Properties

Label 1792.2.b.a.897.1
Level $1792$
Weight $2$
Character 1792.897
Analytic conductor $14.309$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3091920422\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 897.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1792.897
Dual form 1792.2.b.a.897.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{3} -4.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} -4.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} -8.00000 q^{15} -2.00000 q^{17} +2.00000i q^{19} +2.00000i q^{21} -8.00000 q^{23} -11.0000 q^{25} -4.00000i q^{27} -2.00000i q^{29} +4.00000 q^{31} +4.00000i q^{35} -6.00000i q^{37} +2.00000 q^{41} +8.00000i q^{43} +4.00000i q^{45} -4.00000 q^{47} +1.00000 q^{49} +4.00000i q^{51} -10.0000i q^{53} +4.00000 q^{57} +6.00000i q^{59} -4.00000i q^{61} +1.00000 q^{63} +12.0000i q^{67} +16.0000i q^{69} +14.0000 q^{73} +22.0000i q^{75} -8.00000 q^{79} -11.0000 q^{81} -6.00000i q^{83} +8.00000i q^{85} -4.00000 q^{87} -10.0000 q^{89} -8.00000i q^{93} +8.00000 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{7} - 2q^{9} - 16q^{15} - 4q^{17} - 16q^{23} - 22q^{25} + 8q^{31} + 4q^{41} - 8q^{47} + 2q^{49} + 8q^{57} + 2q^{63} + 28q^{73} - 16q^{79} - 22q^{81} - 8q^{87} - 20q^{89} + 16q^{95} - 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) − 4.00000i − 1.78885i −0.447214 0.894427i \(-0.647584\pi\)
0.447214 0.894427i \(-0.352416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −8.00000 −2.06559
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) −11.0000 −2.20000
\(26\) 0 0
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) − 2.00000i − 0.371391i −0.982607 0.185695i \(-0.940546\pi\)
0.982607 0.185695i \(-0.0594537\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 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
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 4.00000i 0.596285i
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000i 0.560112i
\(52\) 0 0
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) − 4.00000i − 0.512148i −0.966657 0.256074i \(-0.917571\pi\)
0.966657 0.256074i \(-0.0824290\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 0 0
\(69\) 16.0000i 1.92617i
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) 22.0000i 2.54034i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 8.00000i 0.867722i
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.0000i 1.19404i 0.802225 + 0.597022i \(0.203650\pi\)
−0.802225 + 0.597022i \(0.796350\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 0 0
\(105\) 8.00000 0.780720
\(106\) 0 0
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) − 10.0000i − 0.957826i −0.877862 0.478913i \(-0.841031\pi\)
0.877862 0.478913i \(-0.158969\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 32.0000i 2.98402i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) − 4.00000i − 0.360668i
\(124\) 0 0
\(125\) 24.0000i 2.14663i
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 16.0000 1.40872
\(130\) 0 0
\(131\) − 14.0000i − 1.22319i −0.791173 0.611593i \(-0.790529\pi\)
0.791173 0.611593i \(-0.209471\pi\)
\(132\) 0 0
\(133\) − 2.00000i − 0.173422i
\(134\) 0 0
\(135\) −16.0000 −1.37706
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 18.0000i 1.52674i 0.645961 + 0.763370i \(0.276457\pi\)
−0.645961 + 0.763370i \(0.723543\pi\)
\(140\) 0 0
\(141\) 8.00000i 0.673722i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) − 2.00000i − 0.164957i
\(148\) 0 0
\(149\) − 2.00000i − 0.163846i −0.996639 0.0819232i \(-0.973894\pi\)
0.996639 0.0819232i \(-0.0261062\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) − 16.0000i − 1.28515i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −20.0000 −1.58610
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) − 16.0000i − 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 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
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) − 2.00000i − 0.152944i
\(172\) 0 0
\(173\) − 8.00000i − 0.608229i −0.952636 0.304114i \(-0.901639\pi\)
0.952636 0.304114i \(-0.0983605\pi\)
\(174\) 0 0
\(175\) 11.0000 0.831522
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 4.00000i 0.298974i 0.988764 + 0.149487i \(0.0477622\pi\)
−0.988764 + 0.149487i \(0.952238\pi\)
\(180\) 0 0
\(181\) 8.00000i 0.594635i 0.954779 + 0.297318i \(0.0960920\pi\)
−0.954779 + 0.297318i \(0.903908\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) −24.0000 −1.76452
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 4.00000i 0.290957i
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 24.0000 1.69283
\(202\) 0 0
\(203\) 2.00000i 0.140372i
\(204\) 0 0
\(205\) − 8.00000i − 0.558744i
\(206\) 0 0
\(207\) 8.00000 0.556038
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) − 20.0000i − 1.37686i −0.725304 0.688428i \(-0.758301\pi\)
0.725304 0.688428i \(-0.241699\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 32.0000 2.18238
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) − 28.0000i − 1.89206i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) − 14.0000i − 0.929213i −0.885517 0.464606i \(-0.846196\pi\)
0.885517 0.464606i \(-0.153804\pi\)
\(228\) 0 0
\(229\) − 16.0000i − 1.05731i −0.848837 0.528655i \(-0.822697\pi\)
0.848837 0.528655i \(-0.177303\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 16.0000i 1.04372i
\(236\) 0 0
\(237\) 16.0000i 1.03931i
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 10.0000i 0.641500i
\(244\) 0 0
\(245\) − 4.00000i − 0.255551i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) − 14.0000i − 0.883672i −0.897096 0.441836i \(-0.854327\pi\)
0.897096 0.441836i \(-0.145673\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 16.0000 1.00196
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 6.00000i 0.372822i
\(260\) 0 0
\(261\) 2.00000i 0.123797i
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −40.0000 −2.45718
\(266\) 0 0
\(267\) 20.0000i 1.22398i
\(268\) 0 0
\(269\) 24.0000i 1.46331i 0.681677 + 0.731653i \(0.261251\pi\)
−0.681677 + 0.731653i \(0.738749\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000i 1.32185i 0.750451 + 0.660926i \(0.229836\pi\)
−0.750451 + 0.660926i \(0.770164\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) − 10.0000i − 0.594438i −0.954809 0.297219i \(-0.903941\pi\)
0.954809 0.297219i \(-0.0960592\pi\)
\(284\) 0 0
\(285\) − 16.0000i − 0.947758i
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 4.00000i 0.234484i
\(292\) 0 0
\(293\) − 12.0000i − 0.701047i −0.936554 0.350524i \(-0.886004\pi\)
0.936554 0.350524i \(-0.113996\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 8.00000i − 0.461112i
\(302\) 0 0
\(303\) 24.0000 1.37876
\(304\) 0 0
\(305\) −16.0000 −0.916157
\(306\) 0 0
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) 0 0
\(309\) − 24.0000i − 1.36531i
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) − 4.00000i − 0.225374i
\(316\) 0 0
\(317\) − 6.00000i − 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) − 4.00000i − 0.222566i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −20.0000 −1.10600
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 8.00000i 0.439720i 0.975531 + 0.219860i \(0.0705600\pi\)
−0.975531 + 0.219860i \(0.929440\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 48.0000 2.62252
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) − 12.0000i − 0.651751i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 64.0000 3.44564
\(346\) 0 0
\(347\) − 24.0000i − 1.28839i −0.764862 0.644194i \(-0.777193\pi\)
0.764862 0.644194i \(-0.222807\pi\)
\(348\) 0 0
\(349\) − 8.00000i − 0.428230i −0.976808 0.214115i \(-0.931313\pi\)
0.976808 0.214115i \(-0.0686868\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 4.00000i − 0.211702i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 0 0
\(363\) − 22.0000i − 1.15470i
\(364\) 0 0
\(365\) − 56.0000i − 2.93117i
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 10.0000i 0.519174i
\(372\) 0 0
\(373\) − 34.0000i − 1.76045i −0.474554 0.880227i \(-0.657390\pi\)
0.474554 0.880227i \(-0.342610\pi\)
\(374\) 0 0
\(375\) 48.0000 2.47871
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) − 16.0000i − 0.819705i
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 8.00000i − 0.406663i
\(388\) 0 0
\(389\) 10.0000i 0.507020i 0.967333 + 0.253510i \(0.0815851\pi\)
−0.967333 + 0.253510i \(0.918415\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) −28.0000 −1.41241
\(394\) 0 0
\(395\) 32.0000i 1.61009i
\(396\) 0 0
\(397\) 8.00000i 0.401508i 0.979642 + 0.200754i \(0.0643393\pi\)
−0.979642 + 0.200754i \(0.935661\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 44.0000i 2.18638i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 4.00000i 0.197305i
\(412\) 0 0
\(413\) − 6.00000i − 0.295241i
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) 36.0000 1.76293
\(418\) 0 0
\(419\) − 26.0000i − 1.27018i −0.772437 0.635092i \(-0.780962\pi\)
0.772437 0.635092i \(-0.219038\pi\)
\(420\) 0 0
\(421\) 22.0000i 1.07221i 0.844150 + 0.536107i \(0.180106\pi\)
−0.844150 + 0.536107i \(0.819894\pi\)
\(422\) 0 0
\(423\) 4.00000 0.194487
\(424\) 0 0
\(425\) 22.0000 1.06716
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 16.0000i 0.767141i
\(436\) 0 0
\(437\) − 16.0000i − 0.765384i
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 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
\(445\) 40.0000i 1.89618i
\(446\) 0 0
\(447\) −4.00000 −0.189194
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 32.0000i 1.50349i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 8.00000i 0.373408i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) −32.0000 −1.48396
\(466\) 0 0
\(467\) − 6.00000i − 0.277647i −0.990317 0.138823i \(-0.955668\pi\)
0.990317 0.138823i \(-0.0443321\pi\)
\(468\) 0 0
\(469\) − 12.0000i − 0.554109i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) − 22.0000i − 1.00943i
\(476\) 0 0
\(477\) 10.0000i 0.457869i
\(478\) 0 0
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 16.0000i − 0.728025i
\(484\) 0 0
\(485\) 8.00000i 0.363261i
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) −32.0000 −1.44709
\(490\) 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
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 4.00000i − 0.179065i −0.995984 0.0895323i \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) 0 0
\(501\) 24.0000i 1.07224i
\(502\) 0 0
\(503\) 32.0000 1.42681 0.713405 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) 0 0
\(505\) 48.0000 2.13597
\(506\) 0 0
\(507\) − 26.0000i − 1.15470i
\(508\) 0 0
\(509\) 24.0000i 1.06378i 0.846813 + 0.531891i \(0.178518\pi\)
−0.846813 + 0.531891i \(0.821482\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 0 0
\(513\) 8.00000 0.353209
\(514\) 0 0
\(515\) − 48.0000i − 2.11513i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −16.0000 −0.702322
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) − 34.0000i − 1.48672i −0.668894 0.743358i \(-0.733232\pi\)
0.668894 0.743358i \(-0.266768\pi\)
\(524\) 0 0
\(525\) − 22.0000i − 0.960159i
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) − 6.00000i − 0.260378i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −48.0000 −2.07522
\(536\) 0 0
\(537\) 8.00000 0.345225
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 22.0000i − 0.945854i −0.881102 0.472927i \(-0.843197\pi\)
0.881102 0.472927i \(-0.156803\pi\)
\(542\) 0 0
\(543\) 16.0000 0.686626
\(544\) 0 0
\(545\) −40.0000 −1.71341
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 0 0
\(549\) 4.00000i 0.170716i
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 48.0000i 2.03749i
\(556\) 0 0
\(557\) 26.0000i 1.10166i 0.834619 + 0.550828i \(0.185688\pi\)
−0.834619 + 0.550828i \(0.814312\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 34.0000i − 1.43293i −0.697623 0.716465i \(-0.745759\pi\)
0.697623 0.716465i \(-0.254241\pi\)
\(564\) 0 0
\(565\) − 24.0000i − 1.00969i
\(566\) 0 0
\(567\) 11.0000 0.461957
\(568\) 0 0
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) − 16.0000i − 0.669579i −0.942293 0.334790i \(-0.891335\pi\)
0.942293 0.334790i \(-0.108665\pi\)
\(572\) 0 0
\(573\) 16.0000i 0.668410i
\(574\) 0 0
\(575\) 88.0000 3.66985
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 0 0
\(579\) 36.0000i 1.49611i
\(580\) 0 0
\(581\) 6.00000i 0.248922i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10.0000i 0.412744i 0.978474 + 0.206372i \(0.0661657\pi\)
−0.978474 + 0.206372i \(0.933834\pi\)
\(588\) 0 0
\(589\) 8.00000i 0.329634i
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) 0 0
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) − 8.00000i − 0.327968i
\(596\) 0 0
\(597\) − 8.00000i − 0.327418i
\(598\) 0 0
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 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
\(605\) − 44.0000i − 1.78885i
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 10.0000i 0.403896i 0.979396 + 0.201948i \(0.0647272\pi\)
−0.979396 + 0.201948i \(0.935273\pi\)
\(614\) 0 0
\(615\) −16.0000 −0.645182
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 6.00000i 0.241160i 0.992704 + 0.120580i \(0.0384755\pi\)
−0.992704 + 0.120580i \(0.961525\pi\)
\(620\) 0 0
\(621\) 32.0000i 1.28412i
\(622\) 0 0
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.0000i 0.478471i
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 0 0
\(633\) −40.0000 −1.58986
\(634\) 0 0
\(635\) − 32.0000i − 1.26988i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) 14.0000i 0.552106i 0.961142 + 0.276053i \(0.0890266\pi\)
−0.961142 + 0.276053i \(0.910973\pi\)
\(644\) 0 0
\(645\) − 64.0000i − 2.52000i
\(646\) 0 0
\(647\) 36.0000 1.41531 0.707653 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 8.00000i 0.313545i
\(652\) 0 0
\(653\) 22.0000i 0.860927i 0.902608 + 0.430463i \(0.141650\pi\)
−0.902608 + 0.430463i \(0.858350\pi\)
\(654\) 0 0
\(655\) −56.0000 −2.18810
\(656\) 0 0
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) 40.0000i 1.55818i 0.626913 + 0.779089i \(0.284318\pi\)
−0.626913 + 0.779089i \(0.715682\pi\)
\(660\) 0 0
\(661\) 20.0000i 0.777910i 0.921257 + 0.388955i \(0.127164\pi\)
−0.921257 + 0.388955i \(0.872836\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 16.0000i 0.619522i
\(668\) 0 0
\(669\) 48.0000i 1.85579i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 0 0
\(675\) 44.0000i 1.69356i
\(676\) 0 0
\(677\) − 24.0000i − 0.922395i −0.887298 0.461197i \(-0.847420\pi\)
0.887298 0.461197i \(-0.152580\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) −28.0000 −1.07296
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 8.00000i 0.305664i
\(686\) 0 0
\(687\) −32.0000 −1.22088
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 14.0000i − 0.532585i −0.963892 0.266293i \(-0.914201\pi\)
0.963892 0.266293i \(-0.0857987\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 72.0000 2.73112
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0 0
\(699\) 52.0000i 1.96682i
\(700\) 0 0
\(701\) − 10.0000i − 0.377695i −0.982006 0.188847i \(-0.939525\pi\)
0.982006 0.188847i \(-0.0604752\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) 0 0
\(705\) 32.0000 1.20519
\(706\) 0 0
\(707\) − 12.0000i − 0.451306i
\(708\) 0 0
\(709\) 10.0000i 0.375558i 0.982211 + 0.187779i \(0.0601289\pi\)
−0.982211 + 0.187779i \(0.939871\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 32.0000i 1.19506i
\(718\) 0 0
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) 0 0
\(723\) 4.00000i 0.148762i
\(724\) 0 0
\(725\) 22.0000i 0.817059i
\(726\) 0 0
\(727\) −20.0000 −0.741759 −0.370879 0.928681i \(-0.620944\pi\)
−0.370879 + 0.928681i \(0.620944\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) − 16.0000i − 0.591781i
\(732\) 0 0
\(733\) − 52.0000i − 1.92066i −0.278859 0.960332i \(-0.589956\pi\)
0.278859 0.960332i \(-0.410044\pi\)
\(734\) 0 0
\(735\) −8.00000 −0.295084
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 16.0000i − 0.588570i −0.955718 0.294285i \(-0.904919\pi\)
0.955718 0.294285i \(-0.0950814\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) −8.00000 −0.293097
\(746\) 0 0
\(747\) 6.00000i 0.219529i
\(748\) 0 0
\(749\) 12.0000i 0.438470i
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) −28.0000 −1.02038
\(754\) 0 0
\(755\) 64.0000i 2.32920i
\(756\) 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
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 10.0000i 0.362024i
\(764\) 0 0
\(765\) − 8.00000i − 0.289241i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) − 36.0000i − 1.29651i
\(772\) 0 0
\(773\) 4.00000i 0.143870i 0.997409 + 0.0719350i \(0.0229174\pi\)
−0.997409 + 0.0719350i \(0.977083\pi\)
\(774\) 0 0
\(775\) −44.0000 −1.58053
\(776\) 0 0
\(777\) 12.0000 0.430498
\(778\) 0 0
\(779\) 4.00000i 0.143315i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 22.0000i − 0.784215i −0.919919 0.392108i \(-0.871746\pi\)
0.919919 0.392108i \(-0.128254\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 80.0000i 2.83731i
\(796\) 0 0
\(797\) 24.0000i 0.850124i 0.905164 + 0.425062i \(0.139748\pi\)
−0.905164 + 0.425062i \(0.860252\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) − 32.0000i − 1.12785i
\(806\) 0 0
\(807\) 48.0000 1.68968
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 30.0000i 1.05344i 0.850038 + 0.526721i \(0.176579\pi\)
−0.850038 + 0.526721i \(0.823421\pi\)
\(812\) 0 0
\(813\) − 64.0000i − 2.24458i
\(814\) 0 0
\(815\) −64.0000 −2.24182
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000i 0.209401i 0.994504 + 0.104701i \(0.0333885\pi\)
−0.994504 + 0.104701i \(0.966612\pi\)
\(822\) 0 0
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) 0 0
\(829\) − 20.0000i − 0.694629i −0.937749 0.347314i \(-0.887094\pi\)
0.937749 0.347314i \(-0.112906\pi\)
\(830\) 0 0
\(831\) 44.0000 1.52634
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 48.0000i 1.66111i
\(836\) 0 0
\(837\) − 16.0000i − 0.553041i
\(838\) 0 0
\(839\) −36.0000 −1.24286 −0.621429 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) − 12.0000i − 0.413302i
\(844\) 0 0
\(845\) − 52.0000i − 1.78885i
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 48.0000i 1.64542i
\(852\) 0 0
\(853\) − 32.0000i − 1.09566i −0.836590 0.547830i \(-0.815454\pi\)
0.836590 0.547830i \(-0.184546\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) − 14.0000i − 0.477674i −0.971060 0.238837i \(-0.923234\pi\)
0.971060 0.238837i \(-0.0767661\pi\)
\(860\) 0 0
\(861\) 4.00000i 0.136320i
\(862\) 0 0
\(863\) 8.00000 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 0 0
\(865\) −32.0000 −1.08803
\(866\) 0 0
\(867\) 26.0000i 0.883006i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) − 24.0000i − 0.811348i
\(876\) 0 0
\(877\) − 2.00000i − 0.0675352i −0.999430 0.0337676i \(-0.989249\pi\)
0.999430 0.0337676i \(-0.0107506\pi\)
\(878\) 0 0
\(879\) −24.0000 −0.809500
\(880\) 0 0
\(881\) 10.0000 0.336909 0.168454 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) 0 0
\(883\) 20.0000i 0.673054i 0.941674 + 0.336527i \(0.109252\pi\)
−0.941674 + 0.336527i \(0.890748\pi\)
\(884\) 0 0
\(885\) − 48.0000i − 1.61350i
\(886\) 0 0
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 8.00000i − 0.267710i
\(894\) 0 0
\(895\) 16.0000 0.534821
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 8.00000i − 0.266815i
\(900\) 0 0
\(901\) 20.0000i 0.666297i
\(902\) 0 0
\(903\) −16.0000 −0.532447
\(904\) 0 0
\(905\) 32.0000 1.06372
\(906\) 0 0
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) 0 0
\(909\) − 12.0000i − 0.398015i
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 32.0000i 1.05789i
\(916\) 0 0
\(917\) 14.0000i 0.462321i
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 66.0000i 2.17007i
\(926\) 0 0
\(927\) −12.0000 −0.394132
\(928\) 0 0
\(929\) −2.00000 −0.0656179 −0.0328089 0.999462i \(-0.510445\pi\)
−0.0328089 + 0.999462i \(0.510445\pi\)
\(930\) 0 0
\(931\) 2.00000i 0.0655474i
\(932\) 0 0
\(933\) − 48.0000i − 1.57145i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) 0 0
\(939\) 28.0000i 0.913745i
\(940\) 0 0
\(941\) 12.0000i 0.391189i 0.980685 + 0.195594i \(0.0626636\pi\)
−0.980685 + 0.195594i \(0.937336\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) 0 0
\(945\) 16.0000 0.520480
\(946\) 0 0
\(947\) − 40.0000i − 1.29983i −0.760009 0.649913i \(-0.774805\pi\)
0.760009 0.649913i \(-0.225195\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) −10.0000 −0.323932 −0.161966 0.986796i \(-0.551783\pi\)
−0.161966 + 0.986796i \(0.551783\pi\)
\(954\) 0 0
\(955\) 32.0000i 1.03550i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) 0 0
\(965\) 72.0000i 2.31776i
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) − 26.0000i − 0.834380i −0.908819 0.417190i \(-0.863015\pi\)
0.908819 0.417190i \(-0.136985\pi\)
\(972\) 0 0
\(973\) − 18.0000i − 0.577054i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −54.0000 −1.72761 −0.863807 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000i 0.319275i
\(982\) 0 0
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) −72.0000 −2.29411
\(986\) 0 0
\(987\) − 8.00000i − 0.254643i
\(988\) 0 0
\(989\) − 64.0000i − 2.03508i
\(990\) 0 0
\(991\) −48.0000 −1.52477 −0.762385 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) 0 0
\(993\) 16.0000 0.507745
\(994\) 0 0
\(995\) − 16.0000i − 0.507234i
\(996\) 0 0
\(997\) 52.0000i 1.64686i 0.567420 + 0.823428i \(0.307941\pi\)
−0.567420 + 0.823428i \(0.692059\pi\)
\(998\) 0 0
\(999\) −24.0000 −0.759326
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 1792.2.b.a.897.1 2
4.3 odd 2 1792.2.b.h.897.2 2
8.3 odd 2 1792.2.b.h.897.1 2
8.5 even 2 inner 1792.2.b.a.897.2 2
16.3 odd 4 112.2.a.a.1.1 1
16.5 even 4 448.2.a.c.1.1 1
16.11 odd 4 448.2.a.h.1.1 1
16.13 even 4 56.2.a.b.1.1 1
48.5 odd 4 4032.2.a.d.1.1 1
48.11 even 4 4032.2.a.a.1.1 1
48.29 odd 4 504.2.a.h.1.1 1
48.35 even 4 1008.2.a.m.1.1 1
80.3 even 4 2800.2.g.g.449.1 2
80.13 odd 4 1400.2.g.b.449.2 2
80.19 odd 4 2800.2.a.bd.1.1 1
80.29 even 4 1400.2.a.a.1.1 1
80.67 even 4 2800.2.g.g.449.2 2
80.77 odd 4 1400.2.g.b.449.1 2
112.3 even 12 784.2.i.b.177.1 2
112.13 odd 4 392.2.a.b.1.1 1
112.19 even 12 784.2.i.b.753.1 2
112.27 even 4 3136.2.a.c.1.1 1
112.45 odd 12 392.2.i.e.177.1 2
112.51 odd 12 784.2.i.j.753.1 2
112.61 odd 12 392.2.i.e.361.1 2
112.67 odd 12 784.2.i.j.177.1 2
112.69 odd 4 3136.2.a.w.1.1 1
112.83 even 4 784.2.a.i.1.1 1
112.93 even 12 392.2.i.a.361.1 2
112.109 even 12 392.2.i.a.177.1 2
176.109 odd 4 6776.2.a.h.1.1 1
208.77 even 4 9464.2.a.h.1.1 1
336.83 odd 4 7056.2.a.c.1.1 1
336.125 even 4 3528.2.a.b.1.1 1
336.173 even 12 3528.2.s.ba.361.1 2
336.221 odd 12 3528.2.s.a.3313.1 2
336.269 even 12 3528.2.s.ba.3313.1 2
336.317 odd 12 3528.2.s.a.361.1 2
560.349 odd 4 9800.2.a.bj.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.a.b.1.1 1 16.13 even 4
112.2.a.a.1.1 1 16.3 odd 4
392.2.a.b.1.1 1 112.13 odd 4
392.2.i.a.177.1 2 112.109 even 12
392.2.i.a.361.1 2 112.93 even 12
392.2.i.e.177.1 2 112.45 odd 12
392.2.i.e.361.1 2 112.61 odd 12
448.2.a.c.1.1 1 16.5 even 4
448.2.a.h.1.1 1 16.11 odd 4
504.2.a.h.1.1 1 48.29 odd 4
784.2.a.i.1.1 1 112.83 even 4
784.2.i.b.177.1 2 112.3 even 12
784.2.i.b.753.1 2 112.19 even 12
784.2.i.j.177.1 2 112.67 odd 12
784.2.i.j.753.1 2 112.51 odd 12
1008.2.a.m.1.1 1 48.35 even 4
1400.2.a.a.1.1 1 80.29 even 4
1400.2.g.b.449.1 2 80.77 odd 4
1400.2.g.b.449.2 2 80.13 odd 4
1792.2.b.a.897.1 2 1.1 even 1 trivial
1792.2.b.a.897.2 2 8.5 even 2 inner
1792.2.b.h.897.1 2 8.3 odd 2
1792.2.b.h.897.2 2 4.3 odd 2
2800.2.a.bd.1.1 1 80.19 odd 4
2800.2.g.g.449.1 2 80.3 even 4
2800.2.g.g.449.2 2 80.67 even 4
3136.2.a.c.1.1 1 112.27 even 4
3136.2.a.w.1.1 1 112.69 odd 4
3528.2.a.b.1.1 1 336.125 even 4
3528.2.s.a.361.1 2 336.317 odd 12
3528.2.s.a.3313.1 2 336.221 odd 12
3528.2.s.ba.361.1 2 336.173 even 12
3528.2.s.ba.3313.1 2 336.269 even 12
4032.2.a.a.1.1 1 48.11 even 4
4032.2.a.d.1.1 1 48.5 odd 4
6776.2.a.h.1.1 1 176.109 odd 4
7056.2.a.c.1.1 1 336.83 odd 4
9464.2.a.h.1.1 1 208.77 even 4
9800.2.a.bj.1.1 1 560.349 odd 4