Properties

Label 38.10.a.d.1.1
Level $38$
Weight $10$
Character 38.1
Self dual yes
Analytic conductor $19.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5713617742\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 34433 x^{2} - 2723303 x - 48270488\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-124.888\) of defining polynomial
Character \(\chi\) \(=\) 38.1

$q$-expansion

\(f(q)\) \(=\) \(q-16.0000 q^{2} -140.229 q^{3} +256.000 q^{4} +1263.95 q^{5} +2243.67 q^{6} -3487.42 q^{7} -4096.00 q^{8} -18.7042 q^{9} +O(q^{10})\) \(q-16.0000 q^{2} -140.229 q^{3} +256.000 q^{4} +1263.95 q^{5} +2243.67 q^{6} -3487.42 q^{7} -4096.00 q^{8} -18.7042 q^{9} -20223.2 q^{10} -49259.2 q^{11} -35898.7 q^{12} -68065.3 q^{13} +55798.7 q^{14} -177243. q^{15} +65536.0 q^{16} +505716. q^{17} +299.267 q^{18} +130321. q^{19} +323571. q^{20} +489039. q^{21} +788147. q^{22} -585044. q^{23} +574380. q^{24} -355562. q^{25} +1.08905e6 q^{26} +2.76276e6 q^{27} -892779. q^{28} +2.62021e6 q^{29} +2.83588e6 q^{30} +3.53093e6 q^{31} -1.04858e6 q^{32} +6.90759e6 q^{33} -8.09145e6 q^{34} -4.40791e6 q^{35} -4788.27 q^{36} +1.85431e7 q^{37} -2.08514e6 q^{38} +9.54476e6 q^{39} -5.17713e6 q^{40} +2.18141e7 q^{41} -7.82462e6 q^{42} -1.12704e7 q^{43} -1.26104e7 q^{44} -23641.1 q^{45} +9.36070e6 q^{46} +1.54092e7 q^{47} -9.19008e6 q^{48} -2.81915e7 q^{49} +5.68899e6 q^{50} -7.09162e7 q^{51} -1.74247e7 q^{52} +7.05741e6 q^{53} -4.42041e7 q^{54} -6.22611e7 q^{55} +1.42845e7 q^{56} -1.82748e7 q^{57} -4.19234e7 q^{58} +2.90730e7 q^{59} -4.53741e7 q^{60} +1.44284e8 q^{61} -5.64949e7 q^{62} +65229.3 q^{63} +1.67772e7 q^{64} -8.60310e7 q^{65} -1.10521e8 q^{66} +2.65266e7 q^{67} +1.29463e8 q^{68} +8.20404e7 q^{69} +7.05266e7 q^{70} -4.27081e7 q^{71} +76612.3 q^{72} +3.24644e8 q^{73} -2.96689e8 q^{74} +4.98602e7 q^{75} +3.33622e7 q^{76} +1.71787e8 q^{77} -1.52716e8 q^{78} -8.88210e7 q^{79} +8.28341e7 q^{80} -3.87052e8 q^{81} -3.49026e8 q^{82} +6.29830e7 q^{83} +1.25194e8 q^{84} +6.39198e8 q^{85} +1.80326e8 q^{86} -3.67431e8 q^{87} +2.01766e8 q^{88} -4.54329e7 q^{89} +378258. q^{90} +2.37372e8 q^{91} -1.49771e8 q^{92} -4.95140e8 q^{93} -2.46547e8 q^{94} +1.64719e8 q^{95} +1.47041e8 q^{96} +1.64830e9 q^{97} +4.51064e8 q^{98} +921353. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 64q^{2} + 84q^{3} + 1024q^{4} - 1395q^{5} - 1344q^{6} + 12307q^{7} - 16384q^{8} + 16538q^{9} + O(q^{10}) \) \( 4q - 64q^{2} + 84q^{3} + 1024q^{4} - 1395q^{5} - 1344q^{6} + 12307q^{7} - 16384q^{8} + 16538q^{9} + 22320q^{10} - 104249q^{11} + 21504q^{12} + 120486q^{13} - 196912q^{14} - 591090q^{15} + 262144q^{16} - 412139q^{17} - 264608q^{18} + 521284q^{19} - 357120q^{20} + 2437006q^{21} + 1667984q^{22} + 3010300q^{23} - 344064q^{24} + 9760585q^{25} - 1927776q^{26} + 12387978q^{27} + 3150592q^{28} + 6153240q^{29} + 9457440q^{30} + 12774024q^{31} - 4194304q^{32} - 3258022q^{33} + 6594224q^{34} + 9823425q^{35} + 4233728q^{36} + 20506048q^{37} - 8340544q^{38} + 69881444q^{39} + 5713920q^{40} + 11620300q^{41} - 38992096q^{42} + 7698327q^{43} - 26687744q^{44} - 124015815q^{45} - 48164800q^{46} - 31581083q^{47} + 5505024q^{48} + 18970383q^{49} - 156169360q^{50} - 8594812q^{51} + 30844416q^{52} + 72549422q^{53} - 198207648q^{54} + 21332505q^{55} - 50409472q^{56} + 10946964q^{57} - 98451840q^{58} - 149234120q^{59} - 151319040q^{60} + 129004373q^{61} - 204384384q^{62} + 102967551q^{63} + 67108864q^{64} + 124691700q^{65} + 52128352q^{66} + 132595266q^{67} - 105507584q^{68} - 45529972q^{69} - 157174800q^{70} - 47138482q^{71} - 67739648q^{72} - 39332795q^{73} - 328096768q^{74} + 824627010q^{75} + 133448704q^{76} - 165933719q^{77} - 1118103104q^{78} - 307010840q^{79} - 91422720q^{80} + 1305551744q^{81} - 185924800q^{82} - 746568232q^{83} + 623873536q^{84} - 105005985q^{85} - 123173232q^{86} - 82148208q^{87} + 427003904q^{88} + 286943482q^{89} + 1984253040q^{90} + 3155781114q^{91} + 770636800q^{92} + 1151901596q^{93} + 505297328q^{94} - 181797795q^{95} - 88080384q^{96} + 793519958q^{97} - 303526128q^{98} - 1681833809q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 −0.707107
\(3\) −140.229 −0.999525 −0.499762 0.866163i \(-0.666579\pi\)
−0.499762 + 0.866163i \(0.666579\pi\)
\(4\) 256.000 0.500000
\(5\) 1263.95 0.904407 0.452204 0.891915i \(-0.350638\pi\)
0.452204 + 0.891915i \(0.350638\pi\)
\(6\) 2243.67 0.706771
\(7\) −3487.42 −0.548988 −0.274494 0.961589i \(-0.588510\pi\)
−0.274494 + 0.961589i \(0.588510\pi\)
\(8\) −4096.00 −0.353553
\(9\) −18.7042 −0.000950271 0
\(10\) −20223.2 −0.639512
\(11\) −49259.2 −1.01443 −0.507213 0.861821i \(-0.669324\pi\)
−0.507213 + 0.861821i \(0.669324\pi\)
\(12\) −35898.7 −0.499762
\(13\) −68065.3 −0.660969 −0.330484 0.943811i \(-0.607212\pi\)
−0.330484 + 0.943811i \(0.607212\pi\)
\(14\) 55798.7 0.388193
\(15\) −177243. −0.903977
\(16\) 65536.0 0.250000
\(17\) 505716. 1.46854 0.734271 0.678857i \(-0.237524\pi\)
0.734271 + 0.678857i \(0.237524\pi\)
\(18\) 299.267 0.000671943 0
\(19\) 130321. 0.229416
\(20\) 323571. 0.452204
\(21\) 489039. 0.548727
\(22\) 788147. 0.717307
\(23\) −585044. −0.435926 −0.217963 0.975957i \(-0.569941\pi\)
−0.217963 + 0.975957i \(0.569941\pi\)
\(24\) 574380. 0.353385
\(25\) −355562. −0.182048
\(26\) 1.08905e6 0.467375
\(27\) 2.76276e6 1.00047
\(28\) −892779. −0.274494
\(29\) 2.62021e6 0.687932 0.343966 0.938982i \(-0.388229\pi\)
0.343966 + 0.938982i \(0.388229\pi\)
\(30\) 2.83588e6 0.639209
\(31\) 3.53093e6 0.686691 0.343345 0.939209i \(-0.388440\pi\)
0.343345 + 0.939209i \(0.388440\pi\)
\(32\) −1.04858e6 −0.176777
\(33\) 6.90759e6 1.01394
\(34\) −8.09145e6 −1.03842
\(35\) −4.40791e6 −0.496509
\(36\) −4788.27 −0.000475135 0
\(37\) 1.85431e7 1.62657 0.813287 0.581863i \(-0.197676\pi\)
0.813287 + 0.581863i \(0.197676\pi\)
\(38\) −2.08514e6 −0.162221
\(39\) 9.54476e6 0.660655
\(40\) −5.17713e6 −0.319756
\(41\) 2.18141e7 1.20562 0.602811 0.797884i \(-0.294047\pi\)
0.602811 + 0.797884i \(0.294047\pi\)
\(42\) −7.82462e6 −0.388009
\(43\) −1.12704e7 −0.502725 −0.251362 0.967893i \(-0.580879\pi\)
−0.251362 + 0.967893i \(0.580879\pi\)
\(44\) −1.26104e7 −0.507213
\(45\) −23641.1 −0.000859432 0
\(46\) 9.36070e6 0.308246
\(47\) 1.54092e7 0.460616 0.230308 0.973118i \(-0.426027\pi\)
0.230308 + 0.973118i \(0.426027\pi\)
\(48\) −9.19008e6 −0.249881
\(49\) −2.81915e7 −0.698612
\(50\) 5.68899e6 0.128727
\(51\) −7.09162e7 −1.46784
\(52\) −1.74247e7 −0.330484
\(53\) 7.05741e6 0.122858 0.0614291 0.998111i \(-0.480434\pi\)
0.0614291 + 0.998111i \(0.480434\pi\)
\(54\) −4.42041e7 −0.707442
\(55\) −6.22611e7 −0.917454
\(56\) 1.42845e7 0.194097
\(57\) −1.82748e7 −0.229307
\(58\) −4.19234e7 −0.486442
\(59\) 2.90730e7 0.312360 0.156180 0.987729i \(-0.450082\pi\)
0.156180 + 0.987729i \(0.450082\pi\)
\(60\) −4.53741e7 −0.451989
\(61\) 1.44284e8 1.33424 0.667121 0.744949i \(-0.267526\pi\)
0.667121 + 0.744949i \(0.267526\pi\)
\(62\) −5.64949e7 −0.485564
\(63\) 65229.3 0.000521687 0
\(64\) 1.67772e7 0.125000
\(65\) −8.60310e7 −0.597785
\(66\) −1.10521e8 −0.716967
\(67\) 2.65266e7 0.160822 0.0804110 0.996762i \(-0.474377\pi\)
0.0804110 + 0.996762i \(0.474377\pi\)
\(68\) 1.29463e8 0.734271
\(69\) 8.20404e7 0.435719
\(70\) 7.05266e7 0.351085
\(71\) −4.27081e7 −0.199456 −0.0997281 0.995015i \(-0.531797\pi\)
−0.0997281 + 0.995015i \(0.531797\pi\)
\(72\) 76612.3 0.000335972 0
\(73\) 3.24644e8 1.33800 0.668998 0.743264i \(-0.266723\pi\)
0.668998 + 0.743264i \(0.266723\pi\)
\(74\) −2.96689e8 −1.15016
\(75\) 4.98602e7 0.181961
\(76\) 3.33622e7 0.114708
\(77\) 1.71787e8 0.556908
\(78\) −1.52716e8 −0.467153
\(79\) −8.88210e7 −0.256563 −0.128281 0.991738i \(-0.540946\pi\)
−0.128281 + 0.991738i \(0.540946\pi\)
\(80\) 8.28341e7 0.226102
\(81\) −3.87052e8 −0.999049
\(82\) −3.49026e8 −0.852503
\(83\) 6.29830e7 0.145671 0.0728353 0.997344i \(-0.476795\pi\)
0.0728353 + 0.997344i \(0.476795\pi\)
\(84\) 1.25194e8 0.274363
\(85\) 6.39198e8 1.32816
\(86\) 1.80326e8 0.355480
\(87\) −3.67431e8 −0.687605
\(88\) 2.01766e8 0.358654
\(89\) −4.54329e7 −0.0767565 −0.0383783 0.999263i \(-0.512219\pi\)
−0.0383783 + 0.999263i \(0.512219\pi\)
\(90\) 378258. 0.000607710 0
\(91\) 2.37372e8 0.362864
\(92\) −1.49771e8 −0.217963
\(93\) −4.95140e8 −0.686365
\(94\) −2.46547e8 −0.325705
\(95\) 1.64719e8 0.207485
\(96\) 1.47041e8 0.176693
\(97\) 1.64830e9 1.89044 0.945219 0.326438i \(-0.105848\pi\)
0.945219 + 0.326438i \(0.105848\pi\)
\(98\) 4.51064e8 0.493994
\(99\) 921353. 0.000963980 0
\(100\) −9.10238e7 −0.0910238
\(101\) −8.97226e8 −0.857938 −0.428969 0.903319i \(-0.641123\pi\)
−0.428969 + 0.903319i \(0.641123\pi\)
\(102\) 1.13466e9 1.03792
\(103\) 1.67622e9 1.46745 0.733726 0.679445i \(-0.237779\pi\)
0.733726 + 0.679445i \(0.237779\pi\)
\(104\) 2.78796e8 0.233688
\(105\) 6.18119e8 0.496273
\(106\) −1.12919e8 −0.0868738
\(107\) 3.08053e8 0.227195 0.113597 0.993527i \(-0.463763\pi\)
0.113597 + 0.993527i \(0.463763\pi\)
\(108\) 7.07266e8 0.500237
\(109\) −1.79569e8 −0.121846 −0.0609230 0.998142i \(-0.519404\pi\)
−0.0609230 + 0.998142i \(0.519404\pi\)
\(110\) 9.96177e8 0.648738
\(111\) −2.60028e9 −1.62580
\(112\) −2.28551e8 −0.137247
\(113\) −2.30787e9 −1.33155 −0.665776 0.746152i \(-0.731899\pi\)
−0.665776 + 0.746152i \(0.731899\pi\)
\(114\) 2.92397e8 0.162144
\(115\) −7.39465e8 −0.394255
\(116\) 6.70775e8 0.343966
\(117\) 1.27311e6 0.000628099 0
\(118\) −4.65168e8 −0.220872
\(119\) −1.76364e9 −0.806212
\(120\) 7.25986e8 0.319604
\(121\) 6.85222e7 0.0290601
\(122\) −2.30855e9 −0.943452
\(123\) −3.05899e9 −1.20505
\(124\) 9.03918e8 0.343345
\(125\) −2.91806e9 −1.06905
\(126\) −1.04367e6 −0.000368889 0
\(127\) −4.51313e9 −1.53943 −0.769717 0.638385i \(-0.779603\pi\)
−0.769717 + 0.638385i \(0.779603\pi\)
\(128\) −2.68435e8 −0.0883883
\(129\) 1.58044e9 0.502486
\(130\) 1.37650e9 0.422698
\(131\) −2.53716e9 −0.752708 −0.376354 0.926476i \(-0.622822\pi\)
−0.376354 + 0.926476i \(0.622822\pi\)
\(132\) 1.76834e9 0.506972
\(133\) −4.54484e8 −0.125946
\(134\) −4.24426e8 −0.113718
\(135\) 3.49198e9 0.904836
\(136\) −2.07141e9 −0.519208
\(137\) 4.04090e9 0.980021 0.490010 0.871717i \(-0.336993\pi\)
0.490010 + 0.871717i \(0.336993\pi\)
\(138\) −1.31265e9 −0.308100
\(139\) −1.86522e9 −0.423803 −0.211901 0.977291i \(-0.567966\pi\)
−0.211901 + 0.977291i \(0.567966\pi\)
\(140\) −1.12843e9 −0.248254
\(141\) −2.16082e9 −0.460397
\(142\) 6.83329e8 0.141037
\(143\) 3.35284e9 0.670504
\(144\) −1.22580e6 −0.000237568 0
\(145\) 3.31181e9 0.622171
\(146\) −5.19431e9 −0.946106
\(147\) 3.95328e9 0.698280
\(148\) 4.74702e9 0.813287
\(149\) −7.60481e9 −1.26401 −0.632004 0.774965i \(-0.717768\pi\)
−0.632004 + 0.774965i \(0.717768\pi\)
\(150\) −7.97763e8 −0.128666
\(151\) 1.74653e9 0.273388 0.136694 0.990613i \(-0.456352\pi\)
0.136694 + 0.990613i \(0.456352\pi\)
\(152\) −5.33795e8 −0.0811107
\(153\) −9.45900e6 −0.00139551
\(154\) −2.74860e9 −0.393793
\(155\) 4.46291e9 0.621048
\(156\) 2.44346e9 0.330327
\(157\) 1.06564e10 1.39979 0.699893 0.714248i \(-0.253231\pi\)
0.699893 + 0.714248i \(0.253231\pi\)
\(158\) 1.42114e9 0.181417
\(159\) −9.89657e8 −0.122800
\(160\) −1.32535e9 −0.159878
\(161\) 2.04029e9 0.239318
\(162\) 6.19283e9 0.706434
\(163\) 9.11348e9 1.01121 0.505603 0.862766i \(-0.331270\pi\)
0.505603 + 0.862766i \(0.331270\pi\)
\(164\) 5.58442e9 0.602811
\(165\) 8.73083e9 0.917018
\(166\) −1.00773e9 −0.103005
\(167\) 1.15546e10 1.14956 0.574778 0.818309i \(-0.305088\pi\)
0.574778 + 0.818309i \(0.305088\pi\)
\(168\) −2.00310e9 −0.194004
\(169\) −5.97161e9 −0.563120
\(170\) −1.02272e10 −0.939151
\(171\) −2.43755e6 −0.000218007 0
\(172\) −2.88522e9 −0.251362
\(173\) −2.52786e9 −0.214558 −0.107279 0.994229i \(-0.534214\pi\)
−0.107279 + 0.994229i \(0.534214\pi\)
\(174\) 5.87890e9 0.486210
\(175\) 1.23999e9 0.0999419
\(176\) −3.22825e9 −0.253606
\(177\) −4.07689e9 −0.312212
\(178\) 7.26926e8 0.0542751
\(179\) −1.73068e10 −1.26002 −0.630009 0.776588i \(-0.716949\pi\)
−0.630009 + 0.776588i \(0.716949\pi\)
\(180\) −6.05212e6 −0.000429716 0
\(181\) 1.08688e10 0.752711 0.376355 0.926475i \(-0.377177\pi\)
0.376355 + 0.926475i \(0.377177\pi\)
\(182\) −3.79796e9 −0.256583
\(183\) −2.02329e10 −1.33361
\(184\) 2.39634e9 0.154123
\(185\) 2.34375e10 1.47108
\(186\) 7.92224e9 0.485333
\(187\) −2.49112e10 −1.48973
\(188\) 3.94475e9 0.230308
\(189\) −9.63489e9 −0.549248
\(190\) −2.63550e9 −0.146714
\(191\) −5.33871e9 −0.290259 −0.145130 0.989413i \(-0.546360\pi\)
−0.145130 + 0.989413i \(0.546360\pi\)
\(192\) −2.35266e9 −0.124941
\(193\) 2.75948e10 1.43159 0.715797 0.698309i \(-0.246064\pi\)
0.715797 + 0.698309i \(0.246064\pi\)
\(194\) −2.63727e10 −1.33674
\(195\) 1.20641e10 0.597501
\(196\) −7.21703e9 −0.349306
\(197\) −9.61053e9 −0.454621 −0.227310 0.973822i \(-0.572993\pi\)
−0.227310 + 0.973822i \(0.572993\pi\)
\(198\) −1.47417e7 −0.000681636 0
\(199\) 2.88815e10 1.30551 0.652756 0.757568i \(-0.273612\pi\)
0.652756 + 0.757568i \(0.273612\pi\)
\(200\) 1.45638e9 0.0643635
\(201\) −3.71982e9 −0.160746
\(202\) 1.43556e10 0.606654
\(203\) −9.13778e9 −0.377666
\(204\) −1.81546e10 −0.733922
\(205\) 2.75719e10 1.09037
\(206\) −2.68196e10 −1.03765
\(207\) 1.09428e7 0.000414248 0
\(208\) −4.46073e9 −0.165242
\(209\) −6.41951e9 −0.232725
\(210\) −9.88990e9 −0.350918
\(211\) 2.26589e10 0.786986 0.393493 0.919328i \(-0.371267\pi\)
0.393493 + 0.919328i \(0.371267\pi\)
\(212\) 1.80670e9 0.0614291
\(213\) 5.98893e9 0.199361
\(214\) −4.92884e9 −0.160651
\(215\) −1.42452e10 −0.454668
\(216\) −1.13163e10 −0.353721
\(217\) −1.23138e10 −0.376985
\(218\) 2.87310e9 0.0861581
\(219\) −4.55247e10 −1.33736
\(220\) −1.59388e10 −0.458727
\(221\) −3.44217e10 −0.970660
\(222\) 4.16045e10 1.14961
\(223\) −5.84566e10 −1.58293 −0.791465 0.611214i \(-0.790681\pi\)
−0.791465 + 0.611214i \(0.790681\pi\)
\(224\) 3.65682e9 0.0970483
\(225\) 6.65049e6 0.000172995 0
\(226\) 3.69259e10 0.941549
\(227\) −4.82187e9 −0.120531 −0.0602656 0.998182i \(-0.519195\pi\)
−0.0602656 + 0.998182i \(0.519195\pi\)
\(228\) −4.67836e9 −0.114653
\(229\) −7.15833e9 −0.172009 −0.0860046 0.996295i \(-0.527410\pi\)
−0.0860046 + 0.996295i \(0.527410\pi\)
\(230\) 1.18314e10 0.278780
\(231\) −2.40897e10 −0.556643
\(232\) −1.07324e10 −0.243221
\(233\) 4.52737e10 1.00634 0.503169 0.864188i \(-0.332167\pi\)
0.503169 + 0.864188i \(0.332167\pi\)
\(234\) −2.03697e7 −0.000444133 0
\(235\) 1.94764e10 0.416585
\(236\) 7.44268e9 0.156180
\(237\) 1.24553e10 0.256441
\(238\) 2.82183e10 0.570078
\(239\) −7.05422e10 −1.39849 −0.699244 0.714883i \(-0.746480\pi\)
−0.699244 + 0.714883i \(0.746480\pi\)
\(240\) −1.16158e10 −0.225994
\(241\) 7.83138e10 1.49542 0.747708 0.664028i \(-0.231154\pi\)
0.747708 + 0.664028i \(0.231154\pi\)
\(242\) −1.09635e9 −0.0205486
\(243\) −1.03301e8 −0.00190054
\(244\) 3.69368e10 0.667121
\(245\) −3.56326e10 −0.631830
\(246\) 4.89438e10 0.852098
\(247\) −8.87034e9 −0.151637
\(248\) −1.44627e10 −0.242782
\(249\) −8.83207e9 −0.145601
\(250\) 4.66889e10 0.755934
\(251\) 1.03661e11 1.64848 0.824240 0.566241i \(-0.191603\pi\)
0.824240 + 0.566241i \(0.191603\pi\)
\(252\) 1.66987e7 0.000260844 0
\(253\) 2.88188e10 0.442215
\(254\) 7.22101e10 1.08854
\(255\) −8.96344e10 −1.32753
\(256\) 4.29497e9 0.0625000
\(257\) 1.07294e11 1.53419 0.767093 0.641536i \(-0.221703\pi\)
0.767093 + 0.641536i \(0.221703\pi\)
\(258\) −2.52870e10 −0.355311
\(259\) −6.46674e10 −0.892969
\(260\) −2.20239e10 −0.298892
\(261\) −4.90090e7 −0.000653722 0
\(262\) 4.05945e10 0.532245
\(263\) −1.18240e11 −1.52393 −0.761963 0.647620i \(-0.775764\pi\)
−0.761963 + 0.647620i \(0.775764\pi\)
\(264\) −2.82935e10 −0.358483
\(265\) 8.92019e9 0.111114
\(266\) 7.27174e9 0.0890576
\(267\) 6.37103e9 0.0767201
\(268\) 6.79082e9 0.0804110
\(269\) 1.08142e11 1.25924 0.629620 0.776903i \(-0.283211\pi\)
0.629620 + 0.776903i \(0.283211\pi\)
\(270\) −5.58717e10 −0.639816
\(271\) −1.22183e11 −1.37609 −0.688047 0.725667i \(-0.741532\pi\)
−0.688047 + 0.725667i \(0.741532\pi\)
\(272\) 3.31426e10 0.367135
\(273\) −3.32866e10 −0.362691
\(274\) −6.46544e10 −0.692979
\(275\) 1.75147e10 0.184674
\(276\) 2.10023e10 0.217860
\(277\) 1.18944e11 1.21390 0.606951 0.794740i \(-0.292393\pi\)
0.606951 + 0.794740i \(0.292393\pi\)
\(278\) 2.98436e10 0.299674
\(279\) −6.60431e7 −0.000652542 0
\(280\) 1.80548e10 0.175542
\(281\) −4.73680e10 −0.453218 −0.226609 0.973986i \(-0.572764\pi\)
−0.226609 + 0.973986i \(0.572764\pi\)
\(282\) 3.45731e10 0.325550
\(283\) 1.59829e11 1.48121 0.740603 0.671943i \(-0.234540\pi\)
0.740603 + 0.671943i \(0.234540\pi\)
\(284\) −1.09333e10 −0.0997281
\(285\) −2.30984e10 −0.207387
\(286\) −5.36455e10 −0.474118
\(287\) −7.60750e10 −0.661871
\(288\) 1.96128e7 0.000167986 0
\(289\) 1.37161e11 1.15662
\(290\) −5.29890e10 −0.439941
\(291\) −2.31140e11 −1.88954
\(292\) 8.31089e10 0.668998
\(293\) 1.32731e11 1.05212 0.526062 0.850446i \(-0.323668\pi\)
0.526062 + 0.850446i \(0.323668\pi\)
\(294\) −6.32525e10 −0.493759
\(295\) 3.67467e10 0.282501
\(296\) −7.59524e10 −0.575081
\(297\) −1.36091e11 −1.01491
\(298\) 1.21677e11 0.893789
\(299\) 3.98212e10 0.288134
\(300\) 1.27642e10 0.0909805
\(301\) 3.93045e10 0.275990
\(302\) −2.79444e10 −0.193314
\(303\) 1.25818e11 0.857530
\(304\) 8.54072e9 0.0573539
\(305\) 1.82368e11 1.20670
\(306\) 1.51344e8 0.000986776 0
\(307\) 2.41951e11 1.55455 0.777274 0.629162i \(-0.216602\pi\)
0.777274 + 0.629162i \(0.216602\pi\)
\(308\) 4.39776e10 0.278454
\(309\) −2.35056e11 −1.46676
\(310\) −7.14065e10 −0.439147
\(311\) −1.75276e11 −1.06243 −0.531217 0.847236i \(-0.678265\pi\)
−0.531217 + 0.847236i \(0.678265\pi\)
\(312\) −3.90954e10 −0.233577
\(313\) −2.56460e11 −1.51032 −0.755161 0.655539i \(-0.772442\pi\)
−0.755161 + 0.655539i \(0.772442\pi\)
\(314\) −1.70502e11 −0.989798
\(315\) 8.24464e7 0.000471818 0
\(316\) −2.27382e10 −0.128281
\(317\) −2.91020e10 −0.161866 −0.0809331 0.996720i \(-0.525790\pi\)
−0.0809331 + 0.996720i \(0.525790\pi\)
\(318\) 1.58345e10 0.0868325
\(319\) −1.29070e11 −0.697856
\(320\) 2.12055e10 0.113051
\(321\) −4.31980e10 −0.227087
\(322\) −3.26447e10 −0.169224
\(323\) 6.59054e10 0.336907
\(324\) −9.90853e10 −0.499524
\(325\) 2.42014e10 0.120328
\(326\) −1.45816e11 −0.715031
\(327\) 2.51808e10 0.121788
\(328\) −8.93508e10 −0.426251
\(329\) −5.37382e10 −0.252873
\(330\) −1.39693e11 −0.648430
\(331\) −1.45683e11 −0.667088 −0.333544 0.942734i \(-0.608245\pi\)
−0.333544 + 0.942734i \(0.608245\pi\)
\(332\) 1.61236e10 0.0728353
\(333\) −3.46833e8 −0.00154569
\(334\) −1.84873e11 −0.812859
\(335\) 3.35283e10 0.145449
\(336\) 3.20496e10 0.137182
\(337\) −4.16729e10 −0.176002 −0.0880012 0.996120i \(-0.528048\pi\)
−0.0880012 + 0.996120i \(0.528048\pi\)
\(338\) 9.55457e10 0.398186
\(339\) 3.23631e11 1.33092
\(340\) 1.63635e11 0.664080
\(341\) −1.73931e11 −0.696597
\(342\) 3.90008e7 0.000154154 0
\(343\) 2.39045e11 0.932518
\(344\) 4.61635e10 0.177740
\(345\) 1.03695e11 0.394067
\(346\) 4.04457e10 0.151716
\(347\) −5.79909e10 −0.214722 −0.107361 0.994220i \(-0.534240\pi\)
−0.107361 + 0.994220i \(0.534240\pi\)
\(348\) −9.40624e10 −0.343803
\(349\) −1.06762e11 −0.385214 −0.192607 0.981276i \(-0.561694\pi\)
−0.192607 + 0.981276i \(0.561694\pi\)
\(350\) −1.98399e10 −0.0706696
\(351\) −1.88048e11 −0.661282
\(352\) 5.16520e10 0.179327
\(353\) −4.27202e11 −1.46436 −0.732180 0.681112i \(-0.761497\pi\)
−0.732180 + 0.681112i \(0.761497\pi\)
\(354\) 6.52302e10 0.220767
\(355\) −5.39808e10 −0.180390
\(356\) −1.16308e10 −0.0383783
\(357\) 2.47314e11 0.805828
\(358\) 2.76908e11 0.890968
\(359\) 4.29892e11 1.36595 0.682975 0.730442i \(-0.260686\pi\)
0.682975 + 0.730442i \(0.260686\pi\)
\(360\) 9.68340e7 0.000303855 0
\(361\) 1.69836e10 0.0526316
\(362\) −1.73901e11 −0.532247
\(363\) −9.60883e9 −0.0290463
\(364\) 6.07673e10 0.181432
\(365\) 4.10333e11 1.21009
\(366\) 3.23727e11 0.943004
\(367\) 1.40582e11 0.404512 0.202256 0.979333i \(-0.435173\pi\)
0.202256 + 0.979333i \(0.435173\pi\)
\(368\) −3.83414e10 −0.108982
\(369\) −4.08016e8 −0.00114567
\(370\) −3.74999e11 −1.04021
\(371\) −2.46121e10 −0.0674476
\(372\) −1.26756e11 −0.343182
\(373\) 4.31647e11 1.15462 0.577310 0.816525i \(-0.304102\pi\)
0.577310 + 0.816525i \(0.304102\pi\)
\(374\) 3.98579e11 1.05340
\(375\) 4.09198e11 1.06854
\(376\) −6.31160e10 −0.162852
\(377\) −1.78346e11 −0.454702
\(378\) 1.54158e11 0.388377
\(379\) 7.16332e11 1.78336 0.891678 0.452670i \(-0.149528\pi\)
0.891678 + 0.452670i \(0.149528\pi\)
\(380\) 4.21680e10 0.103743
\(381\) 6.32874e11 1.53870
\(382\) 8.54194e10 0.205244
\(383\) −1.03271e11 −0.245236 −0.122618 0.992454i \(-0.539129\pi\)
−0.122618 + 0.992454i \(0.539129\pi\)
\(384\) 3.76426e10 0.0883463
\(385\) 2.17130e11 0.503671
\(386\) −4.41517e11 −1.01229
\(387\) 2.10803e8 0.000477725 0
\(388\) 4.21964e11 0.945219
\(389\) −3.49935e11 −0.774845 −0.387422 0.921902i \(-0.626634\pi\)
−0.387422 + 0.921902i \(0.626634\pi\)
\(390\) −1.93025e11 −0.422497
\(391\) −2.95866e11 −0.640176
\(392\) 1.15473e11 0.246997
\(393\) 3.55784e11 0.752351
\(394\) 1.53769e11 0.321466
\(395\) −1.12265e11 −0.232037
\(396\) 2.35866e8 0.000481990 0
\(397\) −2.83640e11 −0.573073 −0.286536 0.958069i \(-0.592504\pi\)
−0.286536 + 0.958069i \(0.592504\pi\)
\(398\) −4.62104e11 −0.923137
\(399\) 6.37320e10 0.125887
\(400\) −2.33021e10 −0.0455119
\(401\) −9.94065e11 −1.91984 −0.959920 0.280275i \(-0.909574\pi\)
−0.959920 + 0.280275i \(0.909574\pi\)
\(402\) 5.95171e10 0.113664
\(403\) −2.40334e11 −0.453881
\(404\) −2.29690e11 −0.428969
\(405\) −4.89213e11 −0.903547
\(406\) 1.46204e11 0.267051
\(407\) −9.13417e11 −1.65004
\(408\) 2.90473e11 0.518961
\(409\) 2.41111e11 0.426051 0.213026 0.977047i \(-0.431668\pi\)
0.213026 + 0.977047i \(0.431668\pi\)
\(410\) −4.41151e11 −0.771010
\(411\) −5.66653e11 −0.979555
\(412\) 4.29113e11 0.733726
\(413\) −1.01390e11 −0.171482
\(414\) −1.75084e8 −0.000292918 0
\(415\) 7.96072e10 0.131745
\(416\) 7.13717e10 0.116844
\(417\) 2.61559e11 0.423602
\(418\) 1.02712e11 0.164562
\(419\) 7.03102e11 1.11444 0.557218 0.830366i \(-0.311869\pi\)
0.557218 + 0.830366i \(0.311869\pi\)
\(420\) 1.58238e11 0.248136
\(421\) −2.68270e11 −0.416200 −0.208100 0.978108i \(-0.566728\pi\)
−0.208100 + 0.978108i \(0.566728\pi\)
\(422\) −3.62542e11 −0.556483
\(423\) −2.88216e8 −0.000437710 0
\(424\) −2.89071e10 −0.0434369
\(425\) −1.79813e11 −0.267344
\(426\) −9.58229e10 −0.140970
\(427\) −5.03180e11 −0.732483
\(428\) 7.88615e10 0.113597
\(429\) −4.70168e11 −0.670185
\(430\) 2.27923e11 0.321499
\(431\) 5.95908e10 0.0831825 0.0415912 0.999135i \(-0.486757\pi\)
0.0415912 + 0.999135i \(0.486757\pi\)
\(432\) 1.81060e11 0.250119
\(433\) −1.16290e12 −1.58982 −0.794909 0.606729i \(-0.792481\pi\)
−0.794909 + 0.606729i \(0.792481\pi\)
\(434\) 1.97021e11 0.266569
\(435\) −4.64414e11 −0.621875
\(436\) −4.59696e10 −0.0609230
\(437\) −7.62435e10 −0.100008
\(438\) 7.28395e11 0.945657
\(439\) −1.32465e12 −1.70220 −0.851100 0.525004i \(-0.824064\pi\)
−0.851100 + 0.525004i \(0.824064\pi\)
\(440\) 2.55021e11 0.324369
\(441\) 5.27300e8 0.000663871 0
\(442\) 5.50747e11 0.686360
\(443\) 7.48682e11 0.923593 0.461797 0.886986i \(-0.347205\pi\)
0.461797 + 0.886986i \(0.347205\pi\)
\(444\) −6.65673e11 −0.812900
\(445\) −5.74248e10 −0.0694192
\(446\) 9.35306e11 1.11930
\(447\) 1.06642e12 1.26341
\(448\) −5.85091e10 −0.0686235
\(449\) 8.07450e11 0.937577 0.468788 0.883310i \(-0.344691\pi\)
0.468788 + 0.883310i \(0.344691\pi\)
\(450\) −1.06408e8 −0.000122326 0
\(451\) −1.07455e12 −1.22301
\(452\) −5.90814e11 −0.665776
\(453\) −2.44914e11 −0.273258
\(454\) 7.71500e10 0.0852284
\(455\) 3.00026e11 0.328177
\(456\) 7.48537e10 0.0810722
\(457\) −2.09593e11 −0.224778 −0.112389 0.993664i \(-0.535850\pi\)
−0.112389 + 0.993664i \(0.535850\pi\)
\(458\) 1.14533e11 0.121629
\(459\) 1.39717e12 1.46924
\(460\) −1.89303e11 −0.197127
\(461\) 1.12823e12 1.16343 0.581717 0.813391i \(-0.302381\pi\)
0.581717 + 0.813391i \(0.302381\pi\)
\(462\) 3.85434e11 0.393606
\(463\) −9.75512e11 −0.986548 −0.493274 0.869874i \(-0.664200\pi\)
−0.493274 + 0.869874i \(0.664200\pi\)
\(464\) 1.71718e11 0.171983
\(465\) −6.25831e11 −0.620753
\(466\) −7.24379e11 −0.711589
\(467\) −5.15878e11 −0.501904 −0.250952 0.968000i \(-0.580744\pi\)
−0.250952 + 0.968000i \(0.580744\pi\)
\(468\) 3.25915e8 0.000314050 0
\(469\) −9.25095e10 −0.0882894
\(470\) −3.11622e11 −0.294570
\(471\) −1.49434e12 −1.39912
\(472\) −1.19083e11 −0.110436
\(473\) 5.55170e11 0.509977
\(474\) −1.99285e11 −0.181331
\(475\) −4.63372e10 −0.0417646
\(476\) −4.51492e11 −0.403106
\(477\) −1.32003e8 −0.000116749 0
\(478\) 1.12868e12 0.988880
\(479\) 1.89215e12 1.64228 0.821139 0.570728i \(-0.193339\pi\)
0.821139 + 0.570728i \(0.193339\pi\)
\(480\) 1.85852e11 0.159802
\(481\) −1.26214e12 −1.07511
\(482\) −1.25302e12 −1.05742
\(483\) −2.86109e11 −0.239204
\(484\) 1.75417e10 0.0145300
\(485\) 2.08336e12 1.70973
\(486\) 1.65282e9 0.00134389
\(487\) 1.35254e12 1.08960 0.544802 0.838565i \(-0.316605\pi\)
0.544802 + 0.838565i \(0.316605\pi\)
\(488\) −5.90989e11 −0.471726
\(489\) −1.27798e12 −1.01073
\(490\) 5.70122e11 0.446771
\(491\) 9.23267e11 0.716903 0.358452 0.933548i \(-0.383305\pi\)
0.358452 + 0.933548i \(0.383305\pi\)
\(492\) −7.83100e11 −0.602524
\(493\) 1.32508e12 1.01026
\(494\) 1.41925e11 0.107223
\(495\) 1.16454e9 0.000871830 0
\(496\) 2.31403e11 0.171673
\(497\) 1.48941e11 0.109499
\(498\) 1.41313e11 0.102956
\(499\) 1.98116e12 1.43043 0.715214 0.698905i \(-0.246329\pi\)
0.715214 + 0.698905i \(0.246329\pi\)
\(500\) −7.47023e11 −0.534526
\(501\) −1.62029e12 −1.14901
\(502\) −1.65858e12 −1.16565
\(503\) 2.56937e12 1.78966 0.894830 0.446406i \(-0.147296\pi\)
0.894830 + 0.446406i \(0.147296\pi\)
\(504\) −2.67179e8 −0.000184444 0
\(505\) −1.13405e12 −0.775925
\(506\) −4.61101e11 −0.312693
\(507\) 8.37395e11 0.562853
\(508\) −1.15536e12 −0.769717
\(509\) 1.56900e11 0.103608 0.0518040 0.998657i \(-0.483503\pi\)
0.0518040 + 0.998657i \(0.483503\pi\)
\(510\) 1.43415e12 0.938705
\(511\) −1.13217e12 −0.734544
\(512\) −6.87195e10 −0.0441942
\(513\) 3.60046e11 0.229525
\(514\) −1.71671e12 −1.08483
\(515\) 2.11866e12 1.32717
\(516\) 4.04592e11 0.251243
\(517\) −7.59044e11 −0.467261
\(518\) 1.03468e12 0.631424
\(519\) 3.54480e11 0.214456
\(520\) 3.52383e11 0.211349
\(521\) −1.30379e12 −0.775241 −0.387621 0.921819i \(-0.626703\pi\)
−0.387621 + 0.921819i \(0.626703\pi\)
\(522\) 7.84143e8 0.000462251 0
\(523\) 2.30458e12 1.34690 0.673449 0.739234i \(-0.264812\pi\)
0.673449 + 0.739234i \(0.264812\pi\)
\(524\) −6.49513e11 −0.376354
\(525\) −1.73883e11 −0.0998944
\(526\) 1.89184e12 1.07758
\(527\) 1.78565e12 1.00843
\(528\) 4.52696e11 0.253486
\(529\) −1.45888e12 −0.809968
\(530\) −1.42723e11 −0.0785693
\(531\) −5.43786e8 −0.000296827 0
\(532\) −1.16348e11 −0.0629732
\(533\) −1.48479e12 −0.796878
\(534\) −1.01936e11 −0.0542493
\(535\) 3.89362e11 0.205476
\(536\) −1.08653e11 −0.0568592
\(537\) 2.42692e12 1.25942
\(538\) −1.73027e12 −0.890417
\(539\) 1.38869e12 0.708691
\(540\) 8.93947e11 0.452418
\(541\) 5.93525e11 0.297887 0.148943 0.988846i \(-0.452413\pi\)
0.148943 + 0.988846i \(0.452413\pi\)
\(542\) 1.95492e12 0.973045
\(543\) −1.52413e12 −0.752353
\(544\) −5.30281e11 −0.259604
\(545\) −2.26965e11 −0.110198
\(546\) 5.32585e11 0.256462
\(547\) −2.57843e12 −1.23144 −0.615719 0.787966i \(-0.711134\pi\)
−0.615719 + 0.787966i \(0.711134\pi\)
\(548\) 1.03447e12 0.490010
\(549\) −2.69872e9 −0.00126789
\(550\) −2.80235e11 −0.130584
\(551\) 3.41469e11 0.157822
\(552\) −3.36037e11 −0.154050
\(553\) 3.09756e11 0.140850
\(554\) −1.90310e12 −0.858358
\(555\) −3.28662e12 −1.47039
\(556\) −4.77497e11 −0.211901
\(557\) −3.49163e12 −1.53702 −0.768510 0.639838i \(-0.779001\pi\)
−0.768510 + 0.639838i \(0.779001\pi\)
\(558\) 1.05669e9 0.000461417 0
\(559\) 7.67122e11 0.332285
\(560\) −2.88877e11 −0.124127
\(561\) 3.49328e12 1.48902
\(562\) 7.57888e11 0.320473
\(563\) −1.40676e12 −0.590109 −0.295054 0.955480i \(-0.595338\pi\)
−0.295054 + 0.955480i \(0.595338\pi\)
\(564\) −5.53170e11 −0.230199
\(565\) −2.91702e12 −1.20427
\(566\) −2.55726e12 −1.04737
\(567\) 1.34981e12 0.548466
\(568\) 1.74932e11 0.0705184
\(569\) 2.81635e12 1.12637 0.563185 0.826331i \(-0.309576\pi\)
0.563185 + 0.826331i \(0.309576\pi\)
\(570\) 3.69575e11 0.146644
\(571\) −1.85246e11 −0.0729266 −0.0364633 0.999335i \(-0.511609\pi\)
−0.0364633 + 0.999335i \(0.511609\pi\)
\(572\) 8.58328e11 0.335252
\(573\) 7.48645e11 0.290121
\(574\) 1.21720e12 0.468014
\(575\) 2.08019e11 0.0793593
\(576\) −3.13804e8 −0.000118784 0
\(577\) −2.36440e12 −0.888034 −0.444017 0.896018i \(-0.646447\pi\)
−0.444017 + 0.896018i \(0.646447\pi\)
\(578\) −2.19457e12 −0.817850
\(579\) −3.86961e12 −1.43091
\(580\) 8.47824e11 0.311085
\(581\) −2.19648e11 −0.0799714
\(582\) 3.69823e12 1.33611
\(583\) −3.47642e11 −0.124630
\(584\) −1.32974e12 −0.473053
\(585\) 1.60914e9 0.000568058 0
\(586\) −2.12369e12 −0.743964
\(587\) 8.75403e11 0.304324 0.152162 0.988356i \(-0.451376\pi\)
0.152162 + 0.988356i \(0.451376\pi\)
\(588\) 1.01204e12 0.349140
\(589\) 4.60154e11 0.157538
\(590\) −5.87947e11 −0.199758
\(591\) 1.34768e12 0.454405
\(592\) 1.21524e12 0.406643
\(593\) −2.10579e12 −0.699307 −0.349654 0.936879i \(-0.613701\pi\)
−0.349654 + 0.936879i \(0.613701\pi\)
\(594\) 2.17746e12 0.717648
\(595\) −2.22915e12 −0.729144
\(596\) −1.94683e12 −0.632004
\(597\) −4.05004e12 −1.30489
\(598\) −6.37139e11 −0.203741
\(599\) 5.86647e12 1.86190 0.930950 0.365147i \(-0.118981\pi\)
0.930950 + 0.365147i \(0.118981\pi\)
\(600\) −2.04227e11 −0.0643330
\(601\) −8.97472e11 −0.280599 −0.140299 0.990109i \(-0.544807\pi\)
−0.140299 + 0.990109i \(0.544807\pi\)
\(602\) −6.28872e11 −0.195154
\(603\) −4.96159e8 −0.000152825 0
\(604\) 4.47111e11 0.136694
\(605\) 8.66084e10 0.0262822
\(606\) −2.01308e12 −0.606365
\(607\) −4.20534e12 −1.25734 −0.628669 0.777673i \(-0.716400\pi\)
−0.628669 + 0.777673i \(0.716400\pi\)
\(608\) −1.36651e11 −0.0405554
\(609\) 1.28139e12 0.377487
\(610\) −2.91788e12 −0.853265
\(611\) −1.04883e12 −0.304453
\(612\) −2.42150e9 −0.000697756 0
\(613\) 4.66333e12 1.33390 0.666951 0.745102i \(-0.267599\pi\)
0.666951 + 0.745102i \(0.267599\pi\)
\(614\) −3.87121e12 −1.09923
\(615\) −3.86640e12 −1.08985
\(616\) −7.03641e11 −0.196897
\(617\) 3.80910e12 1.05813 0.529065 0.848581i \(-0.322543\pi\)
0.529065 + 0.848581i \(0.322543\pi\)
\(618\) 3.76089e12 1.03715
\(619\) −3.62001e12 −0.991063 −0.495532 0.868590i \(-0.665027\pi\)
−0.495532 + 0.868590i \(0.665027\pi\)
\(620\) 1.14250e12 0.310524
\(621\) −1.61633e12 −0.436133
\(622\) 2.80442e12 0.751254
\(623\) 1.58443e11 0.0421384
\(624\) 6.25526e11 0.165164
\(625\) −2.99382e12 −0.784811
\(626\) 4.10336e12 1.06796
\(627\) 9.00204e11 0.232615
\(628\) 2.72804e12 0.699893
\(629\) 9.37752e12 2.38869
\(630\) −1.31914e9 −0.000333625 0
\(631\) −7.16358e11 −0.179886 −0.0899431 0.995947i \(-0.528669\pi\)
−0.0899431 + 0.995947i \(0.528669\pi\)
\(632\) 3.63811e11 0.0907086
\(633\) −3.17744e12 −0.786612
\(634\) 4.65632e11 0.114457
\(635\) −5.70436e12 −1.39228
\(636\) −2.53352e11 −0.0613999
\(637\) 1.91887e12 0.461761
\(638\) 2.06511e12 0.493459
\(639\) 7.98820e8 0.000189537 0
\(640\) −3.39288e11 −0.0799391
\(641\) −4.92458e12 −1.15215 −0.576074 0.817397i \(-0.695416\pi\)
−0.576074 + 0.817397i \(0.695416\pi\)
\(642\) 6.91169e11 0.160574
\(643\) 3.16432e12 0.730013 0.365007 0.931005i \(-0.381067\pi\)
0.365007 + 0.931005i \(0.381067\pi\)
\(644\) 5.22315e11 0.119659
\(645\) 1.99759e12 0.454452
\(646\) −1.05449e12 −0.238229
\(647\) 8.63261e12 1.93675 0.968374 0.249504i \(-0.0802677\pi\)
0.968374 + 0.249504i \(0.0802677\pi\)
\(648\) 1.58536e12 0.353217
\(649\) −1.43211e12 −0.316866
\(650\) −3.87223e11 −0.0850846
\(651\) 1.72676e12 0.376806
\(652\) 2.33305e12 0.505603
\(653\) −4.53346e12 −0.975710 −0.487855 0.872925i \(-0.662220\pi\)
−0.487855 + 0.872925i \(0.662220\pi\)
\(654\) −4.02893e11 −0.0861172
\(655\) −3.20684e12 −0.680755
\(656\) 1.42961e12 0.301405
\(657\) −6.07221e9 −0.00127146
\(658\) 8.59812e11 0.178808
\(659\) 2.67609e12 0.552734 0.276367 0.961052i \(-0.410869\pi\)
0.276367 + 0.961052i \(0.410869\pi\)
\(660\) 2.23509e12 0.458509
\(661\) 1.23915e12 0.252475 0.126238 0.992000i \(-0.459710\pi\)
0.126238 + 0.992000i \(0.459710\pi\)
\(662\) 2.33093e12 0.471703
\(663\) 4.82694e12 0.970199
\(664\) −2.57978e11 −0.0515023
\(665\) −5.74444e11 −0.113907
\(666\) 5.54933e9 0.00109296
\(667\) −1.53294e12 −0.299888
\(668\) 2.95797e12 0.574778
\(669\) 8.19734e12 1.58218
\(670\) −5.36452e11 −0.102848
\(671\) −7.10733e12 −1.35349
\(672\) −5.12794e11 −0.0970021
\(673\) −5.79692e11 −0.108926 −0.0544628 0.998516i \(-0.517345\pi\)
−0.0544628 + 0.998516i \(0.517345\pi\)
\(674\) 6.66766e11 0.124453
\(675\) −9.82331e11 −0.182134
\(676\) −1.52873e12 −0.281560
\(677\) 6.09981e12 1.11601 0.558004 0.829838i \(-0.311567\pi\)
0.558004 + 0.829838i \(0.311567\pi\)
\(678\) −5.17810e12 −0.941102
\(679\) −5.74829e12 −1.03783
\(680\) −2.61816e12 −0.469575
\(681\) 6.76168e11 0.120474
\(682\) 2.78289e12 0.492569
\(683\) −1.92035e12 −0.337665 −0.168833 0.985645i \(-0.554000\pi\)
−0.168833 + 0.985645i \(0.554000\pi\)
\(684\) −6.24012e8 −0.000109004 0
\(685\) 5.10748e12 0.886338
\(686\) −3.82473e12 −0.659389
\(687\) 1.00381e12 0.171928
\(688\) −7.38615e11 −0.125681
\(689\) −4.80365e11 −0.0812054
\(690\) −1.65912e12 −0.278648
\(691\) −1.22588e12 −0.204549 −0.102275 0.994756i \(-0.532612\pi\)
−0.102275 + 0.994756i \(0.532612\pi\)
\(692\) −6.47132e11 −0.107279
\(693\) −3.21314e9 −0.000529213 0
\(694\) 9.27854e11 0.151832
\(695\) −2.35754e12 −0.383290
\(696\) 1.50500e12 0.243105
\(697\) 1.10318e13 1.77051
\(698\) 1.70819e12 0.272388
\(699\) −6.34870e12 −1.00586
\(700\) 3.17438e11 0.0499710
\(701\) −7.54657e10 −0.0118037 −0.00590186 0.999983i \(-0.501879\pi\)
−0.00590186 + 0.999983i \(0.501879\pi\)
\(702\) 3.00877e12 0.467597
\(703\) 2.41655e12 0.373162
\(704\) −8.26432e11 −0.126803
\(705\) −2.73116e12 −0.416387
\(706\) 6.83524e12 1.03546
\(707\) 3.12900e12 0.470997
\(708\) −1.04368e12 −0.156106
\(709\) −2.48251e12 −0.368964 −0.184482 0.982836i \(-0.559061\pi\)
−0.184482 + 0.982836i \(0.559061\pi\)
\(710\) 8.63692e11 0.127555
\(711\) 1.66132e9 0.000243804 0
\(712\) 1.86093e11 0.0271375
\(713\) −2.06575e12 −0.299347
\(714\) −3.95703e12 −0.569807
\(715\) 4.23782e12 0.606409
\(716\) −4.43053e12 −0.630009
\(717\) 9.89210e12 1.39782
\(718\) −6.87828e12 −0.965872
\(719\) −5.21696e12 −0.728010 −0.364005 0.931397i \(-0.618591\pi\)
−0.364005 + 0.931397i \(0.618591\pi\)
\(720\) −1.54934e9 −0.000214858 0
\(721\) −5.84569e12 −0.805614
\(722\) −2.71737e11 −0.0372161
\(723\) −1.09819e13 −1.49470
\(724\) 2.78241e12 0.376355
\(725\) −9.31648e11 −0.125236
\(726\) 1.53741e11 0.0205388
\(727\) 1.23462e13 1.63919 0.819595 0.572944i \(-0.194199\pi\)
0.819595 + 0.572944i \(0.194199\pi\)
\(728\) −9.72277e11 −0.128292
\(729\) 7.63283e12 1.00095
\(730\) −6.56533e12 −0.855665
\(731\) −5.69961e12 −0.738273
\(732\) −5.17962e12 −0.666804
\(733\) −7.23773e12 −0.926050 −0.463025 0.886345i \(-0.653236\pi\)
−0.463025 + 0.886345i \(0.653236\pi\)
\(734\) −2.24931e12 −0.286033
\(735\) 4.99674e12 0.631530
\(736\) 6.13463e11 0.0770616
\(737\) −1.30668e12 −0.163142
\(738\) 6.52825e9 0.000810109 0
\(739\) −1.58616e12 −0.195636 −0.0978179 0.995204i \(-0.531186\pi\)
−0.0978179 + 0.995204i \(0.531186\pi\)
\(740\) 5.99999e12 0.735542
\(741\) 1.24388e12 0.151565
\(742\) 3.93794e11 0.0476927
\(743\) −1.32982e13 −1.60082 −0.800408 0.599455i \(-0.795384\pi\)
−0.800408 + 0.599455i \(0.795384\pi\)
\(744\) 2.02809e12 0.242667
\(745\) −9.61208e12 −1.14318
\(746\) −6.90636e12 −0.816440
\(747\) −1.17805e9 −0.000138426 0
\(748\) −6.37726e12 −0.744864
\(749\) −1.07431e12 −0.124727
\(750\) −6.54716e12 −0.755575
\(751\) −7.69495e12 −0.882726 −0.441363 0.897329i \(-0.645505\pi\)
−0.441363 + 0.897329i \(0.645505\pi\)
\(752\) 1.00986e12 0.115154
\(753\) −1.45363e13 −1.64770
\(754\) 2.85353e12 0.321523
\(755\) 2.20752e12 0.247254
\(756\) −2.46653e12 −0.274624
\(757\) 1.46042e13 1.61640 0.808198 0.588911i \(-0.200443\pi\)
0.808198 + 0.588911i \(0.200443\pi\)
\(758\) −1.14613e13 −1.26102
\(759\) −4.04124e12 −0.442005
\(760\) −6.74689e11 −0.0733571
\(761\) −2.14254e12 −0.231578 −0.115789 0.993274i \(-0.536940\pi\)
−0.115789 + 0.993274i \(0.536940\pi\)
\(762\) −1.01260e13 −1.08803
\(763\) 6.26231e11 0.0668920
\(764\) −1.36671e12 −0.145130
\(765\) −1.19557e10 −0.00126211
\(766\) 1.65234e12 0.173408
\(767\) −1.97886e12 −0.206460
\(768\) −6.02281e11 −0.0624703
\(769\) 1.37576e13 1.41865 0.709326 0.704881i \(-0.249000\pi\)
0.709326 + 0.704881i \(0.249000\pi\)
\(770\) −3.47408e12 −0.356149
\(771\) −1.50458e13 −1.53346
\(772\) 7.06428e12 0.715797
\(773\) 1.78934e13 1.80254 0.901270 0.433257i \(-0.142636\pi\)
0.901270 + 0.433257i \(0.142636\pi\)
\(774\) −3.37285e9 −0.000337802 0
\(775\) −1.25546e12 −0.125010
\(776\) −6.75142e12 −0.668371
\(777\) 9.06827e12 0.892545
\(778\) 5.59897e12 0.547898
\(779\) 2.84284e12 0.276588
\(780\) 3.08840e12 0.298750
\(781\) 2.10377e12 0.202334
\(782\) 4.73385e12 0.452673
\(783\) 7.23902e12 0.688259
\(784\) −1.84756e12 −0.174653
\(785\) 1.34691e13 1.26598
\(786\) −5.69255e12 −0.531992
\(787\) 1.90983e13 1.77464 0.887318 0.461158i \(-0.152566\pi\)
0.887318 + 0.461158i \(0.152566\pi\)
\(788\) −2.46030e12 −0.227310
\(789\) 1.65807e13 1.52320
\(790\) 1.79624e12 0.164075
\(791\) 8.04850e12 0.731006
\(792\) −3.77386e9 −0.000340818 0
\(793\) −9.82076e12 −0.881893
\(794\) 4.53823e12 0.405224
\(795\) −1.25087e12 −0.111061
\(796\) 7.39367e12 0.652756
\(797\) −4.39907e10 −0.00386188 −0.00193094 0.999998i \(-0.500615\pi\)
−0.00193094 + 0.999998i \(0.500615\pi\)
\(798\) −1.01971e12 −0.0890153
\(799\) 7.79267e12 0.676434
\(800\) 3.72833e11 0.0321818
\(801\) 8.49785e8 7.29395e−5 0
\(802\) 1.59050e13 1.35753
\(803\) −1.59917e13 −1.35730
\(804\) −9.52273e11 −0.0803728
\(805\) 2.57882e12 0.216441
\(806\) 3.84534e12 0.320943
\(807\) −1.51647e13 −1.25864
\(808\) 3.67504e12 0.303327
\(809\) −1.85839e13 −1.52535 −0.762675 0.646782i \(-0.776114\pi\)
−0.762675 + 0.646782i \(0.776114\pi\)
\(810\) 7.82741e12 0.638904
\(811\) 1.29810e13 1.05369 0.526847 0.849960i \(-0.323374\pi\)
0.526847 + 0.849960i \(0.323374\pi\)
\(812\) −2.33927e12 −0.188833
\(813\) 1.71336e13 1.37544
\(814\) 1.46147e13 1.16675
\(815\) 1.15190e13 0.914543
\(816\) −4.64757e12 −0.366961
\(817\) −1.46877e12 −0.115333
\(818\) −3.85777e12 −0.301264
\(819\) −4.43985e9 −0.000344819 0
\(820\) 7.05842e12 0.545186
\(821\) −4.47787e12 −0.343975 −0.171987 0.985099i \(-0.555019\pi\)
−0.171987 + 0.985099i \(0.555019\pi\)
\(822\) 9.06644e12 0.692650
\(823\) 1.13315e13 0.860973 0.430487 0.902597i \(-0.358342\pi\)
0.430487 + 0.902597i \(0.358342\pi\)
\(824\) −6.86581e12 −0.518823
\(825\) −2.45607e12 −0.184586
\(826\) 1.62223e12 0.121256
\(827\) −1.46669e13 −1.09034 −0.545171 0.838325i \(-0.683535\pi\)
−0.545171 + 0.838325i \(0.683535\pi\)
\(828\) 2.80135e9 0.000207124 0
\(829\) −2.45530e13 −1.80555 −0.902774 0.430115i \(-0.858473\pi\)
−0.902774 + 0.430115i \(0.858473\pi\)
\(830\) −1.27371e12 −0.0931581
\(831\) −1.66794e13 −1.21332
\(832\) −1.14195e12 −0.0826211
\(833\) −1.42569e13 −1.02594
\(834\) −4.18495e12 −0.299532
\(835\) 1.46044e13 1.03967
\(836\) −1.64339e12 −0.116363
\(837\) 9.75510e12 0.687017
\(838\) −1.12496e13 −0.788025
\(839\) 8.71743e12 0.607379 0.303689 0.952771i \(-0.401781\pi\)
0.303689 + 0.952771i \(0.401781\pi\)
\(840\) −2.53182e12 −0.175459
\(841\) −7.64163e12 −0.526749
\(842\) 4.29231e12 0.294298
\(843\) 6.64239e12 0.453002
\(844\) 5.80067e12 0.393493
\(845\) −7.54780e12 −0.509290
\(846\) 4.61146e9 0.000309508 0
\(847\) −2.38965e11 −0.0159536
\(848\) 4.62514e11 0.0307145
\(849\) −2.24127e13 −1.48050
\(850\) 2.87701e12 0.189041
\(851\) −1.08485e13 −0.709066
\(852\) 1.53317e12 0.0996807
\(853\) 1.30680e12 0.0845159 0.0422580 0.999107i \(-0.486545\pi\)
0.0422580 + 0.999107i \(0.486545\pi\)
\(854\) 8.05087e12 0.517944
\(855\) −3.08093e9 −0.000197167 0
\(856\) −1.26178e12 −0.0803254
\(857\) −1.57408e12 −0.0996811 −0.0498405 0.998757i \(-0.515871\pi\)
−0.0498405 + 0.998757i \(0.515871\pi\)
\(858\) 7.52268e12 0.473893
\(859\) 1.39034e13 0.871265 0.435633 0.900125i \(-0.356525\pi\)
0.435633 + 0.900125i \(0.356525\pi\)
\(860\) −3.64676e12 −0.227334
\(861\) 1.06680e13 0.661557
\(862\) −9.53453e11 −0.0588189
\(863\) 2.53344e11 0.0155476 0.00777378 0.999970i \(-0.497526\pi\)
0.00777378 + 0.999970i \(0.497526\pi\)
\(864\) −2.89696e12 −0.176861
\(865\) −3.19508e12 −0.194048
\(866\) 1.86064e13 1.12417
\(867\) −1.92339e13 −1.15607
\(868\) −3.15234e12 −0.188492
\(869\) 4.37525e12 0.260264
\(870\) 7.43062e12 0.439732
\(871\) −1.80554e12 −0.106298
\(872\) 7.35513e11 0.0430791
\(873\) −3.08300e10 −0.00179643
\(874\) 1.21990e12 0.0707166
\(875\) 1.01765e13 0.586897
\(876\) −1.16543e13 −0.668680
\(877\) −4.95849e12 −0.283042 −0.141521 0.989935i \(-0.545199\pi\)
−0.141521 + 0.989935i \(0.545199\pi\)
\(878\) 2.11944e13 1.20364
\(879\) −1.86127e13 −1.05162
\(880\) −4.08034e12 −0.229364
\(881\) −1.32981e13 −0.743702 −0.371851 0.928292i \(-0.621277\pi\)
−0.371851 + 0.928292i \(0.621277\pi\)
\(882\) −8.43679e9 −0.000469428 0
\(883\) 1.27657e13 0.706677 0.353338 0.935496i \(-0.385046\pi\)
0.353338 + 0.935496i \(0.385046\pi\)
\(884\) −8.81196e12 −0.485330
\(885\) −5.15297e12 −0.282366
\(886\) −1.19789e13 −0.653079
\(887\) 1.14862e13 0.623044 0.311522 0.950239i \(-0.399161\pi\)
0.311522 + 0.950239i \(0.399161\pi\)
\(888\) 1.06508e13 0.574807
\(889\) 1.57392e13 0.845131
\(890\) 9.18797e11 0.0490868
\(891\) 1.90659e13 1.01346
\(892\) −1.49649e13 −0.791465
\(893\) 2.00814e12 0.105673
\(894\) −1.70627e13 −0.893365
\(895\) −2.18748e13 −1.13957
\(896\) 9.36146e11 0.0485241
\(897\) −5.58410e12 −0.287997
\(898\) −1.29192e13 −0.662967
\(899\) 9.25179e12 0.472397
\(900\) 1.70253e9 8.64973e−5 0
\(901\) 3.56904e12 0.180422
\(902\) 1.71928e13 0.864801
\(903\) −5.51165e12 −0.275859
\(904\) 9.45303e12 0.470775
\(905\) 1.37376e13 0.680757
\(906\) 3.91863e12 0.193222
\(907\) 1.87998e13 0.922400 0.461200 0.887296i \(-0.347419\pi\)
0.461200 + 0.887296i \(0.347419\pi\)
\(908\) −1.23440e12 −0.0602656
\(909\) 1.67819e10 0.000815273 0
\(910\) −4.80042e12 −0.232056
\(911\) −1.56523e12 −0.0752914 −0.0376457 0.999291i \(-0.511986\pi\)
−0.0376457 + 0.999291i \(0.511986\pi\)
\(912\) −1.19766e12 −0.0573267
\(913\) −3.10249e12 −0.147772
\(914\) 3.35349e12 0.158942
\(915\) −2.55733e13 −1.20613
\(916\) −1.83253e12 −0.0860046
\(917\) 8.84813e12 0.413228
\(918\) −2.23547e13 −1.03891
\(919\) 2.72891e13 1.26203 0.631014 0.775771i \(-0.282639\pi\)
0.631014 + 0.775771i \(0.282639\pi\)
\(920\) 3.02885e12 0.139390
\(921\) −3.39286e13 −1.55381
\(922\) −1.80516e13 −0.822673
\(923\) 2.90694e12 0.131834
\(924\) −6.16695e12 −0.278321
\(925\) −6.59320e12 −0.296114
\(926\) 1.56082e13 0.697595
\(927\) −3.13524e10 −0.00139448
\(928\) −2.74749e12 −0.121610
\(929\) −3.27413e13 −1.44220 −0.721100 0.692831i \(-0.756363\pi\)
−0.721100 + 0.692831i \(0.756363\pi\)
\(930\) 1.00133e13 0.438939
\(931\) −3.67395e12 −0.160273
\(932\) 1.15901e13 0.503169
\(933\) 2.45789e13 1.06193
\(934\) 8.25404e12 0.354900
\(935\) −3.14864e13 −1.34732
\(936\) −5.21464e9 −0.000222067 0
\(937\) −2.41823e13 −1.02487 −0.512435 0.858726i \(-0.671256\pi\)
−0.512435 + 0.858726i \(0.671256\pi\)
\(938\) 1.48015e12 0.0624300
\(939\) 3.59632e13 1.50961
\(940\) 4.98596e12 0.208292
\(941\) 7.48541e12 0.311216 0.155608 0.987819i \(-0.450266\pi\)
0.155608 + 0.987819i \(0.450266\pi\)
\(942\) 2.39094e13 0.989328
\(943\) −1.27622e13 −0.525562
\(944\) 1.90533e12 0.0780900
\(945\) −1.21780e13 −0.496744
\(946\) −8.88272e12 −0.360608
\(947\) −4.31138e13 −1.74197 −0.870986 0.491307i \(-0.836519\pi\)
−0.870986 + 0.491307i \(0.836519\pi\)
\(948\) 3.18856e12 0.128220
\(949\) −2.20970e13 −0.884374
\(950\) 7.41394e11 0.0295320
\(951\) 4.08096e12 0.161789
\(952\) 7.22388e12 0.285039
\(953\) −9.69686e12 −0.380814 −0.190407 0.981705i \(-0.560981\pi\)
−0.190407 + 0.981705i \(0.560981\pi\)
\(954\) 2.11205e9 8.25537e−5 0
\(955\) −6.74785e12 −0.262513
\(956\) −1.80588e13 −0.699244
\(957\) 1.80994e13 0.697525
\(958\) −3.02745e13 −1.16127
\(959\) −1.40923e13 −0.538019
\(960\) −2.97364e12 −0.112997
\(961\) −1.39722e13 −0.528456
\(962\) 2.01942e13 0.760221
\(963\) −5.76187e9 −0.000215896 0
\(964\) 2.00483e13 0.747708
\(965\) 3.48784e13 1.29474
\(966\) 4.57774e12 0.169143
\(967\) 2.84710e13 1.04709 0.523545 0.851998i \(-0.324609\pi\)
0.523545 + 0.851998i \(0.324609\pi\)
\(968\) −2.80667e11 −0.0102743
\(969\) −9.24187e12 −0.336746
\(970\) −3.33337e13 −1.20896
\(971\) 3.67091e13 1.32522 0.662609 0.748965i \(-0.269449\pi\)
0.662609 + 0.748965i \(0.269449\pi\)
\(972\) −2.64451e10 −0.000950271 0
\(973\) 6.50481e12 0.232663
\(974\) −2.16406e13 −0.770467
\(975\) −3.39375e12 −0.120271
\(976\) 9.45582e12 0.333561
\(977\) −4.89209e13 −1.71778 −0.858892 0.512157i \(-0.828847\pi\)
−0.858892 + 0.512157i \(0.828847\pi\)
\(978\) 2.04476e13 0.714691
\(979\) 2.23799e12 0.0778638
\(980\) −9.12195e12 −0.315915
\(981\) 3.35868e9 0.000115787 0
\(982\) −1.47723e13 −0.506927
\(983\) −1.06278e13 −0.363038 −0.181519 0.983387i \(-0.558101\pi\)
−0.181519 + 0.983387i \(0.558101\pi\)
\(984\) 1.25296e13 0.426049
\(985\) −1.21472e13 −0.411162
\(986\) −2.12013e13 −0.714360
\(987\) 7.53568e12 0.252753
\(988\) −2.27081e12 −0.0758183
\(989\) 6.59366e12 0.219151
\(990\) −1.86327e10 −0.000616477 0
\(991\) −9.73679e12 −0.320689 −0.160345 0.987061i \(-0.551261\pi\)
−0.160345 + 0.987061i \(0.551261\pi\)
\(992\) −3.70245e12 −0.121391
\(993\) 2.04291e13 0.666771
\(994\) −2.38305e12 −0.0774275
\(995\) 3.65047e13 1.18072
\(996\) −2.26101e12 −0.0728007
\(997\) 1.04996e13 0.336547 0.168273 0.985740i \(-0.446181\pi\)
0.168273 + 0.985740i \(0.446181\pi\)
\(998\) −3.16985e13 −1.01147
\(999\) 5.12300e13 1.62735
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 38.10.a.d.1.1 4
3.2 odd 2 342.10.a.l.1.2 4
4.3 odd 2 304.10.a.e.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.10.a.d.1.1 4 1.1 even 1 trivial
304.10.a.e.1.4 4 4.3 odd 2
342.10.a.l.1.2 4 3.2 odd 2