Properties

Label 43.8.a.b.1.2
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 1.2
Root \(-20.1662\) 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

\(f(q)\) \(=\) \(q-19.1662 q^{2} -36.5899 q^{3} +239.342 q^{4} -174.722 q^{5} +701.287 q^{6} -1126.43 q^{7} -2133.99 q^{8} -848.182 q^{9} +O(q^{10})\) \(q-19.1662 q^{2} -36.5899 q^{3} +239.342 q^{4} -174.722 q^{5} +701.287 q^{6} -1126.43 q^{7} -2133.99 q^{8} -848.182 q^{9} +3348.75 q^{10} -8178.46 q^{11} -8757.47 q^{12} -13500.8 q^{13} +21589.4 q^{14} +6393.05 q^{15} +10264.6 q^{16} +37508.8 q^{17} +16256.4 q^{18} -26617.7 q^{19} -41818.2 q^{20} +41216.0 q^{21} +156750. q^{22} -20061.6 q^{23} +78082.3 q^{24} -47597.3 q^{25} +258757. q^{26} +111057. q^{27} -269602. q^{28} -27521.8 q^{29} -122530. q^{30} -184114. q^{31} +76416.8 q^{32} +299249. q^{33} -718900. q^{34} +196812. q^{35} -203005. q^{36} +301128. q^{37} +510158. q^{38} +493991. q^{39} +372855. q^{40} -507344. q^{41} -789952. q^{42} -79507.0 q^{43} -1.95745e6 q^{44} +148196. q^{45} +384503. q^{46} -137672. q^{47} -375582. q^{48} +445307. q^{49} +912256. q^{50} -1.37244e6 q^{51} -3.23129e6 q^{52} +577240. q^{53} -2.12853e6 q^{54} +1.42896e6 q^{55} +2.40379e6 q^{56} +973937. q^{57} +527487. q^{58} +1.70368e6 q^{59} +1.53012e6 q^{60} +1.24127e6 q^{61} +3.52877e6 q^{62} +955420. q^{63} -2.77849e6 q^{64} +2.35888e6 q^{65} -5.73545e6 q^{66} -1.24429e6 q^{67} +8.97742e6 q^{68} +734050. q^{69} -3.77214e6 q^{70} -3.10722e6 q^{71} +1.81001e6 q^{72} -3.60450e6 q^{73} -5.77146e6 q^{74} +1.74158e6 q^{75} -6.37071e6 q^{76} +9.21248e6 q^{77} -9.46790e6 q^{78} -3.14376e6 q^{79} -1.79346e6 q^{80} -2.20858e6 q^{81} +9.72384e6 q^{82} +347048. q^{83} +9.86470e6 q^{84} -6.55361e6 q^{85} +1.52384e6 q^{86} +1.00702e6 q^{87} +1.74527e7 q^{88} -4.60056e6 q^{89} -2.84035e6 q^{90} +1.52077e7 q^{91} -4.80156e6 q^{92} +6.73672e6 q^{93} +2.63864e6 q^{94} +4.65069e6 q^{95} -2.79608e6 q^{96} -1.55843e7 q^{97} -8.53482e6 q^{98} +6.93682e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.1662 −1.69406 −0.847032 0.531541i \(-0.821613\pi\)
−0.847032 + 0.531541i \(0.821613\pi\)
\(3\) −36.5899 −0.782414 −0.391207 0.920303i \(-0.627942\pi\)
−0.391207 + 0.920303i \(0.627942\pi\)
\(4\) 239.342 1.86986
\(5\) −174.722 −0.625104 −0.312552 0.949901i \(-0.601184\pi\)
−0.312552 + 0.949901i \(0.601184\pi\)
\(6\) 701.287 1.32546
\(7\) −1126.43 −1.24126 −0.620629 0.784104i \(-0.713123\pi\)
−0.620629 + 0.784104i \(0.713123\pi\)
\(8\) −2133.99 −1.47359
\(9\) −848.182 −0.387829
\(10\) 3348.75 1.05897
\(11\) −8178.46 −1.85267 −0.926333 0.376705i \(-0.877057\pi\)
−0.926333 + 0.376705i \(0.877057\pi\)
\(12\) −8757.47 −1.46300
\(13\) −13500.8 −1.70434 −0.852170 0.523265i \(-0.824714\pi\)
−0.852170 + 0.523265i \(0.824714\pi\)
\(14\) 21589.4 2.10277
\(15\) 6393.05 0.489090
\(16\) 10264.6 0.626504
\(17\) 37508.8 1.85166 0.925832 0.377935i \(-0.123366\pi\)
0.925832 + 0.377935i \(0.123366\pi\)
\(18\) 16256.4 0.657008
\(19\) −26617.7 −0.890292 −0.445146 0.895458i \(-0.646848\pi\)
−0.445146 + 0.895458i \(0.646848\pi\)
\(20\) −41818.2 −1.16885
\(21\) 41216.0 0.971177
\(22\) 156750. 3.13854
\(23\) −20061.6 −0.343809 −0.171905 0.985114i \(-0.554992\pi\)
−0.171905 + 0.985114i \(0.554992\pi\)
\(24\) 78082.3 1.15296
\(25\) −47597.3 −0.609245
\(26\) 258757. 2.88726
\(27\) 111057. 1.08586
\(28\) −269602. −2.32097
\(29\) −27521.8 −0.209548 −0.104774 0.994496i \(-0.533412\pi\)
−0.104774 + 0.994496i \(0.533412\pi\)
\(30\) −122530. −0.828550
\(31\) −184114. −1.11000 −0.554999 0.831851i \(-0.687281\pi\)
−0.554999 + 0.831851i \(0.687281\pi\)
\(32\) 76416.8 0.412253
\(33\) 299249. 1.44955
\(34\) −718900. −3.13684
\(35\) 196812. 0.775915
\(36\) −203005. −0.725184
\(37\) 301128. 0.977338 0.488669 0.872469i \(-0.337483\pi\)
0.488669 + 0.872469i \(0.337483\pi\)
\(38\) 510158. 1.50821
\(39\) 493991. 1.33350
\(40\) 372855. 0.921148
\(41\) −507344. −1.14963 −0.574817 0.818282i \(-0.694927\pi\)
−0.574817 + 0.818282i \(0.694927\pi\)
\(42\) −789952. −1.64524
\(43\) −79507.0 −0.152499
\(44\) −1.95745e6 −3.46422
\(45\) 148196. 0.242434
\(46\) 384503. 0.582435
\(47\) −137672. −0.193420 −0.0967102 0.995313i \(-0.530832\pi\)
−0.0967102 + 0.995313i \(0.530832\pi\)
\(48\) −375582. −0.490185
\(49\) 445307. 0.540721
\(50\) 912256. 1.03210
\(51\) −1.37244e6 −1.44877
\(52\) −3.23129e6 −3.18687
\(53\) 577240. 0.532588 0.266294 0.963892i \(-0.414201\pi\)
0.266294 + 0.963892i \(0.414201\pi\)
\(54\) −2.12853e6 −1.83951
\(55\) 1.42896e6 1.15811
\(56\) 2.40379e6 1.82911
\(57\) 973937. 0.696577
\(58\) 527487. 0.354988
\(59\) 1.70368e6 1.07995 0.539977 0.841680i \(-0.318433\pi\)
0.539977 + 0.841680i \(0.318433\pi\)
\(60\) 1.53012e6 0.914527
\(61\) 1.24127e6 0.700182 0.350091 0.936716i \(-0.386151\pi\)
0.350091 + 0.936716i \(0.386151\pi\)
\(62\) 3.52877e6 1.88041
\(63\) 955420. 0.481396
\(64\) −2.77849e6 −1.32489
\(65\) 2.35888e6 1.06539
\(66\) −5.73545e6 −2.45563
\(67\) −1.24429e6 −0.505430 −0.252715 0.967541i \(-0.581323\pi\)
−0.252715 + 0.967541i \(0.581323\pi\)
\(68\) 8.97742e6 3.46234
\(69\) 734050. 0.269001
\(70\) −3.77214e6 −1.31445
\(71\) −3.10722e6 −1.03031 −0.515154 0.857098i \(-0.672265\pi\)
−0.515154 + 0.857098i \(0.672265\pi\)
\(72\) 1.81001e6 0.571502
\(73\) −3.60450e6 −1.08446 −0.542232 0.840229i \(-0.682421\pi\)
−0.542232 + 0.840229i \(0.682421\pi\)
\(74\) −5.77146e6 −1.65567
\(75\) 1.74158e6 0.476681
\(76\) −6.37071e6 −1.66472
\(77\) 9.21248e6 2.29964
\(78\) −9.46790e6 −2.25903
\(79\) −3.14376e6 −0.717390 −0.358695 0.933455i \(-0.616778\pi\)
−0.358695 + 0.933455i \(0.616778\pi\)
\(80\) −1.79346e6 −0.391630
\(81\) −2.20858e6 −0.461760
\(82\) 9.72384e6 1.94755
\(83\) 347048. 0.0666218 0.0333109 0.999445i \(-0.489395\pi\)
0.0333109 + 0.999445i \(0.489395\pi\)
\(84\) 9.86470e6 1.81596
\(85\) −6.55361e6 −1.15748
\(86\) 1.52384e6 0.258342
\(87\) 1.00702e6 0.163953
\(88\) 1.74527e7 2.73007
\(89\) −4.60056e6 −0.691745 −0.345872 0.938282i \(-0.612417\pi\)
−0.345872 + 0.938282i \(0.612417\pi\)
\(90\) −2.84035e6 −0.410698
\(91\) 1.52077e7 2.11553
\(92\) −4.80156e6 −0.642873
\(93\) 6.73672e6 0.868477
\(94\) 2.63864e6 0.327667
\(95\) 4.65069e6 0.556525
\(96\) −2.79608e6 −0.322552
\(97\) −1.55843e7 −1.73375 −0.866875 0.498526i \(-0.833875\pi\)
−0.866875 + 0.498526i \(0.833875\pi\)
\(98\) −8.53482e6 −0.916016
\(99\) 6.93682e6 0.718518
\(100\) −1.13920e7 −1.13920
\(101\) −9.18925e6 −0.887474 −0.443737 0.896157i \(-0.646348\pi\)
−0.443737 + 0.896157i \(0.646348\pi\)
\(102\) 2.63044e7 2.45431
\(103\) −1.58567e7 −1.42982 −0.714910 0.699216i \(-0.753533\pi\)
−0.714910 + 0.699216i \(0.753533\pi\)
\(104\) 2.88104e7 2.51150
\(105\) −7.20134e6 −0.607087
\(106\) −1.10635e7 −0.902238
\(107\) −833526. −0.0657773 −0.0328886 0.999459i \(-0.510471\pi\)
−0.0328886 + 0.999459i \(0.510471\pi\)
\(108\) 2.65805e7 2.03039
\(109\) 7.34548e6 0.543285 0.271642 0.962398i \(-0.412433\pi\)
0.271642 + 0.962398i \(0.412433\pi\)
\(110\) −2.73876e7 −1.96191
\(111\) −1.10182e7 −0.764683
\(112\) −1.15624e7 −0.777653
\(113\) 3.08279e6 0.200988 0.100494 0.994938i \(-0.467958\pi\)
0.100494 + 0.994938i \(0.467958\pi\)
\(114\) −1.86666e7 −1.18005
\(115\) 3.50519e6 0.214916
\(116\) −6.58710e6 −0.391824
\(117\) 1.14511e7 0.660993
\(118\) −3.26530e7 −1.82951
\(119\) −4.22512e7 −2.29839
\(120\) −1.36427e7 −0.720719
\(121\) 4.74000e7 2.43237
\(122\) −2.37903e7 −1.18615
\(123\) 1.85637e7 0.899489
\(124\) −4.40662e7 −2.07553
\(125\) 2.19664e7 1.00595
\(126\) −1.83117e7 −0.815516
\(127\) −1.91611e6 −0.0830057 −0.0415028 0.999138i \(-0.513215\pi\)
−0.0415028 + 0.999138i \(0.513215\pi\)
\(128\) 4.34716e7 1.83219
\(129\) 2.90915e6 0.119317
\(130\) −4.52106e7 −1.80484
\(131\) −2.07138e7 −0.805026 −0.402513 0.915414i \(-0.631863\pi\)
−0.402513 + 0.915414i \(0.631863\pi\)
\(132\) 7.16226e7 2.71045
\(133\) 2.99830e7 1.10508
\(134\) 2.38483e7 0.856231
\(135\) −1.94041e7 −0.678773
\(136\) −8.00434e7 −2.72860
\(137\) −3.16796e7 −1.05259 −0.526294 0.850303i \(-0.676419\pi\)
−0.526294 + 0.850303i \(0.676419\pi\)
\(138\) −1.40689e7 −0.455705
\(139\) −4.90696e7 −1.54975 −0.774873 0.632117i \(-0.782186\pi\)
−0.774873 + 0.632117i \(0.782186\pi\)
\(140\) 4.71054e7 1.45085
\(141\) 5.03739e6 0.151335
\(142\) 5.95534e7 1.74541
\(143\) 1.10415e8 3.15757
\(144\) −8.70629e6 −0.242976
\(145\) 4.80866e6 0.130989
\(146\) 6.90844e7 1.83715
\(147\) −1.62937e7 −0.423067
\(148\) 7.20724e7 1.82748
\(149\) 8.04857e6 0.199327 0.0996637 0.995021i \(-0.468223\pi\)
0.0996637 + 0.995021i \(0.468223\pi\)
\(150\) −3.33793e7 −0.807529
\(151\) −9.45579e6 −0.223500 −0.111750 0.993736i \(-0.535646\pi\)
−0.111750 + 0.993736i \(0.535646\pi\)
\(152\) 5.68018e7 1.31193
\(153\) −3.18143e7 −0.718129
\(154\) −1.76568e8 −3.89573
\(155\) 3.21688e7 0.693864
\(156\) 1.18232e8 2.49345
\(157\) 5.50699e7 1.13571 0.567853 0.823130i \(-0.307774\pi\)
0.567853 + 0.823130i \(0.307774\pi\)
\(158\) 6.02539e7 1.21530
\(159\) −2.11211e7 −0.416704
\(160\) −1.33517e7 −0.257701
\(161\) 2.25980e7 0.426756
\(162\) 4.23300e7 0.782251
\(163\) 3.56380e7 0.644551 0.322275 0.946646i \(-0.395552\pi\)
0.322275 + 0.946646i \(0.395552\pi\)
\(164\) −1.21429e8 −2.14965
\(165\) −5.22853e7 −0.906120
\(166\) −6.65158e6 −0.112862
\(167\) −4.96680e7 −0.825218 −0.412609 0.910908i \(-0.635382\pi\)
−0.412609 + 0.910908i \(0.635382\pi\)
\(168\) −8.79545e7 −1.43112
\(169\) 1.19522e8 1.90477
\(170\) 1.25608e8 1.96085
\(171\) 2.25766e7 0.345281
\(172\) −1.90293e7 −0.285150
\(173\) 5.97290e7 0.877049 0.438524 0.898719i \(-0.355501\pi\)
0.438524 + 0.898719i \(0.355501\pi\)
\(174\) −1.93007e7 −0.277747
\(175\) 5.36151e7 0.756230
\(176\) −8.39490e7 −1.16070
\(177\) −6.23373e7 −0.844971
\(178\) 8.81751e7 1.17186
\(179\) −5.19556e6 −0.0677090 −0.0338545 0.999427i \(-0.510778\pi\)
−0.0338545 + 0.999427i \(0.510778\pi\)
\(180\) 3.54695e7 0.453316
\(181\) 9.47873e6 0.118816 0.0594080 0.998234i \(-0.481079\pi\)
0.0594080 + 0.998234i \(0.481079\pi\)
\(182\) −2.91473e8 −3.58384
\(183\) −4.54178e7 −0.547832
\(184\) 4.28111e7 0.506634
\(185\) −5.26136e7 −0.610938
\(186\) −1.29117e8 −1.47126
\(187\) −3.06764e8 −3.43052
\(188\) −3.29506e7 −0.361668
\(189\) −1.25098e8 −1.34783
\(190\) −8.91359e7 −0.942790
\(191\) 1.10607e8 1.14859 0.574295 0.818648i \(-0.305276\pi\)
0.574295 + 0.818648i \(0.305276\pi\)
\(192\) 1.01665e8 1.03661
\(193\) −8.62108e7 −0.863199 −0.431599 0.902065i \(-0.642051\pi\)
−0.431599 + 0.902065i \(0.642051\pi\)
\(194\) 2.98691e8 2.93708
\(195\) −8.63110e7 −0.833575
\(196\) 1.06580e8 1.01107
\(197\) −1.14482e8 −1.06686 −0.533429 0.845845i \(-0.679097\pi\)
−0.533429 + 0.845845i \(0.679097\pi\)
\(198\) −1.32952e8 −1.21722
\(199\) −3.94085e7 −0.354490 −0.177245 0.984167i \(-0.556719\pi\)
−0.177245 + 0.984167i \(0.556719\pi\)
\(200\) 1.01572e8 0.897778
\(201\) 4.55285e7 0.395455
\(202\) 1.76123e8 1.50344
\(203\) 3.10014e7 0.260103
\(204\) −3.28482e8 −2.70899
\(205\) 8.86442e7 0.718640
\(206\) 3.03911e8 2.42221
\(207\) 1.70159e7 0.133339
\(208\) −1.38580e8 −1.06778
\(209\) 2.17692e8 1.64941
\(210\) 1.38022e8 1.02844
\(211\) −1.39696e8 −1.02375 −0.511876 0.859059i \(-0.671049\pi\)
−0.511876 + 0.859059i \(0.671049\pi\)
\(212\) 1.38158e8 0.995862
\(213\) 1.13693e8 0.806127
\(214\) 1.59755e7 0.111431
\(215\) 1.38916e7 0.0953275
\(216\) −2.36994e8 −1.60011
\(217\) 2.07393e8 1.37779
\(218\) −1.40785e8 −0.920360
\(219\) 1.31888e8 0.848499
\(220\) 3.42009e8 2.16550
\(221\) −5.06397e8 −3.15586
\(222\) 2.11177e8 1.29542
\(223\) −1.20926e8 −0.730218 −0.365109 0.930965i \(-0.618968\pi\)
−0.365109 + 0.930965i \(0.618968\pi\)
\(224\) −8.60784e7 −0.511712
\(225\) 4.03711e7 0.236283
\(226\) −5.90852e7 −0.340486
\(227\) −1.15625e8 −0.656084 −0.328042 0.944663i \(-0.606389\pi\)
−0.328042 + 0.944663i \(0.606389\pi\)
\(228\) 2.33104e8 1.30250
\(229\) −1.39236e8 −0.766175 −0.383087 0.923712i \(-0.625139\pi\)
−0.383087 + 0.923712i \(0.625139\pi\)
\(230\) −6.71811e7 −0.364082
\(231\) −3.37083e8 −1.79927
\(232\) 5.87311e7 0.308788
\(233\) −3.66295e7 −0.189708 −0.0948539 0.995491i \(-0.530238\pi\)
−0.0948539 + 0.995491i \(0.530238\pi\)
\(234\) −2.19473e8 −1.11976
\(235\) 2.40543e7 0.120908
\(236\) 4.07761e8 2.01936
\(237\) 1.15030e8 0.561295
\(238\) 8.09792e8 3.89363
\(239\) −3.71440e7 −0.175993 −0.0879967 0.996121i \(-0.528046\pi\)
−0.0879967 + 0.996121i \(0.528046\pi\)
\(240\) 6.56224e7 0.306417
\(241\) 3.96773e7 0.182592 0.0912962 0.995824i \(-0.470899\pi\)
0.0912962 + 0.995824i \(0.470899\pi\)
\(242\) −9.08477e8 −4.12060
\(243\) −1.62070e8 −0.724569
\(244\) 2.97087e8 1.30924
\(245\) −7.78049e7 −0.338007
\(246\) −3.55794e8 −1.52379
\(247\) 3.59359e8 1.51736
\(248\) 3.92898e8 1.63568
\(249\) −1.26984e7 −0.0521258
\(250\) −4.21012e8 −1.70414
\(251\) 6.08382e7 0.242839 0.121419 0.992601i \(-0.461255\pi\)
0.121419 + 0.992601i \(0.461255\pi\)
\(252\) 2.28672e8 0.900141
\(253\) 1.64073e8 0.636963
\(254\) 3.67245e7 0.140617
\(255\) 2.39796e8 0.905630
\(256\) −4.77537e8 −1.77896
\(257\) 9.91857e7 0.364488 0.182244 0.983253i \(-0.441664\pi\)
0.182244 + 0.983253i \(0.441664\pi\)
\(258\) −5.57572e7 −0.202131
\(259\) −3.39200e8 −1.21313
\(260\) 5.64577e8 1.99212
\(261\) 2.33435e7 0.0812688
\(262\) 3.97004e8 1.36377
\(263\) −2.73541e8 −0.927209 −0.463605 0.886042i \(-0.653444\pi\)
−0.463605 + 0.886042i \(0.653444\pi\)
\(264\) −6.38593e8 −2.13605
\(265\) −1.00857e8 −0.332923
\(266\) −5.74659e8 −1.87208
\(267\) 1.68334e8 0.541231
\(268\) −2.97811e8 −0.945081
\(269\) 2.75672e8 0.863494 0.431747 0.901995i \(-0.357897\pi\)
0.431747 + 0.901995i \(0.357897\pi\)
\(270\) 3.71901e8 1.14989
\(271\) 4.75651e8 1.45176 0.725882 0.687819i \(-0.241432\pi\)
0.725882 + 0.687819i \(0.241432\pi\)
\(272\) 3.85015e8 1.16008
\(273\) −5.56447e8 −1.65522
\(274\) 6.07177e8 1.78315
\(275\) 3.89272e8 1.12873
\(276\) 1.75689e8 0.502993
\(277\) 2.38870e8 0.675279 0.337639 0.941276i \(-0.390372\pi\)
0.337639 + 0.941276i \(0.390372\pi\)
\(278\) 9.40475e8 2.62537
\(279\) 1.56163e8 0.430489
\(280\) −4.19995e8 −1.14338
\(281\) −5.54874e8 −1.49184 −0.745920 0.666036i \(-0.767990\pi\)
−0.745920 + 0.666036i \(0.767990\pi\)
\(282\) −9.65474e7 −0.256371
\(283\) −2.03982e7 −0.0534981 −0.0267491 0.999642i \(-0.508516\pi\)
−0.0267491 + 0.999642i \(0.508516\pi\)
\(284\) −7.43686e8 −1.92653
\(285\) −1.70168e8 −0.435433
\(286\) −2.11624e9 −5.34913
\(287\) 5.71489e8 1.42699
\(288\) −6.48154e7 −0.159884
\(289\) 9.96573e8 2.42866
\(290\) −9.21635e7 −0.221904
\(291\) 5.70228e8 1.35651
\(292\) −8.62707e8 −2.02779
\(293\) 2.34109e8 0.543728 0.271864 0.962336i \(-0.412360\pi\)
0.271864 + 0.962336i \(0.412360\pi\)
\(294\) 3.12288e8 0.716704
\(295\) −2.97670e8 −0.675084
\(296\) −6.42603e8 −1.44020
\(297\) −9.08274e8 −2.01173
\(298\) −1.54260e8 −0.337673
\(299\) 2.70846e8 0.585967
\(300\) 4.16832e8 0.891325
\(301\) 8.95593e7 0.189290
\(302\) 1.81231e8 0.378624
\(303\) 3.36234e8 0.694371
\(304\) −2.73221e8 −0.557772
\(305\) −2.16877e8 −0.437687
\(306\) 6.09758e8 1.21656
\(307\) 3.42975e8 0.676515 0.338258 0.941054i \(-0.390162\pi\)
0.338258 + 0.941054i \(0.390162\pi\)
\(308\) 2.20493e9 4.29999
\(309\) 5.80193e8 1.11871
\(310\) −6.16553e8 −1.17545
\(311\) −7.24592e8 −1.36594 −0.682971 0.730446i \(-0.739312\pi\)
−0.682971 + 0.730446i \(0.739312\pi\)
\(312\) −1.05417e9 −1.96503
\(313\) 2.90144e8 0.534820 0.267410 0.963583i \(-0.413832\pi\)
0.267410 + 0.963583i \(0.413832\pi\)
\(314\) −1.05548e9 −1.92396
\(315\) −1.66933e8 −0.300923
\(316\) −7.52433e8 −1.34141
\(317\) 6.90711e8 1.21784 0.608918 0.793233i \(-0.291604\pi\)
0.608918 + 0.793233i \(0.291604\pi\)
\(318\) 4.04811e8 0.705923
\(319\) 2.25086e8 0.388222
\(320\) 4.85463e8 0.828193
\(321\) 3.04986e7 0.0514650
\(322\) −4.33117e8 −0.722952
\(323\) −9.98398e8 −1.64852
\(324\) −5.28605e8 −0.863424
\(325\) 6.42599e8 1.03836
\(326\) −6.83044e8 −1.09191
\(327\) −2.68770e8 −0.425073
\(328\) 1.08267e9 1.69409
\(329\) 1.55078e8 0.240085
\(330\) 1.00211e9 1.53503
\(331\) −1.29514e9 −1.96300 −0.981498 0.191470i \(-0.938675\pi\)
−0.981498 + 0.191470i \(0.938675\pi\)
\(332\) 8.30630e7 0.124573
\(333\) −2.55411e8 −0.379040
\(334\) 9.51944e8 1.39797
\(335\) 2.17405e8 0.315946
\(336\) 4.23068e8 0.608446
\(337\) −4.49309e8 −0.639500 −0.319750 0.947502i \(-0.603599\pi\)
−0.319750 + 0.947502i \(0.603599\pi\)
\(338\) −2.29077e9 −3.22681
\(339\) −1.12799e8 −0.157255
\(340\) −1.56855e9 −2.16433
\(341\) 1.50577e9 2.05645
\(342\) −4.32707e8 −0.584929
\(343\) 4.26057e8 0.570084
\(344\) 1.69667e8 0.224721
\(345\) −1.28255e8 −0.168154
\(346\) −1.14477e9 −1.48578
\(347\) 6.18982e7 0.0795288 0.0397644 0.999209i \(-0.487339\pi\)
0.0397644 + 0.999209i \(0.487339\pi\)
\(348\) 2.41021e8 0.306569
\(349\) 7.40278e8 0.932193 0.466096 0.884734i \(-0.345660\pi\)
0.466096 + 0.884734i \(0.345660\pi\)
\(350\) −1.02760e9 −1.28110
\(351\) −1.49935e9 −1.85067
\(352\) −6.24972e8 −0.763767
\(353\) 1.53714e9 1.85995 0.929976 0.367620i \(-0.119827\pi\)
0.929976 + 0.367620i \(0.119827\pi\)
\(354\) 1.19477e9 1.43144
\(355\) 5.42899e8 0.644050
\(356\) −1.10111e9 −1.29346
\(357\) 1.54596e9 1.79829
\(358\) 9.95789e7 0.114704
\(359\) −1.14737e9 −1.30880 −0.654400 0.756149i \(-0.727079\pi\)
−0.654400 + 0.756149i \(0.727079\pi\)
\(360\) −3.16249e8 −0.357248
\(361\) −1.85371e8 −0.207380
\(362\) −1.81671e8 −0.201282
\(363\) −1.73436e9 −1.90312
\(364\) 3.63983e9 3.95573
\(365\) 6.29785e8 0.677903
\(366\) 8.70485e8 0.928063
\(367\) −1.07850e9 −1.13890 −0.569452 0.822024i \(-0.692845\pi\)
−0.569452 + 0.822024i \(0.692845\pi\)
\(368\) −2.05925e8 −0.215398
\(369\) 4.30320e8 0.445861
\(370\) 1.00840e9 1.03497
\(371\) −6.50222e8 −0.661079
\(372\) 1.61238e9 1.62393
\(373\) 4.29210e8 0.428242 0.214121 0.976807i \(-0.431311\pi\)
0.214121 + 0.976807i \(0.431311\pi\)
\(374\) 5.87949e9 5.81152
\(375\) −8.03749e8 −0.787065
\(376\) 2.93790e8 0.285023
\(377\) 3.71565e8 0.357141
\(378\) 2.39765e9 2.28331
\(379\) 4.96839e8 0.468790 0.234395 0.972141i \(-0.424689\pi\)
0.234395 + 0.972141i \(0.424689\pi\)
\(380\) 1.11310e9 1.04062
\(381\) 7.01103e7 0.0649448
\(382\) −2.11991e9 −1.94579
\(383\) −8.74354e7 −0.0795228 −0.0397614 0.999209i \(-0.512660\pi\)
−0.0397614 + 0.999209i \(0.512660\pi\)
\(384\) −1.59062e9 −1.43353
\(385\) −1.60962e9 −1.43751
\(386\) 1.65233e9 1.46232
\(387\) 6.74364e7 0.0591434
\(388\) −3.72997e9 −3.24186
\(389\) −1.92667e9 −1.65953 −0.829763 0.558116i \(-0.811524\pi\)
−0.829763 + 0.558116i \(0.811524\pi\)
\(390\) 1.65425e9 1.41213
\(391\) −7.52485e8 −0.636619
\(392\) −9.50280e8 −0.796802
\(393\) 7.57914e8 0.629863
\(394\) 2.19419e9 1.80733
\(395\) 5.49284e8 0.448443
\(396\) 1.66027e9 1.34352
\(397\) 1.16563e9 0.934961 0.467481 0.884003i \(-0.345162\pi\)
0.467481 + 0.884003i \(0.345162\pi\)
\(398\) 7.55310e8 0.600530
\(399\) −1.09707e9 −0.864631
\(400\) −4.88569e8 −0.381694
\(401\) 7.50534e8 0.581253 0.290626 0.956837i \(-0.406136\pi\)
0.290626 + 0.956837i \(0.406136\pi\)
\(402\) −8.72607e8 −0.669927
\(403\) 2.48568e9 1.89181
\(404\) −2.19937e9 −1.65945
\(405\) 3.85888e8 0.288648
\(406\) −5.94178e8 −0.440631
\(407\) −2.46276e9 −1.81068
\(408\) 2.92878e9 2.13489
\(409\) 8.51195e8 0.615174 0.307587 0.951520i \(-0.400479\pi\)
0.307587 + 0.951520i \(0.400479\pi\)
\(410\) −1.69897e9 −1.21742
\(411\) 1.15915e9 0.823558
\(412\) −3.79516e9 −2.67356
\(413\) −1.91908e9 −1.34050
\(414\) −3.26129e8 −0.225885
\(415\) −6.06369e7 −0.0416456
\(416\) −1.03168e9 −0.702619
\(417\) 1.79545e9 1.21254
\(418\) −4.17231e9 −2.79421
\(419\) −1.60505e9 −1.06596 −0.532978 0.846129i \(-0.678927\pi\)
−0.532978 + 0.846129i \(0.678927\pi\)
\(420\) −1.72358e9 −1.13516
\(421\) −8.58777e8 −0.560910 −0.280455 0.959867i \(-0.590485\pi\)
−0.280455 + 0.959867i \(0.590485\pi\)
\(422\) 2.67743e9 1.73430
\(423\) 1.16771e8 0.0750140
\(424\) −1.23182e9 −0.784817
\(425\) −1.78532e9 −1.12812
\(426\) −2.17905e9 −1.36563
\(427\) −1.39820e9 −0.869107
\(428\) −1.99497e8 −0.122994
\(429\) −4.04008e9 −2.47053
\(430\) −2.66249e8 −0.161491
\(431\) 9.60074e8 0.577610 0.288805 0.957388i \(-0.406742\pi\)
0.288805 + 0.957388i \(0.406742\pi\)
\(432\) 1.13996e9 0.680293
\(433\) −4.31741e8 −0.255574 −0.127787 0.991802i \(-0.540787\pi\)
−0.127787 + 0.991802i \(0.540787\pi\)
\(434\) −3.97492e9 −2.33407
\(435\) −1.75948e8 −0.102488
\(436\) 1.75808e9 1.01586
\(437\) 5.33992e8 0.306090
\(438\) −2.52779e9 −1.43741
\(439\) −1.43295e9 −0.808363 −0.404182 0.914679i \(-0.632444\pi\)
−0.404182 + 0.914679i \(0.632444\pi\)
\(440\) −3.04938e9 −1.70658
\(441\) −3.77701e8 −0.209707
\(442\) 9.70569e9 5.34624
\(443\) 1.62330e9 0.887127 0.443564 0.896243i \(-0.353714\pi\)
0.443564 + 0.896243i \(0.353714\pi\)
\(444\) −2.63712e9 −1.42985
\(445\) 8.03819e8 0.432413
\(446\) 2.31768e9 1.23704
\(447\) −2.94496e8 −0.155956
\(448\) 3.12978e9 1.64453
\(449\) −7.89867e8 −0.411805 −0.205903 0.978573i \(-0.566013\pi\)
−0.205903 + 0.978573i \(0.566013\pi\)
\(450\) −7.73760e8 −0.400279
\(451\) 4.14930e9 2.12989
\(452\) 7.37839e8 0.375818
\(453\) 3.45986e8 0.174870
\(454\) 2.21608e9 1.11145
\(455\) −2.65712e9 −1.32242
\(456\) −2.07837e9 −1.02647
\(457\) 3.17411e9 1.55566 0.777831 0.628474i \(-0.216320\pi\)
0.777831 + 0.628474i \(0.216320\pi\)
\(458\) 2.66862e9 1.29795
\(459\) 4.16561e9 2.01064
\(460\) 8.38938e8 0.401863
\(461\) −1.10733e9 −0.526408 −0.263204 0.964740i \(-0.584779\pi\)
−0.263204 + 0.964740i \(0.584779\pi\)
\(462\) 6.46059e9 3.04807
\(463\) −1.45587e9 −0.681693 −0.340847 0.940119i \(-0.610714\pi\)
−0.340847 + 0.940119i \(0.610714\pi\)
\(464\) −2.82501e8 −0.131283
\(465\) −1.17705e9 −0.542889
\(466\) 7.02046e8 0.321377
\(467\) −2.23002e9 −1.01321 −0.506606 0.862178i \(-0.669100\pi\)
−0.506606 + 0.862178i \(0.669100\pi\)
\(468\) 2.74072e9 1.23596
\(469\) 1.40161e9 0.627369
\(470\) −4.61028e8 −0.204826
\(471\) −2.01500e9 −0.888591
\(472\) −3.63563e9 −1.59141
\(473\) 6.50245e8 0.282529
\(474\) −2.20468e9 −0.950871
\(475\) 1.26693e9 0.542406
\(476\) −1.01125e10 −4.29766
\(477\) −4.89605e8 −0.206553
\(478\) 7.11908e8 0.298144
\(479\) 9.33167e8 0.387958 0.193979 0.981006i \(-0.437861\pi\)
0.193979 + 0.981006i \(0.437861\pi\)
\(480\) 4.88536e8 0.201629
\(481\) −4.06545e9 −1.66572
\(482\) −7.60461e8 −0.309323
\(483\) −8.26857e8 −0.333899
\(484\) 1.13448e10 4.54818
\(485\) 2.72292e9 1.08377
\(486\) 3.10625e9 1.22747
\(487\) −1.72559e9 −0.676994 −0.338497 0.940967i \(-0.609919\pi\)
−0.338497 + 0.940967i \(0.609919\pi\)
\(488\) −2.64885e9 −1.03178
\(489\) −1.30399e9 −0.504305
\(490\) 1.49122e9 0.572606
\(491\) −6.20324e8 −0.236501 −0.118251 0.992984i \(-0.537729\pi\)
−0.118251 + 0.992984i \(0.537729\pi\)
\(492\) 4.44305e9 1.68191
\(493\) −1.03231e9 −0.388012
\(494\) −6.88752e9 −2.57051
\(495\) −1.21202e9 −0.449148
\(496\) −1.88987e9 −0.695418
\(497\) 3.50007e9 1.27888
\(498\) 2.43380e8 0.0883045
\(499\) 3.98852e9 1.43701 0.718506 0.695521i \(-0.244826\pi\)
0.718506 + 0.695521i \(0.244826\pi\)
\(500\) 5.25748e9 1.88097
\(501\) 1.81734e9 0.645662
\(502\) −1.16603e9 −0.411385
\(503\) 1.29341e9 0.453157 0.226579 0.973993i \(-0.427246\pi\)
0.226579 + 0.973993i \(0.427246\pi\)
\(504\) −2.03886e9 −0.709381
\(505\) 1.60556e9 0.554763
\(506\) −3.14464e9 −1.07906
\(507\) −4.37328e9 −1.49032
\(508\) −4.58605e8 −0.155209
\(509\) 2.09129e9 0.702913 0.351456 0.936204i \(-0.385687\pi\)
0.351456 + 0.936204i \(0.385687\pi\)
\(510\) −4.59596e9 −1.53420
\(511\) 4.06023e9 1.34610
\(512\) 3.58818e9 1.18149
\(513\) −2.95608e9 −0.966729
\(514\) −1.90101e9 −0.617466
\(515\) 2.77051e9 0.893787
\(516\) 6.96280e8 0.223105
\(517\) 1.12594e9 0.358343
\(518\) 6.50116e9 2.05512
\(519\) −2.18547e9 −0.686215
\(520\) −5.03382e9 −1.56995
\(521\) 4.31681e9 1.33731 0.668653 0.743575i \(-0.266871\pi\)
0.668653 + 0.743575i \(0.266871\pi\)
\(522\) −4.47405e8 −0.137675
\(523\) 5.49978e9 1.68108 0.840542 0.541746i \(-0.182237\pi\)
0.840542 + 0.541746i \(0.182237\pi\)
\(524\) −4.95767e9 −1.50528
\(525\) −1.96177e9 −0.591685
\(526\) 5.24273e9 1.57075
\(527\) −6.90592e9 −2.05534
\(528\) 3.07168e9 0.908150
\(529\) −3.00236e9 −0.881795
\(530\) 1.93303e9 0.563993
\(531\) −1.44503e9 −0.418838
\(532\) 7.17618e9 2.06634
\(533\) 6.84953e9 1.95937
\(534\) −3.22631e9 −0.916880
\(535\) 1.45635e8 0.0411176
\(536\) 2.65531e9 0.744797
\(537\) 1.90105e8 0.0529765
\(538\) −5.28357e9 −1.46281
\(539\) −3.64193e9 −1.00178
\(540\) −4.64420e9 −1.26921
\(541\) −5.48712e9 −1.48989 −0.744945 0.667126i \(-0.767524\pi\)
−0.744945 + 0.667126i \(0.767524\pi\)
\(542\) −9.11641e9 −2.45938
\(543\) −3.46825e8 −0.0929633
\(544\) 2.86630e9 0.763354
\(545\) −1.28342e9 −0.339610
\(546\) 1.06650e10 2.80404
\(547\) −5.58182e9 −1.45821 −0.729105 0.684402i \(-0.760063\pi\)
−0.729105 + 0.684402i \(0.760063\pi\)
\(548\) −7.58225e9 −1.96819
\(549\) −1.05282e9 −0.271551
\(550\) −7.46085e9 −1.91214
\(551\) 7.32566e8 0.186559
\(552\) −1.56645e9 −0.396397
\(553\) 3.54124e9 0.890466
\(554\) −4.57823e9 −1.14397
\(555\) 1.92512e9 0.478006
\(556\) −1.17444e10 −2.89780
\(557\) −1.55205e9 −0.380551 −0.190276 0.981731i \(-0.560938\pi\)
−0.190276 + 0.981731i \(0.560938\pi\)
\(558\) −2.99304e9 −0.729277
\(559\) 1.07340e9 0.259909
\(560\) 2.02021e9 0.486114
\(561\) 1.12245e10 2.68408
\(562\) 1.06348e10 2.52727
\(563\) −2.49961e9 −0.590328 −0.295164 0.955447i \(-0.595374\pi\)
−0.295164 + 0.955447i \(0.595374\pi\)
\(564\) 1.20566e9 0.282974
\(565\) −5.38631e8 −0.125638
\(566\) 3.90954e8 0.0906293
\(567\) 2.48782e9 0.573163
\(568\) 6.63076e9 1.51825
\(569\) 6.84845e9 1.55847 0.779236 0.626730i \(-0.215607\pi\)
0.779236 + 0.626730i \(0.215607\pi\)
\(570\) 3.26147e9 0.737651
\(571\) −4.81071e7 −0.0108139 −0.00540696 0.999985i \(-0.501721\pi\)
−0.00540696 + 0.999985i \(0.501721\pi\)
\(572\) 2.64270e10 5.90420
\(573\) −4.04709e9 −0.898673
\(574\) −1.09532e10 −2.41742
\(575\) 9.54875e8 0.209464
\(576\) 2.35667e9 0.513830
\(577\) −6.83896e9 −1.48209 −0.741045 0.671455i \(-0.765669\pi\)
−0.741045 + 0.671455i \(0.765669\pi\)
\(578\) −1.91005e10 −4.11431
\(579\) 3.15444e9 0.675379
\(580\) 1.15091e9 0.244931
\(581\) −3.90926e8 −0.0826948
\(582\) −1.09291e10 −2.29801
\(583\) −4.72094e9 −0.986707
\(584\) 7.69196e9 1.59806
\(585\) −2.00076e9 −0.413189
\(586\) −4.48697e9 −0.921111
\(587\) 1.56312e9 0.318976 0.159488 0.987200i \(-0.449016\pi\)
0.159488 + 0.987200i \(0.449016\pi\)
\(588\) −3.89976e9 −0.791075
\(589\) 4.90070e9 0.988222
\(590\) 5.70519e9 1.14364
\(591\) 4.18890e9 0.834725
\(592\) 3.09097e9 0.612306
\(593\) −3.33176e9 −0.656118 −0.328059 0.944657i \(-0.606395\pi\)
−0.328059 + 0.944657i \(0.606395\pi\)
\(594\) 1.74081e10 3.40800
\(595\) 7.38220e9 1.43673
\(596\) 1.92636e9 0.372713
\(597\) 1.44195e9 0.277358
\(598\) −5.19108e9 −0.992667
\(599\) 2.00026e9 0.380271 0.190136 0.981758i \(-0.439107\pi\)
0.190136 + 0.981758i \(0.439107\pi\)
\(600\) −3.71651e9 −0.702434
\(601\) −8.58111e9 −1.61244 −0.806218 0.591618i \(-0.798489\pi\)
−0.806218 + 0.591618i \(0.798489\pi\)
\(602\) −1.71651e9 −0.320670
\(603\) 1.05539e9 0.196020
\(604\) −2.26316e9 −0.417914
\(605\) −8.28183e9 −1.52049
\(606\) −6.44430e9 −1.17631
\(607\) 2.24609e9 0.407631 0.203815 0.979009i \(-0.434666\pi\)
0.203815 + 0.979009i \(0.434666\pi\)
\(608\) −2.03404e9 −0.367026
\(609\) −1.13434e9 −0.203508
\(610\) 4.15669e9 0.741470
\(611\) 1.85867e9 0.329654
\(612\) −7.61449e9 −1.34280
\(613\) −7.36697e9 −1.29175 −0.645873 0.763445i \(-0.723507\pi\)
−0.645873 + 0.763445i \(0.723507\pi\)
\(614\) −6.57351e9 −1.14606
\(615\) −3.24348e9 −0.562274
\(616\) −1.96593e10 −3.38872
\(617\) 8.07677e9 1.38433 0.692165 0.721739i \(-0.256657\pi\)
0.692165 + 0.721739i \(0.256657\pi\)
\(618\) −1.11201e10 −1.89517
\(619\) 4.11289e9 0.696995 0.348498 0.937310i \(-0.386692\pi\)
0.348498 + 0.937310i \(0.386692\pi\)
\(620\) 7.69934e9 1.29743
\(621\) −2.22797e9 −0.373327
\(622\) 1.38876e10 2.31399
\(623\) 5.18222e9 0.858634
\(624\) 5.07064e9 0.835442
\(625\) −1.19481e8 −0.0195757
\(626\) −5.56094e9 −0.906020
\(627\) −7.96531e9 −1.29052
\(628\) 1.31805e10 2.12360
\(629\) 1.12949e10 1.80970
\(630\) 3.19946e9 0.509782
\(631\) −1.93829e9 −0.307126 −0.153563 0.988139i \(-0.549075\pi\)
−0.153563 + 0.988139i \(0.549075\pi\)
\(632\) 6.70876e9 1.05714
\(633\) 5.11145e9 0.800997
\(634\) −1.32383e10 −2.06309
\(635\) 3.34787e8 0.0518872
\(636\) −5.05517e9 −0.779176
\(637\) −6.01198e9 −0.921572
\(638\) −4.31403e9 −0.657674
\(639\) 2.63548e9 0.399583
\(640\) −7.59545e9 −1.14531
\(641\) −4.63296e9 −0.694793 −0.347397 0.937718i \(-0.612934\pi\)
−0.347397 + 0.937718i \(0.612934\pi\)
\(642\) −5.84541e8 −0.0871851
\(643\) 4.81785e9 0.714685 0.357342 0.933973i \(-0.383683\pi\)
0.357342 + 0.933973i \(0.383683\pi\)
\(644\) 5.40864e9 0.797971
\(645\) −5.08292e8 −0.0745855
\(646\) 1.91354e10 2.79270
\(647\) −4.86975e9 −0.706873 −0.353437 0.935458i \(-0.614987\pi\)
−0.353437 + 0.935458i \(0.614987\pi\)
\(648\) 4.71309e9 0.680445
\(649\) −1.39335e10 −2.00080
\(650\) −1.23161e10 −1.75905
\(651\) −7.58846e9 −1.07800
\(652\) 8.52966e9 1.20522
\(653\) −4.32144e9 −0.607340 −0.303670 0.952777i \(-0.598212\pi\)
−0.303670 + 0.952777i \(0.598212\pi\)
\(654\) 5.15129e9 0.720102
\(655\) 3.61915e9 0.503225
\(656\) −5.20771e9 −0.720250
\(657\) 3.05727e9 0.420587
\(658\) −2.97225e9 −0.406719
\(659\) −1.84953e9 −0.251746 −0.125873 0.992046i \(-0.540173\pi\)
−0.125873 + 0.992046i \(0.540173\pi\)
\(660\) −1.25140e10 −1.69431
\(661\) 1.67946e9 0.226186 0.113093 0.993584i \(-0.463924\pi\)
0.113093 + 0.993584i \(0.463924\pi\)
\(662\) 2.48229e10 3.32544
\(663\) 1.85290e10 2.46919
\(664\) −7.40597e8 −0.0981733
\(665\) −5.23869e9 −0.690791
\(666\) 4.89525e9 0.642118
\(667\) 5.52130e8 0.0720445
\(668\) −1.18876e10 −1.54304
\(669\) 4.42466e9 0.571332
\(670\) −4.16682e9 −0.535234
\(671\) −1.01517e10 −1.29720
\(672\) 3.14960e9 0.400371
\(673\) −3.83952e9 −0.485540 −0.242770 0.970084i \(-0.578056\pi\)
−0.242770 + 0.970084i \(0.578056\pi\)
\(674\) 8.61153e9 1.08335
\(675\) −5.28600e9 −0.661552
\(676\) 2.86065e10 3.56165
\(677\) 1.14031e10 1.41241 0.706206 0.708006i \(-0.250405\pi\)
0.706206 + 0.708006i \(0.250405\pi\)
\(678\) 2.16192e9 0.266401
\(679\) 1.75547e10 2.15203
\(680\) 1.39853e10 1.70566
\(681\) 4.23069e9 0.513329
\(682\) −2.88599e10 −3.48377
\(683\) 2.62449e9 0.315190 0.157595 0.987504i \(-0.449626\pi\)
0.157595 + 0.987504i \(0.449626\pi\)
\(684\) 5.40353e9 0.645626
\(685\) 5.53512e9 0.657977
\(686\) −8.16588e9 −0.965759
\(687\) 5.09463e9 0.599466
\(688\) −8.16111e8 −0.0955410
\(689\) −7.79318e9 −0.907711
\(690\) 2.45815e9 0.284863
\(691\) 1.64230e9 0.189356 0.0946778 0.995508i \(-0.469818\pi\)
0.0946778 + 0.995508i \(0.469818\pi\)
\(692\) 1.42956e10 1.63995
\(693\) −7.81386e9 −0.891866
\(694\) −1.18635e9 −0.134727
\(695\) 8.57353e9 0.968752
\(696\) −2.14896e9 −0.241600
\(697\) −1.90299e10 −2.12873
\(698\) −1.41883e10 −1.57919
\(699\) 1.34027e9 0.148430
\(700\) 1.28323e10 1.41404
\(701\) −5.25977e9 −0.576705 −0.288352 0.957524i \(-0.593107\pi\)
−0.288352 + 0.957524i \(0.593107\pi\)
\(702\) 2.87368e10 3.13515
\(703\) −8.01532e9 −0.870116
\(704\) 2.27238e10 2.45457
\(705\) −8.80142e8 −0.0945999
\(706\) −2.94610e10 −3.15088
\(707\) 1.03511e10 1.10158
\(708\) −1.49199e10 −1.57997
\(709\) −9.05667e9 −0.954348 −0.477174 0.878809i \(-0.658339\pi\)
−0.477174 + 0.878809i \(0.658339\pi\)
\(710\) −1.04053e10 −1.09106
\(711\) 2.66648e9 0.278225
\(712\) 9.81755e9 1.01935
\(713\) 3.69362e9 0.381627
\(714\) −2.96302e10 −3.04643
\(715\) −1.92920e10 −1.97381
\(716\) −1.24351e9 −0.126606
\(717\) 1.35909e9 0.137700
\(718\) 2.19907e10 2.21719
\(719\) 1.23375e10 1.23787 0.618935 0.785443i \(-0.287565\pi\)
0.618935 + 0.785443i \(0.287565\pi\)
\(720\) 1.52118e9 0.151886
\(721\) 1.78615e10 1.77478
\(722\) 3.55285e9 0.351315
\(723\) −1.45179e9 −0.142863
\(724\) 2.26865e9 0.222169
\(725\) 1.30996e9 0.127666
\(726\) 3.32410e10 3.22401
\(727\) 8.00292e8 0.0772464 0.0386232 0.999254i \(-0.487703\pi\)
0.0386232 + 0.999254i \(0.487703\pi\)
\(728\) −3.24530e10 −3.11742
\(729\) 1.07603e10 1.02867
\(730\) −1.20706e10 −1.14841
\(731\) −2.98221e9 −0.282376
\(732\) −1.08704e10 −1.02437
\(733\) −8.12305e9 −0.761825 −0.380912 0.924611i \(-0.624390\pi\)
−0.380912 + 0.924611i \(0.624390\pi\)
\(734\) 2.06706e10 1.92938
\(735\) 2.84687e9 0.264461
\(736\) −1.53304e9 −0.141736
\(737\) 1.01764e10 0.936393
\(738\) −8.24759e9 −0.755318
\(739\) −1.58853e10 −1.44791 −0.723954 0.689848i \(-0.757677\pi\)
−0.723954 + 0.689848i \(0.757677\pi\)
\(740\) −1.25926e10 −1.14237
\(741\) −1.31489e10 −1.18720
\(742\) 1.24623e10 1.11991
\(743\) 1.84265e10 1.64810 0.824048 0.566521i \(-0.191711\pi\)
0.824048 + 0.566521i \(0.191711\pi\)
\(744\) −1.43761e10 −1.27978
\(745\) −1.40626e9 −0.124600
\(746\) −8.22631e9 −0.725469
\(747\) −2.94360e8 −0.0258379
\(748\) −7.34215e10 −6.41457
\(749\) 9.38910e8 0.0816465
\(750\) 1.54048e10 1.33334
\(751\) 4.18559e9 0.360592 0.180296 0.983612i \(-0.442294\pi\)
0.180296 + 0.983612i \(0.442294\pi\)
\(752\) −1.41315e9 −0.121179
\(753\) −2.22606e9 −0.190000
\(754\) −7.12147e9 −0.605020
\(755\) 1.65213e9 0.139711
\(756\) −2.99412e10 −2.52024
\(757\) 1.67312e10 1.40182 0.700908 0.713252i \(-0.252778\pi\)
0.700908 + 0.713252i \(0.252778\pi\)
\(758\) −9.52250e9 −0.794161
\(759\) −6.00340e9 −0.498369
\(760\) −9.92452e9 −0.820091
\(761\) −1.16945e10 −0.961911 −0.480955 0.876745i \(-0.659710\pi\)
−0.480955 + 0.876745i \(0.659710\pi\)
\(762\) −1.34374e9 −0.110021
\(763\) −8.27419e9 −0.674356
\(764\) 2.64728e10 2.14770
\(765\) 5.55866e9 0.448905
\(766\) 1.67580e9 0.134717
\(767\) −2.30009e10 −1.84061
\(768\) 1.74730e10 1.39189
\(769\) −1.70901e10 −1.35519 −0.677597 0.735433i \(-0.736979\pi\)
−0.677597 + 0.735433i \(0.736979\pi\)
\(770\) 3.08503e10 2.43524
\(771\) −3.62919e9 −0.285180
\(772\) −2.06338e10 −1.61406
\(773\) −1.12070e9 −0.0872690 −0.0436345 0.999048i \(-0.513894\pi\)
−0.0436345 + 0.999048i \(0.513894\pi\)
\(774\) −1.29250e9 −0.100193
\(775\) 8.76334e9 0.676260
\(776\) 3.32567e10 2.55484
\(777\) 1.24113e10 0.949168
\(778\) 3.69269e10 2.81134
\(779\) 1.35043e10 1.02351
\(780\) −2.06578e10 −1.55867
\(781\) 2.54122e10 1.90882
\(782\) 1.44223e10 1.07847
\(783\) −3.05648e9 −0.227539
\(784\) 4.57092e9 0.338764
\(785\) −9.62192e9 −0.709934
\(786\) −1.45263e10 −1.06703
\(787\) 3.19989e9 0.234004 0.117002 0.993132i \(-0.462672\pi\)
0.117002 + 0.993132i \(0.462672\pi\)
\(788\) −2.74004e10 −1.99487
\(789\) 1.00088e10 0.725461
\(790\) −1.05277e10 −0.759692
\(791\) −3.47255e9 −0.249477
\(792\) −1.48031e10 −1.05880
\(793\) −1.67580e10 −1.19335
\(794\) −2.23406e10 −1.58388
\(795\) 3.69033e9 0.260483
\(796\) −9.43209e9 −0.662846
\(797\) 1.93896e10 1.35664 0.678322 0.734765i \(-0.262708\pi\)
0.678322 + 0.734765i \(0.262708\pi\)
\(798\) 2.10267e10 1.46474
\(799\) −5.16390e9 −0.358150
\(800\) −3.63723e9 −0.251163
\(801\) 3.90212e9 0.268279
\(802\) −1.43849e10 −0.984680
\(803\) 2.94793e10 2.00915
\(804\) 1.08969e10 0.739444
\(805\) −3.94836e9 −0.266767
\(806\) −4.76410e10 −3.20485
\(807\) −1.00868e10 −0.675609
\(808\) 1.96098e10 1.30777
\(809\) −2.20959e10 −1.46721 −0.733604 0.679577i \(-0.762163\pi\)
−0.733604 + 0.679577i \(0.762163\pi\)
\(810\) −7.39598e9 −0.488988
\(811\) 1.03080e10 0.678583 0.339292 0.940681i \(-0.389813\pi\)
0.339292 + 0.940681i \(0.389813\pi\)
\(812\) 7.41993e9 0.486355
\(813\) −1.74040e10 −1.13588
\(814\) 4.72017e10 3.06741
\(815\) −6.22675e9 −0.402911
\(816\) −1.40876e10 −0.907658
\(817\) 2.11629e9 0.135768
\(818\) −1.63141e10 −1.04214
\(819\) −1.28989e10 −0.820462
\(820\) 2.12162e10 1.34375
\(821\) −1.08963e10 −0.687192 −0.343596 0.939117i \(-0.611645\pi\)
−0.343596 + 0.939117i \(0.611645\pi\)
\(822\) −2.22165e10 −1.39516
\(823\) 1.58099e10 0.988622 0.494311 0.869285i \(-0.335420\pi\)
0.494311 + 0.869285i \(0.335420\pi\)
\(824\) 3.38379e10 2.10697
\(825\) −1.42434e10 −0.883132
\(826\) 3.67813e10 2.27090
\(827\) 6.58783e9 0.405017 0.202508 0.979281i \(-0.435091\pi\)
0.202508 + 0.979281i \(0.435091\pi\)
\(828\) 4.07260e9 0.249325
\(829\) 1.31751e10 0.803180 0.401590 0.915819i \(-0.368458\pi\)
0.401590 + 0.915819i \(0.368458\pi\)
\(830\) 1.16218e9 0.0705503
\(831\) −8.74024e9 −0.528347
\(832\) 3.75117e10 2.25806
\(833\) 1.67029e10 1.00123
\(834\) −3.44118e10 −2.05412
\(835\) 8.67808e9 0.515847
\(836\) 5.21026e10 3.08417
\(837\) −2.04472e10 −1.20530
\(838\) 3.07626e10 1.80580
\(839\) 1.44377e10 0.843981 0.421990 0.906600i \(-0.361332\pi\)
0.421990 + 0.906600i \(0.361332\pi\)
\(840\) 1.53676e10 0.894598
\(841\) −1.64924e10 −0.956090
\(842\) 1.64595e10 0.950218
\(843\) 2.03028e10 1.16724
\(844\) −3.34350e10 −1.91427
\(845\) −2.08831e10 −1.19068
\(846\) −2.23805e9 −0.127079
\(847\) −5.33930e10 −3.01920
\(848\) 5.92517e9 0.333668
\(849\) 7.46366e8 0.0418577
\(850\) 3.42177e10 1.91110
\(851\) −6.04109e9 −0.336018
\(852\) 2.72114e10 1.50734
\(853\) −1.27917e10 −0.705679 −0.352840 0.935684i \(-0.614784\pi\)
−0.352840 + 0.935684i \(0.614784\pi\)
\(854\) 2.67982e10 1.47232
\(855\) −3.94463e9 −0.215837
\(856\) 1.77873e9 0.0969288
\(857\) −2.68071e10 −1.45485 −0.727423 0.686189i \(-0.759282\pi\)
−0.727423 + 0.686189i \(0.759282\pi\)
\(858\) 7.74329e10 4.18523
\(859\) 7.43216e9 0.400073 0.200036 0.979788i \(-0.435894\pi\)
0.200036 + 0.979788i \(0.435894\pi\)
\(860\) 3.32484e9 0.178249
\(861\) −2.09107e10 −1.11650
\(862\) −1.84009e10 −0.978508
\(863\) 2.49200e10 1.31980 0.659902 0.751352i \(-0.270598\pi\)
0.659902 + 0.751352i \(0.270598\pi\)
\(864\) 8.48661e9 0.447648
\(865\) −1.04360e10 −0.548247
\(866\) 8.27482e9 0.432958
\(867\) −3.64645e10 −1.90022
\(868\) 4.96376e10 2.57627
\(869\) 2.57111e10 1.32908
\(870\) 3.37225e9 0.173621
\(871\) 1.67989e10 0.861424
\(872\) −1.56752e10 −0.800580
\(873\) 1.32183e10 0.672399
\(874\) −1.02346e10 −0.518537
\(875\) −2.47437e10 −1.24864
\(876\) 3.15663e10 1.58657
\(877\) −3.16610e10 −1.58499 −0.792493 0.609881i \(-0.791217\pi\)
−0.792493 + 0.609881i \(0.791217\pi\)
\(878\) 2.74642e10 1.36942
\(879\) −8.56602e9 −0.425420
\(880\) 1.46677e10 0.725560
\(881\) 9.18139e9 0.452369 0.226185 0.974084i \(-0.427375\pi\)
0.226185 + 0.974084i \(0.427375\pi\)
\(882\) 7.23908e9 0.355258
\(883\) −2.18382e10 −1.06746 −0.533732 0.845653i \(-0.679211\pi\)
−0.533732 + 0.845653i \(0.679211\pi\)
\(884\) −1.21202e11 −5.90101
\(885\) 1.08917e10 0.528195
\(886\) −3.11124e10 −1.50285
\(887\) −3.51106e10 −1.68929 −0.844647 0.535324i \(-0.820189\pi\)
−0.844647 + 0.535324i \(0.820189\pi\)
\(888\) 2.35128e10 1.12683
\(889\) 2.15837e9 0.103031
\(890\) −1.54061e10 −0.732535
\(891\) 1.80628e10 0.855486
\(892\) −2.89426e10 −1.36540
\(893\) 3.66450e9 0.172201
\(894\) 5.64436e9 0.264200
\(895\) 9.07778e8 0.0423252
\(896\) −4.89679e10 −2.27422
\(897\) −9.91022e9 −0.458469
\(898\) 1.51387e10 0.697624
\(899\) 5.06716e9 0.232598
\(900\) 9.66249e9 0.441815
\(901\) 2.16516e10 0.986174
\(902\) −7.95261e10 −3.60817
\(903\) −3.27696e9 −0.148103
\(904\) −6.57863e9 −0.296174
\(905\) −1.65614e9 −0.0742724
\(906\) −6.63122e9 −0.296241
\(907\) 1.81270e10 0.806676 0.403338 0.915051i \(-0.367850\pi\)
0.403338 + 0.915051i \(0.367850\pi\)
\(908\) −2.76738e10 −1.22678
\(909\) 7.79416e9 0.344188
\(910\) 5.09267e10 2.24027
\(911\) 4.43842e10 1.94498 0.972488 0.232952i \(-0.0748386\pi\)
0.972488 + 0.232952i \(0.0748386\pi\)
\(912\) 9.99712e9 0.436408
\(913\) −2.83832e9 −0.123428
\(914\) −6.08354e10 −2.63539
\(915\) 7.93549e9 0.342452
\(916\) −3.33250e10 −1.43264
\(917\) 2.33327e10 0.999245
\(918\) −7.98388e10 −3.40616
\(919\) −3.80235e10 −1.61602 −0.808012 0.589166i \(-0.799456\pi\)
−0.808012 + 0.589166i \(0.799456\pi\)
\(920\) −7.48004e9 −0.316699
\(921\) −1.25494e10 −0.529315
\(922\) 2.12232e10 0.891769
\(923\) 4.19497e10 1.75600
\(924\) −8.06781e10 −3.36437
\(925\) −1.43329e10 −0.595438
\(926\) 2.79034e10 1.15483
\(927\) 1.34493e10 0.554526
\(928\) −2.10313e9 −0.0863868
\(929\) −3.00377e10 −1.22917 −0.614585 0.788851i \(-0.710677\pi\)
−0.614585 + 0.788851i \(0.710677\pi\)
\(930\) 2.25596e10 0.919688
\(931\) −1.18530e10 −0.481400
\(932\) −8.76696e9 −0.354726
\(933\) 2.65127e10 1.06873
\(934\) 4.27409e10 1.71645
\(935\) 5.35985e10 2.14443
\(936\) −2.44365e10 −0.974033
\(937\) −4.22202e10 −1.67661 −0.838303 0.545204i \(-0.816452\pi\)
−0.838303 + 0.545204i \(0.816452\pi\)
\(938\) −2.68635e10 −1.06280
\(939\) −1.06163e10 −0.418451
\(940\) 5.75718e9 0.226080
\(941\) 3.37621e10 1.32089 0.660444 0.750875i \(-0.270368\pi\)
0.660444 + 0.750875i \(0.270368\pi\)
\(942\) 3.86198e10 1.50533
\(943\) 1.01781e10 0.395254
\(944\) 1.74876e10 0.676596
\(945\) 2.18574e10 0.842532
\(946\) −1.24627e10 −0.478622
\(947\) −2.01458e10 −0.770832 −0.385416 0.922743i \(-0.625942\pi\)
−0.385416 + 0.922743i \(0.625942\pi\)
\(948\) 2.75314e10 1.04954
\(949\) 4.86635e10 1.84830
\(950\) −2.42821e10 −0.918871
\(951\) −2.52730e10 −0.952851
\(952\) 9.01635e10 3.38689
\(953\) −2.24395e10 −0.839825 −0.419913 0.907565i \(-0.637939\pi\)
−0.419913 + 0.907565i \(0.637939\pi\)
\(954\) 9.38385e9 0.349914
\(955\) −1.93254e10 −0.717989
\(956\) −8.89011e9 −0.329082
\(957\) −8.23586e9 −0.303750
\(958\) −1.78852e10 −0.657226
\(959\) 3.56850e10 1.30653
\(960\) −1.77630e10 −0.647989
\(961\) 6.38552e9 0.232094
\(962\) 7.79191e10 2.82183
\(963\) 7.06982e8 0.0255103
\(964\) 9.49642e9 0.341421
\(965\) 1.50629e10 0.539589
\(966\) 1.58477e10 0.565647
\(967\) −3.16478e10 −1.12551 −0.562757 0.826622i \(-0.690259\pi\)
−0.562757 + 0.826622i \(0.690259\pi\)
\(968\) −1.01151e11 −3.58432
\(969\) 3.65312e10 1.28983
\(970\) −5.21879e10 −1.83598
\(971\) −2.82843e10 −0.991468 −0.495734 0.868474i \(-0.665101\pi\)
−0.495734 + 0.868474i \(0.665101\pi\)
\(972\) −3.87900e10 −1.35484
\(973\) 5.52736e10 1.92363
\(974\) 3.30728e10 1.14687
\(975\) −2.35126e10 −0.812427
\(976\) 1.27412e10 0.438667
\(977\) 2.65176e10 0.909709 0.454855 0.890566i \(-0.349691\pi\)
0.454855 + 0.890566i \(0.349691\pi\)
\(978\) 2.49925e10 0.854326
\(979\) 3.76255e10 1.28157
\(980\) −1.86219e10 −0.632024
\(981\) −6.23031e9 −0.210702
\(982\) 1.18892e10 0.400648
\(983\) 4.89517e10 1.64373 0.821866 0.569681i \(-0.192933\pi\)
0.821866 + 0.569681i \(0.192933\pi\)
\(984\) −3.96146e10 −1.32548
\(985\) 2.00026e10 0.666898
\(986\) 1.97854e10 0.657318
\(987\) −5.67428e9 −0.187845
\(988\) 8.60094e10 2.83724
\(989\) 1.59503e9 0.0524304
\(990\) 2.32297e10 0.760886
\(991\) −1.75057e10 −0.571374 −0.285687 0.958323i \(-0.592222\pi\)
−0.285687 + 0.958323i \(0.592222\pi\)
\(992\) −1.40694e10 −0.457600
\(993\) 4.73891e10 1.53588
\(994\) −6.70829e10 −2.16650
\(995\) 6.88553e9 0.221593
\(996\) −3.03926e9 −0.0974677
\(997\) −1.49917e10 −0.479090 −0.239545 0.970885i \(-0.576998\pi\)
−0.239545 + 0.970885i \(0.576998\pi\)
\(998\) −7.64447e10 −2.43439
\(999\) 3.34423e10 1.06125
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.1.2 13
3.2 odd 2 387.8.a.d.1.12 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.2 13 1.1 even 1 trivial
387.8.a.d.1.12 13 3.2 odd 2