Properties

Label 177.6.a.c.1.4
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3879361069\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + 15565376 x^{4} + 6775664 x^{3} - 75006848 x^{2} + 21512960 x + 49172480\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(4.10256\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.10256 q^{2} +9.00000 q^{3} -27.5792 q^{4} -81.1594 q^{5} -18.9230 q^{6} -97.6213 q^{7} +125.269 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-2.10256 q^{2} +9.00000 q^{3} -27.5792 q^{4} -81.1594 q^{5} -18.9230 q^{6} -97.6213 q^{7} +125.269 q^{8} +81.0000 q^{9} +170.642 q^{10} -297.163 q^{11} -248.213 q^{12} -814.611 q^{13} +205.254 q^{14} -730.435 q^{15} +619.151 q^{16} -1275.83 q^{17} -170.307 q^{18} +801.964 q^{19} +2238.32 q^{20} -878.592 q^{21} +624.802 q^{22} +255.443 q^{23} +1127.42 q^{24} +3461.85 q^{25} +1712.77 q^{26} +729.000 q^{27} +2692.32 q^{28} +7410.46 q^{29} +1535.78 q^{30} -2821.92 q^{31} -5310.40 q^{32} -2674.46 q^{33} +2682.51 q^{34} +7922.89 q^{35} -2233.92 q^{36} +4781.71 q^{37} -1686.18 q^{38} -7331.50 q^{39} -10166.7 q^{40} -8224.42 q^{41} +1847.29 q^{42} -1932.60 q^{43} +8195.53 q^{44} -6573.91 q^{45} -537.083 q^{46} +23212.7 q^{47} +5572.36 q^{48} -7277.08 q^{49} -7278.75 q^{50} -11482.5 q^{51} +22466.3 q^{52} +15252.2 q^{53} -1532.76 q^{54} +24117.6 q^{55} -12228.9 q^{56} +7217.68 q^{57} -15580.9 q^{58} -3481.00 q^{59} +20144.8 q^{60} +9488.35 q^{61} +5933.26 q^{62} -7907.33 q^{63} -8647.40 q^{64} +66113.3 q^{65} +5623.22 q^{66} +28014.3 q^{67} +35186.5 q^{68} +2298.98 q^{69} -16658.3 q^{70} -26155.1 q^{71} +10146.8 q^{72} +7782.60 q^{73} -10053.8 q^{74} +31156.7 q^{75} -22117.6 q^{76} +29009.4 q^{77} +15414.9 q^{78} -51295.8 q^{79} -50249.9 q^{80} +6561.00 q^{81} +17292.3 q^{82} +17726.9 q^{83} +24230.9 q^{84} +103546. q^{85} +4063.40 q^{86} +66694.1 q^{87} -37225.2 q^{88} +104423. q^{89} +13822.0 q^{90} +79523.3 q^{91} -7044.92 q^{92} -25397.3 q^{93} -48806.1 q^{94} -65086.9 q^{95} -47793.6 q^{96} -144515. q^{97} +15300.5 q^{98} -24070.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + O(q^{10}) \) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + 601q^{10} + 1480q^{11} + 1782q^{12} + 472q^{13} + 1065q^{14} + 1422q^{15} + 6370q^{16} + 1565q^{17} + 1782q^{18} + 3939q^{19} + 8033q^{20} + 3717q^{21} - 1738q^{22} + 7245q^{23} + 6507q^{24} + 9690q^{25} + 3764q^{26} + 8748q^{27} + 12154q^{28} + 10003q^{29} + 5409q^{30} + 7295q^{31} + 11628q^{32} + 13320q^{33} - 16344q^{34} + 11015q^{35} + 16038q^{36} + 6741q^{37} + 3035q^{38} + 4248q^{39} + 5572q^{40} + 34025q^{41} + 9585q^{42} - 6336q^{43} + 41168q^{44} + 12798q^{45} + 2345q^{46} + 66167q^{47} + 57330q^{48} + 28319q^{49} + 31173q^{50} + 14085q^{51} + 16440q^{52} + 62290q^{53} + 16038q^{54} + 55764q^{55} + 107306q^{56} + 35451q^{57} + 37952q^{58} - 41772q^{59} + 72297q^{60} + 68469q^{61} + 99190q^{62} + 33453q^{63} + 68525q^{64} + 80156q^{65} - 15642q^{66} + 113310q^{67} + 33887q^{68} + 65205q^{69} + 32034q^{70} + 84520q^{71} + 58563q^{72} + 135895q^{73} - 31962q^{74} + 87210q^{75} - 61848q^{76} - 3799q^{77} + 33876q^{78} + 14122q^{79} + 77609q^{80} + 78732q^{81} - 1501q^{82} + 114463q^{83} + 109386q^{84} - 101097q^{85} - 203536q^{86} + 90027q^{87} - 244967q^{88} + 189109q^{89} + 48681q^{90} - 168249q^{91} - 71946q^{92} + 65655q^{93} - 472284q^{94} + 21923q^{95} + 104652q^{96} - 76192q^{97} - 17544q^{98} + 119880q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.10256 −0.371683 −0.185842 0.982580i \(-0.559501\pi\)
−0.185842 + 0.982580i \(0.559501\pi\)
\(3\) 9.00000 0.577350
\(4\) −27.5792 −0.861852
\(5\) −81.1594 −1.45182 −0.725912 0.687788i \(-0.758582\pi\)
−0.725912 + 0.687788i \(0.758582\pi\)
\(6\) −18.9230 −0.214591
\(7\) −97.6213 −0.753008 −0.376504 0.926415i \(-0.622874\pi\)
−0.376504 + 0.926415i \(0.622874\pi\)
\(8\) 125.269 0.692019
\(9\) 81.0000 0.333333
\(10\) 170.642 0.539619
\(11\) −297.163 −0.740479 −0.370239 0.928936i \(-0.620724\pi\)
−0.370239 + 0.928936i \(0.620724\pi\)
\(12\) −248.213 −0.497590
\(13\) −814.611 −1.33688 −0.668439 0.743767i \(-0.733037\pi\)
−0.668439 + 0.743767i \(0.733037\pi\)
\(14\) 205.254 0.279880
\(15\) −730.435 −0.838211
\(16\) 619.151 0.604640
\(17\) −1275.83 −1.07071 −0.535355 0.844627i \(-0.679822\pi\)
−0.535355 + 0.844627i \(0.679822\pi\)
\(18\) −170.307 −0.123894
\(19\) 801.964 0.509649 0.254824 0.966987i \(-0.417982\pi\)
0.254824 + 0.966987i \(0.417982\pi\)
\(20\) 2238.32 1.25126
\(21\) −878.592 −0.434749
\(22\) 624.802 0.275224
\(23\) 255.443 0.100687 0.0503436 0.998732i \(-0.483968\pi\)
0.0503436 + 0.998732i \(0.483968\pi\)
\(24\) 1127.42 0.399537
\(25\) 3461.85 1.10779
\(26\) 1712.77 0.496895
\(27\) 729.000 0.192450
\(28\) 2692.32 0.648981
\(29\) 7410.46 1.63625 0.818125 0.575040i \(-0.195013\pi\)
0.818125 + 0.575040i \(0.195013\pi\)
\(30\) 1535.78 0.311549
\(31\) −2821.92 −0.527401 −0.263700 0.964605i \(-0.584943\pi\)
−0.263700 + 0.964605i \(0.584943\pi\)
\(32\) −5310.40 −0.916754
\(33\) −2674.46 −0.427516
\(34\) 2682.51 0.397965
\(35\) 7922.89 1.09323
\(36\) −2233.92 −0.287284
\(37\) 4781.71 0.574221 0.287111 0.957897i \(-0.407305\pi\)
0.287111 + 0.957897i \(0.407305\pi\)
\(38\) −1686.18 −0.189428
\(39\) −7331.50 −0.771847
\(40\) −10166.7 −1.00469
\(41\) −8224.42 −0.764092 −0.382046 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(42\) 1847.29 0.161589
\(43\) −1932.60 −0.159393 −0.0796967 0.996819i \(-0.525395\pi\)
−0.0796967 + 0.996819i \(0.525395\pi\)
\(44\) 8195.53 0.638183
\(45\) −6573.91 −0.483941
\(46\) −537.083 −0.0374237
\(47\) 23212.7 1.53278 0.766392 0.642373i \(-0.222050\pi\)
0.766392 + 0.642373i \(0.222050\pi\)
\(48\) 5572.36 0.349089
\(49\) −7277.08 −0.432979
\(50\) −7278.75 −0.411748
\(51\) −11482.5 −0.618175
\(52\) 22466.3 1.15219
\(53\) 15252.2 0.745836 0.372918 0.927864i \(-0.378357\pi\)
0.372918 + 0.927864i \(0.378357\pi\)
\(54\) −1532.76 −0.0715305
\(55\) 24117.6 1.07505
\(56\) −12228.9 −0.521096
\(57\) 7217.68 0.294246
\(58\) −15580.9 −0.608167
\(59\) −3481.00 −0.130189
\(60\) 20144.8 0.722413
\(61\) 9488.35 0.326487 0.163244 0.986586i \(-0.447804\pi\)
0.163244 + 0.986586i \(0.447804\pi\)
\(62\) 5933.26 0.196026
\(63\) −7907.33 −0.251003
\(64\) −8647.40 −0.263898
\(65\) 66113.3 1.94091
\(66\) 5623.22 0.158900
\(67\) 28014.3 0.762417 0.381209 0.924489i \(-0.375508\pi\)
0.381209 + 0.924489i \(0.375508\pi\)
\(68\) 35186.5 0.922793
\(69\) 2298.98 0.0581317
\(70\) −16658.3 −0.406337
\(71\) −26155.1 −0.615758 −0.307879 0.951425i \(-0.599619\pi\)
−0.307879 + 0.951425i \(0.599619\pi\)
\(72\) 10146.8 0.230673
\(73\) 7782.60 0.170930 0.0854649 0.996341i \(-0.472762\pi\)
0.0854649 + 0.996341i \(0.472762\pi\)
\(74\) −10053.8 −0.213428
\(75\) 31156.7 0.639585
\(76\) −22117.6 −0.439241
\(77\) 29009.4 0.557586
\(78\) 15414.9 0.286882
\(79\) −51295.8 −0.924728 −0.462364 0.886690i \(-0.652999\pi\)
−0.462364 + 0.886690i \(0.652999\pi\)
\(80\) −50249.9 −0.877830
\(81\) 6561.00 0.111111
\(82\) 17292.3 0.284000
\(83\) 17726.9 0.282447 0.141224 0.989978i \(-0.454896\pi\)
0.141224 + 0.989978i \(0.454896\pi\)
\(84\) 24230.9 0.374689
\(85\) 103546. 1.55448
\(86\) 4063.40 0.0592438
\(87\) 66694.1 0.944690
\(88\) −37225.2 −0.512426
\(89\) 104423. 1.39740 0.698701 0.715414i \(-0.253762\pi\)
0.698701 + 0.715414i \(0.253762\pi\)
\(90\) 13822.0 0.179873
\(91\) 79523.3 1.00668
\(92\) −7044.92 −0.0867774
\(93\) −25397.3 −0.304495
\(94\) −48806.1 −0.569710
\(95\) −65086.9 −0.739920
\(96\) −47793.6 −0.529288
\(97\) −144515. −1.55949 −0.779746 0.626096i \(-0.784652\pi\)
−0.779746 + 0.626096i \(0.784652\pi\)
\(98\) 15300.5 0.160931
\(99\) −24070.2 −0.246826
\(100\) −95475.3 −0.954753
\(101\) 5054.57 0.0493038 0.0246519 0.999696i \(-0.492152\pi\)
0.0246519 + 0.999696i \(0.492152\pi\)
\(102\) 24142.6 0.229765
\(103\) −12446.5 −0.115599 −0.0577997 0.998328i \(-0.518408\pi\)
−0.0577997 + 0.998328i \(0.518408\pi\)
\(104\) −102045. −0.925145
\(105\) 71306.0 0.631179
\(106\) −32068.7 −0.277215
\(107\) 62239.2 0.525538 0.262769 0.964859i \(-0.415364\pi\)
0.262769 + 0.964859i \(0.415364\pi\)
\(108\) −20105.3 −0.165863
\(109\) −109466. −0.882496 −0.441248 0.897385i \(-0.645464\pi\)
−0.441248 + 0.897385i \(0.645464\pi\)
\(110\) −50708.6 −0.399576
\(111\) 43035.4 0.331527
\(112\) −60442.3 −0.455298
\(113\) −132747. −0.977977 −0.488988 0.872290i \(-0.662634\pi\)
−0.488988 + 0.872290i \(0.662634\pi\)
\(114\) −15175.6 −0.109366
\(115\) −20731.6 −0.146180
\(116\) −204375. −1.41021
\(117\) −65983.5 −0.445626
\(118\) 7319.00 0.0483890
\(119\) 124549. 0.806253
\(120\) −91500.7 −0.580058
\(121\) −72745.3 −0.451691
\(122\) −19949.8 −0.121350
\(123\) −74019.8 −0.441149
\(124\) 77826.5 0.454541
\(125\) −27338.9 −0.156497
\(126\) 16625.6 0.0932935
\(127\) −223789. −1.23120 −0.615601 0.788058i \(-0.711087\pi\)
−0.615601 + 0.788058i \(0.711087\pi\)
\(128\) 188115. 1.01484
\(129\) −17393.4 −0.0920258
\(130\) −139007. −0.721404
\(131\) 247877. 1.26200 0.630998 0.775785i \(-0.282646\pi\)
0.630998 + 0.775785i \(0.282646\pi\)
\(132\) 73759.7 0.368455
\(133\) −78288.8 −0.383769
\(134\) −58901.7 −0.283378
\(135\) −59165.2 −0.279404
\(136\) −159822. −0.740952
\(137\) 202795. 0.923114 0.461557 0.887111i \(-0.347291\pi\)
0.461557 + 0.887111i \(0.347291\pi\)
\(138\) −4833.75 −0.0216066
\(139\) −192882. −0.846750 −0.423375 0.905955i \(-0.639155\pi\)
−0.423375 + 0.905955i \(0.639155\pi\)
\(140\) −218507. −0.942206
\(141\) 208914. 0.884954
\(142\) 54992.6 0.228867
\(143\) 242072. 0.989930
\(144\) 50151.2 0.201547
\(145\) −601428. −2.37555
\(146\) −16363.4 −0.0635317
\(147\) −65493.7 −0.249981
\(148\) −131876. −0.494893
\(149\) 211494. 0.780429 0.390214 0.920724i \(-0.372401\pi\)
0.390214 + 0.920724i \(0.372401\pi\)
\(150\) −65508.7 −0.237723
\(151\) −122664. −0.437797 −0.218899 0.975748i \(-0.570246\pi\)
−0.218899 + 0.975748i \(0.570246\pi\)
\(152\) 100461. 0.352687
\(153\) −103343. −0.356903
\(154\) −60994.0 −0.207246
\(155\) 229026. 0.765693
\(156\) 202197. 0.665217
\(157\) 120555. 0.390333 0.195166 0.980770i \(-0.437475\pi\)
0.195166 + 0.980770i \(0.437475\pi\)
\(158\) 107852. 0.343706
\(159\) 137270. 0.430609
\(160\) 430989. 1.33096
\(161\) −24936.7 −0.0758182
\(162\) −13794.9 −0.0412981
\(163\) −558933. −1.64775 −0.823875 0.566772i \(-0.808192\pi\)
−0.823875 + 0.566772i \(0.808192\pi\)
\(164\) 226823. 0.658534
\(165\) 217058. 0.620678
\(166\) −37271.8 −0.104981
\(167\) 225611. 0.625992 0.312996 0.949754i \(-0.398667\pi\)
0.312996 + 0.949754i \(0.398667\pi\)
\(168\) −110060. −0.300855
\(169\) 292297. 0.787242
\(170\) −217711. −0.577775
\(171\) 64959.1 0.169883
\(172\) 53299.6 0.137373
\(173\) 151994. 0.386110 0.193055 0.981188i \(-0.438160\pi\)
0.193055 + 0.981188i \(0.438160\pi\)
\(174\) −140228. −0.351125
\(175\) −337951. −0.834177
\(176\) −183989. −0.447723
\(177\) −31329.0 −0.0751646
\(178\) −219556. −0.519391
\(179\) −801580. −1.86988 −0.934942 0.354801i \(-0.884548\pi\)
−0.934942 + 0.354801i \(0.884548\pi\)
\(180\) 181304. 0.417086
\(181\) 678519. 1.53945 0.769725 0.638375i \(-0.220393\pi\)
0.769725 + 0.638375i \(0.220393\pi\)
\(182\) −167202. −0.374166
\(183\) 85395.2 0.188498
\(184\) 31999.0 0.0696774
\(185\) −388081. −0.833668
\(186\) 53399.3 0.113176
\(187\) 379130. 0.792838
\(188\) −640189. −1.32103
\(189\) −71165.9 −0.144916
\(190\) 136849. 0.275016
\(191\) −564563. −1.11977 −0.559886 0.828570i \(-0.689155\pi\)
−0.559886 + 0.828570i \(0.689155\pi\)
\(192\) −77826.6 −0.152361
\(193\) 774264. 1.49622 0.748111 0.663574i \(-0.230961\pi\)
0.748111 + 0.663574i \(0.230961\pi\)
\(194\) 303851. 0.579637
\(195\) 595020. 1.12059
\(196\) 200696. 0.373164
\(197\) 530628. 0.974147 0.487073 0.873361i \(-0.338064\pi\)
0.487073 + 0.873361i \(0.338064\pi\)
\(198\) 50609.0 0.0917412
\(199\) 128471. 0.229970 0.114985 0.993367i \(-0.463318\pi\)
0.114985 + 0.993367i \(0.463318\pi\)
\(200\) 433662. 0.766614
\(201\) 252129. 0.440182
\(202\) −10627.5 −0.0183254
\(203\) −723418. −1.23211
\(204\) 316679. 0.532775
\(205\) 667489. 1.10933
\(206\) 26169.6 0.0429664
\(207\) 20690.9 0.0335624
\(208\) −504367. −0.808329
\(209\) −238314. −0.377384
\(210\) −149925. −0.234599
\(211\) −447265. −0.691607 −0.345803 0.938307i \(-0.612394\pi\)
−0.345803 + 0.938307i \(0.612394\pi\)
\(212\) −420645. −0.642800
\(213\) −235396. −0.355508
\(214\) −130861. −0.195334
\(215\) 156848. 0.231411
\(216\) 91321.0 0.133179
\(217\) 275480. 0.397137
\(218\) 230158. 0.328009
\(219\) 70043.4 0.0986864
\(220\) −665144. −0.926529
\(221\) 1.03931e6 1.43141
\(222\) −90484.5 −0.123223
\(223\) 523227. 0.704576 0.352288 0.935892i \(-0.385404\pi\)
0.352288 + 0.935892i \(0.385404\pi\)
\(224\) 518409. 0.690323
\(225\) 280410. 0.369264
\(226\) 279108. 0.363498
\(227\) 360000. 0.463700 0.231850 0.972752i \(-0.425522\pi\)
0.231850 + 0.972752i \(0.425522\pi\)
\(228\) −199058. −0.253596
\(229\) −1.17199e6 −1.47684 −0.738421 0.674340i \(-0.764428\pi\)
−0.738421 + 0.674340i \(0.764428\pi\)
\(230\) 43589.4 0.0543327
\(231\) 261085. 0.321923
\(232\) 928299. 1.13232
\(233\) 163113. 0.196834 0.0984170 0.995145i \(-0.468622\pi\)
0.0984170 + 0.995145i \(0.468622\pi\)
\(234\) 138734. 0.165632
\(235\) −1.88393e6 −2.22533
\(236\) 96003.4 0.112204
\(237\) −461662. −0.533892
\(238\) −261871. −0.299671
\(239\) 1.32642e6 1.50205 0.751026 0.660273i \(-0.229559\pi\)
0.751026 + 0.660273i \(0.229559\pi\)
\(240\) −452249. −0.506816
\(241\) 1.52608e6 1.69252 0.846259 0.532771i \(-0.178849\pi\)
0.846259 + 0.532771i \(0.178849\pi\)
\(242\) 152951. 0.167886
\(243\) 59049.0 0.0641500
\(244\) −261682. −0.281384
\(245\) 590604. 0.628610
\(246\) 155631. 0.163968
\(247\) −653288. −0.681338
\(248\) −353499. −0.364971
\(249\) 159542. 0.163071
\(250\) 57481.5 0.0581672
\(251\) −739184. −0.740573 −0.370287 0.928918i \(-0.620741\pi\)
−0.370287 + 0.928918i \(0.620741\pi\)
\(252\) 218078. 0.216327
\(253\) −75908.1 −0.0745567
\(254\) 470529. 0.457617
\(255\) 931914. 0.897481
\(256\) −118805. −0.113301
\(257\) 385343. 0.363927 0.181964 0.983305i \(-0.441755\pi\)
0.181964 + 0.983305i \(0.441755\pi\)
\(258\) 36570.6 0.0342044
\(259\) −466797. −0.432393
\(260\) −1.82336e6 −1.67278
\(261\) 600247. 0.545417
\(262\) −521175. −0.469063
\(263\) −907092. −0.808653 −0.404326 0.914615i \(-0.632494\pi\)
−0.404326 + 0.914615i \(0.632494\pi\)
\(264\) −335027. −0.295849
\(265\) −1.23786e6 −1.08282
\(266\) 164607. 0.142641
\(267\) 939807. 0.806790
\(268\) −772613. −0.657091
\(269\) 1.65731e6 1.39644 0.698220 0.715883i \(-0.253976\pi\)
0.698220 + 0.715883i \(0.253976\pi\)
\(270\) 124398. 0.103850
\(271\) 2.23356e6 1.84746 0.923730 0.383045i \(-0.125125\pi\)
0.923730 + 0.383045i \(0.125125\pi\)
\(272\) −789934. −0.647394
\(273\) 715710. 0.581207
\(274\) −426388. −0.343106
\(275\) −1.02873e6 −0.820297
\(276\) −63404.3 −0.0501009
\(277\) −785622. −0.615197 −0.307599 0.951516i \(-0.599525\pi\)
−0.307599 + 0.951516i \(0.599525\pi\)
\(278\) 405546. 0.314723
\(279\) −228576. −0.175800
\(280\) 992491. 0.756539
\(281\) 1.77758e6 1.34296 0.671480 0.741023i \(-0.265659\pi\)
0.671480 + 0.741023i \(0.265659\pi\)
\(282\) −439255. −0.328922
\(283\) 797157. 0.591667 0.295834 0.955239i \(-0.404403\pi\)
0.295834 + 0.955239i \(0.404403\pi\)
\(284\) 721338. 0.530692
\(285\) −585782. −0.427193
\(286\) −508970. −0.367940
\(287\) 802879. 0.575367
\(288\) −430143. −0.305585
\(289\) 207895. 0.146420
\(290\) 1.26454e6 0.882952
\(291\) −1.30063e6 −0.900373
\(292\) −214638. −0.147316
\(293\) −142245. −0.0967986 −0.0483993 0.998828i \(-0.515412\pi\)
−0.0483993 + 0.998828i \(0.515412\pi\)
\(294\) 137704. 0.0929136
\(295\) 282516. 0.189011
\(296\) 599000. 0.397372
\(297\) −216632. −0.142505
\(298\) −444679. −0.290072
\(299\) −208086. −0.134606
\(300\) −859278. −0.551227
\(301\) 188663. 0.120024
\(302\) 257907. 0.162722
\(303\) 45491.1 0.0284656
\(304\) 496537. 0.308154
\(305\) −770069. −0.474002
\(306\) 217284. 0.132655
\(307\) 2.26783e6 1.37330 0.686649 0.726989i \(-0.259081\pi\)
0.686649 + 0.726989i \(0.259081\pi\)
\(308\) −800058. −0.480557
\(309\) −112019. −0.0667414
\(310\) −481540. −0.284595
\(311\) 2.65296e6 1.55536 0.777678 0.628663i \(-0.216398\pi\)
0.777678 + 0.628663i \(0.216398\pi\)
\(312\) −918408. −0.534133
\(313\) 2.58601e6 1.49200 0.746000 0.665946i \(-0.231972\pi\)
0.746000 + 0.665946i \(0.231972\pi\)
\(314\) −253473. −0.145080
\(315\) 641754. 0.364412
\(316\) 1.41470e6 0.796979
\(317\) −1.84274e6 −1.02995 −0.514974 0.857206i \(-0.672198\pi\)
−0.514974 + 0.857206i \(0.672198\pi\)
\(318\) −288618. −0.160050
\(319\) −2.20211e6 −1.21161
\(320\) 701818. 0.383133
\(321\) 560152. 0.303420
\(322\) 52430.8 0.0281804
\(323\) −1.02317e6 −0.545686
\(324\) −180947. −0.0957613
\(325\) −2.82006e6 −1.48098
\(326\) 1.17519e6 0.612441
\(327\) −985193. −0.509509
\(328\) −1.03026e6 −0.528766
\(329\) −2.26605e6 −1.15420
\(330\) −456377. −0.230695
\(331\) −153074. −0.0767949 −0.0383974 0.999263i \(-0.512225\pi\)
−0.0383974 + 0.999263i \(0.512225\pi\)
\(332\) −488895. −0.243428
\(333\) 387319. 0.191407
\(334\) −474360. −0.232671
\(335\) −2.27362e6 −1.10690
\(336\) −543981. −0.262867
\(337\) 630168. 0.302260 0.151130 0.988514i \(-0.451709\pi\)
0.151130 + 0.988514i \(0.451709\pi\)
\(338\) −614572. −0.292605
\(339\) −1.19472e6 −0.564635
\(340\) −2.85572e6 −1.33973
\(341\) 838570. 0.390529
\(342\) −136580. −0.0631426
\(343\) 2.35112e6 1.07904
\(344\) −242094. −0.110303
\(345\) −186584. −0.0843971
\(346\) −319576. −0.143511
\(347\) −985059. −0.439176 −0.219588 0.975593i \(-0.570471\pi\)
−0.219588 + 0.975593i \(0.570471\pi\)
\(348\) −1.83937e6 −0.814182
\(349\) −2.32668e6 −1.02252 −0.511262 0.859425i \(-0.670822\pi\)
−0.511262 + 0.859425i \(0.670822\pi\)
\(350\) 710561. 0.310050
\(351\) −593851. −0.257282
\(352\) 1.57805e6 0.678837
\(353\) 898841. 0.383925 0.191962 0.981402i \(-0.438515\pi\)
0.191962 + 0.981402i \(0.438515\pi\)
\(354\) 65871.0 0.0279374
\(355\) 2.12273e6 0.893973
\(356\) −2.87991e6 −1.20435
\(357\) 1.12094e6 0.465490
\(358\) 1.68537e6 0.695004
\(359\) −1.45729e6 −0.596774 −0.298387 0.954445i \(-0.596449\pi\)
−0.298387 + 0.954445i \(0.596449\pi\)
\(360\) −823507. −0.334897
\(361\) −1.83295e6 −0.740258
\(362\) −1.42663e6 −0.572188
\(363\) −654708. −0.260784
\(364\) −2.19319e6 −0.867608
\(365\) −631632. −0.248160
\(366\) −179548. −0.0700614
\(367\) 4.24268e6 1.64428 0.822139 0.569287i \(-0.192781\pi\)
0.822139 + 0.569287i \(0.192781\pi\)
\(368\) 158158. 0.0608794
\(369\) −666178. −0.254697
\(370\) 815963. 0.309861
\(371\) −1.48894e6 −0.561620
\(372\) 700439. 0.262430
\(373\) −2.66753e6 −0.992742 −0.496371 0.868110i \(-0.665334\pi\)
−0.496371 + 0.868110i \(0.665334\pi\)
\(374\) −797144. −0.294685
\(375\) −246050. −0.0903534
\(376\) 2.90783e6 1.06072
\(377\) −6.03664e6 −2.18747
\(378\) 149631. 0.0538630
\(379\) −3.11327e6 −1.11332 −0.556659 0.830741i \(-0.687917\pi\)
−0.556659 + 0.830741i \(0.687917\pi\)
\(380\) 1.79505e6 0.637701
\(381\) −2.01410e6 −0.710835
\(382\) 1.18703e6 0.416200
\(383\) −1.39854e6 −0.487167 −0.243584 0.969880i \(-0.578323\pi\)
−0.243584 + 0.969880i \(0.578323\pi\)
\(384\) 1.69303e6 0.585918
\(385\) −2.35439e6 −0.809517
\(386\) −1.62794e6 −0.556120
\(387\) −156540. −0.0531311
\(388\) 3.98561e6 1.34405
\(389\) 5.04752e6 1.69124 0.845618 0.533789i \(-0.179232\pi\)
0.845618 + 0.533789i \(0.179232\pi\)
\(390\) −1.25106e6 −0.416503
\(391\) −325902. −0.107807
\(392\) −911591. −0.299630
\(393\) 2.23089e6 0.728613
\(394\) −1.11568e6 −0.362074
\(395\) 4.16314e6 1.34254
\(396\) 663838. 0.212728
\(397\) 5.24113e6 1.66897 0.834486 0.551030i \(-0.185765\pi\)
0.834486 + 0.551030i \(0.185765\pi\)
\(398\) −270117. −0.0854761
\(399\) −704599. −0.221569
\(400\) 2.14341e6 0.669816
\(401\) −4.27332e6 −1.32710 −0.663551 0.748131i \(-0.730951\pi\)
−0.663551 + 0.748131i \(0.730951\pi\)
\(402\) −530115. −0.163608
\(403\) 2.29877e6 0.705070
\(404\) −139401. −0.0424926
\(405\) −532487. −0.161314
\(406\) 1.52103e6 0.457955
\(407\) −1.42095e6 −0.425199
\(408\) −1.43840e6 −0.427789
\(409\) 6.70311e6 1.98138 0.990691 0.136133i \(-0.0434676\pi\)
0.990691 + 0.136133i \(0.0434676\pi\)
\(410\) −1.40344e6 −0.412318
\(411\) 1.82515e6 0.532960
\(412\) 343266. 0.0996296
\(413\) 339820. 0.0980333
\(414\) −43503.7 −0.0124746
\(415\) −1.43871e6 −0.410064
\(416\) 4.32591e6 1.22559
\(417\) −1.73594e6 −0.488871
\(418\) 501069. 0.140267
\(419\) −1.48858e6 −0.414225 −0.207112 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(420\) −1.96657e6 −0.543983
\(421\) 4.24222e6 1.16651 0.583255 0.812289i \(-0.301779\pi\)
0.583255 + 0.812289i \(0.301779\pi\)
\(422\) 940402. 0.257059
\(423\) 1.88023e6 0.510928
\(424\) 1.91063e6 0.516133
\(425\) −4.41675e6 −1.18613
\(426\) 494933. 0.132136
\(427\) −926266. −0.245848
\(428\) −1.71651e6 −0.452936
\(429\) 2.17865e6 0.571536
\(430\) −329783. −0.0860116
\(431\) 6.78108e6 1.75835 0.879176 0.476497i \(-0.158094\pi\)
0.879176 + 0.476497i \(0.158094\pi\)
\(432\) 451361. 0.116363
\(433\) −2.89130e6 −0.741094 −0.370547 0.928814i \(-0.620830\pi\)
−0.370547 + 0.928814i \(0.620830\pi\)
\(434\) −579212. −0.147609
\(435\) −5.41286e6 −1.37152
\(436\) 3.01899e6 0.760580
\(437\) 204856. 0.0513151
\(438\) −147270. −0.0366801
\(439\) −1.15482e6 −0.285991 −0.142996 0.989723i \(-0.545674\pi\)
−0.142996 + 0.989723i \(0.545674\pi\)
\(440\) 3.02118e6 0.743952
\(441\) −589444. −0.144326
\(442\) −2.18521e6 −0.532030
\(443\) −339517. −0.0821964 −0.0410982 0.999155i \(-0.513086\pi\)
−0.0410982 + 0.999155i \(0.513086\pi\)
\(444\) −1.18688e6 −0.285727
\(445\) −8.47492e6 −2.02878
\(446\) −1.10011e6 −0.261879
\(447\) 1.90345e6 0.450581
\(448\) 844170. 0.198717
\(449\) −7.65479e6 −1.79191 −0.895957 0.444141i \(-0.853509\pi\)
−0.895957 + 0.444141i \(0.853509\pi\)
\(450\) −589579. −0.137249
\(451\) 2.44399e6 0.565794
\(452\) 3.66106e6 0.842871
\(453\) −1.10397e6 −0.252762
\(454\) −756920. −0.172350
\(455\) −6.45407e6 −1.46152
\(456\) 904150. 0.203624
\(457\) −4.53237e6 −1.01516 −0.507581 0.861604i \(-0.669460\pi\)
−0.507581 + 0.861604i \(0.669460\pi\)
\(458\) 2.46417e6 0.548917
\(459\) −930083. −0.206058
\(460\) 571762. 0.125985
\(461\) −999991. −0.219151 −0.109576 0.993978i \(-0.534949\pi\)
−0.109576 + 0.993978i \(0.534949\pi\)
\(462\) −548946. −0.119653
\(463\) 7.61544e6 1.65098 0.825492 0.564414i \(-0.190898\pi\)
0.825492 + 0.564414i \(0.190898\pi\)
\(464\) 4.58819e6 0.989342
\(465\) 2.06123e6 0.442073
\(466\) −342955. −0.0731599
\(467\) −2.99420e6 −0.635315 −0.317658 0.948206i \(-0.602896\pi\)
−0.317658 + 0.948206i \(0.602896\pi\)
\(468\) 1.81977e6 0.384063
\(469\) −2.73479e6 −0.574106
\(470\) 3.96107e6 0.827119
\(471\) 1.08499e6 0.225359
\(472\) −436061. −0.0900932
\(473\) 574296. 0.118027
\(474\) 970672. 0.198439
\(475\) 2.77628e6 0.564585
\(476\) −3.43496e6 −0.694870
\(477\) 1.23543e6 0.248612
\(478\) −2.78887e6 −0.558288
\(479\) 6.42416e6 1.27932 0.639658 0.768660i \(-0.279076\pi\)
0.639658 + 0.768660i \(0.279076\pi\)
\(480\) 3.87890e6 0.768433
\(481\) −3.89523e6 −0.767664
\(482\) −3.20866e6 −0.629081
\(483\) −224430. −0.0437737
\(484\) 2.00626e6 0.389291
\(485\) 1.17287e7 2.26411
\(486\) −124154. −0.0238435
\(487\) 9.44440e6 1.80448 0.902240 0.431234i \(-0.141922\pi\)
0.902240 + 0.431234i \(0.141922\pi\)
\(488\) 1.18860e6 0.225935
\(489\) −5.03040e6 −0.951328
\(490\) −1.24178e6 −0.233644
\(491\) −7.87399e6 −1.47398 −0.736989 0.675905i \(-0.763753\pi\)
−0.736989 + 0.675905i \(0.763753\pi\)
\(492\) 2.04141e6 0.380205
\(493\) −9.45451e6 −1.75195
\(494\) 1.37358e6 0.253242
\(495\) 1.95352e6 0.358348
\(496\) −1.74720e6 −0.318887
\(497\) 2.55329e6 0.463671
\(498\) −335447. −0.0606108
\(499\) 8.14323e6 1.46401 0.732007 0.681297i \(-0.238584\pi\)
0.732007 + 0.681297i \(0.238584\pi\)
\(500\) 753985. 0.134877
\(501\) 2.03050e6 0.361416
\(502\) 1.55418e6 0.275259
\(503\) 3.34061e6 0.588717 0.294358 0.955695i \(-0.404894\pi\)
0.294358 + 0.955695i \(0.404894\pi\)
\(504\) −990541. −0.173699
\(505\) −410226. −0.0715805
\(506\) 159601. 0.0277115
\(507\) 2.63068e6 0.454514
\(508\) 6.17193e6 1.06111
\(509\) −1.19251e6 −0.204017 −0.102009 0.994784i \(-0.532527\pi\)
−0.102009 + 0.994784i \(0.532527\pi\)
\(510\) −1.95940e6 −0.333579
\(511\) −759748. −0.128711
\(512\) −5.76987e6 −0.972728
\(513\) 584632. 0.0980819
\(514\) −810206. −0.135266
\(515\) 1.01015e6 0.167830
\(516\) 479696. 0.0793126
\(517\) −6.89795e6 −1.13499
\(518\) 981468. 0.160713
\(519\) 1.36794e6 0.222921
\(520\) 8.28194e6 1.34315
\(521\) 9.55161e6 1.54164 0.770819 0.637055i \(-0.219847\pi\)
0.770819 + 0.637055i \(0.219847\pi\)
\(522\) −1.26205e6 −0.202722
\(523\) 1.46152e6 0.233642 0.116821 0.993153i \(-0.462730\pi\)
0.116821 + 0.993153i \(0.462730\pi\)
\(524\) −6.83626e6 −1.08765
\(525\) −3.04156e6 −0.481612
\(526\) 1.90721e6 0.300563
\(527\) 3.60030e6 0.564693
\(528\) −1.65590e6 −0.258493
\(529\) −6.37109e6 −0.989862
\(530\) 2.60268e6 0.402467
\(531\) −281961. −0.0433963
\(532\) 2.15915e6 0.330752
\(533\) 6.69970e6 1.02150
\(534\) −1.97600e6 −0.299871
\(535\) −5.05129e6 −0.762989
\(536\) 3.50932e6 0.527607
\(537\) −7.21422e6 −1.07958
\(538\) −3.48459e6 −0.519034
\(539\) 2.16248e6 0.320612
\(540\) 1.63173e6 0.240804
\(541\) −1.06338e7 −1.56205 −0.781025 0.624500i \(-0.785303\pi\)
−0.781025 + 0.624500i \(0.785303\pi\)
\(542\) −4.69619e6 −0.686670
\(543\) 6.10667e6 0.888802
\(544\) 6.77519e6 0.981577
\(545\) 8.88419e6 1.28123
\(546\) −1.50482e6 −0.216025
\(547\) 1.81124e6 0.258826 0.129413 0.991591i \(-0.458691\pi\)
0.129413 + 0.991591i \(0.458691\pi\)
\(548\) −5.59293e6 −0.795587
\(549\) 768557. 0.108829
\(550\) 2.16297e6 0.304891
\(551\) 5.94292e6 0.833913
\(552\) 287991. 0.0402283
\(553\) 5.00756e6 0.696328
\(554\) 1.65182e6 0.228659
\(555\) −3.49273e6 −0.481319
\(556\) 5.31954e6 0.729772
\(557\) −1.77561e6 −0.242499 −0.121250 0.992622i \(-0.538690\pi\)
−0.121250 + 0.992622i \(0.538690\pi\)
\(558\) 480594. 0.0653420
\(559\) 1.57431e6 0.213089
\(560\) 4.90546e6 0.661013
\(561\) 3.41217e6 0.457745
\(562\) −3.73746e6 −0.499156
\(563\) −7.13593e6 −0.948810 −0.474405 0.880307i \(-0.657337\pi\)
−0.474405 + 0.880307i \(0.657337\pi\)
\(564\) −5.76170e6 −0.762699
\(565\) 1.07737e7 1.41985
\(566\) −1.67607e6 −0.219913
\(567\) −640493. −0.0836675
\(568\) −3.27642e6 −0.426117
\(569\) 3.82803e6 0.495672 0.247836 0.968802i \(-0.420281\pi\)
0.247836 + 0.968802i \(0.420281\pi\)
\(570\) 1.23164e6 0.158781
\(571\) 1.15161e7 1.47814 0.739070 0.673628i \(-0.235265\pi\)
0.739070 + 0.673628i \(0.235265\pi\)
\(572\) −6.67616e6 −0.853172
\(573\) −5.08107e6 −0.646500
\(574\) −1.68810e6 −0.213854
\(575\) 884305. 0.111541
\(576\) −700439. −0.0879659
\(577\) −1.06710e7 −1.33434 −0.667169 0.744906i \(-0.732494\pi\)
−0.667169 + 0.744906i \(0.732494\pi\)
\(578\) −437112. −0.0544218
\(579\) 6.96838e6 0.863844
\(580\) 1.65869e7 2.04737
\(581\) −1.73052e6 −0.212685
\(582\) 2.73466e6 0.334654
\(583\) −4.53239e6 −0.552276
\(584\) 974918. 0.118287
\(585\) 5.35518e6 0.646970
\(586\) 299079. 0.0359784
\(587\) 3.86670e6 0.463176 0.231588 0.972814i \(-0.425608\pi\)
0.231588 + 0.972814i \(0.425608\pi\)
\(588\) 1.80627e6 0.215446
\(589\) −2.26308e6 −0.268789
\(590\) −594006. −0.0702524
\(591\) 4.77565e6 0.562424
\(592\) 2.96060e6 0.347197
\(593\) −5.82698e6 −0.680467 −0.340234 0.940341i \(-0.610506\pi\)
−0.340234 + 0.940341i \(0.610506\pi\)
\(594\) 455481. 0.0529668
\(595\) −1.01083e7 −1.17054
\(596\) −5.83286e6 −0.672614
\(597\) 1.15624e6 0.132773
\(598\) 437514. 0.0500309
\(599\) −8.74348e6 −0.995674 −0.497837 0.867271i \(-0.665872\pi\)
−0.497837 + 0.867271i \(0.665872\pi\)
\(600\) 3.90296e6 0.442605
\(601\) −5.10457e6 −0.576465 −0.288233 0.957560i \(-0.593068\pi\)
−0.288233 + 0.957560i \(0.593068\pi\)
\(602\) −396674. −0.0446111
\(603\) 2.26916e6 0.254139
\(604\) 3.38297e6 0.377316
\(605\) 5.90397e6 0.655776
\(606\) −95647.7 −0.0105802
\(607\) 1.23037e6 0.135539 0.0677697 0.997701i \(-0.478412\pi\)
0.0677697 + 0.997701i \(0.478412\pi\)
\(608\) −4.25875e6 −0.467222
\(609\) −6.51076e6 −0.711359
\(610\) 1.61912e6 0.176179
\(611\) −1.89093e7 −2.04915
\(612\) 2.85011e6 0.307598
\(613\) −1.66177e7 −1.78616 −0.893079 0.449900i \(-0.851460\pi\)
−0.893079 + 0.449900i \(0.851460\pi\)
\(614\) −4.76824e6 −0.510432
\(615\) 6.00740e6 0.640470
\(616\) 3.63398e6 0.385860
\(617\) −2.06757e6 −0.218649 −0.109324 0.994006i \(-0.534869\pi\)
−0.109324 + 0.994006i \(0.534869\pi\)
\(618\) 235526. 0.0248067
\(619\) −1.03780e7 −1.08864 −0.544321 0.838877i \(-0.683213\pi\)
−0.544321 + 0.838877i \(0.683213\pi\)
\(620\) −6.31635e6 −0.659914
\(621\) 186218. 0.0193772
\(622\) −5.57800e6 −0.578100
\(623\) −1.01939e7 −1.05225
\(624\) −4.53930e6 −0.466689
\(625\) −8.59949e6 −0.880587
\(626\) −5.43723e6 −0.554551
\(627\) −2.14482e6 −0.217883
\(628\) −3.32481e6 −0.336409
\(629\) −6.10067e6 −0.614824
\(630\) −1.34933e6 −0.135446
\(631\) −1.57468e6 −0.157441 −0.0787205 0.996897i \(-0.525083\pi\)
−0.0787205 + 0.996897i \(0.525083\pi\)
\(632\) −6.42577e6 −0.639930
\(633\) −4.02539e6 −0.399299
\(634\) 3.87446e6 0.382814
\(635\) 1.81626e7 1.78749
\(636\) −3.78580e6 −0.371121
\(637\) 5.92799e6 0.578840
\(638\) 4.63007e6 0.450335
\(639\) −2.11856e6 −0.205253
\(640\) −1.52673e7 −1.47337
\(641\) 1.37514e7 1.32191 0.660954 0.750426i \(-0.270152\pi\)
0.660954 + 0.750426i \(0.270152\pi\)
\(642\) −1.17775e6 −0.112776
\(643\) −1.81725e7 −1.73336 −0.866678 0.498867i \(-0.833750\pi\)
−0.866678 + 0.498867i \(0.833750\pi\)
\(644\) 687734. 0.0653440
\(645\) 1.41164e6 0.133605
\(646\) 2.15128e6 0.202822
\(647\) 1.41770e7 1.33144 0.665721 0.746201i \(-0.268124\pi\)
0.665721 + 0.746201i \(0.268124\pi\)
\(648\) 821889. 0.0768910
\(649\) 1.03442e6 0.0964021
\(650\) 5.92935e6 0.550457
\(651\) 2.47932e6 0.229287
\(652\) 1.54150e7 1.42012
\(653\) −7.55832e6 −0.693653 −0.346826 0.937929i \(-0.612741\pi\)
−0.346826 + 0.937929i \(0.612741\pi\)
\(654\) 2.07143e6 0.189376
\(655\) −2.01175e7 −1.83220
\(656\) −5.09216e6 −0.462000
\(657\) 630391. 0.0569766
\(658\) 4.76451e6 0.428996
\(659\) −1.57669e7 −1.41427 −0.707134 0.707080i \(-0.750012\pi\)
−0.707134 + 0.707080i \(0.750012\pi\)
\(660\) −5.98630e6 −0.534932
\(661\) −4.61947e6 −0.411234 −0.205617 0.978633i \(-0.565920\pi\)
−0.205617 + 0.978633i \(0.565920\pi\)
\(662\) 321848. 0.0285434
\(663\) 9.35377e6 0.826424
\(664\) 2.22063e6 0.195459
\(665\) 6.35387e6 0.557166
\(666\) −814360. −0.0711428
\(667\) 1.89295e6 0.164749
\(668\) −6.22217e6 −0.539512
\(669\) 4.70904e6 0.406787
\(670\) 4.78043e6 0.411415
\(671\) −2.81959e6 −0.241757
\(672\) 4.66568e6 0.398558
\(673\) −107564. −0.00915436 −0.00457718 0.999990i \(-0.501457\pi\)
−0.00457718 + 0.999990i \(0.501457\pi\)
\(674\) −1.32496e6 −0.112345
\(675\) 2.52369e6 0.213195
\(676\) −8.06134e6 −0.678486
\(677\) 1.09050e7 0.914441 0.457221 0.889353i \(-0.348845\pi\)
0.457221 + 0.889353i \(0.348845\pi\)
\(678\) 2.51197e6 0.209865
\(679\) 1.41077e7 1.17431
\(680\) 1.29711e7 1.07573
\(681\) 3.24000e6 0.267718
\(682\) −1.76314e6 −0.145153
\(683\) −1.14750e7 −0.941237 −0.470619 0.882337i \(-0.655969\pi\)
−0.470619 + 0.882337i \(0.655969\pi\)
\(684\) −1.79152e6 −0.146414
\(685\) −1.64587e7 −1.34020
\(686\) −4.94337e6 −0.401063
\(687\) −1.05479e7 −0.852655
\(688\) −1.19657e6 −0.0963755
\(689\) −1.24246e7 −0.997092
\(690\) 392304. 0.0313690
\(691\) −1.11344e7 −0.887101 −0.443550 0.896249i \(-0.646281\pi\)
−0.443550 + 0.896249i \(0.646281\pi\)
\(692\) −4.19188e6 −0.332769
\(693\) 2.34976e6 0.185862
\(694\) 2.07114e6 0.163234
\(695\) 1.56542e7 1.22933
\(696\) 8.35469e6 0.653743
\(697\) 1.04930e7 0.818121
\(698\) 4.89198e6 0.380055
\(699\) 1.46802e6 0.113642
\(700\) 9.32043e6 0.718937
\(701\) −1.80281e7 −1.38565 −0.692827 0.721104i \(-0.743635\pi\)
−0.692827 + 0.721104i \(0.743635\pi\)
\(702\) 1.24861e6 0.0956275
\(703\) 3.83476e6 0.292651
\(704\) 2.56969e6 0.195411
\(705\) −1.69554e7 −1.28480
\(706\) −1.88986e6 −0.142698
\(707\) −493434. −0.0371262
\(708\) 864030. 0.0647807
\(709\) −2.61098e7 −1.95069 −0.975343 0.220694i \(-0.929168\pi\)
−0.975343 + 0.220694i \(0.929168\pi\)
\(710\) −4.46317e6 −0.332275
\(711\) −4.15496e6 −0.308243
\(712\) 1.30810e7 0.967029
\(713\) −720840. −0.0531025
\(714\) −2.35684e6 −0.173015
\(715\) −1.96464e7 −1.43720
\(716\) 2.21070e7 1.61156
\(717\) 1.19377e7 0.867210
\(718\) 3.06404e6 0.221811
\(719\) −1.47690e7 −1.06544 −0.532720 0.846291i \(-0.678830\pi\)
−0.532720 + 0.846291i \(0.678830\pi\)
\(720\) −4.07025e6 −0.292610
\(721\) 1.21505e6 0.0870473
\(722\) 3.85389e6 0.275142
\(723\) 1.37347e7 0.977176
\(724\) −1.87131e7 −1.32678
\(725\) 2.56539e7 1.81263
\(726\) 1.37656e6 0.0969290
\(727\) 685611. 0.0481107 0.0240554 0.999711i \(-0.492342\pi\)
0.0240554 + 0.999711i \(0.492342\pi\)
\(728\) 9.96180e6 0.696641
\(729\) 531441. 0.0370370
\(730\) 1.32804e6 0.0922369
\(731\) 2.46567e6 0.170664
\(732\) −2.35514e6 −0.162457
\(733\) −1.49501e7 −1.02774 −0.513871 0.857867i \(-0.671789\pi\)
−0.513871 + 0.857867i \(0.671789\pi\)
\(734\) −8.92048e6 −0.611150
\(735\) 5.31543e6 0.362928
\(736\) −1.35650e6 −0.0923053
\(737\) −8.32481e6 −0.564554
\(738\) 1.40068e6 0.0946667
\(739\) 1.74384e7 1.17461 0.587307 0.809365i \(-0.300188\pi\)
0.587307 + 0.809365i \(0.300188\pi\)
\(740\) 1.07030e7 0.718498
\(741\) −5.87959e6 −0.393371
\(742\) 3.13059e6 0.208745
\(743\) 1.74179e7 1.15751 0.578753 0.815503i \(-0.303540\pi\)
0.578753 + 0.815503i \(0.303540\pi\)
\(744\) −3.18149e6 −0.210716
\(745\) −1.71648e7 −1.13304
\(746\) 5.60863e6 0.368986
\(747\) 1.43588e6 0.0941491
\(748\) −1.04561e7 −0.683309
\(749\) −6.07587e6 −0.395734
\(750\) 517334. 0.0335829
\(751\) 1.11888e7 0.723908 0.361954 0.932196i \(-0.382110\pi\)
0.361954 + 0.932196i \(0.382110\pi\)
\(752\) 1.43722e7 0.926782
\(753\) −6.65265e6 −0.427570
\(754\) 1.26924e7 0.813045
\(755\) 9.95530e6 0.635605
\(756\) 1.96270e6 0.124896
\(757\) 1.99140e7 1.26305 0.631523 0.775357i \(-0.282430\pi\)
0.631523 + 0.775357i \(0.282430\pi\)
\(758\) 6.54583e6 0.413801
\(759\) −683173. −0.0430453
\(760\) −8.15337e6 −0.512039
\(761\) 7.12189e6 0.445793 0.222897 0.974842i \(-0.428449\pi\)
0.222897 + 0.974842i \(0.428449\pi\)
\(762\) 4.23476e6 0.264205
\(763\) 1.06862e7 0.664526
\(764\) 1.55702e7 0.965076
\(765\) 8.38722e6 0.518161
\(766\) 2.94051e6 0.181072
\(767\) 2.83566e6 0.174047
\(768\) −1.06925e6 −0.0654145
\(769\) −1.73482e7 −1.05788 −0.528941 0.848658i \(-0.677411\pi\)
−0.528941 + 0.848658i \(0.677411\pi\)
\(770\) 4.95024e6 0.300884
\(771\) 3.46809e6 0.210114
\(772\) −2.13536e7 −1.28952
\(773\) 1.42390e7 0.857096 0.428548 0.903519i \(-0.359025\pi\)
0.428548 + 0.903519i \(0.359025\pi\)
\(774\) 329135. 0.0197479
\(775\) −9.76908e6 −0.584251
\(776\) −1.81032e7 −1.07920
\(777\) −4.20117e6 −0.249642
\(778\) −1.06127e7 −0.628604
\(779\) −6.59569e6 −0.389418
\(780\) −1.64102e7 −0.965778
\(781\) 7.77232e6 0.455956
\(782\) 685229. 0.0400699
\(783\) 5.40222e6 0.314897
\(784\) −4.50561e6 −0.261796
\(785\) −9.78415e6 −0.566694
\(786\) −4.69058e6 −0.270813
\(787\) 1.80722e7 1.04010 0.520049 0.854137i \(-0.325914\pi\)
0.520049 + 0.854137i \(0.325914\pi\)
\(788\) −1.46343e7 −0.839570
\(789\) −8.16383e6 −0.466876
\(790\) −8.75324e6 −0.499001
\(791\) 1.29589e7 0.736424
\(792\) −3.01524e6 −0.170809
\(793\) −7.72931e6 −0.436474
\(794\) −1.10198e7 −0.620329
\(795\) −1.11408e7 −0.625168
\(796\) −3.54313e6 −0.198200
\(797\) 1.94132e7 1.08256 0.541280 0.840843i \(-0.317940\pi\)
0.541280 + 0.840843i \(0.317940\pi\)
\(798\) 1.48146e6 0.0823536
\(799\) −2.96156e7 −1.64117
\(800\) −1.83838e7 −1.01557
\(801\) 8.45827e6 0.465801
\(802\) 8.98490e6 0.493261
\(803\) −2.31270e6 −0.126570
\(804\) −6.95352e6 −0.379371
\(805\) 2.02384e6 0.110075
\(806\) −4.83329e6 −0.262063
\(807\) 1.49158e7 0.806235
\(808\) 633180. 0.0341192
\(809\) −2.24002e7 −1.20332 −0.601659 0.798753i \(-0.705493\pi\)
−0.601659 + 0.798753i \(0.705493\pi\)
\(810\) 1.11958e6 0.0599576
\(811\) −1.60987e7 −0.859486 −0.429743 0.902951i \(-0.641396\pi\)
−0.429743 + 0.902951i \(0.641396\pi\)
\(812\) 1.99513e7 1.06190
\(813\) 2.01021e7 1.06663
\(814\) 2.98762e6 0.158039
\(815\) 4.53627e7 2.39224
\(816\) −7.10940e6 −0.373773
\(817\) −1.54987e6 −0.0812346
\(818\) −1.40937e7 −0.736446
\(819\) 6.44139e6 0.335560
\(820\) −1.84089e7 −0.956075
\(821\) 3.03172e7 1.56975 0.784877 0.619651i \(-0.212726\pi\)
0.784877 + 0.619651i \(0.212726\pi\)
\(822\) −3.83749e6 −0.198092
\(823\) 7.97674e6 0.410512 0.205256 0.978708i \(-0.434197\pi\)
0.205256 + 0.978708i \(0.434197\pi\)
\(824\) −1.55916e6 −0.0799970
\(825\) −9.25861e6 −0.473599
\(826\) −714491. −0.0364373
\(827\) 1.52413e7 0.774923 0.387461 0.921886i \(-0.373352\pi\)
0.387461 + 0.921886i \(0.373352\pi\)
\(828\) −570638. −0.0289258
\(829\) 1.31859e7 0.666384 0.333192 0.942859i \(-0.391874\pi\)
0.333192 + 0.942859i \(0.391874\pi\)
\(830\) 3.02496e6 0.152414
\(831\) −7.07060e6 −0.355184
\(832\) 7.04426e6 0.352799
\(833\) 9.28435e6 0.463595
\(834\) 3.64991e6 0.181705
\(835\) −1.83104e7 −0.908830
\(836\) 6.57252e6 0.325249
\(837\) −2.05718e6 −0.101498
\(838\) 3.12982e6 0.153960
\(839\) 1.87408e7 0.919144 0.459572 0.888141i \(-0.348003\pi\)
0.459572 + 0.888141i \(0.348003\pi\)
\(840\) 8.93242e6 0.436788
\(841\) 3.44037e7 1.67732
\(842\) −8.91952e6 −0.433572
\(843\) 1.59982e7 0.775358
\(844\) 1.23352e7 0.596062
\(845\) −2.37227e7 −1.14294
\(846\) −3.95329e6 −0.189903
\(847\) 7.10149e6 0.340127
\(848\) 9.44343e6 0.450962
\(849\) 7.17441e6 0.341599
\(850\) 9.28647e6 0.440863
\(851\) 1.22145e6 0.0578167
\(852\) 6.49204e6 0.306395
\(853\) 3.32001e7 1.56231 0.781153 0.624339i \(-0.214632\pi\)
0.781153 + 0.624339i \(0.214632\pi\)
\(854\) 1.94753e6 0.0913774
\(855\) −5.27204e6 −0.246640
\(856\) 7.79663e6 0.363682
\(857\) 1.05094e7 0.488795 0.244397 0.969675i \(-0.421410\pi\)
0.244397 + 0.969675i \(0.421410\pi\)
\(858\) −4.58073e6 −0.212430
\(859\) 1.47534e7 0.682198 0.341099 0.940027i \(-0.389201\pi\)
0.341099 + 0.940027i \(0.389201\pi\)
\(860\) −4.32576e6 −0.199442
\(861\) 7.22591e6 0.332188
\(862\) −1.42576e7 −0.653550
\(863\) 3.15055e7 1.43999 0.719994 0.693980i \(-0.244144\pi\)
0.719994 + 0.693980i \(0.244144\pi\)
\(864\) −3.87128e6 −0.176429
\(865\) −1.23357e7 −0.560564
\(866\) 6.07912e6 0.275452
\(867\) 1.87106e6 0.0845355
\(868\) −7.59752e6 −0.342273
\(869\) 1.52432e7 0.684742
\(870\) 1.13808e7 0.509772
\(871\) −2.28207e7 −1.01926
\(872\) −1.37127e7 −0.610704
\(873\) −1.17057e7 −0.519831
\(874\) −430721. −0.0190729
\(875\) 2.66885e6 0.117843
\(876\) −1.93175e6 −0.0850530
\(877\) 1.78432e7 0.783380 0.391690 0.920097i \(-0.371891\pi\)
0.391690 + 0.920097i \(0.371891\pi\)
\(878\) 2.42808e6 0.106298
\(879\) −1.28021e6 −0.0558867
\(880\) 1.49324e7 0.650015
\(881\) −4.44430e6 −0.192914 −0.0964569 0.995337i \(-0.530751\pi\)
−0.0964569 + 0.995337i \(0.530751\pi\)
\(882\) 1.23934e6 0.0536437
\(883\) 2.25724e7 0.974264 0.487132 0.873328i \(-0.338043\pi\)
0.487132 + 0.873328i \(0.338043\pi\)
\(884\) −2.86633e7 −1.23366
\(885\) 2.54264e6 0.109126
\(886\) 713855. 0.0305510
\(887\) −1.77218e7 −0.756307 −0.378153 0.925743i \(-0.623441\pi\)
−0.378153 + 0.925743i \(0.623441\pi\)
\(888\) 5.39100e6 0.229423
\(889\) 2.18466e7 0.927104
\(890\) 1.78190e7 0.754064
\(891\) −1.94968e6 −0.0822754
\(892\) −1.44302e7 −0.607240
\(893\) 1.86158e7 0.781182
\(894\) −4.00211e6 −0.167473
\(895\) 6.50558e7 2.71474
\(896\) −1.83640e7 −0.764182
\(897\) −1.87278e6 −0.0777150
\(898\) 1.60946e7 0.666025
\(899\) −2.09117e7 −0.862960
\(900\) −7.73350e6 −0.318251
\(901\) −1.94593e7 −0.798574
\(902\) −5.13863e6 −0.210296
\(903\) 1.69796e6 0.0692961
\(904\) −1.66291e7 −0.676779
\(905\) −5.50683e7 −2.23501
\(906\) 2.32116e6 0.0939476
\(907\) −309663. −0.0124989 −0.00624944 0.999980i \(-0.501989\pi\)
−0.00624944 + 0.999980i \(0.501989\pi\)
\(908\) −9.92852e6 −0.399641
\(909\) 409420. 0.0164346
\(910\) 1.35701e7 0.543223
\(911\) 3.21108e7 1.28190 0.640952 0.767581i \(-0.278540\pi\)
0.640952 + 0.767581i \(0.278540\pi\)
\(912\) 4.46883e6 0.177913
\(913\) −5.26777e6 −0.209146
\(914\) 9.52958e6 0.377318
\(915\) −6.93063e6 −0.273665
\(916\) 3.23225e7 1.27282
\(917\) −2.41981e7 −0.950292
\(918\) 1.95555e6 0.0765884
\(919\) −1.92631e7 −0.752380 −0.376190 0.926543i \(-0.622766\pi\)
−0.376190 + 0.926543i \(0.622766\pi\)
\(920\) −2.59702e6 −0.101159
\(921\) 2.04105e7 0.792874
\(922\) 2.10254e6 0.0814549
\(923\) 2.13062e7 0.823194
\(924\) −7.20052e6 −0.277450
\(925\) 1.65536e7 0.636118
\(926\) −1.60119e7 −0.613643
\(927\) −1.00817e6 −0.0385332
\(928\) −3.93525e7 −1.50004
\(929\) −2.58406e7 −0.982344 −0.491172 0.871063i \(-0.663431\pi\)
−0.491172 + 0.871063i \(0.663431\pi\)
\(930\) −4.33386e6 −0.164311
\(931\) −5.83596e6 −0.220667
\(932\) −4.49855e6 −0.169642
\(933\) 2.38766e7 0.897985
\(934\) 6.29549e6 0.236136
\(935\) −3.07700e7 −1.15106
\(936\) −8.26567e6 −0.308382
\(937\) −3.94413e7 −1.46758 −0.733791 0.679375i \(-0.762251\pi\)
−0.733791 + 0.679375i \(0.762251\pi\)
\(938\) 5.75006e6 0.213386
\(939\) 2.32741e7 0.861406
\(940\) 5.19574e7 1.91791
\(941\) −3.84459e7 −1.41539 −0.707695 0.706518i \(-0.750265\pi\)
−0.707695 + 0.706518i \(0.750265\pi\)
\(942\) −2.28126e6 −0.0837621
\(943\) −2.10087e6 −0.0769342
\(944\) −2.15526e6 −0.0787174
\(945\) 5.77579e6 0.210393
\(946\) −1.20749e6 −0.0438688
\(947\) −1.88805e7 −0.684129 −0.342065 0.939676i \(-0.611126\pi\)
−0.342065 + 0.939676i \(0.611126\pi\)
\(948\) 1.27323e7 0.460136
\(949\) −6.33979e6 −0.228512
\(950\) −5.83729e6 −0.209847
\(951\) −1.65846e7 −0.594640
\(952\) 1.56021e7 0.557942
\(953\) −1.50588e7 −0.537103 −0.268552 0.963265i \(-0.586545\pi\)
−0.268552 + 0.963265i \(0.586545\pi\)
\(954\) −2.59756e6 −0.0924049
\(955\) 4.58196e7 1.62571
\(956\) −3.65816e7 −1.29455
\(957\) −1.98190e7 −0.699523
\(958\) −1.35072e7 −0.475500
\(959\) −1.97971e7 −0.695112
\(960\) 6.31636e6 0.221202
\(961\) −2.06659e7 −0.721848
\(962\) 8.18996e6 0.285328
\(963\) 5.04137e6 0.175179
\(964\) −4.20880e7 −1.45870
\(965\) −6.28388e7 −2.17225
\(966\) 471877. 0.0162699
\(967\) 1.72922e7 0.594681 0.297340 0.954772i \(-0.403900\pi\)
0.297340 + 0.954772i \(0.403900\pi\)
\(968\) −9.11272e6 −0.312579
\(969\) −9.20856e6 −0.315052
\(970\) −2.46604e7 −0.841531
\(971\) 2.51318e7 0.855411 0.427705 0.903918i \(-0.359322\pi\)
0.427705 + 0.903918i \(0.359322\pi\)
\(972\) −1.62853e6 −0.0552878
\(973\) 1.88294e7 0.637609
\(974\) −1.98574e7 −0.670695
\(975\) −2.53806e7 −0.855046
\(976\) 5.87472e6 0.197407
\(977\) 3.05606e7 1.02430 0.512148 0.858897i \(-0.328850\pi\)
0.512148 + 0.858897i \(0.328850\pi\)
\(978\) 1.05767e7 0.353593
\(979\) −3.10306e7 −1.03475
\(980\) −1.62884e7 −0.541768
\(981\) −8.86674e6 −0.294165
\(982\) 1.65555e7 0.547853
\(983\) 2.31760e7 0.764989 0.382495 0.923958i \(-0.375065\pi\)
0.382495 + 0.923958i \(0.375065\pi\)
\(984\) −9.27237e6 −0.305283
\(985\) −4.30654e7 −1.41429
\(986\) 1.98787e7 0.651171
\(987\) −2.03945e7 −0.666377
\(988\) 1.80172e7 0.587212
\(989\) −493668. −0.0160489
\(990\) −4.10739e6 −0.133192
\(991\) −3.50375e7 −1.13331 −0.566655 0.823955i \(-0.691763\pi\)
−0.566655 + 0.823955i \(0.691763\pi\)
\(992\) 1.49855e7 0.483497
\(993\) −1.37767e6 −0.0443375
\(994\) −5.36845e6 −0.172339
\(995\) −1.04266e7 −0.333876
\(996\) −4.40005e6 −0.140543
\(997\) −5.65148e6 −0.180063 −0.0900314 0.995939i \(-0.528697\pi\)
−0.0900314 + 0.995939i \(0.528697\pi\)
\(998\) −1.71216e7 −0.544150
\(999\) 3.48587e6 0.110509
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 177.6.a.c.1.4 12
3.2 odd 2 531.6.a.c.1.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.c.1.4 12 1.1 even 1 trivial
531.6.a.c.1.9 12 3.2 odd 2