Properties

Label 29.18.a.b.1.15
Level $29$
Weight $18$
Character 29.1
Self dual yes
Analytic conductor $53.134$
Analytic rank $0$
Dimension $21$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.1344053299\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) \(=\) 29.1

$q$-expansion

\(f(q)\) \(=\) \(q+327.881 q^{2} +6727.77 q^{3} -23565.9 q^{4} +1.45971e6 q^{5} +2.20591e6 q^{6} +2.18126e7 q^{7} -5.07029e7 q^{8} -8.38773e7 q^{9} +O(q^{10})\) \(q+327.881 q^{2} +6727.77 q^{3} -23565.9 q^{4} +1.45971e6 q^{5} +2.20591e6 q^{6} +2.18126e7 q^{7} -5.07029e7 q^{8} -8.38773e7 q^{9} +4.78610e8 q^{10} +9.06517e8 q^{11} -1.58546e8 q^{12} +1.99314e9 q^{13} +7.15194e9 q^{14} +9.82056e9 q^{15} -1.35357e10 q^{16} +2.19793e10 q^{17} -2.75018e10 q^{18} -1.22325e11 q^{19} -3.43992e10 q^{20} +1.46750e11 q^{21} +2.97230e11 q^{22} +1.06484e11 q^{23} -3.41117e11 q^{24} +1.36780e12 q^{25} +6.53514e11 q^{26} -1.43313e12 q^{27} -5.14033e11 q^{28} +5.00246e11 q^{29} +3.21998e12 q^{30} +4.28486e11 q^{31} +2.20763e12 q^{32} +6.09884e12 q^{33} +7.20659e12 q^{34} +3.18400e13 q^{35} +1.97664e12 q^{36} -3.72444e13 q^{37} -4.01080e13 q^{38} +1.34094e13 q^{39} -7.40112e13 q^{40} +6.35703e13 q^{41} +4.81166e13 q^{42} +8.68501e13 q^{43} -2.13629e13 q^{44} -1.22436e14 q^{45} +3.49140e13 q^{46} +2.38379e14 q^{47} -9.10650e13 q^{48} +2.43159e14 q^{49} +4.48476e14 q^{50} +1.47871e14 q^{51} -4.69701e13 q^{52} -4.92045e14 q^{53} -4.69897e14 q^{54} +1.32325e15 q^{55} -1.10596e15 q^{56} -8.22973e14 q^{57} +1.64021e14 q^{58} -8.09233e13 q^{59} -2.31430e14 q^{60} +7.00512e14 q^{61} +1.40492e14 q^{62} -1.82958e15 q^{63} +2.49799e15 q^{64} +2.90940e15 q^{65} +1.99969e15 q^{66} +1.84485e14 q^{67} -5.17961e14 q^{68} +7.16397e14 q^{69} +1.04397e16 q^{70} +2.30383e14 q^{71} +4.25282e15 q^{72} +6.34133e15 q^{73} -1.22117e16 q^{74} +9.20225e15 q^{75} +2.88269e15 q^{76} +1.97735e16 q^{77} +4.39669e15 q^{78} -1.59594e16 q^{79} -1.97581e16 q^{80} +1.19014e15 q^{81} +2.08435e16 q^{82} +3.22884e16 q^{83} -3.45830e15 q^{84} +3.20833e16 q^{85} +2.84765e16 q^{86} +3.36554e15 q^{87} -4.59630e16 q^{88} -3.38978e16 q^{89} -4.01445e16 q^{90} +4.34756e16 q^{91} -2.50938e15 q^{92} +2.88275e15 q^{93} +7.81599e16 q^{94} -1.78558e17 q^{95} +1.48524e16 q^{96} -5.54890e16 q^{97} +7.97273e16 q^{98} -7.60362e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q + 256q^{2} + 23966q^{3} + 1452522q^{4} + 998272q^{5} + 3411526q^{6} + 2193368q^{7} - 138137226q^{8} + 1264832799q^{9} + O(q^{10}) \) \( 21q + 256q^{2} + 23966q^{3} + 1452522q^{4} + 998272q^{5} + 3411526q^{6} + 2193368q^{7} - 138137226q^{8} + 1264832799q^{9} - 224469478q^{10} + 1203139534q^{11} - 5164251122q^{12} + 3854339312q^{13} + 25262272904q^{14} + 28324474306q^{15} + 196520815922q^{16} + 76444714794q^{17} + 75758949126q^{18} + 246497292428q^{19} - 46900976670q^{20} + 360937126704q^{21} - 275001533522q^{22} + 213498528140q^{23} - 451123453870q^{24} + 3898884886997q^{25} - 3609347694206q^{26} - 2718903745978q^{27} - 5946174617200q^{28} + 10505174672181q^{29} - 20237658929454q^{30} + 16670029895798q^{31} - 42141001912046q^{32} - 7157109761394q^{33} + 12785761151136q^{34} + 46677934312888q^{35} + 132137824374868q^{36} + 53445659988410q^{37} + 76581637956388q^{38} + 79233849032530q^{39} + 193617444734146q^{40} - 20814769309298q^{41} + 76690667258352q^{42} + 185498647364454q^{43} + 315429066899678q^{44} - 486270821438526q^{45} + 261474367677132q^{46} + 389503471719450q^{47} - 101509672247630q^{48} + 730079062141437q^{49} + 1482269666368354q^{50} + 718238208473988q^{51} + 1966802817157170q^{52} + 747441265526156q^{53} + 5692893333117030q^{54} + 1639109418219546q^{55} + 5657219329125240q^{56} + 4694352396864932q^{57} + 128063081718016q^{58} + 5280258638332960q^{59} + 15251367906033378q^{60} + 5813675353074254q^{61} + 6242066590947250q^{62} + 10947760075450368q^{63} + 24583792057508902q^{64} + 19190799243789974q^{65} + 41877805444482390q^{66} + 13420580230958268q^{67} + 24771837384165388q^{68} + 30973047049935252q^{69} + 8505088080182440q^{70} + 4824462822979508q^{71} + 1180071997284592q^{72} + 11228916281304662q^{73} - 89132715356772q^{74} + 59161419576630296q^{75} + 57466858643173460q^{76} + 58741564492720064q^{77} + 142050530910210210q^{78} + 71718598015696758q^{79} + 48350023652407550q^{80} + 75805931446703569q^{81} + 188661890754420812q^{82} + 50769377111735608q^{83} + 198832046985593048q^{84} + 53422044849490784q^{85} + 35014892323844118q^{86} + 11988905533023326q^{87} + 37459283979085258q^{88} - 70981414576978018q^{89} + 57211029866143724q^{90} + 112933943315157320q^{91} - 103019729095759724q^{92} - 350358290646906646q^{93} - 150286322409612578q^{94} - 102561321856584476q^{95} - 213770098354021866q^{96} - 130930167251505210q^{97} - 537387515497557296q^{98} - 95267700931431064q^{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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 327.881 0.905653 0.452826 0.891599i \(-0.350416\pi\)
0.452826 + 0.891599i \(0.350416\pi\)
\(3\) 6727.77 0.592026 0.296013 0.955184i \(-0.404343\pi\)
0.296013 + 0.955184i \(0.404343\pi\)
\(4\) −23565.9 −0.179793
\(5\) 1.45971e6 1.67117 0.835584 0.549362i \(-0.185129\pi\)
0.835584 + 0.549362i \(0.185129\pi\)
\(6\) 2.20591e6 0.536169
\(7\) 2.18126e7 1.43013 0.715063 0.699060i \(-0.246398\pi\)
0.715063 + 0.699060i \(0.246398\pi\)
\(8\) −5.07029e7 −1.06848
\(9\) −8.38773e7 −0.649506
\(10\) 4.78610e8 1.51350
\(11\) 9.06517e8 1.27508 0.637541 0.770417i \(-0.279952\pi\)
0.637541 + 0.770417i \(0.279952\pi\)
\(12\) −1.58546e8 −0.106442
\(13\) 1.99314e9 0.677672 0.338836 0.940845i \(-0.389967\pi\)
0.338836 + 0.940845i \(0.389967\pi\)
\(14\) 7.15194e9 1.29520
\(15\) 9.82056e9 0.989374
\(16\) −1.35357e10 −0.787881
\(17\) 2.19793e10 0.764183 0.382092 0.924124i \(-0.375204\pi\)
0.382092 + 0.924124i \(0.375204\pi\)
\(18\) −2.75018e10 −0.588227
\(19\) −1.22325e11 −1.65238 −0.826188 0.563394i \(-0.809495\pi\)
−0.826188 + 0.563394i \(0.809495\pi\)
\(20\) −3.43992e10 −0.300465
\(21\) 1.46750e11 0.846671
\(22\) 2.97230e11 1.15478
\(23\) 1.06484e11 0.283528 0.141764 0.989900i \(-0.454723\pi\)
0.141764 + 0.989900i \(0.454723\pi\)
\(24\) −3.41117e11 −0.632569
\(25\) 1.36780e12 1.79280
\(26\) 6.53514e11 0.613736
\(27\) −1.43313e12 −0.976550
\(28\) −5.14033e11 −0.257127
\(29\) 5.00246e11 0.185695
\(30\) 3.21998e12 0.896029
\(31\) 4.28486e11 0.0902323 0.0451162 0.998982i \(-0.485634\pi\)
0.0451162 + 0.998982i \(0.485634\pi\)
\(32\) 2.20763e12 0.354936
\(33\) 6.09884e12 0.754881
\(34\) 7.20659e12 0.692084
\(35\) 3.18400e13 2.38998
\(36\) 1.97664e12 0.116777
\(37\) −3.72444e13 −1.74320 −0.871598 0.490221i \(-0.836916\pi\)
−0.871598 + 0.490221i \(0.836916\pi\)
\(38\) −4.01080e13 −1.49648
\(39\) 1.34094e13 0.401199
\(40\) −7.40112e13 −1.78561
\(41\) 6.35703e13 1.24334 0.621672 0.783278i \(-0.286454\pi\)
0.621672 + 0.783278i \(0.286454\pi\)
\(42\) 4.81166e13 0.766789
\(43\) 8.68501e13 1.13315 0.566576 0.824009i \(-0.308268\pi\)
0.566576 + 0.824009i \(0.308268\pi\)
\(44\) −2.13629e13 −0.229251
\(45\) −1.22436e14 −1.08543
\(46\) 3.49140e13 0.256778
\(47\) 2.38379e14 1.46028 0.730139 0.683298i \(-0.239455\pi\)
0.730139 + 0.683298i \(0.239455\pi\)
\(48\) −9.10650e13 −0.466446
\(49\) 2.43159e14 1.04526
\(50\) 4.48476e14 1.62366
\(51\) 1.47871e14 0.452416
\(52\) −4.69701e13 −0.121841
\(53\) −4.92045e14 −1.08558 −0.542788 0.839870i \(-0.682631\pi\)
−0.542788 + 0.839870i \(0.682631\pi\)
\(54\) −4.69897e14 −0.884415
\(55\) 1.32325e15 2.13088
\(56\) −1.10596e15 −1.52806
\(57\) −8.22973e14 −0.978249
\(58\) 1.64021e14 0.168175
\(59\) −8.09233e13 −0.0717516 −0.0358758 0.999356i \(-0.511422\pi\)
−0.0358758 + 0.999356i \(0.511422\pi\)
\(60\) −2.31430e14 −0.177883
\(61\) 7.00512e14 0.467856 0.233928 0.972254i \(-0.424842\pi\)
0.233928 + 0.972254i \(0.424842\pi\)
\(62\) 1.40492e14 0.0817191
\(63\) −1.82958e15 −0.928875
\(64\) 2.49799e15 1.10933
\(65\) 2.90940e15 1.13250
\(66\) 1.99969e15 0.683660
\(67\) 1.84485e14 0.0555043 0.0277521 0.999615i \(-0.491165\pi\)
0.0277521 + 0.999615i \(0.491165\pi\)
\(68\) −5.17961e14 −0.137395
\(69\) 7.16397e14 0.167856
\(70\) 1.04397e16 2.16449
\(71\) 2.30383e14 0.0423403 0.0211702 0.999776i \(-0.493261\pi\)
0.0211702 + 0.999776i \(0.493261\pi\)
\(72\) 4.25282e15 0.693986
\(73\) 6.34133e15 0.920314 0.460157 0.887837i \(-0.347793\pi\)
0.460157 + 0.887837i \(0.347793\pi\)
\(74\) −1.22117e16 −1.57873
\(75\) 9.20225e15 1.06139
\(76\) 2.88269e15 0.297086
\(77\) 1.97735e16 1.82353
\(78\) 4.39669e15 0.363347
\(79\) −1.59594e16 −1.18355 −0.591775 0.806103i \(-0.701572\pi\)
−0.591775 + 0.806103i \(0.701572\pi\)
\(80\) −1.97581e16 −1.31668
\(81\) 1.19014e15 0.0713635
\(82\) 2.08435e16 1.12604
\(83\) 3.22884e16 1.57356 0.786779 0.617234i \(-0.211747\pi\)
0.786779 + 0.617234i \(0.211747\pi\)
\(84\) −3.45830e15 −0.152226
\(85\) 3.20833e16 1.27708
\(86\) 2.84765e16 1.02624
\(87\) 3.36554e15 0.109936
\(88\) −4.59630e16 −1.36240
\(89\) −3.38978e16 −0.912761 −0.456380 0.889785i \(-0.650854\pi\)
−0.456380 + 0.889785i \(0.650854\pi\)
\(90\) −4.01445e16 −0.983026
\(91\) 4.34756e16 0.969156
\(92\) −2.50938e15 −0.0509764
\(93\) 2.88275e15 0.0534198
\(94\) 7.81599e16 1.32251
\(95\) −1.78558e17 −2.76140
\(96\) 1.48524e16 0.210131
\(97\) −5.54890e16 −0.718865 −0.359432 0.933171i \(-0.617030\pi\)
−0.359432 + 0.933171i \(0.617030\pi\)
\(98\) 7.97273e16 0.946641
\(99\) −7.60362e16 −0.828173
\(100\) −3.22334e16 −0.322334
\(101\) −5.98576e16 −0.550031 −0.275016 0.961440i \(-0.588683\pi\)
−0.275016 + 0.961440i \(0.588683\pi\)
\(102\) 4.84843e16 0.409732
\(103\) −2.31102e17 −1.79757 −0.898787 0.438385i \(-0.855551\pi\)
−0.898787 + 0.438385i \(0.855551\pi\)
\(104\) −1.01058e17 −0.724081
\(105\) 2.14212e17 1.41493
\(106\) −1.61332e17 −0.983155
\(107\) −1.28964e17 −0.725616 −0.362808 0.931864i \(-0.618182\pi\)
−0.362808 + 0.931864i \(0.618182\pi\)
\(108\) 3.37730e16 0.175577
\(109\) −9.68812e16 −0.465709 −0.232854 0.972512i \(-0.574807\pi\)
−0.232854 + 0.972512i \(0.574807\pi\)
\(110\) 4.33868e17 1.92983
\(111\) −2.50572e17 −1.03202
\(112\) −2.95249e17 −1.12677
\(113\) 1.63216e17 0.577557 0.288779 0.957396i \(-0.406751\pi\)
0.288779 + 0.957396i \(0.406751\pi\)
\(114\) −2.69837e17 −0.885954
\(115\) 1.55435e17 0.473823
\(116\) −1.17887e16 −0.0333868
\(117\) −1.67179e17 −0.440152
\(118\) −2.65332e16 −0.0649820
\(119\) 4.79425e17 1.09288
\(120\) −4.97931e17 −1.05713
\(121\) 3.16326e17 0.625833
\(122\) 2.29685e17 0.423715
\(123\) 4.27686e17 0.736091
\(124\) −1.00976e16 −0.0162232
\(125\) 8.82919e17 1.32491
\(126\) −5.99885e17 −0.841238
\(127\) −1.01659e18 −1.33295 −0.666473 0.745529i \(-0.732197\pi\)
−0.666473 + 0.745529i \(0.732197\pi\)
\(128\) 5.29686e17 0.649731
\(129\) 5.84307e17 0.670855
\(130\) 9.53938e17 1.02566
\(131\) 4.10599e17 0.413629 0.206815 0.978380i \(-0.433690\pi\)
0.206815 + 0.978380i \(0.433690\pi\)
\(132\) −1.43724e17 −0.135723
\(133\) −2.66822e18 −2.36310
\(134\) 6.04893e16 0.0502676
\(135\) −2.09195e18 −1.63198
\(136\) −1.11441e18 −0.816517
\(137\) −8.86837e17 −0.610547 −0.305273 0.952265i \(-0.598748\pi\)
−0.305273 + 0.952265i \(0.598748\pi\)
\(138\) 2.34893e17 0.152019
\(139\) −2.95476e17 −0.179844 −0.0899220 0.995949i \(-0.528662\pi\)
−0.0899220 + 0.995949i \(0.528662\pi\)
\(140\) −7.50337e17 −0.429703
\(141\) 1.60376e18 0.864522
\(142\) 7.55382e16 0.0383456
\(143\) 1.80682e18 0.864087
\(144\) 1.13534e18 0.511733
\(145\) 7.30212e17 0.310328
\(146\) 2.07920e18 0.833485
\(147\) 1.63592e18 0.618819
\(148\) 8.77697e17 0.313415
\(149\) 6.39723e17 0.215729 0.107865 0.994166i \(-0.465599\pi\)
0.107865 + 0.994166i \(0.465599\pi\)
\(150\) 3.01724e18 0.961247
\(151\) −3.05483e18 −0.919779 −0.459890 0.887976i \(-0.652111\pi\)
−0.459890 + 0.887976i \(0.652111\pi\)
\(152\) 6.20222e18 1.76554
\(153\) −1.84356e18 −0.496341
\(154\) 6.48336e18 1.65148
\(155\) 6.25463e17 0.150793
\(156\) −3.16004e17 −0.0721330
\(157\) −3.98069e18 −0.860619 −0.430310 0.902681i \(-0.641596\pi\)
−0.430310 + 0.902681i \(0.641596\pi\)
\(158\) −5.23279e18 −1.07188
\(159\) −3.31037e18 −0.642689
\(160\) 3.22248e18 0.593159
\(161\) 2.32268e18 0.405480
\(162\) 3.90225e17 0.0646306
\(163\) −1.23420e19 −1.93996 −0.969979 0.243187i \(-0.921807\pi\)
−0.969979 + 0.243187i \(0.921807\pi\)
\(164\) −1.49809e18 −0.223545
\(165\) 8.90250e18 1.26153
\(166\) 1.05868e19 1.42510
\(167\) −7.73936e18 −0.989954 −0.494977 0.868906i \(-0.664823\pi\)
−0.494977 + 0.868906i \(0.664823\pi\)
\(168\) −7.44065e18 −0.904653
\(169\) −4.67780e18 −0.540760
\(170\) 1.05195e19 1.15659
\(171\) 1.02603e19 1.07323
\(172\) −2.04670e18 −0.203733
\(173\) 1.56524e19 1.48316 0.741582 0.670862i \(-0.234076\pi\)
0.741582 + 0.670862i \(0.234076\pi\)
\(174\) 1.10350e18 0.0995642
\(175\) 2.98353e19 2.56393
\(176\) −1.22703e19 −1.00461
\(177\) −5.44433e17 −0.0424788
\(178\) −1.11145e19 −0.826644
\(179\) −2.02080e19 −1.43308 −0.716542 0.697544i \(-0.754276\pi\)
−0.716542 + 0.697544i \(0.754276\pi\)
\(180\) 2.88531e18 0.195154
\(181\) 1.00683e19 0.649661 0.324831 0.945772i \(-0.394693\pi\)
0.324831 + 0.945772i \(0.394693\pi\)
\(182\) 1.42548e19 0.877719
\(183\) 4.71289e18 0.276983
\(184\) −5.39902e18 −0.302945
\(185\) −5.43658e19 −2.91317
\(186\) 9.45201e17 0.0483798
\(187\) 1.99246e19 0.974396
\(188\) −5.61760e18 −0.262548
\(189\) −3.12603e19 −1.39659
\(190\) −5.85459e19 −2.50087
\(191\) 3.28029e19 1.34007 0.670036 0.742329i \(-0.266279\pi\)
0.670036 + 0.742329i \(0.266279\pi\)
\(192\) 1.68059e19 0.656752
\(193\) 3.04161e18 0.113728 0.0568638 0.998382i \(-0.481890\pi\)
0.0568638 + 0.998382i \(0.481890\pi\)
\(194\) −1.81938e19 −0.651042
\(195\) 1.95738e19 0.670471
\(196\) −5.73025e18 −0.187930
\(197\) 2.46461e19 0.774079 0.387039 0.922063i \(-0.373498\pi\)
0.387039 + 0.922063i \(0.373498\pi\)
\(198\) −2.49308e19 −0.750037
\(199\) −5.32537e19 −1.53496 −0.767482 0.641070i \(-0.778491\pi\)
−0.767482 + 0.641070i \(0.778491\pi\)
\(200\) −6.93514e19 −1.91558
\(201\) 1.24118e18 0.0328599
\(202\) −1.96262e19 −0.498137
\(203\) 1.09117e19 0.265568
\(204\) −3.48472e18 −0.0813414
\(205\) 9.27939e19 2.07784
\(206\) −7.57739e19 −1.62798
\(207\) −8.93155e18 −0.184153
\(208\) −2.69786e19 −0.533925
\(209\) −1.10889e20 −2.10691
\(210\) 7.02361e19 1.28143
\(211\) 9.36675e19 1.64130 0.820650 0.571431i \(-0.193612\pi\)
0.820650 + 0.571431i \(0.193612\pi\)
\(212\) 1.15955e19 0.195179
\(213\) 1.54996e18 0.0250666
\(214\) −4.22849e19 −0.657156
\(215\) 1.26776e20 1.89369
\(216\) 7.26639e19 1.04343
\(217\) 9.34639e18 0.129043
\(218\) −3.17655e19 −0.421770
\(219\) 4.26630e19 0.544850
\(220\) −3.11835e19 −0.383117
\(221\) 4.38078e19 0.517866
\(222\) −8.21577e19 −0.934648
\(223\) −6.28942e19 −0.688682 −0.344341 0.938845i \(-0.611898\pi\)
−0.344341 + 0.938845i \(0.611898\pi\)
\(224\) 4.81540e19 0.507604
\(225\) −1.14727e20 −1.16444
\(226\) 5.35154e19 0.523066
\(227\) −1.04145e20 −0.980436 −0.490218 0.871600i \(-0.663083\pi\)
−0.490218 + 0.871600i \(0.663083\pi\)
\(228\) 1.93941e19 0.175883
\(229\) −1.51162e20 −1.32081 −0.660404 0.750911i \(-0.729615\pi\)
−0.660404 + 0.750911i \(0.729615\pi\)
\(230\) 5.09641e19 0.429119
\(231\) 1.33031e20 1.07957
\(232\) −2.53639e19 −0.198412
\(233\) 1.43579e20 1.08284 0.541422 0.840751i \(-0.317886\pi\)
0.541422 + 0.840751i \(0.317886\pi\)
\(234\) −5.48150e19 −0.398625
\(235\) 3.47963e20 2.44037
\(236\) 1.90703e18 0.0129005
\(237\) −1.07371e20 −0.700691
\(238\) 1.57194e20 0.989767
\(239\) 1.15025e20 0.698892 0.349446 0.936957i \(-0.386370\pi\)
0.349446 + 0.936957i \(0.386370\pi\)
\(240\) −1.32928e20 −0.779509
\(241\) −6.01272e19 −0.340350 −0.170175 0.985414i \(-0.554433\pi\)
−0.170175 + 0.985414i \(0.554433\pi\)
\(242\) 1.03717e20 0.566788
\(243\) 1.93082e20 1.01880
\(244\) −1.65082e19 −0.0841174
\(245\) 3.54940e20 1.74680
\(246\) 1.40230e20 0.666643
\(247\) −2.43811e20 −1.11977
\(248\) −2.17255e19 −0.0964117
\(249\) 2.17229e20 0.931587
\(250\) 2.89493e20 1.19991
\(251\) −1.05912e20 −0.424343 −0.212172 0.977232i \(-0.568054\pi\)
−0.212172 + 0.977232i \(0.568054\pi\)
\(252\) 4.31157e19 0.167005
\(253\) 9.65291e19 0.361521
\(254\) −3.33320e20 −1.20719
\(255\) 2.15849e20 0.756063
\(256\) −1.53742e20 −0.520899
\(257\) 1.68953e20 0.553775 0.276888 0.960902i \(-0.410697\pi\)
0.276888 + 0.960902i \(0.410697\pi\)
\(258\) 1.91583e20 0.607562
\(259\) −8.12397e20 −2.49299
\(260\) −6.85625e19 −0.203617
\(261\) −4.19593e19 −0.120610
\(262\) 1.34628e20 0.374605
\(263\) 3.52541e20 0.949698 0.474849 0.880067i \(-0.342503\pi\)
0.474849 + 0.880067i \(0.342503\pi\)
\(264\) −3.09228e20 −0.806577
\(265\) −7.18241e20 −1.81418
\(266\) −8.74860e20 −2.14015
\(267\) −2.28057e20 −0.540378
\(268\) −4.34756e18 −0.00997930
\(269\) 8.82157e20 1.96178 0.980892 0.194551i \(-0.0623251\pi\)
0.980892 + 0.194551i \(0.0623251\pi\)
\(270\) −6.85912e20 −1.47801
\(271\) 2.81024e20 0.586820 0.293410 0.955987i \(-0.405210\pi\)
0.293410 + 0.955987i \(0.405210\pi\)
\(272\) −2.97505e20 −0.602085
\(273\) 2.92494e20 0.573765
\(274\) −2.90777e20 −0.552943
\(275\) 1.23993e21 2.28597
\(276\) −1.68825e19 −0.0301794
\(277\) −7.85707e20 −1.36202 −0.681008 0.732276i \(-0.738458\pi\)
−0.681008 + 0.732276i \(0.738458\pi\)
\(278\) −9.68809e19 −0.162876
\(279\) −3.59402e19 −0.0586064
\(280\) −1.61438e21 −2.55365
\(281\) 3.04537e20 0.467343 0.233672 0.972316i \(-0.424926\pi\)
0.233672 + 0.972316i \(0.424926\pi\)
\(282\) 5.25842e20 0.782957
\(283\) 7.30789e20 1.05586 0.527931 0.849287i \(-0.322968\pi\)
0.527931 + 0.849287i \(0.322968\pi\)
\(284\) −5.42917e18 −0.00761251
\(285\) −1.20130e21 −1.63482
\(286\) 5.92421e20 0.782563
\(287\) 1.38663e21 1.77814
\(288\) −1.85170e20 −0.230533
\(289\) −3.44152e20 −0.416024
\(290\) 2.39423e20 0.281050
\(291\) −3.73317e20 −0.425586
\(292\) −1.49439e20 −0.165466
\(293\) −1.31610e21 −1.41552 −0.707759 0.706454i \(-0.750294\pi\)
−0.707759 + 0.706454i \(0.750294\pi\)
\(294\) 5.36387e20 0.560435
\(295\) −1.18124e20 −0.119909
\(296\) 1.88840e21 1.86257
\(297\) −1.29916e21 −1.24518
\(298\) 2.09753e20 0.195376
\(299\) 2.12237e20 0.192139
\(300\) −2.16859e20 −0.190830
\(301\) 1.89443e21 1.62055
\(302\) −1.00162e21 −0.833000
\(303\) −4.02708e20 −0.325633
\(304\) 1.65575e21 1.30188
\(305\) 1.02254e21 0.781866
\(306\) −6.04469e20 −0.449513
\(307\) 1.80622e20 0.130645 0.0653227 0.997864i \(-0.479192\pi\)
0.0653227 + 0.997864i \(0.479192\pi\)
\(308\) −4.65979e20 −0.327858
\(309\) −1.55480e21 −1.06421
\(310\) 2.05078e20 0.136566
\(311\) −1.91012e21 −1.23765 −0.618826 0.785528i \(-0.712391\pi\)
−0.618826 + 0.785528i \(0.712391\pi\)
\(312\) −6.79895e20 −0.428675
\(313\) −6.59463e20 −0.404635 −0.202318 0.979320i \(-0.564847\pi\)
−0.202318 + 0.979320i \(0.564847\pi\)
\(314\) −1.30519e21 −0.779422
\(315\) −2.67065e21 −1.55231
\(316\) 3.76097e20 0.212794
\(317\) 2.78486e21 1.53391 0.766956 0.641700i \(-0.221770\pi\)
0.766956 + 0.641700i \(0.221770\pi\)
\(318\) −1.08541e21 −0.582053
\(319\) 4.53482e20 0.236777
\(320\) 3.64633e21 1.85388
\(321\) −8.67641e20 −0.429583
\(322\) 7.61564e20 0.367224
\(323\) −2.68861e21 −1.26272
\(324\) −2.80467e19 −0.0128307
\(325\) 2.72622e21 1.21493
\(326\) −4.04673e21 −1.75693
\(327\) −6.51794e20 −0.275711
\(328\) −3.22320e21 −1.32849
\(329\) 5.19966e21 2.08838
\(330\) 2.91896e21 1.14251
\(331\) −8.37974e20 −0.319663 −0.159832 0.987144i \(-0.551095\pi\)
−0.159832 + 0.987144i \(0.551095\pi\)
\(332\) −7.60905e20 −0.282915
\(333\) 3.12396e21 1.13222
\(334\) −2.53759e21 −0.896554
\(335\) 2.69294e20 0.0927570
\(336\) −1.98636e21 −0.667076
\(337\) 4.10587e21 1.34447 0.672235 0.740338i \(-0.265334\pi\)
0.672235 + 0.740338i \(0.265334\pi\)
\(338\) −1.53376e21 −0.489741
\(339\) 1.09808e21 0.341929
\(340\) −7.56070e20 −0.229610
\(341\) 3.88430e20 0.115054
\(342\) 3.36415e21 0.971972
\(343\) 2.29652e20 0.0647247
\(344\) −4.40355e21 −1.21075
\(345\) 1.04573e21 0.280515
\(346\) 5.13213e21 1.34323
\(347\) −1.00068e21 −0.255560 −0.127780 0.991803i \(-0.540785\pi\)
−0.127780 + 0.991803i \(0.540785\pi\)
\(348\) −7.93120e19 −0.0197658
\(349\) −3.27750e21 −0.797126 −0.398563 0.917141i \(-0.630491\pi\)
−0.398563 + 0.917141i \(0.630491\pi\)
\(350\) 9.78243e21 2.32203
\(351\) −2.85644e21 −0.661780
\(352\) 2.00125e21 0.452573
\(353\) 4.58335e20 0.101181 0.0505903 0.998719i \(-0.483890\pi\)
0.0505903 + 0.998719i \(0.483890\pi\)
\(354\) −1.78509e20 −0.0384710
\(355\) 3.36291e20 0.0707578
\(356\) 7.98832e20 0.164108
\(357\) 3.22546e21 0.647011
\(358\) −6.62581e21 −1.29788
\(359\) 5.03266e20 0.0962709 0.0481354 0.998841i \(-0.484672\pi\)
0.0481354 + 0.998841i \(0.484672\pi\)
\(360\) 6.20786e21 1.15977
\(361\) 9.48297e21 1.73035
\(362\) 3.30120e21 0.588368
\(363\) 2.12817e21 0.370509
\(364\) −1.02454e21 −0.174248
\(365\) 9.25648e21 1.53800
\(366\) 1.54527e21 0.250850
\(367\) −8.94361e21 −1.41857 −0.709286 0.704921i \(-0.750982\pi\)
−0.709286 + 0.704921i \(0.750982\pi\)
\(368\) −1.44133e21 −0.223386
\(369\) −5.33210e21 −0.807559
\(370\) −1.78255e22 −2.63832
\(371\) −1.07328e22 −1.55251
\(372\) −6.79346e19 −0.00960453
\(373\) 3.36625e21 0.465181 0.232590 0.972575i \(-0.425280\pi\)
0.232590 + 0.972575i \(0.425280\pi\)
\(374\) 6.53290e21 0.882464
\(375\) 5.94008e21 0.784379
\(376\) −1.20865e22 −1.56028
\(377\) 9.97062e20 0.125841
\(378\) −1.02497e22 −1.26482
\(379\) 3.28121e19 0.00395914 0.00197957 0.999998i \(-0.499370\pi\)
0.00197957 + 0.999998i \(0.499370\pi\)
\(380\) 4.20788e21 0.496481
\(381\) −6.83936e21 −0.789138
\(382\) 1.07554e22 1.21364
\(383\) 9.55720e21 1.05473 0.527365 0.849639i \(-0.323180\pi\)
0.527365 + 0.849639i \(0.323180\pi\)
\(384\) 3.56361e21 0.384657
\(385\) 2.88635e22 3.04742
\(386\) 9.97286e20 0.102998
\(387\) −7.28475e21 −0.735989
\(388\) 1.30765e21 0.129247
\(389\) 7.02922e21 0.679728 0.339864 0.940475i \(-0.389619\pi\)
0.339864 + 0.940475i \(0.389619\pi\)
\(390\) 6.41787e21 0.607214
\(391\) 2.34043e21 0.216667
\(392\) −1.23289e22 −1.11684
\(393\) 2.76241e21 0.244879
\(394\) 8.08099e21 0.701046
\(395\) −2.32960e22 −1.97791
\(396\) 1.79186e21 0.148900
\(397\) −1.96775e22 −1.60048 −0.800242 0.599677i \(-0.795296\pi\)
−0.800242 + 0.599677i \(0.795296\pi\)
\(398\) −1.74609e22 −1.39014
\(399\) −1.79512e22 −1.39902
\(400\) −1.85141e22 −1.41252
\(401\) −2.13267e22 −1.59293 −0.796465 0.604685i \(-0.793299\pi\)
−0.796465 + 0.604685i \(0.793299\pi\)
\(402\) 4.06958e20 0.0297597
\(403\) 8.54033e20 0.0611479
\(404\) 1.41060e21 0.0988920
\(405\) 1.73726e21 0.119260
\(406\) 3.57773e21 0.240512
\(407\) −3.37627e22 −2.22272
\(408\) −7.49751e21 −0.483399
\(409\) −3.03623e22 −1.91729 −0.958643 0.284610i \(-0.908136\pi\)
−0.958643 + 0.284610i \(0.908136\pi\)
\(410\) 3.04254e22 1.88180
\(411\) −5.96644e21 −0.361459
\(412\) 5.44612e21 0.323192
\(413\) −1.76515e21 −0.102614
\(414\) −2.92849e21 −0.166779
\(415\) 4.71316e22 2.62968
\(416\) 4.40011e21 0.240531
\(417\) −1.98789e21 −0.106472
\(418\) −3.63586e22 −1.90813
\(419\) 1.69166e22 0.869947 0.434973 0.900443i \(-0.356758\pi\)
0.434973 + 0.900443i \(0.356758\pi\)
\(420\) −5.04809e21 −0.254395
\(421\) −3.38430e22 −1.67136 −0.835681 0.549215i \(-0.814927\pi\)
−0.835681 + 0.549215i \(0.814927\pi\)
\(422\) 3.07118e22 1.48645
\(423\) −1.99946e22 −0.948459
\(424\) 2.49481e22 1.15992
\(425\) 3.00633e22 1.37003
\(426\) 5.08204e20 0.0227016
\(427\) 1.52800e22 0.669093
\(428\) 3.03915e21 0.130461
\(429\) 1.21558e22 0.511562
\(430\) 4.15673e22 1.71502
\(431\) −2.53815e22 −1.02674 −0.513371 0.858167i \(-0.671603\pi\)
−0.513371 + 0.858167i \(0.671603\pi\)
\(432\) 1.93984e22 0.769405
\(433\) 2.00085e22 0.778158 0.389079 0.921204i \(-0.372793\pi\)
0.389079 + 0.921204i \(0.372793\pi\)
\(434\) 3.06451e21 0.116869
\(435\) 4.91270e21 0.183722
\(436\) 2.28309e21 0.0837313
\(437\) −1.30256e22 −0.468495
\(438\) 1.39884e22 0.493444
\(439\) 3.67625e22 1.27191 0.635955 0.771726i \(-0.280606\pi\)
0.635955 + 0.771726i \(0.280606\pi\)
\(440\) −6.70924e22 −2.27680
\(441\) −2.03955e22 −0.678901
\(442\) 1.43638e22 0.469006
\(443\) −2.72263e20 −0.00872082 −0.00436041 0.999990i \(-0.501388\pi\)
−0.00436041 + 0.999990i \(0.501388\pi\)
\(444\) 5.90494e21 0.185550
\(445\) −4.94809e22 −1.52538
\(446\) −2.06218e22 −0.623707
\(447\) 4.30391e21 0.127717
\(448\) 5.44876e22 1.58648
\(449\) 3.00571e22 0.858723 0.429361 0.903133i \(-0.358739\pi\)
0.429361 + 0.903133i \(0.358739\pi\)
\(450\) −3.76170e22 −1.05457
\(451\) 5.76275e22 1.58537
\(452\) −3.84632e21 −0.103841
\(453\) −2.05522e22 −0.544533
\(454\) −3.41473e22 −0.887935
\(455\) 6.34616e22 1.61962
\(456\) 4.17271e22 1.04524
\(457\) −1.65823e22 −0.407715 −0.203858 0.979001i \(-0.565348\pi\)
−0.203858 + 0.979001i \(0.565348\pi\)
\(458\) −4.95630e22 −1.19619
\(459\) −3.14992e22 −0.746263
\(460\) −3.66295e21 −0.0851902
\(461\) −1.23592e22 −0.282185 −0.141093 0.989996i \(-0.545062\pi\)
−0.141093 + 0.989996i \(0.545062\pi\)
\(462\) 4.36185e22 0.977719
\(463\) 4.77741e22 1.05137 0.525683 0.850680i \(-0.323810\pi\)
0.525683 + 0.850680i \(0.323810\pi\)
\(464\) −6.77118e21 −0.146306
\(465\) 4.20797e21 0.0892735
\(466\) 4.70769e22 0.980680
\(467\) 1.00133e21 0.0204825 0.0102412 0.999948i \(-0.496740\pi\)
0.0102412 + 0.999948i \(0.496740\pi\)
\(468\) 3.93973e21 0.0791364
\(469\) 4.02411e21 0.0793780
\(470\) 1.14090e23 2.21013
\(471\) −2.67811e22 −0.509509
\(472\) 4.10304e21 0.0766653
\(473\) 7.87310e22 1.44486
\(474\) −3.52050e22 −0.634583
\(475\) −1.67316e23 −2.96239
\(476\) −1.12981e22 −0.196492
\(477\) 4.12714e22 0.705088
\(478\) 3.77145e22 0.632953
\(479\) 6.34545e22 1.04619 0.523095 0.852274i \(-0.324777\pi\)
0.523095 + 0.852274i \(0.324777\pi\)
\(480\) 2.16801e22 0.351165
\(481\) −7.42333e22 −1.18132
\(482\) −1.97146e22 −0.308239
\(483\) 1.56265e22 0.240055
\(484\) −7.45449e21 −0.112521
\(485\) −8.09976e22 −1.20134
\(486\) 6.33080e22 0.922678
\(487\) 3.63304e22 0.520325 0.260162 0.965565i \(-0.416224\pi\)
0.260162 + 0.965565i \(0.416224\pi\)
\(488\) −3.55180e22 −0.499896
\(489\) −8.30344e22 −1.14851
\(490\) 1.16378e23 1.58200
\(491\) 1.21537e22 0.162373 0.0811866 0.996699i \(-0.474129\pi\)
0.0811866 + 0.996699i \(0.474129\pi\)
\(492\) −1.00788e22 −0.132344
\(493\) 1.09951e22 0.141905
\(494\) −7.99409e22 −1.01412
\(495\) −1.10990e23 −1.38402
\(496\) −5.79985e21 −0.0710923
\(497\) 5.02525e21 0.0605520
\(498\) 7.12253e22 0.843694
\(499\) −3.17479e22 −0.369710 −0.184855 0.982766i \(-0.559182\pi\)
−0.184855 + 0.982766i \(0.559182\pi\)
\(500\) −2.08068e22 −0.238210
\(501\) −5.20686e22 −0.586078
\(502\) −3.47265e22 −0.384308
\(503\) 1.05072e22 0.114329 0.0571646 0.998365i \(-0.481794\pi\)
0.0571646 + 0.998365i \(0.481794\pi\)
\(504\) 9.27650e22 0.992487
\(505\) −8.73744e22 −0.919195
\(506\) 3.16501e22 0.327413
\(507\) −3.14712e22 −0.320144
\(508\) 2.39567e22 0.239655
\(509\) −1.93166e23 −1.90033 −0.950166 0.311744i \(-0.899087\pi\)
−0.950166 + 0.311744i \(0.899087\pi\)
\(510\) 7.07728e22 0.684731
\(511\) 1.38321e23 1.31616
\(512\) −1.19836e23 −1.12149
\(513\) 1.75308e23 1.61363
\(514\) 5.53964e22 0.501528
\(515\) −3.37341e23 −3.00405
\(516\) −1.37697e22 −0.120615
\(517\) 2.16094e23 1.86197
\(518\) −2.66370e23 −2.25778
\(519\) 1.05306e23 0.878071
\(520\) −1.47515e23 −1.21006
\(521\) −2.17012e23 −1.75131 −0.875654 0.482938i \(-0.839570\pi\)
−0.875654 + 0.482938i \(0.839570\pi\)
\(522\) −1.37577e22 −0.109231
\(523\) 7.80323e22 0.609551 0.304775 0.952424i \(-0.401419\pi\)
0.304775 + 0.952424i \(0.401419\pi\)
\(524\) −9.67611e21 −0.0743678
\(525\) 2.00725e23 1.51791
\(526\) 1.15592e23 0.860096
\(527\) 9.41781e21 0.0689540
\(528\) −8.25520e22 −0.594756
\(529\) −1.29711e23 −0.919612
\(530\) −2.35498e23 −1.64302
\(531\) 6.78762e21 0.0466031
\(532\) 6.28790e22 0.424871
\(533\) 1.26705e23 0.842580
\(534\) −7.47756e22 −0.489395
\(535\) −1.88250e23 −1.21263
\(536\) −9.35394e21 −0.0593054
\(537\) −1.35954e23 −0.848422
\(538\) 2.89243e23 1.77670
\(539\) 2.20428e23 1.33279
\(540\) 4.92987e22 0.293419
\(541\) 1.21903e23 0.714229 0.357115 0.934061i \(-0.383761\pi\)
0.357115 + 0.934061i \(0.383761\pi\)
\(542\) 9.21426e22 0.531455
\(543\) 6.77370e22 0.384616
\(544\) 4.85220e22 0.271236
\(545\) −1.41418e23 −0.778278
\(546\) 9.59032e22 0.519632
\(547\) 2.98026e23 1.58987 0.794936 0.606693i \(-0.207504\pi\)
0.794936 + 0.606693i \(0.207504\pi\)
\(548\) 2.08991e22 0.109772
\(549\) −5.87571e22 −0.303875
\(550\) 4.06551e23 2.07030
\(551\) −6.11925e22 −0.306839
\(552\) −3.63234e22 −0.179351
\(553\) −3.48116e23 −1.69262
\(554\) −2.57619e23 −1.23351
\(555\) −3.65761e23 −1.72467
\(556\) 6.96314e21 0.0323347
\(557\) −1.66559e23 −0.761728 −0.380864 0.924631i \(-0.624373\pi\)
−0.380864 + 0.924631i \(0.624373\pi\)
\(558\) −1.17841e22 −0.0530770
\(559\) 1.73104e23 0.767906
\(560\) −4.30976e23 −1.88302
\(561\) 1.34048e23 0.576867
\(562\) 9.98519e22 0.423251
\(563\) 4.20849e23 1.75714 0.878568 0.477618i \(-0.158500\pi\)
0.878568 + 0.477618i \(0.158500\pi\)
\(564\) −3.77939e22 −0.155435
\(565\) 2.38247e23 0.965196
\(566\) 2.39612e23 0.956245
\(567\) 2.59601e22 0.102059
\(568\) −1.16811e22 −0.0452399
\(569\) 6.24767e22 0.238377 0.119188 0.992872i \(-0.461971\pi\)
0.119188 + 0.992872i \(0.461971\pi\)
\(570\) −3.93883e23 −1.48058
\(571\) −3.83744e23 −1.42113 −0.710566 0.703631i \(-0.751561\pi\)
−0.710566 + 0.703631i \(0.751561\pi\)
\(572\) −4.25792e22 −0.155357
\(573\) 2.20690e23 0.793357
\(574\) 4.54651e23 1.61038
\(575\) 1.45648e23 0.508310
\(576\) −2.09525e23 −0.720516
\(577\) −1.10564e23 −0.374646 −0.187323 0.982298i \(-0.559981\pi\)
−0.187323 + 0.982298i \(0.559981\pi\)
\(578\) −1.12841e23 −0.376773
\(579\) 2.04632e22 0.0673297
\(580\) −1.72081e22 −0.0557949
\(581\) 7.04294e23 2.25039
\(582\) −1.22404e23 −0.385433
\(583\) −4.46047e23 −1.38420
\(584\) −3.21524e23 −0.983340
\(585\) −2.44033e23 −0.735568
\(586\) −4.31526e23 −1.28197
\(587\) 4.47105e23 1.30914 0.654570 0.756002i \(-0.272850\pi\)
0.654570 + 0.756002i \(0.272850\pi\)
\(588\) −3.85518e22 −0.111260
\(589\) −5.24144e22 −0.149098
\(590\) −3.87307e22 −0.108596
\(591\) 1.65813e23 0.458274
\(592\) 5.04129e23 1.37343
\(593\) 1.77200e23 0.475882 0.237941 0.971280i \(-0.423527\pi\)
0.237941 + 0.971280i \(0.423527\pi\)
\(594\) −4.25970e23 −1.12770
\(595\) 6.99819e23 1.82638
\(596\) −1.50756e22 −0.0387867
\(597\) −3.58278e23 −0.908738
\(598\) 6.95885e22 0.174011
\(599\) −5.19820e22 −0.128152 −0.0640760 0.997945i \(-0.520410\pi\)
−0.0640760 + 0.997945i \(0.520410\pi\)
\(600\) −4.66580e23 −1.13407
\(601\) 9.35878e22 0.224278 0.112139 0.993693i \(-0.464230\pi\)
0.112139 + 0.993693i \(0.464230\pi\)
\(602\) 6.21147e23 1.46765
\(603\) −1.54741e22 −0.0360503
\(604\) 7.19898e22 0.165370
\(605\) 4.61742e23 1.04587
\(606\) −1.32040e23 −0.294910
\(607\) 1.50667e23 0.331829 0.165915 0.986140i \(-0.446942\pi\)
0.165915 + 0.986140i \(0.446942\pi\)
\(608\) −2.70047e23 −0.586489
\(609\) 7.34112e22 0.157223
\(610\) 3.35272e23 0.708099
\(611\) 4.75123e23 0.989590
\(612\) 4.34451e22 0.0892389
\(613\) 7.36068e23 1.49109 0.745545 0.666455i \(-0.232189\pi\)
0.745545 + 0.666455i \(0.232189\pi\)
\(614\) 5.92225e22 0.118319
\(615\) 6.24296e23 1.23013
\(616\) −1.00257e24 −1.94841
\(617\) 7.03580e23 1.34862 0.674310 0.738449i \(-0.264441\pi\)
0.674310 + 0.738449i \(0.264441\pi\)
\(618\) −5.09790e23 −0.963804
\(619\) 9.58495e23 1.78739 0.893695 0.448675i \(-0.148104\pi\)
0.893695 + 0.448675i \(0.148104\pi\)
\(620\) −1.47396e22 −0.0271116
\(621\) −1.52605e23 −0.276879
\(622\) −6.26294e23 −1.12088
\(623\) −7.39400e23 −1.30536
\(624\) −1.81505e23 −0.316097
\(625\) 2.45253e23 0.421341
\(626\) −2.16226e23 −0.366459
\(627\) −7.46039e23 −1.24735
\(628\) 9.38084e22 0.154734
\(629\) −8.18605e23 −1.33212
\(630\) −8.75656e23 −1.40585
\(631\) −2.26530e23 −0.358819 −0.179410 0.983774i \(-0.557419\pi\)
−0.179410 + 0.983774i \(0.557419\pi\)
\(632\) 8.09187e23 1.26460
\(633\) 6.30173e23 0.971691
\(634\) 9.13105e23 1.38919
\(635\) −1.48392e24 −2.22758
\(636\) 7.80117e22 0.115551
\(637\) 4.84650e23 0.708342
\(638\) 1.48688e23 0.214437
\(639\) −1.93239e22 −0.0275003
\(640\) 7.73186e23 1.08581
\(641\) −5.22645e23 −0.724292 −0.362146 0.932121i \(-0.617956\pi\)
−0.362146 + 0.932121i \(0.617956\pi\)
\(642\) −2.84483e23 −0.389053
\(643\) −6.48185e23 −0.874794 −0.437397 0.899268i \(-0.644100\pi\)
−0.437397 + 0.899268i \(0.644100\pi\)
\(644\) −5.47361e22 −0.0729027
\(645\) 8.52916e23 1.12111
\(646\) −8.81545e23 −1.14358
\(647\) −1.29109e23 −0.165300 −0.0826498 0.996579i \(-0.526338\pi\)
−0.0826498 + 0.996579i \(0.526338\pi\)
\(648\) −6.03436e22 −0.0762507
\(649\) −7.33583e22 −0.0914891
\(650\) 8.93877e23 1.10031
\(651\) 6.28804e22 0.0763970
\(652\) 2.90851e23 0.348792
\(653\) 1.49980e24 1.77529 0.887647 0.460525i \(-0.152339\pi\)
0.887647 + 0.460525i \(0.152339\pi\)
\(654\) −2.13711e23 −0.249699
\(655\) 5.99353e23 0.691244
\(656\) −8.60468e23 −0.979607
\(657\) −5.31894e23 −0.597749
\(658\) 1.70487e24 1.89135
\(659\) 3.48779e23 0.381965 0.190983 0.981593i \(-0.438833\pi\)
0.190983 + 0.981593i \(0.438833\pi\)
\(660\) −2.09795e23 −0.226815
\(661\) −2.57921e23 −0.275280 −0.137640 0.990482i \(-0.543952\pi\)
−0.137640 + 0.990482i \(0.543952\pi\)
\(662\) −2.74756e23 −0.289504
\(663\) 2.94729e23 0.306590
\(664\) −1.63712e24 −1.68132
\(665\) −3.89482e24 −3.94915
\(666\) 1.02429e24 1.02539
\(667\) 5.32680e22 0.0526498
\(668\) 1.82385e23 0.177987
\(669\) −4.23137e23 −0.407718
\(670\) 8.82966e22 0.0840056
\(671\) 6.35026e23 0.596555
\(672\) 3.23969e23 0.300514
\(673\) −8.97036e23 −0.821641 −0.410820 0.911716i \(-0.634758\pi\)
−0.410820 + 0.911716i \(0.634758\pi\)
\(674\) 1.34624e24 1.21762
\(675\) −1.96024e24 −1.75076
\(676\) 1.10237e23 0.0972251
\(677\) 4.13358e23 0.360017 0.180008 0.983665i \(-0.442388\pi\)
0.180008 + 0.983665i \(0.442388\pi\)
\(678\) 3.60039e23 0.309669
\(679\) −1.21036e24 −1.02807
\(680\) −1.62671e24 −1.36454
\(681\) −7.00665e23 −0.580443
\(682\) 1.27359e23 0.104199
\(683\) −8.83551e23 −0.713930 −0.356965 0.934118i \(-0.616188\pi\)
−0.356965 + 0.934118i \(0.616188\pi\)
\(684\) −2.41792e23 −0.192959
\(685\) −1.29452e24 −1.02033
\(686\) 7.52986e22 0.0586181
\(687\) −1.01698e24 −0.781952
\(688\) −1.17558e24 −0.892789
\(689\) −9.80716e23 −0.735665
\(690\) 3.42875e23 0.254049
\(691\) 3.85744e23 0.282316 0.141158 0.989987i \(-0.454917\pi\)
0.141158 + 0.989987i \(0.454917\pi\)
\(692\) −3.68863e23 −0.266663
\(693\) −1.65855e24 −1.18439
\(694\) −3.28103e23 −0.231448
\(695\) −4.31307e23 −0.300549
\(696\) −1.70643e23 −0.117465
\(697\) 1.39723e24 0.950143
\(698\) −1.07463e24 −0.721919
\(699\) 9.65967e23 0.641071
\(700\) −7.03095e23 −0.460978
\(701\) 1.43098e24 0.926893 0.463446 0.886125i \(-0.346613\pi\)
0.463446 + 0.886125i \(0.346613\pi\)
\(702\) −9.36572e23 −0.599343
\(703\) 4.55591e24 2.88042
\(704\) 2.26447e24 1.41449
\(705\) 2.34101e24 1.44476
\(706\) 1.50279e23 0.0916346
\(707\) −1.30565e24 −0.786614
\(708\) 1.28300e22 0.00763740
\(709\) 4.46129e22 0.0262402 0.0131201 0.999914i \(-0.495824\pi\)
0.0131201 + 0.999914i \(0.495824\pi\)
\(710\) 1.10264e23 0.0640820
\(711\) 1.33863e24 0.768722
\(712\) 1.71872e24 0.975269
\(713\) 4.56267e22 0.0255834
\(714\) 1.05757e24 0.585968
\(715\) 2.63742e24 1.44404
\(716\) 4.76218e23 0.257659
\(717\) 7.73861e23 0.413762
\(718\) 1.65012e23 0.0871880
\(719\) 2.71624e24 1.41831 0.709157 0.705050i \(-0.249076\pi\)
0.709157 + 0.705050i \(0.249076\pi\)
\(720\) 1.65726e24 0.855192
\(721\) −5.04093e24 −2.57076
\(722\) 3.10929e24 1.56709
\(723\) −4.04522e23 −0.201496
\(724\) −2.37268e23 −0.116805
\(725\) 6.84237e23 0.332915
\(726\) 6.97786e23 0.335553
\(727\) 1.96738e24 0.935073 0.467537 0.883974i \(-0.345142\pi\)
0.467537 + 0.883974i \(0.345142\pi\)
\(728\) −2.20434e24 −1.03553
\(729\) 1.14532e24 0.531791
\(730\) 3.03503e24 1.39289
\(731\) 1.90890e24 0.865936
\(732\) −1.11063e23 −0.0497997
\(733\) 3.28195e23 0.145462 0.0727309 0.997352i \(-0.476829\pi\)
0.0727309 + 0.997352i \(0.476829\pi\)
\(734\) −2.93244e24 −1.28473
\(735\) 2.38796e24 1.03415
\(736\) 2.35076e23 0.100634
\(737\) 1.67239e23 0.0707725
\(738\) −1.74830e24 −0.731368
\(739\) −1.77329e24 −0.733335 −0.366668 0.930352i \(-0.619501\pi\)
−0.366668 + 0.930352i \(0.619501\pi\)
\(740\) 1.28118e24 0.523769
\(741\) −1.64030e24 −0.662932
\(742\) −3.51908e24 −1.40603
\(743\) 1.63750e24 0.646811 0.323405 0.946260i \(-0.395172\pi\)
0.323405 + 0.946260i \(0.395172\pi\)
\(744\) −1.46164e23 −0.0570782
\(745\) 9.33807e23 0.360520
\(746\) 1.10373e24 0.421292
\(747\) −2.70826e24 −1.02204
\(748\) −4.69540e23 −0.175190
\(749\) −2.81304e24 −1.03772
\(750\) 1.94764e24 0.710375
\(751\) −1.26189e24 −0.455075 −0.227537 0.973769i \(-0.573067\pi\)
−0.227537 + 0.973769i \(0.573067\pi\)
\(752\) −3.22662e24 −1.15053
\(753\) −7.12549e23 −0.251222
\(754\) 3.26918e23 0.113968
\(755\) −4.45916e24 −1.53711
\(756\) 7.36677e23 0.251097
\(757\) 4.84738e24 1.63377 0.816887 0.576797i \(-0.195698\pi\)
0.816887 + 0.576797i \(0.195698\pi\)
\(758\) 1.07585e22 0.00358561
\(759\) 6.49426e23 0.214030
\(760\) 9.05341e24 2.95051
\(761\) −3.16525e24 −1.02009 −0.510045 0.860148i \(-0.670371\pi\)
−0.510045 + 0.860148i \(0.670371\pi\)
\(762\) −2.24250e24 −0.714685
\(763\) −2.11323e24 −0.666022
\(764\) −7.73028e23 −0.240936
\(765\) −2.69106e24 −0.829470
\(766\) 3.13363e24 0.955218
\(767\) −1.61292e23 −0.0486240
\(768\) −1.03434e24 −0.308386
\(769\) −1.29823e24 −0.382804 −0.191402 0.981512i \(-0.561303\pi\)
−0.191402 + 0.981512i \(0.561303\pi\)
\(770\) 9.46379e24 2.75990
\(771\) 1.13667e24 0.327849
\(772\) −7.16781e22 −0.0204475
\(773\) 3.55520e24 1.00309 0.501544 0.865132i \(-0.332766\pi\)
0.501544 + 0.865132i \(0.332766\pi\)
\(774\) −2.38853e24 −0.666550
\(775\) 5.86083e23 0.161769
\(776\) 2.81345e24 0.768094
\(777\) −5.46562e24 −1.47591
\(778\) 2.30475e24 0.615598
\(779\) −7.77622e24 −2.05447
\(780\) −4.61273e23 −0.120546
\(781\) 2.08846e23 0.0539874
\(782\) 7.67383e23 0.196225
\(783\) −7.16919e23 −0.181341
\(784\) −3.29132e24 −0.823539
\(785\) −5.81063e24 −1.43824
\(786\) 9.05743e23 0.221775
\(787\) 1.61751e24 0.391797 0.195898 0.980624i \(-0.437238\pi\)
0.195898 + 0.980624i \(0.437238\pi\)
\(788\) −5.80807e23 −0.139174
\(789\) 2.37181e24 0.562245
\(790\) −7.63833e24 −1.79130
\(791\) 3.56016e24 0.825979
\(792\) 3.85525e24 0.884889
\(793\) 1.39622e24 0.317053
\(794\) −6.45190e24 −1.44948
\(795\) −4.83216e24 −1.07404
\(796\) 1.25497e24 0.275976
\(797\) −4.85450e24 −1.05621 −0.528103 0.849180i \(-0.677096\pi\)
−0.528103 + 0.849180i \(0.677096\pi\)
\(798\) −5.88586e24 −1.26702
\(799\) 5.23939e24 1.11592
\(800\) 3.01959e24 0.636331
\(801\) 2.84326e24 0.592843
\(802\) −6.99262e24 −1.44264
\(803\) 5.74852e24 1.17348
\(804\) −2.92494e22 −0.00590800
\(805\) 3.39043e24 0.677626
\(806\) 2.80021e23 0.0553788
\(807\) 5.93495e24 1.16143
\(808\) 3.03495e24 0.587699
\(809\) 4.71065e24 0.902648 0.451324 0.892360i \(-0.350952\pi\)
0.451324 + 0.892360i \(0.350952\pi\)
\(810\) 5.69614e23 0.108009
\(811\) 3.62987e24 0.681105 0.340553 0.940225i \(-0.389386\pi\)
0.340553 + 0.940225i \(0.389386\pi\)
\(812\) −2.57143e23 −0.0477473
\(813\) 1.89067e24 0.347412
\(814\) −1.10701e25 −2.01301
\(815\) −1.80157e25 −3.24200
\(816\) −2.00154e24 −0.356450
\(817\) −1.06239e25 −1.87239
\(818\) −9.95524e24 −1.73640
\(819\) −3.64661e24 −0.629472
\(820\) −2.18677e24 −0.373581
\(821\) 1.21391e24 0.205243 0.102621 0.994720i \(-0.467277\pi\)
0.102621 + 0.994720i \(0.467277\pi\)
\(822\) −1.95628e24 −0.327357
\(823\) 5.81195e24 0.962550 0.481275 0.876570i \(-0.340174\pi\)
0.481275 + 0.876570i \(0.340174\pi\)
\(824\) 1.17175e25 1.92068
\(825\) 8.34199e24 1.35335
\(826\) −5.78759e23 −0.0929324
\(827\) 4.75053e24 0.754997 0.377498 0.926010i \(-0.376784\pi\)
0.377498 + 0.926010i \(0.376784\pi\)
\(828\) 2.10480e23 0.0331095
\(829\) −8.41454e24 −1.31014 −0.655069 0.755569i \(-0.727360\pi\)
−0.655069 + 0.755569i \(0.727360\pi\)
\(830\) 1.54536e25 2.38158
\(831\) −5.28605e24 −0.806349
\(832\) 4.97885e24 0.751762
\(833\) 5.34446e24 0.798769
\(834\) −6.51792e23 −0.0964268
\(835\) −1.12972e25 −1.65438
\(836\) 2.61321e24 0.378809
\(837\) −6.14077e23 −0.0881163
\(838\) 5.54662e24 0.787869
\(839\) 3.50641e24 0.493045 0.246522 0.969137i \(-0.420712\pi\)
0.246522 + 0.969137i \(0.420712\pi\)
\(840\) −1.08612e25 −1.51183
\(841\) 2.50246e23 0.0344828
\(842\) −1.10965e25 −1.51367
\(843\) 2.04885e24 0.276679
\(844\) −2.20736e24 −0.295095
\(845\) −6.82821e24 −0.903702
\(846\) −6.55584e24 −0.858975
\(847\) 6.89988e24 0.895020
\(848\) 6.66017e24 0.855305
\(849\) 4.91658e24 0.625098
\(850\) 9.85718e24 1.24077
\(851\) −3.96591e24 −0.494245
\(852\) −3.65262e22 −0.00450680
\(853\) 2.01835e24 0.246564 0.123282 0.992372i \(-0.460658\pi\)
0.123282 + 0.992372i \(0.460658\pi\)
\(854\) 5.01002e24 0.605966
\(855\) 1.49770e25 1.79354
\(856\) 6.53885e24 0.775308
\(857\) −1.13064e25 −1.32736 −0.663680 0.748016i \(-0.731006\pi\)
−0.663680 + 0.748016i \(0.731006\pi\)
\(858\) 3.98567e24 0.463297
\(859\) −1.04837e25 −1.20663 −0.603313 0.797505i \(-0.706153\pi\)
−0.603313 + 0.797505i \(0.706153\pi\)
\(860\) −2.98758e24 −0.340473
\(861\) 9.32895e24 1.05270
\(862\) −8.32213e24 −0.929871
\(863\) −6.81348e24 −0.753836 −0.376918 0.926247i \(-0.623016\pi\)
−0.376918 + 0.926247i \(0.623016\pi\)
\(864\) −3.16382e24 −0.346613
\(865\) 2.28479e25 2.47862
\(866\) 6.56042e24 0.704741
\(867\) −2.31537e24 −0.246297
\(868\) −2.20256e23 −0.0232012
\(869\) −1.44675e25 −1.50912
\(870\) 1.61078e24 0.166388
\(871\) 3.67706e23 0.0376137
\(872\) 4.91215e24 0.497602
\(873\) 4.65427e24 0.466907
\(874\) −4.27084e24 −0.424294
\(875\) 1.92588e25 1.89478
\(876\) −1.00539e24 −0.0979603
\(877\) 4.94993e24 0.477642 0.238821 0.971064i \(-0.423239\pi\)
0.238821 + 0.971064i \(0.423239\pi\)
\(878\) 1.20537e25 1.15191
\(879\) −8.85445e24 −0.838023
\(880\) −1.79111e25 −1.67888
\(881\) 4.20081e24 0.389975 0.194988 0.980806i \(-0.437533\pi\)
0.194988 + 0.980806i \(0.437533\pi\)
\(882\) −6.68731e24 −0.614849
\(883\) 1.26234e25 1.14950 0.574751 0.818328i \(-0.305099\pi\)
0.574751 + 0.818328i \(0.305099\pi\)
\(884\) −1.03237e24 −0.0931088
\(885\) −7.94712e23 −0.0709891
\(886\) −8.92700e22 −0.00789803
\(887\) 3.60457e24 0.315866 0.157933 0.987450i \(-0.449517\pi\)
0.157933 + 0.987450i \(0.449517\pi\)
\(888\) 1.27047e25 1.10269
\(889\) −2.21744e25 −1.90628
\(890\) −1.62238e25 −1.38146
\(891\) 1.07888e24 0.0909943
\(892\) 1.48216e24 0.123821
\(893\) −2.91596e25 −2.41293
\(894\) 1.41117e24 0.115667
\(895\) −2.94977e25 −2.39492
\(896\) 1.15538e25 0.929197
\(897\) 1.42788e24 0.113751
\(898\) 9.85516e24 0.777705
\(899\) 2.14348e23 0.0167557
\(900\) 2.70365e24 0.209358
\(901\) −1.08148e25 −0.829579
\(902\) 1.88950e25 1.43579
\(903\) 1.27453e25 0.959407
\(904\) −8.27550e24 −0.617110
\(905\) 1.46967e25 1.08569
\(906\) −6.73869e24 −0.493157
\(907\) 1.71250e25 1.24156 0.620782 0.783983i \(-0.286815\pi\)
0.620782 + 0.783983i \(0.286815\pi\)
\(908\) 2.45427e24 0.176276
\(909\) 5.02069e24 0.357249
\(910\) 2.08079e25 1.46682
\(911\) −1.07652e25 −0.751826 −0.375913 0.926655i \(-0.622671\pi\)
−0.375913 + 0.926655i \(0.622671\pi\)
\(912\) 1.11395e25 0.770744
\(913\) 2.92700e25 2.00642
\(914\) −5.43702e24 −0.369248
\(915\) 6.87943e24 0.462885
\(916\) 3.56225e24 0.237472
\(917\) 8.95622e24 0.591542
\(918\) −1.03280e25 −0.675855
\(919\) 1.37024e25 0.888416 0.444208 0.895924i \(-0.353485\pi\)
0.444208 + 0.895924i \(0.353485\pi\)
\(920\) −7.88098e24 −0.506272
\(921\) 1.21518e24 0.0773454
\(922\) −4.05236e24 −0.255562
\(923\) 4.59186e23 0.0286929
\(924\) −3.13500e24 −0.194100
\(925\) −5.09429e25 −3.12521
\(926\) 1.56642e25 0.952173
\(927\) 1.93842e25 1.16753
\(928\) 1.10436e24 0.0659100
\(929\) −1.43769e25 −0.850219 −0.425109 0.905142i \(-0.639764\pi\)
−0.425109 + 0.905142i \(0.639764\pi\)
\(930\) 1.37972e24 0.0808508
\(931\) −2.97444e25 −1.72716
\(932\) −3.38357e24 −0.194688
\(933\) −1.28509e25 −0.732721
\(934\) 3.28316e23 0.0185500
\(935\) 2.90840e25 1.62838
\(936\) 8.47647e24 0.470295
\(937\) 2.47776e25 1.36230 0.681150 0.732144i \(-0.261480\pi\)
0.681150 + 0.732144i \(0.261480\pi\)
\(938\) 1.31943e24 0.0718889
\(939\) −4.43672e24 −0.239554
\(940\) −8.20005e24 −0.438763
\(941\) −9.20959e24 −0.488347 −0.244174 0.969732i \(-0.578517\pi\)
−0.244174 + 0.969732i \(0.578517\pi\)
\(942\) −8.78104e24 −0.461438
\(943\) 6.76919e24 0.352523
\(944\) 1.09535e24 0.0565317
\(945\) −4.56309e25 −2.33393
\(946\) 2.58144e25 1.30854
\(947\) −1.03683e25 −0.520874 −0.260437 0.965491i \(-0.583867\pi\)
−0.260437 + 0.965491i \(0.583867\pi\)
\(948\) 2.53030e24 0.125980
\(949\) 1.26392e25 0.623671
\(950\) −5.48598e25 −2.68289
\(951\) 1.87359e25 0.908115
\(952\) −2.43082e25 −1.16772
\(953\) 2.35634e25 1.12188 0.560942 0.827855i \(-0.310439\pi\)
0.560942 + 0.827855i \(0.310439\pi\)
\(954\) 1.35321e25 0.638565
\(955\) 4.78825e25 2.23949
\(956\) −2.71066e24 −0.125656
\(957\) 3.05092e24 0.140178
\(958\) 2.08055e25 0.947485
\(959\) −1.93442e25 −0.873159
\(960\) 2.45317e25 1.09754
\(961\) −2.23665e25 −0.991858
\(962\) −2.43397e25 −1.06986
\(963\) 1.08172e25 0.471292
\(964\) 1.41695e24 0.0611927
\(965\) 4.43985e24 0.190058
\(966\) 5.12363e24 0.217406
\(967\) 1.78014e25 0.748736 0.374368 0.927280i \(-0.377860\pi\)
0.374368 + 0.927280i \(0.377860\pi\)
\(968\) −1.60386e25 −0.668692
\(969\) −1.80884e25 −0.747561
\(970\) −2.65576e25 −1.08800
\(971\) −2.84221e25 −1.15423 −0.577116 0.816662i \(-0.695822\pi\)
−0.577116 + 0.816662i \(0.695822\pi\)
\(972\) −4.55014e24 −0.183173
\(973\) −6.44509e24 −0.257199
\(974\) 1.19121e25 0.471233
\(975\) 1.83414e25 0.719271
\(976\) −9.48192e24 −0.368615
\(977\) 2.63061e25 1.01380 0.506901 0.862004i \(-0.330791\pi\)
0.506901 + 0.862004i \(0.330791\pi\)
\(978\) −2.72254e25 −1.04015
\(979\) −3.07290e25 −1.16384
\(980\) −8.36448e24 −0.314063
\(981\) 8.12613e24 0.302480
\(982\) 3.98496e24 0.147054
\(983\) −4.87916e25 −1.78501 −0.892504 0.451039i \(-0.851053\pi\)
−0.892504 + 0.451039i \(0.851053\pi\)
\(984\) −2.16849e25 −0.786501
\(985\) 3.59760e25 1.29362
\(986\) 3.60507e24 0.128517
\(987\) 3.49821e25 1.23638
\(988\) 5.74561e24 0.201327
\(989\) 9.24810e24 0.321280
\(990\) −3.63917e25 −1.25344
\(991\) 1.94268e25 0.663401 0.331701 0.943385i \(-0.392378\pi\)
0.331701 + 0.943385i \(0.392378\pi\)
\(992\) 9.45936e23 0.0320267
\(993\) −5.63770e24 −0.189249
\(994\) 1.64768e24 0.0548390
\(995\) −7.77346e25 −2.56518
\(996\) −5.11919e24 −0.167493
\(997\) −1.27472e25 −0.413530 −0.206765 0.978391i \(-0.566294\pi\)
−0.206765 + 0.978391i \(0.566294\pi\)
\(998\) −1.04096e25 −0.334829
\(999\) 5.33761e25 1.70232
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 29.18.a.b.1.15 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.18.a.b.1.15 21 1.1 even 1 trivial