Properties

Label 43.8.a.b.1.5
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 1.5
Root \(-4.99525\) 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

\(f(q)\) \(=\) \(q-3.99525 q^{2} +84.4924 q^{3} -112.038 q^{4} -383.451 q^{5} -337.568 q^{6} +1003.83 q^{7} +959.011 q^{8} +4951.96 q^{9} +O(q^{10})\) \(q-3.99525 q^{2} +84.4924 q^{3} -112.038 q^{4} -383.451 q^{5} -337.568 q^{6} +1003.83 q^{7} +959.011 q^{8} +4951.96 q^{9} +1531.98 q^{10} +1620.93 q^{11} -9466.36 q^{12} +7962.52 q^{13} -4010.57 q^{14} -32398.7 q^{15} +10509.4 q^{16} +36412.1 q^{17} -19784.3 q^{18} -40353.3 q^{19} +42961.1 q^{20} +84816.3 q^{21} -6476.03 q^{22} +13321.7 q^{23} +81029.1 q^{24} +68909.7 q^{25} -31812.2 q^{26} +233618. q^{27} -112468. q^{28} +119451. q^{29} +129441. q^{30} -103590. q^{31} -164741. q^{32} +136956. q^{33} -145476. q^{34} -384921. q^{35} -554808. q^{36} +191874. q^{37} +161221. q^{38} +672772. q^{39} -367734. q^{40} -240868. q^{41} -338862. q^{42} -79507.0 q^{43} -181606. q^{44} -1.89884e6 q^{45} -53223.7 q^{46} +502478. q^{47} +887962. q^{48} +184140. q^{49} -275311. q^{50} +3.07655e6 q^{51} -892105. q^{52} +1.09942e6 q^{53} -933363. q^{54} -621548. q^{55} +962688. q^{56} -3.40954e6 q^{57} -477236. q^{58} -2.43199e6 q^{59} +3.62988e6 q^{60} -620335. q^{61} +413868. q^{62} +4.97095e6 q^{63} -687019. q^{64} -3.05324e6 q^{65} -547175. q^{66} -4.73308e6 q^{67} -4.07954e6 q^{68} +1.12559e6 q^{69} +1.53786e6 q^{70} -2.81874e6 q^{71} +4.74899e6 q^{72} +2.78345e6 q^{73} -766582. q^{74} +5.82235e6 q^{75} +4.52110e6 q^{76} +1.62715e6 q^{77} -2.68789e6 q^{78} +6.00989e6 q^{79} -4.02983e6 q^{80} +8.90902e6 q^{81} +962329. q^{82} +4.68609e6 q^{83} -9.50265e6 q^{84} -1.39623e7 q^{85} +317650. q^{86} +1.00927e7 q^{87} +1.55449e6 q^{88} +2.94676e6 q^{89} +7.58632e6 q^{90} +7.99305e6 q^{91} -1.49254e6 q^{92} -8.75257e6 q^{93} -2.00753e6 q^{94} +1.54735e7 q^{95} -1.39194e7 q^{96} -1.41877e7 q^{97} -735685. q^{98} +8.02680e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.99525 −0.353133 −0.176567 0.984289i \(-0.556499\pi\)
−0.176567 + 0.984289i \(0.556499\pi\)
\(3\) 84.4924 1.80673 0.903365 0.428873i \(-0.141089\pi\)
0.903365 + 0.428873i \(0.141089\pi\)
\(4\) −112.038 −0.875297
\(5\) −383.451 −1.37188 −0.685938 0.727660i \(-0.740608\pi\)
−0.685938 + 0.727660i \(0.740608\pi\)
\(6\) −337.568 −0.638016
\(7\) 1003.83 1.10616 0.553081 0.833127i \(-0.313452\pi\)
0.553081 + 0.833127i \(0.313452\pi\)
\(8\) 959.011 0.662230
\(9\) 4951.96 2.26427
\(10\) 1531.98 0.484455
\(11\) 1620.93 0.367190 0.183595 0.983002i \(-0.441227\pi\)
0.183595 + 0.983002i \(0.441227\pi\)
\(12\) −9466.36 −1.58142
\(13\) 7962.52 1.00519 0.502596 0.864522i \(-0.332378\pi\)
0.502596 + 0.864522i \(0.332378\pi\)
\(14\) −4010.57 −0.390623
\(15\) −32398.7 −2.47861
\(16\) 10509.4 0.641441
\(17\) 36412.1 1.79753 0.898763 0.438435i \(-0.144467\pi\)
0.898763 + 0.438435i \(0.144467\pi\)
\(18\) −19784.3 −0.799590
\(19\) −40353.3 −1.34971 −0.674856 0.737950i \(-0.735794\pi\)
−0.674856 + 0.737950i \(0.735794\pi\)
\(20\) 42961.1 1.20080
\(21\) 84816.3 1.99854
\(22\) −6476.03 −0.129667
\(23\) 13321.7 0.228304 0.114152 0.993463i \(-0.463585\pi\)
0.114152 + 0.993463i \(0.463585\pi\)
\(24\) 81029.1 1.19647
\(25\) 68909.7 0.882044
\(26\) −31812.2 −0.354967
\(27\) 233618. 2.28420
\(28\) −112468. −0.968220
\(29\) 119451. 0.909487 0.454743 0.890623i \(-0.349731\pi\)
0.454743 + 0.890623i \(0.349731\pi\)
\(30\) 129441. 0.875280
\(31\) −103590. −0.624529 −0.312264 0.949995i \(-0.601087\pi\)
−0.312264 + 0.949995i \(0.601087\pi\)
\(32\) −164741. −0.888744
\(33\) 136956. 0.663413
\(34\) −145476. −0.634766
\(35\) −384921. −1.51752
\(36\) −554808. −1.98191
\(37\) 191874. 0.622743 0.311372 0.950288i \(-0.399212\pi\)
0.311372 + 0.950288i \(0.399212\pi\)
\(38\) 161221. 0.476628
\(39\) 672772. 1.81611
\(40\) −367734. −0.908497
\(41\) −240868. −0.545804 −0.272902 0.962042i \(-0.587983\pi\)
−0.272902 + 0.962042i \(0.587983\pi\)
\(42\) −338862. −0.705750
\(43\) −79507.0 −0.152499
\(44\) −181606. −0.321400
\(45\) −1.89884e6 −3.10630
\(46\) −53223.7 −0.0806218
\(47\) 502478. 0.705951 0.352976 0.935632i \(-0.385170\pi\)
0.352976 + 0.935632i \(0.385170\pi\)
\(48\) 887962. 1.15891
\(49\) 184140. 0.223595
\(50\) −275311. −0.311479
\(51\) 3.07655e6 3.24764
\(52\) −892105. −0.879841
\(53\) 1.09942e6 1.01437 0.507187 0.861836i \(-0.330685\pi\)
0.507187 + 0.861836i \(0.330685\pi\)
\(54\) −933363. −0.806626
\(55\) −621548. −0.503739
\(56\) 962688. 0.732534
\(57\) −3.40954e6 −2.43856
\(58\) −477236. −0.321170
\(59\) −2.43199e6 −1.54163 −0.770813 0.637061i \(-0.780150\pi\)
−0.770813 + 0.637061i \(0.780150\pi\)
\(60\) 3.62988e6 2.16952
\(61\) −620335. −0.349923 −0.174961 0.984575i \(-0.555980\pi\)
−0.174961 + 0.984575i \(0.555980\pi\)
\(62\) 413868. 0.220542
\(63\) 4.97095e6 2.50465
\(64\) −687019. −0.327596
\(65\) −3.05324e6 −1.37900
\(66\) −547175. −0.234273
\(67\) −4.73308e6 −1.92257 −0.961285 0.275555i \(-0.911138\pi\)
−0.961285 + 0.275555i \(0.911138\pi\)
\(68\) −4.07954e6 −1.57337
\(69\) 1.12559e6 0.412484
\(70\) 1.53786e6 0.535886
\(71\) −2.81874e6 −0.934655 −0.467327 0.884084i \(-0.654783\pi\)
−0.467327 + 0.884084i \(0.654783\pi\)
\(72\) 4.74899e6 1.49947
\(73\) 2.78345e6 0.837440 0.418720 0.908115i \(-0.362479\pi\)
0.418720 + 0.908115i \(0.362479\pi\)
\(74\) −766582. −0.219911
\(75\) 5.82235e6 1.59362
\(76\) 4.52110e6 1.18140
\(77\) 1.62715e6 0.406172
\(78\) −2.68789e6 −0.641329
\(79\) 6.00989e6 1.37142 0.685712 0.727873i \(-0.259491\pi\)
0.685712 + 0.727873i \(0.259491\pi\)
\(80\) −4.02983e6 −0.879978
\(81\) 8.90902e6 1.86265
\(82\) 962329. 0.192741
\(83\) 4.68609e6 0.899574 0.449787 0.893136i \(-0.351500\pi\)
0.449787 + 0.893136i \(0.351500\pi\)
\(84\) −9.50265e6 −1.74931
\(85\) −1.39623e7 −2.46598
\(86\) 317650. 0.0538523
\(87\) 1.00927e7 1.64320
\(88\) 1.55449e6 0.243164
\(89\) 2.94676e6 0.443077 0.221539 0.975152i \(-0.428892\pi\)
0.221539 + 0.975152i \(0.428892\pi\)
\(90\) 7.58632e6 1.09694
\(91\) 7.99305e6 1.11191
\(92\) −1.49254e6 −0.199834
\(93\) −8.75257e6 −1.12835
\(94\) −2.00753e6 −0.249295
\(95\) 1.54735e7 1.85164
\(96\) −1.39194e7 −1.60572
\(97\) −1.41877e7 −1.57838 −0.789192 0.614147i \(-0.789500\pi\)
−0.789192 + 0.614147i \(0.789500\pi\)
\(98\) −735685. −0.0789588
\(99\) 8.02680e6 0.831418
\(100\) −7.72051e6 −0.772051
\(101\) 1.46728e7 1.41706 0.708531 0.705680i \(-0.249358\pi\)
0.708531 + 0.705680i \(0.249358\pi\)
\(102\) −1.22916e7 −1.14685
\(103\) −1.29765e7 −1.17011 −0.585057 0.810992i \(-0.698928\pi\)
−0.585057 + 0.810992i \(0.698928\pi\)
\(104\) 7.63615e6 0.665668
\(105\) −3.25229e7 −2.74174
\(106\) −4.39246e6 −0.358209
\(107\) −1.57757e7 −1.24493 −0.622464 0.782648i \(-0.713868\pi\)
−0.622464 + 0.782648i \(0.713868\pi\)
\(108\) −2.61741e7 −1.99935
\(109\) 1.93156e7 1.42861 0.714307 0.699833i \(-0.246742\pi\)
0.714307 + 0.699833i \(0.246742\pi\)
\(110\) 2.48324e6 0.177887
\(111\) 1.62119e7 1.12513
\(112\) 1.05497e7 0.709538
\(113\) −2.22839e6 −0.145283 −0.0726417 0.997358i \(-0.523143\pi\)
−0.0726417 + 0.997358i \(0.523143\pi\)
\(114\) 1.36220e7 0.861138
\(115\) −5.10824e6 −0.313205
\(116\) −1.33830e7 −0.796071
\(117\) 3.94301e7 2.27603
\(118\) 9.71639e6 0.544400
\(119\) 3.65518e7 1.98835
\(120\) −3.10707e7 −1.64141
\(121\) −1.68597e7 −0.865172
\(122\) 2.47839e6 0.123569
\(123\) −2.03515e7 −0.986119
\(124\) 1.16060e7 0.546648
\(125\) 3.53361e6 0.161821
\(126\) −1.98602e7 −0.884476
\(127\) −3.40735e7 −1.47606 −0.738028 0.674770i \(-0.764243\pi\)
−0.738028 + 0.674770i \(0.764243\pi\)
\(128\) 2.38317e7 1.00443
\(129\) −6.71774e6 −0.275524
\(130\) 1.21984e7 0.486970
\(131\) 3.55332e7 1.38097 0.690487 0.723345i \(-0.257396\pi\)
0.690487 + 0.723345i \(0.257396\pi\)
\(132\) −1.53443e7 −0.580683
\(133\) −4.05080e7 −1.49300
\(134\) 1.89098e7 0.678924
\(135\) −8.95812e7 −3.13364
\(136\) 3.49197e7 1.19037
\(137\) −3.77219e7 −1.25335 −0.626674 0.779281i \(-0.715584\pi\)
−0.626674 + 0.779281i \(0.715584\pi\)
\(138\) −4.49699e6 −0.145662
\(139\) 1.44974e7 0.457867 0.228934 0.973442i \(-0.426476\pi\)
0.228934 + 0.973442i \(0.426476\pi\)
\(140\) 4.31258e7 1.32828
\(141\) 4.24556e7 1.27546
\(142\) 1.12616e7 0.330058
\(143\) 1.29067e7 0.369096
\(144\) 5.20420e7 1.45240
\(145\) −4.58036e7 −1.24770
\(146\) −1.11206e7 −0.295728
\(147\) 1.55584e7 0.403976
\(148\) −2.14971e7 −0.545085
\(149\) 2.17133e7 0.537743 0.268871 0.963176i \(-0.413349\pi\)
0.268871 + 0.963176i \(0.413349\pi\)
\(150\) −2.32617e7 −0.562759
\(151\) −1.73559e7 −0.410231 −0.205115 0.978738i \(-0.565757\pi\)
−0.205115 + 0.978738i \(0.565757\pi\)
\(152\) −3.86992e7 −0.893819
\(153\) 1.80312e8 4.07009
\(154\) −6.50086e6 −0.143433
\(155\) 3.97217e7 0.856776
\(156\) −7.53761e7 −1.58963
\(157\) −2.86214e6 −0.0590258 −0.0295129 0.999564i \(-0.509396\pi\)
−0.0295129 + 0.999564i \(0.509396\pi\)
\(158\) −2.40110e7 −0.484296
\(159\) 9.28927e7 1.83270
\(160\) 6.31701e7 1.21925
\(161\) 1.33728e7 0.252541
\(162\) −3.55937e7 −0.657766
\(163\) 2.66759e7 0.482461 0.241231 0.970468i \(-0.422449\pi\)
0.241231 + 0.970468i \(0.422449\pi\)
\(164\) 2.69864e7 0.477740
\(165\) −5.25161e7 −0.910120
\(166\) −1.87221e7 −0.317670
\(167\) −4.84714e7 −0.805337 −0.402669 0.915346i \(-0.631917\pi\)
−0.402669 + 0.915346i \(0.631917\pi\)
\(168\) 8.13398e7 1.32349
\(169\) 653231. 0.0104103
\(170\) 5.57827e7 0.870821
\(171\) −1.99828e8 −3.05611
\(172\) 8.90781e6 0.133482
\(173\) 1.18312e8 1.73726 0.868632 0.495458i \(-0.165000\pi\)
0.868632 + 0.495458i \(0.165000\pi\)
\(174\) −4.03228e7 −0.580267
\(175\) 6.91739e7 0.975684
\(176\) 1.70350e7 0.235531
\(177\) −2.05484e8 −2.78530
\(178\) −1.17730e7 −0.156465
\(179\) −3.21297e7 −0.418718 −0.209359 0.977839i \(-0.567138\pi\)
−0.209359 + 0.977839i \(0.567138\pi\)
\(180\) 2.12742e8 2.71894
\(181\) −1.19731e8 −1.50083 −0.750413 0.660969i \(-0.770145\pi\)
−0.750413 + 0.660969i \(0.770145\pi\)
\(182\) −3.19342e7 −0.392651
\(183\) −5.24136e7 −0.632216
\(184\) 1.27757e7 0.151190
\(185\) −7.35741e7 −0.854327
\(186\) 3.49687e7 0.398459
\(187\) 5.90216e7 0.660033
\(188\) −5.62967e7 −0.617917
\(189\) 2.34514e8 2.52669
\(190\) −6.18205e7 −0.653875
\(191\) −2.39670e7 −0.248884 −0.124442 0.992227i \(-0.539714\pi\)
−0.124442 + 0.992227i \(0.539714\pi\)
\(192\) −5.80479e7 −0.591878
\(193\) 5.81894e7 0.582631 0.291315 0.956627i \(-0.405907\pi\)
0.291315 + 0.956627i \(0.405907\pi\)
\(194\) 5.66836e7 0.557380
\(195\) −2.57975e8 −2.49148
\(196\) −2.06307e7 −0.195712
\(197\) −1.66210e6 −0.0154890 −0.00774452 0.999970i \(-0.502465\pi\)
−0.00774452 + 0.999970i \(0.502465\pi\)
\(198\) −3.20690e7 −0.293601
\(199\) 2.39093e7 0.215070 0.107535 0.994201i \(-0.465704\pi\)
0.107535 + 0.994201i \(0.465704\pi\)
\(200\) 6.60852e7 0.584116
\(201\) −3.99909e8 −3.47357
\(202\) −5.86216e7 −0.500412
\(203\) 1.19909e8 1.00604
\(204\) −3.44690e8 −2.84265
\(205\) 9.23612e7 0.748775
\(206\) 5.18444e7 0.413206
\(207\) 6.59688e7 0.516943
\(208\) 8.36811e7 0.644772
\(209\) −6.54099e7 −0.495600
\(210\) 1.29937e8 0.968201
\(211\) −1.89004e7 −0.138510 −0.0692550 0.997599i \(-0.522062\pi\)
−0.0692550 + 0.997599i \(0.522062\pi\)
\(212\) −1.23177e8 −0.887879
\(213\) −2.38162e8 −1.68867
\(214\) 6.30277e7 0.439626
\(215\) 3.04870e7 0.209209
\(216\) 2.24043e8 1.51266
\(217\) −1.03987e8 −0.690830
\(218\) −7.71705e7 −0.504491
\(219\) 2.35180e8 1.51303
\(220\) 6.96370e7 0.440921
\(221\) 2.89933e8 1.80686
\(222\) −6.47704e7 −0.397320
\(223\) −2.65364e8 −1.60241 −0.801207 0.598387i \(-0.795808\pi\)
−0.801207 + 0.598387i \(0.795808\pi\)
\(224\) −1.65373e8 −0.983095
\(225\) 3.41238e8 1.99719
\(226\) 8.90296e6 0.0513044
\(227\) 2.54830e8 1.44597 0.722986 0.690862i \(-0.242769\pi\)
0.722986 + 0.690862i \(0.242769\pi\)
\(228\) 3.81998e8 2.13447
\(229\) −1.45333e8 −0.799727 −0.399863 0.916575i \(-0.630942\pi\)
−0.399863 + 0.916575i \(0.630942\pi\)
\(230\) 2.04087e7 0.110603
\(231\) 1.37482e8 0.733842
\(232\) 1.14555e8 0.602289
\(233\) 6.81068e7 0.352732 0.176366 0.984325i \(-0.443566\pi\)
0.176366 + 0.984325i \(0.443566\pi\)
\(234\) −1.57533e8 −0.803741
\(235\) −1.92676e8 −0.968478
\(236\) 2.72475e8 1.34938
\(237\) 5.07790e8 2.47779
\(238\) −1.46033e8 −0.702154
\(239\) −2.19022e8 −1.03776 −0.518878 0.854848i \(-0.673650\pi\)
−0.518878 + 0.854848i \(0.673650\pi\)
\(240\) −3.40490e8 −1.58988
\(241\) 1.44105e7 0.0663160 0.0331580 0.999450i \(-0.489444\pi\)
0.0331580 + 0.999450i \(0.489444\pi\)
\(242\) 6.73589e7 0.305521
\(243\) 2.41821e8 1.08112
\(244\) 6.95011e7 0.306286
\(245\) −7.06087e7 −0.306745
\(246\) 8.13095e7 0.348232
\(247\) −3.21314e8 −1.35672
\(248\) −9.93441e7 −0.413581
\(249\) 3.95939e8 1.62529
\(250\) −1.41177e7 −0.0571442
\(251\) −2.09847e8 −0.837615 −0.418808 0.908075i \(-0.637552\pi\)
−0.418808 + 0.908075i \(0.637552\pi\)
\(252\) −5.56935e8 −2.19231
\(253\) 2.15937e7 0.0838309
\(254\) 1.36132e8 0.521245
\(255\) −1.17971e9 −4.45536
\(256\) −7.27492e6 −0.0271012
\(257\) −2.67717e8 −0.983808 −0.491904 0.870649i \(-0.663699\pi\)
−0.491904 + 0.870649i \(0.663699\pi\)
\(258\) 2.68390e7 0.0972966
\(259\) 1.92609e8 0.688855
\(260\) 3.42079e8 1.20703
\(261\) 5.91516e8 2.05932
\(262\) −1.41964e8 −0.487668
\(263\) −1.00008e8 −0.338993 −0.169497 0.985531i \(-0.554214\pi\)
−0.169497 + 0.985531i \(0.554214\pi\)
\(264\) 1.31343e8 0.439332
\(265\) −4.21574e8 −1.39160
\(266\) 1.61839e8 0.527228
\(267\) 2.48979e8 0.800521
\(268\) 5.30285e8 1.68282
\(269\) 8.77731e7 0.274934 0.137467 0.990506i \(-0.456104\pi\)
0.137467 + 0.990506i \(0.456104\pi\)
\(270\) 3.57899e8 1.10659
\(271\) −2.14577e8 −0.654924 −0.327462 0.944864i \(-0.606193\pi\)
−0.327462 + 0.944864i \(0.606193\pi\)
\(272\) 3.82669e8 1.15301
\(273\) 6.75352e8 2.00891
\(274\) 1.50708e8 0.442599
\(275\) 1.11698e8 0.323878
\(276\) −1.26108e8 −0.361046
\(277\) −3.67563e8 −1.03909 −0.519544 0.854443i \(-0.673898\pi\)
−0.519544 + 0.854443i \(0.673898\pi\)
\(278\) −5.79208e7 −0.161688
\(279\) −5.12974e8 −1.41410
\(280\) −3.69144e8 −1.00495
\(281\) −3.86156e8 −1.03822 −0.519111 0.854707i \(-0.673737\pi\)
−0.519111 + 0.854707i \(0.673737\pi\)
\(282\) −1.69621e8 −0.450409
\(283\) 2.07849e7 0.0545125 0.0272562 0.999628i \(-0.491323\pi\)
0.0272562 + 0.999628i \(0.491323\pi\)
\(284\) 3.15806e8 0.818100
\(285\) 1.30739e9 3.34541
\(286\) −5.15655e7 −0.130340
\(287\) −2.41792e8 −0.603747
\(288\) −8.15791e8 −2.01236
\(289\) 9.15506e8 2.23110
\(290\) 1.82997e8 0.440606
\(291\) −1.19876e9 −2.85171
\(292\) −3.11852e8 −0.733008
\(293\) 9.72174e7 0.225791 0.112896 0.993607i \(-0.463987\pi\)
0.112896 + 0.993607i \(0.463987\pi\)
\(294\) −6.21598e7 −0.142657
\(295\) 9.32548e8 2.11492
\(296\) 1.84009e8 0.412399
\(297\) 3.78680e8 0.838734
\(298\) −8.67501e7 −0.189895
\(299\) 1.06075e8 0.229489
\(300\) −6.52324e8 −1.39489
\(301\) −7.98118e7 −0.168688
\(302\) 6.93412e7 0.144866
\(303\) 1.23974e9 2.56025
\(304\) −4.24088e8 −0.865761
\(305\) 2.37868e8 0.480051
\(306\) −7.20389e8 −1.43728
\(307\) −8.13750e8 −1.60512 −0.802558 0.596573i \(-0.796528\pi\)
−0.802558 + 0.596573i \(0.796528\pi\)
\(308\) −1.82302e8 −0.355521
\(309\) −1.09642e9 −2.11408
\(310\) −1.58698e8 −0.302556
\(311\) −4.77734e7 −0.0900585 −0.0450293 0.998986i \(-0.514338\pi\)
−0.0450293 + 0.998986i \(0.514338\pi\)
\(312\) 6.45196e8 1.20268
\(313\) 4.14127e7 0.0763357 0.0381679 0.999271i \(-0.487848\pi\)
0.0381679 + 0.999271i \(0.487848\pi\)
\(314\) 1.14349e7 0.0208440
\(315\) −1.90612e9 −3.43607
\(316\) −6.73336e8 −1.20040
\(317\) 1.29531e8 0.228385 0.114192 0.993459i \(-0.463572\pi\)
0.114192 + 0.993459i \(0.463572\pi\)
\(318\) −3.71129e8 −0.647188
\(319\) 1.93622e8 0.333954
\(320\) 2.63438e8 0.449422
\(321\) −1.33292e9 −2.24925
\(322\) −5.34277e7 −0.0891808
\(323\) −1.46935e9 −2.42614
\(324\) −9.98149e8 −1.63038
\(325\) 5.48695e8 0.886624
\(326\) −1.06577e8 −0.170373
\(327\) 1.63202e9 2.58112
\(328\) −2.30995e8 −0.361447
\(329\) 5.04405e8 0.780897
\(330\) 2.09815e8 0.321394
\(331\) 1.06545e9 1.61486 0.807431 0.589962i \(-0.200857\pi\)
0.807431 + 0.589962i \(0.200857\pi\)
\(332\) −5.25020e8 −0.787395
\(333\) 9.50151e8 1.41006
\(334\) 1.93655e8 0.284391
\(335\) 1.81491e9 2.63753
\(336\) 8.91367e8 1.28194
\(337\) −2.29211e8 −0.326235 −0.163118 0.986607i \(-0.552155\pi\)
−0.163118 + 0.986607i \(0.552155\pi\)
\(338\) −2.60982e6 −0.00367622
\(339\) −1.88282e8 −0.262488
\(340\) 1.56431e9 2.15847
\(341\) −1.67913e8 −0.229321
\(342\) 7.98362e8 1.07922
\(343\) −6.41855e8 −0.858830
\(344\) −7.62481e7 −0.100989
\(345\) −4.31607e8 −0.565877
\(346\) −4.72684e8 −0.613486
\(347\) −5.28413e8 −0.678923 −0.339461 0.940620i \(-0.610245\pi\)
−0.339461 + 0.940620i \(0.610245\pi\)
\(348\) −1.13076e9 −1.43828
\(349\) −4.51049e6 −0.00567982 −0.00283991 0.999996i \(-0.500904\pi\)
−0.00283991 + 0.999996i \(0.500904\pi\)
\(350\) −2.76367e8 −0.344547
\(351\) 1.86019e9 2.29606
\(352\) −2.67034e8 −0.326338
\(353\) 2.41469e8 0.292179 0.146090 0.989271i \(-0.453331\pi\)
0.146090 + 0.989271i \(0.453331\pi\)
\(354\) 8.20961e8 0.983583
\(355\) 1.08085e9 1.28223
\(356\) −3.30149e8 −0.387824
\(357\) 3.08834e9 3.59242
\(358\) 1.28366e8 0.147863
\(359\) −7.82115e8 −0.892154 −0.446077 0.894995i \(-0.647179\pi\)
−0.446077 + 0.894995i \(0.647179\pi\)
\(360\) −1.82100e9 −2.05708
\(361\) 7.34514e8 0.821722
\(362\) 4.78353e8 0.529992
\(363\) −1.42452e9 −1.56313
\(364\) −8.95525e8 −0.973247
\(365\) −1.06732e9 −1.14886
\(366\) 2.09405e8 0.223256
\(367\) −2.09454e8 −0.221186 −0.110593 0.993866i \(-0.535275\pi\)
−0.110593 + 0.993866i \(0.535275\pi\)
\(368\) 1.40003e8 0.146444
\(369\) −1.19277e9 −1.23585
\(370\) 2.93947e8 0.301691
\(371\) 1.10364e9 1.12206
\(372\) 9.80621e8 0.987645
\(373\) 7.65743e8 0.764015 0.382008 0.924159i \(-0.375233\pi\)
0.382008 + 0.924159i \(0.375233\pi\)
\(374\) −2.35806e8 −0.233080
\(375\) 2.98563e8 0.292366
\(376\) 4.81882e8 0.467502
\(377\) 9.51130e8 0.914208
\(378\) −9.36941e8 −0.892259
\(379\) 5.91930e8 0.558513 0.279257 0.960217i \(-0.409912\pi\)
0.279257 + 0.960217i \(0.409912\pi\)
\(380\) −1.73362e9 −1.62073
\(381\) −2.87895e9 −2.66684
\(382\) 9.57543e7 0.0878894
\(383\) 1.93100e9 1.75625 0.878125 0.478432i \(-0.158795\pi\)
0.878125 + 0.478432i \(0.158795\pi\)
\(384\) 2.01359e9 1.81473
\(385\) −6.23932e8 −0.557217
\(386\) −2.32481e8 −0.205746
\(387\) −3.93716e8 −0.345298
\(388\) 1.58957e9 1.38155
\(389\) −2.18340e8 −0.188066 −0.0940330 0.995569i \(-0.529976\pi\)
−0.0940330 + 0.995569i \(0.529976\pi\)
\(390\) 1.03068e9 0.879824
\(391\) 4.85073e8 0.410382
\(392\) 1.76592e8 0.148071
\(393\) 3.00229e9 2.49505
\(394\) 6.64048e6 0.00546969
\(395\) −2.30450e9 −1.88142
\(396\) −8.99306e8 −0.727737
\(397\) 1.18371e9 0.949464 0.474732 0.880130i \(-0.342545\pi\)
0.474732 + 0.880130i \(0.342545\pi\)
\(398\) −9.55235e7 −0.0759485
\(399\) −3.42262e9 −2.69745
\(400\) 7.24198e8 0.565780
\(401\) −1.56318e9 −1.21061 −0.605304 0.795994i \(-0.706948\pi\)
−0.605304 + 0.795994i \(0.706948\pi\)
\(402\) 1.59774e9 1.22663
\(403\) −8.24838e8 −0.627771
\(404\) −1.64391e9 −1.24035
\(405\) −3.41617e9 −2.55533
\(406\) −4.79066e8 −0.355266
\(407\) 3.11014e8 0.228665
\(408\) 2.95044e9 2.15069
\(409\) −2.43312e8 −0.175846 −0.0879230 0.996127i \(-0.528023\pi\)
−0.0879230 + 0.996127i \(0.528023\pi\)
\(410\) −3.69006e8 −0.264417
\(411\) −3.18721e9 −2.26446
\(412\) 1.45386e9 1.02420
\(413\) −2.44131e9 −1.70529
\(414\) −2.63562e8 −0.182550
\(415\) −1.79689e9 −1.23410
\(416\) −1.31175e9 −0.893358
\(417\) 1.22492e9 0.827242
\(418\) 2.61329e8 0.175013
\(419\) −8.76950e8 −0.582406 −0.291203 0.956661i \(-0.594056\pi\)
−0.291203 + 0.956661i \(0.594056\pi\)
\(420\) 3.64380e9 2.39984
\(421\) 4.62306e7 0.0301955 0.0150977 0.999886i \(-0.495194\pi\)
0.0150977 + 0.999886i \(0.495194\pi\)
\(422\) 7.55116e7 0.0489125
\(423\) 2.48825e9 1.59847
\(424\) 1.05436e9 0.671749
\(425\) 2.50915e9 1.58550
\(426\) 9.51517e8 0.596325
\(427\) −6.22714e8 −0.387071
\(428\) 1.76747e9 1.08968
\(429\) 1.09052e9 0.666857
\(430\) −1.21803e8 −0.0738787
\(431\) 2.41795e9 1.45471 0.727355 0.686261i \(-0.240749\pi\)
0.727355 + 0.686261i \(0.240749\pi\)
\(432\) 2.45518e9 1.46518
\(433\) −2.05635e9 −1.21728 −0.608638 0.793448i \(-0.708284\pi\)
−0.608638 + 0.793448i \(0.708284\pi\)
\(434\) 4.15455e8 0.243955
\(435\) −3.87005e9 −2.25426
\(436\) −2.16408e9 −1.25046
\(437\) −5.37576e8 −0.308145
\(438\) −9.39604e8 −0.534300
\(439\) −3.96658e8 −0.223764 −0.111882 0.993722i \(-0.535688\pi\)
−0.111882 + 0.993722i \(0.535688\pi\)
\(440\) −5.96072e8 −0.333591
\(441\) 9.11854e8 0.506280
\(442\) −1.15835e9 −0.638062
\(443\) −2.23365e9 −1.22068 −0.610340 0.792139i \(-0.708967\pi\)
−0.610340 + 0.792139i \(0.708967\pi\)
\(444\) −1.81634e9 −0.984822
\(445\) −1.12994e9 −0.607847
\(446\) 1.06019e9 0.565866
\(447\) 1.83461e9 0.971556
\(448\) −6.89653e8 −0.362375
\(449\) −1.08951e9 −0.568027 −0.284014 0.958820i \(-0.591666\pi\)
−0.284014 + 0.958820i \(0.591666\pi\)
\(450\) −1.36333e9 −0.705274
\(451\) −3.90432e8 −0.200413
\(452\) 2.49664e8 0.127166
\(453\) −1.46644e9 −0.741176
\(454\) −1.01811e9 −0.510621
\(455\) −3.06494e9 −1.52540
\(456\) −3.26979e9 −1.61489
\(457\) 9.19833e8 0.450819 0.225410 0.974264i \(-0.427628\pi\)
0.225410 + 0.974264i \(0.427628\pi\)
\(458\) 5.80643e8 0.282410
\(459\) 8.50654e9 4.10590
\(460\) 5.72317e8 0.274147
\(461\) 3.88554e8 0.184713 0.0923567 0.995726i \(-0.470560\pi\)
0.0923567 + 0.995726i \(0.470560\pi\)
\(462\) −5.49273e8 −0.259144
\(463\) 1.15813e7 0.00542279 0.00271140 0.999996i \(-0.499137\pi\)
0.00271140 + 0.999996i \(0.499137\pi\)
\(464\) 1.25535e9 0.583382
\(465\) 3.35618e9 1.54796
\(466\) −2.72104e8 −0.124561
\(467\) −1.80641e9 −0.820742 −0.410371 0.911919i \(-0.634601\pi\)
−0.410371 + 0.911919i \(0.634601\pi\)
\(468\) −4.41767e9 −1.99220
\(469\) −4.75123e9 −2.12667
\(470\) 7.69788e8 0.342002
\(471\) −2.41829e8 −0.106644
\(472\) −2.33230e9 −1.02091
\(473\) −1.28876e8 −0.0559959
\(474\) −2.02875e9 −0.874991
\(475\) −2.78073e9 −1.19051
\(476\) −4.09519e9 −1.74040
\(477\) 5.44429e9 2.29682
\(478\) 8.75047e8 0.366466
\(479\) 4.53886e8 0.188700 0.0943500 0.995539i \(-0.469923\pi\)
0.0943500 + 0.995539i \(0.469923\pi\)
\(480\) 5.33739e9 2.20285
\(481\) 1.52780e9 0.625976
\(482\) −5.75734e7 −0.0234184
\(483\) 1.12990e9 0.456274
\(484\) 1.88893e9 0.757282
\(485\) 5.44031e9 2.16535
\(486\) −9.66136e8 −0.381778
\(487\) 3.04275e9 1.19375 0.596877 0.802333i \(-0.296408\pi\)
0.596877 + 0.802333i \(0.296408\pi\)
\(488\) −5.94909e8 −0.231729
\(489\) 2.25391e9 0.871677
\(490\) 2.82099e8 0.108322
\(491\) 6.19673e8 0.236253 0.118126 0.992999i \(-0.462311\pi\)
0.118126 + 0.992999i \(0.462311\pi\)
\(492\) 2.28015e9 0.863147
\(493\) 4.34946e9 1.63483
\(494\) 1.28373e9 0.479103
\(495\) −3.07788e9 −1.14060
\(496\) −1.08867e9 −0.400598
\(497\) −2.82955e9 −1.03388
\(498\) −1.58187e9 −0.573943
\(499\) 4.07486e9 1.46812 0.734059 0.679086i \(-0.237624\pi\)
0.734059 + 0.679086i \(0.237624\pi\)
\(500\) −3.95899e8 −0.141641
\(501\) −4.09546e9 −1.45503
\(502\) 8.38390e8 0.295790
\(503\) −3.44575e8 −0.120725 −0.0603623 0.998177i \(-0.519226\pi\)
−0.0603623 + 0.998177i \(0.519226\pi\)
\(504\) 4.76720e9 1.65866
\(505\) −5.62631e9 −1.94403
\(506\) −8.62720e7 −0.0296035
\(507\) 5.51930e7 0.0188086
\(508\) 3.81752e9 1.29199
\(509\) 5.78650e8 0.194493 0.0972464 0.995260i \(-0.468997\pi\)
0.0972464 + 0.995260i \(0.468997\pi\)
\(510\) 4.71322e9 1.57334
\(511\) 2.79412e9 0.926344
\(512\) −3.02139e9 −0.994859
\(513\) −9.42726e9 −3.08301
\(514\) 1.06960e9 0.347415
\(515\) 4.97586e9 1.60525
\(516\) 7.52642e8 0.241165
\(517\) 8.14484e8 0.259218
\(518\) −7.69521e8 −0.243258
\(519\) 9.99642e9 3.13877
\(520\) −2.92809e9 −0.913214
\(521\) 3.77073e9 1.16814 0.584068 0.811705i \(-0.301460\pi\)
0.584068 + 0.811705i \(0.301460\pi\)
\(522\) −2.36325e9 −0.727216
\(523\) 5.11275e8 0.156278 0.0781391 0.996942i \(-0.475102\pi\)
0.0781391 + 0.996942i \(0.475102\pi\)
\(524\) −3.98107e9 −1.20876
\(525\) 5.84467e9 1.76280
\(526\) 3.99558e8 0.119710
\(527\) −3.77194e9 −1.12261
\(528\) 1.43933e9 0.425540
\(529\) −3.22736e9 −0.947877
\(530\) 1.68429e9 0.491419
\(531\) −1.20431e10 −3.49066
\(532\) 4.53843e9 1.30682
\(533\) −1.91792e9 −0.548637
\(534\) −9.94731e8 −0.282691
\(535\) 6.04920e9 1.70789
\(536\) −4.53908e9 −1.27318
\(537\) −2.71472e9 −0.756510
\(538\) −3.50675e8 −0.0970884
\(539\) 2.98479e8 0.0821018
\(540\) 1.00365e10 2.74286
\(541\) 4.17611e9 1.13392 0.566959 0.823746i \(-0.308120\pi\)
0.566959 + 0.823746i \(0.308120\pi\)
\(542\) 8.57289e8 0.231276
\(543\) −1.01163e10 −2.71159
\(544\) −5.99857e9 −1.59754
\(545\) −7.40657e9 −1.95988
\(546\) −2.69820e9 −0.709414
\(547\) 1.14489e9 0.299093 0.149547 0.988755i \(-0.452219\pi\)
0.149547 + 0.988755i \(0.452219\pi\)
\(548\) 4.22629e9 1.09705
\(549\) −3.07188e9 −0.792320
\(550\) −4.46261e8 −0.114372
\(551\) −4.82023e9 −1.22754
\(552\) 1.07945e9 0.273159
\(553\) 6.03294e9 1.51702
\(554\) 1.46851e9 0.366937
\(555\) −6.21645e9 −1.54354
\(556\) −1.62426e9 −0.400770
\(557\) −5.14104e9 −1.26054 −0.630271 0.776375i \(-0.717056\pi\)
−0.630271 + 0.776375i \(0.717056\pi\)
\(558\) 2.04946e9 0.499367
\(559\) −6.33076e8 −0.153290
\(560\) −4.04528e9 −0.973399
\(561\) 4.98688e9 1.19250
\(562\) 1.54279e9 0.366631
\(563\) −9.25393e8 −0.218548 −0.109274 0.994012i \(-0.534853\pi\)
−0.109274 + 0.994012i \(0.534853\pi\)
\(564\) −4.75664e9 −1.11641
\(565\) 8.54478e8 0.199311
\(566\) −8.30409e7 −0.0192502
\(567\) 8.94318e9 2.06040
\(568\) −2.70321e9 −0.618956
\(569\) 3.80230e9 0.865273 0.432637 0.901568i \(-0.357583\pi\)
0.432637 + 0.901568i \(0.357583\pi\)
\(570\) −5.22336e9 −1.18138
\(571\) 3.02988e9 0.681082 0.340541 0.940230i \(-0.389390\pi\)
0.340541 + 0.940230i \(0.389390\pi\)
\(572\) −1.44604e9 −0.323069
\(573\) −2.02503e9 −0.449667
\(574\) 9.66019e8 0.213203
\(575\) 9.17998e8 0.201374
\(576\) −3.40209e9 −0.741767
\(577\) 4.05303e9 0.878344 0.439172 0.898403i \(-0.355272\pi\)
0.439172 + 0.898403i \(0.355272\pi\)
\(578\) −3.65767e9 −0.787875
\(579\) 4.91656e9 1.05266
\(580\) 5.13174e9 1.09211
\(581\) 4.70406e9 0.995075
\(582\) 4.78933e9 1.00703
\(583\) 1.78209e9 0.372468
\(584\) 2.66936e9 0.554578
\(585\) −1.51195e10 −3.12243
\(586\) −3.88407e8 −0.0797344
\(587\) 1.99299e9 0.406697 0.203348 0.979106i \(-0.434818\pi\)
0.203348 + 0.979106i \(0.434818\pi\)
\(588\) −1.74314e9 −0.353598
\(589\) 4.18020e9 0.842933
\(590\) −3.72576e9 −0.746849
\(591\) −1.40434e8 −0.0279845
\(592\) 2.01647e9 0.399453
\(593\) 6.10682e9 1.20261 0.601303 0.799021i \(-0.294648\pi\)
0.601303 + 0.799021i \(0.294648\pi\)
\(594\) −1.51292e9 −0.296185
\(595\) −1.40158e10 −2.72778
\(596\) −2.43272e9 −0.470684
\(597\) 2.02015e9 0.388574
\(598\) −4.23795e8 −0.0810403
\(599\) 6.06702e9 1.15340 0.576702 0.816954i \(-0.304339\pi\)
0.576702 + 0.816954i \(0.304339\pi\)
\(600\) 5.58369e9 1.05534
\(601\) −3.62416e9 −0.681000 −0.340500 0.940245i \(-0.610596\pi\)
−0.340500 + 0.940245i \(0.610596\pi\)
\(602\) 3.18868e8 0.0595694
\(603\) −2.34380e10 −4.35322
\(604\) 1.94452e9 0.359074
\(605\) 6.46489e9 1.18691
\(606\) −4.95308e9 −0.904109
\(607\) 6.80418e9 1.23485 0.617427 0.786629i \(-0.288175\pi\)
0.617427 + 0.786629i \(0.288175\pi\)
\(608\) 6.64784e9 1.19955
\(609\) 1.01314e10 1.81764
\(610\) −9.50343e8 −0.169522
\(611\) 4.00099e9 0.709616
\(612\) −2.02017e10 −3.56253
\(613\) −5.90679e9 −1.03571 −0.517857 0.855467i \(-0.673270\pi\)
−0.517857 + 0.855467i \(0.673270\pi\)
\(614\) 3.25113e9 0.566820
\(615\) 7.80382e9 1.35283
\(616\) 1.56045e9 0.268979
\(617\) −3.23829e9 −0.555031 −0.277516 0.960721i \(-0.589511\pi\)
−0.277516 + 0.960721i \(0.589511\pi\)
\(618\) 4.38046e9 0.746552
\(619\) −3.39944e9 −0.576090 −0.288045 0.957617i \(-0.593005\pi\)
−0.288045 + 0.957617i \(0.593005\pi\)
\(620\) −4.45034e9 −0.749933
\(621\) 3.11220e9 0.521492
\(622\) 1.90867e8 0.0318027
\(623\) 2.95806e9 0.490115
\(624\) 7.07042e9 1.16493
\(625\) −6.73854e9 −1.10404
\(626\) −1.65454e8 −0.0269567
\(627\) −5.52664e9 −0.895416
\(628\) 3.20668e8 0.0516651
\(629\) 6.98653e9 1.11940
\(630\) 7.61540e9 1.21339
\(631\) 1.54777e9 0.245247 0.122623 0.992453i \(-0.460869\pi\)
0.122623 + 0.992453i \(0.460869\pi\)
\(632\) 5.76356e9 0.908198
\(633\) −1.59694e9 −0.250250
\(634\) −5.17509e8 −0.0806503
\(635\) 1.30655e10 2.02497
\(636\) −1.04075e10 −1.60416
\(637\) 1.46622e9 0.224756
\(638\) −7.73567e8 −0.117930
\(639\) −1.39583e10 −2.11631
\(640\) −9.13827e9 −1.37795
\(641\) −7.31802e9 −1.09747 −0.548733 0.835998i \(-0.684889\pi\)
−0.548733 + 0.835998i \(0.684889\pi\)
\(642\) 5.32536e9 0.794285
\(643\) −1.05084e10 −1.55883 −0.779413 0.626511i \(-0.784482\pi\)
−0.779413 + 0.626511i \(0.784482\pi\)
\(644\) −1.49826e9 −0.221049
\(645\) 2.57592e9 0.377984
\(646\) 5.87041e9 0.856751
\(647\) −1.82079e9 −0.264299 −0.132150 0.991230i \(-0.542188\pi\)
−0.132150 + 0.991230i \(0.542188\pi\)
\(648\) 8.54385e9 1.23351
\(649\) −3.94209e9 −0.566070
\(650\) −2.19217e9 −0.313096
\(651\) −8.78613e9 −1.24814
\(652\) −2.98872e9 −0.422297
\(653\) 1.05192e10 1.47838 0.739190 0.673497i \(-0.235208\pi\)
0.739190 + 0.673497i \(0.235208\pi\)
\(654\) −6.52032e9 −0.911479
\(655\) −1.36253e10 −1.89452
\(656\) −2.53138e9 −0.350101
\(657\) 1.37835e10 1.89619
\(658\) −2.01522e9 −0.275761
\(659\) 7.91936e9 1.07793 0.538965 0.842328i \(-0.318815\pi\)
0.538965 + 0.842328i \(0.318815\pi\)
\(660\) 5.88380e9 0.796625
\(661\) 1.09291e10 1.47191 0.735954 0.677031i \(-0.236734\pi\)
0.735954 + 0.677031i \(0.236734\pi\)
\(662\) −4.25674e9 −0.570262
\(663\) 2.44971e10 3.26450
\(664\) 4.49401e9 0.595725
\(665\) 1.55328e10 2.04821
\(666\) −3.79609e9 −0.497939
\(667\) 1.59129e9 0.207640
\(668\) 5.43064e9 0.704909
\(669\) −2.24212e10 −2.89513
\(670\) −7.25100e9 −0.931399
\(671\) −1.00552e9 −0.128488
\(672\) −1.39727e10 −1.77619
\(673\) 8.76741e9 1.10871 0.554356 0.832280i \(-0.312965\pi\)
0.554356 + 0.832280i \(0.312965\pi\)
\(674\) 9.15754e8 0.115204
\(675\) 1.60986e10 2.01476
\(676\) −7.31866e7 −0.00911210
\(677\) −3.98192e8 −0.0493211 −0.0246605 0.999696i \(-0.507850\pi\)
−0.0246605 + 0.999696i \(0.507850\pi\)
\(678\) 7.52232e8 0.0926932
\(679\) −1.42421e10 −1.74595
\(680\) −1.33900e10 −1.63305
\(681\) 2.15312e10 2.61248
\(682\) 6.70852e8 0.0809807
\(683\) 4.45535e9 0.535068 0.267534 0.963548i \(-0.413791\pi\)
0.267534 + 0.963548i \(0.413791\pi\)
\(684\) 2.23883e10 2.67501
\(685\) 1.44645e10 1.71944
\(686\) 2.56437e9 0.303281
\(687\) −1.22796e10 −1.44489
\(688\) −8.35569e8 −0.0978189
\(689\) 8.75416e9 1.01964
\(690\) 1.72438e9 0.199830
\(691\) −1.49029e10 −1.71829 −0.859146 0.511730i \(-0.829005\pi\)
−0.859146 + 0.511730i \(0.829005\pi\)
\(692\) −1.32554e10 −1.52062
\(693\) 8.05757e9 0.919683
\(694\) 2.11114e9 0.239750
\(695\) −5.55906e9 −0.628137
\(696\) 9.67900e9 1.08817
\(697\) −8.77054e9 −0.981096
\(698\) 1.80205e7 0.00200573
\(699\) 5.75451e9 0.637291
\(700\) −7.75011e9 −0.854013
\(701\) 4.40614e9 0.483109 0.241555 0.970387i \(-0.422343\pi\)
0.241555 + 0.970387i \(0.422343\pi\)
\(702\) −7.43192e9 −0.810814
\(703\) −7.74272e9 −0.840524
\(704\) −1.11361e9 −0.120290
\(705\) −1.62796e10 −1.74978
\(706\) −9.64727e8 −0.103178
\(707\) 1.47291e10 1.56750
\(708\) 2.30221e10 2.43797
\(709\) −7.42152e9 −0.782043 −0.391022 0.920381i \(-0.627878\pi\)
−0.391022 + 0.920381i \(0.627878\pi\)
\(710\) −4.31826e9 −0.452798
\(711\) 2.97608e10 3.10528
\(712\) 2.82597e9 0.293419
\(713\) −1.38000e9 −0.142582
\(714\) −1.23387e10 −1.26860
\(715\) −4.94909e9 −0.506354
\(716\) 3.59975e9 0.366503
\(717\) −1.85057e10 −1.87494
\(718\) 3.12474e9 0.315049
\(719\) −7.84543e9 −0.787165 −0.393582 0.919289i \(-0.628764\pi\)
−0.393582 + 0.919289i \(0.628764\pi\)
\(720\) −1.99556e10 −1.99251
\(721\) −1.30263e10 −1.29434
\(722\) −2.93456e9 −0.290177
\(723\) 1.21757e9 0.119815
\(724\) 1.34144e10 1.31367
\(725\) 8.23132e9 0.802207
\(726\) 5.69131e9 0.551994
\(727\) 1.92504e10 1.85810 0.929052 0.369949i \(-0.120625\pi\)
0.929052 + 0.369949i \(0.120625\pi\)
\(728\) 7.66543e9 0.736337
\(729\) 9.48025e8 0.0906303
\(730\) 4.26420e9 0.405702
\(731\) −2.89502e9 −0.274120
\(732\) 5.87232e9 0.553376
\(733\) −1.25342e10 −1.17553 −0.587766 0.809031i \(-0.699992\pi\)
−0.587766 + 0.809031i \(0.699992\pi\)
\(734\) 8.36820e8 0.0781081
\(735\) −5.96590e9 −0.554204
\(736\) −2.19464e9 −0.202904
\(737\) −7.67201e9 −0.705948
\(738\) 4.76542e9 0.436419
\(739\) 3.51649e9 0.320519 0.160259 0.987075i \(-0.448767\pi\)
0.160259 + 0.987075i \(0.448767\pi\)
\(740\) 8.24310e9 0.747789
\(741\) −2.71486e10 −2.45122
\(742\) −4.40930e9 −0.396238
\(743\) 3.44470e9 0.308099 0.154050 0.988063i \(-0.450768\pi\)
0.154050 + 0.988063i \(0.450768\pi\)
\(744\) −8.39382e9 −0.747230
\(745\) −8.32600e9 −0.737716
\(746\) −3.05933e9 −0.269799
\(747\) 2.32053e10 2.03688
\(748\) −6.61267e9 −0.577725
\(749\) −1.58362e10 −1.37709
\(750\) −1.19283e9 −0.103244
\(751\) 5.87692e9 0.506302 0.253151 0.967427i \(-0.418533\pi\)
0.253151 + 0.967427i \(0.418533\pi\)
\(752\) 5.28073e9 0.452826
\(753\) −1.77305e10 −1.51334
\(754\) −3.80000e9 −0.322837
\(755\) 6.65514e9 0.562786
\(756\) −2.62745e10 −2.21161
\(757\) −8.53566e9 −0.715157 −0.357579 0.933883i \(-0.616398\pi\)
−0.357579 + 0.933883i \(0.616398\pi\)
\(758\) −2.36491e9 −0.197230
\(759\) 1.82450e9 0.151460
\(760\) 1.48393e10 1.22621
\(761\) 1.24999e10 1.02815 0.514077 0.857744i \(-0.328134\pi\)
0.514077 + 0.857744i \(0.328134\pi\)
\(762\) 1.15021e10 0.941749
\(763\) 1.93896e10 1.58028
\(764\) 2.68522e9 0.217848
\(765\) −6.91407e10 −5.58365
\(766\) −7.71482e9 −0.620190
\(767\) −1.93647e10 −1.54963
\(768\) −6.14676e8 −0.0489646
\(769\) −9.50817e9 −0.753971 −0.376985 0.926219i \(-0.623039\pi\)
−0.376985 + 0.926219i \(0.623039\pi\)
\(770\) 2.49276e9 0.196772
\(771\) −2.26201e10 −1.77747
\(772\) −6.51943e9 −0.509975
\(773\) −1.06727e10 −0.831089 −0.415544 0.909573i \(-0.636409\pi\)
−0.415544 + 0.909573i \(0.636409\pi\)
\(774\) 1.57299e9 0.121936
\(775\) −7.13836e9 −0.550862
\(776\) −1.36062e10 −1.04525
\(777\) 1.62740e10 1.24457
\(778\) 8.72323e8 0.0664123
\(779\) 9.71982e9 0.736677
\(780\) 2.89030e10 2.18078
\(781\) −4.56899e9 −0.343196
\(782\) −1.93799e9 −0.144920
\(783\) 2.79059e10 2.07745
\(784\) 1.93520e9 0.143423
\(785\) 1.09749e9 0.0809760
\(786\) −1.19949e10 −0.881084
\(787\) 2.51180e10 1.83685 0.918425 0.395596i \(-0.129462\pi\)
0.918425 + 0.395596i \(0.129462\pi\)
\(788\) 1.86218e8 0.0135575
\(789\) −8.44994e9 −0.612469
\(790\) 9.20705e9 0.664394
\(791\) −2.23693e9 −0.160707
\(792\) 7.69779e9 0.550590
\(793\) −4.93943e9 −0.351739
\(794\) −4.72921e9 −0.335287
\(795\) −3.56198e10 −2.51424
\(796\) −2.67875e9 −0.188250
\(797\) 1.35480e10 0.947917 0.473959 0.880547i \(-0.342825\pi\)
0.473959 + 0.880547i \(0.342825\pi\)
\(798\) 1.36742e10 0.952559
\(799\) 1.82963e10 1.26897
\(800\) −1.13523e10 −0.783912
\(801\) 1.45922e10 1.00325
\(802\) 6.24529e9 0.427506
\(803\) 4.51179e9 0.307499
\(804\) 4.48051e10 3.04040
\(805\) −5.12782e9 −0.346455
\(806\) 3.29543e9 0.221687
\(807\) 7.41616e9 0.496732
\(808\) 1.40714e10 0.938421
\(809\) 1.21582e10 0.807328 0.403664 0.914907i \(-0.367736\pi\)
0.403664 + 0.914907i \(0.367736\pi\)
\(810\) 1.36485e10 0.902373
\(811\) 2.69149e10 1.77182 0.885911 0.463855i \(-0.153534\pi\)
0.885911 + 0.463855i \(0.153534\pi\)
\(812\) −1.34343e10 −0.880583
\(813\) −1.81301e10 −1.18327
\(814\) −1.24258e9 −0.0807492
\(815\) −1.02289e10 −0.661877
\(816\) 3.23326e10 2.08317
\(817\) 3.20837e9 0.205829
\(818\) 9.72093e8 0.0620971
\(819\) 3.95813e10 2.51766
\(820\) −1.03480e10 −0.655400
\(821\) −1.50522e10 −0.949292 −0.474646 0.880177i \(-0.657424\pi\)
−0.474646 + 0.880177i \(0.657424\pi\)
\(822\) 1.27337e10 0.799657
\(823\) −1.95484e10 −1.22240 −0.611198 0.791478i \(-0.709312\pi\)
−0.611198 + 0.791478i \(0.709312\pi\)
\(824\) −1.24446e10 −0.774885
\(825\) 9.43763e9 0.585159
\(826\) 9.75364e9 0.602194
\(827\) −2.07950e9 −0.127847 −0.0639233 0.997955i \(-0.520361\pi\)
−0.0639233 + 0.997955i \(0.520361\pi\)
\(828\) −7.39101e9 −0.452478
\(829\) 2.13263e10 1.30009 0.650047 0.759894i \(-0.274749\pi\)
0.650047 + 0.759894i \(0.274749\pi\)
\(830\) 7.17900e9 0.435804
\(831\) −3.10563e10 −1.87735
\(832\) −5.47040e9 −0.329297
\(833\) 6.70493e9 0.401918
\(834\) −4.89387e9 −0.292127
\(835\) 1.85864e10 1.10482
\(836\) 7.32840e9 0.433797
\(837\) −2.42005e10 −1.42655
\(838\) 3.50363e9 0.205667
\(839\) −1.08997e10 −0.637161 −0.318580 0.947896i \(-0.603206\pi\)
−0.318580 + 0.947896i \(0.603206\pi\)
\(840\) −3.11898e10 −1.81566
\(841\) −2.98137e9 −0.172834
\(842\) −1.84703e8 −0.0106630
\(843\) −3.26272e10 −1.87579
\(844\) 2.11756e9 0.121237
\(845\) −2.50482e8 −0.0142816
\(846\) −9.94119e9 −0.564472
\(847\) −1.69244e10 −0.957020
\(848\) 1.15542e10 0.650662
\(849\) 1.75617e9 0.0984893
\(850\) −1.00247e10 −0.559892
\(851\) 2.55609e9 0.142175
\(852\) 2.66832e10 1.47809
\(853\) −2.05206e10 −1.13206 −0.566028 0.824386i \(-0.691521\pi\)
−0.566028 + 0.824386i \(0.691521\pi\)
\(854\) 2.48790e9 0.136688
\(855\) 7.66242e10 4.19261
\(856\) −1.51290e10 −0.824429
\(857\) −2.27910e10 −1.23689 −0.618443 0.785830i \(-0.712236\pi\)
−0.618443 + 0.785830i \(0.712236\pi\)
\(858\) −4.35689e9 −0.235489
\(859\) −5.82236e9 −0.313418 −0.156709 0.987645i \(-0.550088\pi\)
−0.156709 + 0.987645i \(0.550088\pi\)
\(860\) −3.41571e9 −0.183120
\(861\) −2.04296e10 −1.09081
\(862\) −9.66030e9 −0.513707
\(863\) 1.77088e10 0.937891 0.468945 0.883227i \(-0.344634\pi\)
0.468945 + 0.883227i \(0.344634\pi\)
\(864\) −3.84865e10 −2.03007
\(865\) −4.53667e10 −2.38331
\(866\) 8.21562e9 0.429861
\(867\) 7.73533e10 4.03099
\(868\) 1.16505e10 0.604681
\(869\) 9.74163e9 0.503573
\(870\) 1.54618e10 0.796055
\(871\) −3.76873e10 −1.93255
\(872\) 1.85238e10 0.946070
\(873\) −7.02572e10 −3.57389
\(874\) 2.14775e9 0.108816
\(875\) 3.54716e9 0.179000
\(876\) −2.63491e10 −1.32435
\(877\) 2.41157e10 1.20726 0.603630 0.797264i \(-0.293720\pi\)
0.603630 + 0.797264i \(0.293720\pi\)
\(878\) 1.58475e9 0.0790185
\(879\) 8.21413e9 0.407944
\(880\) −6.53209e9 −0.323119
\(881\) 2.64320e10 1.30231 0.651155 0.758945i \(-0.274285\pi\)
0.651155 + 0.758945i \(0.274285\pi\)
\(882\) −3.64308e9 −0.178784
\(883\) 2.34325e10 1.14540 0.572699 0.819766i \(-0.305896\pi\)
0.572699 + 0.819766i \(0.305896\pi\)
\(884\) −3.24835e10 −1.58154
\(885\) 7.87932e10 3.82109
\(886\) 8.92398e9 0.431063
\(887\) −2.01156e10 −0.967832 −0.483916 0.875114i \(-0.660786\pi\)
−0.483916 + 0.875114i \(0.660786\pi\)
\(888\) 1.55473e10 0.745094
\(889\) −3.42041e10 −1.63276
\(890\) 4.51438e9 0.214651
\(891\) 1.44409e10 0.683948
\(892\) 2.97308e10 1.40259
\(893\) −2.02766e10 −0.952831
\(894\) −7.32973e9 −0.343089
\(895\) 1.23202e10 0.574429
\(896\) 2.39230e10 1.11106
\(897\) 8.96250e9 0.414625
\(898\) 4.35286e9 0.200589
\(899\) −1.23739e10 −0.568000
\(900\) −3.82317e10 −1.74813
\(901\) 4.00323e10 1.82336
\(902\) 1.55987e9 0.0707727
\(903\) −6.74349e9 −0.304774
\(904\) −2.13705e9 −0.0962110
\(905\) 4.59108e10 2.05895
\(906\) 5.85880e9 0.261734
\(907\) −2.12612e10 −0.946156 −0.473078 0.881021i \(-0.656857\pi\)
−0.473078 + 0.881021i \(0.656857\pi\)
\(908\) −2.85506e10 −1.26566
\(909\) 7.26593e10 3.20861
\(910\) 1.22452e10 0.538668
\(911\) 5.63102e9 0.246759 0.123379 0.992360i \(-0.460627\pi\)
0.123379 + 0.992360i \(0.460627\pi\)
\(912\) −3.58322e10 −1.56420
\(913\) 7.59583e9 0.330315
\(914\) −3.67496e9 −0.159199
\(915\) 2.00981e10 0.867322
\(916\) 1.62829e10 0.699998
\(917\) 3.56695e10 1.52758
\(918\) −3.39857e10 −1.44993
\(919\) −1.37242e10 −0.583286 −0.291643 0.956527i \(-0.594202\pi\)
−0.291643 + 0.956527i \(0.594202\pi\)
\(920\) −4.89886e9 −0.207414
\(921\) −6.87557e10 −2.90001
\(922\) −1.55237e9 −0.0652285
\(923\) −2.24443e10 −0.939507
\(924\) −1.54032e10 −0.642330
\(925\) 1.32220e10 0.549287
\(926\) −4.62701e7 −0.00191497
\(927\) −6.42593e10 −2.64946
\(928\) −1.96785e10 −0.808301
\(929\) 1.80192e10 0.737363 0.368682 0.929556i \(-0.379809\pi\)
0.368682 + 0.929556i \(0.379809\pi\)
\(930\) −1.34088e10 −0.546637
\(931\) −7.43065e9 −0.301789
\(932\) −7.63055e9 −0.308745
\(933\) −4.03649e9 −0.162711
\(934\) 7.21705e9 0.289831
\(935\) −2.26319e10 −0.905484
\(936\) 3.78139e10 1.50725
\(937\) 1.76207e10 0.699736 0.349868 0.936799i \(-0.386227\pi\)
0.349868 + 0.936799i \(0.386227\pi\)
\(938\) 1.89823e10 0.751000
\(939\) 3.49905e9 0.137918
\(940\) 2.15870e10 0.847706
\(941\) −3.45116e9 −0.135021 −0.0675105 0.997719i \(-0.521506\pi\)
−0.0675105 + 0.997719i \(0.521506\pi\)
\(942\) 9.66166e8 0.0376594
\(943\) −3.20879e9 −0.124609
\(944\) −2.55587e10 −0.988863
\(945\) −8.99246e10 −3.46631
\(946\) 5.14890e8 0.0197740
\(947\) 5.63093e9 0.215454 0.107727 0.994181i \(-0.465643\pi\)
0.107727 + 0.994181i \(0.465643\pi\)
\(948\) −5.68918e10 −2.16880
\(949\) 2.21633e10 0.841787
\(950\) 1.11097e10 0.420407
\(951\) 1.09444e10 0.412629
\(952\) 3.50535e10 1.31675
\(953\) 4.39854e10 1.64620 0.823101 0.567896i \(-0.192242\pi\)
0.823101 + 0.567896i \(0.192242\pi\)
\(954\) −2.17513e10 −0.811084
\(955\) 9.19019e9 0.341439
\(956\) 2.45388e10 0.908344
\(957\) 1.63596e10 0.603365
\(958\) −1.81339e9 −0.0666363
\(959\) −3.78666e10 −1.38641
\(960\) 2.22585e10 0.811983
\(961\) −1.67817e10 −0.609964
\(962\) −6.10393e9 −0.221053
\(963\) −7.81205e10 −2.81886
\(964\) −1.61452e9 −0.0580462
\(965\) −2.23128e10 −0.799298
\(966\) −4.51424e9 −0.161126
\(967\) 3.74348e10 1.33132 0.665660 0.746255i \(-0.268150\pi\)
0.665660 + 0.746255i \(0.268150\pi\)
\(968\) −1.61687e10 −0.572942
\(969\) −1.24149e11 −4.38338
\(970\) −2.17354e10 −0.764656
\(971\) 3.42358e9 0.120009 0.0600044 0.998198i \(-0.480889\pi\)
0.0600044 + 0.998198i \(0.480889\pi\)
\(972\) −2.70932e10 −0.946298
\(973\) 1.45530e10 0.506475
\(974\) −1.21565e10 −0.421554
\(975\) 4.63606e10 1.60189
\(976\) −6.51934e9 −0.224455
\(977\) −5.03847e10 −1.72849 −0.864247 0.503068i \(-0.832205\pi\)
−0.864247 + 0.503068i \(0.832205\pi\)
\(978\) −9.00493e9 −0.307818
\(979\) 4.77650e9 0.162693
\(980\) 7.91086e9 0.268493
\(981\) 9.56500e10 3.23477
\(982\) −2.47575e9 −0.0834288
\(983\) −3.95623e10 −1.32845 −0.664224 0.747534i \(-0.731238\pi\)
−0.664224 + 0.747534i \(0.731238\pi\)
\(984\) −1.95174e10 −0.653038
\(985\) 6.37332e8 0.0212490
\(986\) −1.73772e10 −0.577311
\(987\) 4.26184e10 1.41087
\(988\) 3.59993e10 1.18753
\(989\) −1.05917e9 −0.0348160
\(990\) 1.22969e10 0.402785
\(991\) 1.79657e10 0.586388 0.293194 0.956053i \(-0.405282\pi\)
0.293194 + 0.956053i \(0.405282\pi\)
\(992\) 1.70655e10 0.555046
\(993\) 9.00225e10 2.91762
\(994\) 1.13048e10 0.365097
\(995\) −9.16804e9 −0.295050
\(996\) −4.43602e10 −1.42261
\(997\) 1.97079e9 0.0629807 0.0314904 0.999504i \(-0.489975\pi\)
0.0314904 + 0.999504i \(0.489975\pi\)
\(998\) −1.62801e10 −0.518441
\(999\) 4.48252e10 1.42247
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.1.5 13
3.2 odd 2 387.8.a.d.1.9 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.5 13 1.1 even 1 trivial
387.8.a.d.1.9 13 3.2 odd 2