Properties

Label 43.8.a.b.1.13
Level 43
Weight 8
Character 43.1
Self dual yes
Analytic conductor 13.433
Analytic rank 0
Dimension 13
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.4325560958\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(21.1822\) of \(x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + 193428526 x^{6} + 11160446785 x^{5} + 1754605765 x^{4} - 349352939351 x^{3} - 481872751923 x^{2} + 2098464001560 x + 1551032970660\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+22.1822 q^{2} +2.16379 q^{3} +364.050 q^{4} +122.553 q^{5} +47.9976 q^{6} -90.8173 q^{7} +5236.10 q^{8} -2182.32 q^{9} +O(q^{10})\) \(q+22.1822 q^{2} +2.16379 q^{3} +364.050 q^{4} +122.553 q^{5} +47.9976 q^{6} -90.8173 q^{7} +5236.10 q^{8} -2182.32 q^{9} +2718.48 q^{10} +3410.31 q^{11} +787.727 q^{12} -11275.0 q^{13} -2014.53 q^{14} +265.178 q^{15} +69549.8 q^{16} +9720.37 q^{17} -48408.6 q^{18} +25351.7 q^{19} +44615.2 q^{20} -196.510 q^{21} +75648.2 q^{22} -52795.4 q^{23} +11329.8 q^{24} -63105.9 q^{25} -250104. q^{26} -9454.29 q^{27} -33062.0 q^{28} -216763. q^{29} +5882.23 q^{30} +142113. q^{31} +872545. q^{32} +7379.20 q^{33} +215619. q^{34} -11129.9 q^{35} -794472. q^{36} -74290.8 q^{37} +562357. q^{38} -24396.7 q^{39} +641697. q^{40} -273026. q^{41} -4359.01 q^{42} -79507.0 q^{43} +1.24152e6 q^{44} -267449. q^{45} -1.17112e6 q^{46} -112553. q^{47} +150491. q^{48} -815295. q^{49} -1.39983e6 q^{50} +21032.8 q^{51} -4.10466e6 q^{52} +1.41102e6 q^{53} -209717. q^{54} +417942. q^{55} -475528. q^{56} +54855.8 q^{57} -4.80827e6 q^{58} -1.44713e6 q^{59} +96537.9 q^{60} +3.19474e6 q^{61} +3.15239e6 q^{62} +198192. q^{63} +1.04526e7 q^{64} -1.38178e6 q^{65} +163687. q^{66} -3.52359e6 q^{67} +3.53870e6 q^{68} -114238. q^{69} -246885. q^{70} -327344. q^{71} -1.14268e7 q^{72} +2.80064e6 q^{73} -1.64793e6 q^{74} -136548. q^{75} +9.22928e6 q^{76} -309715. q^{77} -541173. q^{78} +3.35723e6 q^{79} +8.52350e6 q^{80} +4.75227e6 q^{81} -6.05633e6 q^{82} +7.10311e6 q^{83} -71539.2 q^{84} +1.19126e6 q^{85} -1.76364e6 q^{86} -469029. q^{87} +1.78567e7 q^{88} -1.32585e6 q^{89} -5.93259e6 q^{90} +1.02396e6 q^{91} -1.92201e7 q^{92} +307504. q^{93} -2.49668e6 q^{94} +3.10692e6 q^{95} +1.88801e6 q^{96} +1.29568e7 q^{97} -1.80850e7 q^{98} -7.44239e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 22.1822 1.96065 0.980324 0.197397i \(-0.0632489\pi\)
0.980324 + 0.197397i \(0.0632489\pi\)
\(3\) 2.16379 0.0462691 0.0231345 0.999732i \(-0.492635\pi\)
0.0231345 + 0.999732i \(0.492635\pi\)
\(4\) 364.050 2.84414
\(5\) 122.553 0.438457 0.219229 0.975674i \(-0.429646\pi\)
0.219229 + 0.975674i \(0.429646\pi\)
\(6\) 47.9976 0.0907173
\(7\) −90.8173 −0.100075 −0.0500375 0.998747i \(-0.515934\pi\)
−0.0500375 + 0.998747i \(0.515934\pi\)
\(8\) 5236.10 3.61570
\(9\) −2182.32 −0.997859
\(10\) 2718.48 0.859660
\(11\) 3410.31 0.772538 0.386269 0.922386i \(-0.373764\pi\)
0.386269 + 0.922386i \(0.373764\pi\)
\(12\) 787.727 0.131596
\(13\) −11275.0 −1.42336 −0.711680 0.702504i \(-0.752065\pi\)
−0.711680 + 0.702504i \(0.752065\pi\)
\(14\) −2014.53 −0.196212
\(15\) 265.178 0.0202870
\(16\) 69549.8 4.24498
\(17\) 9720.37 0.479857 0.239928 0.970791i \(-0.422876\pi\)
0.239928 + 0.970791i \(0.422876\pi\)
\(18\) −48408.6 −1.95645
\(19\) 25351.7 0.847949 0.423975 0.905674i \(-0.360635\pi\)
0.423975 + 0.905674i \(0.360635\pi\)
\(20\) 44615.2 1.24703
\(21\) −196.510 −0.00463038
\(22\) 75648.2 1.51467
\(23\) −52795.4 −0.904792 −0.452396 0.891817i \(-0.649431\pi\)
−0.452396 + 0.891817i \(0.649431\pi\)
\(24\) 11329.8 0.167295
\(25\) −63105.9 −0.807755
\(26\) −250104. −2.79071
\(27\) −9454.29 −0.0924391
\(28\) −33062.0 −0.284627
\(29\) −216763. −1.65041 −0.825204 0.564835i \(-0.808940\pi\)
−0.825204 + 0.564835i \(0.808940\pi\)
\(30\) 5882.23 0.0397757
\(31\) 142113. 0.856780 0.428390 0.903594i \(-0.359081\pi\)
0.428390 + 0.903594i \(0.359081\pi\)
\(32\) 872545. 4.70721
\(33\) 7379.20 0.0357446
\(34\) 215619. 0.940830
\(35\) −11129.9 −0.0438786
\(36\) −794472. −2.83805
\(37\) −74290.8 −0.241118 −0.120559 0.992706i \(-0.538469\pi\)
−0.120559 + 0.992706i \(0.538469\pi\)
\(38\) 562357. 1.66253
\(39\) −24396.7 −0.0658576
\(40\) 641697. 1.58533
\(41\) −273026. −0.618673 −0.309337 0.950953i \(-0.600107\pi\)
−0.309337 + 0.950953i \(0.600107\pi\)
\(42\) −4359.01 −0.00907853
\(43\) −79507.0 −0.152499
\(44\) 1.24152e6 2.19720
\(45\) −267449. −0.437519
\(46\) −1.17112e6 −1.77398
\(47\) −112553. −0.158130 −0.0790652 0.996869i \(-0.525194\pi\)
−0.0790652 + 0.996869i \(0.525194\pi\)
\(48\) 150491. 0.196411
\(49\) −815295. −0.989985
\(50\) −1.39983e6 −1.58372
\(51\) 21032.8 0.0222025
\(52\) −4.10466e6 −4.04823
\(53\) 1.41102e6 1.30187 0.650936 0.759132i \(-0.274376\pi\)
0.650936 + 0.759132i \(0.274376\pi\)
\(54\) −209717. −0.181240
\(55\) 417942. 0.338725
\(56\) −475528. −0.361841
\(57\) 54855.8 0.0392338
\(58\) −4.80827e6 −3.23587
\(59\) −1.44713e6 −0.917329 −0.458665 0.888609i \(-0.651672\pi\)
−0.458665 + 0.888609i \(0.651672\pi\)
\(60\) 96537.9 0.0576990
\(61\) 3.19474e6 1.80211 0.901054 0.433708i \(-0.142795\pi\)
0.901054 + 0.433708i \(0.142795\pi\)
\(62\) 3.15239e6 1.67984
\(63\) 198192. 0.0998607
\(64\) 1.04526e7 4.98419
\(65\) −1.38178e6 −0.624083
\(66\) 163687. 0.0700826
\(67\) −3.52359e6 −1.43128 −0.715638 0.698472i \(-0.753864\pi\)
−0.715638 + 0.698472i \(0.753864\pi\)
\(68\) 3.53870e6 1.36478
\(69\) −114238. −0.0418639
\(70\) −246885. −0.0860304
\(71\) −327344. −0.108542 −0.0542712 0.998526i \(-0.517284\pi\)
−0.0542712 + 0.998526i \(0.517284\pi\)
\(72\) −1.14268e7 −3.60796
\(73\) 2.80064e6 0.842613 0.421306 0.906918i \(-0.361572\pi\)
0.421306 + 0.906918i \(0.361572\pi\)
\(74\) −1.64793e6 −0.472746
\(75\) −136548. −0.0373741
\(76\) 9.22928e6 2.41168
\(77\) −309715. −0.0773117
\(78\) −541173. −0.129123
\(79\) 3.35723e6 0.766101 0.383051 0.923727i \(-0.374874\pi\)
0.383051 + 0.923727i \(0.374874\pi\)
\(80\) 8.52350e6 1.86124
\(81\) 4.75227e6 0.993582
\(82\) −6.05633e6 −1.21300
\(83\) 7.10311e6 1.36356 0.681781 0.731556i \(-0.261206\pi\)
0.681781 + 0.731556i \(0.261206\pi\)
\(84\) −71539.2 −0.0131694
\(85\) 1.19126e6 0.210397
\(86\) −1.76364e6 −0.298996
\(87\) −469029. −0.0763628
\(88\) 1.78567e7 2.79327
\(89\) −1.32585e6 −0.199356 −0.0996778 0.995020i \(-0.531781\pi\)
−0.0996778 + 0.995020i \(0.531781\pi\)
\(90\) −5.93259e6 −0.857819
\(91\) 1.02396e6 0.142443
\(92\) −1.92201e7 −2.57335
\(93\) 307504. 0.0396424
\(94\) −2.49668e6 −0.310038
\(95\) 3.10692e6 0.371789
\(96\) 1.88801e6 0.217798
\(97\) 1.29568e7 1.44144 0.720718 0.693228i \(-0.243812\pi\)
0.720718 + 0.693228i \(0.243812\pi\)
\(98\) −1.80850e7 −1.94101
\(99\) −7.44239e6 −0.770884
\(100\) −2.29737e7 −2.29737
\(101\) 309915. 0.0299308 0.0149654 0.999888i \(-0.495236\pi\)
0.0149654 + 0.999888i \(0.495236\pi\)
\(102\) 466555. 0.0435313
\(103\) 1.99810e7 1.80172 0.900858 0.434113i \(-0.142938\pi\)
0.900858 + 0.434113i \(0.142938\pi\)
\(104\) −5.90370e7 −5.14645
\(105\) −24082.7 −0.00203022
\(106\) 3.12996e7 2.55251
\(107\) 1.77323e6 0.139934 0.0699669 0.997549i \(-0.477711\pi\)
0.0699669 + 0.997549i \(0.477711\pi\)
\(108\) −3.44183e6 −0.262910
\(109\) 7.12031e6 0.526631 0.263315 0.964710i \(-0.415184\pi\)
0.263315 + 0.964710i \(0.415184\pi\)
\(110\) 9.27088e6 0.664120
\(111\) −160750. −0.0111563
\(112\) −6.31632e6 −0.424816
\(113\) −9.47406e6 −0.617677 −0.308839 0.951114i \(-0.599940\pi\)
−0.308839 + 0.951114i \(0.599940\pi\)
\(114\) 1.21682e6 0.0769237
\(115\) −6.47021e6 −0.396713
\(116\) −7.89123e7 −4.69399
\(117\) 2.46056e7 1.42031
\(118\) −3.21005e7 −1.79856
\(119\) −882778. −0.0480216
\(120\) 1.38850e6 0.0733518
\(121\) −7.85694e6 −0.403185
\(122\) 7.08663e7 3.53330
\(123\) −590772. −0.0286254
\(124\) 5.17363e7 2.43680
\(125\) −1.73082e7 −0.792623
\(126\) 4.39634e6 0.195792
\(127\) 2.94089e7 1.27399 0.636994 0.770868i \(-0.280177\pi\)
0.636994 + 0.770868i \(0.280177\pi\)
\(128\) 1.20176e8 5.06503
\(129\) −172037. −0.00705597
\(130\) −3.06509e7 −1.22361
\(131\) 4.80770e7 1.86848 0.934238 0.356650i \(-0.116081\pi\)
0.934238 + 0.356650i \(0.116081\pi\)
\(132\) 2.68640e6 0.101663
\(133\) −2.30237e6 −0.0848585
\(134\) −7.81609e7 −2.80623
\(135\) −1.15865e6 −0.0405306
\(136\) 5.08968e7 1.73502
\(137\) 5.87744e6 0.195284 0.0976419 0.995222i \(-0.468870\pi\)
0.0976419 + 0.995222i \(0.468870\pi\)
\(138\) −2.53405e6 −0.0820803
\(139\) −5.12513e7 −1.61865 −0.809326 0.587360i \(-0.800167\pi\)
−0.809326 + 0.587360i \(0.800167\pi\)
\(140\) −4.05183e6 −0.124797
\(141\) −243541. −0.00731654
\(142\) −7.26120e6 −0.212813
\(143\) −3.84513e7 −1.09960
\(144\) −1.51780e8 −4.23589
\(145\) −2.65648e7 −0.723633
\(146\) 6.21244e7 1.65207
\(147\) −1.76413e6 −0.0458057
\(148\) −2.70455e7 −0.685771
\(149\) −6.91179e6 −0.171174 −0.0855871 0.996331i \(-0.527277\pi\)
−0.0855871 + 0.996331i \(0.527277\pi\)
\(150\) −3.02893e6 −0.0732774
\(151\) −6.97335e7 −1.64825 −0.824124 0.566410i \(-0.808332\pi\)
−0.824124 + 0.566410i \(0.808332\pi\)
\(152\) 1.32744e8 3.06593
\(153\) −2.12129e7 −0.478829
\(154\) −6.87016e6 −0.151581
\(155\) 1.74164e7 0.375661
\(156\) −8.88162e6 −0.187308
\(157\) −6.10206e7 −1.25843 −0.629213 0.777233i \(-0.716623\pi\)
−0.629213 + 0.777233i \(0.716623\pi\)
\(158\) 7.44707e7 1.50205
\(159\) 3.05316e6 0.0602365
\(160\) 1.06933e8 2.06391
\(161\) 4.79474e6 0.0905470
\(162\) 1.05416e8 1.94806
\(163\) −1.28985e6 −0.0233282 −0.0116641 0.999932i \(-0.503713\pi\)
−0.0116641 + 0.999932i \(0.503713\pi\)
\(164\) −9.93952e7 −1.75959
\(165\) 904340. 0.0156725
\(166\) 1.57563e8 2.67347
\(167\) −8.25906e7 −1.37222 −0.686109 0.727499i \(-0.740683\pi\)
−0.686109 + 0.727499i \(0.740683\pi\)
\(168\) −1.02894e6 −0.0167421
\(169\) 6.43771e7 1.02595
\(170\) 2.64247e7 0.412514
\(171\) −5.53255e7 −0.846134
\(172\) −2.89445e7 −0.433727
\(173\) −6.06382e6 −0.0890400 −0.0445200 0.999008i \(-0.514176\pi\)
−0.0445200 + 0.999008i \(0.514176\pi\)
\(174\) −1.04041e7 −0.149721
\(175\) 5.73110e6 0.0808361
\(176\) 2.37186e8 3.27941
\(177\) −3.13128e6 −0.0424440
\(178\) −2.94102e7 −0.390866
\(179\) −5.06661e7 −0.660286 −0.330143 0.943931i \(-0.607097\pi\)
−0.330143 + 0.943931i \(0.607097\pi\)
\(180\) −9.73645e7 −1.24436
\(181\) −2.08989e6 −0.0261968 −0.0130984 0.999914i \(-0.504169\pi\)
−0.0130984 + 0.999914i \(0.504169\pi\)
\(182\) 2.27138e7 0.279280
\(183\) 6.91274e6 0.0833818
\(184\) −2.76442e8 −3.27146
\(185\) −9.10452e6 −0.105720
\(186\) 6.82110e6 0.0777248
\(187\) 3.31495e7 0.370707
\(188\) −4.09749e7 −0.449744
\(189\) 858613. 0.00925084
\(190\) 6.89182e7 0.728948
\(191\) 1.78113e8 1.84960 0.924800 0.380454i \(-0.124232\pi\)
0.924800 + 0.380454i \(0.124232\pi\)
\(192\) 2.26172e7 0.230614
\(193\) 1.26757e8 1.26917 0.634586 0.772852i \(-0.281170\pi\)
0.634586 + 0.772852i \(0.281170\pi\)
\(194\) 2.87410e8 2.82615
\(195\) −2.98988e6 −0.0288757
\(196\) −2.96808e8 −2.81565
\(197\) −1.27815e8 −1.19110 −0.595550 0.803318i \(-0.703066\pi\)
−0.595550 + 0.803318i \(0.703066\pi\)
\(198\) −1.65088e8 −1.51143
\(199\) −1.29981e8 −1.16921 −0.584606 0.811318i \(-0.698751\pi\)
−0.584606 + 0.811318i \(0.698751\pi\)
\(200\) −3.30429e8 −2.92060
\(201\) −7.62431e6 −0.0662238
\(202\) 6.87459e6 0.0586837
\(203\) 1.96858e7 0.165164
\(204\) 7.65700e6 0.0631470
\(205\) −3.34601e7 −0.271262
\(206\) 4.43222e8 3.53253
\(207\) 1.15216e8 0.902855
\(208\) −7.84173e8 −6.04214
\(209\) 8.64573e7 0.655073
\(210\) −534208. −0.00398055
\(211\) −1.85689e8 −1.36081 −0.680406 0.732836i \(-0.738196\pi\)
−0.680406 + 0.732836i \(0.738196\pi\)
\(212\) 5.13682e8 3.70270
\(213\) −708303. −0.00502216
\(214\) 3.93342e7 0.274361
\(215\) −9.74378e6 −0.0668641
\(216\) −4.95036e7 −0.334232
\(217\) −1.29064e7 −0.0857422
\(218\) 1.57944e8 1.03254
\(219\) 6.06001e6 0.0389869
\(220\) 1.52152e8 0.963380
\(221\) −1.09597e8 −0.683009
\(222\) −3.56578e6 −0.0218735
\(223\) 1.57910e7 0.0953545 0.0476773 0.998863i \(-0.484818\pi\)
0.0476773 + 0.998863i \(0.484818\pi\)
\(224\) −7.92422e7 −0.471073
\(225\) 1.37717e8 0.806026
\(226\) −2.10155e8 −1.21105
\(227\) 3.37705e7 0.191623 0.0958113 0.995400i \(-0.469455\pi\)
0.0958113 + 0.995400i \(0.469455\pi\)
\(228\) 1.99702e7 0.111586
\(229\) 2.60087e8 1.43118 0.715590 0.698520i \(-0.246158\pi\)
0.715590 + 0.698520i \(0.246158\pi\)
\(230\) −1.43523e8 −0.777813
\(231\) −670159. −0.00357714
\(232\) −1.13499e9 −5.96738
\(233\) 1.45083e8 0.751398 0.375699 0.926742i \(-0.377403\pi\)
0.375699 + 0.926742i \(0.377403\pi\)
\(234\) 5.45807e8 2.78473
\(235\) −1.37937e7 −0.0693334
\(236\) −5.26827e8 −2.60901
\(237\) 7.26434e6 0.0354468
\(238\) −1.95819e7 −0.0941535
\(239\) −1.55295e8 −0.735808 −0.367904 0.929864i \(-0.619924\pi\)
−0.367904 + 0.929864i \(0.619924\pi\)
\(240\) 1.84431e7 0.0861180
\(241\) 3.22999e8 1.48642 0.743211 0.669057i \(-0.233302\pi\)
0.743211 + 0.669057i \(0.233302\pi\)
\(242\) −1.74284e8 −0.790504
\(243\) 3.09595e7 0.138411
\(244\) 1.16304e9 5.12544
\(245\) −9.99165e7 −0.434066
\(246\) −1.31046e7 −0.0561244
\(247\) −2.85841e8 −1.20694
\(248\) 7.44119e8 3.09786
\(249\) 1.53696e7 0.0630908
\(250\) −3.83934e8 −1.55405
\(251\) −2.35414e8 −0.939668 −0.469834 0.882755i \(-0.655686\pi\)
−0.469834 + 0.882755i \(0.655686\pi\)
\(252\) 7.21518e7 0.284018
\(253\) −1.80049e8 −0.698986
\(254\) 6.52354e8 2.49784
\(255\) 2.57763e6 0.00973486
\(256\) 1.32783e9 4.94655
\(257\) −4.25190e7 −0.156249 −0.0781245 0.996944i \(-0.524893\pi\)
−0.0781245 + 0.996944i \(0.524893\pi\)
\(258\) −3.81615e6 −0.0138343
\(259\) 6.74689e6 0.0241298
\(260\) −5.03036e8 −1.77498
\(261\) 4.73045e8 1.64687
\(262\) 1.06645e9 3.66342
\(263\) 1.04522e8 0.354292 0.177146 0.984185i \(-0.443314\pi\)
0.177146 + 0.984185i \(0.443314\pi\)
\(264\) 3.86382e7 0.129242
\(265\) 1.72924e8 0.570815
\(266\) −5.10717e7 −0.166378
\(267\) −2.86885e6 −0.00922400
\(268\) −1.28276e9 −4.07074
\(269\) −1.86374e8 −0.583784 −0.291892 0.956451i \(-0.594285\pi\)
−0.291892 + 0.956451i \(0.594285\pi\)
\(270\) −2.57013e7 −0.0794662
\(271\) 3.37243e8 1.02932 0.514660 0.857394i \(-0.327918\pi\)
0.514660 + 0.857394i \(0.327918\pi\)
\(272\) 6.76049e8 2.03698
\(273\) 2.21565e6 0.00659069
\(274\) 1.30375e8 0.382883
\(275\) −2.15211e8 −0.624022
\(276\) −4.15884e7 −0.119067
\(277\) −4.22893e8 −1.19550 −0.597752 0.801681i \(-0.703939\pi\)
−0.597752 + 0.801681i \(0.703939\pi\)
\(278\) −1.13687e9 −3.17360
\(279\) −3.10137e8 −0.854945
\(280\) −5.82772e7 −0.158652
\(281\) 3.41588e8 0.918396 0.459198 0.888334i \(-0.348137\pi\)
0.459198 + 0.888334i \(0.348137\pi\)
\(282\) −5.40228e6 −0.0143452
\(283\) −1.93785e8 −0.508238 −0.254119 0.967173i \(-0.581785\pi\)
−0.254119 + 0.967173i \(0.581785\pi\)
\(284\) −1.19169e8 −0.308710
\(285\) 6.72272e6 0.0172024
\(286\) −8.52934e8 −2.15593
\(287\) 2.47955e7 0.0619137
\(288\) −1.90417e9 −4.69713
\(289\) −3.15853e8 −0.769738
\(290\) −5.89265e8 −1.41879
\(291\) 2.80357e7 0.0666940
\(292\) 1.01957e9 2.39651
\(293\) 4.63942e8 1.07752 0.538762 0.842458i \(-0.318892\pi\)
0.538762 + 0.842458i \(0.318892\pi\)
\(294\) −3.91322e7 −0.0898088
\(295\) −1.77349e8 −0.402210
\(296\) −3.88994e8 −0.871809
\(297\) −3.22421e7 −0.0714127
\(298\) −1.53319e8 −0.335612
\(299\) 5.95268e8 1.28785
\(300\) −4.97102e7 −0.106297
\(301\) 7.22061e6 0.0152613
\(302\) −1.54684e9 −3.23163
\(303\) 670591. 0.00138487
\(304\) 1.76321e9 3.59953
\(305\) 3.91523e8 0.790147
\(306\) −4.70549e8 −0.938816
\(307\) 1.10340e8 0.217644 0.108822 0.994061i \(-0.465292\pi\)
0.108822 + 0.994061i \(0.465292\pi\)
\(308\) −1.12752e8 −0.219885
\(309\) 4.32347e7 0.0833638
\(310\) 3.86333e8 0.736539
\(311\) −1.65769e8 −0.312494 −0.156247 0.987718i \(-0.549940\pi\)
−0.156247 + 0.987718i \(0.549940\pi\)
\(312\) −1.27744e8 −0.238121
\(313\) −1.02978e9 −1.89818 −0.949091 0.315003i \(-0.897994\pi\)
−0.949091 + 0.315003i \(0.897994\pi\)
\(314\) −1.35357e9 −2.46733
\(315\) 2.42890e7 0.0437846
\(316\) 1.22220e9 2.17890
\(317\) 6.40081e8 1.12857 0.564284 0.825581i \(-0.309152\pi\)
0.564284 + 0.825581i \(0.309152\pi\)
\(318\) 6.77257e7 0.118102
\(319\) −7.39228e8 −1.27500
\(320\) 1.28099e9 2.18535
\(321\) 3.83691e6 0.00647461
\(322\) 1.06358e8 0.177531
\(323\) 2.46428e8 0.406894
\(324\) 1.73006e9 2.82588
\(325\) 7.11519e8 1.14973
\(326\) −2.86116e7 −0.0457383
\(327\) 1.54069e7 0.0243667
\(328\) −1.42959e9 −2.23694
\(329\) 1.02218e7 0.0158249
\(330\) 2.00602e7 0.0307282
\(331\) −4.29838e8 −0.651489 −0.325744 0.945458i \(-0.605615\pi\)
−0.325744 + 0.945458i \(0.605615\pi\)
\(332\) 2.58588e9 3.87816
\(333\) 1.62126e8 0.240601
\(334\) −1.83204e9 −2.69044
\(335\) −4.31824e8 −0.627553
\(336\) −1.36672e7 −0.0196559
\(337\) −3.78144e8 −0.538211 −0.269105 0.963111i \(-0.586728\pi\)
−0.269105 + 0.963111i \(0.586728\pi\)
\(338\) 1.42803e9 2.01153
\(339\) −2.04999e7 −0.0285794
\(340\) 4.33676e8 0.598397
\(341\) 4.84651e8 0.661895
\(342\) −1.22724e9 −1.65897
\(343\) 1.48835e8 0.199148
\(344\) −4.16306e8 −0.551389
\(345\) −1.40002e7 −0.0183555
\(346\) −1.34509e8 −0.174576
\(347\) −3.18524e8 −0.409250 −0.204625 0.978840i \(-0.565597\pi\)
−0.204625 + 0.978840i \(0.565597\pi\)
\(348\) −1.70750e8 −0.217186
\(349\) 9.98844e8 1.25779 0.628896 0.777489i \(-0.283507\pi\)
0.628896 + 0.777489i \(0.283507\pi\)
\(350\) 1.27128e8 0.158491
\(351\) 1.06597e8 0.131574
\(352\) 2.97565e9 3.63649
\(353\) 1.22295e8 0.147978 0.0739890 0.997259i \(-0.476427\pi\)
0.0739890 + 0.997259i \(0.476427\pi\)
\(354\) −6.94588e7 −0.0832177
\(355\) −4.01168e7 −0.0475912
\(356\) −4.82674e8 −0.566994
\(357\) −1.91015e6 −0.00222192
\(358\) −1.12389e9 −1.29459
\(359\) −1.04011e9 −1.18645 −0.593225 0.805037i \(-0.702146\pi\)
−0.593225 + 0.805037i \(0.702146\pi\)
\(360\) −1.40039e9 −1.58194
\(361\) −2.51162e8 −0.280982
\(362\) −4.63583e7 −0.0513627
\(363\) −1.70008e7 −0.0186550
\(364\) 3.72774e8 0.405127
\(365\) 3.43226e8 0.369450
\(366\) 1.53340e8 0.163482
\(367\) 3.08148e8 0.325408 0.162704 0.986675i \(-0.447978\pi\)
0.162704 + 0.986675i \(0.447978\pi\)
\(368\) −3.67191e9 −3.84082
\(369\) 5.95831e8 0.617349
\(370\) −2.01958e8 −0.207279
\(371\) −1.28145e8 −0.130285
\(372\) 1.11947e8 0.112748
\(373\) −1.52740e9 −1.52395 −0.761977 0.647604i \(-0.775771\pi\)
−0.761977 + 0.647604i \(0.775771\pi\)
\(374\) 7.35329e8 0.726827
\(375\) −3.74513e7 −0.0366740
\(376\) −5.89339e8 −0.571752
\(377\) 2.44400e9 2.34912
\(378\) 1.90459e7 0.0181376
\(379\) −6.94059e8 −0.654876 −0.327438 0.944873i \(-0.606185\pi\)
−0.327438 + 0.944873i \(0.606185\pi\)
\(380\) 1.13107e9 1.05742
\(381\) 6.36347e7 0.0589463
\(382\) 3.95093e9 3.62641
\(383\) 4.76633e8 0.433500 0.216750 0.976227i \(-0.430454\pi\)
0.216750 + 0.976227i \(0.430454\pi\)
\(384\) 2.60035e8 0.234354
\(385\) −3.79564e7 −0.0338979
\(386\) 2.81174e9 2.48840
\(387\) 1.73510e8 0.152172
\(388\) 4.71691e9 4.09964
\(389\) 7.93506e8 0.683482 0.341741 0.939794i \(-0.388984\pi\)
0.341741 + 0.939794i \(0.388984\pi\)
\(390\) −6.63221e7 −0.0566151
\(391\) −5.13191e8 −0.434171
\(392\) −4.26896e9 −3.57949
\(393\) 1.04028e8 0.0864527
\(394\) −2.83521e9 −2.33533
\(395\) 4.11437e8 0.335903
\(396\) −2.70940e9 −2.19250
\(397\) 1.68006e9 1.34759 0.673795 0.738918i \(-0.264663\pi\)
0.673795 + 0.738918i \(0.264663\pi\)
\(398\) −2.88326e9 −2.29241
\(399\) −4.98186e6 −0.00392632
\(400\) −4.38900e9 −3.42891
\(401\) −7.52355e8 −0.582663 −0.291331 0.956622i \(-0.594098\pi\)
−0.291331 + 0.956622i \(0.594098\pi\)
\(402\) −1.69124e8 −0.129841
\(403\) −1.60233e9 −1.21951
\(404\) 1.12824e8 0.0851272
\(405\) 5.82403e8 0.435643
\(406\) 4.36674e8 0.323829
\(407\) −2.53355e8 −0.186272
\(408\) 1.10130e8 0.0802777
\(409\) −4.62613e8 −0.334338 −0.167169 0.985928i \(-0.553463\pi\)
−0.167169 + 0.985928i \(0.553463\pi\)
\(410\) −7.42218e8 −0.531848
\(411\) 1.27176e7 0.00903560
\(412\) 7.27407e9 5.12433
\(413\) 1.31424e8 0.0918017
\(414\) 2.55575e9 1.77018
\(415\) 8.70504e8 0.597864
\(416\) −9.83795e9 −6.70005
\(417\) −1.10897e8 −0.0748935
\(418\) 1.91781e9 1.28437
\(419\) 1.30611e9 0.867420 0.433710 0.901053i \(-0.357204\pi\)
0.433710 + 0.901053i \(0.357204\pi\)
\(420\) −8.76731e6 −0.00577423
\(421\) −2.16516e8 −0.141417 −0.0707087 0.997497i \(-0.522526\pi\)
−0.0707087 + 0.997497i \(0.522526\pi\)
\(422\) −4.11899e9 −2.66807
\(423\) 2.45627e8 0.157792
\(424\) 7.38825e9 4.70718
\(425\) −6.13413e8 −0.387607
\(426\) −1.57117e7 −0.00984668
\(427\) −2.90137e8 −0.180346
\(428\) 6.45545e8 0.397991
\(429\) −8.32005e7 −0.0508775
\(430\) −2.16138e8 −0.131097
\(431\) 2.46067e9 1.48041 0.740207 0.672379i \(-0.234727\pi\)
0.740207 + 0.672379i \(0.234727\pi\)
\(432\) −6.57544e8 −0.392402
\(433\) −1.42408e9 −0.843001 −0.421500 0.906828i \(-0.638496\pi\)
−0.421500 + 0.906828i \(0.638496\pi\)
\(434\) −2.86291e8 −0.168110
\(435\) −5.74807e7 −0.0334818
\(436\) 2.59215e9 1.49781
\(437\) −1.33845e9 −0.767218
\(438\) 1.34424e8 0.0764396
\(439\) −1.86431e9 −1.05170 −0.525851 0.850577i \(-0.676253\pi\)
−0.525851 + 0.850577i \(0.676253\pi\)
\(440\) 2.18839e9 1.22473
\(441\) 1.77923e9 0.987866
\(442\) −2.43111e9 −1.33914
\(443\) −4.59781e8 −0.251269 −0.125634 0.992077i \(-0.540097\pi\)
−0.125634 + 0.992077i \(0.540097\pi\)
\(444\) −5.85209e7 −0.0317300
\(445\) −1.62486e8 −0.0874089
\(446\) 3.50278e8 0.186957
\(447\) −1.49557e7 −0.00792007
\(448\) −9.49277e8 −0.498792
\(449\) 1.81677e9 0.947192 0.473596 0.880742i \(-0.342956\pi\)
0.473596 + 0.880742i \(0.342956\pi\)
\(450\) 3.05487e9 1.58033
\(451\) −9.31106e8 −0.477948
\(452\) −3.44903e9 −1.75676
\(453\) −1.50889e8 −0.0762629
\(454\) 7.49103e8 0.375704
\(455\) 1.25489e8 0.0624550
\(456\) 2.87230e8 0.141858
\(457\) −1.28773e9 −0.631131 −0.315566 0.948904i \(-0.602194\pi\)
−0.315566 + 0.948904i \(0.602194\pi\)
\(458\) 5.76930e9 2.80604
\(459\) −9.18992e7 −0.0443575
\(460\) −2.35548e9 −1.12831
\(461\) −1.09265e8 −0.0519430 −0.0259715 0.999663i \(-0.508268\pi\)
−0.0259715 + 0.999663i \(0.508268\pi\)
\(462\) −1.48656e7 −0.00701351
\(463\) −1.73669e9 −0.813185 −0.406592 0.913610i \(-0.633283\pi\)
−0.406592 + 0.913610i \(0.633283\pi\)
\(464\) −1.50758e10 −7.00595
\(465\) 3.76853e7 0.0173815
\(466\) 3.21825e9 1.47323
\(467\) −2.48860e9 −1.13070 −0.565349 0.824852i \(-0.691259\pi\)
−0.565349 + 0.824852i \(0.691259\pi\)
\(468\) 8.95767e9 4.03957
\(469\) 3.20003e8 0.143235
\(470\) −3.05974e8 −0.135938
\(471\) −1.32036e8 −0.0582262
\(472\) −7.57731e9 −3.31679
\(473\) −2.71144e8 −0.117811
\(474\) 1.61139e8 0.0694986
\(475\) −1.59984e9 −0.684935
\(476\) −3.21375e8 −0.136580
\(477\) −3.07930e9 −1.29909
\(478\) −3.44478e9 −1.44266
\(479\) 8.40928e8 0.349611 0.174805 0.984603i \(-0.444070\pi\)
0.174805 + 0.984603i \(0.444070\pi\)
\(480\) 2.31380e8 0.0954951
\(481\) 8.37628e8 0.343197
\(482\) 7.16483e9 2.91435
\(483\) 1.03748e7 0.00418953
\(484\) −2.86032e9 −1.14671
\(485\) 1.58788e9 0.632008
\(486\) 6.86749e8 0.271376
\(487\) −1.41392e9 −0.554718 −0.277359 0.960766i \(-0.589459\pi\)
−0.277359 + 0.960766i \(0.589459\pi\)
\(488\) 1.67279e10 6.51588
\(489\) −2.79096e6 −0.00107937
\(490\) −2.21637e9 −0.851050
\(491\) −2.99438e9 −1.14162 −0.570810 0.821082i \(-0.693371\pi\)
−0.570810 + 0.821082i \(0.693371\pi\)
\(492\) −2.15070e8 −0.0814147
\(493\) −2.10701e9 −0.791959
\(494\) −6.34057e9 −2.36638
\(495\) −9.12083e8 −0.338000
\(496\) 9.88395e9 3.63701
\(497\) 2.97285e7 0.0108624
\(498\) 3.40932e8 0.123699
\(499\) −3.88719e9 −1.40050 −0.700251 0.713897i \(-0.746929\pi\)
−0.700251 + 0.713897i \(0.746929\pi\)
\(500\) −6.30104e9 −2.25433
\(501\) −1.78709e8 −0.0634913
\(502\) −5.22200e9 −1.84236
\(503\) −2.25318e9 −0.789420 −0.394710 0.918806i \(-0.629155\pi\)
−0.394710 + 0.918806i \(0.629155\pi\)
\(504\) 1.03775e9 0.361067
\(505\) 3.79809e7 0.0131234
\(506\) −3.99388e9 −1.37047
\(507\) 1.39299e8 0.0474700
\(508\) 1.07063e10 3.62340
\(509\) 2.23850e9 0.752393 0.376196 0.926540i \(-0.377232\pi\)
0.376196 + 0.926540i \(0.377232\pi\)
\(510\) 5.71774e7 0.0190866
\(511\) −2.54347e8 −0.0843244
\(512\) 1.40717e10 4.63341
\(513\) −2.39682e8 −0.0783837
\(514\) −9.43165e8 −0.306349
\(515\) 2.44872e9 0.789976
\(516\) −6.26298e7 −0.0200681
\(517\) −3.83841e8 −0.122162
\(518\) 1.49661e8 0.0473101
\(519\) −1.31208e7 −0.00411980
\(520\) −7.23513e9 −2.25650
\(521\) 3.53193e7 0.0109416 0.00547079 0.999985i \(-0.498259\pi\)
0.00547079 + 0.999985i \(0.498259\pi\)
\(522\) 1.04932e10 3.22894
\(523\) 1.93242e9 0.590671 0.295335 0.955394i \(-0.404569\pi\)
0.295335 + 0.955394i \(0.404569\pi\)
\(524\) 1.75024e10 5.31420
\(525\) 1.24009e7 0.00374021
\(526\) 2.31852e9 0.694642
\(527\) 1.38139e9 0.411131
\(528\) 5.13222e8 0.151735
\(529\) −6.17469e8 −0.181351
\(530\) 3.83584e9 1.11917
\(531\) 3.15810e9 0.915365
\(532\) −8.38178e8 −0.241349
\(533\) 3.07837e9 0.880595
\(534\) −6.36375e7 −0.0180850
\(535\) 2.17314e8 0.0613550
\(536\) −1.84498e10 −5.17507
\(537\) −1.09631e8 −0.0305508
\(538\) −4.13419e9 −1.14460
\(539\) −2.78041e9 −0.764801
\(540\) −4.21805e8 −0.115275
\(541\) 4.16649e9 1.13130 0.565652 0.824644i \(-0.308625\pi\)
0.565652 + 0.824644i \(0.308625\pi\)
\(542\) 7.48079e9 2.01813
\(543\) −4.52208e6 −0.00121210
\(544\) 8.48147e9 2.25878
\(545\) 8.72612e8 0.230905
\(546\) 4.91479e7 0.0129220
\(547\) −2.80094e9 −0.731725 −0.365862 0.930669i \(-0.619226\pi\)
−0.365862 + 0.930669i \(0.619226\pi\)
\(548\) 2.13968e9 0.555414
\(549\) −6.97193e9 −1.79825
\(550\) −4.77385e9 −1.22349
\(551\) −5.49530e9 −1.39946
\(552\) −5.98162e8 −0.151367
\(553\) −3.04894e8 −0.0766675
\(554\) −9.38068e9 −2.34396
\(555\) −1.97003e7 −0.00489155
\(556\) −1.86580e10 −4.60367
\(557\) 4.87182e9 1.19453 0.597266 0.802043i \(-0.296254\pi\)
0.597266 + 0.802043i \(0.296254\pi\)
\(558\) −6.87951e9 −1.67625
\(559\) 8.96441e8 0.217060
\(560\) −7.74081e8 −0.186264
\(561\) 7.17286e7 0.0171523
\(562\) 7.57716e9 1.80065
\(563\) 6.20948e9 1.46648 0.733240 0.679970i \(-0.238007\pi\)
0.733240 + 0.679970i \(0.238007\pi\)
\(564\) −8.86612e7 −0.0208093
\(565\) −1.16107e9 −0.270825
\(566\) −4.29857e9 −0.996475
\(567\) −4.31588e8 −0.0994327
\(568\) −1.71400e9 −0.392457
\(569\) −5.71470e9 −1.30047 −0.650235 0.759733i \(-0.725330\pi\)
−0.650235 + 0.759733i \(0.725330\pi\)
\(570\) 1.49125e8 0.0337277
\(571\) 2.96607e9 0.666737 0.333368 0.942797i \(-0.391815\pi\)
0.333368 + 0.942797i \(0.391815\pi\)
\(572\) −1.39982e10 −3.12741
\(573\) 3.85398e8 0.0855793
\(574\) 5.50019e8 0.121391
\(575\) 3.33170e9 0.730851
\(576\) −2.28109e10 −4.97352
\(577\) 5.24837e9 1.13739 0.568694 0.822549i \(-0.307449\pi\)
0.568694 + 0.822549i \(0.307449\pi\)
\(578\) −7.00631e9 −1.50918
\(579\) 2.74275e8 0.0587235
\(580\) −9.67090e9 −2.05811
\(581\) −6.45085e8 −0.136458
\(582\) 6.21894e8 0.130763
\(583\) 4.81203e9 1.00575
\(584\) 1.46644e10 3.04664
\(585\) 3.01548e9 0.622747
\(586\) 1.02912e10 2.11264
\(587\) 1.22772e9 0.250534 0.125267 0.992123i \(-0.460021\pi\)
0.125267 + 0.992123i \(0.460021\pi\)
\(588\) −6.42230e8 −0.130278
\(589\) 3.60282e9 0.726505
\(590\) −3.93400e9 −0.788591
\(591\) −2.76564e8 −0.0551111
\(592\) −5.16690e9 −1.02354
\(593\) 6.05113e9 1.19164 0.595820 0.803118i \(-0.296827\pi\)
0.595820 + 0.803118i \(0.296827\pi\)
\(594\) −7.15200e8 −0.140015
\(595\) −1.08187e8 −0.0210554
\(596\) −2.51623e9 −0.486843
\(597\) −2.81251e8 −0.0540983
\(598\) 1.32044e10 2.52501
\(599\) 5.84902e9 1.11196 0.555980 0.831196i \(-0.312343\pi\)
0.555980 + 0.831196i \(0.312343\pi\)
\(600\) −7.14978e8 −0.135134
\(601\) −8.45352e9 −1.58846 −0.794231 0.607616i \(-0.792126\pi\)
−0.794231 + 0.607616i \(0.792126\pi\)
\(602\) 1.60169e8 0.0299220
\(603\) 7.68959e9 1.42821
\(604\) −2.53865e10 −4.68784
\(605\) −9.62888e8 −0.176779
\(606\) 1.48752e7 0.00271524
\(607\) −9.03739e8 −0.164015 −0.0820074 0.996632i \(-0.526133\pi\)
−0.0820074 + 0.996632i \(0.526133\pi\)
\(608\) 2.21205e10 3.99147
\(609\) 4.25959e7 0.00764201
\(610\) 8.68484e9 1.54920
\(611\) 1.26904e9 0.225076
\(612\) −7.72256e9 −1.36186
\(613\) −3.64313e9 −0.638797 −0.319399 0.947620i \(-0.603481\pi\)
−0.319399 + 0.947620i \(0.603481\pi\)
\(614\) 2.44757e9 0.426723
\(615\) −7.24006e7 −0.0125510
\(616\) −1.62170e9 −0.279536
\(617\) 8.10529e9 1.38922 0.694609 0.719387i \(-0.255577\pi\)
0.694609 + 0.719387i \(0.255577\pi\)
\(618\) 9.59039e8 0.163447
\(619\) −7.22020e9 −1.22358 −0.611789 0.791021i \(-0.709550\pi\)
−0.611789 + 0.791021i \(0.709550\pi\)
\(620\) 6.34042e9 1.06843
\(621\) 4.99143e8 0.0836382
\(622\) −3.67712e9 −0.612691
\(623\) 1.20410e8 0.0199505
\(624\) −1.69679e9 −0.279564
\(625\) 2.80898e9 0.460224
\(626\) −2.28427e10 −3.72166
\(627\) 1.87075e8 0.0303096
\(628\) −2.22145e10 −3.57914
\(629\) −7.22134e8 −0.115702
\(630\) 5.38782e8 0.0858462
\(631\) 4.23517e8 0.0671070 0.0335535 0.999437i \(-0.489318\pi\)
0.0335535 + 0.999437i \(0.489318\pi\)
\(632\) 1.75788e10 2.76999
\(633\) −4.01793e8 −0.0629635
\(634\) 1.41984e10 2.21272
\(635\) 3.60413e9 0.558590
\(636\) 1.11150e9 0.171321
\(637\) 9.19245e9 1.40911
\(638\) −1.63977e10 −2.49983
\(639\) 7.14368e8 0.108310
\(640\) 1.47278e10 2.22080
\(641\) 3.03264e9 0.454797 0.227398 0.973802i \(-0.426978\pi\)
0.227398 + 0.973802i \(0.426978\pi\)
\(642\) 8.51110e7 0.0126944
\(643\) −2.78734e9 −0.413478 −0.206739 0.978396i \(-0.566285\pi\)
−0.206739 + 0.978396i \(0.566285\pi\)
\(644\) 1.74552e9 0.257528
\(645\) −2.10835e7 −0.00309374
\(646\) 5.46631e9 0.797776
\(647\) −1.73353e9 −0.251632 −0.125816 0.992054i \(-0.540155\pi\)
−0.125816 + 0.992054i \(0.540155\pi\)
\(648\) 2.48834e10 3.59250
\(649\) −4.93516e9 −0.708672
\(650\) 1.57830e10 2.25421
\(651\) −2.79266e7 −0.00396721
\(652\) −4.69568e8 −0.0663486
\(653\) −9.51546e9 −1.33731 −0.668657 0.743571i \(-0.733131\pi\)
−0.668657 + 0.743571i \(0.733131\pi\)
\(654\) 3.41758e8 0.0477745
\(655\) 5.89195e9 0.819247
\(656\) −1.89889e10 −2.62626
\(657\) −6.11190e9 −0.840809
\(658\) 2.26741e8 0.0310270
\(659\) 7.55803e9 1.02875 0.514375 0.857565i \(-0.328024\pi\)
0.514375 + 0.857565i \(0.328024\pi\)
\(660\) 3.29225e8 0.0445747
\(661\) −1.39999e9 −0.188547 −0.0942733 0.995546i \(-0.530053\pi\)
−0.0942733 + 0.995546i \(0.530053\pi\)
\(662\) −9.53475e9 −1.27734
\(663\) −2.37145e8 −0.0316022
\(664\) 3.71926e10 4.93024
\(665\) −2.82162e8 −0.0372068
\(666\) 3.59631e9 0.471734
\(667\) 1.14441e10 1.49328
\(668\) −3.00671e10 −3.90278
\(669\) 3.41683e7 0.00441197
\(670\) −9.57881e9 −1.23041
\(671\) 1.08950e10 1.39220
\(672\) −1.71464e8 −0.0217961
\(673\) −5.04985e8 −0.0638595 −0.0319298 0.999490i \(-0.510165\pi\)
−0.0319298 + 0.999490i \(0.510165\pi\)
\(674\) −8.38806e9 −1.05524
\(675\) 5.96621e8 0.0746682
\(676\) 2.34365e10 2.91796
\(677\) −8.77580e9 −1.08699 −0.543496 0.839412i \(-0.682900\pi\)
−0.543496 + 0.839412i \(0.682900\pi\)
\(678\) −4.54732e8 −0.0560340
\(679\) −1.17670e9 −0.144252
\(680\) 6.23753e9 0.760732
\(681\) 7.30723e7 0.00886620
\(682\) 1.07506e10 1.29774
\(683\) 5.66666e9 0.680541 0.340271 0.940328i \(-0.389481\pi\)
0.340271 + 0.940328i \(0.389481\pi\)
\(684\) −2.01412e10 −2.40652
\(685\) 7.20295e8 0.0856236
\(686\) 3.30148e9 0.390458
\(687\) 5.62774e8 0.0662194
\(688\) −5.52969e9 −0.647353
\(689\) −1.59093e10 −1.85303
\(690\) −3.10555e8 −0.0359887
\(691\) −7.72356e9 −0.890522 −0.445261 0.895401i \(-0.646889\pi\)
−0.445261 + 0.895401i \(0.646889\pi\)
\(692\) −2.20753e9 −0.253242
\(693\) 6.75897e8 0.0771462
\(694\) −7.06555e9 −0.802394
\(695\) −6.28098e9 −0.709709
\(696\) −2.45588e9 −0.276105
\(697\) −2.65392e9 −0.296874
\(698\) 2.21566e10 2.46609
\(699\) 3.13929e8 0.0347665
\(700\) 2.08641e9 0.229909
\(701\) −9.02472e9 −0.989511 −0.494755 0.869032i \(-0.664742\pi\)
−0.494755 + 0.869032i \(0.664742\pi\)
\(702\) 2.36456e9 0.257970
\(703\) −1.88340e9 −0.204455
\(704\) 3.56466e10 3.85047
\(705\) −2.98466e7 −0.00320799
\(706\) 2.71277e9 0.290133
\(707\) −2.81456e7 −0.00299532
\(708\) −1.13994e9 −0.120717
\(709\) 1.01828e10 1.07301 0.536505 0.843897i \(-0.319744\pi\)
0.536505 + 0.843897i \(0.319744\pi\)
\(710\) −8.89878e8 −0.0933096
\(711\) −7.32654e9 −0.764461
\(712\) −6.94226e9 −0.720810
\(713\) −7.50294e9 −0.775207
\(714\) −4.23712e7 −0.00435640
\(715\) −4.71230e9 −0.482127
\(716\) −1.84450e10 −1.87794
\(717\) −3.36026e8 −0.0340452
\(718\) −2.30720e10 −2.32621
\(719\) −3.88344e9 −0.389642 −0.194821 0.980839i \(-0.562413\pi\)
−0.194821 + 0.980839i \(0.562413\pi\)
\(720\) −1.86010e10 −1.85726
\(721\) −1.81462e9 −0.180307
\(722\) −5.57133e9 −0.550907
\(723\) 6.98903e8 0.0687753
\(724\) −7.60823e8 −0.0745073
\(725\) 1.36790e10 1.33313
\(726\) −3.77114e8 −0.0365759
\(727\) 3.21178e9 0.310009 0.155005 0.987914i \(-0.450461\pi\)
0.155005 + 0.987914i \(0.450461\pi\)
\(728\) 5.36158e9 0.515030
\(729\) −1.03262e10 −0.987178
\(730\) 7.61351e9 0.724360
\(731\) −7.72837e8 −0.0731775
\(732\) 2.51658e9 0.237149
\(733\) 2.53300e9 0.237559 0.118779 0.992921i \(-0.462102\pi\)
0.118779 + 0.992921i \(0.462102\pi\)
\(734\) 6.83540e9 0.638010
\(735\) −2.16198e8 −0.0200838
\(736\) −4.60664e10 −4.25904
\(737\) −1.20165e10 −1.10571
\(738\) 1.32168e10 1.21040
\(739\) 8.64031e9 0.787542 0.393771 0.919209i \(-0.371170\pi\)
0.393771 + 0.919209i \(0.371170\pi\)
\(740\) −3.31450e9 −0.300681
\(741\) −6.18499e8 −0.0558439
\(742\) −2.84254e9 −0.255443
\(743\) −7.96383e9 −0.712297 −0.356148 0.934429i \(-0.615910\pi\)
−0.356148 + 0.934429i \(0.615910\pi\)
\(744\) 1.61012e9 0.143335
\(745\) −8.47057e8 −0.0750526
\(746\) −3.38811e10 −2.98794
\(747\) −1.55012e10 −1.36064
\(748\) 1.20681e10 1.05434
\(749\) −1.61040e8 −0.0140039
\(750\) −8.30752e8 −0.0719047
\(751\) 3.65497e9 0.314879 0.157439 0.987529i \(-0.449676\pi\)
0.157439 + 0.987529i \(0.449676\pi\)
\(752\) −7.82804e9 −0.671260
\(753\) −5.09387e8 −0.0434776
\(754\) 5.42132e10 4.60580
\(755\) −8.54602e9 −0.722686
\(756\) 3.12578e8 0.0263107
\(757\) 1.20006e9 0.100546 0.0502732 0.998736i \(-0.483991\pi\)
0.0502732 + 0.998736i \(0.483991\pi\)
\(758\) −1.53958e10 −1.28398
\(759\) −3.89588e8 −0.0323414
\(760\) 1.62681e10 1.34428
\(761\) 2.32910e9 0.191576 0.0957880 0.995402i \(-0.469463\pi\)
0.0957880 + 0.995402i \(0.469463\pi\)
\(762\) 1.41156e9 0.115573
\(763\) −6.46647e8 −0.0527025
\(764\) 6.48418e10 5.26052
\(765\) −2.59970e9 −0.209946
\(766\) 1.05728e10 0.849940
\(767\) 1.63164e10 1.30569
\(768\) 2.87314e9 0.228872
\(769\) −4.56990e9 −0.362381 −0.181190 0.983448i \(-0.557995\pi\)
−0.181190 + 0.983448i \(0.557995\pi\)
\(770\) −8.41956e8 −0.0664618
\(771\) −9.20022e7 −0.00722950
\(772\) 4.61458e10 3.60970
\(773\) 1.34361e10 1.04627 0.523135 0.852250i \(-0.324762\pi\)
0.523135 + 0.852250i \(0.324762\pi\)
\(774\) 3.84882e9 0.298356
\(775\) −8.96819e9 −0.692068
\(776\) 6.78429e10 5.21181
\(777\) 1.45988e7 0.00111646
\(778\) 1.76017e10 1.34007
\(779\) −6.92169e9 −0.524603
\(780\) −1.08847e9 −0.0821265
\(781\) −1.11634e9 −0.0838532
\(782\) −1.13837e10 −0.851255
\(783\) 2.04934e9 0.152562
\(784\) −5.67036e10 −4.20247
\(785\) −7.47823e9 −0.551766
\(786\) 2.30758e9 0.169503
\(787\) 2.05668e10 1.50403 0.752014 0.659147i \(-0.229082\pi\)
0.752014 + 0.659147i \(0.229082\pi\)
\(788\) −4.65308e10 −3.38765
\(789\) 2.26163e8 0.0163928
\(790\) 9.12657e9 0.658586
\(791\) 8.60408e8 0.0618140
\(792\) −3.89691e10 −2.78729
\(793\) −3.60207e10 −2.56505
\(794\) 3.72674e10 2.64215
\(795\) 3.74172e8 0.0264111
\(796\) −4.73194e10 −3.32540
\(797\) 1.49135e10 1.04346 0.521729 0.853111i \(-0.325287\pi\)
0.521729 + 0.853111i \(0.325287\pi\)
\(798\) −1.10508e8 −0.00769813
\(799\) −1.09406e9 −0.0758799
\(800\) −5.50628e10 −3.80227
\(801\) 2.89342e9 0.198929
\(802\) −1.66889e10 −1.14240
\(803\) 9.55107e9 0.650950
\(804\) −2.77563e9 −0.188350
\(805\) 5.87607e8 0.0397010
\(806\) −3.55432e10 −2.39102
\(807\) −4.03275e8 −0.0270112
\(808\) 1.62275e9 0.108221
\(809\) −2.29446e10 −1.52356 −0.761781 0.647835i \(-0.775675\pi\)
−0.761781 + 0.647835i \(0.775675\pi\)
\(810\) 1.29190e10 0.854143
\(811\) 4.15707e9 0.273662 0.136831 0.990594i \(-0.456308\pi\)
0.136831 + 0.990594i \(0.456308\pi\)
\(812\) 7.16660e9 0.469750
\(813\) 7.29723e8 0.0476257
\(814\) −5.61996e9 −0.365215
\(815\) −1.58074e8 −0.0102284
\(816\) 1.46283e9 0.0942493
\(817\) −2.01564e9 −0.129311
\(818\) −1.02618e10 −0.655520
\(819\) −2.23462e9 −0.142138
\(820\) −1.21811e10 −0.771506
\(821\) 1.06838e10 0.673790 0.336895 0.941542i \(-0.390623\pi\)
0.336895 + 0.941542i \(0.390623\pi\)
\(822\) 2.82103e8 0.0177156
\(823\) −1.14304e10 −0.714766 −0.357383 0.933958i \(-0.616331\pi\)
−0.357383 + 0.933958i \(0.616331\pi\)
\(824\) 1.04622e11 6.51447
\(825\) −4.65671e8 −0.0288729
\(826\) 2.91528e9 0.179991
\(827\) −1.17022e10 −0.719448 −0.359724 0.933059i \(-0.617129\pi\)
−0.359724 + 0.933059i \(0.617129\pi\)
\(828\) 4.19445e10 2.56784
\(829\) −1.53161e10 −0.933699 −0.466850 0.884337i \(-0.654611\pi\)
−0.466850 + 0.884337i \(0.654611\pi\)
\(830\) 1.93097e10 1.17220
\(831\) −9.15051e8 −0.0553148
\(832\) −1.17853e11 −7.09430
\(833\) −7.92497e9 −0.475051
\(834\) −2.45994e9 −0.146840
\(835\) −1.01217e10 −0.601659
\(836\) 3.14747e10 1.86312
\(837\) −1.34358e9 −0.0791999
\(838\) 2.89723e10 1.70070
\(839\) 1.05214e10 0.615043 0.307521 0.951541i \(-0.400500\pi\)
0.307521 + 0.951541i \(0.400500\pi\)
\(840\) −1.26100e8 −0.00734068
\(841\) 2.97361e10 1.72385
\(842\) −4.80280e9 −0.277270
\(843\) 7.39124e8 0.0424933
\(844\) −6.76001e10 −3.87034
\(845\) 7.88958e9 0.449837
\(846\) 5.44854e9 0.309374
\(847\) 7.13546e8 0.0403487
\(848\) 9.81363e10 5.52642
\(849\) −4.19310e8 −0.0235157
\(850\) −1.36068e10 −0.759960
\(851\) 3.92221e9 0.218161
\(852\) −2.57857e8 −0.0142837
\(853\) −2.95826e10 −1.63198 −0.815989 0.578068i \(-0.803807\pi\)
−0.815989 + 0.578068i \(0.803807\pi\)
\(854\) −6.43588e9 −0.353594
\(855\) −6.78028e9 −0.370993
\(856\) 9.28482e9 0.505959
\(857\) 6.19614e9 0.336270 0.168135 0.985764i \(-0.446225\pi\)
0.168135 + 0.985764i \(0.446225\pi\)
\(858\) −1.84557e9 −0.0997528
\(859\) 2.78077e10 1.49689 0.748443 0.663199i \(-0.230802\pi\)
0.748443 + 0.663199i \(0.230802\pi\)
\(860\) −3.54722e9 −0.190171
\(861\) 5.36523e7 0.00286469
\(862\) 5.45831e10 2.90257
\(863\) 3.15999e10 1.67359 0.836793 0.547519i \(-0.184428\pi\)
0.836793 + 0.547519i \(0.184428\pi\)
\(864\) −8.24930e9 −0.435130
\(865\) −7.43137e8 −0.0390402
\(866\) −3.15893e10 −1.65283
\(867\) −6.83440e8 −0.0356150
\(868\) −4.69855e9 −0.243863
\(869\) 1.14492e10 0.591842
\(870\) −1.27505e9 −0.0656461
\(871\) 3.97284e10 2.03722
\(872\) 3.72826e10 1.90414
\(873\) −2.82758e10 −1.43835
\(874\) −2.96899e10 −1.50424
\(875\) 1.57188e9 0.0793217
\(876\) 2.20614e9 0.110884
\(877\) 2.48099e10 1.24201 0.621006 0.783806i \(-0.286724\pi\)
0.621006 + 0.783806i \(0.286724\pi\)
\(878\) −4.13545e10 −2.06201
\(879\) 1.00387e9 0.0498560
\(880\) 2.90678e10 1.43788
\(881\) 2.37579e9 0.117056 0.0585279 0.998286i \(-0.481359\pi\)
0.0585279 + 0.998286i \(0.481359\pi\)
\(882\) 3.94673e10 1.93686
\(883\) −1.16100e10 −0.567505 −0.283752 0.958898i \(-0.591579\pi\)
−0.283752 + 0.958898i \(0.591579\pi\)
\(884\) −3.98988e10 −1.94257
\(885\) −3.83747e8 −0.0186099
\(886\) −1.01990e10 −0.492649
\(887\) 1.53556e10 0.738813 0.369407 0.929268i \(-0.379561\pi\)
0.369407 + 0.929268i \(0.379561\pi\)
\(888\) −8.41701e8 −0.0403378
\(889\) −2.67084e9 −0.127494
\(890\) −3.60429e9 −0.171378
\(891\) 1.62067e10 0.767580
\(892\) 5.74869e9 0.271201
\(893\) −2.85342e9 −0.134086
\(894\) −3.31749e8 −0.0155285
\(895\) −6.20926e9 −0.289507
\(896\) −1.09140e10 −0.506883
\(897\) 1.28804e9 0.0595874
\(898\) 4.03000e10 1.85711
\(899\) −3.08049e10 −1.41404
\(900\) 5.01359e10 2.29245
\(901\) 1.37157e10 0.624712
\(902\) −2.06540e10 −0.937088
\(903\) 1.56239e7 0.000706126 0
\(904\) −4.96071e10 −2.23334
\(905\) −2.56121e8 −0.0114862
\(906\) −3.34704e9 −0.149525
\(907\) 3.31807e9 0.147659 0.0738295 0.997271i \(-0.476478\pi\)
0.0738295 + 0.997271i \(0.476478\pi\)
\(908\) 1.22941e10 0.545001
\(909\) −6.76333e8 −0.0298667
\(910\) 2.78363e9 0.122452
\(911\) −3.21961e9 −0.141087 −0.0705437 0.997509i \(-0.522473\pi\)
−0.0705437 + 0.997509i \(0.522473\pi\)
\(912\) 3.81521e9 0.166547
\(913\) 2.42238e10 1.05340
\(914\) −2.85648e10 −1.23743
\(915\) 8.47174e8 0.0365594
\(916\) 9.46846e10 4.07047
\(917\) −4.36622e9 −0.186988
\(918\) −2.03853e9 −0.0869695
\(919\) 7.57125e9 0.321783 0.160892 0.986972i \(-0.448563\pi\)
0.160892 + 0.986972i \(0.448563\pi\)
\(920\) −3.38787e10 −1.43439
\(921\) 2.38752e8 0.0100702
\(922\) −2.42373e9 −0.101842
\(923\) 3.69080e9 0.154495
\(924\) −2.43971e8 −0.0101739
\(925\) 4.68818e9 0.194764
\(926\) −3.85236e10 −1.59437
\(927\) −4.36049e10 −1.79786
\(928\) −1.89135e11 −7.76881
\(929\) −2.89017e10 −1.18268 −0.591342 0.806421i \(-0.701402\pi\)
−0.591342 + 0.806421i \(0.701402\pi\)
\(930\) 8.35943e8 0.0340790
\(931\) −2.06691e10 −0.839457
\(932\) 5.28173e10 2.13708
\(933\) −3.58690e8 −0.0144588
\(934\) −5.52027e10 −2.21690
\(935\) 4.06255e9 0.162539
\(936\) 1.28837e11 5.13543
\(937\) 3.91577e10 1.55499 0.777496 0.628888i \(-0.216489\pi\)
0.777496 + 0.628888i \(0.216489\pi\)
\(938\) 7.09836e9 0.280833
\(939\) −2.22822e9 −0.0878271
\(940\) −5.02158e9 −0.197194
\(941\) −2.41576e10 −0.945127 −0.472564 0.881297i \(-0.656671\pi\)
−0.472564 + 0.881297i \(0.656671\pi\)
\(942\) −2.92884e9 −0.114161
\(943\) 1.44145e10 0.559771
\(944\) −1.00647e11 −3.89404
\(945\) 1.05225e8 0.00405610
\(946\) −6.01456e9 −0.230986
\(947\) 3.20511e10 1.22636 0.613179 0.789944i \(-0.289890\pi\)
0.613179 + 0.789944i \(0.289890\pi\)
\(948\) 2.64458e9 0.100816
\(949\) −3.15773e10 −1.19934
\(950\) −3.54880e10 −1.34292
\(951\) 1.38500e9 0.0522178
\(952\) −4.62231e9 −0.173632
\(953\) 4.93663e9 0.184759 0.0923794 0.995724i \(-0.470553\pi\)
0.0923794 + 0.995724i \(0.470553\pi\)
\(954\) −6.83056e10 −2.54705
\(955\) 2.18281e10 0.810970
\(956\) −5.65350e10 −2.09274
\(957\) −1.59953e9 −0.0589932
\(958\) 1.86536e10 0.685463
\(959\) −5.33773e8 −0.0195430
\(960\) 2.77180e9 0.101114
\(961\) −7.31640e9 −0.265929
\(962\) 1.85804e10 0.672888
\(963\) −3.86976e9 −0.139634
\(964\) 1.17588e11 4.22759
\(965\) 1.55344e10 0.556478
\(966\) 2.30136e8 0.00821419
\(967\) 3.10919e10 1.10574 0.552872 0.833266i \(-0.313532\pi\)
0.552872 + 0.833266i \(0.313532\pi\)
\(968\) −4.11397e10 −1.45780
\(969\) 5.33219e8 0.0188266
\(970\) 3.52228e10 1.23915
\(971\) 3.37297e10 1.18235 0.591174 0.806544i \(-0.298665\pi\)
0.591174 + 0.806544i \(0.298665\pi\)
\(972\) 1.12708e10 0.393661
\(973\) 4.65451e9 0.161986
\(974\) −3.13637e10 −1.08761
\(975\) 1.53958e9 0.0531968
\(976\) 2.22193e11 7.64991
\(977\) −2.60034e10 −0.892072 −0.446036 0.895015i \(-0.647165\pi\)
−0.446036 + 0.895015i \(0.647165\pi\)
\(978\) −6.19095e7 −0.00211627
\(979\) −4.52155e9 −0.154010
\(980\) −3.63746e10 −1.23454
\(981\) −1.55388e10 −0.525503
\(982\) −6.64219e10 −2.23831
\(983\) −4.35623e9 −0.146276 −0.0731381 0.997322i \(-0.523301\pi\)
−0.0731381 + 0.997322i \(0.523301\pi\)
\(984\) −3.09334e9 −0.103501
\(985\) −1.56640e10 −0.522247
\(986\) −4.67381e10 −1.55275
\(987\) 2.21178e7 0.000732203 0
\(988\) −1.04060e11 −3.43269
\(989\) 4.19761e9 0.137980
\(990\) −2.02320e10 −0.662698
\(991\) −3.99586e10 −1.30423 −0.652113 0.758122i \(-0.726117\pi\)
−0.652113 + 0.758122i \(0.726117\pi\)
\(992\) 1.24000e11 4.03304
\(993\) −9.30080e8 −0.0301438
\(994\) 6.59442e8 0.0212973
\(995\) −1.59295e10 −0.512649
\(996\) 5.59531e9 0.179439
\(997\) 1.79270e10 0.572894 0.286447 0.958096i \(-0.407526\pi\)
0.286447 + 0.958096i \(0.407526\pi\)
\(998\) −8.62264e10 −2.74589
\(999\) 7.02366e8 0.0222887
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 43.8.a.b.1.13 13
3.2 odd 2 387.8.a.d.1.1 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.13 13 1.1 even 1 trivial
387.8.a.d.1.1 13 3.2 odd 2