Properties

Label 20.10.a.b.1.1
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: \(\Q(\sqrt{79}) \)
Defining polynomial: \(x^{2} - 79\)
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 1.1
Root \(-8.88819\) of defining polynomial
Character \(\chi\) \(=\) 20.1

$q$-expansion

\(f(q)\) \(=\) \(q-272.211 q^{3} -625.000 q^{5} -10002.6 q^{7} +54415.9 q^{9} +O(q^{10})\) \(q-272.211 q^{3} -625.000 q^{5} -10002.6 q^{7} +54415.9 q^{9} +47093.7 q^{11} +9362.93 q^{13} +170132. q^{15} +108521. q^{17} -665173. q^{19} +2.72281e6 q^{21} +576426. q^{23} +390625. q^{25} -9.45468e6 q^{27} -2.61067e6 q^{29} +3.87896e6 q^{31} -1.28194e7 q^{33} +6.25160e6 q^{35} +1.41599e7 q^{37} -2.54869e6 q^{39} +4.62193e6 q^{41} +8.31227e6 q^{43} -3.40099e7 q^{45} +2.51923e7 q^{47} +5.96977e7 q^{49} -2.95405e7 q^{51} +3.49333e7 q^{53} -2.94335e7 q^{55} +1.81068e8 q^{57} +6.71868e6 q^{59} -4.75982e6 q^{61} -5.44299e8 q^{63} -5.85183e6 q^{65} -1.38772e8 q^{67} -1.56909e8 q^{69} +3.54359e8 q^{71} +2.41596e8 q^{73} -1.06332e8 q^{75} -4.71058e8 q^{77} +2.61173e8 q^{79} +1.50260e9 q^{81} -6.55898e8 q^{83} -6.78253e7 q^{85} +7.10654e8 q^{87} -1.00221e9 q^{89} -9.36534e7 q^{91} -1.05590e9 q^{93} +4.15733e8 q^{95} +1.24869e9 q^{97} +2.56264e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 260q^{3} - 1250q^{5} - 380q^{7} + 34882q^{9} + O(q^{10}) \) \( 2q - 260q^{3} - 1250q^{5} - 380q^{7} + 34882q^{9} + 102720q^{11} + 179140q^{13} + 162500q^{15} + 316020q^{17} + 137272q^{19} + 2840312q^{21} - 665460q^{23} + 781250q^{25} - 9933560q^{27} - 6893748q^{29} + 291832q^{31} - 12140160q^{33} + 237500q^{35} + 11261380q^{37} - 475528q^{39} + 29773452q^{41} - 11708180q^{43} - 21801250q^{45} + 62493300q^{47} + 111937914q^{49} - 27006696q^{51} + 9417780q^{53} - 64200000q^{55} + 190866320q^{57} - 92930856q^{59} + 195673924q^{61} - 732264700q^{63} - 111962500q^{65} - 219767420q^{67} - 172074264q^{69} + 311207016q^{71} - 99224060q^{73} - 101562500q^{75} + 64210560q^{77} + 542261776q^{79} + 1881238378q^{81} - 1256915700q^{83} - 197512500q^{85} + 658353000q^{87} - 462291852q^{89} + 1540037768q^{91} - 1099698160q^{93} - 85795000q^{95} + 1671716740q^{97} + 1476045120q^{99} + O(q^{100}) \)

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
\(3\) −272.211 −1.94026 −0.970131 0.242583i \(-0.922005\pi\)
−0.970131 + 0.242583i \(0.922005\pi\)
\(4\) 0 0
\(5\) −625.000 −0.447214
\(6\) 0 0
\(7\) −10002.6 −1.57460 −0.787300 0.616570i \(-0.788522\pi\)
−0.787300 + 0.616570i \(0.788522\pi\)
\(8\) 0 0
\(9\) 54415.9 2.76461
\(10\) 0 0
\(11\) 47093.7 0.969830 0.484915 0.874561i \(-0.338851\pi\)
0.484915 + 0.874561i \(0.338851\pi\)
\(12\) 0 0
\(13\) 9362.93 0.0909216 0.0454608 0.998966i \(-0.485524\pi\)
0.0454608 + 0.998966i \(0.485524\pi\)
\(14\) 0 0
\(15\) 170132. 0.867711
\(16\) 0 0
\(17\) 108521. 0.315131 0.157566 0.987509i \(-0.449635\pi\)
0.157566 + 0.987509i \(0.449635\pi\)
\(18\) 0 0
\(19\) −665173. −1.17096 −0.585482 0.810685i \(-0.699095\pi\)
−0.585482 + 0.810685i \(0.699095\pi\)
\(20\) 0 0
\(21\) 2.72281e6 3.05514
\(22\) 0 0
\(23\) 576426. 0.429505 0.214752 0.976669i \(-0.431106\pi\)
0.214752 + 0.976669i \(0.431106\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 0 0
\(27\) −9.45468e6 −3.42381
\(28\) 0 0
\(29\) −2.61067e6 −0.685427 −0.342714 0.939440i \(-0.611346\pi\)
−0.342714 + 0.939440i \(0.611346\pi\)
\(30\) 0 0
\(31\) 3.87896e6 0.754375 0.377188 0.926137i \(-0.376891\pi\)
0.377188 + 0.926137i \(0.376891\pi\)
\(32\) 0 0
\(33\) −1.28194e7 −1.88172
\(34\) 0 0
\(35\) 6.25160e6 0.704183
\(36\) 0 0
\(37\) 1.41599e7 1.24209 0.621046 0.783774i \(-0.286708\pi\)
0.621046 + 0.783774i \(0.286708\pi\)
\(38\) 0 0
\(39\) −2.54869e6 −0.176412
\(40\) 0 0
\(41\) 4.62193e6 0.255444 0.127722 0.991810i \(-0.459233\pi\)
0.127722 + 0.991810i \(0.459233\pi\)
\(42\) 0 0
\(43\) 8.31227e6 0.370776 0.185388 0.982665i \(-0.440646\pi\)
0.185388 + 0.982665i \(0.440646\pi\)
\(44\) 0 0
\(45\) −3.40099e7 −1.23637
\(46\) 0 0
\(47\) 2.51923e7 0.753056 0.376528 0.926405i \(-0.377118\pi\)
0.376528 + 0.926405i \(0.377118\pi\)
\(48\) 0 0
\(49\) 5.96977e7 1.47937
\(50\) 0 0
\(51\) −2.95405e7 −0.611437
\(52\) 0 0
\(53\) 3.49333e7 0.608133 0.304066 0.952651i \(-0.401656\pi\)
0.304066 + 0.952651i \(0.401656\pi\)
\(54\) 0 0
\(55\) −2.94335e7 −0.433721
\(56\) 0 0
\(57\) 1.81068e8 2.27198
\(58\) 0 0
\(59\) 6.71868e6 0.0721855 0.0360928 0.999348i \(-0.488509\pi\)
0.0360928 + 0.999348i \(0.488509\pi\)
\(60\) 0 0
\(61\) −4.75982e6 −0.0440156 −0.0220078 0.999758i \(-0.507006\pi\)
−0.0220078 + 0.999758i \(0.507006\pi\)
\(62\) 0 0
\(63\) −5.44299e8 −4.35316
\(64\) 0 0
\(65\) −5.85183e6 −0.0406614
\(66\) 0 0
\(67\) −1.38772e8 −0.841330 −0.420665 0.907216i \(-0.638203\pi\)
−0.420665 + 0.907216i \(0.638203\pi\)
\(68\) 0 0
\(69\) −1.56909e8 −0.833352
\(70\) 0 0
\(71\) 3.54359e8 1.65494 0.827468 0.561513i \(-0.189781\pi\)
0.827468 + 0.561513i \(0.189781\pi\)
\(72\) 0 0
\(73\) 2.41596e8 0.995719 0.497859 0.867258i \(-0.334120\pi\)
0.497859 + 0.867258i \(0.334120\pi\)
\(74\) 0 0
\(75\) −1.06332e8 −0.388052
\(76\) 0 0
\(77\) −4.71058e8 −1.52709
\(78\) 0 0
\(79\) 2.61173e8 0.754409 0.377204 0.926130i \(-0.376885\pi\)
0.377204 + 0.926130i \(0.376885\pi\)
\(80\) 0 0
\(81\) 1.50260e9 3.87847
\(82\) 0 0
\(83\) −6.55898e8 −1.51700 −0.758499 0.651674i \(-0.774067\pi\)
−0.758499 + 0.651674i \(0.774067\pi\)
\(84\) 0 0
\(85\) −6.78253e7 −0.140931
\(86\) 0 0
\(87\) 7.10654e8 1.32991
\(88\) 0 0
\(89\) −1.00221e9 −1.69318 −0.846590 0.532245i \(-0.821348\pi\)
−0.846590 + 0.532245i \(0.821348\pi\)
\(90\) 0 0
\(91\) −9.36534e7 −0.143165
\(92\) 0 0
\(93\) −1.05590e9 −1.46368
\(94\) 0 0
\(95\) 4.15733e8 0.523671
\(96\) 0 0
\(97\) 1.24869e9 1.43213 0.716065 0.698034i \(-0.245942\pi\)
0.716065 + 0.698034i \(0.245942\pi\)
\(98\) 0 0
\(99\) 2.56264e9 2.68120
\(100\) 0 0
\(101\) −9.39590e7 −0.0898446 −0.0449223 0.998990i \(-0.514304\pi\)
−0.0449223 + 0.998990i \(0.514304\pi\)
\(102\) 0 0
\(103\) −3.58300e8 −0.313675 −0.156837 0.987624i \(-0.550130\pi\)
−0.156837 + 0.987624i \(0.550130\pi\)
\(104\) 0 0
\(105\) −1.70176e9 −1.36630
\(106\) 0 0
\(107\) −1.02197e9 −0.753719 −0.376860 0.926270i \(-0.622996\pi\)
−0.376860 + 0.926270i \(0.622996\pi\)
\(108\) 0 0
\(109\) −8.94205e8 −0.606761 −0.303380 0.952869i \(-0.598115\pi\)
−0.303380 + 0.952869i \(0.598115\pi\)
\(110\) 0 0
\(111\) −3.85449e9 −2.40998
\(112\) 0 0
\(113\) 1.11113e9 0.641077 0.320539 0.947235i \(-0.396136\pi\)
0.320539 + 0.947235i \(0.396136\pi\)
\(114\) 0 0
\(115\) −3.60266e8 −0.192080
\(116\) 0 0
\(117\) 5.09492e8 0.251363
\(118\) 0 0
\(119\) −1.08548e9 −0.496206
\(120\) 0 0
\(121\) −1.40134e8 −0.0594306
\(122\) 0 0
\(123\) −1.25814e9 −0.495628
\(124\) 0 0
\(125\) −2.44141e8 −0.0894427
\(126\) 0 0
\(127\) 5.51824e9 1.88228 0.941139 0.338020i \(-0.109757\pi\)
0.941139 + 0.338020i \(0.109757\pi\)
\(128\) 0 0
\(129\) −2.26269e9 −0.719402
\(130\) 0 0
\(131\) 4.66527e9 1.38406 0.692031 0.721867i \(-0.256716\pi\)
0.692031 + 0.721867i \(0.256716\pi\)
\(132\) 0 0
\(133\) 6.65344e9 1.84380
\(134\) 0 0
\(135\) 5.90917e9 1.53117
\(136\) 0 0
\(137\) −6.97325e9 −1.69119 −0.845595 0.533825i \(-0.820754\pi\)
−0.845595 + 0.533825i \(0.820754\pi\)
\(138\) 0 0
\(139\) 6.27736e9 1.42630 0.713149 0.701013i \(-0.247268\pi\)
0.713149 + 0.701013i \(0.247268\pi\)
\(140\) 0 0
\(141\) −6.85762e9 −1.46113
\(142\) 0 0
\(143\) 4.40935e8 0.0881784
\(144\) 0 0
\(145\) 1.63167e9 0.306532
\(146\) 0 0
\(147\) −1.62504e10 −2.87036
\(148\) 0 0
\(149\) −1.62156e9 −0.269523 −0.134761 0.990878i \(-0.543027\pi\)
−0.134761 + 0.990878i \(0.543027\pi\)
\(150\) 0 0
\(151\) 1.17677e10 1.84202 0.921011 0.389537i \(-0.127365\pi\)
0.921011 + 0.389537i \(0.127365\pi\)
\(152\) 0 0
\(153\) 5.90524e9 0.871217
\(154\) 0 0
\(155\) −2.42435e9 −0.337367
\(156\) 0 0
\(157\) −1.71288e9 −0.224998 −0.112499 0.993652i \(-0.535886\pi\)
−0.112499 + 0.993652i \(0.535886\pi\)
\(158\) 0 0
\(159\) −9.50923e9 −1.17994
\(160\) 0 0
\(161\) −5.76574e9 −0.676298
\(162\) 0 0
\(163\) 9.98636e9 1.10806 0.554030 0.832497i \(-0.313089\pi\)
0.554030 + 0.832497i \(0.313089\pi\)
\(164\) 0 0
\(165\) 8.01214e9 0.841532
\(166\) 0 0
\(167\) 1.56153e10 1.55355 0.776776 0.629777i \(-0.216854\pi\)
0.776776 + 0.629777i \(0.216854\pi\)
\(168\) 0 0
\(169\) −1.05168e10 −0.991733
\(170\) 0 0
\(171\) −3.61960e10 −3.23726
\(172\) 0 0
\(173\) 4.61727e8 0.0391902 0.0195951 0.999808i \(-0.493762\pi\)
0.0195951 + 0.999808i \(0.493762\pi\)
\(174\) 0 0
\(175\) −3.90725e9 −0.314920
\(176\) 0 0
\(177\) −1.82890e9 −0.140059
\(178\) 0 0
\(179\) −1.56860e10 −1.14202 −0.571011 0.820943i \(-0.693449\pi\)
−0.571011 + 0.820943i \(0.693449\pi\)
\(180\) 0 0
\(181\) 5.26075e8 0.0364329 0.0182165 0.999834i \(-0.494201\pi\)
0.0182165 + 0.999834i \(0.494201\pi\)
\(182\) 0 0
\(183\) 1.29568e9 0.0854017
\(184\) 0 0
\(185\) −8.84996e9 −0.555480
\(186\) 0 0
\(187\) 5.11063e9 0.305624
\(188\) 0 0
\(189\) 9.45710e10 5.39113
\(190\) 0 0
\(191\) −2.60641e10 −1.41708 −0.708538 0.705673i \(-0.750645\pi\)
−0.708538 + 0.705673i \(0.750645\pi\)
\(192\) 0 0
\(193\) 4.83514e9 0.250843 0.125421 0.992104i \(-0.459972\pi\)
0.125421 + 0.992104i \(0.459972\pi\)
\(194\) 0 0
\(195\) 1.59293e9 0.0788937
\(196\) 0 0
\(197\) 1.09275e10 0.516919 0.258460 0.966022i \(-0.416785\pi\)
0.258460 + 0.966022i \(0.416785\pi\)
\(198\) 0 0
\(199\) −2.24042e10 −1.01272 −0.506361 0.862322i \(-0.669010\pi\)
−0.506361 + 0.862322i \(0.669010\pi\)
\(200\) 0 0
\(201\) 3.77754e10 1.63240
\(202\) 0 0
\(203\) 2.61134e10 1.07927
\(204\) 0 0
\(205\) −2.88871e9 −0.114238
\(206\) 0 0
\(207\) 3.13667e10 1.18741
\(208\) 0 0
\(209\) −3.13255e10 −1.13564
\(210\) 0 0
\(211\) 3.17129e10 1.10145 0.550725 0.834687i \(-0.314351\pi\)
0.550725 + 0.834687i \(0.314351\pi\)
\(212\) 0 0
\(213\) −9.64605e10 −3.21101
\(214\) 0 0
\(215\) −5.19517e9 −0.165816
\(216\) 0 0
\(217\) −3.87995e10 −1.18784
\(218\) 0 0
\(219\) −6.57651e10 −1.93195
\(220\) 0 0
\(221\) 1.01607e9 0.0286522
\(222\) 0 0
\(223\) 1.52662e10 0.413388 0.206694 0.978406i \(-0.433730\pi\)
0.206694 + 0.978406i \(0.433730\pi\)
\(224\) 0 0
\(225\) 2.12562e10 0.552923
\(226\) 0 0
\(227\) −7.53481e9 −0.188346 −0.0941729 0.995556i \(-0.530021\pi\)
−0.0941729 + 0.995556i \(0.530021\pi\)
\(228\) 0 0
\(229\) −1.28244e10 −0.308160 −0.154080 0.988058i \(-0.549241\pi\)
−0.154080 + 0.988058i \(0.549241\pi\)
\(230\) 0 0
\(231\) 1.28227e11 2.96296
\(232\) 0 0
\(233\) 2.06272e10 0.458500 0.229250 0.973368i \(-0.426373\pi\)
0.229250 + 0.973368i \(0.426373\pi\)
\(234\) 0 0
\(235\) −1.57452e10 −0.336777
\(236\) 0 0
\(237\) −7.10943e10 −1.46375
\(238\) 0 0
\(239\) 3.96664e10 0.786380 0.393190 0.919457i \(-0.371372\pi\)
0.393190 + 0.919457i \(0.371372\pi\)
\(240\) 0 0
\(241\) −3.75642e9 −0.0717294 −0.0358647 0.999357i \(-0.511419\pi\)
−0.0358647 + 0.999357i \(0.511419\pi\)
\(242\) 0 0
\(243\) −2.22928e11 −4.10144
\(244\) 0 0
\(245\) −3.73111e10 −0.661592
\(246\) 0 0
\(247\) −6.22797e9 −0.106466
\(248\) 0 0
\(249\) 1.78543e11 2.94337
\(250\) 0 0
\(251\) 7.74656e10 1.23191 0.615953 0.787783i \(-0.288771\pi\)
0.615953 + 0.787783i \(0.288771\pi\)
\(252\) 0 0
\(253\) 2.71460e10 0.416546
\(254\) 0 0
\(255\) 1.84628e10 0.273443
\(256\) 0 0
\(257\) 3.31990e10 0.474707 0.237354 0.971423i \(-0.423720\pi\)
0.237354 + 0.971423i \(0.423720\pi\)
\(258\) 0 0
\(259\) −1.41636e11 −1.95580
\(260\) 0 0
\(261\) −1.42062e11 −1.89494
\(262\) 0 0
\(263\) 1.00754e11 1.29856 0.649281 0.760549i \(-0.275070\pi\)
0.649281 + 0.760549i \(0.275070\pi\)
\(264\) 0 0
\(265\) −2.18333e10 −0.271965
\(266\) 0 0
\(267\) 2.72812e11 3.28521
\(268\) 0 0
\(269\) −7.49946e10 −0.873262 −0.436631 0.899641i \(-0.643828\pi\)
−0.436631 + 0.899641i \(0.643828\pi\)
\(270\) 0 0
\(271\) −5.80768e10 −0.654095 −0.327048 0.945008i \(-0.606054\pi\)
−0.327048 + 0.945008i \(0.606054\pi\)
\(272\) 0 0
\(273\) 2.54935e10 0.277778
\(274\) 0 0
\(275\) 1.83960e10 0.193966
\(276\) 0 0
\(277\) 1.44538e11 1.47511 0.737554 0.675289i \(-0.235981\pi\)
0.737554 + 0.675289i \(0.235981\pi\)
\(278\) 0 0
\(279\) 2.11077e11 2.08556
\(280\) 0 0
\(281\) 1.73877e11 1.66366 0.831828 0.555034i \(-0.187295\pi\)
0.831828 + 0.555034i \(0.187295\pi\)
\(282\) 0 0
\(283\) −3.86637e8 −0.00358315 −0.00179157 0.999998i \(-0.500570\pi\)
−0.00179157 + 0.999998i \(0.500570\pi\)
\(284\) 0 0
\(285\) −1.13167e11 −1.01606
\(286\) 0 0
\(287\) −4.62311e10 −0.402222
\(288\) 0 0
\(289\) −1.06811e11 −0.900692
\(290\) 0 0
\(291\) −3.39908e11 −2.77870
\(292\) 0 0
\(293\) 6.25206e10 0.495586 0.247793 0.968813i \(-0.420295\pi\)
0.247793 + 0.968813i \(0.420295\pi\)
\(294\) 0 0
\(295\) −4.19918e9 −0.0322823
\(296\) 0 0
\(297\) −4.45255e11 −3.32051
\(298\) 0 0
\(299\) 5.39703e9 0.0390512
\(300\) 0 0
\(301\) −8.31440e10 −0.583824
\(302\) 0 0
\(303\) 2.55767e10 0.174322
\(304\) 0 0
\(305\) 2.97489e9 0.0196844
\(306\) 0 0
\(307\) −1.57636e10 −0.101282 −0.0506410 0.998717i \(-0.516126\pi\)
−0.0506410 + 0.998717i \(0.516126\pi\)
\(308\) 0 0
\(309\) 9.75334e10 0.608611
\(310\) 0 0
\(311\) −2.01301e10 −0.122018 −0.0610091 0.998137i \(-0.519432\pi\)
−0.0610091 + 0.998137i \(0.519432\pi\)
\(312\) 0 0
\(313\) −6.03790e10 −0.355579 −0.177790 0.984069i \(-0.556895\pi\)
−0.177790 + 0.984069i \(0.556895\pi\)
\(314\) 0 0
\(315\) 3.40187e11 1.94679
\(316\) 0 0
\(317\) −1.24292e11 −0.691313 −0.345657 0.938361i \(-0.612344\pi\)
−0.345657 + 0.938361i \(0.612344\pi\)
\(318\) 0 0
\(319\) −1.22946e11 −0.664748
\(320\) 0 0
\(321\) 2.78191e11 1.46241
\(322\) 0 0
\(323\) −7.21850e10 −0.369008
\(324\) 0 0
\(325\) 3.65740e9 0.0181843
\(326\) 0 0
\(327\) 2.43412e11 1.17727
\(328\) 0 0
\(329\) −2.51988e11 −1.18576
\(330\) 0 0
\(331\) −1.14638e11 −0.524930 −0.262465 0.964941i \(-0.584535\pi\)
−0.262465 + 0.964941i \(0.584535\pi\)
\(332\) 0 0
\(333\) 7.70526e11 3.43390
\(334\) 0 0
\(335\) 8.67327e10 0.376254
\(336\) 0 0
\(337\) −3.40785e11 −1.43928 −0.719642 0.694345i \(-0.755694\pi\)
−0.719642 + 0.694345i \(0.755694\pi\)
\(338\) 0 0
\(339\) −3.02461e11 −1.24386
\(340\) 0 0
\(341\) 1.82674e11 0.731615
\(342\) 0 0
\(343\) −1.93491e11 −0.754809
\(344\) 0 0
\(345\) 9.80684e10 0.372686
\(346\) 0 0
\(347\) −3.38240e11 −1.25240 −0.626198 0.779664i \(-0.715390\pi\)
−0.626198 + 0.779664i \(0.715390\pi\)
\(348\) 0 0
\(349\) 2.34359e11 0.845603 0.422802 0.906222i \(-0.361047\pi\)
0.422802 + 0.906222i \(0.361047\pi\)
\(350\) 0 0
\(351\) −8.85235e10 −0.311298
\(352\) 0 0
\(353\) 3.56202e11 1.22098 0.610492 0.792022i \(-0.290972\pi\)
0.610492 + 0.792022i \(0.290972\pi\)
\(354\) 0 0
\(355\) −2.21474e11 −0.740110
\(356\) 0 0
\(357\) 2.95481e11 0.962769
\(358\) 0 0
\(359\) 3.37815e11 1.07338 0.536691 0.843779i \(-0.319674\pi\)
0.536691 + 0.843779i \(0.319674\pi\)
\(360\) 0 0
\(361\) 1.19768e11 0.371157
\(362\) 0 0
\(363\) 3.81461e10 0.115311
\(364\) 0 0
\(365\) −1.50997e11 −0.445299
\(366\) 0 0
\(367\) 1.36383e11 0.392431 0.196215 0.980561i \(-0.437135\pi\)
0.196215 + 0.980561i \(0.437135\pi\)
\(368\) 0 0
\(369\) 2.51506e11 0.706204
\(370\) 0 0
\(371\) −3.49423e11 −0.957566
\(372\) 0 0
\(373\) 5.08929e11 1.36134 0.680672 0.732589i \(-0.261688\pi\)
0.680672 + 0.732589i \(0.261688\pi\)
\(374\) 0 0
\(375\) 6.64578e10 0.173542
\(376\) 0 0
\(377\) −2.44436e10 −0.0623201
\(378\) 0 0
\(379\) 9.70726e10 0.241669 0.120834 0.992673i \(-0.461443\pi\)
0.120834 + 0.992673i \(0.461443\pi\)
\(380\) 0 0
\(381\) −1.50213e12 −3.65211
\(382\) 0 0
\(383\) −7.13792e11 −1.69503 −0.847514 0.530772i \(-0.821902\pi\)
−0.847514 + 0.530772i \(0.821902\pi\)
\(384\) 0 0
\(385\) 2.94411e11 0.682937
\(386\) 0 0
\(387\) 4.52320e11 1.02505
\(388\) 0 0
\(389\) 2.48149e11 0.549465 0.274732 0.961521i \(-0.411411\pi\)
0.274732 + 0.961521i \(0.411411\pi\)
\(390\) 0 0
\(391\) 6.25540e10 0.135350
\(392\) 0 0
\(393\) −1.26994e12 −2.68544
\(394\) 0 0
\(395\) −1.63233e11 −0.337382
\(396\) 0 0
\(397\) 7.04445e11 1.42328 0.711640 0.702545i \(-0.247953\pi\)
0.711640 + 0.702545i \(0.247953\pi\)
\(398\) 0 0
\(399\) −1.81114e12 −3.57745
\(400\) 0 0
\(401\) −2.82127e11 −0.544873 −0.272437 0.962174i \(-0.587830\pi\)
−0.272437 + 0.962174i \(0.587830\pi\)
\(402\) 0 0
\(403\) 3.63184e10 0.0685890
\(404\) 0 0
\(405\) −9.39125e11 −1.73451
\(406\) 0 0
\(407\) 6.66844e11 1.20462
\(408\) 0 0
\(409\) 9.15011e11 1.61686 0.808428 0.588595i \(-0.200319\pi\)
0.808428 + 0.588595i \(0.200319\pi\)
\(410\) 0 0
\(411\) 1.89820e12 3.28135
\(412\) 0 0
\(413\) −6.72041e10 −0.113663
\(414\) 0 0
\(415\) 4.09937e11 0.678422
\(416\) 0 0
\(417\) −1.70877e12 −2.76739
\(418\) 0 0
\(419\) 1.19010e12 1.88635 0.943174 0.332299i \(-0.107824\pi\)
0.943174 + 0.332299i \(0.107824\pi\)
\(420\) 0 0
\(421\) 7.44534e11 1.15509 0.577544 0.816360i \(-0.304011\pi\)
0.577544 + 0.816360i \(0.304011\pi\)
\(422\) 0 0
\(423\) 1.37086e12 2.08191
\(424\) 0 0
\(425\) 4.23908e10 0.0630263
\(426\) 0 0
\(427\) 4.76104e10 0.0693069
\(428\) 0 0
\(429\) −1.20027e11 −0.171089
\(430\) 0 0
\(431\) −3.47758e11 −0.485433 −0.242716 0.970097i \(-0.578038\pi\)
−0.242716 + 0.970097i \(0.578038\pi\)
\(432\) 0 0
\(433\) −8.13744e11 −1.11248 −0.556240 0.831022i \(-0.687756\pi\)
−0.556240 + 0.831022i \(0.687756\pi\)
\(434\) 0 0
\(435\) −4.44159e11 −0.594753
\(436\) 0 0
\(437\) −3.83423e11 −0.502935
\(438\) 0 0
\(439\) −1.09730e12 −1.41005 −0.705023 0.709184i \(-0.749063\pi\)
−0.705023 + 0.709184i \(0.749063\pi\)
\(440\) 0 0
\(441\) 3.24851e12 4.08987
\(442\) 0 0
\(443\) 6.59236e10 0.0813251 0.0406625 0.999173i \(-0.487053\pi\)
0.0406625 + 0.999173i \(0.487053\pi\)
\(444\) 0 0
\(445\) 6.26381e11 0.757213
\(446\) 0 0
\(447\) 4.41407e11 0.522945
\(448\) 0 0
\(449\) 2.15220e11 0.249905 0.124952 0.992163i \(-0.460122\pi\)
0.124952 + 0.992163i \(0.460122\pi\)
\(450\) 0 0
\(451\) 2.17664e11 0.247737
\(452\) 0 0
\(453\) −3.20329e12 −3.57400
\(454\) 0 0
\(455\) 5.85334e10 0.0640254
\(456\) 0 0
\(457\) 7.36259e11 0.789601 0.394801 0.918767i \(-0.370814\pi\)
0.394801 + 0.918767i \(0.370814\pi\)
\(458\) 0 0
\(459\) −1.02603e12 −1.07895
\(460\) 0 0
\(461\) −1.23730e12 −1.27591 −0.637957 0.770072i \(-0.720220\pi\)
−0.637957 + 0.770072i \(0.720220\pi\)
\(462\) 0 0
\(463\) −7.36969e11 −0.745306 −0.372653 0.927971i \(-0.621552\pi\)
−0.372653 + 0.927971i \(0.621552\pi\)
\(464\) 0 0
\(465\) 6.59935e11 0.654580
\(466\) 0 0
\(467\) −1.22439e12 −1.19123 −0.595614 0.803271i \(-0.703091\pi\)
−0.595614 + 0.803271i \(0.703091\pi\)
\(468\) 0 0
\(469\) 1.38808e12 1.32476
\(470\) 0 0
\(471\) 4.66265e11 0.436555
\(472\) 0 0
\(473\) 3.91455e11 0.359590
\(474\) 0 0
\(475\) −2.59833e11 −0.234193
\(476\) 0 0
\(477\) 1.90093e12 1.68125
\(478\) 0 0
\(479\) 1.11970e12 0.971833 0.485916 0.874005i \(-0.338486\pi\)
0.485916 + 0.874005i \(0.338486\pi\)
\(480\) 0 0
\(481\) 1.32579e11 0.112933
\(482\) 0 0
\(483\) 1.56950e12 1.31220
\(484\) 0 0
\(485\) −7.80432e11 −0.640468
\(486\) 0 0
\(487\) −2.25337e11 −0.181532 −0.0907659 0.995872i \(-0.528932\pi\)
−0.0907659 + 0.995872i \(0.528932\pi\)
\(488\) 0 0
\(489\) −2.71840e12 −2.14992
\(490\) 0 0
\(491\) −1.60716e11 −0.124794 −0.0623970 0.998051i \(-0.519874\pi\)
−0.0623970 + 0.998051i \(0.519874\pi\)
\(492\) 0 0
\(493\) −2.83312e11 −0.216000
\(494\) 0 0
\(495\) −1.60165e12 −1.19907
\(496\) 0 0
\(497\) −3.54450e12 −2.60586
\(498\) 0 0
\(499\) 8.05187e11 0.581359 0.290680 0.956820i \(-0.406119\pi\)
0.290680 + 0.956820i \(0.406119\pi\)
\(500\) 0 0
\(501\) −4.25065e12 −3.01430
\(502\) 0 0
\(503\) −1.68493e12 −1.17361 −0.586807 0.809727i \(-0.699615\pi\)
−0.586807 + 0.809727i \(0.699615\pi\)
\(504\) 0 0
\(505\) 5.87244e10 0.0401797
\(506\) 0 0
\(507\) 2.86280e12 1.92422
\(508\) 0 0
\(509\) 1.50208e12 0.991890 0.495945 0.868354i \(-0.334822\pi\)
0.495945 + 0.868354i \(0.334822\pi\)
\(510\) 0 0
\(511\) −2.41658e12 −1.56786
\(512\) 0 0
\(513\) 6.28900e12 4.00916
\(514\) 0 0
\(515\) 2.23938e11 0.140280
\(516\) 0 0
\(517\) 1.18640e12 0.730336
\(518\) 0 0
\(519\) −1.25687e11 −0.0760393
\(520\) 0 0
\(521\) 9.24724e11 0.549848 0.274924 0.961466i \(-0.411347\pi\)
0.274924 + 0.961466i \(0.411347\pi\)
\(522\) 0 0
\(523\) −3.81782e10 −0.0223130 −0.0111565 0.999938i \(-0.503551\pi\)
−0.0111565 + 0.999938i \(0.503551\pi\)
\(524\) 0 0
\(525\) 1.06360e12 0.611027
\(526\) 0 0
\(527\) 4.20947e11 0.237727
\(528\) 0 0
\(529\) −1.46889e12 −0.815526
\(530\) 0 0
\(531\) 3.65603e11 0.199565
\(532\) 0 0
\(533\) 4.32748e10 0.0232254
\(534\) 0 0
\(535\) 6.38729e11 0.337073
\(536\) 0 0
\(537\) 4.26991e12 2.21582
\(538\) 0 0
\(539\) 2.81139e12 1.43473
\(540\) 0 0
\(541\) −1.05812e12 −0.531064 −0.265532 0.964102i \(-0.585548\pi\)
−0.265532 + 0.964102i \(0.585548\pi\)
\(542\) 0 0
\(543\) −1.43203e11 −0.0706894
\(544\) 0 0
\(545\) 5.58878e11 0.271352
\(546\) 0 0
\(547\) −1.05156e12 −0.502215 −0.251108 0.967959i \(-0.580795\pi\)
−0.251108 + 0.967959i \(0.580795\pi\)
\(548\) 0 0
\(549\) −2.59010e11 −0.121686
\(550\) 0 0
\(551\) 1.73655e12 0.802611
\(552\) 0 0
\(553\) −2.61240e12 −1.18789
\(554\) 0 0
\(555\) 2.40906e12 1.07778
\(556\) 0 0
\(557\) −3.92684e11 −0.172860 −0.0864301 0.996258i \(-0.527546\pi\)
−0.0864301 + 0.996258i \(0.527546\pi\)
\(558\) 0 0
\(559\) 7.78272e10 0.0337115
\(560\) 0 0
\(561\) −1.39117e12 −0.592990
\(562\) 0 0
\(563\) 2.36287e12 0.991180 0.495590 0.868557i \(-0.334952\pi\)
0.495590 + 0.868557i \(0.334952\pi\)
\(564\) 0 0
\(565\) −6.94454e11 −0.286699
\(566\) 0 0
\(567\) −1.50299e13 −6.10705
\(568\) 0 0
\(569\) −2.50603e12 −1.00226 −0.501130 0.865372i \(-0.667082\pi\)
−0.501130 + 0.865372i \(0.667082\pi\)
\(570\) 0 0
\(571\) −2.99195e12 −1.17785 −0.588927 0.808187i \(-0.700449\pi\)
−0.588927 + 0.808187i \(0.700449\pi\)
\(572\) 0 0
\(573\) 7.09494e12 2.74950
\(574\) 0 0
\(575\) 2.25166e11 0.0859010
\(576\) 0 0
\(577\) −3.06622e11 −0.115163 −0.0575815 0.998341i \(-0.518339\pi\)
−0.0575815 + 0.998341i \(0.518339\pi\)
\(578\) 0 0
\(579\) −1.31618e12 −0.486700
\(580\) 0 0
\(581\) 6.56067e12 2.38867
\(582\) 0 0
\(583\) 1.64514e12 0.589785
\(584\) 0 0
\(585\) −3.18433e11 −0.112413
\(586\) 0 0
\(587\) −2.80823e12 −0.976249 −0.488125 0.872774i \(-0.662319\pi\)
−0.488125 + 0.872774i \(0.662319\pi\)
\(588\) 0 0
\(589\) −2.58018e12 −0.883346
\(590\) 0 0
\(591\) −2.97459e12 −1.00296
\(592\) 0 0
\(593\) −1.34954e12 −0.448166 −0.224083 0.974570i \(-0.571939\pi\)
−0.224083 + 0.974570i \(0.571939\pi\)
\(594\) 0 0
\(595\) 6.78427e11 0.221910
\(596\) 0 0
\(597\) 6.09867e12 1.96494
\(598\) 0 0
\(599\) 3.38598e12 1.07464 0.537321 0.843378i \(-0.319436\pi\)
0.537321 + 0.843378i \(0.319436\pi\)
\(600\) 0 0
\(601\) 4.62837e12 1.44708 0.723540 0.690282i \(-0.242514\pi\)
0.723540 + 0.690282i \(0.242514\pi\)
\(602\) 0 0
\(603\) −7.55142e12 −2.32595
\(604\) 0 0
\(605\) 8.75839e10 0.0265782
\(606\) 0 0
\(607\) 4.78828e12 1.43163 0.715814 0.698291i \(-0.246056\pi\)
0.715814 + 0.698291i \(0.246056\pi\)
\(608\) 0 0
\(609\) −7.10837e12 −2.09407
\(610\) 0 0
\(611\) 2.35874e11 0.0684690
\(612\) 0 0
\(613\) −1.02749e12 −0.293905 −0.146953 0.989144i \(-0.546946\pi\)
−0.146953 + 0.989144i \(0.546946\pi\)
\(614\) 0 0
\(615\) 7.86338e11 0.221652
\(616\) 0 0
\(617\) −4.53079e12 −1.25861 −0.629304 0.777159i \(-0.716660\pi\)
−0.629304 + 0.777159i \(0.716660\pi\)
\(618\) 0 0
\(619\) −3.42902e12 −0.938777 −0.469388 0.882992i \(-0.655526\pi\)
−0.469388 + 0.882992i \(0.655526\pi\)
\(620\) 0 0
\(621\) −5.44992e12 −1.47054
\(622\) 0 0
\(623\) 1.00247e13 2.66608
\(624\) 0 0
\(625\) 1.52588e11 0.0400000
\(626\) 0 0
\(627\) 8.52714e12 2.20343
\(628\) 0 0
\(629\) 1.53664e12 0.391422
\(630\) 0 0
\(631\) −3.52587e12 −0.885389 −0.442695 0.896672i \(-0.645977\pi\)
−0.442695 + 0.896672i \(0.645977\pi\)
\(632\) 0 0
\(633\) −8.63260e12 −2.13710
\(634\) 0 0
\(635\) −3.44890e12 −0.841780
\(636\) 0 0
\(637\) 5.58946e11 0.134506
\(638\) 0 0
\(639\) 1.92828e13 4.57526
\(640\) 0 0
\(641\) −7.00458e12 −1.63878 −0.819390 0.573236i \(-0.805688\pi\)
−0.819390 + 0.573236i \(0.805688\pi\)
\(642\) 0 0
\(643\) −4.31047e12 −0.994432 −0.497216 0.867627i \(-0.665644\pi\)
−0.497216 + 0.867627i \(0.665644\pi\)
\(644\) 0 0
\(645\) 1.41418e12 0.321726
\(646\) 0 0
\(647\) −2.84949e12 −0.639290 −0.319645 0.947537i \(-0.603564\pi\)
−0.319645 + 0.947537i \(0.603564\pi\)
\(648\) 0 0
\(649\) 3.16407e11 0.0700077
\(650\) 0 0
\(651\) 1.05617e13 2.30472
\(652\) 0 0
\(653\) 1.93213e12 0.415841 0.207920 0.978146i \(-0.433331\pi\)
0.207920 + 0.978146i \(0.433331\pi\)
\(654\) 0 0
\(655\) −2.91579e12 −0.618972
\(656\) 0 0
\(657\) 1.31467e13 2.75278
\(658\) 0 0
\(659\) 8.58675e12 1.77355 0.886777 0.462198i \(-0.152939\pi\)
0.886777 + 0.462198i \(0.152939\pi\)
\(660\) 0 0
\(661\) 1.11478e12 0.227133 0.113567 0.993530i \(-0.463772\pi\)
0.113567 + 0.993530i \(0.463772\pi\)
\(662\) 0 0
\(663\) −2.76586e11 −0.0555928
\(664\) 0 0
\(665\) −4.15840e12 −0.824573
\(666\) 0 0
\(667\) −1.50486e12 −0.294394
\(668\) 0 0
\(669\) −4.15562e12 −0.802081
\(670\) 0 0
\(671\) −2.24157e11 −0.0426876
\(672\) 0 0
\(673\) 7.57716e12 1.42377 0.711883 0.702298i \(-0.247843\pi\)
0.711883 + 0.702298i \(0.247843\pi\)
\(674\) 0 0
\(675\) −3.69323e12 −0.684762
\(676\) 0 0
\(677\) −5.08558e12 −0.930447 −0.465223 0.885193i \(-0.654026\pi\)
−0.465223 + 0.885193i \(0.654026\pi\)
\(678\) 0 0
\(679\) −1.24901e13 −2.25503
\(680\) 0 0
\(681\) 2.05106e12 0.365440
\(682\) 0 0
\(683\) −1.78278e11 −0.0313476 −0.0156738 0.999877i \(-0.504989\pi\)
−0.0156738 + 0.999877i \(0.504989\pi\)
\(684\) 0 0
\(685\) 4.35828e12 0.756323
\(686\) 0 0
\(687\) 3.49093e12 0.597911
\(688\) 0 0
\(689\) 3.27078e11 0.0552924
\(690\) 0 0
\(691\) −8.66612e11 −0.144602 −0.0723008 0.997383i \(-0.523034\pi\)
−0.0723008 + 0.997383i \(0.523034\pi\)
\(692\) 0 0
\(693\) −2.56330e13 −4.22182
\(694\) 0 0
\(695\) −3.92335e12 −0.637860
\(696\) 0 0
\(697\) 5.01574e11 0.0804985
\(698\) 0 0
\(699\) −5.61497e12 −0.889610
\(700\) 0 0
\(701\) 5.18099e12 0.810367 0.405183 0.914235i \(-0.367208\pi\)
0.405183 + 0.914235i \(0.367208\pi\)
\(702\) 0 0
\(703\) −9.41882e12 −1.45445
\(704\) 0 0
\(705\) 4.28601e12 0.653435
\(706\) 0 0
\(707\) 9.39831e11 0.141469
\(708\) 0 0
\(709\) −1.22351e13 −1.81844 −0.909218 0.416320i \(-0.863319\pi\)
−0.909218 + 0.416320i \(0.863319\pi\)
\(710\) 0 0
\(711\) 1.42120e13 2.08565
\(712\) 0 0
\(713\) 2.23593e12 0.324008
\(714\) 0 0
\(715\) −2.75584e11 −0.0394346
\(716\) 0 0
\(717\) −1.07976e13 −1.52578
\(718\) 0 0
\(719\) −1.11881e13 −1.56127 −0.780634 0.624989i \(-0.785104\pi\)
−0.780634 + 0.624989i \(0.785104\pi\)
\(720\) 0 0
\(721\) 3.58392e12 0.493913
\(722\) 0 0
\(723\) 1.02254e12 0.139174
\(724\) 0 0
\(725\) −1.01979e12 −0.137085
\(726\) 0 0
\(727\) 7.94265e12 1.05453 0.527267 0.849699i \(-0.323217\pi\)
0.527267 + 0.849699i \(0.323217\pi\)
\(728\) 0 0
\(729\) 3.11078e13 4.07940
\(730\) 0 0
\(731\) 9.02052e11 0.116843
\(732\) 0 0
\(733\) 1.09012e13 1.39478 0.697392 0.716690i \(-0.254344\pi\)
0.697392 + 0.716690i \(0.254344\pi\)
\(734\) 0 0
\(735\) 1.01565e13 1.28366
\(736\) 0 0
\(737\) −6.53530e12 −0.815947
\(738\) 0 0
\(739\) 1.54854e12 0.190996 0.0954978 0.995430i \(-0.469556\pi\)
0.0954978 + 0.995430i \(0.469556\pi\)
\(740\) 0 0
\(741\) 1.69532e12 0.206572
\(742\) 0 0
\(743\) 5.96254e12 0.717764 0.358882 0.933383i \(-0.383158\pi\)
0.358882 + 0.933383i \(0.383158\pi\)
\(744\) 0 0
\(745\) 1.01348e12 0.120534
\(746\) 0 0
\(747\) −3.56913e13 −4.19392
\(748\) 0 0
\(749\) 1.02223e13 1.18681
\(750\) 0 0
\(751\) 9.92051e12 1.13803 0.569016 0.822326i \(-0.307324\pi\)
0.569016 + 0.822326i \(0.307324\pi\)
\(752\) 0 0
\(753\) −2.10870e13 −2.39022
\(754\) 0 0
\(755\) −7.35480e12 −0.823777
\(756\) 0 0
\(757\) 1.23733e13 1.36947 0.684737 0.728791i \(-0.259917\pi\)
0.684737 + 0.728791i \(0.259917\pi\)
\(758\) 0 0
\(759\) −7.38944e12 −0.808209
\(760\) 0 0
\(761\) −8.52217e12 −0.921127 −0.460563 0.887627i \(-0.652353\pi\)
−0.460563 + 0.887627i \(0.652353\pi\)
\(762\) 0 0
\(763\) 8.94434e12 0.955406
\(764\) 0 0
\(765\) −3.69078e12 −0.389620
\(766\) 0 0
\(767\) 6.29066e10 0.00656322
\(768\) 0 0
\(769\) 2.99595e12 0.308934 0.154467 0.987998i \(-0.450634\pi\)
0.154467 + 0.987998i \(0.450634\pi\)
\(770\) 0 0
\(771\) −9.03713e12 −0.921056
\(772\) 0 0
\(773\) 6.44491e11 0.0649245 0.0324623 0.999473i \(-0.489665\pi\)
0.0324623 + 0.999473i \(0.489665\pi\)
\(774\) 0 0
\(775\) 1.51522e12 0.150875
\(776\) 0 0
\(777\) 3.85548e13 3.79476
\(778\) 0 0
\(779\) −3.07438e12 −0.299116
\(780\) 0 0
\(781\) 1.66881e13 1.60501
\(782\) 0 0
\(783\) 2.46831e13 2.34677
\(784\) 0 0
\(785\) 1.07055e12 0.100622
\(786\) 0 0
\(787\) 8.05096e12 0.748103 0.374052 0.927408i \(-0.377968\pi\)
0.374052 + 0.927408i \(0.377968\pi\)
\(788\) 0 0
\(789\) −2.74264e13 −2.51955
\(790\) 0 0
\(791\) −1.11141e13 −1.00944
\(792\) 0 0
\(793\) −4.45659e10 −0.00400196
\(794\) 0 0
\(795\) 5.94327e12 0.527683
\(796\) 0 0
\(797\) −1.11381e13 −0.977799 −0.488899 0.872340i \(-0.662601\pi\)
−0.488899 + 0.872340i \(0.662601\pi\)
\(798\) 0 0
\(799\) 2.73388e12 0.237312
\(800\) 0 0
\(801\) −5.45361e13 −4.68099
\(802\) 0 0
\(803\) 1.13776e13 0.965678
\(804\) 0 0
\(805\) 3.60358e12 0.302450
\(806\) 0 0
\(807\) 2.04144e13 1.69436
\(808\) 0 0
\(809\) 1.69716e13 1.39301 0.696506 0.717550i \(-0.254737\pi\)
0.696506 + 0.717550i \(0.254737\pi\)
\(810\) 0 0
\(811\) −1.17583e13 −0.954444 −0.477222 0.878783i \(-0.658356\pi\)
−0.477222 + 0.878783i \(0.658356\pi\)
\(812\) 0 0
\(813\) 1.58092e13 1.26912
\(814\) 0 0
\(815\) −6.24148e12 −0.495539
\(816\) 0 0
\(817\) −5.52910e12 −0.434165
\(818\) 0 0
\(819\) −5.09623e12 −0.395796
\(820\) 0 0
\(821\) 1.99657e13 1.53370 0.766851 0.641825i \(-0.221823\pi\)
0.766851 + 0.641825i \(0.221823\pi\)
\(822\) 0 0
\(823\) 5.52813e11 0.0420029 0.0210014 0.999779i \(-0.493315\pi\)
0.0210014 + 0.999779i \(0.493315\pi\)
\(824\) 0 0
\(825\) −5.00759e12 −0.376345
\(826\) 0 0
\(827\) −1.57036e13 −1.16741 −0.583707 0.811965i \(-0.698398\pi\)
−0.583707 + 0.811965i \(0.698398\pi\)
\(828\) 0 0
\(829\) 1.05928e13 0.778958 0.389479 0.921035i \(-0.372655\pi\)
0.389479 + 0.921035i \(0.372655\pi\)
\(830\) 0 0
\(831\) −3.93449e13 −2.86209
\(832\) 0 0
\(833\) 6.47843e12 0.466195
\(834\) 0 0
\(835\) −9.75955e12 −0.694770
\(836\) 0 0
\(837\) −3.66743e13 −2.58284
\(838\) 0 0
\(839\) −6.55506e11 −0.0456718 −0.0228359 0.999739i \(-0.507270\pi\)
−0.0228359 + 0.999739i \(0.507270\pi\)
\(840\) 0 0
\(841\) −7.69153e12 −0.530189
\(842\) 0 0
\(843\) −4.73312e13 −3.22793
\(844\) 0 0
\(845\) 6.57302e12 0.443517
\(846\) 0 0
\(847\) 1.40170e12 0.0935794
\(848\) 0 0
\(849\) 1.05247e11 0.00695224
\(850\) 0 0
\(851\) 8.16215e12 0.533484
\(852\) 0 0
\(853\) 2.08805e13 1.35043 0.675213 0.737622i \(-0.264051\pi\)
0.675213 + 0.737622i \(0.264051\pi\)
\(854\) 0 0
\(855\) 2.26225e13 1.44775
\(856\) 0 0
\(857\) −2.19074e13 −1.38732 −0.693660 0.720303i \(-0.744003\pi\)
−0.693660 + 0.720303i \(0.744003\pi\)
\(858\) 0 0
\(859\) 1.47238e13 0.922680 0.461340 0.887223i \(-0.347369\pi\)
0.461340 + 0.887223i \(0.347369\pi\)
\(860\) 0 0
\(861\) 1.25846e13 0.780416
\(862\) 0 0
\(863\) −2.74805e12 −0.168646 −0.0843230 0.996438i \(-0.526873\pi\)
−0.0843230 + 0.996438i \(0.526873\pi\)
\(864\) 0 0
\(865\) −2.88579e11 −0.0175264
\(866\) 0 0
\(867\) 2.90752e13 1.74758
\(868\) 0 0
\(869\) 1.22996e13 0.731648
\(870\) 0 0
\(871\) −1.29932e12 −0.0764950
\(872\) 0 0
\(873\) 6.79486e13 3.95928
\(874\) 0 0
\(875\) 2.44203e12 0.140837
\(876\) 0 0
\(877\) 6.68274e12 0.381466 0.190733 0.981642i \(-0.438913\pi\)
0.190733 + 0.981642i \(0.438913\pi\)
\(878\) 0 0
\(879\) −1.70188e13 −0.961567
\(880\) 0 0
\(881\) 1.42384e13 0.796289 0.398144 0.917323i \(-0.369654\pi\)
0.398144 + 0.917323i \(0.369654\pi\)
\(882\) 0 0
\(883\) −2.46098e13 −1.36234 −0.681170 0.732125i \(-0.738529\pi\)
−0.681170 + 0.732125i \(0.738529\pi\)
\(884\) 0 0
\(885\) 1.14306e12 0.0626362
\(886\) 0 0
\(887\) −9.12625e12 −0.495035 −0.247518 0.968883i \(-0.579615\pi\)
−0.247518 + 0.968883i \(0.579615\pi\)
\(888\) 0 0
\(889\) −5.51966e13 −2.96384
\(890\) 0 0
\(891\) 7.07630e13 3.76146
\(892\) 0 0
\(893\) −1.67572e13 −0.881802
\(894\) 0 0
\(895\) 9.80376e12 0.510727
\(896\) 0 0
\(897\) −1.46913e12 −0.0757696
\(898\) 0 0
\(899\) −1.01267e13 −0.517069
\(900\) 0 0
\(901\) 3.79098e12 0.191642
\(902\) 0 0
\(903\) 2.26327e13 1.13277
\(904\) 0 0
\(905\) −3.28797e11 −0.0162933
\(906\) 0 0
\(907\) 3.32446e13 1.63113 0.815564 0.578668i \(-0.196427\pi\)
0.815564 + 0.578668i \(0.196427\pi\)
\(908\) 0 0
\(909\) −5.11286e12 −0.248386
\(910\) 0 0
\(911\) −1.42660e13 −0.686232 −0.343116 0.939293i \(-0.611482\pi\)
−0.343116 + 0.939293i \(0.611482\pi\)
\(912\) 0 0
\(913\) −3.08887e13 −1.47123
\(914\) 0 0
\(915\) −8.09798e11 −0.0381928
\(916\) 0 0
\(917\) −4.66647e13 −2.17935
\(918\) 0 0
\(919\) 3.23171e13 1.49456 0.747278 0.664512i \(-0.231360\pi\)
0.747278 + 0.664512i \(0.231360\pi\)
\(920\) 0 0
\(921\) 4.29102e12 0.196514
\(922\) 0 0
\(923\) 3.31784e12 0.150469
\(924\) 0 0
\(925\) 5.53123e12 0.248418
\(926\) 0 0
\(927\) −1.94972e13 −0.867190
\(928\) 0 0
\(929\) −3.49040e12 −0.153746 −0.0768732 0.997041i \(-0.524494\pi\)
−0.0768732 + 0.997041i \(0.524494\pi\)
\(930\) 0 0
\(931\) −3.97093e13 −1.73228
\(932\) 0 0
\(933\) 5.47964e12 0.236747
\(934\) 0 0
\(935\) −3.19414e12 −0.136679
\(936\) 0 0
\(937\) −1.48063e13 −0.627507 −0.313754 0.949504i \(-0.601587\pi\)
−0.313754 + 0.949504i \(0.601587\pi\)
\(938\) 0 0
\(939\) 1.64358e13 0.689916
\(940\) 0 0
\(941\) 3.90935e13 1.62537 0.812684 0.582705i \(-0.198006\pi\)
0.812684 + 0.582705i \(0.198006\pi\)
\(942\) 0 0
\(943\) 2.66420e12 0.109714
\(944\) 0 0
\(945\) −5.91069e13 −2.41099
\(946\) 0 0
\(947\) 2.30013e13 0.929346 0.464673 0.885482i \(-0.346172\pi\)
0.464673 + 0.885482i \(0.346172\pi\)
\(948\) 0 0
\(949\) 2.26205e12 0.0905323
\(950\) 0 0
\(951\) 3.38335e13 1.34133
\(952\) 0 0
\(953\) −3.32845e13 −1.30714 −0.653572 0.756864i \(-0.726730\pi\)
−0.653572 + 0.756864i \(0.726730\pi\)
\(954\) 0 0
\(955\) 1.62901e13 0.633735
\(956\) 0 0
\(957\) 3.34673e13 1.28978
\(958\) 0 0
\(959\) 6.97504e13 2.66295
\(960\) 0 0
\(961\) −1.13933e13 −0.430918
\(962\) 0 0
\(963\) −5.56112e13 −2.08374
\(964\) 0 0
\(965\) −3.02196e12 −0.112180
\(966\) 0 0
\(967\) −5.77848e12 −0.212518 −0.106259 0.994339i \(-0.533887\pi\)
−0.106259 + 0.994339i \(0.533887\pi\)
\(968\) 0 0
\(969\) 1.96495e13 0.715971
\(970\) 0 0
\(971\) −4.46200e13 −1.61080 −0.805402 0.592728i \(-0.798051\pi\)
−0.805402 + 0.592728i \(0.798051\pi\)
\(972\) 0 0
\(973\) −6.27897e13 −2.24585
\(974\) 0 0
\(975\) −9.95584e11 −0.0352823
\(976\) 0 0
\(977\) 3.90697e13 1.37188 0.685938 0.727660i \(-0.259392\pi\)
0.685938 + 0.727660i \(0.259392\pi\)
\(978\) 0 0
\(979\) −4.71977e13 −1.64210
\(980\) 0 0
\(981\) −4.86589e13 −1.67746
\(982\) 0 0
\(983\) 1.96137e12 0.0669990 0.0334995 0.999439i \(-0.489335\pi\)
0.0334995 + 0.999439i \(0.489335\pi\)
\(984\) 0 0
\(985\) −6.82969e12 −0.231173
\(986\) 0 0
\(987\) 6.85938e13 2.30069
\(988\) 0 0
\(989\) 4.79141e12 0.159250
\(990\) 0 0
\(991\) −4.33785e13 −1.42871 −0.714354 0.699785i \(-0.753279\pi\)
−0.714354 + 0.699785i \(0.753279\pi\)
\(992\) 0 0
\(993\) 3.12057e13 1.01850
\(994\) 0 0
\(995\) 1.40026e13 0.452903
\(996\) 0 0
\(997\) −2.79360e12 −0.0895440 −0.0447720 0.998997i \(-0.514256\pi\)
−0.0447720 + 0.998997i \(0.514256\pi\)
\(998\) 0 0
\(999\) −1.33878e14 −4.25269
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.1.1 2
3.2 odd 2 180.10.a.e.1.1 2
4.3 odd 2 80.10.a.j.1.2 2
5.2 odd 4 100.10.c.c.49.4 4
5.3 odd 4 100.10.c.c.49.1 4
5.4 even 2 100.10.a.c.1.2 2
8.3 odd 2 320.10.a.l.1.1 2
8.5 even 2 320.10.a.t.1.2 2
20.3 even 4 400.10.c.l.49.4 4
20.7 even 4 400.10.c.l.49.1 4
20.19 odd 2 400.10.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.10.a.b.1.1 2 1.1 even 1 trivial
80.10.a.j.1.2 2 4.3 odd 2
100.10.a.c.1.2 2 5.4 even 2
100.10.c.c.49.1 4 5.3 odd 4
100.10.c.c.49.4 4 5.2 odd 4
180.10.a.e.1.1 2 3.2 odd 2
320.10.a.l.1.1 2 8.3 odd 2
320.10.a.t.1.2 2 8.5 even 2
400.10.a.l.1.1 2 20.19 odd 2
400.10.c.l.49.1 4 20.7 even 4
400.10.c.l.49.4 4 20.3 even 4