Properties

Label 7728.2.a.by.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1229.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.59261\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.59261 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.59261 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.87096 q^{11} +0.721651 q^{13} -2.59261 q^{15} +5.31427 q^{17} +7.31427 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.72165 q^{25} +1.00000 q^{27} -5.31427 q^{29} -2.12904 q^{31} +1.87096 q^{33} +2.59261 q^{35} -6.49950 q^{37} +0.721651 q^{39} +4.12904 q^{41} +1.27835 q^{43} -2.59261 q^{45} -12.9272 q^{47} +1.00000 q^{49} +5.31427 q^{51} +10.3345 q^{53} -4.85069 q^{55} +7.31427 q^{57} +10.0359 q^{59} -7.09211 q^{61} -1.00000 q^{63} -1.87096 q^{65} -2.03592 q^{67} -1.00000 q^{69} +11.9069 q^{71} -5.57234 q^{73} +1.72165 q^{75} -1.87096 q^{77} +11.3143 q^{79} +1.00000 q^{81} -1.87096 q^{83} -13.7778 q^{85} -5.31427 q^{87} -1.90688 q^{89} -0.721651 q^{91} -2.12904 q^{93} -18.9631 q^{95} +7.87096 q^{97} +1.87096 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + q^{5} - 3q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + q^{5} - 3q^{7} + 3q^{9} + 2q^{11} - 3q^{13} + q^{15} + 2q^{17} + 8q^{19} - 3q^{21} - 3q^{23} + 3q^{27} - 2q^{29} - 10q^{31} + 2q^{33} - q^{35} + 12q^{37} - 3q^{39} + 16q^{41} + 9q^{43} + q^{45} - 14q^{47} + 3q^{49} + 2q^{51} + 15q^{53} - 13q^{55} + 8q^{57} + 11q^{59} + 19q^{61} - 3q^{63} - 2q^{65} + 13q^{67} - 3q^{69} + 13q^{71} - 10q^{73} - 2q^{77} + 20q^{79} + 3q^{81} - 2q^{83} - 15q^{85} - 2q^{87} + 17q^{89} + 3q^{91} - 10q^{93} - 13q^{95} + 20q^{97} + 2q^{99} + O(q^{100}) \)

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
\(5\) −2.59261 −1.15945 −0.579726 0.814811i \(-0.696841\pi\)
−0.579726 + 0.814811i \(0.696841\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.87096 0.564117 0.282058 0.959397i \(-0.408983\pi\)
0.282058 + 0.959397i \(0.408983\pi\)
\(12\) 0 0
\(13\) 0.721651 0.200150 0.100075 0.994980i \(-0.468092\pi\)
0.100075 + 0.994980i \(0.468092\pi\)
\(14\) 0 0
\(15\) −2.59261 −0.669410
\(16\) 0 0
\(17\) 5.31427 1.28890 0.644449 0.764647i \(-0.277087\pi\)
0.644449 + 0.764647i \(0.277087\pi\)
\(18\) 0 0
\(19\) 7.31427 1.67801 0.839004 0.544125i \(-0.183138\pi\)
0.839004 + 0.544125i \(0.183138\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.72165 0.344330
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −5.31427 −0.986834 −0.493417 0.869793i \(-0.664252\pi\)
−0.493417 + 0.869793i \(0.664252\pi\)
\(30\) 0 0
\(31\) −2.12904 −0.382386 −0.191193 0.981552i \(-0.561236\pi\)
−0.191193 + 0.981552i \(0.561236\pi\)
\(32\) 0 0
\(33\) 1.87096 0.325693
\(34\) 0 0
\(35\) 2.59261 0.438232
\(36\) 0 0
\(37\) −6.49950 −1.06851 −0.534255 0.845323i \(-0.679408\pi\)
−0.534255 + 0.845323i \(0.679408\pi\)
\(38\) 0 0
\(39\) 0.721651 0.115557
\(40\) 0 0
\(41\) 4.12904 0.644847 0.322424 0.946595i \(-0.395502\pi\)
0.322424 + 0.946595i \(0.395502\pi\)
\(42\) 0 0
\(43\) 1.27835 0.194946 0.0974732 0.995238i \(-0.468924\pi\)
0.0974732 + 0.995238i \(0.468924\pi\)
\(44\) 0 0
\(45\) −2.59261 −0.386484
\(46\) 0 0
\(47\) −12.9272 −1.88562 −0.942810 0.333331i \(-0.891827\pi\)
−0.942810 + 0.333331i \(0.891827\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 5.31427 0.744146
\(52\) 0 0
\(53\) 10.3345 1.41956 0.709779 0.704424i \(-0.248795\pi\)
0.709779 + 0.704424i \(0.248795\pi\)
\(54\) 0 0
\(55\) −4.85069 −0.654067
\(56\) 0 0
\(57\) 7.31427 0.968798
\(58\) 0 0
\(59\) 10.0359 1.30657 0.653283 0.757114i \(-0.273391\pi\)
0.653283 + 0.757114i \(0.273391\pi\)
\(60\) 0 0
\(61\) −7.09211 −0.908052 −0.454026 0.890989i \(-0.650013\pi\)
−0.454026 + 0.890989i \(0.650013\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −1.87096 −0.232064
\(66\) 0 0
\(67\) −2.03592 −0.248727 −0.124363 0.992237i \(-0.539689\pi\)
−0.124363 + 0.992237i \(0.539689\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 11.9069 1.41309 0.706543 0.707670i \(-0.250254\pi\)
0.706543 + 0.707670i \(0.250254\pi\)
\(72\) 0 0
\(73\) −5.57234 −0.652193 −0.326096 0.945336i \(-0.605733\pi\)
−0.326096 + 0.945336i \(0.605733\pi\)
\(74\) 0 0
\(75\) 1.72165 0.198799
\(76\) 0 0
\(77\) −1.87096 −0.213216
\(78\) 0 0
\(79\) 11.3143 1.27295 0.636477 0.771296i \(-0.280391\pi\)
0.636477 + 0.771296i \(0.280391\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.87096 −0.205365 −0.102682 0.994714i \(-0.532743\pi\)
−0.102682 + 0.994714i \(0.532743\pi\)
\(84\) 0 0
\(85\) −13.7778 −1.49442
\(86\) 0 0
\(87\) −5.31427 −0.569749
\(88\) 0 0
\(89\) −1.90688 −0.202129 −0.101064 0.994880i \(-0.532225\pi\)
−0.101064 + 0.994880i \(0.532225\pi\)
\(90\) 0 0
\(91\) −0.721651 −0.0756496
\(92\) 0 0
\(93\) −2.12904 −0.220771
\(94\) 0 0
\(95\) −18.9631 −1.94557
\(96\) 0 0
\(97\) 7.87096 0.799175 0.399588 0.916695i \(-0.369153\pi\)
0.399588 + 0.916695i \(0.369153\pi\)
\(98\) 0 0
\(99\) 1.87096 0.188039
\(100\) 0 0
\(101\) 12.0359 1.19762 0.598809 0.800892i \(-0.295641\pi\)
0.598809 + 0.800892i \(0.295641\pi\)
\(102\) 0 0
\(103\) −10.3705 −1.02183 −0.510916 0.859631i \(-0.670694\pi\)
−0.510916 + 0.859631i \(0.670694\pi\)
\(104\) 0 0
\(105\) 2.59261 0.253013
\(106\) 0 0
\(107\) −14.5354 −1.40519 −0.702596 0.711589i \(-0.747976\pi\)
−0.702596 + 0.711589i \(0.747976\pi\)
\(108\) 0 0
\(109\) 1.14931 0.110084 0.0550421 0.998484i \(-0.482471\pi\)
0.0550421 + 0.998484i \(0.482471\pi\)
\(110\) 0 0
\(111\) −6.49950 −0.616905
\(112\) 0 0
\(113\) 19.7778 1.86054 0.930272 0.366872i \(-0.119571\pi\)
0.930272 + 0.366872i \(0.119571\pi\)
\(114\) 0 0
\(115\) 2.59261 0.241763
\(116\) 0 0
\(117\) 0.721651 0.0667167
\(118\) 0 0
\(119\) −5.31427 −0.487158
\(120\) 0 0
\(121\) −7.49950 −0.681772
\(122\) 0 0
\(123\) 4.12904 0.372303
\(124\) 0 0
\(125\) 8.49950 0.760218
\(126\) 0 0
\(127\) 6.46358 0.573550 0.286775 0.957998i \(-0.407417\pi\)
0.286775 + 0.957998i \(0.407417\pi\)
\(128\) 0 0
\(129\) 1.27835 0.112552
\(130\) 0 0
\(131\) −6.12904 −0.535496 −0.267748 0.963489i \(-0.586280\pi\)
−0.267748 + 0.963489i \(0.586280\pi\)
\(132\) 0 0
\(133\) −7.31427 −0.634227
\(134\) 0 0
\(135\) −2.59261 −0.223137
\(136\) 0 0
\(137\) 7.87096 0.672462 0.336231 0.941780i \(-0.390848\pi\)
0.336231 + 0.941780i \(0.390848\pi\)
\(138\) 0 0
\(139\) −14.9631 −1.26915 −0.634576 0.772861i \(-0.718825\pi\)
−0.634576 + 0.772861i \(0.718825\pi\)
\(140\) 0 0
\(141\) −12.9272 −1.08866
\(142\) 0 0
\(143\) 1.35018 0.112908
\(144\) 0 0
\(145\) 13.7778 1.14419
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −11.1852 −0.916330 −0.458165 0.888867i \(-0.651493\pi\)
−0.458165 + 0.888867i \(0.651493\pi\)
\(150\) 0 0
\(151\) −12.9272 −1.05200 −0.525999 0.850485i \(-0.676308\pi\)
−0.525999 + 0.850485i \(0.676308\pi\)
\(152\) 0 0
\(153\) 5.31427 0.429633
\(154\) 0 0
\(155\) 5.51977 0.443359
\(156\) 0 0
\(157\) 14.3299 1.14365 0.571826 0.820375i \(-0.306235\pi\)
0.571826 + 0.820375i \(0.306235\pi\)
\(158\) 0 0
\(159\) 10.3345 0.819582
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 13.7778 1.07916 0.539582 0.841933i \(-0.318582\pi\)
0.539582 + 0.841933i \(0.318582\pi\)
\(164\) 0 0
\(165\) −4.85069 −0.377626
\(166\) 0 0
\(167\) 20.4995 1.58630 0.793149 0.609027i \(-0.208440\pi\)
0.793149 + 0.609027i \(0.208440\pi\)
\(168\) 0 0
\(169\) −12.4792 −0.959940
\(170\) 0 0
\(171\) 7.31427 0.559336
\(172\) 0 0
\(173\) 22.7576 1.73023 0.865113 0.501577i \(-0.167247\pi\)
0.865113 + 0.501577i \(0.167247\pi\)
\(174\) 0 0
\(175\) −1.72165 −0.130145
\(176\) 0 0
\(177\) 10.0359 0.754346
\(178\) 0 0
\(179\) −16.6644 −1.24556 −0.622780 0.782397i \(-0.713997\pi\)
−0.622780 + 0.782397i \(0.713997\pi\)
\(180\) 0 0
\(181\) 25.1280 1.86775 0.933876 0.357598i \(-0.116404\pi\)
0.933876 + 0.357598i \(0.116404\pi\)
\(182\) 0 0
\(183\) −7.09211 −0.524264
\(184\) 0 0
\(185\) 16.8507 1.23889
\(186\) 0 0
\(187\) 9.94280 0.727089
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −10.5567 −0.763856 −0.381928 0.924192i \(-0.624740\pi\)
−0.381928 + 0.924192i \(0.624740\pi\)
\(192\) 0 0
\(193\) 7.44330 0.535781 0.267890 0.963449i \(-0.413674\pi\)
0.267890 + 0.963449i \(0.413674\pi\)
\(194\) 0 0
\(195\) −1.87096 −0.133982
\(196\) 0 0
\(197\) 7.02028 0.500174 0.250087 0.968223i \(-0.419541\pi\)
0.250087 + 0.968223i \(0.419541\pi\)
\(198\) 0 0
\(199\) −3.66546 −0.259837 −0.129919 0.991525i \(-0.541472\pi\)
−0.129919 + 0.991525i \(0.541472\pi\)
\(200\) 0 0
\(201\) −2.03592 −0.143603
\(202\) 0 0
\(203\) 5.31427 0.372988
\(204\) 0 0
\(205\) −10.7050 −0.747670
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 13.6847 0.946592
\(210\) 0 0
\(211\) −0.685734 −0.0472079 −0.0236039 0.999721i \(-0.507514\pi\)
−0.0236039 + 0.999721i \(0.507514\pi\)
\(212\) 0 0
\(213\) 11.9069 0.815846
\(214\) 0 0
\(215\) −3.31427 −0.226031
\(216\) 0 0
\(217\) 2.12904 0.144528
\(218\) 0 0
\(219\) −5.57234 −0.376544
\(220\) 0 0
\(221\) 3.83505 0.257973
\(222\) 0 0
\(223\) 28.6644 1.91951 0.959757 0.280833i \(-0.0906106\pi\)
0.959757 + 0.280833i \(0.0906106\pi\)
\(224\) 0 0
\(225\) 1.72165 0.114777
\(226\) 0 0
\(227\) −24.1483 −1.60278 −0.801390 0.598143i \(-0.795906\pi\)
−0.801390 + 0.598143i \(0.795906\pi\)
\(228\) 0 0
\(229\) 15.7778 1.04263 0.521315 0.853365i \(-0.325442\pi\)
0.521315 + 0.853365i \(0.325442\pi\)
\(230\) 0 0
\(231\) −1.87096 −0.123100
\(232\) 0 0
\(233\) 19.7778 1.29569 0.647845 0.761772i \(-0.275671\pi\)
0.647845 + 0.761772i \(0.275671\pi\)
\(234\) 0 0
\(235\) 33.5151 2.18629
\(236\) 0 0
\(237\) 11.3143 0.734941
\(238\) 0 0
\(239\) 16.4064 1.06124 0.530620 0.847610i \(-0.321959\pi\)
0.530620 + 0.847610i \(0.321959\pi\)
\(240\) 0 0
\(241\) 14.4995 0.933995 0.466997 0.884259i \(-0.345336\pi\)
0.466997 + 0.884259i \(0.345336\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.59261 −0.165636
\(246\) 0 0
\(247\) 5.27835 0.335853
\(248\) 0 0
\(249\) −1.87096 −0.118567
\(250\) 0 0
\(251\) −7.55569 −0.476911 −0.238455 0.971153i \(-0.576641\pi\)
−0.238455 + 0.971153i \(0.576641\pi\)
\(252\) 0 0
\(253\) −1.87096 −0.117626
\(254\) 0 0
\(255\) −13.7778 −0.862802
\(256\) 0 0
\(257\) −6.66908 −0.416006 −0.208003 0.978128i \(-0.566696\pi\)
−0.208003 + 0.978128i \(0.566696\pi\)
\(258\) 0 0
\(259\) 6.49950 0.403859
\(260\) 0 0
\(261\) −5.31427 −0.328945
\(262\) 0 0
\(263\) 24.0552 1.48331 0.741653 0.670784i \(-0.234042\pi\)
0.741653 + 0.670784i \(0.234042\pi\)
\(264\) 0 0
\(265\) −26.7935 −1.64591
\(266\) 0 0
\(267\) −1.90688 −0.116699
\(268\) 0 0
\(269\) 8.72165 0.531768 0.265884 0.964005i \(-0.414336\pi\)
0.265884 + 0.964005i \(0.414336\pi\)
\(270\) 0 0
\(271\) −22.5400 −1.36921 −0.684605 0.728914i \(-0.740025\pi\)
−0.684605 + 0.728914i \(0.740025\pi\)
\(272\) 0 0
\(273\) −0.721651 −0.0436763
\(274\) 0 0
\(275\) 3.22115 0.194242
\(276\) 0 0
\(277\) 21.9787 1.32057 0.660286 0.751014i \(-0.270435\pi\)
0.660286 + 0.751014i \(0.270435\pi\)
\(278\) 0 0
\(279\) −2.12904 −0.127462
\(280\) 0 0
\(281\) 14.9272 0.890480 0.445240 0.895411i \(-0.353118\pi\)
0.445240 + 0.895411i \(0.353118\pi\)
\(282\) 0 0
\(283\) −32.6644 −1.94170 −0.970850 0.239688i \(-0.922955\pi\)
−0.970850 + 0.239688i \(0.922955\pi\)
\(284\) 0 0
\(285\) −18.9631 −1.12328
\(286\) 0 0
\(287\) −4.12904 −0.243729
\(288\) 0 0
\(289\) 11.2414 0.661260
\(290\) 0 0
\(291\) 7.87096 0.461404
\(292\) 0 0
\(293\) −11.5151 −0.672721 −0.336361 0.941733i \(-0.609196\pi\)
−0.336361 + 0.941733i \(0.609196\pi\)
\(294\) 0 0
\(295\) −26.0193 −1.51490
\(296\) 0 0
\(297\) 1.87096 0.108564
\(298\) 0 0
\(299\) −0.721651 −0.0417342
\(300\) 0 0
\(301\) −1.27835 −0.0736828
\(302\) 0 0
\(303\) 12.0359 0.691445
\(304\) 0 0
\(305\) 18.3871 1.05284
\(306\) 0 0
\(307\) 29.7566 1.69830 0.849148 0.528155i \(-0.177116\pi\)
0.849148 + 0.528155i \(0.177116\pi\)
\(308\) 0 0
\(309\) −10.3705 −0.589955
\(310\) 0 0
\(311\) 15.4626 0.876802 0.438401 0.898780i \(-0.355545\pi\)
0.438401 + 0.898780i \(0.355545\pi\)
\(312\) 0 0
\(313\) 16.6285 0.939900 0.469950 0.882693i \(-0.344272\pi\)
0.469950 + 0.882693i \(0.344272\pi\)
\(314\) 0 0
\(315\) 2.59261 0.146077
\(316\) 0 0
\(317\) 16.9631 0.952741 0.476371 0.879245i \(-0.341952\pi\)
0.476371 + 0.879245i \(0.341952\pi\)
\(318\) 0 0
\(319\) −9.94280 −0.556690
\(320\) 0 0
\(321\) −14.5354 −0.811288
\(322\) 0 0
\(323\) 38.8700 2.16278
\(324\) 0 0
\(325\) 1.24243 0.0689177
\(326\) 0 0
\(327\) 1.14931 0.0635571
\(328\) 0 0
\(329\) 12.9272 0.712697
\(330\) 0 0
\(331\) 7.74193 0.425535 0.212767 0.977103i \(-0.431752\pi\)
0.212767 + 0.977103i \(0.431752\pi\)
\(332\) 0 0
\(333\) −6.49950 −0.356170
\(334\) 0 0
\(335\) 5.27835 0.288387
\(336\) 0 0
\(337\) −13.9787 −0.761469 −0.380735 0.924684i \(-0.624329\pi\)
−0.380735 + 0.924684i \(0.624329\pi\)
\(338\) 0 0
\(339\) 19.7778 1.07419
\(340\) 0 0
\(341\) −3.98335 −0.215710
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 2.59261 0.139582
\(346\) 0 0
\(347\) 9.35482 0.502193 0.251096 0.967962i \(-0.419209\pi\)
0.251096 + 0.967962i \(0.419209\pi\)
\(348\) 0 0
\(349\) −0.0359171 −0.00192260 −0.000961300 1.00000i \(-0.500306\pi\)
−0.000961300 1.00000i \(0.500306\pi\)
\(350\) 0 0
\(351\) 0.721651 0.0385189
\(352\) 0 0
\(353\) 11.0967 0.590620 0.295310 0.955401i \(-0.404577\pi\)
0.295310 + 0.955401i \(0.404577\pi\)
\(354\) 0 0
\(355\) −30.8700 −1.63841
\(356\) 0 0
\(357\) −5.31427 −0.281261
\(358\) 0 0
\(359\) 13.8497 0.730958 0.365479 0.930820i \(-0.380905\pi\)
0.365479 + 0.930820i \(0.380905\pi\)
\(360\) 0 0
\(361\) 34.4985 1.81571
\(362\) 0 0
\(363\) −7.49950 −0.393621
\(364\) 0 0
\(365\) 14.4469 0.756187
\(366\) 0 0
\(367\) −1.53642 −0.0802006 −0.0401003 0.999196i \(-0.512768\pi\)
−0.0401003 + 0.999196i \(0.512768\pi\)
\(368\) 0 0
\(369\) 4.12904 0.214949
\(370\) 0 0
\(371\) −10.3345 −0.536543
\(372\) 0 0
\(373\) 16.6119 0.860131 0.430065 0.902798i \(-0.358491\pi\)
0.430065 + 0.902798i \(0.358491\pi\)
\(374\) 0 0
\(375\) 8.49950 0.438912
\(376\) 0 0
\(377\) −3.83505 −0.197515
\(378\) 0 0
\(379\) −27.6275 −1.41913 −0.709565 0.704640i \(-0.751109\pi\)
−0.709565 + 0.704640i \(0.751109\pi\)
\(380\) 0 0
\(381\) 6.46358 0.331139
\(382\) 0 0
\(383\) 8.24142 0.421117 0.210559 0.977581i \(-0.432472\pi\)
0.210559 + 0.977581i \(0.432472\pi\)
\(384\) 0 0
\(385\) 4.85069 0.247214
\(386\) 0 0
\(387\) 1.27835 0.0649821
\(388\) 0 0
\(389\) −20.4589 −1.03731 −0.518655 0.854984i \(-0.673567\pi\)
−0.518655 + 0.854984i \(0.673567\pi\)
\(390\) 0 0
\(391\) −5.31427 −0.268754
\(392\) 0 0
\(393\) −6.12904 −0.309169
\(394\) 0 0
\(395\) −29.3335 −1.47593
\(396\) 0 0
\(397\) −11.7014 −0.587275 −0.293638 0.955917i \(-0.594866\pi\)
−0.293638 + 0.955917i \(0.594866\pi\)
\(398\) 0 0
\(399\) −7.31427 −0.366171
\(400\) 0 0
\(401\) 4.45894 0.222669 0.111335 0.993783i \(-0.464488\pi\)
0.111335 + 0.993783i \(0.464488\pi\)
\(402\) 0 0
\(403\) −1.53642 −0.0765346
\(404\) 0 0
\(405\) −2.59261 −0.128828
\(406\) 0 0
\(407\) −12.1603 −0.602765
\(408\) 0 0
\(409\) −6.75757 −0.334140 −0.167070 0.985945i \(-0.553431\pi\)
−0.167070 + 0.985945i \(0.553431\pi\)
\(410\) 0 0
\(411\) 7.87096 0.388246
\(412\) 0 0
\(413\) −10.0359 −0.493835
\(414\) 0 0
\(415\) 4.85069 0.238111
\(416\) 0 0
\(417\) −14.9631 −0.732745
\(418\) 0 0
\(419\) 11.4074 0.557287 0.278644 0.960395i \(-0.410115\pi\)
0.278644 + 0.960395i \(0.410115\pi\)
\(420\) 0 0
\(421\) −37.3907 −1.82231 −0.911156 0.412061i \(-0.864809\pi\)
−0.911156 + 0.412061i \(0.864809\pi\)
\(422\) 0 0
\(423\) −12.9272 −0.628540
\(424\) 0 0
\(425\) 9.14931 0.443807
\(426\) 0 0
\(427\) 7.09211 0.343211
\(428\) 0 0
\(429\) 1.35018 0.0651875
\(430\) 0 0
\(431\) 34.0193 1.63865 0.819325 0.573329i \(-0.194348\pi\)
0.819325 + 0.573329i \(0.194348\pi\)
\(432\) 0 0
\(433\) −0.0405518 −0.00194879 −0.000974397 1.00000i \(-0.500310\pi\)
−0.000974397 1.00000i \(0.500310\pi\)
\(434\) 0 0
\(435\) 13.7778 0.660597
\(436\) 0 0
\(437\) −7.31427 −0.349889
\(438\) 0 0
\(439\) −14.8700 −0.709704 −0.354852 0.934922i \(-0.615469\pi\)
−0.354852 + 0.934922i \(0.615469\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 14.3705 0.682761 0.341381 0.939925i \(-0.389106\pi\)
0.341381 + 0.939925i \(0.389106\pi\)
\(444\) 0 0
\(445\) 4.94381 0.234359
\(446\) 0 0
\(447\) −11.1852 −0.529043
\(448\) 0 0
\(449\) 21.4792 1.01367 0.506834 0.862044i \(-0.330816\pi\)
0.506834 + 0.862044i \(0.330816\pi\)
\(450\) 0 0
\(451\) 7.72528 0.363769
\(452\) 0 0
\(453\) −12.9272 −0.607371
\(454\) 0 0
\(455\) 1.87096 0.0877121
\(456\) 0 0
\(457\) 0.705001 0.0329785 0.0164893 0.999864i \(-0.494751\pi\)
0.0164893 + 0.999864i \(0.494751\pi\)
\(458\) 0 0
\(459\) 5.31427 0.248049
\(460\) 0 0
\(461\) 30.9225 1.44021 0.720103 0.693867i \(-0.244095\pi\)
0.720103 + 0.693867i \(0.244095\pi\)
\(462\) 0 0
\(463\) −5.53179 −0.257084 −0.128542 0.991704i \(-0.541030\pi\)
−0.128542 + 0.991704i \(0.541030\pi\)
\(464\) 0 0
\(465\) 5.51977 0.255973
\(466\) 0 0
\(467\) 28.2414 1.30686 0.653429 0.756988i \(-0.273330\pi\)
0.653429 + 0.756988i \(0.273330\pi\)
\(468\) 0 0
\(469\) 2.03592 0.0940099
\(470\) 0 0
\(471\) 14.3299 0.660287
\(472\) 0 0
\(473\) 2.39174 0.109973
\(474\) 0 0
\(475\) 12.5926 0.577789
\(476\) 0 0
\(477\) 10.3345 0.473186
\(478\) 0 0
\(479\) 26.9418 1.23100 0.615501 0.788136i \(-0.288954\pi\)
0.615501 + 0.788136i \(0.288954\pi\)
\(480\) 0 0
\(481\) −4.69037 −0.213862
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −20.4064 −0.926606
\(486\) 0 0
\(487\) −0.757568 −0.0343287 −0.0171643 0.999853i \(-0.505464\pi\)
−0.0171643 + 0.999853i \(0.505464\pi\)
\(488\) 0 0
\(489\) 13.7778 0.623056
\(490\) 0 0
\(491\) −32.5760 −1.47013 −0.735066 0.677995i \(-0.762849\pi\)
−0.735066 + 0.677995i \(0.762849\pi\)
\(492\) 0 0
\(493\) −28.2414 −1.27193
\(494\) 0 0
\(495\) −4.85069 −0.218022
\(496\) 0 0
\(497\) −11.9069 −0.534097
\(498\) 0 0
\(499\) 27.8902 1.24854 0.624269 0.781209i \(-0.285397\pi\)
0.624269 + 0.781209i \(0.285397\pi\)
\(500\) 0 0
\(501\) 20.4995 0.915850
\(502\) 0 0
\(503\) −15.0349 −0.670373 −0.335187 0.942152i \(-0.608799\pi\)
−0.335187 + 0.942152i \(0.608799\pi\)
\(504\) 0 0
\(505\) −31.2045 −1.38858
\(506\) 0 0
\(507\) −12.4792 −0.554222
\(508\) 0 0
\(509\) −34.2966 −1.52017 −0.760085 0.649823i \(-0.774843\pi\)
−0.760085 + 0.649823i \(0.774843\pi\)
\(510\) 0 0
\(511\) 5.57234 0.246506
\(512\) 0 0
\(513\) 7.31427 0.322933
\(514\) 0 0
\(515\) 26.8866 1.18477
\(516\) 0 0
\(517\) −24.1862 −1.06371
\(518\) 0 0
\(519\) 22.7576 0.998946
\(520\) 0 0
\(521\) 7.11340 0.311644 0.155822 0.987785i \(-0.450197\pi\)
0.155822 + 0.987785i \(0.450197\pi\)
\(522\) 0 0
\(523\) −23.7419 −1.03816 −0.519081 0.854725i \(-0.673726\pi\)
−0.519081 + 0.854725i \(0.673726\pi\)
\(524\) 0 0
\(525\) −1.72165 −0.0751390
\(526\) 0 0
\(527\) −11.3143 −0.492857
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 10.0359 0.435522
\(532\) 0 0
\(533\) 2.97972 0.129066
\(534\) 0 0
\(535\) 37.6847 1.62925
\(536\) 0 0
\(537\) −16.6644 −0.719124
\(538\) 0 0
\(539\) 1.87096 0.0805881
\(540\) 0 0
\(541\) 1.55569 0.0668843 0.0334421 0.999441i \(-0.489353\pi\)
0.0334421 + 0.999441i \(0.489353\pi\)
\(542\) 0 0
\(543\) 25.1280 1.07835
\(544\) 0 0
\(545\) −2.97972 −0.127637
\(546\) 0 0
\(547\) 10.5521 0.451174 0.225587 0.974223i \(-0.427570\pi\)
0.225587 + 0.974223i \(0.427570\pi\)
\(548\) 0 0
\(549\) −7.09211 −0.302684
\(550\) 0 0
\(551\) −38.8700 −1.65592
\(552\) 0 0
\(553\) −11.3143 −0.481132
\(554\) 0 0
\(555\) 16.8507 0.715272
\(556\) 0 0
\(557\) 32.7004 1.38556 0.692779 0.721149i \(-0.256386\pi\)
0.692779 + 0.721149i \(0.256386\pi\)
\(558\) 0 0
\(559\) 0.922522 0.0390185
\(560\) 0 0
\(561\) 9.94280 0.419785
\(562\) 0 0
\(563\) −33.0921 −1.39467 −0.697333 0.716747i \(-0.745630\pi\)
−0.697333 + 0.716747i \(0.745630\pi\)
\(564\) 0 0
\(565\) −51.2763 −2.15721
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 17.1447 0.718742 0.359371 0.933195i \(-0.382991\pi\)
0.359371 + 0.933195i \(0.382991\pi\)
\(570\) 0 0
\(571\) 12.0885 0.505887 0.252944 0.967481i \(-0.418601\pi\)
0.252944 + 0.967481i \(0.418601\pi\)
\(572\) 0 0
\(573\) −10.5567 −0.441012
\(574\) 0 0
\(575\) −1.72165 −0.0717978
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 7.44330 0.309333
\(580\) 0 0
\(581\) 1.87096 0.0776206
\(582\) 0 0
\(583\) 19.3356 0.800797
\(584\) 0 0
\(585\) −1.87096 −0.0773548
\(586\) 0 0
\(587\) −36.1317 −1.49131 −0.745656 0.666331i \(-0.767864\pi\)
−0.745656 + 0.666331i \(0.767864\pi\)
\(588\) 0 0
\(589\) −15.5723 −0.641647
\(590\) 0 0
\(591\) 7.02028 0.288776
\(592\) 0 0
\(593\) 34.3299 1.40976 0.704880 0.709326i \(-0.251001\pi\)
0.704880 + 0.709326i \(0.251001\pi\)
\(594\) 0 0
\(595\) 13.7778 0.564837
\(596\) 0 0
\(597\) −3.66546 −0.150017
\(598\) 0 0
\(599\) 18.5354 0.757336 0.378668 0.925532i \(-0.376382\pi\)
0.378668 + 0.925532i \(0.376382\pi\)
\(600\) 0 0
\(601\) −11.8497 −0.483358 −0.241679 0.970356i \(-0.577698\pi\)
−0.241679 + 0.970356i \(0.577698\pi\)
\(602\) 0 0
\(603\) −2.03592 −0.0829090
\(604\) 0 0
\(605\) 19.4433 0.790483
\(606\) 0 0
\(607\) 36.5760 1.48457 0.742286 0.670083i \(-0.233741\pi\)
0.742286 + 0.670083i \(0.233741\pi\)
\(608\) 0 0
\(609\) 5.31427 0.215345
\(610\) 0 0
\(611\) −9.32890 −0.377407
\(612\) 0 0
\(613\) 4.54005 0.183371 0.0916854 0.995788i \(-0.470775\pi\)
0.0916854 + 0.995788i \(0.470775\pi\)
\(614\) 0 0
\(615\) −10.7050 −0.431667
\(616\) 0 0
\(617\) −28.7935 −1.15918 −0.579591 0.814907i \(-0.696788\pi\)
−0.579591 + 0.814907i \(0.696788\pi\)
\(618\) 0 0
\(619\) −44.1317 −1.77380 −0.886900 0.461961i \(-0.847146\pi\)
−0.886900 + 0.461961i \(0.847146\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 1.90688 0.0763976
\(624\) 0 0
\(625\) −30.6442 −1.22577
\(626\) 0 0
\(627\) 13.6847 0.546515
\(628\) 0 0
\(629\) −34.5400 −1.37720
\(630\) 0 0
\(631\) 15.3861 0.612511 0.306255 0.951949i \(-0.400924\pi\)
0.306255 + 0.951949i \(0.400924\pi\)
\(632\) 0 0
\(633\) −0.685734 −0.0272555
\(634\) 0 0
\(635\) −16.7576 −0.665004
\(636\) 0 0
\(637\) 0.721651 0.0285929
\(638\) 0 0
\(639\) 11.9069 0.471029
\(640\) 0 0
\(641\) 22.5760 0.891697 0.445848 0.895108i \(-0.352902\pi\)
0.445848 + 0.895108i \(0.352902\pi\)
\(642\) 0 0
\(643\) −31.0183 −1.22324 −0.611620 0.791151i \(-0.709482\pi\)
−0.611620 + 0.791151i \(0.709482\pi\)
\(644\) 0 0
\(645\) −3.31427 −0.130499
\(646\) 0 0
\(647\) 20.1483 0.792112 0.396056 0.918226i \(-0.370379\pi\)
0.396056 + 0.918226i \(0.370379\pi\)
\(648\) 0 0
\(649\) 18.7768 0.737055
\(650\) 0 0
\(651\) 2.12904 0.0834435
\(652\) 0 0
\(653\) −23.3335 −0.913112 −0.456556 0.889695i \(-0.650917\pi\)
−0.456556 + 0.889695i \(0.650917\pi\)
\(654\) 0 0
\(655\) 15.8902 0.620883
\(656\) 0 0
\(657\) −5.57234 −0.217398
\(658\) 0 0
\(659\) 34.0146 1.32502 0.662511 0.749052i \(-0.269491\pi\)
0.662511 + 0.749052i \(0.269491\pi\)
\(660\) 0 0
\(661\) 9.50050 0.369527 0.184763 0.982783i \(-0.440848\pi\)
0.184763 + 0.982783i \(0.440848\pi\)
\(662\) 0 0
\(663\) 3.83505 0.148941
\(664\) 0 0
\(665\) 18.9631 0.735356
\(666\) 0 0
\(667\) 5.31427 0.205769
\(668\) 0 0
\(669\) 28.6644 1.10823
\(670\) 0 0
\(671\) −13.2691 −0.512247
\(672\) 0 0
\(673\) −6.61390 −0.254947 −0.127474 0.991842i \(-0.540687\pi\)
−0.127474 + 0.991842i \(0.540687\pi\)
\(674\) 0 0
\(675\) 1.72165 0.0662664
\(676\) 0 0
\(677\) 2.24606 0.0863230 0.0431615 0.999068i \(-0.486257\pi\)
0.0431615 + 0.999068i \(0.486257\pi\)
\(678\) 0 0
\(679\) −7.87096 −0.302060
\(680\) 0 0
\(681\) −24.1483 −0.925365
\(682\) 0 0
\(683\) −9.49849 −0.363449 −0.181725 0.983349i \(-0.558168\pi\)
−0.181725 + 0.983349i \(0.558168\pi\)
\(684\) 0 0
\(685\) −20.4064 −0.779688
\(686\) 0 0
\(687\) 15.7778 0.601962
\(688\) 0 0
\(689\) 7.45793 0.284125
\(690\) 0 0
\(691\) −29.6894 −1.12944 −0.564718 0.825284i \(-0.691015\pi\)
−0.564718 + 0.825284i \(0.691015\pi\)
\(692\) 0 0
\(693\) −1.87096 −0.0710720
\(694\) 0 0
\(695\) 38.7935 1.47152
\(696\) 0 0
\(697\) 21.9428 0.831143
\(698\) 0 0
\(699\) 19.7778 0.748067
\(700\) 0 0
\(701\) −3.96408 −0.149721 −0.0748607 0.997194i \(-0.523851\pi\)
−0.0748607 + 0.997194i \(0.523851\pi\)
\(702\) 0 0
\(703\) −47.5390 −1.79297
\(704\) 0 0
\(705\) 33.5151 1.26225
\(706\) 0 0
\(707\) −12.0359 −0.452657
\(708\) 0 0
\(709\) −15.4646 −0.580785 −0.290392 0.956908i \(-0.593786\pi\)
−0.290392 + 0.956908i \(0.593786\pi\)
\(710\) 0 0
\(711\) 11.3143 0.424318
\(712\) 0 0
\(713\) 2.12904 0.0797330
\(714\) 0 0
\(715\) −3.50050 −0.130911
\(716\) 0 0
\(717\) 16.4064 0.612707
\(718\) 0 0
\(719\) −48.0552 −1.79216 −0.896078 0.443897i \(-0.853596\pi\)
−0.896078 + 0.443897i \(0.853596\pi\)
\(720\) 0 0
\(721\) 10.3705 0.386216
\(722\) 0 0
\(723\) 14.4995 0.539242
\(724\) 0 0
\(725\) −9.14931 −0.339797
\(726\) 0 0
\(727\) −45.8284 −1.69968 −0.849841 0.527040i \(-0.823302\pi\)
−0.849841 + 0.527040i \(0.823302\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.79349 0.251266
\(732\) 0 0
\(733\) −29.2165 −1.07914 −0.539568 0.841942i \(-0.681413\pi\)
−0.539568 + 0.841942i \(0.681413\pi\)
\(734\) 0 0
\(735\) −2.59261 −0.0956300
\(736\) 0 0
\(737\) −3.80913 −0.140311
\(738\) 0 0
\(739\) −0.685734 −0.0252251 −0.0126126 0.999920i \(-0.504015\pi\)
−0.0126126 + 0.999920i \(0.504015\pi\)
\(740\) 0 0
\(741\) 5.27835 0.193905
\(742\) 0 0
\(743\) −32.7456 −1.20132 −0.600659 0.799505i \(-0.705095\pi\)
−0.600659 + 0.799505i \(0.705095\pi\)
\(744\) 0 0
\(745\) 28.9990 1.06244
\(746\) 0 0
\(747\) −1.87096 −0.0684550
\(748\) 0 0
\(749\) 14.5354 0.531112
\(750\) 0 0
\(751\) 22.3751 0.816479 0.408239 0.912875i \(-0.366143\pi\)
0.408239 + 0.912875i \(0.366143\pi\)
\(752\) 0 0
\(753\) −7.55569 −0.275345
\(754\) 0 0
\(755\) 33.5151 1.21974
\(756\) 0 0
\(757\) −42.7576 −1.55405 −0.777025 0.629470i \(-0.783272\pi\)
−0.777025 + 0.629470i \(0.783272\pi\)
\(758\) 0 0
\(759\) −1.87096 −0.0679117
\(760\) 0 0
\(761\) −33.3694 −1.20964 −0.604821 0.796362i \(-0.706755\pi\)
−0.604821 + 0.796362i \(0.706755\pi\)
\(762\) 0 0
\(763\) −1.14931 −0.0416079
\(764\) 0 0
\(765\) −13.7778 −0.498139
\(766\) 0 0
\(767\) 7.24243 0.261509
\(768\) 0 0
\(769\) −22.8461 −0.823850 −0.411925 0.911218i \(-0.635143\pi\)
−0.411925 + 0.911218i \(0.635143\pi\)
\(770\) 0 0
\(771\) −6.66908 −0.240181
\(772\) 0 0
\(773\) 45.2091 1.62606 0.813030 0.582222i \(-0.197817\pi\)
0.813030 + 0.582222i \(0.197817\pi\)
\(774\) 0 0
\(775\) −3.66546 −0.131667
\(776\) 0 0
\(777\) 6.49950 0.233168
\(778\) 0 0
\(779\) 30.2009 1.08206
\(780\) 0 0
\(781\) 22.2773 0.797146
\(782\) 0 0
\(783\) −5.31427 −0.189916
\(784\) 0 0
\(785\) −37.1519 −1.32601
\(786\) 0 0
\(787\) 29.5916 1.05483 0.527413 0.849609i \(-0.323162\pi\)
0.527413 + 0.849609i \(0.323162\pi\)
\(788\) 0 0
\(789\) 24.0552 0.856387
\(790\) 0 0
\(791\) −19.7778 −0.703219
\(792\) 0 0
\(793\) −5.11803 −0.181747
\(794\) 0 0
\(795\) −26.7935 −0.950267
\(796\) 0 0
\(797\) −27.7014 −0.981233 −0.490617 0.871376i \(-0.663228\pi\)
−0.490617 + 0.871376i \(0.663228\pi\)
\(798\) 0 0
\(799\) −68.6983 −2.43037
\(800\) 0 0
\(801\) −1.90688 −0.0673763
\(802\) 0 0
\(803\) −10.4256 −0.367913
\(804\) 0 0
\(805\) −2.59261 −0.0913777
\(806\) 0 0
\(807\) 8.72165 0.307017
\(808\) 0 0
\(809\) 36.8653 1.29612 0.648058 0.761591i \(-0.275582\pi\)
0.648058 + 0.761591i \(0.275582\pi\)
\(810\) 0 0
\(811\) −12.9438 −0.454519 −0.227259 0.973834i \(-0.572976\pi\)
−0.227259 + 0.973834i \(0.572976\pi\)
\(812\) 0 0
\(813\) −22.5400 −0.790514
\(814\) 0 0
\(815\) −35.7206 −1.25124
\(816\) 0 0
\(817\) 9.35018 0.327121
\(818\) 0 0
\(819\) −0.721651 −0.0252165
\(820\) 0 0
\(821\) 6.66908 0.232753 0.116376 0.993205i \(-0.462872\pi\)
0.116376 + 0.993205i \(0.462872\pi\)
\(822\) 0 0
\(823\) −8.92252 −0.311020 −0.155510 0.987834i \(-0.549702\pi\)
−0.155510 + 0.987834i \(0.549702\pi\)
\(824\) 0 0
\(825\) 3.22115 0.112146
\(826\) 0 0
\(827\) −25.5198 −0.887409 −0.443705 0.896173i \(-0.646336\pi\)
−0.443705 + 0.896173i \(0.646336\pi\)
\(828\) 0 0
\(829\) −37.6109 −1.30628 −0.653140 0.757237i \(-0.726549\pi\)
−0.653140 + 0.757237i \(0.726549\pi\)
\(830\) 0 0
\(831\) 21.9787 0.762433
\(832\) 0 0
\(833\) 5.31427 0.184128
\(834\) 0 0
\(835\) −53.1473 −1.83924
\(836\) 0 0
\(837\) −2.12904 −0.0735903
\(838\) 0 0
\(839\) 4.42303 0.152700 0.0763499 0.997081i \(-0.475673\pi\)
0.0763499 + 0.997081i \(0.475673\pi\)
\(840\) 0 0
\(841\) −0.758577 −0.0261578
\(842\) 0 0
\(843\) 14.9272 0.514119
\(844\) 0 0
\(845\) 32.3538 1.11300
\(846\) 0 0
\(847\) 7.49950 0.257686
\(848\) 0 0
\(849\) −32.6644 −1.12104
\(850\) 0 0
\(851\) 6.49950 0.222800
\(852\) 0 0
\(853\) −28.0406 −0.960090 −0.480045 0.877244i \(-0.659380\pi\)
−0.480045 + 0.877244i \(0.659380\pi\)
\(854\) 0 0
\(855\) −18.9631 −0.648523
\(856\) 0 0
\(857\) −23.6681 −0.808486 −0.404243 0.914652i \(-0.632465\pi\)
−0.404243 + 0.914652i \(0.632465\pi\)
\(858\) 0 0
\(859\) 49.5962 1.69220 0.846101 0.533023i \(-0.178944\pi\)
0.846101 + 0.533023i \(0.178944\pi\)
\(860\) 0 0
\(861\) −4.12904 −0.140717
\(862\) 0 0
\(863\) −5.92615 −0.201728 −0.100864 0.994900i \(-0.532161\pi\)
−0.100864 + 0.994900i \(0.532161\pi\)
\(864\) 0 0
\(865\) −59.0016 −2.00611
\(866\) 0 0
\(867\) 11.2414 0.381779
\(868\) 0 0
\(869\) 21.1686 0.718095
\(870\) 0 0
\(871\) −1.46922 −0.0497827
\(872\) 0 0
\(873\) 7.87096 0.266392
\(874\) 0 0
\(875\) −8.49950 −0.287335
\(876\) 0 0
\(877\) 43.0801 1.45471 0.727356 0.686261i \(-0.240749\pi\)
0.727356 + 0.686261i \(0.240749\pi\)
\(878\) 0 0
\(879\) −11.5151 −0.388396
\(880\) 0 0
\(881\) −19.5870 −0.659902 −0.329951 0.943998i \(-0.607032\pi\)
−0.329951 + 0.943998i \(0.607032\pi\)
\(882\) 0 0
\(883\) 17.9382 0.603667 0.301834 0.953361i \(-0.402401\pi\)
0.301834 + 0.953361i \(0.402401\pi\)
\(884\) 0 0
\(885\) −26.0193 −0.874628
\(886\) 0 0
\(887\) −2.53541 −0.0851308 −0.0425654 0.999094i \(-0.513553\pi\)
−0.0425654 + 0.999094i \(0.513553\pi\)
\(888\) 0 0
\(889\) −6.46358 −0.216781
\(890\) 0 0
\(891\) 1.87096 0.0626796
\(892\) 0 0
\(893\) −94.5527 −3.16408
\(894\) 0 0
\(895\) 43.2045 1.44417
\(896\) 0 0
\(897\) −0.721651 −0.0240952
\(898\) 0 0
\(899\) 11.3143 0.377352
\(900\) 0 0
\(901\) 54.9205 1.82967
\(902\) 0 0
\(903\) −1.27835 −0.0425408
\(904\) 0 0
\(905\) −65.1473 −2.16557
\(906\) 0 0
\(907\) −28.7363 −0.954173 −0.477086 0.878856i \(-0.658307\pi\)
−0.477086 + 0.878856i \(0.658307\pi\)
\(908\) 0 0
\(909\) 12.0359 0.399206
\(910\) 0 0
\(911\) −38.7815 −1.28489 −0.642444 0.766333i \(-0.722079\pi\)
−0.642444 + 0.766333i \(0.722079\pi\)
\(912\) 0 0
\(913\) −3.50050 −0.115850
\(914\) 0 0
\(915\) 18.3871 0.607859
\(916\) 0 0
\(917\) 6.12904 0.202399
\(918\) 0 0
\(919\) −23.7253 −0.782625 −0.391312 0.920258i \(-0.627979\pi\)
−0.391312 + 0.920258i \(0.627979\pi\)
\(920\) 0 0
\(921\) 29.7566 0.980512
\(922\) 0 0
\(923\) 8.59261 0.282829
\(924\) 0 0
\(925\) −11.1899 −0.367920
\(926\) 0 0
\(927\) −10.3705 −0.340611
\(928\) 0 0
\(929\) −55.0329 −1.80557 −0.902785 0.430092i \(-0.858481\pi\)
−0.902785 + 0.430092i \(0.858481\pi\)
\(930\) 0 0
\(931\) 7.31427 0.239715
\(932\) 0 0
\(933\) 15.4626 0.506222
\(934\) 0 0
\(935\) −25.7778 −0.843026
\(936\) 0 0
\(937\) 12.3153 0.402323 0.201161 0.979558i \(-0.435528\pi\)
0.201161 + 0.979558i \(0.435528\pi\)
\(938\) 0 0
\(939\) 16.6285 0.542652
\(940\) 0 0
\(941\) −28.7096 −0.935907 −0.467954 0.883753i \(-0.655009\pi\)
−0.467954 + 0.883753i \(0.655009\pi\)
\(942\) 0 0
\(943\) −4.12904 −0.134460
\(944\) 0 0
\(945\) 2.59261 0.0843378
\(946\) 0 0
\(947\) 53.1427 1.72690 0.863452 0.504431i \(-0.168298\pi\)
0.863452 + 0.504431i \(0.168298\pi\)
\(948\) 0 0
\(949\) −4.02129 −0.130536
\(950\) 0 0
\(951\) 16.9631 0.550065
\(952\) 0 0
\(953\) 35.4939 1.14976 0.574879 0.818238i \(-0.305049\pi\)
0.574879 + 0.818238i \(0.305049\pi\)
\(954\) 0 0
\(955\) 27.3694 0.885655
\(956\) 0 0
\(957\) −9.94280 −0.321405
\(958\) 0 0
\(959\) −7.87096 −0.254167
\(960\) 0 0
\(961\) −26.4672 −0.853781
\(962\) 0 0
\(963\) −14.5354 −0.468397
\(964\) 0 0
\(965\) −19.2976 −0.621212
\(966\) 0 0
\(967\) 7.00101 0.225137 0.112569 0.993644i \(-0.464092\pi\)
0.112569 + 0.993644i \(0.464092\pi\)
\(968\) 0 0
\(969\) 38.8700 1.24868
\(970\) 0 0
\(971\) 13.6894 0.439312 0.219656 0.975577i \(-0.429506\pi\)
0.219656 + 0.975577i \(0.429506\pi\)
\(972\) 0 0
\(973\) 14.9631 0.479694
\(974\) 0 0
\(975\) 1.24243 0.0397897
\(976\) 0 0
\(977\) −18.9059 −0.604852 −0.302426 0.953173i \(-0.597797\pi\)
−0.302426 + 0.953173i \(0.597797\pi\)
\(978\) 0 0
\(979\) −3.56770 −0.114024
\(980\) 0 0
\(981\) 1.14931 0.0366947
\(982\) 0 0
\(983\) −36.1270 −1.15227 −0.576136 0.817354i \(-0.695440\pi\)
−0.576136 + 0.817354i \(0.695440\pi\)
\(984\) 0 0
\(985\) −18.2009 −0.579928
\(986\) 0 0
\(987\) 12.9272 0.411476
\(988\) 0 0
\(989\) −1.27835 −0.0406491
\(990\) 0 0
\(991\) −0.317891 −0.0100982 −0.00504908 0.999987i \(-0.501607\pi\)
−0.00504908 + 0.999987i \(0.501607\pi\)
\(992\) 0 0
\(993\) 7.74193 0.245683
\(994\) 0 0
\(995\) 9.50312 0.301269
\(996\) 0 0
\(997\) −51.7566 −1.63915 −0.819573 0.572974i \(-0.805789\pi\)
−0.819573 + 0.572974i \(0.805789\pi\)
\(998\) 0 0
\(999\) −6.49950 −0.205635
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 7728.2.a.by.1.1 3
4.3 odd 2 3864.2.a.n.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.n.1.1 3 4.3 odd 2
7728.2.a.by.1.1 3 1.1 even 1 trivial