Properties

Label 384.7.b.d.319.5
Level $384$
Weight $7$
Character 384.319
Analytic conductor $88.341$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(88.3407681100\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1731891456.1
Defining polynomial: \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.5
Root \(1.35234 - 0.780776i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.7.b.d.319.8

$q$-expansion

\(f(q)\) \(=\) \(q+15.5885 q^{3} -20.0000i q^{5} -155.727i q^{7} +243.000 q^{9} +O(q^{10})\) \(q+15.5885 q^{3} -20.0000i q^{5} -155.727i q^{7} +243.000 q^{9} -2306.46 q^{11} -3221.18i q^{13} -311.769i q^{15} -6566.73 q^{17} +3926.40 q^{19} -2427.54i q^{21} +17392.0i q^{23} +15225.0 q^{25} +3788.00 q^{27} +44713.3i q^{29} -30636.4i q^{31} -35954.2 q^{33} -3114.54 q^{35} +44158.1i q^{37} -50213.3i q^{39} +9008.18 q^{41} -25270.3 q^{43} -4860.00i q^{45} +175606. i q^{47} +93398.1 q^{49} -102365. q^{51} -96206.3i q^{53} +46129.2i q^{55} +61206.5 q^{57} +35018.3 q^{59} +86135.7i q^{61} -37841.7i q^{63} -64423.6 q^{65} +424605. q^{67} +271114. i q^{69} +36572.4i q^{71} -375121. q^{73} +237334. q^{75} +359179. i q^{77} +520323. i q^{79} +59049.0 q^{81} -594208. q^{83} +131335. i q^{85} +697011. i q^{87} +901101. q^{89} -501625. q^{91} -477574. i q^{93} -78528.0i q^{95} -1.25744e6 q^{97} -560470. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 1944 q^{9} + 4464 q^{17} + 121800 q^{25} - 116640 q^{33} - 98928 q^{41} - 278776 q^{49} - 23328 q^{57} - 230400 q^{65} + 76912 q^{73} + 472392 q^{81} + 6638832 q^{89} - 4929680 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 15.5885 0.577350
\(4\) 0 0
\(5\) − 20.0000i − 0.160000i −0.996795 0.0800000i \(-0.974508\pi\)
0.996795 0.0800000i \(-0.0254920\pi\)
\(6\) 0 0
\(7\) − 155.727i − 0.454015i −0.973893 0.227007i \(-0.927106\pi\)
0.973893 0.227007i \(-0.0728942\pi\)
\(8\) 0 0
\(9\) 243.000 0.333333
\(10\) 0 0
\(11\) −2306.46 −1.73288 −0.866439 0.499282i \(-0.833597\pi\)
−0.866439 + 0.499282i \(0.833597\pi\)
\(12\) 0 0
\(13\) − 3221.18i − 1.46617i −0.680136 0.733086i \(-0.738079\pi\)
0.680136 0.733086i \(-0.261921\pi\)
\(14\) 0 0
\(15\) − 311.769i − 0.0923760i
\(16\) 0 0
\(17\) −6566.73 −1.33660 −0.668301 0.743891i \(-0.732978\pi\)
−0.668301 + 0.743891i \(0.732978\pi\)
\(18\) 0 0
\(19\) 3926.40 0.572445 0.286223 0.958163i \(-0.407600\pi\)
0.286223 + 0.958163i \(0.407600\pi\)
\(20\) 0 0
\(21\) − 2427.54i − 0.262126i
\(22\) 0 0
\(23\) 17392.0i 1.42944i 0.699411 + 0.714720i \(0.253446\pi\)
−0.699411 + 0.714720i \(0.746554\pi\)
\(24\) 0 0
\(25\) 15225.0 0.974400
\(26\) 0 0
\(27\) 3788.00 0.192450
\(28\) 0 0
\(29\) 44713.3i 1.83334i 0.399648 + 0.916669i \(0.369132\pi\)
−0.399648 + 0.916669i \(0.630868\pi\)
\(30\) 0 0
\(31\) − 30636.4i − 1.02838i −0.857677 0.514188i \(-0.828093\pi\)
0.857677 0.514188i \(-0.171907\pi\)
\(32\) 0 0
\(33\) −35954.2 −1.00048
\(34\) 0 0
\(35\) −3114.54 −0.0726424
\(36\) 0 0
\(37\) 44158.1i 0.871776i 0.900001 + 0.435888i \(0.143566\pi\)
−0.900001 + 0.435888i \(0.856434\pi\)
\(38\) 0 0
\(39\) − 50213.3i − 0.846495i
\(40\) 0 0
\(41\) 9008.18 0.130703 0.0653515 0.997862i \(-0.479183\pi\)
0.0653515 + 0.997862i \(0.479183\pi\)
\(42\) 0 0
\(43\) −25270.3 −0.317838 −0.158919 0.987292i \(-0.550801\pi\)
−0.158919 + 0.987292i \(0.550801\pi\)
\(44\) 0 0
\(45\) − 4860.00i − 0.0533333i
\(46\) 0 0
\(47\) 175606.i 1.69140i 0.533658 + 0.845700i \(0.320817\pi\)
−0.533658 + 0.845700i \(0.679183\pi\)
\(48\) 0 0
\(49\) 93398.1 0.793871
\(50\) 0 0
\(51\) −102365. −0.771688
\(52\) 0 0
\(53\) − 96206.3i − 0.646214i −0.946363 0.323107i \(-0.895273\pi\)
0.946363 0.323107i \(-0.104727\pi\)
\(54\) 0 0
\(55\) 46129.2i 0.277261i
\(56\) 0 0
\(57\) 61206.5 0.330501
\(58\) 0 0
\(59\) 35018.3 0.170506 0.0852529 0.996359i \(-0.472830\pi\)
0.0852529 + 0.996359i \(0.472830\pi\)
\(60\) 0 0
\(61\) 86135.7i 0.379484i 0.981834 + 0.189742i \(0.0607652\pi\)
−0.981834 + 0.189742i \(0.939235\pi\)
\(62\) 0 0
\(63\) − 37841.7i − 0.151338i
\(64\) 0 0
\(65\) −64423.6 −0.234588
\(66\) 0 0
\(67\) 424605. 1.41176 0.705880 0.708332i \(-0.250552\pi\)
0.705880 + 0.708332i \(0.250552\pi\)
\(68\) 0 0
\(69\) 271114.i 0.825287i
\(70\) 0 0
\(71\) 36572.4i 0.102183i 0.998694 + 0.0510915i \(0.0162700\pi\)
−0.998694 + 0.0510915i \(0.983730\pi\)
\(72\) 0 0
\(73\) −375121. −0.964280 −0.482140 0.876094i \(-0.660140\pi\)
−0.482140 + 0.876094i \(0.660140\pi\)
\(74\) 0 0
\(75\) 237334. 0.562570
\(76\) 0 0
\(77\) 359179.i 0.786753i
\(78\) 0 0
\(79\) 520323.i 1.05534i 0.849450 + 0.527670i \(0.176934\pi\)
−0.849450 + 0.527670i \(0.823066\pi\)
\(80\) 0 0
\(81\) 59049.0 0.111111
\(82\) 0 0
\(83\) −594208. −1.03921 −0.519606 0.854406i \(-0.673921\pi\)
−0.519606 + 0.854406i \(0.673921\pi\)
\(84\) 0 0
\(85\) 131335.i 0.213856i
\(86\) 0 0
\(87\) 697011.i 1.05848i
\(88\) 0 0
\(89\) 901101. 1.27821 0.639107 0.769118i \(-0.279304\pi\)
0.639107 + 0.769118i \(0.279304\pi\)
\(90\) 0 0
\(91\) −501625. −0.665664
\(92\) 0 0
\(93\) − 477574.i − 0.593733i
\(94\) 0 0
\(95\) − 78528.0i − 0.0915912i
\(96\) 0 0
\(97\) −1.25744e6 −1.37775 −0.688875 0.724880i \(-0.741895\pi\)
−0.688875 + 0.724880i \(0.741895\pi\)
\(98\) 0 0
\(99\) −560470. −0.577626
\(100\) 0 0
\(101\) − 391040.i − 0.379539i −0.981829 0.189770i \(-0.939226\pi\)
0.981829 0.189770i \(-0.0607741\pi\)
\(102\) 0 0
\(103\) 1.36080e6i 1.24533i 0.782490 + 0.622663i \(0.213949\pi\)
−0.782490 + 0.622663i \(0.786051\pi\)
\(104\) 0 0
\(105\) −48550.9 −0.0419401
\(106\) 0 0
\(107\) 1.28620e6 1.04992 0.524960 0.851127i \(-0.324080\pi\)
0.524960 + 0.851127i \(0.324080\pi\)
\(108\) 0 0
\(109\) − 1.75763e6i − 1.35721i −0.734504 0.678605i \(-0.762585\pi\)
0.734504 0.678605i \(-0.237415\pi\)
\(110\) 0 0
\(111\) 688357.i 0.503320i
\(112\) 0 0
\(113\) −730749. −0.506446 −0.253223 0.967408i \(-0.581491\pi\)
−0.253223 + 0.967408i \(0.581491\pi\)
\(114\) 0 0
\(115\) 347840. 0.228710
\(116\) 0 0
\(117\) − 782747.i − 0.488724i
\(118\) 0 0
\(119\) 1.02262e6i 0.606837i
\(120\) 0 0
\(121\) 3.54820e6 2.00287
\(122\) 0 0
\(123\) 140424. 0.0754614
\(124\) 0 0
\(125\) − 617000.i − 0.315904i
\(126\) 0 0
\(127\) 2.59678e6i 1.26772i 0.773448 + 0.633860i \(0.218531\pi\)
−0.773448 + 0.633860i \(0.781469\pi\)
\(128\) 0 0
\(129\) −393925. −0.183504
\(130\) 0 0
\(131\) −605543. −0.269359 −0.134679 0.990889i \(-0.543000\pi\)
−0.134679 + 0.990889i \(0.543000\pi\)
\(132\) 0 0
\(133\) − 611447.i − 0.259899i
\(134\) 0 0
\(135\) − 75759.9i − 0.0307920i
\(136\) 0 0
\(137\) −1.75932e6 −0.684199 −0.342100 0.939664i \(-0.611138\pi\)
−0.342100 + 0.939664i \(0.611138\pi\)
\(138\) 0 0
\(139\) −303680. −0.113076 −0.0565382 0.998400i \(-0.518006\pi\)
−0.0565382 + 0.998400i \(0.518006\pi\)
\(140\) 0 0
\(141\) 2.73743e6i 0.976531i
\(142\) 0 0
\(143\) 7.42953e6i 2.54070i
\(144\) 0 0
\(145\) 894265. 0.293334
\(146\) 0 0
\(147\) 1.45593e6 0.458341
\(148\) 0 0
\(149\) 1.32218e6i 0.399696i 0.979827 + 0.199848i \(0.0640449\pi\)
−0.979827 + 0.199848i \(0.935955\pi\)
\(150\) 0 0
\(151\) 4.67518e6i 1.35790i 0.734184 + 0.678950i \(0.237565\pi\)
−0.734184 + 0.678950i \(0.762435\pi\)
\(152\) 0 0
\(153\) −1.59571e6 −0.445534
\(154\) 0 0
\(155\) −612727. −0.164540
\(156\) 0 0
\(157\) 2.44769e6i 0.632495i 0.948677 + 0.316247i \(0.102423\pi\)
−0.948677 + 0.316247i \(0.897577\pi\)
\(158\) 0 0
\(159\) − 1.49971e6i − 0.373092i
\(160\) 0 0
\(161\) 2.70840e6 0.648987
\(162\) 0 0
\(163\) −6.62033e6 −1.52868 −0.764340 0.644813i \(-0.776935\pi\)
−0.764340 + 0.644813i \(0.776935\pi\)
\(164\) 0 0
\(165\) 719084.i 0.160076i
\(166\) 0 0
\(167\) 7.27537e6i 1.56209i 0.624476 + 0.781044i \(0.285313\pi\)
−0.624476 + 0.781044i \(0.714687\pi\)
\(168\) 0 0
\(169\) −5.54920e6 −1.14966
\(170\) 0 0
\(171\) 954116. 0.190815
\(172\) 0 0
\(173\) − 3.72601e6i − 0.719623i −0.933025 0.359812i \(-0.882841\pi\)
0.933025 0.359812i \(-0.117159\pi\)
\(174\) 0 0
\(175\) − 2.37094e6i − 0.442392i
\(176\) 0 0
\(177\) 545881. 0.0984415
\(178\) 0 0
\(179\) 201811. 0.0351873 0.0175937 0.999845i \(-0.494399\pi\)
0.0175937 + 0.999845i \(0.494399\pi\)
\(180\) 0 0
\(181\) − 5.43440e6i − 0.916466i −0.888832 0.458233i \(-0.848483\pi\)
0.888832 0.458233i \(-0.151517\pi\)
\(182\) 0 0
\(183\) 1.34272e6i 0.219095i
\(184\) 0 0
\(185\) 883162. 0.139484
\(186\) 0 0
\(187\) 1.51459e7 2.31617
\(188\) 0 0
\(189\) − 589893.i − 0.0873752i
\(190\) 0 0
\(191\) 6.82608e6i 0.979650i 0.871821 + 0.489825i \(0.162939\pi\)
−0.871821 + 0.489825i \(0.837061\pi\)
\(192\) 0 0
\(193\) 1.30841e6 0.182001 0.0910004 0.995851i \(-0.470994\pi\)
0.0910004 + 0.995851i \(0.470994\pi\)
\(194\) 0 0
\(195\) −1.00427e6 −0.135439
\(196\) 0 0
\(197\) 7.61375e6i 0.995863i 0.867216 + 0.497932i \(0.165907\pi\)
−0.867216 + 0.497932i \(0.834093\pi\)
\(198\) 0 0
\(199\) 6.47692e6i 0.821882i 0.911662 + 0.410941i \(0.134800\pi\)
−0.911662 + 0.410941i \(0.865200\pi\)
\(200\) 0 0
\(201\) 6.61894e6 0.815080
\(202\) 0 0
\(203\) 6.96307e6 0.832362
\(204\) 0 0
\(205\) − 180164.i − 0.0209125i
\(206\) 0 0
\(207\) 4.22625e6i 0.476480i
\(208\) 0 0
\(209\) −9.05609e6 −0.991978
\(210\) 0 0
\(211\) −1.58058e7 −1.68256 −0.841278 0.540603i \(-0.818196\pi\)
−0.841278 + 0.540603i \(0.818196\pi\)
\(212\) 0 0
\(213\) 570108.i 0.0589954i
\(214\) 0 0
\(215\) 505406.i 0.0508540i
\(216\) 0 0
\(217\) −4.77091e6 −0.466898
\(218\) 0 0
\(219\) −5.84756e6 −0.556727
\(220\) 0 0
\(221\) 2.11526e7i 1.95969i
\(222\) 0 0
\(223\) 2.07099e7i 1.86751i 0.357907 + 0.933757i \(0.383491\pi\)
−0.357907 + 0.933757i \(0.616509\pi\)
\(224\) 0 0
\(225\) 3.69967e6 0.324800
\(226\) 0 0
\(227\) −7.06179e6 −0.603722 −0.301861 0.953352i \(-0.597608\pi\)
−0.301861 + 0.953352i \(0.597608\pi\)
\(228\) 0 0
\(229\) 4.64766e6i 0.387015i 0.981099 + 0.193507i \(0.0619863\pi\)
−0.981099 + 0.193507i \(0.938014\pi\)
\(230\) 0 0
\(231\) 5.59904e6i 0.454232i
\(232\) 0 0
\(233\) 1.13777e7 0.899469 0.449735 0.893162i \(-0.351519\pi\)
0.449735 + 0.893162i \(0.351519\pi\)
\(234\) 0 0
\(235\) 3.51213e6 0.270624
\(236\) 0 0
\(237\) 8.11104e6i 0.609300i
\(238\) 0 0
\(239\) − 1.93765e7i − 1.41932i −0.704542 0.709662i \(-0.748848\pi\)
0.704542 0.709662i \(-0.251152\pi\)
\(240\) 0 0
\(241\) 6.84883e6 0.489289 0.244644 0.969613i \(-0.421329\pi\)
0.244644 + 0.969613i \(0.421329\pi\)
\(242\) 0 0
\(243\) 920483. 0.0641500
\(244\) 0 0
\(245\) − 1.86796e6i − 0.127019i
\(246\) 0 0
\(247\) − 1.26477e7i − 0.839304i
\(248\) 0 0
\(249\) −9.26279e6 −0.599989
\(250\) 0 0
\(251\) 129527. 0.00819104 0.00409552 0.999992i \(-0.498696\pi\)
0.00409552 + 0.999992i \(0.498696\pi\)
\(252\) 0 0
\(253\) − 4.01140e7i − 2.47705i
\(254\) 0 0
\(255\) 2.04730e6i 0.123470i
\(256\) 0 0
\(257\) 1.18895e7 0.700427 0.350213 0.936670i \(-0.386109\pi\)
0.350213 + 0.936670i \(0.386109\pi\)
\(258\) 0 0
\(259\) 6.87661e6 0.395799
\(260\) 0 0
\(261\) 1.08653e7i 0.611113i
\(262\) 0 0
\(263\) − 1.35674e7i − 0.745811i −0.927869 0.372906i \(-0.878362\pi\)
0.927869 0.372906i \(-0.121638\pi\)
\(264\) 0 0
\(265\) −1.92413e6 −0.103394
\(266\) 0 0
\(267\) 1.40468e7 0.737977
\(268\) 0 0
\(269\) − 1.71231e7i − 0.879680i −0.898076 0.439840i \(-0.855035\pi\)
0.898076 0.439840i \(-0.144965\pi\)
\(270\) 0 0
\(271\) 320201.i 0.0160885i 0.999968 + 0.00804423i \(0.00256058\pi\)
−0.999968 + 0.00804423i \(0.997439\pi\)
\(272\) 0 0
\(273\) −7.81956e6 −0.384321
\(274\) 0 0
\(275\) −3.51159e7 −1.68852
\(276\) 0 0
\(277\) − 1.50949e7i − 0.710216i −0.934825 0.355108i \(-0.884444\pi\)
0.934825 0.355108i \(-0.115556\pi\)
\(278\) 0 0
\(279\) − 7.44463e6i − 0.342792i
\(280\) 0 0
\(281\) −4.32580e7 −1.94961 −0.974804 0.223063i \(-0.928394\pi\)
−0.974804 + 0.223063i \(0.928394\pi\)
\(282\) 0 0
\(283\) 3.22002e7 1.42069 0.710345 0.703854i \(-0.248539\pi\)
0.710345 + 0.703854i \(0.248539\pi\)
\(284\) 0 0
\(285\) − 1.22413e6i − 0.0528802i
\(286\) 0 0
\(287\) − 1.40282e6i − 0.0593411i
\(288\) 0 0
\(289\) 1.89843e7 0.786505
\(290\) 0 0
\(291\) −1.96015e7 −0.795444
\(292\) 0 0
\(293\) − 3.01307e7i − 1.19786i −0.800801 0.598931i \(-0.795592\pi\)
0.800801 0.598931i \(-0.204408\pi\)
\(294\) 0 0
\(295\) − 700366.i − 0.0272809i
\(296\) 0 0
\(297\) −8.73687e6 −0.333493
\(298\) 0 0
\(299\) 5.60228e7 2.09581
\(300\) 0 0
\(301\) 3.93527e6i 0.144303i
\(302\) 0 0
\(303\) − 6.09571e6i − 0.219127i
\(304\) 0 0
\(305\) 1.72271e6 0.0607175
\(306\) 0 0
\(307\) −2.09501e7 −0.724053 −0.362026 0.932168i \(-0.617915\pi\)
−0.362026 + 0.932168i \(0.617915\pi\)
\(308\) 0 0
\(309\) 2.12128e7i 0.718990i
\(310\) 0 0
\(311\) 4.94101e6i 0.164261i 0.996622 + 0.0821305i \(0.0261724\pi\)
−0.996622 + 0.0821305i \(0.973828\pi\)
\(312\) 0 0
\(313\) 2.67449e7 0.872182 0.436091 0.899903i \(-0.356363\pi\)
0.436091 + 0.899903i \(0.356363\pi\)
\(314\) 0 0
\(315\) −756834. −0.0242141
\(316\) 0 0
\(317\) 4.74499e7i 1.48956i 0.667311 + 0.744779i \(0.267445\pi\)
−0.667311 + 0.744779i \(0.732555\pi\)
\(318\) 0 0
\(319\) − 1.03129e8i − 3.17695i
\(320\) 0 0
\(321\) 2.00498e7 0.606172
\(322\) 0 0
\(323\) −2.57836e7 −0.765131
\(324\) 0 0
\(325\) − 4.90425e7i − 1.42864i
\(326\) 0 0
\(327\) − 2.73987e7i − 0.783585i
\(328\) 0 0
\(329\) 2.73467e7 0.767921
\(330\) 0 0
\(331\) −4.84318e7 −1.33551 −0.667754 0.744382i \(-0.732744\pi\)
−0.667754 + 0.744382i \(0.732744\pi\)
\(332\) 0 0
\(333\) 1.07304e7i 0.290592i
\(334\) 0 0
\(335\) − 8.49210e6i − 0.225881i
\(336\) 0 0
\(337\) −5.96066e7 −1.55742 −0.778709 0.627386i \(-0.784125\pi\)
−0.778709 + 0.627386i \(0.784125\pi\)
\(338\) 0 0
\(339\) −1.13913e7 −0.292397
\(340\) 0 0
\(341\) 7.06616e7i 1.78205i
\(342\) 0 0
\(343\) − 3.28657e7i − 0.814444i
\(344\) 0 0
\(345\) 5.42229e6 0.132046
\(346\) 0 0
\(347\) 2.95157e7 0.706423 0.353211 0.935544i \(-0.385090\pi\)
0.353211 + 0.935544i \(0.385090\pi\)
\(348\) 0 0
\(349\) − 5.55247e7i − 1.30620i −0.757272 0.653100i \(-0.773468\pi\)
0.757272 0.653100i \(-0.226532\pi\)
\(350\) 0 0
\(351\) − 1.22018e7i − 0.282165i
\(352\) 0 0
\(353\) −1.81396e7 −0.412386 −0.206193 0.978511i \(-0.566107\pi\)
−0.206193 + 0.978511i \(0.566107\pi\)
\(354\) 0 0
\(355\) 731449. 0.0163493
\(356\) 0 0
\(357\) 1.59410e7i 0.350358i
\(358\) 0 0
\(359\) − 8.83261e6i − 0.190900i −0.995434 0.0954499i \(-0.969571\pi\)
0.995434 0.0954499i \(-0.0304290\pi\)
\(360\) 0 0
\(361\) −3.16293e7 −0.672307
\(362\) 0 0
\(363\) 5.53110e7 1.15636
\(364\) 0 0
\(365\) 7.50242e6i 0.154285i
\(366\) 0 0
\(367\) 4.83247e7i 0.977623i 0.872389 + 0.488811i \(0.162569\pi\)
−0.872389 + 0.488811i \(0.837431\pi\)
\(368\) 0 0
\(369\) 2.18899e6 0.0435677
\(370\) 0 0
\(371\) −1.49819e7 −0.293391
\(372\) 0 0
\(373\) − 8.98844e7i − 1.73204i −0.500010 0.866020i \(-0.666670\pi\)
0.500010 0.866020i \(-0.333330\pi\)
\(374\) 0 0
\(375\) − 9.61808e6i − 0.182387i
\(376\) 0 0
\(377\) 1.44030e8 2.68799
\(378\) 0 0
\(379\) 8.60279e7 1.58024 0.790118 0.612955i \(-0.210019\pi\)
0.790118 + 0.612955i \(0.210019\pi\)
\(380\) 0 0
\(381\) 4.04798e7i 0.731919i
\(382\) 0 0
\(383\) − 6.65042e7i − 1.18373i −0.806037 0.591865i \(-0.798392\pi\)
0.806037 0.591865i \(-0.201608\pi\)
\(384\) 0 0
\(385\) 7.18357e6 0.125880
\(386\) 0 0
\(387\) −6.14069e6 −0.105946
\(388\) 0 0
\(389\) 5.66889e7i 0.963050i 0.876432 + 0.481525i \(0.159917\pi\)
−0.876432 + 0.481525i \(0.840083\pi\)
\(390\) 0 0
\(391\) − 1.14208e8i − 1.91059i
\(392\) 0 0
\(393\) −9.43949e6 −0.155514
\(394\) 0 0
\(395\) 1.04065e7 0.168854
\(396\) 0 0
\(397\) − 3.21267e7i − 0.513446i −0.966485 0.256723i \(-0.917357\pi\)
0.966485 0.256723i \(-0.0826429\pi\)
\(398\) 0 0
\(399\) − 9.53152e6i − 0.150053i
\(400\) 0 0
\(401\) −7.28625e7 −1.12998 −0.564991 0.825097i \(-0.691120\pi\)
−0.564991 + 0.825097i \(0.691120\pi\)
\(402\) 0 0
\(403\) −9.86853e7 −1.50778
\(404\) 0 0
\(405\) − 1.18098e6i − 0.0177778i
\(406\) 0 0
\(407\) − 1.01849e8i − 1.51068i
\(408\) 0 0
\(409\) −2.07185e7 −0.302823 −0.151412 0.988471i \(-0.548382\pi\)
−0.151412 + 0.988471i \(0.548382\pi\)
\(410\) 0 0
\(411\) −2.74251e7 −0.395023
\(412\) 0 0
\(413\) − 5.45330e6i − 0.0774121i
\(414\) 0 0
\(415\) 1.18842e7i 0.166274i
\(416\) 0 0
\(417\) −4.73390e6 −0.0652846
\(418\) 0 0
\(419\) −6.79561e7 −0.923818 −0.461909 0.886927i \(-0.652835\pi\)
−0.461909 + 0.886927i \(0.652835\pi\)
\(420\) 0 0
\(421\) 522580.i 0.00700336i 0.999994 + 0.00350168i \(0.00111462\pi\)
−0.999994 + 0.00350168i \(0.998885\pi\)
\(422\) 0 0
\(423\) 4.26723e7i 0.563800i
\(424\) 0 0
\(425\) −9.99784e7 −1.30239
\(426\) 0 0
\(427\) 1.34137e7 0.172291
\(428\) 0 0
\(429\) 1.15815e8i 1.46687i
\(430\) 0 0
\(431\) − 1.10533e8i − 1.38057i −0.723538 0.690285i \(-0.757485\pi\)
0.723538 0.690285i \(-0.242515\pi\)
\(432\) 0 0
\(433\) 6.65239e7 0.819435 0.409717 0.912213i \(-0.365627\pi\)
0.409717 + 0.912213i \(0.365627\pi\)
\(434\) 0 0
\(435\) 1.39402e7 0.169356
\(436\) 0 0
\(437\) 6.82879e7i 0.818276i
\(438\) 0 0
\(439\) 1.06995e8i 1.26465i 0.774702 + 0.632327i \(0.217900\pi\)
−0.774702 + 0.632327i \(0.782100\pi\)
\(440\) 0 0
\(441\) 2.26957e7 0.264624
\(442\) 0 0
\(443\) −1.01597e7 −0.116861 −0.0584304 0.998291i \(-0.518610\pi\)
−0.0584304 + 0.998291i \(0.518610\pi\)
\(444\) 0 0
\(445\) − 1.80220e7i − 0.204514i
\(446\) 0 0
\(447\) 2.06107e7i 0.230765i
\(448\) 0 0
\(449\) −4.39511e7 −0.485546 −0.242773 0.970083i \(-0.578057\pi\)
−0.242773 + 0.970083i \(0.578057\pi\)
\(450\) 0 0
\(451\) −2.07770e7 −0.226492
\(452\) 0 0
\(453\) 7.28789e7i 0.783984i
\(454\) 0 0
\(455\) 1.00325e7i 0.106506i
\(456\) 0 0
\(457\) −5.92987e6 −0.0621293 −0.0310646 0.999517i \(-0.509890\pi\)
−0.0310646 + 0.999517i \(0.509890\pi\)
\(458\) 0 0
\(459\) −2.48747e7 −0.257229
\(460\) 0 0
\(461\) 4.31145e7i 0.440069i 0.975492 + 0.220035i \(0.0706170\pi\)
−0.975492 + 0.220035i \(0.929383\pi\)
\(462\) 0 0
\(463\) − 8.09555e7i − 0.815649i −0.913060 0.407825i \(-0.866288\pi\)
0.913060 0.407825i \(-0.133712\pi\)
\(464\) 0 0
\(465\) −9.55147e6 −0.0949973
\(466\) 0 0
\(467\) 1.20309e8 1.18127 0.590633 0.806940i \(-0.298878\pi\)
0.590633 + 0.806940i \(0.298878\pi\)
\(468\) 0 0
\(469\) − 6.61225e7i − 0.640960i
\(470\) 0 0
\(471\) 3.81557e7i 0.365171i
\(472\) 0 0
\(473\) 5.82850e7 0.550774
\(474\) 0 0
\(475\) 5.97795e7 0.557791
\(476\) 0 0
\(477\) − 2.33781e7i − 0.215405i
\(478\) 0 0
\(479\) 1.54238e8i 1.40341i 0.712468 + 0.701705i \(0.247578\pi\)
−0.712468 + 0.701705i \(0.752422\pi\)
\(480\) 0 0
\(481\) 1.42241e8 1.27817
\(482\) 0 0
\(483\) 4.22198e7 0.374693
\(484\) 0 0
\(485\) 2.51487e7i 0.220440i
\(486\) 0 0
\(487\) 6.53601e7i 0.565882i 0.959137 + 0.282941i \(0.0913101\pi\)
−0.959137 + 0.282941i \(0.908690\pi\)
\(488\) 0 0
\(489\) −1.03201e8 −0.882584
\(490\) 0 0
\(491\) 4.51212e7 0.381185 0.190593 0.981669i \(-0.438959\pi\)
0.190593 + 0.981669i \(0.438959\pi\)
\(492\) 0 0
\(493\) − 2.93620e8i − 2.45044i
\(494\) 0 0
\(495\) 1.12094e7i 0.0924202i
\(496\) 0 0
\(497\) 5.69532e6 0.0463926
\(498\) 0 0
\(499\) −5.03581e7 −0.405292 −0.202646 0.979252i \(-0.564954\pi\)
−0.202646 + 0.979252i \(0.564954\pi\)
\(500\) 0 0
\(501\) 1.13412e8i 0.901872i
\(502\) 0 0
\(503\) − 4.41054e6i − 0.0346567i −0.999850 0.0173284i \(-0.994484\pi\)
0.999850 0.0173284i \(-0.00551607\pi\)
\(504\) 0 0
\(505\) −7.82080e6 −0.0607263
\(506\) 0 0
\(507\) −8.65035e7 −0.663758
\(508\) 0 0
\(509\) 2.44005e7i 0.185031i 0.995711 + 0.0925156i \(0.0294908\pi\)
−0.995711 + 0.0925156i \(0.970509\pi\)
\(510\) 0 0
\(511\) 5.84165e7i 0.437797i
\(512\) 0 0
\(513\) 1.48732e7 0.110167
\(514\) 0 0
\(515\) 2.72160e7 0.199252
\(516\) 0 0
\(517\) − 4.05029e8i − 2.93099i
\(518\) 0 0
\(519\) − 5.80827e7i − 0.415475i
\(520\) 0 0
\(521\) 6.53917e7 0.462391 0.231195 0.972907i \(-0.425736\pi\)
0.231195 + 0.972907i \(0.425736\pi\)
\(522\) 0 0
\(523\) 1.19428e8 0.834833 0.417417 0.908715i \(-0.362936\pi\)
0.417417 + 0.908715i \(0.362936\pi\)
\(524\) 0 0
\(525\) − 3.69594e7i − 0.255415i
\(526\) 0 0
\(527\) 2.01181e8i 1.37453i
\(528\) 0 0
\(529\) −1.54446e8 −1.04330
\(530\) 0 0
\(531\) 8.50945e6 0.0568352
\(532\) 0 0
\(533\) − 2.90170e7i − 0.191633i
\(534\) 0 0
\(535\) − 2.57240e7i − 0.167987i
\(536\) 0 0
\(537\) 3.14593e6 0.0203154
\(538\) 0 0
\(539\) −2.15419e8 −1.37568
\(540\) 0 0
\(541\) − 3.06540e7i − 0.193596i −0.995304 0.0967979i \(-0.969140\pi\)
0.995304 0.0967979i \(-0.0308600\pi\)
\(542\) 0 0
\(543\) − 8.47140e7i − 0.529122i
\(544\) 0 0
\(545\) −3.51525e7 −0.217153
\(546\) 0 0
\(547\) 1.60733e8 0.982071 0.491035 0.871140i \(-0.336619\pi\)
0.491035 + 0.871140i \(0.336619\pi\)
\(548\) 0 0
\(549\) 2.09310e7i 0.126495i
\(550\) 0 0
\(551\) 1.75562e8i 1.04949i
\(552\) 0 0
\(553\) 8.10284e7 0.479140
\(554\) 0 0
\(555\) 1.37671e7 0.0805313
\(556\) 0 0
\(557\) 3.11683e8i 1.80363i 0.432125 + 0.901814i \(0.357764\pi\)
−0.432125 + 0.901814i \(0.642236\pi\)
\(558\) 0 0
\(559\) 8.14003e7i 0.466005i
\(560\) 0 0
\(561\) 2.36101e8 1.33724
\(562\) 0 0
\(563\) −2.81653e8 −1.57830 −0.789149 0.614201i \(-0.789478\pi\)
−0.789149 + 0.614201i \(0.789478\pi\)
\(564\) 0 0
\(565\) 1.46150e7i 0.0810313i
\(566\) 0 0
\(567\) − 9.19553e6i − 0.0504461i
\(568\) 0 0
\(569\) 8.82677e7 0.479143 0.239571 0.970879i \(-0.422993\pi\)
0.239571 + 0.970879i \(0.422993\pi\)
\(570\) 0 0
\(571\) −1.66103e8 −0.892217 −0.446108 0.894979i \(-0.647190\pi\)
−0.446108 + 0.894979i \(0.647190\pi\)
\(572\) 0 0
\(573\) 1.06408e8i 0.565601i
\(574\) 0 0
\(575\) 2.64793e8i 1.39285i
\(576\) 0 0
\(577\) −5.26455e7 −0.274053 −0.137026 0.990567i \(-0.543754\pi\)
−0.137026 + 0.990567i \(0.543754\pi\)
\(578\) 0 0
\(579\) 2.03962e7 0.105078
\(580\) 0 0
\(581\) 9.25343e7i 0.471818i
\(582\) 0 0
\(583\) 2.21896e8i 1.11981i
\(584\) 0 0
\(585\) −1.56549e7 −0.0781959
\(586\) 0 0
\(587\) 3.88069e7 0.191865 0.0959323 0.995388i \(-0.469417\pi\)
0.0959323 + 0.995388i \(0.469417\pi\)
\(588\) 0 0
\(589\) − 1.20291e8i − 0.588689i
\(590\) 0 0
\(591\) 1.18687e8i 0.574962i
\(592\) 0 0
\(593\) −3.26100e8 −1.56382 −0.781909 0.623392i \(-0.785754\pi\)
−0.781909 + 0.623392i \(0.785754\pi\)
\(594\) 0 0
\(595\) 2.04523e7 0.0970939
\(596\) 0 0
\(597\) 1.00965e8i 0.474514i
\(598\) 0 0
\(599\) 3.03039e8i 1.41000i 0.709210 + 0.704998i \(0.249052\pi\)
−0.709210 + 0.704998i \(0.750948\pi\)
\(600\) 0 0
\(601\) −3.98916e8 −1.83763 −0.918815 0.394689i \(-0.870852\pi\)
−0.918815 + 0.394689i \(0.870852\pi\)
\(602\) 0 0
\(603\) 1.03179e8 0.470586
\(604\) 0 0
\(605\) − 7.09641e7i − 0.320459i
\(606\) 0 0
\(607\) 2.46327e8i 1.10140i 0.834703 + 0.550701i \(0.185639\pi\)
−0.834703 + 0.550701i \(0.814361\pi\)
\(608\) 0 0
\(609\) 1.08543e8 0.480565
\(610\) 0 0
\(611\) 5.65660e8 2.47989
\(612\) 0 0
\(613\) 2.16670e8i 0.940628i 0.882499 + 0.470314i \(0.155859\pi\)
−0.882499 + 0.470314i \(0.844141\pi\)
\(614\) 0 0
\(615\) − 2.80847e6i − 0.0120738i
\(616\) 0 0
\(617\) 1.27942e8 0.544699 0.272350 0.962198i \(-0.412199\pi\)
0.272350 + 0.962198i \(0.412199\pi\)
\(618\) 0 0
\(619\) −1.96738e8 −0.829499 −0.414749 0.909936i \(-0.636131\pi\)
−0.414749 + 0.909936i \(0.636131\pi\)
\(620\) 0 0
\(621\) 6.58808e7i 0.275096i
\(622\) 0 0
\(623\) − 1.40326e8i − 0.580328i
\(624\) 0 0
\(625\) 2.25551e8 0.923855
\(626\) 0 0
\(627\) −1.41171e8 −0.572719
\(628\) 0 0
\(629\) − 2.89974e8i − 1.16522i
\(630\) 0 0
\(631\) 9.81163e7i 0.390529i 0.980751 + 0.195264i \(0.0625565\pi\)
−0.980751 + 0.195264i \(0.937443\pi\)
\(632\) 0 0
\(633\) −2.46388e8 −0.971424
\(634\) 0 0
\(635\) 5.19355e7 0.202835
\(636\) 0 0
\(637\) − 3.00852e8i − 1.16395i
\(638\) 0 0
\(639\) 8.88710e6i 0.0340610i
\(640\) 0 0
\(641\) 2.85929e8 1.08564 0.542819 0.839850i \(-0.317357\pi\)
0.542819 + 0.839850i \(0.317357\pi\)
\(642\) 0 0
\(643\) 2.98146e8 1.12149 0.560745 0.827988i \(-0.310515\pi\)
0.560745 + 0.827988i \(0.310515\pi\)
\(644\) 0 0
\(645\) 7.87850e6i 0.0293606i
\(646\) 0 0
\(647\) − 3.84615e7i − 0.142008i −0.997476 0.0710042i \(-0.977380\pi\)
0.997476 0.0710042i \(-0.0226204\pi\)
\(648\) 0 0
\(649\) −8.07684e7 −0.295466
\(650\) 0 0
\(651\) −7.43711e7 −0.269564
\(652\) 0 0
\(653\) 9.67190e7i 0.347354i 0.984803 + 0.173677i \(0.0555648\pi\)
−0.984803 + 0.173677i \(0.944435\pi\)
\(654\) 0 0
\(655\) 1.21109e7i 0.0430974i
\(656\) 0 0
\(657\) −9.11545e7 −0.321427
\(658\) 0 0
\(659\) 6.49525e7 0.226955 0.113477 0.993541i \(-0.463801\pi\)
0.113477 + 0.993541i \(0.463801\pi\)
\(660\) 0 0
\(661\) − 3.29739e8i − 1.14174i −0.821042 0.570868i \(-0.806607\pi\)
0.821042 0.570868i \(-0.193393\pi\)
\(662\) 0 0
\(663\) 3.29737e8i 1.13143i
\(664\) 0 0
\(665\) −1.22289e7 −0.0415838
\(666\) 0 0
\(667\) −7.77653e8 −2.62065
\(668\) 0 0
\(669\) 3.22836e8i 1.07821i
\(670\) 0 0
\(671\) − 1.98669e8i − 0.657600i
\(672\) 0 0
\(673\) 5.17571e8 1.69795 0.848975 0.528433i \(-0.177220\pi\)
0.848975 + 0.528433i \(0.177220\pi\)
\(674\) 0 0
\(675\) 5.76722e7 0.187523
\(676\) 0 0
\(677\) 3.25473e8i 1.04894i 0.851430 + 0.524469i \(0.175736\pi\)
−0.851430 + 0.524469i \(0.824264\pi\)
\(678\) 0 0
\(679\) 1.95817e8i 0.625519i
\(680\) 0 0
\(681\) −1.10082e8 −0.348559
\(682\) 0 0
\(683\) −5.37538e8 −1.68713 −0.843563 0.537031i \(-0.819546\pi\)
−0.843563 + 0.537031i \(0.819546\pi\)
\(684\) 0 0
\(685\) 3.51864e7i 0.109472i
\(686\) 0 0
\(687\) 7.24498e7i 0.223443i
\(688\) 0 0
\(689\) −3.09898e8 −0.947461
\(690\) 0 0
\(691\) 4.64528e8 1.40792 0.703960 0.710239i \(-0.251413\pi\)
0.703960 + 0.710239i \(0.251413\pi\)
\(692\) 0 0
\(693\) 8.72804e7i 0.262251i
\(694\) 0 0
\(695\) 6.07360e6i 0.0180922i
\(696\) 0 0
\(697\) −5.91543e7 −0.174698
\(698\) 0 0
\(699\) 1.77361e8 0.519309
\(700\) 0 0
\(701\) 3.59130e7i 0.104255i 0.998640 + 0.0521275i \(0.0166002\pi\)
−0.998640 + 0.0521275i \(0.983400\pi\)
\(702\) 0 0
\(703\) 1.73382e8i 0.499044i
\(704\) 0 0
\(705\) 5.47486e7 0.156245
\(706\) 0 0
\(707\) −6.08955e7 −0.172317
\(708\) 0 0
\(709\) 1.86788e8i 0.524094i 0.965055 + 0.262047i \(0.0843976\pi\)
−0.965055 + 0.262047i \(0.915602\pi\)
\(710\) 0 0
\(711\) 1.26439e8i 0.351780i
\(712\) 0 0
\(713\) 5.32827e8 1.47000
\(714\) 0 0
\(715\) 1.48591e8 0.406512
\(716\) 0 0
\(717\) − 3.02050e8i − 0.819447i
\(718\) 0 0
\(719\) 3.76368e7i 0.101257i 0.998718 + 0.0506286i \(0.0161225\pi\)
−0.998718 + 0.0506286i \(0.983878\pi\)
\(720\) 0 0
\(721\) 2.11914e8 0.565397
\(722\) 0 0
\(723\) 1.06763e8 0.282491
\(724\) 0 0
\(725\) 6.80760e8i 1.78640i
\(726\) 0 0
\(727\) − 2.58872e8i − 0.673724i −0.941554 0.336862i \(-0.890634\pi\)
0.941554 0.336862i \(-0.109366\pi\)
\(728\) 0 0
\(729\) 1.43489e7 0.0370370
\(730\) 0 0
\(731\) 1.65943e8 0.424822
\(732\) 0 0
\(733\) 5.31872e8i 1.35050i 0.737588 + 0.675251i \(0.235965\pi\)
−0.737588 + 0.675251i \(0.764035\pi\)
\(734\) 0 0
\(735\) − 2.91186e7i − 0.0733346i
\(736\) 0 0
\(737\) −9.79335e8 −2.44641
\(738\) 0 0
\(739\) 1.70943e8 0.423564 0.211782 0.977317i \(-0.432073\pi\)
0.211782 + 0.977317i \(0.432073\pi\)
\(740\) 0 0
\(741\) − 1.97157e8i − 0.484572i
\(742\) 0 0
\(743\) − 4.23377e8i − 1.03219i −0.856531 0.516096i \(-0.827385\pi\)
0.856531 0.516096i \(-0.172615\pi\)
\(744\) 0 0
\(745\) 2.64435e7 0.0639514
\(746\) 0 0
\(747\) −1.44393e8 −0.346404
\(748\) 0 0
\(749\) − 2.00296e8i − 0.476679i
\(750\) 0 0
\(751\) 3.04170e8i 0.718118i 0.933315 + 0.359059i \(0.116902\pi\)
−0.933315 + 0.359059i \(0.883098\pi\)
\(752\) 0 0
\(753\) 2.01912e6 0.00472910
\(754\) 0 0
\(755\) 9.35037e7 0.217264
\(756\) 0 0
\(757\) − 3.06435e8i − 0.706401i −0.935548 0.353201i \(-0.885093\pi\)
0.935548 0.353201i \(-0.114907\pi\)
\(758\) 0 0
\(759\) − 6.25315e8i − 1.43012i
\(760\) 0 0
\(761\) 4.23923e8 0.961907 0.480954 0.876746i \(-0.340291\pi\)
0.480954 + 0.876746i \(0.340291\pi\)
\(762\) 0 0
\(763\) −2.73710e8 −0.616193
\(764\) 0 0
\(765\) 3.19143e7i 0.0712854i
\(766\) 0 0
\(767\) − 1.12800e8i − 0.249991i
\(768\) 0 0
\(769\) −5.19668e8 −1.14274 −0.571369 0.820693i \(-0.693588\pi\)
−0.571369 + 0.820693i \(0.693588\pi\)
\(770\) 0 0
\(771\) 1.85338e8 0.404391
\(772\) 0 0
\(773\) − 1.89126e7i − 0.0409462i −0.999790 0.0204731i \(-0.993483\pi\)
0.999790 0.0204731i \(-0.00651724\pi\)
\(774\) 0 0
\(775\) − 4.66439e8i − 1.00205i
\(776\) 0 0
\(777\) 1.07196e8 0.228515
\(778\) 0 0
\(779\) 3.53697e7 0.0748203
\(780\) 0 0
\(781\) − 8.43529e7i − 0.177071i
\(782\) 0 0
\(783\) 1.69374e8i 0.352826i
\(784\) 0 0
\(785\) 4.89537e7 0.101199
\(786\) 0 0
\(787\) −2.65036e8 −0.543726 −0.271863 0.962336i \(-0.587640\pi\)
−0.271863 + 0.962336i \(0.587640\pi\)
\(788\) 0 0
\(789\) − 2.11495e8i − 0.430594i
\(790\) 0 0
\(791\) 1.13797e8i 0.229934i
\(792\) 0 0
\(793\) 2.77459e8 0.556390
\(794\) 0 0
\(795\) −2.99942e7 −0.0596947
\(796\) 0 0
\(797\) − 6.82061e8i − 1.34725i −0.739073 0.673626i \(-0.764736\pi\)
0.739073 0.673626i \(-0.235264\pi\)
\(798\) 0 0
\(799\) − 1.15316e9i − 2.26073i
\(800\) 0 0
\(801\) 2.18968e8 0.426071
\(802\) 0 0
\(803\) 8.65203e8 1.67098
\(804\) 0 0
\(805\) − 5.41681e7i − 0.103838i
\(806\) 0 0
\(807\) − 2.66922e8i − 0.507884i
\(808\) 0 0
\(809\) 5.50511e8 1.03973 0.519865 0.854248i \(-0.325982\pi\)
0.519865 + 0.854248i \(0.325982\pi\)
\(810\) 0 0
\(811\) 3.11549e8 0.584068 0.292034 0.956408i \(-0.405668\pi\)
0.292034 + 0.956408i \(0.405668\pi\)
\(812\) 0 0
\(813\) 4.99143e6i 0.00928867i
\(814\) 0 0
\(815\) 1.32407e8i 0.244589i
\(816\) 0 0
\(817\) −9.92214e7 −0.181945
\(818\) 0 0
\(819\) −1.21895e8 −0.221888
\(820\) 0 0
\(821\) − 7.17579e8i − 1.29670i −0.761342 0.648351i \(-0.775459\pi\)
0.761342 0.648351i \(-0.224541\pi\)
\(822\) 0 0
\(823\) 5.41195e8i 0.970854i 0.874277 + 0.485427i \(0.161336\pi\)
−0.874277 + 0.485427i \(0.838664\pi\)
\(824\) 0 0
\(825\) −5.47402e8 −0.974866
\(826\) 0 0
\(827\) −8.79442e8 −1.55486 −0.777429 0.628971i \(-0.783476\pi\)
−0.777429 + 0.628971i \(0.783476\pi\)
\(828\) 0 0
\(829\) − 4.29123e8i − 0.753213i −0.926373 0.376607i \(-0.877091\pi\)
0.926373 0.376607i \(-0.122909\pi\)
\(830\) 0 0
\(831\) − 2.35306e8i − 0.410043i
\(832\) 0 0
\(833\) −6.13320e8 −1.06109
\(834\) 0 0
\(835\) 1.45507e8 0.249934
\(836\) 0 0
\(837\) − 1.16050e8i − 0.197911i
\(838\) 0 0
\(839\) 5.86636e8i 0.993305i 0.867949 + 0.496653i \(0.165438\pi\)
−0.867949 + 0.496653i \(0.834562\pi\)
\(840\) 0 0
\(841\) −1.40445e9 −2.36113
\(842\) 0 0
\(843\) −6.74325e8 −1.12561
\(844\) 0 0
\(845\) 1.10984e8i 0.183946i
\(846\) 0 0
\(847\) − 5.52552e8i − 0.909332i
\(848\) 0 0
\(849\) 5.01951e8 0.820236
\(850\) 0 0
\(851\) −7.67997e8 −1.24615
\(852\) 0 0
\(853\) 7.00029e8i 1.12790i 0.825810 + 0.563948i \(0.190718\pi\)
−0.825810 + 0.563948i \(0.809282\pi\)
\(854\) 0 0
\(855\) − 1.90823e7i − 0.0305304i
\(856\) 0 0
\(857\) 9.52114e8 1.51268 0.756339 0.654180i \(-0.226986\pi\)
0.756339 + 0.654180i \(0.226986\pi\)
\(858\) 0 0
\(859\) −4.16968e8 −0.657844 −0.328922 0.944357i \(-0.606685\pi\)
−0.328922 + 0.944357i \(0.606685\pi\)
\(860\) 0 0
\(861\) − 2.18678e7i − 0.0342606i
\(862\) 0 0
\(863\) − 4.62686e8i − 0.719869i −0.932978 0.359935i \(-0.882799\pi\)
0.932978 0.359935i \(-0.117201\pi\)
\(864\) 0 0
\(865\) −7.45201e7 −0.115140
\(866\) 0 0
\(867\) 2.95936e8 0.454089
\(868\) 0 0
\(869\) − 1.20011e9i − 1.82877i
\(870\) 0 0
\(871\) − 1.36773e9i − 2.06988i
\(872\) 0 0
\(873\) −3.05557e8 −0.459250
\(874\) 0 0
\(875\) −9.60836e7 −0.143425
\(876\) 0 0
\(877\) 6.73329e8i 0.998225i 0.866537 + 0.499113i \(0.166341\pi\)
−0.866537 + 0.499113i \(0.833659\pi\)
\(878\) 0 0
\(879\) − 4.69692e8i − 0.691586i
\(880\) 0 0
\(881\) 1.77155e8 0.259074 0.129537 0.991575i \(-0.458651\pi\)
0.129537 + 0.991575i \(0.458651\pi\)
\(882\) 0 0
\(883\) −1.35707e8 −0.197115 −0.0985574 0.995131i \(-0.531423\pi\)
−0.0985574 + 0.995131i \(0.531423\pi\)
\(884\) 0 0
\(885\) − 1.09176e7i − 0.0157506i
\(886\) 0 0
\(887\) − 1.09553e9i − 1.56984i −0.619598 0.784919i \(-0.712704\pi\)
0.619598 0.784919i \(-0.287296\pi\)
\(888\) 0 0
\(889\) 4.04389e8 0.575564
\(890\) 0 0
\(891\) −1.36194e8 −0.192542
\(892\) 0 0
\(893\) 6.89501e8i 0.968234i
\(894\) 0 0
\(895\) − 4.03623e6i − 0.00562997i
\(896\) 0 0
\(897\) 8.73309e8 1.21001
\(898\) 0 0
\(899\) 1.36985e9 1.88536
\(900\) 0 0
\(901\) 6.31761e8i 0.863731i
\(902\) 0 0
\(903\) 6.13448e7i 0.0833133i
\(904\) 0 0
\(905\) −1.08688e8 −0.146635
\(906\) 0 0
\(907\) 1.34159e9 1.79803 0.899015 0.437917i \(-0.144284\pi\)
0.899015 + 0.437917i \(0.144284\pi\)
\(908\) 0 0
\(909\) − 9.50227e7i − 0.126513i
\(910\) 0 0
\(911\) − 1.16107e9i − 1.53569i −0.640638 0.767843i \(-0.721330\pi\)
0.640638 0.767843i \(-0.278670\pi\)
\(912\) 0 0
\(913\) 1.37052e9 1.80083
\(914\) 0 0
\(915\) 2.68545e7 0.0350553
\(916\) 0 0
\(917\) 9.42995e7i 0.122293i
\(918\) 0 0
\(919\) − 1.03538e9i − 1.33399i −0.745061 0.666996i \(-0.767580\pi\)
0.745061 0.666996i \(-0.232420\pi\)
\(920\) 0 0
\(921\) −3.26579e8 −0.418032
\(922\) 0 0
\(923\) 1.17806e8 0.149818
\(924\) 0 0
\(925\) 6.72307e8i 0.849459i
\(926\) 0 0
\(927\) 3.30675e8i 0.415109i
\(928\) 0 0
\(929\) −1.47698e9 −1.84217 −0.921083 0.389367i \(-0.872694\pi\)
−0.921083 + 0.389367i \(0.872694\pi\)
\(930\) 0 0
\(931\) 3.66718e8 0.454447
\(932\) 0 0
\(933\) 7.70227e7i 0.0948361i
\(934\) 0 0
\(935\) − 3.02918e8i − 0.370587i
\(936\) 0 0
\(937\) 4.90868e8 0.596686 0.298343 0.954459i \(-0.403566\pi\)
0.298343 + 0.954459i \(0.403566\pi\)
\(938\) 0 0
\(939\) 4.16911e8 0.503555
\(940\) 0 0
\(941\) − 1.54536e8i − 0.185464i −0.995691 0.0927321i \(-0.970440\pi\)
0.995691 0.0927321i \(-0.0295600\pi\)
\(942\) 0 0
\(943\) 1.56670e8i 0.186832i
\(944\) 0 0
\(945\) −1.17979e7 −0.0139800
\(946\) 0 0
\(947\) −1.24667e9 −1.46792 −0.733961 0.679192i \(-0.762331\pi\)
−0.733961 + 0.679192i \(0.762331\pi\)
\(948\) 0 0
\(949\) 1.20833e9i 1.41380i
\(950\) 0 0
\(951\) 7.39671e8i 0.859997i
\(952\) 0 0
\(953\) −4.21385e8 −0.486856 −0.243428 0.969919i \(-0.578272\pi\)
−0.243428 + 0.969919i \(0.578272\pi\)
\(954\) 0 0
\(955\) 1.36522e8 0.156744
\(956\) 0 0
\(957\) − 1.60763e9i − 1.83421i
\(958\) 0 0
\(959\) 2.73973e8i 0.310637i
\(960\) 0 0
\(961\) −5.10826e7 −0.0575576
\(962\) 0 0
\(963\) 3.12546e8 0.349973
\(964\) 0 0
\(965\) − 2.61683e7i − 0.0291201i
\(966\) 0 0
\(967\) 1.35010e9i 1.49309i 0.665334 + 0.746546i \(0.268289\pi\)
−0.665334 + 0.746546i \(0.731711\pi\)
\(968\) 0 0
\(969\) −4.01927e8 −0.441749
\(970\) 0 0
\(971\) −2.75696e8 −0.301143 −0.150571 0.988599i \(-0.548111\pi\)
−0.150571 + 0.988599i \(0.548111\pi\)
\(972\) 0 0
\(973\) 4.72912e7i 0.0513383i
\(974\) 0 0
\(975\) − 7.64497e8i − 0.824825i
\(976\) 0 0
\(977\) −4.86639e8 −0.521823 −0.260911 0.965363i \(-0.584023\pi\)
−0.260911 + 0.965363i \(0.584023\pi\)
\(978\) 0 0
\(979\) −2.07836e9 −2.21499
\(980\) 0 0
\(981\) − 4.27103e8i − 0.452403i
\(982\) 0 0
\(983\) − 1.34935e9i − 1.42058i −0.703909 0.710290i \(-0.748564\pi\)
0.703909 0.710290i \(-0.251436\pi\)
\(984\) 0 0
\(985\) 1.52275e8 0.159338
\(986\) 0 0
\(987\) 4.26292e8 0.443359
\(988\) 0 0
\(989\) − 4.39501e8i − 0.454330i
\(990\) 0 0
\(991\) 5.76213e8i 0.592056i 0.955179 + 0.296028i \(0.0956621\pi\)
−0.955179 + 0.296028i \(0.904338\pi\)
\(992\) 0 0
\(993\) −7.54976e8 −0.771055
\(994\) 0 0
\(995\) 1.29538e8 0.131501
\(996\) 0 0
\(997\) 8.84682e7i 0.0892692i 0.999003 + 0.0446346i \(0.0142124\pi\)
−0.999003 + 0.0446346i \(0.985788\pi\)
\(998\) 0 0
\(999\) 1.67271e8i 0.167773i
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 384.7.b.d.319.5 yes 8
4.3 odd 2 inner 384.7.b.d.319.2 8
8.3 odd 2 inner 384.7.b.d.319.8 yes 8
8.5 even 2 inner 384.7.b.d.319.3 yes 8
16.3 odd 4 768.7.g.c.511.1 4
16.5 even 4 768.7.g.e.511.2 4
16.11 odd 4 768.7.g.e.511.3 4
16.13 even 4 768.7.g.c.511.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.7.b.d.319.2 8 4.3 odd 2 inner
384.7.b.d.319.3 yes 8 8.5 even 2 inner
384.7.b.d.319.5 yes 8 1.1 even 1 trivial
384.7.b.d.319.8 yes 8 8.3 odd 2 inner
768.7.g.c.511.1 4 16.3 odd 4
768.7.g.c.511.4 4 16.13 even 4
768.7.g.e.511.2 4 16.5 even 4
768.7.g.e.511.3 4 16.11 odd 4