Properties

Label 384.3.g.b.127.3
Level $384$
Weight $3$
Character 384.127
Analytic conductor $10.463$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.3
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 384.127
Dual form 384.3.g.b.127.7

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{3} +2.63567 q^{5} +12.5558i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} +2.63567 q^{5} +12.5558i q^{7} -3.00000 q^{9} +5.79796i q^{11} -8.78710 q^{13} -4.56512i q^{15} -30.1810 q^{17} +17.4125i q^{19} +21.7473 q^{21} +2.48425i q^{23} -18.0532 q^{25} +5.19615i q^{27} +26.4175 q^{29} +38.0082i q^{31} +10.0424 q^{33} +33.0931i q^{35} +47.7930 q^{37} +15.2197i q^{39} +53.3294 q^{41} -30.7044i q^{43} -7.90702 q^{45} +16.2238i q^{47} -108.649 q^{49} +52.2750i q^{51} +49.8432 q^{53} +15.2815i q^{55} +30.1593 q^{57} +107.412i q^{59} -62.6220 q^{61} -37.6675i q^{63} -23.1599 q^{65} -60.9540i q^{67} +4.30285 q^{69} -19.9465i q^{71} +5.13929 q^{73} +31.2691i q^{75} -72.7982 q^{77} -6.83349i q^{79} +9.00000 q^{81} -159.213i q^{83} -79.5472 q^{85} -45.7565i q^{87} +39.4473 q^{89} -110.329i q^{91} +65.8321 q^{93} +45.8936i q^{95} -60.5944 q^{97} -17.3939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{5} - 24 q^{9} + O(q^{10}) \) \( 8 q + 16 q^{5} - 24 q^{9} - 48 q^{13} + 16 q^{17} - 8 q^{25} + 80 q^{29} + 16 q^{37} + 80 q^{41} - 48 q^{45} - 88 q^{49} - 176 q^{53} + 96 q^{57} + 272 q^{61} - 160 q^{65} - 16 q^{73} - 320 q^{77} + 72 q^{81} - 32 q^{85} - 240 q^{89} + 192 q^{93} + 400 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.73205i − 0.577350i
\(4\) 0 0
\(5\) 2.63567 0.527135 0.263567 0.964641i \(-0.415101\pi\)
0.263567 + 0.964641i \(0.415101\pi\)
\(6\) 0 0
\(7\) 12.5558i 1.79369i 0.442344 + 0.896845i \(0.354147\pi\)
−0.442344 + 0.896845i \(0.645853\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) 5.79796i 0.527087i 0.964647 + 0.263544i \(0.0848913\pi\)
−0.964647 + 0.263544i \(0.915109\pi\)
\(12\) 0 0
\(13\) −8.78710 −0.675931 −0.337965 0.941159i \(-0.609739\pi\)
−0.337965 + 0.941159i \(0.609739\pi\)
\(14\) 0 0
\(15\) − 4.56512i − 0.304341i
\(16\) 0 0
\(17\) −30.1810 −1.77535 −0.887676 0.460469i \(-0.847681\pi\)
−0.887676 + 0.460469i \(0.847681\pi\)
\(18\) 0 0
\(19\) 17.4125i 0.916445i 0.888838 + 0.458222i \(0.151514\pi\)
−0.888838 + 0.458222i \(0.848486\pi\)
\(20\) 0 0
\(21\) 21.7473 1.03559
\(22\) 0 0
\(23\) 2.48425i 0.108011i 0.998541 + 0.0540054i \(0.0171988\pi\)
−0.998541 + 0.0540054i \(0.982801\pi\)
\(24\) 0 0
\(25\) −18.0532 −0.722129
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) 26.4175 0.910950 0.455475 0.890249i \(-0.349469\pi\)
0.455475 + 0.890249i \(0.349469\pi\)
\(30\) 0 0
\(31\) 38.0082i 1.22607i 0.790056 + 0.613035i \(0.210051\pi\)
−0.790056 + 0.613035i \(0.789949\pi\)
\(32\) 0 0
\(33\) 10.0424 0.304314
\(34\) 0 0
\(35\) 33.0931i 0.945517i
\(36\) 0 0
\(37\) 47.7930 1.29170 0.645851 0.763463i \(-0.276503\pi\)
0.645851 + 0.763463i \(0.276503\pi\)
\(38\) 0 0
\(39\) 15.2197i 0.390249i
\(40\) 0 0
\(41\) 53.3294 1.30072 0.650358 0.759628i \(-0.274619\pi\)
0.650358 + 0.759628i \(0.274619\pi\)
\(42\) 0 0
\(43\) − 30.7044i − 0.714057i −0.934094 0.357028i \(-0.883790\pi\)
0.934094 0.357028i \(-0.116210\pi\)
\(44\) 0 0
\(45\) −7.90702 −0.175712
\(46\) 0 0
\(47\) 16.2238i 0.345186i 0.984993 + 0.172593i \(0.0552146\pi\)
−0.984993 + 0.172593i \(0.944785\pi\)
\(48\) 0 0
\(49\) −108.649 −2.21733
\(50\) 0 0
\(51\) 52.2750i 1.02500i
\(52\) 0 0
\(53\) 49.8432 0.940437 0.470219 0.882550i \(-0.344175\pi\)
0.470219 + 0.882550i \(0.344175\pi\)
\(54\) 0 0
\(55\) 15.2815i 0.277846i
\(56\) 0 0
\(57\) 30.1593 0.529110
\(58\) 0 0
\(59\) 107.412i 1.82054i 0.414012 + 0.910271i \(0.364127\pi\)
−0.414012 + 0.910271i \(0.635873\pi\)
\(60\) 0 0
\(61\) −62.6220 −1.02659 −0.513295 0.858212i \(-0.671575\pi\)
−0.513295 + 0.858212i \(0.671575\pi\)
\(62\) 0 0
\(63\) − 37.6675i − 0.597897i
\(64\) 0 0
\(65\) −23.1599 −0.356307
\(66\) 0 0
\(67\) − 60.9540i − 0.909762i −0.890552 0.454881i \(-0.849682\pi\)
0.890552 0.454881i \(-0.150318\pi\)
\(68\) 0 0
\(69\) 4.30285 0.0623601
\(70\) 0 0
\(71\) − 19.9465i − 0.280937i −0.990085 0.140469i \(-0.955139\pi\)
0.990085 0.140469i \(-0.0448609\pi\)
\(72\) 0 0
\(73\) 5.13929 0.0704013 0.0352006 0.999380i \(-0.488793\pi\)
0.0352006 + 0.999380i \(0.488793\pi\)
\(74\) 0 0
\(75\) 31.2691i 0.416921i
\(76\) 0 0
\(77\) −72.7982 −0.945431
\(78\) 0 0
\(79\) − 6.83349i − 0.0864999i −0.999064 0.0432500i \(-0.986229\pi\)
0.999064 0.0432500i \(-0.0137712\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) − 159.213i − 1.91823i −0.283016 0.959115i \(-0.591335\pi\)
0.283016 0.959115i \(-0.408665\pi\)
\(84\) 0 0
\(85\) −79.5472 −0.935850
\(86\) 0 0
\(87\) − 45.7565i − 0.525937i
\(88\) 0 0
\(89\) 39.4473 0.443229 0.221614 0.975134i \(-0.428867\pi\)
0.221614 + 0.975134i \(0.428867\pi\)
\(90\) 0 0
\(91\) − 110.329i − 1.21241i
\(92\) 0 0
\(93\) 65.8321 0.707872
\(94\) 0 0
\(95\) 45.8936i 0.483090i
\(96\) 0 0
\(97\) −60.5944 −0.624684 −0.312342 0.949970i \(-0.601114\pi\)
−0.312342 + 0.949970i \(0.601114\pi\)
\(98\) 0 0
\(99\) − 17.3939i − 0.175696i
\(100\) 0 0
\(101\) 169.076 1.67402 0.837011 0.547186i \(-0.184301\pi\)
0.837011 + 0.547186i \(0.184301\pi\)
\(102\) 0 0
\(103\) 163.390i 1.58631i 0.609020 + 0.793155i \(0.291563\pi\)
−0.609020 + 0.793155i \(0.708437\pi\)
\(104\) 0 0
\(105\) 57.3189 0.545894
\(106\) 0 0
\(107\) − 37.4288i − 0.349802i −0.984586 0.174901i \(-0.944040\pi\)
0.984586 0.174901i \(-0.0559605\pi\)
\(108\) 0 0
\(109\) −210.117 −1.92768 −0.963838 0.266489i \(-0.914137\pi\)
−0.963838 + 0.266489i \(0.914137\pi\)
\(110\) 0 0
\(111\) − 82.7799i − 0.745765i
\(112\) 0 0
\(113\) 190.809 1.68857 0.844286 0.535893i \(-0.180025\pi\)
0.844286 + 0.535893i \(0.180025\pi\)
\(114\) 0 0
\(115\) 6.54768i 0.0569363i
\(116\) 0 0
\(117\) 26.3613 0.225310
\(118\) 0 0
\(119\) − 378.947i − 3.18443i
\(120\) 0 0
\(121\) 87.3837 0.722179
\(122\) 0 0
\(123\) − 92.3692i − 0.750969i
\(124\) 0 0
\(125\) −113.474 −0.907794
\(126\) 0 0
\(127\) 34.4377i 0.271163i 0.990766 + 0.135581i \(0.0432902\pi\)
−0.990766 + 0.135581i \(0.956710\pi\)
\(128\) 0 0
\(129\) −53.1817 −0.412261
\(130\) 0 0
\(131\) 92.5491i 0.706482i 0.935532 + 0.353241i \(0.114920\pi\)
−0.935532 + 0.353241i \(0.885080\pi\)
\(132\) 0 0
\(133\) −218.628 −1.64382
\(134\) 0 0
\(135\) 13.6954i 0.101447i
\(136\) 0 0
\(137\) −20.0745 −0.146529 −0.0732647 0.997313i \(-0.523342\pi\)
−0.0732647 + 0.997313i \(0.523342\pi\)
\(138\) 0 0
\(139\) − 70.0557i − 0.503998i −0.967728 0.251999i \(-0.918912\pi\)
0.967728 0.251999i \(-0.0810879\pi\)
\(140\) 0 0
\(141\) 28.1004 0.199293
\(142\) 0 0
\(143\) − 50.9472i − 0.356274i
\(144\) 0 0
\(145\) 69.6281 0.480193
\(146\) 0 0
\(147\) 188.186i 1.28017i
\(148\) 0 0
\(149\) −165.919 −1.11355 −0.556774 0.830664i \(-0.687961\pi\)
−0.556774 + 0.830664i \(0.687961\pi\)
\(150\) 0 0
\(151\) − 59.7106i − 0.395434i −0.980259 0.197717i \(-0.936647\pi\)
0.980259 0.197717i \(-0.0633528\pi\)
\(152\) 0 0
\(153\) 90.5429 0.591784
\(154\) 0 0
\(155\) 100.177i 0.646304i
\(156\) 0 0
\(157\) 138.133 0.879829 0.439915 0.898040i \(-0.355009\pi\)
0.439915 + 0.898040i \(0.355009\pi\)
\(158\) 0 0
\(159\) − 86.3309i − 0.542962i
\(160\) 0 0
\(161\) −31.1918 −0.193738
\(162\) 0 0
\(163\) − 178.296i − 1.09384i −0.837185 0.546920i \(-0.815800\pi\)
0.837185 0.546920i \(-0.184200\pi\)
\(164\) 0 0
\(165\) 26.4684 0.160414
\(166\) 0 0
\(167\) 98.6284i 0.590589i 0.955406 + 0.295295i \(0.0954178\pi\)
−0.955406 + 0.295295i \(0.904582\pi\)
\(168\) 0 0
\(169\) −91.7869 −0.543118
\(170\) 0 0
\(171\) − 52.2374i − 0.305482i
\(172\) 0 0
\(173\) 166.380 0.961733 0.480867 0.876794i \(-0.340322\pi\)
0.480867 + 0.876794i \(0.340322\pi\)
\(174\) 0 0
\(175\) − 226.673i − 1.29528i
\(176\) 0 0
\(177\) 186.043 1.05109
\(178\) 0 0
\(179\) − 206.496i − 1.15361i −0.816882 0.576805i \(-0.804299\pi\)
0.816882 0.576805i \(-0.195701\pi\)
\(180\) 0 0
\(181\) 181.330 1.00182 0.500911 0.865499i \(-0.332998\pi\)
0.500911 + 0.865499i \(0.332998\pi\)
\(182\) 0 0
\(183\) 108.465i 0.592702i
\(184\) 0 0
\(185\) 125.967 0.680901
\(186\) 0 0
\(187\) − 174.988i − 0.935765i
\(188\) 0 0
\(189\) −65.2420 −0.345196
\(190\) 0 0
\(191\) 185.390i 0.970627i 0.874340 + 0.485314i \(0.161295\pi\)
−0.874340 + 0.485314i \(0.838705\pi\)
\(192\) 0 0
\(193\) 15.5439 0.0805382 0.0402691 0.999189i \(-0.487178\pi\)
0.0402691 + 0.999189i \(0.487178\pi\)
\(194\) 0 0
\(195\) 40.1142i 0.205714i
\(196\) 0 0
\(197\) −114.464 −0.581034 −0.290517 0.956870i \(-0.593827\pi\)
−0.290517 + 0.956870i \(0.593827\pi\)
\(198\) 0 0
\(199\) 299.009i 1.50256i 0.659984 + 0.751280i \(0.270563\pi\)
−0.659984 + 0.751280i \(0.729437\pi\)
\(200\) 0 0
\(201\) −105.576 −0.525251
\(202\) 0 0
\(203\) 331.694i 1.63396i
\(204\) 0 0
\(205\) 140.559 0.685653
\(206\) 0 0
\(207\) − 7.45275i − 0.0360036i
\(208\) 0 0
\(209\) −100.957 −0.483046
\(210\) 0 0
\(211\) 139.706i 0.662111i 0.943611 + 0.331056i \(0.107405\pi\)
−0.943611 + 0.331056i \(0.892595\pi\)
\(212\) 0 0
\(213\) −34.5484 −0.162199
\(214\) 0 0
\(215\) − 80.9269i − 0.376404i
\(216\) 0 0
\(217\) −477.224 −2.19919
\(218\) 0 0
\(219\) − 8.90152i − 0.0406462i
\(220\) 0 0
\(221\) 265.203 1.20001
\(222\) 0 0
\(223\) − 66.1141i − 0.296476i −0.988952 0.148238i \(-0.952640\pi\)
0.988952 0.148238i \(-0.0473601\pi\)
\(224\) 0 0
\(225\) 54.1597 0.240710
\(226\) 0 0
\(227\) − 49.9611i − 0.220093i −0.993926 0.110046i \(-0.964900\pi\)
0.993926 0.110046i \(-0.0351000\pi\)
\(228\) 0 0
\(229\) −129.255 −0.564431 −0.282215 0.959351i \(-0.591069\pi\)
−0.282215 + 0.959351i \(0.591069\pi\)
\(230\) 0 0
\(231\) 126.090i 0.545845i
\(232\) 0 0
\(233\) −196.318 −0.842567 −0.421284 0.906929i \(-0.638420\pi\)
−0.421284 + 0.906929i \(0.638420\pi\)
\(234\) 0 0
\(235\) 42.7606i 0.181960i
\(236\) 0 0
\(237\) −11.8360 −0.0499407
\(238\) 0 0
\(239\) 153.654i 0.642905i 0.946926 + 0.321452i \(0.104171\pi\)
−0.946926 + 0.321452i \(0.895829\pi\)
\(240\) 0 0
\(241\) −41.3543 −0.171595 −0.0857974 0.996313i \(-0.527344\pi\)
−0.0857974 + 0.996313i \(0.527344\pi\)
\(242\) 0 0
\(243\) − 15.5885i − 0.0641500i
\(244\) 0 0
\(245\) −286.363 −1.16883
\(246\) 0 0
\(247\) − 153.005i − 0.619453i
\(248\) 0 0
\(249\) −275.765 −1.10749
\(250\) 0 0
\(251\) 35.4495i 0.141233i 0.997504 + 0.0706165i \(0.0224967\pi\)
−0.997504 + 0.0706165i \(0.977503\pi\)
\(252\) 0 0
\(253\) −14.4036 −0.0569312
\(254\) 0 0
\(255\) 137.780i 0.540313i
\(256\) 0 0
\(257\) 66.0626 0.257053 0.128527 0.991706i \(-0.458975\pi\)
0.128527 + 0.991706i \(0.458975\pi\)
\(258\) 0 0
\(259\) 600.081i 2.31691i
\(260\) 0 0
\(261\) −79.2526 −0.303650
\(262\) 0 0
\(263\) 93.5910i 0.355859i 0.984043 + 0.177930i \(0.0569400\pi\)
−0.984043 + 0.177930i \(0.943060\pi\)
\(264\) 0 0
\(265\) 131.370 0.495737
\(266\) 0 0
\(267\) − 68.3248i − 0.255898i
\(268\) 0 0
\(269\) 305.548 1.13587 0.567933 0.823074i \(-0.307743\pi\)
0.567933 + 0.823074i \(0.307743\pi\)
\(270\) 0 0
\(271\) 191.602i 0.707017i 0.935431 + 0.353509i \(0.115012\pi\)
−0.935431 + 0.353509i \(0.884988\pi\)
\(272\) 0 0
\(273\) −191.096 −0.699985
\(274\) 0 0
\(275\) − 104.672i − 0.380625i
\(276\) 0 0
\(277\) 188.436 0.680276 0.340138 0.940376i \(-0.389526\pi\)
0.340138 + 0.940376i \(0.389526\pi\)
\(278\) 0 0
\(279\) − 114.024i − 0.408690i
\(280\) 0 0
\(281\) −296.087 −1.05369 −0.526844 0.849962i \(-0.676625\pi\)
−0.526844 + 0.849962i \(0.676625\pi\)
\(282\) 0 0
\(283\) 260.681i 0.921136i 0.887625 + 0.460568i \(0.152354\pi\)
−0.887625 + 0.460568i \(0.847646\pi\)
\(284\) 0 0
\(285\) 79.4900 0.278912
\(286\) 0 0
\(287\) 669.595i 2.33308i
\(288\) 0 0
\(289\) 621.891 2.15187
\(290\) 0 0
\(291\) 104.953i 0.360662i
\(292\) 0 0
\(293\) 314.756 1.07425 0.537127 0.843501i \(-0.319510\pi\)
0.537127 + 0.843501i \(0.319510\pi\)
\(294\) 0 0
\(295\) 283.103i 0.959672i
\(296\) 0 0
\(297\) −30.1271 −0.101438
\(298\) 0 0
\(299\) − 21.8294i − 0.0730079i
\(300\) 0 0
\(301\) 385.520 1.28080
\(302\) 0 0
\(303\) − 292.849i − 0.966497i
\(304\) 0 0
\(305\) −165.051 −0.541152
\(306\) 0 0
\(307\) 168.120i 0.547621i 0.961784 + 0.273811i \(0.0882841\pi\)
−0.961784 + 0.273811i \(0.911716\pi\)
\(308\) 0 0
\(309\) 283.000 0.915856
\(310\) 0 0
\(311\) − 468.033i − 1.50493i −0.658633 0.752464i \(-0.728865\pi\)
0.658633 0.752464i \(-0.271135\pi\)
\(312\) 0 0
\(313\) 225.370 0.720033 0.360017 0.932946i \(-0.382771\pi\)
0.360017 + 0.932946i \(0.382771\pi\)
\(314\) 0 0
\(315\) − 99.2793i − 0.315172i
\(316\) 0 0
\(317\) −162.613 −0.512976 −0.256488 0.966547i \(-0.582565\pi\)
−0.256488 + 0.966547i \(0.582565\pi\)
\(318\) 0 0
\(319\) 153.168i 0.480150i
\(320\) 0 0
\(321\) −64.8285 −0.201958
\(322\) 0 0
\(323\) − 525.525i − 1.62701i
\(324\) 0 0
\(325\) 158.635 0.488109
\(326\) 0 0
\(327\) 363.933i 1.11294i
\(328\) 0 0
\(329\) −203.703 −0.619158
\(330\) 0 0
\(331\) 254.527i 0.768964i 0.923133 + 0.384482i \(0.125620\pi\)
−0.923133 + 0.384482i \(0.874380\pi\)
\(332\) 0 0
\(333\) −143.379 −0.430567
\(334\) 0 0
\(335\) − 160.655i − 0.479567i
\(336\) 0 0
\(337\) −94.7347 −0.281112 −0.140556 0.990073i \(-0.544889\pi\)
−0.140556 + 0.990073i \(0.544889\pi\)
\(338\) 0 0
\(339\) − 330.490i − 0.974897i
\(340\) 0 0
\(341\) −220.370 −0.646246
\(342\) 0 0
\(343\) − 748.942i − 2.18351i
\(344\) 0 0
\(345\) 11.3409 0.0328722
\(346\) 0 0
\(347\) 173.315i 0.499467i 0.968315 + 0.249734i \(0.0803431\pi\)
−0.968315 + 0.249734i \(0.919657\pi\)
\(348\) 0 0
\(349\) 195.247 0.559448 0.279724 0.960081i \(-0.409757\pi\)
0.279724 + 0.960081i \(0.409757\pi\)
\(350\) 0 0
\(351\) − 45.6591i − 0.130083i
\(352\) 0 0
\(353\) 296.107 0.838830 0.419415 0.907795i \(-0.362235\pi\)
0.419415 + 0.907795i \(0.362235\pi\)
\(354\) 0 0
\(355\) − 52.5726i − 0.148092i
\(356\) 0 0
\(357\) −656.356 −1.83853
\(358\) 0 0
\(359\) 448.964i 1.25060i 0.780386 + 0.625298i \(0.215023\pi\)
−0.780386 + 0.625298i \(0.784977\pi\)
\(360\) 0 0
\(361\) 57.8065 0.160129
\(362\) 0 0
\(363\) − 151.353i − 0.416950i
\(364\) 0 0
\(365\) 13.5455 0.0371110
\(366\) 0 0
\(367\) − 623.564i − 1.69908i −0.527521 0.849542i \(-0.676878\pi\)
0.527521 0.849542i \(-0.323122\pi\)
\(368\) 0 0
\(369\) −159.988 −0.433572
\(370\) 0 0
\(371\) 625.823i 1.68685i
\(372\) 0 0
\(373\) −297.116 −0.796558 −0.398279 0.917264i \(-0.630392\pi\)
−0.398279 + 0.917264i \(0.630392\pi\)
\(374\) 0 0
\(375\) 196.543i 0.524115i
\(376\) 0 0
\(377\) −232.134 −0.615739
\(378\) 0 0
\(379\) 224.021i 0.591084i 0.955330 + 0.295542i \(0.0955003\pi\)
−0.955330 + 0.295542i \(0.904500\pi\)
\(380\) 0 0
\(381\) 59.6478 0.156556
\(382\) 0 0
\(383\) − 573.847i − 1.49830i −0.662403 0.749148i \(-0.730463\pi\)
0.662403 0.749148i \(-0.269537\pi\)
\(384\) 0 0
\(385\) −191.872 −0.498370
\(386\) 0 0
\(387\) 92.1133i 0.238019i
\(388\) 0 0
\(389\) 148.849 0.382646 0.191323 0.981527i \(-0.438722\pi\)
0.191323 + 0.981527i \(0.438722\pi\)
\(390\) 0 0
\(391\) − 74.9771i − 0.191757i
\(392\) 0 0
\(393\) 160.300 0.407887
\(394\) 0 0
\(395\) − 18.0109i − 0.0455971i
\(396\) 0 0
\(397\) 245.952 0.619526 0.309763 0.950814i \(-0.399750\pi\)
0.309763 + 0.950814i \(0.399750\pi\)
\(398\) 0 0
\(399\) 378.675i 0.949059i
\(400\) 0 0
\(401\) 64.0509 0.159728 0.0798640 0.996806i \(-0.474551\pi\)
0.0798640 + 0.996806i \(0.474551\pi\)
\(402\) 0 0
\(403\) − 333.981i − 0.828738i
\(404\) 0 0
\(405\) 23.7211 0.0585705
\(406\) 0 0
\(407\) 277.102i 0.680840i
\(408\) 0 0
\(409\) 555.403 1.35795 0.678977 0.734160i \(-0.262424\pi\)
0.678977 + 0.734160i \(0.262424\pi\)
\(410\) 0 0
\(411\) 34.7701i 0.0845988i
\(412\) 0 0
\(413\) −1348.65 −3.26549
\(414\) 0 0
\(415\) − 419.634i − 1.01117i
\(416\) 0 0
\(417\) −121.340 −0.290983
\(418\) 0 0
\(419\) − 793.486i − 1.89376i −0.321583 0.946881i \(-0.604215\pi\)
0.321583 0.946881i \(-0.395785\pi\)
\(420\) 0 0
\(421\) 87.1175 0.206930 0.103465 0.994633i \(-0.467007\pi\)
0.103465 + 0.994633i \(0.467007\pi\)
\(422\) 0 0
\(423\) − 48.6713i − 0.115062i
\(424\) 0 0
\(425\) 544.864 1.28203
\(426\) 0 0
\(427\) − 786.272i − 1.84139i
\(428\) 0 0
\(429\) −88.2432 −0.205695
\(430\) 0 0
\(431\) 432.932i 1.00448i 0.864727 + 0.502242i \(0.167491\pi\)
−0.864727 + 0.502242i \(0.832509\pi\)
\(432\) 0 0
\(433\) 391.344 0.903798 0.451899 0.892069i \(-0.350747\pi\)
0.451899 + 0.892069i \(0.350747\pi\)
\(434\) 0 0
\(435\) − 120.599i − 0.277240i
\(436\) 0 0
\(437\) −43.2569 −0.0989860
\(438\) 0 0
\(439\) 12.3066i 0.0280332i 0.999902 + 0.0140166i \(0.00446177\pi\)
−0.999902 + 0.0140166i \(0.995538\pi\)
\(440\) 0 0
\(441\) 325.947 0.739109
\(442\) 0 0
\(443\) − 567.554i − 1.28116i −0.767892 0.640580i \(-0.778694\pi\)
0.767892 0.640580i \(-0.221306\pi\)
\(444\) 0 0
\(445\) 103.970 0.233641
\(446\) 0 0
\(447\) 287.380i 0.642908i
\(448\) 0 0
\(449\) −407.897 −0.908457 −0.454229 0.890885i \(-0.650085\pi\)
−0.454229 + 0.890885i \(0.650085\pi\)
\(450\) 0 0
\(451\) 309.201i 0.685591i
\(452\) 0 0
\(453\) −103.422 −0.228304
\(454\) 0 0
\(455\) − 290.792i − 0.639104i
\(456\) 0 0
\(457\) 231.789 0.507197 0.253598 0.967310i \(-0.418386\pi\)
0.253598 + 0.967310i \(0.418386\pi\)
\(458\) 0 0
\(459\) − 156.825i − 0.341667i
\(460\) 0 0
\(461\) −438.217 −0.950578 −0.475289 0.879830i \(-0.657657\pi\)
−0.475289 + 0.879830i \(0.657657\pi\)
\(462\) 0 0
\(463\) − 185.141i − 0.399873i −0.979809 0.199936i \(-0.935926\pi\)
0.979809 0.199936i \(-0.0640735\pi\)
\(464\) 0 0
\(465\) 173.512 0.373144
\(466\) 0 0
\(467\) 82.9765i 0.177680i 0.996046 + 0.0888399i \(0.0283159\pi\)
−0.996046 + 0.0888399i \(0.971684\pi\)
\(468\) 0 0
\(469\) 765.329 1.63183
\(470\) 0 0
\(471\) − 239.254i − 0.507970i
\(472\) 0 0
\(473\) 178.023 0.376370
\(474\) 0 0
\(475\) − 314.351i − 0.661791i
\(476\) 0 0
\(477\) −149.530 −0.313479
\(478\) 0 0
\(479\) 173.847i 0.362936i 0.983397 + 0.181468i \(0.0580849\pi\)
−0.983397 + 0.181468i \(0.941915\pi\)
\(480\) 0 0
\(481\) −419.962 −0.873101
\(482\) 0 0
\(483\) 54.0258i 0.111855i
\(484\) 0 0
\(485\) −159.707 −0.329293
\(486\) 0 0
\(487\) − 250.667i − 0.514716i −0.966316 0.257358i \(-0.917148\pi\)
0.966316 0.257358i \(-0.0828520\pi\)
\(488\) 0 0
\(489\) −308.817 −0.631529
\(490\) 0 0
\(491\) − 283.478i − 0.577349i −0.957427 0.288674i \(-0.906786\pi\)
0.957427 0.288674i \(-0.0932145\pi\)
\(492\) 0 0
\(493\) −797.307 −1.61726
\(494\) 0 0
\(495\) − 45.8446i − 0.0926153i
\(496\) 0 0
\(497\) 250.445 0.503914
\(498\) 0 0
\(499\) 192.802i 0.386377i 0.981162 + 0.193188i \(0.0618829\pi\)
−0.981162 + 0.193188i \(0.938117\pi\)
\(500\) 0 0
\(501\) 170.829 0.340977
\(502\) 0 0
\(503\) − 398.251i − 0.791752i −0.918304 0.395876i \(-0.870441\pi\)
0.918304 0.395876i \(-0.129559\pi\)
\(504\) 0 0
\(505\) 445.630 0.882436
\(506\) 0 0
\(507\) 158.980i 0.313569i
\(508\) 0 0
\(509\) 550.212 1.08097 0.540483 0.841355i \(-0.318242\pi\)
0.540483 + 0.841355i \(0.318242\pi\)
\(510\) 0 0
\(511\) 64.5281i 0.126278i
\(512\) 0 0
\(513\) −90.4778 −0.176370
\(514\) 0 0
\(515\) 430.643i 0.836199i
\(516\) 0 0
\(517\) −94.0647 −0.181943
\(518\) 0 0
\(519\) − 288.178i − 0.555257i
\(520\) 0 0
\(521\) −164.181 −0.315126 −0.157563 0.987509i \(-0.550364\pi\)
−0.157563 + 0.987509i \(0.550364\pi\)
\(522\) 0 0
\(523\) 643.556i 1.23051i 0.788328 + 0.615255i \(0.210947\pi\)
−0.788328 + 0.615255i \(0.789053\pi\)
\(524\) 0 0
\(525\) −392.610 −0.747828
\(526\) 0 0
\(527\) − 1147.12i − 2.17670i
\(528\) 0 0
\(529\) 522.828 0.988334
\(530\) 0 0
\(531\) − 322.236i − 0.606848i
\(532\) 0 0
\(533\) −468.610 −0.879194
\(534\) 0 0
\(535\) − 98.6501i − 0.184393i
\(536\) 0 0
\(537\) −357.662 −0.666037
\(538\) 0 0
\(539\) − 629.942i − 1.16872i
\(540\) 0 0
\(541\) −727.713 −1.34513 −0.672563 0.740040i \(-0.734807\pi\)
−0.672563 + 0.740040i \(0.734807\pi\)
\(542\) 0 0
\(543\) − 314.072i − 0.578402i
\(544\) 0 0
\(545\) −553.799 −1.01615
\(546\) 0 0
\(547\) − 97.2958i − 0.177872i −0.996037 0.0889359i \(-0.971653\pi\)
0.996037 0.0889359i \(-0.0283466\pi\)
\(548\) 0 0
\(549\) 187.866 0.342197
\(550\) 0 0
\(551\) 459.994i 0.834835i
\(552\) 0 0
\(553\) 85.8002 0.155154
\(554\) 0 0
\(555\) − 218.181i − 0.393119i
\(556\) 0 0
\(557\) 520.548 0.934556 0.467278 0.884110i \(-0.345235\pi\)
0.467278 + 0.884110i \(0.345235\pi\)
\(558\) 0 0
\(559\) 269.803i 0.482653i
\(560\) 0 0
\(561\) −303.088 −0.540264
\(562\) 0 0
\(563\) 525.371i 0.933163i 0.884478 + 0.466581i \(0.154515\pi\)
−0.884478 + 0.466581i \(0.845485\pi\)
\(564\) 0 0
\(565\) 502.909 0.890105
\(566\) 0 0
\(567\) 113.003i 0.199299i
\(568\) 0 0
\(569\) 824.774 1.44951 0.724757 0.689004i \(-0.241952\pi\)
0.724757 + 0.689004i \(0.241952\pi\)
\(570\) 0 0
\(571\) 1088.90i 1.90701i 0.301381 + 0.953504i \(0.402552\pi\)
−0.301381 + 0.953504i \(0.597448\pi\)
\(572\) 0 0
\(573\) 321.105 0.560392
\(574\) 0 0
\(575\) − 44.8487i − 0.0779978i
\(576\) 0 0
\(577\) −688.463 −1.19318 −0.596589 0.802547i \(-0.703478\pi\)
−0.596589 + 0.802547i \(0.703478\pi\)
\(578\) 0 0
\(579\) − 26.9228i − 0.0464988i
\(580\) 0 0
\(581\) 1999.05 3.44071
\(582\) 0 0
\(583\) 288.989i 0.495692i
\(584\) 0 0
\(585\) 69.4798 0.118769
\(586\) 0 0
\(587\) 441.108i 0.751462i 0.926729 + 0.375731i \(0.122608\pi\)
−0.926729 + 0.375731i \(0.877392\pi\)
\(588\) 0 0
\(589\) −661.815 −1.12363
\(590\) 0 0
\(591\) 198.257i 0.335460i
\(592\) 0 0
\(593\) −481.317 −0.811664 −0.405832 0.913948i \(-0.633018\pi\)
−0.405832 + 0.913948i \(0.633018\pi\)
\(594\) 0 0
\(595\) − 998.782i − 1.67862i
\(596\) 0 0
\(597\) 517.899 0.867503
\(598\) 0 0
\(599\) 541.322i 0.903709i 0.892092 + 0.451854i \(0.149237\pi\)
−0.892092 + 0.451854i \(0.850763\pi\)
\(600\) 0 0
\(601\) 64.7772 0.107782 0.0538912 0.998547i \(-0.482838\pi\)
0.0538912 + 0.998547i \(0.482838\pi\)
\(602\) 0 0
\(603\) 182.862i 0.303254i
\(604\) 0 0
\(605\) 230.315 0.380686
\(606\) 0 0
\(607\) 608.260i 1.00208i 0.865426 + 0.501038i \(0.167048\pi\)
−0.865426 + 0.501038i \(0.832952\pi\)
\(608\) 0 0
\(609\) 574.511 0.943368
\(610\) 0 0
\(611\) − 142.560i − 0.233322i
\(612\) 0 0
\(613\) −924.407 −1.50800 −0.754002 0.656872i \(-0.771879\pi\)
−0.754002 + 0.656872i \(0.771879\pi\)
\(614\) 0 0
\(615\) − 243.455i − 0.395862i
\(616\) 0 0
\(617\) 652.781 1.05799 0.528996 0.848624i \(-0.322569\pi\)
0.528996 + 0.848624i \(0.322569\pi\)
\(618\) 0 0
\(619\) 246.098i 0.397574i 0.980043 + 0.198787i \(0.0637001\pi\)
−0.980043 + 0.198787i \(0.936300\pi\)
\(620\) 0 0
\(621\) −12.9085 −0.0207867
\(622\) 0 0
\(623\) 495.294i 0.795015i
\(624\) 0 0
\(625\) 152.249 0.243599
\(626\) 0 0
\(627\) 174.862i 0.278887i
\(628\) 0 0
\(629\) −1442.44 −2.29323
\(630\) 0 0
\(631\) 563.715i 0.893368i 0.894692 + 0.446684i \(0.147395\pi\)
−0.894692 + 0.446684i \(0.852605\pi\)
\(632\) 0 0
\(633\) 241.977 0.382270
\(634\) 0 0
\(635\) 90.7665i 0.142939i
\(636\) 0 0
\(637\) 954.709 1.49876
\(638\) 0 0
\(639\) 59.8396i 0.0936457i
\(640\) 0 0
\(641\) −163.485 −0.255046 −0.127523 0.991836i \(-0.540703\pi\)
−0.127523 + 0.991836i \(0.540703\pi\)
\(642\) 0 0
\(643\) − 307.908i − 0.478862i −0.970913 0.239431i \(-0.923039\pi\)
0.970913 0.239431i \(-0.0769609\pi\)
\(644\) 0 0
\(645\) −140.170 −0.217317
\(646\) 0 0
\(647\) 909.931i 1.40638i 0.711000 + 0.703192i \(0.248243\pi\)
−0.711000 + 0.703192i \(0.751757\pi\)
\(648\) 0 0
\(649\) −622.771 −0.959585
\(650\) 0 0
\(651\) 826.576i 1.26970i
\(652\) 0 0
\(653\) 299.487 0.458632 0.229316 0.973352i \(-0.426351\pi\)
0.229316 + 0.973352i \(0.426351\pi\)
\(654\) 0 0
\(655\) 243.929i 0.372411i
\(656\) 0 0
\(657\) −15.4179 −0.0234671
\(658\) 0 0
\(659\) 141.105i 0.214120i 0.994253 + 0.107060i \(0.0341437\pi\)
−0.994253 + 0.107060i \(0.965856\pi\)
\(660\) 0 0
\(661\) 1176.74 1.78024 0.890121 0.455725i \(-0.150620\pi\)
0.890121 + 0.455725i \(0.150620\pi\)
\(662\) 0 0
\(663\) − 459.345i − 0.692829i
\(664\) 0 0
\(665\) −576.232 −0.866514
\(666\) 0 0
\(667\) 65.6278i 0.0983925i
\(668\) 0 0
\(669\) −114.513 −0.171170
\(670\) 0 0
\(671\) − 363.080i − 0.541103i
\(672\) 0 0
\(673\) −230.780 −0.342912 −0.171456 0.985192i \(-0.554847\pi\)
−0.171456 + 0.985192i \(0.554847\pi\)
\(674\) 0 0
\(675\) − 93.8073i − 0.138974i
\(676\) 0 0
\(677\) 768.499 1.13515 0.567577 0.823320i \(-0.307881\pi\)
0.567577 + 0.823320i \(0.307881\pi\)
\(678\) 0 0
\(679\) − 760.813i − 1.12049i
\(680\) 0 0
\(681\) −86.5352 −0.127071
\(682\) 0 0
\(683\) − 963.083i − 1.41008i −0.709169 0.705039i \(-0.750930\pi\)
0.709169 0.705039i \(-0.249070\pi\)
\(684\) 0 0
\(685\) −52.9099 −0.0772408
\(686\) 0 0
\(687\) 223.876i 0.325874i
\(688\) 0 0
\(689\) −437.977 −0.635670
\(690\) 0 0
\(691\) 376.440i 0.544776i 0.962188 + 0.272388i \(0.0878134\pi\)
−0.962188 + 0.272388i \(0.912187\pi\)
\(692\) 0 0
\(693\) 218.395 0.315144
\(694\) 0 0
\(695\) − 184.644i − 0.265675i
\(696\) 0 0
\(697\) −1609.53 −2.30923
\(698\) 0 0
\(699\) 340.033i 0.486456i
\(700\) 0 0
\(701\) −543.266 −0.774988 −0.387494 0.921872i \(-0.626659\pi\)
−0.387494 + 0.921872i \(0.626659\pi\)
\(702\) 0 0
\(703\) 832.193i 1.18377i
\(704\) 0 0
\(705\) 74.0635 0.105055
\(706\) 0 0
\(707\) 2122.89i 3.00268i
\(708\) 0 0
\(709\) −403.852 −0.569608 −0.284804 0.958586i \(-0.591929\pi\)
−0.284804 + 0.958586i \(0.591929\pi\)
\(710\) 0 0
\(711\) 20.5005i 0.0288333i
\(712\) 0 0
\(713\) −94.4218 −0.132429
\(714\) 0 0
\(715\) − 134.280i − 0.187805i
\(716\) 0 0
\(717\) 266.137 0.371181
\(718\) 0 0
\(719\) 183.791i 0.255620i 0.991799 + 0.127810i \(0.0407948\pi\)
−0.991799 + 0.127810i \(0.959205\pi\)
\(720\) 0 0
\(721\) −2051.50 −2.84535
\(722\) 0 0
\(723\) 71.6278i 0.0990703i
\(724\) 0 0
\(725\) −476.922 −0.657823
\(726\) 0 0
\(727\) 843.226i 1.15987i 0.814663 + 0.579935i \(0.196922\pi\)
−0.814663 + 0.579935i \(0.803078\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) 926.690i 1.26770i
\(732\) 0 0
\(733\) 286.176 0.390418 0.195209 0.980762i \(-0.437461\pi\)
0.195209 + 0.980762i \(0.437461\pi\)
\(734\) 0 0
\(735\) 495.996i 0.674824i
\(736\) 0 0
\(737\) 353.409 0.479524
\(738\) 0 0
\(739\) − 821.243i − 1.11129i −0.831420 0.555645i \(-0.812471\pi\)
0.831420 0.555645i \(-0.187529\pi\)
\(740\) 0 0
\(741\) −265.012 −0.357641
\(742\) 0 0
\(743\) − 1227.08i − 1.65153i −0.564018 0.825763i \(-0.690745\pi\)
0.564018 0.825763i \(-0.309255\pi\)
\(744\) 0 0
\(745\) −437.308 −0.586990
\(746\) 0 0
\(747\) 477.639i 0.639410i
\(748\) 0 0
\(749\) 469.949 0.627436
\(750\) 0 0
\(751\) 966.507i 1.28696i 0.765463 + 0.643480i \(0.222510\pi\)
−0.765463 + 0.643480i \(0.777490\pi\)
\(752\) 0 0
\(753\) 61.4003 0.0815409
\(754\) 0 0
\(755\) − 157.378i − 0.208447i
\(756\) 0 0
\(757\) −876.359 −1.15767 −0.578837 0.815443i \(-0.696493\pi\)
−0.578837 + 0.815443i \(0.696493\pi\)
\(758\) 0 0
\(759\) 24.9477i 0.0328692i
\(760\) 0 0
\(761\) 217.575 0.285907 0.142953 0.989729i \(-0.454340\pi\)
0.142953 + 0.989729i \(0.454340\pi\)
\(762\) 0 0
\(763\) − 2638.19i − 3.45765i
\(764\) 0 0
\(765\) 238.642 0.311950
\(766\) 0 0
\(767\) − 943.840i − 1.23056i
\(768\) 0 0
\(769\) 497.583 0.647053 0.323526 0.946219i \(-0.395132\pi\)
0.323526 + 0.946219i \(0.395132\pi\)
\(770\) 0 0
\(771\) − 114.424i − 0.148410i
\(772\) 0 0
\(773\) 664.490 0.859624 0.429812 0.902918i \(-0.358580\pi\)
0.429812 + 0.902918i \(0.358580\pi\)
\(774\) 0 0
\(775\) − 686.170i − 0.885380i
\(776\) 0 0
\(777\) 1039.37 1.33767
\(778\) 0 0
\(779\) 928.595i 1.19203i
\(780\) 0 0
\(781\) 115.649 0.148078
\(782\) 0 0
\(783\) 137.270i 0.175312i
\(784\) 0 0
\(785\) 364.074 0.463789
\(786\) 0 0
\(787\) 214.321i 0.272326i 0.990686 + 0.136163i \(0.0434772\pi\)
−0.990686 + 0.136163i \(0.956523\pi\)
\(788\) 0 0
\(789\) 162.104 0.205456
\(790\) 0 0
\(791\) 2395.76i 3.02878i
\(792\) 0 0
\(793\) 550.266 0.693904
\(794\) 0 0
\(795\) − 227.540i − 0.286214i
\(796\) 0 0
\(797\) 487.535 0.611713 0.305857 0.952078i \(-0.401057\pi\)
0.305857 + 0.952078i \(0.401057\pi\)
\(798\) 0 0
\(799\) − 489.649i − 0.612827i
\(800\) 0 0
\(801\) −118.342 −0.147743
\(802\) 0 0
\(803\) 29.7974i 0.0371076i
\(804\) 0 0
\(805\) −82.2115 −0.102126
\(806\) 0 0
\(807\) − 529.225i − 0.655793i
\(808\) 0 0
\(809\) −1232.31 −1.52325 −0.761623 0.648021i \(-0.775597\pi\)
−0.761623 + 0.648021i \(0.775597\pi\)
\(810\) 0 0
\(811\) 38.1721i 0.0470679i 0.999723 + 0.0235340i \(0.00749178\pi\)
−0.999723 + 0.0235340i \(0.992508\pi\)
\(812\) 0 0
\(813\) 331.864 0.408197
\(814\) 0 0
\(815\) − 469.930i − 0.576601i
\(816\) 0 0
\(817\) 534.640 0.654394
\(818\) 0 0
\(819\) 330.988i 0.404137i
\(820\) 0 0
\(821\) 490.147 0.597012 0.298506 0.954408i \(-0.403512\pi\)
0.298506 + 0.954408i \(0.403512\pi\)
\(822\) 0 0
\(823\) 582.258i 0.707482i 0.935343 + 0.353741i \(0.115091\pi\)
−0.935343 + 0.353741i \(0.884909\pi\)
\(824\) 0 0
\(825\) −181.297 −0.219754
\(826\) 0 0
\(827\) − 852.873i − 1.03128i −0.856804 0.515642i \(-0.827553\pi\)
0.856804 0.515642i \(-0.172447\pi\)
\(828\) 0 0
\(829\) 692.476 0.835315 0.417657 0.908605i \(-0.362851\pi\)
0.417657 + 0.908605i \(0.362851\pi\)
\(830\) 0 0
\(831\) − 326.381i − 0.392757i
\(832\) 0 0
\(833\) 3279.13 3.93653
\(834\) 0 0
\(835\) 259.952i 0.311320i
\(836\) 0 0
\(837\) −197.496 −0.235957
\(838\) 0 0
\(839\) − 761.022i − 0.907059i −0.891241 0.453529i \(-0.850165\pi\)
0.891241 0.453529i \(-0.149835\pi\)
\(840\) 0 0
\(841\) −143.113 −0.170170
\(842\) 0 0
\(843\) 512.837i 0.608347i
\(844\) 0 0
\(845\) −241.920 −0.286296
\(846\) 0 0
\(847\) 1097.17i 1.29537i
\(848\) 0 0
\(849\) 451.513 0.531818
\(850\) 0 0
\(851\) 118.730i 0.139518i
\(852\) 0 0
\(853\) −1245.38 −1.46000 −0.729999 0.683448i \(-0.760480\pi\)
−0.729999 + 0.683448i \(0.760480\pi\)
\(854\) 0 0
\(855\) − 137.681i − 0.161030i
\(856\) 0 0
\(857\) −1517.00 −1.77013 −0.885064 0.465469i \(-0.845886\pi\)
−0.885064 + 0.465469i \(0.845886\pi\)
\(858\) 0 0
\(859\) 1311.75i 1.52706i 0.645770 + 0.763532i \(0.276537\pi\)
−0.645770 + 0.763532i \(0.723463\pi\)
\(860\) 0 0
\(861\) 1159.77 1.34701
\(862\) 0 0
\(863\) − 1304.34i − 1.51140i −0.654920 0.755698i \(-0.727298\pi\)
0.654920 0.755698i \(-0.272702\pi\)
\(864\) 0 0
\(865\) 438.523 0.506963
\(866\) 0 0
\(867\) − 1077.15i − 1.24238i
\(868\) 0 0
\(869\) 39.6203 0.0455930
\(870\) 0 0
\(871\) 535.609i 0.614936i
\(872\) 0 0
\(873\) 181.783 0.208228
\(874\) 0 0
\(875\) − 1424.76i − 1.62830i
\(876\) 0 0
\(877\) −1207.07 −1.37637 −0.688183 0.725537i \(-0.741591\pi\)
−0.688183 + 0.725537i \(0.741591\pi\)
\(878\) 0 0
\(879\) − 545.174i − 0.620221i
\(880\) 0 0
\(881\) −375.974 −0.426758 −0.213379 0.976970i \(-0.568447\pi\)
−0.213379 + 0.976970i \(0.568447\pi\)
\(882\) 0 0
\(883\) − 395.955i − 0.448420i −0.974541 0.224210i \(-0.928020\pi\)
0.974541 0.224210i \(-0.0719802\pi\)
\(884\) 0 0
\(885\) 490.349 0.554067
\(886\) 0 0
\(887\) 960.050i 1.08236i 0.840908 + 0.541178i \(0.182021\pi\)
−0.840908 + 0.541178i \(0.817979\pi\)
\(888\) 0 0
\(889\) −432.394 −0.486382
\(890\) 0 0
\(891\) 52.1816i 0.0585652i
\(892\) 0 0
\(893\) −282.496 −0.316344
\(894\) 0 0
\(895\) − 544.256i − 0.608108i
\(896\) 0 0
\(897\) −37.8095 −0.0421511
\(898\) 0 0
\(899\) 1004.08i 1.11689i
\(900\) 0 0
\(901\) −1504.32 −1.66961
\(902\) 0 0
\(903\) − 667.740i − 0.739468i
\(904\) 0 0
\(905\) 477.926 0.528095
\(906\) 0 0
\(907\) 1301.18i 1.43460i 0.696766 + 0.717298i \(0.254622\pi\)
−0.696766 + 0.717298i \(0.745378\pi\)
\(908\) 0 0
\(909\) −507.229 −0.558008
\(910\) 0 0
\(911\) − 1274.60i − 1.39912i −0.714573 0.699561i \(-0.753379\pi\)
0.714573 0.699561i \(-0.246621\pi\)
\(912\) 0 0
\(913\) 923.111 1.01107
\(914\) 0 0
\(915\) 285.877i 0.312434i
\(916\) 0 0
\(917\) −1162.03 −1.26721
\(918\) 0 0
\(919\) − 957.109i − 1.04147i −0.853719 0.520734i \(-0.825658\pi\)
0.853719 0.520734i \(-0.174342\pi\)
\(920\) 0 0
\(921\) 291.192 0.316169
\(922\) 0 0
\(923\) 175.272i 0.189894i
\(924\) 0 0
\(925\) −862.817 −0.932775
\(926\) 0 0
\(927\) − 490.170i − 0.528770i
\(928\) 0 0
\(929\) −912.989 −0.982765 −0.491383 0.870944i \(-0.663508\pi\)
−0.491383 + 0.870944i \(0.663508\pi\)
\(930\) 0 0
\(931\) − 1891.84i − 2.03206i
\(932\) 0 0
\(933\) −810.657 −0.868871
\(934\) 0 0
\(935\) − 461.212i − 0.493274i
\(936\) 0 0
\(937\) 751.958 0.802516 0.401258 0.915965i \(-0.368573\pi\)
0.401258 + 0.915965i \(0.368573\pi\)
\(938\) 0 0
\(939\) − 390.353i − 0.415711i
\(940\) 0 0
\(941\) 417.550 0.443730 0.221865 0.975077i \(-0.428786\pi\)
0.221865 + 0.975077i \(0.428786\pi\)
\(942\) 0 0
\(943\) 132.484i 0.140492i
\(944\) 0 0
\(945\) −171.957 −0.181965
\(946\) 0 0
\(947\) − 928.909i − 0.980897i −0.871470 0.490448i \(-0.836833\pi\)
0.871470 0.490448i \(-0.163167\pi\)
\(948\) 0 0
\(949\) −45.1595 −0.0475864
\(950\) 0 0
\(951\) 281.655i 0.296167i
\(952\) 0 0
\(953\) 676.685 0.710058 0.355029 0.934855i \(-0.384471\pi\)
0.355029 + 0.934855i \(0.384471\pi\)
\(954\) 0 0
\(955\) 488.627i 0.511652i
\(956\) 0 0
\(957\) 265.294 0.277215
\(958\) 0 0
\(959\) − 252.053i − 0.262828i
\(960\) 0 0
\(961\) −483.620 −0.503247
\(962\) 0 0
\(963\) 112.286i 0.116601i
\(964\) 0 0
\(965\) 40.9686 0.0424545
\(966\) 0 0
\(967\) − 208.530i − 0.215646i −0.994170 0.107823i \(-0.965612\pi\)
0.994170 0.107823i \(-0.0343880\pi\)
\(968\) 0 0
\(969\) −910.236 −0.939356
\(970\) 0 0
\(971\) − 932.159i − 0.959999i −0.877269 0.480000i \(-0.840637\pi\)
0.877269 0.480000i \(-0.159363\pi\)
\(972\) 0 0
\(973\) 879.608 0.904016
\(974\) 0 0
\(975\) − 274.765i − 0.281810i
\(976\) 0 0
\(977\) 1567.22 1.60412 0.802059 0.597245i \(-0.203738\pi\)
0.802059 + 0.597245i \(0.203738\pi\)
\(978\) 0 0
\(979\) 228.714i 0.233620i
\(980\) 0 0
\(981\) 630.350 0.642559
\(982\) 0 0
\(983\) − 1201.11i − 1.22188i −0.791676 0.610941i \(-0.790791\pi\)
0.791676 0.610941i \(-0.209209\pi\)
\(984\) 0 0
\(985\) −301.689 −0.306283
\(986\) 0 0
\(987\) 352.824i 0.357471i
\(988\) 0 0
\(989\) 76.2775 0.0771259
\(990\) 0 0
\(991\) 568.813i 0.573979i 0.957934 + 0.286989i \(0.0926545\pi\)
−0.957934 + 0.286989i \(0.907346\pi\)
\(992\) 0 0
\(993\) 440.854 0.443961
\(994\) 0 0
\(995\) 788.091i 0.792052i
\(996\) 0 0
\(997\) 66.4659 0.0666659 0.0333330 0.999444i \(-0.489388\pi\)
0.0333330 + 0.999444i \(0.489388\pi\)
\(998\) 0 0
\(999\) 248.340i 0.248588i
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 384.3.g.b.127.3 yes 8
3.2 odd 2 1152.3.g.c.127.4 8
4.3 odd 2 inner 384.3.g.b.127.7 yes 8
8.3 odd 2 384.3.g.a.127.2 8
8.5 even 2 384.3.g.a.127.6 yes 8
12.11 even 2 1152.3.g.c.127.3 8
16.3 odd 4 768.3.b.f.127.2 8
16.5 even 4 768.3.b.f.127.3 8
16.11 odd 4 768.3.b.e.127.7 8
16.13 even 4 768.3.b.e.127.6 8
24.5 odd 2 1152.3.g.f.127.6 8
24.11 even 2 1152.3.g.f.127.5 8
48.5 odd 4 2304.3.b.q.127.3 8
48.11 even 4 2304.3.b.t.127.3 8
48.29 odd 4 2304.3.b.t.127.6 8
48.35 even 4 2304.3.b.q.127.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.g.a.127.2 8 8.3 odd 2
384.3.g.a.127.6 yes 8 8.5 even 2
384.3.g.b.127.3 yes 8 1.1 even 1 trivial
384.3.g.b.127.7 yes 8 4.3 odd 2 inner
768.3.b.e.127.6 8 16.13 even 4
768.3.b.e.127.7 8 16.11 odd 4
768.3.b.f.127.2 8 16.3 odd 4
768.3.b.f.127.3 8 16.5 even 4
1152.3.g.c.127.3 8 12.11 even 2
1152.3.g.c.127.4 8 3.2 odd 2
1152.3.g.f.127.5 8 24.11 even 2
1152.3.g.f.127.6 8 24.5 odd 2
2304.3.b.q.127.3 8 48.5 odd 4
2304.3.b.q.127.6 8 48.35 even 4
2304.3.b.t.127.3 8 48.11 even 4
2304.3.b.t.127.6 8 48.29 odd 4