Properties

Label 43.8.a.b.1.4
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 1.4
Root \(-8.80627\) 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

\(f(q)\) \(=\) \(q-7.80627 q^{2} -2.71381 q^{3} -67.0621 q^{4} -402.005 q^{5} +21.1847 q^{6} -356.766 q^{7} +1522.71 q^{8} -2179.64 q^{9} +O(q^{10})\) \(q-7.80627 q^{2} -2.71381 q^{3} -67.0621 q^{4} -402.005 q^{5} +21.1847 q^{6} -356.766 q^{7} +1522.71 q^{8} -2179.64 q^{9} +3138.16 q^{10} +338.626 q^{11} +181.994 q^{12} -1901.99 q^{13} +2785.01 q^{14} +1090.96 q^{15} -3302.71 q^{16} -13710.6 q^{17} +17014.8 q^{18} +32375.1 q^{19} +26959.3 q^{20} +968.193 q^{21} -2643.41 q^{22} +103235. q^{23} -4132.34 q^{24} +83482.7 q^{25} +14847.4 q^{26} +11850.2 q^{27} +23925.5 q^{28} +97.3595 q^{29} -8516.35 q^{30} -2919.43 q^{31} -169125. q^{32} -918.966 q^{33} +107029. q^{34} +143421. q^{35} +146171. q^{36} -280235. q^{37} -252729. q^{38} +5161.63 q^{39} -612136. q^{40} -68080.9 q^{41} -7557.98 q^{42} -79507.0 q^{43} -22709.0 q^{44} +876223. q^{45} -805883. q^{46} +145265. q^{47} +8962.93 q^{48} -696261. q^{49} -651689. q^{50} +37208.1 q^{51} +127551. q^{52} -450286. q^{53} -92505.9 q^{54} -136129. q^{55} -543250. q^{56} -87859.8 q^{57} -760.015 q^{58} +2.16940e6 q^{59} -73162.3 q^{60} -240945. q^{61} +22789.9 q^{62} +777619. q^{63} +1.74298e6 q^{64} +764608. q^{65} +7173.70 q^{66} +763129. q^{67} +919465. q^{68} -280161. q^{69} -1.11959e6 q^{70} +256406. q^{71} -3.31895e6 q^{72} -4.25607e6 q^{73} +2.18759e6 q^{74} -226556. q^{75} -2.17114e6 q^{76} -120810. q^{77} -40293.1 q^{78} +6.60338e6 q^{79} +1.32771e6 q^{80} +4.73470e6 q^{81} +531458. q^{82} +1.57834e6 q^{83} -64929.1 q^{84} +5.51174e6 q^{85} +620653. q^{86} -264.215 q^{87} +515629. q^{88} +1.01079e7 q^{89} -6.84004e6 q^{90} +678564. q^{91} -6.92319e6 q^{92} +7922.77 q^{93} -1.13398e6 q^{94} -1.30149e7 q^{95} +458972. q^{96} -1.90764e6 q^{97} +5.43520e6 q^{98} -738081. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.80627 −0.689983 −0.344992 0.938606i \(-0.612118\pi\)
−0.344992 + 0.938606i \(0.612118\pi\)
\(3\) −2.71381 −0.0580303 −0.0290151 0.999579i \(-0.509237\pi\)
−0.0290151 + 0.999579i \(0.509237\pi\)
\(4\) −67.0621 −0.523923
\(5\) −402.005 −1.43826 −0.719128 0.694878i \(-0.755458\pi\)
−0.719128 + 0.694878i \(0.755458\pi\)
\(6\) 21.1847 0.0400399
\(7\) −356.766 −0.393133 −0.196567 0.980490i \(-0.562979\pi\)
−0.196567 + 0.980490i \(0.562979\pi\)
\(8\) 1522.71 1.05148
\(9\) −2179.64 −0.996632
\(10\) 3138.16 0.992372
\(11\) 338.626 0.0767090 0.0383545 0.999264i \(-0.487788\pi\)
0.0383545 + 0.999264i \(0.487788\pi\)
\(12\) 181.994 0.0304034
\(13\) −1901.99 −0.240108 −0.120054 0.992767i \(-0.538307\pi\)
−0.120054 + 0.992767i \(0.538307\pi\)
\(14\) 2785.01 0.271255
\(15\) 1090.96 0.0834624
\(16\) −3302.71 −0.201582
\(17\) −13710.6 −0.676841 −0.338421 0.940995i \(-0.609893\pi\)
−0.338421 + 0.940995i \(0.609893\pi\)
\(18\) 17014.8 0.687660
\(19\) 32375.1 1.08286 0.541431 0.840745i \(-0.317883\pi\)
0.541431 + 0.840745i \(0.317883\pi\)
\(20\) 26959.3 0.753535
\(21\) 968.193 0.0228136
\(22\) −2643.41 −0.0529279
\(23\) 103235. 1.76922 0.884609 0.466334i \(-0.154426\pi\)
0.884609 + 0.466334i \(0.154426\pi\)
\(24\) −4132.34 −0.0610178
\(25\) 83482.7 1.06858
\(26\) 14847.4 0.165670
\(27\) 11850.2 0.115865
\(28\) 23925.5 0.205972
\(29\) 97.3595 0.000741285 0 0.000370643 1.00000i \(-0.499882\pi\)
0.000370643 1.00000i \(0.499882\pi\)
\(30\) −8516.35 −0.0575876
\(31\) −2919.43 −0.0176008 −0.00880039 0.999961i \(-0.502801\pi\)
−0.00880039 + 0.999961i \(0.502801\pi\)
\(32\) −169125. −0.912394
\(33\) −918.966 −0.00445144
\(34\) 107029. 0.467009
\(35\) 143421. 0.565426
\(36\) 146171. 0.522159
\(37\) −280235. −0.909529 −0.454764 0.890612i \(-0.650277\pi\)
−0.454764 + 0.890612i \(0.650277\pi\)
\(38\) −252729. −0.747157
\(39\) 5161.63 0.0139335
\(40\) −612136. −1.51230
\(41\) −68080.9 −0.154270 −0.0771350 0.997021i \(-0.524577\pi\)
−0.0771350 + 0.997021i \(0.524577\pi\)
\(42\) −7557.98 −0.0157410
\(43\) −79507.0 −0.152499
\(44\) −22709.0 −0.0401896
\(45\) 876223. 1.43341
\(46\) −805883. −1.22073
\(47\) 145265. 0.204089 0.102044 0.994780i \(-0.467462\pi\)
0.102044 + 0.994780i \(0.467462\pi\)
\(48\) 8962.93 0.0116978
\(49\) −696261. −0.845446
\(50\) −651689. −0.737301
\(51\) 37208.1 0.0392773
\(52\) 127551. 0.125798
\(53\) −450286. −0.415454 −0.207727 0.978187i \(-0.566607\pi\)
−0.207727 + 0.978187i \(0.566607\pi\)
\(54\) −92505.9 −0.0799450
\(55\) −136129. −0.110327
\(56\) −543250. −0.413372
\(57\) −87859.8 −0.0628388
\(58\) −760.015 −0.000511474 0
\(59\) 2.16940e6 1.37518 0.687588 0.726101i \(-0.258670\pi\)
0.687588 + 0.726101i \(0.258670\pi\)
\(60\) −73162.3 −0.0437279
\(61\) −240945. −0.135914 −0.0679570 0.997688i \(-0.521648\pi\)
−0.0679570 + 0.997688i \(0.521648\pi\)
\(62\) 22789.9 0.0121442
\(63\) 777619. 0.391809
\(64\) 1.74298e6 0.831118
\(65\) 764608. 0.345336
\(66\) 7173.70 0.00307142
\(67\) 763129. 0.309982 0.154991 0.987916i \(-0.450465\pi\)
0.154991 + 0.987916i \(0.450465\pi\)
\(68\) 919465. 0.354613
\(69\) −280161. −0.102668
\(70\) −1.11959e6 −0.390135
\(71\) 256406. 0.0850206 0.0425103 0.999096i \(-0.486464\pi\)
0.0425103 + 0.999096i \(0.486464\pi\)
\(72\) −3.31895e6 −1.04794
\(73\) −4.25607e6 −1.28050 −0.640249 0.768168i \(-0.721169\pi\)
−0.640249 + 0.768168i \(0.721169\pi\)
\(74\) 2.18759e6 0.627560
\(75\) −226556. −0.0620099
\(76\) −2.17114e6 −0.567337
\(77\) −120810. −0.0301568
\(78\) −40293.1 −0.00961390
\(79\) 6.60338e6 1.50686 0.753428 0.657531i \(-0.228399\pi\)
0.753428 + 0.657531i \(0.228399\pi\)
\(80\) 1.32771e6 0.289926
\(81\) 4.73470e6 0.989909
\(82\) 531458. 0.106444
\(83\) 1.57834e6 0.302989 0.151495 0.988458i \(-0.451591\pi\)
0.151495 + 0.988458i \(0.451591\pi\)
\(84\) −64929.1 −0.0119526
\(85\) 5.51174e6 0.973470
\(86\) 620653. 0.105221
\(87\) −264.215 −4.30170e−5 0
\(88\) 515629. 0.0806580
\(89\) 1.01079e7 1.51983 0.759914 0.650024i \(-0.225241\pi\)
0.759914 + 0.650024i \(0.225241\pi\)
\(90\) −6.84004e6 −0.989030
\(91\) 678564. 0.0943944
\(92\) −6.92319e6 −0.926934
\(93\) 7922.77 0.00102138
\(94\) −1.13398e6 −0.140818
\(95\) −1.30149e7 −1.55743
\(96\) 458972. 0.0529465
\(97\) −1.90764e6 −0.212224 −0.106112 0.994354i \(-0.533840\pi\)
−0.106112 + 0.994354i \(0.533840\pi\)
\(98\) 5.43520e6 0.583344
\(99\) −738081. −0.0764506
\(100\) −5.59853e6 −0.559853
\(101\) −8.79838e6 −0.849724 −0.424862 0.905258i \(-0.639677\pi\)
−0.424862 + 0.905258i \(0.639677\pi\)
\(102\) −290456. −0.0271007
\(103\) 1.98046e7 1.78581 0.892907 0.450242i \(-0.148662\pi\)
0.892907 + 0.450242i \(0.148662\pi\)
\(104\) −2.89617e6 −0.252469
\(105\) −389218. −0.0328118
\(106\) 3.51506e6 0.286657
\(107\) 1.19023e7 0.939264 0.469632 0.882862i \(-0.344387\pi\)
0.469632 + 0.882862i \(0.344387\pi\)
\(108\) −794700. −0.0607044
\(109\) −1.61101e7 −1.19153 −0.595766 0.803158i \(-0.703151\pi\)
−0.595766 + 0.803158i \(0.703151\pi\)
\(110\) 1.06266e6 0.0761238
\(111\) 760504. 0.0527802
\(112\) 1.17829e6 0.0792484
\(113\) 7.15228e6 0.466305 0.233153 0.972440i \(-0.425096\pi\)
0.233153 + 0.972440i \(0.425096\pi\)
\(114\) 685857. 0.0433577
\(115\) −4.15011e7 −2.54459
\(116\) −6529.14 −0.000388376 0
\(117\) 4.14564e6 0.239299
\(118\) −1.69349e7 −0.948849
\(119\) 4.89149e6 0.266089
\(120\) 1.66122e6 0.0877591
\(121\) −1.93725e7 −0.994116
\(122\) 1.88088e6 0.0937784
\(123\) 184758. 0.00895234
\(124\) 195783. 0.00922146
\(125\) −2.15382e6 −0.0986336
\(126\) −6.07030e6 −0.270342
\(127\) −1.95885e7 −0.848569 −0.424284 0.905529i \(-0.639474\pi\)
−0.424284 + 0.905529i \(0.639474\pi\)
\(128\) 8.04179e6 0.338936
\(129\) 215767. 0.00884953
\(130\) −5.96874e6 −0.238276
\(131\) 3.84474e7 1.49423 0.747115 0.664695i \(-0.231438\pi\)
0.747115 + 0.664695i \(0.231438\pi\)
\(132\) 61627.8 0.00233221
\(133\) −1.15503e7 −0.425709
\(134\) −5.95719e6 −0.213882
\(135\) −4.76384e6 −0.166644
\(136\) −2.08773e7 −0.711686
\(137\) 5.35857e7 1.78044 0.890219 0.455533i \(-0.150551\pi\)
0.890219 + 0.455533i \(0.150551\pi\)
\(138\) 2.18701e6 0.0708393
\(139\) −2.44467e7 −0.772091 −0.386046 0.922480i \(-0.626159\pi\)
−0.386046 + 0.922480i \(0.626159\pi\)
\(140\) −9.61815e6 −0.296240
\(141\) −394222. −0.0118433
\(142\) −2.00158e6 −0.0586628
\(143\) −644063. −0.0184184
\(144\) 7.19871e6 0.200903
\(145\) −39139.0 −0.00106616
\(146\) 3.32240e7 0.883522
\(147\) 1.88952e6 0.0490615
\(148\) 1.87932e7 0.476523
\(149\) 6.83645e7 1.69308 0.846542 0.532321i \(-0.178680\pi\)
0.846542 + 0.532321i \(0.178680\pi\)
\(150\) 1.76856e6 0.0427858
\(151\) 4.48140e7 1.05924 0.529620 0.848235i \(-0.322335\pi\)
0.529620 + 0.848235i \(0.322335\pi\)
\(152\) 4.92978e7 1.13861
\(153\) 2.98842e7 0.674562
\(154\) 943077. 0.0208077
\(155\) 1.17362e6 0.0253144
\(156\) −346150. −0.00730009
\(157\) 4.45418e7 0.918585 0.459293 0.888285i \(-0.348103\pi\)
0.459293 + 0.888285i \(0.348103\pi\)
\(158\) −5.15478e7 −1.03970
\(159\) 1.22199e6 0.0241089
\(160\) 6.79889e7 1.31225
\(161\) −3.68308e7 −0.695538
\(162\) −3.69604e7 −0.683021
\(163\) −2.04717e7 −0.370251 −0.185126 0.982715i \(-0.559269\pi\)
−0.185126 + 0.982715i \(0.559269\pi\)
\(164\) 4.56565e6 0.0808256
\(165\) 369429. 0.00640231
\(166\) −1.23209e7 −0.209057
\(167\) 9.99656e6 0.166090 0.0830449 0.996546i \(-0.473536\pi\)
0.0830449 + 0.996546i \(0.473536\pi\)
\(168\) 1.47428e6 0.0239881
\(169\) −5.91310e7 −0.942348
\(170\) −4.30262e7 −0.671678
\(171\) −7.05659e7 −1.07922
\(172\) 5.33191e6 0.0798975
\(173\) 9.15039e7 1.34363 0.671813 0.740721i \(-0.265516\pi\)
0.671813 + 0.740721i \(0.265516\pi\)
\(174\) 2062.53 2.96810e−5 0
\(175\) −2.97838e7 −0.420094
\(176\) −1.11838e6 −0.0154631
\(177\) −5.88734e6 −0.0798019
\(178\) −7.89047e7 −1.04866
\(179\) −7.48516e7 −0.975474 −0.487737 0.872991i \(-0.662177\pi\)
−0.487737 + 0.872991i \(0.662177\pi\)
\(180\) −5.87614e7 −0.750998
\(181\) −1.08411e7 −0.135894 −0.0679470 0.997689i \(-0.521645\pi\)
−0.0679470 + 0.997689i \(0.521645\pi\)
\(182\) −5.29706e6 −0.0651305
\(183\) 653879. 0.00788712
\(184\) 1.57197e8 1.86030
\(185\) 1.12656e8 1.30813
\(186\) −61847.3 −0.000704734 0
\(187\) −4.64278e6 −0.0519198
\(188\) −9.74179e6 −0.106927
\(189\) −4.22775e6 −0.0455504
\(190\) 1.01598e8 1.07460
\(191\) −1.37400e8 −1.42682 −0.713410 0.700747i \(-0.752850\pi\)
−0.713410 + 0.700747i \(0.752850\pi\)
\(192\) −4.73011e6 −0.0482300
\(193\) −1.16548e8 −1.16695 −0.583477 0.812129i \(-0.698308\pi\)
−0.583477 + 0.812129i \(0.698308\pi\)
\(194\) 1.48915e7 0.146431
\(195\) −2.07500e6 −0.0200400
\(196\) 4.66928e7 0.442949
\(197\) −1.79080e8 −1.66884 −0.834422 0.551126i \(-0.814198\pi\)
−0.834422 + 0.551126i \(0.814198\pi\)
\(198\) 5.76166e6 0.0527497
\(199\) −1.30708e8 −1.17575 −0.587875 0.808952i \(-0.700035\pi\)
−0.587875 + 0.808952i \(0.700035\pi\)
\(200\) 1.27120e8 1.12359
\(201\) −2.07099e6 −0.0179883
\(202\) 6.86825e7 0.586295
\(203\) −34734.5 −0.000291424 0
\(204\) −2.49525e6 −0.0205783
\(205\) 2.73688e7 0.221880
\(206\) −1.54600e8 −1.23218
\(207\) −2.25015e8 −1.76326
\(208\) 6.28172e6 0.0484013
\(209\) 1.09631e7 0.0830653
\(210\) 3.03834e6 0.0226396
\(211\) 3.03753e7 0.222604 0.111302 0.993787i \(-0.464498\pi\)
0.111302 + 0.993787i \(0.464498\pi\)
\(212\) 3.01972e7 0.217666
\(213\) −695837. −0.00493377
\(214\) −9.29126e7 −0.648076
\(215\) 3.19622e7 0.219332
\(216\) 1.80444e7 0.121830
\(217\) 1.04155e6 0.00691945
\(218\) 1.25760e8 0.822137
\(219\) 1.15502e7 0.0743076
\(220\) 9.12912e6 0.0578029
\(221\) 2.60775e7 0.162515
\(222\) −5.93670e6 −0.0364175
\(223\) 8.68403e7 0.524390 0.262195 0.965015i \(-0.415554\pi\)
0.262195 + 0.965015i \(0.415554\pi\)
\(224\) 6.03379e7 0.358692
\(225\) −1.81962e8 −1.06498
\(226\) −5.58326e7 −0.321743
\(227\) 1.67604e8 0.951028 0.475514 0.879708i \(-0.342262\pi\)
0.475514 + 0.879708i \(0.342262\pi\)
\(228\) 5.89206e6 0.0329227
\(229\) 1.15619e8 0.636218 0.318109 0.948054i \(-0.396952\pi\)
0.318109 + 0.948054i \(0.396952\pi\)
\(230\) 3.23969e8 1.75572
\(231\) 327855. 0.00175001
\(232\) 148250. 0.000779448 0
\(233\) 1.02835e8 0.532592 0.266296 0.963891i \(-0.414200\pi\)
0.266296 + 0.963891i \(0.414200\pi\)
\(234\) −3.23620e7 −0.165112
\(235\) −5.83973e7 −0.293532
\(236\) −1.45485e8 −0.720486
\(237\) −1.79203e7 −0.0874432
\(238\) −3.81843e7 −0.183597
\(239\) 2.55890e8 1.21244 0.606221 0.795296i \(-0.292685\pi\)
0.606221 + 0.795296i \(0.292685\pi\)
\(240\) −3.60314e6 −0.0168245
\(241\) −1.95565e6 −0.00899976 −0.00449988 0.999990i \(-0.501432\pi\)
−0.00449988 + 0.999990i \(0.501432\pi\)
\(242\) 1.51227e8 0.685923
\(243\) −3.87655e7 −0.173310
\(244\) 1.61583e7 0.0712085
\(245\) 2.79900e8 1.21597
\(246\) −1.44227e6 −0.00617696
\(247\) −6.15771e7 −0.260004
\(248\) −4.44544e6 −0.0185069
\(249\) −4.28331e6 −0.0175825
\(250\) 1.68133e7 0.0680556
\(251\) −2.99606e8 −1.19589 −0.597946 0.801536i \(-0.704016\pi\)
−0.597946 + 0.801536i \(0.704016\pi\)
\(252\) −5.21488e7 −0.205278
\(253\) 3.49582e7 0.135715
\(254\) 1.52913e8 0.585498
\(255\) −1.49578e7 −0.0564908
\(256\) −2.85878e8 −1.06498
\(257\) −3.65657e8 −1.34372 −0.671859 0.740679i \(-0.734504\pi\)
−0.671859 + 0.740679i \(0.734504\pi\)
\(258\) −1.68433e6 −0.00610603
\(259\) 9.99782e7 0.357566
\(260\) −5.12763e7 −0.180930
\(261\) −212208. −0.000738789 0
\(262\) −3.00131e8 −1.03099
\(263\) 4.38062e8 1.48488 0.742438 0.669914i \(-0.233669\pi\)
0.742438 + 0.669914i \(0.233669\pi\)
\(264\) −1.39932e6 −0.00468061
\(265\) 1.81017e8 0.597529
\(266\) 9.01649e7 0.293732
\(267\) −2.74308e7 −0.0881960
\(268\) −5.11771e7 −0.162407
\(269\) −6.37770e7 −0.199771 −0.0998853 0.994999i \(-0.531848\pi\)
−0.0998853 + 0.994999i \(0.531848\pi\)
\(270\) 3.71878e7 0.114981
\(271\) 3.49200e8 1.06582 0.532908 0.846173i \(-0.321099\pi\)
0.532908 + 0.846173i \(0.321099\pi\)
\(272\) 4.52823e7 0.136439
\(273\) −1.84149e6 −0.00547773
\(274\) −4.18304e8 −1.22847
\(275\) 2.82694e7 0.0819695
\(276\) 1.87882e7 0.0537902
\(277\) 2.62910e8 0.743238 0.371619 0.928385i \(-0.378803\pi\)
0.371619 + 0.928385i \(0.378803\pi\)
\(278\) 1.90838e8 0.532730
\(279\) 6.36329e6 0.0175415
\(280\) 2.18389e8 0.594535
\(281\) −3.13673e8 −0.843345 −0.421673 0.906748i \(-0.638557\pi\)
−0.421673 + 0.906748i \(0.638557\pi\)
\(282\) 3.07740e6 0.00817170
\(283\) 3.05065e7 0.0800093 0.0400046 0.999199i \(-0.487263\pi\)
0.0400046 + 0.999199i \(0.487263\pi\)
\(284\) −1.71951e7 −0.0445442
\(285\) 3.53200e7 0.0903783
\(286\) 5.02773e6 0.0127084
\(287\) 2.42889e7 0.0606487
\(288\) 3.68630e8 0.909321
\(289\) −2.22357e8 −0.541886
\(290\) 305529. 0.000735631 0
\(291\) 5.17696e6 0.0123154
\(292\) 2.85421e8 0.670882
\(293\) 6.50794e8 1.51149 0.755747 0.654863i \(-0.227274\pi\)
0.755747 + 0.654863i \(0.227274\pi\)
\(294\) −1.47501e7 −0.0338516
\(295\) −8.72110e8 −1.97785
\(296\) −4.26716e8 −0.956353
\(297\) 4.01279e6 0.00888789
\(298\) −5.33672e8 −1.16820
\(299\) −1.96353e8 −0.424803
\(300\) 1.51933e7 0.0324884
\(301\) 2.83654e7 0.0599523
\(302\) −3.49830e8 −0.730858
\(303\) 2.38771e7 0.0493097
\(304\) −1.06926e8 −0.218285
\(305\) 9.68611e7 0.195479
\(306\) −2.33284e8 −0.465436
\(307\) −1.64397e8 −0.324273 −0.162136 0.986768i \(-0.551838\pi\)
−0.162136 + 0.986768i \(0.551838\pi\)
\(308\) 8.10179e6 0.0157999
\(309\) −5.37459e7 −0.103631
\(310\) −9.16163e6 −0.0174665
\(311\) 6.67878e7 0.125903 0.0629515 0.998017i \(-0.479949\pi\)
0.0629515 + 0.998017i \(0.479949\pi\)
\(312\) 7.85966e6 0.0146508
\(313\) 3.66247e8 0.675101 0.337550 0.941307i \(-0.390402\pi\)
0.337550 + 0.941307i \(0.390402\pi\)
\(314\) −3.47706e8 −0.633808
\(315\) −3.12606e8 −0.563522
\(316\) −4.42837e8 −0.789476
\(317\) 8.45942e8 1.49153 0.745767 0.666206i \(-0.232083\pi\)
0.745767 + 0.666206i \(0.232083\pi\)
\(318\) −9.53919e6 −0.0166348
\(319\) 32968.5 5.68632e−5 0
\(320\) −7.00686e8 −1.19536
\(321\) −3.23006e7 −0.0545058
\(322\) 2.87511e8 0.479910
\(323\) −4.43883e8 −0.732926
\(324\) −3.17519e8 −0.518636
\(325\) −1.58783e8 −0.256574
\(326\) 1.59807e8 0.255467
\(327\) 4.37197e7 0.0691449
\(328\) −1.03667e8 −0.162212
\(329\) −5.18256e7 −0.0802341
\(330\) −2.88386e6 −0.00441749
\(331\) −1.56474e8 −0.237162 −0.118581 0.992944i \(-0.537835\pi\)
−0.118581 + 0.992944i \(0.537835\pi\)
\(332\) −1.05847e8 −0.158743
\(333\) 6.10810e8 0.906466
\(334\) −7.80358e7 −0.114599
\(335\) −3.06781e8 −0.445833
\(336\) −3.19766e6 −0.00459881
\(337\) 1.10367e9 1.57085 0.785424 0.618958i \(-0.212445\pi\)
0.785424 + 0.618958i \(0.212445\pi\)
\(338\) 4.61592e8 0.650205
\(339\) −1.94099e7 −0.0270598
\(340\) −3.69629e8 −0.510024
\(341\) −988595. −0.00135014
\(342\) 5.50856e8 0.744641
\(343\) 5.42214e8 0.725506
\(344\) −1.21066e8 −0.160349
\(345\) 1.12626e8 0.147663
\(346\) −7.14304e8 −0.927079
\(347\) −7.51448e7 −0.0965485 −0.0482743 0.998834i \(-0.515372\pi\)
−0.0482743 + 0.998834i \(0.515372\pi\)
\(348\) 17718.8 2.25376e−5 0
\(349\) 4.40370e8 0.554534 0.277267 0.960793i \(-0.410571\pi\)
0.277267 + 0.960793i \(0.410571\pi\)
\(350\) 2.32500e8 0.289858
\(351\) −2.25390e7 −0.0278201
\(352\) −5.72701e7 −0.0699888
\(353\) −8.89528e7 −0.107634 −0.0538168 0.998551i \(-0.517139\pi\)
−0.0538168 + 0.998551i \(0.517139\pi\)
\(354\) 4.59582e7 0.0550619
\(355\) −1.03076e8 −0.122281
\(356\) −6.77855e8 −0.796273
\(357\) −1.32746e7 −0.0154412
\(358\) 5.84312e8 0.673061
\(359\) −1.09248e9 −1.24619 −0.623095 0.782146i \(-0.714125\pi\)
−0.623095 + 0.782146i \(0.714125\pi\)
\(360\) 1.33423e9 1.50721
\(361\) 1.54275e8 0.172591
\(362\) 8.46289e7 0.0937645
\(363\) 5.25732e7 0.0576888
\(364\) −4.55060e7 −0.0494554
\(365\) 1.71096e9 1.84168
\(366\) −5.10436e6 −0.00544198
\(367\) −5.91559e8 −0.624693 −0.312346 0.949968i \(-0.601115\pi\)
−0.312346 + 0.949968i \(0.601115\pi\)
\(368\) −3.40957e8 −0.356642
\(369\) 1.48391e8 0.153751
\(370\) −8.79422e8 −0.902591
\(371\) 1.60647e8 0.163329
\(372\) −531318. −0.000535124 0
\(373\) 1.17638e9 1.17373 0.586863 0.809686i \(-0.300363\pi\)
0.586863 + 0.809686i \(0.300363\pi\)
\(374\) 3.62428e7 0.0358238
\(375\) 5.84506e6 0.00572374
\(376\) 2.21196e8 0.214595
\(377\) −185177. −0.000177988 0
\(378\) 3.30029e7 0.0314290
\(379\) 1.58878e9 1.49908 0.749541 0.661958i \(-0.230274\pi\)
0.749541 + 0.661958i \(0.230274\pi\)
\(380\) 8.72810e8 0.815975
\(381\) 5.31593e7 0.0492427
\(382\) 1.07258e9 0.984482
\(383\) −1.52083e9 −1.38320 −0.691599 0.722281i \(-0.743094\pi\)
−0.691599 + 0.722281i \(0.743094\pi\)
\(384\) −2.18239e7 −0.0196686
\(385\) 4.85662e7 0.0433732
\(386\) 9.09805e8 0.805179
\(387\) 1.73296e8 0.151985
\(388\) 1.27930e8 0.111189
\(389\) 1.81957e9 1.56728 0.783639 0.621217i \(-0.213361\pi\)
0.783639 + 0.621217i \(0.213361\pi\)
\(390\) 1.61980e7 0.0138272
\(391\) −1.41542e9 −1.19748
\(392\) −1.06020e9 −0.888971
\(393\) −1.04339e8 −0.0867106
\(394\) 1.39795e9 1.15147
\(395\) −2.65459e9 −2.16724
\(396\) 4.94973e7 0.0400542
\(397\) 2.11509e9 1.69653 0.848265 0.529571i \(-0.177647\pi\)
0.848265 + 0.529571i \(0.177647\pi\)
\(398\) 1.02034e9 0.811248
\(399\) 3.13453e7 0.0247040
\(400\) −2.75719e8 −0.215406
\(401\) −2.18744e9 −1.69407 −0.847034 0.531539i \(-0.821614\pi\)
−0.847034 + 0.531539i \(0.821614\pi\)
\(402\) 1.61667e7 0.0124116
\(403\) 5.55272e6 0.00422609
\(404\) 5.90038e8 0.445190
\(405\) −1.90337e9 −1.42374
\(406\) 271147. 0.000201078 0
\(407\) −9.48949e7 −0.0697690
\(408\) 5.66570e7 0.0412993
\(409\) −1.16511e9 −0.842045 −0.421022 0.907050i \(-0.638329\pi\)
−0.421022 + 0.907050i \(0.638329\pi\)
\(410\) −2.13648e8 −0.153093
\(411\) −1.45421e8 −0.103319
\(412\) −1.32814e9 −0.935629
\(413\) −7.73968e8 −0.540627
\(414\) 1.75653e9 1.21662
\(415\) −6.34499e8 −0.435776
\(416\) 3.21673e8 0.219073
\(417\) 6.63437e7 0.0448047
\(418\) −8.55805e7 −0.0573136
\(419\) −9.59686e8 −0.637353 −0.318677 0.947864i \(-0.603238\pi\)
−0.318677 + 0.947864i \(0.603238\pi\)
\(420\) 2.61018e7 0.0171909
\(421\) −1.02361e7 −0.00668572 −0.00334286 0.999994i \(-0.501064\pi\)
−0.00334286 + 0.999994i \(0.501064\pi\)
\(422\) −2.37118e8 −0.153593
\(423\) −3.16625e8 −0.203401
\(424\) −6.85655e8 −0.436843
\(425\) −1.14460e9 −0.723258
\(426\) 5.43189e6 0.00340422
\(427\) 8.59610e7 0.0534323
\(428\) −7.98194e8 −0.492102
\(429\) 1.74786e6 0.00106883
\(430\) −2.49505e8 −0.151335
\(431\) −3.14206e9 −1.89036 −0.945179 0.326552i \(-0.894113\pi\)
−0.945179 + 0.326552i \(0.894113\pi\)
\(432\) −3.91378e7 −0.0233563
\(433\) −2.31364e9 −1.36958 −0.684792 0.728739i \(-0.740107\pi\)
−0.684792 + 0.728739i \(0.740107\pi\)
\(434\) −8.13063e6 −0.00477431
\(435\) 106216. 6.18694e−5 0
\(436\) 1.08038e9 0.624271
\(437\) 3.34225e9 1.91582
\(438\) −9.01636e7 −0.0512710
\(439\) 1.64208e9 0.926337 0.463168 0.886270i \(-0.346713\pi\)
0.463168 + 0.886270i \(0.346713\pi\)
\(440\) −2.07285e8 −0.116007
\(441\) 1.51760e9 0.842599
\(442\) −2.03568e8 −0.112133
\(443\) 7.12250e8 0.389242 0.194621 0.980879i \(-0.437652\pi\)
0.194621 + 0.980879i \(0.437652\pi\)
\(444\) −5.10010e7 −0.0276528
\(445\) −4.06341e9 −2.18590
\(446\) −6.77899e8 −0.361820
\(447\) −1.85528e8 −0.0982502
\(448\) −6.21835e8 −0.326740
\(449\) 2.15899e9 1.12561 0.562805 0.826590i \(-0.309722\pi\)
0.562805 + 0.826590i \(0.309722\pi\)
\(450\) 1.42044e9 0.734819
\(451\) −2.30540e7 −0.0118339
\(452\) −4.79647e8 −0.244308
\(453\) −1.21617e8 −0.0614680
\(454\) −1.30836e9 −0.656193
\(455\) −2.72786e8 −0.135763
\(456\) −1.33785e8 −0.0660739
\(457\) −1.05753e9 −0.518308 −0.259154 0.965836i \(-0.583444\pi\)
−0.259154 + 0.965836i \(0.583444\pi\)
\(458\) −9.02555e8 −0.438980
\(459\) −1.62474e8 −0.0784223
\(460\) 2.78315e9 1.33317
\(461\) −2.93548e9 −1.39549 −0.697743 0.716348i \(-0.745812\pi\)
−0.697743 + 0.716348i \(0.745812\pi\)
\(462\) −2.55933e6 −0.00120748
\(463\) 2.80303e9 1.31248 0.656242 0.754550i \(-0.272145\pi\)
0.656242 + 0.754550i \(0.272145\pi\)
\(464\) −321551. −0.000149429 0
\(465\) −3.18499e6 −0.00146900
\(466\) −8.02756e8 −0.367479
\(467\) 1.35420e9 0.615281 0.307640 0.951503i \(-0.400461\pi\)
0.307640 + 0.951503i \(0.400461\pi\)
\(468\) −2.78016e8 −0.125374
\(469\) −2.72258e8 −0.121864
\(470\) 4.55865e8 0.202532
\(471\) −1.20878e8 −0.0533058
\(472\) 3.30337e9 1.44597
\(473\) −2.69231e7 −0.0116980
\(474\) 1.39891e8 0.0603344
\(475\) 2.70276e9 1.15712
\(476\) −3.28034e8 −0.139410
\(477\) 9.81460e8 0.414055
\(478\) −1.99755e9 −0.836565
\(479\) 1.17660e9 0.489165 0.244583 0.969628i \(-0.421349\pi\)
0.244583 + 0.969628i \(0.421349\pi\)
\(480\) −1.84509e8 −0.0761505
\(481\) 5.33004e8 0.218385
\(482\) 1.52663e7 0.00620969
\(483\) 9.99518e7 0.0403623
\(484\) 1.29916e9 0.520840
\(485\) 7.66879e8 0.305233
\(486\) 3.02614e8 0.119581
\(487\) −1.73453e9 −0.680504 −0.340252 0.940334i \(-0.610512\pi\)
−0.340252 + 0.940334i \(0.610512\pi\)
\(488\) −3.66889e8 −0.142911
\(489\) 5.55562e7 0.0214858
\(490\) −2.18498e9 −0.838997
\(491\) 7.94788e8 0.303016 0.151508 0.988456i \(-0.451587\pi\)
0.151508 + 0.988456i \(0.451587\pi\)
\(492\) −1.23903e7 −0.00469033
\(493\) −1.33486e6 −0.000501732 0
\(494\) 4.80687e8 0.179398
\(495\) 2.96712e8 0.109956
\(496\) 9.64204e6 0.00354799
\(497\) −9.14769e7 −0.0334244
\(498\) 3.34367e7 0.0121317
\(499\) −2.73492e8 −0.0985354 −0.0492677 0.998786i \(-0.515689\pi\)
−0.0492677 + 0.998786i \(0.515689\pi\)
\(500\) 1.44440e8 0.0516764
\(501\) −2.71287e7 −0.00963824
\(502\) 2.33880e9 0.825146
\(503\) 3.20927e9 1.12439 0.562197 0.827004i \(-0.309956\pi\)
0.562197 + 0.827004i \(0.309956\pi\)
\(504\) 1.18409e9 0.411980
\(505\) 3.53699e9 1.22212
\(506\) −2.72893e8 −0.0936409
\(507\) 1.60470e8 0.0546847
\(508\) 1.31364e9 0.444585
\(509\) −2.06801e9 −0.695090 −0.347545 0.937663i \(-0.612985\pi\)
−0.347545 + 0.937663i \(0.612985\pi\)
\(510\) 1.16765e8 0.0389777
\(511\) 1.51842e9 0.503406
\(512\) 1.20229e9 0.395881
\(513\) 3.83652e8 0.125466
\(514\) 2.85442e9 0.927143
\(515\) −7.96155e9 −2.56846
\(516\) −1.44698e7 −0.00463648
\(517\) 4.91906e7 0.0156554
\(518\) −7.80457e8 −0.246715
\(519\) −2.48324e8 −0.0779709
\(520\) 1.16428e9 0.363115
\(521\) −3.73056e8 −0.115569 −0.0577846 0.998329i \(-0.518404\pi\)
−0.0577846 + 0.998329i \(0.518404\pi\)
\(522\) 1.65655e6 0.000509752 0
\(523\) 3.23366e9 0.988414 0.494207 0.869344i \(-0.335459\pi\)
0.494207 + 0.869344i \(0.335459\pi\)
\(524\) −2.57837e9 −0.782862
\(525\) 8.08274e7 0.0243782
\(526\) −3.41963e9 −1.02454
\(527\) 4.00273e7 0.0119129
\(528\) 3.03508e6 0.000897329 0
\(529\) 7.25272e9 2.13013
\(530\) −1.41307e9 −0.412285
\(531\) −4.72851e9 −1.37055
\(532\) 7.74589e8 0.223039
\(533\) 1.29489e8 0.0370414
\(534\) 2.14132e8 0.0608538
\(535\) −4.78478e9 −1.35090
\(536\) 1.16202e9 0.325940
\(537\) 2.03133e8 0.0566070
\(538\) 4.97861e8 0.137838
\(539\) −2.35772e8 −0.0648533
\(540\) 3.19473e8 0.0873085
\(541\) −6.33940e9 −1.72131 −0.860653 0.509192i \(-0.829944\pi\)
−0.860653 + 0.509192i \(0.829944\pi\)
\(542\) −2.72595e9 −0.735395
\(543\) 2.94208e7 0.00788596
\(544\) 2.31881e9 0.617545
\(545\) 6.47633e9 1.71373
\(546\) 1.43752e7 0.00377954
\(547\) 2.18850e9 0.571731 0.285865 0.958270i \(-0.407719\pi\)
0.285865 + 0.958270i \(0.407719\pi\)
\(548\) −3.59357e9 −0.932812
\(549\) 5.25173e8 0.135456
\(550\) −2.20679e8 −0.0565576
\(551\) 3.15202e6 0.000802710 0
\(552\) −4.26603e8 −0.107954
\(553\) −2.35586e9 −0.592395
\(554\) −2.05235e9 −0.512822
\(555\) −3.05726e8 −0.0759114
\(556\) 1.63945e9 0.404516
\(557\) −2.65106e9 −0.650019 −0.325009 0.945711i \(-0.605367\pi\)
−0.325009 + 0.945711i \(0.605367\pi\)
\(558\) −4.96736e7 −0.0121034
\(559\) 1.51221e8 0.0366161
\(560\) −4.73680e8 −0.113979
\(561\) 1.25996e7 0.00301292
\(562\) 2.44862e9 0.581894
\(563\) −4.33945e9 −1.02484 −0.512419 0.858736i \(-0.671251\pi\)
−0.512419 + 0.858736i \(0.671251\pi\)
\(564\) 2.64374e7 0.00620499
\(565\) −2.87525e9 −0.670666
\(566\) −2.38142e8 −0.0552051
\(567\) −1.68918e9 −0.389166
\(568\) 3.90432e8 0.0893976
\(569\) 6.16980e8 0.140404 0.0702018 0.997533i \(-0.477636\pi\)
0.0702018 + 0.997533i \(0.477636\pi\)
\(570\) −2.75718e8 −0.0623595
\(571\) 5.73913e9 1.29009 0.645045 0.764145i \(-0.276839\pi\)
0.645045 + 0.764145i \(0.276839\pi\)
\(572\) 4.31923e7 0.00964983
\(573\) 3.72877e8 0.0827988
\(574\) −1.89606e8 −0.0418466
\(575\) 8.61837e9 1.89055
\(576\) −3.79906e9 −0.828319
\(577\) 5.29584e9 1.14768 0.573838 0.818969i \(-0.305454\pi\)
0.573838 + 0.818969i \(0.305454\pi\)
\(578\) 1.73578e9 0.373892
\(579\) 3.16289e8 0.0677187
\(580\) 2.62474e6 0.000558585 0
\(581\) −5.63097e8 −0.119115
\(582\) −4.04128e7 −0.00849744
\(583\) −1.52479e8 −0.0318691
\(584\) −6.48075e9 −1.34642
\(585\) −1.66657e9 −0.344173
\(586\) −5.08027e9 −1.04291
\(587\) 4.99464e8 0.101923 0.0509613 0.998701i \(-0.483771\pi\)
0.0509613 + 0.998701i \(0.483771\pi\)
\(588\) −1.26715e8 −0.0257044
\(589\) −9.45168e7 −0.0190592
\(590\) 6.80793e9 1.36469
\(591\) 4.85989e8 0.0968435
\(592\) 9.25536e8 0.183344
\(593\) −1.20268e9 −0.236842 −0.118421 0.992963i \(-0.537783\pi\)
−0.118421 + 0.992963i \(0.537783\pi\)
\(594\) −3.13249e7 −0.00613250
\(595\) −1.96640e9 −0.382704
\(596\) −4.58467e9 −0.887046
\(597\) 3.54715e8 0.0682291
\(598\) 1.53278e9 0.293107
\(599\) −5.47286e9 −1.04045 −0.520224 0.854030i \(-0.674151\pi\)
−0.520224 + 0.854030i \(0.674151\pi\)
\(600\) −3.44979e8 −0.0652023
\(601\) −4.60823e9 −0.865912 −0.432956 0.901415i \(-0.642529\pi\)
−0.432956 + 0.901415i \(0.642529\pi\)
\(602\) −2.21428e8 −0.0413661
\(603\) −1.66334e9 −0.308938
\(604\) −3.00532e9 −0.554960
\(605\) 7.78784e9 1.42979
\(606\) −1.86391e8 −0.0340229
\(607\) 7.85259e9 1.42512 0.712562 0.701610i \(-0.247535\pi\)
0.712562 + 0.701610i \(0.247535\pi\)
\(608\) −5.47543e9 −0.987997
\(609\) 94262.8 1.69114e−5 0
\(610\) −7.56124e8 −0.134877
\(611\) −2.76293e8 −0.0490033
\(612\) −2.00410e9 −0.353418
\(613\) −1.04035e10 −1.82419 −0.912095 0.409980i \(-0.865536\pi\)
−0.912095 + 0.409980i \(0.865536\pi\)
\(614\) 1.28333e9 0.223743
\(615\) −7.42737e7 −0.0128757
\(616\) −1.83959e8 −0.0317094
\(617\) 2.89619e9 0.496397 0.248198 0.968709i \(-0.420162\pi\)
0.248198 + 0.968709i \(0.420162\pi\)
\(618\) 4.19555e8 0.0715038
\(619\) −1.69036e9 −0.286458 −0.143229 0.989690i \(-0.545749\pi\)
−0.143229 + 0.989690i \(0.545749\pi\)
\(620\) −7.87057e7 −0.0132628
\(621\) 1.22336e9 0.204991
\(622\) −5.21364e8 −0.0868710
\(623\) −3.60614e9 −0.597495
\(624\) −1.70474e7 −0.00280874
\(625\) −5.65624e9 −0.926718
\(626\) −2.85902e9 −0.465808
\(627\) −2.97516e7 −0.00482030
\(628\) −2.98707e9 −0.481268
\(629\) 3.84220e9 0.615606
\(630\) 2.44029e9 0.388821
\(631\) −3.25816e9 −0.516261 −0.258131 0.966110i \(-0.583106\pi\)
−0.258131 + 0.966110i \(0.583106\pi\)
\(632\) 1.00550e10 1.58443
\(633\) −8.24328e7 −0.0129178
\(634\) −6.60366e9 −1.02913
\(635\) 7.87465e9 1.22046
\(636\) −8.19493e7 −0.0126312
\(637\) 1.32428e9 0.202998
\(638\) −257361. −3.92347e−5 0
\(639\) −5.58872e8 −0.0847343
\(640\) −3.23284e9 −0.487477
\(641\) 1.62115e9 0.243119 0.121560 0.992584i \(-0.461210\pi\)
0.121560 + 0.992584i \(0.461210\pi\)
\(642\) 2.52147e8 0.0376081
\(643\) −4.87271e8 −0.0722823 −0.0361412 0.999347i \(-0.511507\pi\)
−0.0361412 + 0.999347i \(0.511507\pi\)
\(644\) 2.46995e9 0.364408
\(645\) −8.67392e7 −0.0127279
\(646\) 3.46507e9 0.505707
\(647\) 1.32422e10 1.92219 0.961095 0.276219i \(-0.0890817\pi\)
0.961095 + 0.276219i \(0.0890817\pi\)
\(648\) 7.20957e9 1.04087
\(649\) 7.34617e8 0.105488
\(650\) 1.23950e9 0.177032
\(651\) −2.82657e6 −0.000401538 0
\(652\) 1.37287e9 0.193983
\(653\) 1.05133e10 1.47755 0.738774 0.673953i \(-0.235405\pi\)
0.738774 + 0.673953i \(0.235405\pi\)
\(654\) −3.41288e8 −0.0477088
\(655\) −1.54560e10 −2.14908
\(656\) 2.24852e8 0.0310980
\(657\) 9.27668e9 1.27619
\(658\) 4.04565e8 0.0553602
\(659\) 5.82980e9 0.793514 0.396757 0.917924i \(-0.370136\pi\)
0.396757 + 0.917924i \(0.370136\pi\)
\(660\) −2.47747e7 −0.00335432
\(661\) 4.98936e9 0.671954 0.335977 0.941870i \(-0.390934\pi\)
0.335977 + 0.941870i \(0.390934\pi\)
\(662\) 1.22148e9 0.163638
\(663\) −7.07693e7 −0.00943078
\(664\) 2.40335e9 0.318587
\(665\) 4.64328e9 0.612279
\(666\) −4.76815e9 −0.625446
\(667\) 1.00509e7 0.00131149
\(668\) −6.70391e8 −0.0870183
\(669\) −2.35668e8 −0.0304305
\(670\) 2.39482e9 0.307617
\(671\) −8.15904e7 −0.0104258
\(672\) −1.63745e8 −0.0208150
\(673\) −9.45755e9 −1.19599 −0.597993 0.801502i \(-0.704035\pi\)
−0.597993 + 0.801502i \(0.704035\pi\)
\(674\) −8.61554e9 −1.08386
\(675\) 9.89287e8 0.123811
\(676\) 3.96545e9 0.493718
\(677\) −1.17413e10 −1.45431 −0.727155 0.686474i \(-0.759158\pi\)
−0.727155 + 0.686474i \(0.759158\pi\)
\(678\) 1.51519e8 0.0186708
\(679\) 6.80579e8 0.0834324
\(680\) 8.39277e9 1.02359
\(681\) −4.54844e8 −0.0551884
\(682\) 7.71724e6 0.000931572 0
\(683\) 8.01180e9 0.962183 0.481091 0.876670i \(-0.340241\pi\)
0.481091 + 0.876670i \(0.340241\pi\)
\(684\) 4.73230e9 0.565426
\(685\) −2.15417e10 −2.56072
\(686\) −4.23267e9 −0.500587
\(687\) −3.13768e8 −0.0369199
\(688\) 2.62589e8 0.0307409
\(689\) 8.56440e8 0.0997538
\(690\) −8.79189e8 −0.101885
\(691\) 9.40691e9 1.08461 0.542305 0.840181i \(-0.317552\pi\)
0.542305 + 0.840181i \(0.317552\pi\)
\(692\) −6.13645e9 −0.703956
\(693\) 2.63322e8 0.0300553
\(694\) 5.86600e8 0.0666169
\(695\) 9.82769e9 1.11046
\(696\) −402322. −4.52316e−5 0
\(697\) 9.33433e8 0.104416
\(698\) −3.43765e9 −0.382619
\(699\) −2.79074e8 −0.0309064
\(700\) 1.99736e9 0.220097
\(701\) 5.77903e9 0.633639 0.316819 0.948486i \(-0.397385\pi\)
0.316819 + 0.948486i \(0.397385\pi\)
\(702\) 1.75945e8 0.0191954
\(703\) −9.07264e9 −0.984895
\(704\) 5.90219e8 0.0637542
\(705\) 1.58479e8 0.0170337
\(706\) 6.94389e8 0.0742654
\(707\) 3.13896e9 0.334055
\(708\) 3.94818e8 0.0418100
\(709\) 1.04068e10 1.09662 0.548311 0.836274i \(-0.315271\pi\)
0.548311 + 0.836274i \(0.315271\pi\)
\(710\) 8.04642e8 0.0843721
\(711\) −1.43930e10 −1.50178
\(712\) 1.53913e10 1.59807
\(713\) −3.01388e8 −0.0311396
\(714\) 1.03625e8 0.0106542
\(715\) 2.58916e8 0.0264904
\(716\) 5.01971e9 0.511073
\(717\) −6.94437e8 −0.0703583
\(718\) 8.52822e9 0.859851
\(719\) 1.83550e10 1.84164 0.920819 0.389991i \(-0.127522\pi\)
0.920819 + 0.389991i \(0.127522\pi\)
\(720\) −2.89391e9 −0.288949
\(721\) −7.06560e9 −0.702063
\(722\) −1.20431e9 −0.119085
\(723\) 5.30725e6 0.000522259 0
\(724\) 7.27030e8 0.0711980
\(725\) 8.12784e6 0.000792122 0
\(726\) −4.10401e8 −0.0398043
\(727\) 1.38023e9 0.133223 0.0666116 0.997779i \(-0.478781\pi\)
0.0666116 + 0.997779i \(0.478781\pi\)
\(728\) 1.03325e9 0.0992539
\(729\) −1.02496e10 −0.979852
\(730\) −1.33562e10 −1.27073
\(731\) 1.09009e9 0.103217
\(732\) −4.38505e7 −0.00413225
\(733\) 2.24467e9 0.210518 0.105259 0.994445i \(-0.466433\pi\)
0.105259 + 0.994445i \(0.466433\pi\)
\(734\) 4.61787e9 0.431028
\(735\) −7.59595e8 −0.0705629
\(736\) −1.74597e10 −1.61422
\(737\) 2.58415e8 0.0237784
\(738\) −1.15838e9 −0.106085
\(739\) 4.41123e9 0.402072 0.201036 0.979584i \(-0.435569\pi\)
0.201036 + 0.979584i \(0.435569\pi\)
\(740\) −7.55494e9 −0.685362
\(741\) 1.67108e8 0.0150881
\(742\) −1.25405e9 −0.112694
\(743\) 1.31411e10 1.17536 0.587679 0.809094i \(-0.300042\pi\)
0.587679 + 0.809094i \(0.300042\pi\)
\(744\) 1.20641e7 0.00107396
\(745\) −2.74828e10 −2.43509
\(746\) −9.18315e9 −0.809852
\(747\) −3.44020e9 −0.301969
\(748\) 3.11355e8 0.0272020
\(749\) −4.24633e9 −0.369256
\(750\) −4.56281e7 −0.00394928
\(751\) 8.90398e9 0.767087 0.383543 0.923523i \(-0.374704\pi\)
0.383543 + 0.923523i \(0.374704\pi\)
\(752\) −4.79769e8 −0.0411405
\(753\) 8.13072e8 0.0693980
\(754\) 1.44554e6 0.000122809 0
\(755\) −1.80154e10 −1.52346
\(756\) 2.83522e8 0.0238649
\(757\) −1.98575e10 −1.66375 −0.831877 0.554960i \(-0.812734\pi\)
−0.831877 + 0.554960i \(0.812734\pi\)
\(758\) −1.24024e10 −1.03434
\(759\) −9.48698e7 −0.00787557
\(760\) −1.98179e10 −1.63761
\(761\) 4.10456e9 0.337614 0.168807 0.985649i \(-0.446009\pi\)
0.168807 + 0.985649i \(0.446009\pi\)
\(762\) −4.14976e8 −0.0339766
\(763\) 5.74753e9 0.468431
\(764\) 9.21432e9 0.747544
\(765\) −1.20136e10 −0.970192
\(766\) 1.18720e10 0.954384
\(767\) −4.12618e9 −0.330190
\(768\) 7.75818e8 0.0618010
\(769\) 1.69512e10 1.34418 0.672091 0.740468i \(-0.265396\pi\)
0.672091 + 0.740468i \(0.265396\pi\)
\(770\) −3.79121e8 −0.0299268
\(771\) 9.92323e8 0.0779763
\(772\) 7.81595e9 0.611394
\(773\) 7.71790e8 0.0600995 0.0300498 0.999548i \(-0.490433\pi\)
0.0300498 + 0.999548i \(0.490433\pi\)
\(774\) −1.35280e9 −0.104867
\(775\) −2.43722e8 −0.0188078
\(776\) −2.90478e9 −0.223150
\(777\) −2.71322e8 −0.0207497
\(778\) −1.42041e10 −1.08140
\(779\) −2.20412e9 −0.167053
\(780\) 1.39154e8 0.0104994
\(781\) 8.68258e7 0.00652184
\(782\) 1.10492e10 0.826240
\(783\) 1.15373e6 8.58891e−5 0
\(784\) 2.29955e9 0.170426
\(785\) −1.79060e10 −1.32116
\(786\) 8.14497e8 0.0598289
\(787\) −1.55107e10 −1.13428 −0.567139 0.823622i \(-0.691950\pi\)
−0.567139 + 0.823622i \(0.691950\pi\)
\(788\) 1.20095e10 0.874346
\(789\) −1.18882e9 −0.0861678
\(790\) 2.07224e10 1.49536
\(791\) −2.55169e9 −0.183320
\(792\) −1.12388e9 −0.0803864
\(793\) 4.58275e8 0.0326340
\(794\) −1.65110e10 −1.17058
\(795\) −4.91246e8 −0.0346748
\(796\) 8.76553e9 0.616003
\(797\) 1.44069e10 1.00801 0.504006 0.863700i \(-0.331859\pi\)
0.504006 + 0.863700i \(0.331859\pi\)
\(798\) −2.44690e8 −0.0170454
\(799\) −1.99168e9 −0.138136
\(800\) −1.41190e10 −0.974964
\(801\) −2.20315e10 −1.51471
\(802\) 1.70757e10 1.16888
\(803\) −1.44122e9 −0.0982256
\(804\) 1.38885e8 0.00942450
\(805\) 1.48062e10 1.00036
\(806\) −4.33460e7 −0.00291593
\(807\) 1.73079e8 0.0115927
\(808\) −1.33974e10 −0.893469
\(809\) −1.32032e10 −0.876714 −0.438357 0.898801i \(-0.644440\pi\)
−0.438357 + 0.898801i \(0.644440\pi\)
\(810\) 1.48582e10 0.982358
\(811\) 2.59038e10 1.70526 0.852629 0.522517i \(-0.175007\pi\)
0.852629 + 0.522517i \(0.175007\pi\)
\(812\) 2.32937e6 0.000152684 0
\(813\) −9.47663e8 −0.0618496
\(814\) 7.40775e8 0.0481394
\(815\) 8.22971e9 0.532516
\(816\) −1.22888e8 −0.00791758
\(817\) −2.57405e9 −0.165135
\(818\) 9.09516e9 0.580997
\(819\) −1.47902e9 −0.0940765
\(820\) −1.83541e9 −0.116248
\(821\) −1.80408e9 −0.113777 −0.0568885 0.998381i \(-0.518118\pi\)
−0.0568885 + 0.998381i \(0.518118\pi\)
\(822\) 1.13520e9 0.0712886
\(823\) 2.15004e10 1.34445 0.672227 0.740345i \(-0.265338\pi\)
0.672227 + 0.740345i \(0.265338\pi\)
\(824\) 3.01566e10 1.87775
\(825\) −7.67178e7 −0.00475672
\(826\) 6.04181e9 0.373024
\(827\) 1.95017e10 1.19895 0.599477 0.800392i \(-0.295375\pi\)
0.599477 + 0.800392i \(0.295375\pi\)
\(828\) 1.50900e10 0.923812
\(829\) 1.50758e10 0.919049 0.459524 0.888165i \(-0.348020\pi\)
0.459524 + 0.888165i \(0.348020\pi\)
\(830\) 4.95307e9 0.300678
\(831\) −7.13487e8 −0.0431303
\(832\) −3.31513e9 −0.199558
\(833\) 9.54619e9 0.572233
\(834\) −5.17896e8 −0.0309145
\(835\) −4.01866e9 −0.238880
\(836\) −7.35206e8 −0.0435198
\(837\) −3.45958e7 −0.00203932
\(838\) 7.49157e9 0.439763
\(839\) −3.01700e10 −1.76363 −0.881816 0.471594i \(-0.843679\pi\)
−0.881816 + 0.471594i \(0.843679\pi\)
\(840\) −5.92665e8 −0.0345010
\(841\) −1.72499e10 −0.999999
\(842\) 7.99060e7 0.00461304
\(843\) 8.51249e8 0.0489396
\(844\) −2.03703e9 −0.116627
\(845\) 2.37709e10 1.35534
\(846\) 2.47166e9 0.140344
\(847\) 6.91144e9 0.390820
\(848\) 1.48717e9 0.0837480
\(849\) −8.27889e7 −0.00464296
\(850\) 8.93507e9 0.499036
\(851\) −2.89302e10 −1.60915
\(852\) 4.66643e7 0.00258492
\(853\) −1.64145e9 −0.0905534 −0.0452767 0.998974i \(-0.514417\pi\)
−0.0452767 + 0.998974i \(0.514417\pi\)
\(854\) −6.71035e8 −0.0368674
\(855\) 2.83678e10 1.55219
\(856\) 1.81237e10 0.987619
\(857\) −2.57589e10 −1.39796 −0.698981 0.715141i \(-0.746363\pi\)
−0.698981 + 0.715141i \(0.746363\pi\)
\(858\) −1.36443e7 −0.000737472 0
\(859\) −2.14742e10 −1.15595 −0.577977 0.816053i \(-0.696158\pi\)
−0.577977 + 0.816053i \(0.696158\pi\)
\(860\) −2.14345e9 −0.114913
\(861\) −6.59154e7 −0.00351946
\(862\) 2.45278e10 1.30432
\(863\) −3.38483e9 −0.179266 −0.0896332 0.995975i \(-0.528569\pi\)
−0.0896332 + 0.995975i \(0.528569\pi\)
\(864\) −2.00416e9 −0.105715
\(865\) −3.67850e10 −1.93248
\(866\) 1.80609e10 0.944990
\(867\) 6.03434e8 0.0314458
\(868\) −6.98487e7 −0.00362526
\(869\) 2.23608e9 0.115589
\(870\) −829148. −4.26889e−5 0
\(871\) −1.45146e9 −0.0744290
\(872\) −2.45310e10 −1.25287
\(873\) 4.15795e9 0.211510
\(874\) −2.60905e10 −1.32188
\(875\) 7.68410e8 0.0387762
\(876\) −7.74578e8 −0.0389315
\(877\) −2.37382e10 −1.18836 −0.594182 0.804331i \(-0.702524\pi\)
−0.594182 + 0.804331i \(0.702524\pi\)
\(878\) −1.28185e10 −0.639157
\(879\) −1.76613e9 −0.0877124
\(880\) 4.49596e8 0.0222399
\(881\) −1.49518e10 −0.736678 −0.368339 0.929692i \(-0.620073\pi\)
−0.368339 + 0.929692i \(0.620073\pi\)
\(882\) −1.18468e10 −0.581379
\(883\) 2.73527e10 1.33702 0.668510 0.743703i \(-0.266932\pi\)
0.668510 + 0.743703i \(0.266932\pi\)
\(884\) −1.74881e9 −0.0851453
\(885\) 2.36674e9 0.114775
\(886\) −5.56002e9 −0.268570
\(887\) 3.74606e10 1.80236 0.901181 0.433443i \(-0.142701\pi\)
0.901181 + 0.433443i \(0.142701\pi\)
\(888\) 1.15803e9 0.0554974
\(889\) 6.98849e9 0.333601
\(890\) 3.17201e10 1.50823
\(891\) 1.60329e9 0.0759349
\(892\) −5.82370e9 −0.274740
\(893\) 4.70297e9 0.221000
\(894\) 1.44828e9 0.0677910
\(895\) 3.00907e10 1.40298
\(896\) −2.86903e9 −0.133247
\(897\) 5.32863e8 0.0246514
\(898\) −1.68536e10 −0.776652
\(899\) −284234. −1.30472e−5 0
\(900\) 1.22028e10 0.557968
\(901\) 6.17372e9 0.281197
\(902\) 1.79965e8 0.00816519
\(903\) −7.69781e7 −0.00347905
\(904\) 1.08908e10 0.490311
\(905\) 4.35819e9 0.195450
\(906\) 9.49372e8 0.0424119
\(907\) −4.76936e9 −0.212243 −0.106122 0.994353i \(-0.533843\pi\)
−0.106122 + 0.994353i \(0.533843\pi\)
\(908\) −1.12399e10 −0.498265
\(909\) 1.91773e10 0.846863
\(910\) 2.12944e9 0.0936743
\(911\) −4.59972e9 −0.201566 −0.100783 0.994908i \(-0.532135\pi\)
−0.100783 + 0.994908i \(0.532135\pi\)
\(912\) 2.90176e8 0.0126671
\(913\) 5.34467e8 0.0232420
\(914\) 8.25540e9 0.357624
\(915\) −2.62862e8 −0.0113437
\(916\) −7.75368e9 −0.333329
\(917\) −1.37167e10 −0.587432
\(918\) 1.26832e9 0.0541101
\(919\) 1.59138e10 0.676348 0.338174 0.941084i \(-0.390191\pi\)
0.338174 + 0.941084i \(0.390191\pi\)
\(920\) −6.31941e10 −2.67559
\(921\) 4.46143e8 0.0188176
\(922\) 2.29151e10 0.962862
\(923\) −4.87682e8 −0.0204141
\(924\) −2.19867e7 −0.000916871 0
\(925\) −2.33948e10 −0.971903
\(926\) −2.18812e10 −0.905592
\(927\) −4.31668e10 −1.77980
\(928\) −1.64659e7 −0.000676344 0
\(929\) −2.22650e10 −0.911102 −0.455551 0.890210i \(-0.650558\pi\)
−0.455551 + 0.890210i \(0.650558\pi\)
\(930\) 2.48629e7 0.00101359
\(931\) −2.25415e10 −0.915502
\(932\) −6.89632e9 −0.279037
\(933\) −1.81249e8 −0.00730619
\(934\) −1.05712e10 −0.424533
\(935\) 1.86642e9 0.0746739
\(936\) 6.31260e9 0.251619
\(937\) −2.34525e10 −0.931322 −0.465661 0.884963i \(-0.654183\pi\)
−0.465661 + 0.884963i \(0.654183\pi\)
\(938\) 2.12532e9 0.0840842
\(939\) −9.93923e8 −0.0391763
\(940\) 3.91625e9 0.153788
\(941\) 2.99250e10 1.17077 0.585383 0.810757i \(-0.300944\pi\)
0.585383 + 0.810757i \(0.300944\pi\)
\(942\) 9.43606e8 0.0367801
\(943\) −7.02836e9 −0.272937
\(944\) −7.16492e9 −0.277210
\(945\) 1.69957e9 0.0655132
\(946\) 2.10169e8 0.00807143
\(947\) −2.94339e10 −1.12622 −0.563109 0.826383i \(-0.690395\pi\)
−0.563109 + 0.826383i \(0.690395\pi\)
\(948\) 1.20177e9 0.0458135
\(949\) 8.09500e9 0.307457
\(950\) −2.10985e10 −0.798396
\(951\) −2.29573e9 −0.0865542
\(952\) 7.44830e9 0.279787
\(953\) 3.72084e10 1.39257 0.696283 0.717767i \(-0.254836\pi\)
0.696283 + 0.717767i \(0.254836\pi\)
\(954\) −7.66154e9 −0.285691
\(955\) 5.52353e10 2.05213
\(956\) −1.71605e10 −0.635226
\(957\) −89470.1 −3.29979e−6 0
\(958\) −9.18488e9 −0.337516
\(959\) −1.91175e10 −0.699949
\(960\) 1.90153e9 0.0693671
\(961\) −2.75041e10 −0.999690
\(962\) −4.16077e9 −0.150682
\(963\) −2.59427e10 −0.936101
\(964\) 1.31150e8 0.00471518
\(965\) 4.68528e10 1.67838
\(966\) −7.80251e8 −0.0278493
\(967\) −1.49635e10 −0.532159 −0.266080 0.963951i \(-0.585728\pi\)
−0.266080 + 0.963951i \(0.585728\pi\)
\(968\) −2.94987e10 −1.04529
\(969\) 1.20461e9 0.0425319
\(970\) −5.98647e9 −0.210605
\(971\) 3.15367e10 1.10548 0.552738 0.833355i \(-0.313583\pi\)
0.552738 + 0.833355i \(0.313583\pi\)
\(972\) 2.59970e9 0.0908010
\(973\) 8.72174e9 0.303535
\(974\) 1.35402e10 0.469536
\(975\) 4.30907e8 0.0148891
\(976\) 7.95773e8 0.0273978
\(977\) 1.36797e10 0.469295 0.234648 0.972080i \(-0.424606\pi\)
0.234648 + 0.972080i \(0.424606\pi\)
\(978\) −4.33687e8 −0.0148248
\(979\) 3.42279e9 0.116584
\(980\) −1.87707e10 −0.637073
\(981\) 3.51141e10 1.18752
\(982\) −6.20433e9 −0.209076
\(983\) 1.28389e10 0.431112 0.215556 0.976491i \(-0.430844\pi\)
0.215556 + 0.976491i \(0.430844\pi\)
\(984\) 2.81333e8 0.00941321
\(985\) 7.19910e10 2.40022
\(986\) 1.04203e7 0.000346187 0
\(987\) 1.40645e8 0.00465601
\(988\) 4.12949e9 0.136222
\(989\) −8.20794e9 −0.269803
\(990\) −2.31622e9 −0.0758675
\(991\) −5.51139e10 −1.79889 −0.899443 0.437038i \(-0.856028\pi\)
−0.899443 + 0.437038i \(0.856028\pi\)
\(992\) 4.93748e8 0.0160588
\(993\) 4.24641e8 0.0137626
\(994\) 7.14093e8 0.0230623
\(995\) 5.25451e10 1.69103
\(996\) 2.87248e8 0.00921190
\(997\) 5.87820e10 1.87850 0.939250 0.343234i \(-0.111522\pi\)
0.939250 + 0.343234i \(0.111522\pi\)
\(998\) 2.13495e9 0.0679878
\(999\) −3.32084e9 −0.105383
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.1.4 13
3.2 odd 2 387.8.a.d.1.10 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.4 13 1.1 even 1 trivial
387.8.a.d.1.10 13 3.2 odd 2