Properties

Label 177.12.a.b.1.6
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $27$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(1\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-63.4822 q^{2} +243.000 q^{3} +1982.00 q^{4} -1374.17 q^{5} -15426.2 q^{6} -16112.8 q^{7} +4190.11 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-63.4822 q^{2} +243.000 q^{3} +1982.00 q^{4} -1374.17 q^{5} -15426.2 q^{6} -16112.8 q^{7} +4190.11 q^{8} +59049.0 q^{9} +87235.4 q^{10} +391891. q^{11} +481625. q^{12} -1.24240e6 q^{13} +1.02288e6 q^{14} -333923. q^{15} -4.32512e6 q^{16} +824345. q^{17} -3.74856e6 q^{18} +4.80225e6 q^{19} -2.72360e6 q^{20} -3.91541e6 q^{21} -2.48781e7 q^{22} +2.79540e7 q^{23} +1.01820e6 q^{24} -4.69398e7 q^{25} +7.88702e7 q^{26} +1.43489e7 q^{27} -3.19355e7 q^{28} +4.28560e7 q^{29} +2.11982e7 q^{30} -1.46002e8 q^{31} +2.65987e8 q^{32} +9.52295e7 q^{33} -5.23313e7 q^{34} +2.21418e7 q^{35} +1.17035e8 q^{36} +6.91833e8 q^{37} -3.04857e8 q^{38} -3.01903e8 q^{39} -5.75793e6 q^{40} -1.20424e9 q^{41} +2.48559e8 q^{42} +3.82608e8 q^{43} +7.76726e8 q^{44} -8.11434e7 q^{45} -1.77458e9 q^{46} -1.39550e7 q^{47} -1.05101e9 q^{48} -1.71770e9 q^{49} +2.97984e9 q^{50} +2.00316e8 q^{51} -2.46243e9 q^{52} -4.64080e9 q^{53} -9.10901e8 q^{54} -5.38525e8 q^{55} -6.75145e7 q^{56} +1.16695e9 q^{57} -2.72059e9 q^{58} -7.14924e8 q^{59} -6.61835e8 q^{60} -4.97725e9 q^{61} +9.26851e9 q^{62} -9.51446e8 q^{63} -8.02761e9 q^{64} +1.70727e9 q^{65} -6.04538e9 q^{66} +1.85064e10 q^{67} +1.63385e9 q^{68} +6.79282e9 q^{69} -1.40561e9 q^{70} +4.49987e9 q^{71} +2.47422e8 q^{72} +2.44140e10 q^{73} -4.39191e10 q^{74} -1.14064e10 q^{75} +9.51803e9 q^{76} -6.31447e9 q^{77} +1.91655e10 q^{78} -1.48608e10 q^{79} +5.94346e9 q^{80} +3.48678e9 q^{81} +7.64476e10 q^{82} +5.49701e10 q^{83} -7.76033e9 q^{84} -1.13279e9 q^{85} -2.42888e10 q^{86} +1.04140e10 q^{87} +1.64207e9 q^{88} +4.35958e10 q^{89} +5.15116e9 q^{90} +2.00185e10 q^{91} +5.54047e10 q^{92} -3.54784e10 q^{93} +8.85893e8 q^{94} -6.59911e9 q^{95} +6.46349e10 q^{96} +1.25330e11 q^{97} +1.09044e11 q^{98} +2.31408e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} - 383719q^{10} - 1816556q^{11} + 6352506q^{12} - 3951804q^{13} - 6207867q^{14} - 4176684q^{15} + 28295194q^{16} - 17723275q^{17} - 7558272q^{18} - 19573013q^{19} - 48468099q^{20} - 30758697q^{21} - 1729910q^{22} - 88593797q^{23} - 86458671q^{24} + 345714963q^{25} - 6676346q^{26} + 387420489q^{27} + 126954286q^{28} - 276632427q^{29} - 93243717q^{30} - 357680917q^{31} - 859842334q^{32} - 441423108q^{33} + 232730000q^{34} - 510315139q^{35} + 1543658958q^{36} - 660238257q^{37} - 2067286961q^{38} - 960288372q^{39} - 3388951110q^{40} - 1671147569q^{41} - 1508511681q^{42} - 1883107790q^{43} - 3895687630q^{44} - 1014934212q^{45} - 1720344243q^{46} - 5818572501q^{47} + 6875732142q^{48} - 18858180q^{49} - 21474519647q^{50} - 4306755825q^{51} - 42214560062q^{52} - 11444513368q^{53} - 1836660096q^{54} - 24401486484q^{55} - 50583585764q^{56} - 4756242159q^{57} - 45017395090q^{58} - 19302956073q^{59} - 11777748057q^{60} + 408637955q^{61} - 28543084070q^{62} - 7474363371q^{63} + 33067284293q^{64} - 21656714730q^{65} - 420368130q^{66} - 49803132690q^{67} - 16500749319q^{68} - 21528292671q^{69} - 45808890782q^{70} - 34127492216q^{71} - 21009457053q^{72} - 55734362153q^{73} - 40367816298q^{74} + 84008736009q^{75} - 14840406404q^{76} - 99723443615q^{77} - 1622352078q^{78} - 76484916442q^{79} + 93882788915q^{80} + 94143178827q^{81} + 52951239205q^{82} - 140433865655q^{83} + 30849891498q^{84} + 34329063335q^{85} + 175223869508q^{86} - 67221679761q^{87} + 268823645069q^{88} - 1191878597q^{89} - 22658223231q^{90} + 201632581559q^{91} - 206501888812q^{92} - 86916462831q^{93} + 319770144384q^{94} - 81387074885q^{95} - 208941687162q^{96} - 144896178730q^{97} + 135739195260q^{98} - 107265815244q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −63.4822 −1.40277 −0.701386 0.712781i \(-0.747435\pi\)
−0.701386 + 0.712781i \(0.747435\pi\)
\(3\) 243.000 0.577350
\(4\) 1982.00 0.967771
\(5\) −1374.17 −0.196655 −0.0983276 0.995154i \(-0.531349\pi\)
−0.0983276 + 0.995154i \(0.531349\pi\)
\(6\) −15426.2 −0.809891
\(7\) −16112.8 −0.362353 −0.181177 0.983451i \(-0.557991\pi\)
−0.181177 + 0.983451i \(0.557991\pi\)
\(8\) 4190.11 0.0452096
\(9\) 59049.0 0.333333
\(10\) 87235.4 0.275863
\(11\) 391891. 0.733678 0.366839 0.930284i \(-0.380440\pi\)
0.366839 + 0.930284i \(0.380440\pi\)
\(12\) 481625. 0.558743
\(13\) −1.24240e6 −0.928052 −0.464026 0.885822i \(-0.653595\pi\)
−0.464026 + 0.885822i \(0.653595\pi\)
\(14\) 1.02288e6 0.508299
\(15\) −333923. −0.113539
\(16\) −4.32512e6 −1.03119
\(17\) 824345. 0.140812 0.0704061 0.997518i \(-0.477570\pi\)
0.0704061 + 0.997518i \(0.477570\pi\)
\(18\) −3.74856e6 −0.467591
\(19\) 4.80225e6 0.444938 0.222469 0.974940i \(-0.428588\pi\)
0.222469 + 0.974940i \(0.428588\pi\)
\(20\) −2.72360e6 −0.190317
\(21\) −3.91541e6 −0.209205
\(22\) −2.48781e7 −1.02918
\(23\) 2.79540e7 0.905609 0.452804 0.891610i \(-0.350424\pi\)
0.452804 + 0.891610i \(0.350424\pi\)
\(24\) 1.01820e6 0.0261018
\(25\) −4.69398e7 −0.961327
\(26\) 7.88702e7 1.30185
\(27\) 1.43489e7 0.192450
\(28\) −3.19355e7 −0.350675
\(29\) 4.28560e7 0.387992 0.193996 0.981002i \(-0.437855\pi\)
0.193996 + 0.981002i \(0.437855\pi\)
\(30\) 2.11982e7 0.159269
\(31\) −1.46002e8 −0.915943 −0.457972 0.888967i \(-0.651424\pi\)
−0.457972 + 0.888967i \(0.651424\pi\)
\(32\) 2.65987e8 1.40132
\(33\) 9.52295e7 0.423589
\(34\) −5.23313e7 −0.197527
\(35\) 2.21418e7 0.0712587
\(36\) 1.17035e8 0.322590
\(37\) 6.91833e8 1.64018 0.820090 0.572235i \(-0.193923\pi\)
0.820090 + 0.572235i \(0.193923\pi\)
\(38\) −3.04857e8 −0.624147
\(39\) −3.01903e8 −0.535811
\(40\) −5.75793e6 −0.00889070
\(41\) −1.20424e9 −1.62331 −0.811653 0.584139i \(-0.801432\pi\)
−0.811653 + 0.584139i \(0.801432\pi\)
\(42\) 2.48559e8 0.293467
\(43\) 3.82608e8 0.396897 0.198448 0.980111i \(-0.436410\pi\)
0.198448 + 0.980111i \(0.436410\pi\)
\(44\) 7.76726e8 0.710033
\(45\) −8.11434e7 −0.0655517
\(46\) −1.77458e9 −1.27036
\(47\) −1.39550e7 −0.00887545 −0.00443773 0.999990i \(-0.501413\pi\)
−0.00443773 + 0.999990i \(0.501413\pi\)
\(48\) −1.05101e9 −0.595358
\(49\) −1.71770e9 −0.868700
\(50\) 2.97984e9 1.34852
\(51\) 2.00316e8 0.0812979
\(52\) −2.46243e9 −0.898142
\(53\) −4.64080e9 −1.52432 −0.762158 0.647391i \(-0.775860\pi\)
−0.762158 + 0.647391i \(0.775860\pi\)
\(54\) −9.10901e8 −0.269964
\(55\) −5.38525e8 −0.144282
\(56\) −6.75145e7 −0.0163818
\(57\) 1.16695e9 0.256885
\(58\) −2.72059e9 −0.544264
\(59\) −7.14924e8 −0.130189
\(60\) −6.61835e8 −0.109880
\(61\) −4.97725e9 −0.754528 −0.377264 0.926106i \(-0.623135\pi\)
−0.377264 + 0.926106i \(0.623135\pi\)
\(62\) 9.26851e9 1.28486
\(63\) −9.51446e8 −0.120784
\(64\) −8.02761e9 −0.934537
\(65\) 1.70727e9 0.182506
\(66\) −6.04538e9 −0.594199
\(67\) 1.85064e10 1.67460 0.837299 0.546745i \(-0.184133\pi\)
0.837299 + 0.546745i \(0.184133\pi\)
\(68\) 1.63385e9 0.136274
\(69\) 6.79282e9 0.522853
\(70\) −1.40561e9 −0.0999597
\(71\) 4.49987e9 0.295991 0.147996 0.988988i \(-0.452718\pi\)
0.147996 + 0.988988i \(0.452718\pi\)
\(72\) 2.47422e8 0.0150699
\(73\) 2.44140e10 1.37836 0.689182 0.724588i \(-0.257970\pi\)
0.689182 + 0.724588i \(0.257970\pi\)
\(74\) −4.39191e10 −2.30080
\(75\) −1.14064e10 −0.555022
\(76\) 9.51803e9 0.430598
\(77\) −6.31447e9 −0.265851
\(78\) 1.91655e10 0.751621
\(79\) −1.48608e10 −0.543368 −0.271684 0.962387i \(-0.587581\pi\)
−0.271684 + 0.962387i \(0.587581\pi\)
\(80\) 5.94346e9 0.202789
\(81\) 3.48678e9 0.111111
\(82\) 7.64476e10 2.27713
\(83\) 5.49701e10 1.53178 0.765891 0.642971i \(-0.222298\pi\)
0.765891 + 0.642971i \(0.222298\pi\)
\(84\) −7.76033e9 −0.202462
\(85\) −1.13279e9 −0.0276914
\(86\) −2.42888e10 −0.556756
\(87\) 1.04140e10 0.224007
\(88\) 1.64207e9 0.0331693
\(89\) 4.35958e10 0.827560 0.413780 0.910377i \(-0.364208\pi\)
0.413780 + 0.910377i \(0.364208\pi\)
\(90\) 5.15116e9 0.0919542
\(91\) 2.00185e10 0.336282
\(92\) 5.54047e10 0.876422
\(93\) −3.54784e10 −0.528820
\(94\) 8.85893e8 0.0124502
\(95\) −6.59911e9 −0.0874994
\(96\) 6.46349e10 0.809050
\(97\) 1.25330e11 1.48187 0.740937 0.671574i \(-0.234382\pi\)
0.740937 + 0.671574i \(0.234382\pi\)
\(98\) 1.09044e11 1.21859
\(99\) 2.31408e10 0.244559
\(100\) −9.30344e10 −0.930344
\(101\) −4.85151e10 −0.459313 −0.229657 0.973272i \(-0.573760\pi\)
−0.229657 + 0.973272i \(0.573760\pi\)
\(102\) −1.27165e10 −0.114043
\(103\) −1.20559e11 −1.02469 −0.512346 0.858779i \(-0.671223\pi\)
−0.512346 + 0.858779i \(0.671223\pi\)
\(104\) −5.20578e9 −0.0419568
\(105\) 5.38045e9 0.0411412
\(106\) 2.94608e11 2.13827
\(107\) −9.35045e10 −0.644498 −0.322249 0.946655i \(-0.604439\pi\)
−0.322249 + 0.946655i \(0.604439\pi\)
\(108\) 2.84395e10 0.186248
\(109\) 1.39138e11 0.866162 0.433081 0.901355i \(-0.357426\pi\)
0.433081 + 0.901355i \(0.357426\pi\)
\(110\) 3.41868e10 0.202394
\(111\) 1.68115e11 0.946958
\(112\) 6.96899e10 0.373655
\(113\) −1.28480e11 −0.656001 −0.328000 0.944678i \(-0.606375\pi\)
−0.328000 + 0.944678i \(0.606375\pi\)
\(114\) −7.40804e10 −0.360352
\(115\) −3.84135e10 −0.178093
\(116\) 8.49403e10 0.375487
\(117\) −7.33623e10 −0.309351
\(118\) 4.53850e10 0.182625
\(119\) −1.32825e10 −0.0510237
\(120\) −1.39918e9 −0.00513305
\(121\) −1.31733e11 −0.461717
\(122\) 3.15967e11 1.05843
\(123\) −2.92629e11 −0.937217
\(124\) −2.89375e11 −0.886423
\(125\) 1.31601e11 0.385705
\(126\) 6.03999e10 0.169433
\(127\) −1.20433e10 −0.0323464 −0.0161732 0.999869i \(-0.505148\pi\)
−0.0161732 + 0.999869i \(0.505148\pi\)
\(128\) −3.51309e10 −0.0903722
\(129\) 9.29737e10 0.229148
\(130\) −1.08381e11 −0.256015
\(131\) 5.85420e11 1.32579 0.662896 0.748711i \(-0.269327\pi\)
0.662896 + 0.748711i \(0.269327\pi\)
\(132\) 1.88744e11 0.409937
\(133\) −7.73777e10 −0.161225
\(134\) −1.17483e12 −2.34908
\(135\) −1.97178e10 −0.0378463
\(136\) 3.45410e9 0.00636606
\(137\) −4.31517e11 −0.763897 −0.381948 0.924184i \(-0.624747\pi\)
−0.381948 + 0.924184i \(0.624747\pi\)
\(138\) −4.31223e11 −0.733445
\(139\) 5.71359e11 0.933960 0.466980 0.884268i \(-0.345342\pi\)
0.466980 + 0.884268i \(0.345342\pi\)
\(140\) 4.38849e10 0.0689621
\(141\) −3.39106e9 −0.00512425
\(142\) −2.85662e11 −0.415208
\(143\) −4.86884e11 −0.680891
\(144\) −2.55394e11 −0.343730
\(145\) −5.88914e10 −0.0763006
\(146\) −1.54986e12 −1.93353
\(147\) −4.17402e11 −0.501544
\(148\) 1.37121e12 1.58732
\(149\) −1.13013e12 −1.26068 −0.630338 0.776321i \(-0.717084\pi\)
−0.630338 + 0.776321i \(0.717084\pi\)
\(150\) 7.24102e11 0.778570
\(151\) 7.30726e11 0.757497 0.378749 0.925500i \(-0.376354\pi\)
0.378749 + 0.925500i \(0.376354\pi\)
\(152\) 2.01219e10 0.0201155
\(153\) 4.86768e10 0.0469374
\(154\) 4.00857e11 0.372928
\(155\) 2.00631e11 0.180125
\(156\) −5.98370e11 −0.518542
\(157\) −4.75618e11 −0.397933 −0.198967 0.980006i \(-0.563759\pi\)
−0.198967 + 0.980006i \(0.563759\pi\)
\(158\) 9.43398e11 0.762221
\(159\) −1.12771e12 −0.880064
\(160\) −3.65512e11 −0.275576
\(161\) −4.50417e11 −0.328150
\(162\) −2.21349e11 −0.155864
\(163\) 1.66878e12 1.13597 0.567985 0.823039i \(-0.307723\pi\)
0.567985 + 0.823039i \(0.307723\pi\)
\(164\) −2.38679e12 −1.57099
\(165\) −1.30862e11 −0.0833010
\(166\) −3.48962e12 −2.14874
\(167\) −4.86847e10 −0.0290036 −0.0145018 0.999895i \(-0.504616\pi\)
−0.0145018 + 0.999895i \(0.504616\pi\)
\(168\) −1.64060e10 −0.00945806
\(169\) −2.48609e11 −0.138720
\(170\) 7.19121e10 0.0388448
\(171\) 2.83568e11 0.148313
\(172\) 7.58327e11 0.384105
\(173\) −3.95225e12 −1.93906 −0.969528 0.244981i \(-0.921218\pi\)
−0.969528 + 0.244981i \(0.921218\pi\)
\(174\) −6.61104e11 −0.314231
\(175\) 7.56332e11 0.348340
\(176\) −1.69498e12 −0.756561
\(177\) −1.73727e11 −0.0751646
\(178\) −2.76756e12 −1.16088
\(179\) 2.86039e12 1.16341 0.581706 0.813399i \(-0.302386\pi\)
0.581706 + 0.813399i \(0.302386\pi\)
\(180\) −1.60826e11 −0.0634391
\(181\) −4.07848e12 −1.56051 −0.780254 0.625462i \(-0.784910\pi\)
−0.780254 + 0.625462i \(0.784910\pi\)
\(182\) −1.27082e12 −0.471728
\(183\) −1.20947e12 −0.435627
\(184\) 1.17130e11 0.0409422
\(185\) −9.50696e11 −0.322550
\(186\) 2.25225e12 0.741814
\(187\) 3.23054e11 0.103311
\(188\) −2.76587e10 −0.00858941
\(189\) −2.31201e11 −0.0697349
\(190\) 4.18926e11 0.122742
\(191\) −3.61284e12 −1.02841 −0.514204 0.857668i \(-0.671912\pi\)
−0.514204 + 0.857668i \(0.671912\pi\)
\(192\) −1.95071e12 −0.539555
\(193\) −3.43973e12 −0.924612 −0.462306 0.886720i \(-0.652978\pi\)
−0.462306 + 0.886720i \(0.652978\pi\)
\(194\) −7.95624e12 −2.07873
\(195\) 4.14866e11 0.105370
\(196\) −3.40448e12 −0.840703
\(197\) −4.20044e12 −1.00863 −0.504313 0.863521i \(-0.668254\pi\)
−0.504313 + 0.863521i \(0.668254\pi\)
\(198\) −1.46903e12 −0.343061
\(199\) −1.56899e12 −0.356393 −0.178197 0.983995i \(-0.557026\pi\)
−0.178197 + 0.983995i \(0.557026\pi\)
\(200\) −1.96683e11 −0.0434612
\(201\) 4.49706e12 0.966830
\(202\) 3.07985e12 0.644312
\(203\) −6.90530e11 −0.140590
\(204\) 3.97025e11 0.0786778
\(205\) 1.65483e12 0.319232
\(206\) 7.65333e12 1.43741
\(207\) 1.65065e12 0.301870
\(208\) 5.37352e12 0.956997
\(209\) 1.88196e12 0.326441
\(210\) −3.41563e11 −0.0577118
\(211\) −1.07129e13 −1.76341 −0.881707 0.471796i \(-0.843606\pi\)
−0.881707 + 0.471796i \(0.843606\pi\)
\(212\) −9.19804e12 −1.47519
\(213\) 1.09347e12 0.170891
\(214\) 5.93588e12 0.904085
\(215\) −5.25768e11 −0.0780518
\(216\) 6.01235e10 0.00870059
\(217\) 2.35250e12 0.331895
\(218\) −8.83277e12 −1.21503
\(219\) 5.93261e12 0.795799
\(220\) −1.06735e12 −0.139632
\(221\) −1.02416e12 −0.130681
\(222\) −1.06723e13 −1.32837
\(223\) −9.90267e12 −1.20247 −0.601237 0.799071i \(-0.705325\pi\)
−0.601237 + 0.799071i \(0.705325\pi\)
\(224\) −4.28580e12 −0.507771
\(225\) −2.77175e12 −0.320442
\(226\) 8.15620e12 0.920220
\(227\) −1.03526e13 −1.14001 −0.570005 0.821641i \(-0.693059\pi\)
−0.570005 + 0.821641i \(0.693059\pi\)
\(228\) 2.31288e12 0.248606
\(229\) −1.12813e13 −1.18376 −0.591882 0.806024i \(-0.701615\pi\)
−0.591882 + 0.806024i \(0.701615\pi\)
\(230\) 2.43858e12 0.249824
\(231\) −1.53442e12 −0.153489
\(232\) 1.79571e11 0.0175409
\(233\) −1.19919e13 −1.14401 −0.572007 0.820249i \(-0.693835\pi\)
−0.572007 + 0.820249i \(0.693835\pi\)
\(234\) 4.65720e12 0.433948
\(235\) 1.91765e10 0.00174540
\(236\) −1.41698e12 −0.125993
\(237\) −3.61118e12 −0.313714
\(238\) 8.43204e11 0.0715747
\(239\) 1.78812e13 1.48323 0.741615 0.670826i \(-0.234060\pi\)
0.741615 + 0.670826i \(0.234060\pi\)
\(240\) 1.44426e12 0.117080
\(241\) 1.29149e13 1.02329 0.511645 0.859197i \(-0.329036\pi\)
0.511645 + 0.859197i \(0.329036\pi\)
\(242\) 8.36271e12 0.647683
\(243\) 8.47289e11 0.0641500
\(244\) −9.86489e12 −0.730210
\(245\) 2.36042e12 0.170834
\(246\) 1.85768e13 1.31470
\(247\) −5.96630e12 −0.412926
\(248\) −6.11763e11 −0.0414094
\(249\) 1.33577e13 0.884375
\(250\) −8.35435e12 −0.541057
\(251\) −1.76652e13 −1.11921 −0.559606 0.828759i \(-0.689048\pi\)
−0.559606 + 0.828759i \(0.689048\pi\)
\(252\) −1.88576e12 −0.116892
\(253\) 1.09549e13 0.664425
\(254\) 7.64537e11 0.0453746
\(255\) −2.75268e11 −0.0159877
\(256\) 1.86707e13 1.06131
\(257\) −4.11262e12 −0.228816 −0.114408 0.993434i \(-0.536497\pi\)
−0.114408 + 0.993434i \(0.536497\pi\)
\(258\) −5.90218e12 −0.321443
\(259\) −1.11474e13 −0.594325
\(260\) 3.38379e12 0.176624
\(261\) 2.53060e12 0.129331
\(262\) −3.71638e13 −1.85979
\(263\) 2.38482e13 1.16869 0.584344 0.811506i \(-0.301352\pi\)
0.584344 + 0.811506i \(0.301352\pi\)
\(264\) 3.99022e11 0.0191503
\(265\) 6.37724e12 0.299765
\(266\) 4.91211e12 0.226162
\(267\) 1.05938e13 0.477792
\(268\) 3.66796e13 1.62063
\(269\) 2.08896e13 0.904256 0.452128 0.891953i \(-0.350665\pi\)
0.452128 + 0.891953i \(0.350665\pi\)
\(270\) 1.25173e12 0.0530898
\(271\) 2.73735e13 1.13762 0.568812 0.822467i \(-0.307403\pi\)
0.568812 + 0.822467i \(0.307403\pi\)
\(272\) −3.56540e12 −0.145204
\(273\) 4.86450e12 0.194153
\(274\) 2.73937e13 1.07157
\(275\) −1.83953e13 −0.705304
\(276\) 1.34633e13 0.506003
\(277\) 2.07159e13 0.763246 0.381623 0.924318i \(-0.375365\pi\)
0.381623 + 0.924318i \(0.375365\pi\)
\(278\) −3.62712e13 −1.31013
\(279\) −8.62125e12 −0.305314
\(280\) 9.27764e10 0.00322158
\(281\) −2.37571e13 −0.808924 −0.404462 0.914555i \(-0.632541\pi\)
−0.404462 + 0.914555i \(0.632541\pi\)
\(282\) 2.15272e11 0.00718815
\(283\) −3.77963e13 −1.23772 −0.618862 0.785500i \(-0.712406\pi\)
−0.618862 + 0.785500i \(0.712406\pi\)
\(284\) 8.91871e12 0.286452
\(285\) −1.60358e12 −0.0505178
\(286\) 3.09085e13 0.955135
\(287\) 1.94036e13 0.588210
\(288\) 1.57063e13 0.467105
\(289\) −3.35924e13 −0.980172
\(290\) 3.73856e12 0.107032
\(291\) 3.04552e13 0.855560
\(292\) 4.83885e13 1.33394
\(293\) −2.72248e13 −0.736535 −0.368268 0.929720i \(-0.620049\pi\)
−0.368268 + 0.929720i \(0.620049\pi\)
\(294\) 2.64976e13 0.703553
\(295\) 9.82428e11 0.0256023
\(296\) 2.89885e12 0.0741519
\(297\) 5.62321e12 0.141196
\(298\) 7.17431e13 1.76844
\(299\) −3.47300e13 −0.840452
\(300\) −2.26074e13 −0.537135
\(301\) −6.16489e12 −0.143817
\(302\) −4.63881e13 −1.06260
\(303\) −1.17892e13 −0.265185
\(304\) −2.07703e13 −0.458816
\(305\) 6.83959e12 0.148382
\(306\) −3.09011e12 −0.0658425
\(307\) −1.75539e13 −0.367378 −0.183689 0.982984i \(-0.558804\pi\)
−0.183689 + 0.982984i \(0.558804\pi\)
\(308\) −1.25152e13 −0.257283
\(309\) −2.92957e13 −0.591606
\(310\) −1.27365e13 −0.252674
\(311\) 4.35889e13 0.849560 0.424780 0.905297i \(-0.360351\pi\)
0.424780 + 0.905297i \(0.360351\pi\)
\(312\) −1.26501e12 −0.0242238
\(313\) 6.40792e13 1.20566 0.602828 0.797871i \(-0.294041\pi\)
0.602828 + 0.797871i \(0.294041\pi\)
\(314\) 3.01933e13 0.558210
\(315\) 1.30745e12 0.0237529
\(316\) −2.94541e13 −0.525856
\(317\) −8.58063e13 −1.50554 −0.752771 0.658282i \(-0.771283\pi\)
−0.752771 + 0.658282i \(0.771283\pi\)
\(318\) 7.15898e13 1.23453
\(319\) 1.67949e13 0.284661
\(320\) 1.10313e13 0.183782
\(321\) −2.27216e13 −0.372101
\(322\) 2.85935e13 0.460320
\(323\) 3.95871e12 0.0626527
\(324\) 6.91079e12 0.107530
\(325\) 5.83179e13 0.892161
\(326\) −1.05938e14 −1.59351
\(327\) 3.38105e13 0.500079
\(328\) −5.04588e12 −0.0733890
\(329\) 2.24854e11 0.00321605
\(330\) 8.30739e12 0.116852
\(331\) −1.07052e12 −0.0148095 −0.00740474 0.999973i \(-0.502357\pi\)
−0.00740474 + 0.999973i \(0.502357\pi\)
\(332\) 1.08950e14 1.48241
\(333\) 4.08520e13 0.546727
\(334\) 3.09062e12 0.0406855
\(335\) −2.54309e13 −0.329318
\(336\) 1.69347e13 0.215730
\(337\) −4.69489e13 −0.588384 −0.294192 0.955746i \(-0.595051\pi\)
−0.294192 + 0.955746i \(0.595051\pi\)
\(338\) 1.57823e13 0.194593
\(339\) −3.12207e13 −0.378742
\(340\) −2.24519e12 −0.0267990
\(341\) −5.72167e13 −0.672007
\(342\) −1.80015e13 −0.208049
\(343\) 5.95373e13 0.677130
\(344\) 1.60317e12 0.0179435
\(345\) −9.33449e12 −0.102822
\(346\) 2.50897e14 2.72005
\(347\) −1.53442e14 −1.63731 −0.818656 0.574284i \(-0.805281\pi\)
−0.818656 + 0.574284i \(0.805281\pi\)
\(348\) 2.06405e13 0.216788
\(349\) −1.46212e14 −1.51162 −0.755811 0.654789i \(-0.772757\pi\)
−0.755811 + 0.654789i \(0.772757\pi\)
\(350\) −4.80136e13 −0.488642
\(351\) −1.78270e13 −0.178604
\(352\) 1.04238e14 1.02811
\(353\) −1.86216e14 −1.80824 −0.904121 0.427277i \(-0.859473\pi\)
−0.904121 + 0.427277i \(0.859473\pi\)
\(354\) 1.10286e13 0.105439
\(355\) −6.18358e12 −0.0582082
\(356\) 8.64067e13 0.800889
\(357\) −3.22765e12 −0.0294586
\(358\) −1.81584e14 −1.63200
\(359\) −1.52519e14 −1.34991 −0.674956 0.737858i \(-0.735838\pi\)
−0.674956 + 0.737858i \(0.735838\pi\)
\(360\) −3.40000e11 −0.00296357
\(361\) −9.34287e13 −0.802030
\(362\) 2.58911e14 2.18904
\(363\) −3.20111e13 −0.266572
\(364\) 3.96766e13 0.325445
\(365\) −3.35491e13 −0.271063
\(366\) 7.67800e13 0.611086
\(367\) 1.95702e14 1.53437 0.767187 0.641424i \(-0.221656\pi\)
0.767187 + 0.641424i \(0.221656\pi\)
\(368\) −1.20904e14 −0.933855
\(369\) −7.11090e13 −0.541102
\(370\) 6.03523e13 0.452464
\(371\) 7.47763e13 0.552341
\(372\) −7.03180e13 −0.511777
\(373\) −2.51032e14 −1.80024 −0.900119 0.435644i \(-0.856521\pi\)
−0.900119 + 0.435644i \(0.856521\pi\)
\(374\) −2.05082e13 −0.144922
\(375\) 3.19791e13 0.222687
\(376\) −5.84729e10 −0.000401256 0
\(377\) −5.32441e13 −0.360076
\(378\) 1.46772e13 0.0978222
\(379\) −2.50484e14 −1.64537 −0.822686 0.568496i \(-0.807526\pi\)
−0.822686 + 0.568496i \(0.807526\pi\)
\(380\) −1.30794e13 −0.0846794
\(381\) −2.92653e12 −0.0186752
\(382\) 2.29351e14 1.44262
\(383\) −2.39133e14 −1.48268 −0.741338 0.671131i \(-0.765809\pi\)
−0.741338 + 0.671131i \(0.765809\pi\)
\(384\) −8.53682e12 −0.0521764
\(385\) 8.67715e12 0.0522809
\(386\) 2.18362e14 1.29702
\(387\) 2.25926e13 0.132299
\(388\) 2.48404e14 1.43412
\(389\) −1.82634e14 −1.03958 −0.519791 0.854294i \(-0.673990\pi\)
−0.519791 + 0.854294i \(0.673990\pi\)
\(390\) −2.63366e13 −0.147810
\(391\) 2.30437e13 0.127521
\(392\) −7.19737e12 −0.0392736
\(393\) 1.42257e14 0.765447
\(394\) 2.66653e14 1.41487
\(395\) 2.04213e13 0.106856
\(396\) 4.58649e13 0.236678
\(397\) 2.44665e14 1.24516 0.622578 0.782558i \(-0.286085\pi\)
0.622578 + 0.782558i \(0.286085\pi\)
\(398\) 9.96033e13 0.499939
\(399\) −1.88028e13 −0.0930832
\(400\) 2.03020e14 0.991311
\(401\) 3.83666e14 1.84782 0.923910 0.382609i \(-0.124974\pi\)
0.923910 + 0.382609i \(0.124974\pi\)
\(402\) −2.85483e14 −1.35624
\(403\) 1.81392e14 0.850042
\(404\) −9.61566e13 −0.444510
\(405\) −4.79144e12 −0.0218506
\(406\) 4.38364e13 0.197216
\(407\) 2.71123e14 1.20336
\(408\) 8.39346e11 0.00367545
\(409\) 1.75721e14 0.759180 0.379590 0.925155i \(-0.376065\pi\)
0.379590 + 0.925155i \(0.376065\pi\)
\(410\) −1.05052e14 −0.447810
\(411\) −1.04859e14 −0.441036
\(412\) −2.38947e14 −0.991668
\(413\) 1.15194e13 0.0471744
\(414\) −1.04787e14 −0.423454
\(415\) −7.55383e13 −0.301233
\(416\) −3.30462e14 −1.30049
\(417\) 1.38840e14 0.539222
\(418\) −1.19471e14 −0.457923
\(419\) 4.20566e13 0.159095 0.0795475 0.996831i \(-0.474652\pi\)
0.0795475 + 0.996831i \(0.474652\pi\)
\(420\) 1.06640e13 0.0398153
\(421\) 2.80170e14 1.03245 0.516226 0.856453i \(-0.327337\pi\)
0.516226 + 0.856453i \(0.327337\pi\)
\(422\) 6.80080e14 2.47367
\(423\) −8.24027e11 −0.00295848
\(424\) −1.94454e13 −0.0689137
\(425\) −3.86946e13 −0.135366
\(426\) −6.94158e13 −0.239721
\(427\) 8.01975e13 0.273406
\(428\) −1.85326e14 −0.623727
\(429\) −1.18313e14 −0.393113
\(430\) 3.33770e13 0.109489
\(431\) −5.24917e14 −1.70007 −0.850033 0.526729i \(-0.823418\pi\)
−0.850033 + 0.526729i \(0.823418\pi\)
\(432\) −6.20608e13 −0.198453
\(433\) 1.62531e14 0.513161 0.256580 0.966523i \(-0.417404\pi\)
0.256580 + 0.966523i \(0.417404\pi\)
\(434\) −1.49342e14 −0.465573
\(435\) −1.43106e13 −0.0440522
\(436\) 2.75770e14 0.838246
\(437\) 1.34242e14 0.402940
\(438\) −3.76616e14 −1.11633
\(439\) 6.65170e14 1.94705 0.973526 0.228574i \(-0.0734064\pi\)
0.973526 + 0.228574i \(0.0734064\pi\)
\(440\) −2.25648e12 −0.00652291
\(441\) −1.01429e14 −0.289567
\(442\) 6.50163e13 0.183316
\(443\) −6.13790e14 −1.70923 −0.854613 0.519266i \(-0.826206\pi\)
−0.854613 + 0.519266i \(0.826206\pi\)
\(444\) 3.33204e14 0.916439
\(445\) −5.99080e13 −0.162744
\(446\) 6.28644e14 1.68680
\(447\) −2.74621e14 −0.727851
\(448\) 1.29347e14 0.338633
\(449\) 4.50418e14 1.16482 0.582412 0.812894i \(-0.302109\pi\)
0.582412 + 0.812894i \(0.302109\pi\)
\(450\) 1.75957e14 0.449508
\(451\) −4.71929e14 −1.19098
\(452\) −2.54647e14 −0.634859
\(453\) 1.77566e14 0.437341
\(454\) 6.57209e14 1.59918
\(455\) −2.75089e13 −0.0661317
\(456\) 4.88963e12 0.0116137
\(457\) −2.60319e14 −0.610896 −0.305448 0.952209i \(-0.598806\pi\)
−0.305448 + 0.952209i \(0.598806\pi\)
\(458\) 7.16165e14 1.66055
\(459\) 1.18285e13 0.0270993
\(460\) −7.61355e13 −0.172353
\(461\) 3.15031e14 0.704691 0.352345 0.935870i \(-0.385384\pi\)
0.352345 + 0.935870i \(0.385384\pi\)
\(462\) 9.74081e13 0.215310
\(463\) 1.15008e14 0.251208 0.125604 0.992080i \(-0.459913\pi\)
0.125604 + 0.992080i \(0.459913\pi\)
\(464\) −1.85357e14 −0.400093
\(465\) 4.87534e13 0.103995
\(466\) 7.61274e14 1.60479
\(467\) −1.62777e14 −0.339117 −0.169559 0.985520i \(-0.554234\pi\)
−0.169559 + 0.985520i \(0.554234\pi\)
\(468\) −1.45404e14 −0.299381
\(469\) −2.98190e14 −0.606796
\(470\) −1.21737e12 −0.00244841
\(471\) −1.15575e14 −0.229747
\(472\) −2.99561e12 −0.00588579
\(473\) 1.49941e14 0.291194
\(474\) 2.29246e14 0.440069
\(475\) −2.25416e14 −0.427731
\(476\) −2.63259e13 −0.0493793
\(477\) −2.74034e14 −0.508105
\(478\) −1.13514e15 −2.08063
\(479\) 2.72531e14 0.493823 0.246911 0.969038i \(-0.420584\pi\)
0.246911 + 0.969038i \(0.420584\pi\)
\(480\) −8.88194e13 −0.159104
\(481\) −8.59531e14 −1.52217
\(482\) −8.19869e14 −1.43544
\(483\) −1.09451e14 −0.189458
\(484\) −2.61094e14 −0.446836
\(485\) −1.72225e14 −0.291418
\(486\) −5.37878e13 −0.0899879
\(487\) −4.45484e13 −0.0736924 −0.0368462 0.999321i \(-0.511731\pi\)
−0.0368462 + 0.999321i \(0.511731\pi\)
\(488\) −2.08552e13 −0.0341119
\(489\) 4.05513e14 0.655853
\(490\) −1.49845e14 −0.239642
\(491\) 2.32630e14 0.367890 0.183945 0.982937i \(-0.441113\pi\)
0.183945 + 0.982937i \(0.441113\pi\)
\(492\) −5.79990e14 −0.907011
\(493\) 3.53281e13 0.0546339
\(494\) 3.78754e14 0.579241
\(495\) −3.17994e13 −0.0480939
\(496\) 6.31475e14 0.944511
\(497\) −7.25055e13 −0.107253
\(498\) −8.47979e14 −1.24058
\(499\) −4.19440e14 −0.606900 −0.303450 0.952847i \(-0.598139\pi\)
−0.303450 + 0.952847i \(0.598139\pi\)
\(500\) 2.60833e14 0.373274
\(501\) −1.18304e13 −0.0167452
\(502\) 1.12143e15 1.57000
\(503\) 1.42016e14 0.196659 0.0983294 0.995154i \(-0.468650\pi\)
0.0983294 + 0.995154i \(0.468650\pi\)
\(504\) −3.98666e12 −0.00546061
\(505\) 6.66680e13 0.0903264
\(506\) −6.95443e14 −0.932038
\(507\) −6.04120e13 −0.0800902
\(508\) −2.38698e13 −0.0313039
\(509\) −2.48643e14 −0.322574 −0.161287 0.986908i \(-0.551564\pi\)
−0.161287 + 0.986908i \(0.551564\pi\)
\(510\) 1.74746e13 0.0224271
\(511\) −3.93379e14 −0.499455
\(512\) −1.11331e15 −1.39840
\(513\) 6.89070e13 0.0856284
\(514\) 2.61078e14 0.320977
\(515\) 1.65668e14 0.201511
\(516\) 1.84274e14 0.221763
\(517\) −5.46883e12 −0.00651173
\(518\) 7.07660e14 0.833702
\(519\) −9.60396e14 −1.11951
\(520\) 7.15363e12 0.00825103
\(521\) −1.12001e15 −1.27825 −0.639125 0.769103i \(-0.720703\pi\)
−0.639125 + 0.769103i \(0.720703\pi\)
\(522\) −1.60648e14 −0.181421
\(523\) 3.76657e14 0.420908 0.210454 0.977604i \(-0.432506\pi\)
0.210454 + 0.977604i \(0.432506\pi\)
\(524\) 1.16030e15 1.28306
\(525\) 1.83789e14 0.201114
\(526\) −1.51394e15 −1.63940
\(527\) −1.20356e14 −0.128976
\(528\) −4.11879e14 −0.436801
\(529\) −1.71384e14 −0.179873
\(530\) −4.04842e14 −0.420502
\(531\) −4.22156e13 −0.0433963
\(532\) −1.53362e14 −0.156029
\(533\) 1.49614e15 1.50651
\(534\) −6.72517e14 −0.670234
\(535\) 1.28491e14 0.126744
\(536\) 7.75439e13 0.0757079
\(537\) 6.95074e14 0.671696
\(538\) −1.32612e15 −1.26847
\(539\) −6.73153e14 −0.637346
\(540\) −3.90807e13 −0.0366266
\(541\) −6.25461e14 −0.580250 −0.290125 0.956989i \(-0.593697\pi\)
−0.290125 + 0.956989i \(0.593697\pi\)
\(542\) −1.73773e15 −1.59583
\(543\) −9.91071e14 −0.900960
\(544\) 2.19265e14 0.197322
\(545\) −1.91199e14 −0.170335
\(546\) −3.08809e14 −0.272352
\(547\) 1.69694e14 0.148162 0.0740809 0.997252i \(-0.476398\pi\)
0.0740809 + 0.997252i \(0.476398\pi\)
\(548\) −8.55265e14 −0.739277
\(549\) −2.93902e14 −0.251509
\(550\) 1.16777e15 0.989382
\(551\) 2.05805e14 0.172632
\(552\) 2.84627e13 0.0236380
\(553\) 2.39450e14 0.196891
\(554\) −1.31509e15 −1.07066
\(555\) −2.31019e14 −0.186224
\(556\) 1.13243e15 0.903859
\(557\) 9.00925e14 0.712009 0.356004 0.934484i \(-0.384139\pi\)
0.356004 + 0.934484i \(0.384139\pi\)
\(558\) 5.47297e14 0.428287
\(559\) −4.75351e14 −0.368341
\(560\) −9.57658e13 −0.0734812
\(561\) 7.85020e13 0.0596465
\(562\) 1.50815e15 1.13474
\(563\) 2.55254e15 1.90185 0.950923 0.309426i \(-0.100137\pi\)
0.950923 + 0.309426i \(0.100137\pi\)
\(564\) −6.72106e12 −0.00495910
\(565\) 1.76554e14 0.129006
\(566\) 2.39939e15 1.73624
\(567\) −5.61819e13 −0.0402615
\(568\) 1.88549e13 0.0133816
\(569\) 1.45933e15 1.02574 0.512869 0.858467i \(-0.328583\pi\)
0.512869 + 0.858467i \(0.328583\pi\)
\(570\) 1.01799e14 0.0708650
\(571\) 6.41354e14 0.442180 0.221090 0.975253i \(-0.429039\pi\)
0.221090 + 0.975253i \(0.429039\pi\)
\(572\) −9.65003e14 −0.658947
\(573\) −8.77920e14 −0.593751
\(574\) −1.23179e15 −0.825126
\(575\) −1.31215e15 −0.870586
\(576\) −4.74023e14 −0.311512
\(577\) −1.66562e15 −1.08420 −0.542098 0.840315i \(-0.682370\pi\)
−0.542098 + 0.840315i \(0.682370\pi\)
\(578\) 2.13252e15 1.37496
\(579\) −8.35855e14 −0.533825
\(580\) −1.16722e14 −0.0738415
\(581\) −8.85723e14 −0.555046
\(582\) −1.93337e15 −1.20016
\(583\) −1.81869e15 −1.11836
\(584\) 1.02298e14 0.0623153
\(585\) 1.00812e14 0.0608354
\(586\) 1.72829e15 1.03319
\(587\) 8.16234e14 0.483398 0.241699 0.970351i \(-0.422295\pi\)
0.241699 + 0.970351i \(0.422295\pi\)
\(588\) −8.27289e14 −0.485380
\(589\) −7.01136e14 −0.407538
\(590\) −6.23667e13 −0.0359143
\(591\) −1.02071e15 −0.582330
\(592\) −2.99226e15 −1.69134
\(593\) −2.77779e15 −1.55561 −0.777803 0.628509i \(-0.783666\pi\)
−0.777803 + 0.628509i \(0.783666\pi\)
\(594\) −3.56974e14 −0.198066
\(595\) 1.82524e13 0.0100341
\(596\) −2.23991e15 −1.22005
\(597\) −3.81266e14 −0.205764
\(598\) 2.20474e15 1.17896
\(599\) 8.71268e14 0.461641 0.230820 0.972996i \(-0.425859\pi\)
0.230820 + 0.972996i \(0.425859\pi\)
\(600\) −4.77939e13 −0.0250923
\(601\) −6.04340e14 −0.314392 −0.157196 0.987567i \(-0.550246\pi\)
−0.157196 + 0.987567i \(0.550246\pi\)
\(602\) 3.91361e14 0.201742
\(603\) 1.09278e15 0.558199
\(604\) 1.44829e15 0.733084
\(605\) 1.81024e14 0.0907990
\(606\) 7.48402e14 0.371994
\(607\) 1.64594e14 0.0810732 0.0405366 0.999178i \(-0.487093\pi\)
0.0405366 + 0.999178i \(0.487093\pi\)
\(608\) 1.27734e15 0.623499
\(609\) −1.67799e14 −0.0811697
\(610\) −4.34192e14 −0.208146
\(611\) 1.73376e13 0.00823688
\(612\) 9.64771e13 0.0454246
\(613\) −1.47340e15 −0.687525 −0.343763 0.939057i \(-0.611702\pi\)
−0.343763 + 0.939057i \(0.611702\pi\)
\(614\) 1.11436e15 0.515348
\(615\) 4.02123e14 0.184309
\(616\) −2.64583e13 −0.0120190
\(617\) −2.66219e15 −1.19859 −0.599295 0.800529i \(-0.704552\pi\)
−0.599295 + 0.800529i \(0.704552\pi\)
\(618\) 1.85976e15 0.829889
\(619\) −2.51176e15 −1.11091 −0.555456 0.831546i \(-0.687456\pi\)
−0.555456 + 0.831546i \(0.687456\pi\)
\(620\) 3.97650e14 0.174320
\(621\) 4.01109e14 0.174284
\(622\) −2.76712e15 −1.19174
\(623\) −7.02451e14 −0.299869
\(624\) 1.30577e15 0.552523
\(625\) 2.11114e15 0.885476
\(626\) −4.06789e15 −1.69126
\(627\) 4.57316e14 0.188471
\(628\) −9.42673e14 −0.385109
\(629\) 5.70309e14 0.230957
\(630\) −8.29997e13 −0.0333199
\(631\) 2.30101e15 0.915707 0.457853 0.889028i \(-0.348618\pi\)
0.457853 + 0.889028i \(0.348618\pi\)
\(632\) −6.22685e13 −0.0245654
\(633\) −2.60324e15 −1.01811
\(634\) 5.44717e15 2.11193
\(635\) 1.65496e13 0.00636109
\(636\) −2.23512e15 −0.851701
\(637\) 2.13407e15 0.806198
\(638\) −1.06618e15 −0.399314
\(639\) 2.65713e14 0.0986637
\(640\) 4.82759e13 0.0177722
\(641\) −1.77314e15 −0.647180 −0.323590 0.946197i \(-0.604890\pi\)
−0.323590 + 0.946197i \(0.604890\pi\)
\(642\) 1.44242e15 0.521973
\(643\) 2.86869e15 1.02926 0.514628 0.857413i \(-0.327930\pi\)
0.514628 + 0.857413i \(0.327930\pi\)
\(644\) −8.92725e14 −0.317574
\(645\) −1.27762e14 −0.0450632
\(646\) −2.51308e14 −0.0878875
\(647\) −4.03348e14 −0.139864 −0.0699321 0.997552i \(-0.522278\pi\)
−0.0699321 + 0.997552i \(0.522278\pi\)
\(648\) 1.46100e13 0.00502329
\(649\) −2.80172e14 −0.0955167
\(650\) −3.70215e15 −1.25150
\(651\) 5.71657e14 0.191620
\(652\) 3.30751e15 1.09936
\(653\) −9.72103e14 −0.320398 −0.160199 0.987085i \(-0.551214\pi\)
−0.160199 + 0.987085i \(0.551214\pi\)
\(654\) −2.14636e15 −0.701497
\(655\) −8.04467e14 −0.260724
\(656\) 5.20847e15 1.67394
\(657\) 1.44162e15 0.459455
\(658\) −1.42742e13 −0.00451139
\(659\) 6.57482e14 0.206070 0.103035 0.994678i \(-0.467145\pi\)
0.103035 + 0.994678i \(0.467145\pi\)
\(660\) −2.59367e14 −0.0806164
\(661\) −5.07978e15 −1.56580 −0.782901 0.622147i \(-0.786261\pi\)
−0.782901 + 0.622147i \(0.786261\pi\)
\(662\) 6.79589e13 0.0207743
\(663\) −2.48872e14 −0.0754487
\(664\) 2.30331e14 0.0692512
\(665\) 1.06330e14 0.0317057
\(666\) −2.59338e15 −0.766933
\(667\) 1.19799e15 0.351369
\(668\) −9.64929e13 −0.0280689
\(669\) −2.40635e15 −0.694249
\(670\) 1.61441e15 0.461959
\(671\) −1.95054e15 −0.553581
\(672\) −1.04145e15 −0.293162
\(673\) 2.44434e15 0.682462 0.341231 0.939979i \(-0.389156\pi\)
0.341231 + 0.939979i \(0.389156\pi\)
\(674\) 2.98042e15 0.825369
\(675\) −6.73535e14 −0.185007
\(676\) −4.92742e14 −0.134250
\(677\) −2.97668e15 −0.804442 −0.402221 0.915543i \(-0.631762\pi\)
−0.402221 + 0.915543i \(0.631762\pi\)
\(678\) 1.98196e15 0.531289
\(679\) −2.01942e15 −0.536962
\(680\) −4.74652e12 −0.00125192
\(681\) −2.51569e15 −0.658186
\(682\) 3.63225e15 0.942674
\(683\) 6.09320e15 1.56867 0.784335 0.620338i \(-0.213004\pi\)
0.784335 + 0.620338i \(0.213004\pi\)
\(684\) 5.62030e14 0.143533
\(685\) 5.92978e14 0.150224
\(686\) −3.77956e15 −0.949859
\(687\) −2.74136e15 −0.683447
\(688\) −1.65483e15 −0.409276
\(689\) 5.76571e15 1.41464
\(690\) 5.92574e14 0.144236
\(691\) 7.54517e15 1.82196 0.910982 0.412447i \(-0.135326\pi\)
0.910982 + 0.412447i \(0.135326\pi\)
\(692\) −7.83333e15 −1.87656
\(693\) −3.72863e14 −0.0886169
\(694\) 9.74083e15 2.29678
\(695\) −7.85145e14 −0.183668
\(696\) 4.36358e13 0.0101273
\(697\) −9.92707e14 −0.228581
\(698\) 9.28187e15 2.12046
\(699\) −2.91404e15 −0.660496
\(700\) 1.49905e15 0.337113
\(701\) 9.37328e13 0.0209143 0.0104571 0.999945i \(-0.496671\pi\)
0.0104571 + 0.999945i \(0.496671\pi\)
\(702\) 1.13170e15 0.250540
\(703\) 3.32235e15 0.729779
\(704\) −3.14595e15 −0.685649
\(705\) 4.65989e12 0.00100771
\(706\) 1.18214e16 2.53655
\(707\) 7.81714e14 0.166434
\(708\) −3.44325e14 −0.0727421
\(709\) 7.32883e15 1.53632 0.768158 0.640261i \(-0.221174\pi\)
0.768158 + 0.640261i \(0.221174\pi\)
\(710\) 3.92548e14 0.0816529
\(711\) −8.77517e14 −0.181123
\(712\) 1.82671e14 0.0374137
\(713\) −4.08133e15 −0.829486
\(714\) 2.04899e14 0.0413237
\(715\) 6.69062e14 0.133901
\(716\) 5.66928e15 1.12592
\(717\) 4.34513e15 0.856343
\(718\) 9.68228e15 1.89362
\(719\) −2.09644e15 −0.406887 −0.203444 0.979087i \(-0.565213\pi\)
−0.203444 + 0.979087i \(0.565213\pi\)
\(720\) 3.50955e14 0.0675963
\(721\) 1.94254e15 0.371301
\(722\) 5.93106e15 1.12507
\(723\) 3.13833e15 0.590796
\(724\) −8.08353e15 −1.51022
\(725\) −2.01165e15 −0.372987
\(726\) 2.03214e15 0.373940
\(727\) 2.30351e15 0.420678 0.210339 0.977628i \(-0.432543\pi\)
0.210339 + 0.977628i \(0.432543\pi\)
\(728\) 8.38798e13 0.0152032
\(729\) 2.05891e14 0.0370370
\(730\) 2.12977e15 0.380239
\(731\) 3.15401e14 0.0558879
\(732\) −2.39717e15 −0.421587
\(733\) −6.47539e15 −1.13030 −0.565150 0.824988i \(-0.691182\pi\)
−0.565150 + 0.824988i \(0.691182\pi\)
\(734\) −1.24236e16 −2.15238
\(735\) 5.73582e14 0.0986313
\(736\) 7.43540e15 1.26904
\(737\) 7.25249e15 1.22862
\(738\) 4.51416e15 0.759043
\(739\) −6.76817e15 −1.12961 −0.564803 0.825226i \(-0.691048\pi\)
−0.564803 + 0.825226i \(0.691048\pi\)
\(740\) −1.88427e15 −0.312155
\(741\) −1.44981e15 −0.238403
\(742\) −4.74697e15 −0.774809
\(743\) 2.76218e15 0.447521 0.223760 0.974644i \(-0.428167\pi\)
0.223760 + 0.974644i \(0.428167\pi\)
\(744\) −1.48658e14 −0.0239077
\(745\) 1.55299e15 0.247919
\(746\) 1.59361e16 2.52532
\(747\) 3.24593e15 0.510594
\(748\) 6.40291e14 0.0999812
\(749\) 1.50662e15 0.233536
\(750\) −2.03011e15 −0.312379
\(751\) 8.20164e15 1.25280 0.626399 0.779502i \(-0.284528\pi\)
0.626399 + 0.779502i \(0.284528\pi\)
\(752\) 6.03570e13 0.00915228
\(753\) −4.29264e15 −0.646178
\(754\) 3.38006e15 0.505105
\(755\) −1.00414e15 −0.148966
\(756\) −4.58240e14 −0.0674874
\(757\) 5.71284e15 0.835266 0.417633 0.908616i \(-0.362860\pi\)
0.417633 + 0.908616i \(0.362860\pi\)
\(758\) 1.59013e16 2.30808
\(759\) 2.66204e15 0.383606
\(760\) −2.76510e13 −0.00395581
\(761\) −8.05028e15 −1.14339 −0.571696 0.820465i \(-0.693715\pi\)
−0.571696 + 0.820465i \(0.693715\pi\)
\(762\) 1.85783e14 0.0261971
\(763\) −2.24190e15 −0.313856
\(764\) −7.16063e15 −0.995263
\(765\) −6.68902e13 −0.00923048
\(766\) 1.51807e16 2.07986
\(767\) 8.88220e14 0.120822
\(768\) 4.53699e15 0.612747
\(769\) −3.93277e15 −0.527356 −0.263678 0.964611i \(-0.584936\pi\)
−0.263678 + 0.964611i \(0.584936\pi\)
\(770\) −5.50845e14 −0.0733382
\(771\) −9.99366e14 −0.132107
\(772\) −6.81754e15 −0.894813
\(773\) −7.74208e15 −1.00895 −0.504476 0.863426i \(-0.668314\pi\)
−0.504476 + 0.863426i \(0.668314\pi\)
\(774\) −1.43423e15 −0.185585
\(775\) 6.85329e15 0.880521
\(776\) 5.25148e14 0.0669949
\(777\) −2.70881e15 −0.343133
\(778\) 1.15940e16 1.45830
\(779\) −5.78304e15 −0.722271
\(780\) 8.22262e14 0.101974
\(781\) 1.76346e15 0.217162
\(782\) −1.46287e15 −0.178883
\(783\) 6.14936e14 0.0746690
\(784\) 7.42928e15 0.895795
\(785\) 6.53581e14 0.0782557
\(786\) −9.03080e15 −1.07375
\(787\) 8.87344e15 1.04769 0.523843 0.851815i \(-0.324498\pi\)
0.523843 + 0.851815i \(0.324498\pi\)
\(788\) −8.32524e15 −0.976119
\(789\) 5.79510e15 0.674742
\(790\) −1.29639e15 −0.149895
\(791\) 2.07018e15 0.237704
\(792\) 9.69624e13 0.0110564
\(793\) 6.18372e15 0.700241
\(794\) −1.55319e16 −1.74667
\(795\) 1.54967e15 0.173069
\(796\) −3.10974e15 −0.344907
\(797\) −5.75633e15 −0.634052 −0.317026 0.948417i \(-0.602684\pi\)
−0.317026 + 0.948417i \(0.602684\pi\)
\(798\) 1.19364e15 0.130575
\(799\) −1.15037e13 −0.00124977
\(800\) −1.24854e16 −1.34712
\(801\) 2.57429e15 0.275853
\(802\) −2.43560e16 −2.59207
\(803\) 9.56764e15 1.01128
\(804\) 8.91314e15 0.935670
\(805\) 6.18950e14 0.0645325
\(806\) −1.15152e16 −1.19242
\(807\) 5.07616e15 0.522073
\(808\) −2.03284e14 −0.0207654
\(809\) 4.45104e15 0.451590 0.225795 0.974175i \(-0.427502\pi\)
0.225795 + 0.974175i \(0.427502\pi\)
\(810\) 3.04171e14 0.0306514
\(811\) −1.29435e16 −1.29550 −0.647751 0.761852i \(-0.724290\pi\)
−0.647751 + 0.761852i \(0.724290\pi\)
\(812\) −1.36863e15 −0.136059
\(813\) 6.65176e15 0.656808
\(814\) −1.72115e16 −1.68805
\(815\) −2.29319e15 −0.223394
\(816\) −8.66391e14 −0.0838336
\(817\) 1.83738e15 0.176595
\(818\) −1.11552e16 −1.06496
\(819\) 1.18207e15 0.112094
\(820\) 3.27986e15 0.308943
\(821\) 4.69898e15 0.439659 0.219830 0.975538i \(-0.429450\pi\)
0.219830 + 0.975538i \(0.429450\pi\)
\(822\) 6.65666e15 0.618673
\(823\) −4.08560e15 −0.377187 −0.188593 0.982055i \(-0.560393\pi\)
−0.188593 + 0.982055i \(0.560393\pi\)
\(824\) −5.05154e14 −0.0463259
\(825\) −4.47005e15 −0.407208
\(826\) −7.31280e14 −0.0661749
\(827\) −1.25404e16 −1.12728 −0.563640 0.826021i \(-0.690599\pi\)
−0.563640 + 0.826021i \(0.690599\pi\)
\(828\) 3.27159e15 0.292141
\(829\) −1.41486e16 −1.25506 −0.627529 0.778593i \(-0.715934\pi\)
−0.627529 + 0.778593i \(0.715934\pi\)
\(830\) 4.79534e15 0.422561
\(831\) 5.03396e15 0.440660
\(832\) 9.97349e15 0.867299
\(833\) −1.41598e15 −0.122324
\(834\) −8.81390e15 −0.756406
\(835\) 6.69011e13 0.00570371
\(836\) 3.73003e15 0.315921
\(837\) −2.09496e15 −0.176273
\(838\) −2.66985e15 −0.223174
\(839\) −4.25727e15 −0.353542 −0.176771 0.984252i \(-0.556565\pi\)
−0.176771 + 0.984252i \(0.556565\pi\)
\(840\) 2.25447e13 0.00185998
\(841\) −1.03639e16 −0.849463
\(842\) −1.77858e16 −1.44829
\(843\) −5.77296e15 −0.467033
\(844\) −2.12330e16 −1.70658
\(845\) 3.41631e14 0.0272801
\(846\) 5.23111e13 0.00415008
\(847\) 2.12259e15 0.167304
\(848\) 2.00720e16 1.57186
\(849\) −9.18450e15 −0.714600
\(850\) 2.45642e15 0.189888
\(851\) 1.93395e16 1.48536
\(852\) 2.16725e15 0.165383
\(853\) −1.06217e16 −0.805329 −0.402665 0.915348i \(-0.631916\pi\)
−0.402665 + 0.915348i \(0.631916\pi\)
\(854\) −5.09112e15 −0.383526
\(855\) −3.89671e14 −0.0291665
\(856\) −3.91794e14 −0.0291375
\(857\) −2.07068e16 −1.53010 −0.765049 0.643972i \(-0.777285\pi\)
−0.765049 + 0.643972i \(0.777285\pi\)
\(858\) 7.51077e15 0.551448
\(859\) 1.44569e15 0.105466 0.0527331 0.998609i \(-0.483207\pi\)
0.0527331 + 0.998609i \(0.483207\pi\)
\(860\) −1.04207e15 −0.0755363
\(861\) 4.71508e15 0.339603
\(862\) 3.33229e16 2.38481
\(863\) 3.39961e15 0.241752 0.120876 0.992668i \(-0.461430\pi\)
0.120876 + 0.992668i \(0.461430\pi\)
\(864\) 3.81663e15 0.269683
\(865\) 5.43106e15 0.381326
\(866\) −1.03178e16 −0.719848
\(867\) −8.16294e15 −0.565903
\(868\) 4.66264e15 0.321198
\(869\) −5.82382e15 −0.398657
\(870\) 9.08469e14 0.0617952
\(871\) −2.29923e16 −1.55411
\(872\) 5.83002e14 0.0391588
\(873\) 7.40062e15 0.493958
\(874\) −8.52198e15 −0.565233
\(875\) −2.12047e15 −0.139762
\(876\) 1.17584e16 0.770151
\(877\) −1.87390e16 −1.21968 −0.609842 0.792523i \(-0.708767\pi\)
−0.609842 + 0.792523i \(0.708767\pi\)
\(878\) −4.22265e16 −2.73127
\(879\) −6.61564e15 −0.425239
\(880\) 2.32919e15 0.148782
\(881\) 1.90703e15 0.121057 0.0605285 0.998166i \(-0.480721\pi\)
0.0605285 + 0.998166i \(0.480721\pi\)
\(882\) 6.43892e15 0.406196
\(883\) 4.45112e15 0.279052 0.139526 0.990218i \(-0.455442\pi\)
0.139526 + 0.990218i \(0.455442\pi\)
\(884\) −2.02989e15 −0.126469
\(885\) 2.38730e14 0.0147815
\(886\) 3.89648e16 2.39766
\(887\) −6.97407e15 −0.426488 −0.213244 0.976999i \(-0.568403\pi\)
−0.213244 + 0.976999i \(0.568403\pi\)
\(888\) 7.04422e14 0.0428116
\(889\) 1.94052e14 0.0117208
\(890\) 3.80310e15 0.228293
\(891\) 1.36644e15 0.0815198
\(892\) −1.96271e16 −1.16372
\(893\) −6.70152e13 −0.00394903
\(894\) 1.74336e16 1.02101
\(895\) −3.93066e15 −0.228791
\(896\) 5.66058e14 0.0327467
\(897\) −8.43938e15 −0.485235
\(898\) −2.85935e16 −1.63398
\(899\) −6.25704e15 −0.355378
\(900\) −5.49359e15 −0.310115
\(901\) −3.82562e15 −0.214642
\(902\) 2.99591e16 1.67068
\(903\) −1.49807e15 −0.0830327
\(904\) −5.38346e14 −0.0296575
\(905\) 5.60453e15 0.306882
\(906\) −1.12723e16 −0.613490
\(907\) −3.26700e16 −1.76729 −0.883647 0.468153i \(-0.844920\pi\)
−0.883647 + 0.468153i \(0.844920\pi\)
\(908\) −2.05189e16 −1.10327
\(909\) −2.86477e15 −0.153104
\(910\) 1.74632e15 0.0927678
\(911\) −1.41035e16 −0.744690 −0.372345 0.928094i \(-0.621446\pi\)
−0.372345 + 0.928094i \(0.621446\pi\)
\(912\) −5.04719e15 −0.264897
\(913\) 2.15423e16 1.12383
\(914\) 1.65257e16 0.856948
\(915\) 1.66202e15 0.0856683
\(916\) −2.23596e16 −1.14561
\(917\) −9.43277e15 −0.480405
\(918\) −7.50897e14 −0.0380142
\(919\) 2.92523e14 0.0147206 0.00736030 0.999973i \(-0.497657\pi\)
0.00736030 + 0.999973i \(0.497657\pi\)
\(920\) −1.60957e14 −0.00805150
\(921\) −4.26560e15 −0.212106
\(922\) −1.99989e16 −0.988521
\(923\) −5.59062e15 −0.274695
\(924\) −3.04120e15 −0.148542
\(925\) −3.24745e16 −1.57675
\(926\) −7.30099e15 −0.352388
\(927\) −7.11886e15 −0.341564
\(928\) 1.13991e16 0.543699
\(929\) 4.93609e15 0.234044 0.117022 0.993129i \(-0.462665\pi\)
0.117022 + 0.993129i \(0.462665\pi\)
\(930\) −3.09497e15 −0.145882
\(931\) −8.24884e15 −0.386518
\(932\) −2.37679e16 −1.10714
\(933\) 1.05921e16 0.490494
\(934\) 1.03334e16 0.475704
\(935\) −4.43931e14 −0.0203166
\(936\) −3.07396e14 −0.0139856
\(937\) 3.91617e16 1.77131 0.885654 0.464345i \(-0.153710\pi\)
0.885654 + 0.464345i \(0.153710\pi\)
\(938\) 1.89298e16 0.851197
\(939\) 1.55712e16 0.696085
\(940\) 3.80077e13 0.00168915
\(941\) −7.71148e15 −0.340718 −0.170359 0.985382i \(-0.554493\pi\)
−0.170359 + 0.985382i \(0.554493\pi\)
\(942\) 7.33698e15 0.322283
\(943\) −3.36632e16 −1.47008
\(944\) 3.09214e15 0.134250
\(945\) 3.17710e14 0.0137137
\(946\) −9.51857e15 −0.408480
\(947\) 4.41750e15 0.188474 0.0942371 0.995550i \(-0.469959\pi\)
0.0942371 + 0.995550i \(0.469959\pi\)
\(948\) −7.15734e15 −0.303603
\(949\) −3.03319e16 −1.27919
\(950\) 1.43099e16 0.600009
\(951\) −2.08509e16 −0.869225
\(952\) −5.56552e13 −0.00230676
\(953\) 2.33786e16 0.963403 0.481702 0.876335i \(-0.340019\pi\)
0.481702 + 0.876335i \(0.340019\pi\)
\(954\) 1.73963e16 0.712756
\(955\) 4.96466e15 0.202242
\(956\) 3.54405e16 1.43543
\(957\) 4.08115e15 0.164349
\(958\) −1.73009e16 −0.692721
\(959\) 6.95295e15 0.276800
\(960\) 2.68061e15 0.106106
\(961\) −4.09199e15 −0.161048
\(962\) 5.45650e16 2.13526
\(963\) −5.52135e15 −0.214833
\(964\) 2.55973e16 0.990310
\(965\) 4.72678e15 0.181830
\(966\) 6.94822e15 0.265766
\(967\) −1.00913e16 −0.383797 −0.191898 0.981415i \(-0.561464\pi\)
−0.191898 + 0.981415i \(0.561464\pi\)
\(968\) −5.51976e14 −0.0208740
\(969\) 9.61967e14 0.0361726
\(970\) 1.09332e16 0.408794
\(971\) 4.57498e16 1.70092 0.850460 0.526041i \(-0.176324\pi\)
0.850460 + 0.526041i \(0.176324\pi\)
\(972\) 1.67932e15 0.0620826
\(973\) −9.20621e15 −0.338423
\(974\) 2.82803e15 0.103374
\(975\) 1.41712e16 0.515089
\(976\) 2.15272e16 0.778062
\(977\) −1.62337e16 −0.583442 −0.291721 0.956503i \(-0.594228\pi\)
−0.291721 + 0.956503i \(0.594228\pi\)
\(978\) −2.57429e16 −0.920012
\(979\) 1.70848e16 0.607163
\(980\) 4.67834e15 0.165329
\(981\) 8.21594e15 0.288721
\(982\) −1.47679e16 −0.516065
\(983\) 2.21897e16 0.771094 0.385547 0.922688i \(-0.374013\pi\)
0.385547 + 0.922688i \(0.374013\pi\)
\(984\) −1.22615e15 −0.0423712
\(985\) 5.77211e15 0.198352
\(986\) −2.24271e15 −0.0766390
\(987\) 5.46395e13 0.00185679
\(988\) −1.18252e16 −0.399618
\(989\) 1.06954e16 0.359433
\(990\) 2.01869e15 0.0674648
\(991\) 2.43549e16 0.809435 0.404717 0.914442i \(-0.367370\pi\)
0.404717 + 0.914442i \(0.367370\pi\)
\(992\) −3.88346e16 −1.28353
\(993\) −2.60136e14 −0.00855026
\(994\) 4.60281e15 0.150452
\(995\) 2.15607e15 0.0700866
\(996\) 2.64750e16 0.855872
\(997\) −2.23125e16 −0.717339 −0.358669 0.933465i \(-0.616769\pi\)
−0.358669 + 0.933465i \(0.616769\pi\)
\(998\) 2.66270e16 0.851343
\(999\) 9.92704e15 0.315653
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 177.12.a.b.1.6 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.b.1.6 27 1.1 even 1 trivial