Properties

Label 5376.2.c.bc.2689.2
Level $5376$
Weight $2$
Character 5376.2689
Analytic conductor $42.928$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5376.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(42.9275761266\)
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 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2689.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5376.2689
Dual form 5376.2.c.bc.2689.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -6.00000i q^{13} +2.00000 q^{15} +2.00000 q^{17} +4.00000i q^{19} +1.00000i q^{21} -8.00000 q^{23} +1.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} +4.00000 q^{33} -2.00000i q^{35} -10.0000i q^{37} +6.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} +2.00000i q^{45} +1.00000 q^{49} +2.00000i q^{51} +6.00000i q^{53} -8.00000 q^{55} -4.00000 q^{57} +4.00000i q^{59} -6.00000i q^{61} -1.00000 q^{63} -12.0000 q^{65} -4.00000i q^{67} -8.00000i q^{69} -8.00000 q^{71} -10.0000 q^{73} +1.00000i q^{75} -4.00000i q^{77} +1.00000 q^{81} +4.00000i q^{83} -4.00000i q^{85} -2.00000 q^{87} +6.00000 q^{89} -6.00000i q^{91} +8.00000 q^{95} -14.0000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{7} - 2q^{9} + 4q^{15} + 4q^{17} - 16q^{23} + 2q^{25} + 8q^{33} + 12q^{39} + 12q^{41} + 2q^{49} - 16q^{55} - 8q^{57} - 2q^{63} - 24q^{65} - 16q^{71} - 20q^{73} + 2q^{81} - 4q^{87} + 12q^{89} + 16q^{95} - 28q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times\).

\(n\) \(1793\) \(2815\) \(4609\) \(5125\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(-0.852416\pi\)
0.894427 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 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
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 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 \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 1.00000i 0.218218i
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) − 2.00000i − 0.338062i
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 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
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.00000i 0.280056i
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 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 \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) − 6.00000i − 0.768221i −0.923287 0.384111i \(-0.874508\pi\)
0.923287 0.384111i \(-0.125492\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 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
\(69\) − 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
\(77\) − 4.00000i − 0.455842i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) − 4.00000i − 0.433861i
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) − 6.00000i − 0.628971i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 4.00000i 0.402015i
\(100\) 0 0
\(101\) − 2.00000i − 0.199007i −0.995037 0.0995037i \(-0.968274\pi\)
0.995037 0.0995037i \(-0.0317255\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 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 \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 16.0000i 1.49201i
\(116\) 0 0
\(117\) 6.00000i 0.554700i
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 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
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 20.0000i 1.74741i 0.486458 + 0.873704i \(0.338289\pi\)
−0.486458 + 0.873704i \(0.661711\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 4.00000i 0.339276i 0.985506 + 0.169638i \(0.0542598\pi\)
−0.985506 + 0.169638i \(0.945740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −24.0000 −2.00698
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) 0 0
\(165\) − 8.00000i − 0.622799i
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) − 4.00000i − 0.305888i
\(172\) 0 0
\(173\) − 22.0000i − 1.67263i −0.548250 0.836315i \(-0.684706\pi\)
0.548250 0.836315i \(-0.315294\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 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
\(181\) − 18.0000i − 1.33793i −0.743294 0.668965i \(-0.766738\pi\)
0.743294 0.668965i \(-0.233262\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 0 0
\(185\) −20.0000 −1.47043
\(186\) 0 0
\(187\) − 8.00000i − 0.585018i
\(188\) 0 0
\(189\) − 1.00000i − 0.0727393i
\(190\) 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
\(195\) − 12.0000i − 0.859338i
\(196\) 0 0
\(197\) − 10.0000i − 0.712470i −0.934396 0.356235i \(-0.884060\pi\)
0.934396 0.356235i \(-0.115940\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 2.00000i 0.140372i
\(204\) 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
\(221\) − 12.0000i − 0.807207i
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) − 2.00000i − 0.132164i −0.997814 0.0660819i \(-0.978950\pi\)
0.997814 0.0660819i \(-0.0210498\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) − 2.00000i − 0.127775i
\(246\) 0 0
\(247\) 24.0000 1.52708
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) − 12.0000i − 0.757433i −0.925513 0.378717i \(-0.876365\pi\)
0.925513 0.378717i \(-0.123635\pi\)
\(252\) 0 0
\(253\) 32.0000i 2.01182i
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) 0 0
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) − 10.0000i − 0.621370i
\(260\) 0 0
\(261\) − 2.00000i − 0.123797i
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 0 0
\(269\) − 22.0000i − 1.34136i −0.741745 0.670682i \(-0.766002\pi\)
0.741745 0.670682i \(-0.233998\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 6.00000 0.363137
\(274\) 0 0
\(275\) − 4.00000i − 0.241209i
\(276\) 0 0
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 0 0
\(285\) 8.00000i 0.473879i
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) − 14.0000i − 0.820695i
\(292\) 0 0
\(293\) 30.0000i 1.75262i 0.481749 + 0.876309i \(0.340002\pi\)
−0.481749 + 0.876309i \(0.659998\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 48.0000i 2.77591i
\(300\) 0 0
\(301\) − 4.00000i − 0.230556i
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) − 28.0000i − 1.59804i −0.601302 0.799022i \(-0.705351\pi\)
0.601302 0.799022i \(-0.294649\pi\)
\(308\) 0 0
\(309\) − 8.00000i − 0.455104i
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 2.00000i 0.112687i
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) − 6.00000i − 0.332820i
\(326\) 0 0
\(327\) −2.00000 −0.110600
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 4.00000i − 0.219860i −0.993939 0.109930i \(-0.964937\pi\)
0.993939 0.109930i \(-0.0350627\pi\)
\(332\) 0 0
\(333\) 10.0000i 0.547997i
\(334\) 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
\(339\) − 14.0000i − 0.760376i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 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
\(349\) − 22.0000i − 1.17763i −0.808267 0.588817i \(-0.799594\pi\)
0.808267 0.588817i \(-0.200406\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 16.0000i 0.849192i
\(356\) 0 0
\(357\) 2.00000i 0.105851i
\(358\) 0 0
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 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
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 6.00000i 0.311504i
\(372\) 0 0
\(373\) 22.0000i 1.13912i 0.821951 + 0.569558i \(0.192886\pi\)
−0.821951 + 0.569558i \(0.807114\pi\)
\(374\) 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
\(381\) 0 0
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) − 26.0000i − 1.31825i −0.752032 0.659126i \(-0.770926\pi\)
0.752032 0.659126i \(-0.229074\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) −20.0000 −1.00887
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 2.00000i − 0.0993808i
\(406\) 0 0
\(407\) −40.0000 −1.98273
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) − 10.0000i − 0.493264i
\(412\) 0 0
\(413\) 4.00000i 0.196827i
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 36.0000i 1.75872i 0.476162 + 0.879358i \(0.342028\pi\)
−0.476162 + 0.879358i \(0.657972\pi\)
\(420\) 0 0
\(421\) 6.00000i 0.292422i 0.989253 + 0.146211i \(0.0467079\pi\)
−0.989253 + 0.146211i \(0.953292\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) − 6.00000i − 0.290360i
\(428\) 0 0
\(429\) − 24.0000i − 1.15873i
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 4.00000i 0.191785i
\(436\) 0 0
\(437\) − 32.0000i − 1.53077i
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) 0 0
\(445\) − 12.0000i − 0.568855i
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 0 0
\(451\) − 24.0000i − 1.13012i
\(452\) 0 0
\(453\) 8.00000i 0.375873i
\(454\) 0 0
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) − 2.00000i − 0.0933520i
\(460\) 0 0
\(461\) − 22.0000i − 1.02464i −0.858794 0.512321i \(-0.828786\pi\)
0.858794 0.512321i \(-0.171214\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 28.0000i − 1.29569i −0.761774 0.647843i \(-0.775671\pi\)
0.761774 0.647843i \(-0.224329\pi\)
\(468\) 0 0
\(469\) − 4.00000i − 0.184703i
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 4.00000i 0.183533i
\(476\) 0 0
\(477\) − 6.00000i − 0.274721i
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −60.0000 −2.73576
\(482\) 0 0
\(483\) − 8.00000i − 0.364013i
\(484\) 0 0
\(485\) 28.0000i 1.27141i
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 12.0000i 0.541552i 0.962642 + 0.270776i \(0.0872803\pi\)
−0.962642 + 0.270776i \(0.912720\pi\)
\(492\) 0 0
\(493\) 4.00000i 0.180151i
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 44.0000i 1.96971i 0.173379 + 0.984855i \(0.444532\pi\)
−0.173379 + 0.984855i \(0.555468\pi\)
\(500\) 0 0
\(501\) 8.00000i 0.357414i
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) − 23.0000i − 1.02147i
\(508\) 0 0
\(509\) − 6.00000i − 0.265945i −0.991120 0.132973i \(-0.957548\pi\)
0.991120 0.132973i \(-0.0424523\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) 16.0000i 0.705044i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 22.0000 0.965693
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 0 0
\(525\) 1.00000i 0.0436436i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) − 4.00000i − 0.173585i
\(532\) 0 0
\(533\) − 36.0000i − 1.55933i
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) − 4.00000i − 0.172292i
\(540\) 0 0
\(541\) − 30.0000i − 1.28980i −0.764267 0.644900i \(-0.776899\pi\)
0.764267 0.644900i \(-0.223101\pi\)
\(542\) 0 0
\(543\) 18.0000 0.772454
\(544\) 0 0
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) 12.0000i 0.513083i 0.966533 + 0.256541i \(0.0825830\pi\)
−0.966533 + 0.256541i \(0.917417\pi\)
\(548\) 0 0
\(549\) 6.00000i 0.256074i
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) − 20.0000i − 0.848953i
\(556\) 0 0
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) − 44.0000i − 1.85438i −0.374593 0.927189i \(-0.622217\pi\)
0.374593 0.927189i \(-0.377783\pi\)
\(564\) 0 0
\(565\) 28.0000i 1.17797i
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 12.0000i 0.502184i 0.967963 + 0.251092i \(0.0807897\pi\)
−0.967963 + 0.251092i \(0.919210\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 0 0
\(579\) 2.00000i 0.0831172i
\(580\) 0 0
\(581\) 4.00000i 0.165948i
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 0 0
\(585\) 12.0000 0.496139
\(586\) 0 0
\(587\) − 28.0000i − 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) − 4.00000i − 0.163984i
\(596\) 0 0
\(597\) − 8.00000i − 0.327418i
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) 10.0000i 0.406558i
\(606\) 0 0
\(607\) 48.0000 1.94826 0.974130 0.225989i \(-0.0725612\pi\)
0.974130 + 0.225989i \(0.0725612\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 42.0000i − 1.69636i −0.529705 0.848182i \(-0.677697\pi\)
0.529705 0.848182i \(-0.322303\pi\)
\(614\) 0 0
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) − 44.0000i − 1.76851i −0.467005 0.884255i \(-0.654667\pi\)
0.467005 0.884255i \(-0.345333\pi\)
\(620\) 0 0
\(621\) 8.00000i 0.321029i
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 16.0000i 0.638978i
\(628\) 0 0
\(629\) − 20.0000i − 0.797452i
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 6.00000i − 0.237729i
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 0 0
\(645\) − 8.00000i − 0.315000i
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.0000i 0.704394i 0.935926 + 0.352197i \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) 0 0
\(655\) 40.0000 1.56293
\(656\) 0 0
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) 28.0000i 1.09073i 0.838200 + 0.545363i \(0.183608\pi\)
−0.838200 + 0.545363i \(0.816392\pi\)
\(660\) 0 0
\(661\) − 2.00000i − 0.0777910i −0.999243 0.0388955i \(-0.987616\pi\)
0.999243 0.0388955i \(-0.0123839\pi\)
\(662\) 0 0
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) − 16.0000i − 0.619522i
\(668\) 0 0
\(669\) − 16.0000i − 0.618596i
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 0 0
\(675\) − 1.00000i − 0.0384900i
\(676\) 0 0
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 20.0000i 0.764161i
\(686\) 0 0
\(687\) 2.00000 0.0763048
\(688\) 0 0
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) 4.00000i 0.152167i 0.997101 + 0.0760836i \(0.0242416\pi\)
−0.997101 + 0.0760836i \(0.975758\pi\)
\(692\) 0 0
\(693\) 4.00000i 0.151947i
\(694\) 0 0
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) 22.0000i 0.832116i
\(700\) 0 0
\(701\) 2.00000i 0.0755390i 0.999286 + 0.0377695i \(0.0120253\pi\)
−0.999286 + 0.0377695i \(0.987975\pi\)
\(702\) 0 0
\(703\) 40.0000 1.50863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 2.00000i − 0.0752177i
\(708\) 0 0
\(709\) − 10.0000i − 0.375558i −0.982211 0.187779i \(-0.939871\pi\)
0.982211 0.187779i \(-0.0601289\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 48.0000i 1.79510i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 2.00000i 0.0743808i
\(724\) 0 0
\(725\) 2.00000i 0.0742781i
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 8.00000i − 0.295891i
\(732\) 0 0
\(733\) − 6.00000i − 0.221615i −0.993842 0.110808i \(-0.964656\pi\)
0.993842 0.110808i \(-0.0353437\pi\)
\(734\) 0 0
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 12.0000i 0.441427i 0.975339 + 0.220714i \(0.0708386\pi\)
−0.975339 + 0.220714i \(0.929161\pi\)
\(740\) 0 0
\(741\) 24.0000i 0.881662i
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) 0 0
\(747\) − 4.00000i − 0.146352i
\(748\) 0 0
\(749\) 12.0000i 0.438470i
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) − 16.0000i − 0.582300i
\(756\) 0 0
\(757\) 6.00000i 0.218074i 0.994038 + 0.109037i \(0.0347767\pi\)
−0.994038 + 0.109037i \(0.965223\pi\)
\(758\) 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
\(763\) 2.00000i 0.0724049i
\(764\) 0 0
\(765\) 4.00000i 0.144620i
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) − 30.0000i − 1.08042i
\(772\) 0 0
\(773\) − 2.00000i − 0.0719350i −0.999353 0.0359675i \(-0.988549\pi\)
0.999353 0.0359675i \(-0.0114513\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 10.0000 0.358748
\(778\) 0 0
\(779\) 24.0000i 0.859889i
\(780\) 0 0
\(781\) 32.0000i 1.14505i
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) 20.0000 0.713831
\(786\) 0 0
\(787\) 36.0000i 1.28326i 0.767014 + 0.641631i \(0.221742\pi\)
−0.767014 + 0.641631i \(0.778258\pi\)
\(788\) 0 0
\(789\) 24.0000i 0.854423i
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) −36.0000 −1.27840
\(794\) 0 0
\(795\) 12.0000i 0.425596i
\(796\) 0 0
\(797\) − 6.00000i − 0.212531i −0.994338 0.106265i \(-0.966111\pi\)
0.994338 0.106265i \(-0.0338893\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 40.0000i 1.41157i
\(804\) 0 0
\(805\) 16.0000i 0.563926i
\(806\) 0 0
\(807\) 22.0000 0.774437
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) − 44.0000i − 1.54505i −0.634985 0.772524i \(-0.718994\pi\)
0.634985 0.772524i \(-0.281006\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −40.0000 −1.40114
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 6.00000i 0.209657i
\(820\) 0 0
\(821\) 38.0000i 1.32621i 0.748527 + 0.663105i \(0.230762\pi\)
−0.748527 + 0.663105i \(0.769238\pi\)
\(822\) 0 0
\(823\) 56.0000 1.95204 0.976019 0.217687i \(-0.0698512\pi\)
0.976019 + 0.217687i \(0.0698512\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) − 36.0000i − 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) 0 0
\(829\) 26.0000i 0.903017i 0.892267 + 0.451509i \(0.149114\pi\)
−0.892267 + 0.451509i \(0.850886\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) − 16.0000i − 0.553703i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −56.0000 −1.93333 −0.966667 0.256036i \(-0.917584\pi\)
−0.966667 + 0.256036i \(0.917584\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) − 26.0000i − 0.895488i
\(844\) 0 0
\(845\) 46.0000i 1.58245i
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 80.0000i 2.74236i
\(852\) 0 0
\(853\) 14.0000i 0.479351i 0.970853 + 0.239675i \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) 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
\(859\) 20.0000i 0.682391i 0.939992 + 0.341196i \(0.110832\pi\)
−0.939992 + 0.341196i \(0.889168\pi\)
\(860\) 0 0
\(861\) 6.00000i 0.204479i
\(862\) 0 0
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 0 0
\(865\) −44.0000 −1.49604
\(866\) 0 0
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) − 12.0000i − 0.405674i
\(876\) 0 0
\(877\) 2.00000i 0.0675352i 0.999430 + 0.0337676i \(0.0107506\pi\)
−0.999430 + 0.0337676i \(0.989249\pi\)
\(878\) 0 0
\(879\) −30.0000 −1.01187
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) − 20.0000i − 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) 0 0
\(885\) 8.00000i 0.268917i
\(886\) 0 0
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 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
\(897\) −48.0000 −1.60267
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000i 0.399778i
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) −36.0000 −1.19668
\(906\) 0 0
\(907\) 12.0000i 0.398453i 0.979953 + 0.199227i \(0.0638430\pi\)
−0.979953 + 0.199227i \(0.936157\pi\)
\(908\) 0 0
\(909\) 2.00000i 0.0663358i
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 0 0
\(915\) − 12.0000i − 0.396708i
\(916\) 0 0
\(917\) 20.0000i 0.660458i
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) 0 0
\(923\) 48.0000i 1.57994i
\(924\) 0 0
\(925\) − 10.0000i − 0.328798i
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 4.00000i 0.131095i
\(932\) 0 0
\(933\) 8.00000i 0.261908i
\(934\) 0 0
\(935\) −16.0000 −0.523256
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) − 10.0000i − 0.326338i
\(940\) 0 0
\(941\) 26.0000i 0.847576i 0.905761 + 0.423788i \(0.139300\pi\)
−0.905761 + 0.423788i \(0.860700\pi\)
\(942\) 0 0
\(943\) −48.0000 −1.56310
\(944\) 0 0
\(945\) −2.00000 −0.0650600
\(946\) 0 0
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) 0 0
\(949\) 60.0000i 1.94768i
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 8.00000i 0.258603i
\(958\) 0 0
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 12.0000i − 0.386695i
\(964\) 0 0
\(965\) − 4.00000i − 0.128765i
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) − 12.0000i − 0.385098i −0.981287 0.192549i \(-0.938325\pi\)
0.981287 0.192549i \(-0.0616755\pi\)
\(972\) 0 0
\(973\) 4.00000i 0.128234i
\(974\) 0 0
\(975\) 6.00000 0.192154
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 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
\(985\) −20.0000 −0.637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0000i 1.01754i
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 16.0000i 0.507234i
\(996\) 0 0
\(997\) 14.0000i 0.443384i 0.975117 + 0.221692i \(0.0711580\pi\)
−0.975117 + 0.221692i \(0.928842\pi\)
\(998\) 0 0
\(999\) −10.0000 −0.316386
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 5376.2.c.bc.2689.2 2
4.3 odd 2 5376.2.c.e.2689.1 2
8.3 odd 2 5376.2.c.e.2689.2 2
8.5 even 2 inner 5376.2.c.bc.2689.1 2
16.3 odd 4 336.2.a.d.1.1 1
16.5 even 4 1344.2.a.q.1.1 1
16.11 odd 4 1344.2.a.i.1.1 1
16.13 even 4 42.2.a.a.1.1 1
48.5 odd 4 4032.2.a.e.1.1 1
48.11 even 4 4032.2.a.m.1.1 1
48.29 odd 4 126.2.a.a.1.1 1
48.35 even 4 1008.2.a.j.1.1 1
80.13 odd 4 1050.2.g.a.799.1 2
80.19 odd 4 8400.2.a.k.1.1 1
80.29 even 4 1050.2.a.i.1.1 1
80.77 odd 4 1050.2.g.a.799.2 2
112.3 even 12 2352.2.q.n.961.1 2
112.13 odd 4 294.2.a.g.1.1 1
112.19 even 12 2352.2.q.n.1537.1 2
112.27 even 4 9408.2.a.bw.1.1 1
112.45 odd 12 294.2.e.a.79.1 2
112.51 odd 12 2352.2.q.i.1537.1 2
112.61 odd 12 294.2.e.a.67.1 2
112.67 odd 12 2352.2.q.i.961.1 2
112.69 odd 4 9408.2.a.n.1.1 1
112.83 even 4 2352.2.a.l.1.1 1
112.93 even 12 294.2.e.c.67.1 2
112.109 even 12 294.2.e.c.79.1 2
144.13 even 12 1134.2.f.g.379.1 2
144.29 odd 12 1134.2.f.j.757.1 2
144.61 even 12 1134.2.f.g.757.1 2
144.77 odd 12 1134.2.f.j.379.1 2
176.109 odd 4 5082.2.a.d.1.1 1
208.77 even 4 7098.2.a.f.1.1 1
240.29 odd 4 3150.2.a.bo.1.1 1
240.77 even 4 3150.2.g.r.2899.1 2
240.173 even 4 3150.2.g.r.2899.2 2
336.83 odd 4 7056.2.a.k.1.1 1
336.125 even 4 882.2.a.b.1.1 1
336.173 even 12 882.2.g.j.361.1 2
336.221 odd 12 882.2.g.h.667.1 2
336.269 even 12 882.2.g.j.667.1 2
336.317 odd 12 882.2.g.h.361.1 2
560.349 odd 4 7350.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 1 16.13 even 4
126.2.a.a.1.1 1 48.29 odd 4
294.2.a.g.1.1 1 112.13 odd 4
294.2.e.a.67.1 2 112.61 odd 12
294.2.e.a.79.1 2 112.45 odd 12
294.2.e.c.67.1 2 112.93 even 12
294.2.e.c.79.1 2 112.109 even 12
336.2.a.d.1.1 1 16.3 odd 4
882.2.a.b.1.1 1 336.125 even 4
882.2.g.h.361.1 2 336.317 odd 12
882.2.g.h.667.1 2 336.221 odd 12
882.2.g.j.361.1 2 336.173 even 12
882.2.g.j.667.1 2 336.269 even 12
1008.2.a.j.1.1 1 48.35 even 4
1050.2.a.i.1.1 1 80.29 even 4
1050.2.g.a.799.1 2 80.13 odd 4
1050.2.g.a.799.2 2 80.77 odd 4
1134.2.f.g.379.1 2 144.13 even 12
1134.2.f.g.757.1 2 144.61 even 12
1134.2.f.j.379.1 2 144.77 odd 12
1134.2.f.j.757.1 2 144.29 odd 12
1344.2.a.i.1.1 1 16.11 odd 4
1344.2.a.q.1.1 1 16.5 even 4
2352.2.a.l.1.1 1 112.83 even 4
2352.2.q.i.961.1 2 112.67 odd 12
2352.2.q.i.1537.1 2 112.51 odd 12
2352.2.q.n.961.1 2 112.3 even 12
2352.2.q.n.1537.1 2 112.19 even 12
3150.2.a.bo.1.1 1 240.29 odd 4
3150.2.g.r.2899.1 2 240.77 even 4
3150.2.g.r.2899.2 2 240.173 even 4
4032.2.a.e.1.1 1 48.5 odd 4
4032.2.a.m.1.1 1 48.11 even 4
5082.2.a.d.1.1 1 176.109 odd 4
5376.2.c.e.2689.1 2 4.3 odd 2
5376.2.c.e.2689.2 2 8.3 odd 2
5376.2.c.bc.2689.1 2 8.5 even 2 inner
5376.2.c.bc.2689.2 2 1.1 even 1 trivial
7056.2.a.k.1.1 1 336.83 odd 4
7098.2.a.f.1.1 1 208.77 even 4
7350.2.a.f.1.1 1 560.349 odd 4
8400.2.a.k.1.1 1 80.19 odd 4
9408.2.a.n.1.1 1 112.69 odd 4
9408.2.a.bw.1.1 1 112.27 even 4