Properties

Label 43.6.a.b.1.6
Level 43
Weight 6
Character 43.1
Self dual yes
Analytic conductor 6.897
Analytic rank 0
Dimension 10
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.89650425196\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.48720\) of \(x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + 8437093 x^{2} - 5752252 x - 22734604\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.48720 q^{2} -27.5943 q^{3} -25.8138 q^{4} +101.308 q^{5} -68.6327 q^{6} +15.7005 q^{7} -143.795 q^{8} +518.448 q^{9} +O(q^{10})\) \(q+2.48720 q^{2} -27.5943 q^{3} -25.8138 q^{4} +101.308 q^{5} -68.6327 q^{6} +15.7005 q^{7} -143.795 q^{8} +518.448 q^{9} +251.974 q^{10} +394.739 q^{11} +712.316 q^{12} +666.939 q^{13} +39.0503 q^{14} -2795.53 q^{15} +468.396 q^{16} +172.038 q^{17} +1289.48 q^{18} -1280.86 q^{19} -2615.15 q^{20} -433.245 q^{21} +981.796 q^{22} +569.882 q^{23} +3967.92 q^{24} +7138.34 q^{25} +1658.81 q^{26} -7600.80 q^{27} -405.290 q^{28} -6328.41 q^{29} -6953.05 q^{30} +7795.01 q^{31} +5766.42 q^{32} -10892.6 q^{33} +427.892 q^{34} +1590.59 q^{35} -13383.1 q^{36} +16252.3 q^{37} -3185.75 q^{38} -18403.7 q^{39} -14567.6 q^{40} +7454.95 q^{41} -1077.57 q^{42} +1849.00 q^{43} -10189.7 q^{44} +52523.0 q^{45} +1417.41 q^{46} -5628.68 q^{47} -12925.1 q^{48} -16560.5 q^{49} +17754.5 q^{50} -4747.27 q^{51} -17216.2 q^{52} +22460.1 q^{53} -18904.7 q^{54} +39990.3 q^{55} -2257.65 q^{56} +35344.4 q^{57} -15740.0 q^{58} -9061.48 q^{59} +72163.4 q^{60} -18280.8 q^{61} +19387.8 q^{62} +8139.89 q^{63} -646.417 q^{64} +67566.3 q^{65} -27092.0 q^{66} -27428.5 q^{67} -4440.95 q^{68} -15725.5 q^{69} +3956.11 q^{70} -12860.9 q^{71} -74550.0 q^{72} -63446.5 q^{73} +40422.7 q^{74} -196978. q^{75} +33063.8 q^{76} +6197.60 q^{77} -45773.8 q^{78} -1911.26 q^{79} +47452.4 q^{80} +83756.3 q^{81} +18542.0 q^{82} -52124.9 q^{83} +11183.7 q^{84} +17428.8 q^{85} +4598.84 q^{86} +174628. q^{87} -56761.4 q^{88} +66185.2 q^{89} +130635. q^{90} +10471.3 q^{91} -14710.8 q^{92} -215098. q^{93} -13999.7 q^{94} -129761. q^{95} -159121. q^{96} +37497.8 q^{97} -41189.3 q^{98} +204652. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 8q^{2} + 28q^{3} + 202q^{4} + 138q^{5} + 75q^{6} + 60q^{7} + 294q^{8} + 1356q^{9} + O(q^{10}) \) \( 10q + 8q^{2} + 28q^{3} + 202q^{4} + 138q^{5} + 75q^{6} + 60q^{7} + 294q^{8} + 1356q^{9} - 17q^{10} + 745q^{11} + 4627q^{12} + 1917q^{13} + 1936q^{14} + 1688q^{15} + 5354q^{16} + 4017q^{17} - 2725q^{18} - 2404q^{19} + 1311q^{20} - 228q^{21} - 5836q^{22} + 1733q^{23} - 10711q^{24} + 7120q^{25} - 1484q^{26} - 2324q^{27} - 15028q^{28} + 6996q^{29} - 48420q^{30} - 4899q^{31} - 7554q^{32} - 15734q^{33} - 27033q^{34} + 7084q^{35} + 4433q^{36} + 1466q^{37} + 13905q^{38} - 26542q^{39} - 93211q^{40} + 10297q^{41} - 37642q^{42} + 18490q^{43} - 36140q^{44} + 73822q^{45} + 17991q^{46} + 48592q^{47} + 83607q^{48} + 29458q^{49} + 983q^{50} + 92972q^{51} + 14232q^{52} + 127165q^{53} - 92002q^{54} + 106672q^{55} - 7780q^{56} + 34060q^{57} - 10305q^{58} + 99372q^{59} + 111372q^{60} + 17408q^{61} + 28265q^{62} + 2244q^{63} + 47202q^{64} + 54484q^{65} - 150292q^{66} - 2021q^{67} + 192151q^{68} + 1654q^{69} - 33194q^{70} + 11286q^{71} - 298365q^{72} + 49892q^{73} - 125431q^{74} - 44662q^{75} - 249803q^{76} + 98144q^{77} - 28494q^{78} - 91524q^{79} + 12251q^{80} - 26450q^{81} - 158909q^{82} - 105203q^{83} - 357682q^{84} - 87212q^{85} + 14792q^{86} + 181200q^{87} - 461824q^{88} - 62682q^{89} - 522670q^{90} - 295304q^{91} + 183783q^{92} - 238430q^{93} + 7259q^{94} - 305340q^{95} - 162399q^{96} + 108383q^{97} + 354656q^{98} - 270499q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.48720 0.439679 0.219840 0.975536i \(-0.429447\pi\)
0.219840 + 0.975536i \(0.429447\pi\)
\(3\) −27.5943 −1.77018 −0.885089 0.465422i \(-0.845903\pi\)
−0.885089 + 0.465422i \(0.845903\pi\)
\(4\) −25.8138 −0.806682
\(5\) 101.308 1.81226 0.906128 0.423004i \(-0.139024\pi\)
0.906128 + 0.423004i \(0.139024\pi\)
\(6\) −68.6327 −0.778311
\(7\) 15.7005 0.121107 0.0605534 0.998165i \(-0.480713\pi\)
0.0605534 + 0.998165i \(0.480713\pi\)
\(8\) −143.795 −0.794361
\(9\) 518.448 2.13353
\(10\) 251.974 0.796811
\(11\) 394.739 0.983623 0.491811 0.870702i \(-0.336335\pi\)
0.491811 + 0.870702i \(0.336335\pi\)
\(12\) 712.316 1.42797
\(13\) 666.939 1.09453 0.547265 0.836959i \(-0.315669\pi\)
0.547265 + 0.836959i \(0.315669\pi\)
\(14\) 39.0503 0.0532481
\(15\) −2795.53 −3.20801
\(16\) 468.396 0.457418
\(17\) 172.038 0.144378 0.0721890 0.997391i \(-0.477002\pi\)
0.0721890 + 0.997391i \(0.477002\pi\)
\(18\) 1289.48 0.938069
\(19\) −1280.86 −0.813985 −0.406992 0.913432i \(-0.633422\pi\)
−0.406992 + 0.913432i \(0.633422\pi\)
\(20\) −2615.15 −1.46191
\(21\) −433.245 −0.214380
\(22\) 981.796 0.432479
\(23\) 569.882 0.224629 0.112314 0.993673i \(-0.464174\pi\)
0.112314 + 0.993673i \(0.464174\pi\)
\(24\) 3967.92 1.40616
\(25\) 7138.34 2.28427
\(26\) 1658.81 0.481242
\(27\) −7600.80 −2.00655
\(28\) −405.290 −0.0976946
\(29\) −6328.41 −1.39733 −0.698666 0.715448i \(-0.746223\pi\)
−0.698666 + 0.715448i \(0.746223\pi\)
\(30\) −6953.05 −1.41050
\(31\) 7795.01 1.45684 0.728421 0.685129i \(-0.240254\pi\)
0.728421 + 0.685129i \(0.240254\pi\)
\(32\) 5766.42 0.995478
\(33\) −10892.6 −1.74119
\(34\) 427.892 0.0634800
\(35\) 1590.59 0.219476
\(36\) −13383.1 −1.72108
\(37\) 16252.3 1.95169 0.975844 0.218469i \(-0.0701061\pi\)
0.975844 + 0.218469i \(0.0701061\pi\)
\(38\) −3185.75 −0.357892
\(39\) −18403.7 −1.93751
\(40\) −14567.6 −1.43958
\(41\) 7454.95 0.692604 0.346302 0.938123i \(-0.387437\pi\)
0.346302 + 0.938123i \(0.387437\pi\)
\(42\) −1077.57 −0.0942587
\(43\) 1849.00 0.152499
\(44\) −10189.7 −0.793471
\(45\) 52523.0 3.86650
\(46\) 1417.41 0.0987646
\(47\) −5628.68 −0.371673 −0.185837 0.982581i \(-0.559500\pi\)
−0.185837 + 0.982581i \(0.559500\pi\)
\(48\) −12925.1 −0.809712
\(49\) −16560.5 −0.985333
\(50\) 17754.5 1.00435
\(51\) −4747.27 −0.255575
\(52\) −17216.2 −0.882938
\(53\) 22460.1 1.09830 0.549151 0.835723i \(-0.314951\pi\)
0.549151 + 0.835723i \(0.314951\pi\)
\(54\) −18904.7 −0.882238
\(55\) 39990.3 1.78258
\(56\) −2257.65 −0.0962024
\(57\) 35344.4 1.44090
\(58\) −15740.0 −0.614378
\(59\) −9061.48 −0.338898 −0.169449 0.985539i \(-0.554199\pi\)
−0.169449 + 0.985539i \(0.554199\pi\)
\(60\) 72163.4 2.58785
\(61\) −18280.8 −0.629029 −0.314514 0.949253i \(-0.601842\pi\)
−0.314514 + 0.949253i \(0.601842\pi\)
\(62\) 19387.8 0.640544
\(63\) 8139.89 0.258385
\(64\) −646.417 −0.0197271
\(65\) 67566.3 1.98357
\(66\) −27092.0 −0.765564
\(67\) −27428.5 −0.746474 −0.373237 0.927736i \(-0.621752\pi\)
−0.373237 + 0.927736i \(0.621752\pi\)
\(68\) −4440.95 −0.116467
\(69\) −15725.5 −0.397633
\(70\) 3956.11 0.0964992
\(71\) −12860.9 −0.302780 −0.151390 0.988474i \(-0.548375\pi\)
−0.151390 + 0.988474i \(0.548375\pi\)
\(72\) −74550.0 −1.69479
\(73\) −63446.5 −1.39348 −0.696740 0.717324i \(-0.745367\pi\)
−0.696740 + 0.717324i \(0.745367\pi\)
\(74\) 40422.7 0.858117
\(75\) −196978. −4.04356
\(76\) 33063.8 0.656627
\(77\) 6197.60 0.119123
\(78\) −45773.8 −0.851884
\(79\) −1911.26 −0.0344549 −0.0172275 0.999852i \(-0.505484\pi\)
−0.0172275 + 0.999852i \(0.505484\pi\)
\(80\) 47452.4 0.828959
\(81\) 83756.3 1.41842
\(82\) 18542.0 0.304524
\(83\) −52124.9 −0.830520 −0.415260 0.909703i \(-0.636309\pi\)
−0.415260 + 0.909703i \(0.636309\pi\)
\(84\) 11183.7 0.172937
\(85\) 17428.8 0.261650
\(86\) 4598.84 0.0670505
\(87\) 174628. 2.47353
\(88\) −56761.4 −0.781351
\(89\) 66185.2 0.885698 0.442849 0.896596i \(-0.353968\pi\)
0.442849 + 0.896596i \(0.353968\pi\)
\(90\) 130635. 1.70002
\(91\) 10471.3 0.132555
\(92\) −14710.8 −0.181204
\(93\) −215098. −2.57887
\(94\) −13999.7 −0.163417
\(95\) −129761. −1.47515
\(96\) −159121. −1.76217
\(97\) 37497.8 0.404648 0.202324 0.979319i \(-0.435151\pi\)
0.202324 + 0.979319i \(0.435151\pi\)
\(98\) −41189.3 −0.433231
\(99\) 204652. 2.09859
\(100\) −184268. −1.84268
\(101\) −35537.7 −0.346646 −0.173323 0.984865i \(-0.555450\pi\)
−0.173323 + 0.984865i \(0.555450\pi\)
\(102\) −11807.4 −0.112371
\(103\) 177955. 1.65279 0.826395 0.563091i \(-0.190388\pi\)
0.826395 + 0.563091i \(0.190388\pi\)
\(104\) −95902.2 −0.869451
\(105\) −43891.2 −0.388512
\(106\) 55862.8 0.482901
\(107\) 43589.8 0.368065 0.184033 0.982920i \(-0.441085\pi\)
0.184033 + 0.982920i \(0.441085\pi\)
\(108\) 196206. 1.61865
\(109\) −49084.5 −0.395711 −0.197856 0.980231i \(-0.563398\pi\)
−0.197856 + 0.980231i \(0.563398\pi\)
\(110\) 99464.0 0.783762
\(111\) −448471. −3.45484
\(112\) 7354.05 0.0553964
\(113\) −34530.0 −0.254390 −0.127195 0.991878i \(-0.540597\pi\)
−0.127195 + 0.991878i \(0.540597\pi\)
\(114\) 87908.6 0.633533
\(115\) 57733.7 0.407085
\(116\) 163360. 1.12720
\(117\) 345773. 2.33521
\(118\) −22537.7 −0.149006
\(119\) 2701.08 0.0174852
\(120\) 401983. 2.54832
\(121\) −5231.89 −0.0324859
\(122\) −45468.0 −0.276571
\(123\) −205715. −1.22603
\(124\) −201219. −1.17521
\(125\) 406584. 2.32742
\(126\) 20245.5 0.113606
\(127\) −10008.6 −0.0550637 −0.0275318 0.999621i \(-0.508765\pi\)
−0.0275318 + 0.999621i \(0.508765\pi\)
\(128\) −186133. −1.00415
\(129\) −51021.9 −0.269950
\(130\) 168051. 0.872133
\(131\) 336036. 1.71083 0.855415 0.517943i \(-0.173302\pi\)
0.855415 + 0.517943i \(0.173302\pi\)
\(132\) 281179. 1.40458
\(133\) −20110.1 −0.0985791
\(134\) −68220.2 −0.328209
\(135\) −770023. −3.63638
\(136\) −24738.1 −0.114688
\(137\) −1217.37 −0.00554141 −0.00277071 0.999996i \(-0.500882\pi\)
−0.00277071 + 0.999996i \(0.500882\pi\)
\(138\) −39112.5 −0.174831
\(139\) −157629. −0.691989 −0.345995 0.938237i \(-0.612458\pi\)
−0.345995 + 0.938237i \(0.612458\pi\)
\(140\) −41059.2 −0.177048
\(141\) 155320. 0.657928
\(142\) −31987.8 −0.133126
\(143\) 263267. 1.07660
\(144\) 242839. 0.975915
\(145\) −641119. −2.53232
\(146\) −157804. −0.612684
\(147\) 456976. 1.74422
\(148\) −419534. −1.57439
\(149\) −501641. −1.85109 −0.925545 0.378639i \(-0.876392\pi\)
−0.925545 + 0.378639i \(0.876392\pi\)
\(150\) −489924. −1.77787
\(151\) −430865. −1.53780 −0.768899 0.639370i \(-0.779195\pi\)
−0.768899 + 0.639370i \(0.779195\pi\)
\(152\) 184180. 0.646598
\(153\) 89192.5 0.308035
\(154\) 15414.7 0.0523761
\(155\) 789698. 2.64017
\(156\) 475071. 1.56296
\(157\) 157952. 0.511418 0.255709 0.966754i \(-0.417691\pi\)
0.255709 + 0.966754i \(0.417691\pi\)
\(158\) −4753.68 −0.0151491
\(159\) −619772. −1.94419
\(160\) 584186. 1.80406
\(161\) 8947.43 0.0272041
\(162\) 208319. 0.623650
\(163\) 405846. 1.19645 0.598223 0.801330i \(-0.295874\pi\)
0.598223 + 0.801330i \(0.295874\pi\)
\(164\) −192441. −0.558712
\(165\) −1.10351e6 −3.15548
\(166\) −129645. −0.365162
\(167\) 166792. 0.462789 0.231394 0.972860i \(-0.425671\pi\)
0.231394 + 0.972860i \(0.425671\pi\)
\(168\) 62298.3 0.170295
\(169\) 73514.3 0.197995
\(170\) 43349.0 0.115042
\(171\) −664057. −1.73666
\(172\) −47729.8 −0.123018
\(173\) −287751. −0.730973 −0.365487 0.930817i \(-0.619097\pi\)
−0.365487 + 0.930817i \(0.619097\pi\)
\(174\) 434336. 1.08756
\(175\) 112076. 0.276640
\(176\) 184894. 0.449927
\(177\) 250045. 0.599910
\(178\) 164616. 0.389423
\(179\) 522579. 1.21904 0.609522 0.792769i \(-0.291361\pi\)
0.609522 + 0.792769i \(0.291361\pi\)
\(180\) −1.35582e6 −3.11904
\(181\) 81268.9 0.184386 0.0921930 0.995741i \(-0.470612\pi\)
0.0921930 + 0.995741i \(0.470612\pi\)
\(182\) 26044.2 0.0582817
\(183\) 504447. 1.11349
\(184\) −81946.0 −0.178436
\(185\) 1.64649e6 3.53696
\(186\) −534993. −1.13388
\(187\) 67910.0 0.142014
\(188\) 145298. 0.299822
\(189\) −119336. −0.243007
\(190\) −322742. −0.648592
\(191\) −608799. −1.20751 −0.603755 0.797170i \(-0.706329\pi\)
−0.603755 + 0.797170i \(0.706329\pi\)
\(192\) 17837.5 0.0349205
\(193\) 184959. 0.357423 0.178712 0.983902i \(-0.442807\pi\)
0.178712 + 0.983902i \(0.442807\pi\)
\(194\) 93264.7 0.177915
\(195\) −1.86445e6 −3.51127
\(196\) 427490. 0.794851
\(197\) 200949. 0.368910 0.184455 0.982841i \(-0.440948\pi\)
0.184455 + 0.982841i \(0.440948\pi\)
\(198\) 509010. 0.922706
\(199\) −233653. −0.418253 −0.209127 0.977889i \(-0.567062\pi\)
−0.209127 + 0.977889i \(0.567062\pi\)
\(200\) −1.02646e6 −1.81453
\(201\) 756871. 1.32139
\(202\) −88389.5 −0.152413
\(203\) −99359.2 −0.169226
\(204\) 122545. 0.206168
\(205\) 755248. 1.25518
\(206\) 442610. 0.726697
\(207\) 295454. 0.479252
\(208\) 312392. 0.500658
\(209\) −505604. −0.800654
\(210\) −109166. −0.170821
\(211\) 200513. 0.310054 0.155027 0.987910i \(-0.450454\pi\)
0.155027 + 0.987910i \(0.450454\pi\)
\(212\) −579781. −0.885981
\(213\) 354889. 0.535974
\(214\) 108417. 0.161831
\(215\) 187319. 0.276366
\(216\) 1.09295e6 1.59392
\(217\) 122386. 0.176433
\(218\) −122083. −0.173986
\(219\) 1.75076e6 2.46671
\(220\) −1.03230e6 −1.43797
\(221\) 114739. 0.158026
\(222\) −1.11544e6 −1.51902
\(223\) 347051. 0.467338 0.233669 0.972316i \(-0.424927\pi\)
0.233669 + 0.972316i \(0.424927\pi\)
\(224\) 90535.7 0.120559
\(225\) 3.70086e6 4.87356
\(226\) −85883.2 −0.111850
\(227\) −212111. −0.273212 −0.136606 0.990625i \(-0.543619\pi\)
−0.136606 + 0.990625i \(0.543619\pi\)
\(228\) −912374. −1.16235
\(229\) −470629. −0.593049 −0.296524 0.955025i \(-0.595828\pi\)
−0.296524 + 0.955025i \(0.595828\pi\)
\(230\) 143595. 0.178987
\(231\) −171019. −0.210870
\(232\) 909991. 1.10999
\(233\) −952248. −1.14911 −0.574553 0.818467i \(-0.694824\pi\)
−0.574553 + 0.818467i \(0.694824\pi\)
\(234\) 860007. 1.02674
\(235\) −570231. −0.673567
\(236\) 233911. 0.273383
\(237\) 52739.9 0.0609914
\(238\) 6718.12 0.00768786
\(239\) −1.63690e6 −1.85365 −0.926824 0.375497i \(-0.877472\pi\)
−0.926824 + 0.375497i \(0.877472\pi\)
\(240\) −1.30942e6 −1.46740
\(241\) −695944. −0.771848 −0.385924 0.922531i \(-0.626117\pi\)
−0.385924 + 0.922531i \(0.626117\pi\)
\(242\) −13012.8 −0.0142834
\(243\) −464206. −0.504307
\(244\) 471897. 0.507426
\(245\) −1.67771e6 −1.78568
\(246\) −511654. −0.539061
\(247\) −854253. −0.890931
\(248\) −1.12088e6 −1.15726
\(249\) 1.43835e6 1.47017
\(250\) 1.01126e6 1.02332
\(251\) 657886. 0.659123 0.329561 0.944134i \(-0.393099\pi\)
0.329561 + 0.944134i \(0.393099\pi\)
\(252\) −210122. −0.208434
\(253\) 224955. 0.220950
\(254\) −24893.5 −0.0242104
\(255\) −480937. −0.463167
\(256\) −442266. −0.421778
\(257\) −599821. −0.566486 −0.283243 0.959048i \(-0.591410\pi\)
−0.283243 + 0.959048i \(0.591410\pi\)
\(258\) −126902. −0.118691
\(259\) 255169. 0.236363
\(260\) −1.74415e6 −1.60011
\(261\) −3.28095e6 −2.98125
\(262\) 835788. 0.752217
\(263\) −1.62868e6 −1.45194 −0.725968 0.687728i \(-0.758608\pi\)
−0.725968 + 0.687728i \(0.758608\pi\)
\(264\) 1.56629e6 1.38313
\(265\) 2.27539e6 1.99040
\(266\) −50017.8 −0.0433432
\(267\) −1.82634e6 −1.56784
\(268\) 708034. 0.602167
\(269\) 1.99405e6 1.68017 0.840087 0.542451i \(-0.182504\pi\)
0.840087 + 0.542451i \(0.182504\pi\)
\(270\) −1.91520e6 −1.59884
\(271\) −1.89207e6 −1.56500 −0.782501 0.622650i \(-0.786056\pi\)
−0.782501 + 0.622650i \(0.786056\pi\)
\(272\) 80581.8 0.0660411
\(273\) −288948. −0.234646
\(274\) −3027.84 −0.00243644
\(275\) 2.81778e6 2.24686
\(276\) 405936. 0.320763
\(277\) −2.11477e6 −1.65601 −0.828007 0.560717i \(-0.810525\pi\)
−0.828007 + 0.560717i \(0.810525\pi\)
\(278\) −392055. −0.304253
\(279\) 4.04131e6 3.10822
\(280\) −228718. −0.174343
\(281\) −80948.7 −0.0611567 −0.0305784 0.999532i \(-0.509735\pi\)
−0.0305784 + 0.999532i \(0.509735\pi\)
\(282\) 386311. 0.289277
\(283\) 1.01158e6 0.750814 0.375407 0.926860i \(-0.377503\pi\)
0.375407 + 0.926860i \(0.377503\pi\)
\(284\) 331990. 0.244247
\(285\) 3.58067e6 2.61128
\(286\) 654798. 0.473361
\(287\) 117046. 0.0838791
\(288\) 2.98959e6 2.12388
\(289\) −1.39026e6 −0.979155
\(290\) −1.59459e6 −1.11341
\(291\) −1.03473e6 −0.716298
\(292\) 1.63780e6 1.12409
\(293\) 786392. 0.535143 0.267572 0.963538i \(-0.413779\pi\)
0.267572 + 0.963538i \(0.413779\pi\)
\(294\) 1.13659e6 0.766895
\(295\) −918001. −0.614170
\(296\) −2.33699e6 −1.55034
\(297\) −3.00034e6 −1.97369
\(298\) −1.24768e6 −0.813886
\(299\) 380076. 0.245863
\(300\) 5.08475e6 3.26187
\(301\) 29030.2 0.0184686
\(302\) −1.07165e6 −0.676138
\(303\) 980641. 0.613625
\(304\) −599948. −0.372332
\(305\) −1.85199e6 −1.13996
\(306\) 221840. 0.135437
\(307\) 106306. 0.0643744 0.0321872 0.999482i \(-0.489753\pi\)
0.0321872 + 0.999482i \(0.489753\pi\)
\(308\) −159984. −0.0960947
\(309\) −4.91056e6 −2.92573
\(310\) 1.96414e6 1.16083
\(311\) −82617.8 −0.0484365 −0.0242183 0.999707i \(-0.507710\pi\)
−0.0242183 + 0.999707i \(0.507710\pi\)
\(312\) 2.64636e6 1.53908
\(313\) 1.38431e6 0.798681 0.399341 0.916803i \(-0.369239\pi\)
0.399341 + 0.916803i \(0.369239\pi\)
\(314\) 392859. 0.224860
\(315\) 824637. 0.468259
\(316\) 49336.9 0.0277942
\(317\) 964944. 0.539329 0.269665 0.962954i \(-0.413087\pi\)
0.269665 + 0.962954i \(0.413087\pi\)
\(318\) −1.54150e6 −0.854820
\(319\) −2.49807e6 −1.37445
\(320\) −65487.3 −0.0357505
\(321\) −1.20283e6 −0.651541
\(322\) 22254.1 0.0119611
\(323\) −220356. −0.117522
\(324\) −2.16207e6 −1.14421
\(325\) 4.76084e6 2.50020
\(326\) 1.00942e6 0.526052
\(327\) 1.35446e6 0.700479
\(328\) −1.07198e6 −0.550178
\(329\) −88373.0 −0.0450122
\(330\) −2.74464e6 −1.38740
\(331\) −3.21355e6 −1.61218 −0.806092 0.591790i \(-0.798422\pi\)
−0.806092 + 0.591790i \(0.798422\pi\)
\(332\) 1.34554e6 0.669966
\(333\) 8.42597e6 4.16398
\(334\) 414844. 0.203479
\(335\) −2.77873e6 −1.35280
\(336\) −202930. −0.0980615
\(337\) 835398. 0.400699 0.200350 0.979724i \(-0.435792\pi\)
0.200350 + 0.979724i \(0.435792\pi\)
\(338\) 182845. 0.0870544
\(339\) 952834. 0.450316
\(340\) −449904. −0.211068
\(341\) 3.07700e6 1.43298
\(342\) −1.65164e6 −0.763574
\(343\) −523886. −0.240437
\(344\) −265876. −0.121139
\(345\) −1.59312e6 −0.720612
\(346\) −715695. −0.321394
\(347\) 2.19943e6 0.980589 0.490295 0.871557i \(-0.336889\pi\)
0.490295 + 0.871557i \(0.336889\pi\)
\(348\) −4.50782e6 −1.99535
\(349\) 2.90995e6 1.27886 0.639428 0.768851i \(-0.279171\pi\)
0.639428 + 0.768851i \(0.279171\pi\)
\(350\) 278754. 0.121633
\(351\) −5.06927e6 −2.19623
\(352\) 2.27623e6 0.979175
\(353\) 2.63861e6 1.12704 0.563518 0.826104i \(-0.309448\pi\)
0.563518 + 0.826104i \(0.309448\pi\)
\(354\) 621914. 0.263768
\(355\) −1.30292e6 −0.548714
\(356\) −1.70849e6 −0.714477
\(357\) −74534.4 −0.0309518
\(358\) 1.29976e6 0.535988
\(359\) −37197.0 −0.0152325 −0.00761626 0.999971i \(-0.502424\pi\)
−0.00761626 + 0.999971i \(0.502424\pi\)
\(360\) −7.55253e6 −3.07140
\(361\) −835506. −0.337428
\(362\) 202132. 0.0810707
\(363\) 144371. 0.0575059
\(364\) −270304. −0.106930
\(365\) −6.42765e6 −2.52534
\(366\) 1.25466e6 0.489580
\(367\) 2.27179e6 0.880447 0.440224 0.897888i \(-0.354899\pi\)
0.440224 + 0.897888i \(0.354899\pi\)
\(368\) 266930. 0.102749
\(369\) 3.86500e6 1.47769
\(370\) 4.09515e6 1.55513
\(371\) 352635. 0.133012
\(372\) 5.55251e6 2.08033
\(373\) 2.42341e6 0.901892 0.450946 0.892551i \(-0.351087\pi\)
0.450946 + 0.892551i \(0.351087\pi\)
\(374\) 168906. 0.0624404
\(375\) −1.12194e7 −4.11996
\(376\) 809373. 0.295243
\(377\) −4.22066e6 −1.52942
\(378\) −296814. −0.106845
\(379\) −2.23180e6 −0.798101 −0.399050 0.916929i \(-0.630660\pi\)
−0.399050 + 0.916929i \(0.630660\pi\)
\(380\) 3.34963e6 1.18998
\(381\) 276182. 0.0974725
\(382\) −1.51421e6 −0.530917
\(383\) −1.13758e6 −0.396263 −0.198131 0.980175i \(-0.563487\pi\)
−0.198131 + 0.980175i \(0.563487\pi\)
\(384\) 5.13623e6 1.77753
\(385\) 627868. 0.215882
\(386\) 460031. 0.157152
\(387\) 958610. 0.325360
\(388\) −967963. −0.326422
\(389\) 1.97034e6 0.660188 0.330094 0.943948i \(-0.392920\pi\)
0.330094 + 0.943948i \(0.392920\pi\)
\(390\) −4.63726e6 −1.54383
\(391\) 98041.1 0.0324315
\(392\) 2.38131e6 0.782710
\(393\) −9.27268e6 −3.02847
\(394\) 499800. 0.162202
\(395\) −193626. −0.0624412
\(396\) −5.28284e6 −1.69289
\(397\) −5.13128e6 −1.63399 −0.816996 0.576644i \(-0.804362\pi\)
−0.816996 + 0.576644i \(0.804362\pi\)
\(398\) −581143. −0.183897
\(399\) 554925. 0.174503
\(400\) 3.34357e6 1.04487
\(401\) −2.92332e6 −0.907852 −0.453926 0.891039i \(-0.649977\pi\)
−0.453926 + 0.891039i \(0.649977\pi\)
\(402\) 1.88249e6 0.580989
\(403\) 5.19880e6 1.59456
\(404\) 917365. 0.279633
\(405\) 8.48520e6 2.57054
\(406\) −247126. −0.0744053
\(407\) 6.41542e6 1.91973
\(408\) 682631. 0.203019
\(409\) −2.10623e6 −0.622582 −0.311291 0.950315i \(-0.600761\pi\)
−0.311291 + 0.950315i \(0.600761\pi\)
\(410\) 1.87845e6 0.551875
\(411\) 33592.5 0.00980929
\(412\) −4.59370e6 −1.33328
\(413\) −142270. −0.0410428
\(414\) 734854. 0.210717
\(415\) −5.28068e6 −1.50511
\(416\) 3.84585e6 1.08958
\(417\) 4.34967e6 1.22494
\(418\) −1.25754e6 −0.352031
\(419\) −787697. −0.219192 −0.109596 0.993976i \(-0.534956\pi\)
−0.109596 + 0.993976i \(0.534956\pi\)
\(420\) 1.13300e6 0.313406
\(421\) −2.73861e6 −0.753051 −0.376526 0.926406i \(-0.622881\pi\)
−0.376526 + 0.926406i \(0.622881\pi\)
\(422\) 498717. 0.136324
\(423\) −2.91817e6 −0.792977
\(424\) −3.22964e6 −0.872448
\(425\) 1.22806e6 0.329798
\(426\) 882681. 0.235657
\(427\) −287018. −0.0761796
\(428\) −1.12522e6 −0.296912
\(429\) −7.26468e6 −1.90578
\(430\) 465900. 0.121513
\(431\) 2.23642e6 0.579911 0.289955 0.957040i \(-0.406360\pi\)
0.289955 + 0.957040i \(0.406360\pi\)
\(432\) −3.56019e6 −0.917832
\(433\) −7.32309e6 −1.87704 −0.938522 0.345219i \(-0.887804\pi\)
−0.938522 + 0.345219i \(0.887804\pi\)
\(434\) 304398. 0.0775741
\(435\) 1.76913e7 4.48266
\(436\) 1.26706e6 0.319213
\(437\) −729937. −0.182844
\(438\) 4.35450e6 1.08456
\(439\) 4.05522e6 1.00428 0.502138 0.864788i \(-0.332547\pi\)
0.502138 + 0.864788i \(0.332547\pi\)
\(440\) −5.75039e6 −1.41601
\(441\) −8.58575e6 −2.10224
\(442\) 285378. 0.0694808
\(443\) 2.23385e6 0.540810 0.270405 0.962747i \(-0.412842\pi\)
0.270405 + 0.962747i \(0.412842\pi\)
\(444\) 1.15768e7 2.78695
\(445\) 6.70510e6 1.60511
\(446\) 863185. 0.205479
\(447\) 1.38425e7 3.27676
\(448\) −10149.1 −0.00238908
\(449\) −3.90186e6 −0.913389 −0.456695 0.889623i \(-0.650967\pi\)
−0.456695 + 0.889623i \(0.650967\pi\)
\(450\) 9.20478e6 2.14280
\(451\) 2.94276e6 0.681262
\(452\) 891352. 0.205212
\(453\) 1.18894e7 2.72218
\(454\) −527564. −0.120126
\(455\) 1.06083e6 0.240223
\(456\) −5.08233e6 −1.14459
\(457\) −184098. −0.0412343 −0.0206172 0.999787i \(-0.506563\pi\)
−0.0206172 + 0.999787i \(0.506563\pi\)
\(458\) −1.17055e6 −0.260751
\(459\) −1.30762e6 −0.289702
\(460\) −1.49033e6 −0.328388
\(461\) −4.89953e6 −1.07375 −0.536873 0.843663i \(-0.680395\pi\)
−0.536873 + 0.843663i \(0.680395\pi\)
\(462\) −425358. −0.0927150
\(463\) −2.72830e6 −0.591479 −0.295740 0.955269i \(-0.595566\pi\)
−0.295740 + 0.955269i \(0.595566\pi\)
\(464\) −2.96420e6 −0.639165
\(465\) −2.17912e7 −4.67357
\(466\) −2.36843e6 −0.505238
\(467\) 3.18682e6 0.676185 0.338093 0.941113i \(-0.390218\pi\)
0.338093 + 0.941113i \(0.390218\pi\)
\(468\) −8.92572e6 −1.88377
\(469\) −430641. −0.0904031
\(470\) −1.41828e6 −0.296154
\(471\) −4.35859e6 −0.905302
\(472\) 1.30299e6 0.269207
\(473\) 729873. 0.150001
\(474\) 131175. 0.0268167
\(475\) −9.14319e6 −1.85936
\(476\) −69725.1 −0.0141050
\(477\) 1.16444e7 2.34326
\(478\) −4.07130e6 −0.815010
\(479\) 4.93825e6 0.983410 0.491705 0.870762i \(-0.336374\pi\)
0.491705 + 0.870762i \(0.336374\pi\)
\(480\) −1.61202e7 −3.19351
\(481\) 1.08393e7 2.13618
\(482\) −1.73095e6 −0.339365
\(483\) −246898. −0.0481560
\(484\) 135055. 0.0262058
\(485\) 3.79884e6 0.733325
\(486\) −1.15457e6 −0.221733
\(487\) 534531. 0.102129 0.0510646 0.998695i \(-0.483739\pi\)
0.0510646 + 0.998695i \(0.483739\pi\)
\(488\) 2.62868e6 0.499676
\(489\) −1.11991e7 −2.11792
\(490\) −4.17281e6 −0.785124
\(491\) 4.65073e6 0.870597 0.435299 0.900286i \(-0.356643\pi\)
0.435299 + 0.900286i \(0.356643\pi\)
\(492\) 5.31028e6 0.989019
\(493\) −1.08872e6 −0.201744
\(494\) −2.12470e6 −0.391724
\(495\) 2.07329e7 3.80318
\(496\) 3.65115e6 0.666386
\(497\) −201923. −0.0366687
\(498\) 3.57747e6 0.646403
\(499\) −215791. −0.0387955 −0.0193977 0.999812i \(-0.506175\pi\)
−0.0193977 + 0.999812i \(0.506175\pi\)
\(500\) −1.04955e7 −1.87749
\(501\) −4.60250e6 −0.819219
\(502\) 1.63629e6 0.289803
\(503\) −6.45815e6 −1.13812 −0.569060 0.822296i \(-0.692693\pi\)
−0.569060 + 0.822296i \(0.692693\pi\)
\(504\) −1.17047e6 −0.205251
\(505\) −3.60026e6 −0.628211
\(506\) 559508. 0.0971471
\(507\) −2.02858e6 −0.350487
\(508\) 258361. 0.0444189
\(509\) 3.88028e6 0.663847 0.331924 0.943306i \(-0.392302\pi\)
0.331924 + 0.943306i \(0.392302\pi\)
\(510\) −1.19619e6 −0.203645
\(511\) −996142. −0.168760
\(512\) 4.85626e6 0.818705
\(513\) 9.73553e6 1.63330
\(514\) −1.49188e6 −0.249072
\(515\) 1.80283e7 2.99528
\(516\) 1.31707e6 0.217764
\(517\) −2.22186e6 −0.365587
\(518\) 634657. 0.103924
\(519\) 7.94030e6 1.29395
\(520\) −9.71568e6 −1.57567
\(521\) 2.45313e6 0.395937 0.197969 0.980208i \(-0.436566\pi\)
0.197969 + 0.980208i \(0.436566\pi\)
\(522\) −8.16038e6 −1.31079
\(523\) −1.91833e6 −0.306668 −0.153334 0.988174i \(-0.549001\pi\)
−0.153334 + 0.988174i \(0.549001\pi\)
\(524\) −8.67436e6 −1.38010
\(525\) −3.09265e6 −0.489703
\(526\) −4.05087e6 −0.638386
\(527\) 1.34104e6 0.210336
\(528\) −5.10204e6 −0.796451
\(529\) −6.11158e6 −0.949542
\(530\) 5.65936e6 0.875139
\(531\) −4.69790e6 −0.723049
\(532\) 519118. 0.0795220
\(533\) 4.97200e6 0.758076
\(534\) −4.54247e6 −0.689348
\(535\) 4.41600e6 0.667029
\(536\) 3.94407e6 0.592970
\(537\) −1.44202e7 −2.15792
\(538\) 4.95959e6 0.738738
\(539\) −6.53708e6 −0.969196
\(540\) 1.98772e7 2.93340
\(541\) −6.89363e6 −1.01264 −0.506320 0.862346i \(-0.668995\pi\)
−0.506320 + 0.862346i \(0.668995\pi\)
\(542\) −4.70597e6 −0.688099
\(543\) −2.24256e6 −0.326396
\(544\) 992042. 0.143725
\(545\) −4.97266e6 −0.717130
\(546\) −718672. −0.103169
\(547\) 4.78180e6 0.683319 0.341659 0.939824i \(-0.389011\pi\)
0.341659 + 0.939824i \(0.389011\pi\)
\(548\) 31424.9 0.00447016
\(549\) −9.47764e6 −1.34205
\(550\) 7.00840e6 0.987898
\(551\) 8.10578e6 1.13741
\(552\) 2.26125e6 0.315864
\(553\) −30007.7 −0.00417273
\(554\) −5.25986e6 −0.728115
\(555\) −4.54338e7 −6.26104
\(556\) 4.06901e6 0.558215
\(557\) −1.16813e7 −1.59534 −0.797672 0.603091i \(-0.793935\pi\)
−0.797672 + 0.603091i \(0.793935\pi\)
\(558\) 1.00515e7 1.36662
\(559\) 1.23317e6 0.166914
\(560\) 745026. 0.100392
\(561\) −1.87393e6 −0.251389
\(562\) −201336. −0.0268893
\(563\) 9.15272e6 1.21697 0.608484 0.793566i \(-0.291778\pi\)
0.608484 + 0.793566i \(0.291778\pi\)
\(564\) −4.00939e6 −0.530739
\(565\) −3.49817e6 −0.461021
\(566\) 2.51599e6 0.330117
\(567\) 1.31502e6 0.171780
\(568\) 1.84933e6 0.240516
\(569\) −4.46016e6 −0.577523 −0.288762 0.957401i \(-0.593243\pi\)
−0.288762 + 0.957401i \(0.593243\pi\)
\(570\) 8.90586e6 1.14812
\(571\) 3.82289e6 0.490684 0.245342 0.969437i \(-0.421100\pi\)
0.245342 + 0.969437i \(0.421100\pi\)
\(572\) −6.79593e6 −0.868478
\(573\) 1.67994e7 2.13751
\(574\) 291118. 0.0368799
\(575\) 4.06801e6 0.513112
\(576\) −335134. −0.0420883
\(577\) 1.15777e7 1.44772 0.723858 0.689949i \(-0.242367\pi\)
0.723858 + 0.689949i \(0.242367\pi\)
\(578\) −3.45786e6 −0.430514
\(579\) −5.10383e6 −0.632702
\(580\) 1.65497e7 2.04278
\(581\) −818387. −0.100582
\(582\) −2.57358e6 −0.314941
\(583\) 8.86588e6 1.08032
\(584\) 9.12327e6 1.10693
\(585\) 3.50296e7 4.23200
\(586\) 1.95592e6 0.235291
\(587\) 2.16236e6 0.259020 0.129510 0.991578i \(-0.458660\pi\)
0.129510 + 0.991578i \(0.458660\pi\)
\(588\) −1.17963e7 −1.40703
\(589\) −9.98429e6 −1.18585
\(590\) −2.28325e6 −0.270038
\(591\) −5.54505e6 −0.653036
\(592\) 7.61251e6 0.892738
\(593\) 1.42792e7 1.66751 0.833753 0.552137i \(-0.186188\pi\)
0.833753 + 0.552137i \(0.186188\pi\)
\(594\) −7.46244e6 −0.867790
\(595\) 273641. 0.0316876
\(596\) 1.29493e7 1.49324
\(597\) 6.44751e6 0.740382
\(598\) 945326. 0.108101
\(599\) 1.13983e7 1.29799 0.648997 0.760791i \(-0.275189\pi\)
0.648997 + 0.760791i \(0.275189\pi\)
\(600\) 2.83244e7 3.21205
\(601\) −665939. −0.0752053 −0.0376026 0.999293i \(-0.511972\pi\)
−0.0376026 + 0.999293i \(0.511972\pi\)
\(602\) 72204.0 0.00812026
\(603\) −1.42202e7 −1.59263
\(604\) 1.11223e7 1.24051
\(605\) −530033. −0.0588728
\(606\) 2.43905e6 0.269798
\(607\) −1.35927e7 −1.49738 −0.748692 0.662918i \(-0.769318\pi\)
−0.748692 + 0.662918i \(0.769318\pi\)
\(608\) −7.38596e6 −0.810304
\(609\) 2.74175e6 0.299561
\(610\) −4.60628e6 −0.501217
\(611\) −3.75398e6 −0.406808
\(612\) −2.30240e6 −0.248486
\(613\) −1.08635e7 −1.16767 −0.583833 0.811874i \(-0.698448\pi\)
−0.583833 + 0.811874i \(0.698448\pi\)
\(614\) 264405. 0.0283041
\(615\) −2.08406e7 −2.22189
\(616\) −891182. −0.0946269
\(617\) −1.62403e6 −0.171744 −0.0858720 0.996306i \(-0.527368\pi\)
−0.0858720 + 0.996306i \(0.527368\pi\)
\(618\) −1.22135e7 −1.28638
\(619\) 3.82440e6 0.401177 0.200589 0.979676i \(-0.435715\pi\)
0.200589 + 0.979676i \(0.435715\pi\)
\(620\) −2.03851e7 −2.12978
\(621\) −4.33156e6 −0.450729
\(622\) −205487. −0.0212965
\(623\) 1.03914e6 0.107264
\(624\) −8.62024e6 −0.886253
\(625\) 1.88830e7 1.93362
\(626\) 3.44307e6 0.351164
\(627\) 1.39518e7 1.41730
\(628\) −4.07735e6 −0.412552
\(629\) 2.79601e6 0.281781
\(630\) 2.05104e6 0.205884
\(631\) −4.03571e6 −0.403503 −0.201751 0.979437i \(-0.564663\pi\)
−0.201751 + 0.979437i \(0.564663\pi\)
\(632\) 274829. 0.0273697
\(633\) −5.53303e6 −0.548850
\(634\) 2.40001e6 0.237132
\(635\) −1.01396e6 −0.0997894
\(636\) 1.59987e7 1.56834
\(637\) −1.10448e7 −1.07848
\(638\) −6.21321e6 −0.604316
\(639\) −6.66773e6 −0.645990
\(640\) −1.88568e7 −1.81978
\(641\) 1.03975e7 0.999505 0.499753 0.866168i \(-0.333424\pi\)
0.499753 + 0.866168i \(0.333424\pi\)
\(642\) −2.99168e6 −0.286469
\(643\) −703463. −0.0670986 −0.0335493 0.999437i \(-0.510681\pi\)
−0.0335493 + 0.999437i \(0.510681\pi\)
\(644\) −230967. −0.0219450
\(645\) −5.16894e6 −0.489218
\(646\) −548069. −0.0516718
\(647\) −1.39550e7 −1.31060 −0.655300 0.755369i \(-0.727458\pi\)
−0.655300 + 0.755369i \(0.727458\pi\)
\(648\) −1.20437e7 −1.12674
\(649\) −3.57692e6 −0.333348
\(650\) 1.18412e7 1.09929
\(651\) −3.37715e6 −0.312319
\(652\) −1.04765e7 −0.965151
\(653\) 4.10881e6 0.377079 0.188540 0.982066i \(-0.439625\pi\)
0.188540 + 0.982066i \(0.439625\pi\)
\(654\) 3.36880e6 0.307986
\(655\) 3.40431e7 3.10046
\(656\) 3.49187e6 0.316810
\(657\) −3.28937e7 −2.97303
\(658\) −219802. −0.0197909
\(659\) 4.28147e6 0.384042 0.192021 0.981391i \(-0.438496\pi\)
0.192021 + 0.981391i \(0.438496\pi\)
\(660\) 2.84857e7 2.54547
\(661\) −6.69686e6 −0.596166 −0.298083 0.954540i \(-0.596347\pi\)
−0.298083 + 0.954540i \(0.596347\pi\)
\(662\) −7.99274e6 −0.708844
\(663\) −3.16614e6 −0.279734
\(664\) 7.49528e6 0.659732
\(665\) −2.03732e6 −0.178650
\(666\) 2.09571e7 1.83082
\(667\) −3.60645e6 −0.313881
\(668\) −4.30553e6 −0.373323
\(669\) −9.57663e6 −0.827271
\(670\) −6.91126e6 −0.594799
\(671\) −7.21615e6 −0.618727
\(672\) −2.49827e6 −0.213411
\(673\) −1.29636e7 −1.10329 −0.551644 0.834080i \(-0.685999\pi\)
−0.551644 + 0.834080i \(0.685999\pi\)
\(674\) 2.07780e6 0.176179
\(675\) −5.42571e7 −4.58350
\(676\) −1.89768e6 −0.159719
\(677\) 1.00392e7 0.841833 0.420916 0.907099i \(-0.361709\pi\)
0.420916 + 0.907099i \(0.361709\pi\)
\(678\) 2.36989e6 0.197995
\(679\) 588735. 0.0490055
\(680\) −2.50617e6 −0.207844
\(681\) 5.85308e6 0.483633
\(682\) 7.65312e6 0.630053
\(683\) 1.37778e7 1.13013 0.565063 0.825048i \(-0.308852\pi\)
0.565063 + 0.825048i \(0.308852\pi\)
\(684\) 1.71419e7 1.40093
\(685\) −123329. −0.0100425
\(686\) −1.30301e6 −0.105715
\(687\) 1.29867e7 1.04980
\(688\) 866065. 0.0697556
\(689\) 1.49795e7 1.20212
\(690\) −3.96242e6 −0.316838
\(691\) −2.12787e7 −1.69532 −0.847658 0.530544i \(-0.821988\pi\)
−0.847658 + 0.530544i \(0.821988\pi\)
\(692\) 7.42795e6 0.589663
\(693\) 3.21313e6 0.254153
\(694\) 5.47044e6 0.431145
\(695\) −1.59691e7 −1.25406
\(696\) −2.51106e7 −1.96487
\(697\) 1.28253e6 0.0999969
\(698\) 7.23763e6 0.562287
\(699\) 2.62767e7 2.03412
\(700\) −2.89310e6 −0.223161
\(701\) 2.07790e7 1.59709 0.798547 0.601933i \(-0.205602\pi\)
0.798547 + 0.601933i \(0.205602\pi\)
\(702\) −1.26083e7 −0.965636
\(703\) −2.08169e7 −1.58864
\(704\) −255166. −0.0194040
\(705\) 1.57351e7 1.19233
\(706\) 6.56274e6 0.495534
\(707\) −557960. −0.0419812
\(708\) −6.45463e6 −0.483936
\(709\) 4.16247e6 0.310982 0.155491 0.987837i \(-0.450304\pi\)
0.155491 + 0.987837i \(0.450304\pi\)
\(710\) −3.24062e6 −0.241258
\(711\) −990888. −0.0735107
\(712\) −9.51708e6 −0.703564
\(713\) 4.44224e6 0.327249
\(714\) −185382. −0.0136089
\(715\) 2.66711e7 1.95108
\(716\) −1.34898e7 −0.983381
\(717\) 4.51692e7 3.28129
\(718\) −92516.4 −0.00669742
\(719\) −1.69124e7 −1.22007 −0.610033 0.792376i \(-0.708844\pi\)
−0.610033 + 0.792376i \(0.708844\pi\)
\(720\) 2.46016e7 1.76861
\(721\) 2.79399e6 0.200164
\(722\) −2.07807e6 −0.148360
\(723\) 1.92041e7 1.36631
\(724\) −2.09786e6 −0.148741
\(725\) −4.51743e7 −3.19188
\(726\) 359079. 0.0252841
\(727\) 1.27084e7 0.891776 0.445888 0.895089i \(-0.352888\pi\)
0.445888 + 0.895089i \(0.352888\pi\)
\(728\) −1.50571e6 −0.105296
\(729\) −7.54333e6 −0.525708
\(730\) −1.59869e7 −1.11034
\(731\) 318098. 0.0220174
\(732\) −1.30217e7 −0.898235
\(733\) 1.32015e6 0.0907535 0.0453767 0.998970i \(-0.485551\pi\)
0.0453767 + 0.998970i \(0.485551\pi\)
\(734\) 5.65040e6 0.387114
\(735\) 4.62954e7 3.16096
\(736\) 3.28618e6 0.223613
\(737\) −1.08271e7 −0.734249
\(738\) 9.61304e6 0.649711
\(739\) 2.42511e7 1.63351 0.816753 0.576988i \(-0.195772\pi\)
0.816753 + 0.576988i \(0.195772\pi\)
\(740\) −4.25022e7 −2.85320
\(741\) 2.35725e7 1.57711
\(742\) 877073. 0.0584825
\(743\) −1.31970e7 −0.877009 −0.438504 0.898729i \(-0.644492\pi\)
−0.438504 + 0.898729i \(0.644492\pi\)
\(744\) 3.09300e7 2.04855
\(745\) −5.08203e7 −3.35465
\(746\) 6.02750e6 0.396543
\(747\) −2.70240e7 −1.77194
\(748\) −1.75302e6 −0.114560
\(749\) 684381. 0.0445752
\(750\) −2.79050e7 −1.81146
\(751\) −1.06321e7 −0.687891 −0.343945 0.938990i \(-0.611763\pi\)
−0.343945 + 0.938990i \(0.611763\pi\)
\(752\) −2.63645e6 −0.170010
\(753\) −1.81539e7 −1.16676
\(754\) −1.04976e7 −0.672455
\(755\) −4.36502e7 −2.78688
\(756\) 3.08053e6 0.196029
\(757\) 2.44667e7 1.55180 0.775900 0.630856i \(-0.217296\pi\)
0.775900 + 0.630856i \(0.217296\pi\)
\(758\) −5.55094e6 −0.350908
\(759\) −6.20748e6 −0.391121
\(760\) 1.86590e7 1.17180
\(761\) 2.20641e7 1.38110 0.690550 0.723285i \(-0.257369\pi\)
0.690550 + 0.723285i \(0.257369\pi\)
\(762\) 686919. 0.0428566
\(763\) −770652. −0.0479233
\(764\) 1.57154e7 0.974076
\(765\) 9.03593e6 0.558238
\(766\) −2.82938e6 −0.174228
\(767\) −6.04345e6 −0.370934
\(768\) 1.22040e7 0.746621
\(769\) 2.58590e7 1.57687 0.788434 0.615119i \(-0.210892\pi\)
0.788434 + 0.615119i \(0.210892\pi\)
\(770\) 1.56163e6 0.0949188
\(771\) 1.65517e7 1.00278
\(772\) −4.77450e6 −0.288327
\(773\) −1.57927e7 −0.950620 −0.475310 0.879818i \(-0.657664\pi\)
−0.475310 + 0.879818i \(0.657664\pi\)
\(774\) 2.38426e6 0.143054
\(775\) 5.56435e7 3.32782
\(776\) −5.39199e6 −0.321436
\(777\) −7.04122e6 −0.418404
\(778\) 4.90064e6 0.290271
\(779\) −9.54872e6 −0.563770
\(780\) 4.81286e7 2.83248
\(781\) −5.07672e6 −0.297821
\(782\) 243848. 0.0142594
\(783\) 4.81010e7 2.80382
\(784\) −7.75687e6 −0.450709
\(785\) 1.60018e7 0.926821
\(786\) −2.30630e7 −1.33156
\(787\) −2.14880e7 −1.23669 −0.618344 0.785908i \(-0.712196\pi\)
−0.618344 + 0.785908i \(0.712196\pi\)
\(788\) −5.18726e6 −0.297593
\(789\) 4.49425e7 2.57019
\(790\) −481587. −0.0274541
\(791\) −542139. −0.0308084
\(792\) −2.94278e7 −1.66704
\(793\) −1.21922e7 −0.688491
\(794\) −1.27625e7 −0.718432
\(795\) −6.27879e7 −3.52337
\(796\) 6.03148e6 0.337397
\(797\) −1.54431e7 −0.861170 −0.430585 0.902550i \(-0.641693\pi\)
−0.430585 + 0.902550i \(0.641693\pi\)
\(798\) 1.38021e6 0.0767251
\(799\) −968344. −0.0536615
\(800\) 4.11627e7 2.27394
\(801\) 3.43136e7 1.88966
\(802\) −7.27088e6 −0.399164
\(803\) −2.50448e7 −1.37066
\(804\) −1.95377e7 −1.06594
\(805\) 906448. 0.0493007
\(806\) 1.29305e7 0.701094
\(807\) −5.50244e7 −2.97421
\(808\) 5.11014e6 0.275362
\(809\) −5.05884e6 −0.271756 −0.135878 0.990726i \(-0.543386\pi\)
−0.135878 + 0.990726i \(0.543386\pi\)
\(810\) 2.11044e7 1.13021
\(811\) 2.95840e7 1.57944 0.789722 0.613465i \(-0.210225\pi\)
0.789722 + 0.613465i \(0.210225\pi\)
\(812\) 2.56484e6 0.136512
\(813\) 5.22105e7 2.77033
\(814\) 1.59564e7 0.844063
\(815\) 4.11156e7 2.16826
\(816\) −2.22360e6 −0.116905
\(817\) −2.36830e6 −0.124132
\(818\) −5.23861e6 −0.273737
\(819\) 5.42881e6 0.282810
\(820\) −1.94958e7 −1.01253
\(821\) 4.56831e6 0.236536 0.118268 0.992982i \(-0.462266\pi\)
0.118268 + 0.992982i \(0.462266\pi\)
\(822\) 83551.3 0.00431294
\(823\) 3.99029e6 0.205355 0.102677 0.994715i \(-0.467259\pi\)
0.102677 + 0.994715i \(0.467259\pi\)
\(824\) −2.55890e7 −1.31291
\(825\) −7.77549e7 −3.97734
\(826\) −353853. −0.0180457
\(827\) 7.46365e6 0.379479 0.189739 0.981835i \(-0.439236\pi\)
0.189739 + 0.981835i \(0.439236\pi\)
\(828\) −7.62680e6 −0.386604
\(829\) −1.22180e7 −0.617467 −0.308734 0.951149i \(-0.599905\pi\)
−0.308734 + 0.951149i \(0.599905\pi\)
\(830\) −1.31341e7 −0.661768
\(831\) 5.83557e7 2.93144
\(832\) −431121. −0.0215919
\(833\) −2.84903e6 −0.142260
\(834\) 1.08185e7 0.538582
\(835\) 1.68973e7 0.838692
\(836\) 1.30516e7 0.645874
\(837\) −5.92484e7 −2.92323
\(838\) −1.95916e6 −0.0963740
\(839\) −1.83694e7 −0.900929 −0.450464 0.892794i \(-0.648742\pi\)
−0.450464 + 0.892794i \(0.648742\pi\)
\(840\) 6.31133e6 0.308619
\(841\) 1.95376e7 0.952536
\(842\) −6.81147e6 −0.331101
\(843\) 2.23373e6 0.108258
\(844\) −5.17601e6 −0.250115
\(845\) 7.44760e6 0.358818
\(846\) −7.25809e6 −0.348655
\(847\) −82143.3 −0.00393426
\(848\) 1.05202e7 0.502383
\(849\) −2.79138e7 −1.32907
\(850\) 3.05444e6 0.145006
\(851\) 9.26189e6 0.438405
\(852\) −9.16105e6 −0.432361
\(853\) −3.51699e7 −1.65500 −0.827501 0.561465i \(-0.810238\pi\)
−0.827501 + 0.561465i \(0.810238\pi\)
\(854\) −713871. −0.0334946
\(855\) −6.72744e7 −3.14727
\(856\) −6.26798e6 −0.292377
\(857\) −1.23215e7 −0.573076 −0.286538 0.958069i \(-0.592504\pi\)
−0.286538 + 0.958069i \(0.592504\pi\)
\(858\) −1.80687e7 −0.837933
\(859\) 2.32476e6 0.107497 0.0537484 0.998555i \(-0.482883\pi\)
0.0537484 + 0.998555i \(0.482883\pi\)
\(860\) −4.83541e6 −0.222940
\(861\) −3.22982e6 −0.148481
\(862\) 5.56244e6 0.254975
\(863\) 1.53407e7 0.701163 0.350582 0.936532i \(-0.385984\pi\)
0.350582 + 0.936532i \(0.385984\pi\)
\(864\) −4.38294e7 −1.99748
\(865\) −2.91515e7 −1.32471
\(866\) −1.82140e7 −0.825297
\(867\) 3.83633e7 1.73328
\(868\) −3.15924e6 −0.142326
\(869\) −754449. −0.0338907
\(870\) 4.40018e7 1.97093
\(871\) −1.82931e7 −0.817038
\(872\) 7.05809e6 0.314337
\(873\) 1.94407e7 0.863328
\(874\) −1.81550e6 −0.0803929
\(875\) 6.38358e6 0.281867
\(876\) −4.51939e7 −1.98985
\(877\) 1.21207e7 0.532142 0.266071 0.963953i \(-0.414274\pi\)
0.266071 + 0.963953i \(0.414274\pi\)
\(878\) 1.00861e7 0.441559
\(879\) −2.17000e7 −0.947299
\(880\) 1.87313e7 0.815383
\(881\) 208397. 0.00904589 0.00452294 0.999990i \(-0.498560\pi\)
0.00452294 + 0.999990i \(0.498560\pi\)
\(882\) −2.13545e7 −0.924310
\(883\) −3.57371e7 −1.54247 −0.771237 0.636548i \(-0.780362\pi\)
−0.771237 + 0.636548i \(0.780362\pi\)
\(884\) −2.96184e6 −0.127477
\(885\) 2.53316e7 1.08719
\(886\) 5.55604e6 0.237783
\(887\) 4.64203e7 1.98107 0.990533 0.137276i \(-0.0438346\pi\)
0.990533 + 0.137276i \(0.0438346\pi\)
\(888\) 6.44878e7 2.74439
\(889\) −157140. −0.00666858
\(890\) 1.66769e7 0.705734
\(891\) 3.30619e7 1.39519
\(892\) −8.95871e6 −0.376993
\(893\) 7.20952e6 0.302537
\(894\) 3.44290e7 1.44072
\(895\) 5.29415e7 2.20922
\(896\) −2.92239e6 −0.121610
\(897\) −1.04880e7 −0.435221
\(898\) −9.70472e6 −0.401598
\(899\) −4.93300e7 −2.03569
\(900\) −9.55333e7 −3.93141
\(901\) 3.86398e6 0.158571
\(902\) 7.31924e6 0.299537
\(903\) −801070. −0.0326927
\(904\) 4.96523e6 0.202078
\(905\) 8.23320e6 0.334155
\(906\) 2.95715e7 1.19688
\(907\) −2.81640e7 −1.13678 −0.568390 0.822759i \(-0.692434\pi\)
−0.568390 + 0.822759i \(0.692434\pi\)
\(908\) 5.47541e6 0.220395
\(909\) −1.84245e7 −0.739580
\(910\) 2.63849e6 0.105621
\(911\) 8.25253e6 0.329451 0.164726 0.986339i \(-0.447326\pi\)
0.164726 + 0.986339i \(0.447326\pi\)
\(912\) 1.65552e7 0.659093
\(913\) −2.05758e7 −0.816919
\(914\) −457889. −0.0181299
\(915\) 5.11045e7 2.01793
\(916\) 1.21487e7 0.478402
\(917\) 5.27593e6 0.207193
\(918\) −3.25232e6 −0.127376
\(919\) −1.14244e6 −0.0446217 −0.0223108 0.999751i \(-0.507102\pi\)
−0.0223108 + 0.999751i \(0.507102\pi\)
\(920\) −8.30179e6 −0.323372
\(921\) −2.93346e6 −0.113954
\(922\) −1.21861e7 −0.472104
\(923\) −8.57746e6 −0.331401
\(924\) 4.41465e6 0.170105
\(925\) 1.16014e8 4.45818
\(926\) −6.78583e6 −0.260061
\(927\) 9.22605e7 3.52628
\(928\) −3.64923e7 −1.39101
\(929\) −3.37164e7 −1.28175 −0.640873 0.767647i \(-0.721428\pi\)
−0.640873 + 0.767647i \(0.721428\pi\)
\(930\) −5.41991e7 −2.05487
\(931\) 2.12116e7 0.802046
\(932\) 2.45812e7 0.926963
\(933\) 2.27979e6 0.0857413
\(934\) 7.92627e6 0.297305
\(935\) 6.87984e6 0.257365
\(936\) −4.97203e7 −1.85500
\(937\) −4.06327e7 −1.51191 −0.755955 0.654623i \(-0.772827\pi\)
−0.755955 + 0.654623i \(0.772827\pi\)
\(938\) −1.07109e6 −0.0397484
\(939\) −3.81992e7 −1.41381
\(940\) 1.47198e7 0.543355
\(941\) 4.51496e7 1.66219 0.831093 0.556134i \(-0.187716\pi\)
0.831093 + 0.556134i \(0.187716\pi\)
\(942\) −1.08407e7 −0.398042
\(943\) 4.24844e6 0.155579
\(944\) −4.24436e6 −0.155018
\(945\) −1.20897e7 −0.440390
\(946\) 1.81534e6 0.0659524
\(947\) −1.85178e7 −0.670986 −0.335493 0.942043i \(-0.608903\pi\)
−0.335493 + 0.942043i \(0.608903\pi\)
\(948\) −1.36142e6 −0.0492007
\(949\) −4.23149e7 −1.52520
\(950\) −2.27410e7 −0.817523
\(951\) −2.66270e7 −0.954709
\(952\) −388400. −0.0138895
\(953\) −3.17478e7 −1.13235 −0.566176 0.824284i \(-0.691578\pi\)
−0.566176 + 0.824284i \(0.691578\pi\)
\(954\) 2.89619e7 1.03028
\(955\) −6.16763e7 −2.18832
\(956\) 4.22546e7 1.49530
\(957\) 6.89326e7 2.43302
\(958\) 1.22824e7 0.432385
\(959\) −19113.3 −0.000671103 0
\(960\) 1.80708e6 0.0632848
\(961\) 3.21331e7 1.12239
\(962\) 2.69595e7 0.939234
\(963\) 2.25990e7 0.785279
\(964\) 1.79650e7 0.622636
\(965\) 1.87379e7 0.647742
\(966\) −614086. −0.0211732
\(967\) 4.45707e7 1.53279 0.766396 0.642368i \(-0.222048\pi\)
0.766396 + 0.642368i \(0.222048\pi\)
\(968\) 752318. 0.0258055
\(969\) 6.08057e6 0.208034
\(970\) 9.44847e6 0.322428
\(971\) 2.22017e7 0.755679 0.377840 0.925871i \(-0.376667\pi\)
0.377840 + 0.925871i \(0.376667\pi\)
\(972\) 1.19829e7 0.406815
\(973\) −2.47485e6 −0.0838045
\(974\) 1.32949e6 0.0449041
\(975\) −1.31372e8 −4.42580
\(976\) −8.56265e6 −0.287729
\(977\) −3.20865e6 −0.107544 −0.0537719 0.998553i \(-0.517124\pi\)
−0.0537719 + 0.998553i \(0.517124\pi\)
\(978\) −2.78543e7 −0.931206
\(979\) 2.61259e7 0.871193
\(980\) 4.33082e7 1.44047
\(981\) −2.54478e7 −0.844262
\(982\) 1.15673e7 0.382783
\(983\) −4.20453e7 −1.38782 −0.693910 0.720061i \(-0.744114\pi\)
−0.693910 + 0.720061i \(0.744114\pi\)
\(984\) 2.95806e7 0.973913
\(985\) 2.03578e7 0.668558
\(986\) −2.70788e6 −0.0887027
\(987\) 2.43860e6 0.0796795
\(988\) 2.20515e7 0.718698
\(989\) 1.05371e6 0.0342556
\(990\) 5.15669e7 1.67218
\(991\) 1.08440e7 0.350755 0.175377 0.984501i \(-0.443885\pi\)
0.175377 + 0.984501i \(0.443885\pi\)
\(992\) 4.49494e7 1.45025
\(993\) 8.86757e7 2.85385
\(994\) −502224. −0.0161225
\(995\) −2.36710e7 −0.757981
\(996\) −3.71294e7 −1.18596
\(997\) 3.29287e7 1.04915 0.524574 0.851365i \(-0.324225\pi\)
0.524574 + 0.851365i \(0.324225\pi\)
\(998\) −536715. −0.0170576
\(999\) −1.23530e8 −3.91616
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 43.6.a.b.1.6 10
3.2 odd 2 387.6.a.e.1.5 10
4.3 odd 2 688.6.a.h.1.10 10
5.4 even 2 1075.6.a.b.1.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.b.1.6 10 1.1 even 1 trivial
387.6.a.e.1.5 10 3.2 odd 2
688.6.a.h.1.10 10 4.3 odd 2
1075.6.a.b.1.5 10 5.4 even 2