Properties

Label 20.10.a.b.1.2
Level 20
Weight 10
Character 20.1
Self dual yes
Analytic conductor 10.301
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 20.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.3007167233\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{79}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(8.88819\) of \(x^{2} - 79\)
Character \(\chi\) \(=\) 20.1

$q$-expansion

\(f(q)\) \(=\) \(q+12.2111 q^{3} -625.000 q^{5} +9622.57 q^{7} -19533.9 q^{9} +O(q^{10})\) \(q+12.2111 q^{3} -625.000 q^{5} +9622.57 q^{7} -19533.9 q^{9} +55626.3 q^{11} +169777. q^{13} -7631.94 q^{15} +207499. q^{17} +802445. q^{19} +117502. q^{21} -1.24189e6 q^{23} +390625. q^{25} -478882. q^{27} -4.28308e6 q^{29} -3.58713e6 q^{31} +679259. q^{33} -6.01410e6 q^{35} -2.89856e6 q^{37} +2.07317e6 q^{39} +2.51515e7 q^{41} -2.00204e7 q^{43} +1.22087e7 q^{45} +3.73010e7 q^{47} +5.22402e7 q^{49} +2.53380e6 q^{51} -2.55155e7 q^{53} -3.47665e7 q^{55} +9.79875e6 q^{57} -9.96495e7 q^{59} +2.00434e8 q^{61} -1.87966e8 q^{63} -1.06111e8 q^{65} -8.09951e7 q^{67} -1.51648e7 q^{69} -4.31522e7 q^{71} -3.40820e8 q^{73} +4.76997e6 q^{75} +5.35268e8 q^{77} +2.81089e8 q^{79} +3.78638e8 q^{81} -6.01017e8 q^{83} -1.29687e8 q^{85} -5.23011e7 q^{87} +5.39917e8 q^{89} +1.63369e9 q^{91} -4.38028e7 q^{93} -5.01528e8 q^{95} +4.23026e8 q^{97} -1.08660e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 260q^{3} - 1250q^{5} - 380q^{7} + 34882q^{9} + O(q^{10}) \) \( 2q - 260q^{3} - 1250q^{5} - 380q^{7} + 34882q^{9} + 102720q^{11} + 179140q^{13} + 162500q^{15} + 316020q^{17} + 137272q^{19} + 2840312q^{21} - 665460q^{23} + 781250q^{25} - 9933560q^{27} - 6893748q^{29} + 291832q^{31} - 12140160q^{33} + 237500q^{35} + 11261380q^{37} - 475528q^{39} + 29773452q^{41} - 11708180q^{43} - 21801250q^{45} + 62493300q^{47} + 111937914q^{49} - 27006696q^{51} + 9417780q^{53} - 64200000q^{55} + 190866320q^{57} - 92930856q^{59} + 195673924q^{61} - 732264700q^{63} - 111962500q^{65} - 219767420q^{67} - 172074264q^{69} + 311207016q^{71} - 99224060q^{73} - 101562500q^{75} + 64210560q^{77} + 542261776q^{79} + 1881238378q^{81} - 1256915700q^{83} - 197512500q^{85} + 658353000q^{87} - 462291852q^{89} + 1540037768q^{91} - 1099698160q^{93} - 85795000q^{95} + 1671716740q^{97} + 1476045120q^{99} + O(q^{100}) \)

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
\(3\) 12.2111 0.0870381 0.0435191 0.999053i \(-0.486143\pi\)
0.0435191 + 0.999053i \(0.486143\pi\)
\(4\) 0 0
\(5\) −625.000 −0.447214
\(6\) 0 0
\(7\) 9622.57 1.51478 0.757390 0.652962i \(-0.226474\pi\)
0.757390 + 0.652962i \(0.226474\pi\)
\(8\) 0 0
\(9\) −19533.9 −0.992424
\(10\) 0 0
\(11\) 55626.3 1.14555 0.572774 0.819713i \(-0.305867\pi\)
0.572774 + 0.819713i \(0.305867\pi\)
\(12\) 0 0
\(13\) 169777. 1.64867 0.824335 0.566102i \(-0.191549\pi\)
0.824335 + 0.566102i \(0.191549\pi\)
\(14\) 0 0
\(15\) −7631.94 −0.0389246
\(16\) 0 0
\(17\) 207499. 0.602555 0.301278 0.953536i \(-0.402587\pi\)
0.301278 + 0.953536i \(0.402587\pi\)
\(18\) 0 0
\(19\) 802445. 1.41262 0.706308 0.707904i \(-0.250359\pi\)
0.706308 + 0.707904i \(0.250359\pi\)
\(20\) 0 0
\(21\) 117502. 0.131844
\(22\) 0 0
\(23\) −1.24189e6 −0.925351 −0.462675 0.886528i \(-0.653110\pi\)
−0.462675 + 0.886528i \(0.653110\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 0 0
\(27\) −478882. −0.173417
\(28\) 0 0
\(29\) −4.28308e6 −1.12451 −0.562257 0.826963i \(-0.690067\pi\)
−0.562257 + 0.826963i \(0.690067\pi\)
\(30\) 0 0
\(31\) −3.58713e6 −0.697620 −0.348810 0.937193i \(-0.613414\pi\)
−0.348810 + 0.937193i \(0.613414\pi\)
\(32\) 0 0
\(33\) 679259. 0.0997064
\(34\) 0 0
\(35\) −6.01410e6 −0.677430
\(36\) 0 0
\(37\) −2.89856e6 −0.254258 −0.127129 0.991886i \(-0.540576\pi\)
−0.127129 + 0.991886i \(0.540576\pi\)
\(38\) 0 0
\(39\) 2.07317e6 0.143497
\(40\) 0 0
\(41\) 2.51515e7 1.39007 0.695035 0.718975i \(-0.255389\pi\)
0.695035 + 0.718975i \(0.255389\pi\)
\(42\) 0 0
\(43\) −2.00204e7 −0.893029 −0.446515 0.894776i \(-0.647335\pi\)
−0.446515 + 0.894776i \(0.647335\pi\)
\(44\) 0 0
\(45\) 1.22087e7 0.443826
\(46\) 0 0
\(47\) 3.73010e7 1.11501 0.557507 0.830172i \(-0.311758\pi\)
0.557507 + 0.830172i \(0.311758\pi\)
\(48\) 0 0
\(49\) 5.22402e7 1.29456
\(50\) 0 0
\(51\) 2.53380e6 0.0524453
\(52\) 0 0
\(53\) −2.55155e7 −0.444184 −0.222092 0.975026i \(-0.571289\pi\)
−0.222092 + 0.975026i \(0.571289\pi\)
\(54\) 0 0
\(55\) −3.47665e7 −0.512305
\(56\) 0 0
\(57\) 9.79875e6 0.122951
\(58\) 0 0
\(59\) −9.96495e7 −1.07063 −0.535317 0.844651i \(-0.679808\pi\)
−0.535317 + 0.844651i \(0.679808\pi\)
\(60\) 0 0
\(61\) 2.00434e8 1.85347 0.926737 0.375710i \(-0.122601\pi\)
0.926737 + 0.375710i \(0.122601\pi\)
\(62\) 0 0
\(63\) −1.87966e8 −1.50331
\(64\) 0 0
\(65\) −1.06111e8 −0.737308
\(66\) 0 0
\(67\) −8.09951e7 −0.491046 −0.245523 0.969391i \(-0.578960\pi\)
−0.245523 + 0.969391i \(0.578960\pi\)
\(68\) 0 0
\(69\) −1.51648e7 −0.0805408
\(70\) 0 0
\(71\) −4.31522e7 −0.201530 −0.100765 0.994910i \(-0.532129\pi\)
−0.100765 + 0.994910i \(0.532129\pi\)
\(72\) 0 0
\(73\) −3.40820e8 −1.40466 −0.702332 0.711850i \(-0.747858\pi\)
−0.702332 + 0.711850i \(0.747858\pi\)
\(74\) 0 0
\(75\) 4.76997e6 0.0174076
\(76\) 0 0
\(77\) 5.35268e8 1.73525
\(78\) 0 0
\(79\) 2.81089e8 0.811935 0.405967 0.913888i \(-0.366935\pi\)
0.405967 + 0.913888i \(0.366935\pi\)
\(80\) 0 0
\(81\) 3.78638e8 0.977330
\(82\) 0 0
\(83\) −6.01017e8 −1.39007 −0.695033 0.718978i \(-0.744610\pi\)
−0.695033 + 0.718978i \(0.744610\pi\)
\(84\) 0 0
\(85\) −1.29687e8 −0.269471
\(86\) 0 0
\(87\) −5.23011e7 −0.0978756
\(88\) 0 0
\(89\) 5.39917e8 0.912162 0.456081 0.889938i \(-0.349253\pi\)
0.456081 + 0.889938i \(0.349253\pi\)
\(90\) 0 0
\(91\) 1.63369e9 2.49737
\(92\) 0 0
\(93\) −4.38028e7 −0.0607195
\(94\) 0 0
\(95\) −5.01528e8 −0.631741
\(96\) 0 0
\(97\) 4.23026e8 0.485170 0.242585 0.970130i \(-0.422005\pi\)
0.242585 + 0.970130i \(0.422005\pi\)
\(98\) 0 0
\(99\) −1.08660e9 −1.13687
\(100\) 0 0
\(101\) −1.09286e9 −1.04501 −0.522503 0.852637i \(-0.675002\pi\)
−0.522503 + 0.852637i \(0.675002\pi\)
\(102\) 0 0
\(103\) −8.50494e7 −0.0744567 −0.0372284 0.999307i \(-0.511853\pi\)
−0.0372284 + 0.999307i \(0.511853\pi\)
\(104\) 0 0
\(105\) −7.34389e7 −0.0589623
\(106\) 0 0
\(107\) 5.23881e8 0.386372 0.193186 0.981162i \(-0.438118\pi\)
0.193186 + 0.981162i \(0.438118\pi\)
\(108\) 0 0
\(109\) −5.59417e8 −0.379591 −0.189796 0.981824i \(-0.560783\pi\)
−0.189796 + 0.981824i \(0.560783\pi\)
\(110\) 0 0
\(111\) −3.53947e7 −0.0221302
\(112\) 0 0
\(113\) −2.63801e9 −1.52203 −0.761016 0.648733i \(-0.775299\pi\)
−0.761016 + 0.648733i \(0.775299\pi\)
\(114\) 0 0
\(115\) 7.76179e8 0.413829
\(116\) 0 0
\(117\) −3.31641e9 −1.63618
\(118\) 0 0
\(119\) 1.99668e9 0.912739
\(120\) 0 0
\(121\) 7.36341e8 0.312281
\(122\) 0 0
\(123\) 3.07128e8 0.120989
\(124\) 0 0
\(125\) −2.44141e8 −0.0894427
\(126\) 0 0
\(127\) 3.33268e9 1.13678 0.568391 0.822758i \(-0.307566\pi\)
0.568391 + 0.822758i \(0.307566\pi\)
\(128\) 0 0
\(129\) −2.44472e8 −0.0777276
\(130\) 0 0
\(131\) −3.78884e9 −1.12405 −0.562024 0.827121i \(-0.689977\pi\)
−0.562024 + 0.827121i \(0.689977\pi\)
\(132\) 0 0
\(133\) 7.72158e9 2.13980
\(134\) 0 0
\(135\) 2.99301e8 0.0775544
\(136\) 0 0
\(137\) −2.15185e9 −0.521879 −0.260940 0.965355i \(-0.584032\pi\)
−0.260940 + 0.965355i \(0.584032\pi\)
\(138\) 0 0
\(139\) 2.75086e8 0.0625032 0.0312516 0.999512i \(-0.490051\pi\)
0.0312516 + 0.999512i \(0.490051\pi\)
\(140\) 0 0
\(141\) 4.55487e8 0.0970487
\(142\) 0 0
\(143\) 9.44408e9 1.88863
\(144\) 0 0
\(145\) 2.67692e9 0.502898
\(146\) 0 0
\(147\) 6.37911e8 0.112676
\(148\) 0 0
\(149\) −6.16582e9 −1.02483 −0.512416 0.858737i \(-0.671249\pi\)
−0.512416 + 0.858737i \(0.671249\pi\)
\(150\) 0 0
\(151\) −3.28376e9 −0.514014 −0.257007 0.966410i \(-0.582736\pi\)
−0.257007 + 0.966410i \(0.582736\pi\)
\(152\) 0 0
\(153\) −4.05327e9 −0.597990
\(154\) 0 0
\(155\) 2.24195e9 0.311985
\(156\) 0 0
\(157\) −1.89255e9 −0.248599 −0.124299 0.992245i \(-0.539668\pi\)
−0.124299 + 0.992245i \(0.539668\pi\)
\(158\) 0 0
\(159\) −3.11573e8 −0.0386610
\(160\) 0 0
\(161\) −1.19501e10 −1.40170
\(162\) 0 0
\(163\) 4.14293e8 0.0459688 0.0229844 0.999736i \(-0.492683\pi\)
0.0229844 + 0.999736i \(0.492683\pi\)
\(164\) 0 0
\(165\) −4.24537e8 −0.0445900
\(166\) 0 0
\(167\) −1.56225e10 −1.55427 −0.777136 0.629332i \(-0.783328\pi\)
−0.777136 + 0.629332i \(0.783328\pi\)
\(168\) 0 0
\(169\) 1.82198e10 1.71812
\(170\) 0 0
\(171\) −1.56749e10 −1.40191
\(172\) 0 0
\(173\) 1.16058e10 0.985069 0.492535 0.870293i \(-0.336070\pi\)
0.492535 + 0.870293i \(0.336070\pi\)
\(174\) 0 0
\(175\) 3.75882e9 0.302956
\(176\) 0 0
\(177\) −1.21683e9 −0.0931860
\(178\) 0 0
\(179\) −2.45556e10 −1.78777 −0.893886 0.448294i \(-0.852032\pi\)
−0.893886 + 0.448294i \(0.852032\pi\)
\(180\) 0 0
\(181\) 3.21032e9 0.222328 0.111164 0.993802i \(-0.464542\pi\)
0.111164 + 0.993802i \(0.464542\pi\)
\(182\) 0 0
\(183\) 2.44752e9 0.161323
\(184\) 0 0
\(185\) 1.81160e9 0.113708
\(186\) 0 0
\(187\) 1.15424e10 0.690256
\(188\) 0 0
\(189\) −4.60807e9 −0.262689
\(190\) 0 0
\(191\) 3.41548e10 1.85695 0.928477 0.371390i \(-0.121119\pi\)
0.928477 + 0.371390i \(0.121119\pi\)
\(192\) 0 0
\(193\) 2.50875e10 1.30152 0.650759 0.759284i \(-0.274451\pi\)
0.650759 + 0.759284i \(0.274451\pi\)
\(194\) 0 0
\(195\) −1.29573e9 −0.0641739
\(196\) 0 0
\(197\) 9.64109e9 0.456067 0.228033 0.973653i \(-0.426771\pi\)
0.228033 + 0.973653i \(0.426771\pi\)
\(198\) 0 0
\(199\) 3.34318e9 0.151120 0.0755598 0.997141i \(-0.475926\pi\)
0.0755598 + 0.997141i \(0.475926\pi\)
\(200\) 0 0
\(201\) −9.89040e8 −0.0427397
\(202\) 0 0
\(203\) −4.12142e10 −1.70339
\(204\) 0 0
\(205\) −1.57197e10 −0.621659
\(206\) 0 0
\(207\) 2.42589e10 0.918341
\(208\) 0 0
\(209\) 4.46371e10 1.61822
\(210\) 0 0
\(211\) −1.76087e10 −0.611585 −0.305792 0.952098i \(-0.598921\pi\)
−0.305792 + 0.952098i \(0.598921\pi\)
\(212\) 0 0
\(213\) −5.26936e8 −0.0175408
\(214\) 0 0
\(215\) 1.25128e10 0.399375
\(216\) 0 0
\(217\) −3.45174e10 −1.05674
\(218\) 0 0
\(219\) −4.16179e9 −0.122259
\(220\) 0 0
\(221\) 3.52287e10 0.993415
\(222\) 0 0
\(223\) −5.00503e10 −1.35530 −0.677649 0.735386i \(-0.737001\pi\)
−0.677649 + 0.735386i \(0.737001\pi\)
\(224\) 0 0
\(225\) −7.63043e9 −0.198485
\(226\) 0 0
\(227\) −8.68117e9 −0.217001 −0.108501 0.994096i \(-0.534605\pi\)
−0.108501 + 0.994096i \(0.534605\pi\)
\(228\) 0 0
\(229\) −2.96014e10 −0.711299 −0.355649 0.934619i \(-0.615740\pi\)
−0.355649 + 0.934619i \(0.615740\pi\)
\(230\) 0 0
\(231\) 6.53622e9 0.151033
\(232\) 0 0
\(233\) −7.42617e10 −1.65068 −0.825341 0.564634i \(-0.809017\pi\)
−0.825341 + 0.564634i \(0.809017\pi\)
\(234\) 0 0
\(235\) −2.33131e10 −0.498649
\(236\) 0 0
\(237\) 3.43240e9 0.0706693
\(238\) 0 0
\(239\) −1.65577e10 −0.328253 −0.164126 0.986439i \(-0.552481\pi\)
−0.164126 + 0.986439i \(0.552481\pi\)
\(240\) 0 0
\(241\) 3.61072e10 0.689473 0.344737 0.938699i \(-0.387968\pi\)
0.344737 + 0.938699i \(0.387968\pi\)
\(242\) 0 0
\(243\) 1.40494e10 0.258482
\(244\) 0 0
\(245\) −3.26501e10 −0.578945
\(246\) 0 0
\(247\) 1.36237e11 2.32894
\(248\) 0 0
\(249\) −7.33909e9 −0.120989
\(250\) 0 0
\(251\) 2.32959e10 0.370465 0.185232 0.982695i \(-0.440696\pi\)
0.185232 + 0.982695i \(0.440696\pi\)
\(252\) 0 0
\(253\) −6.90815e10 −1.06003
\(254\) 0 0
\(255\) −1.58362e9 −0.0234542
\(256\) 0 0
\(257\) −4.57624e10 −0.654350 −0.327175 0.944964i \(-0.606097\pi\)
−0.327175 + 0.944964i \(0.606097\pi\)
\(258\) 0 0
\(259\) −2.78916e10 −0.385145
\(260\) 0 0
\(261\) 8.36651e10 1.11599
\(262\) 0 0
\(263\) 9.93911e10 1.28099 0.640496 0.767961i \(-0.278729\pi\)
0.640496 + 0.767961i \(0.278729\pi\)
\(264\) 0 0
\(265\) 1.59472e10 0.198645
\(266\) 0 0
\(267\) 6.59299e9 0.0793929
\(268\) 0 0
\(269\) 1.57475e11 1.83369 0.916845 0.399242i \(-0.130727\pi\)
0.916845 + 0.399242i \(0.130727\pi\)
\(270\) 0 0
\(271\) 2.79292e10 0.314555 0.157278 0.987554i \(-0.449728\pi\)
0.157278 + 0.987554i \(0.449728\pi\)
\(272\) 0 0
\(273\) 1.99492e10 0.217367
\(274\) 0 0
\(275\) 2.17290e10 0.229110
\(276\) 0 0
\(277\) −1.68307e11 −1.71768 −0.858842 0.512240i \(-0.828816\pi\)
−0.858842 + 0.512240i \(0.828816\pi\)
\(278\) 0 0
\(279\) 7.00705e10 0.692335
\(280\) 0 0
\(281\) −8.51877e10 −0.815076 −0.407538 0.913188i \(-0.633613\pi\)
−0.407538 + 0.913188i \(0.633613\pi\)
\(282\) 0 0
\(283\) −2.02954e11 −1.88087 −0.940436 0.339970i \(-0.889583\pi\)
−0.940436 + 0.339970i \(0.889583\pi\)
\(284\) 0 0
\(285\) −6.12422e9 −0.0549856
\(286\) 0 0
\(287\) 2.42022e11 2.10565
\(288\) 0 0
\(289\) −7.55318e10 −0.636927
\(290\) 0 0
\(291\) 5.16561e9 0.0422283
\(292\) 0 0
\(293\) 1.31572e11 1.04294 0.521470 0.853270i \(-0.325384\pi\)
0.521470 + 0.853270i \(0.325384\pi\)
\(294\) 0 0
\(295\) 6.22810e10 0.478802
\(296\) 0 0
\(297\) −2.66384e10 −0.198657
\(298\) 0 0
\(299\) −2.10844e11 −1.52560
\(300\) 0 0
\(301\) −1.92648e11 −1.35274
\(302\) 0 0
\(303\) −1.33450e10 −0.0909554
\(304\) 0 0
\(305\) −1.25271e11 −0.828899
\(306\) 0 0
\(307\) −1.88785e11 −1.21296 −0.606479 0.795099i \(-0.707419\pi\)
−0.606479 + 0.795099i \(0.707419\pi\)
\(308\) 0 0
\(309\) −1.03855e9 −0.00648057
\(310\) 0 0
\(311\) −7.10454e10 −0.430640 −0.215320 0.976544i \(-0.569079\pi\)
−0.215320 + 0.976544i \(0.569079\pi\)
\(312\) 0 0
\(313\) 3.35032e11 1.97305 0.986523 0.163623i \(-0.0523182\pi\)
0.986523 + 0.163623i \(0.0523182\pi\)
\(314\) 0 0
\(315\) 1.17479e11 0.672299
\(316\) 0 0
\(317\) −1.98163e11 −1.10219 −0.551094 0.834443i \(-0.685789\pi\)
−0.551094 + 0.834443i \(0.685789\pi\)
\(318\) 0 0
\(319\) −2.38252e11 −1.28818
\(320\) 0 0
\(321\) 6.39717e9 0.0336291
\(322\) 0 0
\(323\) 1.66507e11 0.851179
\(324\) 0 0
\(325\) 6.63192e10 0.329734
\(326\) 0 0
\(327\) −6.83110e9 −0.0330389
\(328\) 0 0
\(329\) 3.58931e11 1.68900
\(330\) 0 0
\(331\) −1.60576e11 −0.735284 −0.367642 0.929967i \(-0.619835\pi\)
−0.367642 + 0.929967i \(0.619835\pi\)
\(332\) 0 0
\(333\) 5.66202e10 0.252332
\(334\) 0 0
\(335\) 5.06219e10 0.219602
\(336\) 0 0
\(337\) 8.77234e10 0.370494 0.185247 0.982692i \(-0.440692\pi\)
0.185247 + 0.982692i \(0.440692\pi\)
\(338\) 0 0
\(339\) −3.22130e10 −0.132475
\(340\) 0 0
\(341\) −1.99539e11 −0.799157
\(342\) 0 0
\(343\) 1.14379e11 0.446194
\(344\) 0 0
\(345\) 9.47800e9 0.0360189
\(346\) 0 0
\(347\) 1.89987e11 0.703464 0.351732 0.936101i \(-0.385593\pi\)
0.351732 + 0.936101i \(0.385593\pi\)
\(348\) 0 0
\(349\) 4.85678e11 1.75240 0.876201 0.481946i \(-0.160070\pi\)
0.876201 + 0.481946i \(0.160070\pi\)
\(350\) 0 0
\(351\) −8.13031e10 −0.285907
\(352\) 0 0
\(353\) 5.15816e10 0.176811 0.0884054 0.996085i \(-0.471823\pi\)
0.0884054 + 0.996085i \(0.471823\pi\)
\(354\) 0 0
\(355\) 2.69701e10 0.0901270
\(356\) 0 0
\(357\) 2.43816e10 0.0794431
\(358\) 0 0
\(359\) −1.79131e11 −0.569176 −0.284588 0.958650i \(-0.591857\pi\)
−0.284588 + 0.958650i \(0.591857\pi\)
\(360\) 0 0
\(361\) 3.21231e11 0.995485
\(362\) 0 0
\(363\) 8.99154e9 0.0271803
\(364\) 0 0
\(365\) 2.13012e11 0.628184
\(366\) 0 0
\(367\) −2.76314e10 −0.0795071 −0.0397536 0.999210i \(-0.512657\pi\)
−0.0397536 + 0.999210i \(0.512657\pi\)
\(368\) 0 0
\(369\) −4.91307e11 −1.37954
\(370\) 0 0
\(371\) −2.45525e11 −0.672842
\(372\) 0 0
\(373\) 2.35196e11 0.629129 0.314565 0.949236i \(-0.398142\pi\)
0.314565 + 0.949236i \(0.398142\pi\)
\(374\) 0 0
\(375\) −2.98123e9 −0.00778493
\(376\) 0 0
\(377\) −7.27168e11 −1.85395
\(378\) 0 0
\(379\) −3.11965e11 −0.776658 −0.388329 0.921521i \(-0.626948\pi\)
−0.388329 + 0.921521i \(0.626948\pi\)
\(380\) 0 0
\(381\) 4.06958e10 0.0989434
\(382\) 0 0
\(383\) −4.78535e10 −0.113637 −0.0568185 0.998385i \(-0.518096\pi\)
−0.0568185 + 0.998385i \(0.518096\pi\)
\(384\) 0 0
\(385\) −3.34543e11 −0.776029
\(386\) 0 0
\(387\) 3.91077e11 0.886264
\(388\) 0 0
\(389\) 1.19528e11 0.264665 0.132332 0.991205i \(-0.457753\pi\)
0.132332 + 0.991205i \(0.457753\pi\)
\(390\) 0 0
\(391\) −2.57691e11 −0.557575
\(392\) 0 0
\(393\) −4.62659e10 −0.0978351
\(394\) 0 0
\(395\) −1.75680e11 −0.363108
\(396\) 0 0
\(397\) 7.08892e10 0.143226 0.0716132 0.997432i \(-0.477185\pi\)
0.0716132 + 0.997432i \(0.477185\pi\)
\(398\) 0 0
\(399\) 9.42891e10 0.186245
\(400\) 0 0
\(401\) 5.62486e11 1.08633 0.543165 0.839626i \(-0.317226\pi\)
0.543165 + 0.839626i \(0.317226\pi\)
\(402\) 0 0
\(403\) −6.09012e11 −1.15015
\(404\) 0 0
\(405\) −2.36649e11 −0.437075
\(406\) 0 0
\(407\) −1.61236e11 −0.291265
\(408\) 0 0
\(409\) 2.23341e11 0.394651 0.197325 0.980338i \(-0.436774\pi\)
0.197325 + 0.980338i \(0.436774\pi\)
\(410\) 0 0
\(411\) −2.62765e10 −0.0454234
\(412\) 0 0
\(413\) −9.58884e11 −1.62178
\(414\) 0 0
\(415\) 3.75636e11 0.621657
\(416\) 0 0
\(417\) 3.35911e9 0.00544016
\(418\) 0 0
\(419\) 3.54170e11 0.561370 0.280685 0.959800i \(-0.409438\pi\)
0.280685 + 0.959800i \(0.409438\pi\)
\(420\) 0 0
\(421\) 5.29294e11 0.821159 0.410580 0.911825i \(-0.365326\pi\)
0.410580 + 0.911825i \(0.365326\pi\)
\(422\) 0 0
\(423\) −7.28634e11 −1.10657
\(424\) 0 0
\(425\) 8.10545e10 0.120511
\(426\) 0 0
\(427\) 1.92869e12 2.80761
\(428\) 0 0
\(429\) 1.15323e11 0.164383
\(430\) 0 0
\(431\) 1.26760e12 1.76944 0.884718 0.466127i \(-0.154351\pi\)
0.884718 + 0.466127i \(0.154351\pi\)
\(432\) 0 0
\(433\) 3.28861e11 0.449591 0.224795 0.974406i \(-0.427829\pi\)
0.224795 + 0.974406i \(0.427829\pi\)
\(434\) 0 0
\(435\) 3.26882e10 0.0437713
\(436\) 0 0
\(437\) −9.96545e11 −1.30717
\(438\) 0 0
\(439\) 2.69396e11 0.346178 0.173089 0.984906i \(-0.444625\pi\)
0.173089 + 0.984906i \(0.444625\pi\)
\(440\) 0 0
\(441\) −1.02045e12 −1.28475
\(442\) 0 0
\(443\) 4.20444e11 0.518671 0.259335 0.965787i \(-0.416497\pi\)
0.259335 + 0.965787i \(0.416497\pi\)
\(444\) 0 0
\(445\) −3.37448e11 −0.407931
\(446\) 0 0
\(447\) −7.52915e10 −0.0891995
\(448\) 0 0
\(449\) 1.19773e11 0.139075 0.0695376 0.997579i \(-0.477848\pi\)
0.0695376 + 0.997579i \(0.477848\pi\)
\(450\) 0 0
\(451\) 1.39909e12 1.59239
\(452\) 0 0
\(453\) −4.00984e10 −0.0447388
\(454\) 0 0
\(455\) −1.02106e12 −1.11686
\(456\) 0 0
\(457\) 1.34943e12 1.44719 0.723597 0.690223i \(-0.242488\pi\)
0.723597 + 0.690223i \(0.242488\pi\)
\(458\) 0 0
\(459\) −9.93677e10 −0.104493
\(460\) 0 0
\(461\) 2.09834e11 0.216383 0.108191 0.994130i \(-0.465494\pi\)
0.108191 + 0.994130i \(0.465494\pi\)
\(462\) 0 0
\(463\) −6.60348e11 −0.667819 −0.333909 0.942605i \(-0.608368\pi\)
−0.333909 + 0.942605i \(0.608368\pi\)
\(464\) 0 0
\(465\) 2.73767e10 0.0271546
\(466\) 0 0
\(467\) −8.28966e11 −0.806512 −0.403256 0.915087i \(-0.632122\pi\)
−0.403256 + 0.915087i \(0.632122\pi\)
\(468\) 0 0
\(469\) −7.79381e11 −0.743827
\(470\) 0 0
\(471\) −2.31102e10 −0.0216376
\(472\) 0 0
\(473\) −1.11366e12 −1.02301
\(474\) 0 0
\(475\) 3.13455e11 0.282523
\(476\) 0 0
\(477\) 4.98417e11 0.440819
\(478\) 0 0
\(479\) −6.48710e11 −0.563042 −0.281521 0.959555i \(-0.590839\pi\)
−0.281521 + 0.959555i \(0.590839\pi\)
\(480\) 0 0
\(481\) −4.92110e11 −0.419188
\(482\) 0 0
\(483\) −1.45924e11 −0.122002
\(484\) 0 0
\(485\) −2.64391e11 −0.216975
\(486\) 0 0
\(487\) −1.75979e12 −1.41768 −0.708842 0.705367i \(-0.750782\pi\)
−0.708842 + 0.705367i \(0.750782\pi\)
\(488\) 0 0
\(489\) 5.05898e9 0.00400104
\(490\) 0 0
\(491\) 3.02135e11 0.234603 0.117302 0.993096i \(-0.462576\pi\)
0.117302 + 0.993096i \(0.462576\pi\)
\(492\) 0 0
\(493\) −8.88736e11 −0.677582
\(494\) 0 0
\(495\) 6.79124e11 0.508424
\(496\) 0 0
\(497\) −4.15235e11 −0.305274
\(498\) 0 0
\(499\) 5.11758e11 0.369498 0.184749 0.982786i \(-0.440853\pi\)
0.184749 + 0.982786i \(0.440853\pi\)
\(500\) 0 0
\(501\) −1.90768e11 −0.135281
\(502\) 0 0
\(503\) 1.69213e12 1.17863 0.589314 0.807904i \(-0.299398\pi\)
0.589314 + 0.807904i \(0.299398\pi\)
\(504\) 0 0
\(505\) 6.83038e11 0.467341
\(506\) 0 0
\(507\) 2.22483e11 0.149542
\(508\) 0 0
\(509\) 7.10216e11 0.468986 0.234493 0.972118i \(-0.424657\pi\)
0.234493 + 0.972118i \(0.424657\pi\)
\(510\) 0 0
\(511\) −3.27956e12 −2.12776
\(512\) 0 0
\(513\) −3.84276e11 −0.244972
\(514\) 0 0
\(515\) 5.31559e10 0.0332981
\(516\) 0 0
\(517\) 2.07492e12 1.27730
\(518\) 0 0
\(519\) 1.41719e11 0.0857386
\(520\) 0 0
\(521\) −8.88794e11 −0.528483 −0.264242 0.964456i \(-0.585122\pi\)
−0.264242 + 0.964456i \(0.585122\pi\)
\(522\) 0 0
\(523\) 1.01046e12 0.590555 0.295277 0.955412i \(-0.404588\pi\)
0.295277 + 0.955412i \(0.404588\pi\)
\(524\) 0 0
\(525\) 4.58993e10 0.0263687
\(526\) 0 0
\(527\) −7.44327e11 −0.420355
\(528\) 0 0
\(529\) −2.58873e11 −0.143726
\(530\) 0 0
\(531\) 1.94654e12 1.06252
\(532\) 0 0
\(533\) 4.27015e12 2.29177
\(534\) 0 0
\(535\) −3.27426e11 −0.172791
\(536\) 0 0
\(537\) −2.99851e11 −0.155604
\(538\) 0 0
\(539\) 2.90593e12 1.48298
\(540\) 0 0
\(541\) 3.39951e12 1.70619 0.853097 0.521753i \(-0.174722\pi\)
0.853097 + 0.521753i \(0.174722\pi\)
\(542\) 0 0
\(543\) 3.92015e10 0.0193510
\(544\) 0 0
\(545\) 3.49636e11 0.169758
\(546\) 0 0
\(547\) −3.10308e12 −1.48201 −0.741004 0.671501i \(-0.765650\pi\)
−0.741004 + 0.671501i \(0.765650\pi\)
\(548\) 0 0
\(549\) −3.91525e12 −1.83943
\(550\) 0 0
\(551\) −3.43693e12 −1.58851
\(552\) 0 0
\(553\) 2.70479e12 1.22990
\(554\) 0 0
\(555\) 2.21217e10 0.00989691
\(556\) 0 0
\(557\) −4.92558e11 −0.216825 −0.108412 0.994106i \(-0.534577\pi\)
−0.108412 + 0.994106i \(0.534577\pi\)
\(558\) 0 0
\(559\) −3.39901e12 −1.47231
\(560\) 0 0
\(561\) 1.40946e11 0.0600786
\(562\) 0 0
\(563\) 1.67786e12 0.703832 0.351916 0.936032i \(-0.385530\pi\)
0.351916 + 0.936032i \(0.385530\pi\)
\(564\) 0 0
\(565\) 1.64876e12 0.680673
\(566\) 0 0
\(567\) 3.64347e12 1.48044
\(568\) 0 0
\(569\) 3.40646e12 1.36238 0.681190 0.732107i \(-0.261463\pi\)
0.681190 + 0.732107i \(0.261463\pi\)
\(570\) 0 0
\(571\) −4.05234e12 −1.59530 −0.797651 0.603119i \(-0.793924\pi\)
−0.797651 + 0.603119i \(0.793924\pi\)
\(572\) 0 0
\(573\) 4.17068e11 0.161626
\(574\) 0 0
\(575\) −4.85112e11 −0.185070
\(576\) 0 0
\(577\) −4.55977e11 −0.171258 −0.0856291 0.996327i \(-0.527290\pi\)
−0.0856291 + 0.996327i \(0.527290\pi\)
\(578\) 0 0
\(579\) 3.06347e11 0.113282
\(580\) 0 0
\(581\) −5.78333e12 −2.10565
\(582\) 0 0
\(583\) −1.41933e12 −0.508834
\(584\) 0 0
\(585\) 2.07275e12 0.731722
\(586\) 0 0
\(587\) −2.83858e12 −0.986801 −0.493401 0.869802i \(-0.664246\pi\)
−0.493401 + 0.869802i \(0.664246\pi\)
\(588\) 0 0
\(589\) −2.87847e12 −0.985470
\(590\) 0 0
\(591\) 1.17728e11 0.0396952
\(592\) 0 0
\(593\) −4.08687e12 −1.35720 −0.678601 0.734507i \(-0.737414\pi\)
−0.678601 + 0.734507i \(0.737414\pi\)
\(594\) 0 0
\(595\) −1.24792e12 −0.408189
\(596\) 0 0
\(597\) 4.08240e10 0.0131532
\(598\) 0 0
\(599\) 4.47524e12 1.42035 0.710175 0.704025i \(-0.248616\pi\)
0.710175 + 0.704025i \(0.248616\pi\)
\(600\) 0 0
\(601\) −4.83907e11 −0.151296 −0.0756478 0.997135i \(-0.524102\pi\)
−0.0756478 + 0.997135i \(0.524102\pi\)
\(602\) 0 0
\(603\) 1.58215e12 0.487326
\(604\) 0 0
\(605\) −4.60213e11 −0.139656
\(606\) 0 0
\(607\) −4.96609e12 −1.48479 −0.742395 0.669962i \(-0.766310\pi\)
−0.742395 + 0.669962i \(0.766310\pi\)
\(608\) 0 0
\(609\) −5.03271e11 −0.148260
\(610\) 0 0
\(611\) 6.33285e12 1.83829
\(612\) 0 0
\(613\) −2.43460e12 −0.696396 −0.348198 0.937421i \(-0.613206\pi\)
−0.348198 + 0.937421i \(0.613206\pi\)
\(614\) 0 0
\(615\) −1.91955e11 −0.0541080
\(616\) 0 0
\(617\) 1.82614e12 0.507284 0.253642 0.967298i \(-0.418372\pi\)
0.253642 + 0.967298i \(0.418372\pi\)
\(618\) 0 0
\(619\) −1.13058e12 −0.309523 −0.154761 0.987952i \(-0.549461\pi\)
−0.154761 + 0.987952i \(0.549461\pi\)
\(620\) 0 0
\(621\) 5.94716e11 0.160471
\(622\) 0 0
\(623\) 5.19539e12 1.38173
\(624\) 0 0
\(625\) 1.52588e11 0.0400000
\(626\) 0 0
\(627\) 5.45068e11 0.140847
\(628\) 0 0
\(629\) −6.01450e11 −0.153205
\(630\) 0 0
\(631\) 1.96900e12 0.494440 0.247220 0.968959i \(-0.420483\pi\)
0.247220 + 0.968959i \(0.420483\pi\)
\(632\) 0 0
\(633\) −2.15022e11 −0.0532312
\(634\) 0 0
\(635\) −2.08293e12 −0.508385
\(636\) 0 0
\(637\) 8.86918e12 2.13430
\(638\) 0 0
\(639\) 8.42930e11 0.200003
\(640\) 0 0
\(641\) 2.69141e12 0.629679 0.314840 0.949145i \(-0.398049\pi\)
0.314840 + 0.949145i \(0.398049\pi\)
\(642\) 0 0
\(643\) −2.27990e12 −0.525977 −0.262989 0.964799i \(-0.584708\pi\)
−0.262989 + 0.964799i \(0.584708\pi\)
\(644\) 0 0
\(645\) 1.52795e11 0.0347608
\(646\) 0 0
\(647\) −4.86871e12 −1.09231 −0.546153 0.837685i \(-0.683908\pi\)
−0.546153 + 0.837685i \(0.683908\pi\)
\(648\) 0 0
\(649\) −5.54314e12 −1.22646
\(650\) 0 0
\(651\) −4.21495e11 −0.0919768
\(652\) 0 0
\(653\) 2.30610e12 0.496329 0.248164 0.968718i \(-0.420173\pi\)
0.248164 + 0.968718i \(0.420173\pi\)
\(654\) 0 0
\(655\) 2.36802e12 0.502690
\(656\) 0 0
\(657\) 6.65754e12 1.39402
\(658\) 0 0
\(659\) 7.40069e12 1.52858 0.764289 0.644873i \(-0.223090\pi\)
0.764289 + 0.644873i \(0.223090\pi\)
\(660\) 0 0
\(661\) −2.00760e12 −0.409045 −0.204522 0.978862i \(-0.565564\pi\)
−0.204522 + 0.978862i \(0.565564\pi\)
\(662\) 0 0
\(663\) 4.30181e11 0.0864650
\(664\) 0 0
\(665\) −4.82599e12 −0.956949
\(666\) 0 0
\(667\) 5.31909e12 1.04057
\(668\) 0 0
\(669\) −6.11170e11 −0.117963
\(670\) 0 0
\(671\) 1.11494e13 2.12324
\(672\) 0 0
\(673\) −1.01154e13 −1.90071 −0.950355 0.311167i \(-0.899280\pi\)
−0.950355 + 0.311167i \(0.899280\pi\)
\(674\) 0 0
\(675\) −1.87063e11 −0.0346834
\(676\) 0 0
\(677\) −8.91166e12 −1.63046 −0.815229 0.579139i \(-0.803389\pi\)
−0.815229 + 0.579139i \(0.803389\pi\)
\(678\) 0 0
\(679\) 4.07059e12 0.734926
\(680\) 0 0
\(681\) −1.06007e11 −0.0188874
\(682\) 0 0
\(683\) −9.26863e12 −1.62976 −0.814878 0.579633i \(-0.803196\pi\)
−0.814878 + 0.579633i \(0.803196\pi\)
\(684\) 0 0
\(685\) 1.34491e12 0.233391
\(686\) 0 0
\(687\) −3.61466e11 −0.0619101
\(688\) 0 0
\(689\) −4.33195e12 −0.732313
\(690\) 0 0
\(691\) 1.03490e11 0.0172682 0.00863409 0.999963i \(-0.497252\pi\)
0.00863409 + 0.999963i \(0.497252\pi\)
\(692\) 0 0
\(693\) −1.04559e13 −1.72211
\(694\) 0 0
\(695\) −1.71929e11 −0.0279523
\(696\) 0 0
\(697\) 5.21893e12 0.837594
\(698\) 0 0
\(699\) −9.06818e11 −0.143672
\(700\) 0 0
\(701\) 4.77133e12 0.746291 0.373145 0.927773i \(-0.378279\pi\)
0.373145 + 0.927773i \(0.378279\pi\)
\(702\) 0 0
\(703\) −2.32594e12 −0.359169
\(704\) 0 0
\(705\) −2.84679e11 −0.0434015
\(706\) 0 0
\(707\) −1.05161e13 −1.58295
\(708\) 0 0
\(709\) 7.58922e12 1.12795 0.563974 0.825793i \(-0.309272\pi\)
0.563974 + 0.825793i \(0.309272\pi\)
\(710\) 0 0
\(711\) −5.49075e12 −0.805784
\(712\) 0 0
\(713\) 4.45480e12 0.645543
\(714\) 0 0
\(715\) −5.90255e12 −0.844622
\(716\) 0 0
\(717\) −2.02188e11 −0.0285705
\(718\) 0 0
\(719\) 1.50766e12 0.210389 0.105194 0.994452i \(-0.466454\pi\)
0.105194 + 0.994452i \(0.466454\pi\)
\(720\) 0 0
\(721\) −8.18394e11 −0.112786
\(722\) 0 0
\(723\) 4.40909e11 0.0600105
\(724\) 0 0
\(725\) −1.67308e12 −0.224903
\(726\) 0 0
\(727\) 1.12985e13 1.50009 0.750045 0.661387i \(-0.230032\pi\)
0.750045 + 0.661387i \(0.230032\pi\)
\(728\) 0 0
\(729\) −7.28117e12 −0.954833
\(730\) 0 0
\(731\) −4.15423e12 −0.538100
\(732\) 0 0
\(733\) 1.07062e12 0.136984 0.0684919 0.997652i \(-0.478181\pi\)
0.0684919 + 0.997652i \(0.478181\pi\)
\(734\) 0 0
\(735\) −3.98694e11 −0.0503903
\(736\) 0 0
\(737\) −4.50546e12 −0.562517
\(738\) 0 0
\(739\) −1.44855e13 −1.78662 −0.893311 0.449439i \(-0.851624\pi\)
−0.893311 + 0.449439i \(0.851624\pi\)
\(740\) 0 0
\(741\) 1.66360e12 0.202707
\(742\) 0 0
\(743\) 1.24339e13 1.49678 0.748388 0.663261i \(-0.230828\pi\)
0.748388 + 0.663261i \(0.230828\pi\)
\(744\) 0 0
\(745\) 3.85364e12 0.458319
\(746\) 0 0
\(747\) 1.17402e13 1.37954
\(748\) 0 0
\(749\) 5.04108e12 0.585269
\(750\) 0 0
\(751\) 8.79097e12 1.00846 0.504228 0.863570i \(-0.331777\pi\)
0.504228 + 0.863570i \(0.331777\pi\)
\(752\) 0 0
\(753\) 2.84468e11 0.0322446
\(754\) 0 0
\(755\) 2.05235e12 0.229874
\(756\) 0 0
\(757\) 7.93877e12 0.878662 0.439331 0.898325i \(-0.355216\pi\)
0.439331 + 0.898325i \(0.355216\pi\)
\(758\) 0 0
\(759\) −8.43562e11 −0.0922633
\(760\) 0 0
\(761\) 1.12763e13 1.21881 0.609405 0.792859i \(-0.291408\pi\)
0.609405 + 0.792859i \(0.291408\pi\)
\(762\) 0 0
\(763\) −5.38303e12 −0.574998
\(764\) 0 0
\(765\) 2.53329e12 0.267429
\(766\) 0 0
\(767\) −1.69182e13 −1.76512
\(768\) 0 0
\(769\) 3.80743e12 0.392612 0.196306 0.980543i \(-0.437105\pi\)
0.196306 + 0.980543i \(0.437105\pi\)
\(770\) 0 0
\(771\) −5.58810e11 −0.0569534
\(772\) 0 0
\(773\) −3.89254e12 −0.392126 −0.196063 0.980591i \(-0.562816\pi\)
−0.196063 + 0.980591i \(0.562816\pi\)
\(774\) 0 0
\(775\) −1.40122e12 −0.139524
\(776\) 0 0
\(777\) −3.40588e11 −0.0335223
\(778\) 0 0
\(779\) 2.01827e13 1.96364
\(780\) 0 0
\(781\) −2.40040e12 −0.230862
\(782\) 0 0
\(783\) 2.05109e12 0.195010
\(784\) 0 0
\(785\) 1.18284e12 0.111177
\(786\) 0 0
\(787\) −2.70256e12 −0.251124 −0.125562 0.992086i \(-0.540073\pi\)
−0.125562 + 0.992086i \(0.540073\pi\)
\(788\) 0 0
\(789\) 1.21368e12 0.111495
\(790\) 0 0
\(791\) −2.53844e13 −2.30554
\(792\) 0 0
\(793\) 3.40291e13 3.05577
\(794\) 0 0
\(795\) 1.94733e11 0.0172897
\(796\) 0 0
\(797\) −1.58748e12 −0.139363 −0.0696813 0.997569i \(-0.522198\pi\)
−0.0696813 + 0.997569i \(0.522198\pi\)
\(798\) 0 0
\(799\) 7.73994e12 0.671857
\(800\) 0 0
\(801\) −1.05467e13 −0.905252
\(802\) 0 0
\(803\) −1.89586e13 −1.60911
\(804\) 0 0
\(805\) 7.46883e12 0.626861
\(806\) 0 0
\(807\) 1.92294e12 0.159601
\(808\) 0 0
\(809\) −4.55705e12 −0.374038 −0.187019 0.982356i \(-0.559883\pi\)
−0.187019 + 0.982356i \(0.559883\pi\)
\(810\) 0 0
\(811\) −2.77014e12 −0.224858 −0.112429 0.993660i \(-0.535863\pi\)
−0.112429 + 0.993660i \(0.535863\pi\)
\(812\) 0 0
\(813\) 3.41047e11 0.0273783
\(814\) 0 0
\(815\) −2.58933e11 −0.0205579
\(816\) 0 0
\(817\) −1.60653e13 −1.26151
\(818\) 0 0
\(819\) −3.19123e13 −2.47846
\(820\) 0 0
\(821\) −1.21734e13 −0.935124 −0.467562 0.883960i \(-0.654868\pi\)
−0.467562 + 0.883960i \(0.654868\pi\)
\(822\) 0 0
\(823\) 1.75355e13 1.33235 0.666175 0.745796i \(-0.267931\pi\)
0.666175 + 0.745796i \(0.267931\pi\)
\(824\) 0 0
\(825\) 2.65336e11 0.0199413
\(826\) 0 0
\(827\) −9.15138e12 −0.680318 −0.340159 0.940368i \(-0.610481\pi\)
−0.340159 + 0.940368i \(0.610481\pi\)
\(828\) 0 0
\(829\) −5.76678e12 −0.424071 −0.212035 0.977262i \(-0.568009\pi\)
−0.212035 + 0.977262i \(0.568009\pi\)
\(830\) 0 0
\(831\) −2.05522e12 −0.149504
\(832\) 0 0
\(833\) 1.08398e13 0.780044
\(834\) 0 0
\(835\) 9.76408e12 0.695092
\(836\) 0 0
\(837\) 1.71781e12 0.120979
\(838\) 0 0
\(839\) 8.69274e11 0.0605658 0.0302829 0.999541i \(-0.490359\pi\)
0.0302829 + 0.999541i \(0.490359\pi\)
\(840\) 0 0
\(841\) 3.83759e12 0.264531
\(842\) 0 0
\(843\) −1.04024e12 −0.0709427
\(844\) 0 0
\(845\) −1.13873e13 −0.768365
\(846\) 0 0
\(847\) 7.08549e12 0.473037
\(848\) 0 0
\(849\) −2.47830e12 −0.163708
\(850\) 0 0
\(851\) 3.59968e12 0.235278
\(852\) 0 0
\(853\) −9.19943e12 −0.594963 −0.297482 0.954728i \(-0.596147\pi\)
−0.297482 + 0.954728i \(0.596147\pi\)
\(854\) 0 0
\(855\) 9.79680e12 0.626955
\(856\) 0 0
\(857\) −6.57679e12 −0.416486 −0.208243 0.978077i \(-0.566775\pi\)
−0.208243 + 0.978077i \(0.566775\pi\)
\(858\) 0 0
\(859\) −1.80030e13 −1.12817 −0.564085 0.825717i \(-0.690771\pi\)
−0.564085 + 0.825717i \(0.690771\pi\)
\(860\) 0 0
\(861\) 2.95536e12 0.183272
\(862\) 0 0
\(863\) 3.08737e13 1.89470 0.947350 0.320200i \(-0.103750\pi\)
0.947350 + 0.320200i \(0.103750\pi\)
\(864\) 0 0
\(865\) −7.25361e12 −0.440536
\(866\) 0 0
\(867\) −9.22328e11 −0.0554370
\(868\) 0 0
\(869\) 1.56359e13 0.930110
\(870\) 0 0
\(871\) −1.37511e13 −0.809573
\(872\) 0 0
\(873\) −8.26334e12 −0.481495
\(874\) 0 0
\(875\) −2.34926e12 −0.135486
\(876\) 0 0
\(877\) 2.77535e12 0.158424 0.0792118 0.996858i \(-0.474760\pi\)
0.0792118 + 0.996858i \(0.474760\pi\)
\(878\) 0 0
\(879\) 1.60664e12 0.0907755
\(880\) 0 0
\(881\) −1.64369e13 −0.919241 −0.459620 0.888116i \(-0.652014\pi\)
−0.459620 + 0.888116i \(0.652014\pi\)
\(882\) 0 0
\(883\) −5.38106e12 −0.297882 −0.148941 0.988846i \(-0.547586\pi\)
−0.148941 + 0.988846i \(0.547586\pi\)
\(884\) 0 0
\(885\) 7.60520e11 0.0416741
\(886\) 0 0
\(887\) 1.61191e13 0.874347 0.437174 0.899377i \(-0.355980\pi\)
0.437174 + 0.899377i \(0.355980\pi\)
\(888\) 0 0
\(889\) 3.20690e13 1.72198
\(890\) 0 0
\(891\) 2.10622e13 1.11958
\(892\) 0 0
\(893\) 2.99320e13 1.57509
\(894\) 0 0
\(895\) 1.53473e13 0.799516
\(896\) 0 0
\(897\) −2.57464e12 −0.132785
\(898\) 0 0
\(899\) 1.53639e13 0.784483
\(900\) 0 0
\(901\) −5.29446e12 −0.267645
\(902\) 0 0
\(903\) −2.35245e12 −0.117740
\(904\) 0 0
\(905\) −2.00645e12 −0.0994281
\(906\) 0 0
\(907\) 1.50156e13 0.736734 0.368367 0.929681i \(-0.379917\pi\)
0.368367 + 0.929681i \(0.379917\pi\)
\(908\) 0 0
\(909\) 2.13478e13 1.03709
\(910\) 0 0
\(911\) −2.12904e12 −0.102412 −0.0512061 0.998688i \(-0.516307\pi\)
−0.0512061 + 0.998688i \(0.516307\pi\)
\(912\) 0 0
\(913\) −3.34324e13 −1.59239
\(914\) 0 0
\(915\) −1.52970e12 −0.0721458
\(916\) 0 0
\(917\) −3.64583e13 −1.70269
\(918\) 0 0
\(919\) −2.82600e13 −1.30693 −0.653464 0.756957i \(-0.726685\pi\)
−0.653464 + 0.756957i \(0.726685\pi\)
\(920\) 0 0
\(921\) −2.30528e12 −0.105574
\(922\) 0 0
\(923\) −7.32625e12 −0.332257
\(924\) 0 0
\(925\) −1.13225e12 −0.0508517
\(926\) 0 0
\(927\) 1.66135e12 0.0738927
\(928\) 0 0
\(929\) −1.74749e11 −0.00769739 −0.00384869 0.999993i \(-0.501225\pi\)
−0.00384869 + 0.999993i \(0.501225\pi\)
\(930\) 0 0
\(931\) 4.19199e13 1.82872
\(932\) 0 0
\(933\) −8.67543e11 −0.0374821
\(934\) 0 0
\(935\) −7.21402e12 −0.308692
\(936\) 0 0
\(937\) 6.04012e12 0.255987 0.127993 0.991775i \(-0.459146\pi\)
0.127993 + 0.991775i \(0.459146\pi\)
\(938\) 0 0
\(939\) 4.09112e12 0.171730
\(940\) 0 0
\(941\) 2.13848e13 0.889104 0.444552 0.895753i \(-0.353363\pi\)
0.444552 + 0.895753i \(0.353363\pi\)
\(942\) 0 0
\(943\) −3.12353e13 −1.28630
\(944\) 0 0
\(945\) 2.88004e12 0.117478
\(946\) 0 0
\(947\) 4.58008e13 1.85054 0.925269 0.379312i \(-0.123839\pi\)
0.925269 + 0.379312i \(0.123839\pi\)
\(948\) 0 0
\(949\) −5.78634e13 −2.31583
\(950\) 0 0
\(951\) −2.41979e12 −0.0959323
\(952\) 0 0
\(953\) −1.57835e13 −0.619847 −0.309924 0.950761i \(-0.600304\pi\)
−0.309924 + 0.950761i \(0.600304\pi\)
\(954\) 0 0
\(955\) −2.13467e13 −0.830455
\(956\) 0 0
\(957\) −2.90932e12 −0.112121
\(958\) 0 0
\(959\) −2.07063e13 −0.790533
\(960\) 0 0
\(961\) −1.35722e13 −0.513326
\(962\) 0 0
\(963\) −1.02334e13 −0.383445
\(964\) 0 0
\(965\) −1.56797e13 −0.582056
\(966\) 0 0
\(967\) 1.61550e13 0.594140 0.297070 0.954856i \(-0.403991\pi\)
0.297070 + 0.954856i \(0.403991\pi\)
\(968\) 0 0
\(969\) 2.03324e12 0.0740851
\(970\) 0 0
\(971\) −5.05363e12 −0.182439 −0.0912194 0.995831i \(-0.529076\pi\)
−0.0912194 + 0.995831i \(0.529076\pi\)
\(972\) 0 0
\(973\) 2.64703e12 0.0946786
\(974\) 0 0
\(975\) 8.09831e11 0.0286994
\(976\) 0 0
\(977\) −3.22726e13 −1.13321 −0.566603 0.823991i \(-0.691743\pi\)
−0.566603 + 0.823991i \(0.691743\pi\)
\(978\) 0 0
\(979\) 3.00336e13 1.04493
\(980\) 0 0
\(981\) 1.09276e13 0.376716
\(982\) 0 0
\(983\) −2.11820e13 −0.723561 −0.361781 0.932263i \(-0.617831\pi\)
−0.361781 + 0.932263i \(0.617831\pi\)
\(984\) 0 0
\(985\) −6.02568e12 −0.203959
\(986\) 0 0
\(987\) 4.38295e12 0.147007
\(988\) 0 0
\(989\) 2.48631e13 0.826365
\(990\) 0 0
\(991\) −2.18946e13 −0.721116 −0.360558 0.932737i \(-0.617414\pi\)
−0.360558 + 0.932737i \(0.617414\pi\)
\(992\) 0 0
\(993\) −1.96081e12 −0.0639977
\(994\) 0 0
\(995\) −2.08949e12 −0.0675828
\(996\) 0 0
\(997\) −4.15332e13 −1.33127 −0.665636 0.746277i \(-0.731840\pi\)
−0.665636 + 0.746277i \(0.731840\pi\)
\(998\) 0 0
\(999\) 1.38807e12 0.0440927
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 20.10.a.b.1.2 2
3.2 odd 2 180.10.a.e.1.2 2
4.3 odd 2 80.10.a.j.1.1 2
5.2 odd 4 100.10.c.c.49.2 4
5.3 odd 4 100.10.c.c.49.3 4
5.4 even 2 100.10.a.c.1.1 2
8.3 odd 2 320.10.a.l.1.2 2
8.5 even 2 320.10.a.t.1.1 2
20.3 even 4 400.10.c.l.49.2 4
20.7 even 4 400.10.c.l.49.3 4
20.19 odd 2 400.10.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.10.a.b.1.2 2 1.1 even 1 trivial
80.10.a.j.1.1 2 4.3 odd 2
100.10.a.c.1.1 2 5.4 even 2
100.10.c.c.49.2 4 5.2 odd 4
100.10.c.c.49.3 4 5.3 odd 4
180.10.a.e.1.2 2 3.2 odd 2
320.10.a.l.1.2 2 8.3 odd 2
320.10.a.t.1.1 2 8.5 even 2
400.10.a.l.1.2 2 20.19 odd 2
400.10.c.l.49.2 4 20.3 even 4
400.10.c.l.49.3 4 20.7 even 4