Properties

Label 1280.2.d.j.641.1
Level $1280$
Weight $2$
Character 1280.641
Analytic conductor $10.221$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 641.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1280.641
Dual form 1280.2.d.j.641.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{5} +4.00000 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{5} +4.00000 q^{7} +3.00000 q^{9} -4.00000i q^{11} -2.00000i q^{13} +2.00000 q^{17} +4.00000i q^{19} -4.00000 q^{23} -1.00000 q^{25} -2.00000i q^{29} -8.00000 q^{31} -4.00000i q^{35} -6.00000i q^{37} +6.00000 q^{41} +8.00000i q^{43} -3.00000i q^{45} +4.00000 q^{47} +9.00000 q^{49} -6.00000i q^{53} -4.00000 q^{55} +4.00000i q^{59} -2.00000i q^{61} +12.0000 q^{63} -2.00000 q^{65} +8.00000i q^{67} +6.00000 q^{73} -16.0000i q^{77} +9.00000 q^{81} -16.0000i q^{83} -2.00000i q^{85} +6.00000 q^{89} -8.00000i q^{91} +4.00000 q^{95} -14.0000 q^{97} -12.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{7} + 6q^{9} + O(q^{10}) \) \( 2q + 8q^{7} + 6q^{9} + 4q^{17} - 8q^{23} - 2q^{25} - 16q^{31} + 12q^{41} + 8q^{47} + 18q^{49} - 8q^{55} + 24q^{63} - 4q^{65} + 12q^{73} + 18q^{81} + 12q^{89} + 8q^{95} - 28q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 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
\(15\) 0 0
\(16\) 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
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 2.00000i − 0.371391i −0.982607 0.185695i \(-0.940546\pi\)
0.982607 0.185695i \(-0.0594537\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 4.00000i − 0.676123i
\(36\) 0 0
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) − 3.00000i − 0.447214i
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) − 2.00000i − 0.256074i −0.991769 0.128037i \(-0.959132\pi\)
0.991769 0.128037i \(-0.0408676\pi\)
\(62\) 0 0
\(63\) 12.0000 1.51186
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 16.0000i − 1.82337i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 16.0000i − 1.75623i −0.478451 0.878114i \(-0.658802\pi\)
0.478451 0.878114i \(-0.341198\pi\)
\(84\) 0 0
\(85\) − 2.00000i − 0.216930i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) − 8.00000i − 0.838628i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) − 12.0000i − 1.20605i
\(100\) 0 0
\(101\) − 6.00000i − 0.597022i −0.954406 0.298511i \(-0.903510\pi\)
0.954406 0.298511i \(-0.0964900\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 14.0000i 1.34096i 0.741929 + 0.670478i \(0.233911\pi\)
−0.741929 + 0.670478i \(0.766089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 4.00000i 0.373002i
\(116\) 0 0
\(117\) − 6.00000i − 0.554700i
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000i 1.04844i 0.851581 + 0.524222i \(0.175644\pi\)
−0.851581 + 0.524222i \(0.824356\pi\)
\(132\) 0 0
\(133\) 16.0000i 1.38738i
\(134\) 0 0
\(135\) 0 0
\(136\) 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
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.0000i 0.819232i 0.912258 + 0.409616i \(0.134337\pi\)
−0.912258 + 0.409616i \(0.865663\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 8.00000i 0.642575i
\(156\) 0 0
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 12.0000i 0.917663i
\(172\) 0 0
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 20.0000i 1.49487i 0.664335 + 0.747435i \(0.268715\pi\)
−0.664335 + 0.747435i \(0.731285\pi\)
\(180\) 0 0
\(181\) 10.0000i 0.743294i 0.928374 + 0.371647i \(0.121207\pi\)
−0.928374 + 0.371647i \(0.878793\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) − 8.00000i − 0.585018i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 22.0000i − 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 8.00000i − 0.561490i
\(204\) 0 0
\(205\) − 6.00000i − 0.419058i
\(206\) 0 0
\(207\) −12.0000 −0.834058
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) − 4.00000i − 0.275371i −0.990476 0.137686i \(-0.956034\pi\)
0.990476 0.137686i \(-0.0439664\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) −32.0000 −2.17230
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 4.00000i − 0.269069i
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) − 24.0000i − 1.59294i −0.604681 0.796468i \(-0.706699\pi\)
0.604681 0.796468i \(-0.293301\pi\)
\(228\) 0 0
\(229\) 26.0000i 1.71813i 0.511868 + 0.859064i \(0.328954\pi\)
−0.511868 + 0.859064i \(0.671046\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) − 4.00000i − 0.260931i
\(236\) 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
\(243\) 0 0
\(244\) 0 0
\(245\) − 9.00000i − 0.574989i
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000i 0.757433i 0.925513 + 0.378717i \(0.123635\pi\)
−0.925513 + 0.378717i \(0.876365\pi\)
\(252\) 0 0
\(253\) 16.0000i 1.00591i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) − 24.0000i − 1.49129i
\(260\) 0 0
\(261\) − 6.00000i − 0.371391i
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 14.0000i 0.853595i 0.904347 + 0.426798i \(0.140358\pi\)
−0.904347 + 0.426798i \(0.859642\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.00000i 0.241209i
\(276\) 0 0
\(277\) 10.0000i 0.600842i 0.953807 + 0.300421i \(0.0971271\pi\)
−0.953807 + 0.300421i \(0.902873\pi\)
\(278\) 0 0
\(279\) −24.0000 −1.43684
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) − 8.00000i − 0.475551i −0.971320 0.237775i \(-0.923582\pi\)
0.971320 0.237775i \(-0.0764182\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 26.0000i 1.51894i 0.650545 + 0.759468i \(0.274541\pi\)
−0.650545 + 0.759468i \(0.725459\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.00000i 0.462652i
\(300\) 0 0
\(301\) 32.0000i 1.84445i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) − 8.00000i − 0.456584i −0.973593 0.228292i \(-0.926686\pi\)
0.973593 0.228292i \(-0.0733141\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) 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
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) 2.00000i 0.110940i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 12.0000i 0.659580i 0.944054 + 0.329790i \(0.106978\pi\)
−0.944054 + 0.329790i \(0.893022\pi\)
\(332\) 0 0
\(333\) − 18.0000i − 0.986394i
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 32.0000i 1.73290i
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.0000i 0.858925i 0.903085 + 0.429463i \(0.141297\pi\)
−0.903085 + 0.429463i \(0.858703\pi\)
\(348\) 0 0
\(349\) 30.0000i 1.60586i 0.596071 + 0.802932i \(0.296728\pi\)
−0.596071 + 0.802932i \(0.703272\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(363\) 0 0
\(364\) 0 0
\(365\) − 6.00000i − 0.314054i
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) − 24.0000i − 1.24602i
\(372\) 0 0
\(373\) − 22.0000i − 1.13912i −0.821951 0.569558i \(-0.807114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) 20.0000i 1.02733i 0.857991 + 0.513665i \(0.171713\pi\)
−0.857991 + 0.513665i \(0.828287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) 0 0
\(387\) 24.0000i 1.21999i
\(388\) 0 0
\(389\) − 6.00000i − 0.304212i −0.988364 0.152106i \(-0.951394\pi\)
0.988364 0.152106i \(-0.0486055\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 2.00000i − 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 16.0000i 0.797017i
\(404\) 0 0
\(405\) − 9.00000i − 0.447214i
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 16.0000i 0.787309i
\(414\) 0 0
\(415\) −16.0000 −0.785409
\(416\) 0 0
\(417\) 0 0
\(418\) 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 \(-0.953292\pi\)
0.989253 0.146211i \(-0.0467079\pi\)
\(422\) 0 0
\(423\) 12.0000 0.583460
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) − 8.00000i − 0.387147i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −40.0000 −1.92673 −0.963366 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 16.0000i − 0.765384i
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 27.0000 1.28571
\(442\) 0 0
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) 0 0
\(445\) − 6.00000i − 0.284427i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) − 24.0000i − 1.13012i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 18.0000i − 0.838344i −0.907907 0.419172i \(-0.862320\pi\)
0.907907 0.419172i \(-0.137680\pi\)
\(462\) 0 0
\(463\) 12.0000 0.557687 0.278844 0.960337i \(-0.410049\pi\)
0.278844 + 0.960337i \(0.410049\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.00000i 0.370196i 0.982720 + 0.185098i \(0.0592602\pi\)
−0.982720 + 0.185098i \(0.940740\pi\)
\(468\) 0 0
\(469\) 32.0000i 1.47762i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 32.0000 1.47136
\(474\) 0 0
\(475\) − 4.00000i − 0.183533i
\(476\) 0 0
\(477\) − 18.0000i − 0.824163i
\(478\) 0 0
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 14.0000i 0.635707i
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 36.0000i − 1.62466i −0.583200 0.812329i \(-0.698200\pi\)
0.583200 0.812329i \(-0.301800\pi\)
\(492\) 0 0
\(493\) − 4.00000i − 0.180151i
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 28.0000i − 1.25345i −0.779240 0.626726i \(-0.784395\pi\)
0.779240 0.626726i \(-0.215605\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 2.00000i − 0.0886484i −0.999017 0.0443242i \(-0.985887\pi\)
0.999017 0.0443242i \(-0.0141135\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.00000i 0.176261i
\(516\) 0 0
\(517\) − 16.0000i − 0.703679i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) 8.00000i 0.349816i 0.984585 + 0.174908i \(0.0559627\pi\)
−0.984585 + 0.174908i \(0.944037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 12.0000i 0.520756i
\(532\) 0 0
\(533\) − 12.0000i − 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 36.0000i − 1.55063i
\(540\) 0 0
\(541\) − 2.00000i − 0.0859867i −0.999075 0.0429934i \(-0.986311\pi\)
0.999075 0.0429934i \(-0.0136894\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 0 0
\(549\) − 6.00000i − 0.256074i
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.0000i 0.593199i 0.955002 + 0.296600i \(0.0958526\pi\)
−0.955002 + 0.296600i \(0.904147\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16.0000i 0.674320i 0.941447 + 0.337160i \(0.109466\pi\)
−0.941447 + 0.337160i \(0.890534\pi\)
\(564\) 0 0
\(565\) − 18.0000i − 0.757266i
\(566\) 0 0
\(567\) 36.0000 1.51186
\(568\) 0 0
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) − 4.00000i − 0.167395i −0.996491 0.0836974i \(-0.973327\pi\)
0.996491 0.0836974i \(-0.0266729\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 64.0000i − 2.65517i
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) 48.0000i 1.98117i 0.136892 + 0.990586i \(0.456289\pi\)
−0.136892 + 0.990586i \(0.543711\pi\)
\(588\) 0 0
\(589\) − 32.0000i − 1.31854i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) − 8.00000i − 0.327968i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 24.0000i 0.977356i
\(604\) 0 0
\(605\) 5.00000i 0.203279i
\(606\) 0 0
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 8.00000i − 0.323645i
\(612\) 0 0
\(613\) 42.0000i 1.69636i 0.529705 + 0.848182i \(0.322303\pi\)
−0.529705 + 0.848182i \(0.677697\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 4.00000i 0.160774i 0.996764 + 0.0803868i \(0.0256155\pi\)
−0.996764 + 0.0803868i \(0.974384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 0 0
\(635\) 12.0000i 0.476205i
\(636\) 0 0
\(637\) − 18.0000i − 0.713186i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) − 48.0000i − 1.89294i −0.322799 0.946468i \(-0.604624\pi\)
0.322799 0.946468i \(-0.395376\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 34.0000i − 1.33052i −0.746611 0.665261i \(-0.768320\pi\)
0.746611 0.665261i \(-0.231680\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) 18.0000 0.702247
\(658\) 0 0
\(659\) − 12.0000i − 0.467454i −0.972302 0.233727i \(-0.924908\pi\)
0.972302 0.233727i \(-0.0750921\pi\)
\(660\) 0 0
\(661\) 42.0000i 1.63361i 0.576913 + 0.816805i \(0.304257\pi\)
−0.576913 + 0.816805i \(0.695743\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) 8.00000i 0.309761i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 22.0000i − 0.845529i −0.906240 0.422764i \(-0.861060\pi\)
0.906240 0.422764i \(-0.138940\pi\)
\(678\) 0 0
\(679\) −56.0000 −2.14908
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 24.0000i − 0.918334i −0.888350 0.459167i \(-0.848148\pi\)
0.888350 0.459167i \(-0.151852\pi\)
\(684\) 0 0
\(685\) 10.0000i 0.382080i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 44.0000i 1.67384i 0.547326 + 0.836919i \(0.315646\pi\)
−0.547326 + 0.836919i \(0.684354\pi\)
\(692\) 0 0
\(693\) − 48.0000i − 1.82337i
\(694\) 0 0
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 34.0000i − 1.28416i −0.766637 0.642081i \(-0.778071\pi\)
0.766637 0.642081i \(-0.221929\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 24.0000i − 0.902613i
\(708\) 0 0
\(709\) − 38.0000i − 1.42712i −0.700594 0.713560i \(-0.747082\pi\)
0.700594 0.713560i \(-0.252918\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 8.00000i 0.299183i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −32.0000 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.00000i 0.0742781i
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 16.0000i 0.591781i
\(732\) 0 0
\(733\) 30.0000i 1.10808i 0.832492 + 0.554038i \(0.186914\pi\)
−0.832492 + 0.554038i \(0.813086\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 32.0000 1.17874
\(738\) 0 0
\(739\) 36.0000i 1.32428i 0.749380 + 0.662141i \(0.230352\pi\)
−0.749380 + 0.662141i \(0.769648\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 0 0
\(747\) − 48.0000i − 1.75623i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 16.0000i − 0.582300i
\(756\) 0 0
\(757\) 10.0000i 0.363456i 0.983349 + 0.181728i \(0.0581691\pi\)
−0.983349 + 0.181728i \(0.941831\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 56.0000i 2.02734i
\(764\) 0 0
\(765\) − 6.00000i − 0.216930i
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 42.0000i 1.51064i 0.655359 + 0.755318i \(0.272517\pi\)
−0.655359 + 0.755318i \(0.727483\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.0000i 0.859889i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 8.00000i 0.285169i 0.989783 + 0.142585i \(0.0455413\pi\)
−0.989783 + 0.142585i \(0.954459\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 72.0000 2.56003
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 18.0000i − 0.637593i −0.947823 0.318796i \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) − 24.0000i − 0.846942i
\(804\) 0 0
\(805\) 16.0000i 0.563926i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) 28.0000i 0.983213i 0.870817 + 0.491606i \(0.163590\pi\)
−0.870817 + 0.491606i \(0.836410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 0 0
\(819\) − 24.0000i − 0.838628i
\(820\) 0 0
\(821\) − 22.0000i − 0.767805i −0.923374 0.383903i \(-0.874580\pi\)
0.923374 0.383903i \(-0.125420\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 48.0000i − 1.66912i −0.550914 0.834562i \(-0.685721\pi\)
0.550914 0.834562i \(-0.314279\pi\)
\(828\) 0 0
\(829\) − 2.00000i − 0.0694629i −0.999397 0.0347314i \(-0.988942\pi\)
0.999397 0.0347314i \(-0.0110576\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 12.0000i 0.415277i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 9.00000i − 0.309609i
\(846\) 0 0
\(847\) −20.0000 −0.687208
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 24.0000i 0.822709i
\(852\) 0 0
\(853\) − 38.0000i − 1.30110i −0.759465 0.650548i \(-0.774539\pi\)
0.759465 0.650548i \(-0.225461\pi\)
\(854\) 0 0
\(855\) 12.0000 0.410391
\(856\) 0 0
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) 4.00000i 0.136478i 0.997669 + 0.0682391i \(0.0217381\pi\)
−0.997669 + 0.0682391i \(0.978262\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 0 0
\(873\) −42.0000 −1.42148
\(874\) 0 0
\(875\) 4.00000i 0.135225i
\(876\) 0 0
\(877\) 46.0000i 1.55331i 0.629926 + 0.776655i \(0.283085\pi\)
−0.629926 + 0.776655i \(0.716915\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) − 16.0000i − 0.538443i −0.963078 0.269221i \(-0.913234\pi\)
0.963078 0.269221i \(-0.0867663\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 20.0000 0.671534 0.335767 0.941945i \(-0.391004\pi\)
0.335767 + 0.941945i \(0.391004\pi\)
\(888\) 0 0
\(889\) −48.0000 −1.60987
\(890\) 0 0
\(891\) − 36.0000i − 1.20605i
\(892\) 0 0
\(893\) 16.0000i 0.535420i
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 16.0000i 0.533630i
\(900\) 0 0
\(901\) − 12.0000i − 0.399778i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) − 48.0000i − 1.59381i −0.604102 0.796907i \(-0.706468\pi\)
0.604102 0.796907i \(-0.293532\pi\)
\(908\) 0 0
\(909\) − 18.0000i − 0.597022i
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) −64.0000 −2.11809
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 48.0000i 1.58510i
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 6.00000i 0.197279i
\(926\) 0 0
\(927\) −12.0000 −0.394132
\(928\) 0 0
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) 36.0000i 1.17985i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 46.0000i 1.49956i 0.661689 + 0.749779i \(0.269840\pi\)
−0.661689 + 0.749779i \(0.730160\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 8.00000i − 0.259965i −0.991516 0.129983i \(-0.958508\pi\)
0.991516 0.129983i \(-0.0414921\pi\)
\(948\) 0 0
\(949\) − 12.0000i − 0.389536i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) − 8.00000i − 0.258874i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −40.0000 −1.29167
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 14.0000i 0.450676i
\(966\) 0 0
\(967\) 36.0000 1.15768 0.578841 0.815440i \(-0.303505\pi\)
0.578841 + 0.815440i \(0.303505\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 20.0000i − 0.641831i −0.947108 0.320915i \(-0.896010\pi\)
0.947108 0.320915i \(-0.103990\pi\)
\(972\) 0 0
\(973\) − 48.0000i − 1.53881i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 0 0
\(979\) − 24.0000i − 0.767043i
\(980\) 0 0
\(981\) 42.0000i 1.34096i
\(982\) 0 0
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) −22.0000 −0.700978
\(986\) 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 \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.00000i 0.253617i
\(996\) 0 0
\(997\) 42.0000i 1.33015i 0.746775 + 0.665077i \(0.231601\pi\)
−0.746775 + 0.665077i \(0.768399\pi\)
\(998\) 0 0
\(999\) 0 0
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 1280.2.d.j.641.1 2
4.3 odd 2 1280.2.d.a.641.1 2
8.3 odd 2 1280.2.d.a.641.2 2
8.5 even 2 inner 1280.2.d.j.641.2 2
16.3 odd 4 320.2.a.d.1.1 1
16.5 even 4 40.2.a.a.1.1 1
16.11 odd 4 80.2.a.a.1.1 1
16.13 even 4 320.2.a.c.1.1 1
48.5 odd 4 360.2.a.a.1.1 1
48.11 even 4 720.2.a.e.1.1 1
48.29 odd 4 2880.2.a.t.1.1 1
48.35 even 4 2880.2.a.bg.1.1 1
80.3 even 4 1600.2.c.m.449.1 2
80.13 odd 4 1600.2.c.k.449.2 2
80.19 odd 4 1600.2.a.k.1.1 1
80.27 even 4 400.2.c.d.49.2 2
80.29 even 4 1600.2.a.o.1.1 1
80.37 odd 4 200.2.c.b.49.1 2
80.43 even 4 400.2.c.d.49.1 2
80.53 odd 4 200.2.c.b.49.2 2
80.59 odd 4 400.2.a.e.1.1 1
80.67 even 4 1600.2.c.m.449.2 2
80.69 even 4 200.2.a.c.1.1 1
80.77 odd 4 1600.2.c.k.449.1 2
112.5 odd 12 1960.2.q.i.361.1 2
112.27 even 4 3920.2.a.s.1.1 1
112.37 even 12 1960.2.q.h.361.1 2
112.53 even 12 1960.2.q.h.961.1 2
112.69 odd 4 1960.2.a.g.1.1 1
112.101 odd 12 1960.2.q.i.961.1 2
144.5 odd 12 3240.2.q.x.2161.1 2
144.85 even 12 3240.2.q.k.2161.1 2
144.101 odd 12 3240.2.q.x.1081.1 2
144.133 even 12 3240.2.q.k.1081.1 2
176.21 odd 4 4840.2.a.f.1.1 1
176.43 even 4 9680.2.a.q.1.1 1
208.181 even 4 6760.2.a.i.1.1 1
240.53 even 4 1800.2.f.a.649.2 2
240.59 even 4 3600.2.a.h.1.1 1
240.107 odd 4 3600.2.f.t.2449.2 2
240.149 odd 4 1800.2.a.v.1.1 1
240.197 even 4 1800.2.f.a.649.1 2
240.203 odd 4 3600.2.f.t.2449.1 2
560.69 odd 4 9800.2.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.a.a.1.1 1 16.5 even 4
80.2.a.a.1.1 1 16.11 odd 4
200.2.a.c.1.1 1 80.69 even 4
200.2.c.b.49.1 2 80.37 odd 4
200.2.c.b.49.2 2 80.53 odd 4
320.2.a.c.1.1 1 16.13 even 4
320.2.a.d.1.1 1 16.3 odd 4
360.2.a.a.1.1 1 48.5 odd 4
400.2.a.e.1.1 1 80.59 odd 4
400.2.c.d.49.1 2 80.43 even 4
400.2.c.d.49.2 2 80.27 even 4
720.2.a.e.1.1 1 48.11 even 4
1280.2.d.a.641.1 2 4.3 odd 2
1280.2.d.a.641.2 2 8.3 odd 2
1280.2.d.j.641.1 2 1.1 even 1 trivial
1280.2.d.j.641.2 2 8.5 even 2 inner
1600.2.a.k.1.1 1 80.19 odd 4
1600.2.a.o.1.1 1 80.29 even 4
1600.2.c.k.449.1 2 80.77 odd 4
1600.2.c.k.449.2 2 80.13 odd 4
1600.2.c.m.449.1 2 80.3 even 4
1600.2.c.m.449.2 2 80.67 even 4
1800.2.a.v.1.1 1 240.149 odd 4
1800.2.f.a.649.1 2 240.197 even 4
1800.2.f.a.649.2 2 240.53 even 4
1960.2.a.g.1.1 1 112.69 odd 4
1960.2.q.h.361.1 2 112.37 even 12
1960.2.q.h.961.1 2 112.53 even 12
1960.2.q.i.361.1 2 112.5 odd 12
1960.2.q.i.961.1 2 112.101 odd 12
2880.2.a.t.1.1 1 48.29 odd 4
2880.2.a.bg.1.1 1 48.35 even 4
3240.2.q.k.1081.1 2 144.133 even 12
3240.2.q.k.2161.1 2 144.85 even 12
3240.2.q.x.1081.1 2 144.101 odd 12
3240.2.q.x.2161.1 2 144.5 odd 12
3600.2.a.h.1.1 1 240.59 even 4
3600.2.f.t.2449.1 2 240.203 odd 4
3600.2.f.t.2449.2 2 240.107 odd 4
3920.2.a.s.1.1 1 112.27 even 4
4840.2.a.f.1.1 1 176.21 odd 4
6760.2.a.i.1.1 1 208.181 even 4
9680.2.a.q.1.1 1 176.43 even 4
9800.2.a.x.1.1 1 560.69 odd 4