Properties

Label 3072.2.d.d.1537.3
Level $3072$
Weight $2$
Character 3072.1537
Analytic conductor $24.530$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 768)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1537.3
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 3072.1537
Dual form 3072.2.d.d.1537.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -2.82843i q^{5} -4.24264 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -2.82843i q^{5} -4.24264 q^{7} -1.00000 q^{9} +4.00000i q^{11} +4.24264i q^{13} +2.82843 q^{15} +6.00000 q^{17} -2.00000i q^{19} -4.24264i q^{21} -2.82843 q^{23} -3.00000 q^{25} -1.00000i q^{27} -5.65685i q^{29} -4.24264 q^{31} -4.00000 q^{33} +12.0000i q^{35} -4.24264i q^{37} -4.24264 q^{39} +10.0000 q^{41} -6.00000i q^{43} +2.82843i q^{45} -2.82843 q^{47} +11.0000 q^{49} +6.00000i q^{51} +5.65685i q^{53} +11.3137 q^{55} +2.00000 q^{57} -4.24264i q^{61} +4.24264 q^{63} +12.0000 q^{65} -4.00000i q^{67} -2.82843i q^{69} -2.82843 q^{71} -16.0000 q^{73} -3.00000i q^{75} -16.9706i q^{77} -4.24264 q^{79} +1.00000 q^{81} -16.0000i q^{83} -16.9706i q^{85} +5.65685 q^{87} -14.0000 q^{89} -18.0000i q^{91} -4.24264i q^{93} -5.65685 q^{95} -4.00000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} + 24q^{17} - 12q^{25} - 16q^{33} + 40q^{41} + 44q^{49} + 8q^{57} + 48q^{65} - 64q^{73} + 4q^{81} - 56q^{89} - 16q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) − 2.82843i − 1.26491i −0.774597 0.632456i \(-0.782047\pi\)
0.774597 0.632456i \(-0.217953\pi\)
\(6\) 0 0
\(7\) −4.24264 −1.60357 −0.801784 0.597614i \(-0.796115\pi\)
−0.801784 + 0.597614i \(0.796115\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) 4.24264i 1.17670i 0.808608 + 0.588348i \(0.200222\pi\)
−0.808608 + 0.588348i \(0.799778\pi\)
\(14\) 0 0
\(15\) 2.82843 0.730297
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) − 2.00000i − 0.458831i −0.973329 0.229416i \(-0.926318\pi\)
0.973329 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) − 4.24264i − 0.925820i
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 5.65685i − 1.05045i −0.850963 0.525226i \(-0.823981\pi\)
0.850963 0.525226i \(-0.176019\pi\)
\(30\) 0 0
\(31\) −4.24264 −0.762001 −0.381000 0.924575i \(-0.624420\pi\)
−0.381000 + 0.924575i \(0.624420\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 12.0000i 2.02837i
\(36\) 0 0
\(37\) − 4.24264i − 0.697486i −0.937218 0.348743i \(-0.886609\pi\)
0.937218 0.348743i \(-0.113391\pi\)
\(38\) 0 0
\(39\) −4.24264 −0.679366
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) − 6.00000i − 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) 0 0
\(45\) 2.82843i 0.421637i
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) 11.0000 1.57143
\(50\) 0 0
\(51\) 6.00000i 0.840168i
\(52\) 0 0
\(53\) 5.65685i 0.777029i 0.921443 + 0.388514i \(0.127012\pi\)
−0.921443 + 0.388514i \(0.872988\pi\)
\(54\) 0 0
\(55\) 11.3137 1.52554
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) − 4.24264i − 0.543214i −0.962408 0.271607i \(-0.912445\pi\)
0.962408 0.271607i \(-0.0875552\pi\)
\(62\) 0 0
\(63\) 4.24264 0.534522
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) − 2.82843i − 0.340503i
\(70\) 0 0
\(71\) −2.82843 −0.335673 −0.167836 0.985815i \(-0.553678\pi\)
−0.167836 + 0.985815i \(0.553678\pi\)
\(72\) 0 0
\(73\) −16.0000 −1.87266 −0.936329 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) − 3.00000i − 0.346410i
\(76\) 0 0
\(77\) − 16.9706i − 1.93398i
\(78\) 0 0
\(79\) −4.24264 −0.477334 −0.238667 0.971101i \(-0.576710\pi\)
−0.238667 + 0.971101i \(0.576710\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 16.0000i − 1.75623i −0.478451 0.878114i \(-0.658802\pi\)
0.478451 0.878114i \(-0.341198\pi\)
\(84\) 0 0
\(85\) − 16.9706i − 1.84072i
\(86\) 0 0
\(87\) 5.65685 0.606478
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) − 18.0000i − 1.88691i
\(92\) 0 0
\(93\) − 4.24264i − 0.439941i
\(94\) 0 0
\(95\) −5.65685 −0.580381
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) − 4.00000i − 0.402015i
\(100\) 0 0
\(101\) − 11.3137i − 1.12576i −0.826540 0.562878i \(-0.809694\pi\)
0.826540 0.562878i \(-0.190306\pi\)
\(102\) 0 0
\(103\) 18.3848 1.81151 0.905753 0.423806i \(-0.139306\pi\)
0.905753 + 0.423806i \(0.139306\pi\)
\(104\) 0 0
\(105\) −12.0000 −1.17108
\(106\) 0 0
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) − 9.89949i − 0.948200i −0.880471 0.474100i \(-0.842774\pi\)
0.880471 0.474100i \(-0.157226\pi\)
\(110\) 0 0
\(111\) 4.24264 0.402694
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 8.00000i 0.746004i
\(116\) 0 0
\(117\) − 4.24264i − 0.392232i
\(118\) 0 0
\(119\) −25.4558 −2.33353
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 10.0000i 0.901670i
\(124\) 0 0
\(125\) − 5.65685i − 0.505964i
\(126\) 0 0
\(127\) −4.24264 −0.376473 −0.188237 0.982124i \(-0.560277\pi\)
−0.188237 + 0.982124i \(0.560277\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) − 20.0000i − 1.74741i −0.486458 0.873704i \(-0.661711\pi\)
0.486458 0.873704i \(-0.338289\pi\)
\(132\) 0 0
\(133\) 8.48528i 0.735767i
\(134\) 0 0
\(135\) −2.82843 −0.243432
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) − 12.0000i − 1.01783i −0.860818 0.508913i \(-0.830047\pi\)
0.860818 0.508913i \(-0.169953\pi\)
\(140\) 0 0
\(141\) − 2.82843i − 0.238197i
\(142\) 0 0
\(143\) −16.9706 −1.41915
\(144\) 0 0
\(145\) −16.0000 −1.32873
\(146\) 0 0
\(147\) 11.0000i 0.907265i
\(148\) 0 0
\(149\) 8.48528i 0.695141i 0.937654 + 0.347571i \(0.112993\pi\)
−0.937654 + 0.347571i \(0.887007\pi\)
\(150\) 0 0
\(151\) 1.41421 0.115087 0.0575435 0.998343i \(-0.481673\pi\)
0.0575435 + 0.998343i \(0.481673\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 12.0000i 0.963863i
\(156\) 0 0
\(157\) − 7.07107i − 0.564333i −0.959366 0.282166i \(-0.908947\pi\)
0.959366 0.282166i \(-0.0910530\pi\)
\(158\) 0 0
\(159\) −5.65685 −0.448618
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) − 10.0000i − 0.783260i −0.920123 0.391630i \(-0.871911\pi\)
0.920123 0.391630i \(-0.128089\pi\)
\(164\) 0 0
\(165\) 11.3137i 0.880771i
\(166\) 0 0
\(167\) 11.3137 0.875481 0.437741 0.899101i \(-0.355779\pi\)
0.437741 + 0.899101i \(0.355779\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 2.00000i 0.152944i
\(172\) 0 0
\(173\) − 2.82843i − 0.215041i −0.994203 0.107521i \(-0.965709\pi\)
0.994203 0.107521i \(-0.0342912\pi\)
\(174\) 0 0
\(175\) 12.7279 0.962140
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) − 1.41421i − 0.105118i −0.998618 0.0525588i \(-0.983262\pi\)
0.998618 0.0525588i \(-0.0167377\pi\)
\(182\) 0 0
\(183\) 4.24264 0.313625
\(184\) 0 0
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 24.0000i 1.75505i
\(188\) 0 0
\(189\) 4.24264i 0.308607i
\(190\) 0 0
\(191\) 22.6274 1.63726 0.818631 0.574320i \(-0.194733\pi\)
0.818631 + 0.574320i \(0.194733\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 0 0
\(195\) 12.0000i 0.859338i
\(196\) 0 0
\(197\) − 5.65685i − 0.403034i −0.979485 0.201517i \(-0.935413\pi\)
0.979485 0.201517i \(-0.0645872\pi\)
\(198\) 0 0
\(199\) −7.07107 −0.501255 −0.250627 0.968084i \(-0.580637\pi\)
−0.250627 + 0.968084i \(0.580637\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 24.0000i 1.68447i
\(204\) 0 0
\(205\) − 28.2843i − 1.97546i
\(206\) 0 0
\(207\) 2.82843 0.196589
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 12.0000i 0.826114i 0.910705 + 0.413057i \(0.135539\pi\)
−0.910705 + 0.413057i \(0.864461\pi\)
\(212\) 0 0
\(213\) − 2.82843i − 0.193801i
\(214\) 0 0
\(215\) −16.9706 −1.15738
\(216\) 0 0
\(217\) 18.0000 1.22192
\(218\) 0 0
\(219\) − 16.0000i − 1.08118i
\(220\) 0 0
\(221\) 25.4558i 1.71235i
\(222\) 0 0
\(223\) 21.2132 1.42054 0.710271 0.703929i \(-0.248573\pi\)
0.710271 + 0.703929i \(0.248573\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) − 15.5563i − 1.02799i −0.857792 0.513996i \(-0.828165\pi\)
0.857792 0.513996i \(-0.171835\pi\)
\(230\) 0 0
\(231\) 16.9706 1.11658
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 8.00000i 0.521862i
\(236\) 0 0
\(237\) − 4.24264i − 0.275589i
\(238\) 0 0
\(239\) 11.3137 0.731823 0.365911 0.930650i \(-0.380757\pi\)
0.365911 + 0.930650i \(0.380757\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) − 31.1127i − 1.98772i
\(246\) 0 0
\(247\) 8.48528 0.539906
\(248\) 0 0
\(249\) 16.0000 1.01396
\(250\) 0 0
\(251\) 4.00000i 0.252478i 0.992000 + 0.126239i \(0.0402906\pi\)
−0.992000 + 0.126239i \(0.959709\pi\)
\(252\) 0 0
\(253\) − 11.3137i − 0.711287i
\(254\) 0 0
\(255\) 16.9706 1.06274
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 18.0000i 1.11847i
\(260\) 0 0
\(261\) 5.65685i 0.350150i
\(262\) 0 0
\(263\) −22.6274 −1.39527 −0.697633 0.716455i \(-0.745763\pi\)
−0.697633 + 0.716455i \(0.745763\pi\)
\(264\) 0 0
\(265\) 16.0000 0.982872
\(266\) 0 0
\(267\) − 14.0000i − 0.856786i
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −1.41421 −0.0859074 −0.0429537 0.999077i \(-0.513677\pi\)
−0.0429537 + 0.999077i \(0.513677\pi\)
\(272\) 0 0
\(273\) 18.0000 1.08941
\(274\) 0 0
\(275\) − 12.0000i − 0.723627i
\(276\) 0 0
\(277\) 1.41421i 0.0849719i 0.999097 + 0.0424859i \(0.0135278\pi\)
−0.999097 + 0.0424859i \(0.986472\pi\)
\(278\) 0 0
\(279\) 4.24264 0.254000
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 0 0
\(285\) − 5.65685i − 0.335083i
\(286\) 0 0
\(287\) −42.4264 −2.50435
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) − 4.00000i − 0.234484i
\(292\) 0 0
\(293\) − 5.65685i − 0.330477i −0.986254 0.165238i \(-0.947161\pi\)
0.986254 0.165238i \(-0.0528394\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) − 12.0000i − 0.693978i
\(300\) 0 0
\(301\) 25.4558i 1.46725i
\(302\) 0 0
\(303\) 11.3137 0.649956
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) 18.3848i 1.04587i
\(310\) 0 0
\(311\) 28.2843 1.60385 0.801927 0.597422i \(-0.203808\pi\)
0.801927 + 0.597422i \(0.203808\pi\)
\(312\) 0 0
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 0 0
\(315\) − 12.0000i − 0.676123i
\(316\) 0 0
\(317\) − 16.9706i − 0.953162i −0.879131 0.476581i \(-0.841876\pi\)
0.879131 0.476581i \(-0.158124\pi\)
\(318\) 0 0
\(319\) 22.6274 1.26689
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) − 12.0000i − 0.667698i
\(324\) 0 0
\(325\) − 12.7279i − 0.706018i
\(326\) 0 0
\(327\) 9.89949 0.547443
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 4.00000i 0.219860i 0.993939 + 0.109930i \(0.0350627\pi\)
−0.993939 + 0.109930i \(0.964937\pi\)
\(332\) 0 0
\(333\) 4.24264i 0.232495i
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) − 6.00000i − 0.325875i
\(340\) 0 0
\(341\) − 16.9706i − 0.919007i
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) −8.00000 −0.430706
\(346\) 0 0
\(347\) 24.0000i 1.28839i 0.764862 + 0.644194i \(0.222807\pi\)
−0.764862 + 0.644194i \(0.777193\pi\)
\(348\) 0 0
\(349\) 4.24264i 0.227103i 0.993532 + 0.113552i \(0.0362227\pi\)
−0.993532 + 0.113552i \(0.963777\pi\)
\(350\) 0 0
\(351\) 4.24264 0.226455
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 8.00000i 0.424596i
\(356\) 0 0
\(357\) − 25.4558i − 1.34727i
\(358\) 0 0
\(359\) −2.82843 −0.149279 −0.0746393 0.997211i \(-0.523781\pi\)
−0.0746393 + 0.997211i \(0.523781\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 0 0
\(363\) − 5.00000i − 0.262432i
\(364\) 0 0
\(365\) 45.2548i 2.36875i
\(366\) 0 0
\(367\) 18.3848 0.959678 0.479839 0.877357i \(-0.340695\pi\)
0.479839 + 0.877357i \(0.340695\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) − 24.0000i − 1.24602i
\(372\) 0 0
\(373\) 18.3848i 0.951928i 0.879465 + 0.475964i \(0.157901\pi\)
−0.879465 + 0.475964i \(0.842099\pi\)
\(374\) 0 0
\(375\) 5.65685 0.292119
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) − 38.0000i − 1.95193i −0.217930 0.975964i \(-0.569930\pi\)
0.217930 0.975964i \(-0.430070\pi\)
\(380\) 0 0
\(381\) − 4.24264i − 0.217357i
\(382\) 0 0
\(383\) −28.2843 −1.44526 −0.722629 0.691236i \(-0.757067\pi\)
−0.722629 + 0.691236i \(0.757067\pi\)
\(384\) 0 0
\(385\) −48.0000 −2.44631
\(386\) 0 0
\(387\) 6.00000i 0.304997i
\(388\) 0 0
\(389\) 2.82843i 0.143407i 0.997426 + 0.0717035i \(0.0228435\pi\)
−0.997426 + 0.0717035i \(0.977156\pi\)
\(390\) 0 0
\(391\) −16.9706 −0.858238
\(392\) 0 0
\(393\) 20.0000 1.00887
\(394\) 0 0
\(395\) 12.0000i 0.603786i
\(396\) 0 0
\(397\) 32.5269i 1.63248i 0.577714 + 0.816239i \(0.303945\pi\)
−0.577714 + 0.816239i \(0.696055\pi\)
\(398\) 0 0
\(399\) −8.48528 −0.424795
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) − 18.0000i − 0.896644i
\(404\) 0 0
\(405\) − 2.82843i − 0.140546i
\(406\) 0 0
\(407\) 16.9706 0.841200
\(408\) 0 0
\(409\) 28.0000 1.38451 0.692255 0.721653i \(-0.256617\pi\)
0.692255 + 0.721653i \(0.256617\pi\)
\(410\) 0 0
\(411\) − 10.0000i − 0.493264i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −45.2548 −2.22147
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) − 16.0000i − 0.781651i −0.920465 0.390826i \(-0.872190\pi\)
0.920465 0.390826i \(-0.127810\pi\)
\(420\) 0 0
\(421\) 35.3553i 1.72311i 0.507661 + 0.861557i \(0.330510\pi\)
−0.507661 + 0.861557i \(0.669490\pi\)
\(422\) 0 0
\(423\) 2.82843 0.137523
\(424\) 0 0
\(425\) −18.0000 −0.873128
\(426\) 0 0
\(427\) 18.0000i 0.871081i
\(428\) 0 0
\(429\) − 16.9706i − 0.819346i
\(430\) 0 0
\(431\) −36.7696 −1.77113 −0.885564 0.464518i \(-0.846227\pi\)
−0.885564 + 0.464518i \(0.846227\pi\)
\(432\) 0 0
\(433\) 8.00000 0.384455 0.192228 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) 0 0
\(435\) − 16.0000i − 0.767141i
\(436\) 0 0
\(437\) 5.65685i 0.270604i
\(438\) 0 0
\(439\) −1.41421 −0.0674967 −0.0337484 0.999430i \(-0.510744\pi\)
−0.0337484 + 0.999430i \(0.510744\pi\)
\(440\) 0 0
\(441\) −11.0000 −0.523810
\(442\) 0 0
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) 0 0
\(445\) 39.5980i 1.87712i
\(446\) 0 0
\(447\) −8.48528 −0.401340
\(448\) 0 0
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) 40.0000i 1.88353i
\(452\) 0 0
\(453\) 1.41421i 0.0664455i
\(454\) 0 0
\(455\) −50.9117 −2.38678
\(456\) 0 0
\(457\) −12.0000 −0.561336 −0.280668 0.959805i \(-0.590556\pi\)
−0.280668 + 0.959805i \(0.590556\pi\)
\(458\) 0 0
\(459\) − 6.00000i − 0.280056i
\(460\) 0 0
\(461\) 8.48528i 0.395199i 0.980283 + 0.197599i \(0.0633145\pi\)
−0.980283 + 0.197599i \(0.936685\pi\)
\(462\) 0 0
\(463\) 12.7279 0.591517 0.295758 0.955263i \(-0.404428\pi\)
0.295758 + 0.955263i \(0.404428\pi\)
\(464\) 0 0
\(465\) −12.0000 −0.556487
\(466\) 0 0
\(467\) 24.0000i 1.11059i 0.831654 + 0.555294i \(0.187394\pi\)
−0.831654 + 0.555294i \(0.812606\pi\)
\(468\) 0 0
\(469\) 16.9706i 0.783628i
\(470\) 0 0
\(471\) 7.07107 0.325818
\(472\) 0 0
\(473\) 24.0000 1.10352
\(474\) 0 0
\(475\) 6.00000i 0.275299i
\(476\) 0 0
\(477\) − 5.65685i − 0.259010i
\(478\) 0 0
\(479\) −25.4558 −1.16311 −0.581554 0.813508i \(-0.697555\pi\)
−0.581554 + 0.813508i \(0.697555\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) 0 0
\(483\) 12.0000i 0.546019i
\(484\) 0 0
\(485\) 11.3137i 0.513729i
\(486\) 0 0
\(487\) 43.8406 1.98661 0.993304 0.115529i \(-0.0368564\pi\)
0.993304 + 0.115529i \(0.0368564\pi\)
\(488\) 0 0
\(489\) 10.0000 0.452216
\(490\) 0 0
\(491\) 8.00000i 0.361035i 0.983572 + 0.180517i \(0.0577772\pi\)
−0.983572 + 0.180517i \(0.942223\pi\)
\(492\) 0 0
\(493\) − 33.9411i − 1.52863i
\(494\) 0 0
\(495\) −11.3137 −0.508513
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) − 20.0000i − 0.895323i −0.894203 0.447661i \(-0.852257\pi\)
0.894203 0.447661i \(-0.147743\pi\)
\(500\) 0 0
\(501\) 11.3137i 0.505459i
\(502\) 0 0
\(503\) 14.1421 0.630567 0.315283 0.948998i \(-0.397900\pi\)
0.315283 + 0.948998i \(0.397900\pi\)
\(504\) 0 0
\(505\) −32.0000 −1.42398
\(506\) 0 0
\(507\) − 5.00000i − 0.222058i
\(508\) 0 0
\(509\) − 22.6274i − 1.00294i −0.865174 0.501471i \(-0.832792\pi\)
0.865174 0.501471i \(-0.167208\pi\)
\(510\) 0 0
\(511\) 67.8823 3.00293
\(512\) 0 0
\(513\) −2.00000 −0.0883022
\(514\) 0 0
\(515\) − 52.0000i − 2.29139i
\(516\) 0 0
\(517\) − 11.3137i − 0.497576i
\(518\) 0 0
\(519\) 2.82843 0.124154
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) − 22.0000i − 0.961993i −0.876723 0.480996i \(-0.840275\pi\)
0.876723 0.480996i \(-0.159725\pi\)
\(524\) 0 0
\(525\) 12.7279i 0.555492i
\(526\) 0 0
\(527\) −25.4558 −1.10887
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 42.4264i 1.83769i
\(534\) 0 0
\(535\) 11.3137 0.489134
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 44.0000i 1.89521i
\(540\) 0 0
\(541\) − 41.0122i − 1.76325i −0.471949 0.881626i \(-0.656449\pi\)
0.471949 0.881626i \(-0.343551\pi\)
\(542\) 0 0
\(543\) 1.41421 0.0606897
\(544\) 0 0
\(545\) −28.0000 −1.19939
\(546\) 0 0
\(547\) 34.0000i 1.45374i 0.686778 + 0.726868i \(0.259025\pi\)
−0.686778 + 0.726868i \(0.740975\pi\)
\(548\) 0 0
\(549\) 4.24264i 0.181071i
\(550\) 0 0
\(551\) −11.3137 −0.481980
\(552\) 0 0
\(553\) 18.0000 0.765438
\(554\) 0 0
\(555\) − 12.0000i − 0.509372i
\(556\) 0 0
\(557\) − 19.7990i − 0.838910i −0.907776 0.419455i \(-0.862221\pi\)
0.907776 0.419455i \(-0.137779\pi\)
\(558\) 0 0
\(559\) 25.4558 1.07667
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 0 0
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 0 0
\(565\) 16.9706i 0.713957i
\(566\) 0 0
\(567\) −4.24264 −0.178174
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) − 20.0000i − 0.836974i −0.908223 0.418487i \(-0.862561\pi\)
0.908223 0.418487i \(-0.137439\pi\)
\(572\) 0 0
\(573\) 22.6274i 0.945274i
\(574\) 0 0
\(575\) 8.48528 0.353861
\(576\) 0 0
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) 0 0
\(579\) 18.0000i 0.748054i
\(580\) 0 0
\(581\) 67.8823i 2.81623i
\(582\) 0 0
\(583\) −22.6274 −0.937132
\(584\) 0 0
\(585\) −12.0000 −0.496139
\(586\) 0 0
\(587\) − 24.0000i − 0.990586i −0.868726 0.495293i \(-0.835061\pi\)
0.868726 0.495293i \(-0.164939\pi\)
\(588\) 0 0
\(589\) 8.48528i 0.349630i
\(590\) 0 0
\(591\) 5.65685 0.232692
\(592\) 0 0
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 72.0000i 2.95171i
\(596\) 0 0
\(597\) − 7.07107i − 0.289400i
\(598\) 0 0
\(599\) −48.0833 −1.96463 −0.982314 0.187239i \(-0.940046\pi\)
−0.982314 + 0.187239i \(0.940046\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) 14.1421i 0.574960i
\(606\) 0 0
\(607\) −4.24264 −0.172203 −0.0861017 0.996286i \(-0.527441\pi\)
−0.0861017 + 0.996286i \(0.527441\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) − 12.0000i − 0.485468i
\(612\) 0 0
\(613\) 18.3848i 0.742554i 0.928522 + 0.371277i \(0.121080\pi\)
−0.928522 + 0.371277i \(0.878920\pi\)
\(614\) 0 0
\(615\) 28.2843 1.14053
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) − 36.0000i − 1.44696i −0.690344 0.723481i \(-0.742541\pi\)
0.690344 0.723481i \(-0.257459\pi\)
\(620\) 0 0
\(621\) 2.82843i 0.113501i
\(622\) 0 0
\(623\) 59.3970 2.37969
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 8.00000i 0.319489i
\(628\) 0 0
\(629\) − 25.4558i − 1.01499i
\(630\) 0 0
\(631\) −35.3553 −1.40747 −0.703737 0.710461i \(-0.748487\pi\)
−0.703737 + 0.710461i \(0.748487\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) 12.0000i 0.476205i
\(636\) 0 0
\(637\) 46.6690i 1.84909i
\(638\) 0 0
\(639\) 2.82843 0.111891
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) − 14.0000i − 0.552106i −0.961142 0.276053i \(-0.910973\pi\)
0.961142 0.276053i \(-0.0890266\pi\)
\(644\) 0 0
\(645\) − 16.9706i − 0.668215i
\(646\) 0 0
\(647\) −14.1421 −0.555985 −0.277992 0.960583i \(-0.589669\pi\)
−0.277992 + 0.960583i \(0.589669\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 18.0000i 0.705476i
\(652\) 0 0
\(653\) − 8.48528i − 0.332055i −0.986121 0.166027i \(-0.946906\pi\)
0.986121 0.166027i \(-0.0530940\pi\)
\(654\) 0 0
\(655\) −56.5685 −2.21032
\(656\) 0 0
\(657\) 16.0000 0.624219
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) − 12.7279i − 0.495059i −0.968880 0.247529i \(-0.920381\pi\)
0.968880 0.247529i \(-0.0796187\pi\)
\(662\) 0 0
\(663\) −25.4558 −0.988623
\(664\) 0 0
\(665\) 24.0000 0.930680
\(666\) 0 0
\(667\) 16.0000i 0.619522i
\(668\) 0 0
\(669\) 21.2132i 0.820150i
\(670\) 0 0
\(671\) 16.9706 0.655141
\(672\) 0 0
\(673\) −40.0000 −1.54189 −0.770943 0.636904i \(-0.780215\pi\)
−0.770943 + 0.636904i \(0.780215\pi\)
\(674\) 0 0
\(675\) 3.00000i 0.115470i
\(676\) 0 0
\(677\) 36.7696i 1.41317i 0.707629 + 0.706584i \(0.249765\pi\)
−0.707629 + 0.706584i \(0.750235\pi\)
\(678\) 0 0
\(679\) 16.9706 0.651270
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 0 0
\(685\) 28.2843i 1.08069i
\(686\) 0 0
\(687\) 15.5563 0.593512
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 22.0000i 0.836919i 0.908235 + 0.418460i \(0.137430\pi\)
−0.908235 + 0.418460i \(0.862570\pi\)
\(692\) 0 0
\(693\) 16.9706i 0.644658i
\(694\) 0 0
\(695\) −33.9411 −1.28746
\(696\) 0 0
\(697\) 60.0000 2.27266
\(698\) 0 0
\(699\) − 18.0000i − 0.680823i
\(700\) 0 0
\(701\) 5.65685i 0.213656i 0.994277 + 0.106828i \(0.0340695\pi\)
−0.994277 + 0.106828i \(0.965931\pi\)
\(702\) 0 0
\(703\) −8.48528 −0.320028
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) 48.0000i 1.80523i
\(708\) 0 0
\(709\) − 18.3848i − 0.690455i −0.938519 0.345227i \(-0.887802\pi\)
0.938519 0.345227i \(-0.112198\pi\)
\(710\) 0 0
\(711\) 4.24264 0.159111
\(712\) 0 0
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 48.0000i 1.79510i
\(716\) 0 0
\(717\) 11.3137i 0.422518i
\(718\) 0 0
\(719\) 8.48528 0.316448 0.158224 0.987403i \(-0.449423\pi\)
0.158224 + 0.987403i \(0.449423\pi\)
\(720\) 0 0
\(721\) −78.0000 −2.90487
\(722\) 0 0
\(723\) 18.0000i 0.669427i
\(724\) 0 0
\(725\) 16.9706i 0.630271i
\(726\) 0 0
\(727\) 26.8701 0.996555 0.498278 0.867018i \(-0.333966\pi\)
0.498278 + 0.867018i \(0.333966\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 36.0000i − 1.33151i
\(732\) 0 0
\(733\) 35.3553i 1.30588i 0.757410 + 0.652940i \(0.226464\pi\)
−0.757410 + 0.652940i \(0.773536\pi\)
\(734\) 0 0
\(735\) 31.1127 1.14761
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) − 36.0000i − 1.32428i −0.749380 0.662141i \(-0.769648\pi\)
0.749380 0.662141i \(-0.230352\pi\)
\(740\) 0 0
\(741\) 8.48528i 0.311715i
\(742\) 0 0
\(743\) 22.6274 0.830119 0.415060 0.909794i \(-0.363761\pi\)
0.415060 + 0.909794i \(0.363761\pi\)
\(744\) 0 0
\(745\) 24.0000 0.879292
\(746\) 0 0
\(747\) 16.0000i 0.585409i
\(748\) 0 0
\(749\) − 16.9706i − 0.620091i
\(750\) 0 0
\(751\) −12.7279 −0.464448 −0.232224 0.972662i \(-0.574600\pi\)
−0.232224 + 0.972662i \(0.574600\pi\)
\(752\) 0 0
\(753\) −4.00000 −0.145768
\(754\) 0 0
\(755\) − 4.00000i − 0.145575i
\(756\) 0 0
\(757\) 38.1838i 1.38781i 0.720065 + 0.693906i \(0.244112\pi\)
−0.720065 + 0.693906i \(0.755888\pi\)
\(758\) 0 0
\(759\) 11.3137 0.410662
\(760\) 0 0
\(761\) 34.0000 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 0 0
\(763\) 42.0000i 1.52050i
\(764\) 0 0
\(765\) 16.9706i 0.613572i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) − 14.0000i − 0.504198i
\(772\) 0 0
\(773\) 5.65685i 0.203463i 0.994812 + 0.101731i \(0.0324382\pi\)
−0.994812 + 0.101731i \(0.967562\pi\)
\(774\) 0 0
\(775\) 12.7279 0.457200
\(776\) 0 0
\(777\) −18.0000 −0.645746
\(778\) 0 0
\(779\) − 20.0000i − 0.716574i
\(780\) 0 0
\(781\) − 11.3137i − 0.404836i
\(782\) 0 0
\(783\) −5.65685 −0.202159
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) − 38.0000i − 1.35455i −0.735728 0.677277i \(-0.763160\pi\)
0.735728 0.677277i \(-0.236840\pi\)
\(788\) 0 0
\(789\) − 22.6274i − 0.805557i
\(790\) 0 0
\(791\) 25.4558 0.905106
\(792\) 0 0
\(793\) 18.0000 0.639199
\(794\) 0 0
\(795\) 16.0000i 0.567462i
\(796\) 0 0
\(797\) 31.1127i 1.10207i 0.834483 + 0.551034i \(0.185767\pi\)
−0.834483 + 0.551034i \(0.814233\pi\)
\(798\) 0 0
\(799\) −16.9706 −0.600375
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) 0 0
\(803\) − 64.0000i − 2.25851i
\(804\) 0 0
\(805\) − 33.9411i − 1.19627i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) − 2.00000i − 0.0702295i −0.999383 0.0351147i \(-0.988820\pi\)
0.999383 0.0351147i \(-0.0111797\pi\)
\(812\) 0 0
\(813\) − 1.41421i − 0.0495986i
\(814\) 0 0
\(815\) −28.2843 −0.990755
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) 0 0
\(819\) 18.0000i 0.628971i
\(820\) 0 0
\(821\) − 28.2843i − 0.987128i −0.869710 0.493564i \(-0.835694\pi\)
0.869710 0.493564i \(-0.164306\pi\)
\(822\) 0 0
\(823\) 4.24264 0.147889 0.0739446 0.997262i \(-0.476441\pi\)
0.0739446 + 0.997262i \(0.476441\pi\)
\(824\) 0 0
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) 0 0
\(829\) − 21.2132i − 0.736765i −0.929674 0.368383i \(-0.879912\pi\)
0.929674 0.368383i \(-0.120088\pi\)
\(830\) 0 0
\(831\) −1.41421 −0.0490585
\(832\) 0 0
\(833\) 66.0000 2.28676
\(834\) 0 0
\(835\) − 32.0000i − 1.10741i
\(836\) 0 0
\(837\) 4.24264i 0.146647i
\(838\) 0 0
\(839\) −31.1127 −1.07413 −0.537065 0.843541i \(-0.680467\pi\)
−0.537065 + 0.843541i \(0.680467\pi\)
\(840\) 0 0
\(841\) −3.00000 −0.103448
\(842\) 0 0
\(843\) − 6.00000i − 0.206651i
\(844\) 0 0
\(845\) 14.1421i 0.486504i
\(846\) 0 0
\(847\) 21.2132 0.728894
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 12.0000i 0.411355i
\(852\) 0 0
\(853\) − 21.2132i − 0.726326i −0.931726 0.363163i \(-0.881697\pi\)
0.931726 0.363163i \(-0.118303\pi\)
\(854\) 0 0
\(855\) 5.65685 0.193460
\(856\) 0 0
\(857\) −30.0000 −1.02478 −0.512390 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 0 0
\(859\) − 10.0000i − 0.341196i −0.985341 0.170598i \(-0.945430\pi\)
0.985341 0.170598i \(-0.0545699\pi\)
\(860\) 0 0
\(861\) − 42.4264i − 1.44589i
\(862\) 0 0
\(863\) −28.2843 −0.962808 −0.481404 0.876499i \(-0.659873\pi\)
−0.481404 + 0.876499i \(0.659873\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.272008
\(866\) 0 0
\(867\) 19.0000i 0.645274i
\(868\) 0 0
\(869\) − 16.9706i − 0.575687i
\(870\) 0 0
\(871\) 16.9706 0.575026
\(872\) 0 0
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) 24.0000i 0.811348i
\(876\) 0 0
\(877\) − 15.5563i − 0.525301i −0.964891 0.262650i \(-0.915403\pi\)
0.964891 0.262650i \(-0.0845966\pi\)
\(878\) 0 0
\(879\) 5.65685 0.190801
\(880\) 0 0
\(881\) 58.0000 1.95407 0.977035 0.213080i \(-0.0683494\pi\)
0.977035 + 0.213080i \(0.0683494\pi\)
\(882\) 0 0
\(883\) − 34.0000i − 1.14419i −0.820187 0.572096i \(-0.806131\pi\)
0.820187 0.572096i \(-0.193869\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.9706 −0.569816 −0.284908 0.958555i \(-0.591963\pi\)
−0.284908 + 0.958555i \(0.591963\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) 4.00000i 0.134005i
\(892\) 0 0
\(893\) 5.65685i 0.189299i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) 0 0
\(899\) 24.0000i 0.800445i
\(900\) 0 0
\(901\) 33.9411i 1.13074i
\(902\) 0 0
\(903\) −25.4558 −0.847117
\(904\) 0 0
\(905\) −4.00000 −0.132964
\(906\) 0 0
\(907\) − 58.0000i − 1.92586i −0.269754 0.962929i \(-0.586942\pi\)
0.269754 0.962929i \(-0.413058\pi\)
\(908\) 0 0
\(909\) 11.3137i 0.375252i
\(910\) 0 0
\(911\) −22.6274 −0.749680 −0.374840 0.927090i \(-0.622302\pi\)
−0.374840 + 0.927090i \(0.622302\pi\)
\(912\) 0 0
\(913\) 64.0000 2.11809
\(914\) 0 0
\(915\) − 12.0000i − 0.396708i
\(916\) 0 0
\(917\) 84.8528i 2.80209i
\(918\) 0 0
\(919\) −18.3848 −0.606458 −0.303229 0.952918i \(-0.598065\pi\)
−0.303229 + 0.952918i \(0.598065\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) − 12.0000i − 0.394985i
\(924\) 0 0
\(925\) 12.7279i 0.418491i
\(926\) 0 0
\(927\) −18.3848 −0.603835
\(928\) 0 0
\(929\) −46.0000 −1.50921 −0.754606 0.656179i \(-0.772172\pi\)
−0.754606 + 0.656179i \(0.772172\pi\)
\(930\) 0 0
\(931\) − 22.0000i − 0.721021i
\(932\) 0 0
\(933\) 28.2843i 0.925985i
\(934\) 0 0
\(935\) 67.8823 2.21999
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) − 26.0000i − 0.848478i
\(940\) 0 0
\(941\) 36.7696i 1.19865i 0.800505 + 0.599327i \(0.204565\pi\)
−0.800505 + 0.599327i \(0.795435\pi\)
\(942\) 0 0
\(943\) −28.2843 −0.921063
\(944\) 0 0
\(945\) 12.0000 0.390360
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) − 67.8823i − 2.20355i
\(950\) 0 0
\(951\) 16.9706 0.550308
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) − 64.0000i − 2.07099i
\(956\) 0 0
\(957\) 22.6274i 0.731441i
\(958\) 0 0
\(959\) 42.4264 1.37002
\(960\) 0 0
\(961\) −13.0000 −0.419355
\(962\) 0 0
\(963\) − 4.00000i − 0.128898i
\(964\) 0 0
\(965\) − 50.9117i − 1.63891i
\(966\) 0 0
\(967\) 35.3553 1.13695 0.568476 0.822700i \(-0.307533\pi\)
0.568476 + 0.822700i \(0.307533\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) 32.0000i 1.02693i 0.858111 + 0.513464i \(0.171638\pi\)
−0.858111 + 0.513464i \(0.828362\pi\)
\(972\) 0 0
\(973\) 50.9117i 1.63215i
\(974\) 0 0
\(975\) 12.7279 0.407620
\(976\) 0 0
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 0 0
\(979\) − 56.0000i − 1.78977i
\(980\) 0 0
\(981\) 9.89949i 0.316067i
\(982\) 0 0
\(983\) −22.6274 −0.721703 −0.360851 0.932623i \(-0.617514\pi\)
−0.360851 + 0.932623i \(0.617514\pi\)
\(984\) 0 0
\(985\) −16.0000 −0.509802
\(986\) 0 0
\(987\) 12.0000i 0.381964i
\(988\) 0 0
\(989\) 16.9706i 0.539633i
\(990\) 0 0
\(991\) −46.6690 −1.48249 −0.741246 0.671234i \(-0.765765\pi\)
−0.741246 + 0.671234i \(0.765765\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 20.0000i 0.634043i
\(996\) 0 0
\(997\) 46.6690i 1.47802i 0.673693 + 0.739012i \(0.264707\pi\)
−0.673693 + 0.739012i \(0.735293\pi\)
\(998\) 0 0
\(999\) −4.24264 −0.134231
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 3072.2.d.d.1537.3 4
4.3 odd 2 inner 3072.2.d.d.1537.1 4
8.3 odd 2 inner 3072.2.d.d.1537.4 4
8.5 even 2 inner 3072.2.d.d.1537.2 4
16.3 odd 4 3072.2.a.f.1.1 2
16.5 even 4 3072.2.a.f.1.2 2
16.11 odd 4 3072.2.a.d.1.2 2
16.13 even 4 3072.2.a.d.1.1 2
32.3 odd 8 768.2.j.d.193.2 yes 4
32.5 even 8 768.2.j.d.577.1 yes 4
32.11 odd 8 768.2.j.a.577.1 yes 4
32.13 even 8 768.2.j.a.193.2 yes 4
32.19 odd 8 768.2.j.a.193.1 4
32.21 even 8 768.2.j.a.577.2 yes 4
32.27 odd 8 768.2.j.d.577.2 yes 4
32.29 even 8 768.2.j.d.193.1 yes 4
48.5 odd 4 9216.2.a.q.1.1 2
48.11 even 4 9216.2.a.e.1.1 2
48.29 odd 4 9216.2.a.e.1.2 2
48.35 even 4 9216.2.a.q.1.2 2
96.5 odd 8 2304.2.k.a.577.1 4
96.11 even 8 2304.2.k.d.577.2 4
96.29 odd 8 2304.2.k.a.1729.2 4
96.35 even 8 2304.2.k.a.1729.1 4
96.53 odd 8 2304.2.k.d.577.1 4
96.59 even 8 2304.2.k.a.577.2 4
96.77 odd 8 2304.2.k.d.1729.2 4
96.83 even 8 2304.2.k.d.1729.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
768.2.j.a.193.1 4 32.19 odd 8
768.2.j.a.193.2 yes 4 32.13 even 8
768.2.j.a.577.1 yes 4 32.11 odd 8
768.2.j.a.577.2 yes 4 32.21 even 8
768.2.j.d.193.1 yes 4 32.29 even 8
768.2.j.d.193.2 yes 4 32.3 odd 8
768.2.j.d.577.1 yes 4 32.5 even 8
768.2.j.d.577.2 yes 4 32.27 odd 8
2304.2.k.a.577.1 4 96.5 odd 8
2304.2.k.a.577.2 4 96.59 even 8
2304.2.k.a.1729.1 4 96.35 even 8
2304.2.k.a.1729.2 4 96.29 odd 8
2304.2.k.d.577.1 4 96.53 odd 8
2304.2.k.d.577.2 4 96.11 even 8
2304.2.k.d.1729.1 4 96.83 even 8
2304.2.k.d.1729.2 4 96.77 odd 8
3072.2.a.d.1.1 2 16.13 even 4
3072.2.a.d.1.2 2 16.11 odd 4
3072.2.a.f.1.1 2 16.3 odd 4
3072.2.a.f.1.2 2 16.5 even 4
3072.2.d.d.1537.1 4 4.3 odd 2 inner
3072.2.d.d.1537.2 4 8.5 even 2 inner
3072.2.d.d.1537.3 4 1.1 even 1 trivial
3072.2.d.d.1537.4 4 8.3 odd 2 inner
9216.2.a.e.1.1 2 48.11 even 4
9216.2.a.e.1.2 2 48.29 odd 4
9216.2.a.q.1.1 2 48.5 odd 4
9216.2.a.q.1.2 2 48.35 even 4