Properties

Label 177.10.a.a.1.15
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 1.15
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+15.9818 q^{2} +81.0000 q^{3} -256.580 q^{4} +61.0966 q^{5} +1294.53 q^{6} +4757.03 q^{7} -12283.3 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q+15.9818 q^{2} +81.0000 q^{3} -256.580 q^{4} +61.0966 q^{5} +1294.53 q^{6} +4757.03 q^{7} -12283.3 q^{8} +6561.00 q^{9} +976.437 q^{10} +5603.63 q^{11} -20783.0 q^{12} -73495.1 q^{13} +76026.1 q^{14} +4948.83 q^{15} -64941.2 q^{16} +430505. q^{17} +104857. q^{18} -718411. q^{19} -15676.2 q^{20} +385319. q^{21} +89556.4 q^{22} +3945.26 q^{23} -994950. q^{24} -1.94939e6 q^{25} -1.17459e6 q^{26} +531441. q^{27} -1.22056e6 q^{28} +627200. q^{29} +79091.4 q^{30} +1.02481e6 q^{31} +5.25119e6 q^{32} +453894. q^{33} +6.88027e6 q^{34} +290638. q^{35} -1.68342e6 q^{36} -1.31994e7 q^{37} -1.14815e7 q^{38} -5.95311e6 q^{39} -750470. q^{40} -7.00235e6 q^{41} +6.15811e6 q^{42} +3.11873e7 q^{43} -1.43778e6 q^{44} +400855. q^{45} +63052.5 q^{46} -7.14985e6 q^{47} -5.26024e6 q^{48} -1.77243e7 q^{49} -3.11549e7 q^{50} +3.48709e7 q^{51} +1.88574e7 q^{52} -1.00893e8 q^{53} +8.49341e6 q^{54} +342363. q^{55} -5.84322e7 q^{56} -5.81913e7 q^{57} +1.00238e7 q^{58} +1.21174e7 q^{59} -1.26977e6 q^{60} -4.94621e7 q^{61} +1.63783e7 q^{62} +3.12109e7 q^{63} +1.17174e8 q^{64} -4.49030e6 q^{65} +7.25407e6 q^{66} -2.76538e8 q^{67} -1.10459e8 q^{68} +319566. q^{69} +4.64494e6 q^{70} -2.12153e8 q^{71} -8.05910e7 q^{72} -1.63627e8 q^{73} -2.10951e8 q^{74} -1.57901e8 q^{75} +1.84330e8 q^{76} +2.66566e7 q^{77} -9.51416e7 q^{78} -5.43394e8 q^{79} -3.96769e6 q^{80} +4.30467e7 q^{81} -1.11911e8 q^{82} +5.64040e8 q^{83} -9.88654e7 q^{84} +2.63024e7 q^{85} +4.98430e8 q^{86} +5.08032e7 q^{87} -6.88313e7 q^{88} -3.74114e8 q^{89} +6.40640e6 q^{90} -3.49618e8 q^{91} -1.01228e6 q^{92} +8.30095e7 q^{93} -1.14268e8 q^{94} -4.38925e7 q^{95} +4.25346e8 q^{96} +1.37988e9 q^{97} -2.83267e8 q^{98} +3.67654e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} - 54663q^{10} - 151769q^{11} + 421686q^{12} - 153611q^{13} - 286771q^{14} - 240084q^{15} + 805530q^{16} - 723621q^{17} - 433026q^{18} - 549388q^{19} - 527311q^{20} - 2492775q^{21} + 2973158q^{22} + 169962q^{23} - 1994301q^{24} + 8035779q^{25} - 2337392q^{26} + 11160261q^{27} - 22659054q^{28} - 16845442q^{29} - 4427703q^{30} - 19307976q^{31} - 44923568q^{32} - 12293289q^{33} - 35547496q^{34} - 34882596q^{35} + 34156566q^{36} - 41561129q^{37} - 52335371q^{38} - 12442491q^{39} - 125735038q^{40} - 68169291q^{41} - 23228451q^{42} - 25719587q^{43} - 126277032q^{44} - 19446804q^{45} - 292814271q^{46} - 174095332q^{47} + 65247930q^{48} + 7479350q^{49} - 227877439q^{50} - 58613301q^{51} - 232397708q^{52} - 228390500q^{53} - 35075106q^{54} - 29426208q^{55} + 326778474q^{56} - 44500428q^{57} + 480343762q^{58} + 254464581q^{59} - 42712191q^{60} - 183928964q^{61} - 21753862q^{62} - 201914775q^{63} + 310571245q^{64} + 5308466q^{65} + 240825798q^{66} - 82724114q^{67} - 138336205q^{68} + 13766922q^{69} + 1030274876q^{70} - 404721965q^{71} - 161538381q^{72} + 154162574q^{73} + 36352054q^{74} + 650898099q^{75} + 1068940636q^{76} - 448535481q^{77} - 189328752q^{78} + 272529635q^{79} - 345587859q^{80} + 903981141q^{81} - 38412637q^{82} + 432518643q^{83} - 1835383374q^{84} - 126211490q^{85} - 3699273072q^{86} - 1364480802q^{87} + 170111045q^{88} - 1255621070q^{89} - 358643943q^{90} + 1448885849q^{91} + 1568933320q^{92} - 1563946056q^{93} - 1908445164q^{94} - 2896546490q^{95} - 3638809008q^{96} + 1007235486q^{97} - 9506868248q^{98} - 995756409q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 15.9818 0.706305 0.353152 0.935566i \(-0.385110\pi\)
0.353152 + 0.935566i \(0.385110\pi\)
\(3\) 81.0000 0.577350
\(4\) −256.580 −0.501134
\(5\) 61.0966 0.0437172 0.0218586 0.999761i \(-0.493042\pi\)
0.0218586 + 0.999761i \(0.493042\pi\)
\(6\) 1294.53 0.407785
\(7\) 4757.03 0.748850 0.374425 0.927257i \(-0.377840\pi\)
0.374425 + 0.927257i \(0.377840\pi\)
\(8\) −12283.3 −1.06026
\(9\) 6561.00 0.333333
\(10\) 976.437 0.0308776
\(11\) 5603.63 0.115399 0.0576996 0.998334i \(-0.481623\pi\)
0.0576996 + 0.998334i \(0.481623\pi\)
\(12\) −20783.0 −0.289330
\(13\) −73495.1 −0.713696 −0.356848 0.934162i \(-0.616149\pi\)
−0.356848 + 0.934162i \(0.616149\pi\)
\(14\) 76026.1 0.528916
\(15\) 4948.83 0.0252401
\(16\) −64941.2 −0.247731
\(17\) 430505. 1.25014 0.625069 0.780569i \(-0.285071\pi\)
0.625069 + 0.780569i \(0.285071\pi\)
\(18\) 104857. 0.235435
\(19\) −718411. −1.26468 −0.632341 0.774690i \(-0.717906\pi\)
−0.632341 + 0.774690i \(0.717906\pi\)
\(20\) −15676.2 −0.0219082
\(21\) 385319. 0.432348
\(22\) 89556.4 0.0815070
\(23\) 3945.26 0.00293968 0.00146984 0.999999i \(-0.499532\pi\)
0.00146984 + 0.999999i \(0.499532\pi\)
\(24\) −994950. −0.612140
\(25\) −1.94939e6 −0.998089
\(26\) −1.17459e6 −0.504087
\(27\) 531441. 0.192450
\(28\) −1.22056e6 −0.375274
\(29\) 627200. 0.164670 0.0823351 0.996605i \(-0.473762\pi\)
0.0823351 + 0.996605i \(0.473762\pi\)
\(30\) 79091.4 0.0178272
\(31\) 1.02481e6 0.199304 0.0996518 0.995022i \(-0.468227\pi\)
0.0996518 + 0.995022i \(0.468227\pi\)
\(32\) 5.25119e6 0.885284
\(33\) 453894. 0.0666257
\(34\) 6.88027e6 0.882978
\(35\) 290638. 0.0327376
\(36\) −1.68342e6 −0.167045
\(37\) −1.31994e7 −1.15783 −0.578917 0.815386i \(-0.696525\pi\)
−0.578917 + 0.815386i \(0.696525\pi\)
\(38\) −1.14815e7 −0.893251
\(39\) −5.95311e6 −0.412053
\(40\) −750470. −0.0463515
\(41\) −7.00235e6 −0.387005 −0.193503 0.981100i \(-0.561985\pi\)
−0.193503 + 0.981100i \(0.561985\pi\)
\(42\) 6.15811e6 0.305370
\(43\) 3.11873e7 1.39113 0.695567 0.718461i \(-0.255153\pi\)
0.695567 + 0.718461i \(0.255153\pi\)
\(44\) −1.43778e6 −0.0578304
\(45\) 400855. 0.0145724
\(46\) 63052.5 0.00207631
\(47\) −7.14985e6 −0.213726 −0.106863 0.994274i \(-0.534081\pi\)
−0.106863 + 0.994274i \(0.534081\pi\)
\(48\) −5.26024e6 −0.143028
\(49\) −1.77243e7 −0.439224
\(50\) −3.11549e7 −0.704955
\(51\) 3.48709e7 0.721768
\(52\) 1.88574e7 0.357657
\(53\) −1.00893e8 −1.75639 −0.878195 0.478303i \(-0.841252\pi\)
−0.878195 + 0.478303i \(0.841252\pi\)
\(54\) 8.49341e6 0.135928
\(55\) 342363. 0.00504493
\(56\) −5.84322e7 −0.793973
\(57\) −5.81913e7 −0.730165
\(58\) 1.00238e7 0.116307
\(59\) 1.21174e7 0.130189
\(60\) −1.26977e6 −0.0126487
\(61\) −4.94621e7 −0.457391 −0.228696 0.973498i \(-0.573446\pi\)
−0.228696 + 0.973498i \(0.573446\pi\)
\(62\) 1.63783e7 0.140769
\(63\) 3.12109e7 0.249617
\(64\) 1.17174e8 0.873011
\(65\) −4.49030e6 −0.0312008
\(66\) 7.25407e6 0.0470581
\(67\) −2.76538e8 −1.67656 −0.838280 0.545240i \(-0.816438\pi\)
−0.838280 + 0.545240i \(0.816438\pi\)
\(68\) −1.10459e8 −0.626486
\(69\) 319566. 0.00169723
\(70\) 4.64494e6 0.0231227
\(71\) −2.12153e8 −0.990799 −0.495400 0.868665i \(-0.664978\pi\)
−0.495400 + 0.868665i \(0.664978\pi\)
\(72\) −8.05910e7 −0.353419
\(73\) −1.63627e8 −0.674377 −0.337189 0.941437i \(-0.609476\pi\)
−0.337189 + 0.941437i \(0.609476\pi\)
\(74\) −2.10951e8 −0.817784
\(75\) −1.57901e8 −0.576247
\(76\) 1.84330e8 0.633775
\(77\) 2.66566e7 0.0864166
\(78\) −9.51416e7 −0.291035
\(79\) −5.43394e8 −1.56961 −0.784807 0.619741i \(-0.787238\pi\)
−0.784807 + 0.619741i \(0.787238\pi\)
\(80\) −3.96769e6 −0.0108301
\(81\) 4.30467e7 0.111111
\(82\) −1.11911e8 −0.273344
\(83\) 5.64040e8 1.30454 0.652272 0.757985i \(-0.273816\pi\)
0.652272 + 0.757985i \(0.273816\pi\)
\(84\) −9.88654e7 −0.216664
\(85\) 2.63024e7 0.0546525
\(86\) 4.98430e8 0.982565
\(87\) 5.08032e7 0.0950724
\(88\) −6.88313e7 −0.122353
\(89\) −3.74114e8 −0.632047 −0.316023 0.948751i \(-0.602348\pi\)
−0.316023 + 0.948751i \(0.602348\pi\)
\(90\) 6.40640e6 0.0102925
\(91\) −3.49618e8 −0.534451
\(92\) −1.01228e6 −0.00147317
\(93\) 8.30095e7 0.115068
\(94\) −1.14268e8 −0.150955
\(95\) −4.38925e7 −0.0552884
\(96\) 4.25346e8 0.511119
\(97\) 1.37988e9 1.58259 0.791296 0.611433i \(-0.209407\pi\)
0.791296 + 0.611433i \(0.209407\pi\)
\(98\) −2.83267e8 −0.310226
\(99\) 3.67654e7 0.0384664
\(100\) 5.00176e8 0.500176
\(101\) 7.62944e8 0.729536 0.364768 0.931099i \(-0.381148\pi\)
0.364768 + 0.931099i \(0.381148\pi\)
\(102\) 5.57301e8 0.509788
\(103\) −1.73748e9 −1.52108 −0.760538 0.649293i \(-0.775065\pi\)
−0.760538 + 0.649293i \(0.775065\pi\)
\(104\) 9.02765e8 0.756702
\(105\) 2.35417e7 0.0189011
\(106\) −1.61246e9 −1.24055
\(107\) −9.32433e8 −0.687687 −0.343843 0.939027i \(-0.611729\pi\)
−0.343843 + 0.939027i \(0.611729\pi\)
\(108\) −1.36357e8 −0.0964432
\(109\) −1.28116e9 −0.869331 −0.434665 0.900592i \(-0.643133\pi\)
−0.434665 + 0.900592i \(0.643133\pi\)
\(110\) 5.47159e6 0.00356325
\(111\) −1.06915e9 −0.668476
\(112\) −3.08927e8 −0.185513
\(113\) 2.83825e8 0.163756 0.0818782 0.996642i \(-0.473908\pi\)
0.0818782 + 0.996642i \(0.473908\pi\)
\(114\) −9.30004e8 −0.515719
\(115\) 241042. 0.000128515 0
\(116\) −1.60927e8 −0.0825218
\(117\) −4.82202e8 −0.237899
\(118\) 1.93658e8 0.0919530
\(119\) 2.04792e9 0.936165
\(120\) −6.07881e7 −0.0267610
\(121\) −2.32655e9 −0.986683
\(122\) −7.90495e8 −0.323058
\(123\) −5.67191e8 −0.223438
\(124\) −2.62946e8 −0.0998778
\(125\) −2.38431e8 −0.0873508
\(126\) 4.98807e8 0.176305
\(127\) 1.68868e9 0.576012 0.288006 0.957629i \(-0.407008\pi\)
0.288006 + 0.957629i \(0.407008\pi\)
\(128\) −8.15957e8 −0.268672
\(129\) 2.52617e9 0.803172
\(130\) −7.17634e7 −0.0220373
\(131\) −3.62798e9 −1.07633 −0.538164 0.842840i \(-0.680882\pi\)
−0.538164 + 0.842840i \(0.680882\pi\)
\(132\) −1.16460e8 −0.0333884
\(133\) −3.41750e9 −0.947057
\(134\) −4.41960e9 −1.18416
\(135\) 3.24692e7 0.00841337
\(136\) −5.28804e9 −1.32547
\(137\) 5.88802e9 1.42800 0.713998 0.700148i \(-0.246883\pi\)
0.713998 + 0.700148i \(0.246883\pi\)
\(138\) 5.10725e6 0.00119876
\(139\) −2.41235e9 −0.548117 −0.274058 0.961713i \(-0.588366\pi\)
−0.274058 + 0.961713i \(0.588366\pi\)
\(140\) −7.45721e7 −0.0164059
\(141\) −5.79138e8 −0.123395
\(142\) −3.39059e9 −0.699806
\(143\) −4.11840e8 −0.0823600
\(144\) −4.26080e8 −0.0825771
\(145\) 3.83198e7 0.00719892
\(146\) −2.61507e9 −0.476316
\(147\) −1.43567e9 −0.253586
\(148\) 3.38671e9 0.580230
\(149\) −2.91781e9 −0.484975 −0.242487 0.970155i \(-0.577963\pi\)
−0.242487 + 0.970155i \(0.577963\pi\)
\(150\) −2.52355e9 −0.407006
\(151\) 8.01296e9 1.25429 0.627144 0.778904i \(-0.284224\pi\)
0.627144 + 0.778904i \(0.284224\pi\)
\(152\) 8.82448e9 1.34089
\(153\) 2.82454e9 0.416713
\(154\) 4.26022e8 0.0610365
\(155\) 6.26123e7 0.00871299
\(156\) 1.52745e9 0.206494
\(157\) −4.74620e9 −0.623444 −0.311722 0.950173i \(-0.600906\pi\)
−0.311722 + 0.950173i \(0.600906\pi\)
\(158\) −8.68444e9 −1.10863
\(159\) −8.17236e9 −1.01405
\(160\) 3.20830e8 0.0387021
\(161\) 1.87677e7 0.00220138
\(162\) 6.87966e8 0.0784783
\(163\) 7.60267e9 0.843571 0.421786 0.906696i \(-0.361403\pi\)
0.421786 + 0.906696i \(0.361403\pi\)
\(164\) 1.79667e9 0.193941
\(165\) 2.77314e7 0.00291269
\(166\) 9.01440e9 0.921405
\(167\) −7.86896e9 −0.782876 −0.391438 0.920204i \(-0.628022\pi\)
−0.391438 + 0.920204i \(0.628022\pi\)
\(168\) −4.73301e9 −0.458401
\(169\) −5.20297e9 −0.490638
\(170\) 4.20361e8 0.0386013
\(171\) −4.71349e9 −0.421561
\(172\) −8.00204e9 −0.697145
\(173\) −1.07654e10 −0.913741 −0.456871 0.889533i \(-0.651030\pi\)
−0.456871 + 0.889533i \(0.651030\pi\)
\(174\) 8.11929e8 0.0671501
\(175\) −9.27331e9 −0.747418
\(176\) −3.63907e8 −0.0285880
\(177\) 9.81506e8 0.0751646
\(178\) −5.97904e9 −0.446417
\(179\) −4.69084e9 −0.341517 −0.170758 0.985313i \(-0.554622\pi\)
−0.170758 + 0.985313i \(0.554622\pi\)
\(180\) −1.02852e8 −0.00730272
\(181\) 6.97407e9 0.482984 0.241492 0.970403i \(-0.422363\pi\)
0.241492 + 0.970403i \(0.422363\pi\)
\(182\) −5.58755e9 −0.377485
\(183\) −4.00643e9 −0.264075
\(184\) −4.84609e7 −0.00311682
\(185\) −8.06439e8 −0.0506173
\(186\) 1.32665e9 0.0812731
\(187\) 2.41239e9 0.144265
\(188\) 1.83451e9 0.107105
\(189\) 2.52808e9 0.144116
\(190\) −7.01483e8 −0.0390504
\(191\) −1.76966e10 −0.962142 −0.481071 0.876682i \(-0.659752\pi\)
−0.481071 + 0.876682i \(0.659752\pi\)
\(192\) 9.49106e9 0.504033
\(193\) −1.86734e10 −0.968761 −0.484380 0.874858i \(-0.660955\pi\)
−0.484380 + 0.874858i \(0.660955\pi\)
\(194\) 2.20531e10 1.11779
\(195\) −3.63715e8 −0.0180138
\(196\) 4.54771e9 0.220110
\(197\) 3.63127e10 1.71775 0.858875 0.512185i \(-0.171164\pi\)
0.858875 + 0.512185i \(0.171164\pi\)
\(198\) 5.87580e8 0.0271690
\(199\) −5.95324e9 −0.269100 −0.134550 0.990907i \(-0.542959\pi\)
−0.134550 + 0.990907i \(0.542959\pi\)
\(200\) 2.39450e10 1.05823
\(201\) −2.23996e10 −0.967962
\(202\) 1.21933e10 0.515274
\(203\) 2.98361e9 0.123313
\(204\) −8.94719e9 −0.361702
\(205\) −4.27820e8 −0.0169188
\(206\) −2.77681e10 −1.07434
\(207\) 2.58848e7 0.000979894 0
\(208\) 4.77287e9 0.176805
\(209\) −4.02571e9 −0.145943
\(210\) 3.76240e8 0.0133499
\(211\) 3.77367e10 1.31067 0.655334 0.755340i \(-0.272528\pi\)
0.655334 + 0.755340i \(0.272528\pi\)
\(212\) 2.58873e10 0.880186
\(213\) −1.71844e10 −0.572038
\(214\) −1.49020e10 −0.485716
\(215\) 1.90544e9 0.0608165
\(216\) −6.52787e9 −0.204047
\(217\) 4.87504e9 0.149248
\(218\) −2.04754e10 −0.614012
\(219\) −1.32538e10 −0.389352
\(220\) −8.78437e7 −0.00252818
\(221\) −3.16400e10 −0.892219
\(222\) −1.70870e10 −0.472148
\(223\) 1.69256e10 0.458324 0.229162 0.973388i \(-0.426401\pi\)
0.229162 + 0.973388i \(0.426401\pi\)
\(224\) 2.49800e10 0.662945
\(225\) −1.27900e10 −0.332696
\(226\) 4.53605e9 0.115662
\(227\) 7.85769e10 1.96417 0.982084 0.188442i \(-0.0603437\pi\)
0.982084 + 0.188442i \(0.0603437\pi\)
\(228\) 1.49307e10 0.365910
\(229\) −4.19589e10 −1.00824 −0.504121 0.863633i \(-0.668183\pi\)
−0.504121 + 0.863633i \(0.668183\pi\)
\(230\) 3.85230e6 9.07704e−5 0
\(231\) 2.15919e9 0.0498927
\(232\) −7.70411e9 −0.174593
\(233\) −3.02106e10 −0.671519 −0.335759 0.941948i \(-0.608993\pi\)
−0.335759 + 0.941948i \(0.608993\pi\)
\(234\) −7.70647e9 −0.168029
\(235\) −4.36832e8 −0.00934348
\(236\) −3.10908e9 −0.0652421
\(237\) −4.40149e10 −0.906217
\(238\) 3.27296e10 0.661218
\(239\) 6.22144e10 1.23339 0.616695 0.787202i \(-0.288471\pi\)
0.616695 + 0.787202i \(0.288471\pi\)
\(240\) −3.21383e8 −0.00625277
\(241\) 1.51274e10 0.288861 0.144430 0.989515i \(-0.453865\pi\)
0.144430 + 0.989515i \(0.453865\pi\)
\(242\) −3.71825e10 −0.696899
\(243\) 3.48678e9 0.0641500
\(244\) 1.26910e10 0.229214
\(245\) −1.08289e9 −0.0192017
\(246\) −9.06476e9 −0.157815
\(247\) 5.27997e10 0.902599
\(248\) −1.25881e10 −0.211313
\(249\) 4.56872e10 0.753178
\(250\) −3.81056e9 −0.0616963
\(251\) 2.58692e10 0.411387 0.205694 0.978616i \(-0.434055\pi\)
0.205694 + 0.978616i \(0.434055\pi\)
\(252\) −8.00810e9 −0.125091
\(253\) 2.21078e7 0.000339237 0
\(254\) 2.69883e10 0.406840
\(255\) 2.13049e9 0.0315536
\(256\) −7.30334e10 −1.06278
\(257\) 3.89821e10 0.557399 0.278699 0.960378i \(-0.410097\pi\)
0.278699 + 0.960378i \(0.410097\pi\)
\(258\) 4.03728e10 0.567284
\(259\) −6.27900e10 −0.867044
\(260\) 1.15212e9 0.0156358
\(261\) 4.11506e9 0.0548901
\(262\) −5.79819e10 −0.760215
\(263\) −5.55292e10 −0.715682 −0.357841 0.933783i \(-0.616487\pi\)
−0.357841 + 0.933783i \(0.616487\pi\)
\(264\) −5.57534e9 −0.0706405
\(265\) −6.16424e9 −0.0767844
\(266\) −5.46180e10 −0.668911
\(267\) −3.03032e10 −0.364912
\(268\) 7.09544e10 0.840181
\(269\) −1.80216e10 −0.209849 −0.104925 0.994480i \(-0.533460\pi\)
−0.104925 + 0.994480i \(0.533460\pi\)
\(270\) 5.18919e8 0.00594241
\(271\) 1.52363e11 1.71600 0.858001 0.513648i \(-0.171706\pi\)
0.858001 + 0.513648i \(0.171706\pi\)
\(272\) −2.79575e10 −0.309698
\(273\) −2.83191e10 −0.308566
\(274\) 9.41015e10 1.00860
\(275\) −1.09237e10 −0.115179
\(276\) −8.19944e7 −0.000850537 0
\(277\) −4.45229e8 −0.00454386 −0.00227193 0.999997i \(-0.500723\pi\)
−0.00227193 + 0.999997i \(0.500723\pi\)
\(278\) −3.85538e10 −0.387138
\(279\) 6.72377e9 0.0664345
\(280\) −3.57001e9 −0.0347103
\(281\) 1.75664e11 1.68075 0.840377 0.542002i \(-0.182334\pi\)
0.840377 + 0.542002i \(0.182334\pi\)
\(282\) −9.25569e9 −0.0871541
\(283\) −3.05315e10 −0.282950 −0.141475 0.989942i \(-0.545184\pi\)
−0.141475 + 0.989942i \(0.545184\pi\)
\(284\) 5.44342e10 0.496523
\(285\) −3.55529e9 −0.0319207
\(286\) −6.58196e9 −0.0581712
\(287\) −3.33104e10 −0.289809
\(288\) 3.44530e10 0.295095
\(289\) 6.67466e10 0.562845
\(290\) 6.12421e8 0.00508463
\(291\) 1.11770e11 0.913710
\(292\) 4.19836e10 0.337953
\(293\) 1.77930e11 1.41041 0.705205 0.709003i \(-0.250855\pi\)
0.705205 + 0.709003i \(0.250855\pi\)
\(294\) −2.29446e10 −0.179109
\(295\) 7.40330e8 0.00569149
\(296\) 1.62133e11 1.22760
\(297\) 2.97800e9 0.0222086
\(298\) −4.66320e10 −0.342540
\(299\) −2.89957e8 −0.00209804
\(300\) 4.05143e10 0.288777
\(301\) 1.48359e11 1.04175
\(302\) 1.28062e11 0.885909
\(303\) 6.17985e10 0.421198
\(304\) 4.66545e10 0.313301
\(305\) −3.02196e9 −0.0199959
\(306\) 4.51414e10 0.294326
\(307\) 2.97466e11 1.91124 0.955620 0.294602i \(-0.0951870\pi\)
0.955620 + 0.294602i \(0.0951870\pi\)
\(308\) −6.83957e9 −0.0433063
\(309\) −1.40735e11 −0.878194
\(310\) 1.00066e9 0.00615403
\(311\) 1.82887e11 1.10856 0.554282 0.832329i \(-0.312993\pi\)
0.554282 + 0.832329i \(0.312993\pi\)
\(312\) 7.31240e10 0.436882
\(313\) 8.49405e10 0.500225 0.250112 0.968217i \(-0.419532\pi\)
0.250112 + 0.968217i \(0.419532\pi\)
\(314\) −7.58530e10 −0.440341
\(315\) 1.90688e9 0.0109125
\(316\) 1.39424e11 0.786586
\(317\) −2.29248e11 −1.27509 −0.637543 0.770415i \(-0.720049\pi\)
−0.637543 + 0.770415i \(0.720049\pi\)
\(318\) −1.30609e11 −0.716230
\(319\) 3.51460e9 0.0190028
\(320\) 7.15891e9 0.0381656
\(321\) −7.55271e10 −0.397036
\(322\) 2.99943e8 0.00155484
\(323\) −3.09279e11 −1.58103
\(324\) −1.10449e10 −0.0556815
\(325\) 1.43271e11 0.712332
\(326\) 1.21505e11 0.595818
\(327\) −1.03774e11 −0.501908
\(328\) 8.60123e10 0.410325
\(329\) −3.40120e10 −0.160048
\(330\) 4.43199e8 0.00205725
\(331\) 3.80256e11 1.74121 0.870603 0.491986i \(-0.163729\pi\)
0.870603 + 0.491986i \(0.163729\pi\)
\(332\) −1.44722e11 −0.653751
\(333\) −8.66013e10 −0.385945
\(334\) −1.25760e11 −0.552949
\(335\) −1.68956e10 −0.0732944
\(336\) −2.50231e10 −0.107106
\(337\) −2.10397e11 −0.888599 −0.444299 0.895878i \(-0.646547\pi\)
−0.444299 + 0.895878i \(0.646547\pi\)
\(338\) −8.31530e10 −0.346540
\(339\) 2.29899e10 0.0945448
\(340\) −6.74868e9 −0.0273882
\(341\) 5.74265e9 0.0229995
\(342\) −7.53303e10 −0.297750
\(343\) −2.76278e11 −1.07776
\(344\) −3.83084e11 −1.47496
\(345\) 1.95244e7 7.41979e−5 0
\(346\) −1.72051e11 −0.645380
\(347\) −1.31135e10 −0.0485552 −0.0242776 0.999705i \(-0.507729\pi\)
−0.0242776 + 0.999705i \(0.507729\pi\)
\(348\) −1.30351e10 −0.0476440
\(349\) 4.66318e11 1.68255 0.841275 0.540608i \(-0.181806\pi\)
0.841275 + 0.540608i \(0.181806\pi\)
\(350\) −1.48205e11 −0.527905
\(351\) −3.90583e10 −0.137351
\(352\) 2.94257e10 0.102161
\(353\) −8.93581e10 −0.306301 −0.153150 0.988203i \(-0.548942\pi\)
−0.153150 + 0.988203i \(0.548942\pi\)
\(354\) 1.56863e10 0.0530891
\(355\) −1.29618e10 −0.0433150
\(356\) 9.59904e10 0.316740
\(357\) 1.65882e11 0.540495
\(358\) −7.49683e10 −0.241215
\(359\) 3.45879e11 1.09900 0.549502 0.835492i \(-0.314817\pi\)
0.549502 + 0.835492i \(0.314817\pi\)
\(360\) −4.92384e9 −0.0154505
\(361\) 1.93426e11 0.599422
\(362\) 1.11458e11 0.341134
\(363\) −1.88450e11 −0.569662
\(364\) 8.97053e10 0.267832
\(365\) −9.99708e9 −0.0294819
\(366\) −6.40301e10 −0.186517
\(367\) 7.78994e10 0.224149 0.112074 0.993700i \(-0.464250\pi\)
0.112074 + 0.993700i \(0.464250\pi\)
\(368\) −2.56210e8 −0.000728251 0
\(369\) −4.59425e10 −0.129002
\(370\) −1.28884e10 −0.0357512
\(371\) −4.79952e11 −1.31527
\(372\) −2.12986e10 −0.0576645
\(373\) 3.52486e11 0.942870 0.471435 0.881901i \(-0.343736\pi\)
0.471435 + 0.881901i \(0.343736\pi\)
\(374\) 3.85545e10 0.101895
\(375\) −1.93129e10 −0.0504320
\(376\) 8.78240e10 0.226604
\(377\) −4.60962e10 −0.117525
\(378\) 4.04034e10 0.101790
\(379\) −7.71787e11 −1.92142 −0.960708 0.277562i \(-0.910473\pi\)
−0.960708 + 0.277562i \(0.910473\pi\)
\(380\) 1.12619e10 0.0277069
\(381\) 1.36783e11 0.332561
\(382\) −2.82824e11 −0.679566
\(383\) −5.61364e11 −1.33306 −0.666530 0.745478i \(-0.732221\pi\)
−0.666530 + 0.745478i \(0.732221\pi\)
\(384\) −6.60925e10 −0.155118
\(385\) 1.62863e9 0.00377789
\(386\) −2.98436e11 −0.684240
\(387\) 2.04620e11 0.463712
\(388\) −3.54051e11 −0.793091
\(389\) −1.29451e10 −0.0286636 −0.0143318 0.999897i \(-0.504562\pi\)
−0.0143318 + 0.999897i \(0.504562\pi\)
\(390\) −5.81283e9 −0.0127232
\(391\) 1.69845e9 0.00367501
\(392\) 2.17713e11 0.465691
\(393\) −2.93867e11 −0.621418
\(394\) 5.80344e11 1.21326
\(395\) −3.31995e10 −0.0686191
\(396\) −9.43329e9 −0.0192768
\(397\) −4.55778e11 −0.920866 −0.460433 0.887695i \(-0.652306\pi\)
−0.460433 + 0.887695i \(0.652306\pi\)
\(398\) −9.51438e10 −0.190067
\(399\) −2.76817e11 −0.546784
\(400\) 1.26596e11 0.247258
\(401\) −7.04177e11 −1.35998 −0.679989 0.733222i \(-0.738016\pi\)
−0.679989 + 0.733222i \(0.738016\pi\)
\(402\) −3.57987e11 −0.683676
\(403\) −7.53184e10 −0.142242
\(404\) −1.95757e11 −0.365595
\(405\) 2.63001e9 0.00485746
\(406\) 4.76836e10 0.0870967
\(407\) −7.39646e10 −0.133613
\(408\) −4.28331e11 −0.765260
\(409\) −3.04678e11 −0.538376 −0.269188 0.963088i \(-0.586755\pi\)
−0.269188 + 0.963088i \(0.586755\pi\)
\(410\) −6.83736e9 −0.0119498
\(411\) 4.76930e11 0.824454
\(412\) 4.45802e11 0.762263
\(413\) 5.76426e10 0.0974919
\(414\) 4.13688e8 0.000692103 0
\(415\) 3.44609e10 0.0570310
\(416\) −3.85937e11 −0.631824
\(417\) −1.95400e11 −0.316455
\(418\) −6.43383e10 −0.103080
\(419\) −1.10294e12 −1.74818 −0.874092 0.485760i \(-0.838543\pi\)
−0.874092 + 0.485760i \(0.838543\pi\)
\(420\) −6.04034e9 −0.00947196
\(421\) 3.72442e11 0.577815 0.288908 0.957357i \(-0.406708\pi\)
0.288908 + 0.957357i \(0.406708\pi\)
\(422\) 6.03102e11 0.925730
\(423\) −4.69102e10 −0.0712419
\(424\) 1.23931e12 1.86223
\(425\) −8.39223e11 −1.24775
\(426\) −2.74638e11 −0.404033
\(427\) −2.35292e11 −0.342517
\(428\) 2.39244e11 0.344623
\(429\) −3.33590e10 −0.0475505
\(430\) 3.04524e10 0.0429550
\(431\) −8.30400e11 −1.15915 −0.579575 0.814919i \(-0.696782\pi\)
−0.579575 + 0.814919i \(0.696782\pi\)
\(432\) −3.45124e10 −0.0476759
\(433\) −1.15753e12 −1.58248 −0.791238 0.611508i \(-0.790563\pi\)
−0.791238 + 0.611508i \(0.790563\pi\)
\(434\) 7.79122e10 0.105415
\(435\) 3.10390e9 0.00415630
\(436\) 3.28721e11 0.435651
\(437\) −2.83432e9 −0.00371776
\(438\) −2.11820e11 −0.275001
\(439\) 4.39794e11 0.565144 0.282572 0.959246i \(-0.408812\pi\)
0.282572 + 0.959246i \(0.408812\pi\)
\(440\) −4.20536e9 −0.00534892
\(441\) −1.16289e11 −0.146408
\(442\) −5.05666e11 −0.630178
\(443\) −7.14220e11 −0.881080 −0.440540 0.897733i \(-0.645213\pi\)
−0.440540 + 0.897733i \(0.645213\pi\)
\(444\) 2.74324e11 0.334996
\(445\) −2.28571e10 −0.0276313
\(446\) 2.70503e11 0.323717
\(447\) −2.36343e11 −0.280000
\(448\) 5.57398e11 0.653754
\(449\) −1.44352e12 −1.67616 −0.838078 0.545551i \(-0.816320\pi\)
−0.838078 + 0.545551i \(0.816320\pi\)
\(450\) −2.04407e11 −0.234985
\(451\) −3.92386e10 −0.0446601
\(452\) −7.28241e10 −0.0820639
\(453\) 6.49050e11 0.724163
\(454\) 1.25580e12 1.38730
\(455\) −2.13605e10 −0.0233647
\(456\) 7.14783e11 0.774163
\(457\) 9.30712e11 0.998142 0.499071 0.866561i \(-0.333675\pi\)
0.499071 + 0.866561i \(0.333675\pi\)
\(458\) −6.70581e11 −0.712126
\(459\) 2.28788e11 0.240589
\(460\) −6.18467e7 −6.44030e−5 0
\(461\) −4.17109e11 −0.430125 −0.215063 0.976600i \(-0.568996\pi\)
−0.215063 + 0.976600i \(0.568996\pi\)
\(462\) 3.45078e10 0.0352394
\(463\) 6.97753e11 0.705646 0.352823 0.935690i \(-0.385222\pi\)
0.352823 + 0.935690i \(0.385222\pi\)
\(464\) −4.07312e10 −0.0407940
\(465\) 5.07160e9 0.00503045
\(466\) −4.82822e11 −0.474297
\(467\) 1.07830e12 1.04909 0.524545 0.851383i \(-0.324235\pi\)
0.524545 + 0.851383i \(0.324235\pi\)
\(468\) 1.23724e11 0.119219
\(469\) −1.31550e12 −1.25549
\(470\) −6.98138e9 −0.00659934
\(471\) −3.84442e11 −0.359945
\(472\) −1.48842e11 −0.138034
\(473\) 1.74762e11 0.160536
\(474\) −7.03439e11 −0.640065
\(475\) 1.40046e12 1.26227
\(476\) −5.25457e11 −0.469144
\(477\) −6.61961e11 −0.585463
\(478\) 9.94302e11 0.871150
\(479\) −2.23813e11 −0.194257 −0.0971283 0.995272i \(-0.530966\pi\)
−0.0971283 + 0.995272i \(0.530966\pi\)
\(480\) 2.59872e10 0.0223447
\(481\) 9.70092e11 0.826343
\(482\) 2.41764e11 0.204024
\(483\) 1.52018e9 0.00127097
\(484\) 5.96947e11 0.494460
\(485\) 8.43061e10 0.0691865
\(486\) 5.57253e10 0.0453095
\(487\) 7.00426e11 0.564264 0.282132 0.959376i \(-0.408958\pi\)
0.282132 + 0.959376i \(0.408958\pi\)
\(488\) 6.07559e11 0.484953
\(489\) 6.15816e11 0.487036
\(490\) −1.73066e10 −0.0135622
\(491\) 7.68487e11 0.596719 0.298359 0.954454i \(-0.403561\pi\)
0.298359 + 0.954454i \(0.403561\pi\)
\(492\) 1.45530e11 0.111972
\(493\) 2.70013e11 0.205861
\(494\) 8.43837e11 0.637510
\(495\) 2.24624e9 0.00168164
\(496\) −6.65523e10 −0.0493737
\(497\) −1.00922e12 −0.741960
\(498\) 7.30167e11 0.531973
\(499\) 2.27431e12 1.64209 0.821044 0.570865i \(-0.193392\pi\)
0.821044 + 0.570865i \(0.193392\pi\)
\(500\) 6.11766e10 0.0437744
\(501\) −6.37385e11 −0.451994
\(502\) 4.13437e11 0.290565
\(503\) −1.50660e11 −0.104940 −0.0524700 0.998623i \(-0.516709\pi\)
−0.0524700 + 0.998623i \(0.516709\pi\)
\(504\) −3.83374e11 −0.264658
\(505\) 4.66133e10 0.0318932
\(506\) 3.53323e8 0.000239604 0
\(507\) −4.21440e11 −0.283270
\(508\) −4.33283e11 −0.288659
\(509\) 6.47058e11 0.427281 0.213640 0.976912i \(-0.431468\pi\)
0.213640 + 0.976912i \(0.431468\pi\)
\(510\) 3.40492e10 0.0222865
\(511\) −7.78380e11 −0.505007
\(512\) −7.49439e11 −0.481971
\(513\) −3.81793e11 −0.243388
\(514\) 6.23006e11 0.393693
\(515\) −1.06154e11 −0.0664972
\(516\) −6.48165e11 −0.402497
\(517\) −4.00651e10 −0.0246638
\(518\) −1.00350e12 −0.612397
\(519\) −8.71999e11 −0.527549
\(520\) 5.51559e10 0.0330809
\(521\) −1.54003e12 −0.915713 −0.457857 0.889026i \(-0.651383\pi\)
−0.457857 + 0.889026i \(0.651383\pi\)
\(522\) 6.57663e10 0.0387691
\(523\) 2.04280e12 1.19390 0.596949 0.802279i \(-0.296379\pi\)
0.596949 + 0.802279i \(0.296379\pi\)
\(524\) 9.30870e11 0.539384
\(525\) −7.51138e11 −0.431522
\(526\) −8.87459e11 −0.505490
\(527\) 4.41185e11 0.249157
\(528\) −2.94765e10 −0.0165053
\(529\) −1.80114e12 −0.999991
\(530\) −9.85160e10 −0.0542332
\(531\) 7.95020e10 0.0433963
\(532\) 8.76864e11 0.474602
\(533\) 5.14639e11 0.276204
\(534\) −4.84302e11 −0.257739
\(535\) −5.69685e10 −0.0300637
\(536\) 3.39682e12 1.77759
\(537\) −3.79958e11 −0.197175
\(538\) −2.88018e11 −0.148217
\(539\) −9.93204e10 −0.0506861
\(540\) −8.33097e9 −0.00421623
\(541\) −3.00028e12 −1.50582 −0.752912 0.658121i \(-0.771352\pi\)
−0.752912 + 0.658121i \(0.771352\pi\)
\(542\) 2.43504e12 1.21202
\(543\) 5.64899e11 0.278851
\(544\) 2.26066e12 1.10673
\(545\) −7.82747e10 −0.0380047
\(546\) −4.52591e11 −0.217941
\(547\) 2.43565e12 1.16325 0.581625 0.813457i \(-0.302417\pi\)
0.581625 + 0.813457i \(0.302417\pi\)
\(548\) −1.51075e12 −0.715617
\(549\) −3.24521e11 −0.152464
\(550\) −1.74581e11 −0.0813512
\(551\) −4.50587e11 −0.208256
\(552\) −3.92534e9 −0.00179950
\(553\) −2.58494e12 −1.17540
\(554\) −7.11559e9 −0.00320935
\(555\) −6.53216e10 −0.0292239
\(556\) 6.18961e11 0.274680
\(557\) 1.08394e12 0.477151 0.238575 0.971124i \(-0.423320\pi\)
0.238575 + 0.971124i \(0.423320\pi\)
\(558\) 1.07458e11 0.0469230
\(559\) −2.29211e12 −0.992848
\(560\) −1.88744e10 −0.00811012
\(561\) 1.95404e11 0.0832914
\(562\) 2.80743e12 1.18712
\(563\) −3.18803e12 −1.33732 −0.668659 0.743569i \(-0.733131\pi\)
−0.668659 + 0.743569i \(0.733131\pi\)
\(564\) 1.48595e11 0.0618372
\(565\) 1.73408e10 0.00715897
\(566\) −4.87950e11 −0.199849
\(567\) 2.04774e11 0.0832055
\(568\) 2.60594e12 1.05050
\(569\) 2.82225e12 1.12873 0.564366 0.825525i \(-0.309121\pi\)
0.564366 + 0.825525i \(0.309121\pi\)
\(570\) −5.68201e10 −0.0225458
\(571\) −3.48287e12 −1.37112 −0.685559 0.728017i \(-0.740442\pi\)
−0.685559 + 0.728017i \(0.740442\pi\)
\(572\) 1.05670e11 0.0412734
\(573\) −1.43342e12 −0.555493
\(574\) −5.32362e11 −0.204693
\(575\) −7.69086e9 −0.00293406
\(576\) 7.68776e11 0.291004
\(577\) 8.09796e11 0.304148 0.152074 0.988369i \(-0.451405\pi\)
0.152074 + 0.988369i \(0.451405\pi\)
\(578\) 1.06673e12 0.397540
\(579\) −1.51255e12 −0.559314
\(580\) −9.83211e9 −0.00360762
\(581\) 2.68315e12 0.976907
\(582\) 1.78630e12 0.645358
\(583\) −5.65369e11 −0.202686
\(584\) 2.00989e12 0.715014
\(585\) −2.94609e10 −0.0104003
\(586\) 2.84365e12 0.996179
\(587\) 2.65276e12 0.922203 0.461101 0.887348i \(-0.347454\pi\)
0.461101 + 0.887348i \(0.347454\pi\)
\(588\) 3.68364e11 0.127081
\(589\) −7.36233e11 −0.252056
\(590\) 1.18318e10 0.00401993
\(591\) 2.94133e12 0.991744
\(592\) 8.57186e11 0.286832
\(593\) −1.08368e12 −0.359879 −0.179940 0.983678i \(-0.557590\pi\)
−0.179940 + 0.983678i \(0.557590\pi\)
\(594\) 4.75940e10 0.0156860
\(595\) 1.25121e11 0.0409265
\(596\) 7.48654e11 0.243037
\(597\) −4.82212e11 −0.155365
\(598\) −4.63405e9 −0.00148186
\(599\) −1.57285e12 −0.499191 −0.249595 0.968350i \(-0.580298\pi\)
−0.249595 + 0.968350i \(0.580298\pi\)
\(600\) 1.93955e12 0.610970
\(601\) 8.26489e10 0.0258406 0.0129203 0.999917i \(-0.495887\pi\)
0.0129203 + 0.999917i \(0.495887\pi\)
\(602\) 2.37105e12 0.735793
\(603\) −1.81437e12 −0.558853
\(604\) −2.05597e12 −0.628566
\(605\) −1.42144e11 −0.0431350
\(606\) 9.87654e11 0.297494
\(607\) −1.82660e12 −0.546127 −0.273063 0.961996i \(-0.588037\pi\)
−0.273063 + 0.961996i \(0.588037\pi\)
\(608\) −3.77251e12 −1.11960
\(609\) 2.41672e11 0.0711949
\(610\) −4.82966e10 −0.0141232
\(611\) 5.25479e11 0.152535
\(612\) −7.24723e11 −0.208829
\(613\) −8.02251e11 −0.229476 −0.114738 0.993396i \(-0.536603\pi\)
−0.114738 + 0.993396i \(0.536603\pi\)
\(614\) 4.75406e12 1.34992
\(615\) −3.46534e10 −0.00976806
\(616\) −3.27433e11 −0.0916239
\(617\) 1.88160e11 0.0522691 0.0261345 0.999658i \(-0.491680\pi\)
0.0261345 + 0.999658i \(0.491680\pi\)
\(618\) −2.24921e12 −0.620272
\(619\) 3.64562e12 0.998076 0.499038 0.866580i \(-0.333687\pi\)
0.499038 + 0.866580i \(0.333687\pi\)
\(620\) −1.60651e10 −0.00436637
\(621\) 2.09667e9 0.000565742 0
\(622\) 2.92287e12 0.782984
\(623\) −1.77967e12 −0.473308
\(624\) 3.86602e11 0.102078
\(625\) 3.79284e12 0.994270
\(626\) 1.35751e12 0.353311
\(627\) −3.26083e11 −0.0842604
\(628\) 1.21778e12 0.312429
\(629\) −5.68241e12 −1.44745
\(630\) 3.04754e10 0.00770757
\(631\) −6.95665e12 −1.74690 −0.873450 0.486914i \(-0.838123\pi\)
−0.873450 + 0.486914i \(0.838123\pi\)
\(632\) 6.67469e12 1.66419
\(633\) 3.05667e12 0.756714
\(634\) −3.66381e12 −0.900599
\(635\) 1.03173e11 0.0251816
\(636\) 2.09687e12 0.508176
\(637\) 1.30265e12 0.313473
\(638\) 5.61698e10 0.0134218
\(639\) −1.39193e12 −0.330266
\(640\) −4.98522e10 −0.0117456
\(641\) −6.18919e12 −1.44801 −0.724007 0.689792i \(-0.757702\pi\)
−0.724007 + 0.689792i \(0.757702\pi\)
\(642\) −1.20706e12 −0.280428
\(643\) 2.77548e11 0.0640307 0.0320154 0.999487i \(-0.489807\pi\)
0.0320154 + 0.999487i \(0.489807\pi\)
\(644\) −4.81543e9 −0.00110319
\(645\) 1.54340e11 0.0351124
\(646\) −4.94286e12 −1.11669
\(647\) −6.51803e12 −1.46234 −0.731168 0.682197i \(-0.761024\pi\)
−0.731168 + 0.682197i \(0.761024\pi\)
\(648\) −5.28757e11 −0.117806
\(649\) 6.79013e10 0.0150237
\(650\) 2.28973e12 0.503124
\(651\) 3.94879e11 0.0861686
\(652\) −1.95070e12 −0.422742
\(653\) 7.85123e12 1.68977 0.844887 0.534945i \(-0.179668\pi\)
0.844887 + 0.534945i \(0.179668\pi\)
\(654\) −1.65850e12 −0.354500
\(655\) −2.21658e11 −0.0470540
\(656\) 4.54742e11 0.0958732
\(657\) −1.07356e12 −0.224792
\(658\) −5.43575e11 −0.113043
\(659\) 3.89784e11 0.0805081 0.0402540 0.999189i \(-0.487183\pi\)
0.0402540 + 0.999189i \(0.487183\pi\)
\(660\) −7.11534e9 −0.00145965
\(661\) 1.65637e12 0.337482 0.168741 0.985660i \(-0.446030\pi\)
0.168741 + 0.985660i \(0.446030\pi\)
\(662\) 6.07720e12 1.22982
\(663\) −2.56284e12 −0.515123
\(664\) −6.92829e12 −1.38315
\(665\) −2.08798e11 −0.0414027
\(666\) −1.38405e12 −0.272595
\(667\) 2.47447e9 0.000484078 0
\(668\) 2.01902e12 0.392326
\(669\) 1.37098e12 0.264614
\(670\) −2.70022e11 −0.0517682
\(671\) −2.77167e11 −0.0527826
\(672\) 2.02338e12 0.382751
\(673\) 8.32730e11 0.156472 0.0782360 0.996935i \(-0.475071\pi\)
0.0782360 + 0.996935i \(0.475071\pi\)
\(674\) −3.36254e12 −0.627622
\(675\) −1.03599e12 −0.192082
\(676\) 1.33498e12 0.245875
\(677\) −1.03042e13 −1.88524 −0.942620 0.333868i \(-0.891646\pi\)
−0.942620 + 0.333868i \(0.891646\pi\)
\(678\) 3.67420e11 0.0667774
\(679\) 6.56414e12 1.18512
\(680\) −3.23081e11 −0.0579457
\(681\) 6.36473e12 1.13401
\(682\) 9.17782e10 0.0162446
\(683\) 3.22944e12 0.567850 0.283925 0.958847i \(-0.408363\pi\)
0.283925 + 0.958847i \(0.408363\pi\)
\(684\) 1.20939e12 0.211258
\(685\) 3.59738e11 0.0624279
\(686\) −4.41544e12 −0.761229
\(687\) −3.39867e12 −0.582108
\(688\) −2.02534e12 −0.344627
\(689\) 7.41517e12 1.25353
\(690\) 3.12036e8 5.24063e−5 0
\(691\) −6.35783e12 −1.06086 −0.530430 0.847729i \(-0.677969\pi\)
−0.530430 + 0.847729i \(0.677969\pi\)
\(692\) 2.76220e12 0.457907
\(693\) 1.74894e11 0.0288055
\(694\) −2.09578e11 −0.0342948
\(695\) −1.47386e11 −0.0239621
\(696\) −6.24033e11 −0.100801
\(697\) −3.01455e12 −0.483810
\(698\) 7.45263e12 1.18839
\(699\) −2.44706e12 −0.387702
\(700\) 2.37935e12 0.374557
\(701\) −6.24586e11 −0.0976925 −0.0488463 0.998806i \(-0.515554\pi\)
−0.0488463 + 0.998806i \(0.515554\pi\)
\(702\) −6.24224e11 −0.0970116
\(703\) 9.48259e12 1.46429
\(704\) 6.56598e11 0.100745
\(705\) −3.53834e10 −0.00539446
\(706\) −1.42811e12 −0.216342
\(707\) 3.62935e12 0.546312
\(708\) −2.51835e11 −0.0376675
\(709\) −5.99601e12 −0.891157 −0.445579 0.895243i \(-0.647002\pi\)
−0.445579 + 0.895243i \(0.647002\pi\)
\(710\) −2.07154e11 −0.0305936
\(711\) −3.56521e12 −0.523204
\(712\) 4.59537e12 0.670132
\(713\) 4.04314e9 0.000585889 0
\(714\) 2.65110e12 0.381754
\(715\) −2.51620e10 −0.00360054
\(716\) 1.20358e12 0.171146
\(717\) 5.03937e12 0.712099
\(718\) 5.52779e12 0.776232
\(719\) −5.44093e12 −0.759265 −0.379632 0.925137i \(-0.623949\pi\)
−0.379632 + 0.925137i \(0.623949\pi\)
\(720\) −2.60320e10 −0.00361004
\(721\) −8.26522e12 −1.13906
\(722\) 3.09131e12 0.423375
\(723\) 1.22532e12 0.166774
\(724\) −1.78941e12 −0.242039
\(725\) −1.22266e12 −0.164356
\(726\) −3.01178e12 −0.402355
\(727\) 3.33589e11 0.0442902 0.0221451 0.999755i \(-0.492950\pi\)
0.0221451 + 0.999755i \(0.492950\pi\)
\(728\) 4.29448e12 0.566656
\(729\) 2.82430e11 0.0370370
\(730\) −1.59772e11 −0.0208232
\(731\) 1.34263e13 1.73911
\(732\) 1.02797e12 0.132337
\(733\) 6.61739e12 0.846680 0.423340 0.905971i \(-0.360858\pi\)
0.423340 + 0.905971i \(0.360858\pi\)
\(734\) 1.24498e12 0.158317
\(735\) −8.77144e10 −0.0110861
\(736\) 2.07173e10 0.00260245
\(737\) −1.54962e12 −0.193474
\(738\) −7.34245e11 −0.0911145
\(739\) −5.63499e12 −0.695014 −0.347507 0.937677i \(-0.612972\pi\)
−0.347507 + 0.937677i \(0.612972\pi\)
\(740\) 2.06917e11 0.0253660
\(741\) 4.27677e12 0.521116
\(742\) −7.67053e12 −0.928983
\(743\) −1.64104e13 −1.97547 −0.987735 0.156140i \(-0.950095\pi\)
−0.987735 + 0.156140i \(0.950095\pi\)
\(744\) −1.01963e12 −0.122002
\(745\) −1.78268e11 −0.0212017
\(746\) 5.63337e12 0.665953
\(747\) 3.70067e12 0.434848
\(748\) −6.18973e11 −0.0722960
\(749\) −4.43561e12 −0.514974
\(750\) −3.08655e11 −0.0356204
\(751\) 4.91387e10 0.00563694 0.00281847 0.999996i \(-0.499103\pi\)
0.00281847 + 0.999996i \(0.499103\pi\)
\(752\) 4.64320e11 0.0529465
\(753\) 2.09540e12 0.237515
\(754\) −7.36702e11 −0.0830081
\(755\) 4.89565e11 0.0548339
\(756\) −6.48656e11 −0.0722215
\(757\) −4.10791e12 −0.454663 −0.227332 0.973817i \(-0.573000\pi\)
−0.227332 + 0.973817i \(0.573000\pi\)
\(758\) −1.23346e13 −1.35710
\(759\) 1.79073e9 0.000195858 0
\(760\) 5.39146e11 0.0586199
\(761\) −3.72963e12 −0.403121 −0.201560 0.979476i \(-0.564601\pi\)
−0.201560 + 0.979476i \(0.564601\pi\)
\(762\) 2.18605e12 0.234889
\(763\) −6.09453e12 −0.650998
\(764\) 4.54060e12 0.482162
\(765\) 1.72570e11 0.0182175
\(766\) −8.97163e12 −0.941547
\(767\) −8.90567e11 −0.0929153
\(768\) −5.91570e12 −0.613594
\(769\) −5.30984e12 −0.547537 −0.273768 0.961796i \(-0.588270\pi\)
−0.273768 + 0.961796i \(0.588270\pi\)
\(770\) 2.60285e10 0.00266834
\(771\) 3.15755e12 0.321814
\(772\) 4.79124e12 0.485479
\(773\) 1.53805e13 1.54940 0.774700 0.632329i \(-0.217901\pi\)
0.774700 + 0.632329i \(0.217901\pi\)
\(774\) 3.27020e12 0.327522
\(775\) −1.99775e12 −0.198923
\(776\) −1.69496e13 −1.67796
\(777\) −5.08599e12 −0.500588
\(778\) −2.06886e11 −0.0202453
\(779\) 5.03057e12 0.489439
\(780\) 9.33221e10 0.00902732
\(781\) −1.18883e12 −0.114337
\(782\) 2.71444e10 0.00259567
\(783\) 3.33320e11 0.0316908
\(784\) 1.15104e12 0.108810
\(785\) −2.89977e11 −0.0272552
\(786\) −4.69653e12 −0.438910
\(787\) 2.05993e12 0.191410 0.0957052 0.995410i \(-0.469489\pi\)
0.0957052 + 0.995410i \(0.469489\pi\)
\(788\) −9.31712e12 −0.860823
\(789\) −4.49786e12 −0.413199
\(790\) −5.30590e11 −0.0484660
\(791\) 1.35017e12 0.122629
\(792\) −4.51602e11 −0.0407843
\(793\) 3.63522e12 0.326439
\(794\) −7.28418e12 −0.650412
\(795\) −4.99303e11 −0.0443315
\(796\) 1.52748e12 0.134855
\(797\) −6.13048e12 −0.538186 −0.269093 0.963114i \(-0.586724\pi\)
−0.269093 + 0.963114i \(0.586724\pi\)
\(798\) −4.42406e12 −0.386196
\(799\) −3.07805e12 −0.267187
\(800\) −1.02366e13 −0.883592
\(801\) −2.45456e12 −0.210682
\(802\) −1.12540e13 −0.960559
\(803\) −9.16908e11 −0.0778226
\(804\) 5.74730e12 0.485078
\(805\) 1.14664e9 9.62381e−5 0
\(806\) −1.20373e12 −0.100466
\(807\) −1.45975e12 −0.121156
\(808\) −9.37150e12 −0.773496
\(809\) 1.02968e13 0.845147 0.422574 0.906329i \(-0.361127\pi\)
0.422574 + 0.906329i \(0.361127\pi\)
\(810\) 4.20324e10 0.00343085
\(811\) 1.95327e13 1.58551 0.792754 0.609541i \(-0.208646\pi\)
0.792754 + 0.609541i \(0.208646\pi\)
\(812\) −7.65536e11 −0.0617964
\(813\) 1.23414e13 0.990734
\(814\) −1.18209e12 −0.0943716
\(815\) 4.64497e11 0.0368786
\(816\) −2.26456e12 −0.178804
\(817\) −2.24053e13 −1.75934
\(818\) −4.86931e12 −0.380258
\(819\) −2.29385e12 −0.178150
\(820\) 1.09770e11 0.00847857
\(821\) −4.94835e12 −0.380116 −0.190058 0.981773i \(-0.560868\pi\)
−0.190058 + 0.981773i \(0.560868\pi\)
\(822\) 7.62222e12 0.582315
\(823\) −7.29946e12 −0.554615 −0.277308 0.960781i \(-0.589442\pi\)
−0.277308 + 0.960781i \(0.589442\pi\)
\(824\) 2.13420e13 1.61273
\(825\) −8.84818e11 −0.0664984
\(826\) 9.21236e11 0.0688590
\(827\) −6.55140e12 −0.487034 −0.243517 0.969897i \(-0.578301\pi\)
−0.243517 + 0.969897i \(0.578301\pi\)
\(828\) −6.64155e9 −0.000491058 0
\(829\) 6.85363e12 0.503994 0.251997 0.967728i \(-0.418913\pi\)
0.251997 + 0.967728i \(0.418913\pi\)
\(830\) 5.50749e11 0.0402812
\(831\) −3.60636e10 −0.00262340
\(832\) −8.61169e12 −0.623065
\(833\) −7.63039e12 −0.549091
\(834\) −3.12286e12 −0.223514
\(835\) −4.80767e11 −0.0342251
\(836\) 1.03292e12 0.0731371
\(837\) 5.44625e11 0.0383560
\(838\) −1.76270e13 −1.23475
\(839\) −1.69365e12 −0.118003 −0.0590017 0.998258i \(-0.518792\pi\)
−0.0590017 + 0.998258i \(0.518792\pi\)
\(840\) −2.89171e11 −0.0200400
\(841\) −1.41138e13 −0.972884
\(842\) 5.95231e12 0.408114
\(843\) 1.42288e13 0.970384
\(844\) −9.68249e12 −0.656820
\(845\) −3.17884e11 −0.0214493
\(846\) −7.49711e11 −0.0503185
\(847\) −1.10674e13 −0.738877
\(848\) 6.55214e12 0.435113
\(849\) −2.47305e12 −0.163361
\(850\) −1.34123e13 −0.881291
\(851\) −5.20751e10 −0.00340367
\(852\) 4.40917e12 0.286668
\(853\) 5.29571e12 0.342495 0.171247 0.985228i \(-0.445220\pi\)
0.171247 + 0.985228i \(0.445220\pi\)
\(854\) −3.76041e12 −0.241922
\(855\) −2.87978e11 −0.0184295
\(856\) 1.14534e13 0.729125
\(857\) 1.96625e12 0.124516 0.0622579 0.998060i \(-0.480170\pi\)
0.0622579 + 0.998060i \(0.480170\pi\)
\(858\) −5.33139e11 −0.0335852
\(859\) 9.30080e12 0.582842 0.291421 0.956595i \(-0.405872\pi\)
0.291421 + 0.956595i \(0.405872\pi\)
\(860\) −4.88898e11 −0.0304772
\(861\) −2.69814e12 −0.167321
\(862\) −1.32713e13 −0.818713
\(863\) 1.38352e13 0.849059 0.424530 0.905414i \(-0.360439\pi\)
0.424530 + 0.905414i \(0.360439\pi\)
\(864\) 2.79070e12 0.170373
\(865\) −6.57730e11 −0.0399462
\(866\) −1.84995e13 −1.11771
\(867\) 5.40648e12 0.324959
\(868\) −1.25084e12 −0.0747934
\(869\) −3.04498e12 −0.181132
\(870\) 4.96061e10 0.00293561
\(871\) 2.03242e13 1.19655
\(872\) 1.57370e13 0.921715
\(873\) 9.05340e12 0.527531
\(874\) −4.52976e10 −0.00262587
\(875\) −1.13422e12 −0.0654126
\(876\) 3.40067e12 0.195117
\(877\) −5.14801e11 −0.0293860 −0.0146930 0.999892i \(-0.504677\pi\)
−0.0146930 + 0.999892i \(0.504677\pi\)
\(878\) 7.02872e12 0.399164
\(879\) 1.44123e13 0.814301
\(880\) −2.22335e10 −0.00124979
\(881\) −1.48523e13 −0.830621 −0.415310 0.909680i \(-0.636327\pi\)
−0.415310 + 0.909680i \(0.636327\pi\)
\(882\) −1.85851e12 −0.103409
\(883\) 7.61722e12 0.421671 0.210835 0.977522i \(-0.432382\pi\)
0.210835 + 0.977522i \(0.432382\pi\)
\(884\) 8.11821e12 0.447121
\(885\) 5.99667e10 0.00328598
\(886\) −1.14146e13 −0.622311
\(887\) −1.19196e13 −0.646556 −0.323278 0.946304i \(-0.604785\pi\)
−0.323278 + 0.946304i \(0.604785\pi\)
\(888\) 1.31328e13 0.708757
\(889\) 8.03311e12 0.431346
\(890\) −3.65299e11 −0.0195161
\(891\) 2.41218e11 0.0128221
\(892\) −4.34279e12 −0.229682
\(893\) 5.13653e12 0.270295
\(894\) −3.77719e12 −0.197765
\(895\) −2.86595e11 −0.0149302
\(896\) −3.88153e12 −0.201195
\(897\) −2.34865e10 −0.00121130
\(898\) −2.30701e13 −1.18388
\(899\) 6.42760e11 0.0328194
\(900\) 3.28165e12 0.166725
\(901\) −4.34351e13 −2.19573
\(902\) −6.27106e11 −0.0315436
\(903\) 1.20171e13 0.601455
\(904\) −3.48632e12 −0.173624
\(905\) 4.26092e11 0.0211147
\(906\) 1.03730e13 0.511480
\(907\) 8.17267e12 0.400988 0.200494 0.979695i \(-0.435745\pi\)
0.200494 + 0.979695i \(0.435745\pi\)
\(908\) −2.01613e13 −0.984311
\(909\) 5.00568e12 0.243179
\(910\) −3.41380e11 −0.0165026
\(911\) −3.55337e12 −0.170926 −0.0854630 0.996341i \(-0.527237\pi\)
−0.0854630 + 0.996341i \(0.527237\pi\)
\(912\) 3.77901e12 0.180885
\(913\) 3.16067e12 0.150543
\(914\) 1.48745e13 0.704992
\(915\) −2.44779e11 −0.0115446
\(916\) 1.07658e13 0.505264
\(917\) −1.72584e13 −0.806007
\(918\) 3.65645e12 0.169929
\(919\) −2.00027e12 −0.0925060 −0.0462530 0.998930i \(-0.514728\pi\)
−0.0462530 + 0.998930i \(0.514728\pi\)
\(920\) −2.96080e9 −0.000136259 0
\(921\) 2.40948e13 1.10346
\(922\) −6.66617e12 −0.303799
\(923\) 1.55922e13 0.707130
\(924\) −5.54006e11 −0.0250029
\(925\) 2.57308e13 1.15562
\(926\) 1.11514e13 0.498401
\(927\) −1.13996e13 −0.507026
\(928\) 3.29355e12 0.145780
\(929\) 1.77193e13 0.780507 0.390253 0.920707i \(-0.372387\pi\)
0.390253 + 0.920707i \(0.372387\pi\)
\(930\) 8.10535e10 0.00355303
\(931\) 1.27333e13 0.555479
\(932\) 7.75146e12 0.336521
\(933\) 1.48138e13 0.640030
\(934\) 1.72332e13 0.740977
\(935\) 1.47389e11 0.00630685
\(936\) 5.92304e12 0.252234
\(937\) 2.49060e13 1.05554 0.527771 0.849387i \(-0.323028\pi\)
0.527771 + 0.849387i \(0.323028\pi\)
\(938\) −2.10241e13 −0.886759
\(939\) 6.88018e12 0.288805
\(940\) 1.12082e11 0.00468233
\(941\) −3.15204e13 −1.31050 −0.655252 0.755410i \(-0.727438\pi\)
−0.655252 + 0.755410i \(0.727438\pi\)
\(942\) −6.14409e12 −0.254231
\(943\) −2.76261e10 −0.00113767
\(944\) −7.86917e11 −0.0322519
\(945\) 1.54457e11 0.00630035
\(946\) 2.79302e12 0.113387
\(947\) −3.38266e13 −1.36673 −0.683367 0.730075i \(-0.739485\pi\)
−0.683367 + 0.730075i \(0.739485\pi\)
\(948\) 1.12934e13 0.454136
\(949\) 1.20258e13 0.481301
\(950\) 2.23820e13 0.891544
\(951\) −1.85691e13 −0.736171
\(952\) −2.51553e13 −0.992577
\(953\) 1.17005e13 0.459503 0.229751 0.973249i \(-0.426209\pi\)
0.229751 + 0.973249i \(0.426209\pi\)
\(954\) −1.05794e13 −0.413515
\(955\) −1.08120e12 −0.0420621
\(956\) −1.59630e13 −0.618094
\(957\) 2.84683e11 0.0109713
\(958\) −3.57695e12 −0.137204
\(959\) 2.80095e13 1.06935
\(960\) 5.79872e11 0.0220349
\(961\) −2.53894e13 −0.960278
\(962\) 1.55039e13 0.583650
\(963\) −6.11769e12 −0.229229
\(964\) −3.88140e12 −0.144758
\(965\) −1.14088e12 −0.0423515
\(966\) 2.42954e10 0.000897690 0
\(967\) 5.41005e13 1.98968 0.994838 0.101475i \(-0.0323563\pi\)
0.994838 + 0.101475i \(0.0323563\pi\)
\(968\) 2.85778e13 1.04614
\(969\) −2.50516e13 −0.912807
\(970\) 1.34737e12 0.0488667
\(971\) 1.68712e12 0.0609059 0.0304529 0.999536i \(-0.490305\pi\)
0.0304529 + 0.999536i \(0.490305\pi\)
\(972\) −8.94641e11 −0.0321477
\(973\) −1.14756e13 −0.410457
\(974\) 1.11941e13 0.398542
\(975\) 1.16049e13 0.411265
\(976\) 3.21213e12 0.113310
\(977\) −9.67341e12 −0.339668 −0.169834 0.985473i \(-0.554323\pi\)
−0.169834 + 0.985473i \(0.554323\pi\)
\(978\) 9.84188e12 0.343996
\(979\) −2.09640e12 −0.0729376
\(980\) 2.77849e11 0.00962260
\(981\) −8.40571e12 −0.289777
\(982\) 1.22818e13 0.421465
\(983\) 3.21482e13 1.09816 0.549080 0.835770i \(-0.314978\pi\)
0.549080 + 0.835770i \(0.314978\pi\)
\(984\) 6.96700e12 0.236901
\(985\) 2.21858e12 0.0750952
\(986\) 4.31530e12 0.145400
\(987\) −2.75498e12 −0.0924039
\(988\) −1.35474e13 −0.452323
\(989\) 1.23042e11 0.00408949
\(990\) 3.58991e10 0.00118775
\(991\) −3.41521e13 −1.12483 −0.562413 0.826856i \(-0.690127\pi\)
−0.562413 + 0.826856i \(0.690127\pi\)
\(992\) 5.38146e12 0.176440
\(993\) 3.08007e13 1.00529
\(994\) −1.61291e13 −0.524050
\(995\) −3.63723e11 −0.0117643
\(996\) −1.17225e13 −0.377443
\(997\) 3.99927e13 1.28190 0.640948 0.767584i \(-0.278541\pi\)
0.640948 + 0.767584i \(0.278541\pi\)
\(998\) 3.63476e13 1.15981
\(999\) −7.01471e12 −0.222825
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.1.15 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.a.1.15 21 1.1 even 1 trivial