Properties

Label 3840.2.k.p.1921.1
Level $3840$
Weight $2$
Character 3840.1921
Analytic conductor $30.663$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.6625543762\)
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 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1921.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3840.1921
Dual form 3840.2.k.p.1921.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} +4.00000i q^{11} +2.00000i q^{13} +1.00000 q^{15} -2.00000 q^{17} -8.00000i q^{19} +4.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} -6.00000i q^{29} +4.00000 q^{33} -2.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} +4.00000i q^{43} -1.00000i q^{45} +12.0000 q^{47} -7.00000 q^{49} +2.00000i q^{51} +6.00000i q^{53} -4.00000 q^{55} -8.00000 q^{57} +12.0000i q^{59} +14.0000i q^{61} -2.00000 q^{65} +12.0000i q^{67} -4.00000i q^{69} -2.00000 q^{73} +1.00000i q^{75} +8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -2.00000i q^{85} -6.00000 q^{87} -2.00000 q^{89} +8.00000 q^{95} -14.0000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 2q^{15} - 4q^{17} + 8q^{23} - 2q^{25} + 8q^{33} + 4q^{39} + 12q^{41} + 24q^{47} - 14q^{49} - 8q^{55} - 16q^{57} - 4q^{65} - 4q^{73} + 16q^{79} + 2q^{81} - 12q^{87} - 4q^{89} + 16q^{95} - 28q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) 1.00000i 0.447214i
\(6\) 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 \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) − 8.00000i − 1.83533i −0.397360 0.917663i \(-0.630073\pi\)
0.397360 0.917663i \(-0.369927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 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 \(-0.811919\pi\)
0.830455 0.557086i \(-0.188081\pi\)
\(30\) 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
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 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 \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) − 1.00000i − 0.149071i
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 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
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) 12.0000i 1.56227i 0.624364 + 0.781133i \(0.285358\pi\)
−0.624364 + 0.781133i \(0.714642\pi\)
\(60\) 0 0
\(61\) 14.0000i 1.79252i 0.443533 + 0.896258i \(0.353725\pi\)
−0.443533 + 0.896258i \(0.646275\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 0 0
\(69\) − 4.00000i − 0.481543i
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 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 \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 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
\(85\) − 2.00000i − 0.216930i
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 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
\(101\) 14.0000i 1.39305i 0.717532 + 0.696526i \(0.245272\pi\)
−0.717532 + 0.696526i \(0.754728\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) 14.0000i 1.34096i 0.741929 + 0.670478i \(0.233911\pi\)
−0.741929 + 0.670478i \(0.766089\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 4.00000i 0.373002i
\(116\) 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
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 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
\(133\) 0 0
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 16.0000i 1.35710i 0.734553 + 0.678551i \(0.237392\pi\)
−0.734553 + 0.678551i \(0.762608\pi\)
\(140\) 0 0
\(141\) − 12.0000i − 1.01058i
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 7.00000i 0.577350i
\(148\) 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
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.0000i 1.56652i 0.621694 + 0.783260i \(0.286445\pi\)
−0.621694 + 0.783260i \(0.713555\pi\)
\(164\) 0 0
\(165\) 4.00000i 0.311400i
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 8.00000i 0.611775i
\(172\) 0 0
\(173\) − 14.0000i − 1.06440i −0.846619 0.532200i \(-0.821365\pi\)
0.846619 0.532200i \(-0.178635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) − 12.0000i − 0.896922i −0.893802 0.448461i \(-0.851972\pi\)
0.893802 0.448461i \(-0.148028\pi\)
\(180\) 0 0
\(181\) − 22.0000i − 1.63525i −0.575753 0.817624i \(-0.695291\pi\)
0.575753 0.817624i \(-0.304709\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) − 8.00000i − 0.585018i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 0 0
\(195\) 2.00000i 0.143223i
\(196\) 0 0
\(197\) − 26.0000i − 1.85242i −0.377004 0.926212i \(-0.623046\pi\)
0.377004 0.926212i \(-0.376954\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 6.00000i 0.419058i
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 32.0000 2.21349
\(210\) 0 0
\(211\) 8.00000i 0.550743i 0.961338 + 0.275371i \(0.0888008\pi\)
−0.961338 + 0.275371i \(0.911199\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2.00000i 0.135147i
\(220\) 0 0
\(221\) − 4.00000i − 0.269069i
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 0 0
\(229\) 18.0000i 1.18947i 0.803921 + 0.594737i \(0.202744\pi\)
−0.803921 + 0.594737i \(0.797256\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 12.0000i 0.782794i
\(236\) 0 0
\(237\) − 8.00000i − 0.519656i
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 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
\(245\) − 7.00000i − 0.447214i
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 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
\(255\) −2.00000 −0.125245
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000i 0.371391i
\(262\) 0 0
\(263\) 20.0000 1.23325 0.616626 0.787256i \(-0.288499\pi\)
0.616626 + 0.787256i \(0.288499\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 2.00000i 0.122398i
\(268\) 0 0
\(269\) 10.0000i 0.609711i 0.952399 + 0.304855i \(0.0986081\pi\)
−0.952399 + 0.304855i \(0.901392\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 4.00000i − 0.241209i
\(276\) 0 0
\(277\) 30.0000i 1.80253i 0.433273 + 0.901263i \(0.357359\pi\)
−0.433273 + 0.901263i \(0.642641\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) − 8.00000i − 0.473879i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 14.0000i 0.820695i
\(292\) 0 0
\(293\) − 26.0000i − 1.51894i −0.650545 0.759468i \(-0.725459\pi\)
0.650545 0.759468i \(-0.274541\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 8.00000i 0.462652i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) −14.0000 −0.801638
\(306\) 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
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 16.0000i 0.890264i
\(324\) 0 0
\(325\) − 2.00000i − 0.110940i
\(326\) 0 0
\(327\) 14.0000 0.774202
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) − 6.00000i − 0.325875i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) − 4.00000i − 0.214731i −0.994220 0.107366i \(-0.965758\pi\)
0.994220 0.107366i \(-0.0342415\pi\)
\(348\) 0 0
\(349\) − 2.00000i − 0.107058i −0.998566 0.0535288i \(-0.982953\pi\)
0.998566 0.0535288i \(-0.0170469\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) −45.0000 −2.36842
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) − 2.00000i − 0.104685i
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 22.0000i 1.13912i 0.821951 + 0.569558i \(0.192886\pi\)
−0.821951 + 0.569558i \(0.807114\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) − 16.0000i − 0.821865i −0.911666 0.410932i \(-0.865203\pi\)
0.911666 0.410932i \(-0.134797\pi\)
\(380\) 0 0
\(381\) − 16.0000i − 0.819705i
\(382\) 0 0
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 4.00000i − 0.203331i
\(388\) 0 0
\(389\) 6.00000i 0.304212i 0.988364 + 0.152106i \(0.0486055\pi\)
−0.988364 + 0.152106i \(0.951394\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) 20.0000 1.00887
\(394\) 0 0
\(395\) 8.00000i 0.402524i
\(396\) 0 0
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) − 2.00000i − 0.0986527i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) 12.0000i 0.586238i 0.956076 + 0.293119i \(0.0946933\pi\)
−0.956076 + 0.293119i \(0.905307\pi\)
\(420\) 0 0
\(421\) 2.00000i 0.0974740i 0.998812 + 0.0487370i \(0.0155196\pi\)
−0.998812 + 0.0487370i \(0.984480\pi\)
\(422\) 0 0
\(423\) −12.0000 −0.583460
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 8.00000i 0.386244i
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) − 6.00000i − 0.287678i
\(436\) 0 0
\(437\) − 32.0000i − 1.53077i
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) − 28.0000i − 1.33032i −0.746701 0.665160i \(-0.768363\pi\)
0.746701 0.665160i \(-0.231637\pi\)
\(444\) 0 0
\(445\) − 2.00000i − 0.0948091i
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 24.0000i 1.13012i
\(452\) 0 0
\(453\) − 8.00000i − 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 0 0
\(459\) − 2.00000i − 0.0933520i
\(460\) 0 0
\(461\) − 14.0000i − 0.652045i −0.945362 0.326023i \(-0.894291\pi\)
0.945362 0.326023i \(-0.105709\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 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
\(471\) −14.0000 −0.645086
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 8.00000i 0.367065i
\(476\) 0 0
\(477\) − 6.00000i − 0.274721i
\(478\) 0 0
\(479\) 40.0000 1.82765 0.913823 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 14.0000i − 0.635707i
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 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
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 8.00000i − 0.358129i −0.983837 0.179065i \(-0.942693\pi\)
0.983837 0.179065i \(-0.0573071\pi\)
\(500\) 0 0
\(501\) − 12.0000i − 0.536120i
\(502\) 0 0
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) −14.0000 −0.622992
\(506\) 0 0
\(507\) − 9.00000i − 0.399704i
\(508\) 0 0
\(509\) − 38.0000i − 1.68432i −0.539227 0.842160i \(-0.681284\pi\)
0.539227 0.842160i \(-0.318716\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 8.00000 0.353209
\(514\) 0 0
\(515\) 8.00000i 0.352522i
\(516\) 0 0
\(517\) 48.0000i 2.11104i
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 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
\(527\) 0 0
\(528\) 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
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) − 28.0000i − 1.20605i
\(540\) 0 0
\(541\) 30.0000i 1.28980i 0.764267 + 0.644900i \(0.223101\pi\)
−0.764267 + 0.644900i \(0.776899\pi\)
\(542\) 0 0
\(543\) −22.0000 −0.944110
\(544\) 0 0
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 0 0
\(549\) − 14.0000i − 0.597505i
\(550\) 0 0
\(551\) −48.0000 −2.04487
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) − 2.00000i − 0.0848953i
\(556\) 0 0
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) − 36.0000i − 1.51722i −0.651546 0.758610i \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) 0 0
\(565\) 6.00000i 0.252422i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) 32.0000i 1.33916i 0.742741 + 0.669579i \(0.233526\pi\)
−0.742741 + 0.669579i \(0.766474\pi\)
\(572\) 0 0
\(573\) 16.0000i 0.668410i
\(574\) 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
\(579\) 22.0000i 0.914289i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −26.0000 −1.06950
\(592\) 0 0
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 16.0000i − 0.654836i
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) − 12.0000i − 0.488678i
\(604\) 0 0
\(605\) − 5.00000i − 0.203279i
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000i 0.970936i
\(612\) 0 0
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) 0 0
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) − 16.0000i − 0.643094i −0.946894 0.321547i \(-0.895797\pi\)
0.946894 0.321547i \(-0.104203\pi\)
\(620\) 0 0
\(621\) 4.00000i 0.160514i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) − 32.0000i − 1.27796i
\(628\) 0 0
\(629\) 4.00000i 0.159490i
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 0 0
\(633\) 8.00000 0.317971
\(634\) 0 0
\(635\) 16.0000i 0.634941i
\(636\) 0 0
\(637\) − 14.0000i − 0.554700i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) − 12.0000i − 0.473234i −0.971603 0.236617i \(-0.923961\pi\)
0.971603 0.236617i \(-0.0760386\pi\)
\(644\) 0 0
\(645\) 4.00000i 0.157500i
\(646\) 0 0
\(647\) 20.0000 0.786281 0.393141 0.919478i \(-0.371389\pi\)
0.393141 + 0.919478i \(0.371389\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 14.0000i − 0.547862i −0.961749 0.273931i \(-0.911676\pi\)
0.961749 0.273931i \(-0.0883240\pi\)
\(654\) 0 0
\(655\) −20.0000 −0.781465
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) − 20.0000i − 0.779089i −0.921008 0.389545i \(-0.872632\pi\)
0.921008 0.389545i \(-0.127368\pi\)
\(660\) 0 0
\(661\) 26.0000i 1.01128i 0.862744 + 0.505641i \(0.168744\pi\)
−0.862744 + 0.505641i \(0.831256\pi\)
\(662\) 0 0
\(663\) −4.00000 −0.155347
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 24.0000i − 0.929284i
\(668\) 0 0
\(669\) 24.0000i 0.927894i
\(670\) 0 0
\(671\) −56.0000 −2.16186
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) 0 0
\(675\) − 1.00000i − 0.0384900i
\(676\) 0 0
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 2.00000i 0.0764161i
\(686\) 0 0
\(687\) 18.0000 0.686743
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) − 16.0000i − 0.608669i −0.952565 0.304334i \(-0.901566\pi\)
0.952565 0.304334i \(-0.0984340\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.0000 −0.606915
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 0 0
\(699\) 6.00000i 0.226941i
\(700\) 0 0
\(701\) 10.0000i 0.377695i 0.982006 + 0.188847i \(0.0604752\pi\)
−0.982006 + 0.188847i \(0.939525\pi\)
\(702\) 0 0
\(703\) −16.0000 −0.603451
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.00000i 0.0751116i 0.999295 + 0.0375558i \(0.0119572\pi\)
−0.999295 + 0.0375558i \(0.988043\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) − 8.00000i − 0.299183i
\(716\) 0 0
\(717\) − 8.00000i − 0.298765i
\(718\) 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
\(723\) − 2.00000i − 0.0743808i
\(724\) 0 0
\(725\) 6.00000i 0.222834i
\(726\) 0 0
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 8.00000i − 0.295891i
\(732\) 0 0
\(733\) 10.0000i 0.369358i 0.982799 + 0.184679i \(0.0591246\pi\)
−0.982799 + 0.184679i \(0.940875\pi\)
\(734\) 0 0
\(735\) −7.00000 −0.258199
\(736\) 0 0
\(737\) −48.0000 −1.76810
\(738\) 0 0
\(739\) 8.00000i 0.294285i 0.989115 + 0.147142i \(0.0470076\pi\)
−0.989115 + 0.147142i \(0.952992\pi\)
\(740\) 0 0
\(741\) − 16.0000i − 0.587775i
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) − 4.00000i − 0.146352i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 8.00000i 0.291150i
\(756\) 0 0
\(757\) − 34.0000i − 1.23575i −0.786276 0.617876i \(-0.787994\pi\)
0.786276 0.617876i \(-0.212006\pi\)
\(758\) 0 0
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.00000i 0.0723102i
\(766\) 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
\(771\) − 6.00000i − 0.216085i
\(772\) 0 0
\(773\) − 10.0000i − 0.359675i −0.983696 0.179838i \(-0.942443\pi\)
0.983696 0.179838i \(-0.0575572\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 48.0000i − 1.71978i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) − 20.0000i − 0.712923i −0.934310 0.356462i \(-0.883983\pi\)
0.934310 0.356462i \(-0.116017\pi\)
\(788\) 0 0
\(789\) − 20.0000i − 0.712019i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −28.0000 −0.994309
\(794\) 0 0
\(795\) 6.00000i 0.212798i
\(796\) 0 0
\(797\) 34.0000i 1.20434i 0.798367 + 0.602171i \(0.205697\pi\)
−0.798367 + 0.602171i \(0.794303\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) 0 0
\(803\) − 8.00000i − 0.282314i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 10.0000 0.352017
\(808\) 0 0
\(809\) 22.0000 0.773479 0.386739 0.922189i \(-0.373601\pi\)
0.386739 + 0.922189i \(0.373601\pi\)
\(810\) 0 0
\(811\) 24.0000i 0.842754i 0.906886 + 0.421377i \(0.138453\pi\)
−0.906886 + 0.421377i \(0.861547\pi\)
\(812\) 0 0
\(813\) 8.00000i 0.280572i
\(814\) 0 0
\(815\) −20.0000 −0.700569
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000i 1.04701i 0.852023 + 0.523504i \(0.175375\pi\)
−0.852023 + 0.523504i \(0.824625\pi\)
\(822\) 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
\(827\) − 36.0000i − 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) 0 0
\(829\) 30.0000i 1.04194i 0.853574 + 0.520972i \(0.174430\pi\)
−0.853574 + 0.520972i \(0.825570\pi\)
\(830\) 0 0
\(831\) 30.0000 1.04069
\(832\) 0 0
\(833\) 14.0000 0.485071
\(834\) 0 0
\(835\) 12.0000i 0.415277i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) 2.00000i 0.0688837i
\(844\) 0 0
\(845\) 9.00000i 0.309609i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) − 8.00000i − 0.274236i
\(852\) 0 0
\(853\) 30.0000i 1.02718i 0.858036 + 0.513590i \(0.171685\pi\)
−0.858036 + 0.513590i \(0.828315\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 0 0
\(859\) − 32.0000i − 1.09183i −0.837842 0.545913i \(-0.816183\pi\)
0.837842 0.545913i \(-0.183817\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 0 0
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) 32.0000i 1.08553i
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 14.0000i − 0.472746i −0.971662 0.236373i \(-0.924041\pi\)
0.971662 0.236373i \(-0.0759588\pi\)
\(878\) 0 0
\(879\) −26.0000 −0.876958
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) − 44.0000i − 1.48072i −0.672212 0.740359i \(-0.734656\pi\)
0.672212 0.740359i \(-0.265344\pi\)
\(884\) 0 0
\(885\) 12.0000i 0.403376i
\(886\) 0 0
\(887\) −20.0000 −0.671534 −0.335767 0.941945i \(-0.608996\pi\)
−0.335767 + 0.941945i \(0.608996\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000i 0.134005i
\(892\) 0 0
\(893\) − 96.0000i − 3.21252i
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) − 12.0000i − 0.399778i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 22.0000 0.731305
\(906\) 0 0
\(907\) − 36.0000i − 1.19536i −0.801735 0.597680i \(-0.796089\pi\)
0.801735 0.597680i \(-0.203911\pi\)
\(908\) 0 0
\(909\) − 14.0000i − 0.464351i
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 0 0
\(915\) 14.0000i 0.462826i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000i 0.0657596i
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 56.0000i 1.83533i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) − 14.0000i − 0.456873i
\(940\) 0 0
\(941\) 34.0000i 1.10837i 0.832394 + 0.554184i \(0.186970\pi\)
−0.832394 + 0.554184i \(0.813030\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.00000i 0.129983i 0.997886 + 0.0649913i \(0.0207020\pi\)
−0.997886 + 0.0649913i \(0.979298\pi\)
\(948\) 0 0
\(949\) − 4.00000i − 0.129845i
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 0 0
\(955\) − 16.0000i − 0.517748i
\(956\) 0 0
\(957\) − 24.0000i − 0.775810i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) 0 0
\(965\) − 22.0000i − 0.708205i
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 16.0000 0.513994
\(970\) 0 0
\(971\) 28.0000i 0.898563i 0.893390 + 0.449281i \(0.148320\pi\)
−0.893390 + 0.449281i \(0.851680\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −2.00000 −0.0640513
\(976\) 0 0
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) − 8.00000i − 0.255681i
\(980\) 0 0
\(981\) − 14.0000i − 0.446986i
\(982\) 0 0
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 0 0
\(985\) 26.0000 0.828429
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.0000i 0.508770i
\(990\) 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
\(995\) 16.0000i 0.507234i
\(996\) 0 0
\(997\) 6.00000i 0.190022i 0.995476 + 0.0950110i \(0.0302886\pi\)
−0.995476 + 0.0950110i \(0.969711\pi\)
\(998\) 0 0
\(999\) 2.00000 0.0632772
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 3840.2.k.p.1921.1 2
4.3 odd 2 3840.2.k.k.1921.2 2
8.3 odd 2 3840.2.k.k.1921.1 2
8.5 even 2 inner 3840.2.k.p.1921.2 2
16.3 odd 4 960.2.a.f.1.1 1
16.5 even 4 480.2.a.b.1.1 1
16.11 odd 4 480.2.a.e.1.1 yes 1
16.13 even 4 960.2.a.o.1.1 1
48.5 odd 4 1440.2.a.k.1.1 1
48.11 even 4 1440.2.a.j.1.1 1
48.29 odd 4 2880.2.a.i.1.1 1
48.35 even 4 2880.2.a.j.1.1 1
80.3 even 4 4800.2.f.j.3649.1 2
80.13 odd 4 4800.2.f.ba.3649.2 2
80.19 odd 4 4800.2.a.ca.1.1 1
80.27 even 4 2400.2.f.n.1249.1 2
80.29 even 4 4800.2.a.u.1.1 1
80.37 odd 4 2400.2.f.e.1249.2 2
80.43 even 4 2400.2.f.n.1249.2 2
80.53 odd 4 2400.2.f.e.1249.1 2
80.59 odd 4 2400.2.a.j.1.1 1
80.67 even 4 4800.2.f.j.3649.2 2
80.69 even 4 2400.2.a.y.1.1 1
80.77 odd 4 4800.2.f.ba.3649.1 2
240.53 even 4 7200.2.f.bb.6049.2 2
240.59 even 4 7200.2.a.u.1.1 1
240.107 odd 4 7200.2.f.b.6049.1 2
240.149 odd 4 7200.2.a.bg.1.1 1
240.197 even 4 7200.2.f.bb.6049.1 2
240.203 odd 4 7200.2.f.b.6049.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.2.a.b.1.1 1 16.5 even 4
480.2.a.e.1.1 yes 1 16.11 odd 4
960.2.a.f.1.1 1 16.3 odd 4
960.2.a.o.1.1 1 16.13 even 4
1440.2.a.j.1.1 1 48.11 even 4
1440.2.a.k.1.1 1 48.5 odd 4
2400.2.a.j.1.1 1 80.59 odd 4
2400.2.a.y.1.1 1 80.69 even 4
2400.2.f.e.1249.1 2 80.53 odd 4
2400.2.f.e.1249.2 2 80.37 odd 4
2400.2.f.n.1249.1 2 80.27 even 4
2400.2.f.n.1249.2 2 80.43 even 4
2880.2.a.i.1.1 1 48.29 odd 4
2880.2.a.j.1.1 1 48.35 even 4
3840.2.k.k.1921.1 2 8.3 odd 2
3840.2.k.k.1921.2 2 4.3 odd 2
3840.2.k.p.1921.1 2 1.1 even 1 trivial
3840.2.k.p.1921.2 2 8.5 even 2 inner
4800.2.a.u.1.1 1 80.29 even 4
4800.2.a.ca.1.1 1 80.19 odd 4
4800.2.f.j.3649.1 2 80.3 even 4
4800.2.f.j.3649.2 2 80.67 even 4
4800.2.f.ba.3649.1 2 80.77 odd 4
4800.2.f.ba.3649.2 2 80.13 odd 4
7200.2.a.u.1.1 1 240.59 even 4
7200.2.a.bg.1.1 1 240.149 odd 4
7200.2.f.b.6049.1 2 240.107 odd 4
7200.2.f.b.6049.2 2 240.203 odd 4
7200.2.f.bb.6049.1 2 240.197 even 4
7200.2.f.bb.6049.2 2 240.53 even 4