Properties

Label 177.12.a.b.1.2
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 1.2
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-86.7335 q^{2} +243.000 q^{3} +5474.69 q^{4} -11727.0 q^{5} -21076.2 q^{6} +24236.6 q^{7} -297209. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-86.7335 q^{2} +243.000 q^{3} +5474.69 q^{4} -11727.0 q^{5} -21076.2 q^{6} +24236.6 q^{7} -297209. q^{8} +59049.0 q^{9} +1.01712e6 q^{10} -225721. q^{11} +1.33035e6 q^{12} -1.93565e6 q^{13} -2.10212e6 q^{14} -2.84965e6 q^{15} +1.45658e7 q^{16} -7.58845e6 q^{17} -5.12152e6 q^{18} +2.02438e7 q^{19} -6.42015e7 q^{20} +5.88949e6 q^{21} +1.95775e7 q^{22} +4.13605e6 q^{23} -7.22217e7 q^{24} +8.86937e7 q^{25} +1.67885e8 q^{26} +1.43489e7 q^{27} +1.32688e8 q^{28} +1.28733e8 q^{29} +2.47160e8 q^{30} -1.26556e8 q^{31} -6.54656e8 q^{32} -5.48501e7 q^{33} +6.58173e8 q^{34} -2.84222e8 q^{35} +3.23275e8 q^{36} +1.58528e8 q^{37} -1.75581e9 q^{38} -4.70363e8 q^{39} +3.48536e9 q^{40} +7.42011e8 q^{41} -5.10816e8 q^{42} -1.64490e8 q^{43} -1.23575e9 q^{44} -6.92466e8 q^{45} -3.58734e8 q^{46} -1.37797e9 q^{47} +3.53948e9 q^{48} -1.38992e9 q^{49} -7.69271e9 q^{50} -1.84399e9 q^{51} -1.05971e10 q^{52} +4.06680e8 q^{53} -1.24453e9 q^{54} +2.64702e9 q^{55} -7.20332e9 q^{56} +4.91924e9 q^{57} -1.11655e10 q^{58} -7.14924e8 q^{59} -1.56010e10 q^{60} +1.25207e9 q^{61} +1.09767e10 q^{62} +1.43115e9 q^{63} +2.69499e10 q^{64} +2.26993e10 q^{65} +4.75734e9 q^{66} -3.89914e9 q^{67} -4.15444e10 q^{68} +1.00506e9 q^{69} +2.46515e10 q^{70} +2.09101e10 q^{71} -1.75499e10 q^{72} +1.32307e10 q^{73} -1.37497e10 q^{74} +2.15526e10 q^{75} +1.10829e11 q^{76} -5.47069e9 q^{77} +4.07962e10 q^{78} +3.04105e10 q^{79} -1.70812e11 q^{80} +3.48678e9 q^{81} -6.43572e10 q^{82} -6.10863e10 q^{83} +3.22431e10 q^{84} +8.89895e10 q^{85} +1.42668e10 q^{86} +3.12822e10 q^{87} +6.70861e10 q^{88} +6.05385e10 q^{89} +6.00600e10 q^{90} -4.69135e10 q^{91} +2.26436e10 q^{92} -3.07532e10 q^{93} +1.19516e11 q^{94} -2.37398e11 q^{95} -1.59081e11 q^{96} -1.06180e11 q^{97} +1.20552e11 q^{98} -1.33286e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} - 383719q^{10} - 1816556q^{11} + 6352506q^{12} - 3951804q^{13} - 6207867q^{14} - 4176684q^{15} + 28295194q^{16} - 17723275q^{17} - 7558272q^{18} - 19573013q^{19} - 48468099q^{20} - 30758697q^{21} - 1729910q^{22} - 88593797q^{23} - 86458671q^{24} + 345714963q^{25} - 6676346q^{26} + 387420489q^{27} + 126954286q^{28} - 276632427q^{29} - 93243717q^{30} - 357680917q^{31} - 859842334q^{32} - 441423108q^{33} + 232730000q^{34} - 510315139q^{35} + 1543658958q^{36} - 660238257q^{37} - 2067286961q^{38} - 960288372q^{39} - 3388951110q^{40} - 1671147569q^{41} - 1508511681q^{42} - 1883107790q^{43} - 3895687630q^{44} - 1014934212q^{45} - 1720344243q^{46} - 5818572501q^{47} + 6875732142q^{48} - 18858180q^{49} - 21474519647q^{50} - 4306755825q^{51} - 42214560062q^{52} - 11444513368q^{53} - 1836660096q^{54} - 24401486484q^{55} - 50583585764q^{56} - 4756242159q^{57} - 45017395090q^{58} - 19302956073q^{59} - 11777748057q^{60} + 408637955q^{61} - 28543084070q^{62} - 7474363371q^{63} + 33067284293q^{64} - 21656714730q^{65} - 420368130q^{66} - 49803132690q^{67} - 16500749319q^{68} - 21528292671q^{69} - 45808890782q^{70} - 34127492216q^{71} - 21009457053q^{72} - 55734362153q^{73} - 40367816298q^{74} + 84008736009q^{75} - 14840406404q^{76} - 99723443615q^{77} - 1622352078q^{78} - 76484916442q^{79} + 93882788915q^{80} + 94143178827q^{81} + 52951239205q^{82} - 140433865655q^{83} + 30849891498q^{84} + 34329063335q^{85} + 175223869508q^{86} - 67221679761q^{87} + 268823645069q^{88} - 1191878597q^{89} - 22658223231q^{90} + 201632581559q^{91} - 206501888812q^{92} - 86916462831q^{93} + 319770144384q^{94} - 81387074885q^{95} - 208941687162q^{96} - 144896178730q^{97} + 135739195260q^{98} - 107265815244q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −86.7335 −1.91656 −0.958278 0.285837i \(-0.907728\pi\)
−0.958278 + 0.285837i \(0.907728\pi\)
\(3\) 243.000 0.577350
\(4\) 5474.69 2.67319
\(5\) −11727.0 −1.67823 −0.839114 0.543956i \(-0.816926\pi\)
−0.839114 + 0.543956i \(0.816926\pi\)
\(6\) −21076.2 −1.10652
\(7\) 24236.6 0.545045 0.272522 0.962149i \(-0.412142\pi\)
0.272522 + 0.962149i \(0.412142\pi\)
\(8\) −297209. −3.20676
\(9\) 59049.0 0.333333
\(10\) 1.01712e6 3.21642
\(11\) −225721. −0.422582 −0.211291 0.977423i \(-0.567767\pi\)
−0.211291 + 0.977423i \(0.567767\pi\)
\(12\) 1.33035e6 1.54337
\(13\) −1.93565e6 −1.44590 −0.722950 0.690901i \(-0.757214\pi\)
−0.722950 + 0.690901i \(0.757214\pi\)
\(14\) −2.10212e6 −1.04461
\(15\) −2.84965e6 −0.968925
\(16\) 1.45658e7 3.47275
\(17\) −7.58845e6 −1.29624 −0.648118 0.761540i \(-0.724444\pi\)
−0.648118 + 0.761540i \(0.724444\pi\)
\(18\) −5.12152e6 −0.638852
\(19\) 2.02438e7 1.87563 0.937815 0.347136i \(-0.112846\pi\)
0.937815 + 0.347136i \(0.112846\pi\)
\(20\) −6.42015e7 −4.48622
\(21\) 5.88949e6 0.314682
\(22\) 1.95775e7 0.809903
\(23\) 4.13605e6 0.133993 0.0669965 0.997753i \(-0.478658\pi\)
0.0669965 + 0.997753i \(0.478658\pi\)
\(24\) −7.22217e7 −1.85142
\(25\) 8.86937e7 1.81645
\(26\) 1.67885e8 2.77115
\(27\) 1.43489e7 0.192450
\(28\) 1.32688e8 1.45701
\(29\) 1.28733e8 1.16547 0.582736 0.812662i \(-0.301982\pi\)
0.582736 + 0.812662i \(0.301982\pi\)
\(30\) 2.47160e8 1.85700
\(31\) −1.26556e8 −0.793953 −0.396976 0.917829i \(-0.629941\pi\)
−0.396976 + 0.917829i \(0.629941\pi\)
\(32\) −6.54656e8 −3.44896
\(33\) −5.48501e7 −0.243978
\(34\) 6.58173e8 2.48431
\(35\) −2.84222e8 −0.914709
\(36\) 3.23275e8 0.891063
\(37\) 1.58528e8 0.375834 0.187917 0.982185i \(-0.439826\pi\)
0.187917 + 0.982185i \(0.439826\pi\)
\(38\) −1.75581e9 −3.59475
\(39\) −4.70363e8 −0.834790
\(40\) 3.48536e9 5.38168
\(41\) 7.42011e8 1.00023 0.500114 0.865959i \(-0.333291\pi\)
0.500114 + 0.865959i \(0.333291\pi\)
\(42\) −5.10816e8 −0.603105
\(43\) −1.64490e8 −0.170633 −0.0853166 0.996354i \(-0.527190\pi\)
−0.0853166 + 0.996354i \(0.527190\pi\)
\(44\) −1.23575e9 −1.12964
\(45\) −6.92466e8 −0.559409
\(46\) −3.58734e8 −0.256805
\(47\) −1.37797e9 −0.876401 −0.438200 0.898877i \(-0.644384\pi\)
−0.438200 + 0.898877i \(0.644384\pi\)
\(48\) 3.53948e9 2.00499
\(49\) −1.38992e9 −0.702926
\(50\) −7.69271e9 −3.48132
\(51\) −1.84399e9 −0.748382
\(52\) −1.05971e10 −3.86516
\(53\) 4.06680e8 0.133578 0.0667891 0.997767i \(-0.478725\pi\)
0.0667891 + 0.997767i \(0.478725\pi\)
\(54\) −1.24453e9 −0.368841
\(55\) 2.64702e9 0.709189
\(56\) −7.20332e9 −1.74783
\(57\) 4.91924e9 1.08290
\(58\) −1.11655e10 −2.23369
\(59\) −7.14924e8 −0.130189
\(60\) −1.56010e10 −2.59012
\(61\) 1.25207e9 0.189808 0.0949042 0.995486i \(-0.469746\pi\)
0.0949042 + 0.995486i \(0.469746\pi\)
\(62\) 1.09767e10 1.52166
\(63\) 1.43115e9 0.181682
\(64\) 2.69499e10 3.13738
\(65\) 2.26993e10 2.42655
\(66\) 4.75734e9 0.467598
\(67\) −3.89914e9 −0.352823 −0.176412 0.984316i \(-0.556449\pi\)
−0.176412 + 0.984316i \(0.556449\pi\)
\(68\) −4.15444e10 −3.46508
\(69\) 1.00506e9 0.0773609
\(70\) 2.46515e10 1.75309
\(71\) 2.09101e10 1.37542 0.687709 0.725986i \(-0.258617\pi\)
0.687709 + 0.725986i \(0.258617\pi\)
\(72\) −1.75499e10 −1.06892
\(73\) 1.32307e10 0.746975 0.373487 0.927635i \(-0.378162\pi\)
0.373487 + 0.927635i \(0.378162\pi\)
\(74\) −1.37497e10 −0.720308
\(75\) 2.15526e10 1.04873
\(76\) 1.10829e11 5.01391
\(77\) −5.47069e9 −0.230326
\(78\) 4.07962e10 1.59992
\(79\) 3.04105e10 1.11192 0.555961 0.831208i \(-0.312350\pi\)
0.555961 + 0.831208i \(0.312350\pi\)
\(80\) −1.70812e11 −5.82807
\(81\) 3.48678e9 0.111111
\(82\) −6.43572e10 −1.91699
\(83\) −6.10863e10 −1.70221 −0.851107 0.524993i \(-0.824068\pi\)
−0.851107 + 0.524993i \(0.824068\pi\)
\(84\) 3.22431e10 0.841204
\(85\) 8.89895e10 2.17538
\(86\) 1.42668e10 0.327028
\(87\) 3.12822e10 0.672885
\(88\) 6.70861e10 1.35512
\(89\) 6.05385e10 1.14918 0.574588 0.818443i \(-0.305162\pi\)
0.574588 + 0.818443i \(0.305162\pi\)
\(90\) 6.00600e10 1.07214
\(91\) −4.69135e10 −0.788080
\(92\) 2.26436e10 0.358189
\(93\) −3.07532e10 −0.458389
\(94\) 1.19516e11 1.67967
\(95\) −2.37398e11 −3.14773
\(96\) −1.59081e11 −1.99126
\(97\) −1.06180e11 −1.25544 −0.627720 0.778439i \(-0.716012\pi\)
−0.627720 + 0.778439i \(0.716012\pi\)
\(98\) 1.20552e11 1.34720
\(99\) −1.33286e10 −0.140861
\(100\) 4.85571e11 4.85571
\(101\) 1.20010e11 1.13619 0.568094 0.822964i \(-0.307681\pi\)
0.568094 + 0.822964i \(0.307681\pi\)
\(102\) 1.59936e11 1.43432
\(103\) 8.50039e10 0.722493 0.361247 0.932470i \(-0.382351\pi\)
0.361247 + 0.932470i \(0.382351\pi\)
\(104\) 5.75292e11 4.63665
\(105\) −6.90659e10 −0.528107
\(106\) −3.52728e10 −0.256010
\(107\) 1.46709e11 1.01122 0.505612 0.862761i \(-0.331267\pi\)
0.505612 + 0.862761i \(0.331267\pi\)
\(108\) 7.85558e10 0.514455
\(109\) 2.04883e11 1.27544 0.637721 0.770267i \(-0.279877\pi\)
0.637721 + 0.770267i \(0.279877\pi\)
\(110\) −2.29585e11 −1.35920
\(111\) 3.85223e10 0.216988
\(112\) 3.53024e11 1.89280
\(113\) 2.74469e10 0.140140 0.0700699 0.997542i \(-0.477678\pi\)
0.0700699 + 0.997542i \(0.477678\pi\)
\(114\) −4.26663e11 −2.07543
\(115\) −4.85033e10 −0.224871
\(116\) 7.04775e11 3.11553
\(117\) −1.14298e11 −0.481966
\(118\) 6.20079e10 0.249514
\(119\) −1.83918e11 −0.706506
\(120\) 8.46942e11 3.10711
\(121\) −2.34362e11 −0.821424
\(122\) −1.08597e11 −0.363779
\(123\) 1.80309e11 0.577482
\(124\) −6.92857e11 −2.12239
\(125\) −4.67503e11 −1.37018
\(126\) −1.24128e11 −0.348203
\(127\) 3.40084e11 0.913411 0.456706 0.889618i \(-0.349029\pi\)
0.456706 + 0.889618i \(0.349029\pi\)
\(128\) −9.96721e11 −2.56400
\(129\) −3.99711e10 −0.0985151
\(130\) −1.96879e12 −4.65062
\(131\) −8.54131e11 −1.93434 −0.967169 0.254133i \(-0.918210\pi\)
−0.967169 + 0.254133i \(0.918210\pi\)
\(132\) −3.00287e11 −0.652199
\(133\) 4.90640e11 1.02230
\(134\) 3.38186e11 0.676206
\(135\) −1.68269e11 −0.322975
\(136\) 2.25535e12 4.15672
\(137\) 4.87116e11 0.862321 0.431161 0.902275i \(-0.358104\pi\)
0.431161 + 0.902275i \(0.358104\pi\)
\(138\) −8.71722e10 −0.148267
\(139\) 4.36340e11 0.713253 0.356627 0.934247i \(-0.383927\pi\)
0.356627 + 0.934247i \(0.383927\pi\)
\(140\) −1.55603e12 −2.44519
\(141\) −3.34848e11 −0.505990
\(142\) −1.81360e12 −2.63607
\(143\) 4.36916e11 0.611012
\(144\) 8.60094e11 1.15758
\(145\) −1.50965e12 −1.95593
\(146\) −1.14754e12 −1.43162
\(147\) −3.37749e11 −0.405835
\(148\) 8.67892e11 1.00468
\(149\) 1.56344e11 0.174405 0.0872023 0.996191i \(-0.472207\pi\)
0.0872023 + 0.996191i \(0.472207\pi\)
\(150\) −1.86933e12 −2.00994
\(151\) −9.22244e11 −0.956032 −0.478016 0.878351i \(-0.658644\pi\)
−0.478016 + 0.878351i \(0.658644\pi\)
\(152\) −6.01663e12 −6.01470
\(153\) −4.48090e11 −0.432079
\(154\) 4.74492e11 0.441433
\(155\) 1.48412e12 1.33243
\(156\) −2.57509e12 −2.23155
\(157\) −1.30773e12 −1.09413 −0.547067 0.837089i \(-0.684256\pi\)
−0.547067 + 0.837089i \(0.684256\pi\)
\(158\) −2.63761e12 −2.13106
\(159\) 9.88233e10 0.0771214
\(160\) 7.67713e12 5.78814
\(161\) 1.00244e11 0.0730322
\(162\) −3.02421e11 −0.212951
\(163\) −1.36308e12 −0.927873 −0.463937 0.885868i \(-0.653563\pi\)
−0.463937 + 0.885868i \(0.653563\pi\)
\(164\) 4.06228e12 2.67380
\(165\) 6.43225e11 0.409451
\(166\) 5.29822e12 3.26239
\(167\) 1.84641e12 1.09999 0.549995 0.835168i \(-0.314630\pi\)
0.549995 + 0.835168i \(0.314630\pi\)
\(168\) −1.75041e12 −1.00911
\(169\) 1.95458e12 1.09063
\(170\) −7.71837e12 −4.16924
\(171\) 1.19538e12 0.625210
\(172\) −9.00533e11 −0.456135
\(173\) 1.76237e12 0.864655 0.432327 0.901717i \(-0.357693\pi\)
0.432327 + 0.901717i \(0.357693\pi\)
\(174\) −2.71321e12 −1.28962
\(175\) 2.14963e12 0.990045
\(176\) −3.28779e12 −1.46752
\(177\) −1.73727e11 −0.0751646
\(178\) −5.25071e12 −2.20246
\(179\) 4.00844e12 1.63036 0.815180 0.579208i \(-0.196638\pi\)
0.815180 + 0.579208i \(0.196638\pi\)
\(180\) −3.79104e12 −1.49541
\(181\) 9.05666e11 0.346526 0.173263 0.984876i \(-0.444569\pi\)
0.173263 + 0.984876i \(0.444569\pi\)
\(182\) 4.06897e12 1.51040
\(183\) 3.04254e11 0.109586
\(184\) −1.22927e12 −0.429684
\(185\) −1.85905e12 −0.630735
\(186\) 2.66733e12 0.878528
\(187\) 1.71287e12 0.547766
\(188\) −7.54398e12 −2.34278
\(189\) 3.47768e11 0.104894
\(190\) 2.05904e13 6.03281
\(191\) −6.55756e12 −1.86663 −0.933316 0.359055i \(-0.883099\pi\)
−0.933316 + 0.359055i \(0.883099\pi\)
\(192\) 6.54882e12 1.81137
\(193\) −6.69275e12 −1.79903 −0.899517 0.436885i \(-0.856082\pi\)
−0.899517 + 0.436885i \(0.856082\pi\)
\(194\) 9.20932e12 2.40612
\(195\) 5.51593e12 1.40097
\(196\) −7.60936e12 −1.87906
\(197\) −3.16829e12 −0.760783 −0.380391 0.924826i \(-0.624211\pi\)
−0.380391 + 0.924826i \(0.624211\pi\)
\(198\) 1.15603e12 0.269968
\(199\) 4.98278e12 1.13183 0.565913 0.824465i \(-0.308524\pi\)
0.565913 + 0.824465i \(0.308524\pi\)
\(200\) −2.63605e13 −5.82491
\(201\) −9.47490e11 −0.203703
\(202\) −1.04089e13 −2.17757
\(203\) 3.12005e12 0.635234
\(204\) −1.00953e13 −2.00057
\(205\) −8.70154e12 −1.67861
\(206\) −7.37268e12 −1.38470
\(207\) 2.44229e11 0.0446643
\(208\) −2.81942e13 −5.02125
\(209\) −4.56944e12 −0.792608
\(210\) 5.99032e12 1.01215
\(211\) 4.55874e12 0.750397 0.375199 0.926944i \(-0.377575\pi\)
0.375199 + 0.926944i \(0.377575\pi\)
\(212\) 2.22645e12 0.357080
\(213\) 5.08115e12 0.794098
\(214\) −1.27246e13 −1.93807
\(215\) 1.92897e12 0.286361
\(216\) −4.26462e12 −0.617142
\(217\) −3.06729e12 −0.432740
\(218\) −1.77702e13 −2.44446
\(219\) 3.21505e12 0.431266
\(220\) 1.44916e13 1.89580
\(221\) 1.46886e13 1.87423
\(222\) −3.34117e12 −0.415870
\(223\) −8.99847e12 −1.09268 −0.546339 0.837564i \(-0.683979\pi\)
−0.546339 + 0.837564i \(0.683979\pi\)
\(224\) −1.58666e13 −1.87984
\(225\) 5.23727e12 0.605482
\(226\) −2.38056e12 −0.268586
\(227\) −1.33843e12 −0.147385 −0.0736925 0.997281i \(-0.523478\pi\)
−0.0736925 + 0.997281i \(0.523478\pi\)
\(228\) 2.69313e13 2.89478
\(229\) −1.15610e13 −1.21311 −0.606555 0.795042i \(-0.707449\pi\)
−0.606555 + 0.795042i \(0.707449\pi\)
\(230\) 4.20686e12 0.430978
\(231\) −1.32938e12 −0.132979
\(232\) −3.82606e13 −3.73739
\(233\) −5.98856e12 −0.571301 −0.285650 0.958334i \(-0.592210\pi\)
−0.285650 + 0.958334i \(0.592210\pi\)
\(234\) 9.91347e12 0.923716
\(235\) 1.61595e13 1.47080
\(236\) −3.91399e12 −0.348020
\(237\) 7.38975e12 0.641969
\(238\) 1.59519e13 1.35406
\(239\) 1.71328e13 1.42115 0.710576 0.703620i \(-0.248434\pi\)
0.710576 + 0.703620i \(0.248434\pi\)
\(240\) −4.15074e13 −3.36484
\(241\) 1.29158e13 1.02336 0.511679 0.859177i \(-0.329024\pi\)
0.511679 + 0.859177i \(0.329024\pi\)
\(242\) 2.03270e13 1.57431
\(243\) 8.47289e11 0.0641500
\(244\) 6.85471e12 0.507394
\(245\) 1.62995e13 1.17967
\(246\) −1.56388e13 −1.10678
\(247\) −3.91849e13 −2.71197
\(248\) 3.76137e13 2.54602
\(249\) −1.48440e13 −0.982773
\(250\) 4.05481e13 2.62604
\(251\) −2.60915e13 −1.65308 −0.826540 0.562879i \(-0.809694\pi\)
−0.826540 + 0.562879i \(0.809694\pi\)
\(252\) 7.83508e12 0.485669
\(253\) −9.33591e11 −0.0566231
\(254\) −2.94967e13 −1.75060
\(255\) 2.16245e13 1.25596
\(256\) 3.12557e13 1.77668
\(257\) 2.28259e13 1.26998 0.634989 0.772521i \(-0.281005\pi\)
0.634989 + 0.772521i \(0.281005\pi\)
\(258\) 3.46683e12 0.188810
\(259\) 3.84218e12 0.204846
\(260\) 1.24272e14 6.48662
\(261\) 7.60157e12 0.388491
\(262\) 7.40818e13 3.70727
\(263\) −2.36115e13 −1.15709 −0.578545 0.815651i \(-0.696379\pi\)
−0.578545 + 0.815651i \(0.696379\pi\)
\(264\) 1.63019e13 0.782379
\(265\) −4.76912e12 −0.224175
\(266\) −4.25549e13 −1.95930
\(267\) 1.47108e13 0.663477
\(268\) −2.13466e13 −0.943163
\(269\) −3.13089e13 −1.35528 −0.677642 0.735392i \(-0.736998\pi\)
−0.677642 + 0.735392i \(0.736998\pi\)
\(270\) 1.45946e13 0.619000
\(271\) −4.19841e13 −1.74483 −0.872417 0.488763i \(-0.837448\pi\)
−0.872417 + 0.488763i \(0.837448\pi\)
\(272\) −1.10532e14 −4.50151
\(273\) −1.14000e13 −0.454998
\(274\) −4.22492e13 −1.65269
\(275\) −2.00200e13 −0.767599
\(276\) 5.50239e12 0.206800
\(277\) −3.49366e13 −1.28719 −0.643595 0.765366i \(-0.722558\pi\)
−0.643595 + 0.765366i \(0.722558\pi\)
\(278\) −3.78453e13 −1.36699
\(279\) −7.47303e12 −0.264651
\(280\) 8.44732e13 2.93325
\(281\) −5.31784e13 −1.81072 −0.905359 0.424647i \(-0.860398\pi\)
−0.905359 + 0.424647i \(0.860398\pi\)
\(282\) 2.90425e13 0.969759
\(283\) −1.01159e13 −0.331267 −0.165633 0.986187i \(-0.552967\pi\)
−0.165633 + 0.986187i \(0.552967\pi\)
\(284\) 1.14476e14 3.67675
\(285\) −5.76878e13 −1.81734
\(286\) −3.78952e13 −1.17104
\(287\) 1.79838e13 0.545169
\(288\) −3.86568e13 −1.14965
\(289\) 2.33127e13 0.680228
\(290\) 1.30937e14 3.74864
\(291\) −2.58016e13 −0.724829
\(292\) 7.24338e13 1.99680
\(293\) −1.50814e13 −0.408008 −0.204004 0.978970i \(-0.565396\pi\)
−0.204004 + 0.978970i \(0.565396\pi\)
\(294\) 2.92942e13 0.777805
\(295\) 8.38390e12 0.218487
\(296\) −4.71159e13 −1.20521
\(297\) −3.23884e12 −0.0813260
\(298\) −1.35603e13 −0.334256
\(299\) −8.00593e12 −0.193740
\(300\) 1.17994e14 2.80344
\(301\) −3.98668e12 −0.0930027
\(302\) 7.99894e13 1.83229
\(303\) 2.91624e13 0.655978
\(304\) 2.94866e14 6.51359
\(305\) −1.46830e13 −0.318542
\(306\) 3.88644e13 0.828103
\(307\) 1.31999e13 0.276256 0.138128 0.990414i \(-0.455892\pi\)
0.138128 + 0.990414i \(0.455892\pi\)
\(308\) −2.99504e13 −0.615706
\(309\) 2.06559e13 0.417132
\(310\) −1.28723e14 −2.55368
\(311\) −5.37958e13 −1.04850 −0.524248 0.851566i \(-0.675653\pi\)
−0.524248 + 0.851566i \(0.675653\pi\)
\(312\) 1.39796e14 2.67697
\(313\) 2.38417e13 0.448583 0.224291 0.974522i \(-0.427993\pi\)
0.224291 + 0.974522i \(0.427993\pi\)
\(314\) 1.13424e14 2.09697
\(315\) −1.67830e13 −0.304903
\(316\) 1.66488e14 2.97238
\(317\) 5.46736e13 0.959294 0.479647 0.877461i \(-0.340765\pi\)
0.479647 + 0.877461i \(0.340765\pi\)
\(318\) −8.57128e12 −0.147808
\(319\) −2.90577e13 −0.492508
\(320\) −3.16041e14 −5.26524
\(321\) 3.56504e13 0.583830
\(322\) −8.69447e12 −0.139970
\(323\) −1.53619e14 −2.43126
\(324\) 1.90891e13 0.297021
\(325\) −1.71680e14 −2.62640
\(326\) 1.18224e14 1.77832
\(327\) 4.97866e13 0.736377
\(328\) −2.20532e14 −3.20749
\(329\) −3.33974e13 −0.477677
\(330\) −5.57892e13 −0.784735
\(331\) −4.00410e13 −0.553925 −0.276962 0.960881i \(-0.589328\pi\)
−0.276962 + 0.960881i \(0.589328\pi\)
\(332\) −3.34428e14 −4.55034
\(333\) 9.36092e12 0.125278
\(334\) −1.60146e14 −2.10819
\(335\) 4.57251e13 0.592118
\(336\) 8.57850e13 1.09281
\(337\) 1.09478e14 1.37203 0.686013 0.727589i \(-0.259359\pi\)
0.686013 + 0.727589i \(0.259359\pi\)
\(338\) −1.69527e14 −2.09024
\(339\) 6.66959e12 0.0809098
\(340\) 4.87190e14 5.81520
\(341\) 2.85664e13 0.335511
\(342\) −1.03679e14 −1.19825
\(343\) −8.16104e13 −0.928171
\(344\) 4.88879e13 0.547180
\(345\) −1.17863e13 −0.129829
\(346\) −1.52856e14 −1.65716
\(347\) 6.14955e13 0.656193 0.328096 0.944644i \(-0.393593\pi\)
0.328096 + 0.944644i \(0.393593\pi\)
\(348\) 1.71260e14 1.79875
\(349\) −1.61145e14 −1.66601 −0.833005 0.553266i \(-0.813381\pi\)
−0.833005 + 0.553266i \(0.813381\pi\)
\(350\) −1.86445e14 −1.89748
\(351\) −2.77744e13 −0.278263
\(352\) 1.47769e14 1.45747
\(353\) −5.18807e13 −0.503785 −0.251892 0.967755i \(-0.581053\pi\)
−0.251892 + 0.967755i \(0.581053\pi\)
\(354\) 1.50679e13 0.144057
\(355\) −2.45212e14 −2.30826
\(356\) 3.31429e14 3.07196
\(357\) −4.46921e13 −0.407902
\(358\) −3.47665e14 −3.12468
\(359\) −2.44194e13 −0.216130 −0.108065 0.994144i \(-0.534466\pi\)
−0.108065 + 0.994144i \(0.534466\pi\)
\(360\) 2.05807e14 1.79389
\(361\) 2.93321e14 2.51799
\(362\) −7.85515e13 −0.664137
\(363\) −5.69499e13 −0.474249
\(364\) −2.56837e14 −2.10669
\(365\) −1.55156e14 −1.25359
\(366\) −2.63890e13 −0.210028
\(367\) 1.30968e14 1.02684 0.513418 0.858139i \(-0.328379\pi\)
0.513418 + 0.858139i \(0.328379\pi\)
\(368\) 6.02447e13 0.465324
\(369\) 4.38150e13 0.333409
\(370\) 1.61242e14 1.20884
\(371\) 9.85653e12 0.0728061
\(372\) −1.68364e14 −1.22536
\(373\) 2.23070e14 1.59972 0.799858 0.600189i \(-0.204908\pi\)
0.799858 + 0.600189i \(0.204908\pi\)
\(374\) −1.48563e14 −1.04983
\(375\) −1.13603e14 −0.791076
\(376\) 4.09546e14 2.81041
\(377\) −2.49182e14 −1.68515
\(378\) −3.01632e13 −0.201035
\(379\) 9.13403e13 0.599994 0.299997 0.953940i \(-0.403014\pi\)
0.299997 + 0.953940i \(0.403014\pi\)
\(380\) −1.29968e15 −8.41449
\(381\) 8.26405e13 0.527358
\(382\) 5.68760e14 3.57751
\(383\) −1.79938e14 −1.11565 −0.557826 0.829958i \(-0.688364\pi\)
−0.557826 + 0.829958i \(0.688364\pi\)
\(384\) −2.42203e14 −1.48033
\(385\) 6.41547e13 0.386540
\(386\) 5.80485e14 3.44795
\(387\) −9.71298e12 −0.0568777
\(388\) −5.81300e14 −3.35603
\(389\) −2.49341e14 −1.41929 −0.709643 0.704561i \(-0.751144\pi\)
−0.709643 + 0.704561i \(0.751144\pi\)
\(390\) −4.78416e14 −2.68503
\(391\) −3.13862e13 −0.173687
\(392\) 4.13095e14 2.25412
\(393\) −2.07554e14 −1.11679
\(394\) 2.74797e14 1.45808
\(395\) −3.56623e14 −1.86606
\(396\) −7.29698e13 −0.376548
\(397\) 2.47278e14 1.25845 0.629226 0.777222i \(-0.283372\pi\)
0.629226 + 0.777222i \(0.283372\pi\)
\(398\) −4.32174e14 −2.16921
\(399\) 1.19226e14 0.590226
\(400\) 1.29189e15 6.30807
\(401\) 3.49823e13 0.168483 0.0842413 0.996445i \(-0.473153\pi\)
0.0842413 + 0.996445i \(0.473153\pi\)
\(402\) 8.21791e13 0.390407
\(403\) 2.44969e14 1.14798
\(404\) 6.57018e14 3.03724
\(405\) −4.08894e13 −0.186470
\(406\) −2.70613e14 −1.21746
\(407\) −3.57830e13 −0.158821
\(408\) 5.48051e14 2.39988
\(409\) −2.42776e14 −1.04888 −0.524441 0.851447i \(-0.675726\pi\)
−0.524441 + 0.851447i \(0.675726\pi\)
\(410\) 7.54715e14 3.21715
\(411\) 1.18369e14 0.497861
\(412\) 4.65370e14 1.93136
\(413\) −1.73273e13 −0.0709588
\(414\) −2.11829e13 −0.0856017
\(415\) 7.16357e14 2.85670
\(416\) 1.26718e15 4.98685
\(417\) 1.06031e14 0.411797
\(418\) 3.96323e14 1.51908
\(419\) 1.65501e14 0.626069 0.313035 0.949742i \(-0.398654\pi\)
0.313035 + 0.949742i \(0.398654\pi\)
\(420\) −3.78114e14 −1.41173
\(421\) −5.33194e14 −1.96487 −0.982435 0.186604i \(-0.940252\pi\)
−0.982435 + 0.186604i \(0.940252\pi\)
\(422\) −3.95395e14 −1.43818
\(423\) −8.13680e13 −0.292134
\(424\) −1.20869e14 −0.428353
\(425\) −6.73048e14 −2.35454
\(426\) −4.40706e14 −1.52193
\(427\) 3.03460e13 0.103454
\(428\) 8.03189e14 2.70319
\(429\) 1.06171e14 0.352768
\(430\) −1.67306e14 −0.548828
\(431\) 2.86684e14 0.928494 0.464247 0.885706i \(-0.346325\pi\)
0.464247 + 0.885706i \(0.346325\pi\)
\(432\) 2.09003e14 0.668331
\(433\) 3.80518e14 1.20141 0.600706 0.799470i \(-0.294886\pi\)
0.600706 + 0.799470i \(0.294886\pi\)
\(434\) 2.66037e14 0.829370
\(435\) −3.66845e14 −1.12925
\(436\) 1.12167e15 3.40950
\(437\) 8.37293e13 0.251321
\(438\) −2.78852e14 −0.826546
\(439\) −4.85503e14 −1.42114 −0.710570 0.703626i \(-0.751563\pi\)
−0.710570 + 0.703626i \(0.751563\pi\)
\(440\) −7.86717e14 −2.27420
\(441\) −8.20731e13 −0.234309
\(442\) −1.27399e15 −3.59206
\(443\) 6.11464e14 1.70275 0.851373 0.524560i \(-0.175770\pi\)
0.851373 + 0.524560i \(0.175770\pi\)
\(444\) 2.10898e14 0.580050
\(445\) −7.09933e14 −1.92858
\(446\) 7.80468e14 2.09418
\(447\) 3.79917e13 0.100693
\(448\) 6.53173e14 1.71001
\(449\) −3.00652e14 −0.777516 −0.388758 0.921340i \(-0.627096\pi\)
−0.388758 + 0.921340i \(0.627096\pi\)
\(450\) −4.54247e14 −1.16044
\(451\) −1.67487e14 −0.422679
\(452\) 1.50263e14 0.374620
\(453\) −2.24105e14 −0.551965
\(454\) 1.16087e14 0.282472
\(455\) 5.50153e14 1.32258
\(456\) −1.46204e15 −3.47259
\(457\) 8.18072e12 0.0191979 0.00959893 0.999954i \(-0.496945\pi\)
0.00959893 + 0.999954i \(0.496945\pi\)
\(458\) 1.00272e15 2.32499
\(459\) −1.08886e14 −0.249461
\(460\) −2.65541e14 −0.601122
\(461\) −3.92650e14 −0.878316 −0.439158 0.898410i \(-0.644723\pi\)
−0.439158 + 0.898410i \(0.644723\pi\)
\(462\) 1.15302e14 0.254862
\(463\) −4.44783e13 −0.0971522 −0.0485761 0.998819i \(-0.515468\pi\)
−0.0485761 + 0.998819i \(0.515468\pi\)
\(464\) 1.87510e15 4.04739
\(465\) 3.60642e14 0.769281
\(466\) 5.19408e14 1.09493
\(467\) −1.85784e14 −0.387048 −0.193524 0.981096i \(-0.561992\pi\)
−0.193524 + 0.981096i \(0.561992\pi\)
\(468\) −6.25747e14 −1.28839
\(469\) −9.45017e13 −0.192304
\(470\) −1.40157e15 −2.81887
\(471\) −3.17779e14 −0.631698
\(472\) 2.12482e14 0.417485
\(473\) 3.71288e13 0.0721066
\(474\) −6.40939e14 −1.23037
\(475\) 1.79550e15 3.40698
\(476\) −1.00689e15 −1.88863
\(477\) 2.40141e13 0.0445261
\(478\) −1.48599e15 −2.72372
\(479\) −4.25471e14 −0.770948 −0.385474 0.922719i \(-0.625962\pi\)
−0.385474 + 0.922719i \(0.625962\pi\)
\(480\) 1.86554e15 3.34179
\(481\) −3.06854e14 −0.543419
\(482\) −1.12023e15 −1.96132
\(483\) 2.43592e13 0.0421652
\(484\) −1.28306e15 −2.19582
\(485\) 1.24516e15 2.10692
\(486\) −7.34883e13 −0.122947
\(487\) −4.34881e12 −0.00719385 −0.00359692 0.999994i \(-0.501145\pi\)
−0.00359692 + 0.999994i \(0.501145\pi\)
\(488\) −3.72127e14 −0.608670
\(489\) −3.31228e14 −0.535708
\(490\) −1.41371e15 −2.26090
\(491\) −1.49931e14 −0.237107 −0.118553 0.992948i \(-0.537826\pi\)
−0.118553 + 0.992948i \(0.537826\pi\)
\(492\) 9.87134e14 1.54372
\(493\) −9.76886e14 −1.51073
\(494\) 3.39864e15 5.19765
\(495\) 1.56304e14 0.236396
\(496\) −1.84339e15 −2.75720
\(497\) 5.06789e14 0.749664
\(498\) 1.28747e15 1.88354
\(499\) 6.62725e14 0.958916 0.479458 0.877565i \(-0.340833\pi\)
0.479458 + 0.877565i \(0.340833\pi\)
\(500\) −2.55943e15 −3.66276
\(501\) 4.48679e14 0.635079
\(502\) 2.26301e15 3.16822
\(503\) −1.25716e15 −1.74087 −0.870433 0.492286i \(-0.836161\pi\)
−0.870433 + 0.492286i \(0.836161\pi\)
\(504\) −4.25349e14 −0.582609
\(505\) −1.40735e15 −1.90678
\(506\) 8.09735e13 0.108521
\(507\) 4.74962e14 0.629673
\(508\) 1.86186e15 2.44172
\(509\) −7.08145e14 −0.918702 −0.459351 0.888255i \(-0.651918\pi\)
−0.459351 + 0.888255i \(0.651918\pi\)
\(510\) −1.87556e15 −2.40711
\(511\) 3.20666e14 0.407134
\(512\) −6.69628e14 −0.841102
\(513\) 2.90476e14 0.360965
\(514\) −1.97977e15 −2.43398
\(515\) −9.96838e14 −1.21251
\(516\) −2.18829e14 −0.263349
\(517\) 3.11037e14 0.370351
\(518\) −3.33245e14 −0.392600
\(519\) 4.28255e14 0.499209
\(520\) −6.74643e15 −7.78136
\(521\) 1.38595e15 1.58176 0.790881 0.611971i \(-0.209623\pi\)
0.790881 + 0.611971i \(0.209623\pi\)
\(522\) −6.59310e14 −0.744564
\(523\) −3.11808e14 −0.348440 −0.174220 0.984707i \(-0.555740\pi\)
−0.174220 + 0.984707i \(0.555740\pi\)
\(524\) −4.67611e15 −5.17085
\(525\) 5.22361e14 0.571603
\(526\) 2.04791e15 2.21763
\(527\) 9.60367e14 1.02915
\(528\) −7.98934e14 −0.847275
\(529\) −9.35703e14 −0.982046
\(530\) 4.13643e14 0.429643
\(531\) −4.22156e13 −0.0433963
\(532\) 2.68610e15 2.73281
\(533\) −1.43627e15 −1.44623
\(534\) −1.27592e15 −1.27159
\(535\) −1.72046e15 −1.69706
\(536\) 1.15886e15 1.13142
\(537\) 9.74050e14 0.941288
\(538\) 2.71553e15 2.59748
\(539\) 3.13732e14 0.297044
\(540\) −9.21222e14 −0.863373
\(541\) −1.49757e15 −1.38932 −0.694660 0.719338i \(-0.744445\pi\)
−0.694660 + 0.719338i \(0.744445\pi\)
\(542\) 3.64143e15 3.34407
\(543\) 2.20077e14 0.200067
\(544\) 4.96783e15 4.47067
\(545\) −2.40266e15 −2.14048
\(546\) 9.88760e14 0.872029
\(547\) 3.15839e14 0.275762 0.137881 0.990449i \(-0.455971\pi\)
0.137881 + 0.990449i \(0.455971\pi\)
\(548\) 2.66681e15 2.30515
\(549\) 7.39337e13 0.0632695
\(550\) 1.73640e15 1.47115
\(551\) 2.60605e15 2.18599
\(552\) −2.98712e14 −0.248078
\(553\) 7.37047e14 0.606048
\(554\) 3.03018e15 2.46697
\(555\) −4.51750e14 −0.364155
\(556\) 2.38883e15 1.90666
\(557\) −8.63856e14 −0.682712 −0.341356 0.939934i \(-0.610886\pi\)
−0.341356 + 0.939934i \(0.610886\pi\)
\(558\) 6.48162e14 0.507219
\(559\) 3.18395e14 0.246718
\(560\) −4.13991e15 −3.17656
\(561\) 4.16227e14 0.316253
\(562\) 4.61235e15 3.47034
\(563\) 2.02055e15 1.50548 0.752739 0.658319i \(-0.228732\pi\)
0.752739 + 0.658319i \(0.228732\pi\)
\(564\) −1.83319e15 −1.35261
\(565\) −3.21869e14 −0.235187
\(566\) 8.77385e14 0.634892
\(567\) 8.45077e13 0.0605605
\(568\) −6.21466e15 −4.41064
\(569\) −8.80440e14 −0.618846 −0.309423 0.950925i \(-0.600136\pi\)
−0.309423 + 0.950925i \(0.600136\pi\)
\(570\) 5.00346e15 3.48304
\(571\) −1.19192e15 −0.821767 −0.410883 0.911688i \(-0.634780\pi\)
−0.410883 + 0.911688i \(0.634780\pi\)
\(572\) 2.39198e15 1.63335
\(573\) −1.59349e15 −1.07770
\(574\) −1.55980e15 −1.04485
\(575\) 3.66841e14 0.243391
\(576\) 1.59136e15 1.04579
\(577\) 2.00994e14 0.130832 0.0654162 0.997858i \(-0.479162\pi\)
0.0654162 + 0.997858i \(0.479162\pi\)
\(578\) −2.02199e15 −1.30370
\(579\) −1.62634e15 −1.03867
\(580\) −8.26487e15 −5.22856
\(581\) −1.48052e15 −0.927782
\(582\) 2.23786e15 1.38918
\(583\) −9.17960e13 −0.0564478
\(584\) −3.93227e15 −2.39537
\(585\) 1.34037e15 0.808849
\(586\) 1.30806e15 0.781971
\(587\) −9.19761e14 −0.544710 −0.272355 0.962197i \(-0.587803\pi\)
−0.272355 + 0.962197i \(0.587803\pi\)
\(588\) −1.84907e15 −1.08487
\(589\) −2.56198e15 −1.48916
\(590\) −7.27164e14 −0.418742
\(591\) −7.69894e14 −0.439238
\(592\) 2.30908e15 1.30518
\(593\) 7.80639e14 0.437169 0.218585 0.975818i \(-0.429856\pi\)
0.218585 + 0.975818i \(0.429856\pi\)
\(594\) 2.80916e14 0.155866
\(595\) 2.15680e15 1.18568
\(596\) 8.55937e14 0.466216
\(597\) 1.21082e15 0.653460
\(598\) 6.94382e14 0.371315
\(599\) −5.49675e14 −0.291245 −0.145622 0.989340i \(-0.546518\pi\)
−0.145622 + 0.989340i \(0.546518\pi\)
\(600\) −6.40561e15 −3.36302
\(601\) −1.50512e15 −0.783000 −0.391500 0.920178i \(-0.628044\pi\)
−0.391500 + 0.920178i \(0.628044\pi\)
\(602\) 3.45778e14 0.178245
\(603\) −2.30240e14 −0.117608
\(604\) −5.04900e15 −2.55565
\(605\) 2.74836e15 1.37854
\(606\) −2.52936e15 −1.25722
\(607\) −2.78314e15 −1.37087 −0.685436 0.728133i \(-0.740388\pi\)
−0.685436 + 0.728133i \(0.740388\pi\)
\(608\) −1.32527e16 −6.46898
\(609\) 7.58173e14 0.366753
\(610\) 1.27351e15 0.610503
\(611\) 2.66727e15 1.26719
\(612\) −2.45316e15 −1.15503
\(613\) −1.81488e15 −0.846868 −0.423434 0.905927i \(-0.639175\pi\)
−0.423434 + 0.905927i \(0.639175\pi\)
\(614\) −1.14488e15 −0.529460
\(615\) −2.11447e15 −0.969146
\(616\) 1.62594e15 0.738601
\(617\) 2.32132e15 1.04512 0.522559 0.852603i \(-0.324977\pi\)
0.522559 + 0.852603i \(0.324977\pi\)
\(618\) −1.79156e15 −0.799457
\(619\) 1.23097e15 0.544440 0.272220 0.962235i \(-0.412242\pi\)
0.272220 + 0.962235i \(0.412242\pi\)
\(620\) 8.12512e15 3.56185
\(621\) 5.93477e13 0.0257870
\(622\) 4.66590e15 2.00950
\(623\) 1.46725e15 0.626352
\(624\) −6.85120e15 −2.89902
\(625\) 1.15164e15 0.483033
\(626\) −2.06787e15 −0.859735
\(627\) −1.11037e15 −0.457612
\(628\) −7.15942e15 −2.92483
\(629\) −1.20298e15 −0.487170
\(630\) 1.45565e15 0.584364
\(631\) 1.68636e15 0.671101 0.335551 0.942022i \(-0.391078\pi\)
0.335551 + 0.942022i \(0.391078\pi\)
\(632\) −9.03827e15 −3.56567
\(633\) 1.10777e15 0.433242
\(634\) −4.74203e15 −1.83854
\(635\) −3.98816e15 −1.53291
\(636\) 5.41027e14 0.206160
\(637\) 2.69039e15 1.01636
\(638\) 2.52028e15 0.943919
\(639\) 1.23472e15 0.458473
\(640\) 1.16885e16 4.30298
\(641\) 1.29835e15 0.473885 0.236943 0.971524i \(-0.423855\pi\)
0.236943 + 0.971524i \(0.423855\pi\)
\(642\) −3.09208e15 −1.11894
\(643\) −2.08702e15 −0.748800 −0.374400 0.927267i \(-0.622151\pi\)
−0.374400 + 0.927267i \(0.622151\pi\)
\(644\) 5.48803e14 0.195229
\(645\) 4.68740e14 0.165331
\(646\) 1.33239e16 4.65964
\(647\) −4.43825e15 −1.53900 −0.769499 0.638649i \(-0.779494\pi\)
−0.769499 + 0.638649i \(0.779494\pi\)
\(648\) −1.03630e15 −0.356307
\(649\) 1.61373e14 0.0550155
\(650\) 1.48904e16 5.03364
\(651\) −7.45352e14 −0.249842
\(652\) −7.46243e15 −2.48038
\(653\) −2.02446e15 −0.667247 −0.333624 0.942706i \(-0.608271\pi\)
−0.333624 + 0.942706i \(0.608271\pi\)
\(654\) −4.31817e15 −1.41131
\(655\) 1.00164e16 3.24626
\(656\) 1.08080e16 3.47354
\(657\) 7.81257e14 0.248992
\(658\) 2.89667e15 0.915496
\(659\) 5.05072e15 1.58301 0.791504 0.611164i \(-0.209298\pi\)
0.791504 + 0.611164i \(0.209298\pi\)
\(660\) 3.52146e15 1.09454
\(661\) −5.90345e15 −1.81969 −0.909845 0.414947i \(-0.863800\pi\)
−0.909845 + 0.414947i \(0.863800\pi\)
\(662\) 3.47289e15 1.06163
\(663\) 3.56932e15 1.08209
\(664\) 1.81554e16 5.45859
\(665\) −5.75372e15 −1.71565
\(666\) −8.11905e14 −0.240103
\(667\) 5.32447e14 0.156165
\(668\) 1.01086e16 2.94048
\(669\) −2.18663e15 −0.630858
\(670\) −3.96589e15 −1.13483
\(671\) −2.82619e14 −0.0802097
\(672\) −3.85559e15 −1.08533
\(673\) −2.41929e15 −0.675470 −0.337735 0.941241i \(-0.609661\pi\)
−0.337735 + 0.941241i \(0.609661\pi\)
\(674\) −9.49541e15 −2.62957
\(675\) 1.27266e15 0.349575
\(676\) 1.07007e16 2.91545
\(677\) −4.11801e15 −1.11288 −0.556442 0.830887i \(-0.687834\pi\)
−0.556442 + 0.830887i \(0.687834\pi\)
\(678\) −5.78477e14 −0.155068
\(679\) −2.57343e15 −0.684271
\(680\) −2.64485e16 −6.97592
\(681\) −3.25238e14 −0.0850928
\(682\) −2.47766e15 −0.643025
\(683\) −1.64718e15 −0.424059 −0.212029 0.977263i \(-0.568007\pi\)
−0.212029 + 0.977263i \(0.568007\pi\)
\(684\) 6.54431e15 1.67130
\(685\) −5.71239e15 −1.44717
\(686\) 7.07835e15 1.77889
\(687\) −2.80932e15 −0.700389
\(688\) −2.39593e15 −0.592566
\(689\) −7.87190e14 −0.193141
\(690\) 1.02227e15 0.248825
\(691\) −8.24714e14 −0.199147 −0.0995736 0.995030i \(-0.531748\pi\)
−0.0995736 + 0.995030i \(0.531748\pi\)
\(692\) 9.64841e15 2.31139
\(693\) −3.23039e14 −0.0767754
\(694\) −5.33372e15 −1.25763
\(695\) −5.11695e15 −1.19700
\(696\) −9.29734e15 −2.15778
\(697\) −5.63071e15 −1.29653
\(698\) 1.39767e16 3.19300
\(699\) −1.45522e15 −0.329841
\(700\) 1.17686e16 2.64658
\(701\) 6.73359e14 0.150244 0.0751221 0.997174i \(-0.476065\pi\)
0.0751221 + 0.997174i \(0.476065\pi\)
\(702\) 2.40897e15 0.533308
\(703\) 3.20921e15 0.704926
\(704\) −6.08314e15 −1.32580
\(705\) 3.92675e15 0.849166
\(706\) 4.49979e15 0.965532
\(707\) 2.90863e15 0.619273
\(708\) −9.51100e14 −0.200929
\(709\) −2.01792e15 −0.423008 −0.211504 0.977377i \(-0.567836\pi\)
−0.211504 + 0.977377i \(0.567836\pi\)
\(710\) 2.12681e16 4.42392
\(711\) 1.79571e15 0.370641
\(712\) −1.79926e16 −3.68513
\(713\) −5.23443e14 −0.106384
\(714\) 3.87630e15 0.781767
\(715\) −5.12370e15 −1.02542
\(716\) 2.19449e16 4.35826
\(717\) 4.16328e15 0.820502
\(718\) 2.11798e15 0.414226
\(719\) 3.13698e15 0.608839 0.304419 0.952538i \(-0.401538\pi\)
0.304419 + 0.952538i \(0.401538\pi\)
\(720\) −1.00863e16 −1.94269
\(721\) 2.06020e15 0.393791
\(722\) −2.54407e16 −4.82586
\(723\) 3.13854e15 0.590836
\(724\) 4.95824e15 0.926329
\(725\) 1.14178e16 2.11702
\(726\) 4.93946e15 0.908926
\(727\) 9.93240e15 1.81391 0.906953 0.421231i \(-0.138402\pi\)
0.906953 + 0.421231i \(0.138402\pi\)
\(728\) 1.39431e16 2.52718
\(729\) 2.05891e14 0.0370370
\(730\) 1.34572e16 2.40258
\(731\) 1.24823e15 0.221181
\(732\) 1.66570e15 0.292944
\(733\) −5.38590e15 −0.940127 −0.470064 0.882633i \(-0.655769\pi\)
−0.470064 + 0.882633i \(0.655769\pi\)
\(734\) −1.13593e16 −1.96799
\(735\) 3.96078e15 0.681083
\(736\) −2.70769e15 −0.462137
\(737\) 8.80115e14 0.149097
\(738\) −3.80023e15 −0.638998
\(739\) 8.41325e15 1.40417 0.702085 0.712094i \(-0.252253\pi\)
0.702085 + 0.712094i \(0.252253\pi\)
\(740\) −1.01777e16 −1.68608
\(741\) −9.52192e15 −1.56576
\(742\) −8.54891e14 −0.139537
\(743\) −9.16171e15 −1.48436 −0.742178 0.670203i \(-0.766207\pi\)
−0.742178 + 0.670203i \(0.766207\pi\)
\(744\) 9.14012e15 1.46994
\(745\) −1.83345e15 −0.292691
\(746\) −1.93476e16 −3.06595
\(747\) −3.60708e15 −0.567405
\(748\) 9.37743e15 1.46428
\(749\) 3.55573e15 0.551162
\(750\) 9.85319e15 1.51614
\(751\) 5.67309e15 0.866563 0.433281 0.901259i \(-0.357356\pi\)
0.433281 + 0.901259i \(0.357356\pi\)
\(752\) −2.00712e16 −3.04352
\(753\) −6.34024e15 −0.954406
\(754\) 2.16124e16 3.22969
\(755\) 1.08151e16 1.60444
\(756\) 1.90392e15 0.280401
\(757\) 5.88431e13 0.00860337 0.00430168 0.999991i \(-0.498631\pi\)
0.00430168 + 0.999991i \(0.498631\pi\)
\(758\) −7.92226e15 −1.14992
\(759\) −2.26863e14 −0.0326914
\(760\) 7.05569e16 10.0940
\(761\) −1.03107e16 −1.46444 −0.732222 0.681066i \(-0.761517\pi\)
−0.732222 + 0.681066i \(0.761517\pi\)
\(762\) −7.16770e15 −1.01071
\(763\) 4.96567e15 0.695173
\(764\) −3.59006e16 −4.98986
\(765\) 5.25474e15 0.725126
\(766\) 1.56066e16 2.13821
\(767\) 1.38384e15 0.188240
\(768\) 7.59513e15 1.02577
\(769\) 2.13146e15 0.285813 0.142906 0.989736i \(-0.454355\pi\)
0.142906 + 0.989736i \(0.454355\pi\)
\(770\) −5.56436e15 −0.740825
\(771\) 5.54670e15 0.733222
\(772\) −3.66407e16 −4.80916
\(773\) −1.93099e15 −0.251648 −0.125824 0.992053i \(-0.540157\pi\)
−0.125824 + 0.992053i \(0.540157\pi\)
\(774\) 8.42440e14 0.109009
\(775\) −1.12248e16 −1.44217
\(776\) 3.15575e16 4.02590
\(777\) 9.33649e14 0.118268
\(778\) 2.16262e16 2.72014
\(779\) 1.50211e16 1.87606
\(780\) 3.01980e16 3.74505
\(781\) −4.71983e15 −0.581227
\(782\) 2.72223e15 0.332880
\(783\) 1.84718e15 0.224295
\(784\) −2.02452e16 −2.44109
\(785\) 1.53357e16 1.83621
\(786\) 1.80019e16 2.14039
\(787\) −3.71511e15 −0.438642 −0.219321 0.975653i \(-0.570384\pi\)
−0.219321 + 0.975653i \(0.570384\pi\)
\(788\) −1.73454e16 −2.03372
\(789\) −5.73759e15 −0.668046
\(790\) 3.09312e16 3.57641
\(791\) 6.65219e14 0.0763825
\(792\) 3.96137e15 0.451707
\(793\) −2.42357e15 −0.274444
\(794\) −2.14472e16 −2.41190
\(795\) −1.15890e15 −0.129427
\(796\) 2.72792e16 3.02559
\(797\) 7.65519e15 0.843209 0.421604 0.906780i \(-0.361467\pi\)
0.421604 + 0.906780i \(0.361467\pi\)
\(798\) −1.03408e16 −1.13120
\(799\) 1.04567e16 1.13602
\(800\) −5.80639e16 −6.26486
\(801\) 3.57474e15 0.383059
\(802\) −3.03414e15 −0.322906
\(803\) −2.98643e15 −0.315658
\(804\) −5.18722e15 −0.544535
\(805\) −1.17555e15 −0.122565
\(806\) −2.12470e16 −2.20016
\(807\) −7.60807e15 −0.782474
\(808\) −3.56680e16 −3.64348
\(809\) 1.66989e16 1.69422 0.847111 0.531416i \(-0.178340\pi\)
0.847111 + 0.531416i \(0.178340\pi\)
\(810\) 3.54648e15 0.357380
\(811\) 7.06461e15 0.707088 0.353544 0.935418i \(-0.384976\pi\)
0.353544 + 0.935418i \(0.384976\pi\)
\(812\) 1.70813e16 1.69810
\(813\) −1.02021e16 −1.00738
\(814\) 3.10358e15 0.304389
\(815\) 1.59848e16 1.55718
\(816\) −2.68592e16 −2.59895
\(817\) −3.32990e15 −0.320045
\(818\) 2.10568e16 2.01024
\(819\) −2.77020e15 −0.262693
\(820\) −4.76383e16 −4.48724
\(821\) 6.78885e15 0.635197 0.317599 0.948225i \(-0.397124\pi\)
0.317599 + 0.948225i \(0.397124\pi\)
\(822\) −1.02666e16 −0.954179
\(823\) −1.37578e15 −0.127014 −0.0635068 0.997981i \(-0.520228\pi\)
−0.0635068 + 0.997981i \(0.520228\pi\)
\(824\) −2.52639e16 −2.31686
\(825\) −4.86486e15 −0.443173
\(826\) 1.50286e15 0.135996
\(827\) 1.88493e16 1.69439 0.847196 0.531280i \(-0.178289\pi\)
0.847196 + 0.531280i \(0.178289\pi\)
\(828\) 1.33708e15 0.119396
\(829\) 7.56471e15 0.671031 0.335515 0.942035i \(-0.391090\pi\)
0.335515 + 0.942035i \(0.391090\pi\)
\(830\) −6.21321e16 −5.47503
\(831\) −8.48961e15 −0.743159
\(832\) −5.21655e16 −4.53634
\(833\) 1.05473e16 0.911158
\(834\) −9.19640e15 −0.789232
\(835\) −2.16529e16 −1.84603
\(836\) −2.50163e16 −2.11879
\(837\) −1.81595e15 −0.152796
\(838\) −1.43544e16 −1.19990
\(839\) 1.36055e16 1.12986 0.564928 0.825140i \(-0.308904\pi\)
0.564928 + 0.825140i \(0.308904\pi\)
\(840\) 2.05270e16 1.69351
\(841\) 4.37174e15 0.358324
\(842\) 4.62458e16 3.76579
\(843\) −1.29224e16 −1.04542
\(844\) 2.49577e16 2.00595
\(845\) −2.29212e16 −1.83032
\(846\) 7.05732e15 0.559890
\(847\) −5.68013e15 −0.447713
\(848\) 5.92361e15 0.463884
\(849\) −2.45816e15 −0.191257
\(850\) 5.83758e16 4.51262
\(851\) 6.55679e14 0.0503592
\(852\) 2.78177e16 2.12277
\(853\) −7.49852e15 −0.568533 −0.284267 0.958745i \(-0.591750\pi\)
−0.284267 + 0.958745i \(0.591750\pi\)
\(854\) −2.63201e15 −0.198276
\(855\) −1.40181e16 −1.04924
\(856\) −4.36033e16 −3.24275
\(857\) 1.40143e16 1.03557 0.517783 0.855512i \(-0.326757\pi\)
0.517783 + 0.855512i \(0.326757\pi\)
\(858\) −9.20854e15 −0.676099
\(859\) −1.43539e16 −1.04715 −0.523575 0.851980i \(-0.675402\pi\)
−0.523575 + 0.851980i \(0.675402\pi\)
\(860\) 1.05605e16 0.765498
\(861\) 4.37007e15 0.314754
\(862\) −2.48651e16 −1.77951
\(863\) −2.01996e16 −1.43643 −0.718213 0.695823i \(-0.755040\pi\)
−0.718213 + 0.695823i \(0.755040\pi\)
\(864\) −9.39360e15 −0.663753
\(865\) −2.06672e16 −1.45109
\(866\) −3.30037e16 −2.30257
\(867\) 5.66499e15 0.392730
\(868\) −1.67925e16 −1.15680
\(869\) −6.86428e15 −0.469879
\(870\) 3.18177e16 2.16428
\(871\) 7.54736e15 0.510147
\(872\) −6.08931e16 −4.09004
\(873\) −6.26980e15 −0.418480
\(874\) −7.26213e15 −0.481672
\(875\) −1.13307e16 −0.746811
\(876\) 1.76014e16 1.15286
\(877\) −1.06436e16 −0.692774 −0.346387 0.938092i \(-0.612592\pi\)
−0.346387 + 0.938092i \(0.612592\pi\)
\(878\) 4.21094e16 2.72370
\(879\) −3.66477e15 −0.235564
\(880\) 3.85559e16 2.46284
\(881\) −6.05995e15 −0.384682 −0.192341 0.981328i \(-0.561608\pi\)
−0.192341 + 0.981328i \(0.561608\pi\)
\(882\) 7.11848e15 0.449066
\(883\) 3.73590e15 0.234213 0.117107 0.993119i \(-0.462638\pi\)
0.117107 + 0.993119i \(0.462638\pi\)
\(884\) 8.04154e16 5.01016
\(885\) 2.03729e15 0.126143
\(886\) −5.30343e16 −3.26341
\(887\) 1.30168e16 0.796023 0.398011 0.917380i \(-0.369700\pi\)
0.398011 + 0.917380i \(0.369700\pi\)
\(888\) −1.14492e16 −0.695829
\(889\) 8.24248e15 0.497850
\(890\) 6.15749e16 3.69623
\(891\) −7.87039e14 −0.0469536
\(892\) −4.92638e16 −2.92093
\(893\) −2.78954e16 −1.64380
\(894\) −3.29515e15 −0.192983
\(895\) −4.70068e16 −2.73611
\(896\) −2.41571e16 −1.39750
\(897\) −1.94544e15 −0.111856
\(898\) 2.60766e16 1.49015
\(899\) −1.62920e16 −0.925330
\(900\) 2.86725e16 1.61857
\(901\) −3.08607e15 −0.173149
\(902\) 1.45267e16 0.810088
\(903\) −9.68763e14 −0.0536951
\(904\) −8.15746e15 −0.449395
\(905\) −1.06207e16 −0.581549
\(906\) 1.94374e16 1.05787
\(907\) 3.18539e16 1.72315 0.861575 0.507631i \(-0.169479\pi\)
0.861575 + 0.507631i \(0.169479\pi\)
\(908\) −7.32749e15 −0.393988
\(909\) 7.08647e15 0.378729
\(910\) −4.77167e16 −2.53479
\(911\) 4.37440e15 0.230976 0.115488 0.993309i \(-0.463157\pi\)
0.115488 + 0.993309i \(0.463157\pi\)
\(912\) 7.16526e16 3.76063
\(913\) 1.37884e16 0.719325
\(914\) −7.09542e14 −0.0367938
\(915\) −3.56797e15 −0.183910
\(916\) −6.32929e16 −3.24287
\(917\) −2.07012e16 −1.05430
\(918\) 9.44406e15 0.478106
\(919\) 1.89945e16 0.955858 0.477929 0.878399i \(-0.341388\pi\)
0.477929 + 0.878399i \(0.341388\pi\)
\(920\) 1.44156e16 0.721107
\(921\) 3.20759e15 0.159496
\(922\) 3.40559e16 1.68334
\(923\) −4.04746e16 −1.98872
\(924\) −7.27794e15 −0.355478
\(925\) 1.40604e16 0.682683
\(926\) 3.85776e15 0.186198
\(927\) 5.01939e15 0.240831
\(928\) −8.42760e16 −4.01967
\(929\) 3.75792e16 1.78181 0.890904 0.454191i \(-0.150072\pi\)
0.890904 + 0.454191i \(0.150072\pi\)
\(930\) −3.12797e16 −1.47437
\(931\) −2.81372e16 −1.31843
\(932\) −3.27855e16 −1.52719
\(933\) −1.30724e16 −0.605349
\(934\) 1.61137e16 0.741799
\(935\) −2.00868e16 −0.919277
\(936\) 3.39704e16 1.54555
\(937\) 1.34406e16 0.607928 0.303964 0.952684i \(-0.401690\pi\)
0.303964 + 0.952684i \(0.401690\pi\)
\(938\) 8.19646e15 0.368562
\(939\) 5.79353e15 0.258989
\(940\) 8.84680e16 3.93172
\(941\) −8.15918e15 −0.360499 −0.180249 0.983621i \(-0.557690\pi\)
−0.180249 + 0.983621i \(0.557690\pi\)
\(942\) 2.75620e16 1.21069
\(943\) 3.06899e15 0.134024
\(944\) −1.04134e16 −0.452114
\(945\) −4.07827e15 −0.176036
\(946\) −3.22031e15 −0.138196
\(947\) −2.94524e15 −0.125660 −0.0628298 0.998024i \(-0.520013\pi\)
−0.0628298 + 0.998024i \(0.520013\pi\)
\(948\) 4.04566e16 1.71610
\(949\) −2.56099e16 −1.08005
\(950\) −1.55730e17 −6.52967
\(951\) 1.32857e16 0.553849
\(952\) 5.46621e16 2.26560
\(953\) −4.33689e16 −1.78718 −0.893588 0.448889i \(-0.851820\pi\)
−0.893588 + 0.448889i \(0.851820\pi\)
\(954\) −2.08282e15 −0.0853367
\(955\) 7.69003e16 3.13263
\(956\) 9.37969e16 3.79901
\(957\) −7.06103e15 −0.284349
\(958\) 3.69026e16 1.47757
\(959\) 1.18060e16 0.470004
\(960\) −7.67979e16 −3.03989
\(961\) −9.39196e15 −0.369639
\(962\) 2.66145e16 1.04149
\(963\) 8.66305e15 0.337075
\(964\) 7.07100e16 2.73563
\(965\) 7.84857e16 3.01919
\(966\) −2.11276e15 −0.0808119
\(967\) −3.73342e16 −1.41991 −0.709957 0.704246i \(-0.751285\pi\)
−0.709957 + 0.704246i \(0.751285\pi\)
\(968\) 6.96544e16 2.63411
\(969\) −3.73294e16 −1.40369
\(970\) −1.07997e17 −4.03802
\(971\) −2.32292e15 −0.0863632 −0.0431816 0.999067i \(-0.513749\pi\)
−0.0431816 + 0.999067i \(0.513749\pi\)
\(972\) 4.63864e15 0.171485
\(973\) 1.05754e16 0.388755
\(974\) 3.77187e14 0.0137874
\(975\) −4.17182e16 −1.51635
\(976\) 1.82374e16 0.659157
\(977\) −3.64377e16 −1.30958 −0.654789 0.755812i \(-0.727242\pi\)
−0.654789 + 0.755812i \(0.727242\pi\)
\(978\) 2.87285e16 1.02671
\(979\) −1.36648e16 −0.485621
\(980\) 8.92347e16 3.15348
\(981\) 1.20982e16 0.425147
\(982\) 1.30041e16 0.454429
\(983\) 1.95246e16 0.678480 0.339240 0.940700i \(-0.389830\pi\)
0.339240 + 0.940700i \(0.389830\pi\)
\(984\) −5.35893e16 −1.85185
\(985\) 3.71544e16 1.27677
\(986\) 8.47287e16 2.89539
\(987\) −8.11556e15 −0.275787
\(988\) −2.14525e17 −7.24961
\(989\) −6.80339e14 −0.0228637
\(990\) −1.35568e16 −0.453067
\(991\) −5.04520e16 −1.67677 −0.838384 0.545080i \(-0.816499\pi\)
−0.838384 + 0.545080i \(0.816499\pi\)
\(992\) 8.28509e16 2.73831
\(993\) −9.72996e15 −0.319809
\(994\) −4.39555e16 −1.43677
\(995\) −5.84329e16 −1.89946
\(996\) −8.12661e16 −2.62714
\(997\) −4.06996e16 −1.30848 −0.654239 0.756288i \(-0.727011\pi\)
−0.654239 + 0.756288i \(0.727011\pi\)
\(998\) −5.74804e16 −1.83782
\(999\) 2.27470e15 0.0723293
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.1.2 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.b.1.2 27 1.1 even 1 trivial