Properties

Label 384.9.b.a.319.5
Level $384$
Weight $9$
Character 384.319
Analytic conductor $156.433$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(156.433386263\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} + 200 x^{6} - 586 x^{5} + 10949 x^{4} - 20926 x^{3} + 78946 x^{2} - 68580 x + 222900\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.5
Root \(-0.366025 + 2.01178i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.9.b.a.319.8

$q$-expansion

\(f(q)\) \(=\) \(q+46.7654 q^{3} -721.475i q^{5} +1935.65i q^{7} +2187.00 q^{9} +O(q^{10})\) \(q+46.7654 q^{3} -721.475i q^{5} +1935.65i q^{7} +2187.00 q^{9} -14414.4 q^{11} +2280.40i q^{13} -33740.1i q^{15} -78065.2 q^{17} +48459.6 q^{19} +90521.3i q^{21} +222858. i q^{23} -129901. q^{25} +102276. q^{27} -1.12756e6i q^{29} +828931. i q^{31} -674093. q^{33} +1.39652e6 q^{35} -931641. i q^{37} +106644. i q^{39} +4.58800e6 q^{41} -2.91683e6 q^{43} -1.57787e6i q^{45} +7.52086e6i q^{47} +2.01807e6 q^{49} -3.65075e6 q^{51} +4.79808e6i q^{53} +1.03996e7i q^{55} +2.26623e6 q^{57} +8.36767e6 q^{59} +3.86342e6i q^{61} +4.23326e6i q^{63} +1.64525e6 q^{65} +352024. q^{67} +1.04220e7i q^{69} -2.68851e7i q^{71} +8.13722e6 q^{73} -6.07489e6 q^{75} -2.79011e7i q^{77} +4.31004e7i q^{79} +4.78297e6 q^{81} +3.58735e7 q^{83} +5.63221e7i q^{85} -5.27308e7i q^{87} +2.79238e7 q^{89} -4.41405e6 q^{91} +3.87652e7i q^{93} -3.49624e7i q^{95} +1.07429e8 q^{97} -3.15242e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 17496 q^{9} + O(q^{10}) \) \( 8 q + 17496 q^{9} + 93040 q^{17} + 395912 q^{25} + 1065312 q^{33} + 2978576 q^{41} + 16144520 q^{49} + 11671776 q^{57} + 77025024 q^{65} + 191388656 q^{73} + 38263752 q^{81} + 422872432 q^{89} + 31368560 q^{97} + O(q^{100}) \)

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
\(3\) 46.7654 0.577350
\(4\) 0 0
\(5\) − 721.475i − 1.15436i −0.816617 0.577180i \(-0.804153\pi\)
0.816617 0.577180i \(-0.195847\pi\)
\(6\) 0 0
\(7\) 1935.65i 0.806184i 0.915159 + 0.403092i \(0.132065\pi\)
−0.915159 + 0.403092i \(0.867935\pi\)
\(8\) 0 0
\(9\) 2187.00 0.333333
\(10\) 0 0
\(11\) −14414.4 −0.984521 −0.492260 0.870448i \(-0.663829\pi\)
−0.492260 + 0.870448i \(0.663829\pi\)
\(12\) 0 0
\(13\) 2280.40i 0.0798431i 0.999203 + 0.0399216i \(0.0127108\pi\)
−0.999203 + 0.0399216i \(0.987289\pi\)
\(14\) 0 0
\(15\) − 33740.1i − 0.666470i
\(16\) 0 0
\(17\) −78065.2 −0.934678 −0.467339 0.884078i \(-0.654787\pi\)
−0.467339 + 0.884078i \(0.654787\pi\)
\(18\) 0 0
\(19\) 48459.6 0.371848 0.185924 0.982564i \(-0.440472\pi\)
0.185924 + 0.982564i \(0.440472\pi\)
\(20\) 0 0
\(21\) 90521.3i 0.465451i
\(22\) 0 0
\(23\) 222858.i 0.796375i 0.917304 + 0.398187i \(0.130361\pi\)
−0.917304 + 0.398187i \(0.869639\pi\)
\(24\) 0 0
\(25\) −129901. −0.332548
\(26\) 0 0
\(27\) 102276. 0.192450
\(28\) 0 0
\(29\) − 1.12756e6i − 1.59422i −0.603835 0.797109i \(-0.706361\pi\)
0.603835 0.797109i \(-0.293639\pi\)
\(30\) 0 0
\(31\) 828931.i 0.897576i 0.893638 + 0.448788i \(0.148144\pi\)
−0.893638 + 0.448788i \(0.851856\pi\)
\(32\) 0 0
\(33\) −674093. −0.568413
\(34\) 0 0
\(35\) 1.39652e6 0.930627
\(36\) 0 0
\(37\) − 931641.i − 0.497097i −0.968619 0.248549i \(-0.920046\pi\)
0.968619 0.248549i \(-0.0799536\pi\)
\(38\) 0 0
\(39\) 106644.i 0.0460974i
\(40\) 0 0
\(41\) 4.58800e6 1.62363 0.811816 0.583913i \(-0.198479\pi\)
0.811816 + 0.583913i \(0.198479\pi\)
\(42\) 0 0
\(43\) −2.91683e6 −0.853174 −0.426587 0.904447i \(-0.640284\pi\)
−0.426587 + 0.904447i \(0.640284\pi\)
\(44\) 0 0
\(45\) − 1.57787e6i − 0.384787i
\(46\) 0 0
\(47\) 7.52086e6i 1.54126i 0.637282 + 0.770630i \(0.280059\pi\)
−0.637282 + 0.770630i \(0.719941\pi\)
\(48\) 0 0
\(49\) 2.01807e6 0.350067
\(50\) 0 0
\(51\) −3.65075e6 −0.539637
\(52\) 0 0
\(53\) 4.79808e6i 0.608085i 0.952659 + 0.304042i \(0.0983364\pi\)
−0.952659 + 0.304042i \(0.901664\pi\)
\(54\) 0 0
\(55\) 1.03996e7i 1.13649i
\(56\) 0 0
\(57\) 2.26623e6 0.214686
\(58\) 0 0
\(59\) 8.36767e6 0.690552 0.345276 0.938501i \(-0.387785\pi\)
0.345276 + 0.938501i \(0.387785\pi\)
\(60\) 0 0
\(61\) 3.86342e6i 0.279031i 0.990220 + 0.139515i \(0.0445545\pi\)
−0.990220 + 0.139515i \(0.955446\pi\)
\(62\) 0 0
\(63\) 4.23326e6i 0.268728i
\(64\) 0 0
\(65\) 1.64525e6 0.0921677
\(66\) 0 0
\(67\) 352024. 0.0174692 0.00873459 0.999962i \(-0.497220\pi\)
0.00873459 + 0.999962i \(0.497220\pi\)
\(68\) 0 0
\(69\) 1.04220e7i 0.459787i
\(70\) 0 0
\(71\) − 2.68851e7i − 1.05798i −0.848628 0.528990i \(-0.822571\pi\)
0.848628 0.528990i \(-0.177429\pi\)
\(72\) 0 0
\(73\) 8.13722e6 0.286540 0.143270 0.989684i \(-0.454238\pi\)
0.143270 + 0.989684i \(0.454238\pi\)
\(74\) 0 0
\(75\) −6.07489e6 −0.191997
\(76\) 0 0
\(77\) − 2.79011e7i − 0.793705i
\(78\) 0 0
\(79\) 4.31004e7i 1.10655i 0.832997 + 0.553277i \(0.186623\pi\)
−0.832997 + 0.553277i \(0.813377\pi\)
\(80\) 0 0
\(81\) 4.78297e6 0.111111
\(82\) 0 0
\(83\) 3.58735e7 0.755896 0.377948 0.925827i \(-0.376630\pi\)
0.377948 + 0.925827i \(0.376630\pi\)
\(84\) 0 0
\(85\) 5.63221e7i 1.07896i
\(86\) 0 0
\(87\) − 5.27308e7i − 0.920423i
\(88\) 0 0
\(89\) 2.79238e7 0.445055 0.222528 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(90\) 0 0
\(91\) −4.41405e6 −0.0643683
\(92\) 0 0
\(93\) 3.87652e7i 0.518216i
\(94\) 0 0
\(95\) − 3.49624e7i − 0.429246i
\(96\) 0 0
\(97\) 1.07429e8 1.21349 0.606745 0.794897i \(-0.292475\pi\)
0.606745 + 0.794897i \(0.292475\pi\)
\(98\) 0 0
\(99\) −3.15242e7 −0.328174
\(100\) 0 0
\(101\) − 1.03404e8i − 0.993696i −0.867838 0.496848i \(-0.834491\pi\)
0.867838 0.496848i \(-0.165509\pi\)
\(102\) 0 0
\(103\) − 1.38564e8i − 1.23112i −0.788088 0.615562i \(-0.788929\pi\)
0.788088 0.615562i \(-0.211071\pi\)
\(104\) 0 0
\(105\) 6.53089e7 0.537298
\(106\) 0 0
\(107\) 5.74166e7 0.438028 0.219014 0.975722i \(-0.429716\pi\)
0.219014 + 0.975722i \(0.429716\pi\)
\(108\) 0 0
\(109\) − 6.98643e7i − 0.494936i −0.968896 0.247468i \(-0.920401\pi\)
0.968896 0.247468i \(-0.0795986\pi\)
\(110\) 0 0
\(111\) − 4.35685e7i − 0.286999i
\(112\) 0 0
\(113\) −3.79469e7 −0.232735 −0.116368 0.993206i \(-0.537125\pi\)
−0.116368 + 0.993206i \(0.537125\pi\)
\(114\) 0 0
\(115\) 1.60787e8 0.919303
\(116\) 0 0
\(117\) 4.98723e6i 0.0266144i
\(118\) 0 0
\(119\) − 1.51107e8i − 0.753523i
\(120\) 0 0
\(121\) −6.58494e6 −0.0307193
\(122\) 0 0
\(123\) 2.14559e8 0.937405
\(124\) 0 0
\(125\) − 1.88106e8i − 0.770480i
\(126\) 0 0
\(127\) 3.59422e8i 1.38162i 0.723035 + 0.690811i \(0.242746\pi\)
−0.723035 + 0.690811i \(0.757254\pi\)
\(128\) 0 0
\(129\) −1.36407e8 −0.492580
\(130\) 0 0
\(131\) −2.43800e8 −0.827845 −0.413923 0.910312i \(-0.635842\pi\)
−0.413923 + 0.910312i \(0.635842\pi\)
\(132\) 0 0
\(133\) 9.38007e7i 0.299778i
\(134\) 0 0
\(135\) − 7.37895e7i − 0.222157i
\(136\) 0 0
\(137\) 2.47050e8 0.701299 0.350650 0.936507i \(-0.385961\pi\)
0.350650 + 0.936507i \(0.385961\pi\)
\(138\) 0 0
\(139\) 2.81853e7 0.0755029 0.0377515 0.999287i \(-0.487980\pi\)
0.0377515 + 0.999287i \(0.487980\pi\)
\(140\) 0 0
\(141\) 3.51716e8i 0.889847i
\(142\) 0 0
\(143\) − 3.28705e7i − 0.0786072i
\(144\) 0 0
\(145\) −8.13507e8 −1.84030
\(146\) 0 0
\(147\) 9.43756e7 0.202111
\(148\) 0 0
\(149\) − 7.49164e8i − 1.51996i −0.649947 0.759979i \(-0.725209\pi\)
0.649947 0.759979i \(-0.274791\pi\)
\(150\) 0 0
\(151\) 4.67117e8i 0.898500i 0.893406 + 0.449250i \(0.148309\pi\)
−0.893406 + 0.449250i \(0.851691\pi\)
\(152\) 0 0
\(153\) −1.70729e8 −0.311559
\(154\) 0 0
\(155\) 5.98053e8 1.03613
\(156\) 0 0
\(157\) 9.89031e8i 1.62784i 0.580978 + 0.813919i \(0.302670\pi\)
−0.580978 + 0.813919i \(0.697330\pi\)
\(158\) 0 0
\(159\) 2.24384e8i 0.351078i
\(160\) 0 0
\(161\) −4.31375e8 −0.642025
\(162\) 0 0
\(163\) 1.27344e9 1.80397 0.901983 0.431772i \(-0.142112\pi\)
0.901983 + 0.431772i \(0.142112\pi\)
\(164\) 0 0
\(165\) 4.86342e8i 0.656154i
\(166\) 0 0
\(167\) 1.11871e9i 1.43831i 0.694851 + 0.719154i \(0.255470\pi\)
−0.694851 + 0.719154i \(0.744530\pi\)
\(168\) 0 0
\(169\) 8.10531e8 0.993625
\(170\) 0 0
\(171\) 1.05981e8 0.123949
\(172\) 0 0
\(173\) − 3.83147e8i − 0.427741i −0.976862 0.213871i \(-0.931393\pi\)
0.976862 0.213871i \(-0.0686071\pi\)
\(174\) 0 0
\(175\) − 2.51444e8i − 0.268095i
\(176\) 0 0
\(177\) 3.91317e8 0.398691
\(178\) 0 0
\(179\) 1.05972e9 1.03224 0.516119 0.856517i \(-0.327376\pi\)
0.516119 + 0.856517i \(0.327376\pi\)
\(180\) 0 0
\(181\) 6.48989e8i 0.604676i 0.953201 + 0.302338i \(0.0977672\pi\)
−0.953201 + 0.302338i \(0.902233\pi\)
\(182\) 0 0
\(183\) 1.80674e8i 0.161099i
\(184\) 0 0
\(185\) −6.72156e8 −0.573830
\(186\) 0 0
\(187\) 1.12526e9 0.920210
\(188\) 0 0
\(189\) 1.97970e8i 0.155150i
\(190\) 0 0
\(191\) − 9.60451e8i − 0.721675i −0.932629 0.360838i \(-0.882491\pi\)
0.932629 0.360838i \(-0.117509\pi\)
\(192\) 0 0
\(193\) −3.58783e8 −0.258585 −0.129292 0.991607i \(-0.541271\pi\)
−0.129292 + 0.991607i \(0.541271\pi\)
\(194\) 0 0
\(195\) 7.69408e7 0.0532131
\(196\) 0 0
\(197\) − 4.30337e8i − 0.285722i −0.989743 0.142861i \(-0.954370\pi\)
0.989743 0.142861i \(-0.0456302\pi\)
\(198\) 0 0
\(199\) − 1.29888e9i − 0.828240i −0.910222 0.414120i \(-0.864089\pi\)
0.910222 0.414120i \(-0.135911\pi\)
\(200\) 0 0
\(201\) 1.64625e7 0.0100858
\(202\) 0 0
\(203\) 2.18256e9 1.28523
\(204\) 0 0
\(205\) − 3.31013e9i − 1.87426i
\(206\) 0 0
\(207\) 4.87391e8i 0.265458i
\(208\) 0 0
\(209\) −6.98514e8 −0.366092
\(210\) 0 0
\(211\) −2.04305e8 −0.103074 −0.0515371 0.998671i \(-0.516412\pi\)
−0.0515371 + 0.998671i \(0.516412\pi\)
\(212\) 0 0
\(213\) − 1.25729e9i − 0.610826i
\(214\) 0 0
\(215\) 2.10442e9i 0.984870i
\(216\) 0 0
\(217\) −1.60452e9 −0.723612
\(218\) 0 0
\(219\) 3.80540e8 0.165434
\(220\) 0 0
\(221\) − 1.78020e8i − 0.0746276i
\(222\) 0 0
\(223\) − 3.90139e9i − 1.57761i −0.614642 0.788806i \(-0.710699\pi\)
0.614642 0.788806i \(-0.289301\pi\)
\(224\) 0 0
\(225\) −2.84095e8 −0.110849
\(226\) 0 0
\(227\) 3.87794e9 1.46049 0.730243 0.683188i \(-0.239407\pi\)
0.730243 + 0.683188i \(0.239407\pi\)
\(228\) 0 0
\(229\) 4.23158e9i 1.53872i 0.638813 + 0.769362i \(0.279426\pi\)
−0.638813 + 0.769362i \(0.720574\pi\)
\(230\) 0 0
\(231\) − 1.30481e9i − 0.458246i
\(232\) 0 0
\(233\) 1.60146e9 0.543367 0.271684 0.962387i \(-0.412420\pi\)
0.271684 + 0.962387i \(0.412420\pi\)
\(234\) 0 0
\(235\) 5.42612e9 1.77917
\(236\) 0 0
\(237\) 2.01561e9i 0.638870i
\(238\) 0 0
\(239\) − 4.56209e9i − 1.39821i −0.715020 0.699104i \(-0.753582\pi\)
0.715020 0.699104i \(-0.246418\pi\)
\(240\) 0 0
\(241\) 1.07036e9 0.317293 0.158646 0.987335i \(-0.449287\pi\)
0.158646 + 0.987335i \(0.449287\pi\)
\(242\) 0 0
\(243\) 2.23677e8 0.0641500
\(244\) 0 0
\(245\) − 1.45598e9i − 0.404103i
\(246\) 0 0
\(247\) 1.10507e8i 0.0296895i
\(248\) 0 0
\(249\) 1.67764e9 0.436417
\(250\) 0 0
\(251\) 7.33963e9 1.84918 0.924591 0.380962i \(-0.124407\pi\)
0.924591 + 0.380962i \(0.124407\pi\)
\(252\) 0 0
\(253\) − 3.21236e9i − 0.784047i
\(254\) 0 0
\(255\) 2.63393e9i 0.622935i
\(256\) 0 0
\(257\) 3.12468e9 0.716263 0.358131 0.933671i \(-0.383414\pi\)
0.358131 + 0.933671i \(0.383414\pi\)
\(258\) 0 0
\(259\) 1.80333e9 0.400752
\(260\) 0 0
\(261\) − 2.46598e9i − 0.531406i
\(262\) 0 0
\(263\) − 1.11900e9i − 0.233888i −0.993138 0.116944i \(-0.962690\pi\)
0.993138 0.116944i \(-0.0373099\pi\)
\(264\) 0 0
\(265\) 3.46170e9 0.701949
\(266\) 0 0
\(267\) 1.30587e9 0.256953
\(268\) 0 0
\(269\) 7.46571e9i 1.42581i 0.701260 + 0.712905i \(0.252621\pi\)
−0.701260 + 0.712905i \(0.747379\pi\)
\(270\) 0 0
\(271\) 5.57691e9i 1.03399i 0.855988 + 0.516996i \(0.172950\pi\)
−0.855988 + 0.516996i \(0.827050\pi\)
\(272\) 0 0
\(273\) −2.06425e8 −0.0371630
\(274\) 0 0
\(275\) 1.87245e9 0.327400
\(276\) 0 0
\(277\) − 3.93505e8i − 0.0668392i −0.999441 0.0334196i \(-0.989360\pi\)
0.999441 0.0334196i \(-0.0106398\pi\)
\(278\) 0 0
\(279\) 1.81287e9i 0.299192i
\(280\) 0 0
\(281\) 5.79797e9 0.929931 0.464965 0.885329i \(-0.346067\pi\)
0.464965 + 0.885329i \(0.346067\pi\)
\(282\) 0 0
\(283\) 1.02243e10 1.59400 0.797000 0.603979i \(-0.206419\pi\)
0.797000 + 0.603979i \(0.206419\pi\)
\(284\) 0 0
\(285\) − 1.63503e9i − 0.247825i
\(286\) 0 0
\(287\) 8.88075e9i 1.30895i
\(288\) 0 0
\(289\) −8.81576e8 −0.126377
\(290\) 0 0
\(291\) 5.02397e9 0.700609
\(292\) 0 0
\(293\) 8.81507e9i 1.19607i 0.801471 + 0.598033i \(0.204051\pi\)
−0.801471 + 0.598033i \(0.795949\pi\)
\(294\) 0 0
\(295\) − 6.03707e9i − 0.797146i
\(296\) 0 0
\(297\) −1.47424e9 −0.189471
\(298\) 0 0
\(299\) −5.08206e8 −0.0635850
\(300\) 0 0
\(301\) − 5.64596e9i − 0.687815i
\(302\) 0 0
\(303\) − 4.83574e9i − 0.573710i
\(304\) 0 0
\(305\) 2.78736e9 0.322102
\(306\) 0 0
\(307\) 3.21061e9 0.361438 0.180719 0.983535i \(-0.442157\pi\)
0.180719 + 0.983535i \(0.442157\pi\)
\(308\) 0 0
\(309\) − 6.48001e9i − 0.710790i
\(310\) 0 0
\(311\) 5.08177e9i 0.543218i 0.962408 + 0.271609i \(0.0875557\pi\)
−0.962408 + 0.271609i \(0.912444\pi\)
\(312\) 0 0
\(313\) −1.25663e10 −1.30927 −0.654635 0.755945i \(-0.727178\pi\)
−0.654635 + 0.755945i \(0.727178\pi\)
\(314\) 0 0
\(315\) 3.05419e9 0.310209
\(316\) 0 0
\(317\) 1.32195e10i 1.30911i 0.756012 + 0.654557i \(0.227145\pi\)
−0.756012 + 0.654557i \(0.772855\pi\)
\(318\) 0 0
\(319\) 1.62531e10i 1.56954i
\(320\) 0 0
\(321\) 2.68511e9 0.252896
\(322\) 0 0
\(323\) −3.78301e9 −0.347558
\(324\) 0 0
\(325\) − 2.96227e8i − 0.0265517i
\(326\) 0 0
\(327\) − 3.26723e9i − 0.285752i
\(328\) 0 0
\(329\) −1.45577e10 −1.24254
\(330\) 0 0
\(331\) 2.19088e10 1.82519 0.912594 0.408867i \(-0.134076\pi\)
0.912594 + 0.408867i \(0.134076\pi\)
\(332\) 0 0
\(333\) − 2.03750e9i − 0.165699i
\(334\) 0 0
\(335\) − 2.53976e8i − 0.0201657i
\(336\) 0 0
\(337\) 8.12340e9 0.629823 0.314911 0.949121i \(-0.398025\pi\)
0.314911 + 0.949121i \(0.398025\pi\)
\(338\) 0 0
\(339\) −1.77460e9 −0.134370
\(340\) 0 0
\(341\) − 1.19485e10i − 0.883682i
\(342\) 0 0
\(343\) 1.50649e10i 1.08840i
\(344\) 0 0
\(345\) 7.51925e9 0.530760
\(346\) 0 0
\(347\) 1.53179e10 1.05653 0.528265 0.849080i \(-0.322843\pi\)
0.528265 + 0.849080i \(0.322843\pi\)
\(348\) 0 0
\(349\) − 1.16917e10i − 0.788087i −0.919092 0.394044i \(-0.871076\pi\)
0.919092 0.394044i \(-0.128924\pi\)
\(350\) 0 0
\(351\) 2.33230e8i 0.0153658i
\(352\) 0 0
\(353\) −4.62886e9 −0.298109 −0.149055 0.988829i \(-0.547623\pi\)
−0.149055 + 0.988829i \(0.547623\pi\)
\(354\) 0 0
\(355\) −1.93969e10 −1.22129
\(356\) 0 0
\(357\) − 7.06657e9i − 0.435047i
\(358\) 0 0
\(359\) − 3.20147e10i − 1.92740i −0.266988 0.963700i \(-0.586028\pi\)
0.266988 0.963700i \(-0.413972\pi\)
\(360\) 0 0
\(361\) −1.46352e10 −0.861729
\(362\) 0 0
\(363\) −3.07947e8 −0.0177358
\(364\) 0 0
\(365\) − 5.87080e9i − 0.330770i
\(366\) 0 0
\(367\) − 1.27393e10i − 0.702236i −0.936331 0.351118i \(-0.885802\pi\)
0.936331 0.351118i \(-0.114198\pi\)
\(368\) 0 0
\(369\) 1.00340e10 0.541211
\(370\) 0 0
\(371\) −9.28740e9 −0.490228
\(372\) 0 0
\(373\) 6.72758e9i 0.347555i 0.984785 + 0.173777i \(0.0555974\pi\)
−0.984785 + 0.173777i \(0.944403\pi\)
\(374\) 0 0
\(375\) − 8.79683e9i − 0.444837i
\(376\) 0 0
\(377\) 2.57129e9 0.127287
\(378\) 0 0
\(379\) −1.22538e10 −0.593900 −0.296950 0.954893i \(-0.595969\pi\)
−0.296950 + 0.954893i \(0.595969\pi\)
\(380\) 0 0
\(381\) 1.68085e10i 0.797680i
\(382\) 0 0
\(383\) − 5.30566e9i − 0.246572i −0.992371 0.123286i \(-0.960657\pi\)
0.992371 0.123286i \(-0.0393433\pi\)
\(384\) 0 0
\(385\) −2.01300e10 −0.916222
\(386\) 0 0
\(387\) −6.37911e9 −0.284391
\(388\) 0 0
\(389\) − 7.15446e8i − 0.0312449i −0.999878 0.0156224i \(-0.995027\pi\)
0.999878 0.0156224i \(-0.00497298\pi\)
\(390\) 0 0
\(391\) − 1.73975e10i − 0.744354i
\(392\) 0 0
\(393\) −1.14014e10 −0.477957
\(394\) 0 0
\(395\) 3.10959e10 1.27736
\(396\) 0 0
\(397\) − 3.32363e9i − 0.133798i −0.997760 0.0668992i \(-0.978689\pi\)
0.997760 0.0668992i \(-0.0213106\pi\)
\(398\) 0 0
\(399\) 4.38662e9i 0.173077i
\(400\) 0 0
\(401\) 2.67738e10 1.03546 0.517728 0.855545i \(-0.326778\pi\)
0.517728 + 0.855545i \(0.326778\pi\)
\(402\) 0 0
\(403\) −1.89029e9 −0.0716653
\(404\) 0 0
\(405\) − 3.45079e9i − 0.128262i
\(406\) 0 0
\(407\) 1.34290e10i 0.489403i
\(408\) 0 0
\(409\) −9.94258e9 −0.355309 −0.177654 0.984093i \(-0.556851\pi\)
−0.177654 + 0.984093i \(0.556851\pi\)
\(410\) 0 0
\(411\) 1.15534e10 0.404895
\(412\) 0 0
\(413\) 1.61969e10i 0.556712i
\(414\) 0 0
\(415\) − 2.58819e10i − 0.872576i
\(416\) 0 0
\(417\) 1.31810e9 0.0435916
\(418\) 0 0
\(419\) −4.71310e10 −1.52915 −0.764576 0.644534i \(-0.777051\pi\)
−0.764576 + 0.644534i \(0.777051\pi\)
\(420\) 0 0
\(421\) 4.18839e10i 1.33327i 0.745383 + 0.666636i \(0.232267\pi\)
−0.745383 + 0.666636i \(0.767733\pi\)
\(422\) 0 0
\(423\) 1.64481e10i 0.513754i
\(424\) 0 0
\(425\) 1.01408e10 0.310825
\(426\) 0 0
\(427\) −7.47822e9 −0.224950
\(428\) 0 0
\(429\) − 1.53720e9i − 0.0453839i
\(430\) 0 0
\(431\) − 6.88707e9i − 0.199584i −0.995008 0.0997919i \(-0.968182\pi\)
0.995008 0.0997919i \(-0.0318177\pi\)
\(432\) 0 0
\(433\) −6.00742e10 −1.70898 −0.854489 0.519470i \(-0.826129\pi\)
−0.854489 + 0.519470i \(0.826129\pi\)
\(434\) 0 0
\(435\) −3.80440e10 −1.06250
\(436\) 0 0
\(437\) 1.07996e10i 0.296130i
\(438\) 0 0
\(439\) 4.83040e10i 1.30054i 0.759702 + 0.650272i \(0.225345\pi\)
−0.759702 + 0.650272i \(0.774655\pi\)
\(440\) 0 0
\(441\) 4.41351e9 0.116689
\(442\) 0 0
\(443\) 1.58299e10 0.411020 0.205510 0.978655i \(-0.434115\pi\)
0.205510 + 0.978655i \(0.434115\pi\)
\(444\) 0 0
\(445\) − 2.01463e10i − 0.513754i
\(446\) 0 0
\(447\) − 3.50349e10i − 0.877548i
\(448\) 0 0
\(449\) −1.96933e10 −0.484543 −0.242272 0.970208i \(-0.577893\pi\)
−0.242272 + 0.970208i \(0.577893\pi\)
\(450\) 0 0
\(451\) −6.61331e10 −1.59850
\(452\) 0 0
\(453\) 2.18449e10i 0.518749i
\(454\) 0 0
\(455\) 3.18463e9i 0.0743042i
\(456\) 0 0
\(457\) 1.00151e10 0.229609 0.114804 0.993388i \(-0.463376\pi\)
0.114804 + 0.993388i \(0.463376\pi\)
\(458\) 0 0
\(459\) −7.98419e9 −0.179879
\(460\) 0 0
\(461\) − 6.66757e10i − 1.47626i −0.674656 0.738132i \(-0.735708\pi\)
0.674656 0.738132i \(-0.264292\pi\)
\(462\) 0 0
\(463\) − 2.84891e10i − 0.619947i −0.950745 0.309974i \(-0.899680\pi\)
0.950745 0.309974i \(-0.100320\pi\)
\(464\) 0 0
\(465\) 2.79682e10 0.598208
\(466\) 0 0
\(467\) −7.61916e10 −1.60192 −0.800958 0.598721i \(-0.795676\pi\)
−0.800958 + 0.598721i \(0.795676\pi\)
\(468\) 0 0
\(469\) 6.81394e8i 0.0140834i
\(470\) 0 0
\(471\) 4.62524e10i 0.939833i
\(472\) 0 0
\(473\) 4.20443e10 0.839967
\(474\) 0 0
\(475\) −6.29497e9 −0.123657
\(476\) 0 0
\(477\) 1.04934e10i 0.202695i
\(478\) 0 0
\(479\) 8.65203e10i 1.64352i 0.569832 + 0.821761i \(0.307008\pi\)
−0.569832 + 0.821761i \(0.692992\pi\)
\(480\) 0 0
\(481\) 2.12451e9 0.0396898
\(482\) 0 0
\(483\) −2.01734e10 −0.370673
\(484\) 0 0
\(485\) − 7.75076e10i − 1.40080i
\(486\) 0 0
\(487\) − 6.89487e9i − 0.122577i −0.998120 0.0612886i \(-0.980479\pi\)
0.998120 0.0612886i \(-0.0195210\pi\)
\(488\) 0 0
\(489\) 5.95529e10 1.04152
\(490\) 0 0
\(491\) −3.99344e10 −0.687103 −0.343551 0.939134i \(-0.611630\pi\)
−0.343551 + 0.939134i \(0.611630\pi\)
\(492\) 0 0
\(493\) 8.80233e10i 1.49008i
\(494\) 0 0
\(495\) 2.27439e10i 0.378830i
\(496\) 0 0
\(497\) 5.20401e10 0.852928
\(498\) 0 0
\(499\) −1.20114e11 −1.93727 −0.968636 0.248484i \(-0.920068\pi\)
−0.968636 + 0.248484i \(0.920068\pi\)
\(500\) 0 0
\(501\) 5.23169e10i 0.830407i
\(502\) 0 0
\(503\) 5.58763e10i 0.872882i 0.899733 + 0.436441i \(0.143761\pi\)
−0.899733 + 0.436441i \(0.856239\pi\)
\(504\) 0 0
\(505\) −7.46037e10 −1.14708
\(506\) 0 0
\(507\) 3.79048e10 0.573670
\(508\) 0 0
\(509\) − 8.15450e10i − 1.21486i −0.794373 0.607430i \(-0.792201\pi\)
0.794373 0.607430i \(-0.207799\pi\)
\(510\) 0 0
\(511\) 1.57508e10i 0.231004i
\(512\) 0 0
\(513\) 4.95624e9 0.0715621
\(514\) 0 0
\(515\) −9.99706e10 −1.42116
\(516\) 0 0
\(517\) − 1.08408e11i − 1.51740i
\(518\) 0 0
\(519\) − 1.79180e10i − 0.246957i
\(520\) 0 0
\(521\) 1.07597e11 1.46033 0.730164 0.683272i \(-0.239444\pi\)
0.730164 + 0.683272i \(0.239444\pi\)
\(522\) 0 0
\(523\) 9.80227e10 1.31015 0.655074 0.755565i \(-0.272638\pi\)
0.655074 + 0.755565i \(0.272638\pi\)
\(524\) 0 0
\(525\) − 1.17589e10i − 0.154785i
\(526\) 0 0
\(527\) − 6.47107e10i − 0.838945i
\(528\) 0 0
\(529\) 2.86452e10 0.365787
\(530\) 0 0
\(531\) 1.83001e10 0.230184
\(532\) 0 0
\(533\) 1.04625e10i 0.129636i
\(534\) 0 0
\(535\) − 4.14246e10i − 0.505642i
\(536\) 0 0
\(537\) 4.95583e10 0.595963
\(538\) 0 0
\(539\) −2.90891e10 −0.344648
\(540\) 0 0
\(541\) 6.58826e10i 0.769098i 0.923105 + 0.384549i \(0.125643\pi\)
−0.923105 + 0.384549i \(0.874357\pi\)
\(542\) 0 0
\(543\) 3.03502e10i 0.349110i
\(544\) 0 0
\(545\) −5.04054e10 −0.571335
\(546\) 0 0
\(547\) −5.39006e10 −0.602066 −0.301033 0.953614i \(-0.597332\pi\)
−0.301033 + 0.953614i \(0.597332\pi\)
\(548\) 0 0
\(549\) 8.44930e9i 0.0930103i
\(550\) 0 0
\(551\) − 5.46411e10i − 0.592807i
\(552\) 0 0
\(553\) −8.34272e10 −0.892087
\(554\) 0 0
\(555\) −3.14336e10 −0.331301
\(556\) 0 0
\(557\) 9.99973e10i 1.03888i 0.854506 + 0.519442i \(0.173860\pi\)
−0.854506 + 0.519442i \(0.826140\pi\)
\(558\) 0 0
\(559\) − 6.65154e9i − 0.0681200i
\(560\) 0 0
\(561\) 5.26232e10 0.531283
\(562\) 0 0
\(563\) −1.53496e11 −1.52779 −0.763894 0.645342i \(-0.776715\pi\)
−0.763894 + 0.645342i \(0.776715\pi\)
\(564\) 0 0
\(565\) 2.73777e10i 0.268661i
\(566\) 0 0
\(567\) 9.25815e9i 0.0895760i
\(568\) 0 0
\(569\) −3.48473e10 −0.332445 −0.166223 0.986088i \(-0.553157\pi\)
−0.166223 + 0.986088i \(0.553157\pi\)
\(570\) 0 0
\(571\) −1.37057e11 −1.28931 −0.644653 0.764475i \(-0.722998\pi\)
−0.644653 + 0.764475i \(0.722998\pi\)
\(572\) 0 0
\(573\) − 4.49159e10i − 0.416659i
\(574\) 0 0
\(575\) − 2.89496e10i − 0.264833i
\(576\) 0 0
\(577\) −1.95056e11 −1.75977 −0.879886 0.475184i \(-0.842381\pi\)
−0.879886 + 0.475184i \(0.842381\pi\)
\(578\) 0 0
\(579\) −1.67786e10 −0.149294
\(580\) 0 0
\(581\) 6.94386e10i 0.609391i
\(582\) 0 0
\(583\) − 6.91613e10i − 0.598672i
\(584\) 0 0
\(585\) 3.59817e9 0.0307226
\(586\) 0 0
\(587\) 1.06974e10 0.0901000 0.0450500 0.998985i \(-0.485655\pi\)
0.0450500 + 0.998985i \(0.485655\pi\)
\(588\) 0 0
\(589\) 4.01696e10i 0.333762i
\(590\) 0 0
\(591\) − 2.01249e10i − 0.164962i
\(592\) 0 0
\(593\) 7.88202e9 0.0637410 0.0318705 0.999492i \(-0.489854\pi\)
0.0318705 + 0.999492i \(0.489854\pi\)
\(594\) 0 0
\(595\) −1.09020e11 −0.869837
\(596\) 0 0
\(597\) − 6.07426e10i − 0.478185i
\(598\) 0 0
\(599\) − 1.53927e11i − 1.19566i −0.801623 0.597829i \(-0.796030\pi\)
0.801623 0.597829i \(-0.203970\pi\)
\(600\) 0 0
\(601\) −1.29745e11 −0.994474 −0.497237 0.867615i \(-0.665652\pi\)
−0.497237 + 0.867615i \(0.665652\pi\)
\(602\) 0 0
\(603\) 7.69876e8 0.00582306
\(604\) 0 0
\(605\) 4.75087e9i 0.0354611i
\(606\) 0 0
\(607\) 7.20655e10i 0.530851i 0.964131 + 0.265425i \(0.0855124\pi\)
−0.964131 + 0.265425i \(0.914488\pi\)
\(608\) 0 0
\(609\) 1.02068e11 0.742030
\(610\) 0 0
\(611\) −1.71506e10 −0.123059
\(612\) 0 0
\(613\) − 1.45320e11i − 1.02916i −0.857441 0.514582i \(-0.827947\pi\)
0.857441 0.514582i \(-0.172053\pi\)
\(614\) 0 0
\(615\) − 1.54799e11i − 1.08210i
\(616\) 0 0
\(617\) 2.45016e11 1.69065 0.845326 0.534250i \(-0.179406\pi\)
0.845326 + 0.534250i \(0.179406\pi\)
\(618\) 0 0
\(619\) 1.98540e11 1.35234 0.676168 0.736747i \(-0.263639\pi\)
0.676168 + 0.736747i \(0.263639\pi\)
\(620\) 0 0
\(621\) 2.27930e10i 0.153262i
\(622\) 0 0
\(623\) 5.40506e10i 0.358797i
\(624\) 0 0
\(625\) −1.86456e11 −1.22196
\(626\) 0 0
\(627\) −3.26663e10 −0.211363
\(628\) 0 0
\(629\) 7.27287e10i 0.464626i
\(630\) 0 0
\(631\) − 4.07591e10i − 0.257103i −0.991703 0.128552i \(-0.958967\pi\)
0.991703 0.128552i \(-0.0410328\pi\)
\(632\) 0 0
\(633\) −9.55441e9 −0.0595099
\(634\) 0 0
\(635\) 2.59314e11 1.59489
\(636\) 0 0
\(637\) 4.60199e9i 0.0279504i
\(638\) 0 0
\(639\) − 5.87977e10i − 0.352660i
\(640\) 0 0
\(641\) 1.31764e11 0.780484 0.390242 0.920712i \(-0.372391\pi\)
0.390242 + 0.920712i \(0.372391\pi\)
\(642\) 0 0
\(643\) −3.05733e11 −1.78854 −0.894269 0.447529i \(-0.852304\pi\)
−0.894269 + 0.447529i \(0.852304\pi\)
\(644\) 0 0
\(645\) 9.84140e10i 0.568615i
\(646\) 0 0
\(647\) 1.54416e11i 0.881200i 0.897704 + 0.440600i \(0.145234\pi\)
−0.897704 + 0.440600i \(0.854766\pi\)
\(648\) 0 0
\(649\) −1.20615e11 −0.679863
\(650\) 0 0
\(651\) −7.50359e10 −0.417778
\(652\) 0 0
\(653\) − 1.78985e10i − 0.0984385i −0.998788 0.0492192i \(-0.984327\pi\)
0.998788 0.0492192i \(-0.0156733\pi\)
\(654\) 0 0
\(655\) 1.75896e11i 0.955632i
\(656\) 0 0
\(657\) 1.77961e10 0.0955132
\(658\) 0 0
\(659\) 1.97373e10 0.104652 0.0523259 0.998630i \(-0.483337\pi\)
0.0523259 + 0.998630i \(0.483337\pi\)
\(660\) 0 0
\(661\) 7.60626e10i 0.398442i 0.979955 + 0.199221i \(0.0638412\pi\)
−0.979955 + 0.199221i \(0.936159\pi\)
\(662\) 0 0
\(663\) − 8.32517e9i − 0.0430863i
\(664\) 0 0
\(665\) 6.76749e10 0.346052
\(666\) 0 0
\(667\) 2.51286e11 1.26960
\(668\) 0 0
\(669\) − 1.82450e11i − 0.910835i
\(670\) 0 0
\(671\) − 5.56887e10i − 0.274712i
\(672\) 0 0
\(673\) −4.07843e11 −1.98807 −0.994037 0.109048i \(-0.965220\pi\)
−0.994037 + 0.109048i \(0.965220\pi\)
\(674\) 0 0
\(675\) −1.32858e10 −0.0639989
\(676\) 0 0
\(677\) 2.18653e11i 1.04088i 0.853898 + 0.520441i \(0.174232\pi\)
−0.853898 + 0.520441i \(0.825768\pi\)
\(678\) 0 0
\(679\) 2.07946e11i 0.978297i
\(680\) 0 0
\(681\) 1.81353e11 0.843212
\(682\) 0 0
\(683\) −4.94065e10 −0.227039 −0.113520 0.993536i \(-0.536213\pi\)
−0.113520 + 0.993536i \(0.536213\pi\)
\(684\) 0 0
\(685\) − 1.78241e11i − 0.809552i
\(686\) 0 0
\(687\) 1.97891e11i 0.888382i
\(688\) 0 0
\(689\) −1.09415e10 −0.0485514
\(690\) 0 0
\(691\) 2.61217e11 1.14575 0.572875 0.819643i \(-0.305828\pi\)
0.572875 + 0.819643i \(0.305828\pi\)
\(692\) 0 0
\(693\) − 6.10198e10i − 0.264568i
\(694\) 0 0
\(695\) − 2.03350e10i − 0.0871576i
\(696\) 0 0
\(697\) −3.58163e11 −1.51757
\(698\) 0 0
\(699\) 7.48931e10 0.313713
\(700\) 0 0
\(701\) − 3.02799e10i − 0.125396i −0.998033 0.0626978i \(-0.980030\pi\)
0.998033 0.0626978i \(-0.0199704\pi\)
\(702\) 0 0
\(703\) − 4.51469e10i − 0.184844i
\(704\) 0 0
\(705\) 2.53754e11 1.02720
\(706\) 0 0
\(707\) 2.00155e11 0.801102
\(708\) 0 0
\(709\) − 4.16090e11i − 1.64665i −0.567569 0.823326i \(-0.692116\pi\)
0.567569 0.823326i \(-0.307884\pi\)
\(710\) 0 0
\(711\) 9.42606e10i 0.368852i
\(712\) 0 0
\(713\) −1.84734e11 −0.714807
\(714\) 0 0
\(715\) −2.37153e10 −0.0907410
\(716\) 0 0
\(717\) − 2.13348e11i − 0.807256i
\(718\) 0 0
\(719\) − 1.05973e11i − 0.396533i −0.980148 0.198266i \(-0.936469\pi\)
0.980148 0.198266i \(-0.0635311\pi\)
\(720\) 0 0
\(721\) 2.68212e11 0.992514
\(722\) 0 0
\(723\) 5.00556e10 0.183189
\(724\) 0 0
\(725\) 1.46472e11i 0.530154i
\(726\) 0 0
\(727\) 1.26439e11i 0.452631i 0.974054 + 0.226316i \(0.0726680\pi\)
−0.974054 + 0.226316i \(0.927332\pi\)
\(728\) 0 0
\(729\) 1.04604e10 0.0370370
\(730\) 0 0
\(731\) 2.27703e11 0.797443
\(732\) 0 0
\(733\) 1.65523e11i 0.573380i 0.958023 + 0.286690i \(0.0925550\pi\)
−0.958023 + 0.286690i \(0.907445\pi\)
\(734\) 0 0
\(735\) − 6.80896e10i − 0.233309i
\(736\) 0 0
\(737\) −5.07420e9 −0.0171988
\(738\) 0 0
\(739\) 1.19272e11 0.399908 0.199954 0.979805i \(-0.435921\pi\)
0.199954 + 0.979805i \(0.435921\pi\)
\(740\) 0 0
\(741\) 5.16791e9i 0.0171412i
\(742\) 0 0
\(743\) 3.81265e11i 1.25104i 0.780207 + 0.625521i \(0.215114\pi\)
−0.780207 + 0.625521i \(0.784886\pi\)
\(744\) 0 0
\(745\) −5.40503e11 −1.75458
\(746\) 0 0
\(747\) 7.84554e10 0.251965
\(748\) 0 0
\(749\) 1.11138e11i 0.353131i
\(750\) 0 0
\(751\) 1.41653e11i 0.445314i 0.974897 + 0.222657i \(0.0714730\pi\)
−0.974897 + 0.222657i \(0.928527\pi\)
\(752\) 0 0
\(753\) 3.43241e11 1.06763
\(754\) 0 0
\(755\) 3.37014e11 1.03719
\(756\) 0 0
\(757\) − 5.57665e10i − 0.169820i −0.996389 0.0849102i \(-0.972940\pi\)
0.996389 0.0849102i \(-0.0270603\pi\)
\(758\) 0 0
\(759\) − 1.50227e11i − 0.452670i
\(760\) 0 0
\(761\) 2.13907e11 0.637802 0.318901 0.947788i \(-0.396686\pi\)
0.318901 + 0.947788i \(0.396686\pi\)
\(762\) 0 0
\(763\) 1.35233e11 0.399010
\(764\) 0 0
\(765\) 1.23177e11i 0.359652i
\(766\) 0 0
\(767\) 1.90816e10i 0.0551358i
\(768\) 0 0
\(769\) −2.63198e11 −0.752622 −0.376311 0.926493i \(-0.622808\pi\)
−0.376311 + 0.926493i \(0.622808\pi\)
\(770\) 0 0
\(771\) 1.46127e11 0.413535
\(772\) 0 0
\(773\) 2.77626e11i 0.777576i 0.921327 + 0.388788i \(0.127106\pi\)
−0.921327 + 0.388788i \(0.872894\pi\)
\(774\) 0 0
\(775\) − 1.07679e11i − 0.298487i
\(776\) 0 0
\(777\) 8.43333e10 0.231374
\(778\) 0 0
\(779\) 2.22332e11 0.603744
\(780\) 0 0
\(781\) 3.87531e11i 1.04160i
\(782\) 0 0
\(783\) − 1.15322e11i − 0.306808i
\(784\) 0 0
\(785\) 7.13562e11 1.87911
\(786\) 0 0
\(787\) 5.72082e11 1.49128 0.745640 0.666349i \(-0.232144\pi\)
0.745640 + 0.666349i \(0.232144\pi\)
\(788\) 0 0
\(789\) − 5.23306e10i − 0.135035i
\(790\) 0 0
\(791\) − 7.34519e10i − 0.187628i
\(792\) 0 0
\(793\) −8.81014e9 −0.0222787
\(794\) 0 0
\(795\) 1.61887e11 0.405270
\(796\) 0 0
\(797\) 2.56832e11i 0.636525i 0.948003 + 0.318263i \(0.103099\pi\)
−0.948003 + 0.318263i \(0.896901\pi\)
\(798\) 0 0
\(799\) − 5.87118e11i − 1.44058i
\(800\) 0 0
\(801\) 6.10693e10 0.148352
\(802\) 0 0
\(803\) −1.17293e11 −0.282104
\(804\) 0 0
\(805\) 3.11227e11i 0.741128i
\(806\) 0 0
\(807\) 3.49137e11i 0.823192i
\(808\) 0 0
\(809\) 7.79597e11 1.82002 0.910010 0.414587i \(-0.136074\pi\)
0.910010 + 0.414587i \(0.136074\pi\)
\(810\) 0 0
\(811\) −3.07882e10 −0.0711706 −0.0355853 0.999367i \(-0.511330\pi\)
−0.0355853 + 0.999367i \(0.511330\pi\)
\(812\) 0 0
\(813\) 2.60806e11i 0.596975i
\(814\) 0 0
\(815\) − 9.18756e11i − 2.08243i
\(816\) 0 0
\(817\) −1.41348e11 −0.317251
\(818\) 0 0
\(819\) −9.65353e9 −0.0214561
\(820\) 0 0
\(821\) − 7.99169e11i − 1.75900i −0.475897 0.879501i \(-0.657877\pi\)
0.475897 0.879501i \(-0.342123\pi\)
\(822\) 0 0
\(823\) − 6.70207e11i − 1.46086i −0.682986 0.730432i \(-0.739319\pi\)
0.682986 0.730432i \(-0.260681\pi\)
\(824\) 0 0
\(825\) 8.75657e10 0.189025
\(826\) 0 0
\(827\) −2.99262e11 −0.639778 −0.319889 0.947455i \(-0.603646\pi\)
−0.319889 + 0.947455i \(0.603646\pi\)
\(828\) 0 0
\(829\) − 5.98250e11i − 1.26667i −0.773876 0.633337i \(-0.781685\pi\)
0.773876 0.633337i \(-0.218315\pi\)
\(830\) 0 0
\(831\) − 1.84024e10i − 0.0385896i
\(832\) 0 0
\(833\) −1.57541e11 −0.327200
\(834\) 0 0
\(835\) 8.07122e11 1.66032
\(836\) 0 0
\(837\) 8.47796e10i 0.172739i
\(838\) 0 0
\(839\) 6.61857e11i 1.33572i 0.744286 + 0.667862i \(0.232790\pi\)
−0.744286 + 0.667862i \(0.767210\pi\)
\(840\) 0 0
\(841\) −7.71147e11 −1.54153
\(842\) 0 0
\(843\) 2.71144e11 0.536896
\(844\) 0 0
\(845\) − 5.84778e11i − 1.14700i
\(846\) 0 0
\(847\) − 1.27461e10i − 0.0247654i
\(848\) 0 0
\(849\) 4.78144e11 0.920296
\(850\) 0 0
\(851\) 2.07624e11 0.395876
\(852\) 0 0
\(853\) 9.28630e11i 1.75407i 0.480429 + 0.877034i \(0.340481\pi\)
−0.480429 + 0.877034i \(0.659519\pi\)
\(854\) 0 0
\(855\) − 7.64627e10i − 0.143082i
\(856\) 0 0
\(857\) 6.18265e11 1.14618 0.573088 0.819494i \(-0.305745\pi\)
0.573088 + 0.819494i \(0.305745\pi\)
\(858\) 0 0
\(859\) 1.65298e11 0.303595 0.151797 0.988412i \(-0.451494\pi\)
0.151797 + 0.988412i \(0.451494\pi\)
\(860\) 0 0
\(861\) 4.15312e11i 0.755721i
\(862\) 0 0
\(863\) 8.18737e11i 1.47605i 0.674773 + 0.738025i \(0.264241\pi\)
−0.674773 + 0.738025i \(0.735759\pi\)
\(864\) 0 0
\(865\) −2.76431e11 −0.493768
\(866\) 0 0
\(867\) −4.12272e10 −0.0729638
\(868\) 0 0
\(869\) − 6.21265e11i − 1.08943i
\(870\) 0 0
\(871\) 8.02755e8i 0.00139479i
\(872\) 0 0
\(873\) 2.34948e11 0.404497
\(874\) 0 0
\(875\) 3.64106e11 0.621149
\(876\) 0 0
\(877\) − 2.52430e11i − 0.426719i −0.976974 0.213360i \(-0.931559\pi\)
0.976974 0.213360i \(-0.0684406\pi\)
\(878\) 0 0
\(879\) 4.12240e11i 0.690550i
\(880\) 0 0
\(881\) −4.66961e11 −0.775135 −0.387567 0.921841i \(-0.626685\pi\)
−0.387567 + 0.921841i \(0.626685\pi\)
\(882\) 0 0
\(883\) 4.96639e11 0.816955 0.408478 0.912768i \(-0.366060\pi\)
0.408478 + 0.912768i \(0.366060\pi\)
\(884\) 0 0
\(885\) − 2.82326e11i − 0.460233i
\(886\) 0 0
\(887\) − 1.11639e12i − 1.80352i −0.432241 0.901758i \(-0.642277\pi\)
0.432241 0.901758i \(-0.357723\pi\)
\(888\) 0 0
\(889\) −6.95714e11 −1.11384
\(890\) 0 0
\(891\) −6.89435e10 −0.109391
\(892\) 0 0
\(893\) 3.64458e11i 0.573114i
\(894\) 0 0
\(895\) − 7.64564e11i − 1.19158i
\(896\) 0 0
\(897\) −2.37664e10 −0.0367108
\(898\) 0 0
\(899\) 9.34670e11 1.43093
\(900\) 0 0
\(901\) − 3.74563e11i − 0.568363i
\(902\) 0 0
\(903\) − 2.64035e11i − 0.397110i
\(904\) 0 0
\(905\) 4.68229e11 0.698014
\(906\) 0 0
\(907\) −5.12696e11 −0.757583 −0.378792 0.925482i \(-0.623660\pi\)
−0.378792 + 0.925482i \(0.623660\pi\)
\(908\) 0 0
\(909\) − 2.26145e11i − 0.331232i
\(910\) 0 0
\(911\) 8.66460e11i 1.25798i 0.777412 + 0.628992i \(0.216532\pi\)
−0.777412 + 0.628992i \(0.783468\pi\)
\(912\) 0 0
\(913\) −5.17094e11 −0.744195
\(914\) 0 0
\(915\) 1.30352e11 0.185966
\(916\) 0 0
\(917\) − 4.71912e11i − 0.667396i
\(918\) 0 0
\(919\) 8.36874e11i 1.17327i 0.809851 + 0.586635i \(0.199548\pi\)
−0.809851 + 0.586635i \(0.800452\pi\)
\(920\) 0 0
\(921\) 1.50145e11 0.208677
\(922\) 0 0
\(923\) 6.13087e10 0.0844725
\(924\) 0 0
\(925\) 1.21021e11i 0.165309i
\(926\) 0 0
\(927\) − 3.03040e11i − 0.410375i
\(928\) 0 0
\(929\) −5.54104e11 −0.743924 −0.371962 0.928248i \(-0.621315\pi\)
−0.371962 + 0.928248i \(0.621315\pi\)
\(930\) 0 0
\(931\) 9.77945e10 0.130171
\(932\) 0 0
\(933\) 2.37651e11i 0.313627i
\(934\) 0 0
\(935\) − 8.11848e11i − 1.06225i
\(936\) 0 0
\(937\) −6.88881e11 −0.893687 −0.446844 0.894612i \(-0.647452\pi\)
−0.446844 + 0.894612i \(0.647452\pi\)
\(938\) 0 0
\(939\) −5.87666e11 −0.755907
\(940\) 0 0
\(941\) − 5.89653e10i − 0.0752035i −0.999293 0.0376018i \(-0.988028\pi\)
0.999293 0.0376018i \(-0.0119718\pi\)
\(942\) 0 0
\(943\) 1.02247e12i 1.29302i
\(944\) 0 0
\(945\) 1.42831e11 0.179099
\(946\) 0 0
\(947\) 1.00015e12 1.24355 0.621776 0.783195i \(-0.286412\pi\)
0.621776 + 0.783195i \(0.286412\pi\)
\(948\) 0 0
\(949\) 1.85561e10i 0.0228782i
\(950\) 0 0
\(951\) 6.18214e11i 0.755818i
\(952\) 0 0
\(953\) 4.67568e11 0.566856 0.283428 0.958993i \(-0.408528\pi\)
0.283428 + 0.958993i \(0.408528\pi\)
\(954\) 0 0
\(955\) −6.92942e11 −0.833073
\(956\) 0 0
\(957\) 7.60081e11i 0.906175i
\(958\) 0 0
\(959\) 4.78203e11i 0.565377i
\(960\) 0 0
\(961\) 1.65765e11 0.194357
\(962\) 0 0
\(963\) 1.25570e11 0.146009
\(964\) 0 0
\(965\) 2.58853e11i 0.298500i
\(966\) 0 0
\(967\) 9.44768e11i 1.08049i 0.841509 + 0.540243i \(0.181668\pi\)
−0.841509 + 0.540243i \(0.818332\pi\)
\(968\) 0 0
\(969\) −1.76914e11 −0.200663
\(970\) 0 0
\(971\) −9.80297e11 −1.10276 −0.551380 0.834255i \(-0.685898\pi\)
−0.551380 + 0.834255i \(0.685898\pi\)
\(972\) 0 0
\(973\) 5.45569e10i 0.0608693i
\(974\) 0 0
\(975\) − 1.38532e10i − 0.0153296i
\(976\) 0 0
\(977\) −1.82458e11 −0.200256 −0.100128 0.994975i \(-0.531925\pi\)
−0.100128 + 0.994975i \(0.531925\pi\)
\(978\) 0 0
\(979\) −4.02504e11 −0.438166
\(980\) 0 0
\(981\) − 1.52793e11i − 0.164979i
\(982\) 0 0
\(983\) 5.08194e11i 0.544271i 0.962259 + 0.272136i \(0.0877299\pi\)
−0.962259 + 0.272136i \(0.912270\pi\)
\(984\) 0 0
\(985\) −3.10478e11 −0.329826
\(986\) 0 0
\(987\) −6.80798e11 −0.717381
\(988\) 0 0
\(989\) − 6.50040e11i − 0.679446i
\(990\) 0 0
\(991\) 8.63948e11i 0.895762i 0.894093 + 0.447881i \(0.147821\pi\)
−0.894093 + 0.447881i \(0.852179\pi\)
\(992\) 0 0
\(993\) 1.02458e12 1.05377
\(994\) 0 0
\(995\) −9.37109e11 −0.956088
\(996\) 0 0
\(997\) 1.83564e12i 1.85783i 0.370293 + 0.928915i \(0.379257\pi\)
−0.370293 + 0.928915i \(0.620743\pi\)
\(998\) 0 0
\(999\) − 9.52843e10i − 0.0956664i
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 384.9.b.a.319.5 yes 8
4.3 odd 2 inner 384.9.b.a.319.1 8
8.3 odd 2 inner 384.9.b.a.319.8 yes 8
8.5 even 2 inner 384.9.b.a.319.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.9.b.a.319.1 8 4.3 odd 2 inner
384.9.b.a.319.4 yes 8 8.5 even 2 inner
384.9.b.a.319.5 yes 8 1.1 even 1 trivial
384.9.b.a.319.8 yes 8 8.3 odd 2 inner