Properties

Label 177.10.a.a.1.7
Level $177$
Weight $10$
Character 177.1
Self dual yes
Analytic conductor $91.161$
Analytic rank $1$
Dimension $21$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(91.1613430010\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-24.8760 q^{2} +81.0000 q^{3} +106.816 q^{4} -222.031 q^{5} -2014.96 q^{6} -2494.49 q^{7} +10079.4 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-24.8760 q^{2} +81.0000 q^{3} +106.816 q^{4} -222.031 q^{5} -2014.96 q^{6} -2494.49 q^{7} +10079.4 q^{8} +6561.00 q^{9} +5523.25 q^{10} -17730.3 q^{11} +8652.13 q^{12} -12233.5 q^{13} +62052.9 q^{14} -17984.5 q^{15} -305424. q^{16} -170541. q^{17} -163212. q^{18} +800159. q^{19} -23716.6 q^{20} -202053. q^{21} +441059. q^{22} -44122.7 q^{23} +816428. q^{24} -1.90383e6 q^{25} +304322. q^{26} +531441. q^{27} -266452. q^{28} +2.97244e6 q^{29} +447383. q^{30} +3.78495e6 q^{31} +2.43711e6 q^{32} -1.43615e6 q^{33} +4.24237e6 q^{34} +553854. q^{35} +700823. q^{36} -1.62591e7 q^{37} -1.99048e7 q^{38} -990917. q^{39} -2.23793e6 q^{40} +4.70674e6 q^{41} +5.02629e6 q^{42} -2.93262e6 q^{43} -1.89389e6 q^{44} -1.45675e6 q^{45} +1.09760e6 q^{46} +5.72676e7 q^{47} -2.47394e7 q^{48} -3.41311e7 q^{49} +4.73596e7 q^{50} -1.38138e7 q^{51} -1.30674e6 q^{52} -7.27248e7 q^{53} -1.32201e7 q^{54} +3.93667e6 q^{55} -2.51428e7 q^{56} +6.48129e7 q^{57} -7.39425e7 q^{58} +1.21174e7 q^{59} -1.92104e6 q^{60} +1.75038e8 q^{61} -9.41546e7 q^{62} -1.63663e7 q^{63} +9.57516e7 q^{64} +2.71623e6 q^{65} +3.57258e7 q^{66} +2.04648e8 q^{67} -1.82166e7 q^{68} -3.57394e6 q^{69} -1.37777e7 q^{70} +4.00362e8 q^{71} +6.61306e7 q^{72} -2.17223e8 q^{73} +4.04463e8 q^{74} -1.54210e8 q^{75} +8.54702e7 q^{76} +4.42280e7 q^{77} +2.46501e7 q^{78} +3.33758e8 q^{79} +6.78137e7 q^{80} +4.30467e7 q^{81} -1.17085e8 q^{82} -3.31607e8 q^{83} -2.15826e7 q^{84} +3.78653e7 q^{85} +7.29519e7 q^{86} +2.40768e8 q^{87} -1.78710e8 q^{88} -9.11608e8 q^{89} +3.62380e7 q^{90} +3.05164e7 q^{91} -4.71303e6 q^{92} +3.06581e8 q^{93} -1.42459e9 q^{94} -1.77660e8 q^{95} +1.97406e8 q^{96} -4.51337e8 q^{97} +8.49047e8 q^{98} -1.16328e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} - 54663q^{10} - 151769q^{11} + 421686q^{12} - 153611q^{13} - 286771q^{14} - 240084q^{15} + 805530q^{16} - 723621q^{17} - 433026q^{18} - 549388q^{19} - 527311q^{20} - 2492775q^{21} + 2973158q^{22} + 169962q^{23} - 1994301q^{24} + 8035779q^{25} - 2337392q^{26} + 11160261q^{27} - 22659054q^{28} - 16845442q^{29} - 4427703q^{30} - 19307976q^{31} - 44923568q^{32} - 12293289q^{33} - 35547496q^{34} - 34882596q^{35} + 34156566q^{36} - 41561129q^{37} - 52335371q^{38} - 12442491q^{39} - 125735038q^{40} - 68169291q^{41} - 23228451q^{42} - 25719587q^{43} - 126277032q^{44} - 19446804q^{45} - 292814271q^{46} - 174095332q^{47} + 65247930q^{48} + 7479350q^{49} - 227877439q^{50} - 58613301q^{51} - 232397708q^{52} - 228390500q^{53} - 35075106q^{54} - 29426208q^{55} + 326778474q^{56} - 44500428q^{57} + 480343762q^{58} + 254464581q^{59} - 42712191q^{60} - 183928964q^{61} - 21753862q^{62} - 201914775q^{63} + 310571245q^{64} + 5308466q^{65} + 240825798q^{66} - 82724114q^{67} - 138336205q^{68} + 13766922q^{69} + 1030274876q^{70} - 404721965q^{71} - 161538381q^{72} + 154162574q^{73} + 36352054q^{74} + 650898099q^{75} + 1068940636q^{76} - 448535481q^{77} - 189328752q^{78} + 272529635q^{79} - 345587859q^{80} + 903981141q^{81} - 38412637q^{82} + 432518643q^{83} - 1835383374q^{84} - 126211490q^{85} - 3699273072q^{86} - 1364480802q^{87} + 170111045q^{88} - 1255621070q^{89} - 358643943q^{90} + 1448885849q^{91} + 1568933320q^{92} - 1563946056q^{93} - 1908445164q^{94} - 2896546490q^{95} - 3638809008q^{96} + 1007235486q^{97} - 9506868248q^{98} - 995756409q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −24.8760 −1.09938 −0.549688 0.835370i \(-0.685253\pi\)
−0.549688 + 0.835370i \(0.685253\pi\)
\(3\) 81.0000 0.577350
\(4\) 106.816 0.208626
\(5\) −222.031 −0.158872 −0.0794362 0.996840i \(-0.525312\pi\)
−0.0794362 + 0.996840i \(0.525312\pi\)
\(6\) −2014.96 −0.634725
\(7\) −2494.49 −0.392681 −0.196341 0.980536i \(-0.562906\pi\)
−0.196341 + 0.980536i \(0.562906\pi\)
\(8\) 10079.4 0.870017
\(9\) 6561.00 0.333333
\(10\) 5523.25 0.174660
\(11\) −17730.3 −0.365131 −0.182565 0.983194i \(-0.558440\pi\)
−0.182565 + 0.983194i \(0.558440\pi\)
\(12\) 8652.13 0.120450
\(13\) −12233.5 −0.118797 −0.0593987 0.998234i \(-0.518918\pi\)
−0.0593987 + 0.998234i \(0.518918\pi\)
\(14\) 62052.9 0.431704
\(15\) −17984.5 −0.0917251
\(16\) −305424. −1.16510
\(17\) −170541. −0.495231 −0.247616 0.968858i \(-0.579647\pi\)
−0.247616 + 0.968858i \(0.579647\pi\)
\(18\) −163212. −0.366458
\(19\) 800159. 1.40859 0.704296 0.709906i \(-0.251263\pi\)
0.704296 + 0.709906i \(0.251263\pi\)
\(20\) −23716.6 −0.0331449
\(21\) −202053. −0.226715
\(22\) 441059. 0.401416
\(23\) −44122.7 −0.0328766 −0.0164383 0.999865i \(-0.505233\pi\)
−0.0164383 + 0.999865i \(0.505233\pi\)
\(24\) 816428. 0.502305
\(25\) −1.90383e6 −0.974760
\(26\) 304322. 0.130603
\(27\) 531441. 0.192450
\(28\) −266452. −0.0819235
\(29\) 2.97244e6 0.780409 0.390204 0.920728i \(-0.372404\pi\)
0.390204 + 0.920728i \(0.372404\pi\)
\(30\) 447383. 0.100840
\(31\) 3.78495e6 0.736093 0.368047 0.929807i \(-0.380027\pi\)
0.368047 + 0.929807i \(0.380027\pi\)
\(32\) 2.43711e6 0.410866
\(33\) −1.43615e6 −0.210808
\(34\) 4.24237e6 0.544445
\(35\) 553854. 0.0623862
\(36\) 700823. 0.0695420
\(37\) −1.62591e7 −1.42623 −0.713115 0.701047i \(-0.752716\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(38\) −1.99048e7 −1.54857
\(39\) −990917. −0.0685877
\(40\) −2.23793e6 −0.138222
\(41\) 4.70674e6 0.260131 0.130066 0.991505i \(-0.458481\pi\)
0.130066 + 0.991505i \(0.458481\pi\)
\(42\) 5.02629e6 0.249244
\(43\) −2.93262e6 −0.130812 −0.0654060 0.997859i \(-0.520834\pi\)
−0.0654060 + 0.997859i \(0.520834\pi\)
\(44\) −1.89389e6 −0.0761758
\(45\) −1.45675e6 −0.0529575
\(46\) 1.09760e6 0.0361437
\(47\) 5.72676e7 1.71186 0.855931 0.517089i \(-0.172985\pi\)
0.855931 + 0.517089i \(0.172985\pi\)
\(48\) −2.47394e7 −0.672671
\(49\) −3.41311e7 −0.845801
\(50\) 4.73596e7 1.07163
\(51\) −1.38138e7 −0.285922
\(52\) −1.30674e6 −0.0247842
\(53\) −7.27248e7 −1.26602 −0.633011 0.774143i \(-0.718181\pi\)
−0.633011 + 0.774143i \(0.718181\pi\)
\(54\) −1.32201e7 −0.211575
\(55\) 3.93667e6 0.0580093
\(56\) −2.51428e7 −0.341639
\(57\) 6.48129e7 0.813251
\(58\) −7.39425e7 −0.857962
\(59\) 1.21174e7 0.130189
\(60\) −1.92104e6 −0.0191362
\(61\) 1.75038e8 1.61863 0.809314 0.587377i \(-0.199839\pi\)
0.809314 + 0.587377i \(0.199839\pi\)
\(62\) −9.41546e7 −0.809243
\(63\) −1.63663e7 −0.130894
\(64\) 9.57516e7 0.713405
\(65\) 2.71623e6 0.0188736
\(66\) 3.57258e7 0.231758
\(67\) 2.04648e8 1.24071 0.620357 0.784320i \(-0.286988\pi\)
0.620357 + 0.784320i \(0.286988\pi\)
\(68\) −1.82166e7 −0.103318
\(69\) −3.57394e6 −0.0189813
\(70\) −1.37777e7 −0.0685859
\(71\) 4.00362e8 1.86978 0.934890 0.354938i \(-0.115498\pi\)
0.934890 + 0.354938i \(0.115498\pi\)
\(72\) 6.61306e7 0.290006
\(73\) −2.17223e8 −0.895269 −0.447634 0.894217i \(-0.647733\pi\)
−0.447634 + 0.894217i \(0.647733\pi\)
\(74\) 4.04463e8 1.56796
\(75\) −1.54210e8 −0.562778
\(76\) 8.54702e7 0.293869
\(77\) 4.42280e7 0.143380
\(78\) 2.46501e7 0.0754036
\(79\) 3.33758e8 0.964073 0.482036 0.876151i \(-0.339897\pi\)
0.482036 + 0.876151i \(0.339897\pi\)
\(80\) 6.78137e7 0.185103
\(81\) 4.30467e7 0.111111
\(82\) −1.17085e8 −0.285982
\(83\) −3.31607e8 −0.766959 −0.383479 0.923549i \(-0.625274\pi\)
−0.383479 + 0.923549i \(0.625274\pi\)
\(84\) −2.15826e7 −0.0472985
\(85\) 3.78653e7 0.0786786
\(86\) 7.29519e7 0.143811
\(87\) 2.40768e8 0.450569
\(88\) −1.78710e8 −0.317670
\(89\) −9.11608e8 −1.54011 −0.770057 0.637975i \(-0.779772\pi\)
−0.770057 + 0.637975i \(0.779772\pi\)
\(90\) 3.62380e7 0.0582202
\(91\) 3.05164e7 0.0466495
\(92\) −4.71303e6 −0.00685890
\(93\) 3.06581e8 0.424984
\(94\) −1.42459e9 −1.88198
\(95\) −1.77660e8 −0.223786
\(96\) 1.97406e8 0.237214
\(97\) −4.51337e8 −0.517640 −0.258820 0.965926i \(-0.583334\pi\)
−0.258820 + 0.965926i \(0.583334\pi\)
\(98\) 8.49047e8 0.929853
\(99\) −1.16328e8 −0.121710
\(100\) −2.03360e8 −0.203360
\(101\) −9.39227e8 −0.898099 −0.449050 0.893507i \(-0.648237\pi\)
−0.449050 + 0.893507i \(0.648237\pi\)
\(102\) 3.43632e8 0.314335
\(103\) 4.99025e8 0.436872 0.218436 0.975851i \(-0.429904\pi\)
0.218436 + 0.975851i \(0.429904\pi\)
\(104\) −1.23306e8 −0.103356
\(105\) 4.48621e7 0.0360187
\(106\) 1.80910e9 1.39183
\(107\) −1.84221e9 −1.35866 −0.679331 0.733832i \(-0.737730\pi\)
−0.679331 + 0.733832i \(0.737730\pi\)
\(108\) 5.67666e7 0.0401501
\(109\) 1.61168e9 1.09360 0.546801 0.837263i \(-0.315846\pi\)
0.546801 + 0.837263i \(0.315846\pi\)
\(110\) −9.79288e7 −0.0637740
\(111\) −1.31699e9 −0.823435
\(112\) 7.61877e8 0.457513
\(113\) −2.32034e9 −1.33875 −0.669375 0.742925i \(-0.733438\pi\)
−0.669375 + 0.742925i \(0.733438\pi\)
\(114\) −1.61229e9 −0.894068
\(115\) 9.79660e6 0.00522318
\(116\) 3.17506e8 0.162813
\(117\) −8.02642e7 −0.0395991
\(118\) −3.01432e8 −0.143126
\(119\) 4.25412e8 0.194468
\(120\) −1.81272e8 −0.0798024
\(121\) −2.04358e9 −0.866679
\(122\) −4.35424e9 −1.77948
\(123\) 3.81246e8 0.150187
\(124\) 4.04295e8 0.153568
\(125\) 8.56363e8 0.313735
\(126\) 4.07129e8 0.143901
\(127\) −6.43612e8 −0.219537 −0.109768 0.993957i \(-0.535011\pi\)
−0.109768 + 0.993957i \(0.535011\pi\)
\(128\) −3.62972e9 −1.19517
\(129\) −2.37542e8 −0.0755243
\(130\) −6.75689e7 −0.0207492
\(131\) −1.48253e9 −0.439827 −0.219914 0.975519i \(-0.570578\pi\)
−0.219914 + 0.975519i \(0.570578\pi\)
\(132\) −1.53405e8 −0.0439801
\(133\) −1.99599e9 −0.553128
\(134\) −5.09083e9 −1.36401
\(135\) −1.17996e8 −0.0305750
\(136\) −1.71894e9 −0.430859
\(137\) −4.89641e9 −1.18750 −0.593752 0.804648i \(-0.702354\pi\)
−0.593752 + 0.804648i \(0.702354\pi\)
\(138\) 8.89053e7 0.0208676
\(139\) 1.02222e9 0.232262 0.116131 0.993234i \(-0.462951\pi\)
0.116131 + 0.993234i \(0.462951\pi\)
\(140\) 5.91607e7 0.0130154
\(141\) 4.63868e9 0.988344
\(142\) −9.95942e9 −2.05559
\(143\) 2.16904e8 0.0433766
\(144\) −2.00389e9 −0.388367
\(145\) −6.59974e8 −0.123986
\(146\) 5.40365e9 0.984236
\(147\) −2.76462e9 −0.488324
\(148\) −1.73674e9 −0.297549
\(149\) −8.58555e9 −1.42702 −0.713510 0.700645i \(-0.752896\pi\)
−0.713510 + 0.700645i \(0.752896\pi\)
\(150\) 3.83613e9 0.618704
\(151\) −7.12323e8 −0.111501 −0.0557507 0.998445i \(-0.517755\pi\)
−0.0557507 + 0.998445i \(0.517755\pi\)
\(152\) 8.06509e9 1.22550
\(153\) −1.11892e9 −0.165077
\(154\) −1.10022e9 −0.157629
\(155\) −8.40377e8 −0.116945
\(156\) −1.05846e8 −0.0143092
\(157\) −1.19629e10 −1.57141 −0.785706 0.618601i \(-0.787700\pi\)
−0.785706 + 0.618601i \(0.787700\pi\)
\(158\) −8.30257e9 −1.05988
\(159\) −5.89071e9 −0.730938
\(160\) −5.41114e8 −0.0652753
\(161\) 1.10063e8 0.0129100
\(162\) −1.07083e9 −0.122153
\(163\) −5.38110e9 −0.597073 −0.298536 0.954398i \(-0.596498\pi\)
−0.298536 + 0.954398i \(0.596498\pi\)
\(164\) 5.02757e8 0.0542701
\(165\) 3.18871e8 0.0334917
\(166\) 8.24906e9 0.843175
\(167\) −1.56592e10 −1.55792 −0.778960 0.627074i \(-0.784253\pi\)
−0.778960 + 0.627074i \(0.784253\pi\)
\(168\) −2.03657e9 −0.197246
\(169\) −1.04548e10 −0.985887
\(170\) −9.41939e8 −0.0864973
\(171\) 5.24984e9 0.469531
\(172\) −3.13252e8 −0.0272908
\(173\) −5.95860e9 −0.505751 −0.252875 0.967499i \(-0.581376\pi\)
−0.252875 + 0.967499i \(0.581376\pi\)
\(174\) −5.98934e9 −0.495345
\(175\) 4.74907e9 0.382770
\(176\) 5.41526e9 0.425415
\(177\) 9.81506e8 0.0751646
\(178\) 2.26772e10 1.69316
\(179\) 4.50143e9 0.327727 0.163863 0.986483i \(-0.447604\pi\)
0.163863 + 0.986483i \(0.447604\pi\)
\(180\) −1.55604e8 −0.0110483
\(181\) 1.42253e10 0.985163 0.492582 0.870266i \(-0.336053\pi\)
0.492582 + 0.870266i \(0.336053\pi\)
\(182\) −7.59127e8 −0.0512853
\(183\) 1.41780e10 0.934515
\(184\) −4.44728e8 −0.0286032
\(185\) 3.61003e9 0.226589
\(186\) −7.62652e9 −0.467217
\(187\) 3.02374e9 0.180824
\(188\) 6.11713e9 0.357139
\(189\) −1.32567e9 −0.0755715
\(190\) 4.41948e9 0.246025
\(191\) −3.48150e10 −1.89285 −0.946425 0.322924i \(-0.895334\pi\)
−0.946425 + 0.322924i \(0.895334\pi\)
\(192\) 7.75588e9 0.411885
\(193\) 7.57616e9 0.393044 0.196522 0.980499i \(-0.437035\pi\)
0.196522 + 0.980499i \(0.437035\pi\)
\(194\) 1.12275e10 0.569081
\(195\) 2.20014e8 0.0108967
\(196\) −3.64577e9 −0.176456
\(197\) 3.25173e10 1.53821 0.769105 0.639122i \(-0.220702\pi\)
0.769105 + 0.639122i \(0.220702\pi\)
\(198\) 2.89379e9 0.133805
\(199\) −2.52469e10 −1.14122 −0.570611 0.821221i \(-0.693293\pi\)
−0.570611 + 0.821221i \(0.693293\pi\)
\(200\) −1.91893e10 −0.848057
\(201\) 1.65765e10 0.716326
\(202\) 2.33642e10 0.987348
\(203\) −7.41472e9 −0.306452
\(204\) −1.47554e9 −0.0596507
\(205\) −1.04504e9 −0.0413277
\(206\) −1.24137e10 −0.480287
\(207\) −2.89489e8 −0.0109589
\(208\) 3.73642e9 0.138411
\(209\) −1.41871e10 −0.514321
\(210\) −1.11599e9 −0.0395981
\(211\) 2.07497e10 0.720675 0.360338 0.932822i \(-0.382661\pi\)
0.360338 + 0.932822i \(0.382661\pi\)
\(212\) −7.76821e9 −0.264125
\(213\) 3.24293e10 1.07952
\(214\) 4.58268e10 1.49368
\(215\) 6.51132e8 0.0207824
\(216\) 5.35658e9 0.167435
\(217\) −9.44152e9 −0.289050
\(218\) −4.00921e10 −1.20228
\(219\) −1.75951e10 −0.516884
\(220\) 4.20502e8 0.0121022
\(221\) 2.08632e9 0.0588322
\(222\) 3.27615e10 0.905264
\(223\) 5.08398e10 1.37668 0.688338 0.725390i \(-0.258341\pi\)
0.688338 + 0.725390i \(0.258341\pi\)
\(224\) −6.07934e9 −0.161339
\(225\) −1.24910e10 −0.324920
\(226\) 5.77209e10 1.47179
\(227\) −4.19962e10 −1.04977 −0.524884 0.851174i \(-0.675891\pi\)
−0.524884 + 0.851174i \(0.675891\pi\)
\(228\) 6.92308e9 0.169665
\(229\) −5.74889e10 −1.38142 −0.690708 0.723134i \(-0.742701\pi\)
−0.690708 + 0.723134i \(0.742701\pi\)
\(230\) −2.43700e8 −0.00574224
\(231\) 3.58247e9 0.0827805
\(232\) 2.99603e10 0.678969
\(233\) 8.28802e10 1.84225 0.921126 0.389264i \(-0.127270\pi\)
0.921126 + 0.389264i \(0.127270\pi\)
\(234\) 1.99666e9 0.0435343
\(235\) −1.27152e10 −0.271968
\(236\) 1.29433e9 0.0271608
\(237\) 2.70344e10 0.556608
\(238\) −1.05825e10 −0.213793
\(239\) −3.92640e10 −0.778403 −0.389201 0.921153i \(-0.627249\pi\)
−0.389201 + 0.921153i \(0.627249\pi\)
\(240\) 5.49291e9 0.106869
\(241\) 4.09839e10 0.782595 0.391297 0.920264i \(-0.372026\pi\)
0.391297 + 0.920264i \(0.372026\pi\)
\(242\) 5.08363e10 0.952806
\(243\) 3.48678e9 0.0641500
\(244\) 1.86969e10 0.337688
\(245\) 7.57817e9 0.134375
\(246\) −9.48388e9 −0.165112
\(247\) −9.78878e9 −0.167337
\(248\) 3.81499e10 0.640414
\(249\) −2.68601e10 −0.442804
\(250\) −2.13029e10 −0.344912
\(251\) −4.86170e10 −0.773137 −0.386569 0.922261i \(-0.626340\pi\)
−0.386569 + 0.922261i \(0.626340\pi\)
\(252\) −1.74819e9 −0.0273078
\(253\) 7.82308e8 0.0120043
\(254\) 1.60105e10 0.241353
\(255\) 3.06709e9 0.0454251
\(256\) 4.12682e10 0.600531
\(257\) −1.19445e11 −1.70793 −0.853965 0.520331i \(-0.825809\pi\)
−0.853965 + 0.520331i \(0.825809\pi\)
\(258\) 5.90910e9 0.0830296
\(259\) 4.05582e10 0.560054
\(260\) 2.90138e8 0.00393753
\(261\) 1.95022e10 0.260136
\(262\) 3.68794e10 0.483535
\(263\) −1.23073e11 −1.58621 −0.793107 0.609083i \(-0.791538\pi\)
−0.793107 + 0.609083i \(0.791538\pi\)
\(264\) −1.44755e10 −0.183407
\(265\) 1.61472e10 0.201136
\(266\) 4.96522e10 0.608095
\(267\) −7.38402e10 −0.889186
\(268\) 2.18598e10 0.258845
\(269\) −1.84342e10 −0.214654 −0.107327 0.994224i \(-0.534229\pi\)
−0.107327 + 0.994224i \(0.534229\pi\)
\(270\) 2.93528e9 0.0336134
\(271\) 1.45903e11 1.64325 0.821625 0.570028i \(-0.193068\pi\)
0.821625 + 0.570028i \(0.193068\pi\)
\(272\) 5.20873e10 0.576994
\(273\) 2.47183e9 0.0269331
\(274\) 1.21803e11 1.30551
\(275\) 3.37554e10 0.355915
\(276\) −3.81755e8 −0.00395999
\(277\) −1.10202e11 −1.12469 −0.562345 0.826903i \(-0.690101\pi\)
−0.562345 + 0.826903i \(0.690101\pi\)
\(278\) −2.54288e10 −0.255344
\(279\) 2.48331e10 0.245364
\(280\) 5.58249e9 0.0542771
\(281\) 8.05599e10 0.770798 0.385399 0.922750i \(-0.374064\pi\)
0.385399 + 0.922750i \(0.374064\pi\)
\(282\) −1.15392e11 −1.08656
\(283\) −2.74018e8 −0.00253945 −0.00126973 0.999999i \(-0.500404\pi\)
−0.00126973 + 0.999999i \(0.500404\pi\)
\(284\) 4.27653e10 0.390084
\(285\) −1.43905e10 −0.129203
\(286\) −5.39571e9 −0.0476872
\(287\) −1.17409e10 −0.102149
\(288\) 1.59899e10 0.136955
\(289\) −8.95037e10 −0.754746
\(290\) 1.64175e10 0.136307
\(291\) −3.65583e10 −0.298860
\(292\) −2.32030e10 −0.186776
\(293\) −6.56715e10 −0.520562 −0.260281 0.965533i \(-0.583815\pi\)
−0.260281 + 0.965533i \(0.583815\pi\)
\(294\) 6.87728e10 0.536851
\(295\) −2.69043e9 −0.0206834
\(296\) −1.63882e11 −1.24085
\(297\) −9.42260e9 −0.0702695
\(298\) 2.13574e11 1.56883
\(299\) 5.39776e8 0.00390565
\(300\) −1.64722e10 −0.117410
\(301\) 7.31538e9 0.0513674
\(302\) 1.77198e10 0.122582
\(303\) −7.60774e10 −0.518518
\(304\) −2.44388e11 −1.64115
\(305\) −3.88638e10 −0.257155
\(306\) 2.78342e10 0.181482
\(307\) 2.68465e10 0.172490 0.0862452 0.996274i \(-0.472513\pi\)
0.0862452 + 0.996274i \(0.472513\pi\)
\(308\) 4.72428e9 0.0299128
\(309\) 4.04210e10 0.252228
\(310\) 2.09052e10 0.128566
\(311\) −2.14410e11 −1.29964 −0.649821 0.760088i \(-0.725156\pi\)
−0.649821 + 0.760088i \(0.725156\pi\)
\(312\) −9.98780e9 −0.0596725
\(313\) 2.76816e11 1.63020 0.815102 0.579318i \(-0.196681\pi\)
0.815102 + 0.579318i \(0.196681\pi\)
\(314\) 2.97591e11 1.72757
\(315\) 3.63383e9 0.0207954
\(316\) 3.56508e10 0.201130
\(317\) 9.32046e10 0.518407 0.259203 0.965823i \(-0.416540\pi\)
0.259203 + 0.965823i \(0.416540\pi\)
\(318\) 1.46537e11 0.803575
\(319\) −5.27022e10 −0.284951
\(320\) −2.12598e10 −0.113340
\(321\) −1.49219e11 −0.784424
\(322\) −2.73794e9 −0.0141929
\(323\) −1.36460e11 −0.697578
\(324\) 4.59810e9 0.0231807
\(325\) 2.32905e10 0.115799
\(326\) 1.33860e11 0.656407
\(327\) 1.30546e11 0.631391
\(328\) 4.74409e10 0.226319
\(329\) −1.42853e11 −0.672216
\(330\) −7.93223e9 −0.0368199
\(331\) −2.31416e11 −1.05966 −0.529831 0.848103i \(-0.677745\pi\)
−0.529831 + 0.848103i \(0.677745\pi\)
\(332\) −3.54211e10 −0.160007
\(333\) −1.06676e11 −0.475410
\(334\) 3.89538e11 1.71274
\(335\) −4.54383e10 −0.197115
\(336\) 6.17120e10 0.264145
\(337\) −3.10705e11 −1.31224 −0.656121 0.754655i \(-0.727804\pi\)
−0.656121 + 0.754655i \(0.727804\pi\)
\(338\) 2.60075e11 1.08386
\(339\) −1.87948e11 −0.772927
\(340\) 4.04464e9 0.0164144
\(341\) −6.71083e10 −0.268770
\(342\) −1.30595e11 −0.516190
\(343\) 1.85801e11 0.724812
\(344\) −2.95589e10 −0.113809
\(345\) 7.93525e8 0.00301561
\(346\) 1.48226e11 0.556010
\(347\) 4.06988e11 1.50695 0.753475 0.657476i \(-0.228376\pi\)
0.753475 + 0.657476i \(0.228376\pi\)
\(348\) 2.57180e10 0.0940004
\(349\) 3.09331e11 1.11612 0.558058 0.829802i \(-0.311547\pi\)
0.558058 + 0.829802i \(0.311547\pi\)
\(350\) −1.18138e11 −0.420808
\(351\) −6.50140e9 −0.0228626
\(352\) −4.32107e10 −0.150020
\(353\) 1.60790e11 0.551154 0.275577 0.961279i \(-0.411131\pi\)
0.275577 + 0.961279i \(0.411131\pi\)
\(354\) −2.44160e10 −0.0826341
\(355\) −8.88928e10 −0.297057
\(356\) −9.73747e10 −0.321308
\(357\) 3.44583e10 0.112276
\(358\) −1.11978e11 −0.360294
\(359\) 4.04099e10 0.128399 0.0641996 0.997937i \(-0.479551\pi\)
0.0641996 + 0.997937i \(0.479551\pi\)
\(360\) −1.46831e10 −0.0460739
\(361\) 3.17567e11 0.984131
\(362\) −3.53869e11 −1.08306
\(363\) −1.65530e11 −0.500378
\(364\) 3.25965e9 0.00973230
\(365\) 4.82303e10 0.142234
\(366\) −3.52693e11 −1.02738
\(367\) 4.75777e11 1.36901 0.684505 0.729009i \(-0.260019\pi\)
0.684505 + 0.729009i \(0.260019\pi\)
\(368\) 1.34761e10 0.0383045
\(369\) 3.08809e10 0.0867104
\(370\) −8.98033e10 −0.249106
\(371\) 1.81411e11 0.497143
\(372\) 3.27479e10 0.0886626
\(373\) −6.63497e11 −1.77480 −0.887400 0.461001i \(-0.847490\pi\)
−0.887400 + 0.461001i \(0.847490\pi\)
\(374\) −7.52185e10 −0.198794
\(375\) 6.93654e10 0.181135
\(376\) 5.77221e11 1.48935
\(377\) −3.63635e10 −0.0927106
\(378\) 3.29775e10 0.0830815
\(379\) −1.62805e11 −0.405313 −0.202657 0.979250i \(-0.564957\pi\)
−0.202657 + 0.979250i \(0.564957\pi\)
\(380\) −1.89770e10 −0.0466877
\(381\) −5.21326e10 −0.126750
\(382\) 8.66058e11 2.08095
\(383\) 3.31529e11 0.787276 0.393638 0.919266i \(-0.371216\pi\)
0.393638 + 0.919266i \(0.371216\pi\)
\(384\) −2.94007e11 −0.690029
\(385\) −9.81998e9 −0.0227792
\(386\) −1.88465e11 −0.432103
\(387\) −1.92409e10 −0.0436040
\(388\) −4.82102e10 −0.107993
\(389\) −6.01100e11 −1.33099 −0.665494 0.746404i \(-0.731779\pi\)
−0.665494 + 0.746404i \(0.731779\pi\)
\(390\) −5.47308e9 −0.0119796
\(391\) 7.52471e9 0.0162815
\(392\) −3.44020e11 −0.735862
\(393\) −1.20085e11 −0.253934
\(394\) −8.08900e11 −1.69107
\(395\) −7.41046e10 −0.153165
\(396\) −1.24258e10 −0.0253919
\(397\) −2.47379e11 −0.499811 −0.249905 0.968270i \(-0.580399\pi\)
−0.249905 + 0.968270i \(0.580399\pi\)
\(398\) 6.28043e11 1.25463
\(399\) −1.61675e11 −0.319348
\(400\) 5.81475e11 1.13569
\(401\) −5.01181e11 −0.967931 −0.483966 0.875087i \(-0.660804\pi\)
−0.483966 + 0.875087i \(0.660804\pi\)
\(402\) −4.12358e11 −0.787511
\(403\) −4.63034e10 −0.0874460
\(404\) −1.00325e11 −0.187367
\(405\) −9.55771e9 −0.0176525
\(406\) 1.84449e11 0.336906
\(407\) 2.88279e11 0.520761
\(408\) −1.39234e11 −0.248757
\(409\) 3.05359e11 0.539580 0.269790 0.962919i \(-0.413046\pi\)
0.269790 + 0.962919i \(0.413046\pi\)
\(410\) 2.59965e10 0.0454347
\(411\) −3.96609e11 −0.685606
\(412\) 5.33040e10 0.0911429
\(413\) −3.02266e10 −0.0511227
\(414\) 7.20133e9 0.0120479
\(415\) 7.36270e10 0.121849
\(416\) −2.98145e10 −0.0488098
\(417\) 8.28000e10 0.134097
\(418\) 3.52917e11 0.565431
\(419\) 8.08727e11 1.28186 0.640928 0.767601i \(-0.278550\pi\)
0.640928 + 0.767601i \(0.278550\pi\)
\(420\) 4.79202e9 0.00751444
\(421\) −1.20078e12 −1.86292 −0.931459 0.363846i \(-0.881463\pi\)
−0.931459 + 0.363846i \(0.881463\pi\)
\(422\) −5.16169e11 −0.792293
\(423\) 3.75733e11 0.570621
\(424\) −7.33019e11 −1.10146
\(425\) 3.24680e11 0.482731
\(426\) −8.06713e11 −1.18680
\(427\) −4.36629e11 −0.635605
\(428\) −1.96778e11 −0.283452
\(429\) 1.75692e10 0.0250435
\(430\) −1.61976e10 −0.0228477
\(431\) 8.17289e11 1.14085 0.570424 0.821350i \(-0.306779\pi\)
0.570424 + 0.821350i \(0.306779\pi\)
\(432\) −1.62315e11 −0.224224
\(433\) 4.47154e11 0.611310 0.305655 0.952142i \(-0.401125\pi\)
0.305655 + 0.952142i \(0.401125\pi\)
\(434\) 2.34867e11 0.317774
\(435\) −5.34579e10 −0.0715831
\(436\) 1.72154e11 0.228154
\(437\) −3.53052e10 −0.0463097
\(438\) 4.37696e11 0.568249
\(439\) 4.14846e11 0.533086 0.266543 0.963823i \(-0.414119\pi\)
0.266543 + 0.963823i \(0.414119\pi\)
\(440\) 3.96791e10 0.0504691
\(441\) −2.23934e11 −0.281934
\(442\) −5.18992e10 −0.0646786
\(443\) −1.52484e12 −1.88108 −0.940540 0.339684i \(-0.889680\pi\)
−0.940540 + 0.339684i \(0.889680\pi\)
\(444\) −1.40676e11 −0.171790
\(445\) 2.02405e11 0.244682
\(446\) −1.26469e12 −1.51348
\(447\) −6.95429e11 −0.823890
\(448\) −2.38851e11 −0.280141
\(449\) 2.41329e10 0.0280221 0.0140111 0.999902i \(-0.495540\pi\)
0.0140111 + 0.999902i \(0.495540\pi\)
\(450\) 3.10727e11 0.357209
\(451\) −8.34518e10 −0.0949820
\(452\) −2.47851e11 −0.279298
\(453\) −5.76982e10 −0.0643754
\(454\) 1.04470e12 1.15409
\(455\) −6.77559e9 −0.00741132
\(456\) 6.53272e11 0.707542
\(457\) 1.37101e12 1.47034 0.735170 0.677883i \(-0.237102\pi\)
0.735170 + 0.677883i \(0.237102\pi\)
\(458\) 1.43010e12 1.51869
\(459\) −9.06323e10 −0.0953073
\(460\) 1.04644e9 0.00108969
\(461\) 8.56735e11 0.883471 0.441735 0.897145i \(-0.354363\pi\)
0.441735 + 0.897145i \(0.354363\pi\)
\(462\) −8.91175e10 −0.0910069
\(463\) −1.77854e12 −1.79866 −0.899328 0.437274i \(-0.855944\pi\)
−0.899328 + 0.437274i \(0.855944\pi\)
\(464\) −9.07856e11 −0.909255
\(465\) −6.80705e10 −0.0675182
\(466\) −2.06173e12 −2.02533
\(467\) −1.26353e12 −1.22931 −0.614654 0.788797i \(-0.710704\pi\)
−0.614654 + 0.788797i \(0.710704\pi\)
\(468\) −8.57354e9 −0.00826141
\(469\) −5.10492e11 −0.487205
\(470\) 3.16303e11 0.298995
\(471\) −9.68999e11 −0.907255
\(472\) 1.22135e11 0.113267
\(473\) 5.19962e10 0.0477635
\(474\) −6.72508e11 −0.611921
\(475\) −1.52336e12 −1.37304
\(476\) 4.54410e10 0.0405710
\(477\) −4.77147e11 −0.422007
\(478\) 9.76733e11 0.855757
\(479\) −1.36613e12 −1.18572 −0.592862 0.805304i \(-0.702002\pi\)
−0.592862 + 0.805304i \(0.702002\pi\)
\(480\) −4.38303e10 −0.0376867
\(481\) 1.98907e11 0.169433
\(482\) −1.01952e12 −0.860365
\(483\) 8.91514e9 0.00745360
\(484\) −2.18288e11 −0.180812
\(485\) 1.00211e11 0.0822388
\(486\) −8.67373e10 −0.0705250
\(487\) 1.70385e12 1.37263 0.686313 0.727306i \(-0.259228\pi\)
0.686313 + 0.727306i \(0.259228\pi\)
\(488\) 1.76427e12 1.40823
\(489\) −4.35869e11 −0.344720
\(490\) −1.88515e11 −0.147728
\(491\) 1.97165e12 1.53096 0.765478 0.643462i \(-0.222503\pi\)
0.765478 + 0.643462i \(0.222503\pi\)
\(492\) 4.07233e10 0.0313329
\(493\) −5.06922e11 −0.386483
\(494\) 2.43506e11 0.183966
\(495\) 2.58285e10 0.0193364
\(496\) −1.15602e12 −0.857623
\(497\) −9.98698e11 −0.734227
\(498\) 6.68174e11 0.486808
\(499\) 2.47891e12 1.78981 0.894907 0.446253i \(-0.147242\pi\)
0.894907 + 0.446253i \(0.147242\pi\)
\(500\) 9.14737e10 0.0654532
\(501\) −1.26839e12 −0.899466
\(502\) 1.20940e12 0.849968
\(503\) −1.22607e12 −0.854006 −0.427003 0.904250i \(-0.640431\pi\)
−0.427003 + 0.904250i \(0.640431\pi\)
\(504\) −1.64962e11 −0.113880
\(505\) 2.08537e11 0.142683
\(506\) −1.94607e10 −0.0131972
\(507\) −8.46842e11 −0.569202
\(508\) −6.87484e10 −0.0458011
\(509\) −1.41506e12 −0.934427 −0.467213 0.884145i \(-0.654742\pi\)
−0.467213 + 0.884145i \(0.654742\pi\)
\(510\) −7.62970e10 −0.0499392
\(511\) 5.41861e11 0.351555
\(512\) 8.31829e11 0.534957
\(513\) 4.25237e11 0.271084
\(514\) 2.97132e12 1.87766
\(515\) −1.10799e11 −0.0694070
\(516\) −2.53734e10 −0.0157563
\(517\) −1.01537e12 −0.625054
\(518\) −1.00893e12 −0.615710
\(519\) −4.82647e11 −0.291995
\(520\) 2.73778e10 0.0164204
\(521\) 2.78794e11 0.165773 0.0828865 0.996559i \(-0.473586\pi\)
0.0828865 + 0.996559i \(0.473586\pi\)
\(522\) −4.85137e11 −0.285987
\(523\) 2.29079e12 1.33884 0.669418 0.742886i \(-0.266544\pi\)
0.669418 + 0.742886i \(0.266544\pi\)
\(524\) −1.58358e11 −0.0917593
\(525\) 3.84675e11 0.220992
\(526\) 3.06156e12 1.74384
\(527\) −6.45489e11 −0.364536
\(528\) 4.38636e11 0.245613
\(529\) −1.79921e12 −0.998919
\(530\) −4.01677e11 −0.221124
\(531\) 7.95020e10 0.0433963
\(532\) −2.13204e11 −0.115397
\(533\) −5.75801e10 −0.0309029
\(534\) 1.83685e12 0.977549
\(535\) 4.09027e11 0.215854
\(536\) 2.06272e12 1.07944
\(537\) 3.64616e11 0.189213
\(538\) 4.58570e11 0.235986
\(539\) 6.05155e11 0.308828
\(540\) −1.26040e10 −0.00637874
\(541\) 4.25469e11 0.213541 0.106770 0.994284i \(-0.465949\pi\)
0.106770 + 0.994284i \(0.465949\pi\)
\(542\) −3.62950e12 −1.80655
\(543\) 1.15225e12 0.568784
\(544\) −4.15627e11 −0.203474
\(545\) −3.57843e11 −0.173743
\(546\) −6.14893e10 −0.0296096
\(547\) −1.78323e12 −0.851657 −0.425829 0.904804i \(-0.640017\pi\)
−0.425829 + 0.904804i \(0.640017\pi\)
\(548\) −5.23017e11 −0.247744
\(549\) 1.14842e12 0.539542
\(550\) −8.39700e11 −0.391284
\(551\) 2.37843e12 1.09928
\(552\) −3.60230e10 −0.0165141
\(553\) −8.32555e11 −0.378573
\(554\) 2.74140e12 1.23646
\(555\) 2.92413e11 0.130821
\(556\) 1.09190e11 0.0484560
\(557\) −8.32903e11 −0.366645 −0.183323 0.983053i \(-0.558685\pi\)
−0.183323 + 0.983053i \(0.558685\pi\)
\(558\) −6.17748e11 −0.269748
\(559\) 3.58763e10 0.0155401
\(560\) −1.69160e11 −0.0726863
\(561\) 2.44923e11 0.104399
\(562\) −2.00401e12 −0.847396
\(563\) 8.32713e10 0.0349307 0.0174654 0.999847i \(-0.494440\pi\)
0.0174654 + 0.999847i \(0.494440\pi\)
\(564\) 4.95487e11 0.206194
\(565\) 5.15188e11 0.212690
\(566\) 6.81647e9 0.00279181
\(567\) −1.07380e11 −0.0436312
\(568\) 4.03539e12 1.62674
\(569\) −3.63936e12 −1.45553 −0.727763 0.685829i \(-0.759440\pi\)
−0.727763 + 0.685829i \(0.759440\pi\)
\(570\) 3.57978e11 0.142043
\(571\) −1.64875e12 −0.649073 −0.324536 0.945873i \(-0.605208\pi\)
−0.324536 + 0.945873i \(0.605208\pi\)
\(572\) 2.31689e10 0.00904949
\(573\) −2.82001e12 −1.09284
\(574\) 2.92067e11 0.112300
\(575\) 8.40019e10 0.0320468
\(576\) 6.28226e11 0.237802
\(577\) −3.55485e11 −0.133515 −0.0667574 0.997769i \(-0.521265\pi\)
−0.0667574 + 0.997769i \(0.521265\pi\)
\(578\) 2.22650e12 0.829749
\(579\) 6.13669e11 0.226924
\(580\) −7.04961e10 −0.0258666
\(581\) 8.27189e11 0.301170
\(582\) 9.09424e11 0.328559
\(583\) 1.28943e12 0.462264
\(584\) −2.18947e12 −0.778899
\(585\) 1.78212e10 0.00629121
\(586\) 1.63365e12 0.572293
\(587\) 2.57122e12 0.893856 0.446928 0.894570i \(-0.352518\pi\)
0.446928 + 0.894570i \(0.352518\pi\)
\(588\) −2.95307e11 −0.101877
\(589\) 3.02856e12 1.03686
\(590\) 6.69272e10 0.0227389
\(591\) 2.63390e12 0.888086
\(592\) 4.96594e12 1.66170
\(593\) 2.41018e12 0.800392 0.400196 0.916430i \(-0.368942\pi\)
0.400196 + 0.916430i \(0.368942\pi\)
\(594\) 2.34397e11 0.0772525
\(595\) −9.44546e10 −0.0308956
\(596\) −9.17078e11 −0.297713
\(597\) −2.04500e12 −0.658884
\(598\) −1.34275e10 −0.00429378
\(599\) −4.24751e12 −1.34807 −0.674036 0.738698i \(-0.735441\pi\)
−0.674036 + 0.738698i \(0.735441\pi\)
\(600\) −1.55434e12 −0.489626
\(601\) 2.12128e12 0.663227 0.331613 0.943415i \(-0.392407\pi\)
0.331613 + 0.943415i \(0.392407\pi\)
\(602\) −1.81978e11 −0.0564721
\(603\) 1.34270e12 0.413571
\(604\) −7.60878e10 −0.0232621
\(605\) 4.53739e11 0.137691
\(606\) 1.89250e12 0.570046
\(607\) −7.37312e11 −0.220446 −0.110223 0.993907i \(-0.535156\pi\)
−0.110223 + 0.993907i \(0.535156\pi\)
\(608\) 1.95008e12 0.578743
\(609\) −6.00592e11 −0.176930
\(610\) 9.66776e11 0.282710
\(611\) −7.00586e11 −0.203365
\(612\) −1.19519e11 −0.0344393
\(613\) −1.75322e12 −0.501494 −0.250747 0.968053i \(-0.580676\pi\)
−0.250747 + 0.968053i \(0.580676\pi\)
\(614\) −6.67834e11 −0.189632
\(615\) −8.46484e10 −0.0238606
\(616\) 4.45790e11 0.124743
\(617\) −3.50243e11 −0.0972940 −0.0486470 0.998816i \(-0.515491\pi\)
−0.0486470 + 0.998816i \(0.515491\pi\)
\(618\) −1.00551e12 −0.277294
\(619\) 3.58750e11 0.0982163 0.0491082 0.998793i \(-0.484362\pi\)
0.0491082 + 0.998793i \(0.484362\pi\)
\(620\) −8.97661e10 −0.0243977
\(621\) −2.34486e10 −0.00632710
\(622\) 5.33367e12 1.42879
\(623\) 2.27399e12 0.604774
\(624\) 3.02650e11 0.0799116
\(625\) 3.52827e12 0.924916
\(626\) −6.88608e12 −1.79221
\(627\) −1.14915e12 −0.296943
\(628\) −1.27784e12 −0.327837
\(629\) 2.77284e12 0.706314
\(630\) −9.03953e10 −0.0228620
\(631\) −1.63671e12 −0.410998 −0.205499 0.978657i \(-0.565882\pi\)
−0.205499 + 0.978657i \(0.565882\pi\)
\(632\) 3.36406e12 0.838760
\(633\) 1.68072e12 0.416082
\(634\) −2.31856e12 −0.569924
\(635\) 1.42902e11 0.0348784
\(636\) −6.29225e11 −0.152493
\(637\) 4.17545e11 0.100479
\(638\) 1.31102e12 0.313269
\(639\) 2.62678e12 0.623260
\(640\) 8.05910e11 0.189879
\(641\) 1.03690e11 0.0242592 0.0121296 0.999926i \(-0.496139\pi\)
0.0121296 + 0.999926i \(0.496139\pi\)
\(642\) 3.71197e12 0.862376
\(643\) 5.90450e12 1.36218 0.681089 0.732201i \(-0.261507\pi\)
0.681089 + 0.732201i \(0.261507\pi\)
\(644\) 1.17566e10 0.00269336
\(645\) 5.27417e10 0.0119987
\(646\) 3.39457e12 0.766900
\(647\) −1.37816e12 −0.309194 −0.154597 0.987978i \(-0.549408\pi\)
−0.154597 + 0.987978i \(0.549408\pi\)
\(648\) 4.33883e11 0.0966686
\(649\) −2.14844e11 −0.0475360
\(650\) −5.79376e11 −0.127306
\(651\) −7.64763e11 −0.166883
\(652\) −5.74790e11 −0.124565
\(653\) 3.68278e12 0.792624 0.396312 0.918116i \(-0.370290\pi\)
0.396312 + 0.918116i \(0.370290\pi\)
\(654\) −3.24746e12 −0.694136
\(655\) 3.29167e11 0.0698764
\(656\) −1.43755e12 −0.303079
\(657\) −1.42520e12 −0.298423
\(658\) 3.55362e12 0.739018
\(659\) 2.25561e11 0.0465887 0.0232943 0.999729i \(-0.492585\pi\)
0.0232943 + 0.999729i \(0.492585\pi\)
\(660\) 3.40606e10 0.00698723
\(661\) 4.85476e12 0.989148 0.494574 0.869135i \(-0.335324\pi\)
0.494574 + 0.869135i \(0.335324\pi\)
\(662\) 5.75671e12 1.16497
\(663\) 1.68992e11 0.0339668
\(664\) −3.34238e12 −0.667267
\(665\) 4.43171e11 0.0878768
\(666\) 2.65368e12 0.522654
\(667\) −1.31152e11 −0.0256572
\(668\) −1.67266e12 −0.325022
\(669\) 4.11802e12 0.794825
\(670\) 1.13032e12 0.216704
\(671\) −3.10347e12 −0.591011
\(672\) −4.92427e11 −0.0931494
\(673\) −8.95636e12 −1.68292 −0.841460 0.540319i \(-0.818304\pi\)
−0.841460 + 0.540319i \(0.818304\pi\)
\(674\) 7.72911e12 1.44265
\(675\) −1.01177e12 −0.187593
\(676\) −1.11675e12 −0.205682
\(677\) −4.88016e12 −0.892863 −0.446432 0.894818i \(-0.647305\pi\)
−0.446432 + 0.894818i \(0.647305\pi\)
\(678\) 4.67539e12 0.849737
\(679\) 1.12585e12 0.203268
\(680\) 3.81658e11 0.0684517
\(681\) −3.40169e12 −0.606084
\(682\) 1.66939e12 0.295480
\(683\) 4.85322e12 0.853369 0.426685 0.904400i \(-0.359681\pi\)
0.426685 + 0.904400i \(0.359681\pi\)
\(684\) 5.60770e11 0.0979562
\(685\) 1.08716e12 0.188662
\(686\) −4.62200e12 −0.796840
\(687\) −4.65660e12 −0.797561
\(688\) 8.95693e11 0.152409
\(689\) 8.89682e11 0.150400
\(690\) −1.97397e10 −0.00331528
\(691\) −2.42661e12 −0.404901 −0.202450 0.979293i \(-0.564890\pi\)
−0.202450 + 0.979293i \(0.564890\pi\)
\(692\) −6.36476e11 −0.105513
\(693\) 2.90180e11 0.0477934
\(694\) −1.01242e13 −1.65670
\(695\) −2.26965e11 −0.0369001
\(696\) 2.42678e12 0.392003
\(697\) −8.02690e11 −0.128825
\(698\) −7.69493e12 −1.22703
\(699\) 6.71330e12 1.06363
\(700\) 5.07279e11 0.0798557
\(701\) −2.58871e12 −0.404904 −0.202452 0.979292i \(-0.564891\pi\)
−0.202452 + 0.979292i \(0.564891\pi\)
\(702\) 1.61729e11 0.0251345
\(703\) −1.30099e13 −2.00898
\(704\) −1.69770e12 −0.260486
\(705\) −1.02993e12 −0.157021
\(706\) −3.99982e12 −0.605926
\(707\) 2.34289e12 0.352667
\(708\) 1.04841e11 0.0156813
\(709\) −7.50290e12 −1.11512 −0.557559 0.830137i \(-0.688262\pi\)
−0.557559 + 0.830137i \(0.688262\pi\)
\(710\) 2.21130e12 0.326577
\(711\) 2.18979e12 0.321358
\(712\) −9.18842e12 −1.33993
\(713\) −1.67002e11 −0.0242002
\(714\) −8.57186e11 −0.123434
\(715\) −4.81595e10 −0.00689135
\(716\) 4.80827e11 0.0683722
\(717\) −3.18039e12 −0.449411
\(718\) −1.00524e12 −0.141159
\(719\) −2.42649e12 −0.338609 −0.169304 0.985564i \(-0.554152\pi\)
−0.169304 + 0.985564i \(0.554152\pi\)
\(720\) 4.44925e11 0.0617008
\(721\) −1.24481e12 −0.171552
\(722\) −7.89980e12 −1.08193
\(723\) 3.31970e12 0.451831
\(724\) 1.51950e12 0.205531
\(725\) −5.65901e12 −0.760711
\(726\) 4.11774e12 0.550103
\(727\) 1.31300e13 1.74325 0.871627 0.490169i \(-0.163065\pi\)
0.871627 + 0.490169i \(0.163065\pi\)
\(728\) 3.07586e11 0.0405859
\(729\) 2.82430e11 0.0370370
\(730\) −1.19978e12 −0.156368
\(731\) 5.00131e11 0.0647822
\(732\) 1.51445e12 0.194964
\(733\) −3.38940e11 −0.0433666 −0.0216833 0.999765i \(-0.506903\pi\)
−0.0216833 + 0.999765i \(0.506903\pi\)
\(734\) −1.18354e13 −1.50505
\(735\) 6.13832e11 0.0775812
\(736\) −1.07532e11 −0.0135079
\(737\) −3.62847e12 −0.453023
\(738\) −7.68194e11 −0.0953273
\(739\) −7.80178e12 −0.962262 −0.481131 0.876649i \(-0.659774\pi\)
−0.481131 + 0.876649i \(0.659774\pi\)
\(740\) 3.85611e11 0.0472723
\(741\) −7.92891e11 −0.0966121
\(742\) −4.51279e12 −0.546547
\(743\) −5.12727e11 −0.0617215 −0.0308607 0.999524i \(-0.509825\pi\)
−0.0308607 + 0.999524i \(0.509825\pi\)
\(744\) 3.09014e12 0.369743
\(745\) 1.90626e12 0.226714
\(746\) 1.65052e13 1.95117
\(747\) −2.17567e12 −0.255653
\(748\) 3.22985e11 0.0377246
\(749\) 4.59536e12 0.533521
\(750\) −1.72554e12 −0.199135
\(751\) −9.33550e12 −1.07092 −0.535461 0.844560i \(-0.679862\pi\)
−0.535461 + 0.844560i \(0.679862\pi\)
\(752\) −1.74909e13 −1.99449
\(753\) −3.93798e12 −0.446371
\(754\) 9.04578e11 0.101924
\(755\) 1.58158e11 0.0177145
\(756\) −1.41604e11 −0.0157662
\(757\) −8.67301e12 −0.959928 −0.479964 0.877288i \(-0.659350\pi\)
−0.479964 + 0.877288i \(0.659350\pi\)
\(758\) 4.04993e12 0.445591
\(759\) 6.33669e10 0.00693066
\(760\) −1.79070e12 −0.194698
\(761\) 2.14892e12 0.232268 0.116134 0.993234i \(-0.462950\pi\)
0.116134 + 0.993234i \(0.462950\pi\)
\(762\) 1.29685e12 0.139345
\(763\) −4.02031e12 −0.429437
\(764\) −3.71881e12 −0.394897
\(765\) 2.48434e11 0.0262262
\(766\) −8.24712e12 −0.865511
\(767\) −1.48238e11 −0.0154661
\(768\) 3.34272e12 0.346717
\(769\) −1.38294e13 −1.42605 −0.713023 0.701140i \(-0.752675\pi\)
−0.713023 + 0.701140i \(0.752675\pi\)
\(770\) 2.44282e11 0.0250428
\(771\) −9.67507e12 −0.986074
\(772\) 8.09259e11 0.0819992
\(773\) −1.73146e13 −1.74424 −0.872118 0.489295i \(-0.837254\pi\)
−0.872118 + 0.489295i \(0.837254\pi\)
\(774\) 4.78637e11 0.0479372
\(775\) −7.20590e12 −0.717514
\(776\) −4.54918e12 −0.450356
\(777\) 3.28522e12 0.323347
\(778\) 1.49530e13 1.46325
\(779\) 3.76614e12 0.366419
\(780\) 2.35011e10 0.00227333
\(781\) −7.09854e12 −0.682714
\(782\) −1.87185e11 −0.0178995
\(783\) 1.57968e12 0.150190
\(784\) 1.04245e13 0.985444
\(785\) 2.65615e12 0.249654
\(786\) 2.98723e12 0.279169
\(787\) −1.42737e13 −1.32633 −0.663165 0.748474i \(-0.730787\pi\)
−0.663165 + 0.748474i \(0.730787\pi\)
\(788\) 3.47338e12 0.320911
\(789\) −9.96891e12 −0.915801
\(790\) 1.84343e12 0.168385
\(791\) 5.78807e12 0.525702
\(792\) −1.17252e12 −0.105890
\(793\) −2.14133e12 −0.192289
\(794\) 6.15380e12 0.549479
\(795\) 1.30792e12 0.116126
\(796\) −2.69679e12 −0.238088
\(797\) 3.64156e12 0.319687 0.159843 0.987142i \(-0.448901\pi\)
0.159843 + 0.987142i \(0.448901\pi\)
\(798\) 4.02183e12 0.351084
\(799\) −9.76646e12 −0.847768
\(800\) −4.63984e12 −0.400496
\(801\) −5.98106e12 −0.513372
\(802\) 1.24674e13 1.06412
\(803\) 3.85143e12 0.326890
\(804\) 1.77064e12 0.149444
\(805\) −2.44375e10 −0.00205105
\(806\) 1.15184e12 0.0961360
\(807\) −1.49317e12 −0.123931
\(808\) −9.46680e12 −0.781362
\(809\) 1.26684e13 1.03981 0.519903 0.854225i \(-0.325968\pi\)
0.519903 + 0.854225i \(0.325968\pi\)
\(810\) 2.37758e11 0.0194067
\(811\) 2.26177e13 1.83592 0.917961 0.396670i \(-0.129834\pi\)
0.917961 + 0.396670i \(0.129834\pi\)
\(812\) −7.92014e11 −0.0639338
\(813\) 1.18182e13 0.948731
\(814\) −7.17124e12 −0.572512
\(815\) 1.19477e12 0.0948584
\(816\) 4.21907e12 0.333128
\(817\) −2.34656e12 −0.184261
\(818\) −7.59611e12 −0.593200
\(819\) 2.00218e11 0.0155498
\(820\) −1.11628e11 −0.00862203
\(821\) −1.76896e12 −0.135886 −0.0679428 0.997689i \(-0.521644\pi\)
−0.0679428 + 0.997689i \(0.521644\pi\)
\(822\) 9.86606e12 0.753739
\(823\) −2.02207e13 −1.53638 −0.768189 0.640224i \(-0.778842\pi\)
−0.768189 + 0.640224i \(0.778842\pi\)
\(824\) 5.02985e12 0.380086
\(825\) 2.73419e12 0.205488
\(826\) 7.51918e11 0.0562031
\(827\) −1.25520e13 −0.933120 −0.466560 0.884489i \(-0.654507\pi\)
−0.466560 + 0.884489i \(0.654507\pi\)
\(828\) −3.09222e10 −0.00228630
\(829\) 4.15017e12 0.305190 0.152595 0.988289i \(-0.451237\pi\)
0.152595 + 0.988289i \(0.451237\pi\)
\(830\) −1.83155e12 −0.133957
\(831\) −8.92640e12 −0.649340
\(832\) −1.17138e12 −0.0847507
\(833\) 5.82075e12 0.418867
\(834\) −2.05974e12 −0.147423
\(835\) 3.47683e12 0.247511
\(836\) −1.51541e12 −0.107301
\(837\) 2.01148e12 0.141661
\(838\) −2.01179e13 −1.40924
\(839\) 1.24366e13 0.866506 0.433253 0.901272i \(-0.357366\pi\)
0.433253 + 0.901272i \(0.357366\pi\)
\(840\) 4.52181e11 0.0313369
\(841\) −5.67174e12 −0.390962
\(842\) 2.98706e13 2.04805
\(843\) 6.52535e12 0.445021
\(844\) 2.21640e12 0.150352
\(845\) 2.32130e12 0.156630
\(846\) −9.34674e12 −0.627326
\(847\) 5.09770e12 0.340329
\(848\) 2.22119e13 1.47504
\(849\) −2.21954e10 −0.00146615
\(850\) −8.07675e12 −0.530703
\(851\) 7.17397e11 0.0468896
\(852\) 3.46399e12 0.225215
\(853\) −1.12775e13 −0.729358 −0.364679 0.931133i \(-0.618821\pi\)
−0.364679 + 0.931133i \(0.618821\pi\)
\(854\) 1.08616e13 0.698768
\(855\) −1.16563e12 −0.0745955
\(856\) −1.85683e13 −1.18206
\(857\) 6.20340e12 0.392840 0.196420 0.980520i \(-0.437068\pi\)
0.196420 + 0.980520i \(0.437068\pi\)
\(858\) −4.37053e11 −0.0275322
\(859\) −2.33530e12 −0.146343 −0.0731716 0.997319i \(-0.523312\pi\)
−0.0731716 + 0.997319i \(0.523312\pi\)
\(860\) 6.95517e10 0.00433575
\(861\) −9.51013e11 −0.0589756
\(862\) −2.03309e13 −1.25422
\(863\) −8.44146e12 −0.518047 −0.259023 0.965871i \(-0.583401\pi\)
−0.259023 + 0.965871i \(0.583401\pi\)
\(864\) 1.29518e12 0.0790712
\(865\) 1.32299e12 0.0803499
\(866\) −1.11234e13 −0.672059
\(867\) −7.24980e12 −0.435753
\(868\) −1.00851e12 −0.0603033
\(869\) −5.91762e12 −0.352013
\(870\) 1.32982e12 0.0786966
\(871\) −2.50357e12 −0.147394
\(872\) 1.62447e13 0.951452
\(873\) −2.96122e12 −0.172547
\(874\) 8.78252e11 0.0509117
\(875\) −2.13619e12 −0.123198
\(876\) −1.87944e12 −0.107835
\(877\) 1.55888e13 0.889846 0.444923 0.895569i \(-0.353231\pi\)
0.444923 + 0.895569i \(0.353231\pi\)
\(878\) −1.03197e13 −0.586061
\(879\) −5.31939e12 −0.300547
\(880\) −1.20236e12 −0.0675867
\(881\) 5.38270e11 0.0301029 0.0150515 0.999887i \(-0.495209\pi\)
0.0150515 + 0.999887i \(0.495209\pi\)
\(882\) 5.57060e12 0.309951
\(883\) 5.19066e12 0.287342 0.143671 0.989625i \(-0.454109\pi\)
0.143671 + 0.989625i \(0.454109\pi\)
\(884\) 2.22853e11 0.0122739
\(885\) −2.17925e11 −0.0119416
\(886\) 3.79319e13 2.06801
\(887\) 1.69013e13 0.916776 0.458388 0.888752i \(-0.348427\pi\)
0.458388 + 0.888752i \(0.348427\pi\)
\(888\) −1.32744e13 −0.716402
\(889\) 1.60548e12 0.0862080
\(890\) −5.03504e12 −0.268997
\(891\) −7.63231e11 −0.0405701
\(892\) 5.43053e12 0.287210
\(893\) 4.58232e13 2.41132
\(894\) 1.72995e13 0.905764
\(895\) −9.99457e11 −0.0520667
\(896\) 9.05429e12 0.469319
\(897\) 4.37219e10 0.00225493
\(898\) −6.00331e11 −0.0308068
\(899\) 1.12505e13 0.574454
\(900\) −1.33425e12 −0.0677867
\(901\) 1.24025e13 0.626973
\(902\) 2.07595e12 0.104421
\(903\) 5.92546e11 0.0296570
\(904\) −2.33876e13 −1.16473
\(905\) −3.15846e12 −0.156515
\(906\) 1.43530e12 0.0707727
\(907\) 3.37968e12 0.165822 0.0829112 0.996557i \(-0.473578\pi\)
0.0829112 + 0.996557i \(0.473578\pi\)
\(908\) −4.48588e12 −0.219009
\(909\) −6.16227e12 −0.299366
\(910\) 1.68550e11 0.00814783
\(911\) 3.39086e13 1.63109 0.815544 0.578696i \(-0.196438\pi\)
0.815544 + 0.578696i \(0.196438\pi\)
\(912\) −1.97954e13 −0.947520
\(913\) 5.87948e12 0.280040
\(914\) −3.41053e13 −1.61645
\(915\) −3.14796e12 −0.148469
\(916\) −6.14076e12 −0.288199
\(917\) 3.69815e12 0.172712
\(918\) 2.25457e12 0.104778
\(919\) 1.99191e13 0.921191 0.460596 0.887610i \(-0.347636\pi\)
0.460596 + 0.887610i \(0.347636\pi\)
\(920\) 9.87434e10 0.00454426
\(921\) 2.17457e12 0.0995874
\(922\) −2.13121e13 −0.971266
\(923\) −4.89785e12 −0.222125
\(924\) 3.82666e11 0.0172702
\(925\) 3.09546e13 1.39023
\(926\) 4.42429e13 1.97740
\(927\) 3.27410e12 0.145624
\(928\) 7.24417e12 0.320644
\(929\) −9.51657e12 −0.419189 −0.209594 0.977788i \(-0.567214\pi\)
−0.209594 + 0.977788i \(0.567214\pi\)
\(930\) 1.69332e12 0.0742278
\(931\) −2.73103e13 −1.19139
\(932\) 8.85297e12 0.384342
\(933\) −1.73672e13 −0.750348
\(934\) 3.14317e13 1.35147
\(935\) −6.71363e11 −0.0287280
\(936\) −8.09012e11 −0.0344519
\(937\) −1.61966e13 −0.686431 −0.343216 0.939257i \(-0.611516\pi\)
−0.343216 + 0.939257i \(0.611516\pi\)
\(938\) 1.26990e13 0.535621
\(939\) 2.24221e13 0.941198
\(940\) −1.35819e12 −0.0567395
\(941\) 1.72664e12 0.0717874 0.0358937 0.999356i \(-0.488572\pi\)
0.0358937 + 0.999356i \(0.488572\pi\)
\(942\) 2.41048e13 0.997413
\(943\) −2.07674e11 −0.00855222
\(944\) −3.70094e12 −0.151683
\(945\) 2.94341e11 0.0120062
\(946\) −1.29346e12 −0.0525100
\(947\) 1.75876e13 0.710610 0.355305 0.934750i \(-0.384377\pi\)
0.355305 + 0.934750i \(0.384377\pi\)
\(948\) 2.88772e12 0.116123
\(949\) 2.65741e12 0.106356
\(950\) 3.78953e13 1.50948
\(951\) 7.54958e12 0.299302
\(952\) 4.28787e12 0.169190
\(953\) 3.21974e13 1.26445 0.632226 0.774784i \(-0.282141\pi\)
0.632226 + 0.774784i \(0.282141\pi\)
\(954\) 1.18695e13 0.463944
\(955\) 7.73001e12 0.300722
\(956\) −4.19405e12 −0.162395
\(957\) −4.26888e12 −0.164517
\(958\) 3.39840e13 1.30355
\(959\) 1.22140e13 0.466311
\(960\) −1.72205e12 −0.0654371
\(961\) −1.21138e13 −0.458167
\(962\) −4.94801e12 −0.186270
\(963\) −1.20867e13 −0.452887
\(964\) 4.37776e12 0.163270
\(965\) −1.68214e12 −0.0624439
\(966\) −2.21773e11 −0.00819430
\(967\) −4.18164e13 −1.53790 −0.768950 0.639309i \(-0.779220\pi\)
−0.768950 + 0.639309i \(0.779220\pi\)
\(968\) −2.05980e13 −0.754026
\(969\) −1.10532e13 −0.402747
\(970\) −2.49284e12 −0.0904112
\(971\) −3.40994e13 −1.23101 −0.615503 0.788135i \(-0.711047\pi\)
−0.615503 + 0.788135i \(0.711047\pi\)
\(972\) 3.72446e11 0.0133834
\(973\) −2.54992e12 −0.0912051
\(974\) −4.23851e13 −1.50903
\(975\) 1.88653e12 0.0668565
\(976\) −5.34607e13 −1.88586
\(977\) 4.01527e13 1.40990 0.704952 0.709255i \(-0.250969\pi\)
0.704952 + 0.709255i \(0.250969\pi\)
\(978\) 1.08427e13 0.378977
\(979\) 1.61631e13 0.562344
\(980\) 8.09473e11 0.0280340
\(981\) 1.05742e13 0.364534
\(982\) −4.90468e13 −1.68309
\(983\) 1.78216e13 0.608775 0.304387 0.952548i \(-0.401548\pi\)
0.304387 + 0.952548i \(0.401548\pi\)
\(984\) 3.84271e12 0.130665
\(985\) −7.21984e12 −0.244379
\(986\) 1.26102e13 0.424890
\(987\) −1.15711e13 −0.388104
\(988\) −1.04560e12 −0.0349108
\(989\) 1.29395e11 0.00430065
\(990\) −6.42511e11 −0.0212580
\(991\) −8.05274e12 −0.265223 −0.132612 0.991168i \(-0.542336\pi\)
−0.132612 + 0.991168i \(0.542336\pi\)
\(992\) 9.22435e12 0.302436
\(993\) −1.87447e13 −0.611796
\(994\) 2.48436e13 0.807191
\(995\) 5.60560e12 0.181309
\(996\) −2.86911e12 −0.0923803
\(997\) 7.38050e12 0.236569 0.118285 0.992980i \(-0.462261\pi\)
0.118285 + 0.992980i \(0.462261\pi\)
\(998\) −6.16653e13 −1.96768
\(999\) −8.64077e12 −0.274478
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 177.10.a.a.1.7 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.a.1.7 21 1.1 even 1 trivial