346814
\(364\) 0 0
\(365\) 0.928203 0.0485844
\(366\) 0 0
\(367\) 12.7321 0.664608 0.332304 0.943172i \(-0.392174\pi\)
0.332304 + 0.943172i \(0.392174\pi\)
\(368\) 0 0
\(369\) 11.6603 0.607009
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 16.5885 0.858918 0.429459 0.903086i \(-0.358704\pi\)
0.429459 + 0.903086i \(0.358704\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 12.9282 0.665836
\(378\) 0 0
\(379\) −23.8564 −1.22542 −0.612711 0.790307i \(-0.709921\pi\)
−0.612711 + 0.790307i \(0.709921\pi\)
\(380\) 0 0
\(381\) −1.07180 −0.0549098
\(382\) 0 0
\(383\) −34.6410 −1.77007 −0.885037 0.465521i \(-0.845867\pi\)
−0.885037 + 0.465521i \(0.845867\pi\)
\(384\) 0 0
\(385\) −19.8564 −1.01198
\(386\) 0 0
\(387\) 4.73205 0.240544
\(388\) 0 0
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −41.5692 −2.10225
\(392\) 0 0
\(393\) −20.5885 −1.03855
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −16.9282 −0.849602 −0.424801 0.905287i \(-0.639656\pi\)
−0.424801 + 0.905287i \(0.639656\pi\)
\(398\) 0 0
\(399\) 4.73205 0.236899
\(400\) 0 0
\(401\) 4.33975 0.216717 0.108358 0.994112i \(-0.465441\pi\)
0.108358 + 0.994112i \(0.465441\pi\)
\(402\) 0 0
\(403\) −2.53590 −0.126322
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −11.4641 −0.568254
\(408\) 0 0
\(409\) −0.535898 −0.0264985 −0.0132492 0.999912i \(-0.504217\pi\)
−0.0132492 + 0.999912i \(0.504217\pi\)
\(410\) 0 0
\(411\) −22.7846 −1.12388
\(412\) 0 0
\(413\) 44.7846 2.20371
\(414\) 0 0
\(415\) −8.53590 −0.419011
\(416\) 0 0
\(417\) −17.4641 −0.855221
\(418\) 0 0
\(419\) −1.26795 −0.0619434 −0.0309717 0.999520i \(-0.509860\pi\)
−0.0309717 + 0.999520i \(0.509860\pi\)
\(420\) 0 0
\(421\) −15.0718 −0.734554 −0.367277 0.930112i \(-0.619710\pi\)
−0.367277 + 0.930112i \(0.619710\pi\)
\(422\) 0 0
\(423\) 10.0000 0.486217
\(424\) 0 0
\(425\) −6.92820 −0.336067
\(426\) 0 0
\(427\) 25.8564 1.25128
\(428\) 0 0
\(429\) −5.32051 −0.256877
\(430\) 0 0
\(431\) 34.2487 1.64970 0.824851 0.565350i \(-0.191259\pi\)
0.824851 + 0.565350i \(0.191259\pi\)
\(432\) 0 0
\(433\) 10.0526 0.483095 0.241548 0.970389i \(-0.422345\pi\)
0.241548 + 0.970389i \(0.422345\pi\)
\(434\) 0 0
\(435\) 10.1962 0.488868
\(436\) 0 0
\(437\) −6.00000 −0.287019
\(438\) 0 0
\(439\) −5.85641 −0.279511 −0.139756 0.990186i \(-0.544632\pi\)
−0.139756 + 0.990186i \(0.544632\pi\)
\(440\) 0 0
\(441\) 15.3923 0.732967
\(442\) 0 0
\(443\) −7.07180 −0.335991 −0.167996 0.985788i \(-0.553729\pi\)
−0.167996 + 0.985788i \(0.553729\pi\)
\(444\) 0 0
\(445\) −7.66025 −0.363131
\(446\) 0 0
\(447\) −8.92820 −0.422290
\(448\) 0 0
\(449\) 9.80385 0.462672 0.231336 0.972874i \(-0.425690\pi\)
0.231336 + 0.972874i \(0.425690\pi\)
\(450\) 0 0
\(451\) −48.9282 −2.30394
\(452\) 0 0
\(453\) −2.00000 −0.0939682
\(454\) 0 0
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) −19.4641 −0.910492 −0.455246 0.890366i \(-0.650449\pi\)
−0.455246 + 0.890366i \(0.650449\pi\)
\(458\) 0 0
\(459\) 6.92820 0.323381
\(460\) 0 0
\(461\) 10.7846 0.502289 0.251145 0.967950i \(-0.419193\pi\)
0.251145 + 0.967950i \(0.419193\pi\)
\(462\) 0 0
\(463\) −17.1244 −0.795836 −0.397918 0.917421i \(-0.630267\pi\)
−0.397918 + 0.917421i \(0.630267\pi\)
\(464\) 0 0
\(465\) −2.00000 −0.0927478
\(466\) 0 0
\(467\) −19.0718 −0.882538 −0.441269 0.897375i \(-0.645471\pi\)
−0.441269 + 0.897375i \(0.645471\pi\)
\(468\) 0 0
\(469\) 70.6410 3.26190
\(470\) 0 0
\(471\) 3.46410 0.159617
\(472\) 0 0
\(473\) −19.8564 −0.912999
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.41154 −0.155877 −0.0779387 0.996958i \(-0.524834\pi\)
−0.0779387 + 0.996958i \(0.524834\pi\)
\(480\) 0 0
\(481\) −3.46410 −0.157949
\(482\) 0 0
\(483\) −28.3923 −1.29189
\(484\) 0 0
\(485\) 1.66025 0.0753883
\(486\) 0 0
\(487\) −20.3923 −0.924064 −0.462032 0.886863i \(-0.652879\pi\)
−0.462032 + 0.886863i \(0.652879\pi\)
\(488\) 0 0
\(489\) 25.1244 1.13616
\(490\) 0 0
\(491\) 8.19615 0.369887 0.184944 0.982749i \(-0.440790\pi\)
0.184944 + 0.982749i \(0.440790\pi\)
\(492\) 0 0
\(493\) 70.6410 3.18151
\(494\) 0 0
\(495\) −4.19615 −0.188603
\(496\) 0 0
\(497\) −58.6410 −2.63041
\(498\) 0 0
\(499\) 7.32051 0.327711 0.163855 0.986484i \(-0.447607\pi\)
0.163855 + 0.986484i \(0.447607\pi\)
\(500\) 0 0
\(501\) 14.3923 0.643001
\(502\) 0 0
\(503\) −39.1769 −1.74681 −0.873406 0.486993i \(-0.838094\pi\)
−0.873406 + 0.486993i \(0.838094\pi\)
\(504\) 0 0
\(505\) −3.46410 −0.154150
\(506\) 0 0
\(507\) 11.3923 0.505950
\(508\) 0 0
\(509\) 6.19615 0.274640 0.137320 0.990527i \(-0.456151\pi\)
0.137320 + 0.990527i \(0.456151\pi\)
\(510\) 0 0
\(511\) 4.39230 0.194304
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 13.8564 0.610586
\(516\) 0 0
\(517\) −41.9615 −1.84547
\(518\) 0 0
\(519\) −12.3923 −0.543962
\(520\) 0 0
\(521\) 14.8756 0.651714 0.325857 0.945419i \(-0.394347\pi\)
0.325857 + 0.945419i \(0.394347\pi\)
\(522\) 0 0
\(523\) 6.53590 0.285795 0.142897 0.989738i \(-0.454358\pi\)
0.142897 + 0.989738i \(0.454358\pi\)
\(524\) 0 0
\(525\) −4.73205 −0.206524
\(526\) 0 0
\(527\) −13.8564 −0.603595
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 9.46410 0.410707
\(532\) 0 0
\(533\) −14.7846 −0.640393
\(534\) 0 0
\(535\) 10.9282 0.472467
\(536\) 0 0
\(537\) 21.4641 0.926244
\(538\) 0 0
\(539\) −64.5885 −2.78202
\(540\) 0 0
\(541\) −21.7128 −0.933507 −0.466753 0.884388i \(-0.654576\pi\)
−0.466753 + 0.884388i \(0.654576\pi\)
\(542\) 0 0
\(543\) −4.53590 −0.194654
\(544\) 0 0
\(545\) −10.3923 −0.445157
\(546\) 0 0
\(547\) 18.2487 0.780259 0.390129 0.920760i \(-0.372430\pi\)
0.390129 + 0.920760i \(0.372430\pi\)
\(548\) 0 0
\(549\) 5.46410 0.233202
\(550\) 0 0
\(551\) 10.1962 0.434371
\(552\) 0 0
\(553\) 37.8564 1.60982
\(554\) 0 0
\(555\) −2.73205 −0.115969
\(556\) 0 0
\(557\) 19.8564 0.841343 0.420671 0.907213i \(-0.361795\pi\)
0.420671 + 0.907213i \(0.361795\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) −29.0718 −1.22741
\(562\) 0 0
\(563\) −17.6077 −0.742076 −0.371038 0.928618i \(-0.620998\pi\)
−0.371038 + 0.928618i \(0.620998\pi\)
\(564\) 0 0
\(565\) 2.53590 0.106686
\(566\) 0 0
\(567\) 4.73205 0.198727
\(568\) 0 0
\(569\) 9.80385 0.410999 0.205499 0.978657i \(-0.434118\pi\)
0.205499 + 0.978657i \(0.434118\pi\)
\(570\) 0 0
\(571\) −7.60770 −0.318372 −0.159186 0.987249i \(-0.550887\pi\)
−0.159186 + 0.987249i \(0.550887\pi\)
\(572\) 0 0
\(573\) −13.2679 −0.554277
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) 25.3205 1.05411 0.527053 0.849832i \(-0.323297\pi\)
0.527053 + 0.849832i \(0.323297\pi\)
\(578\) 0 0
\(579\) 11.8038 0.490551
\(580\) 0 0
\(581\) −40.3923 −1.67576
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −1.26795 −0.0524232
\(586\) 0 0
\(587\) −22.7846 −0.940421 −0.470211 0.882554i \(-0.655822\pi\)
−0.470211 + 0.882554i \(0.655822\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) −1.07180 −0.0440878
\(592\) 0 0
\(593\) 2.78461 0.114350 0.0571751 0.998364i \(-0.481791\pi\)
0.0571751 + 0.998364i \(0.481791\pi\)
\(594\) 0 0
\(595\) −32.7846 −1.34404
\(596\) 0 0
\(597\) −2.53590 −0.103787
\(598\) 0 0
\(599\) 28.0000 1.14405 0.572024 0.820237i \(-0.306158\pi\)
0.572024 + 0.820237i \(0.306158\pi\)
\(600\) 0 0
\(601\) −26.3923 −1.07656 −0.538282 0.842765i \(-0.680927\pi\)
−0.538282 + 0.842765i \(0.680927\pi\)
\(602\) 0 0
\(603\) 14.9282 0.607923
\(604\) 0 0
\(605\) 6.60770 0.268641
\(606\) 0 0
\(607\) 10.5359 0.427639 0.213819 0.976873i \(-0.431410\pi\)
0.213819 + 0.976873i \(0.431410\pi\)
\(608\) 0 0
\(609\) 48.2487 1.95514
\(610\) 0 0
\(611\) −12.6795 −0.512957
\(612\) 0 0
\(613\) −27.1769 −1.09767 −0.548833 0.835932i \(-0.684928\pi\)
−0.548833 + 0.835932i \(0.684928\pi\)
\(614\) 0 0
\(615\) −11.6603 −0.470187
\(616\) 0 0
\(617\) 4.78461 0.192621 0.0963106 0.995351i \(-0.469296\pi\)
0.0963106 + 0.995351i \(0.469296\pi\)
\(618\) 0 0
\(619\) 41.1769 1.65504 0.827520 0.561436i \(-0.189751\pi\)
0.827520 + 0.561436i \(0.189751\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) −36.2487 −1.45227
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −4.19615 −0.167578
\(628\) 0 0
\(629\) −18.9282 −0.754717
\(630\) 0 0
\(631\) 19.7128 0.784755 0.392377 0.919804i \(-0.371653\pi\)
0.392377 + 0.919804i \(0.371653\pi\)
\(632\) 0 0
\(633\) −6.92820 −0.275371
\(634\) 0 0
\(635\) 1.07180 0.0425330
\(636\) 0 0
\(637\) −19.5167 −0.773278
\(638\) 0 0
\(639\) −12.3923 −0.490232
\(640\) 0 0
\(641\) −37.5167 −1.48182 −0.740909 0.671605i \(-0.765605\pi\)
−0.740909 + 0.671605i \(0.765605\pi\)
\(642\) 0 0
\(643\) 30.9808 1.22176 0.610881 0.791722i \(-0.290815\pi\)
0.610881 + 0.791722i \(0.290815\pi\)
\(644\) 0 0
\(645\) −4.73205 −0.186324
\(646\) 0 0
\(647\) 6.67949 0.262598 0.131299 0.991343i \(-0.458085\pi\)
0.131299 + 0.991343i \(0.458085\pi\)
\(648\) 0 0
\(649\) −39.7128 −1.55886
\(650\) 0 0
\(651\) −9.46410 −0.370927
\(652\) 0 0
\(653\) −7.21539 −0.282360 −0.141180 0.989984i \(-0.545090\pi\)
−0.141180 + 0.989984i \(0.545090\pi\)
\(654\) 0 0
\(655\) 20.5885 0.804458
\(656\) 0 0
\(657\) 0.928203 0.0362127
\(658\) 0 0
\(659\) −43.7128 −1.70281 −0.851405 0.524509i \(-0.824249\pi\)
−0.851405 + 0.524509i \(0.824249\pi\)
\(660\) 0 0
\(661\) 34.3923 1.33771 0.668853 0.743395i \(-0.266786\pi\)
0.668853 + 0.743395i \(0.266786\pi\)
\(662\) 0 0
\(663\) −8.78461 −0.341166
\(664\) 0 0
\(665\) −4.73205 −0.183501
\(666\) 0 0
\(667\) −61.1769 −2.36878
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) −22.9282 −0.885133
\(672\) 0 0
\(673\) −39.5167 −1.52326 −0.761628 0.648015i \(-0.775599\pi\)
−0.761628 + 0.648015i \(0.775599\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −40.0000 −1.53732 −0.768662 0.639655i \(-0.779077\pi\)
−0.768662 + 0.639655i \(0.779077\pi\)
\(678\) 0 0
\(679\) 7.85641 0.301501
\(680\) 0 0
\(681\) 23.4641 0.899146
\(682\) 0 0
\(683\) 8.00000 0.306111 0.153056 0.988218i \(-0.451089\pi\)
0.153056 + 0.988218i \(0.451089\pi\)
\(684\) 0 0
\(685\) 22.7846 0.870555
\(686\) 0 0
\(687\) −17.4641 −0.666297
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −8.39230 −0.319258 −0.159629 0.987177i \(-0.551030\pi\)
−0.159629 + 0.987177i \(0.551030\pi\)
\(692\) 0 0
\(693\) −19.8564 −0.754283
\(694\) 0 0
\(695\) 17.4641 0.662451
\(696\) 0 0
\(697\) −80.7846 −3.05994
\(698\) 0 0
\(699\) −0.143594 −0.00543121
\(700\) 0 0
\(701\) 24.9282 0.941525 0.470763 0.882260i \(-0.343979\pi\)
0.470763 + 0.882260i \(0.343979\pi\)
\(702\) 0 0
\(703\) −2.73205 −0.103041
\(704\) 0 0
\(705\) −10.0000 −0.376622
\(706\) 0 0
\(707\) −16.3923 −0.616496
\(708\) 0 0
\(709\) 38.2487 1.43646 0.718230 0.695806i \(-0.244952\pi\)
0.718230 + 0.695806i \(0.244952\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 5.32051 0.198976
\(716\) 0 0
\(717\) −22.0526 −0.823568
\(718\) 0 0
\(719\) 28.9808 1.08080 0.540400 0.841408i \(-0.318273\pi\)
0.540400 + 0.841408i \(0.318273\pi\)
\(720\) 0 0
\(721\) 65.5692 2.44193
\(722\) 0 0
\(723\) 8.92820 0.332043
\(724\) 0 0
\(725\) −10.1962 −0.378676
\(726\) 0 0
\(727\) 1.41154 0.0523512 0.0261756 0.999657i \(-0.491667\pi\)
0.0261756 + 0.999657i \(0.491667\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −32.7846 −1.21258
\(732\) 0 0
\(733\) −34.7846 −1.28480 −0.642399 0.766370i \(-0.722061\pi\)
−0.642399 + 0.766370i \(0.722061\pi\)
\(734\) 0 0
\(735\) −15.3923 −0.567754
\(736\) 0 0
\(737\) −62.6410 −2.30741
\(738\) 0 0
\(739\) 22.9282 0.843428 0.421714 0.906729i \(-0.361429\pi\)
0.421714 + 0.906729i \(0.361429\pi\)
\(740\) 0 0
\(741\) −1.26795 −0.0465793
\(742\) 0 0
\(743\) 26.6410 0.977364 0.488682 0.872462i \(-0.337478\pi\)
0.488682 + 0.872462i \(0.337478\pi\)
\(744\) 0 0
\(745\) 8.92820 0.327104
\(746\) 0 0
\(747\) −8.53590 −0.312312
\(748\) 0 0
\(749\) 51.7128 1.88955
\(750\) 0 0
\(751\) 8.14359 0.297164 0.148582 0.988900i \(-0.452529\pi\)
0.148582 + 0.988900i \(0.452529\pi\)
\(752\) 0 0
\(753\) −28.1962 −1.02752
\(754\) 0 0
\(755\) 2.00000 0.0727875
\(756\) 0 0
\(757\) −11.4641 −0.416670 −0.208335 0.978058i \(-0.566804\pi\)
−0.208335 + 0.978058i \(0.566804\pi\)
\(758\) 0 0
\(759\) 25.1769 0.913864
\(760\) 0 0
\(761\) −27.0718 −0.981352 −0.490676 0.871342i \(-0.663250\pi\)
−0.490676 + 0.871342i \(0.663250\pi\)
\(762\) 0 0
\(763\) −49.1769 −1.78032
\(764\) 0 0
\(765\) −6.92820 −0.250490
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) −38.7846 −1.39861 −0.699304 0.714824i \(-0.746507\pi\)
−0.699304 + 0.714824i \(0.746507\pi\)
\(770\) 0 0
\(771\) −27.3205 −0.983924
\(772\) 0 0
\(773\) 1.85641 0.0667703 0.0333851 0.999443i \(-0.489371\pi\)
0.0333851 + 0.999443i \(0.489371\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) −12.9282 −0.463797
\(778\) 0 0
\(779\) −11.6603 −0.417772
\(780\) 0 0
\(781\) 52.0000 1.86071
\(782\) 0 0
\(783\) 10.1962 0.364381
\(784\) 0 0
\(785\) −3.46410 −0.123639
\(786\) 0 0
\(787\) −1.46410 −0.0521896 −0.0260948 0.999659i \(-0.508307\pi\)
−0.0260948 + 0.999659i \(0.508307\pi\)
\(788\) 0 0
\(789\) 3.46410 0.123325
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −6.92820 −0.246028
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.24871 −0.0796534 −0.0398267 0.999207i \(-0.512681\pi\)
−0.0398267 + 0.999207i \(0.512681\pi\)
\(798\) 0 0
\(799\) −69.2820 −2.45102
\(800\) 0 0
\(801\) −7.66025 −0.270662
\(802\) 0 0
\(803\) −3.89488 −0.137447
\(804\) 0 0
\(805\) 28.3923 1.00070
\(806\) 0 0
\(807\) 25.5167 0.898229
\(808\) 0 0
\(809\) 42.4974 1.49413 0.747065 0.664751i \(-0.231462\pi\)
0.747065 + 0.664751i \(0.231462\pi\)
\(810\) 0 0
\(811\) 26.6410 0.935493 0.467746 0.883863i \(-0.345066\pi\)
0.467746 + 0.883863i \(0.345066\pi\)
\(812\) 0 0
\(813\) 23.3205 0.817886
\(814\) 0 0
\(815\) −25.1244 −0.880068
\(816\) 0 0
\(817\) −4.73205 −0.165554
\(818\) 0 0
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) −21.3205 −0.744091 −0.372045 0.928215i \(-0.621343\pi\)
−0.372045 + 0.928215i \(0.621343\pi\)
\(822\) 0 0
\(823\) −10.9808 −0.382765 −0.191383 0.981516i \(-0.561297\pi\)
−0.191383 + 0.981516i \(0.561297\pi\)
\(824\) 0 0
\(825\) 4.19615 0.146091
\(826\) 0 0
\(827\) −26.1051 −0.907764 −0.453882 0.891062i \(-0.649961\pi\)
−0.453882 + 0.891062i \(0.649961\pi\)
\(828\) 0 0
\(829\) 17.6077 0.611541 0.305770 0.952105i \(-0.401086\pi\)
0.305770 + 0.952105i \(0.401086\pi\)
\(830\) 0 0
\(831\) −11.8564 −0.411294
\(832\) 0 0
\(833\) −106.641 −3.69489
\(834\) 0 0
\(835\) −14.3923 −0.498066
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) −25.4641 −0.879118 −0.439559 0.898214i \(-0.644865\pi\)
−0.439559 + 0.898214i \(0.644865\pi\)
\(840\) 0 0
\(841\) 74.9615 2.58488
\(842\) 0 0
\(843\) 8.05256 0.277345
\(844\) 0 0
\(845\) −11.3923 −0.391907
\(846\) 0 0
\(847\) 31.2679 1.07438
\(848\) 0 0
\(849\) −13.8038 −0.473747
\(850\) 0 0
\(851\) 16.3923 0.561921
\(852\) 0 0
\(853\) −24.5359 −0.840093 −0.420047 0.907503i \(-0.637986\pi\)
−0.420047 + 0.907503i \(0.637986\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) −21.0718 −0.719799 −0.359899 0.932991i \(-0.617189\pi\)
−0.359899 + 0.932991i \(0.617189\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) −55.1769 −1.88042
\(862\) 0 0
\(863\) −6.14359 −0.209130 −0.104565 0.994518i \(-0.533345\pi\)
−0.104565 + 0.994518i \(0.533345\pi\)
\(864\) 0 0
\(865\) 12.3923 0.421351
\(866\) 0 0
\(867\) −31.0000 −1.05282
\(868\) 0 0
\(869\) −33.5692 −1.13876
\(870\) 0 0
\(871\) −18.9282 −0.641358
\(872\) 0 0
\(873\) 1.66025 0.0561911
\(874\) 0 0
\(875\) 4.73205 0.159973
\(876\) 0 0
\(877\) −35.8038 −1.20901 −0.604505 0.796601i \(-0.706629\pi\)
−0.604505 + 0.796601i \(0.706629\pi\)
\(878\) 0 0
\(879\) 24.3923 0.822732
\(880\) 0 0
\(881\) −4.24871 −0.143143 −0.0715714 0.997435i \(-0.522801\pi\)
−0.0715714 + 0.997435i \(0.522801\pi\)
\(882\) 0 0
\(883\) −35.2679 −1.18686 −0.593430 0.804885i \(-0.702227\pi\)
−0.593430 + 0.804885i \(0.702227\pi\)
\(884\) 0 0
\(885\) −9.46410 −0.318132
\(886\) 0 0
\(887\) 5.07180 0.170294 0.0851471 0.996368i \(-0.472864\pi\)
0.0851471 + 0.996368i \(0.472864\pi\)
\(888\) 0 0
\(889\) 5.07180 0.170103
\(890\) 0 0
\(891\) −4.19615 −0.140576
\(892\) 0 0
\(893\) −10.0000 −0.334637
\(894\) 0 0
\(895\) −21.4641 −0.717466
\(896\) 0 0
\(897\) 7.60770 0.254014
\(898\) 0 0
\(899\) −20.3923 −0.680121
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −22.3923 −0.745169
\(904\) 0 0
\(905\) 4.53590 0.150778
\(906\) 0 0
\(907\) 47.0333 1.56172 0.780858 0.624709i \(-0.214782\pi\)
0.780858 + 0.624709i \(0.214782\pi\)
\(908\) 0 0
\(909\) −3.46410 −0.114897
\(910\) 0 0
\(911\) −17.0718 −0.565614 −0.282807 0.959177i \(-0.591266\pi\)
−0.282807 + 0.959177i \(0.591266\pi\)
\(912\) 0 0
\(913\) 35.8179 1.18540
\(914\) 0 0
\(915\) −5.46410 −0.180638
\(916\) 0 0
\(917\) 97.4256 3.21728
\(918\) 0 0
\(919\) 22.9282 0.756332 0.378166 0.925738i \(-0.376555\pi\)
0.378166 + 0.925738i \(0.376555\pi\)
\(920\) 0 0
\(921\) 7.32051 0.241219
\(922\) 0 0
\(923\) 15.7128 0.517194
\(924\) 0 0
\(925\) 2.73205 0.0898293
\(926\) 0 0
\(927\) 13.8564 0.455104
\(928\) 0 0
\(929\) 60.2487 1.97670 0.988348 0.152211i \(-0.0486393\pi\)
0.988348 + 0.152211i \(0.0486393\pi\)
\(930\) 0 0
\(931\) −15.3923 −0.504462
\(932\) 0 0
\(933\) 21.2679 0.696281
\(934\) 0 0
\(935\) 29.0718 0.950749
\(936\) 0 0
\(937\) 43.1769 1.41053 0.705264 0.708945i \(-0.250828\pi\)
0.705264 + 0.708945i \(0.250828\pi\)
\(938\) 0 0
\(939\) 13.3205 0.434698
\(940\) 0 0
\(941\) −15.9474 −0.519872 −0.259936 0.965626i \(-0.583701\pi\)
−0.259936 + 0.965626i \(0.583701\pi\)
\(942\) 0 0
\(943\) 69.9615 2.27826
\(944\) 0 0
\(945\) −4.73205 −0.153934
\(946\) 0 0
\(947\) 18.6795 0.607002 0.303501 0.952831i \(-0.401844\pi\)
0.303501 + 0.952831i \(0.401844\pi\)
\(948\) 0 0
\(949\) −1.17691 −0.0382043
\(950\) 0 0
\(951\) −1.07180 −0.0347554
\(952\) 0 0
\(953\) 17.1769 0.556415 0.278207 0.960521i \(-0.410260\pi\)
0.278207 + 0.960521i \(0.410260\pi\)
\(954\) 0 0
\(955\) 13.2679 0.429341
\(956\) 0 0
\(957\) −42.7846 −1.38303
\(958\) 0 0
\(959\) 107.818 3.48162
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 10.9282 0.352156
\(964\) 0 0
\(965\) −11.8038 −0.379979
\(966\) 0 0
\(967\) 46.9808 1.51080 0.755400 0.655264i \(-0.227443\pi\)
0.755400 + 0.655264i \(0.227443\pi\)
\(968\) 0 0
\(969\) −6.92820 −0.222566
\(970\) 0 0
\(971\) −60.4974 −1.94145 −0.970727 0.240184i \(-0.922792\pi\)
−0.970727 + 0.240184i \(0.922792\pi\)
\(972\) 0 0
\(973\) 82.6410 2.64935
\(974\) 0 0
\(975\) 1.26795 0.0406069
\(976\) 0 0
\(977\) 17.8564 0.571277 0.285639 0.958337i \(-0.407794\pi\)
0.285639 + 0.958337i \(0.407794\pi\)
\(978\) 0 0
\(979\) 32.1436 1.02731
\(980\) 0 0
\(981\) −10.3923 −0.331801
\(982\) 0 0
\(983\) −33.6077 −1.07192 −0.535960 0.844244i \(-0.680050\pi\)
−0.535960 + 0.844244i \(0.680050\pi\)
\(984\) 0 0
\(985\) 1.07180 0.0341503
\(986\) 0 0
\(987\) −47.3205 −1.50623
\(988\) 0 0
\(989\) 28.3923 0.902823
\(990\) 0 0
\(991\) 16.7846 0.533181 0.266590 0.963810i \(-0.414103\pi\)
0.266590 + 0.963810i \(0.414103\pi\)
\(992\) 0 0
\(993\) −16.9282 −0.537200
\(994\) 0 0
\(995\) 2.53590 0.0803934
\(996\) 0 0
\(997\) −37.6077 −1.19105 −0.595524 0.803338i \(-0.703055\pi\)
−0.595524 + 0.803338i \(0.703055\pi\)
\(998\) 0 0
\(999\) −2.73205 −0.0864383
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4560.2.a.bi.1.2 2
4.3 odd 2 2280.2.a.q.1.1 2
12.11 even 2 6840.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.q.1.1 2 4.3 odd 2
4560.2.a.bi.1.2 2 1.1 even 1 trivial
6840.2.a.v.1.1 2 12.11 even 2