Properties

Label 38.8.a.d.1.2
Level $38$
Weight $8$
Character 38.1
Self dual yes
Analytic conductor $11.871$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.8706309684\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{633}) \)
Defining polynomial: \(x^{2} - x - 158\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-12.0797\) of defining polynomial
Character \(\chi\) \(=\) 38.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +3.23924 q^{3} +64.0000 q^{4} -312.472 q^{5} +25.9139 q^{6} -766.767 q^{7} +512.000 q^{8} -2176.51 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +3.23924 q^{3} +64.0000 q^{4} -312.472 q^{5} +25.9139 q^{6} -766.767 q^{7} +512.000 q^{8} -2176.51 q^{9} -2499.78 q^{10} +252.042 q^{11} +207.311 q^{12} -1065.19 q^{13} -6134.14 q^{14} -1012.17 q^{15} +4096.00 q^{16} -18769.7 q^{17} -17412.1 q^{18} -6859.00 q^{19} -19998.2 q^{20} -2483.74 q^{21} +2016.33 q^{22} +11559.9 q^{23} +1658.49 q^{24} +19513.8 q^{25} -8521.55 q^{26} -14134.4 q^{27} -49073.1 q^{28} -46290.0 q^{29} -8097.37 q^{30} +46848.2 q^{31} +32768.0 q^{32} +816.422 q^{33} -150158. q^{34} +239593. q^{35} -139296. q^{36} -182916. q^{37} -54872.0 q^{38} -3450.42 q^{39} -159986. q^{40} +819661. q^{41} -19869.9 q^{42} +477471. q^{43} +16130.7 q^{44} +680098. q^{45} +92478.8 q^{46} +992580. q^{47} +13267.9 q^{48} -235611. q^{49} +156111. q^{50} -60799.7 q^{51} -68172.4 q^{52} -852516. q^{53} -113075. q^{54} -78756.0 q^{55} -392585. q^{56} -22217.9 q^{57} -370320. q^{58} -1.92404e6 q^{59} -64779.0 q^{60} +209564. q^{61} +374786. q^{62} +1.66887e6 q^{63} +262144. q^{64} +332843. q^{65} +6531.38 q^{66} -2.32447e6 q^{67} -1.20126e6 q^{68} +37445.1 q^{69} +1.91675e6 q^{70} -5.37237e6 q^{71} -1.11437e6 q^{72} -3.71614e6 q^{73} -1.46333e6 q^{74} +63209.9 q^{75} -438976. q^{76} -193257. q^{77} -27603.3 q^{78} -1.30851e6 q^{79} -1.27989e6 q^{80} +4.71424e6 q^{81} +6.55729e6 q^{82} -6.51683e6 q^{83} -158959. q^{84} +5.86502e6 q^{85} +3.81976e6 q^{86} -149944. q^{87} +129045. q^{88} +3.99586e6 q^{89} +5.44078e6 q^{90} +816756. q^{91} +739831. q^{92} +151753. q^{93} +7.94064e6 q^{94} +2.14325e6 q^{95} +106143. q^{96} -6.90391e6 q^{97} -1.88489e6 q^{98} -548570. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 16q^{2} - 69q^{3} + 128q^{4} + 155q^{5} - 552q^{6} - 2238q^{7} + 1024q^{8} + 855q^{9} + O(q^{10}) \) \( 2q + 16q^{2} - 69q^{3} + 128q^{4} + 155q^{5} - 552q^{6} - 2238q^{7} + 1024q^{8} + 855q^{9} + 1240q^{10} - 3295q^{11} - 4416q^{12} - 13427q^{13} - 17904q^{14} - 34782q^{15} + 8192q^{16} - 32256q^{17} + 6840q^{18} - 13718q^{19} + 9920q^{20} + 103797q^{21} - 26360q^{22} - 82525q^{23} - 35328q^{24} + 159919q^{25} - 107416q^{26} - 75141q^{27} - 143232q^{28} - 12749q^{29} - 278256q^{30} + 258944q^{31} + 65536q^{32} + 257052q^{33} - 258048q^{34} - 448167q^{35} + 54720q^{36} - 149260q^{37} - 109744q^{38} + 889557q^{39} + 79360q^{40} + 339130q^{41} + 830376q^{42} - 83869q^{43} - 210880q^{44} + 2097243q^{45} - 660200q^{46} + 1471025q^{47} - 282624q^{48} + 1105372q^{49} + 1279352q^{50} + 913437q^{51} - 859328q^{52} - 945643q^{53} - 601128q^{54} - 1736899q^{55} - 1145856q^{56} + 473271q^{57} - 101992q^{58} - 969009q^{59} - 2226048q^{60} - 1506755q^{61} + 2071552q^{62} - 2791179q^{63} + 524288q^{64} - 5445956q^{65} + 2056416q^{66} - 1848219q^{67} - 2064384q^{68} + 6834063q^{69} - 3585336q^{70} - 3417184q^{71} + 437760q^{72} - 2499822q^{73} - 1194080q^{74} - 10079553q^{75} - 877952q^{76} + 5025267q^{77} + 7116456q^{78} + 2636926q^{79} + 634880q^{80} + 2491398q^{81} + 2713040q^{82} - 10059354q^{83} + 6643008q^{84} - 439425q^{85} - 670952q^{86} - 2572923q^{87} - 1687040q^{88} - 3506160q^{89} + 16777944q^{90} + 19003851q^{91} - 5281600q^{92} - 15169884q^{93} + 11768200q^{94} - 1063145q^{95} - 2260992q^{96} + 5893526q^{97} + 8842976q^{98} - 11301453q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 3.23924 0.0692657 0.0346329 0.999400i \(-0.488974\pi\)
0.0346329 + 0.999400i \(0.488974\pi\)
\(4\) 64.0000 0.500000
\(5\) −312.472 −1.11793 −0.558967 0.829190i \(-0.688802\pi\)
−0.558967 + 0.829190i \(0.688802\pi\)
\(6\) 25.9139 0.0489783
\(7\) −766.767 −0.844929 −0.422465 0.906379i \(-0.638835\pi\)
−0.422465 + 0.906379i \(0.638835\pi\)
\(8\) 512.000 0.353553
\(9\) −2176.51 −0.995202
\(10\) −2499.78 −0.790499
\(11\) 252.042 0.0570950 0.0285475 0.999592i \(-0.490912\pi\)
0.0285475 + 0.999592i \(0.490912\pi\)
\(12\) 207.311 0.0346329
\(13\) −1065.19 −0.134471 −0.0672353 0.997737i \(-0.521418\pi\)
−0.0672353 + 0.997737i \(0.521418\pi\)
\(14\) −6134.14 −0.597455
\(15\) −1012.17 −0.0774345
\(16\) 4096.00 0.250000
\(17\) −18769.7 −0.926589 −0.463295 0.886204i \(-0.653333\pi\)
−0.463295 + 0.886204i \(0.653333\pi\)
\(18\) −17412.1 −0.703714
\(19\) −6859.00 −0.229416
\(20\) −19998.2 −0.558967
\(21\) −2483.74 −0.0585246
\(22\) 2016.33 0.0403722
\(23\) 11559.9 0.198109 0.0990546 0.995082i \(-0.468418\pi\)
0.0990546 + 0.995082i \(0.468418\pi\)
\(24\) 1658.49 0.0244891
\(25\) 19513.8 0.249777
\(26\) −8521.55 −0.0950850
\(27\) −14134.4 −0.138199
\(28\) −49073.1 −0.422465
\(29\) −46290.0 −0.352448 −0.176224 0.984350i \(-0.556388\pi\)
−0.176224 + 0.984350i \(0.556388\pi\)
\(30\) −8097.37 −0.0547545
\(31\) 46848.2 0.282441 0.141220 0.989978i \(-0.454897\pi\)
0.141220 + 0.989978i \(0.454897\pi\)
\(32\) 32768.0 0.176777
\(33\) 816.422 0.00395472
\(34\) −150158. −0.655197
\(35\) 239593. 0.944575
\(36\) −139296. −0.497601
\(37\) −182916. −0.593672 −0.296836 0.954928i \(-0.595932\pi\)
−0.296836 + 0.954928i \(0.595932\pi\)
\(38\) −54872.0 −0.162221
\(39\) −3450.42 −0.00931420
\(40\) −159986. −0.395249
\(41\) 819661. 1.85734 0.928669 0.370910i \(-0.120954\pi\)
0.928669 + 0.370910i \(0.120954\pi\)
\(42\) −19869.9 −0.0413832
\(43\) 477471. 0.915814 0.457907 0.889000i \(-0.348599\pi\)
0.457907 + 0.889000i \(0.348599\pi\)
\(44\) 16130.7 0.0285475
\(45\) 680098. 1.11257
\(46\) 92478.8 0.140084
\(47\) 992580. 1.39451 0.697257 0.716821i \(-0.254404\pi\)
0.697257 + 0.716821i \(0.254404\pi\)
\(48\) 13267.9 0.0173164
\(49\) −235611. −0.286095
\(50\) 156111. 0.176619
\(51\) −60799.7 −0.0641809
\(52\) −68172.4 −0.0672353
\(53\) −852516. −0.786569 −0.393285 0.919417i \(-0.628661\pi\)
−0.393285 + 0.919417i \(0.628661\pi\)
\(54\) −113075. −0.0977215
\(55\) −78756.0 −0.0638284
\(56\) −392585. −0.298728
\(57\) −22217.9 −0.0158906
\(58\) −370320. −0.249218
\(59\) −1.92404e6 −1.21964 −0.609821 0.792539i \(-0.708759\pi\)
−0.609821 + 0.792539i \(0.708759\pi\)
\(60\) −64779.0 −0.0387173
\(61\) 209564. 0.118212 0.0591062 0.998252i \(-0.481175\pi\)
0.0591062 + 0.998252i \(0.481175\pi\)
\(62\) 374786. 0.199716
\(63\) 1.66887e6 0.840875
\(64\) 262144. 0.125000
\(65\) 332843. 0.150329
\(66\) 6531.38 0.00279641
\(67\) −2.32447e6 −0.944198 −0.472099 0.881546i \(-0.656503\pi\)
−0.472099 + 0.881546i \(0.656503\pi\)
\(68\) −1.20126e6 −0.463295
\(69\) 37445.1 0.0137222
\(70\) 1.91675e6 0.667916
\(71\) −5.37237e6 −1.78140 −0.890700 0.454591i \(-0.849785\pi\)
−0.890700 + 0.454591i \(0.849785\pi\)
\(72\) −1.11437e6 −0.351857
\(73\) −3.71614e6 −1.11805 −0.559027 0.829150i \(-0.688825\pi\)
−0.559027 + 0.829150i \(0.688825\pi\)
\(74\) −1.46333e6 −0.419790
\(75\) 63209.9 0.0173010
\(76\) −438976. −0.114708
\(77\) −193257. −0.0482412
\(78\) −27603.3 −0.00658613
\(79\) −1.30851e6 −0.298596 −0.149298 0.988792i \(-0.547701\pi\)
−0.149298 + 0.988792i \(0.547701\pi\)
\(80\) −1.27989e6 −0.279484
\(81\) 4.71424e6 0.985630
\(82\) 6.55729e6 1.31334
\(83\) −6.51683e6 −1.25102 −0.625508 0.780218i \(-0.715108\pi\)
−0.625508 + 0.780218i \(0.715108\pi\)
\(84\) −158959. −0.0292623
\(85\) 5.86502e6 1.03587
\(86\) 3.81976e6 0.647578
\(87\) −149944. −0.0244125
\(88\) 129045. 0.0201861
\(89\) 3.99586e6 0.600822 0.300411 0.953810i \(-0.402876\pi\)
0.300411 + 0.953810i \(0.402876\pi\)
\(90\) 5.44078e6 0.786706
\(91\) 816756. 0.113618
\(92\) 739831. 0.0990546
\(93\) 151753. 0.0195635
\(94\) 7.94064e6 0.986070
\(95\) 2.14325e6 0.256472
\(96\) 106143. 0.0122446
\(97\) −6.90391e6 −0.768058 −0.384029 0.923321i \(-0.625464\pi\)
−0.384029 + 0.923321i \(0.625464\pi\)
\(98\) −1.88489e6 −0.202299
\(99\) −548570. −0.0568210
\(100\) 1.24888e6 0.124888
\(101\) 2.81610e6 0.271971 0.135986 0.990711i \(-0.456580\pi\)
0.135986 + 0.990711i \(0.456580\pi\)
\(102\) −486397. −0.0453827
\(103\) 7.86273e6 0.708995 0.354497 0.935057i \(-0.384652\pi\)
0.354497 + 0.935057i \(0.384652\pi\)
\(104\) −545379. −0.0475425
\(105\) 776100. 0.0654267
\(106\) −6.82013e6 −0.556188
\(107\) 611626. 0.0482662 0.0241331 0.999709i \(-0.492317\pi\)
0.0241331 + 0.999709i \(0.492317\pi\)
\(108\) −904604. −0.0690996
\(109\) 5.23210e6 0.386976 0.193488 0.981103i \(-0.438020\pi\)
0.193488 + 0.981103i \(0.438020\pi\)
\(110\) −630048. −0.0451335
\(111\) −592510. −0.0411211
\(112\) −3.14068e6 −0.211232
\(113\) −3.58527e6 −0.233747 −0.116874 0.993147i \(-0.537287\pi\)
−0.116874 + 0.993147i \(0.537287\pi\)
\(114\) −177743. −0.0112364
\(115\) −3.61213e6 −0.221473
\(116\) −2.96256e6 −0.176224
\(117\) 2.31840e6 0.133825
\(118\) −1.53923e7 −0.862418
\(119\) 1.43920e7 0.782902
\(120\) −518232. −0.0273772
\(121\) −1.94236e7 −0.996740
\(122\) 1.67651e6 0.0835888
\(123\) 2.65508e6 0.128650
\(124\) 2.99829e6 0.141220
\(125\) 1.83144e7 0.838700
\(126\) 1.33510e7 0.594589
\(127\) −2.99465e7 −1.29728 −0.648639 0.761096i \(-0.724661\pi\)
−0.648639 + 0.761096i \(0.724661\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 1.54664e6 0.0634345
\(130\) 2.66275e6 0.106299
\(131\) −8.11292e6 −0.315303 −0.157651 0.987495i \(-0.550392\pi\)
−0.157651 + 0.987495i \(0.550392\pi\)
\(132\) 52251.0 0.00197736
\(133\) 5.25926e6 0.193840
\(134\) −1.85958e7 −0.667649
\(135\) 4.41662e6 0.154498
\(136\) −9.61011e6 −0.327599
\(137\) 1.44157e7 0.478977 0.239488 0.970899i \(-0.423020\pi\)
0.239488 + 0.970899i \(0.423020\pi\)
\(138\) 299561. 0.00970305
\(139\) −3.77446e7 −1.19207 −0.596036 0.802957i \(-0.703259\pi\)
−0.596036 + 0.802957i \(0.703259\pi\)
\(140\) 1.53340e7 0.472288
\(141\) 3.21520e6 0.0965920
\(142\) −4.29789e7 −1.25964
\(143\) −268473. −0.00767759
\(144\) −8.91497e6 −0.248801
\(145\) 1.44643e7 0.394013
\(146\) −2.97292e7 −0.790583
\(147\) −763200. −0.0198165
\(148\) −1.17067e7 −0.296836
\(149\) 7.30679e7 1.80957 0.904784 0.425872i \(-0.140033\pi\)
0.904784 + 0.425872i \(0.140033\pi\)
\(150\) 505679. 0.0122336
\(151\) −4.14056e7 −0.978678 −0.489339 0.872094i \(-0.662762\pi\)
−0.489339 + 0.872094i \(0.662762\pi\)
\(152\) −3.51181e6 −0.0811107
\(153\) 4.08525e7 0.922144
\(154\) −1.54606e6 −0.0341117
\(155\) −1.46388e7 −0.315750
\(156\) −220827. −0.00465710
\(157\) 2.93945e7 0.606203 0.303101 0.952958i \(-0.401978\pi\)
0.303101 + 0.952958i \(0.401978\pi\)
\(158\) −1.04681e7 −0.211139
\(159\) −2.76150e6 −0.0544823
\(160\) −1.02391e7 −0.197625
\(161\) −8.86371e6 −0.167388
\(162\) 3.77139e7 0.696946
\(163\) 9.03924e6 0.163484 0.0817420 0.996654i \(-0.473952\pi\)
0.0817420 + 0.996654i \(0.473952\pi\)
\(164\) 5.24583e7 0.928669
\(165\) −255109. −0.00442112
\(166\) −5.21346e7 −0.884602
\(167\) −8.16728e7 −1.35697 −0.678484 0.734615i \(-0.737363\pi\)
−0.678484 + 0.734615i \(0.737363\pi\)
\(168\) −1.27168e6 −0.0206916
\(169\) −6.16139e7 −0.981918
\(170\) 4.69202e7 0.732468
\(171\) 1.49287e7 0.228315
\(172\) 3.05581e7 0.457907
\(173\) 1.33191e8 1.95575 0.977874 0.209197i \(-0.0670849\pi\)
0.977874 + 0.209197i \(0.0670849\pi\)
\(174\) −1.19956e6 −0.0172623
\(175\) −1.49626e7 −0.211044
\(176\) 1.03236e6 0.0142737
\(177\) −6.23243e6 −0.0844794
\(178\) 3.19669e7 0.424845
\(179\) −4.90910e7 −0.639759 −0.319879 0.947458i \(-0.603642\pi\)
−0.319879 + 0.947458i \(0.603642\pi\)
\(180\) 4.35263e7 0.556285
\(181\) 8.11979e7 1.01782 0.508908 0.860821i \(-0.330049\pi\)
0.508908 + 0.860821i \(0.330049\pi\)
\(182\) 6.53405e6 0.0803401
\(183\) 678828. 0.00818806
\(184\) 5.91864e6 0.0700422
\(185\) 5.71563e7 0.663686
\(186\) 1.21402e6 0.0138335
\(187\) −4.73076e6 −0.0529036
\(188\) 6.35251e7 0.697257
\(189\) 1.08378e7 0.116768
\(190\) 1.71460e7 0.181353
\(191\) 7.40713e7 0.769190 0.384595 0.923086i \(-0.374341\pi\)
0.384595 + 0.923086i \(0.374341\pi\)
\(192\) 849147. 0.00865821
\(193\) −9.70125e7 −0.971353 −0.485676 0.874139i \(-0.661427\pi\)
−0.485676 + 0.874139i \(0.661427\pi\)
\(194\) −5.52313e7 −0.543099
\(195\) 1.07816e6 0.0104127
\(196\) −1.50791e7 −0.143047
\(197\) −4.34770e7 −0.405161 −0.202581 0.979266i \(-0.564933\pi\)
−0.202581 + 0.979266i \(0.564933\pi\)
\(198\) −4.38856e6 −0.0401785
\(199\) 3.89651e7 0.350502 0.175251 0.984524i \(-0.443926\pi\)
0.175251 + 0.984524i \(0.443926\pi\)
\(200\) 9.99108e6 0.0883095
\(201\) −7.52952e6 −0.0654005
\(202\) 2.25288e7 0.192313
\(203\) 3.54937e7 0.297793
\(204\) −3.89118e6 −0.0320904
\(205\) −2.56121e8 −2.07638
\(206\) 6.29018e7 0.501335
\(207\) −2.51601e7 −0.197159
\(208\) −4.36304e6 −0.0336176
\(209\) −1.72875e6 −0.0130985
\(210\) 6.20880e6 0.0462637
\(211\) −3.85264e7 −0.282339 −0.141169 0.989985i \(-0.545086\pi\)
−0.141169 + 0.989985i \(0.545086\pi\)
\(212\) −5.45610e7 −0.393285
\(213\) −1.74024e7 −0.123390
\(214\) 4.89301e6 0.0341293
\(215\) −1.49196e8 −1.02382
\(216\) −7.23683e6 −0.0488608
\(217\) −3.59217e7 −0.238642
\(218\) 4.18568e7 0.273633
\(219\) −1.20375e7 −0.0774428
\(220\) −5.04038e6 −0.0319142
\(221\) 1.99934e7 0.124599
\(222\) −4.74008e6 −0.0290770
\(223\) −2.03401e8 −1.22825 −0.614124 0.789209i \(-0.710491\pi\)
−0.614124 + 0.789209i \(0.710491\pi\)
\(224\) −2.51254e7 −0.149364
\(225\) −4.24720e7 −0.248579
\(226\) −2.86821e7 −0.165284
\(227\) 2.24944e8 1.27639 0.638195 0.769875i \(-0.279681\pi\)
0.638195 + 0.769875i \(0.279681\pi\)
\(228\) −1.42195e6 −0.00794532
\(229\) −2.38882e8 −1.31450 −0.657248 0.753674i \(-0.728280\pi\)
−0.657248 + 0.753674i \(0.728280\pi\)
\(230\) −2.88971e7 −0.156605
\(231\) −626006. −0.00334146
\(232\) −2.37005e7 −0.124609
\(233\) 3.23019e8 1.67295 0.836475 0.548005i \(-0.184613\pi\)
0.836475 + 0.548005i \(0.184613\pi\)
\(234\) 1.85472e7 0.0946288
\(235\) −3.10153e8 −1.55897
\(236\) −1.23139e8 −0.609821
\(237\) −4.23859e6 −0.0206825
\(238\) 1.15136e8 0.553595
\(239\) −5.18160e7 −0.245511 −0.122756 0.992437i \(-0.539173\pi\)
−0.122756 + 0.992437i \(0.539173\pi\)
\(240\) −4.14585e6 −0.0193586
\(241\) 1.15926e8 0.533484 0.266742 0.963768i \(-0.414053\pi\)
0.266742 + 0.963768i \(0.414053\pi\)
\(242\) −1.55389e8 −0.704802
\(243\) 4.61825e7 0.206469
\(244\) 1.34121e7 0.0591062
\(245\) 7.36219e7 0.319835
\(246\) 2.12406e7 0.0909692
\(247\) 7.30617e6 0.0308497
\(248\) 2.39863e7 0.0998579
\(249\) −2.11096e7 −0.0866525
\(250\) 1.46515e8 0.593050
\(251\) −4.03761e7 −0.161163 −0.0805817 0.996748i \(-0.525678\pi\)
−0.0805817 + 0.996748i \(0.525678\pi\)
\(252\) 1.06808e8 0.420438
\(253\) 2.91356e6 0.0113110
\(254\) −2.39572e8 −0.917314
\(255\) 1.89982e7 0.0717500
\(256\) 1.67772e7 0.0625000
\(257\) 2.67661e8 0.983602 0.491801 0.870708i \(-0.336339\pi\)
0.491801 + 0.870708i \(0.336339\pi\)
\(258\) 1.23731e7 0.0448549
\(259\) 1.40254e8 0.501611
\(260\) 2.13020e7 0.0751646
\(261\) 1.00751e8 0.350757
\(262\) −6.49034e7 −0.222953
\(263\) 5.12778e8 1.73814 0.869068 0.494692i \(-0.164719\pi\)
0.869068 + 0.494692i \(0.164719\pi\)
\(264\) 418008. 0.00139821
\(265\) 2.66387e8 0.879333
\(266\) 4.20740e7 0.137066
\(267\) 1.29435e7 0.0416163
\(268\) −1.48766e8 −0.472099
\(269\) 5.75792e8 1.80357 0.901784 0.432187i \(-0.142258\pi\)
0.901784 + 0.432187i \(0.142258\pi\)
\(270\) 3.53329e7 0.109246
\(271\) 5.24839e8 1.60189 0.800946 0.598737i \(-0.204330\pi\)
0.800946 + 0.598737i \(0.204330\pi\)
\(272\) −7.68809e7 −0.231647
\(273\) 2.64567e6 0.00786984
\(274\) 1.15326e8 0.338688
\(275\) 4.91829e6 0.0142610
\(276\) 2.39649e6 0.00686109
\(277\) −4.64620e8 −1.31347 −0.656733 0.754123i \(-0.728062\pi\)
−0.656733 + 0.754123i \(0.728062\pi\)
\(278\) −3.01957e8 −0.842923
\(279\) −1.01966e8 −0.281086
\(280\) 1.22672e8 0.333958
\(281\) −1.75429e8 −0.471660 −0.235830 0.971794i \(-0.575781\pi\)
−0.235830 + 0.971794i \(0.575781\pi\)
\(282\) 2.57216e7 0.0683009
\(283\) 1.44948e8 0.380154 0.190077 0.981769i \(-0.439126\pi\)
0.190077 + 0.981769i \(0.439126\pi\)
\(284\) −3.43832e8 −0.890700
\(285\) 6.94248e6 0.0177647
\(286\) −2.14779e6 −0.00542888
\(287\) −6.28489e8 −1.56932
\(288\) −7.13198e7 −0.175929
\(289\) −5.80353e7 −0.141433
\(290\) 1.15715e8 0.278609
\(291\) −2.23634e7 −0.0532001
\(292\) −2.37833e8 −0.559027
\(293\) −4.24522e8 −0.985970 −0.492985 0.870038i \(-0.664094\pi\)
−0.492985 + 0.870038i \(0.664094\pi\)
\(294\) −6.10560e6 −0.0140124
\(295\) 6.01210e8 1.36348
\(296\) −9.36532e7 −0.209895
\(297\) −3.56247e6 −0.00789047
\(298\) 5.84543e8 1.27956
\(299\) −1.23135e7 −0.0266399
\(300\) 4.04543e6 0.00865049
\(301\) −3.66109e8 −0.773798
\(302\) −3.31245e8 −0.692030
\(303\) 9.12200e6 0.0188383
\(304\) −2.80945e7 −0.0573539
\(305\) −6.54830e7 −0.132154
\(306\) 3.26820e8 0.652054
\(307\) 4.27528e8 0.843297 0.421649 0.906759i \(-0.361452\pi\)
0.421649 + 0.906759i \(0.361452\pi\)
\(308\) −1.23685e7 −0.0241206
\(309\) 2.54692e7 0.0491090
\(310\) −1.17110e8 −0.223269
\(311\) −1.70275e8 −0.320989 −0.160495 0.987037i \(-0.551309\pi\)
−0.160495 + 0.987037i \(0.551309\pi\)
\(312\) −1.76661e6 −0.00329307
\(313\) −4.02143e8 −0.741269 −0.370634 0.928779i \(-0.620860\pi\)
−0.370634 + 0.928779i \(0.620860\pi\)
\(314\) 2.35156e8 0.428650
\(315\) −5.21477e8 −0.940043
\(316\) −8.37450e7 −0.149298
\(317\) −1.90630e8 −0.336112 −0.168056 0.985777i \(-0.553749\pi\)
−0.168056 + 0.985777i \(0.553749\pi\)
\(318\) −2.20920e7 −0.0385248
\(319\) −1.16670e7 −0.0201230
\(320\) −8.19127e7 −0.139742
\(321\) 1.98120e6 0.00334319
\(322\) −7.09097e7 −0.118361
\(323\) 1.28742e8 0.212574
\(324\) 3.01711e8 0.492815
\(325\) −2.07860e7 −0.0335876
\(326\) 7.23139e7 0.115601
\(327\) 1.69480e7 0.0268041
\(328\) 4.19666e8 0.656668
\(329\) −7.61077e8 −1.17827
\(330\) −2.04087e6 −0.00312620
\(331\) −4.12008e8 −0.624464 −0.312232 0.950006i \(-0.601077\pi\)
−0.312232 + 0.950006i \(0.601077\pi\)
\(332\) −4.17077e8 −0.625508
\(333\) 3.98119e8 0.590824
\(334\) −6.53382e8 −0.959522
\(335\) 7.26333e8 1.05555
\(336\) −1.01734e7 −0.0146312
\(337\) −6.99272e8 −0.995271 −0.497636 0.867386i \(-0.665798\pi\)
−0.497636 + 0.867386i \(0.665798\pi\)
\(338\) −4.92911e8 −0.694321
\(339\) −1.16135e7 −0.0161907
\(340\) 3.75361e8 0.517933
\(341\) 1.18077e7 0.0161259
\(342\) 1.19429e8 0.161443
\(343\) 8.12125e8 1.08666
\(344\) 2.44465e8 0.323789
\(345\) −1.17005e7 −0.0153405
\(346\) 1.06553e9 1.38292
\(347\) −9.29885e8 −1.19475 −0.597374 0.801963i \(-0.703789\pi\)
−0.597374 + 0.801963i \(0.703789\pi\)
\(348\) −9.59644e6 −0.0122063
\(349\) −4.43776e8 −0.558823 −0.279412 0.960171i \(-0.590139\pi\)
−0.279412 + 0.960171i \(0.590139\pi\)
\(350\) −1.19700e8 −0.149231
\(351\) 1.50559e7 0.0185837
\(352\) 8.25890e6 0.0100931
\(353\) −1.49388e9 −1.80761 −0.903804 0.427946i \(-0.859237\pi\)
−0.903804 + 0.427946i \(0.859237\pi\)
\(354\) −4.98594e7 −0.0597360
\(355\) 1.67872e9 1.99149
\(356\) 2.55735e8 0.300411
\(357\) 4.66192e7 0.0542283
\(358\) −3.92728e8 −0.452378
\(359\) −1.77937e8 −0.202972 −0.101486 0.994837i \(-0.532360\pi\)
−0.101486 + 0.994837i \(0.532360\pi\)
\(360\) 3.48210e8 0.393353
\(361\) 4.70459e7 0.0526316
\(362\) 6.49583e8 0.719705
\(363\) −6.29178e7 −0.0690399
\(364\) 5.22724e7 0.0568090
\(365\) 1.16119e9 1.24991
\(366\) 5.43063e6 0.00578983
\(367\) −1.37515e9 −1.45218 −0.726088 0.687601i \(-0.758664\pi\)
−0.726088 + 0.687601i \(0.758664\pi\)
\(368\) 4.73492e7 0.0495273
\(369\) −1.78400e9 −1.84843
\(370\) 4.57250e8 0.469297
\(371\) 6.53681e8 0.664595
\(372\) 9.71216e6 0.00978173
\(373\) 1.55845e9 1.55493 0.777466 0.628925i \(-0.216505\pi\)
0.777466 + 0.628925i \(0.216505\pi\)
\(374\) −3.78461e7 −0.0374085
\(375\) 5.93245e7 0.0580932
\(376\) 5.08201e8 0.493035
\(377\) 4.93079e7 0.0473938
\(378\) 8.67026e7 0.0825678
\(379\) 8.84549e8 0.834613 0.417306 0.908766i \(-0.362974\pi\)
0.417306 + 0.908766i \(0.362974\pi\)
\(380\) 1.37168e8 0.128236
\(381\) −9.70038e7 −0.0898569
\(382\) 5.92571e8 0.543899
\(383\) 1.44573e9 1.31490 0.657449 0.753499i \(-0.271635\pi\)
0.657449 + 0.753499i \(0.271635\pi\)
\(384\) 6.79317e6 0.00612228
\(385\) 6.03875e7 0.0539305
\(386\) −7.76100e8 −0.686850
\(387\) −1.03922e9 −0.911420
\(388\) −4.41850e8 −0.384029
\(389\) −1.02164e9 −0.879986 −0.439993 0.898001i \(-0.645019\pi\)
−0.439993 + 0.898001i \(0.645019\pi\)
\(390\) 8.62527e6 0.00736286
\(391\) −2.16975e8 −0.183566
\(392\) −1.20633e8 −0.101150
\(393\) −2.62797e7 −0.0218397
\(394\) −3.47816e8 −0.286492
\(395\) 4.08874e8 0.333811
\(396\) −3.51085e7 −0.0284105
\(397\) 1.80050e8 0.144419 0.0722097 0.997389i \(-0.476995\pi\)
0.0722097 + 0.997389i \(0.476995\pi\)
\(398\) 3.11721e8 0.247842
\(399\) 1.70360e7 0.0134265
\(400\) 7.99286e7 0.0624442
\(401\) 5.06670e8 0.392392 0.196196 0.980565i \(-0.437141\pi\)
0.196196 + 0.980565i \(0.437141\pi\)
\(402\) −6.02362e7 −0.0462452
\(403\) −4.99025e7 −0.0379799
\(404\) 1.80230e8 0.135986
\(405\) −1.47307e9 −1.10187
\(406\) 2.83949e8 0.210572
\(407\) −4.61026e7 −0.0338957
\(408\) −3.11294e7 −0.0226914
\(409\) 1.14898e9 0.830388 0.415194 0.909733i \(-0.363714\pi\)
0.415194 + 0.909733i \(0.363714\pi\)
\(410\) −2.04897e9 −1.46822
\(411\) 4.66959e7 0.0331767
\(412\) 5.03215e8 0.354497
\(413\) 1.47529e9 1.03051
\(414\) −2.01281e8 −0.139412
\(415\) 2.03633e9 1.39855
\(416\) −3.49043e7 −0.0237713
\(417\) −1.22264e8 −0.0825698
\(418\) −1.38300e7 −0.00926203
\(419\) 2.02289e9 1.34346 0.671729 0.740797i \(-0.265552\pi\)
0.671729 + 0.740797i \(0.265552\pi\)
\(420\) 4.96704e7 0.0327133
\(421\) 2.55503e9 1.66882 0.834408 0.551147i \(-0.185810\pi\)
0.834408 + 0.551147i \(0.185810\pi\)
\(422\) −3.08211e8 −0.199644
\(423\) −2.16036e9 −1.38782
\(424\) −4.36488e8 −0.278094
\(425\) −3.66269e8 −0.231441
\(426\) −1.39219e8 −0.0872499
\(427\) −1.60687e8 −0.0998811
\(428\) 3.91441e7 0.0241331
\(429\) −869648. −0.000531794 0
\(430\) −1.19357e9 −0.723950
\(431\) 1.47316e9 0.886295 0.443147 0.896449i \(-0.353862\pi\)
0.443147 + 0.896449i \(0.353862\pi\)
\(432\) −5.78946e7 −0.0345498
\(433\) −9.33243e8 −0.552443 −0.276221 0.961094i \(-0.589082\pi\)
−0.276221 + 0.961094i \(0.589082\pi\)
\(434\) −2.87373e8 −0.168746
\(435\) 4.68534e7 0.0272916
\(436\) 3.34855e8 0.193488
\(437\) −7.92890e7 −0.0454494
\(438\) −9.62998e7 −0.0547603
\(439\) −1.94411e9 −1.09672 −0.548358 0.836244i \(-0.684747\pi\)
−0.548358 + 0.836244i \(0.684747\pi\)
\(440\) −4.03231e7 −0.0225668
\(441\) 5.12809e8 0.284722
\(442\) 1.59947e8 0.0881047
\(443\) −2.35592e9 −1.28750 −0.643751 0.765235i \(-0.722623\pi\)
−0.643751 + 0.765235i \(0.722623\pi\)
\(444\) −3.79206e7 −0.0205606
\(445\) −1.24860e9 −0.671679
\(446\) −1.62721e9 −0.868503
\(447\) 2.36684e8 0.125341
\(448\) −2.01003e8 −0.105616
\(449\) −1.83374e9 −0.956042 −0.478021 0.878349i \(-0.658646\pi\)
−0.478021 + 0.878349i \(0.658646\pi\)
\(450\) −3.39776e8 −0.175772
\(451\) 2.06589e8 0.106045
\(452\) −2.29457e8 −0.116874
\(453\) −1.34123e8 −0.0677889
\(454\) 1.79955e9 0.902544
\(455\) −2.55213e8 −0.127018
\(456\) −1.13756e7 −0.00561819
\(457\) −3.76683e9 −1.84616 −0.923079 0.384611i \(-0.874336\pi\)
−0.923079 + 0.384611i \(0.874336\pi\)
\(458\) −1.91106e9 −0.929489
\(459\) 2.65300e8 0.128054
\(460\) −2.31176e8 −0.110737
\(461\) 3.30560e9 1.57144 0.785720 0.618583i \(-0.212293\pi\)
0.785720 + 0.618583i \(0.212293\pi\)
\(462\) −5.00805e6 −0.00236277
\(463\) −3.66326e9 −1.71528 −0.857638 0.514253i \(-0.828069\pi\)
−0.857638 + 0.514253i \(0.828069\pi\)
\(464\) −1.89604e8 −0.0881119
\(465\) −4.74184e7 −0.0218707
\(466\) 2.58416e9 1.18295
\(467\) −2.22674e9 −1.01172 −0.505860 0.862616i \(-0.668825\pi\)
−0.505860 + 0.862616i \(0.668825\pi\)
\(468\) 1.48378e8 0.0669127
\(469\) 1.78233e9 0.797780
\(470\) −2.48123e9 −1.10236
\(471\) 9.52159e7 0.0419891
\(472\) −9.85110e8 −0.431209
\(473\) 1.20342e8 0.0522883
\(474\) −3.39087e7 −0.0146247
\(475\) −1.33845e8 −0.0573028
\(476\) 9.21090e8 0.391451
\(477\) 1.85551e9 0.782796
\(478\) −4.14528e8 −0.173603
\(479\) 1.43508e9 0.596625 0.298313 0.954468i \(-0.403576\pi\)
0.298313 + 0.954468i \(0.403576\pi\)
\(480\) −3.31668e7 −0.0136886
\(481\) 1.94842e8 0.0798314
\(482\) 9.27409e8 0.377230
\(483\) −2.87117e7 −0.0115943
\(484\) −1.24311e9 −0.498370
\(485\) 2.15728e9 0.858638
\(486\) 3.69460e8 0.145996
\(487\) −2.11642e9 −0.830330 −0.415165 0.909746i \(-0.636276\pi\)
−0.415165 + 0.909746i \(0.636276\pi\)
\(488\) 1.07297e8 0.0417944
\(489\) 2.92802e7 0.0113238
\(490\) 5.88975e8 0.226157
\(491\) 4.07105e9 1.55211 0.776053 0.630668i \(-0.217219\pi\)
0.776053 + 0.630668i \(0.217219\pi\)
\(492\) 1.69925e8 0.0643249
\(493\) 8.68852e8 0.326574
\(494\) 5.84493e7 0.0218140
\(495\) 1.71413e8 0.0635222
\(496\) 1.91890e8 0.0706102
\(497\) 4.11935e9 1.50516
\(498\) −1.68876e8 −0.0612726
\(499\) 3.93039e9 1.41607 0.708034 0.706178i \(-0.249582\pi\)
0.708034 + 0.706178i \(0.249582\pi\)
\(500\) 1.17212e9 0.419350
\(501\) −2.64558e8 −0.0939914
\(502\) −3.23009e8 −0.113960
\(503\) −1.70030e9 −0.595714 −0.297857 0.954610i \(-0.596272\pi\)
−0.297857 + 0.954610i \(0.596272\pi\)
\(504\) 8.54464e8 0.297294
\(505\) −8.79952e8 −0.304046
\(506\) 2.33085e7 0.00799812
\(507\) −1.99582e8 −0.0680132
\(508\) −1.91658e9 −0.648639
\(509\) −2.62955e8 −0.0883830 −0.0441915 0.999023i \(-0.514071\pi\)
−0.0441915 + 0.999023i \(0.514071\pi\)
\(510\) 1.51986e8 0.0507349
\(511\) 2.84942e9 0.944676
\(512\) 1.34218e8 0.0441942
\(513\) 9.69481e7 0.0317051
\(514\) 2.14129e9 0.695512
\(515\) −2.45688e9 −0.792609
\(516\) 9.89850e7 0.0317172
\(517\) 2.50171e8 0.0796197
\(518\) 1.12203e9 0.354693
\(519\) 4.31436e8 0.135466
\(520\) 1.70416e8 0.0531494
\(521\) −1.86990e9 −0.579277 −0.289639 0.957136i \(-0.593535\pi\)
−0.289639 + 0.957136i \(0.593535\pi\)
\(522\) 8.06005e8 0.248022
\(523\) −5.49460e9 −1.67950 −0.839750 0.542972i \(-0.817299\pi\)
−0.839750 + 0.542972i \(0.817299\pi\)
\(524\) −5.19227e8 −0.157651
\(525\) −4.84673e7 −0.0146181
\(526\) 4.10222e9 1.22905
\(527\) −8.79329e8 −0.261706
\(528\) 3.34407e6 0.000988681 0
\(529\) −3.27120e9 −0.960753
\(530\) 2.13110e9 0.621782
\(531\) 4.18769e9 1.21379
\(532\) 3.36592e8 0.0969200
\(533\) −8.73098e8 −0.249757
\(534\) 1.03548e8 0.0294272
\(535\) −1.91116e8 −0.0539584
\(536\) −1.19013e9 −0.333824
\(537\) −1.59017e8 −0.0443133
\(538\) 4.60634e9 1.27532
\(539\) −5.93838e7 −0.0163346
\(540\) 2.82663e8 0.0772488
\(541\) 6.97892e9 1.89495 0.947475 0.319830i \(-0.103626\pi\)
0.947475 + 0.319830i \(0.103626\pi\)
\(542\) 4.19871e9 1.13271
\(543\) 2.63019e8 0.0704998
\(544\) −6.15047e8 −0.163799
\(545\) −1.63489e9 −0.432613
\(546\) 2.11653e7 0.00556481
\(547\) −5.87859e9 −1.53574 −0.767870 0.640606i \(-0.778683\pi\)
−0.767870 + 0.640606i \(0.778683\pi\)
\(548\) 9.22606e8 0.239488
\(549\) −4.56118e8 −0.117645
\(550\) 3.93464e7 0.0100841
\(551\) 3.17503e8 0.0808570
\(552\) 1.91719e7 0.00485152
\(553\) 1.00333e9 0.252292
\(554\) −3.71696e9 −0.928761
\(555\) 1.85143e8 0.0459707
\(556\) −2.41565e9 −0.596036
\(557\) −2.03155e9 −0.498119 −0.249060 0.968488i \(-0.580122\pi\)
−0.249060 + 0.968488i \(0.580122\pi\)
\(558\) −8.15724e8 −0.198758
\(559\) −5.08599e8 −0.123150
\(560\) 9.81374e8 0.236144
\(561\) −1.53240e7 −0.00366440
\(562\) −1.40343e9 −0.333514
\(563\) 1.68161e9 0.397142 0.198571 0.980087i \(-0.436370\pi\)
0.198571 + 0.980087i \(0.436370\pi\)
\(564\) 2.05773e8 0.0482960
\(565\) 1.12030e9 0.261314
\(566\) 1.15958e9 0.268810
\(567\) −3.61472e9 −0.832787
\(568\) −2.75065e9 −0.629820
\(569\) −4.57746e9 −1.04167 −0.520837 0.853656i \(-0.674380\pi\)
−0.520837 + 0.853656i \(0.674380\pi\)
\(570\) 5.55399e7 0.0125615
\(571\) −5.79221e9 −1.30202 −0.651010 0.759069i \(-0.725655\pi\)
−0.651010 + 0.759069i \(0.725655\pi\)
\(572\) −1.71823e7 −0.00383879
\(573\) 2.39935e8 0.0532785
\(574\) −5.02791e9 −1.10968
\(575\) 2.25577e8 0.0494831
\(576\) −5.70558e8 −0.124400
\(577\) −3.74692e9 −0.812005 −0.406003 0.913872i \(-0.633078\pi\)
−0.406003 + 0.913872i \(0.633078\pi\)
\(578\) −4.64282e8 −0.100008
\(579\) −3.14246e8 −0.0672814
\(580\) 9.25718e8 0.197007
\(581\) 4.99689e9 1.05702
\(582\) −1.78907e8 −0.0376181
\(583\) −2.14869e8 −0.0449091
\(584\) −1.90267e9 −0.395292
\(585\) −7.24436e8 −0.149608
\(586\) −3.39618e9 −0.697186
\(587\) −5.69193e9 −1.16152 −0.580760 0.814075i \(-0.697244\pi\)
−0.580760 + 0.814075i \(0.697244\pi\)
\(588\) −4.88448e7 −0.00990827
\(589\) −3.21332e8 −0.0647963
\(590\) 4.80968e9 0.964126
\(591\) −1.40832e8 −0.0280638
\(592\) −7.49226e8 −0.148418
\(593\) 1.45596e9 0.286720 0.143360 0.989671i \(-0.454209\pi\)
0.143360 + 0.989671i \(0.454209\pi\)
\(594\) −2.84997e7 −0.00557941
\(595\) −4.49711e9 −0.875233
\(596\) 4.67635e9 0.904784
\(597\) 1.26217e8 0.0242778
\(598\) −9.85079e7 −0.0188372
\(599\) −6.21735e9 −1.18198 −0.590992 0.806677i \(-0.701264\pi\)
−0.590992 + 0.806677i \(0.701264\pi\)
\(600\) 3.23635e7 0.00611682
\(601\) 9.98776e9 1.87675 0.938377 0.345613i \(-0.112329\pi\)
0.938377 + 0.345613i \(0.112329\pi\)
\(602\) −2.92887e9 −0.547158
\(603\) 5.05924e9 0.939668
\(604\) −2.64996e9 −0.489339
\(605\) 6.06935e9 1.11429
\(606\) 7.29760e7 0.0133207
\(607\) 3.17827e9 0.576807 0.288404 0.957509i \(-0.406876\pi\)
0.288404 + 0.957509i \(0.406876\pi\)
\(608\) −2.24756e8 −0.0405554
\(609\) 1.14972e8 0.0206269
\(610\) −5.23864e8 −0.0934467
\(611\) −1.05729e9 −0.187521
\(612\) 2.61456e9 0.461072
\(613\) −6.34969e8 −0.111337 −0.0556687 0.998449i \(-0.517729\pi\)
−0.0556687 + 0.998449i \(0.517729\pi\)
\(614\) 3.42023e9 0.596301
\(615\) −8.29637e8 −0.143822
\(616\) −9.89477e7 −0.0170558
\(617\) −4.48835e7 −0.00769287 −0.00384643 0.999993i \(-0.501224\pi\)
−0.00384643 + 0.999993i \(0.501224\pi\)
\(618\) 2.03754e8 0.0347253
\(619\) −9.79132e8 −0.165930 −0.0829648 0.996552i \(-0.526439\pi\)
−0.0829648 + 0.996552i \(0.526439\pi\)
\(620\) −9.36881e8 −0.157875
\(621\) −1.63392e8 −0.0273785
\(622\) −1.36220e9 −0.226974
\(623\) −3.06390e9 −0.507652
\(624\) −1.41329e7 −0.00232855
\(625\) −7.24724e9 −1.18739
\(626\) −3.21715e9 −0.524156
\(627\) −5.59984e6 −0.000907276 0
\(628\) 1.88125e9 0.303101
\(629\) 3.43330e9 0.550090
\(630\) −4.17181e9 −0.664711
\(631\) −2.42563e9 −0.384345 −0.192172 0.981361i \(-0.561553\pi\)
−0.192172 + 0.981361i \(0.561553\pi\)
\(632\) −6.69960e8 −0.105570
\(633\) −1.24796e8 −0.0195564
\(634\) −1.52504e9 −0.237667
\(635\) 9.35744e9 1.45027
\(636\) −1.76736e8 −0.0272411
\(637\) 2.50972e8 0.0384713
\(638\) −9.33361e7 −0.0142291
\(639\) 1.16930e10 1.77285
\(640\) −6.55302e8 −0.0988124
\(641\) 9.23132e8 0.138440 0.0692199 0.997601i \(-0.477949\pi\)
0.0692199 + 0.997601i \(0.477949\pi\)
\(642\) 1.58496e7 0.00236399
\(643\) 6.86334e9 1.01812 0.509058 0.860732i \(-0.329994\pi\)
0.509058 + 0.860732i \(0.329994\pi\)
\(644\) −5.67278e8 −0.0836942
\(645\) −4.83282e8 −0.0709156
\(646\) 1.02993e9 0.150313
\(647\) 5.02399e8 0.0729262 0.0364631 0.999335i \(-0.488391\pi\)
0.0364631 + 0.999335i \(0.488391\pi\)
\(648\) 2.41369e9 0.348473
\(649\) −4.84939e8 −0.0696355
\(650\) −1.66288e8 −0.0237500
\(651\) −1.16359e8 −0.0165297
\(652\) 5.78512e8 0.0817420
\(653\) 4.23523e9 0.595225 0.297612 0.954687i \(-0.403810\pi\)
0.297612 + 0.954687i \(0.403810\pi\)
\(654\) 1.35584e8 0.0189534
\(655\) 2.53506e9 0.352488
\(656\) 3.35733e9 0.464334
\(657\) 8.08822e9 1.11269
\(658\) −6.08862e9 −0.833160
\(659\) −1.34554e10 −1.83146 −0.915731 0.401791i \(-0.868388\pi\)
−0.915731 + 0.401791i \(0.868388\pi\)
\(660\) −1.63270e7 −0.00221056
\(661\) 1.23518e10 1.66351 0.831753 0.555146i \(-0.187338\pi\)
0.831753 + 0.555146i \(0.187338\pi\)
\(662\) −3.29606e9 −0.441563
\(663\) 6.47634e7 0.00863043
\(664\) −3.33662e9 −0.442301
\(665\) −1.64337e9 −0.216700
\(666\) 3.18495e9 0.417776
\(667\) −5.35106e8 −0.0698231
\(668\) −5.22706e9 −0.678484
\(669\) −6.58864e8 −0.0850755
\(670\) 5.81067e9 0.746387
\(671\) 5.28189e7 0.00674933
\(672\) −8.13872e7 −0.0103458
\(673\) −1.52638e9 −0.193024 −0.0965119 0.995332i \(-0.530769\pi\)
−0.0965119 + 0.995332i \(0.530769\pi\)
\(674\) −5.59417e9 −0.703763
\(675\) −2.75817e8 −0.0345189
\(676\) −3.94329e9 −0.490959
\(677\) −1.55415e9 −0.192500 −0.0962502 0.995357i \(-0.530685\pi\)
−0.0962502 + 0.995357i \(0.530685\pi\)
\(678\) −9.29082e7 −0.0114485
\(679\) 5.29369e9 0.648954
\(680\) 3.00289e9 0.366234
\(681\) 7.28646e8 0.0884100
\(682\) 9.44616e7 0.0114028
\(683\) −3.21824e9 −0.386497 −0.193249 0.981150i \(-0.561902\pi\)
−0.193249 + 0.981150i \(0.561902\pi\)
\(684\) 9.55434e8 0.114158
\(685\) −4.50451e9 −0.535465
\(686\) 6.49700e9 0.768384
\(687\) −7.73796e8 −0.0910495
\(688\) 1.95572e9 0.228953
\(689\) 9.08095e8 0.105770
\(690\) −9.36044e7 −0.0108474
\(691\) 9.15999e9 1.05614 0.528071 0.849200i \(-0.322916\pi\)
0.528071 + 0.849200i \(0.322916\pi\)
\(692\) 8.52421e9 0.977874
\(693\) 4.20626e8 0.0480098
\(694\) −7.43908e9 −0.844814
\(695\) 1.17941e10 1.33266
\(696\) −7.67715e7 −0.00863113
\(697\) −1.53848e10 −1.72099
\(698\) −3.55021e9 −0.395148
\(699\) 1.04634e9 0.115878
\(700\) −9.57604e8 −0.105522
\(701\) −7.97717e9 −0.874653 −0.437327 0.899303i \(-0.644075\pi\)
−0.437327 + 0.899303i \(0.644075\pi\)
\(702\) 1.20447e8 0.0131407
\(703\) 1.25462e9 0.136198
\(704\) 6.60712e7 0.00713687
\(705\) −1.00466e9 −0.107984
\(706\) −1.19510e10 −1.27817
\(707\) −2.15929e9 −0.229796
\(708\) −3.98875e8 −0.0422397
\(709\) 7.97010e9 0.839851 0.419925 0.907559i \(-0.362056\pi\)
0.419925 + 0.907559i \(0.362056\pi\)
\(710\) 1.34297e10 1.40819
\(711\) 2.84799e9 0.297163
\(712\) 2.04588e9 0.212423
\(713\) 5.41559e8 0.0559541
\(714\) 3.72953e8 0.0383452
\(715\) 8.38904e7 0.00858304
\(716\) −3.14182e9 −0.319879
\(717\) −1.67844e8 −0.0170055
\(718\) −1.42350e9 −0.143523
\(719\) 1.00263e10 1.00598 0.502991 0.864292i \(-0.332233\pi\)
0.502991 + 0.864292i \(0.332233\pi\)
\(720\) 2.78568e9 0.278143
\(721\) −6.02888e9 −0.599050
\(722\) 3.76367e8 0.0372161
\(723\) 3.75512e8 0.0369522
\(724\) 5.19666e9 0.508908
\(725\) −9.03295e8 −0.0880333
\(726\) −5.03342e8 −0.0488186
\(727\) −1.65401e10 −1.59650 −0.798248 0.602329i \(-0.794240\pi\)
−0.798248 + 0.602329i \(0.794240\pi\)
\(728\) 4.18179e8 0.0401701
\(729\) −1.01604e10 −0.971329
\(730\) 9.28953e9 0.883820
\(731\) −8.96200e9 −0.848583
\(732\) 4.34450e7 0.00409403
\(733\) 9.81994e9 0.920969 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(734\) −1.10012e10 −1.02684
\(735\) 2.38479e8 0.0221536
\(736\) 3.78793e8 0.0350211
\(737\) −5.85864e8 −0.0539089
\(738\) −1.42720e10 −1.30703
\(739\) 5.41304e9 0.493385 0.246692 0.969094i \(-0.420656\pi\)
0.246692 + 0.969094i \(0.420656\pi\)
\(740\) 3.65800e9 0.331843
\(741\) 2.36664e7 0.00213682
\(742\) 5.22945e9 0.469940
\(743\) −8.26105e9 −0.738881 −0.369440 0.929254i \(-0.620451\pi\)
−0.369440 + 0.929254i \(0.620451\pi\)
\(744\) 7.76973e7 0.00691673
\(745\) −2.28317e10 −2.02298
\(746\) 1.24676e10 1.09950
\(747\) 1.41839e10 1.24501
\(748\) −3.02768e8 −0.0264518
\(749\) −4.68975e8 −0.0407815
\(750\) 4.74596e8 0.0410781
\(751\) −1.44332e10 −1.24343 −0.621715 0.783243i \(-0.713564\pi\)
−0.621715 + 0.783243i \(0.713564\pi\)
\(752\) 4.06561e9 0.348628
\(753\) −1.30788e8 −0.0111631
\(754\) 3.94463e8 0.0335125
\(755\) 1.29381e10 1.09410
\(756\) 6.93620e8 0.0583842
\(757\) −8.31294e9 −0.696496 −0.348248 0.937402i \(-0.613223\pi\)
−0.348248 + 0.937402i \(0.613223\pi\)
\(758\) 7.07640e9 0.590160
\(759\) 9.43772e6 0.000783467 0
\(760\) 1.09734e9 0.0906764
\(761\) −1.96636e10 −1.61739 −0.808697 0.588226i \(-0.799827\pi\)
−0.808697 + 0.588226i \(0.799827\pi\)
\(762\) −7.76030e8 −0.0635384
\(763\) −4.01180e9 −0.326967
\(764\) 4.74057e9 0.384595
\(765\) −1.27653e10 −1.03090
\(766\) 1.15659e10 0.929774
\(767\) 2.04948e9 0.164006
\(768\) 5.43454e7 0.00432911
\(769\) 1.27102e10 1.00788 0.503942 0.863738i \(-0.331883\pi\)
0.503942 + 0.863738i \(0.331883\pi\)
\(770\) 4.83100e8 0.0381346
\(771\) 8.67018e8 0.0681299
\(772\) −6.20880e9 −0.485676
\(773\) −2.87614e9 −0.223966 −0.111983 0.993710i \(-0.535720\pi\)
−0.111983 + 0.993710i \(0.535720\pi\)
\(774\) −8.31375e9 −0.644471
\(775\) 9.14188e8 0.0705472
\(776\) −3.53480e9 −0.271549
\(777\) 4.54317e8 0.0347444
\(778\) −8.17315e9 −0.622244
\(779\) −5.62206e9 −0.426102
\(780\) 6.90022e7 0.00520633
\(781\) −1.35406e9 −0.101709
\(782\) −1.73580e9 −0.129801
\(783\) 6.54283e8 0.0487079
\(784\) −9.65063e8 −0.0715236
\(785\) −9.18498e9 −0.677695
\(786\) −2.10237e8 −0.0154430
\(787\) 5.61813e9 0.410847 0.205424 0.978673i \(-0.434143\pi\)
0.205424 + 0.978673i \(0.434143\pi\)
\(788\) −2.78253e9 −0.202581
\(789\) 1.66101e9 0.120393
\(790\) 3.27100e9 0.236040
\(791\) 2.74906e9 0.197500
\(792\) −2.80868e8 −0.0200893
\(793\) −2.23227e8 −0.0158961
\(794\) 1.44040e9 0.102120
\(795\) 8.62892e8 0.0609076
\(796\) 2.49377e9 0.175251
\(797\) −4.98926e9 −0.349085 −0.174543 0.984650i \(-0.555845\pi\)
−0.174543 + 0.984650i \(0.555845\pi\)
\(798\) 1.36288e8 0.00949395
\(799\) −1.86305e10 −1.29214
\(800\) 6.39429e8 0.0441547
\(801\) −8.69703e9 −0.597939
\(802\) 4.05336e9 0.277463
\(803\) −9.36623e8 −0.0638352
\(804\) −4.81889e8 −0.0327003
\(805\) 2.76966e9 0.187129
\(806\) −3.99220e8 −0.0268559
\(807\) 1.86513e9 0.124925
\(808\) 1.44184e9 0.0961563
\(809\) 2.60176e10 1.72762 0.863809 0.503820i \(-0.168073\pi\)
0.863809 + 0.503820i \(0.168073\pi\)
\(810\) −1.17845e10 −0.779139
\(811\) 1.36033e10 0.895513 0.447756 0.894156i \(-0.352223\pi\)
0.447756 + 0.894156i \(0.352223\pi\)
\(812\) 2.27160e9 0.148897
\(813\) 1.70008e9 0.110956
\(814\) −3.68820e8 −0.0239679
\(815\) −2.82451e9 −0.182764
\(816\) −2.49035e8 −0.0160452
\(817\) −3.27497e9 −0.210102
\(818\) 9.19184e9 0.587173
\(819\) −1.77768e9 −0.113073
\(820\) −1.63918e10 −1.03819
\(821\) 5.59803e9 0.353048 0.176524 0.984296i \(-0.443515\pi\)
0.176524 + 0.984296i \(0.443515\pi\)
\(822\) 3.73568e8 0.0234595
\(823\) 2.83609e10 1.77346 0.886728 0.462292i \(-0.152973\pi\)
0.886728 + 0.462292i \(0.152973\pi\)
\(824\) 4.02572e9 0.250667
\(825\) 1.59315e7 0.000987799 0
\(826\) 1.18023e10 0.728682
\(827\) −1.94087e10 −1.19324 −0.596619 0.802524i \(-0.703490\pi\)
−0.596619 + 0.802524i \(0.703490\pi\)
\(828\) −1.61025e9 −0.0985794
\(829\) 1.25701e10 0.766301 0.383150 0.923686i \(-0.374839\pi\)
0.383150 + 0.923686i \(0.374839\pi\)
\(830\) 1.62906e10 0.988927
\(831\) −1.50501e9 −0.0909781
\(832\) −2.79234e8 −0.0168088
\(833\) 4.42236e9 0.265092
\(834\) −9.78109e8 −0.0583856
\(835\) 2.55205e10 1.51700
\(836\) −1.10640e8 −0.00654924
\(837\) −6.62173e8 −0.0390331
\(838\) 1.61831e10 0.949968
\(839\) 1.95535e10 1.14303 0.571515 0.820592i \(-0.306356\pi\)
0.571515 + 0.820592i \(0.306356\pi\)
\(840\) 3.97363e8 0.0231318
\(841\) −1.51071e10 −0.875781
\(842\) 2.04402e10 1.18003
\(843\) −5.68256e8 −0.0326699
\(844\) −2.46569e9 −0.141169
\(845\) 1.92526e10 1.09772
\(846\) −1.72829e10 −0.981339
\(847\) 1.48934e10 0.842175
\(848\) −3.49191e9 −0.196642
\(849\) 4.69521e8 0.0263316
\(850\) −2.93016e9 −0.163653
\(851\) −2.11449e9 −0.117612
\(852\) −1.11375e9 −0.0616950
\(853\) 1.14196e10 0.629986 0.314993 0.949094i \(-0.397998\pi\)
0.314993 + 0.949094i \(0.397998\pi\)
\(854\) −1.28550e9 −0.0706266
\(855\) −4.66479e9 −0.255241
\(856\) 3.13153e8 0.0170647
\(857\) −1.94401e10 −1.05503 −0.527516 0.849545i \(-0.676877\pi\)
−0.527516 + 0.849545i \(0.676877\pi\)
\(858\) −6.95719e6 −0.000376035 0
\(859\) 1.07841e10 0.580509 0.290254 0.956950i \(-0.406260\pi\)
0.290254 + 0.956950i \(0.406260\pi\)
\(860\) −9.54856e9 −0.511910
\(861\) −2.03583e9 −0.108700
\(862\) 1.17852e10 0.626705
\(863\) −8.45236e8 −0.0447651 −0.0223826 0.999749i \(-0.507125\pi\)
−0.0223826 + 0.999749i \(0.507125\pi\)
\(864\) −4.63157e8 −0.0244304
\(865\) −4.16184e10 −2.18640
\(866\) −7.46595e9 −0.390636
\(867\) −1.87990e8 −0.00979643
\(868\) −2.29899e9 −0.119321
\(869\) −3.29800e8 −0.0170483
\(870\) 3.74828e8 0.0192981
\(871\) 2.47602e9 0.126967
\(872\) 2.67884e9 0.136817
\(873\) 1.50264e10 0.764373
\(874\) −6.34312e8 −0.0321376
\(875\) −1.40428e10 −0.708642
\(876\) −7.70398e8 −0.0387214
\(877\) 2.17563e10 1.08915 0.544574 0.838713i \(-0.316691\pi\)
0.544574 + 0.838713i \(0.316691\pi\)
\(878\) −1.55529e10 −0.775495
\(879\) −1.37513e9 −0.0682939
\(880\) −3.22584e8 −0.0159571
\(881\) 2.18056e10 1.07436 0.537182 0.843466i \(-0.319489\pi\)
0.537182 + 0.843466i \(0.319489\pi\)
\(882\) 4.10248e9 0.201329
\(883\) 2.25662e9 0.110305 0.0551526 0.998478i \(-0.482435\pi\)
0.0551526 + 0.998478i \(0.482435\pi\)
\(884\) 1.27958e9 0.0622995
\(885\) 1.94746e9 0.0944424
\(886\) −1.88474e10 −0.910401
\(887\) −1.72271e10 −0.828857 −0.414428 0.910082i \(-0.636018\pi\)
−0.414428 + 0.910082i \(0.636018\pi\)
\(888\) −3.03365e8 −0.0145385
\(889\) 2.29620e10 1.09611
\(890\) −9.98877e9 −0.474949
\(891\) 1.18818e9 0.0562745
\(892\) −1.30177e10 −0.614124
\(893\) −6.80810e9 −0.319923
\(894\) 1.89347e9 0.0886294
\(895\) 1.53396e10 0.715208
\(896\) −1.60803e9 −0.0746819
\(897\) −3.98863e7 −0.00184523
\(898\) −1.46700e10 −0.676023
\(899\) −2.16861e9 −0.0995455
\(900\) −2.71821e9 −0.124289
\(901\) 1.60015e10 0.728827
\(902\) 1.65271e9 0.0749849
\(903\) −1.18591e9 −0.0535976
\(904\) −1.83566e9 −0.0826422
\(905\) −2.53721e10 −1.13785
\(906\) −1.07298e9 −0.0479340
\(907\) 2.32557e10 1.03491 0.517456 0.855710i \(-0.326879\pi\)
0.517456 + 0.855710i \(0.326879\pi\)
\(908\) 1.43964e10 0.638195
\(909\) −6.12926e9 −0.270666
\(910\) −2.04171e9 −0.0898150
\(911\) −4.22775e10 −1.85266 −0.926329 0.376715i \(-0.877054\pi\)
−0.926329 + 0.376715i \(0.877054\pi\)
\(912\) −9.10046e7 −0.00397266
\(913\) −1.64251e9 −0.0714267
\(914\) −3.01346e10 −1.30543
\(915\) −2.12115e8 −0.00915372
\(916\) −1.52885e10 −0.657248
\(917\) 6.22072e9 0.266408
\(918\) 2.12240e9 0.0905477
\(919\) 2.74897e10 1.16833 0.584164 0.811635i \(-0.301422\pi\)
0.584164 + 0.811635i \(0.301422\pi\)
\(920\) −1.84941e9 −0.0783026
\(921\) 1.38487e9 0.0584116
\(922\) 2.64448e10 1.11118
\(923\) 5.72261e9 0.239546
\(924\) −4.00644e7 −0.00167073
\(925\) −3.56940e9 −0.148286
\(926\) −2.93061e10 −1.21288
\(927\) −1.71133e10 −0.705593
\(928\) −1.51683e9 −0.0623045
\(929\) −2.42814e10 −0.993614 −0.496807 0.867861i \(-0.665494\pi\)
−0.496807 + 0.867861i \(0.665494\pi\)
\(930\) −3.79347e8 −0.0154649
\(931\) 1.61606e9 0.0656346
\(932\) 2.06732e10 0.836475
\(933\) −5.51563e8 −0.0222336
\(934\) −1.78139e10 −0.715393
\(935\) 1.47823e9 0.0591427
\(936\) 1.18702e9 0.0473144
\(937\) −3.53708e10 −1.40461 −0.702305 0.711876i \(-0.747846\pi\)
−0.702305 + 0.711876i \(0.747846\pi\)
\(938\) 1.42586e10 0.564116
\(939\) −1.30264e9 −0.0513445
\(940\) −1.98498e10 −0.779487
\(941\) −3.60255e10 −1.40944 −0.704719 0.709486i \(-0.748927\pi\)
−0.704719 + 0.709486i \(0.748927\pi\)
\(942\) 7.61727e8 0.0296908
\(943\) 9.47516e9 0.367956
\(944\) −7.88088e9 −0.304911
\(945\) −3.38652e9 −0.130539
\(946\) 9.62740e8 0.0369734
\(947\) −2.20930e10 −0.845336 −0.422668 0.906285i \(-0.638906\pi\)
−0.422668 + 0.906285i \(0.638906\pi\)
\(948\) −2.71270e8 −0.0103412
\(949\) 3.95842e9 0.150345
\(950\) −1.07076e9 −0.0405192
\(951\) −6.17496e8 −0.0232810
\(952\) 7.36872e9 0.276798
\(953\) −2.36409e9 −0.0884787 −0.0442393 0.999021i \(-0.514086\pi\)
−0.0442393 + 0.999021i \(0.514086\pi\)
\(954\) 1.48441e10 0.553520
\(955\) −2.31452e10 −0.859903
\(956\) −3.31623e9 −0.122756
\(957\) −3.77922e7 −0.00139383
\(958\) 1.14806e10 0.421878
\(959\) −1.10535e10 −0.404702
\(960\) −2.65335e8 −0.00967931
\(961\) −2.53179e10 −0.920227
\(962\) 1.55873e9 0.0564493
\(963\) −1.33121e9 −0.0480346
\(964\) 7.41927e9 0.266742
\(965\) 3.03137e10 1.08591
\(966\) −2.29693e8 −0.00819839
\(967\) 2.69098e10 0.957014 0.478507 0.878084i \(-0.341178\pi\)
0.478507 + 0.878084i \(0.341178\pi\)
\(968\) −9.94491e9 −0.352401
\(969\) 4.17025e8 0.0147241
\(970\) 1.72582e10 0.607149
\(971\) −4.71594e9 −0.165311 −0.0826553 0.996578i \(-0.526340\pi\)
−0.0826553 + 0.996578i \(0.526340\pi\)
\(972\) 2.95568e9 0.103235
\(973\) 2.89413e10 1.00722
\(974\) −1.69314e10 −0.587132
\(975\) −6.73308e7 −0.00232647
\(976\) 8.58375e8 0.0295531
\(977\) 1.79514e10 0.615839 0.307919 0.951412i \(-0.400367\pi\)
0.307919 + 0.951412i \(0.400367\pi\)
\(978\) 2.34242e8 0.00800716
\(979\) 1.00712e9 0.0343039
\(980\) 4.71180e9 0.159917
\(981\) −1.13877e10 −0.385119
\(982\) 3.25684e10 1.09750
\(983\) 2.39450e10 0.804039 0.402019 0.915631i \(-0.368308\pi\)
0.402019 + 0.915631i \(0.368308\pi\)
\(984\) 1.35940e9 0.0454846
\(985\) 1.35854e10 0.452944
\(986\) 6.95082e9 0.230923
\(987\) −2.46531e9 −0.0816134
\(988\) 4.67595e8 0.0154248
\(989\) 5.51949e9 0.181431
\(990\) 1.37130e9 0.0449170
\(991\) 1.07956e10 0.352362 0.176181 0.984358i \(-0.443626\pi\)
0.176181 + 0.984358i \(0.443626\pi\)
\(992\) 1.53512e9 0.0499289
\(993\) −1.33459e9 −0.0432540
\(994\) 3.29548e10 1.06431
\(995\) −1.21755e10 −0.391838
\(996\) −1.35101e9 −0.0433263
\(997\) −3.54933e10 −1.13426 −0.567131 0.823627i \(-0.691947\pi\)
−0.567131 + 0.823627i \(0.691947\pi\)
\(998\) 3.14431e10 1.00131
\(999\) 2.58542e9 0.0820450
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 38.8.a.d.1.2 2
3.2 odd 2 342.8.a.g.1.2 2
4.3 odd 2 304.8.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.8.a.d.1.2 2 1.1 even 1 trivial
304.8.a.d.1.1 2 4.3 odd 2
342.8.a.g.1.2 2 3.2 odd 2