Properties

Label 768.2.d.e.385.1
Level $768$
Weight $2$
Character 768.385
Analytic conductor $6.133$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 385.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 768.385
Dual form 768.2.d.e.385.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +2.00000i q^{5} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +2.00000i q^{5} -1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{13} +2.00000 q^{15} +2.00000 q^{17} -4.00000i q^{19} +8.00000 q^{23} +1.00000 q^{25} +1.00000i q^{27} +6.00000i q^{29} +8.00000 q^{31} -4.00000 q^{33} -6.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} -2.00000i q^{45} -7.00000 q^{49} -2.00000i q^{51} +2.00000i q^{53} +8.00000 q^{55} -4.00000 q^{57} -4.00000i q^{59} -2.00000i q^{61} +4.00000 q^{65} -4.00000i q^{67} -8.00000i q^{69} -8.00000 q^{71} -10.0000 q^{73} -1.00000i q^{75} -8.00000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +4.00000i q^{85} +6.00000 q^{87} +6.00000 q^{89} -8.00000i q^{93} +8.00000 q^{95} +2.00000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 4q^{15} + 4q^{17} + 16q^{23} + 2q^{25} + 16q^{31} - 8q^{33} - 4q^{39} + 12q^{41} - 14q^{49} + 16q^{55} - 8q^{57} + 8q^{65} - 16q^{71} - 20q^{73} - 16q^{79} + 2q^{81} + 12q^{87} + 12q^{89} + 16q^{95} + 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 4.00000i − 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) − 2.00000i − 0.298142i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) − 2.00000i − 0.280056i
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) − 4.00000i − 0.520756i −0.965507 0.260378i \(-0.916153\pi\)
0.965507 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) − 2.00000i − 0.256074i −0.991769 0.128037i \(-0.959132\pi\)
0.991769 0.128037i \(-0.0408676\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) − 8.00000i − 0.963087i
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) − 1.00000i − 0.115470i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 4.00000i 0.433861i
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 4.00000i 0.402015i
\(100\) 0 0
\(101\) 18.0000i 1.79107i 0.444994 + 0.895533i \(0.353206\pi\)
−0.444994 + 0.895533i \(0.646794\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) − 2.00000i − 0.191565i −0.995402 0.0957826i \(-0.969465\pi\)
0.995402 0.0957826i \(-0.0305354\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 16.0000i 1.49201i
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) − 6.00000i − 0.541002i
\(124\) 0 0
\(125\) 12.0000i 1.07331i
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) − 4.00000i − 0.349482i −0.984614 0.174741i \(-0.944091\pi\)
0.984614 0.174741i \(-0.0559088\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 12.0000i 1.01783i 0.860818 + 0.508913i \(0.169953\pi\)
−0.860818 + 0.508913i \(0.830047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) 7.00000i 0.577350i
\(148\) 0 0
\(149\) − 14.0000i − 1.14692i −0.819232 0.573462i \(-0.805600\pi\)
0.819232 0.573462i \(-0.194400\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 16.0000i 1.28515i
\(156\) 0 0
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.0000i 0.939913i 0.882690 + 0.469956i \(0.155730\pi\)
−0.882690 + 0.469956i \(0.844270\pi\)
\(164\) 0 0
\(165\) − 8.00000i − 0.622799i
\(166\) 0 0
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 4.00000i 0.305888i
\(172\) 0 0
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) 12.0000i 0.896922i 0.893802 + 0.448461i \(0.148028\pi\)
−0.893802 + 0.448461i \(0.851972\pi\)
\(180\) 0 0
\(181\) − 6.00000i − 0.445976i −0.974821 0.222988i \(-0.928419\pi\)
0.974821 0.222988i \(-0.0715812\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) − 8.00000i − 0.585018i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) − 4.00000i − 0.286446i
\(196\) 0 0
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 12.0000i 0.838116i
\(206\) 0 0
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) − 20.0000i − 1.37686i −0.725304 0.688428i \(-0.758301\pi\)
0.725304 0.688428i \(-0.241699\pi\)
\(212\) 0 0
\(213\) 8.00000i 0.548151i
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 10.0000i 0.675737i
\(220\) 0 0
\(221\) − 4.00000i − 0.269069i
\(222\) 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
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 0 0
\(229\) − 22.0000i − 1.45380i −0.686743 0.726900i \(-0.740960\pi\)
0.686743 0.726900i \(-0.259040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000i 0.519656i
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 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
\(245\) − 14.0000i − 0.894427i
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) − 20.0000i − 1.26239i −0.775625 0.631194i \(-0.782565\pi\)
0.775625 0.631194i \(-0.217435\pi\)
\(252\) 0 0
\(253\) − 32.0000i − 2.01182i
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) − 6.00000i − 0.371391i
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) − 6.00000i − 0.367194i
\(268\) 0 0
\(269\) − 10.0000i − 0.609711i −0.952399 0.304855i \(-0.901392\pi\)
0.952399 0.304855i \(-0.0986081\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 4.00000i − 0.241209i
\(276\) 0 0
\(277\) 26.0000i 1.56219i 0.624413 + 0.781094i \(0.285338\pi\)
−0.624413 + 0.781094i \(0.714662\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) 28.0000i 1.66443i 0.554455 + 0.832214i \(0.312927\pi\)
−0.554455 + 0.832214i \(0.687073\pi\)
\(284\) 0 0
\(285\) − 8.00000i − 0.473879i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) − 2.00000i − 0.117242i
\(292\) 0 0
\(293\) 18.0000i 1.05157i 0.850617 + 0.525786i \(0.176229\pi\)
−0.850617 + 0.525786i \(0.823771\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) − 16.0000i − 0.925304i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 18.0000 1.03407
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) 16.0000i 0.910208i
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) − 8.00000i − 0.445132i
\(324\) 0 0
\(325\) − 2.00000i − 0.110940i
\(326\) 0 0
\(327\) −2.00000 −0.110600
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 20.0000i − 1.09930i −0.835395 0.549650i \(-0.814761\pi\)
0.835395 0.549650i \(-0.185239\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) − 18.0000i − 0.977626i
\(340\) 0 0
\(341\) − 32.0000i − 1.73290i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 16.0000 0.861411
\(346\) 0 0
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 0 0
\(349\) 30.0000i 1.60586i 0.596071 + 0.802932i \(0.296728\pi\)
−0.596071 + 0.802932i \(0.703272\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 0 0
\(355\) − 16.0000i − 0.849192i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) − 20.0000i − 1.04685i
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) − 20.0000i − 1.02733i −0.857991 0.513665i \(-0.828287\pi\)
0.857991 0.513665i \(-0.171713\pi\)
\(380\) 0 0
\(381\) 8.00000i 0.409852i
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) 2.00000i 0.101404i 0.998714 + 0.0507020i \(0.0161459\pi\)
−0.998714 + 0.0507020i \(0.983854\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) − 16.0000i − 0.805047i
\(396\) 0 0
\(397\) 14.0000i 0.702640i 0.936255 + 0.351320i \(0.114267\pi\)
−0.936255 + 0.351320i \(0.885733\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) − 16.0000i − 0.797017i
\(404\) 0 0
\(405\) 2.00000i 0.0993808i
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) − 6.00000i − 0.295958i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 12.0000i 0.586238i 0.956076 + 0.293119i \(0.0946933\pi\)
−0.956076 + 0.293119i \(0.905307\pi\)
\(420\) 0 0
\(421\) 10.0000i 0.487370i 0.969854 + 0.243685i \(0.0783563\pi\)
−0.969854 + 0.243685i \(0.921644\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 8.00000i 0.386244i
\(430\) 0 0
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 12.0000i 0.575356i
\(436\) 0 0
\(437\) − 32.0000i − 1.53077i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) − 20.0000i − 0.950229i −0.879924 0.475114i \(-0.842407\pi\)
0.879924 0.475114i \(-0.157593\pi\)
\(444\) 0 0
\(445\) 12.0000i 0.568855i
\(446\) 0 0
\(447\) −14.0000 −0.662177
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) − 24.0000i − 1.13012i
\(452\) 0 0
\(453\) − 16.0000i − 0.751746i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 0 0
\(459\) 2.00000i 0.0933520i
\(460\) 0 0
\(461\) − 26.0000i − 1.21094i −0.795868 0.605470i \(-0.792985\pi\)
0.795868 0.605470i \(-0.207015\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 0 0
\(465\) 16.0000 0.741982
\(466\) 0 0
\(467\) − 36.0000i − 1.66588i −0.553362 0.832941i \(-0.686655\pi\)
0.553362 0.832941i \(-0.313345\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) − 4.00000i − 0.183533i
\(476\) 0 0
\(477\) − 2.00000i − 0.0915737i
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.00000i 0.181631i
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 12.0000i 0.541552i 0.962642 + 0.270776i \(0.0872803\pi\)
−0.962642 + 0.270776i \(0.912720\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) −8.00000 −0.359573
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12.0000i 0.537194i 0.963253 + 0.268597i \(0.0865599\pi\)
−0.963253 + 0.268597i \(0.913440\pi\)
\(500\) 0 0
\(501\) 24.0000i 1.07224i
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −36.0000 −1.60198
\(506\) 0 0
\(507\) − 9.00000i − 0.399704i
\(508\) 0 0
\(509\) 6.00000i 0.265945i 0.991120 + 0.132973i \(0.0424523\pi\)
−0.991120 + 0.132973i \(0.957548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) − 32.0000i − 1.41009i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 0 0
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 4.00000i 0.173585i
\(532\) 0 0
\(533\) − 12.0000i − 0.519778i
\(534\) 0 0
\(535\) −24.0000 −1.03761
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 28.0000i 1.20605i
\(540\) 0 0
\(541\) − 18.0000i − 0.773880i −0.922105 0.386940i \(-0.873532\pi\)
0.922105 0.386940i \(-0.126468\pi\)
\(542\) 0 0
\(543\) −6.00000 −0.257485
\(544\) 0 0
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) 44.0000i 1.88130i 0.339372 + 0.940652i \(0.389785\pi\)
−0.339372 + 0.940652i \(0.610215\pi\)
\(548\) 0 0
\(549\) 2.00000i 0.0853579i
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) − 12.0000i − 0.509372i
\(556\) 0 0
\(557\) − 26.0000i − 1.10166i −0.834619 0.550828i \(-0.814312\pi\)
0.834619 0.550828i \(-0.185688\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) 28.0000i 1.18006i 0.807382 + 0.590030i \(0.200884\pi\)
−0.807382 + 0.590030i \(0.799116\pi\)
\(564\) 0 0
\(565\) 36.0000i 1.51453i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) − 36.0000i − 1.50655i −0.657704 0.753277i \(-0.728472\pi\)
0.657704 0.753277i \(-0.271528\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) − 2.00000i − 0.0831172i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) 44.0000i 1.81607i 0.418890 + 0.908037i \(0.362419\pi\)
−0.418890 + 0.908037i \(0.637581\pi\)
\(588\) 0 0
\(589\) − 32.0000i − 1.31854i
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 0 0
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.0000i 0.654836i
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) − 10.0000i − 0.406558i
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 38.0000i − 1.53481i −0.641165 0.767403i \(-0.721549\pi\)
0.641165 0.767403i \(-0.278451\pi\)
\(614\) 0 0
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) 44.0000i 1.76851i 0.467005 + 0.884255i \(0.345333\pi\)
−0.467005 + 0.884255i \(0.654667\pi\)
\(620\) 0 0
\(621\) 8.00000i 0.321029i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 16.0000i 0.638978i
\(628\) 0 0
\(629\) − 12.0000i − 0.478471i
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) − 16.0000i − 0.634941i
\(636\) 0 0
\(637\) 14.0000i 0.554700i
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −14.0000 −0.552967 −0.276483 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(642\) 0 0
\(643\) 12.0000i 0.473234i 0.971603 + 0.236617i \(0.0760386\pi\)
−0.971603 + 0.236617i \(0.923961\pi\)
\(644\) 0 0
\(645\) − 8.00000i − 0.315000i
\(646\) 0 0
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) 0 0
\(655\) 8.00000 0.312586
\(656\) 0 0
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) 12.0000i 0.467454i 0.972302 + 0.233727i \(0.0750921\pi\)
−0.972302 + 0.233727i \(0.924908\pi\)
\(660\) 0 0
\(661\) 10.0000i 0.388955i 0.980907 + 0.194477i \(0.0623011\pi\)
−0.980907 + 0.194477i \(0.937699\pi\)
\(662\) 0 0
\(663\) −4.00000 −0.155347
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 48.0000i 1.85857i
\(668\) 0 0
\(669\) 8.00000i 0.309298i
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 2.00000i 0.0768662i 0.999261 + 0.0384331i \(0.0122367\pi\)
−0.999261 + 0.0384331i \(0.987763\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) − 4.00000i − 0.153056i −0.997067 0.0765279i \(-0.975617\pi\)
0.997067 0.0765279i \(-0.0243834\pi\)
\(684\) 0 0
\(685\) 12.0000i 0.458496i
\(686\) 0 0
\(687\) −22.0000 −0.839352
\(688\) 0 0
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) − 4.00000i − 0.152167i −0.997101 0.0760836i \(-0.975758\pi\)
0.997101 0.0760836i \(-0.0242416\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −24.0000 −0.910372
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) 10.0000i 0.378235i
\(700\) 0 0
\(701\) 6.00000i 0.226617i 0.993560 + 0.113308i \(0.0361448\pi\)
−0.993560 + 0.113308i \(0.963855\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 10.0000i 0.375558i 0.982211 + 0.187779i \(0.0601289\pi\)
−0.982211 + 0.187779i \(0.939871\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 64.0000 2.39682
\(714\) 0 0
\(715\) − 16.0000i − 0.598366i
\(716\) 0 0
\(717\) 16.0000i 0.597531i
\(718\) 0 0
\(719\) −32.0000 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 18.0000i − 0.669427i
\(724\) 0 0
\(725\) 6.00000i 0.222834i
\(726\) 0 0
\(727\) −48.0000 −1.78022 −0.890111 0.455744i \(-0.849373\pi\)
−0.890111 + 0.455744i \(0.849373\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 8.00000i − 0.295891i
\(732\) 0 0
\(733\) 14.0000i 0.517102i 0.965998 + 0.258551i \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) 0 0
\(735\) −14.0000 −0.516398
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) − 4.00000i − 0.147142i −0.997290 0.0735712i \(-0.976560\pi\)
0.997290 0.0735712i \(-0.0234396\pi\)
\(740\) 0 0
\(741\) 8.00000i 0.293887i
\(742\) 0 0
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 28.0000 1.02584
\(746\) 0 0
\(747\) 4.00000i 0.146352i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 0 0
\(753\) −20.0000 −0.728841
\(754\) 0 0
\(755\) 32.0000i 1.16460i
\(756\) 0 0
\(757\) − 38.0000i − 1.38113i −0.723269 0.690567i \(-0.757361\pi\)
0.723269 0.690567i \(-0.242639\pi\)
\(758\) 0 0
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) − 4.00000i − 0.144620i
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) − 2.00000i − 0.0720282i
\(772\) 0 0
\(773\) 18.0000i 0.647415i 0.946157 + 0.323708i \(0.104929\pi\)
−0.946157 + 0.323708i \(0.895071\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 24.0000i − 0.859889i
\(780\) 0 0
\(781\) 32.0000i 1.14505i
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) 28.0000i 0.998092i 0.866575 + 0.499046i \(0.166316\pi\)
−0.866575 + 0.499046i \(0.833684\pi\)
\(788\) 0 0
\(789\) − 8.00000i − 0.284808i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 0 0
\(795\) 4.00000i 0.141865i
\(796\) 0 0
\(797\) 22.0000i 0.779280i 0.920967 + 0.389640i \(0.127401\pi\)
−0.920967 + 0.389640i \(0.872599\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 40.0000i 1.41157i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −10.0000 −0.352017
\(808\) 0 0
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) − 4.00000i − 0.140459i −0.997531 0.0702295i \(-0.977627\pi\)
0.997531 0.0702295i \(-0.0223732\pi\)
\(812\) 0 0
\(813\) − 8.00000i − 0.280572i
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 30.0000i − 1.04701i −0.852023 0.523504i \(-0.824625\pi\)
0.852023 0.523504i \(-0.175375\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 28.0000i 0.973655i 0.873498 + 0.486828i \(0.161846\pi\)
−0.873498 + 0.486828i \(0.838154\pi\)
\(828\) 0 0
\(829\) − 50.0000i − 1.73657i −0.496064 0.868286i \(-0.665222\pi\)
0.496064 0.868286i \(-0.334778\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) − 48.0000i − 1.66111i
\(836\) 0 0
\(837\) 8.00000i 0.276520i
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) 26.0000i 0.895488i
\(844\) 0 0
\(845\) 18.0000i 0.619219i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) − 48.0000i − 1.64542i
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 12.0000i 0.409435i 0.978821 + 0.204717i \(0.0656275\pi\)
−0.978821 + 0.204717i \(0.934372\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 0 0
\(865\) −12.0000 −0.408012
\(866\) 0 0
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) 32.0000i 1.08553i
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 18.0000i − 0.607817i −0.952701 0.303908i \(-0.901708\pi\)
0.952701 0.303908i \(-0.0982917\pi\)
\(878\) 0 0
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 50.0000 1.68454 0.842271 0.539054i \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) 0 0
\(883\) − 4.00000i − 0.134611i −0.997732 0.0673054i \(-0.978560\pi\)
0.997732 0.0673054i \(-0.0214402\pi\)
\(884\) 0 0
\(885\) − 8.00000i − 0.268917i
\(886\) 0 0
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 4.00000i − 0.134005i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 0 0
\(899\) 48.0000i 1.60089i
\(900\) 0 0
\(901\) 4.00000i 0.133259i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12.0000 0.398893
\(906\) 0 0
\(907\) − 4.00000i − 0.132818i −0.997792 0.0664089i \(-0.978846\pi\)
0.997792 0.0664089i \(-0.0211542\pi\)
\(908\) 0 0
\(909\) − 18.0000i − 0.597022i
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 0 0
\(915\) − 4.00000i − 0.132236i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) 16.0000i 0.526646i
\(924\) 0 0
\(925\) − 6.00000i − 0.197279i
\(926\) 0 0
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) 50.0000 1.64045 0.820223 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) 28.0000i 0.917663i
\(932\) 0 0
\(933\) − 24.0000i − 0.785725i
\(934\) 0 0
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) − 6.00000i − 0.195803i
\(940\) 0 0
\(941\) 6.00000i 0.195594i 0.995206 + 0.0977972i \(0.0311797\pi\)
−0.995206 + 0.0977972i \(0.968820\pi\)
\(942\) 0 0
\(943\) 48.0000 1.56310
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) 0 0
\(949\) 20.0000i 0.649227i
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 24.0000i − 0.775810i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) − 12.0000i − 0.386695i
\(964\) 0 0
\(965\) 4.00000i 0.128765i
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) − 36.0000i − 1.15529i −0.816286 0.577647i \(-0.803971\pi\)
0.816286 0.577647i \(-0.196029\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −2.00000 −0.0640513
\(976\) 0 0
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 0 0
\(979\) − 24.0000i − 0.767043i
\(980\) 0 0
\(981\) 2.00000i 0.0638551i
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) −36.0000 −1.14706
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 32.0000i − 1.01754i
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) − 32.0000i − 1.01447i
\(996\) 0 0
\(997\) 26.0000i 0.823428i 0.911313 + 0.411714i \(0.135070\pi\)
−0.911313 + 0.411714i \(0.864930\pi\)
\(998\) 0 0
\(999\) 6.00000 0.189832
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 768.2.d.e.385.1 2
3.2 odd 2 2304.2.d.i.1153.1 2
4.3 odd 2 768.2.d.d.385.2 2
8.3 odd 2 768.2.d.d.385.1 2
8.5 even 2 inner 768.2.d.e.385.2 2
12.11 even 2 2304.2.d.k.1153.1 2
16.3 odd 4 192.2.a.b.1.1 1
16.5 even 4 24.2.a.a.1.1 1
16.11 odd 4 48.2.a.a.1.1 1
16.13 even 4 192.2.a.d.1.1 1
24.5 odd 2 2304.2.d.i.1153.2 2
24.11 even 2 2304.2.d.k.1153.2 2
48.5 odd 4 72.2.a.a.1.1 1
48.11 even 4 144.2.a.b.1.1 1
48.29 odd 4 576.2.a.d.1.1 1
48.35 even 4 576.2.a.b.1.1 1
80.3 even 4 4800.2.f.bg.3649.1 2
80.13 odd 4 4800.2.f.d.3649.2 2
80.19 odd 4 4800.2.a.cc.1.1 1
80.27 even 4 1200.2.f.b.49.1 2
80.29 even 4 4800.2.a.q.1.1 1
80.37 odd 4 600.2.f.e.49.2 2
80.43 even 4 1200.2.f.b.49.2 2
80.53 odd 4 600.2.f.e.49.1 2
80.59 odd 4 1200.2.a.d.1.1 1
80.67 even 4 4800.2.f.bg.3649.2 2
80.69 even 4 600.2.a.h.1.1 1
80.77 odd 4 4800.2.f.d.3649.1 2
112.5 odd 12 1176.2.q.a.361.1 2
112.11 odd 12 2352.2.q.l.961.1 2
112.13 odd 4 9408.2.a.h.1.1 1
112.27 even 4 2352.2.a.i.1.1 1
112.37 even 12 1176.2.q.i.361.1 2
112.53 even 12 1176.2.q.i.961.1 2
112.59 even 12 2352.2.q.r.961.1 2
112.69 odd 4 1176.2.a.i.1.1 1
112.75 even 12 2352.2.q.r.1537.1 2
112.83 even 4 9408.2.a.cc.1.1 1
112.101 odd 12 1176.2.q.a.961.1 2
112.107 odd 12 2352.2.q.l.1537.1 2
144.5 odd 12 648.2.i.b.217.1 2
144.11 even 12 1296.2.i.e.433.1 2
144.43 odd 12 1296.2.i.m.433.1 2
144.59 even 12 1296.2.i.e.865.1 2
144.85 even 12 648.2.i.g.217.1 2
144.101 odd 12 648.2.i.b.433.1 2
144.133 even 12 648.2.i.g.433.1 2
144.139 odd 12 1296.2.i.m.865.1 2
176.21 odd 4 2904.2.a.c.1.1 1
176.43 even 4 5808.2.a.s.1.1 1
208.5 odd 4 4056.2.c.e.337.2 2
208.21 odd 4 4056.2.c.e.337.1 2
208.155 odd 4 8112.2.a.be.1.1 1
208.181 even 4 4056.2.a.i.1.1 1
240.53 even 4 1800.2.f.c.649.1 2
240.59 even 4 3600.2.a.v.1.1 1
240.107 odd 4 3600.2.f.r.2449.2 2
240.149 odd 4 1800.2.a.m.1.1 1
240.197 even 4 1800.2.f.c.649.2 2
240.203 odd 4 3600.2.f.r.2449.1 2
272.101 even 4 6936.2.a.p.1.1 1
304.37 odd 4 8664.2.a.j.1.1 1
336.5 even 12 3528.2.s.y.361.1 2
336.53 odd 12 3528.2.s.j.3313.1 2
336.101 even 12 3528.2.s.y.3313.1 2
336.149 odd 12 3528.2.s.j.361.1 2
336.251 odd 4 7056.2.a.q.1.1 1
336.293 even 4 3528.2.a.d.1.1 1
528.197 even 4 8712.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.a.a.1.1 1 16.5 even 4
48.2.a.a.1.1 1 16.11 odd 4
72.2.a.a.1.1 1 48.5 odd 4
144.2.a.b.1.1 1 48.11 even 4
192.2.a.b.1.1 1 16.3 odd 4
192.2.a.d.1.1 1 16.13 even 4
576.2.a.b.1.1 1 48.35 even 4
576.2.a.d.1.1 1 48.29 odd 4
600.2.a.h.1.1 1 80.69 even 4
600.2.f.e.49.1 2 80.53 odd 4
600.2.f.e.49.2 2 80.37 odd 4
648.2.i.b.217.1 2 144.5 odd 12
648.2.i.b.433.1 2 144.101 odd 12
648.2.i.g.217.1 2 144.85 even 12
648.2.i.g.433.1 2 144.133 even 12
768.2.d.d.385.1 2 8.3 odd 2
768.2.d.d.385.2 2 4.3 odd 2
768.2.d.e.385.1 2 1.1 even 1 trivial
768.2.d.e.385.2 2 8.5 even 2 inner
1176.2.a.i.1.1 1 112.69 odd 4
1176.2.q.a.361.1 2 112.5 odd 12
1176.2.q.a.961.1 2 112.101 odd 12
1176.2.q.i.361.1 2 112.37 even 12
1176.2.q.i.961.1 2 112.53 even 12
1200.2.a.d.1.1 1 80.59 odd 4
1200.2.f.b.49.1 2 80.27 even 4
1200.2.f.b.49.2 2 80.43 even 4
1296.2.i.e.433.1 2 144.11 even 12
1296.2.i.e.865.1 2 144.59 even 12
1296.2.i.m.433.1 2 144.43 odd 12
1296.2.i.m.865.1 2 144.139 odd 12
1800.2.a.m.1.1 1 240.149 odd 4
1800.2.f.c.649.1 2 240.53 even 4
1800.2.f.c.649.2 2 240.197 even 4
2304.2.d.i.1153.1 2 3.2 odd 2
2304.2.d.i.1153.2 2 24.5 odd 2
2304.2.d.k.1153.1 2 12.11 even 2
2304.2.d.k.1153.2 2 24.11 even 2
2352.2.a.i.1.1 1 112.27 even 4
2352.2.q.l.961.1 2 112.11 odd 12
2352.2.q.l.1537.1 2 112.107 odd 12
2352.2.q.r.961.1 2 112.59 even 12
2352.2.q.r.1537.1 2 112.75 even 12
2904.2.a.c.1.1 1 176.21 odd 4
3528.2.a.d.1.1 1 336.293 even 4
3528.2.s.j.361.1 2 336.149 odd 12
3528.2.s.j.3313.1 2 336.53 odd 12
3528.2.s.y.361.1 2 336.5 even 12
3528.2.s.y.3313.1 2 336.101 even 12
3600.2.a.v.1.1 1 240.59 even 4
3600.2.f.r.2449.1 2 240.203 odd 4
3600.2.f.r.2449.2 2 240.107 odd 4
4056.2.a.i.1.1 1 208.181 even 4
4056.2.c.e.337.1 2 208.21 odd 4
4056.2.c.e.337.2 2 208.5 odd 4
4800.2.a.q.1.1 1 80.29 even 4
4800.2.a.cc.1.1 1 80.19 odd 4
4800.2.f.d.3649.1 2 80.77 odd 4
4800.2.f.d.3649.2 2 80.13 odd 4
4800.2.f.bg.3649.1 2 80.3 even 4
4800.2.f.bg.3649.2 2 80.67 even 4
5808.2.a.s.1.1 1 176.43 even 4
6936.2.a.p.1.1 1 272.101 even 4
7056.2.a.q.1.1 1 336.251 odd 4
8112.2.a.be.1.1 1 208.155 odd 4
8664.2.a.j.1.1 1 304.37 odd 4
8712.2.a.u.1.1 1 528.197 even 4
9408.2.a.h.1.1 1 112.13 odd 4
9408.2.a.cc.1.1 1 112.83 even 4