Properties

Label 43.8.a.b.1.1
Level 43
Weight 8
Character 43.1
Self dual yes
Analytic conductor 13.433
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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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 1.1
Root \(-21.3781\) 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

\(f(q)\) \(=\) \(q-20.3781 q^{2} -17.6937 q^{3} +287.267 q^{4} +405.085 q^{5} +360.564 q^{6} +52.5965 q^{7} -3245.57 q^{8} -1873.93 q^{9} +O(q^{10})\) \(q-20.3781 q^{2} -17.6937 q^{3} +287.267 q^{4} +405.085 q^{5} +360.564 q^{6} +52.5965 q^{7} -3245.57 q^{8} -1873.93 q^{9} -8254.88 q^{10} +2544.97 q^{11} -5082.81 q^{12} +7138.95 q^{13} -1071.82 q^{14} -7167.45 q^{15} +29368.3 q^{16} -22507.9 q^{17} +38187.2 q^{18} -6396.31 q^{19} +116368. q^{20} -930.624 q^{21} -51861.7 q^{22} +99842.9 q^{23} +57426.0 q^{24} +85969.2 q^{25} -145478. q^{26} +71852.8 q^{27} +15109.2 q^{28} +5608.53 q^{29} +146059. q^{30} -163095. q^{31} -183038. q^{32} -45029.9 q^{33} +458669. q^{34} +21306.1 q^{35} -538320. q^{36} +198188. q^{37} +130345. q^{38} -126314. q^{39} -1.31473e6 q^{40} +742519. q^{41} +18964.4 q^{42} -79507.0 q^{43} +731087. q^{44} -759103. q^{45} -2.03461e6 q^{46} +786601. q^{47} -519633. q^{48} -820777. q^{49} -1.75189e6 q^{50} +398248. q^{51} +2.05079e6 q^{52} +2.08070e6 q^{53} -1.46422e6 q^{54} +1.03093e6 q^{55} -170705. q^{56} +113174. q^{57} -114291. q^{58} +607486. q^{59} -2.05897e6 q^{60} +2.59833e6 q^{61} +3.32357e6 q^{62} -98562.3 q^{63} -29175.0 q^{64} +2.89188e6 q^{65} +917624. q^{66} -755375. q^{67} -6.46580e6 q^{68} -1.76659e6 q^{69} -434177. q^{70} -383737. q^{71} +6.08198e6 q^{72} +2.94854e6 q^{73} -4.03869e6 q^{74} -1.52111e6 q^{75} -1.83745e6 q^{76} +133856. q^{77} +2.57404e6 q^{78} -7.28812e6 q^{79} +1.18967e7 q^{80} +2.82695e6 q^{81} -1.51311e7 q^{82} -1.91613e6 q^{83} -267338. q^{84} -9.11764e6 q^{85} +1.62020e6 q^{86} -99235.4 q^{87} -8.25988e6 q^{88} +1.05744e7 q^{89} +1.54691e7 q^{90} +375483. q^{91} +2.86816e7 q^{92} +2.88575e6 q^{93} -1.60294e7 q^{94} -2.59105e6 q^{95} +3.23861e6 q^{96} +1.18419e7 q^{97} +1.67259e7 q^{98} -4.76911e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −20.3781 −1.80119 −0.900594 0.434662i \(-0.856868\pi\)
−0.900594 + 0.434662i \(0.856868\pi\)
\(3\) −17.6937 −0.378350 −0.189175 0.981943i \(-0.560581\pi\)
−0.189175 + 0.981943i \(0.560581\pi\)
\(4\) 287.267 2.24428
\(5\) 405.085 1.44928 0.724639 0.689129i \(-0.242007\pi\)
0.724639 + 0.689129i \(0.242007\pi\)
\(6\) 360.564 0.681479
\(7\) 52.5965 0.0579580 0.0289790 0.999580i \(-0.490774\pi\)
0.0289790 + 0.999580i \(0.490774\pi\)
\(8\) −3245.57 −2.24117
\(9\) −1873.93 −0.856851
\(10\) −8254.88 −2.61042
\(11\) 2544.97 0.576512 0.288256 0.957553i \(-0.406925\pi\)
0.288256 + 0.957553i \(0.406925\pi\)
\(12\) −5082.81 −0.849121
\(13\) 7138.95 0.901224 0.450612 0.892720i \(-0.351206\pi\)
0.450612 + 0.892720i \(0.351206\pi\)
\(14\) −1071.82 −0.104393
\(15\) −7167.45 −0.548334
\(16\) 29368.3 1.79250
\(17\) −22507.9 −1.11113 −0.555565 0.831473i \(-0.687498\pi\)
−0.555565 + 0.831473i \(0.687498\pi\)
\(18\) 38187.2 1.54335
\(19\) −6396.31 −0.213940 −0.106970 0.994262i \(-0.534115\pi\)
−0.106970 + 0.994262i \(0.534115\pi\)
\(20\) 116368. 3.25258
\(21\) −930.624 −0.0219284
\(22\) −51861.7 −1.03841
\(23\) 99842.9 1.71108 0.855539 0.517739i \(-0.173226\pi\)
0.855539 + 0.517739i \(0.173226\pi\)
\(24\) 57426.0 0.847948
\(25\) 85969.2 1.10041
\(26\) −145478. −1.62327
\(27\) 71852.8 0.702539
\(28\) 15109.2 0.130074
\(29\) 5608.53 0.0427028 0.0213514 0.999772i \(-0.493203\pi\)
0.0213514 + 0.999772i \(0.493203\pi\)
\(30\) 146059. 0.987652
\(31\) −163095. −0.983274 −0.491637 0.870800i \(-0.663601\pi\)
−0.491637 + 0.870800i \(0.663601\pi\)
\(32\) −183038. −0.987453
\(33\) −45029.9 −0.218123
\(34\) 458669. 2.00135
\(35\) 21306.1 0.0839972
\(36\) −538320. −1.92301
\(37\) 198188. 0.643236 0.321618 0.946869i \(-0.395773\pi\)
0.321618 + 0.946869i \(0.395773\pi\)
\(38\) 130345. 0.385346
\(39\) −126314. −0.340978
\(40\) −1.31473e6 −3.24808
\(41\) 742519. 1.68254 0.841268 0.540619i \(-0.181810\pi\)
0.841268 + 0.540619i \(0.181810\pi\)
\(42\) 18964.4 0.0394972
\(43\) −79507.0 −0.152499
\(44\) 731087. 1.29385
\(45\) −759103. −1.24182
\(46\) −2.03461e6 −3.08197
\(47\) 786601. 1.10513 0.552563 0.833471i \(-0.313650\pi\)
0.552563 + 0.833471i \(0.313650\pi\)
\(48\) −519633. −0.678192
\(49\) −820777. −0.996641
\(50\) −1.75189e6 −1.98204
\(51\) 398248. 0.420396
\(52\) 2.05079e6 2.02259
\(53\) 2.08070e6 1.91975 0.959875 0.280427i \(-0.0904759\pi\)
0.959875 + 0.280427i \(0.0904759\pi\)
\(54\) −1.46422e6 −1.26541
\(55\) 1.03093e6 0.835526
\(56\) −170705. −0.129894
\(57\) 113174. 0.0809442
\(58\) −114291. −0.0769157
\(59\) 607486. 0.385083 0.192542 0.981289i \(-0.438327\pi\)
0.192542 + 0.981289i \(0.438327\pi\)
\(60\) −2.05897e6 −1.23061
\(61\) 2.59833e6 1.46568 0.732841 0.680400i \(-0.238194\pi\)
0.732841 + 0.680400i \(0.238194\pi\)
\(62\) 3.32357e6 1.77106
\(63\) −98562.3 −0.0496614
\(64\) −29175.0 −0.0139117
\(65\) 2.89188e6 1.30612
\(66\) 917624. 0.392881
\(67\) −755375. −0.306832 −0.153416 0.988162i \(-0.549027\pi\)
−0.153416 + 0.988162i \(0.549027\pi\)
\(68\) −6.46580e6 −2.49368
\(69\) −1.76659e6 −0.647386
\(70\) −434177. −0.151295
\(71\) −383737. −0.127242 −0.0636209 0.997974i \(-0.520265\pi\)
−0.0636209 + 0.997974i \(0.520265\pi\)
\(72\) 6.08198e6 1.92035
\(73\) 2.94854e6 0.887109 0.443555 0.896247i \(-0.353717\pi\)
0.443555 + 0.896247i \(0.353717\pi\)
\(74\) −4.03869e6 −1.15859
\(75\) −1.52111e6 −0.416338
\(76\) −1.83745e6 −0.480141
\(77\) 133856. 0.0334135
\(78\) 2.57404e6 0.614165
\(79\) −7.28812e6 −1.66311 −0.831555 0.555443i \(-0.812549\pi\)
−0.831555 + 0.555443i \(0.812549\pi\)
\(80\) 1.18967e7 2.59783
\(81\) 2.82695e6 0.591046
\(82\) −1.51311e7 −3.03056
\(83\) −1.91613e6 −0.367834 −0.183917 0.982942i \(-0.558878\pi\)
−0.183917 + 0.982942i \(0.558878\pi\)
\(84\) −267338. −0.0492134
\(85\) −9.11764e6 −1.61034
\(86\) 1.62020e6 0.274679
\(87\) −99235.4 −0.0161566
\(88\) −8.25988e6 −1.29206
\(89\) 1.05744e7 1.58997 0.794986 0.606627i \(-0.207478\pi\)
0.794986 + 0.606627i \(0.207478\pi\)
\(90\) 1.54691e7 2.23674
\(91\) 375483. 0.0522331
\(92\) 2.86816e7 3.84013
\(93\) 2.88575e6 0.372022
\(94\) −1.60294e7 −1.99054
\(95\) −2.59105e6 −0.310059
\(96\) 3.23861e6 0.373603
\(97\) 1.18419e7 1.31741 0.658705 0.752402i \(-0.271105\pi\)
0.658705 + 0.752402i \(0.271105\pi\)
\(98\) 1.67259e7 1.79514
\(99\) −4.76911e6 −0.493985
\(100\) 2.46961e7 2.46961
\(101\) −725420. −0.0700591 −0.0350296 0.999386i \(-0.511153\pi\)
−0.0350296 + 0.999386i \(0.511153\pi\)
\(102\) −8.11554e6 −0.757211
\(103\) −3.68761e6 −0.332518 −0.166259 0.986082i \(-0.553169\pi\)
−0.166259 + 0.986082i \(0.553169\pi\)
\(104\) −2.31699e7 −2.01980
\(105\) −376982. −0.0317803
\(106\) −4.24008e7 −3.45783
\(107\) −2.07350e7 −1.63629 −0.818145 0.575012i \(-0.804997\pi\)
−0.818145 + 0.575012i \(0.804997\pi\)
\(108\) 2.06410e7 1.57669
\(109\) −3.76731e6 −0.278637 −0.139319 0.990248i \(-0.544491\pi\)
−0.139319 + 0.990248i \(0.544491\pi\)
\(110\) −2.10084e7 −1.50494
\(111\) −3.50667e6 −0.243368
\(112\) 1.54467e6 0.103890
\(113\) 2.24379e6 0.146287 0.0731437 0.997321i \(-0.476697\pi\)
0.0731437 + 0.997321i \(0.476697\pi\)
\(114\) −2.30628e6 −0.145796
\(115\) 4.04449e7 2.47983
\(116\) 1.61115e6 0.0958368
\(117\) −1.33779e7 −0.772215
\(118\) −1.23794e7 −0.693607
\(119\) −1.18384e6 −0.0643988
\(120\) 2.32624e7 1.22891
\(121\) −1.30103e7 −0.667634
\(122\) −5.29491e7 −2.63997
\(123\) −1.31379e7 −0.636587
\(124\) −4.68519e7 −2.20674
\(125\) 3.17758e6 0.145516
\(126\) 2.00851e6 0.0894495
\(127\) 3.84940e7 1.66755 0.833777 0.552101i \(-0.186174\pi\)
0.833777 + 0.552101i \(0.186174\pi\)
\(128\) 2.40234e7 1.01251
\(129\) 1.40677e6 0.0576978
\(130\) −5.89311e7 −2.35257
\(131\) −1.04514e6 −0.0406188 −0.0203094 0.999794i \(-0.506465\pi\)
−0.0203094 + 0.999794i \(0.506465\pi\)
\(132\) −1.29356e7 −0.489529
\(133\) −336423. −0.0123995
\(134\) 1.53931e7 0.552662
\(135\) 2.91065e7 1.01817
\(136\) 7.30511e7 2.49024
\(137\) 3.36576e7 1.11831 0.559154 0.829064i \(-0.311126\pi\)
0.559154 + 0.829064i \(0.311126\pi\)
\(138\) 3.59997e7 1.16606
\(139\) −3.30283e7 −1.04312 −0.521559 0.853215i \(-0.674650\pi\)
−0.521559 + 0.853215i \(0.674650\pi\)
\(140\) 6.12054e6 0.188513
\(141\) −1.39179e7 −0.418124
\(142\) 7.81984e6 0.229186
\(143\) 1.81684e7 0.519566
\(144\) −5.50343e7 −1.53591
\(145\) 2.27193e6 0.0618881
\(146\) −6.00857e7 −1.59785
\(147\) 1.45225e7 0.377079
\(148\) 5.69328e7 1.44360
\(149\) 3.23900e6 0.0802157 0.0401079 0.999195i \(-0.487230\pi\)
0.0401079 + 0.999195i \(0.487230\pi\)
\(150\) 3.09974e7 0.749903
\(151\) 6.37837e7 1.50761 0.753807 0.657096i \(-0.228215\pi\)
0.753807 + 0.657096i \(0.228215\pi\)
\(152\) 2.07597e7 0.479477
\(153\) 4.21784e7 0.952073
\(154\) −2.72774e6 −0.0601840
\(155\) −6.60674e7 −1.42504
\(156\) −3.62859e7 −0.765248
\(157\) −5.75867e7 −1.18761 −0.593804 0.804610i \(-0.702375\pi\)
−0.593804 + 0.804610i \(0.702375\pi\)
\(158\) 1.48518e8 2.99557
\(159\) −3.68153e7 −0.726337
\(160\) −7.41460e7 −1.43109
\(161\) 5.25138e6 0.0991706
\(162\) −5.76080e7 −1.06458
\(163\) 9.59822e7 1.73594 0.867969 0.496619i \(-0.165425\pi\)
0.867969 + 0.496619i \(0.165425\pi\)
\(164\) 2.13302e8 3.77607
\(165\) −1.82410e7 −0.316121
\(166\) 3.90471e7 0.662538
\(167\) −5.45343e7 −0.906070 −0.453035 0.891493i \(-0.649659\pi\)
−0.453035 + 0.891493i \(0.649659\pi\)
\(168\) 3.02040e6 0.0491454
\(169\) −1.17839e7 −0.187796
\(170\) 1.85800e8 2.90052
\(171\) 1.19863e7 0.183315
\(172\) −2.28398e7 −0.342249
\(173\) −5.48114e6 −0.0804841 −0.0402420 0.999190i \(-0.512813\pi\)
−0.0402420 + 0.999190i \(0.512813\pi\)
\(174\) 2.02223e6 0.0291010
\(175\) 4.52168e6 0.0637773
\(176\) 7.47415e7 1.03340
\(177\) −1.07487e7 −0.145696
\(178\) −2.15486e8 −2.86384
\(179\) −7.23802e7 −0.943266 −0.471633 0.881795i \(-0.656335\pi\)
−0.471633 + 0.881795i \(0.656335\pi\)
\(180\) −2.18066e8 −2.78698
\(181\) −7.78319e7 −0.975624 −0.487812 0.872949i \(-0.662205\pi\)
−0.487812 + 0.872949i \(0.662205\pi\)
\(182\) −7.65164e6 −0.0940816
\(183\) −4.59740e7 −0.554541
\(184\) −3.24047e8 −3.83482
\(185\) 8.02829e7 0.932228
\(186\) −5.88061e7 −0.670081
\(187\) −5.72821e7 −0.640580
\(188\) 2.25965e8 2.48021
\(189\) 3.77920e6 0.0407178
\(190\) 5.28008e7 0.558474
\(191\) 9.82338e6 0.102010 0.0510052 0.998698i \(-0.483757\pi\)
0.0510052 + 0.998698i \(0.483757\pi\)
\(192\) 516213. 0.00526350
\(193\) 7.35110e7 0.736040 0.368020 0.929818i \(-0.380036\pi\)
0.368020 + 0.929818i \(0.380036\pi\)
\(194\) −2.41316e8 −2.37290
\(195\) −5.11680e7 −0.494171
\(196\) −2.35782e8 −2.23674
\(197\) −3.73612e7 −0.348168 −0.174084 0.984731i \(-0.555696\pi\)
−0.174084 + 0.984731i \(0.555696\pi\)
\(198\) 9.71854e7 0.889760
\(199\) 2.37151e7 0.213323 0.106662 0.994295i \(-0.465984\pi\)
0.106662 + 0.994295i \(0.465984\pi\)
\(200\) −2.79019e8 −2.46620
\(201\) 1.33654e7 0.116090
\(202\) 1.47827e7 0.126190
\(203\) 294989. 0.00247497
\(204\) 1.14404e8 0.943484
\(205\) 3.00784e8 2.43846
\(206\) 7.51466e7 0.598927
\(207\) −1.87099e8 −1.46614
\(208\) 2.09659e8 1.61544
\(209\) −1.62784e7 −0.123339
\(210\) 7.68219e6 0.0572423
\(211\) 9.22491e7 0.676042 0.338021 0.941139i \(-0.390242\pi\)
0.338021 + 0.941139i \(0.390242\pi\)
\(212\) 5.97719e8 4.30845
\(213\) 6.78972e6 0.0481419
\(214\) 4.22540e8 2.94727
\(215\) −3.22071e7 −0.221013
\(216\) −2.33203e8 −1.57451
\(217\) −8.57822e6 −0.0569886
\(218\) 7.67707e7 0.501877
\(219\) −5.21705e7 −0.335638
\(220\) 2.96153e8 1.87515
\(221\) −1.60683e8 −1.00138
\(222\) 7.14592e7 0.438352
\(223\) −2.83890e8 −1.71428 −0.857141 0.515082i \(-0.827762\pi\)
−0.857141 + 0.515082i \(0.827762\pi\)
\(224\) −9.62715e6 −0.0572308
\(225\) −1.61101e8 −0.942884
\(226\) −4.57241e7 −0.263491
\(227\) 4.18699e7 0.237581 0.118790 0.992919i \(-0.462098\pi\)
0.118790 + 0.992919i \(0.462098\pi\)
\(228\) 3.25113e7 0.181661
\(229\) −8.20861e7 −0.451695 −0.225848 0.974163i \(-0.572515\pi\)
−0.225848 + 0.974163i \(0.572515\pi\)
\(230\) −8.24191e8 −4.46663
\(231\) −2.36841e6 −0.0126420
\(232\) −1.82029e7 −0.0957043
\(233\) −1.98224e8 −1.02662 −0.513312 0.858202i \(-0.671582\pi\)
−0.513312 + 0.858202i \(0.671582\pi\)
\(234\) 2.72617e8 1.39090
\(235\) 3.18640e8 1.60163
\(236\) 1.74511e8 0.864233
\(237\) 1.28954e8 0.629237
\(238\) 2.41244e7 0.115994
\(239\) −4.10134e8 −1.94327 −0.971635 0.236487i \(-0.924004\pi\)
−0.971635 + 0.236487i \(0.924004\pi\)
\(240\) −2.10496e8 −0.982888
\(241\) −2.36729e8 −1.08941 −0.544704 0.838628i \(-0.683358\pi\)
−0.544704 + 0.838628i \(0.683358\pi\)
\(242\) 2.65125e8 1.20253
\(243\) −2.07161e8 −0.926161
\(244\) 7.46415e8 3.28940
\(245\) −3.32485e8 −1.44441
\(246\) 2.67725e8 1.14661
\(247\) −4.56629e7 −0.192808
\(248\) 5.29336e8 2.20369
\(249\) 3.39034e7 0.139170
\(250\) −6.47530e7 −0.262102
\(251\) −1.19914e8 −0.478644 −0.239322 0.970940i \(-0.576925\pi\)
−0.239322 + 0.970940i \(0.576925\pi\)
\(252\) −2.83137e7 −0.111454
\(253\) 2.54097e8 0.986457
\(254\) −7.84435e8 −3.00358
\(255\) 1.61325e8 0.609270
\(256\) −4.85817e8 −1.80981
\(257\) 4.20140e8 1.54393 0.771965 0.635664i \(-0.219274\pi\)
0.771965 + 0.635664i \(0.219274\pi\)
\(258\) −2.86673e7 −0.103925
\(259\) 1.04240e7 0.0372807
\(260\) 8.30744e8 2.93130
\(261\) −1.05100e7 −0.0365899
\(262\) 2.12981e7 0.0731620
\(263\) −2.09315e7 −0.0709506 −0.0354753 0.999371i \(-0.511295\pi\)
−0.0354753 + 0.999371i \(0.511295\pi\)
\(264\) 1.46148e8 0.488852
\(265\) 8.42863e8 2.78225
\(266\) 6.85567e6 0.0223339
\(267\) −1.87100e8 −0.601566
\(268\) −2.16995e8 −0.688616
\(269\) 4.20833e8 1.31819 0.659093 0.752062i \(-0.270940\pi\)
0.659093 + 0.752062i \(0.270940\pi\)
\(270\) −5.93136e8 −1.83392
\(271\) −1.41459e8 −0.431757 −0.215878 0.976420i \(-0.569261\pi\)
−0.215878 + 0.976420i \(0.569261\pi\)
\(272\) −6.61020e8 −1.99170
\(273\) −6.64368e6 −0.0197624
\(274\) −6.85879e8 −2.01428
\(275\) 2.18789e8 0.634398
\(276\) −5.07483e8 −1.45291
\(277\) −4.46880e8 −1.26332 −0.631658 0.775248i \(-0.717625\pi\)
−0.631658 + 0.775248i \(0.717625\pi\)
\(278\) 6.73053e8 1.87885
\(279\) 3.05629e8 0.842520
\(280\) −6.91503e7 −0.188252
\(281\) 4.25457e8 1.14389 0.571944 0.820293i \(-0.306189\pi\)
0.571944 + 0.820293i \(0.306189\pi\)
\(282\) 2.83619e8 0.753120
\(283\) −3.36158e7 −0.0881638 −0.0440819 0.999028i \(-0.514036\pi\)
−0.0440819 + 0.999028i \(0.514036\pi\)
\(284\) −1.10235e8 −0.285566
\(285\) 4.58452e7 0.117311
\(286\) −3.70238e8 −0.935837
\(287\) 3.90539e7 0.0975164
\(288\) 3.43001e8 0.846100
\(289\) 9.62690e7 0.234609
\(290\) −4.62977e7 −0.111472
\(291\) −2.09527e8 −0.498442
\(292\) 8.47019e8 1.99092
\(293\) −4.64940e8 −1.07984 −0.539921 0.841715i \(-0.681546\pi\)
−0.539921 + 0.841715i \(0.681546\pi\)
\(294\) −2.95942e8 −0.679190
\(295\) 2.46084e8 0.558092
\(296\) −6.43231e8 −1.44160
\(297\) 1.82863e8 0.405023
\(298\) −6.60048e7 −0.144484
\(299\) 7.12773e8 1.54206
\(300\) −4.36965e8 −0.934378
\(301\) −4.18179e6 −0.00883851
\(302\) −1.29979e9 −2.71550
\(303\) 1.28353e7 0.0265068
\(304\) −1.87849e8 −0.383487
\(305\) 1.05255e9 2.12418
\(306\) −8.59516e8 −1.71486
\(307\) −9.90322e8 −1.95340 −0.976702 0.214600i \(-0.931155\pi\)
−0.976702 + 0.214600i \(0.931155\pi\)
\(308\) 3.84526e7 0.0749891
\(309\) 6.52474e7 0.125808
\(310\) 1.34633e9 2.56676
\(311\) −1.57593e8 −0.297081 −0.148540 0.988906i \(-0.547457\pi\)
−0.148540 + 0.988906i \(0.547457\pi\)
\(312\) 4.09961e8 0.764191
\(313\) 8.72665e8 1.60858 0.804290 0.594237i \(-0.202546\pi\)
0.804290 + 0.594237i \(0.202546\pi\)
\(314\) 1.17351e9 2.13911
\(315\) −3.99262e7 −0.0719732
\(316\) −2.09364e9 −3.73248
\(317\) 1.63595e8 0.288444 0.144222 0.989545i \(-0.453932\pi\)
0.144222 + 0.989545i \(0.453932\pi\)
\(318\) 7.50226e8 1.30827
\(319\) 1.42735e7 0.0246187
\(320\) −1.18184e7 −0.0201620
\(321\) 3.66878e8 0.619090
\(322\) −1.07013e8 −0.178625
\(323\) 1.43968e8 0.237715
\(324\) 8.12092e8 1.32647
\(325\) 6.13730e8 0.991712
\(326\) −1.95594e9 −3.12675
\(327\) 6.66576e7 0.105422
\(328\) −2.40990e9 −3.77086
\(329\) 4.13724e7 0.0640509
\(330\) 3.71716e8 0.569394
\(331\) 6.41815e8 0.972773 0.486387 0.873744i \(-0.338315\pi\)
0.486387 + 0.873744i \(0.338315\pi\)
\(332\) −5.50442e8 −0.825521
\(333\) −3.71391e8 −0.551158
\(334\) 1.11131e9 1.63200
\(335\) −3.05991e8 −0.444685
\(336\) −2.73309e7 −0.0393066
\(337\) −2.16017e8 −0.307457 −0.153728 0.988113i \(-0.549128\pi\)
−0.153728 + 0.988113i \(0.549128\pi\)
\(338\) 2.40134e8 0.338256
\(339\) −3.97008e7 −0.0553478
\(340\) −2.61920e9 −3.61404
\(341\) −4.15072e8 −0.566870
\(342\) −2.44257e8 −0.330184
\(343\) −8.64854e7 −0.115721
\(344\) 2.58045e8 0.341776
\(345\) −7.15619e8 −0.938242
\(346\) 1.11695e8 0.144967
\(347\) −1.18322e9 −1.52024 −0.760118 0.649785i \(-0.774859\pi\)
−0.760118 + 0.649785i \(0.774859\pi\)
\(348\) −2.85071e7 −0.0362598
\(349\) −4.71635e8 −0.593905 −0.296953 0.954892i \(-0.595970\pi\)
−0.296953 + 0.954892i \(0.595970\pi\)
\(350\) −9.21432e7 −0.114875
\(351\) 5.12954e8 0.633145
\(352\) −4.65827e8 −0.569279
\(353\) −2.14674e8 −0.259757 −0.129879 0.991530i \(-0.541459\pi\)
−0.129879 + 0.991530i \(0.541459\pi\)
\(354\) 2.19037e8 0.262426
\(355\) −1.55446e8 −0.184409
\(356\) 3.03767e9 3.56834
\(357\) 2.09464e7 0.0243653
\(358\) 1.47497e9 1.69900
\(359\) 5.07607e8 0.579024 0.289512 0.957174i \(-0.406507\pi\)
0.289512 + 0.957174i \(0.406507\pi\)
\(360\) 2.46372e9 2.78313
\(361\) −8.52959e8 −0.954230
\(362\) 1.58607e9 1.75728
\(363\) 2.30200e8 0.252599
\(364\) 1.07864e8 0.117226
\(365\) 1.19441e9 1.28567
\(366\) 9.36863e8 0.998832
\(367\) −7.17068e8 −0.757232 −0.378616 0.925554i \(-0.623600\pi\)
−0.378616 + 0.925554i \(0.623600\pi\)
\(368\) 2.93222e9 3.06711
\(369\) −1.39143e9 −1.44168
\(370\) −1.63601e9 −1.67912
\(371\) 1.09438e8 0.111265
\(372\) 8.28981e8 0.834919
\(373\) 1.47530e9 1.47197 0.735985 0.676998i \(-0.236720\pi\)
0.735985 + 0.676998i \(0.236720\pi\)
\(374\) 1.16730e9 1.15380
\(375\) −5.62230e7 −0.0550560
\(376\) −2.55297e9 −2.47678
\(377\) 4.00390e7 0.0384847
\(378\) −7.70130e7 −0.0733403
\(379\) 1.57868e9 1.48956 0.744778 0.667313i \(-0.232556\pi\)
0.744778 + 0.667313i \(0.232556\pi\)
\(380\) −7.44325e8 −0.695857
\(381\) −6.81100e8 −0.630919
\(382\) −2.00182e8 −0.183740
\(383\) 6.74713e8 0.613653 0.306827 0.951765i \(-0.400733\pi\)
0.306827 + 0.951765i \(0.400733\pi\)
\(384\) −4.25062e8 −0.383083
\(385\) 5.42233e7 0.0484254
\(386\) −1.49802e9 −1.32575
\(387\) 1.48991e8 0.130669
\(388\) 3.40179e9 2.95663
\(389\) −5.60683e8 −0.482941 −0.241470 0.970408i \(-0.577630\pi\)
−0.241470 + 0.970408i \(0.577630\pi\)
\(390\) 1.04271e9 0.890095
\(391\) −2.24726e9 −1.90123
\(392\) 2.66389e9 2.23365
\(393\) 1.84924e7 0.0153681
\(394\) 7.61350e8 0.627116
\(395\) −2.95231e9 −2.41031
\(396\) −1.37001e9 −1.10864
\(397\) 3.50283e8 0.280965 0.140482 0.990083i \(-0.455135\pi\)
0.140482 + 0.990083i \(0.455135\pi\)
\(398\) −4.83268e8 −0.384236
\(399\) 5.95256e6 0.00469136
\(400\) 2.52477e9 1.97248
\(401\) 1.19625e9 0.926440 0.463220 0.886243i \(-0.346694\pi\)
0.463220 + 0.886243i \(0.346694\pi\)
\(402\) −2.72361e8 −0.209100
\(403\) −1.16433e9 −0.886150
\(404\) −2.08389e8 −0.157232
\(405\) 1.14516e9 0.856590
\(406\) −6.01131e6 −0.00445788
\(407\) 5.04382e8 0.370834
\(408\) −1.29254e9 −0.942180
\(409\) 1.57416e9 1.13767 0.568836 0.822451i \(-0.307394\pi\)
0.568836 + 0.822451i \(0.307394\pi\)
\(410\) −6.12940e9 −4.39213
\(411\) −5.95527e8 −0.423112
\(412\) −1.05933e9 −0.746262
\(413\) 3.19516e7 0.0223186
\(414\) 3.81272e9 2.64079
\(415\) −7.76197e8 −0.533094
\(416\) −1.30670e9 −0.889916
\(417\) 5.84391e8 0.394664
\(418\) 3.31724e8 0.222157
\(419\) 2.03488e9 1.35142 0.675710 0.737167i \(-0.263837\pi\)
0.675710 + 0.737167i \(0.263837\pi\)
\(420\) −1.08295e8 −0.0713239
\(421\) −1.11312e9 −0.727032 −0.363516 0.931588i \(-0.618424\pi\)
−0.363516 + 0.931588i \(0.618424\pi\)
\(422\) −1.87986e9 −1.21768
\(423\) −1.47404e9 −0.946929
\(424\) −6.75307e9 −4.30250
\(425\) −1.93499e9 −1.22269
\(426\) −1.38362e8 −0.0867126
\(427\) 1.36663e8 0.0849480
\(428\) −5.95648e9 −3.67229
\(429\) −3.21466e8 −0.196578
\(430\) 6.56320e8 0.398085
\(431\) −2.21785e9 −1.33432 −0.667161 0.744913i \(-0.732491\pi\)
−0.667161 + 0.744913i \(0.732491\pi\)
\(432\) 2.11020e9 1.25930
\(433\) 1.61665e9 0.956992 0.478496 0.878090i \(-0.341182\pi\)
0.478496 + 0.878090i \(0.341182\pi\)
\(434\) 1.74808e8 0.102647
\(435\) −4.01988e7 −0.0234154
\(436\) −1.08223e9 −0.625338
\(437\) −6.38626e8 −0.366068
\(438\) 1.06314e9 0.604546
\(439\) −2.27405e9 −1.28285 −0.641424 0.767187i \(-0.721656\pi\)
−0.641424 + 0.767187i \(0.721656\pi\)
\(440\) −3.34596e9 −1.87256
\(441\) 1.53808e9 0.853973
\(442\) 3.27442e9 1.80367
\(443\) −1.06718e9 −0.583209 −0.291605 0.956539i \(-0.594189\pi\)
−0.291605 + 0.956539i \(0.594189\pi\)
\(444\) −1.00735e9 −0.546186
\(445\) 4.28353e9 2.30431
\(446\) 5.78513e9 3.08774
\(447\) −5.73099e7 −0.0303496
\(448\) −1.53450e6 −0.000806296 0
\(449\) −1.72761e9 −0.900707 −0.450354 0.892850i \(-0.648702\pi\)
−0.450354 + 0.892850i \(0.648702\pi\)
\(450\) 3.28293e9 1.69831
\(451\) 1.88969e9 0.970002
\(452\) 6.44567e8 0.328309
\(453\) −1.12857e9 −0.570405
\(454\) −8.53229e8 −0.427927
\(455\) 1.52103e8 0.0757003
\(456\) −3.67315e8 −0.181410
\(457\) 3.85685e8 0.189028 0.0945139 0.995524i \(-0.469870\pi\)
0.0945139 + 0.995524i \(0.469870\pi\)
\(458\) 1.67276e9 0.813587
\(459\) −1.61726e9 −0.780612
\(460\) 1.16185e10 5.56542
\(461\) −2.53341e9 −1.20435 −0.602174 0.798365i \(-0.705699\pi\)
−0.602174 + 0.798365i \(0.705699\pi\)
\(462\) 4.82638e7 0.0227706
\(463\) −5.21077e8 −0.243988 −0.121994 0.992531i \(-0.538929\pi\)
−0.121994 + 0.992531i \(0.538929\pi\)
\(464\) 1.64713e8 0.0765447
\(465\) 1.16897e9 0.539163
\(466\) 4.03944e9 1.84914
\(467\) −1.97610e9 −0.897842 −0.448921 0.893571i \(-0.648192\pi\)
−0.448921 + 0.893571i \(0.648192\pi\)
\(468\) −3.84304e9 −1.73306
\(469\) −3.97300e7 −0.0177834
\(470\) −6.49329e9 −2.88484
\(471\) 1.01892e9 0.449331
\(472\) −1.97164e9 −0.863039
\(473\) −2.02343e8 −0.0879173
\(474\) −2.62783e9 −1.13337
\(475\) −5.49886e8 −0.235421
\(476\) −3.40078e8 −0.144529
\(477\) −3.89910e9 −1.64494
\(478\) 8.35775e9 3.50019
\(479\) −1.86520e9 −0.775444 −0.387722 0.921776i \(-0.626738\pi\)
−0.387722 + 0.921776i \(0.626738\pi\)
\(480\) 1.31192e9 0.541454
\(481\) 1.41485e9 0.579700
\(482\) 4.82408e9 1.96223
\(483\) −9.29162e7 −0.0375212
\(484\) −3.73743e9 −1.49835
\(485\) 4.79699e9 1.90929
\(486\) 4.22156e9 1.66819
\(487\) 2.65232e9 1.04058 0.520288 0.853991i \(-0.325824\pi\)
0.520288 + 0.853991i \(0.325824\pi\)
\(488\) −8.43306e9 −3.28485
\(489\) −1.69828e9 −0.656791
\(490\) 6.77541e9 2.60165
\(491\) 3.14043e9 1.19730 0.598651 0.801010i \(-0.295703\pi\)
0.598651 + 0.801010i \(0.295703\pi\)
\(492\) −3.77409e9 −1.42868
\(493\) −1.26236e8 −0.0474483
\(494\) 9.30525e8 0.347283
\(495\) −1.93190e9 −0.715922
\(496\) −4.78982e9 −1.76252
\(497\) −2.01832e7 −0.00737468
\(498\) −6.90887e8 −0.250671
\(499\) −1.74081e8 −0.0627191 −0.0313596 0.999508i \(-0.509984\pi\)
−0.0313596 + 0.999508i \(0.509984\pi\)
\(500\) 9.12814e8 0.326578
\(501\) 9.64911e8 0.342811
\(502\) 2.44363e9 0.862128
\(503\) 7.08807e8 0.248336 0.124168 0.992261i \(-0.460374\pi\)
0.124168 + 0.992261i \(0.460374\pi\)
\(504\) 3.19891e8 0.111300
\(505\) −2.93857e8 −0.101535
\(506\) −5.17802e9 −1.77679
\(507\) 2.08501e8 0.0710526
\(508\) 1.10581e10 3.74245
\(509\) 5.15128e9 1.73142 0.865711 0.500545i \(-0.166867\pi\)
0.865711 + 0.500545i \(0.166867\pi\)
\(510\) −3.28749e9 −1.09741
\(511\) 1.55083e8 0.0514151
\(512\) 6.82504e9 2.24730
\(513\) −4.59593e8 −0.150301
\(514\) −8.56165e9 −2.78091
\(515\) −1.49380e9 −0.481911
\(516\) 4.04119e8 0.129490
\(517\) 2.00188e9 0.637119
\(518\) −2.12421e8 −0.0671495
\(519\) 9.69815e7 0.0304511
\(520\) −9.38581e9 −2.92725
\(521\) −3.29977e9 −1.02224 −0.511118 0.859510i \(-0.670769\pi\)
−0.511118 + 0.859510i \(0.670769\pi\)
\(522\) 2.14174e8 0.0659053
\(523\) 4.25818e9 1.30157 0.650786 0.759261i \(-0.274440\pi\)
0.650786 + 0.759261i \(0.274440\pi\)
\(524\) −3.00236e8 −0.0911598
\(525\) −8.00050e7 −0.0241301
\(526\) 4.26545e8 0.127795
\(527\) 3.67093e9 1.09255
\(528\) −1.32245e9 −0.390986
\(529\) 6.56378e9 1.92779
\(530\) −1.71760e10 −5.01136
\(531\) −1.13839e9 −0.329959
\(532\) −9.66435e7 −0.0278280
\(533\) 5.30081e9 1.51634
\(534\) 3.81273e9 1.08353
\(535\) −8.39944e9 −2.37144
\(536\) 2.45162e9 0.687664
\(537\) 1.28067e9 0.356885
\(538\) −8.57578e9 −2.37430
\(539\) −2.08885e9 −0.574576
\(540\) 8.36136e9 2.28507
\(541\) −7.50896e8 −0.203887 −0.101943 0.994790i \(-0.532506\pi\)
−0.101943 + 0.994790i \(0.532506\pi\)
\(542\) 2.88268e9 0.777675
\(543\) 1.37713e9 0.369127
\(544\) 4.11981e9 1.09719
\(545\) −1.52608e9 −0.403822
\(546\) 1.35386e8 0.0355958
\(547\) −6.03569e7 −0.0157678 −0.00788390 0.999969i \(-0.502510\pi\)
−0.00788390 + 0.999969i \(0.502510\pi\)
\(548\) 9.66874e9 2.50979
\(549\) −4.86910e9 −1.25587
\(550\) −4.45851e9 −1.14267
\(551\) −3.58739e7 −0.00913583
\(552\) 5.73358e9 1.45090
\(553\) −3.83329e8 −0.0963905
\(554\) 9.10657e9 2.27547
\(555\) −1.42050e9 −0.352708
\(556\) −9.48794e9 −2.34105
\(557\) 1.40575e9 0.344678 0.172339 0.985038i \(-0.444867\pi\)
0.172339 + 0.985038i \(0.444867\pi\)
\(558\) −6.22815e9 −1.51754
\(559\) −5.67596e8 −0.137435
\(560\) 6.25723e8 0.150565
\(561\) 1.01353e9 0.242363
\(562\) −8.67001e9 −2.06036
\(563\) −5.50487e9 −1.30007 −0.650036 0.759903i \(-0.725246\pi\)
−0.650036 + 0.759903i \(0.725246\pi\)
\(564\) −3.99814e9 −0.938386
\(565\) 9.08926e8 0.212011
\(566\) 6.85026e8 0.158800
\(567\) 1.48688e8 0.0342558
\(568\) 1.24544e9 0.285171
\(569\) 1.17013e9 0.266283 0.133141 0.991097i \(-0.457494\pi\)
0.133141 + 0.991097i \(0.457494\pi\)
\(570\) −9.34239e8 −0.211298
\(571\) −1.88234e9 −0.423127 −0.211563 0.977364i \(-0.567855\pi\)
−0.211563 + 0.977364i \(0.567855\pi\)
\(572\) 5.21919e9 1.16605
\(573\) −1.73812e8 −0.0385956
\(574\) −7.95844e8 −0.175645
\(575\) 8.58342e9 1.88288
\(576\) 5.46721e7 0.0119203
\(577\) −6.06026e9 −1.31334 −0.656668 0.754180i \(-0.728035\pi\)
−0.656668 + 0.754180i \(0.728035\pi\)
\(578\) −1.96178e9 −0.422574
\(579\) −1.30068e9 −0.278481
\(580\) 6.52652e8 0.138894
\(581\) −1.00782e8 −0.0213189
\(582\) 4.26976e9 0.897787
\(583\) 5.29533e9 1.10676
\(584\) −9.56969e9 −1.98817
\(585\) −5.41920e9 −1.11915
\(586\) 9.47460e9 1.94500
\(587\) 7.70428e9 1.57217 0.786083 0.618120i \(-0.212106\pi\)
0.786083 + 0.618120i \(0.212106\pi\)
\(588\) 4.17185e9 0.846269
\(589\) 1.04321e9 0.210362
\(590\) −5.01472e9 −1.00523
\(591\) 6.61056e8 0.131729
\(592\) 5.82044e9 1.15300
\(593\) −8.40982e9 −1.65613 −0.828067 0.560629i \(-0.810559\pi\)
−0.828067 + 0.560629i \(0.810559\pi\)
\(594\) −3.72641e9 −0.729522
\(595\) −4.79556e8 −0.0933318
\(596\) 9.30460e8 0.180026
\(597\) −4.19607e8 −0.0807109
\(598\) −1.45250e10 −2.77755
\(599\) 2.66302e9 0.506267 0.253134 0.967431i \(-0.418539\pi\)
0.253134 + 0.967431i \(0.418539\pi\)
\(600\) 4.93687e9 0.933087
\(601\) 5.05119e9 0.949146 0.474573 0.880216i \(-0.342602\pi\)
0.474573 + 0.880216i \(0.342602\pi\)
\(602\) 8.52169e7 0.0159198
\(603\) 1.41552e9 0.262909
\(604\) 1.83230e10 3.38350
\(605\) −5.27028e9 −0.967587
\(606\) −2.61560e8 −0.0477438
\(607\) 1.98881e9 0.360938 0.180469 0.983581i \(-0.442238\pi\)
0.180469 + 0.983581i \(0.442238\pi\)
\(608\) 1.17077e9 0.211256
\(609\) −5.21943e6 −0.000936403 0
\(610\) −2.14489e10 −3.82605
\(611\) 5.61550e9 0.995966
\(612\) 1.21165e10 2.13671
\(613\) 3.81895e9 0.669626 0.334813 0.942285i \(-0.391327\pi\)
0.334813 + 0.942285i \(0.391327\pi\)
\(614\) 2.01809e10 3.51845
\(615\) −5.32197e9 −0.922591
\(616\) −4.34440e8 −0.0748855
\(617\) −8.24101e9 −1.41248 −0.706241 0.707972i \(-0.749610\pi\)
−0.706241 + 0.707972i \(0.749610\pi\)
\(618\) −1.32962e9 −0.226604
\(619\) 2.93974e9 0.498185 0.249093 0.968480i \(-0.419868\pi\)
0.249093 + 0.968480i \(0.419868\pi\)
\(620\) −1.89790e10 −3.19818
\(621\) 7.17399e9 1.20210
\(622\) 3.21144e9 0.535098
\(623\) 5.56175e8 0.0921516
\(624\) −3.70963e9 −0.611202
\(625\) −5.42915e9 −0.889513
\(626\) −1.77833e10 −2.89735
\(627\) 2.88025e8 0.0466653
\(628\) −1.65428e10 −2.66532
\(629\) −4.46080e9 −0.714719
\(630\) 8.13620e8 0.129637
\(631\) −5.09030e9 −0.806567 −0.403283 0.915075i \(-0.632131\pi\)
−0.403283 + 0.915075i \(0.632131\pi\)
\(632\) 2.36541e10 3.72732
\(633\) −1.63223e9 −0.255780
\(634\) −3.33375e9 −0.519542
\(635\) 1.55934e10 2.41675
\(636\) −1.05758e10 −1.63010
\(637\) −5.85948e9 −0.898196
\(638\) −2.90868e8 −0.0443428
\(639\) 7.19098e8 0.109027
\(640\) 9.73153e9 1.46741
\(641\) −1.31257e9 −0.196843 −0.0984216 0.995145i \(-0.531379\pi\)
−0.0984216 + 0.995145i \(0.531379\pi\)
\(642\) −7.47628e9 −1.11510
\(643\) −6.98129e9 −1.03561 −0.517806 0.855498i \(-0.673251\pi\)
−0.517806 + 0.855498i \(0.673251\pi\)
\(644\) 1.50855e9 0.222566
\(645\) 5.69862e8 0.0836201
\(646\) −2.93379e9 −0.428169
\(647\) −4.67412e9 −0.678476 −0.339238 0.940700i \(-0.610169\pi\)
−0.339238 + 0.940700i \(0.610169\pi\)
\(648\) −9.17507e9 −1.32464
\(649\) 1.54604e9 0.222005
\(650\) −1.25067e10 −1.78626
\(651\) 1.51780e8 0.0215616
\(652\) 2.75725e10 3.89592
\(653\) −2.18418e9 −0.306968 −0.153484 0.988151i \(-0.549049\pi\)
−0.153484 + 0.988151i \(0.549049\pi\)
\(654\) −1.35836e9 −0.189885
\(655\) −4.23373e8 −0.0588679
\(656\) 2.18065e10 3.01594
\(657\) −5.52537e9 −0.760121
\(658\) −8.43091e8 −0.115368
\(659\) 3.50457e9 0.477019 0.238510 0.971140i \(-0.423341\pi\)
0.238510 + 0.971140i \(0.423341\pi\)
\(660\) −5.24003e9 −0.709463
\(661\) −8.46214e9 −1.13966 −0.569830 0.821763i \(-0.692991\pi\)
−0.569830 + 0.821763i \(0.692991\pi\)
\(662\) −1.30790e10 −1.75215
\(663\) 2.84307e9 0.378870
\(664\) 6.21893e9 0.824380
\(665\) −1.36280e8 −0.0179704
\(666\) 7.56824e9 0.992739
\(667\) 5.59972e8 0.0730677
\(668\) −1.56659e10 −2.03347
\(669\) 5.02305e9 0.648598
\(670\) 6.23552e9 0.800961
\(671\) 6.61268e9 0.844984
\(672\) 1.70340e8 0.0216533
\(673\) 5.78313e9 0.731324 0.365662 0.930748i \(-0.380843\pi\)
0.365662 + 0.930748i \(0.380843\pi\)
\(674\) 4.40203e9 0.553787
\(675\) 6.17713e9 0.773078
\(676\) −3.38514e9 −0.421466
\(677\) −2.36319e9 −0.292710 −0.146355 0.989232i \(-0.546754\pi\)
−0.146355 + 0.989232i \(0.546754\pi\)
\(678\) 8.09028e8 0.0996918
\(679\) 6.22843e8 0.0763544
\(680\) 2.95919e10 3.60904
\(681\) −7.40832e8 −0.0898886
\(682\) 8.45839e9 1.02104
\(683\) −5.06343e9 −0.608096 −0.304048 0.952657i \(-0.598338\pi\)
−0.304048 + 0.952657i \(0.598338\pi\)
\(684\) 3.44326e9 0.411409
\(685\) 1.36342e10 1.62074
\(686\) 1.76241e9 0.208436
\(687\) 1.45240e9 0.170899
\(688\) −2.33499e9 −0.273354
\(689\) 1.48540e10 1.73012
\(690\) 1.45830e10 1.68995
\(691\) −2.24876e8 −0.0259281 −0.0129640 0.999916i \(-0.504127\pi\)
−0.0129640 + 0.999916i \(0.504127\pi\)
\(692\) −1.57455e9 −0.180628
\(693\) −2.50838e8 −0.0286304
\(694\) 2.41117e10 2.73823
\(695\) −1.33793e10 −1.51177
\(696\) 3.22075e8 0.0362097
\(697\) −1.67126e10 −1.86951
\(698\) 9.61103e9 1.06973
\(699\) 3.50732e9 0.388423
\(700\) 1.29893e9 0.143134
\(701\) 7.33460e9 0.804198 0.402099 0.915596i \(-0.368281\pi\)
0.402099 + 0.915596i \(0.368281\pi\)
\(702\) −1.04530e10 −1.14041
\(703\) −1.26767e9 −0.137614
\(704\) −7.42496e7 −0.00802028
\(705\) −5.63792e9 −0.605978
\(706\) 4.37465e9 0.467871
\(707\) −3.81545e7 −0.00406049
\(708\) −3.08774e9 −0.326982
\(709\) 9.29450e8 0.0979409 0.0489704 0.998800i \(-0.484406\pi\)
0.0489704 + 0.998800i \(0.484406\pi\)
\(710\) 3.16770e9 0.332155
\(711\) 1.36575e10 1.42504
\(712\) −3.43199e10 −3.56341
\(713\) −1.62839e10 −1.68246
\(714\) −4.26849e8 −0.0438864
\(715\) 7.35976e9 0.752996
\(716\) −2.07925e10 −2.11695
\(717\) 7.25677e9 0.735235
\(718\) −1.03441e10 −1.04293
\(719\) 1.46064e10 1.46552 0.732759 0.680489i \(-0.238232\pi\)
0.732759 + 0.680489i \(0.238232\pi\)
\(720\) −2.22936e10 −2.22595
\(721\) −1.93955e8 −0.0192721
\(722\) 1.73817e10 1.71875
\(723\) 4.18860e9 0.412178
\(724\) −2.23586e10 −2.18957
\(725\) 4.82161e8 0.0469904
\(726\) −4.69104e9 −0.454978
\(727\) 6.09535e9 0.588339 0.294170 0.955753i \(-0.404957\pi\)
0.294170 + 0.955753i \(0.404957\pi\)
\(728\) −1.21866e9 −0.117064
\(729\) −2.51710e9 −0.240633
\(730\) −2.43398e10 −2.31573
\(731\) 1.78954e9 0.169446
\(732\) −1.32068e10 −1.24454
\(733\) −5.59722e9 −0.524939 −0.262469 0.964940i \(-0.584537\pi\)
−0.262469 + 0.964940i \(0.584537\pi\)
\(734\) 1.46125e10 1.36392
\(735\) 5.88287e9 0.546492
\(736\) −1.82750e10 −1.68961
\(737\) −1.92241e9 −0.176892
\(738\) 2.83548e10 2.59674
\(739\) −8.34763e9 −0.760865 −0.380433 0.924809i \(-0.624225\pi\)
−0.380433 + 0.924809i \(0.624225\pi\)
\(740\) 2.30627e10 2.09218
\(741\) 8.07945e8 0.0729488
\(742\) −2.23013e9 −0.200409
\(743\) −1.62616e10 −1.45446 −0.727230 0.686393i \(-0.759193\pi\)
−0.727230 + 0.686393i \(0.759193\pi\)
\(744\) −9.36589e9 −0.833765
\(745\) 1.31207e9 0.116255
\(746\) −3.00638e10 −2.65129
\(747\) 3.59070e9 0.315179
\(748\) −1.64553e10 −1.43764
\(749\) −1.09059e9 −0.0948361
\(750\) 1.14572e9 0.0991661
\(751\) −1.06354e10 −0.916253 −0.458127 0.888887i \(-0.651479\pi\)
−0.458127 + 0.888887i \(0.651479\pi\)
\(752\) 2.31011e10 1.98094
\(753\) 2.12172e9 0.181095
\(754\) −8.15919e8 −0.0693182
\(755\) 2.58378e10 2.18495
\(756\) 1.08564e9 0.0913819
\(757\) −1.33873e10 −1.12165 −0.560826 0.827934i \(-0.689516\pi\)
−0.560826 + 0.827934i \(0.689516\pi\)
\(758\) −3.21705e10 −2.68297
\(759\) −4.49591e9 −0.373226
\(760\) 8.40944e9 0.694895
\(761\) −3.98898e9 −0.328107 −0.164054 0.986451i \(-0.552457\pi\)
−0.164054 + 0.986451i \(0.552457\pi\)
\(762\) 1.38795e10 1.13640
\(763\) −1.98147e8 −0.0161492
\(764\) 2.82194e9 0.228939
\(765\) 1.70859e10 1.37982
\(766\) −1.37494e10 −1.10530
\(767\) 4.33681e9 0.347046
\(768\) 8.59589e9 0.684741
\(769\) −1.33758e10 −1.06067 −0.530333 0.847789i \(-0.677933\pi\)
−0.530333 + 0.847789i \(0.677933\pi\)
\(770\) −1.10497e9 −0.0872233
\(771\) −7.43381e9 −0.584146
\(772\) 2.11173e10 1.65188
\(773\) 1.88395e10 1.46704 0.733518 0.679670i \(-0.237877\pi\)
0.733518 + 0.679670i \(0.237877\pi\)
\(774\) −3.03615e9 −0.235359
\(775\) −1.40211e10 −1.08200
\(776\) −3.84337e10 −2.95254
\(777\) −1.84438e8 −0.0141051
\(778\) 1.14257e10 0.869867
\(779\) −4.74938e9 −0.359962
\(780\) −1.46989e10 −1.10906
\(781\) −9.76600e8 −0.0733564
\(782\) 4.57949e10 3.42447
\(783\) 4.02989e8 0.0300004
\(784\) −2.41048e10 −1.78648
\(785\) −2.33275e10 −1.72117
\(786\) −3.76841e8 −0.0276808
\(787\) 1.86675e8 0.0136513 0.00682566 0.999977i \(-0.497827\pi\)
0.00682566 + 0.999977i \(0.497827\pi\)
\(788\) −1.07326e10 −0.781385
\(789\) 3.70356e8 0.0268441
\(790\) 6.01625e10 4.34142
\(791\) 1.18015e8 0.00847853
\(792\) 1.54785e10 1.10711
\(793\) 1.85493e10 1.32091
\(794\) −7.13810e9 −0.506070
\(795\) −1.49133e10 −1.05266
\(796\) 6.81257e9 0.478757
\(797\) −1.16617e10 −0.815938 −0.407969 0.912996i \(-0.633763\pi\)
−0.407969 + 0.912996i \(0.633763\pi\)
\(798\) −1.21302e8 −0.00845002
\(799\) −1.77048e10 −1.22794
\(800\) −1.57356e10 −1.08660
\(801\) −1.98157e10 −1.36237
\(802\) −2.43773e10 −1.66869
\(803\) 7.50395e9 0.511429
\(804\) 3.83943e9 0.260538
\(805\) 2.12726e9 0.143726
\(806\) 2.37268e10 1.59612
\(807\) −7.44608e9 −0.498735
\(808\) 2.35440e9 0.157015
\(809\) 2.35019e10 1.56057 0.780286 0.625423i \(-0.215073\pi\)
0.780286 + 0.625423i \(0.215073\pi\)
\(810\) −2.33362e10 −1.54288
\(811\) −2.42645e10 −1.59734 −0.798670 0.601769i \(-0.794463\pi\)
−0.798670 + 0.601769i \(0.794463\pi\)
\(812\) 8.47406e7 0.00555451
\(813\) 2.50294e9 0.163355
\(814\) −1.02783e10 −0.667941
\(815\) 3.88810e10 2.51586
\(816\) 1.16959e10 0.753559
\(817\) 5.08552e8 0.0326255
\(818\) −3.20784e10 −2.04916
\(819\) −7.03631e8 −0.0447560
\(820\) 8.64053e10 5.47258
\(821\) 2.94003e10 1.85417 0.927087 0.374847i \(-0.122305\pi\)
0.927087 + 0.374847i \(0.122305\pi\)
\(822\) 1.21357e10 0.762103
\(823\) 1.83212e10 1.14566 0.572829 0.819675i \(-0.305846\pi\)
0.572829 + 0.819675i \(0.305846\pi\)
\(824\) 1.19684e10 0.745231
\(825\) −3.87118e9 −0.240024
\(826\) −6.51114e8 −0.0402001
\(827\) −2.71066e10 −1.66650 −0.833251 0.552896i \(-0.813523\pi\)
−0.833251 + 0.552896i \(0.813523\pi\)
\(828\) −5.37474e10 −3.29042
\(829\) −9.94304e9 −0.606147 −0.303074 0.952967i \(-0.598013\pi\)
−0.303074 + 0.952967i \(0.598013\pi\)
\(830\) 1.58174e10 0.960201
\(831\) 7.90695e9 0.477975
\(832\) −2.08279e8 −0.0125376
\(833\) 1.84740e10 1.10740
\(834\) −1.19088e10 −0.710863
\(835\) −2.20910e10 −1.31315
\(836\) −4.67626e9 −0.276807
\(837\) −1.17188e10 −0.690789
\(838\) −4.14671e10 −2.43416
\(839\) −2.24029e10 −1.30960 −0.654798 0.755804i \(-0.727246\pi\)
−0.654798 + 0.755804i \(0.727246\pi\)
\(840\) 1.22352e9 0.0712253
\(841\) −1.72184e10 −0.998176
\(842\) 2.26832e10 1.30952
\(843\) −7.52789e9 −0.432790
\(844\) 2.65002e10 1.51723
\(845\) −4.77350e9 −0.272169
\(846\) 3.00381e10 1.70560
\(847\) −6.84295e8 −0.0386947
\(848\) 6.11068e10 3.44115
\(849\) 5.94786e8 0.0333568
\(850\) 3.94314e10 2.20230
\(851\) 1.97876e10 1.10063
\(852\) 1.95046e9 0.108044
\(853\) 5.08263e9 0.280393 0.140197 0.990124i \(-0.455227\pi\)
0.140197 + 0.990124i \(0.455227\pi\)
\(854\) −2.78493e9 −0.153007
\(855\) 4.85546e9 0.265674
\(856\) 6.72968e10 3.66721
\(857\) 1.75834e10 0.954267 0.477134 0.878831i \(-0.341676\pi\)
0.477134 + 0.878831i \(0.341676\pi\)
\(858\) 6.55087e9 0.354074
\(859\) 3.92669e7 0.00211374 0.00105687 0.999999i \(-0.499664\pi\)
0.00105687 + 0.999999i \(0.499664\pi\)
\(860\) −9.25206e9 −0.496014
\(861\) −6.91006e8 −0.0368953
\(862\) 4.51955e10 2.40337
\(863\) 2.83133e10 1.49952 0.749761 0.661709i \(-0.230168\pi\)
0.749761 + 0.661709i \(0.230168\pi\)
\(864\) −1.31518e10 −0.693725
\(865\) −2.22033e9 −0.116644
\(866\) −3.29443e10 −1.72372
\(867\) −1.70335e9 −0.0887641
\(868\) −2.46424e9 −0.127898
\(869\) −1.85481e10 −0.958803
\(870\) 8.19176e8 0.0421755
\(871\) −5.39258e9 −0.276524
\(872\) 1.22271e10 0.624474
\(873\) −2.21910e10 −1.12882
\(874\) 1.30140e10 0.659357
\(875\) 1.67129e8 0.00843382
\(876\) −1.49869e10 −0.753263
\(877\) −2.63453e10 −1.31888 −0.659440 0.751757i \(-0.729207\pi\)
−0.659440 + 0.751757i \(0.729207\pi\)
\(878\) 4.63409e10 2.31065
\(879\) 8.22650e9 0.408558
\(880\) 3.02767e10 1.49768
\(881\) −7.36199e9 −0.362727 −0.181363 0.983416i \(-0.558051\pi\)
−0.181363 + 0.983416i \(0.558051\pi\)
\(882\) −3.13432e10 −1.53817
\(883\) 2.45713e10 1.20106 0.600531 0.799601i \(-0.294956\pi\)
0.600531 + 0.799601i \(0.294956\pi\)
\(884\) −4.61590e10 −2.24736
\(885\) −4.35413e9 −0.211154
\(886\) 2.17471e10 1.05047
\(887\) 3.76588e9 0.181190 0.0905950 0.995888i \(-0.471123\pi\)
0.0905950 + 0.995888i \(0.471123\pi\)
\(888\) 1.13811e10 0.545431
\(889\) 2.02465e9 0.0966481
\(890\) −8.72902e10 −4.15050
\(891\) 7.19452e9 0.340745
\(892\) −8.15522e10 −3.84732
\(893\) −5.03134e9 −0.236431
\(894\) 1.16787e9 0.0546653
\(895\) −2.93202e10 −1.36705
\(896\) 1.26355e9 0.0586831
\(897\) −1.26116e10 −0.583439
\(898\) 3.52054e10 1.62234
\(899\) −9.14723e8 −0.0419885
\(900\) −4.62790e10 −2.11609
\(901\) −4.68324e10 −2.13309
\(902\) −3.85083e10 −1.74716
\(903\) 7.39911e7 0.00334405
\(904\) −7.28236e9 −0.327856
\(905\) −3.15286e10 −1.41395
\(906\) 2.29981e10 1.02741
\(907\) −6.47573e9 −0.288180 −0.144090 0.989565i \(-0.546025\pi\)
−0.144090 + 0.989565i \(0.546025\pi\)
\(908\) 1.20279e10 0.533197
\(909\) 1.35939e9 0.0600302
\(910\) −3.09957e9 −0.136350
\(911\) −1.27336e10 −0.558004 −0.279002 0.960291i \(-0.590004\pi\)
−0.279002 + 0.960291i \(0.590004\pi\)
\(912\) 3.32374e9 0.145092
\(913\) −4.87650e9 −0.212061
\(914\) −7.85952e9 −0.340474
\(915\) −1.86234e10 −0.803683
\(916\) −2.35807e10 −1.01373
\(917\) −5.49709e7 −0.00235418
\(918\) 3.29567e10 1.40603
\(919\) 6.63174e8 0.0281853 0.0140927 0.999901i \(-0.495514\pi\)
0.0140927 + 0.999901i \(0.495514\pi\)
\(920\) −1.31267e11 −5.55773
\(921\) 1.75224e10 0.739070
\(922\) 5.16261e10 2.16926
\(923\) −2.73948e9 −0.114673
\(924\) −6.80368e8 −0.0283721
\(925\) 1.70380e10 0.707821
\(926\) 1.06186e10 0.439468
\(927\) 6.91035e9 0.284919
\(928\) −1.02657e9 −0.0421670
\(929\) 3.43993e10 1.40765 0.703825 0.710373i \(-0.251474\pi\)
0.703825 + 0.710373i \(0.251474\pi\)
\(930\) −2.38215e10 −0.971133
\(931\) 5.24994e9 0.213221
\(932\) −5.69434e10 −2.30403
\(933\) 2.78839e9 0.112400
\(934\) 4.02692e10 1.61718
\(935\) −2.32041e10 −0.928378
\(936\) 4.34189e10 1.73067
\(937\) −9.38341e9 −0.372625 −0.186313 0.982491i \(-0.559654\pi\)
−0.186313 + 0.982491i \(0.559654\pi\)
\(938\) 8.09623e8 0.0320312
\(939\) −1.54407e10 −0.608606
\(940\) 9.15350e10 3.59451
\(941\) 2.39000e10 0.935050 0.467525 0.883980i \(-0.345146\pi\)
0.467525 + 0.883980i \(0.345146\pi\)
\(942\) −2.07637e10 −0.809330
\(943\) 7.41353e10 2.87895
\(944\) 1.78409e10 0.690261
\(945\) 1.53090e9 0.0590114
\(946\) 4.12337e9 0.158356
\(947\) 4.09342e10 1.56625 0.783126 0.621863i \(-0.213624\pi\)
0.783126 + 0.621863i \(0.213624\pi\)
\(948\) 3.70442e10 1.41218
\(949\) 2.10495e10 0.799484
\(950\) 1.12056e10 0.424037
\(951\) −2.89459e9 −0.109133
\(952\) 3.84223e9 0.144329
\(953\) −2.39304e10 −0.895621 −0.447811 0.894128i \(-0.647796\pi\)
−0.447811 + 0.894128i \(0.647796\pi\)
\(954\) 7.94564e10 2.96285
\(955\) 3.97931e9 0.147841
\(956\) −1.17818e11 −4.36123
\(957\) −2.52551e8 −0.00931446
\(958\) 3.80092e10 1.39672
\(959\) 1.77027e9 0.0648149
\(960\) 2.09110e8 0.00762827
\(961\) −9.12637e8 −0.0331716
\(962\) −2.88320e10 −1.04415
\(963\) 3.88560e10 1.40206
\(964\) −6.80044e10 −2.44493
\(965\) 2.97782e10 1.06673
\(966\) 1.89346e9 0.0675827
\(967\) −9.99102e9 −0.355318 −0.177659 0.984092i \(-0.556852\pi\)
−0.177659 + 0.984092i \(0.556852\pi\)
\(968\) 4.22258e10 1.49628
\(969\) −2.54732e9 −0.0899394
\(970\) −9.77535e10 −3.43899
\(971\) 2.91846e10 1.02303 0.511513 0.859275i \(-0.329085\pi\)
0.511513 + 0.859275i \(0.329085\pi\)
\(972\) −5.95107e10 −2.07856
\(973\) −1.73717e9 −0.0604571
\(974\) −5.40492e10 −1.87427
\(975\) −1.08591e10 −0.375214
\(976\) 7.63086e10 2.62724
\(977\) −1.72546e9 −0.0591935 −0.0295968 0.999562i \(-0.509422\pi\)
−0.0295968 + 0.999562i \(0.509422\pi\)
\(978\) 3.46077e10 1.18300
\(979\) 2.69115e10 0.916639
\(980\) −9.55120e10 −3.24165
\(981\) 7.05969e9 0.238751
\(982\) −6.39960e10 −2.15657
\(983\) 5.02005e10 1.68566 0.842831 0.538178i \(-0.180887\pi\)
0.842831 + 0.538178i \(0.180887\pi\)
\(984\) 4.26399e10 1.42670
\(985\) −1.51345e10 −0.504592
\(986\) 2.57246e9 0.0854633
\(987\) −7.32030e8 −0.0242336
\(988\) −1.31175e10 −0.432714
\(989\) −7.93821e9 −0.260937
\(990\) 3.93684e10 1.28951
\(991\) 3.49646e10 1.14122 0.570612 0.821220i \(-0.306706\pi\)
0.570612 + 0.821220i \(0.306706\pi\)
\(992\) 2.98526e10 0.970937
\(993\) −1.13561e10 −0.368049
\(994\) 4.11296e8 0.0132832
\(995\) 9.60663e9 0.309165
\(996\) 9.73933e9 0.312336
\(997\) −2.66912e9 −0.0852972 −0.0426486 0.999090i \(-0.513580\pi\)
−0.0426486 + 0.999090i \(0.513580\pi\)
\(998\) 3.54745e9 0.112969
\(999\) 1.42403e10 0.451899
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.1.1 13
3.2 odd 2 387.8.a.d.1.13 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.1 13 1.1 even 1 trivial
387.8.a.d.1.13 13 3.2 odd 2