Properties

Label 384.9.b.a.319.3
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 319.3
Root \(1.36603 - 2.01178i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.9.b.a.319.2

$q$-expansion

\(f(q)\) \(=\) \(q-46.7654 q^{3} +402.176i q^{5} -1935.65i q^{7} +2187.00 q^{9} +O(q^{10})\) \(q-46.7654 q^{3} +402.176i q^{5} -1935.65i q^{7} +2187.00 q^{9} -20109.3 q^{11} -43789.3i q^{13} -18807.9i q^{15} +101325. q^{17} -13935.8 q^{19} +90521.3i q^{21} -433050. i q^{23} +228879. q^{25} -102276. q^{27} -467978. i q^{29} +1.48318e6i q^{31} +940421. q^{33} +778471. q^{35} -2.57554e6i q^{37} +2.04782e6i q^{39} -3.84335e6 q^{41} -1.88197e6 q^{43} +879559. i q^{45} +5.93142e6i q^{47} +2.01807e6 q^{49} -4.73851e6 q^{51} +8.07128e6i q^{53} -8.08750e6i q^{55} +651715. q^{57} +1.78704e7 q^{59} -2.11682e7i q^{61} -4.23326e6i q^{63} +1.76110e7 q^{65} -7.11867e6 q^{67} +2.02518e7i q^{69} +1.36430e7i q^{71} +3.97099e7 q^{73} -1.07036e7 q^{75} +3.89246e7i q^{77} +829701. i q^{79} +4.78297e6 q^{81} +1.45656e7 q^{83} +4.07506e7i q^{85} +2.18851e7i q^{87} +7.77943e7 q^{89} -8.47607e7 q^{91} -6.93615e7i q^{93} -5.60466e6i q^{95} -9.95872e7 q^{97} -4.39791e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 17496 q^{9} + O(q^{10}) \) \( 8 q + 17496 q^{9} + 93040 q^{17} + 395912 q^{25} + 1065312 q^{33} + 2978576 q^{41} + 16144520 q^{49} + 11671776 q^{57} + 77025024 q^{65} + 191388656 q^{73} + 38263752 q^{81} + 422872432 q^{89} + 31368560 q^{97} + O(q^{100}) \)

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
\(5\) 402.176i 0.643482i 0.946828 + 0.321741i \(0.104268\pi\)
−0.946828 + 0.321741i \(0.895732\pi\)
\(6\) 0 0
\(7\) − 1935.65i − 0.806184i −0.915159 0.403092i \(-0.867935\pi\)
0.915159 0.403092i \(-0.132065\pi\)
\(8\) 0 0
\(9\) 2187.00 0.333333
\(10\) 0 0
\(11\) −20109.3 −1.37350 −0.686748 0.726896i \(-0.740962\pi\)
−0.686748 + 0.726896i \(0.740962\pi\)
\(12\) 0 0
\(13\) − 43789.3i − 1.53319i −0.642134 0.766593i \(-0.721951\pi\)
0.642134 0.766593i \(-0.278049\pi\)
\(14\) 0 0
\(15\) − 18807.9i − 0.371514i
\(16\) 0 0
\(17\) 101325. 1.21317 0.606585 0.795018i \(-0.292539\pi\)
0.606585 + 0.795018i \(0.292539\pi\)
\(18\) 0 0
\(19\) −13935.8 −0.106935 −0.0534674 0.998570i \(-0.517027\pi\)
−0.0534674 + 0.998570i \(0.517027\pi\)
\(20\) 0 0
\(21\) 90521.3i 0.465451i
\(22\) 0 0
\(23\) − 433050.i − 1.54749i −0.633499 0.773743i \(-0.718382\pi\)
0.633499 0.773743i \(-0.281618\pi\)
\(24\) 0 0
\(25\) 228879. 0.585931
\(26\) 0 0
\(27\) −102276. −0.192450
\(28\) 0 0
\(29\) − 467978.i − 0.661657i −0.943691 0.330829i \(-0.892672\pi\)
0.943691 0.330829i \(-0.107328\pi\)
\(30\) 0 0
\(31\) 1.48318e6i 1.60601i 0.595975 + 0.803003i \(0.296766\pi\)
−0.595975 + 0.803003i \(0.703234\pi\)
\(32\) 0 0
\(33\) 940421. 0.792988
\(34\) 0 0
\(35\) 778471. 0.518765
\(36\) 0 0
\(37\) − 2.57554e6i − 1.37424i −0.726545 0.687119i \(-0.758875\pi\)
0.726545 0.687119i \(-0.241125\pi\)
\(38\) 0 0
\(39\) 2.04782e6i 0.885185i
\(40\) 0 0
\(41\) −3.84335e6 −1.36011 −0.680057 0.733160i \(-0.738045\pi\)
−0.680057 + 0.733160i \(0.738045\pi\)
\(42\) 0 0
\(43\) −1.88197e6 −0.550475 −0.275238 0.961376i \(-0.588757\pi\)
−0.275238 + 0.961376i \(0.588757\pi\)
\(44\) 0 0
\(45\) 879559.i 0.214494i
\(46\) 0 0
\(47\) 5.93142e6i 1.21553i 0.794115 + 0.607767i \(0.207935\pi\)
−0.794115 + 0.607767i \(0.792065\pi\)
\(48\) 0 0
\(49\) 2.01807e6 0.350067
\(50\) 0 0
\(51\) −4.73851e6 −0.700425
\(52\) 0 0
\(53\) 8.07128e6i 1.02291i 0.859309 + 0.511457i \(0.170894\pi\)
−0.859309 + 0.511457i \(0.829106\pi\)
\(54\) 0 0
\(55\) − 8.08750e6i − 0.883819i
\(56\) 0 0
\(57\) 651715. 0.0617388
\(58\) 0 0
\(59\) 1.78704e7 1.47477 0.737386 0.675471i \(-0.236060\pi\)
0.737386 + 0.675471i \(0.236060\pi\)
\(60\) 0 0
\(61\) − 2.11682e7i − 1.52885i −0.644715 0.764423i \(-0.723024\pi\)
0.644715 0.764423i \(-0.276976\pi\)
\(62\) 0 0
\(63\) − 4.23326e6i − 0.268728i
\(64\) 0 0
\(65\) 1.76110e7 0.986576
\(66\) 0 0
\(67\) −7.11867e6 −0.353264 −0.176632 0.984277i \(-0.556520\pi\)
−0.176632 + 0.984277i \(0.556520\pi\)
\(68\) 0 0
\(69\) 2.02518e7i 0.893442i
\(70\) 0 0
\(71\) 1.36430e7i 0.536879i 0.963297 + 0.268439i \(0.0865079\pi\)
−0.963297 + 0.268439i \(0.913492\pi\)
\(72\) 0 0
\(73\) 3.97099e7 1.39832 0.699162 0.714963i \(-0.253557\pi\)
0.699162 + 0.714963i \(0.253557\pi\)
\(74\) 0 0
\(75\) −1.07036e7 −0.338288
\(76\) 0 0
\(77\) 3.89246e7i 1.10729i
\(78\) 0 0
\(79\) 829701.i 0.0213017i 0.999943 + 0.0106508i \(0.00339033\pi\)
−0.999943 + 0.0106508i \(0.996610\pi\)
\(80\) 0 0
\(81\) 4.78297e6 0.111111
\(82\) 0 0
\(83\) 1.45656e7 0.306914 0.153457 0.988155i \(-0.450959\pi\)
0.153457 + 0.988155i \(0.450959\pi\)
\(84\) 0 0
\(85\) 4.07506e7i 0.780653i
\(86\) 0 0
\(87\) 2.18851e7i 0.382008i
\(88\) 0 0
\(89\) 7.77943e7 1.23990 0.619952 0.784640i \(-0.287152\pi\)
0.619952 + 0.784640i \(0.287152\pi\)
\(90\) 0 0
\(91\) −8.47607e7 −1.23603
\(92\) 0 0
\(93\) − 6.93615e7i − 0.927228i
\(94\) 0 0
\(95\) − 5.60466e6i − 0.0688105i
\(96\) 0 0
\(97\) −9.95872e7 −1.12491 −0.562454 0.826829i \(-0.690143\pi\)
−0.562454 + 0.826829i \(0.690143\pi\)
\(98\) 0 0
\(99\) −4.39791e7 −0.457832
\(100\) 0 0
\(101\) − 9.82839e7i − 0.944489i −0.881468 0.472244i \(-0.843444\pi\)
0.881468 0.472244i \(-0.156556\pi\)
\(102\) 0 0
\(103\) − 1.83029e8i − 1.62619i −0.582130 0.813096i \(-0.697780\pi\)
0.582130 0.813096i \(-0.302220\pi\)
\(104\) 0 0
\(105\) −3.64055e7 −0.299509
\(106\) 0 0
\(107\) −4.13285e7 −0.315293 −0.157647 0.987496i \(-0.550391\pi\)
−0.157647 + 0.987496i \(0.550391\pi\)
\(108\) 0 0
\(109\) − 3.44974e7i − 0.244388i −0.992506 0.122194i \(-0.961007\pi\)
0.992506 0.122194i \(-0.0389930\pi\)
\(110\) 0 0
\(111\) 1.20446e8i 0.793416i
\(112\) 0 0
\(113\) −1.89315e7 −0.116110 −0.0580552 0.998313i \(-0.518490\pi\)
−0.0580552 + 0.998313i \(0.518490\pi\)
\(114\) 0 0
\(115\) 1.74162e8 0.995779
\(116\) 0 0
\(117\) − 9.57672e7i − 0.511062i
\(118\) 0 0
\(119\) − 1.96130e8i − 0.978039i
\(120\) 0 0
\(121\) 1.90027e8 0.886490
\(122\) 0 0
\(123\) 1.79736e8 0.785262
\(124\) 0 0
\(125\) 2.49150e8i 1.02052i
\(126\) 0 0
\(127\) − 2.21326e8i − 0.850779i −0.905010 0.425389i \(-0.860137\pi\)
0.905010 0.425389i \(-0.139863\pi\)
\(128\) 0 0
\(129\) 8.80108e7 0.317817
\(130\) 0 0
\(131\) −5.49555e8 −1.86606 −0.933030 0.359799i \(-0.882845\pi\)
−0.933030 + 0.359799i \(0.882845\pi\)
\(132\) 0 0
\(133\) 2.69749e7i 0.0862091i
\(134\) 0 0
\(135\) − 4.11329e7i − 0.123838i
\(136\) 0 0
\(137\) −1.68597e8 −0.478595 −0.239298 0.970946i \(-0.576917\pi\)
−0.239298 + 0.970946i \(0.576917\pi\)
\(138\) 0 0
\(139\) −4.01732e8 −1.07616 −0.538080 0.842894i \(-0.680850\pi\)
−0.538080 + 0.842894i \(0.680850\pi\)
\(140\) 0 0
\(141\) − 2.77385e8i − 0.701789i
\(142\) 0 0
\(143\) 8.80574e8i 2.10582i
\(144\) 0 0
\(145\) 1.88209e8 0.425764
\(146\) 0 0
\(147\) −9.43756e7 −0.202111
\(148\) 0 0
\(149\) 4.43104e7i 0.0899002i 0.998989 + 0.0449501i \(0.0143129\pi\)
−0.998989 + 0.0449501i \(0.985687\pi\)
\(150\) 0 0
\(151\) 7.21307e8i 1.38743i 0.720247 + 0.693717i \(0.244028\pi\)
−0.720247 + 0.693717i \(0.755972\pi\)
\(152\) 0 0
\(153\) 2.21598e8 0.404390
\(154\) 0 0
\(155\) −5.96499e8 −1.03343
\(156\) 0 0
\(157\) 2.39344e8i 0.393934i 0.980410 + 0.196967i \(0.0631091\pi\)
−0.980410 + 0.196967i \(0.936891\pi\)
\(158\) 0 0
\(159\) − 3.77456e8i − 0.590579i
\(160\) 0 0
\(161\) −8.38233e8 −1.24756
\(162\) 0 0
\(163\) −3.87688e7 −0.0549202 −0.0274601 0.999623i \(-0.508742\pi\)
−0.0274601 + 0.999623i \(0.508742\pi\)
\(164\) 0 0
\(165\) 3.78215e8i 0.510273i
\(166\) 0 0
\(167\) − 2.25595e7i − 0.0290044i −0.999895 0.0145022i \(-0.995384\pi\)
0.999895 0.0145022i \(-0.00461635\pi\)
\(168\) 0 0
\(169\) −1.10177e9 −1.35066
\(170\) 0 0
\(171\) −3.04777e7 −0.0356449
\(172\) 0 0
\(173\) 1.04991e9i 1.17210i 0.810273 + 0.586052i \(0.199319\pi\)
−0.810273 + 0.586052i \(0.800681\pi\)
\(174\) 0 0
\(175\) − 4.43030e8i − 0.472369i
\(176\) 0 0
\(177\) −8.35714e8 −0.851460
\(178\) 0 0
\(179\) −7.39481e8 −0.720302 −0.360151 0.932894i \(-0.617275\pi\)
−0.360151 + 0.932894i \(0.617275\pi\)
\(180\) 0 0
\(181\) − 1.75347e9i − 1.63374i −0.576821 0.816870i \(-0.695707\pi\)
0.576821 0.816870i \(-0.304293\pi\)
\(182\) 0 0
\(183\) 9.89937e8i 0.882680i
\(184\) 0 0
\(185\) 1.03582e9 0.884296
\(186\) 0 0
\(187\) −2.03758e9 −1.66628
\(188\) 0 0
\(189\) 1.97970e8i 0.155150i
\(190\) 0 0
\(191\) − 1.91581e9i − 1.43953i −0.694219 0.719764i \(-0.744250\pi\)
0.694219 0.719764i \(-0.255750\pi\)
\(192\) 0 0
\(193\) −1.58868e9 −1.14501 −0.572504 0.819902i \(-0.694028\pi\)
−0.572504 + 0.819902i \(0.694028\pi\)
\(194\) 0 0
\(195\) −8.23585e8 −0.569600
\(196\) 0 0
\(197\) 2.10608e9i 1.39833i 0.714961 + 0.699165i \(0.246445\pi\)
−0.714961 + 0.699165i \(0.753555\pi\)
\(198\) 0 0
\(199\) 1.88646e8i 0.120291i 0.998190 + 0.0601456i \(0.0191565\pi\)
−0.998190 + 0.0601456i \(0.980843\pi\)
\(200\) 0 0
\(201\) 3.32907e8 0.203957
\(202\) 0 0
\(203\) −9.05840e8 −0.533418
\(204\) 0 0
\(205\) − 1.54570e9i − 0.875208i
\(206\) 0 0
\(207\) − 9.47081e8i − 0.515829i
\(208\) 0 0
\(209\) 2.80241e8 0.146874
\(210\) 0 0
\(211\) −3.09009e9 −1.55898 −0.779490 0.626414i \(-0.784522\pi\)
−0.779490 + 0.626414i \(0.784522\pi\)
\(212\) 0 0
\(213\) − 6.38019e8i − 0.309967i
\(214\) 0 0
\(215\) − 7.56881e8i − 0.354221i
\(216\) 0 0
\(217\) 2.87092e9 1.29474
\(218\) 0 0
\(219\) −1.85705e9 −0.807323
\(220\) 0 0
\(221\) − 4.43696e9i − 1.86002i
\(222\) 0 0
\(223\) 3.32988e9i 1.34651i 0.739411 + 0.673255i \(0.235104\pi\)
−0.739411 + 0.673255i \(0.764896\pi\)
\(224\) 0 0
\(225\) 5.00559e8 0.195310
\(226\) 0 0
\(227\) 2.70537e8 0.101888 0.0509439 0.998702i \(-0.483777\pi\)
0.0509439 + 0.998702i \(0.483777\pi\)
\(228\) 0 0
\(229\) 7.89958e8i 0.287251i 0.989632 + 0.143626i \(0.0458762\pi\)
−0.989632 + 0.143626i \(0.954124\pi\)
\(230\) 0 0
\(231\) − 1.82033e9i − 0.639295i
\(232\) 0 0
\(233\) 1.01558e9 0.344579 0.172289 0.985046i \(-0.444884\pi\)
0.172289 + 0.985046i \(0.444884\pi\)
\(234\) 0 0
\(235\) −2.38547e9 −0.782174
\(236\) 0 0
\(237\) − 3.88013e7i − 0.0122985i
\(238\) 0 0
\(239\) 1.85103e9i 0.567313i 0.958926 + 0.283656i \(0.0915475\pi\)
−0.958926 + 0.283656i \(0.908452\pi\)
\(240\) 0 0
\(241\) 9.15721e8 0.271453 0.135727 0.990746i \(-0.456663\pi\)
0.135727 + 0.990746i \(0.456663\pi\)
\(242\) 0 0
\(243\) −2.23677e8 −0.0641500
\(244\) 0 0
\(245\) 8.11617e8i 0.225261i
\(246\) 0 0
\(247\) 6.10241e8i 0.163951i
\(248\) 0 0
\(249\) −6.81166e8 −0.177197
\(250\) 0 0
\(251\) −6.39220e9 −1.61048 −0.805240 0.592949i \(-0.797964\pi\)
−0.805240 + 0.592949i \(0.797964\pi\)
\(252\) 0 0
\(253\) 8.70836e9i 2.12547i
\(254\) 0 0
\(255\) − 1.90572e9i − 0.450710i
\(256\) 0 0
\(257\) 6.28995e8 0.144183 0.0720916 0.997398i \(-0.477033\pi\)
0.0720916 + 0.997398i \(0.477033\pi\)
\(258\) 0 0
\(259\) −4.98535e9 −1.10789
\(260\) 0 0
\(261\) − 1.02347e9i − 0.220552i
\(262\) 0 0
\(263\) − 4.04520e9i − 0.845507i −0.906245 0.422753i \(-0.861064\pi\)
0.906245 0.422753i \(-0.138936\pi\)
\(264\) 0 0
\(265\) −3.24607e9 −0.658226
\(266\) 0 0
\(267\) −3.63808e9 −0.715859
\(268\) 0 0
\(269\) − 5.27436e9i − 1.00730i −0.863907 0.503652i \(-0.831989\pi\)
0.863907 0.503652i \(-0.168011\pi\)
\(270\) 0 0
\(271\) − 9.68764e9i − 1.79614i −0.439851 0.898071i \(-0.644969\pi\)
0.439851 0.898071i \(-0.355031\pi\)
\(272\) 0 0
\(273\) 3.96387e9 0.713622
\(274\) 0 0
\(275\) −4.60262e9 −0.804774
\(276\) 0 0
\(277\) − 1.41820e8i − 0.0240889i −0.999927 0.0120445i \(-0.996166\pi\)
0.999927 0.0120445i \(-0.00383397\pi\)
\(278\) 0 0
\(279\) 3.24371e9i 0.535335i
\(280\) 0 0
\(281\) −5.45914e9 −0.875587 −0.437793 0.899076i \(-0.644240\pi\)
−0.437793 + 0.899076i \(0.644240\pi\)
\(282\) 0 0
\(283\) −8.18396e9 −1.27590 −0.637952 0.770077i \(-0.720218\pi\)
−0.637952 + 0.770077i \(0.720218\pi\)
\(284\) 0 0
\(285\) 2.62104e8i 0.0397278i
\(286\) 0 0
\(287\) 7.43938e9i 1.09650i
\(288\) 0 0
\(289\) 3.29105e9 0.471783
\(290\) 0 0
\(291\) 4.65723e9 0.649466
\(292\) 0 0
\(293\) 2.39451e9i 0.324897i 0.986717 + 0.162448i \(0.0519391\pi\)
−0.986717 + 0.162448i \(0.948061\pi\)
\(294\) 0 0
\(295\) 7.18703e9i 0.948989i
\(296\) 0 0
\(297\) 2.05670e9 0.264329
\(298\) 0 0
\(299\) −1.89630e10 −2.37258
\(300\) 0 0
\(301\) 3.64282e9i 0.443785i
\(302\) 0 0
\(303\) 4.59628e9i 0.545301i
\(304\) 0 0
\(305\) 8.51333e9 0.983784
\(306\) 0 0
\(307\) 9.66896e8 0.108849 0.0544247 0.998518i \(-0.482668\pi\)
0.0544247 + 0.998518i \(0.482668\pi\)
\(308\) 0 0
\(309\) 8.55944e9i 0.938882i
\(310\) 0 0
\(311\) 2.31046e9i 0.246978i 0.992346 + 0.123489i \(0.0394083\pi\)
−0.992346 + 0.123489i \(0.960592\pi\)
\(312\) 0 0
\(313\) 5.76591e8 0.0600745 0.0300373 0.999549i \(-0.490437\pi\)
0.0300373 + 0.999549i \(0.490437\pi\)
\(314\) 0 0
\(315\) 1.70252e9 0.172922
\(316\) 0 0
\(317\) 3.90592e9i 0.386800i 0.981120 + 0.193400i \(0.0619515\pi\)
−0.981120 + 0.193400i \(0.938049\pi\)
\(318\) 0 0
\(319\) 9.41072e9i 0.908783i
\(320\) 0 0
\(321\) 1.93274e9 0.182035
\(322\) 0 0
\(323\) −1.41205e9 −0.129730
\(324\) 0 0
\(325\) − 1.00225e10i − 0.898341i
\(326\) 0 0
\(327\) 1.61328e9i 0.141098i
\(328\) 0 0
\(329\) 1.14811e10 0.979944
\(330\) 0 0
\(331\) 5.78587e9 0.482011 0.241005 0.970524i \(-0.422523\pi\)
0.241005 + 0.970524i \(0.422523\pi\)
\(332\) 0 0
\(333\) − 5.63271e9i − 0.458079i
\(334\) 0 0
\(335\) − 2.86296e9i − 0.227319i
\(336\) 0 0
\(337\) −1.20243e10 −0.932267 −0.466134 0.884714i \(-0.654353\pi\)
−0.466134 + 0.884714i \(0.654353\pi\)
\(338\) 0 0
\(339\) 8.85339e8 0.0670364
\(340\) 0 0
\(341\) − 2.98258e10i − 2.20584i
\(342\) 0 0
\(343\) − 1.50649e10i − 1.08840i
\(344\) 0 0
\(345\) −8.14477e9 −0.574913
\(346\) 0 0
\(347\) −6.76753e9 −0.466780 −0.233390 0.972383i \(-0.574982\pi\)
−0.233390 + 0.972383i \(0.574982\pi\)
\(348\) 0 0
\(349\) − 5.68848e9i − 0.383438i −0.981450 0.191719i \(-0.938594\pi\)
0.981450 0.191719i \(-0.0614062\pi\)
\(350\) 0 0
\(351\) 4.47859e9i 0.295062i
\(352\) 0 0
\(353\) 1.78794e9 0.115147 0.0575736 0.998341i \(-0.481664\pi\)
0.0575736 + 0.998341i \(0.481664\pi\)
\(354\) 0 0
\(355\) −5.48688e9 −0.345471
\(356\) 0 0
\(357\) 9.17210e9i 0.564671i
\(358\) 0 0
\(359\) − 5.27395e9i − 0.317511i −0.987318 0.158755i \(-0.949252\pi\)
0.987318 0.158755i \(-0.0507481\pi\)
\(360\) 0 0
\(361\) −1.67894e10 −0.988565
\(362\) 0 0
\(363\) −8.88668e9 −0.511815
\(364\) 0 0
\(365\) 1.59704e10i 0.899796i
\(366\) 0 0
\(367\) − 6.40472e9i − 0.353050i −0.984296 0.176525i \(-0.943514\pi\)
0.984296 0.176525i \(-0.0564856\pi\)
\(368\) 0 0
\(369\) −8.40542e9 −0.453371
\(370\) 0 0
\(371\) 1.56232e10 0.824657
\(372\) 0 0
\(373\) − 8.98900e9i − 0.464383i −0.972670 0.232191i \(-0.925410\pi\)
0.972670 0.232191i \(-0.0745895\pi\)
\(374\) 0 0
\(375\) − 1.16516e10i − 0.589196i
\(376\) 0 0
\(377\) −2.04924e10 −1.01444
\(378\) 0 0
\(379\) −3.67818e10 −1.78269 −0.891346 0.453324i \(-0.850238\pi\)
−0.891346 + 0.453324i \(0.850238\pi\)
\(380\) 0 0
\(381\) 1.03504e10i 0.491197i
\(382\) 0 0
\(383\) − 1.51464e9i − 0.0703907i −0.999380 0.0351953i \(-0.988795\pi\)
0.999380 0.0351953i \(-0.0112053\pi\)
\(384\) 0 0
\(385\) −1.56546e10 −0.712521
\(386\) 0 0
\(387\) −4.11586e9 −0.183492
\(388\) 0 0
\(389\) 3.35930e10i 1.46707i 0.679652 + 0.733534i \(0.262131\pi\)
−0.679652 + 0.733534i \(0.737869\pi\)
\(390\) 0 0
\(391\) − 4.38789e10i − 1.87737i
\(392\) 0 0
\(393\) 2.57001e10 1.07737
\(394\) 0 0
\(395\) −3.33686e8 −0.0137072
\(396\) 0 0
\(397\) − 8.19714e9i − 0.329990i −0.986294 0.164995i \(-0.947239\pi\)
0.986294 0.164995i \(-0.0527608\pi\)
\(398\) 0 0
\(399\) − 1.26149e9i − 0.0497729i
\(400\) 0 0
\(401\) −1.33618e10 −0.516758 −0.258379 0.966044i \(-0.583188\pi\)
−0.258379 + 0.966044i \(0.583188\pi\)
\(402\) 0 0
\(403\) 6.49474e10 2.46230
\(404\) 0 0
\(405\) 1.92360e9i 0.0714980i
\(406\) 0 0
\(407\) 5.17925e10i 1.88751i
\(408\) 0 0
\(409\) −4.12559e10 −1.47432 −0.737162 0.675716i \(-0.763835\pi\)
−0.737162 + 0.675716i \(0.763835\pi\)
\(410\) 0 0
\(411\) 7.88451e9 0.276317
\(412\) 0 0
\(413\) − 3.45907e10i − 1.18894i
\(414\) 0 0
\(415\) 5.85794e9i 0.197493i
\(416\) 0 0
\(417\) 1.87871e10 0.621322
\(418\) 0 0
\(419\) 5.55204e10 1.80134 0.900671 0.434502i \(-0.143076\pi\)
0.900671 + 0.434502i \(0.143076\pi\)
\(420\) 0 0
\(421\) 3.81918e10i 1.21574i 0.794036 + 0.607871i \(0.207976\pi\)
−0.794036 + 0.607871i \(0.792024\pi\)
\(422\) 0 0
\(423\) 1.29720e10i 0.405178i
\(424\) 0 0
\(425\) 2.31913e10 0.710835
\(426\) 0 0
\(427\) −4.09741e10 −1.23253
\(428\) 0 0
\(429\) − 4.11804e10i − 1.21580i
\(430\) 0 0
\(431\) − 4.58425e10i − 1.32849i −0.747514 0.664246i \(-0.768753\pi\)
0.747514 0.664246i \(-0.231247\pi\)
\(432\) 0 0
\(433\) 5.43662e10 1.54660 0.773299 0.634042i \(-0.218605\pi\)
0.773299 + 0.634042i \(0.218605\pi\)
\(434\) 0 0
\(435\) −8.80168e9 −0.245815
\(436\) 0 0
\(437\) 6.03492e9i 0.165480i
\(438\) 0 0
\(439\) 6.56595e10i 1.76783i 0.467651 + 0.883913i \(0.345100\pi\)
−0.467651 + 0.883913i \(0.654900\pi\)
\(440\) 0 0
\(441\) 4.41351e9 0.116689
\(442\) 0 0
\(443\) 3.38733e10 0.879514 0.439757 0.898117i \(-0.355064\pi\)
0.439757 + 0.898117i \(0.355064\pi\)
\(444\) 0 0
\(445\) 3.12870e10i 0.797855i
\(446\) 0 0
\(447\) − 2.07219e9i − 0.0519039i
\(448\) 0 0
\(449\) 1.81346e10 0.446193 0.223097 0.974796i \(-0.428383\pi\)
0.223097 + 0.974796i \(0.428383\pi\)
\(450\) 0 0
\(451\) 7.72874e10 1.86811
\(452\) 0 0
\(453\) − 3.37322e10i − 0.801036i
\(454\) 0 0
\(455\) − 3.40887e10i − 0.795362i
\(456\) 0 0
\(457\) −2.66606e10 −0.611231 −0.305615 0.952155i \(-0.598862\pi\)
−0.305615 + 0.952155i \(0.598862\pi\)
\(458\) 0 0
\(459\) −1.03631e10 −0.233475
\(460\) 0 0
\(461\) 7.04638e10i 1.56014i 0.625695 + 0.780068i \(0.284816\pi\)
−0.625695 + 0.780068i \(0.715184\pi\)
\(462\) 0 0
\(463\) 4.09104e10i 0.890245i 0.895470 + 0.445123i \(0.146840\pi\)
−0.895470 + 0.445123i \(0.853160\pi\)
\(464\) 0 0
\(465\) 2.78955e10 0.596654
\(466\) 0 0
\(467\) 5.39912e10 1.13515 0.567577 0.823320i \(-0.307881\pi\)
0.567577 + 0.823320i \(0.307881\pi\)
\(468\) 0 0
\(469\) 1.37792e10i 0.284796i
\(470\) 0 0
\(471\) − 1.11930e10i − 0.227438i
\(472\) 0 0
\(473\) 3.78451e10 0.756075
\(474\) 0 0
\(475\) −3.18963e9 −0.0626564
\(476\) 0 0
\(477\) 1.76519e10i 0.340971i
\(478\) 0 0
\(479\) − 4.17582e10i − 0.793231i −0.917985 0.396616i \(-0.870185\pi\)
0.917985 0.396616i \(-0.129815\pi\)
\(480\) 0 0
\(481\) −1.12781e11 −2.10696
\(482\) 0 0
\(483\) 3.92003e10 0.720279
\(484\) 0 0
\(485\) − 4.00516e10i − 0.723857i
\(486\) 0 0
\(487\) 5.40468e10i 0.960847i 0.877037 + 0.480423i \(0.159517\pi\)
−0.877037 + 0.480423i \(0.840483\pi\)
\(488\) 0 0
\(489\) 1.81304e9 0.0317082
\(490\) 0 0
\(491\) −3.85347e10 −0.663019 −0.331510 0.943452i \(-0.607558\pi\)
−0.331510 + 0.943452i \(0.607558\pi\)
\(492\) 0 0
\(493\) − 4.74179e10i − 0.802703i
\(494\) 0 0
\(495\) − 1.76874e10i − 0.294606i
\(496\) 0 0
\(497\) 2.64080e10 0.432823
\(498\) 0 0
\(499\) 5.84419e10 0.942588 0.471294 0.881976i \(-0.343787\pi\)
0.471294 + 0.881976i \(0.343787\pi\)
\(500\) 0 0
\(501\) 1.05500e9i 0.0167457i
\(502\) 0 0
\(503\) 1.70744e10i 0.266731i 0.991067 + 0.133365i \(0.0425783\pi\)
−0.991067 + 0.133365i \(0.957422\pi\)
\(504\) 0 0
\(505\) 3.95274e10 0.607761
\(506\) 0 0
\(507\) 5.15248e10 0.779802
\(508\) 0 0
\(509\) − 1.01105e11i − 1.50627i −0.657867 0.753134i \(-0.728541\pi\)
0.657867 0.753134i \(-0.271459\pi\)
\(510\) 0 0
\(511\) − 7.68645e10i − 1.12731i
\(512\) 0 0
\(513\) 1.42530e9 0.0205796
\(514\) 0 0
\(515\) 7.36100e10 1.04642
\(516\) 0 0
\(517\) − 1.19277e11i − 1.66953i
\(518\) 0 0
\(519\) − 4.90993e10i − 0.676715i
\(520\) 0 0
\(521\) 6.85240e10 0.930019 0.465010 0.885306i \(-0.346051\pi\)
0.465010 + 0.885306i \(0.346051\pi\)
\(522\) 0 0
\(523\) 1.17769e11 1.57407 0.787036 0.616907i \(-0.211615\pi\)
0.787036 + 0.616907i \(0.211615\pi\)
\(524\) 0 0
\(525\) 2.07185e10i 0.272722i
\(526\) 0 0
\(527\) 1.50284e11i 1.94836i
\(528\) 0 0
\(529\) −1.09221e11 −1.39471
\(530\) 0 0
\(531\) 3.90825e10 0.491591
\(532\) 0 0
\(533\) 1.68298e11i 2.08531i
\(534\) 0 0
\(535\) − 1.66213e10i − 0.202885i
\(536\) 0 0
\(537\) 3.45821e10 0.415867
\(538\) 0 0
\(539\) −4.05820e10 −0.480815
\(540\) 0 0
\(541\) 1.06808e10i 0.124685i 0.998055 + 0.0623424i \(0.0198571\pi\)
−0.998055 + 0.0623424i \(0.980143\pi\)
\(542\) 0 0
\(543\) 8.20015e10i 0.943240i
\(544\) 0 0
\(545\) 1.38740e10 0.157259
\(546\) 0 0
\(547\) 1.66204e11 1.85649 0.928245 0.371971i \(-0.121318\pi\)
0.928245 + 0.371971i \(0.121318\pi\)
\(548\) 0 0
\(549\) − 4.62948e10i − 0.509615i
\(550\) 0 0
\(551\) 6.52166e9i 0.0707541i
\(552\) 0 0
\(553\) 1.60601e9 0.0171731
\(554\) 0 0
\(555\) −4.84406e10 −0.510549
\(556\) 0 0
\(557\) 7.12167e10i 0.739880i 0.929056 + 0.369940i \(0.120622\pi\)
−0.929056 + 0.369940i \(0.879378\pi\)
\(558\) 0 0
\(559\) 8.24099e10i 0.843980i
\(560\) 0 0
\(561\) 9.52884e10 0.962030
\(562\) 0 0
\(563\) 5.48999e10 0.546435 0.273217 0.961952i \(-0.411912\pi\)
0.273217 + 0.961952i \(0.411912\pi\)
\(564\) 0 0
\(565\) − 7.61380e9i − 0.0747149i
\(566\) 0 0
\(567\) − 9.25815e9i − 0.0895760i
\(568\) 0 0
\(569\) 1.65851e11 1.58222 0.791112 0.611671i \(-0.209502\pi\)
0.791112 + 0.611671i \(0.209502\pi\)
\(570\) 0 0
\(571\) −1.77889e11 −1.67342 −0.836709 0.547648i \(-0.815523\pi\)
−0.836709 + 0.547648i \(0.815523\pi\)
\(572\) 0 0
\(573\) 8.95938e10i 0.831112i
\(574\) 0 0
\(575\) − 9.91163e10i − 0.906721i
\(576\) 0 0
\(577\) −1.39402e11 −1.25767 −0.628834 0.777540i \(-0.716468\pi\)
−0.628834 + 0.777540i \(0.716468\pi\)
\(578\) 0 0
\(579\) 7.42954e10 0.661070
\(580\) 0 0
\(581\) − 2.81939e10i − 0.247429i
\(582\) 0 0
\(583\) − 1.62308e11i − 1.40497i
\(584\) 0 0
\(585\) 3.85153e10 0.328859
\(586\) 0 0
\(587\) −8.09518e10 −0.681827 −0.340914 0.940095i \(-0.610736\pi\)
−0.340914 + 0.940095i \(0.610736\pi\)
\(588\) 0 0
\(589\) − 2.06694e10i − 0.171738i
\(590\) 0 0
\(591\) − 9.84915e10i − 0.807326i
\(592\) 0 0
\(593\) 1.00987e11 0.816674 0.408337 0.912831i \(-0.366109\pi\)
0.408337 + 0.912831i \(0.366109\pi\)
\(594\) 0 0
\(595\) 7.88788e10 0.629350
\(596\) 0 0
\(597\) − 8.82208e9i − 0.0694502i
\(598\) 0 0
\(599\) − 1.02923e11i − 0.799474i −0.916630 0.399737i \(-0.869102\pi\)
0.916630 0.399737i \(-0.130898\pi\)
\(600\) 0 0
\(601\) −2.21864e11 −1.70055 −0.850273 0.526342i \(-0.823563\pi\)
−0.850273 + 0.526342i \(0.823563\pi\)
\(602\) 0 0
\(603\) −1.55685e10 −0.117755
\(604\) 0 0
\(605\) 7.64243e10i 0.570440i
\(606\) 0 0
\(607\) 5.54725e10i 0.408623i 0.978906 + 0.204311i \(0.0654955\pi\)
−0.978906 + 0.204311i \(0.934504\pi\)
\(608\) 0 0
\(609\) 4.23619e10 0.307969
\(610\) 0 0
\(611\) 2.59733e11 1.86364
\(612\) 0 0
\(613\) 1.49374e11i 1.05787i 0.848662 + 0.528935i \(0.177409\pi\)
−0.848662 + 0.528935i \(0.822591\pi\)
\(614\) 0 0
\(615\) 7.22855e10i 0.505301i
\(616\) 0 0
\(617\) −1.45992e11 −1.00737 −0.503686 0.863887i \(-0.668023\pi\)
−0.503686 + 0.863887i \(0.668023\pi\)
\(618\) 0 0
\(619\) 2.29296e11 1.56183 0.780915 0.624637i \(-0.214753\pi\)
0.780915 + 0.624637i \(0.214753\pi\)
\(620\) 0 0
\(621\) 4.42906e10i 0.297814i
\(622\) 0 0
\(623\) − 1.50582e11i − 0.999591i
\(624\) 0 0
\(625\) −1.07960e10 −0.0707528
\(626\) 0 0
\(627\) −1.31056e10 −0.0847980
\(628\) 0 0
\(629\) − 2.60967e11i − 1.66718i
\(630\) 0 0
\(631\) − 4.64365e10i − 0.292915i −0.989217 0.146458i \(-0.953213\pi\)
0.989217 0.146458i \(-0.0467872\pi\)
\(632\) 0 0
\(633\) 1.44509e11 0.900078
\(634\) 0 0
\(635\) 8.90118e10 0.547460
\(636\) 0 0
\(637\) − 8.83697e10i − 0.536717i
\(638\) 0 0
\(639\) 2.98372e10i 0.178960i
\(640\) 0 0
\(641\) −2.85844e11 −1.69316 −0.846578 0.532264i \(-0.821341\pi\)
−0.846578 + 0.532264i \(0.821341\pi\)
\(642\) 0 0
\(643\) −2.95497e11 −1.72866 −0.864328 0.502928i \(-0.832256\pi\)
−0.864328 + 0.502928i \(0.832256\pi\)
\(644\) 0 0
\(645\) 3.53958e10i 0.204509i
\(646\) 0 0
\(647\) 8.99469e10i 0.513297i 0.966505 + 0.256649i \(0.0826183\pi\)
−0.966505 + 0.256649i \(0.917382\pi\)
\(648\) 0 0
\(649\) −3.59361e11 −2.02559
\(650\) 0 0
\(651\) −1.34259e11 −0.747517
\(652\) 0 0
\(653\) − 2.81927e10i − 0.155055i −0.996990 0.0775273i \(-0.975298\pi\)
0.996990 0.0775273i \(-0.0247025\pi\)
\(654\) 0 0
\(655\) − 2.21018e11i − 1.20078i
\(656\) 0 0
\(657\) 8.68456e10 0.466108
\(658\) 0 0
\(659\) 1.19224e11 0.632152 0.316076 0.948734i \(-0.397635\pi\)
0.316076 + 0.948734i \(0.397635\pi\)
\(660\) 0 0
\(661\) 2.48610e11i 1.30230i 0.758947 + 0.651152i \(0.225714\pi\)
−0.758947 + 0.651152i \(0.774286\pi\)
\(662\) 0 0
\(663\) 2.07496e11i 1.07388i
\(664\) 0 0
\(665\) −1.08487e10 −0.0554740
\(666\) 0 0
\(667\) −2.02658e11 −1.02391
\(668\) 0 0
\(669\) − 1.55723e11i − 0.777408i
\(670\) 0 0
\(671\) 4.25678e11i 2.09986i
\(672\) 0 0
\(673\) −6.89295e10 −0.336004 −0.168002 0.985787i \(-0.553732\pi\)
−0.168002 + 0.985787i \(0.553732\pi\)
\(674\) 0 0
\(675\) −2.34088e10 −0.112763
\(676\) 0 0
\(677\) − 1.65608e11i − 0.788365i −0.919032 0.394182i \(-0.871028\pi\)
0.919032 0.394182i \(-0.128972\pi\)
\(678\) 0 0
\(679\) 1.92766e11i 0.906883i
\(680\) 0 0
\(681\) −1.26517e10 −0.0588250
\(682\) 0 0
\(683\) −2.44222e11 −1.12228 −0.561141 0.827720i \(-0.689638\pi\)
−0.561141 + 0.827720i \(0.689638\pi\)
\(684\) 0 0
\(685\) − 6.78058e10i − 0.307967i
\(686\) 0 0
\(687\) − 3.69427e10i − 0.165845i
\(688\) 0 0
\(689\) 3.53436e11 1.56831
\(690\) 0 0
\(691\) 7.38454e10 0.323900 0.161950 0.986799i \(-0.448222\pi\)
0.161950 + 0.986799i \(0.448222\pi\)
\(692\) 0 0
\(693\) 8.51282e10i 0.369097i
\(694\) 0 0
\(695\) − 1.61567e11i − 0.692490i
\(696\) 0 0
\(697\) −3.89429e11 −1.65005
\(698\) 0 0
\(699\) −4.74937e10 −0.198943
\(700\) 0 0
\(701\) − 2.12728e11i − 0.880953i −0.897764 0.440476i \(-0.854810\pi\)
0.897764 0.440476i \(-0.145190\pi\)
\(702\) 0 0
\(703\) 3.58923e10i 0.146954i
\(704\) 0 0
\(705\) 1.11558e11 0.451588
\(706\) 0 0
\(707\) −1.90243e11 −0.761432
\(708\) 0 0
\(709\) 1.44756e11i 0.572863i 0.958101 + 0.286432i \(0.0924691\pi\)
−0.958101 + 0.286432i \(0.907531\pi\)
\(710\) 0 0
\(711\) 1.81456e9i 0.00710055i
\(712\) 0 0
\(713\) 6.42291e11 2.48527
\(714\) 0 0
\(715\) −3.54146e11 −1.35506
\(716\) 0 0
\(717\) − 8.65642e10i − 0.327538i
\(718\) 0 0
\(719\) 4.70245e11i 1.75958i 0.475363 + 0.879790i \(0.342317\pi\)
−0.475363 + 0.879790i \(0.657683\pi\)
\(720\) 0 0
\(721\) −3.54281e11 −1.31101
\(722\) 0 0
\(723\) −4.28240e10 −0.156724
\(724\) 0 0
\(725\) − 1.07110e11i − 0.387686i
\(726\) 0 0
\(727\) − 8.92240e10i − 0.319407i −0.987165 0.159703i \(-0.948946\pi\)
0.987165 0.159703i \(-0.0510537\pi\)
\(728\) 0 0
\(729\) 1.04604e10 0.0370370
\(730\) 0 0
\(731\) −1.90691e11 −0.667820
\(732\) 0 0
\(733\) 2.06258e11i 0.714487i 0.934011 + 0.357244i \(0.116283\pi\)
−0.934011 + 0.357244i \(0.883717\pi\)
\(734\) 0 0
\(735\) − 3.79556e10i − 0.130055i
\(736\) 0 0
\(737\) 1.43152e11 0.485207
\(738\) 0 0
\(739\) −3.46428e10 −0.116154 −0.0580772 0.998312i \(-0.518497\pi\)
−0.0580772 + 0.998312i \(0.518497\pi\)
\(740\) 0 0
\(741\) − 2.85381e10i − 0.0946570i
\(742\) 0 0
\(743\) − 3.92994e9i − 0.0128953i −0.999979 0.00644764i \(-0.997948\pi\)
0.999979 0.00644764i \(-0.00205236\pi\)
\(744\) 0 0
\(745\) −1.78206e10 −0.0578491
\(746\) 0 0
\(747\) 3.18550e10 0.102305
\(748\) 0 0
\(749\) 7.99975e10i 0.254184i
\(750\) 0 0
\(751\) − 3.31276e11i − 1.04143i −0.853730 0.520716i \(-0.825665\pi\)
0.853730 0.520716i \(-0.174335\pi\)
\(752\) 0 0
\(753\) 2.98934e11 0.929811
\(754\) 0 0
\(755\) −2.90092e11 −0.892789
\(756\) 0 0
\(757\) − 6.32528e10i − 0.192618i −0.995351 0.0963089i \(-0.969296\pi\)
0.995351 0.0963089i \(-0.0307037\pi\)
\(758\) 0 0
\(759\) − 4.07250e11i − 1.22714i
\(760\) 0 0
\(761\) 2.86012e11 0.852797 0.426399 0.904535i \(-0.359782\pi\)
0.426399 + 0.904535i \(0.359782\pi\)
\(762\) 0 0
\(763\) −6.67748e10 −0.197022
\(764\) 0 0
\(765\) 8.91215e10i 0.260218i
\(766\) 0 0
\(767\) − 7.82530e11i − 2.26110i
\(768\) 0 0
\(769\) −1.53604e11 −0.439235 −0.219617 0.975586i \(-0.570481\pi\)
−0.219617 + 0.975586i \(0.570481\pi\)
\(770\) 0 0
\(771\) −2.94152e10 −0.0832443
\(772\) 0 0
\(773\) 2.38819e11i 0.668883i 0.942416 + 0.334442i \(0.108548\pi\)
−0.942416 + 0.334442i \(0.891452\pi\)
\(774\) 0 0
\(775\) 3.39469e11i 0.941009i
\(776\) 0 0
\(777\) 2.33142e11 0.639640
\(778\) 0 0
\(779\) 5.35604e10 0.145443
\(780\) 0 0
\(781\) − 2.74352e11i − 0.737400i
\(782\) 0 0
\(783\) 4.78628e10i 0.127336i
\(784\) 0 0
\(785\) −9.62582e10 −0.253489
\(786\) 0 0
\(787\) −2.90235e11 −0.756572 −0.378286 0.925689i \(-0.623486\pi\)
−0.378286 + 0.925689i \(0.623486\pi\)
\(788\) 0 0
\(789\) 1.89175e11i 0.488153i
\(790\) 0 0
\(791\) 3.66447e10i 0.0936064i
\(792\) 0 0
\(793\) −9.26939e11 −2.34400
\(794\) 0 0
\(795\) 1.51804e11 0.380027
\(796\) 0 0
\(797\) 5.71943e11i 1.41749i 0.705465 + 0.708744i \(0.250738\pi\)
−0.705465 + 0.708744i \(0.749262\pi\)
\(798\) 0 0
\(799\) 6.01002e11i 1.47465i
\(800\) 0 0
\(801\) 1.70136e11 0.413301
\(802\) 0 0
\(803\) −7.98541e11 −1.92059
\(804\) 0 0
\(805\) − 3.37117e11i − 0.802781i
\(806\) 0 0
\(807\) 2.46657e11i 0.581567i
\(808\) 0 0
\(809\) −5.26381e10 −0.122887 −0.0614435 0.998111i \(-0.519570\pi\)
−0.0614435 + 0.998111i \(0.519570\pi\)
\(810\) 0 0
\(811\) −1.17986e11 −0.272739 −0.136369 0.990658i \(-0.543543\pi\)
−0.136369 + 0.990658i \(0.543543\pi\)
\(812\) 0 0
\(813\) 4.53046e11i 1.03700i
\(814\) 0 0
\(815\) − 1.55919e10i − 0.0353402i
\(816\) 0 0
\(817\) 2.62268e10 0.0588649
\(818\) 0 0
\(819\) −1.85372e11 −0.412010
\(820\) 0 0
\(821\) 1.94904e11i 0.428992i 0.976725 + 0.214496i \(0.0688109\pi\)
−0.976725 + 0.214496i \(0.931189\pi\)
\(822\) 0 0
\(823\) 1.25481e11i 0.273513i 0.990605 + 0.136756i \(0.0436678\pi\)
−0.990605 + 0.136756i \(0.956332\pi\)
\(824\) 0 0
\(825\) 2.15243e11 0.464637
\(826\) 0 0
\(827\) −5.47785e11 −1.17108 −0.585542 0.810642i \(-0.699118\pi\)
−0.585542 + 0.810642i \(0.699118\pi\)
\(828\) 0 0
\(829\) − 1.15721e11i − 0.245015i −0.992468 0.122507i \(-0.960906\pi\)
0.992468 0.122507i \(-0.0390935\pi\)
\(830\) 0 0
\(831\) 6.63225e9i 0.0139078i
\(832\) 0 0
\(833\) 2.04481e11 0.424691
\(834\) 0 0
\(835\) 9.07289e9 0.0186638
\(836\) 0 0
\(837\) − 1.51694e11i − 0.309076i
\(838\) 0 0
\(839\) − 2.81036e11i − 0.567172i −0.958947 0.283586i \(-0.908476\pi\)
0.958947 0.283586i \(-0.0915241\pi\)
\(840\) 0 0
\(841\) 2.81243e11 0.562210
\(842\) 0 0
\(843\) 2.55299e11 0.505520
\(844\) 0 0
\(845\) − 4.43106e11i − 0.869123i
\(846\) 0 0
\(847\) − 3.67826e11i − 0.714674i
\(848\) 0 0
\(849\) 3.82726e11 0.736643
\(850\) 0 0
\(851\) −1.11534e12 −2.12661
\(852\) 0 0
\(853\) − 8.36511e11i − 1.58007i −0.613063 0.790034i \(-0.710063\pi\)
0.613063 0.790034i \(-0.289937\pi\)
\(854\) 0 0
\(855\) − 1.22574e10i − 0.0229368i
\(856\) 0 0
\(857\) 8.69801e9 0.0161249 0.00806244 0.999967i \(-0.497434\pi\)
0.00806244 + 0.999967i \(0.497434\pi\)
\(858\) 0 0
\(859\) −2.46238e11 −0.452254 −0.226127 0.974098i \(-0.572606\pi\)
−0.226127 + 0.974098i \(0.572606\pi\)
\(860\) 0 0
\(861\) − 3.47906e11i − 0.633066i
\(862\) 0 0
\(863\) − 8.52034e11i − 1.53608i −0.640402 0.768040i \(-0.721232\pi\)
0.640402 0.768040i \(-0.278768\pi\)
\(864\) 0 0
\(865\) −4.22247e11 −0.754228
\(866\) 0 0
\(867\) −1.53907e11 −0.272384
\(868\) 0 0
\(869\) − 1.66848e10i − 0.0292577i
\(870\) 0 0
\(871\) 3.11722e11i 0.541620i
\(872\) 0 0
\(873\) −2.17797e11 −0.374969
\(874\) 0 0
\(875\) 4.82267e11 0.822725
\(876\) 0 0
\(877\) 9.14656e11i 1.54618i 0.634298 + 0.773089i \(0.281289\pi\)
−0.634298 + 0.773089i \(0.718711\pi\)
\(878\) 0 0
\(879\) − 1.11980e11i − 0.187579i
\(880\) 0 0
\(881\) 6.17443e11 1.02493 0.512464 0.858709i \(-0.328733\pi\)
0.512464 + 0.858709i \(0.328733\pi\)
\(882\) 0 0
\(883\) −6.59286e11 −1.08450 −0.542252 0.840216i \(-0.682428\pi\)
−0.542252 + 0.840216i \(0.682428\pi\)
\(884\) 0 0
\(885\) − 3.36104e11i − 0.547899i
\(886\) 0 0
\(887\) 9.49677e11i 1.53420i 0.641529 + 0.767099i \(0.278300\pi\)
−0.641529 + 0.767099i \(0.721700\pi\)
\(888\) 0 0
\(889\) −4.28409e11 −0.685885
\(890\) 0 0
\(891\) −9.61824e10 −0.152611
\(892\) 0 0
\(893\) − 8.26593e10i − 0.129983i
\(894\) 0 0
\(895\) − 2.97401e11i − 0.463501i
\(896\) 0 0
\(897\) 8.86810e11 1.36981
\(898\) 0 0
\(899\) 6.94095e11 1.06263
\(900\) 0 0
\(901\) 8.17824e11i 1.24097i
\(902\) 0 0
\(903\) − 1.70358e11i − 0.256219i
\(904\) 0 0
\(905\) 7.05202e11 1.05128
\(906\) 0 0
\(907\) −3.18987e11 −0.471350 −0.235675 0.971832i \(-0.575730\pi\)
−0.235675 + 0.971832i \(0.575730\pi\)
\(908\) 0 0
\(909\) − 2.14947e11i − 0.314830i
\(910\) 0 0
\(911\) 1.05774e12i 1.53570i 0.640628 + 0.767852i \(0.278674\pi\)
−0.640628 + 0.767852i \(0.721326\pi\)
\(912\) 0 0
\(913\) −2.92905e11 −0.421545
\(914\) 0 0
\(915\) −3.98129e11 −0.567988
\(916\) 0 0
\(917\) 1.06374e12i 1.50439i
\(918\) 0 0
\(919\) − 1.29009e12i − 1.80867i −0.426828 0.904333i \(-0.640369\pi\)
0.426828 0.904333i \(-0.359631\pi\)
\(920\) 0 0
\(921\) −4.52173e10 −0.0628443
\(922\) 0 0
\(923\) 5.97417e11 0.823134
\(924\) 0 0
\(925\) − 5.89489e11i − 0.805209i
\(926\) 0 0
\(927\) − 4.00285e11i − 0.542064i
\(928\) 0 0
\(929\) −5.16906e11 −0.693983 −0.346991 0.937868i \(-0.612797\pi\)
−0.346991 + 0.937868i \(0.612797\pi\)
\(930\) 0 0
\(931\) −2.81234e10 −0.0374343
\(932\) 0 0
\(933\) − 1.08050e11i − 0.142593i
\(934\) 0 0
\(935\) − 8.19468e11i − 1.07222i
\(936\) 0 0
\(937\) −7.96069e11 −1.03274 −0.516371 0.856365i \(-0.672718\pi\)
−0.516371 + 0.856365i \(0.672718\pi\)
\(938\) 0 0
\(939\) −2.69645e10 −0.0346841
\(940\) 0 0
\(941\) − 5.93891e11i − 0.757440i −0.925511 0.378720i \(-0.876364\pi\)
0.925511 0.378720i \(-0.123636\pi\)
\(942\) 0 0
\(943\) 1.66437e12i 2.10476i
\(944\) 0 0
\(945\) −7.96188e10 −0.0998363
\(946\) 0 0
\(947\) −7.57641e10 −0.0942028 −0.0471014 0.998890i \(-0.514998\pi\)
−0.0471014 + 0.998890i \(0.514998\pi\)
\(948\) 0 0
\(949\) − 1.73887e12i − 2.14389i
\(950\) 0 0
\(951\) − 1.82662e11i − 0.223319i
\(952\) 0 0
\(953\) 6.44761e11 0.781676 0.390838 0.920459i \(-0.372185\pi\)
0.390838 + 0.920459i \(0.372185\pi\)
\(954\) 0 0
\(955\) 7.70495e11 0.926309
\(956\) 0 0
\(957\) − 4.40096e11i − 0.524686i
\(958\) 0 0
\(959\) 3.26345e11i 0.385836i
\(960\) 0 0
\(961\) −1.34693e12 −1.57925
\(962\) 0 0
\(963\) −9.03854e10 −0.105098
\(964\) 0 0
\(965\) − 6.38931e11i − 0.736791i
\(966\) 0 0
\(967\) 1.70360e11i 0.194833i 0.995244 + 0.0974166i \(0.0310579\pi\)
−0.995244 + 0.0974166i \(0.968942\pi\)
\(968\) 0 0
\(969\) 6.60352e10 0.0748997
\(970\) 0 0
\(971\) 4.23154e11 0.476016 0.238008 0.971263i \(-0.423505\pi\)
0.238008 + 0.971263i \(0.423505\pi\)
\(972\) 0 0
\(973\) 7.77612e11i 0.867584i
\(974\) 0 0
\(975\) 4.68705e11i 0.518658i
\(976\) 0 0
\(977\) −1.56737e12 −1.72025 −0.860127 0.510080i \(-0.829616\pi\)
−0.860127 + 0.510080i \(0.829616\pi\)
\(978\) 0 0
\(979\) −1.56439e12 −1.70300
\(980\) 0 0
\(981\) − 7.54458e10i − 0.0814627i
\(982\) 0 0
\(983\) 9.58136e10i 0.102616i 0.998683 + 0.0513078i \(0.0163389\pi\)
−0.998683 + 0.0513078i \(0.983661\pi\)
\(984\) 0 0
\(985\) −8.47014e11 −0.899799
\(986\) 0 0
\(987\) −5.36920e11 −0.565771
\(988\) 0 0
\(989\) 8.14985e11i 0.851853i
\(990\) 0 0
\(991\) 6.32891e11i 0.656198i 0.944643 + 0.328099i \(0.106408\pi\)
−0.944643 + 0.328099i \(0.893592\pi\)
\(992\) 0 0
\(993\) −2.70578e11 −0.278289
\(994\) 0 0
\(995\) −7.58687e10 −0.0774052
\(996\) 0 0
\(997\) 1.96393e12i 1.98767i 0.110851 + 0.993837i \(0.464642\pi\)
−0.110851 + 0.993837i \(0.535358\pi\)
\(998\) 0 0
\(999\) 2.63416e11i 0.264472i
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.319.3 yes 8
4.3 odd 2 inner 384.9.b.a.319.7 yes 8
8.3 odd 2 inner 384.9.b.a.319.2 8
8.5 even 2 inner 384.9.b.a.319.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.9.b.a.319.2 8 8.3 odd 2 inner
384.9.b.a.319.3 yes 8 1.1 even 1 trivial
384.9.b.a.319.6 yes 8 8.5 even 2 inner
384.9.b.a.319.7 yes 8 4.3 odd 2 inner